Brett A. Bednarcyk

Ohio Aerospace Institute, Brook Park, Ohio

Steven M. Arnold

Glenn Research Center, Cleveland, Ohio

MAC/GMC 4.0 User's Manual—

Example Problem Manual

NASA/TM—2002-212077/VOL3

December 2002
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Brett A. Bednarcyk

Ohio Aerospace Institute, Brook Park, Ohio

Steven M. Arnold

Glenn Research Center, Cleveland, Ohio

MAC/GMC 4.0 User's Manual—

Example Problem Manual

NASA/TM—2002-212077/VOL3

December 2002

National Aeronautics and

Space Administration

Glenn Research Center
Available from

NASA Center for Aerospace Information

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Table of Contents 

Table of Contents 

Introduction ...................................................................................................................................................3

Section 1 : Basic Examples ...........................................................................................................................6

Example 1a: Effective Properties of a Composite....................................................................................7

Example 1b: Effective Properties of a Laminate....................................................................................12

Example 1c: Tensile Response of Monolithic Ti-21S............................................................................18

Example 1d: Tensile/Thermal Response of SiC/Ti-21S.........................................................................23

Section 2 : Constituent Materials ................................................................................................................28

Example 2a: Bodner-Partom Viscoplastic Constitutive Model..............................................................29

Example 2b: Strain-Rate Dependence of Ti-21S ...................................................................................32

Example 2c: Incremental Plasticity........................................................................................................35

Example 2d: Shape Memory Alloy (SMA)............................................................................................38

Example 2e: User-Defined Material Properties ñ Input File..................................................................41

Example 2f: User-Defined Material Properties ñ USRFUN..................................................................50

Example 2g: User-Defined Material Constitutive Model.......................................................................55

Example 2h: External Material Database ...............................................................................................60

Section 3 : Analysis Type and Architecture................................................................................................64

Example 3a: Doubly Periodic GMC Internal RUC Library...................................................................65

Example 3b: Doubly Periodic RUC Library with Interface...................................................................70

Example 3c: User-Defined Doubly Periodic Architecture.....................................................................75

Example 3d: Triply Periodic GMC Internal RUC Library.....................................................................78

Example 3e: User-Defined Triply Periodic GMC Architecture.............................................................82

Example 3f: High-Fidelity Generalized Method of Cells......................................................................86

Example 3g: Laminate Theory ñ Single Ply...........................................................................................92

Example 3h: Cross-Ply and Quasi-Isotropic Laminates.........................................................................99

Example 3i: Asymmetic Laminate ......................................................................................................104

Section 4 : Applied Loading......................................................................................................................110

Example 4a: MAC/GMC Internal Loading Options ............................................................................111

Example 4b: SiC/Ti-21S Composite with Residual Stresses ...............................................................115

Example 4c: General loading for RUC analysis ..................................................................................120

Example 4d: Internal Loading Options for Laminate...........................................................................125

Example 4e: General loading Option for Laminates............................................................................129

Example 4f: RUC Yield Surface Analysis...........................................................................................134

Example 4g: Laminate Yield Surface Analysis....................................................................................143

Example 4h: Predictor-Corrector Time Integration..............................................................................148

Section 5 : Damage and Failure ................................................................................................................152

Example 5a: Composite Allowables Calculation.................................................................................153

Example 5b: Subcell Static Failure Analysis .......................................................................................158

Example 5c: Laminate Static Failure Analysis ....................................................................................165

Example 5d: Fatigue Damage Analysis................................................................................................171

Example 5e: Fiber-Matrix Debonding .................................................................................................184

Example 5f: Fiber Breakage ................................................................................................................189

Section 6 : Output and Data Visualization ................................................................................................194

Example 6a: Local x-y Plots ................................................................................................................195

Example 6b: PATRAN Post-Processing ..............................................................................................200

Example 6c: MATLAB Post-Processing .............................................................................................206

NASA/TM—2002-212077/VOL3 1
MAC/GMC 4.0 Example Problem Manual 

Section 7 : Advanced Topics.....................................................................................................................213

Example 7a: Implicitly Integrated Multimechanism GVIPS ...............................................................214

Example 7b: Electromagnetic RUC Analysis ......................................................................................219

Example 7d: Woven Composite Analysis ñ Single Step......................................................................237

Example 7e: Woven Composite Analysis ñ Two Step.........................................................................242

References .................................................................................................................................................253

Appendix...................................................................................................................................................255

Complete output file for Example 1a ....................................................................................................256

usrfun.F90: User-defined material property function file...............................................................261

usrmat.F90: User-defined material constitutive model file.............................................................265

usrformde.F90: User-defined stiffness matrix calculation file ......................................................268

Index..........................................................................................................................................................269

NASA/TM—2002-212077/VOL3 2

Example 7c: Electromagnetic Laminate Analysis................................................................................229
Introduction 

Introduction 

This document is the third volume in the three volume set of Userís Manuals for the Micromechanics 

Analysis Code with Generalized Method of Cells Version 4.0 (MAC/GMC 4.0). Volume 1 is the Theory 

Manual, Volume 2 is the Keywords Manual, and this document is the Example Problems Manual. 

MAC/GMC 4.0 is composite material and laminate analysis software developed at the NASA Glenn 

Research Center. It is based on the generalized method of cells (GMC) micromechanics theory (Paley 

and Aboudi, 1992), which provides access to the local stress and strain fields in the composite material. 

This access grants GMC the ability to accommodate arbitrary local models for inelastic material behavior 

and various types of damage and failure. The MAC/GMC 4.0 software package has been built around 

GMC to provide the theory with a user-friendly framework, along with a library of local inelastic, 

damage, and failure models. Further, application of simulated thermo-mechanical loading, generation of 

output results, and selection of architectures to represent the composite material, have been automated in 

MAC/GMC 4.0. Finally, classical lamination theory has been implemented within MAC/GMC 4.0 

wherein GMC is used to model the composite material response of each ply. Thus, the full range of GMC 

composite material capabilities is available for analysis of arbitrary laminate configurations as well. 

Although the focus of previous versions of MAC/GMC has been on metallic based composites (due to the 

need for inelastic constitutive models for the metallic matrix), the code is fully capable of analyzing 

polymeric and ceramic based composites as well. 

The many new features incorporated within version 4.0 of the MAC/GMC software are enumerated in the 

MAC/GMC 4.0 Keywords Manual Introduction. The two most important new features, from the 

standpoint of analysis capabilities are: 1) the ability to model electromagnetic (ìsmartî) composites and 

laminates, and 2) the incorporation of the new high-fidelity generalized method of cells (HFGMC) 

micromechanics theory. Electromagnetic materials and structures are those that are capable of responding 

electrically or magnetically to imposed thermo-mechanical loads and responding mechanically to 

imposed electrical or magnetic loads. These so-called ìsmartî materials have potential to enable the next 

generation of multi-functional structures that can automatically sense and react to various stimuli. 

Example Problems 7b and 7c illustrate the usage of the MAC/GMC 4.0 electromagnetic capabilities. The 

HFGMC micromechanics theory involves a formulation that enables the micromechanics model to be 

significantly more accurate in its predicted local stress and strain fields compared to GMC. Because this 

improved accuracy naturally places additional computation demands on the MAC/GMC 4.0 code, the 

code may now be considered to have variable fidelity. When computational efficiency is at a premium, 

the standard GMC micromechanics capabilities can be employed. Conversely, when local filed accuracy 

is more important, the HFGMC micromechanics capabilities can be employed. Example Problem 3f 

presents the usage of the HFGMC capabilities of MAC/GMC 4.0. 

Format of this Manual

It is the objective of this manual to illustrate, through example, how to use the MAC/GMC 4.0 software. 

Forty-three individual example problems are presented that describe the usage of most of the codeís 

capabilities. The manual is broken into seven sections. The first section presents four basic examples 

that outline the fundamental procedures used to execute MAC/GMC 4.0 simulations. For new users of 

MAC/GMC, the first step towards learning to use the code should be gaining a good understanding of 

these basic examples. Section 7 deals with advanced topics. Sections 2 ñ 6 contain examples that 

highlight particular capabilities of the code. 

NASA/TM—2002-212077/VOL3 3
MAC/GMC 4.0 Example Problem Manual 

The execution of MAC/GMC 4.0 is controlled by an ASCII input file that contains a number of keywords 

and specifiers. The keywords can be broken into groups that are related to particular capabilities of the 

code. The MAC/GMC 4.0 Keywords are listed with their associated groups and applicable example 

problems in Table I.1. These twenty-two keywords control all aspects of the MAC/GMC 4.0 execution. 

Each is described in detail in the MAC/GMC 4.0 Keywords Manual. The examples presented in Sections 

2 ñ 6 in this MAC/GMC 4.0 Example Problems Manual illustrate capabilities that correspond to the 

keyword groups 2 ñ 6 given in Table I.1. 

Table I.1 MAC/GMC 4.0 keywords and their associated example problems. The problems that 

specifically highlight a particular keyword are hyperlinked (typed in blue and underlined). 

1) Flag-Type Keywords 

*CHECK None 

*CONDUCTIVITY Example 1a

*ELECTROMAG Examples 7b, 7c

2) Material Keywords 

*MDBPATH Example 2h

*CONSTITUENTS All example problems (required keyword) Section 2 Examples

3) Analysis Type and Architecture Keywords 

*RUC Examples 1a, 1c, 1d, 2a-h, 3a, 3b, 3c, 3d, 3e, 3f, 4a-c, 4f, 4h, 5a, 5b, 5d-f, 6a-c, 

7a, 7b, 7d 

*LAMINATE Examples 1b, 3g, 3h, 3i, 4d, 4e, 4g, 5c, 5d, 6c, 7c 

4) Loading Keywords 

*MECH Examples 1c, 1d, 2a-h, 3a-i, 4a, 4b, 4c, 4d, 4e, 4h, 5b-f, 6a, 6b, 7a-c 

*THERM Examples 1d, 2e-h, 3a, 3b, 3d-i, 4a, 4b-f, 4h, 5b-f, 6a-c, 7a-c 

*SOLVER Examples 1c, 1d, 2a, 2b, 2c-h, 3a-e, 3f-i, 4a, 4b-f, 4h, 5b-f, 6a-c, 7a-c 

*SURF Examples 4f, 4g 

5) Damage and Failure Keywords 

*ALLOWABLES Examples 5a, 5b

*FAILURE_SUBCELL Examples 5b, 5c, 5d

*FAILURE_CELL Examples 5c, 5d

*DAMAGE Example 5d

*DEBOND Examples 5e, 5f

*CURTIN Example 5f

6) Results and Data Output Keywords 

*PRINT All example problems 

*XYPLOT Examples 1c, 1d, 2a-h, 3a-i, 4a-d, 4h, 5b, 5d-f, 6a-c, 7a-c 

*PATRAN Example 6b

*MATLAB Examples 3f, 6c

7) End of File Keyword 

*END All example problems (required keyword) 

NASA/TM—2002-212077/VOL3 4
Introduction 

 

Each example problem presented in this manual is associated with a MAC/GMC 4.0 example problem 

input file (all of which are distributed with the MAC/GMC 4.0 software). A general description is given 

for each example problem, followed by a listing of the input file for that example problem. An 

ìAnnotated Input Dataî section is given for each example problem that describes the meaning and 

purpose of each line of data within the input file. Finally, the results of the example problems are 

presented and discussed. The user should be able to reproduce these results using the provided 

MAC/GMC 4.0 example problem input files, upon successful installation of the software package. 

This Example Problem Manual is heavily cross-referenced to the MAC/GMC 4.0 Keywords Manual 

(which gives an in-depth description of each keyword) and the MAC/GMC 4.0 Theory Manual (which 

presents the mathematical models upon which the MAC/GMC 4.0 code is based). Throughout the 

Example Problem Manual, when the annotated input data associated with a particular keyword is given, a 

cross-reference to the Section # in the MAC/GMC 4.0 Keywords Manual is given as: ì[KM_#]î. 

NASA/TM—2002-212077/VOL3 5
MAC/GMC 4.0 Example Problem Manual 

Section 1 : Basic Examples 

In this first section, four basic MAC/GMC 4.0 example problems are presented that introduce many of the 

fundamental concepts associated with the code. Examples that appear in the subsequent sections of this 

manual rely on the userís knowledge and understanding of these basic concepts, and they also borrow 

from the basic problem configurations represented by these four examples. The first two example 

problems do not involve applied loading as the code is used only to determine effective properties for a 

composite and a laminate. The subsequent two problems involve application of mechanical and thermal 

loading on a monolithic metallic alloy and a composite in which the same alloy serves as the matrix 

material. 

NASA/TM—2002-212077/VOL3 6
Section 1: Basic Examples Example 1a: Effective Properties of a Composite 

Example 1a: Effective Properties of a Composite

This example problem determines the effective thermal and mechanical properties of a continuous 

graphite fiber/epoxy matrix composite material. The simplest repeating unit cell (RUC) available within 

MAC/GMC 4.0 that can represent the architecture of this composite, namely a 2×2 RUC (see Figure 1.1), 

is employed. As Figure 1.1 shows, the doubly periodic RUC consists of four subcells in the x2-x3 plane, 

one of which represents the fiber and three of which represent the matrix. The term ìdoubly periodicî 

indicates that the RUC repeats infinitely in the two in-plane (x2-x3) directions and is infinite in the out-ofplane (continuously reinforced, x1) direction. The RUC thus represents a continuum (as opposed to a 

structure with boundaries). Based on the properties and arrangement of the fiber and matrix constituents, 

MAC/GMC 4.0 uses the doubly periodic generalized method of cells (GMC) theory to homogenize the 

composite material and determine the effective (anisotropic) properties of this continuously reinforced 

homogenized material. The constituent material properties in this problem are temperature-independent 

and are read from the MAC/GMC input file. 

Figure 1.1 MAC/GMC 4.0 2×2 doubly periodic repeating unit cell.

MAC/GMC Input File: example_1a.mac

MAC/GMC 4.0 Example 1a - graphite/epoxy effective properties

*CONDUCTIVITY

NTEMP=1 TEMP=21.

*CONSTITUENTS

NMATS=2

# -- Graphite fiber

M=1 CMOD=6 MATID=U MATDB=1 &

EL=388.2E9,7.6E9,0.41,0.45,14.9E9,-0.68E-6,9.74E-6

K=500.,10.

# -- Epoxy matrix

M=2 CMOD=6 MATID=U MATDB=1 &

EL=3.45E9,3.45E9,0.35,0.35,1.278E9,45.E-6,45E-6

K=0.19,0.19

*RUC

MOD=2 ARCHID=1 VF=0.65 F=1 M=2

*PRINT

NPL=-1

*END

! Note: The lines of the input file starting with the ì#î character are comments and thus ignored by the 

code. 

NASA/TM—2002-212077/VOL3 7
MAC/GMC 4.0 Example Problem Manual 

Annotated Input Data

1) Flags: 

a) Determine effective thermal conductivity properties (*CONDUCTIVITY) [KM_1]: 

NTEMP=1 TEMP=21.

Number of conductivity temps: 1 (N TEMP=1) 

Conductivity temps: 21. (TEMP=21.) 

*CONDUCTIVITY is a flag-type keyword that indicates to MAC/GMC 4.0 that effective thermal 

conductivities should be calculated. The thermal conductivity calculation can be performed at any 

desired temperature (specified by NTEMP= and TEMP=). This feature is useful when the constituent 

properties (specified under *CONSTITUENTS) are temperature-dependent. In this example problem, 

the material properties are temperature-independent, so even if additional conductivity temperatures 

were specified, the calculated effective thermal conductivities would be the same for each 

temperature. Note that, in the presence of the *CONDUCTIVITY keyword, thermal conductivities 

for the constituent materials must be specified under *CONSTITUENTS. 

2) Constituent materials (*CONSTITUENTS) [KM_2]: 

NMATS=2

# -- Graphite fiber

M=1 CMOD=6 MATID=U MATDB=1 &

EL=388.2E9,7.6E9,0.41,0.45,14.9E9,-0.68E-6,9.74E-6

K=500.,10.

# -- Epoxy matrix

M=2 CMOD=6 MATID=U MATDB=1 &

EL=3.45E9,3.45E9,0.35,0.35,1.278E9,45.E-6,45.E-6

K=0.19,0.19

Number of materials: 2 (NMATS=2) 

Constitutive models: Elastic (CMOD=6) 

Materials: User-defined (Graphite) (MATID=U) 

 User-defined (Epoxy) (MATID=U) 

Material property source: Read from input file (MATDB=1) 

Material properties: See Table 1.1 (EL=... and K=...) 

Table 1.1 Constituent material properties for example 1a.

EA

(GPa)

ET

(GPa)

νA νT GA

(GPa)

αA

(10-6/ °C)

αT

(10-6/ °C)

κA

(W/mK)

κT 

(W/mK)

Graphite 388.2 7.6 0.41 0.45 14.9 -0.68 9.74 500. 10. 

Epoxy 3.45 3.45 0.35 0.35 1.278 45. 45. 0.19 0.19 

This example problem employs user-defined material properties. In the present case of elastic 

material constitutive behavior (as specified by CMOD=6), the temperature-independent elastic 

properties and coefficients of thermal expansion (CTEs) are simply listed as EL=EA, ET, νA, νT, GA, 

αA, αT (axial and transverse elastic modulus, axial and transverse Poisson ratio, axial shear modulus, 

NASA/TM—2002-212077/VOL3 8
Section 1: Basic Examples Example 1a: Effective Properties of a Composite 

and axial and transverse CTE, respectively). The temperature-independent thermal conductivities 

are specified on a separate line as K=κA, κT (axial and transverse thermal conductivity, respectively). 

Note that, while the K=... specification is required when the *CONDUCTIVITY keyword is present, in 

the absence of *CONDUCTIVITY, K=... should be omitted. In addition, the employed elastic 

constitutive model (CMOD=6) allows for transversely isotropic material properties. That is, it allows 

for specification of different properties in the axial and transverse direction (as specified for the 

graphite fiber, for example). Care should be exercised, however, as CMOD=6 requires that the axial 

direction be associated with the x1-axis (see Figure 1.1). For transversely isotropic elastic constituent 

behavior with axes other than the x1-axis, constitutive model 9 (CMOD=9) can be employed (see 

Keywords Manual Section 2). The isotropic simplification of the employed elastic model (CMOD=6) 

is obtained by setting EA = ET = E, νA = νT = ν, GA = E/2(1+ν) = G, αA = αT = α, and κA = κT. 

In addition to direct specification of the constituent material properties within the MAC/GMC 4.0 

input file, several other options exist for material property specification. These are: material 

properties taken from the codeís internal database, material properties read from an external database, 

and material properties determined from the user-defined subroutine (usrfun.F90). Also, for 

different constitutive models (specified by CMOD=), and for temperature-dependent properties, the 

input requirements are different. These options and requirements are described in Section 2 of the 

MAC/GMC 4.0 Keywords Manual, and example problems involving these options and requirements 

are presented in Section 2 of this Example Problem Manual. 

The units of the material properties employed by MAC/GMC 4.0 are arbitrary. However, the units 

must be consistent in order for the MAC/GMC 4.0 results to be meaningful. It is the userís 

responsibility to ensure that the constituent material property units are consistent. This is particularly 

important when employing constituent materials from the internal material property database. For 

more information on constitutive material property units, see the MAC/GMC 4.0 Keywords Manual 

Section 2. 

3) Analysis type (*RUC) → Repeating Unit Cell Analysis [KM_3]: 

MOD=2 ARCHID=1 VF=0.65 F=1 M=2

Analysis model: Doubly periodic GMC (MOD=2) 

RUC architecture: square fiber, square pack (ARCHID=1) 

Fiber volume fraction: 0.65 (VF=0.65) 

Material assignment: graphite fiber (F=1) 

epoxy matrix (M=2) 

By including *RUC within the input file, repeating unit cell analysis, which represents a continuum 

has been selected. In this example, a repeating unit cell architecture is selected from the MAC/GMC 

internal library by specifying ARCHID=1. This RUC, which is the simplest representation of a 

continuous fiber composite, is shown in Figure 1.1. The additional information required for this RUC 

architecture is the fiber volume fraction and which of the two constituent materials occupy the 

subcells associated with the fiber (F=) and the matrix (M=). Based on the specified fiber volume 

fraction, MAC/GMC 4.0 determines the dimensions of each subcell. Then, given the information on 

which materials occupy which subcells, the code has obtained all of the data (the constituent 

properties and their arrangement) required to homogenize the composite and determine its effective 

properties via the generalized method of cells theory. 

NASA/TM—2002-212077/VOL3 9
MAC/GMC 4.0 Example Problem Manual 

! Note: While only VF=, F=, and M= are required for ARCHID=1, other RUC architectures contained 

within the MAC/GMC 4.0 RUC architecture library require unique input data. Example 

problems dealing with other RUC architectures are presented within Section 3 of this Example 

Manual. For additional information on RUC architectures, see the MAC/GMC 4.0 Keywords 

Manual Section 3. 

4) Loading: None 

As stated earlier, this example problem involves determining effective properties only. Therefore, no 

simulated thermal or mechanical loading need be specified. See Example Problem 1c for an 

introduction on specifying applied loading histories. 

5) Damage and Failure: None 

6) Output: 

a) Output file print level (*PRINT) [KM_6]: 

NPL=-1

Print level: -1 (effective properties only) (NPL=-1) 

The print level generally indicates how much information will be written to the output file. A level of 

0 prints the least amount of information while a level of 10 prints the most. In addition, a special case 

of NPL=-1 indicates that the code should execute only to determine effective properties (as opposed 

to determining the response to simulated applied loading). See the MAC/GMC 4.0 Keywords 

Manual Section 6 for details on the various print levels. 

! Note: If NPL≠-1 at least one of the keywords *MECH and *THERM must be present in the input file. 

b) x-y plots (*XYPLOT): None 

Because no simulated mechanical loading is applied in the present example, no x-y plot data will be 

written to output files. See Example Problem 1c for an introduction to generating x-y plots. 

7) End of file keyword: (*END) 

The *END end of file keyword must be included in all MAC/GMC 4.0 input files. 

Results

In this example problem, since MAC/GMC 4.0 is employed only to determine the effective properties of 

the graphite/epoxy composite, all results are contained within the output file. Recall that (as described in 

the Keywords Manual), if the code is executed using the command: 

mac4 example_1a

NASA/TM—2002-212077/VOL3 10
Section 1: Basic Examples Example 1a: Effective Properties of a Composite 

then the relevant output is written to the ASCII file example_1a.out. The entirety of this output file 

is provided in the appendix, and the results given below were taken directly from this output file. 

The results include the effective thermal conductivities of the 0.65 fiber volume fraction graphite/epoxy 

composite, the effective stiffness matrix (CG), the inverse of the effective compliance matrix (CI), the 

effective engineering moduli (i.e., effective elastic properties), and the effective coefficients of thermal 

expansion. Recall from Figure 1.1 that the axial fiber direction is associated with the x1-axis, while the x2

and x3 directions are transverse to the continuous fiber. The high thermal conductivity, high stiffness, and 

low CTE associated with the axial direction of the fiber constituent are evident in the effective properties 

of the composite associated with the x1 direction. Transverse to the fiber direction, on the other hand, the 

thermal conductivity and stiffness are low while the CTE is high. 

-------------------------------------------------

Effective Thermal Conductivities

At Temperature = 21.0

K11 = 325.0665

K22 = 0.7694

K33 = 0.7694

-------------------------------------------------

--------------------- EFFECTIVE PROPERTIES AT TEMPERATURE = 21.00 ----------------------------

CG - Effective/Macro Stiffness Matrix

0.2571E+12 0.4500E+10 0.4500E+10

0.4500E+10 0.7828E+10 0.3707E+10

0.4500E+10 0.3707E+10 0.7828E+10

0.1916E+10

0.4166E+10

0.4166E+10

CI - Effective/Macro Compliance Matrix

0.3944E-11 -0.1539E-11 -0.1539E-11

-0.1539E-11 0.1653E-09 -0.7738E-10

-0.1539E-11 -0.7738E-10 0.1653E-09

0.5219E-09

0.2400E-09

0.2400E-09

Effective Engineering Moduli

E11S= 0.2535E+12

N12S= 0.3901

E22S= 0.6050E+10

N23S= 0.4682

E33S= 0.6050E+10

G23S= 0.1916E+10

G13S= 0.4166E+10

G12S= 0.4166E+10

Effective Thermal Expansion Coefficients

-0.4724E-06 0.2663E-04 0.2663E-04

------------------------------------------------------------------------------------------------

NASA/TM—2002-212077/VOL3 11
MAC/GMC 4.0 Example Problem Manual 

Example 1b: Effective Properties of a Laminate

This example problem, like Example 1a, involves only determination of effective properties rather than 

determination of the response to an applied loading history. This problem, however, involves a cross-ply 

[90°/0°]s graphite/epoxy laminate rather than a unidirectional graphite/epoxy composite. Since the 

constituent materials are the same, the difference between the input files in this and the previous example 

problem amounts to replacing the *RUC keyword with the *LAMINATE keyword. While the *RUC

keyword specified analysis of a continuum in Example 1a, in the present example the *LAMINATE

keyword specifies analysis of the laminated plate structure (as modeled using lamination theory). 

The geometry and coordinate system employed in the code for the laminate is shown in Figure 1.2. The 

global x-y-z coordinate system is applicable to the laminate as a whole, while the local coordinates (x1-x2-

x3) apply within each layer. As shown, locally, each layer is modeled using a GMC RUC analysis. 

Integration Points (+)

y

x

z

Layer

3

2

1

θ

GMC RUC

x2

x3

Figure 1.2 General laminate geometry and coordinate system employed in MAC/GMC 4.0. The local 

behavior of each layer is modeled using a generalized method of cells (GMC) repeating 

unit cell (RUC) analysis.

NASA/TM—2002-212077/VOL3 12
Section 1: Basic Examples Example 1b: Effective Properties of a Laminate 

MAC/GMC Input File: example_1b.mac

MAC/GMC 4.0 Example 1b - graphite/epoxy laminate

*CONSTITUENTS

NMATS=2

# -- Graphite fiber

M=1 CMOD=6 MATID=U MATDB=1 &

EL=388.2E9,7.6E9,0.41,0.45,14.9E9,-0.68E-6,9.74E-6

# -- Epoxy matrix

M=2 CMOD=6 MATID=U MATDB=1 &

EL=3.45E9,3.45E9,0.35,0.35,1.278E9,45.E-6,45E-6

*LAMINATE

NLY=3

LY=1 MOD=2 THK=0.25 ANG=90 ARCHID=1 VF=0.65 F=1 M=2

LY=2 MOD=2 THK=0.50 ANG=0 ARCHID=1 VF=0.65 F=1 M=2

LY=3 MOD=2 THK=0.25 ANG=90 ARCHID=1 VF=0.65 F=1 M=2

*PRINT

NPL=-1

*END

Annotated Input Data

1) Flags: None 

2) Constituent materials (*CONSTITUENTS) [KM_2]: 

Number of materials: 2 (NMATS=2) 

Constitutive models: Elastic (CMOD=6) 

Materials: User-defined (Graphite) (MATID=U) 

 User-defined (Epoxy) (MATID=U) 

Material property source: Read from input file (MATDB=1) 

Material properties: See Table 1.1 (EL=... and K=...) 

This example problem employs the same user-defined material properties as in Example 1a. 

3) Analysis type (*LAMINATE) → Laminate Analysis [KM_3]: 

NLY=3

LY=1 MOD=2 THK=0.25 ANG=90 ARCHID=1 VF=0.65 F=1 M=2

LY=2 MOD=2 THK=0.50 ANG=0 ARCHID=1 VF=0.65 F=1 M=2

LY=3 MOD=2 THK=0.25 ANG=90 ARCHID=1 VF=0.65 F=1 M=2

Number of layers: 3 (NLY=3) 

Layer Analysis 

Model 

Thickness Fiber 

Angle 

Architecture Volume 

fraction 

Fiber 

material 

Matrix 

material 

(LY=) (MOD) (THK) (ANG) (ARCHID) (VF) (F) (M) 

1 GMC-2D 0.25 90° square fiber, 

square pack 

0.65 graphite epoxy 

2 GMC-2D 0.50 0° square fiber, 

square pack 

0.65 graphite epoxy 

3 GMC-2D 0.25 90° square fiber, 

square pack 

0.65 graphite epoxy 

NASA/TM—2002-212077/VOL3 13
MAC/GMC 4.0 Example Problem Manual 

Because the local behavior of each layer is represented by a GMC repeating unit cell, information 

similar to that specified under *RUC (in Example 1a) is now specified for each layer. Here, each 

layer is represented as the same unidirectional graphite/epoxy composite whose effective properties 

were determined in Example 1a. In addition, for each layer, the thickness (THK) and angle (ANG) in 

degrees must be specified. The thicknesses of the layers is important not only to provide the 

dimensions of the layers with respect to each other, but also to provide the overall thickness of the 

laminate. This overall thickness is important to the calculation of the laminate force and moment 

resultants during simulated applied loading. The angle specifies the orientation of each layerís local 

coordinate system relative to the laminate (global coordinate system). Note that the middle layerís 

thickness is twice that of the outer layers, resulting in a layup equivalent to four equal thickness 

layers. 

Example problems dealing with other laminate configurations are presented within Section 3 of this 

Example Manual. For additional information on the codeís laminate analysis capabilities, see the 

MAC/GMC 4.0 Keywords Manual Section 3 and the MAC/GMC 4.0 Theory Manual Section 3. 

4) Loading: None 

5) Damage and Failure: None 

6) Output: 

a) Output file print level (*PRINT) [KM_6]: 

NPL=-1

Print level: -1 (effective properties only) (NPL=-1) 

b) x-y plots (*XYPLOT): None 

7) End of file keyword: (*END) 

Results

As in Example problem 1a, the results of this example are contained solely within the output file because 

only effective properties (as opposed to the response to applied loading) are determined. The lines below 

are taken directly from the output file. From the GMC repeating unit cell analysis of each layer, the 

effective stiffness matrix (CG), the effective compliance matrix (CI), the effective engineering moduli, 

and the effective CTEs are output (in local coordinates). Then, from the lamination theory analysis, the 

laminate axial stiffness (A), coupling stiffness (B), and bending stiffness (D) matrices are determined, as 

are the effective laminate engineering constants. Examining these results, it is clear that (in the local 

coordinates of each layer) the effective properties of each layer are identical to each other and the 

unidirectional composite results from Example 1a. The laminate results show that, for this symmetric 

cross-ply laminate, there is no coupling between extension and bending (B matrix is zero) and the 

laminate normal and shear behavior is decoupled (A16, A26, D16, and D26 are close to zero). Finally, the 

effective (apparent) elastic moduli of the laminate in the x and y directions are identical as expected. 

NASA/TM—2002-212077/VOL3 14
Section 1: Basic Examples Example 1b: Effective Properties of a Laminate 

--------------------- EFFECTIVE PROPERTIES AT TEMPERATURE = 21.00 ----------------------------

* FOR LAYER NUMBER 1 *

CG - Effective/Macro Stiffness Matrix

0.2571E+12 0.4500E+10 0.4500E+10

0.4500E+10 0.7828E+10 0.3707E+10

0.4500E+10 0.3707E+10 0.7828E+10

0.1916E+10

0.4166E+10

0.4166E+10

CI - Effective/Macro Compliance Matrix

0.3944E-11 -0.1539E-11 -0.1539E-11

-0.1539E-11 0.1653E-09 -0.7738E-10

-0.1539E-11 -0.7738E-10 0.1653E-09

0.5219E-09

0.2400E-09

0.2400E-09

Effective Engineering Moduli

E11S= 0.2535E+12

N12S= 0.3901

E22S= 0.6050E+10

N23S= 0.4682

E33S= 0.6050E+10

G23S= 0.1916E+10

G13S= 0.4166E+10

G12S= 0.4166E+10

Effective Thermal Expansion Coefficients

-0.4724E-06 0.2663E-04 0.2663E-04

--------------------- EFFECTIVE PROPERTIES AT TEMPERATURE = 21.00 ----------------------------

* FOR LAYER NUMBER 2 *

CG - Effective/Macro Stiffness Matrix

0.2571E+12 0.4500E+10 0.4500E+10

0.4500E+10 0.7828E+10 0.3707E+10

0.4500E+10 0.3707E+10 0.7828E+10

0.1916E+10

0.4166E+10

0.4166E+10

CI - Effective/Macro Compliance Matrix

0.3944E-11 -0.1539E-11 -0.1539E-11

-0.1539E-11 0.1653E-09 -0.7738E-10

-0.1539E-11 -0.7738E-10 0.1653E-09

0.5219E-09

0.2400E-09

0.2400E-09

Effective Engineering Moduli

E11S= 0.2535E+12

N12S= 0.3901

NASA/TM—2002-212077/VOL3 15
MAC/GMC 4.0 Example Problem Manual 

E22S= 0.6050E+10

N23S= 0.4682

E33S= 0.6050E+10

G23S= 0.1916E+10

G13S= 0.4166E+10

G12S= 0.4166E+10

Effective Thermal Expansion Coefficients

-0.4724E-06 0.2663E-04 0.2663E-04

--------------------- EFFECTIVE PROPERTIES AT TEMPERATURE = 21.00 ----------------------------

* FOR LAYER NUMBER 3 *

CG - Effective/Macro Stiffness Matrix

0.2571E+12 0.4500E+10 0.4500E+10

0.4500E+10 0.7828E+10 0.3707E+10

0.4500E+10 0.3707E+10 0.7828E+10

0.1916E+10

0.4166E+10

0.4166E+10

CI - Effective/Macro Compliance Matrix

0.3944E-11 -0.1539E-11 -0.1539E-11

-0.1539E-11 0.1653E-09 -0.7738E-10

-0.1539E-11 -0.7738E-10 0.1653E-09

0.5219E-09

0.2400E-09

0.2400E-09

Effective Engineering Moduli

E11S= 0.2535E+12

N12S= 0.3901

E22S= 0.6050E+10

N23S= 0.4682

E33S= 0.6050E+10

G23S= 0.1916E+10

G13S= 0.4166E+10

G12S= 0.4166E+10

Effective Thermal Expansion Coefficients

-0.4724E-06 0.2663E-04 0.2663E-04

----------------------- LAMINATE RESULTS AT TEMPERATURE = 21.00 ------------------------------

-

Laminate Axial Stiffness Matrix [A]

1.303E+11 2.369E+09 -4.748E-01

2.369E+09 1.303E+11 -2.500E+01

-4.748E-01 -2.500E+01 4.166E+09

Laminate Coupling Stiffness Matrix [B]

0.000E+00 0.000E+00 0.000E+00

0.000E+00 0.000E+00 0.000E+00

0.000E+00 0.000E+00 0.000E+00

NASA/TM—2002-212077/VOL3 16
Section 1: Basic Examples Example 1b: Effective Properties of a Laminate 

Laminate Bending Stiffness Matrix [D]

3.093E+09 1.974E+08 -6.924E-02

1.974E+08 1.862E+10 -3.646E+00

-6.924E-02 -3.646E+00 3.472E+08

Laminate Engineering Constants (only valid for symmetric laminates)

Exx= 1.302E+11

Nxy= 1.819E-02

Eyy= 1.302E+11

Gxy= 4.166E+09

NASA/TM—2002-212077/VOL3 17
MAC/GMC 4.0 Example Problem Manual 

Example 1c: Tensile Response of Monolithic Ti-21S

This example problem demonstrates MAC/GMCís capability to apply a loading history to a material and 

determine the response. For the sake of simplicity, the analyzed material is not a composite, but rather 

simply a monolithic titanium alloy commonly used in metal matrix composites; Ti-21S. Unlike the 

previous examples, the material properties for the Ti-21S are taken from the codeís internal material 

property database. An applied strain of 0.02 (i.e., 2%) is applied to the alloy over 200 seconds (sec.), 

resulting in an applied strain rate of 10-4/sec. This example illustrates the basic form of a MAC/GMC 4.0 

input file involving applied mechanical loading and also could be of interest when the response of a 

particular constituent material is desired. 

MAC/GMC Input File: example_1c.mac

MAC/GMC 4.0 Example 1c - monolithic Ti-21S

*CONSTITUENTS

NMATS=1

M=1 CMOD=4 TREF=23. MATID=A

# M=1 CMOD=4 TREF=650. MATID=A

*RUC

MOD=1 M=1

*MECH

LOP=1

NPT=2 TI=0.,200. MAG=0.,0.02 MODE=1

*SOLVER

METHOD=1 NPT=2 TI=0.,200. STP=1.

*PRINT

NPL=6

*XYPLOT

FREQ=5

MACRO=1

NAME=example_1c X=1 Y=7

MICRO=0

*END

Annotated Input Data

1) Flags: None 

2) Constituent materials (*CONSTITUENTS) [KM_2]: 

NMATS=1

M=1 CMOD=4 TREF=23. MATID=A

# M=1 CMOD=4 TREF=650. MATID=A

Number of materials: 1 (NMATS=1) 

Constitutive model: Isotropic GVIPS (CMOD=4) 

Material: Ti-21S (MATID=A) 

Reference Temperature: 23. °C (TREF=23.) 

NASA/TM—2002-212077/VOL3 18
Section 1: Basic Examples Example 1c: Tensile Response of Monolithic Ti-21S 

The reference temperature for a particular constituent material indicates the temperature at which the 

code should evaluate the material properties for that material. This overrides the temperaturedependence of a materialís properties and causes the code to use the temperature indicated by TREF

no matter what temperature the simulation is at. TREF is also useful, as in the current example, when 

no thermal loading is included in the simulation. In this case, without TREF, the code would not 

know at what temperature to take the Ti-21S material properties (and an error would result) since the 

properties stored for Ti-21S in the internal material database vary with temperature. Note that the 

temperature units (°C) are dictated by those used in the internal material database, which are always 

°C. 

In addition, the example input file contains the line: 

# M=1 CMOD=4 TREF=650. MATID=A

Here, the ì#î sign in the first column indicates that this line is a ìcommentî, i.e., it is ignored by the 

code when the input file is read. Thus, by uncommenting this line and commenting out the line 

above, it is possible to execute the exact same case using Ti-21S material properties at 650 °C rather 

than 23 °C. Results for both of these temperatures are given below. 

3) Analysis type (*RUC) → Repeating Unit Cell Analysis [KM_3]: 

MOD=1 M=1

Analysis model: Monolithic material (MOD=1) 

Material assignment: Ti-21S (M=1) 

By including *RUC within the input file, repeating unit cell analysis, which represents a continuum 

has been selected. In this example, a monolithic material is analyzed, and the material to be analyzed 

must be selected from the materials indicated in *CONSTITUENTS. 

4) Loading: 

a) Mechanical (*MECH) [KM_4]: 

LOP=1

NPT=2 TI=0.,200. MAG=0.,0.02 MODE=1

Loading option: 1 (LOP=1) 

Number of points: 2 (NPT=2) 

Time points: 0., 200. sec. (TI=0.,200.) 

Load magnitudes: 0., 0.02 (MAG=0.,0.02) 

Loading mode: strain control (MODE=1) 

The loading option indicates the direction of the applied mechanical loading (see the MAC/GMC 4.0 

Keyword Manual Section 4 for details on the loading options). In this case the mechanical loading is 

applied in the x1-direction, with all other stress components kept at zero. The mechanical loading 

profile is specified through time-magnitude pairs that specify the loading history to be applied (the 

first time must always be zero). In MAC/GMC 4.0, the unit of time is seconds (sec.). Finally, the 

loading mode indicates whether strains (MODE=1) or stresses (MODE=2) are applied. The applied 

mechanical load history for this problem is plotted in Figure 1.3. 

NASA/TM—2002-212077/VOL3 19
MAC/GMC 4.0 Example Problem Manual 

0.000

0.005

0.010

0.015

0.020

0.025

0 50 100 150 200 250

Time (sec.)

Applied Strain

Code stops 

execution when it 

reaches loading 

end time

Figure 1.3 Example 1c: Applied strain vs. time history. 

! Note: The applied global strain rate is 0.02  ̃ 200. sec. = 1×10-4/sec. 

b) Thermal (*THERM): None 

No thermal loading is applied in this example. This necessitates the use of TREF in 

*CONSTITUENTS to indicate the temperature at which the material properties are taken. If a 

thermal loading profile were specified, TREF would not be needed (although it could be specified). 

c) Time integration (*SOLVER) [KM_4]: 

METHOD=1 NPT=2 TI=0.,200. STP=1.

Time integration method: Forward Euler (METHOD=1) 

Number of time points: 2 (NPT=2) 

Time points: 0., 200. sec. (TI=0.,200.) 

Time step size (from 0. ñ 200. sec.): 1. sec. (STP=1.) 

Two explicit methods of time integration are available in MAC/GMC 4.0; Forward Euler 

(METHOD=1) with a specified time step and Predictor/Corrector (METHOD=2) with a self-adaptive 

time step size. In this example, the simpler Forward Euler method is employed. The forward Euler 

time integration is specified by a time point ñ time step size profile. Since the time step size pertains 

to a time segment between two time points, there will always be one fewer time step size specified 

than the number of time points specified. As in *MECH, the first time must always be zero. 

5) Damage and Failure: None 

6) Output: 

a) Output file print level (*PRINT) [KM_6]: 

NPL=6

Print level: 6 (NPL=6) 

NASA/TM—2002-212077/VOL3 20
Section 1: Basic Examples Example 1c: Tensile Response of Monolithic Ti-21S 

The print level indicates how much information will be written to the output file. A level of 0 prints 

the least amount of information while a level of 10 prints the most. A print level of 6 results in an 

intermediate level of writing to the output file including the effective stiffness matrix at each time 

step. See the MAC/GMC 4.0 Keyword Manual Section 6 for details on the various print levels. 

b) x-y plots (*XYPLOT) [KM_6]: 

FREQ=5

MACRO=1

NAME=example_1c X=1 Y=7

MICRO=0

Frequency: 5 (FREQ=5) 

Number of macro plots: 1 (MACRO=1) 

Macro plot name: example_1c (NAME=example_1c) 

Macro plot x-y quantities: ε11, σ11 (X=1 Y=7) 

Number of micro plots: 0 (MICRO=0) 

In this example, a macro (repeating unit cell level) plot file containing the σ11-ε11 stress-strain 

response is generated as indicated by the macro plot x-y quantities X=1, Y=7 (see the MAC/GMC 4.0 

Keyword Manual Section 6 for details on the macro plot x-y quantities). No micro (subcell level) 

plots are generated. The frequency indicates how often data will be written to the x-y plot data file. 

In this case, data will be written every 5 time steps (and thus every 5 seconds of the simulated applied 

loading). The extension ì_macro.dataî is appended to the indicated macro plot file name, so, in 

this example, the file ìEXAMPLE_1c_macro.dataî is written to the path location associated with 

the input file. 

7) End of file keyword: (*END) 

Results

Results are written to both the output file and the x-y plot file in this example. The output file contains all 

input information regarding the case executed, as well as calculated effective properties and detailed 

information at each time step in the loading profile (depending on the print level indicated in *PRINT). 

It is a good idea to check the output file for warnings, which can indicate a problem or unintended result 

of the input data specified. In the present example, the code has generated two warnings based on the use 

of TREF and the lack of thermal loading information. The data written to the x-y plot file have been 

plotted in Figure 1.4. Clearly, the Ti-21S stress-strain response exhibits a significant effect of 

temperature. Given temperature dependent material properties, MAC/GMC 4.0 captures the temperature 

dependence in the results, whether for a monolithic material (as in the present case) of for a composite 

(see Example 1d). 

NASA/TM—2002-212077/VOL3 21
MAC/GMC 4.0 Example Problem Manual 

0

20

40

60

80

100

120

140

160

180

200

0.000 0.005 0.010 0.015 0.020

Strain - ε11

Stress -

σ11 (ksi)

TREF = 23. C

TREF = 650. C

Figure 1.4 Example 1c: plot of the simulated σ11-ε11 stress-strain response of Ti-21S at 23. °C and 650 

°C. Note that the 650 °C results were generated by commenting and uncommenting the 

appropriate lines in the input file under *CONSTITUENTS. Note that the global applied 

strain rate in this example is 1×10-4/sec.

NASA/TM—2002-212077/VOL3 22
Section 1: Basic Examples Example 1d: Tensile/Thermal Response of SiC/Ti-21S 

Example 1d: Tensile/Thermal Response of SiC/Ti-21S

This problem demonstrates several MAC/GMC simulations for a metal matrix composite. The composite 

is 0.25 fiber volume fraction unidirectional SiC/Ti-21S, and it has been represented with the 2×2 doubly 

periodic GMC repeating unit cell shown in Figure 1.1. One subcell represents the SiC fiber while the 

remaining three subcells are associated with the Ti-21S matrix. The material properties for both 

constituents are taken from the internal material database. Several applied loading cases are considered in 

this example. By commenting and uncommenting lines under *MECH and *THERM, results are generated 

for the longitudinal and transverse tensile response of the composite at both 23 °C and 650 °C. 

MAC/GMC Input File: example_1d.mac

MAC/GMC 4.0 Example 1d - SiC/Ti-21S mechanical & thermal loading

*CONSTITUENTS

NMATS=2

M=1 CMOD=6 MATID=E

M=2 CMOD=4 MATID=A

*RUC

MOD=2 ARCHID=1 VF=0.25 F=1 M=2

*MECH

LOP=1

# LOP=2

NPT=2 TI=0.,200. MAG=0.,0.02 MODE=1

*THERM

NPT=2 TI=0.,200. TEMP=23.,23.

# NPT=2 TI=0.,200. TEMP=650.,650.

# NPT=2 TI=0.,200. TEMP=23.,650.

*SOLVER

METHOD=1 NPT=2 TI=0.,200. STP=1.

*PRINT

NPL=6

*XYPLOT

FREQ=5

MACRO=4

NAME=example_1d_11 X=1 Y=7

NAME=example_1d_22 X=2 Y=8

NAME=example_1d_11t X=100 Y=1

NAME=example_1d_22t X=100 Y=2

MICRO=0

*END

Annotated Input Data

1) Flags: None 

2) Constituent materials (*CONSTITUENTS) [KM_2]: 

Number of materials: 2 (NMATS=2) 

Materials: SiC fiber (MATID=E) 

Ti-21S (MATID=A) 

NASA/TM—2002-212077/VOL3 23
MAC/GMC 4.0 Example Problem Manual 

Constitutive models: SiC fiber: linearly elastic (CMOD=6) 

Ti-21S matrix: Isotropic GVIPS (CMOD=4) 

! Note: In contrast to example 1c, TREF is not included in the constituent data for this example. Thus, 

the temperature dependent material property data within the MAC/GMC 4.0 internal material 

database are employed ñ the code determines the material properties at the current temperature 

during the simulation. The temperature history is specified under *THERM. 

3) Analysis type (*RUC) → Repeating Unit Cell Analysis [KM_3]: 

Analysis model: Doubly periodic GMC (MOD=2) 

RUC architecture: square fiber, square pack (ARCHID=1) 

Fiber volume fraction: 0.25 (VF=0.25) 

Material assignment: SiC fiber (F=1) 

Ti-21S matrix (M=2) 

In this case, a repeating unit cell (RUC) architecture is selected from the MAC/GMC 4.0 internal 

library. Further, it is specified which materials from *CONSTITUENTS occupy the subcell 

associated with the fiber and the subcells associated with the matrix in the chosen RUC (see Figure

1.1). 

4) Loading: 

a) Mechanical (*MECH) [KM_4]: 

Loading option: 1 or 2 (LOP=1 or LOP=2) 

Number of points: 2 (NPT=2) 

Time points: 0., 200. sec. (TI=0.,200.) 

Load magnitudes: 0., 0.02 (MAG=0.,0.02) 

Loading mode: strain control (MODE=1) 

Included is the line: 

# LOP=2

By uncommenting this line and commenting the line above (LOP=1), the loading option can be 

switched to apply loading in the transverse (x2) direction rather than the longitudinal (x1) direction. 

As before, in directions other than that of the applied loading, the appropriate stress components are 

kept at zero. 

b) Thermal (*THERM) [KM_4]: 

Number of points: 2 (NPT=2) 

Time points: 0., 200. sec. (TI=0.,200.) 

Temperature points: 23., 23.; 650., 650.; (TEMP=23.,23., TEMP=650.,650., or 

 23., 650. TEMP=23.,650.) 

Much as the applied mechanical loading profile is specified with time-magnitude pairs in *MECH, the 

applied thermal loading profile is specified with time-temperature pairs in *THERM. Included are the 

lines: 

NASA/TM—2002-212077/VOL3 24
Section 1: Basic Examples Example 1d: Tensile/Thermal Response of SiC/Ti-21S 

NPT=2 TI=0.,200. TEMP=23.,23.

# NPT=2 TI=0.,200. TEMP=650.,650.

# NPT=2 TI=0.,200. TEMP=23.,650.

By uncommenting the second line and commenting the first line, the simulation will occur at a 

temperature of 650 °C rather than 23 °C. For the case of applied pure thermal loading, the first two 

lines are commented while the third line is uncommented. This causes the code to apply a simulated 

heat-up from 23 °C to 650 °C. In addition, to eliminate the simulated applied mechanical loading for 

this case, the entirety of the *MECH section must be commented. Thus, for the pure thermal loading 

case, the *MECH ñ *THERM section of the input file should appear as: 

#*MECH

# LOP=1

# LOP=2

# NPT=2 TI=0.,200. MAG=0.,0.02 MODE=1

*THERM

# NPT=2 TI=0.,200. TEMP=23.,23.

# NPT=2 TI=0.,200. TEMP=650.,650.

NPT=2 TI=0.,200. TEMP=23.,650.

c) Time integration (*SOLVER) [KM_4]: 

Time integration method: Forward Euler (METHOD=1) 

Number of time points: 2 (NPT=2) 

Time points: 0., 200. sec. (TI=0.,200.) 

Time step size: 1. sec. (STP=1.) 

5) Damage and Failure: None 

6) Output: 

a) Output file print level (*PRINT) [KM_6]: 

Print level: 6 (NPL=6) 

b) x-y plots (*XYPLOT) [KM_6]: 

Frequency: 5 (FREQ=5) 

Number of macro plots: 4 (MACRO=4) 

Macro plot names: example_1d_11 (NAME=example_1d_11) 

 example_1d_22 (NAME=example_1d_22) 

 example_1d_11t (NAME=example_1d_11t) 

 example_1d_22t (NAME=example_1d_22t) 

Macro plot x-y quantities: ε11, σ11 (X=1 Y=7) 

 ε22, σ22 (X=2 Y=8) 

 temperature, ε11 (X=100 Y=1) 

 temperature, ε22 (X=100 Y=2) 

Number of micro plots: 0 (MICRO=0) 

In this example, four macro (repeating unit cell level) x-y plot files are generated, one for the σ11-ε11

stress-strain response, one for the σ22-ε22 stress-strain response, one for the temperature-ε11 response, 

and one for the temperature-ε22 response. For the case of longitudinal applied loading (LOP=1) and 

 

NASA/TM—2002-212077/VOL3 25
MAC/GMC 4.0 Example Problem Manual 

the thermal loading case, the σ22 values written to the x-y plot file will all be zero. Similarly, for the 

case of transverse applied loading (LOP=2) and the thermal loading case, the σ11 values written to the 

x-y plot file will all be zero. 

7) End of file keyword: (*END) 

Results

Figure 1.5 presents plots of the composite response to the applied global longitudinal (along fiber the 

fiber direction) and transverse (perpendicular to the fiber direction) (see Figure 1.1) strain loading at both 

23 °C and 650 °C. The right hand side of the plot represents the stress-strain response in the direction of 

the applied loading, while the left hand side of the plot shows the strain response normal to the loading 

direction (i.e., the Poisson effect). These results show that the composite is significantly stiffer in the 

longitudinal fiber direction compared to the direction transverse to the fibers. When the loading is 

applied transverse to the fiber direction, the resulting Poisson effect strain is small due to the presence of 

the continuous fibers normal to the loading direction. At elevated temperature, the composite is much 

softer and exhibits more inelastic deformation than at room temperature. Figure 1.6, which is a plot of the 

compositeís longitudinal and transverse thermal response to a globally stress free heat up, shows that the 

composite exhibits greater strain during the thermal loading in the transverse direction compared to the 

longitudinal direction. 

0

50

100

150

200

250

300

350

400

450

-0.010 -0.005 0.000 0.005 0.010 0.015 0.020

Strain

Stress (ksi)

Longitudinal @ 23 C

Transverse @ 23 C

Longitudinal @ 650 C

Transverse @ 650 C

Response in loading direction

Response normal 

to loading 

(Poisson effect)

Figure 1.5 Example 1d: plot of the simulated longitudinal (along the fiber direction) and transverse 

(perpendicular to the fiber direction) stress-strain response of a 25% SiC/Ti-21S composite 

at 23. °C and 650 °C. The strain response normal to the loading direction (i.e., Poisson 

effect) is also plotted. 

NASA/TM—2002-212077/VOL3 26
Section 1: Basic Examples Example 1d: Tensile/Thermal Response of SiC/Ti-21S 

0

0.001

0.002

0.003

0.004

0.005

0.006

0 100 200 300 400 500 600 700

Temperature (°C)

Strain

ε11

ε22

Figure 1.6 Example 1d: plot of the simulated longitudinal and transverse temperature-strain response 

of a 25% SiC/Ti-21S. 

NASA/TM—2002-212077/VOL3 27
MAC/GMC 4.0 Example Problem Manual 

Section 2 : Constituent Materials 

This section presents example problems intended to illustrate MAC/GMCís capabilities with respect to 

constituent materials. Constituent materials are the building blocks of the composite materials that the 

code is designed to analyze. Constituent materials occupy the individual subcells that constitute the 

repeating unit cell used by the micromechanics models to represent the composite. An important concept 

related to constituent materials is that of a constitutive model. By definition, associated with each 

constituent material is a constitutive model. The constitutive model is what determines the reaction of the 

material to the stresses and strains experienced by the material during a MAC/GMC simulation. Then, 

based on the reaction of the constituent materials in each subcell, the micromechanics model determines 

the reaction of the composite as a whole. 

The basic form of the material constitutive equation employed in MAC/GMC 4.0 can be written, 

( ) TI σ C −−= εεε

where σ is the stress vector, C is the material stiffness matrix, ε is the strain vector, I ε is the inelastic 

strain vector, and T ε is the thermal strain vector. The function of the constitutive model is essentially, 

given σ (or ε) for the material at point, to determine ε (or σ) at that point. Thus, the constitutive 

model for a material must include information on the stiffness matrix of the material (C ), as well as the 

inelastic and thermal strains ( I ε and T ε ). Note that MAC/GMC employs this engineering notation 

wherein the stress and strain tensors are reduced to vectors. The ordering of the components within these 

vectors is: σ = (σ11, σ22, σ33, σ23, σ13, σ12) and = ( 11, 22, 33, γ23, γ13, γ12), where γij are the engineering 

shear strain components. 

The stiffness matrix and thermal strains within the constitutive equation written above are fairly straight 

forward. The elastic properties of the material determine the stiffness matrix, while the material 

coefficient of thermal expansion (CTE), in conjunction with the temperature history, determines the 

thermal strains. For a linearly elastic material, this is all that is required from the constitutive model. For 

a material that undergoes inelastic deformation, like the Ti-21S material employed in Example Problems 

1c and 1d, the constitutive model must also determine the inelastic strains. This is usually much more 

complex as inelastic strains typically exhibit history/time/rate dependence, which demands a great deal 

from the constitutive model, particularly for metals. 

MAC/GMC 4.0 includes an internal database of constituent materials, each of which is associated with a 

particular constitutive model. In Example Problem 1d, two internal database materials, each associated 

with a different constitutive model, were employed. The Ti-21S matrix material was associated with the 

isotropic GVIPS viscoplastic constitutive model, while the SiC fiber material was associated with the 

linearly elastic constitutive model. In Example Problems 1a and 1b, although user-defined material 

properties were specified for the graphite and epoxy constituent materials, the internal linear elastic 

constitutive model (CMOD=6) was employed for both materials. Section 2 of this manual presents 

example problems involving many of the other constituent materials and constitutive models contained 

within MAC/GMC 4.0. The code also includes the ability for users to employ their own material 

properties with an internal constitutive model. Further, users may also implement their own material 

constitutive model through MAC/GMCís user-defined subroutines. Finally, the users may employ 

MAC/GMCís external material database to store and keep track of their material parameters using a 

simple ASCII interface. 

NASA/TM—2002-212077/VOL3 28

ε ε ε ε
Section 2: Constituent Materials Example 2a: Bodner-Partom Viscoplastic Constitutive Model 

Example 2a: Bodner-Partom Viscoplastic Constitutive Model

This example problem generates the room-temperature tensile stress-strain response of the monolithic 

constituent materials within the MAC/GMC 4.0 internal material database that are associated with the 

Bodner-Partom viscoplastic constitutive model. The input file is similar to that employed in Example 1c, 

with changes associated with the constituent materials. As in Example 1c, this problem examines the 

response of monolithic inelastic materials and employs an applied strain rate of 10-4/sec. 

MAC/GMC Input File: example_2a.mac

MAC/GMC 4.0 Example 2a - Bodner-Partom Viscoplastic Constitutive Model

*CONSTITUENTS

NMATS=7

# -- Al 2024-T4

M=1 CMOD=1 TREF=23. MATID=A

# -- Al 2024-0

M=2 CMOD=1 TREF=23. MATID=B

# -- Al 6061-0 (a)

M=3 CMOD=1 TREF=23. MATID=C

# -- Al 6061-0 (b)

M=4 CMOD=1 TREF=23. MATID=D

# -- Al pure

M=5 CMOD=1 TREF=23. MATID=E

# -- Ti pure

M=6 CMOD=1 TREF=23. MATID=F

# -- Cu

M=7 CMOD=1 TREF=23. MATID=G

*RUC

MOD=1 M=1

# MOD=1 M=2

# MOD=1 M=3

# MOD=1 M=4

# MOD=1 M=5

# MOD=1 M=6

# MOD=1 M=7

*MECH

LOP=1

NPT=2 TI=0.,200. MAG=0.,0.02 MODE=1

*SOLVER

METHOD=1 NPT=2 TI=0.,200. STP=0.025

*PRINT

NPL=6

*XYPLOT

FREQ=10

MACRO=1

NAME=example_2a X=1 Y=7

MICRO=0

*END

NASA/TM—2002-212077/VOL3 29
MAC/GMC 4.0 Example Problem Manual 

Annotated Input Data

1) Flags: None 

2) Constituent materials (*CONSTITUENTS) [KM_2]: 

Number of materials: 7 (NMATS=7) 

Constitutive model: Bodner-Partom (CMOD=1) 

Materials: Al 2024-T4, Al 2024-0, (MATID=A-G) 

Al 6061-0 (a), Al 6061-0 (b), 

Al pure, Ti pure, Cu 

Reference Temperature: 23. °C (TREF=23.) 

3) Analysis type (*RUC) → Repeating Unit Cell Analysis [KM_3]: 

Analysis model: Monolithic material (MOD=1) 

Material assignment: Each constituent successively (M=1-7) 

! Note: Each material in *CONSTITUENTS is assigned to the monolithic material successively by 

commenting and uncommenting the appropriate lines for separate executions of the code. 

4) Loading: 

a) Mechanical (*MECH) [KM_4]: 

Loading option: 1 (LOP=1) 

Number of points: 2 (NPT=2) 

Time points: 0., 200. sec. (TI=0.,200.) 

Load magnitudes: 0., 0.02 (MAG=0.,0.02) 

Loading mode: strain control (MODE=1) 

b) Thermal (*THERM): None 

c) Time integration (*SOLVER) [KM_4]: 

Time integration method: Forward Euler (METHOD=1) 

Number of time points: 2 (NPT=2) 

Time points: 0., 200. sec. (TI=0.,200.) 

Time step size: 0.025 sec. (STP=0.025) 

! Note: A very small time step size is employed in this example (0.025 sec.) compared to that used in 

Examples 1c and 1d, which employed the isotropic GVIPS constitutive model (1. sec.). The 

smaller time step required for convergence of the forward Euler integration scheme is due to 

the numerically stiff nature of the Bodner-Partom model equations. A smaller time step is 

directly associated with increased execution time, thus, in many cases it is preferable to employ 

the alternative time integration method, the predictor-corrector (METHOD=2) when utilizing the 

Bodner-Partom model as this integration scheme allows for a variable time step. See Example 

4h, which employs this predictor-corrector method, for more information. 

5) Damage and Failure: None 

6) Output: 

a) Output file print level (*PRINT) [KM_6]: 

NASA/TM—2002-212077/VOL3 30
Section 2: Constituent Materials Example 2a: Bodner-Partom Viscoplastic Constitutive Model 

Print level: 0 (NPL=0) 

! Note: A print level of 0 results in minimal output being written to the output file. 

b) x-y plots (*XYPLOT) [KM_6]: 

Frequency: 10 (FREQ=10) 

Number of macro plots: 1 (MACRO=1) 

Macro plot name: example_2a (NAME=example_2a) 

Macro plot x-y quantities: ε11, σ11 (X=1 Y=7) 

Number of micro plots: 0 (MICRO=0) 

7) End of file keyword: (*END) 

Results

This example problem illustrates how MAC/GMC 4.0 can be used to quickly generate the response of the 

materials within the internal material database. By altering the materials in *CONSTITUENTS, the user 

can use the example 2a input file to generate the response of other materials within the codeís internal 

material database at any temperature desired. 

0

50

100

150

200

250

300

350

400

0.000 0.005 0.010 0.015 0.020 0.025

Strain

Stress (MPa)

Al 2024-T4 {MATID=A}

Ti pure {MATID=F}

Al 6061-0 (a) {MATID=C}

Cu {MATID=G}

Al 2024-0 {MATID=B}

Al 6061-0 (b) {MATID=D}

Al pure 

{MATID=E}

Figure 2.1 Example 2a: plots of the room-temperature tensile stress-strain response of the seven 

materials within the MAC/GMC 4.0 internal material database that are associated with the 

Bodner-Partom viscoplastic constitutive model. 

NASA/TM—2002-212077/VOL3 31
MAC/GMC 4.0 Example Problem Manual 

Example 2b: Strain-Rate Dependence of Ti-21S

This example problem examines the elevated temperature strain rate (SR) dependence of two different 

constitutive models for the same material, namely Ti-21S. Ti-21S is associated with both the isotropic 

GVIPS viscoplastic constitutive model and the modified Bodner-Partom (MBP) viscoplastic constitutive 

model in the MAC/GMC 4.0 internal material database (see the Keyword Manual Section 2). Both 

models incorporate strain rate dependence, but, as this example shows, the elevated temperature inelastic 

behavior of Ti-21S predicted by each of these models is different. The applied strain rate is altered from 

10-4/sec. to 10-5/sec. to 10-6/sec. by commenting and uncommenting the appropriate lines under *MECH

and *SOLVER. 

MAC/GMC Input File: example_2b.mac

MAC/GMC 4.0 Example 2b - Strain Rate Dependence of Ti-21S

*CONSTITUENTS

NMATS=2

# -- Ti-21S Isotropic GVIPS

M=1 CMOD=4 TREF=650. MATID=A

# -- Ti-21S MBP

M=2 CMOD=2 TREF=650. MATID=A

*RUC

MOD=1 M=1

# MOD=1 M=2

*MECH

LOP=1

NPT=2 TI=0.,20. MAG=0.,0.02 MODE=1

# NPT=2 TI=0.,200. MAG=0.,0.02 MODE=1

# NPT=2 TI=0.,2000. MAG=0.,0.02 MODE=1

*SOLVER

METHOD=1 NPT=2 TI=0.,20. STP=0.0025

# METHOD=1 NPT=2 TI=0.,200. STP=0.025

# METHOD=1 NPT=2 TI=0.,2000. STP=0.25

*PRINT

NPL=0

*XYPLOT

FREQ=200

MACRO=1

NAME=example_2b X=1 Y=7

MICRO=0

*END

Annotated Input Data

1) Flags: None 

2) Constituent materials (*CONSTITUENTS) [KM_2]: 

Number of materials: 2 (NMATS=2) 

Constitutive models: Isotropic GVIPS (CMOD=4) 

 Modified Bodner-Partom (CMOD=2) 

NASA/TM—2002-212077/VOL3 32
Section 2: Constituent Materials Example 2b: Strain-Rate Dependence of Ti-21S 

Materials: Ti-21S (MATID=A) 

Reference Temperature: 650. °C (TREF=650.) 

3) Analysis type (*RUC) → Repeating Unit Cell Analysis [KM_3]: 

Analysis model: Monolithic material (MOD=1) 

Material assignment: Each constituent successively (M=1,2) 

Each of the two materials in *CONSTITUENTS is assigned to the monolithic material successively 

by commenting and uncommenting the appropriate lines. 

4) Loading: 

a) Mechanical (*MECH) [KM_4]: 

Loading option: 1 (LOP=1) 

Number of points: 2 (NPT=2) 

Time points: 0., 20. sec. (TI=0.,20.) 

 0., 200. sec. (TI=0.,200.) 

 0., 2000. sec. (TI=0.,2000.) 

Load magnitude: 0., 0.02 (MAG=0.,0.02) 

Loading mode: strain control (MODE=1) 

! Note: By altering the time points in the mechanical loading history, the global strain rate is decreased 

from 10-3/sec. to 10-4/sec. to 10-5/sec. 

b) Thermal (*THERM): None 

c) Time integration (*SOLVER) [KM_4]: 

Time integration method: Forward Euler (METHOD=1) 

Number of time points: 2 (NPT=2) 

Time points: 0., 20. sec. (TI=0.,20.) 

 0., 200. sec. (TI=0.,200.) 

 0., 2000. sec. (TI=0.,2000.) 

Time step sizes: 0.0025 sec. (STP=0.0025) 

 0.025 sec. (STP=0.025) 

 0.25 sec. (STP=0.25) 

As in Example 2a, the very small time step sizes employed in this example are due to the stiff nature 

of the modified Bodner-Partom equations. A much larger step size can be used for the GVIPS cases 

presented in this example. 

 

5) Damage and Failure: None 

6) Output: 

a) Output file print level (*PRINT) [KM_6]: 

Print level: 0 (NPL=0) 

b) x-y plots (*XYPLOT): 

Frequency: 200 (FREQ=200) 

Number of macro plots: 1 (MACRO=1) 

NASA/TM—2002-212077/VOL3 33
MAC/GMC 4.0 Example Problem Manual 

Macro plot name: example_2b (NAME=example_2b) 

Macro plot x-y quantities: ε11, σ11 (X=1 Y=7) 

Number of micro plots: 0 (MICRO=0) 

7) End of file keyword: (*END) 

Results

The results for this example problem are shown in Figure 2.2. While the qualitative effect of changing 

the strain rate is similar for both Ti-21S constitutive models, the predicted stress-strain curves at each 

strain rate are somewhat different quantitatively. This demonstrates that the different constitutive models 

within MAC/GMC 4.0 give different results, even for the same material. For an illustration of the impact 

of these types of constitutive model differences, see Bednarcyk and Arnold (2002). 

0

10

20

30

40

50

60

70

80

0.000 0.005 0.010 0.015 0.020 0.025

Strain

Stress (ksi)

Isotropic GVIPS

MBP

SR = 0.001 /s

SR = 0.0001 /s

SR = 0.00001 /s

Figure 2.2 Example 2b: plots of the tensile stress-strain response of Ti-21S at 650 °C as modeled by 

the isotropic GVIPS and modified Bodner-Partom (MBP) constitutive models as a function 

of the applied strain rate (SR).

NASA/TM—2002-212077/VOL3 34
Section 2: Constituent Materials Example 2c: Incremental Plasticity 

Example 2c: Incremental Plasticity

This example problem generates the room-temperature tensile response of the materials associated with 

the incremental plasticity constitutive model within the MAC/GMC 4.0 internal material database. 

Incremental plasticity represents a special case for the code in that, due to the presence of a yield surface, 

global iterations must occur at each increment of the applied loading in order to ensure that the plastically 

deforming material remains on the yield surface. Further, while the incremental plasticity constitutive 

model is history-dependent, the modeled inelastic behavior is independent of strain rate. The inelastic 

behavior of the material is defined by a yield stress and a number of stress-total strain pairs that dictate 

the path of the stress-strain response after yielding. These stress-total strain pairs can be obtained from 

experimental stress-strain curves for the material. In the simplest case, if only one post-yield stress-strain 

pair is specified, bilinear plasticity results. If more than one post-yield stress-strain pair is specified, 

point-wise plasticity results. The MAC/GMC 4.0 internal material database contains materials with 

bilinear incremental plasticity responses and two materials with point-wise incremental plasticity 

responses. This example problem considers the monolithic materials and applies strain at a rate of 

10-4/sec. 

MAC/GMC Input File: example_2c.mac

MAC/GMC 4.0 Example 2c - Incremental Plasticity

*CONSTITUENTS

NMATS=4

# -- OFHC Copper - Bilinear

M=1 CMOD=21 TREF=23. MATID=A

# -- Ti-24-11 - Bilinear

M=2 CMOD=21 TREF=23. MATID=B

# -- Ti-15-3 - Point-wise

M=3 CMOD=21 TREF=23. MATID=C

# -- Ti-24-11 - Point-wise

M=4 CMOD=21 TREF=23. MATID=D

*RUC

MOD=1 M=1

# MOD=1 M=2

# MOD=1 M=3

# MOD=1 M=4

*MECH

LOP=1

NPT=2 TI=0.,20. MAG=0.,0.02 MODE=1

*SOLVER

METHOD=1 NPT=2 TI=0.,20. STP=0.1 ITMAX=20 ERR=0.0001

*PRINT

NPL=6

*XYPLOT

FREQ=5

MACRO=1

NAME=example_2c X=1 Y=7

MICRO=0

*END

NASA/TM—2002-212077/VOL3 35
MAC/GMC 4.0 Example Problem Manual 

Annotated Input Data

1) Flags: None 

2) Constituent materials (*CONSTITUENTS) [KM_2]: 

Number of materials: 4 (NMATS=4) 

Constitutive models: Incremental plasticity (CMOD=21) 

Materials: OFHC Cu ñ bilinear (MATID=A) 

Ti-24-11 ñ bilinear (MATID=B) 

Ti-15-3 ñ point-wise (MATID=C) 

Ti-24-11 ñ point-wise (MATID=D) 

Reference Temperature: 23. °C (TREF=23.) 

3) Analysis type (*RUC) → Repeating Unit Cell Analysis [KM_3]: 

Analysis model: Monolithic material (MOD=1) 

Material assignment: Each constituent successively (M=1-4) 

Each of the materials in *CONSTITUENTS is assigned to the monolithic material successively by 

commenting and uncommenting the appropriate lines. 

4) Loading: 

a) Mechanical (*MECH) [KM_4]: 

Loading option: 1 (LOP=1) 

Number of points: 2 (NPT=2) 

Time points: 0., 20. sec. (TI=0.,20.) 

Load magnitude: 0., 0.02 (MAG=0.,0.02) 

Loading mode: strain control (MODE=1) 

b) Thermal (*THERM): None 

c) Time integration (*SOLVER) [KM_4]: 

Time integration method: Forward Euler (METHOD=1) 

Number of time points: 2 (NPT=2) 

Time points: 0., 20. sec. (TI=0.,20.) 

Time step sizes: 0.1 sec. (STP=0.1) 

Max. number of iterations 20 (ITMAX=20) 

Max. permitted error fraction 0.0001 (ERR=0.0001) 

Due to the necessity of iteration at each time step in the simulation, two additional specifiers are 

required under *SOLVER when incremental plasticity is employed for a constituent material. These 

are the maximum number of iterations permitted at a particular time step (ITMAX) and the maximum 

permitted error fraction for convergence (ERR). The maximum number of iterations places a ceiling 

on the number of iterations at a particular time step. Once this number of iterations is reached, the 

code will advance to the next time step regardless of whether or not convergence has occurred (in 

which case a warning will be written to the output file). Generally, this non-convergent situation 

should be avoided by using a large value for ITMAX. The maximum permitted error fraction is the 

fractional change in the local effective plastic strain increment between iterations that can be 

considered ìsmall enoughî. That is, for example, if the largest fractional change in the effective 

NASA/TM—2002-212077/VOL3 36
Section 2: Constituent Materials Example 2c: Incremental Plasticity 

plastic strain increment over all subcells at a particular time step is 0.00009, and ERR has been set to 

0.0001, convergence is considered to have been achieved for that time step. The appropriate value 

for ERR is case dependent, and there is an inverse relationship between the time step size and ERR. If 

oscillations are present in the results, a lower value of ERR and/or a smaller time step should be 

employed. 

 

5) Damage and Failure: None 

6) Output: 

a) Output file print level (*PRINT) [KM_6]: 

Print level: 6 (NPL=6) 

b) x-y plots (*XYPLOT) [KM_6]: 

Frequency: 5 (FREQ=5) 

Number of macro plots: 1 (MACRO=1) 

Macro plot name: example_2c (NAME=example_2c) 

Macro plot x-y quantities: ε11, σ11 (X=1 Y=7) 

Number of micro plots: 0 (MICRO=0) 

7) End of file keyword: (*END) 

Results

The results for this example problem are plotted in Figure 2.3. The difference between bilinear and pointwise representations of the material stress-strain response is clear. 

 

0

100

200

300

400

500

600

700

800

900

1000

0.000 0.005 0.010 0.015 0.020 0.025

Strain

Stress (MPa)

OFHC Cu - Bilinear [MATID=A]

Ti-24-11 - Bilinear [MATID=B]

Ti-15-3 - Point-wise [MATID=C]

Ti-24-11 Point-wise [MATID=D]

Figure 2.3 Example 2c: plots of the room-temperature tensile stress-strain response of OFHC Cu, Ti-24-

11 (bilinear), Ti-15-3, and Ti-24-11 (point-wise) as modeled by the incremental plasticity 

constitutive model. 

NASA/TM—2002-212077/VOL3 37
MAC/GMC 4.0 Example Problem Manual 

Example 2d: Shape Memory Alloy (SMA)

This example problem generates the tensile response of a NiTi shape memory alloy (SMA), using a 

specially designed constitutive model, which accounts for phase changes within the material. This model 

is a new capability within MAC/GMC 4.0. Cyclic strain-controlled mechanical loading is applied to the 

monolithic SMA at a rate of 0.001/sec. 

MAC/GMC Input File: example_2d.mac

MAC/GMC 4.0 Example 2d - Shape Memory Alloy

*CONSTITUENTS

NMATS=1

M=1 CMOD=30 TREF=71.1 MATID=A

# M=1 CMOD=30 TREF=2.2 MATID=A

*RUC

MOD=1 M=1

*MECH

LOP=1

NPT=3 TI=0.,45.,90. MAG=0.,0.045,0. MODE=1,1

*SOLVER

METHOD=1 NPT=3 TI=0.,45.,90. STP=0.25,0.25

*PRINT

NPL=6

*XYPLOT

FREQ=1

MACRO=1

NAME=example_2d X=1 Y=7

MICRO=0

*END

Annotated Input Data

1) Flags: None 

2) Constituent materials (*CONSTITUENTS) [KM_2]: 

Number of materials: 1 (NMATS=1) 

Constitutive models: Graesser ñ Cozzarelli (CMOD=30) 

 ñ Witting SMA model 

Materials: NiTi SMA (MATID=A) 

Reference Temperatures: 71.1. °C, 2.2 °C (TREF=71.1 or TREF=2.2) 

3) Analysis type (*RUC) → Repeating Unit Cell Analysis [KM_3]: 

Analysis model: Monolithic material (MOD=1) 

Material assignment: SMA (M=1) 

4) Loading: 

a) Mechanical (*MECH) [KM_4]: 

NASA/TM—2002-212077/VOL3 38
Section 2: Constituent Materials Example 2d: Shape Memory Alloy (SMA) 

Loading option: 1 (LOP=1) 

Number of points: 3 (NPT=3) 

Time points: 0., 45., 90. sec. (TI=0.,45.,90.) 

Load magnitude: 0., 0.045, 0. (MAG=0.,0.045,0.) 

Loading mode: strain control (MODE=1,1) 

In order to illustrate the unique features of the SMA constitutive model, a complete mechanical 

loading and unloading cycle has been employed. 

b) Thermal (*THERM): None 

c) Time integration (*SOLVER) [KM_4]: 

Time integration method: Forward Euler (METHOD=1) 

Number of time points: 3 (NPT=3) 

Time points: 0., 45., 90. sec. (TI=0.,45.,90.) 

Time step sizes: 0.25, 0.25 sec. (STP=0.25,0.25) 

5) Damage and Failure: None 

6) Output: 

a) Output file print level (*PRINT) [KM_6]: 

Print level: 6 (NPL=6) 

b) x-y plots (*XYPLOT) [KM_6]: 

Frequency: 5 (FREQ=5) 

Number of macro plots: 1 (MACRO=1) 

Macro plot name: example_2d (NAME=example_2d) 

Macro plot x-y quantities: ε11, σ11 (X=1 Y=7) 

Number of micro plots: 0 (MICRO=0) 

7) End of file keyword: (*END) 

Results

Figure 2.4 shows that, at both temperatures, the implemented constitutive model captures the superelastic 

behavior of the NiTi SMA. That is, upon mechanical unloading, the SMA returns to its original shape. 

This ìshape memoryî behavior is due to stress-induced phase transformations that occur in the NiTi 

during loading and unloading, as modeled by the Graesser ñ Cozzarelli ñ Witting SMA constitutive 

model. Upon complete mechanical unloading, the stress-strain curves return to the origin, indicating that 

no permanent or irreversible deformation has occurred. 

NASA/TM—2002-212077/VOL3 39
MAC/GMC 4.0 Example Problem Manual 

0

100

200

300

400

500

600

700

800

900

1000

0.00 0.01 0.02 0.03 0.04 0.05

Strain

Stress (MPa)

71.1 °C

2.2 °C

Figure 2.4 Example 2d: plots of the tensile stress-strain response of the NiTi SMA as simulated using 

the Graesser ñ Cozzarelli ñ Witting constitutive model at 71.1 °C and 2.2 °C.

NASA/TM—2002-212077/VOL3 40
Section 2: Constituent Materials Example 2e: User-Defined Material Properties ñ Input File 

Example 2e: User-Defined Material Properties ñ Input File

This example problem shows how users can specify their own constituent material properties for internal 

MAC/GMC 4.0 constitutive models through the codeís input file. Note that Example Problems 1a and 1b 

touched on this subject. The present example includes ten materials in order to illustrate the input format 

for the elastic, Bodner-Partom, isotropic GVIPS, and incremental plasticity constitutive models (resulting 

in a large input file). To switch among the materials in separate code executions, the appropriate material 

number is specified under *RUC. A strain rate of 10-4/sec. is employed, and results are generated for at a 

temperature of 500 °C. Another way in which the user can specify constituent material properties is 

through the use of a user-defined subroutine, which can calculate the material properties as a function of 

temperature or other quantity during code execution. This type of problem is illustrated in Example 2f. 

In the present example, the temperature-dependent or temperature-independent material properties are 

typed directly into the input file. It should be noted that the input file format for user-defined material 

properties is slightly different for each material constitutive model (as necessitated by the different 

material parameters needed for each constitutive model). See Section 2 of the MAC/GMC 4.0 Keywords 

Manual for more information on user-defined material properties. 

MAC/GMC Input File: example_2e.mac

MAC/GMC 4.0 Example 2e - User-Defined Material Properties

*CONSTITUENTS

NMATS=10

# -- Elastic; Temp-independent

M=1 CMOD=6 MATID=U MATDB=1 &

EL=58.E3,58.E3,0.20,0.20,24.17E3,6.3E-6,6.3E-6

# -- Elastic; Temp-dependent

M=2 CMOD=6 MATID=U MATDB=1

NTP=4

TEM=18.,200.,400.,600.

EA=45.6E3,43.5E3,40.6E3,29.0E3

ET=45.6E3,43.5E3,40.6E3,29.0E3

NUA=0.41,0.41,0.41,0.41

NUT=0.41,0.41,0.41,0.41

GA=16.154E3,15.429E3,14.400E3,10.286E3

ALPA=4.5E-6,4.8E-6,5.1E-6,5.5E-6

ALPT=4.5E-6,4.8E-6,5.1E-6,5.5E-6

# -- Bodner-Partom; Temp-independent

M=3 CMOD=1 MATID=U MATDB=1 &

EL=9.53E3,9.53E3,0.33,0.33,3.58E3,21.06E-6,21.06E-6 &

VI=1.E4,49.,63.,300.,4.,1.

# -- Bodner-Partom; Temp-dependent

M=4 CMOD=1 MATID=U MATDB=1

NTP=2

TEM=18.,700.

EA=9.53E3,4.12E3

ET=9.53E3,4.12E3

NUA=0.41,0.41

NUT=0.41,0.41

GA=3.58E3,1.46E3

ALPA=21.06E-6,28.92E-6

ALPT=21.06E-6,28.92E-6

NASA/TM—2002-212077/VOL3 41
MAC/GMC 4.0 Example Problem Manual 

V1=1.E4,1.E4

V2=49.,49.

V3=63.,63.

V4=300.,300.

V5=4.,2.5

V6=1.,1.

# -- Isotropic GVIPS; Temp-independent

M=5 CMOD=4 MATID=U MATDB=1 &

EL=14009.,14009.,0.365,0.365,5131.5,5.862E-6,5.862E-6 &

VI=0.000999275,44.960,1.679E-07,2.494561E-05,0.05, &

3.3,1.8,1.35,0.85,3.0183E-7,0.001

# -- Isotropic GVIPS; Temp-dependent

M=6 CMOD=4 MATID=U MATDB=1

NTP=6

TEM=25.,300.,482.,565.,650.,704.

EA=16.5506E+03,15.644E+03,14.009E+03,12.968E+03,11.702E+03,10.793E+03

ET=16.5506E+03,15.644E+03,14.009E+03,12.968E+03,11.702E+03,10.793E+03

NUA=0.365,0.365,0.365,0.365,0.365,0.365

NUT=0.365,0.365,0.365,0.365,0.365,0.365

GA=6062.491,5730.403,5131.502,4750.183,4286.477,3953.480

ALPA=4.2921E-06,5.116E-06,5.862E-06,6.271E-06,6.741E-06,7.07E-06

ALPT=4.2921E-06,5.116E-06,5.862E-06,6.271E-06,6.741E-06,7.07E-06

V1=99.927,20.015,0.000999275,8.499E-08,8.E-8,1.02973E-12

V2=149.964,111.965,44.960,4.786,0.8499,0.10877

V3=0.,0.,1.679E-07,1.685E-07,0.000001,0.00006

V4=9.992748E-06,1.493836E-05,2.494561E-05,7.048586E-05, &

8.498912E-05,9.224075E-05

V5=0.05,0.05,0.05,0.05,0.05,0.05

V6=3.3,3.3,3.3,3.3,3.3,3.3

V7=1.8,1.8,1.8,1.8,1.8,1.8

V8=1.35,1.35,1.35,1.35,1.35,1.35

V9=0.8498912,0.8498912,0.8498912,0.8498912,0.8498912,0.8498912

V10=1.7984E-8,3.6484E-8,3.0183E-7,3.5965E-7,1.5246E-7,1.3429E-7

V11=0.001,0,0,0,0,0

# -- Bilinear plasticity; Temp-independent

M=7 CMOD=21 MATID=U MATDB=1 &

EL=10000.,10000.,0.326,0.326,3770.,12.00E-6,12.00E-6 &

NP=1 VI=5.08,13.1,0.15

# -- Bilinear plasticity; Temp-dependent

M=8 CMOD=21 MATID=U MATDB=1

NTP=3

TEM=21.,400.,800.

EA=10000.,9000.,7300.

ET=10000.,9000.,7300.

NUA=0.326,0.351,0.345

NUT=0.326,0.351,0.345

GA=3771.,3331.,2714.

ALPA=12.00E-6,13.50E-6,22.72E-6

ALPT=12.00E-6,13.50E-6,22.72E-6

NP=1

V1=5.08,4.21,3.05

V2=13.1,10.7,5.10

V3=0.15,0.15,0.15

# -- Point-wise plasticity; Temp-independent

M=9 CMOD=21 MATID=U MATDB=1 &

EL=10000.,10000.,0.326,0.326,3770.,12.00E-6,12.00E-6 &

NP=3 VI=4.20,6.13,6.60,13.1,0.004,0.01,0.15

NASA/TM—2002-212077/VOL3 42
Section 2: Constituent Materials Example 2e: User-Defined Material Properties ñ Input File 

# -- Point-wise plasticity; Temp-dependent

M=10 CMOD=21 MATID=U MATDB=1

NTP=3

TEM=21.,400.,800.

EA=10000.,9000.,7300.

ET=10000.,9000.,7300.

NUA=0.326,0.351,0.345

NUT=0.326,0.351,0.345

GA=3771.,3331.,2714.

ALPA=12.00E-6,13.50E-6,22.72E-6

ALPT=12.00E-6,13.50E-6,22.72E-6

NP=3

V1=4.20,3.81,3.05

V2=6.13,4.77,3.69

V3=6.60,5.10,3.90

V4=13.1,9.70,5.10

V5=0.004,0.004,0.004

V6=0.01,0.01,0.01

V7=0.15,0.15,0.15

*RUC

# -- Alter value of M=* to change simulated material

MOD=1 M=1

*MECH

LOP=1

NPT=2 TI=0.,200. MAG=0.,0.02 MODE=1

*THERM

NPT=2 TI=0.,200. TEMP=500.,500.

*SOLVER

METHOD=1 NPT=2 TI=0.,200. STP=0.1 ITMAX=20 ERR=1.E-6

*PRINT

NPL=6

*XYPLOT

FREQ=1

MACRO=1

NAME=example_2e X=1 Y=7

MICRO=0

*END

Annotated Input Data

1) Flags: None 

2) Constituent materials (*CONSTITUENTS) [KM_2]: 

Number of materials: 10 (NMATS=10) 

Materials: User-Defined (MATID=U) 

Material property source: Read from input file (MATDB=1) 

Constitutive models: 

Linear Elastic (CMOD=6) 

# -- Elastic; Temp-independent

M=1 CMOD=6 MATID=U MATDB=1 &

EL=58.E3,58.E3,0.20,0.20,24.17E3,6.3E-6,6.3E-6

# -- Elastic; Temp-dependent

NASA/TM—2002-212077/VOL3 43
MAC/GMC 4.0 Example Problem Manual 

M=2 CMOD=6 MATID=U MATDB=1

NTP=4

TEM=18.,200.,400.,600.

EA=45.6E3,43.5E3,40.6E3,29.0E3

ET=45.6E3,43.5E3,40.6E3,29.0E3

NUA=0.41,0.41,0.41,0.41

NUT=0.41,0.41,0.41,0.41

GA=16.154E3,15.429E3,14.400E3,10.286E3

ALPA=4.5E-6,4.8E-6,5.1E-6,5.5E-6

ALPT=4.5E-6,4.8E-6,5.1E-6,5.5E-6

The simplest constitutive model for which to input user-defined material properties is the linearly 

elastic model (CMOD=6). Since the material deformation is treated as linearly elastic, all that is 

needed is the thermo-elastic material properties. The standard order for the material properties is: 

ν ν GEE α αTAATATA ,,,,,, . As shown above, for temperature-independent material properties, 

these properties are simply listed as EL=..., on the same line in the input file. For temperaturedependent material properties, the number of input temperatures (NTP=4) is placed on a separate 

line, followed by a separate line specifying the input temperatures (TEM=...). Then, each of the seven 

material properties is placed on its own separate line, with the values at each of the input 

temperatures listed. 

Bodner-Partom (CMOD=1) 

# -- Bodner-Partom; Temp-independent

M=3 CMOD=1 MATID=U MATDB=1 &

EL=9.53E3,9.53E3,0.33,0.33,3.58E3,21.06E-6,21.06E-6 &

VI=1.E4,49.,63.,300.,4.,1.

# -- Bodner-Partom; Temp-dependent

M=4 CMOD=1 MATID=U MATDB=1

NTP=2

TEM=18.,700.

EA=9.53E3,4.12E3

ET=9.53E3,4.12E3

NUA=0.41,0.41

NUT=0.41,0.41

GA=3.58E3,1.46E3

ALPA=21.06E-6,28.92E-6

ALPT=21.06E-6,28.92E-6

V1=1.E4,1.E4

V2=49.,49.

V3=63.,63.

V4=300.,300.

V5=4.,2.5

V6=1.,1.

For the Bodner-Partom viscoplastic constitutive model, the user-defined elastic material properties 

are specified in the same manner as for the elastic material constitutive model. Now, however, six 

viscoplastic material properties must also be specified. In the case of temperature-independence, 

these six material parameters are simply listed on the same line as the elastic properties as VI=.... In 

the case of temperature-dependence, each viscoplastic material property is placed on its own line as 

shown above. Obviously, the order in which these viscoplastic material properties is specified is 

important as the code must know how to interpret the input data. For the Bodner-Partom model, this 

order is: 100 ,,,,, qnmZZD . This order, as well as the proper order for the other MAC/GMC 4.0 

internal constitutive models, is described in Section 2 of the MAC/GMC 4.0 Keywords Manual. 

NASA/TM—2002-212077/VOL3 44
Section 2: Constituent Materials Example 2e: User-Defined Material Properties ñ Input File 

Isotropic GVIPS (CMOD=4) 

# -- Isotropic GVIPS; Temp-independent

M=5 CMOD=4 MATID=U MATDB=1 &

EL=14009.,14009.,0.365,0.365,5131.5,5.862E-6,5.862E-6 &

VI=0.000999275,44.960,1.679E-07,2.494561E-05,0.05, &

3.3,1.8,1.35,0.85,3.0183E-7,0.001

# -- Isotropic GVIPS; Temp-dependent

M=6 CMOD=4 MATID=U MATDB=1

NTP=6

TEM=25.,300.,482.,565.,650.,704.

EA=16.5506E+03,15.644E+03,14.009E+03,12.968E+03,11.702E+03,10.793E+03

ET=16.5506E+03,15.644E+03,14.009E+03,12.968E+03,11.702E+03,10.793E+03

NUA=0.365,0.365,0.365,0.365,0.365,0.365

NUT=0.365,0.365,0.365,0.365,0.365,0.365

GA=6062.491,5730.403,5131.502,4750.183,4286.477,3953.480

ALPA=4.2921E-06,5.116E-06,5.862E-06,6.271E-06,6.741E-06,7.07E-06

ALPT=4.2921E-06,5.116E-06,5.862E-06,6.271E-06,6.741E-06,7.07E-06

V1=99.927,20.015,0.000999275,8.499E-08,8.E-8,1.02973E-12

V2=149.964,111.965,44.960,4.786,0.8499,0.10877

V3=0.,0.,1.679E-07,1.685E-07,0.000001,0.00006

V4=9.992748E-06,1.493836E-05,2.494561E-05,7.048586E-05, &

8.498912E-05,9.224075E-05

V5=0.05,0.05,0.05,0.05,0.05,0.05

V6=3.3,3.3,3.3,3.3,3.3,3.3

V7=1.8,1.8,1.8,1.8,1.8,1.8

V8=1.35,1.35,1.35,1.35,1.35,1.35

V9=0.8498912,0.8498912,0.8498912,0.8498912,0.8498912,0.8498912

V10=1.7984E-8,3.6484E-8,3.0183E-7,3.5965E-7,1.5246E-7,1.3429E-7

V11=0.001,0,0,0,0,0

As with Bodner-Partom viscoplasticity, employing user-defined material properties with the GVIPS 

constitutive model requires specification of viscoplastic material properties in addition to the elastic 

properties. Now, instead of six viscoplastic parameters, there are eleven. As mentioned previously, 

the correct order for specification of these eleven properties is discussed in the MAC/GMC 4.0 

Keywords Manual in Section 2. 

Incremental Plasticity (CMOD=21) 

# -- Bilinear plasticity; Temp-independent

M=7 CMOD=21 MATID=U MATDB=1 &

EL=10000.,10000.,0.326,0.326,3770.,12.00E-6,12.00E-6 &

NP=1 VI=5.08,13.1,0.15

# -- Bilinear plasticity; Temp-dependent

M=8 CMOD=21 MATID=U MATDB=1

NTP=3

TEM=21.,400.,800.

EA=10000.,9000.,7300.

ET=10000.,9000.,7300.

NUA=0.326,0.351,0.345

NUT=0.326,0.351,0.345

GA=3771.,3331.,2714.

ALPA=12.00E-6,13.50E-6,22.72E-6

ALPT=12.00E-6,13.50E-6,22.72E-6

NP=1

V1=5.08,4.21,3.05

NASA/TM—2002-212077/VOL3 45
MAC/GMC 4.0 Example Problem Manual 

V2=13.1,10.7,5.10

V3=0.15,0.15,0.15

# -- Point-wise plasticity; Temp-independent

M=9 CMOD=21 MATID=U MATDB=1 &

EL=10000.,10000.,0.326,0.326,3770.,12.00E-6,12.00E-6 &

NP=3 VI=4.20,6.13,6.60,13.1,0.004,0.01,0.15

# -- Point-wise plasticity; Temp-dependent

M=10 CMOD=21 MATID=U MATDB=1

NTP=3

TEM=21.,400.,800.

EA=10000.,9000.,7300.

ET=10000.,9000.,7300.

NUA=0.326,0.351,0.345

NUT=0.326,0.351,0.345

GA=3771.,3331.,2714.

ALPA=12.00E-6,13.50E-6,22.72E-6

ALPT=12.00E-6,13.50E-6,22.72E-6

NP=3

V1=4.20,3.81,3.05

V2=6.13,4.77,3.69

V3=6.60,5.10,3.90

V4=13.1,9.70,5.10

V5=0.004,0.004,0.004

V6=0.01,0.01,0.01

V7=0.15,0.15,0.15

The incremental plasticity constitutive model is a special case in terms of specifying user-defined 

material properties. The implementation of this model with MAC/GMC 4.0 allows the constituent 

materialís response to be defined by a number of stress ñ total strain point pairs that can be taken 

directly from experimental data for a material. Note that this simplifies the input compared to 

specification of stress ñ inelastic strain point pairs. A maximum of nine such pairs is permitted for 

each temperature. As shown above, the number of these stress-strain point pairs employed by the 

user must be specified as NP=*. This is the number of stress-strain pairs that are specified in addition 

to the materialís yield stress. In the case of temperature-independence, NP=* is placed on the same 

line, while for temperature-dependence, NP=* is placed on its own line. The smallest permitted 

value for NP=* is 1, which corresponds to bilinear plasticity. In this case, three plastic material 

parameters are specified, the first being the yield stress, followed by the stress and strain 

corresponding to one post-yield stress-strain pair. The material response is bilinear in that, at stresses 

below yield, the stress-strain response is linearly elastic, while after yielding, the response follows a 

linear path defined by the yield stress ñ total strain point and the single specified post-yield stress ñ 

total strain pair. The case where NP > 1, is referred to as point-wise plasticity, as the post-yield 

material response will follow a piece-wise linear path defined by the (up to nine) specified stress ñ 

total strain point pairs. The order for the user-defined properties for point-wise plasticity is: 

NPNPY σ σ σ σ ε ε ...,,,,...,,,, ε 2121

where σ Y is the yield stress and ii σ − ε are the stress ñ total strain points. As indicated above, this 

ordering holds true for temperature-dependent material properties as well, as each value on a 

particular line represents a different temperature. For instance, V1=4.20,3.81,3.05 are the 

values of the yield stress at the three input temperatures (21., 400., and 800. °C), 

V2=6.13,4.77,3.69 are the stress values (σ 1 ) of the first stress-strain point at the three input 

NASA/TM—2002-212077/VOL3 46
Section 2: Constituent Materials Example 2e: User-Defined Material Properties ñ Input File 

temperatures, and V5=0.004,0.004,0.004 are the total strain values ( 1 ε ) of the first stressstrain point at the three input temperatures. Note that the total number of incremental plasticity 

material parameters is 2×NP+1. 

3) Analysis type (*RUC) → Repeating Unit Cell Analysis [KM_3]: 

Analysis model: Monolithic material (MOD=1) 

Material assignment: Each constituent successively (M=*) 

4) Loading: 

a) Mechanical (*MECH) [KM_4]: 

Loading option: 1 (LOP=1) 

Number of points: 2 (NPT=2) 

Time points: 0., 200. sec. (TI=0.,200.) 

Load magnitude: 0., 0.02 (MAG=0.,0.02) 

Loading mode: strain control (MODE=1) 

b) Thermal (*THERM) [KM_4]: 

Number of points: 2 (NPT=2) 

Time points: 0., 200. sec. (TI=0.,200.) 

Temperature points: 500., 500. (TEMP=500.,500.) 

c) Time integration (*SOLVER) [KM_4]: 

Time integration method: Forward Euler (METHOD=1) 

Number of time points: 2 (NPT=2) 

Time points: 0., 20. sec. (TI=0.,20.) 

Time step sizes: 0.1 sec. (STP=0.1) 

Max. number of iterations 20 (ITMAX=20) 

Max. permitted error fraction 1.×10-6 (ERR=1.E-6) 

! Note: While ITMAX=20 and ERR=1.E-6 are present in the input file for all ten cases executed in 

this example problem, these data are required only for the incremental plasticity cases. As 

mentioned in Example 2c, if oscillations occur in the results, a smaller value for ERR and/or a 

smaller time step should be employed. 

5) Damage and Failure: None 

6) Output: 

a) Output file print level (*PRINT) [KM_6]: 

Print level: 6 (NPL=6) 

b) x-y plots (*XYPLOT) [KM_6]: 

Frequency: 1 (FREQ=1) 

Number of macro plots: 1 (MACRO=1) 

Macro plot name: example_2e (NAME=example_2e) 

Macro plot x-y quantities: ε11, σ11 (X=1 Y=7) 

Number of micro plots: 0 (MICRO=0) 

7) End of file keyword: (*END) 

NASA/TM—2002-212077/VOL3 47
MAC/GMC 4.0 Example Problem Manual 

Results

The results for this example problem are plotted in Figure 2.5 and Figure 2.6. When temperaturedependent material properties are specified (as they are for material numbers 2, 4, 6, 8, and 10), 

MAC/GMC 4.0 uses linear interpolation to determine the properties at temperature between those 

specified. When temperature-independent material properties are specified, these properties are 

employed regardless of the current temperature during the simulation. Since the stress-strain results 

presented here were generated at a temperature of 500 °C, significant differences are evident between the 

temperature-dependent and temperature-independent materials for each constitutive model. 

 

0

20

40

60

80

100

120

140

0.000 0.005 0.010 0.015 0.020 0.025

Strain

Stress (ksi)

M=1: Linear Elastic

M=2: Linear Elastic

M=5: Isotropic GVIPS

M=3: Bodner-Partom

M=4: Bodner-Partom

M=6: Isotropic GVIPS

Figure 2.5 Example 2e: plots of the tensile stress-strain response of the first 6 user-defined materials at 

500 °C.

NASA/TM—2002-212077/VOL3 48
Section 2: Constituent Materials Example 2e: User-Defined Material Properties ñ Input File 

0

1

2

3

4

5

6

7

8

0.000 0.005 0.010 0.015 0.020 0.025

Strain

Stress (ksi)

M=9: Point-Wise Plasticity

M=10: Point-Wise 

M=8: Bilinear Plasticity

M=7: Bilinear Plasticity

Figure 2.6 Example 2e: plots of the tensile stress-strain response of the incremental plasticity userdefined materials at 500 °C. 

NASA/TM—2002-212077/VOL3 49
MAC/GMC 4.0 Example Problem Manual 

Example 2f: User-Defined Material Properties ñ USRFUN

This example problem illustrates how the user-defined subroutine, usrfun.F90, can be used in 

conjunction with internal MAC/GMC 4.0 constitutive models to calculate user-defined material properties 

during execution of the code. The usrfun.F90 subroutine used in this example is part of the 

MAC/GMC 4.0 executable. The usrfun.F90 source code is also distributed with the MAC/GMC 4.0 

executable, along with a library file from MAC/GMC 4.0. In order for users to employ their own code to 

calculate material properties during code execution, it is necessary not only to include the applicable code 

within the usrfun.F90 source file, but also to compile the file and link it with the MAC/GMC 4.0 

library file. This can be accomplished on a Windows (NT, 2000, XP) PC using the Compaq Visual 

Fortran software package (v. 6.6) (see http://www.compaq.com/fortran/). For additional information, see 

the MAC/GMC 4.0 Keywords Manual Introduction (Using User-Defined Subroutines with MAC/GMC 

4.0). 

It is important to note that in MAC/GMC 4.0, the material properties are associated with constituent 

materials as opposed to subcells. That is, several subcells may contain material #1, and thus all of those 

subcells must have the same material properties. This is relevant when employing the usrfun.F90

subroutine to make material properties a function of field variables, as is done in this example for 

materials #1 and #2. If, for instance, the properties of material #1 are determined from a function of strain 

and material #1 is placed in two subcells, the subcells will in general experience different strain states. It 

is thus impossible for material #1 to have the correct material properties to correspond to the two different 

states of strain in each subcell. For this reason, when material properties are determined from a function 

of field variables (other than temperature, which does not vary within the repeating unit cell), each subcell 

should be given its own material. The present example employs only one subcell, so this is not a problem 

in this case. 

The user-editable portion of the usrfun.F90 subroutine associated with the three materials examined in 

this example problem is given in the Appendix. This subroutine is listed in its entirety in the Keywords 

Manual Appendix. The first material is elastic and has properties that are a function of the axial strain 

and the previous stiffness. The second material is modeled using incremental plasticity and the post-yield 

properties are a function of the strain rate. The third material is modeled using the Bodner-Partom 

viscoplastic constitutive model and the material properties are a function of temperature. The code can be 

executed for each of these monolithic materials by changing the material number under *RUC. The 

thermo-mechanical loading history can also be altered to generate the cases presented in the Results. 

MAC/GMC Input File: example_2f.mac

MAC/GMC 4.0 Example 2f - USRFUN Material Properties

*CONSTITUENTS

NMATS=3

M=1 CMOD=6 MATID=U MATDB=2

M=2 CMOD=21 MATID=U MATDB=2

M=3 CMOD=1 MATID=U MATDB=2

*RUC

# -- Alter value of M=* to change simulated material

MOD=1 M=1

*MECH

LOP=1

NASA/TM—2002-212077/VOL3 50
Section 2: Constituent Materials Example 2f: User-Defined Material Properties ñ USRFUN 

# -- Alter values of TI=* to change simulated rate

# Example: use TI=0.,2000. for rate = 0.00001/sec.

NPT=2 TI=0.,200. MAG=0.,0.02 MODE=1

*THERM

# -- Alter values of TI=* to change simulated rate

# -- Alter values of TEMP=* to change simulated temperature

NPT=2 TI=0.,200. TEMP=23.,23.

*SOLVER

# -- Alter values of TI=* (and STP=*) to change simulated rate

METHOD=1 NPT=2 TI=0.,200. STP=1. ITMAX=50 ERR=1.E-6

*PRINT

NPL=6

*XYPLOT

FREQ=1

MACRO=1

NAME=example_2f X=1 Y=7

MICRO=0

*END

Annotated Input Data

1) Flags: None 

2) Constituent materials (*CONSTITUENTS) [KM_2]: 

Number of materials: 3 (NMATS=3) 

Constitutive models: Linearly elastic (CMOD=6) 

 Incremental plasticity (CMOD=21) 

 Bodner-Partom viscoplasticity (CMOD=1) 

Materials: User-defined (MATID=U) 

Material property source: usrfun.F90 subroutine (MATDB=2) 

3) Analysis type (*RUC) → Repeating Unit Cell Analysis [KM_3]: 

Analysis model: Monolithic material (MOD=1) 

Material assignment: Each constituent successively (M=*) 

4) Loading: 

a) Mechanical (*MECH) [KM_4]: 

Loading option: 1 (LOP=1) 

Number of points: 2 (NPT=2) 

Time points: 0., 200. sec. (TI=0.,200.) 

Load magnitude: 0., 0.02 (MAG=0.,0.02) 

Loading mode: strain control (MODE=1) 

To alter the strain rate as is done to generate Figure 2.8, the second time point can be changed. For 

example, to increase the applied strain rate from 0.0001 /sec. to 0.001 /sec., use TI=0.,20.. 

b) Thermal (*THERM) [KM_4]: 

Number of points: 2 (NPT=2) 

Time points: 0., 200. sec. (TI=0.,200.) 

Temperature points: 23., 23. (TEMP=23.,23.) 

NASA/TM—2002-212077/VOL3 51
MAC/GMC 4.0 Example Problem Manual 

! Note: The ending time point should match up with that given in *MECH. Also, to alter the 

temperature as is done to generate Figure 2.9, both temperature values (TEMP=*,*) should be 

changed. 

c) Time integration (*SOLVER) [KM_4]: 

Time integration method: Forward Euler (METHOD=1) 

Number of time points: 2 (NPT=2) 

Time points: 0., 200. sec. (TI=0.,200.) 

Time step sizes: 0.1 sec. (STP=0.1) 

Max. number of iterations 50 (ITMAX=50) 

Max. permitted error fraction 1.×10-6 (ERR=1.E-6) 

! Note: While ITMAX=50 and ERR=1.E-6 are present in the input file for all cases executed in this 

example problem, these data are required only for the incremental plasticity cases. Also, as was 

the case in *THERM, the ending time point must be altered to match that employed in *MECH. 

Also, as the total time for the simulation is altered, as in Figure 2.8, the time step size can be 

changed accordingly so that the total number of time steps remains constant. 

5) Damage and Failure: None 

6) Output: 

a) Output file print level (*PRINT) [KM_6]: 

Print level: 6 (NPL=6) 

b) x-y plots (*XYPLOT) [KM_6]: 

Frequency: 1 (FREQ=1) 

Number of macro plots: 1 (MACRO=1) 

Macro plot name: example_2f (NAME=example_2f) 

Macro plot x-y quantities: ε11, σ11 (X=1 Y=7) 

Number of micro plots: 0 (MICRO=0) 

7) End of file keyword: (*END) 

Results

Results in the form of simulated stress-strain curves are given in Figure 2.7, Figure 2.8, and Figure 2.9. It 

is important to remember that these results have been generated based on the version of the ursfun.F90

subroutine that is distributed with MAC/GMC 4.0. This version of the subroutine is contained within the 

distributed MAC/GMC 4.0 executable. However, if the usrfun.F90 subroutine source code is altered 

and compiled and linked to the MAC/GMC 4.0 library file, this example problem may not execute as 

expected. 

NASA/TM—2002-212077/VOL3 52
Section 2: Constituent Materials Example 2f: User-Defined Material Properties ñ USRFUN 

0

1000

2000

3000

4000

5000

6000

0.000 0.005 0.010 0.015 0.020 0.025

Strain

Stress (MPa)

Figure 2.7 Example 2f: plot of the tensile stress-strain response for material #1 ñ elastic material with 

stiffness a function of strain and previous stiffness. 

0

20

40

60

80

100

120

140

0.000 0.005 0.010 0.015 0.020 0.025

Strain

Stress (ksi)

SR = 0.001 /s

SR = 0.0001 /s

SR = 0.00001 /s

Figure 2.8 Example 2f: plots of the tensile stress-strain response for material #2 ñ point-wise 

incremental plasticity material with post-yield behavior a function of strain rate. 

NASA/TM—2002-212077/VOL3 53
MAC/GMC 4.0 Example Problem Manual 

0

50

100

150

200

250

300

350

400

0.000 0.005 0.010 0.015 0.020 0.025

Strain

Stress

23 °C

200 °C

300 °C

400 °C

Figure 2.9 Example 2f: plots of the tensile stress-strain response for material #3 ñ Bodner-Partom 

viscoplastic material with elastic and viscoplastic material properties a function of 

temperature.

NASA/TM—2002-212077/VOL3 54
Section 2: Constituent Materials Example 2g: User-Defined Material Constitutive Model 

Example 2g: User-Defined Material Constitutive Model

This example problem illustrates how to employ the user-defined subroutine, usrmat.F90, for 

implementation of a userís own constitutive model. The appropriate code that calculates the inelastic 

strain increments for the userís constitutive model must be placed in the file usrmat.F90. As was the 

case in the previous example, this subroutine must be compiled and linked with the MAC/GMC 4.0 

library file in order to be used. In this example, a power law creep model and the Bodner-Partom model 

have been coded within usrmat.F90. An additional user-defined subroutine must also be implemented 

in conjunction with the usrmat.F90 routine. This subroutine, called usrformde.F90, is employed 

to calculate the stiffness matrix for the user-defined material. In general, since a user-defined constitutive 

model can employ any elastic material parameters desired by the user, the code will not know how to 

determine the required stiffness matrix from the userís elastic material parameters. Thus, the user must 

provide the code to do this in usrformde.F90. The user-defined portions of the usrmat.F90 and 

usrformde.F90 subroutines are given in the Appendix. The full listing of these subroutines is 

provided in the MAC/GMC 4.0 Keywords Manual Appendix. 

Note that the specification of the material properties is somewhat different for a user-defined constitutive 

model. Particularly, the number of elastic and inelastic parameters that code should read in must be 

specified. The present example also includes several types of mechanical loading. In addition to the 

standard applied strain history, this problem involves a stress-controlled creep test simulation in which a 

stress is applied to the material and then held constant as the material creeps. 

MAC/GMC Input File: example_2g.mac

MAC/GMC 4.0 Example 2g - USRMAT Constitutive Model

*CONSTITUENTS

NMATS=2

M=1 CMOD=99 MATID=U MATDB=1 NPE=2 NPV=2 &

EL=55.2E9,0.30 ALP=22.5E-6,22.5E-6 VI=1.5E-28,3.0

M=2 CMOD=99 MATID=U MATDB=1 NPE=2 NPV=6 &

EL=55.2E9,0.30 ALP=22.5E-6,22.5E-6 &

VI=1000.,103.42E6,103.42E6,1700.,10.,1.0

*RUC

# -- Alter value of M=* to change simulated material

MOD=1 M=1

*MECH

LOP=1

NPT=2 TI=0.,200. MAG=0.,0.02 MODE=1

# NPT=3 TI=0.,10.,3600. MAG=0.,87.2E6,87.2E6 MODE=2,2

# NPT=3 TI=0.,10.,3600. MAG=0.,80.E6,80.E6 MODE=2,2

*THERM

NPT=2 TI=0.,200. TEMP=23.,23.

# NPT=3 TI=0.,10.,3600. TEMP=23.,23.,23.

*SOLVER

METHOD=1 NPT=2 TI=0.,200. STP=0.05

# METHOD=1 NPT=3 TI=0.,10.,3600. STP=0.05,0.05

*PRINT

NPL=6

*XYPLOT

NASA/TM—2002-212077/VOL3 55
MAC/GMC 4.0 Example Problem Manual 

FREQ=20

MACRO=2

NAME=example_2g X=1 Y=7

NAME=example_2g X=101 Y=1

MICRO=0

*END

Annotated Input Data

1) Flags: None 

2) Constituent materials (*CONSTITUENTS) [KM_2]: 

NMATS=2

M=1 CMOD=99 MATID=U MATDB=1 NPE=2 NPV=2 &

EL=55.2E9,0.30 ALP=22.5E-6,22.5E-6 VI=1.5E-28,3.0

M=2 CMOD=99 MATID=U MATDB=1 NPE=2 NPV=6 &

EL=55.2E9,0.30 ALP=22.5E-6,22.5E-6 &

VI=1000.,103.42E6,103.42E6,1700.,10.,1.0

Number of materials: 2 (NMATS=2) 

Constitutive models: User-defined (CMOD=99) 

Materials: User-defined (MATID=U) 

Material property source: Input file (MATDB=1) 

No. of Elastic Props: 2 (NPE=2) 

No. of Viscoplastic Props: 2, 6 (NPV=2, NPV=6) 

Since the materials employ user-defined constitutive models, the number of elastic (NPE) and 

viscoplastic (NPV) material parameters must be specified by the user. Then, the elastic and 

viscoplastic material parameters are listed as EL=... and VI=..., respectively. Temperature-dependent 

material parameters may be specified as well, for details, see the MAC/GMC 4.0 Keywords Manual 

Section 2. 

3) Analysis type (*RUC) → Repeating Unit Cell Analysis [KM_3]: 

Analysis model: Monolithic material (MOD=1) 

Material assignment: Each constituent successively (M=*) 

4) Loading: 

a) Mechanical (*MECH) [KM_4]: 

LOP=1

NPT=2 TI=0.,200. MAG=0.,0.02 MODE=1

# NPT=3 TI=0.,10.,3600. MAG=0.,87.2E6,87.2E6 MODE=2,2

# NPT=3 TI=0.,10.,3600. MAG=0.,80.E6,80.E6 MODE=2,2

Loading option: 1 (LOP=1) 

Number of points: 2 or 3 (NPT=2, NPT=3) 

Time points: 0., 200. sec. (TI=0.,200.) 

 0., 10., 3600. sec. (TI=0.,10.,3600.) 

Load magnitude: 0., 0.02 (MAG=0.,0.02) 

 0., 80., 80. MPa (MAG=0.,80.E6,80.E6) 

 0., 87.2, 87.2 MPa (MAG=0.,87.2E6,87.2E6) 

NASA/TM—2002-212077/VOL3 56
Section 2: Constituent Materials Example 2g: User-Defined Material Constitutive Model 

Loading mode: strain control (MODE=1) 

 stress control (MODE=2,2) 

Two types of mechanical loading are specified that can be switched by commenting and 

uncommenting the appropriate lines. The first line specifies mechanical loading in the form of a 

standard strain-controlled (MODE=1) stress-strain test. The second and third lines specify simulated 

stress-controlled (MODE=2) creep tests at two different stress levels. In these simulated creep tests, 

the load is applied quickly in stress control, and then held for 3600 sec. During this hold time, the 

material is free to creep. 

b) Thermal (*THERM) [KM_4]: 

Number of points: 2 or 3 (NPT=2 or NPT=3) 

Time points: 0., 200. sec. (TI=0.,200.) 

 0., 10., 3600. sec. (TI=0.,10.,3600.) 

Temperature points: 23., 23. °C (TEMP=23.,23.) 

 23., 23., 23. °C (TEMP=23.,23.,23.) 

By commenting and uncommenting the appropriate lines, the thermal time points should match those 

in the mechanical loading under *MECH. 

c) Time integration (*SOLVER) [KM_4]: 

Time integration method: Forward Euler (METHOD=1) 

Number of points: 2 or 3 (NPT=2 or NPT=3) 

Time points: 0., 200. sec. (TI=0.,200.) 

 0., 10., 3600. sec. (TI=0.,10.,3600.) 

Time step sizes: 0.05 sec. (STP=0.05) 

 0.05, 0.05 sec. (STP=0.05,0.05) 

Again, the appropriate lines should be commented and uncommented such that the appropriate time 

profile is specified for the desired case. 

5) Damage and Failure: None 

6) Output: 

a) Output file print level (*PRINT) [KM_6]: 

Print level: 6 (NPL=6) 

b) x-y plots (*XYPLOT) [KM_6]: 

Frequency: 20 (FREQ=20) 

Number of macro plots: 2 (MACRO=2) 

Macro plot names: example_2g (NAME=example_2g) 

 example_2g (NAME=example_2g) 

Macro plot x-y quantities: ε11, σ11 (X=1 Y=7) 

 time, ε11 (X=101 Y=7) 

Number of micro plots: 0 (MICRO=0) 

NASA/TM—2002-212077/VOL3 57
MAC/GMC 4.0 Example Problem Manual 

Since both macro x-y plot files are given the same name, the number 2 will be appended to the name 

of the second file. Thus, this input file will cause the code to write x-y plot data to the files: 

example_2g_macro.data and example_2g2_macro.data. See the MAC/GMC 4.0 

Keywords Manuals Section 6 for details. 

7) End of file keyword: (*END) 

Results

Figure 2.10 shows that at an applied strain rate of 10-4/sec., the user-defined power law creep and BodnerPartom materials exhibit similar tensile stress-strain behavior. However, as shown in Figure 2.11, the 

creep behavior of the two materials is completely different at both applied stress levels. Note that because 

this example problem employs the versions of the user-defined subroutines (usrmat.F90 and 

usrformde.F90) that are distributed with MAC/GMC 4.0, this example problem cannot be expected to 

yield the same results if the code is calling altered version of these subroutines. Conversely, the user 

should ensure that the distributed versions of these subroutines are not being called when the intention is 

to have MAC/GMC 4.0 call the userís own versions of these subroutines. 

0

10

20

30

40

50

60

70

80

90

100

0.000 0.005 0.010 0.015 0.020

Strain

Stress (MPa)

Bodner-Partom

Power Law Creep

Figure 2.10 Example 2g: plot of the tensile stress-strain response for user-defined material constitutive 

models. 

NASA/TM—2002-212077/VOL3 58
Section 2: Constituent Materials Example 2g: User-Defined Material Constitutive Model 

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0 500 1000 1500 2000 2500 3000 3500 4000

Time (s)

Strain

80 MPa: Bodner-Partom

87.2 MPa: Bodner-Partom

87.2 MPa: Power Law Creep

80 MPa: Power Law Creep

Figure 2.11 Example 2g: plot of the creep response for user-defined material constitutive models at two 

applied stress levels.

NASA/TM—2002-212077/VOL3 59
MAC/GMC 4.0 Example Problem Manual 

Example 2h: External Material Database

This example problem employs the external material database capabilities of MAC/GMC 4.0. This 

material database provides users with another option for storing their constituent material properties ñ in 

an external ASCII file rather than directly in the input file. Users can thus construct their own database of 

material properties in a single (or multiple) file(s) and access them from MAC/GMC 4.0 at will. To 

reference a particular external material database file from a MAC/GMC 4.0 input file, the additional 

keyword, *MDBPATH, is specified. It is clear from the contents of the external material database ASCII 

file (given below) that the format of this file is similar to that of the *CONSTITUENTS section of the 

input file. The main difference is that the information for each material in the external file begins with the 

line MAT=*, which specifies the name of the material. This name is the referenced in the input file in the 

*CONSTITUENTS section. The only other difference in the form of the external material database is that 

the line following the material name specification line specifies the constitutive model for the material 

(CMOD=*). The material constitutive model is also specified for each material in the input file, and the 

constitutive model specified in both files must match, or an error will occur. For more information on the 

external material database capabilities, see the MAC/GMC 4.0 Keywords Manual Section 2. 

MAC/GMC Input File: example_2h.mac

MAC/GMC 4.0 Example 2h - External Material Database

*MDBPATH

NAME=sample_material_database.mat

*CONSTITUENTS

NMATS=3

M=1 CMOD=1 MATID=U MATDB=3 EXTMAT=copper

M=2 CMOD=99 MATID=U MATDB=3 EXTMAT=Al(6061-0)

M=3 CMOD=6 MATID=U MATDB=3 EXTMAT=SCS-6

*RUC

# -- Alter value of M=* to change simulated material

MOD=1 M=1

*MECH

LOP=1

NPT=2 TI=0.,200. MAG=0.,0.02 MODE=1

*THERM

NPT=2 TI=0.,200. TEMP=23.,23.

*SOLVER

METHOD=1 NPT=2 TI=0.,200. STP=0.02

*PRINT

NPL=6

*XYPLOT

FREQ=20

MACRO=1

NAME=example_2h X=1 Y=7

MICRO=0

*END

NASA/TM—2002-212077/VOL3 60
Section 2: Constituent Materials Example 2h: External Material Database 

External Material Database File:

sample_material_database.mat

Sample Material Database Provided with MAC/GMC 4.0

#

# ------- Bodner-Partom Model Material

MAT=copper

CMOD=1

EL=120.E9,120.E9,0.33,0.33,45.11E9,14.7E-6,14.7E-6 &

VI=1.E4,63.E6,250.E6,8.19,7.5,0.55

#

# ------- Isotropic, Elastic Materials

MAT=SCS-6

CMOD=6

NTP=5

TEM=21.0,316.0,427.0,538.0,860.0

EA=393.E9,382.E9,378.E9,374.E9,368.E9

ET=393.E9,382.E9,378.E9,374.E9,368.E9

NUA=0.25,0.25,0.25,0.25,0.25

NUT=0.25,0.25,0.25,0.25,0.25

GA=157.2E9,152.8E9,151.2E9,149.6E9,147.2E9

ALPA=3.56E-6,3.73E-6,3.91E-6,4.07E-6,4.57E-6

ALPT=3.56E-6,3.73E-6,3.91E-6,4.07E-6,4.57E-6

#

# ------- User constitutive model example

MAT=Al(6061-0)

CMOD=99

NPE=2 NPV=6

NTP=6

TEM=21.0,148.9,204.4,260.0,371.1,400.0

E1=72.5E9,69.4E9,65.8E9,58.5E9,41.5E9,41.5E9

E2=0.33,0.33,0.33,0.33,0.33,0.33

ALPA=22.5E-6,22.5E-6,22.5E-6,22.5E-6,22.5E-6,22.5E-6

ALPT=22.5E-6,22.5E-6,22.5E-6,22.5E-6,22.5E-6,22.5E-6

V1=1.E4, 1.E4, 1.E4, 1.E4, 1.E4, 1.E4

V2=100.E6,100.E6,100.E6,100.E6,100.E6,100.E6

V3=190.E6,190.E6,190.E6,190.E6,190.E6,190.E6

V4=70.0,70.0,70.0,70.0,70.0,70.0

V5=10.0,7.0,4.0,1.6,0.55,0.55

V6=0.4,0.4,0.4,0.4,0.4,0.4

Annotated Input Data

1) Flags: None 

2) Constituent material information

a) External material database file specification (*MDBPATH) [KM_2]: 

Name of external database file: sample_material_database.mat 

(NAME=sample_material_database.mat) 

b) Constituent materials to include for code execution (*CONSTITUENTS) [KM_2]: 

NASA/TM—2002-212077/VOL3 61
MAC/GMC 4.0 Example Problem Manual 

Number of materials: 3 (NMATS=3) 

Constitutive models: Bodner-Partom (CMOD=1) 

User-defined (CMOD=99) 

Linear Elastic (CMOD=6) 

Materials: All User-defined: (MATID=U)

 Copper (EXTMAT=copper) 

 Al 6061-0 (EXTMAT=Al(6061-0)) 

 SCS-6 fiber (EXTMAT=SCS-6) 

Material property source: External material database (MATDB=3) 

3) Analysis type (*RUC) → Repeating Unit Cell Analysis [KM_3]: 

Analysis model: Monolithic material (MOD=1) 

Material assignment: Each constituent successively (M=*) 

4) Loading: 

a) Mechanical (*MECH) [KM_4]: 

Loading option: 1 (LOP=1) 

Number of points: 2 (NPT=2) 

Time points: 0., 200. sec. (TI=0.,200.) 

Load magnitude: 0., 0.02 (MAG=0.,0.02) 

Loading mode: strain control (MODE=1) 

b) Thermal (*THERM) [KM_4]: 

Number of points: 2 (NPT=2) 

Time points: 0., 200. sec. (TI=0.,200.) 

Temperature points: 23., 23. °C (TEMP=23.,23.) 

c) Time integration (*SOLVER) [KM_4]: 

Time integration method: Forward Euler (METHOD=1) 

Number of points: 2 (NPT=2) 

Time points: 0., 200. sec. (TI=0.,200.) 

Time step sizes: 0.02 sec. (STP=0.02) 

5) Damage and Failure: None 

6) Output: 

a) Output file print level (*PRINT) [KM_6]: 

Print level: 6 (NPL=6) 

b) x-y plots (*XYPLOT) [KM_6]: 

Frequency: 20 (FREQ=20) 

Number of macro plots: 1 (MACRO=1) 

Macro plot names: example_2h (NAME=example_2h) 

Macro plot x-y quantities: ε11, σ11 (X=1 Y=7) 

Number of micro plots: 0 (MICRO=0) 

7) End of file keyword: (*END) 

NASA/TM—2002-212077/VOL3 62
Section 2: Constituent Materials Example 2h: External Material Database 

Results

The results for this example problem, in the form of the three stress-strain curves for the three constituent 

materials, are given in Figure 2.12. It is again important to note that, since the Al 6061-0 material 

employs the distributed version of the usrmat.F90 subroutine and the usrformde.F90 subroutine, if 

an altered version of these subroutines is called by MAC/GMC 4.0, this example problem may not 

execute correctly. 

0

20

40

60

80

100

120

140

160

0.000 0.005 0.010 0.015 0.020 0.025

Strain

Stress (MPa)

Copper

Al 6061-0

SCS-6

Figure 2.12 Example 2h: plot of the tensile stress-strain response for three materials taken from the 

external material database. 

NASA/TM—2002-212077/VOL3 63
MAC/GMC 4.0 Example Problem Manual 

Section 3 : Analysis Type and Architecture 

This section presents example problems that demonstrate the variations in the type of micromechanics 

analysis and composite architecture that can be analyzed with MAC/GMC 4.0. Two keywords, only one 

of which may appear in any given input file, specify the analysis type and architecture: *RUC and 

*LAMINATE. A fundamental difference exists between problems using these two keywords. *RUC

indicates that the problem is associated with a repeating unit cell analysis ñ it is a continuum or effective 

medium, without any boundaries, that is simulated. *LAMINATE on the other hand, indicates that 

MAC/GMC is simulating a laminated plate structure (with the use of classical lamination theory) that has 

a finite through-thickness dimension and top and bottom boundaries. The effective medium that 

constitutes each layer of the laminate is then analyzed with its own repeating unit cell analysis on this 

local scale. 

Within repeating unit cell analysis, several variations of the analysis type are possible, as indicated by 

MOD=*. The simplest type of analysis, which was used extensively in the example problems of Section 2, 

is that of a monolithic material. Internally within MAC/GMC, monolithic material analysis actually 

employs the doubly periodic GMC micromechanics model using a single subcell as the repeating unit 

cell. The repeating unit cell analysis type may also be specified as doubly periodic GMC or triply 

periodic GMC. The doubly periodic GMC model simulates unit cell geometries that are repeating in the 

two in-plane coordinate directions, but are constant in the out-of-plane coordinate direction. Triply 

periodic GMC simulates unit cell geometries that are repeating in all three coordinate directions. Hence, 

doubly periodic GMC can be used to analyze continuous fiber composites, while triply periodic GMC can 

be employed for discontinuous or particulate composites. Finally, a new doubly periodic micromechanics 

model is also available in MAC/GMC 4.0. This is high-fidelity GMC (HFGMC). As discussed in the 

MAC/GMC 4.0 Theory Manual, this new micromechanics model is more accurate in terms of the micro 

scale fields compared to GMC. However, GMC is considerably more computationally efficient than 

HFGMC. Example problems using doubly and triply periodic GMC unit cell analyses are presented in 

this section, along with an HFGMC analysis example. 

Laminate analysis within MAC/GMC 4.0 may be thought of as a global or structural scale approach, 

within which GMC is embedded. The laminate is composed of layers that can each be a composite 

material. The composite material within each layer is then modeled using doubly or triply periodic GMC. 

If the laminate is linearly elastic and is not subject to any damage or failure, GMC is merely utilized to 

determine the effective elastic properties of each layer. In the presence of inelasticity, however, the 

inelastic constitutive model (in general) must be evaluated on the level of each individual subcell within 

the GMC analysis of a given ply. Thus, as the simulated loading is applied to the laminate in an 

incremental fashion, localization from the laminate, to the plies, to the repeating unit cells, to the subcells 

must occur. The local inelastic strains can then be calculated for the subcells and then globalized 

(through homogenization) to the repeating unit cell, to the plies, and to the laminate. Thus, this 

localization/globalization process proceeds at each step of the incrementally applied simulated loading on 

the laminate. If the incremental plasticity constitutive model is employed, the localization/globalization 

process occurs for each iteration at each step of the applied loading. Several inelastic laminate analysis 

example problems are given in this section. 

NASA/TM—2002-212077/VOL3 64
Section 3: Analysis Type and Architecture Example 3a: Doubly Periodic GMC Internal RUC Library 

Example 3a: Doubly Periodic GMC Internal RUC Library

This example problem illustrates how to utilize the internal library of doubly periodic repeating unit cell 

architectures within MAC/GMC 4.0. The thermal and mechanical loading conditions, materials, and fiber 

volume fraction remain constant for use with all repeating unit cell architectures with the exception of 

ARCHID=9, which represents a composite with two different sized fibers, each with its own volume 

fraction. Some of the architectures addressed in this example may also be utilized with an interface 

present between the fiber and the matrix. This interface is composed of a third material that must be 

specified in *CONSTITUENTS. Example 3b analyzes doubly periodic unit cell architectures that do 

include this distinct interfacial phase. For more information on the codeís repeating unit cell 

architectures, see the MAC/GMC 4.0 Keywords Manual Section 3. 

MAC/GMC Input File: example_3a.mac

MAC/GMC 4.0 Example 3a - Doubly Periodic GMC RUC Library

*CONSTITUENTS

NMATS=3

M=1 CMOD=6 MATID=E

M=2 CMOD=4 MATID=A

M=3 CMOD=6 MATID=D

*RUC

MOD=2 ARCHID=1 VF=0.25 F=1 M=2

# MOD=2 ARCHID=2 VF=0.25 F=1 M=2

# MOD=2 ARCHID=3 VF=0.25 F=1 M=2

# MOD=2 ARCHID=4 XA=0.2 VF=0.25 F=1 M=2

# MOD=2 ARCHID=6 R=1.0 VF=0.25 F=1 M=2

# MOD=2 ARCHID=7 R=1.0 VF=0.25 F=1 M=2

# MOD=2 ARCHID=9 VF1=0.1 RAD1=50. VF2=0.15 RAD2=71. R=1. F=3,1 M=2

# MOD=2 ARCHID=11 R=2.0 VF=0.25 F=1 M=2

# MOD=2 ARCHID=13 R=1.0 VF=0.25 F=1 M=2

*MECH

LOP=2

NPT=2 TI=0.,200. MAG=0.,0.02 MODE=1

*THERM

NPT=2 TI=0.,200. TEMP=650.,650.

*SOLVER

METHOD=1 NPT=2 TI=0.,200. STP=1.

*PRINT

NPL=6

*XYPLOT

FREQ=5

MACRO=1

NAME=example_3a X=2 Y=8

MICRO=0

*END

NASA/TM—2002-212077/VOL3 65
MAC/GMC 4.0 Example Problem Manual 

ARCHID = 0

ARCHID = 1 ARCHID = 2 ARCHID = 3

ARCHID = 4 ARCHID = 6 ARCHID = 7

ARCHID = 9 ARCHID = 11 ARCHID = 13

Figure 3.1 MAC/GMC 4.0 doubly periodic repeating unit cell (RUC) architecture library.

NASA/TM—2002-212077/VOL3 66
Section 3: Analysis Type and Architecture Example 3a: Doubly Periodic GMC Internal RUC Library 

Annotated Input Data

1) Flags: None 

2) Constituent materials (*CONSTITUENTS) [KM_2]: 

Number of materials: 3 (NMATS=3) 

Materials: SiC fiber (MATID=E) 

Ti-21S (MATID=A) 

Boron fiber (MATID=D) 

Constitutive models: SiC fiber: linearly elastic (CMOD=6) 

Ti-21S matrix: isotropic GVIPS (CMOD=4) 

Boron fiber: linearly elastic (CMOD=6) 

3) Analysis type (*RUC) → Repeating Unit Cell Analysis [KM_3]: 

Analysis model: Doubly periodic GMC (MOD=2) 

RUC architecture: square fiber, square pack (ARCHID=1) 

 Square fiber, hexagonal pack (ARCHID=2) 

 Square fiber, square diagonal pack (ARCHID=3) 

 Cross shaped fiber, square pack (ARCHID=4) 

 7×7 circular fiber approx., rectangular pack (ARCHID=6) 

 14×14 circular fiber approx., rectangular pack (ARCHID=7) 

 2 different size square fibers, rectangular pack (ARCHID=9) 

 Square fiber, rectangular pack (ARCHID=11) 

 26×26 circular fiber approx., rectangular pack (ARCHID=13) 

Fiber volume fractions: 0.25 (VF=0.25) 

ARCHID=9: Boron = 0.15 SiC = 0.10 (VF1=0.15

VF2=0.1)

Unit cell aspect ratio: 1.0 (square pack), 2.0 for ARCHID=11 (R=1.0 or R=2.0) 

Fiber Radii: 50 μm, 71μm (ARCHID=9 only) (RAD1=50 RAD2=71) 

Cross arm length: 0.2 (ARCHID=4 only) (XA=0.2) 

Material assignment: SiC fiber (F=1 or F1=1) 

Ti-21S matrix (M=2) 

Boron fiber (ARCHID=9 only) (F=3) 

All 9 doubly periodic fiber architectures contained in the MAC/GMC 4.0 library are exercised in this 

example problem. Each architecture can be used by commenting and uncommenting the appropriate 

lines of the input file. ARCHID=9 is a special case in that it requires 2 fibers with different radii. 

Note that, while the fiber radii are specified, only the size of the two fibers relative to each other is 

important, not the actual radii themselves. In addition, because ARCHID=9 requires the packing of 

the two fibers to be square or rectangular, not all geometries are possible. As such, the fiber volume 

fraction of the second fiber, the SiC fiber in this example, is adjusted from the specified value to 

allow the packing arrangement to be correct given the value of the first fiberís volume fraction and 

the unit cell aspect ratio, R. Finally, for the cross shaped fiber, ARCHID=4, the cross arm length, 

XA, is the ratio of the length of the arm of the cross to the total length of the cross, where the total 

NASA/TM—2002-212077/VOL3 67
MAC/GMC 4.0 Example Problem Manual 

length of the cross is set to 1.0 (see Figure 3.1). For more information on these fiber architectures, 

see the MAC/GMC 4.0 Keywords Manual Section 3. 

4) Loading: 

a) Mechanical (*MECH) [KM_4]: 

Loading option: 1 (LOP=1) 

Number of points: 2 (NPT=2) 

Time points: 0., 200. sec. (TI=0.,200.) 

Load magnitude: 0., 0.02 (MAG=0.,0.02) 

Loading mode: strain control (MODE=1) 

b) Thermal (*THERM) [KM_4]: 

Number of points: 2 (NPT=2) 

Time points: 0., 200. sec. (TI=0.,200.) 

Temperature points: 650., 650. °C (TEMP=650.,650.) 

c) Time integration (*SOLVER) [KM_4]: 

Time integration method: Forward Euler (METHOD=1) 

Number of points: 2 (NPT=2) 

Time points: 0., 200. sec. (TI=0.,200.) 

Time step sizes: 1. sec. (STP=1.) 

5) Damage and Failure: None 

6) Output: 

a) Output file print level (*PRINT) [KM_6]: 

Print level: 6 (NPL=6) 

b) x-y plots (*XYPLOT) [KM_6]: 

Frequency: 5 (FREQ=5) 

Number of macro plots: 1 (MACRO=1) 

Macro plot names: example_3a (NAME=example_3a) 

Macro plot x-y quantities: ε22, σ22 (X=2 Y=8) 

Number of micro plots: 0 (MICRO=0) 

7) End of file keyword: (*END) 

Results

Figure 3.2 shows the effect of refining the fiber cross-section representation within the repeating unit cell 

for the square fiber packing arrangement. ARCHID=1 is the least refined fiber representation wherein the 

fiber is a square. Then, the fiber representation becomes more circular for ARCHID=4, ARCHID=6, 

ARCHID=7, and ARCHID=13. For the case of the transverse tensile response of a 0.25 fiber volume 

fraction SiC/Ti-21S composite at 650 °C, convergence based on the circular fiber refinement occurs 

quickly ñ the cross shaped, 7×7, 14×14, and 26×26 representations give nearly identical results. 

Figure 3.3 shows that the hexagonal and rectangular packing arrangements also give nearly identical 

results for this example. The large aspect ratio (R=2.0) employed with ARCHID=11, however, does 

NASA/TM—2002-212077/VOL3 68
Section 3: Analysis Type and Architecture Example 3a: Doubly Periodic GMC Internal RUC Library 

have a noticeable effect on the composite response. Since the hybrid composite represented by 

ARCHID=9 is a completely different material system, direct comparison with the other results in this 

example problem is not possible. 

0

10

20

30

40

50

60

70

0.000 0.005 0.010 0.015 0.020 0.025

Strain

Stress (ksi)

ARCHID = 1

ARCHID = 4

ARCHID = 6

ARCHID = 7

ARCHID = 13

Figure 3.2 Example 3a: plot of the transverse tensile stress-strain (σ22-ε22) response for a 0.25 fiber 

volume fraction SiC/Ti-21S composite at 650 °C as represented by repeating unit cell 

architectures 1, 4, 6, 7, and 13 from the MAC/GMC 4.0 architecture library.

0

10

20

30

40

50

60

70

80

0.000 0.005 0.010 0.015 0.020 0.025

Strain

Stress (ksi)

ARCHID = 1

ARCHID = 2

ARCHID = 3

ARCHID = 9

ARCHID = 11, R = 2.

Figure 3.3 Example 3a: plot of the transverse tensile stress-strain (σ22-ε22) response for a 0.25 fiber 

volume fraction SiC/Ti-21S composite at 650 °C as represented by repeating unit cell 

architectures 1, 2, 3, and 11 from the MAC/GMC 4.0 architecture library, plus the response 

of a hybrid 0.15 Boron - 0.07439 SiC / Ti-21S composite.

NASA/TM—2002-212077/VOL3 69
MAC/GMC 4.0 Example Problem Manual 

Example 3b: Doubly Periodic RUC Library with Interface

This example problem illustrates the internal library of doubly periodic repeating unit cell architectures 

within MAC/GMC 4.0 that include a distinct interfacial phase. While the applied loading employed in 

this example is identical to Example 3a, an additional material is present here that was not in this previous 

example. This additional material (with user-input material properties) is used for the interface within the 

analyzed composites. For more information on the codeís repeating unit cell architectures, see the 

MAC/GMC 4.0 Keywords Manual Section 3. 

MAC/GMC Input File: example_3b.mac

MAC/GMC 4.0 Example 3b - Doubly Periodic GMC RUCs w/ interface

*CONSTITUENTS

NMATS=4

M=1 CMOD=6 MATID=E

M=2 CMOD=4 MATID=A

M=3 CMOD=4 MATID=U MATDB=1 &

EL=11700.,11700.,0.365,0.365,4285.7,1.E-6,1.E-6 &

VI=0.8E-8,0.1,0.1E-5,0.85E-3,0.05,3.3,1.8,1.35,0.1,0.,0.

M=4 CMOD=6 MATID=D

*RUC

MOD=2 ARCHID=1 VF=0.25 RAD=71. RITFR=0.1 F=1 M=2 I=3

# MOD=2 ARCHID=2 VF=0.25 RAD=71. RITFR=0.1 F=1 M=2 I=3

# MOD=2 ARCHID=3 VF=0.25 RAD=71. RITFR=0.1 F=1 M=2 I=3

# MOD=2 ARCHID=6 R=1.0 VF=0.25 RITFR=0.1 F=1 M=2 I=3

# MOD=2 ARCHID=9 VF1=0.10 RAD1=50. RITFR1=0.1 &

# VF2=0.15 RAD2=71. RITFR2=0.1 R=1. F=4,1 M=2 I=3,3

# MOD=2 ARCHID=11 R=2.0 VF=0.25 RAD=71. RITFR=0.1 F=1 M=2 I=3

*MECH

LOP=2

NPT=2 TI=0.,200. MAG=0.,0.02 MODE=1

*THERM

NPT=2 TI=0.,200. TEMP=650.,650.

*SOLVER

METHOD=1 NPT=2 TI=0.,200. STP=0.01

*PRINT

NPL=6

*XYPLOT

FREQ=5

MACRO=1

NAME=example_3b X=2 Y=8

MICRO=0

*END

NASA/TM—2002-212077/VOL3 70
Section 3: Analysis Type and Architecture Example 3b: Doubly Periodic RUC Library with Interface 

ARCHID = 1 ARCHID = 2

ARCHID = 3 ARCHID = 6

ARCHID = 9 ARCHID = 11

Figure 3.4 MAC/GMC 4.0 doubly periodic repeating unit cell (RUC) architecture with interface 

library.

NASA/TM—2002-212077/VOL3 71
MAC/GMC 4.0 Example Problem Manual 

Annotated Input Data

1) Flags: None 

2) Constituent materials (*CONSTITUENTS) [KM_2]: 

Number of materials: 4 (NMATS=4) 

Materials: SiC fiber (MATID=E) 

Ti-21S (MATID=A) 

Fictional interface material (MATID=U) 

Boron fiber (MATID=D) 

Constitutive models: SiC fiber: linearly elastic (CMOD=6) 

Ti-21S matrix: isotropic GVIPS (CMOD=4) 

Interface: isotropic GVIPS (CMOD=4) 

Boron fiber: linearly elastic (CMOD=6) 

Source for user props: Read from input file (MATDB=1) 

A material intended to serve as the interface has been specified using user-defined material properties 

in conjunction with the isotropic GVIPS constitutive model. This material has been given elastic 

properties identical to Ti-21S at 650 °C, but its viscoplastic properties cause the material to flow to a 

much greater extent than does the Ti-21S matrix. This situation is intended to simulate the behavior 

of a weak interfacial bond within the composite. An example of an alternative more computationally 

efficient approach to simulating this weak interfacial behavior is examined in Example Problem 5e. 

3) Analysis type (*RUC) → Repeating Unit Cell Analysis [KM_3]: 

Analysis model: Doubly periodic GMC (MOD=2) 

RUC architecture: Square fiber, square pack (ARCHID=1) 

 Square fiber, hexagonal pack (ARCHID=2) 

 Square fiber, square diagonal pack (ARCHID=3) 

 7×7 circular fiber approx., rectangular pack (ARCHID=6) 

 2 square fibers of different size, square pack (ARCHID=9) 

 Square fiber, rectangular pack (ARCHID=11) 

Fiber volume fractions: 0.25 (VF=0.25) 

ARCHID=9: Boron = 0.15 SiC = 0.10 (VF1=0.15

VF2=0.1)

Unit cell aspect ratio: 1.0 (square pack), 2.0 for ARCHID=11 (R=1.0 or R=2.0) 

Fiber Radii: SiC fiber: 71μm (RAD=71 or RAD2=71) 

 Boron fiber: 50 μm (RAD1=50) 

Material assignment: SiC fiber (F=1) 

Ti-21S matrix (M=2) 

Interface material (I=3) 

Boron fiber (ARCHID=9 only) (F=4) 

All six doubly periodic fiber architectures contained in the MAC/GMC 4.0 library that included a 

distinct interface are exercised in this example problem. Each architecture can be used by 

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Section 3: Analysis Type and Architecture Example 3b: Doubly Periodic RUC Library with Interface 

commenting and uncommenting the appropriate lines of the input file. As in Example 3a, 

ARCHID=9 is a special case in that it requires 2 fibers with different radii as well as two interfaces. 

In addition, because ARCHID=9 requires the packing of the two fibers to be square or rectangular, 

not all geometries are possible. As such, the fiber volume fraction of the second fiber, the SiC fiber 

in this example, is adjusted from the user specified value to allow the packing arrangement to be 

correct given the value of the first fiberís volume fraction and the unit cell aspect ratio, R. For more 

information on these fiber architectures, see the MAC/GMC 4.0 Keywords Manual Section 3. 

4) Loading: 

a) Mechanical (*MECH) [KM_4]: 

Loading option: 1 (LOP=1) 

Number of points: 2 (NPT=2) 

Time points: 0., 200. sec. (TI=0.,200.) 

Load magnitude: 0., 0.02 (MAG=0.,0.02) 

Loading mode: strain control (MODE=1) 

b) Thermal (*THERM) [KM_4]: 

Number of points: 2 (NPT=2) 

Time points: 0., 200. sec. (TI=0.,200.) 

Temperature points: 650., 650. °C (TEMP=650.,650.) 

c) Time integration (*SOLVER) [KM_4]: 

Time integration method: Forward Euler (METHOD=1) 

Number of points: 2 (NPT=2) 

Time points: 0., 200. sec. (TI=0.,200.) 

Time step sizes: 1. sec. (STP=1.) 

5) Damage and Failure: None 

6) Output: 

a) Output file print level (*PRINT) [KM_6]: 

Print level: 6 (NPL=6) 

b) x-y plots (*XYPLOT) [KM_6]: 

Frequency: 1 (FREQ=1) 

Number of macro plots: 1 (MACRO=1) 

Macro plot names: example_3b (NAME=example_3b) 

Macro plot x-y quantities: ε22, σ22 (X=2 Y=8) 

Number of micro plots: 0 (MICRO=0) 

7) End of file keyword: (*END) 

NASA/TM—2002-212077/VOL3 73
MAC/GMC 4.0 Example Problem Manual 

Results

Figure 3.5 shows that, when compared to Figure 3.2 and Figure 3.3, that the inclusion of the weak 

interface within the MAC/GMC repeating unit cell architecture has a major impact on the simulated 

transverse response of the composite. As stated earlier, an alternative approach to simulating composites 

with weak fiber-matrix interfaces is presented in Example 5e. 

0

5

10

15

20

25

30

35

40

0.000 0.005 0.010 0.015 0.020 0.025

Strain

Stress (ksi)

ARCHID = 3

ARCHID = 2

ARCHID = 6

ARCHID = 1

ARCHID = 9

ARCHID = 11

Figure 3.5 Example 3b: plot of the transverse tensile stress-strain (σ22-ε22) response for a 0.25 fiber 

volume fraction SiC/Ti-21S composite at 650 °C as represented by repeating unit cell 

architectures MAC/GMC 4.0 architecture library, plus the response of a hybrid 0.15 Boron 

- 0.07439 SiC / Ti-21S composite, all with interfaces.

NASA/TM—2002-212077/VOL3 74
Section 3: Analysis Type and Architecture Example 3c: User-Defined Doubly Periodic Architecture 

Example 3c: User-Defined Doubly Periodic Architecture

This example problem illustrates how users may input their own repeating unit cell architecture through 

the MAC/GMC 4.0 input file. In particular, a B/Al composite with randomly distributed fibers (see 

Figure 3.6) subjected to an applied strain at a (fast) rate of 0.1/s is considered. For more information on 

the user-defined repeating unit cell architecture, see the MAC/GMC 4.0 Keywords Manual Section 3. 

Figure 3.6 Example 3c: Random fiber composite architecture specified through the input file. 

MAC/GMC Input File: example_3c.mac

MAC/GMC 4.0 Example 3c - Random composite architecture using 2-D GMC

*CONSTITUENTS

NMATS=2

M=1 CMOD=6 TREF=21. MATID=A

M=2 CMOD=1 TREF=21. MATID=C

*RUC

MOD=2 ARCHID=99

NB=14 NG=14

H=1.,1.,1.,1.,1.,1.,1.,1.,1.,1.,1.,1.,1.,1.

L=1.,1.,1.,1.,1.,1.,1.,1.,1.,1.,1.,1.,1.,1.

SM=1,2,2,2,2,2,1,1,2,2,1,1,2,1

SM=1,2,1,1,2,2,1,1,2,2,2,2,2,1

SM=2,2,1,1,2,2,2,2,2,2,1,1,2,2

SM=2,2,2,2,2,2,2,1,1,2,1,1,2,2

SM=2,2,1,1,2,2,2,1,1,2,2,2,2,2

SM=1,2,1,1,2,2,2,2,2,2,1,1,2,1

SM=1,2,2,2,2,1,1,2,2,2,1,1,2,1

SM=2,2,2,2,2,1,1,2,2,2,2,2,2,2

SM=2,1,1,2,2,2,2,2,1,1,2,1,1,2

SM=2,1,1,2,1,1,2,2,1,1,2,1,1,2

SM=2,2,2,2,1,1,1,1,2,2,2,2,2,2

SM=1,1,2,2,2,2,1,1,2,2,2,1,1,2

SM=1,1,2,2,2,2,2,2,2,2,2,1,1,2

SM=2,2,2,2,2,2,2,2,2,2,1,1,2,2

*MECH

LOP=2

NPT=2 TI=0.,0.1 MAG=0.,0.01 MODE=1

*SOLVER

METHOD=1 NPT=2 TI=0.,0.1 STP=0.00005

NASA/TM—2002-212077/VOL3 75
MAC/GMC 4.0 Example Problem Manual 

*PRINT

NPL=6

*XYPLOT

FREQ=1

MACRO=1

NAME=example_3c X=2 Y=8

MICRO=0

*END

Annotated Input Data

1) Flags: None 

2) Constituent materials (*CONSTITUENTS) [KM_2]: 

Number of materials: 2 (NMATS=2) 

Materials: Boron fiber (MATID=A) 

Aluminum (6061-0a) (MATID=C) 

Constitutive models: Boron fiber: linearly elastic (CMOD=6) 

Al matrix: Bodner-Partom (CMOD=1) 

Reference Temperature: 21. °C (TREF=21) 

3) Analysis type (*RUC) → Repeating Unit Cell Analysis [KM_3]: 

Analysis model: Doubly periodic GMC (MOD=2) 

Architecture: User-defined (ARCHID=99) 

No. subcells in x2-dir.: 14 (NB=14) 

No. subcells in x3-dir.: 14 (NG=14) 

Subcell heights: 1. (H=1.,1.,...) 

Subcell lengths: 1. (L=1.,1.,...) 

Material assignment: see input file (SM=...) 

The materials that occupy each subcell are specified with SM=.... Each separate SM=... line 

corresponds to a different descending value of β, while the comma separated material numbers on a 

given line correspond to different ascending γ values. For more information on the user-defined 

repeating unit cell architecture, see the MAC/GMC 4.0 Keywords Manual Section 3. 

4) Loading: 

a) Mechanical (*MECH) [KM_4]: 

Loading option: 1 (LOP=1) 

Number of points: 2 (NPT=2) 

Time points: 0., 0.1 sec. (TI=0.,0.1) 

Load magnitude: 0., 0.01 (MAG=0.,0.01) 

Loading mode: strain control (MODE=1) 

b) Thermal (*THERM): None 

c) Time integration (*SOLVER) [KM_4]: 

Time integration method: Forward Euler (METHOD=1) 

Number of points: 2 (NPT=2) 

NASA/TM—2002-212077/VOL3 76
Section 3: Analysis Type and Architecture Example 3c: User-Defined Doubly Periodic Architecture 

Time points: 0., 0.1 sec. (TI=0.,0.1) 

Time step sizes: 0.00005 sec. (STP= 0.00005) 

5) Damage and Failure: None 

6) Output: 

a) Output file print level (*PRINT) [KM_6]: 

Print level: 6 (NPL=6) 

b) x-y plots (*XYPLOT) [KM_6]: 

Frequency: 1 (FREQ=1) 

Number of macro plots: 1 (MACRO=1) 

Macro plot names: example_3c (NAME=example_3c) 

Macro plot x-y quantities: ε22, σ22 (X=2 Y=8) 

Number of micro plots: 0 (MICRO=0) 

7) End of file keyword: (*END) 

Results

The results for this example problem are plotted in Figure 3.7. For this composite with randomly 

distributed strongly bonded fibers, GMC works well. Note, however, that in the presence of a weak 

interface or fiber-matrix debonding, however, the lack of normal-shear field coupling in GMC would 

result in significant inaccuracies in the predicted transverse composite response. Such a composite could 

be modeled more accurately using the high-fidelity GMC (HFGMC) model, which does include the 

normal-shear field coupling. Example Problem 3f illustrates the use of HFGMC within MAC/GMC 4.0. 

0

20

40

60

80

100

120

140

160

180

0.000 0.002 0.004 0.006 0.008 0.010 0.012

strain

stress (ksi)

Figure 3.7 Example 3c: plot of the simulated transverse tensile stress-strain (σ22-ε22) response for a 

0.3469 fiber volume fraction B/Al composite at 21 °C as represented by a user-defined 

repeating unit cell architecture intended to simulate randomly distributed fibers.

NASA/TM—2002-212077/VOL3 77
MAC/GMC 4.0 Example Problem Manual 

Example 3d: Triply Periodic GMC Internal RUC Library

This example problem illustrates how to access the internal library of triply periodic repeating unit cell 

architectures within MAC/GMC 4.0. The loading conditions, materials, and fiber volume fraction remain 

constant for use with all repeating unit cell architectures. For more information on the codeís internal 

repeating unit cell architectures, see the MAC/GMC 4.0 Keywords Manual Section 3. 

ARCHID = 1 ARCHID = 2 ARCHID = 3

ARCHID = 4 ARCHID = 10

Figure 3.8 MAC/GMC triply periodic repeating unit cell architecture library. 

MAC/GMC Input File: example_3d.mac

MAC/GMC 4.0 Example 3d - Triply Periodic GMC RUC Library

*CONSTITUENTS

NMATS=2

M=1 CMOD=6 MATID=E

M=2 CMOD=4 MATID=A

*RUC

MOD=3 ARCHID=1 VF=0.25 ASP=2. F=1 M=2

NASA/TM—2002-212077/VOL3 78
Section 3: Analysis Type and Architecture Example 3d: Triply Periodic GMC Internal RUC Library 

# MOD=3 ARCHID=2 VF=0.25 ASP1=2. ASP2=2. F=1 M=2

# MOD=3 ARCHID=3 VF=0.25 ASP1=2. ASP2=1. DR=1. F=1 M=2

# MOD=3 ARCHID=4 VF=0.25 &

# OPT=3 RA=2. RC=1. D=2. H=1. LL=1. F=1 M=2

# MOD=3 ARCHID=10 VF=0.25 F=1 M=2

*MECH

LOP=1

NPT=2 TI=0.,200. MAG=0.,0.02 MODE=1

*THERM

NPT=2 TI=0.,200. TEMP=650.,650.

*SOLVER

METHOD=1 NPT=2 TI=0.,200. STP=1.

*PRINT

NPL=6

*XYPLOT

FREQ=5

MACRO=1

NAME=example_3d X=1 Y=7

MICRO=0

*END

Annotated Input Data

1) Flags: None 

2) Constituent materials (*CONSTITUENTS) [KM_2]: 

Number of materials: 2 (NMATS=2) 

Materials: SiC fiber (MATID=E) 

Ti-21S (MATID=A) 

Constitutive models: SiC fiber: linearly elastic (CMOD=6) 

Ti-21S matrix: Isotropic GVIPS (CMOD=4) 

3) Analysis type (*RUC) → Repeating Unit Cell Analysis [KM_3]: 

Analysis model: Triply periodic GMC (MOD=3) 

RUC architecture: 2×2×2, short fiber, square array (ARCHID=1) 

 Non-overlapping fibers, square diagonal array (ARCHID=2) 

 Off-set short fibers, square array (ARCHID=3) 

 Ellipsoidal fiber approximation, rect. array (ARCHID=4) 

 Open cell (ARCHID=10) 

Fiber volume fractions: 0.25 (VF=0.25) 

Fiber aspect ratio 2. (ASP=2. or ASP1=2.) 

Unit cell aspect ratio: 2. or 1. (ASP=1. or ASP=2.0) 

D ratio (ARCHID=3 only): 1. (DR=0.25) 

Ellipsoid inclusion: 

RUC dimensions: 2., 1., 1. (D=2. H=1. LL=1.) 

Semi-major axis ratios: a/b = 2., c/b = 1. (RA=2. RC=1.) 

Material assignment: SiC fiber (F=1) 

Ti-21S matrix (M=2) 

NASA/TM—2002-212077/VOL3 79
MAC/GMC 4.0 Example Problem Manual 

All five triply periodic repeating unit cell architectures contained in the MAC/GMC 4.0 library are 

exercised in this example problem. Again, each architecture can be used by commenting and 

uncommenting the appropriate lines of the input file. ARCHID=1 through ARCHID=3 represent 

different packing arrangements for short fiber composites wherein the fiber cross-section is square. 

ARCHID=4 represents an ellipsoidal inclusion, which may be thought of as a three-dimensional 

generalization of the doubly periodic ARCHID=6. Finally, ARCHID=10 represents a threeFor more information on these fiber architectures, see the MAC/GMC 4.0 Keywords Manual 

Section 3. 

4) Loading: 

a) Mechanical (*MECH) [KM_4]: 

Loading option: 1 (LOP=1) 

Number of points: 2 (NPT=2) 

Time points: 0., 200. sec. (TI=0.,200.) 

Load magnitude: 0., 0.02 (MAG=0.,0.02) 

Loading mode: strain control (MODE=1) 

b) Thermal (*THERM) [KM_4]: 

Number of points: 2 (NPT=2) 

Time points: 0., 200. sec. (TI=0.,200.) 

Temperature points: 650., 650. °C (TEMP=650.,650.) 

c) Time integration (*SOLVER) [KM_4]: 

Time integration method: Forward Euler (METHOD=1) 

Number of points: 2 (NPT=2) 

Time points: 0., 200. sec. (TI=0.,200.) 

Time step sizes: 1. sec. (STP=1.) 

5) Damage and Failure: None 

6) Output: 

a) Output file print level (*PRINT) [KM_6]: 

Print level: 6 (NPL=6) 

b) x-y plots (*XYPLOT) [KM_6]: 

Frequency: 5 (FREQ=5) 

Number of macro plots: 1 (MACRO=1) 

Macro plot names: example_3d (NAME=example_3d) 

Macro plot x-y quantities: ε11, σ11 (X=1 Y=7) 

Number of micro plots: 0 (MICRO=0) 

7) End of file keyword: (*END) 

NASA/TM—2002-212077/VOL3 80

dimensional open cell architecture. Diagrams of these architectures are given in Figure 3.8. 
Section 3: Analysis Type and Architecture Example 3d: Triply Periodic GMC Internal RUC Library 

Results

Figure 3.9 shows the simulated response of the triply periodic RUC architectures for this 0.25 fiber 

volume fraction metal matrix composite. It should be noted that, as the fiber volume fraction of triply 

periodic composites rises, the GMC predictions can become less accurate. This has been shown by Pahr 

and Arnold (2002). As the fiber volume fraction rises, the RUC architecture may have thin matrix 

subcells adjacent to or in between larger fiber subcells. This can cause the theory to over-predict the 

stress in these thin matrix subcells, which leads to an over-prediction of the inelastic deformation in these 

subcells and an overall response that is too compliant. 

0

10

20

30

40

50

60

70

80

90

100

0.000 0.005 0.010 0.015 0.020 0.025

strain

stress (ksi)

ARCHID = 1

ARCHID = 2

ARCHID = 3

ARCHID = 4

ARCHID = 10

Figure 3.9 Example 3d: plot of the longitudinal tensile stress-strain (σ11-ε11) response for a 0.25 fiber 

volume fraction triply periodic SiC/Ti-21S composite at 650 °C as represented by the triply 

periodic repeating unit cell architectures from the MAC/GMC 4.0 architecture library. See 

Figure 3.8 for the RUC architectures. 

NASA/TM—2002-212077/VOL3 81
MAC/GMC 4.0 Example Problem Manual 

Example 3e: User-Defined Triply Periodic GMC Architecture

This example problem demonstrates how users can input their own triply periodic repeating unit cell 

architecture. In particular, a periodic [0°/90°] fiber architecture subjected to an applied strain of 0.02 at 

650 °C is analyzed. The geometry of the repeating unit cell is shown in Figure 3.10. It should be noted 

that while the fiber architecture contains 0° and 90° fibers, this problem does not truly simulate a 

laminate. The represented material is periodic, so it repeats in all three coordinate directions to form a 

continuum. Therefore, although the RUC has an appearance similar to that of an asymmetric [0°/90°] 

laminate, no bending will result. A laminate is different in that it has a finite thickness, and the 

lamination theory formulation is based on a plane stress condition. See the MAC/GMC 4.0 Keywords 

Manual Section 3 for additional information on the codeís repeating unit cell architectures. 

Figure 3.10 Geometry of the user-defined repeating unit cell architecture employed in Example 3e to 

simulate the response of a periodic [0°/90°] composite.

MAC/GMC Input File: example_3e.mac

MAC/GMC 4.0 Example 3c - [0/90] architecture using 3-d gmc

*CONSTITUENTS

NMATS=2

M=1 CMOD=6 MATID=E

M=2 CMOD=4 MATID=A

*RUC

MOD=3 ARCHID=99

NA=4 NB=2 NG=2

D=0.67823,0.32177,0.67823,0.32177

NASA/TM—2002-212077/VOL3 82
Section 3: Analysis Type and Architecture Example 3e: User-Defined Triply Periodic GMC Architecture 

H=0.67823,0.32177

L=0.32177,0.67823

# -- gamma = 1

SM=2,2

SM=2,2

SM=2,2

SM=1,2

# -- gamma = 2

SM=2,2

SM=1,1

SM=2,2

SM=1,2

*MECH

LOP=1

# LOP=2

# LOP=3

NPT=2 TI=0.,200. MAG=0.,0.02 MODE=1

*THERM

NPT=2 TI=0.,200. TEMP=650.,650.

*SOLVER

METHOD=1 NPT=2 TI=0.,200. STP=1.

*PRINT

NPL=6

*XYPLOT

FREQ=5

MACRO=3

NAME=example_3e_11 X=1 Y=7

NAME=example_3e_22 X=2 Y=8

NAME=example_3e_33 X=3 Y=9

MICRO=0

*END

Annotated Input Data

1) Flags: None 

2) Constituent materials (*CONSTITUENTS) [KM_2]: 

Number of materials: 2 (NMATS=2) 

Materials: SiC fiber (MATID=E) 

Ti-21S (MATID=A) 

Constitutive models: SiC fiber: linearly elastic (CMOD=6) 

Ti-21S matrix: Isotropic GVIPS (CMOD=4) 

3) Analysis type (*RUC) → Repeating Unit Cell Analysis [KM_3]: 

Analysis model: Triply periodic GMC (MOD=3) 

Architecture: User-defined (ARCHID=99) 

No. subcells in x1-dir.: 4 (NA=4) 

No. subcells in x2-dir.: 2 (NB=2) 

No. subcells in x3-dir.: 2 (NG=2) 

Subcell depths: see input file (D=0.67823,...) 

Subcell heights: see input file (H=0.67823,...) 

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MAC/GMC 4.0 Example Problem Manual 

Subcell lengths: see input file (L=0.32177,...) 

Material assignment: see input file (SM=...) 

The materials that occupy each subcell are specified with SM=.... The order in which the materials are 

specified is: 1) ascending groups for each value of γ, 2) descending lines for each value of α, and 3) 

ascending comma separated material numbers for each value of β. That is, in the present example, 

we have, 

# -- gamma = 1

SM=M(4,1,1),M(4,2,1)

SM=M(3,1,1),M(3,2,1)

SM=M(2,1,1),M(2,2,1)

SM=M(1,1,1),M(1,2,1)

# -- gamma = 2

SM=M(4,1,2),M(4,2,2)

SM=M(3,1,2),M(3,2,2)

SM=M(2,1,2),M(2,2,2)

SM=M(1,1,2),M(1,2,2)

where, M(α,β,γ) is the material number assigned to subcell (α,β,γ). For more information on the 

user-defined repeating unit cell architecture, see the MAC/GMC 4.0 Keywords Manual Section 3. 

4) Loading: 

a) Mechanical (*MECH) [KM_4]: 

Loading option: 1, 2, or 3 (LOP=1, LOP=2, or, LOP=3) 

Number of points: 2 (NPT=2) 

Time points: 0., 200. sec. (TI=0.,200.) 

Load magnitude: 0., 0.02 (MAG=0.,0.02) 

Loading mode: strain control (MODE=1) 

! Note: The desired loading option is selected by commenting and uncommenting the appropriate lines 

of the input file. 

b) Thermal (*THERM) [KM_4]: 

Number of points: 2 (NPT=2) 

Time points: 0., 200. sec. (TI=0.,200.) 

Temperature points: 650., 650. °C (TEMP=650.,650.) 

c) Time integration (*SOLVER) [KM_4]: 

Time integration method: Forward Euler (METHOD=1) 

Number of points: 2 (NPT=2) 

Time points: 0., 200. sec. (TI=0.,200.) 

Time step sizes: 1. sec. (STP=1.) 

5) Damage and Failure: None 

6) Output: 

a) Output file print level (*PRINT) [KM_6]: 

Print level: 6 (NPL=6) 

NASA/TM—2002-212077/VOL3 84
Section 3: Analysis Type and Architecture Example 3e: User-Defined Triply Periodic GMC Architecture 

b) x-y plots (*XYPLOT) [KM_6]: 

Frequency: 5 (FREQ=5) 

Number of macro plots: 1 (MACRO=1) 

Macro plot names: example_3e_11 (NAME=example_3e_11) 

 example_3e_22 (NAME=example_3e_22) 

 example_3e_33 (NAME=example_3e_33) 

Macro plot x-y quantities: ε11, σ11 (X=1 Y=7) 

 ε22, σ22 (X=2 Y=8) 

 ε33, σ33 (X=3 Y=9) 

Number of micro plots: 0 (MICRO=0) 

7) End of file keyword: (*END) 

Results

Figure 3.11 shows that the tensile response of the periodic [0°/90°] composite in the two in-plane 

coordinate directions (x2 and x3) is identical, as expected. In the out-of-plane direction (x1), the overall 

tensile response is more compliant due to the absence of continuous reinforcement in this direction. 

0

50

100

150

200

250

300

350

0.000 0.005 0.010 0.015 0.020 0.025

strain

stress (ksi)

x1 response

x2 response

x3 response

Figure 3.11 Example 3e: plot of the normal tensile stress-strain response in each of the three coordinate 

directions for a 0.46 fiber volume fraction periodic [0°/90°] SiC/Ti-21S composite at 650 

°C as represented by the user-defined triply periodic repeating unit cell architecture shown 

in Figure 3.10.

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MAC/GMC 4.0 Example Problem Manual 

Example 3f: High-Fidelity Generalized Method of Cells

This example problem performs analysis of a SiC/Ti-21S composite using the doubly periodic highfidelity generalized method of cells (HFGMC) and compares the results to those of GMC. In particular, 

transverse strain is applied to the composite, the 26×26 circular fiber approximation RUC is employed, 

and the local field results in the composite are examined. HFGMC is formulated using a higher order 

displacement field than that employed within GMC. This provides HFGMC with normal-shear field 

coupling that is absent in GMC and allows HFGMC to be more accurate in terms of the local fields. In 

addition, while the local fields in GMC are piecewise uniform (i.e., constant within each subcell), 

HFGMCís local fields vary within the subcells. Thus, in HFGMC all field variables must be tracked not 

just once for each subcell (as in GMC), but rather at a number of integration points within each subcell. 

This number of integration points that MAC/GMC 4.0 employs in each subcell, along with the order of 

the polynomial employed to approximate the inelastic strain field in the composite, must be specified by 

the user. Because HFGMC employs a higher order displacement field, it requires solution for a greater 

number of unknowns. This, coupled with the fact that the field variables must be tracked at several 

integration points within each subcell, renders HFGMC significantly more computationally demanding 

than GMC. Thus HFGMCís improved accuracy comes at an increased computational cost over GMC. 

For more information on HFGMC, see the MAC/GMC 4.0 Theory Manual Section 2.1.2. 

MAC/GMC Input File: example_3f.mac

MAC/GMC 4.0 Example 3f - HFGMC doubly periodic analysis

*CONSTITUENTS

NMATS=2

M=1 CMOD=6 MATID=E

M=2 CMOD=4 MATID=A

*RUC

MOD=12 ARCHID=13 VF=0.25 R=1. F=1 M=2

# MOD=2 ARCHID=13 VF=0.25 R=1. F=1 M=2

*MECH

LOP=2

NPT=2 TI=0.,200. MAG=0.,0.02 MODE=1

*THERM

NPT=2 TI=0.,200. TEMP=650.,650.

*SOLVER

METHOD=1 NPT=2 TI=0.,200. STP=1.

NLEG=5 NINTEG=11

*PRINT

NPL=6

*MATLAB

N=1 TIMES=200.

*XYPLOT

FREQ=5

MACRO=1

NAME=example_3f X=2 Y=8

MICRO=0

*END

NASA/TM—2002-212077/VOL3 86
Section 3: Analysis Type and Architecture Example 3f: High-Fidelity Generalized Method of Cells 

Annotated Input Data

1) Flags: None 

2) Constituent materials (*CONSTITUENTS) [KM_2]: 

Number of materials: 2 (NMATS=2) 

Materials: SiC fiber (MATID=E) 

Ti-21S (MATID=A) 

Constitutive models: SiC fiber: linearly elastic (CMOD=6) 

Ti-21S matrix: Isotropic GVIPS (CMOD=4) 

3) Analysis type (*RUC) → Repeating Unit Cell Analysis [KM_3]: 

Analysis model: Doubly periodic HFGMC (MOD=12) 

 Doubly periodic GMC (MOD=2) 

RUC architecture: 26×26 circular fiber approx., rectangular pack (ARCHID=13) 

Fiber volume fractions: 0.25 (VF=0.25) 

Unit cell aspect ratio: 1.0 (square pack) (R=1.0) 

Material assignment: SiC fiber (F=1) 

Ti-21S matrix (M=2) 

By commenting and uncommenting the appropriate lines, HFGMC or GMC may be employed in the 

present example. It should be noted that the present implementation of the doubly periodic HFGMC 

model within MAC/GMC 4.0 requires the number of subcells in each direction to be even. If an 

RUC from the MAC/GMC RUC library that has an odd number of subcells in either direction is 

selected, an error will occur and execution will stop. For more information on HFGMC, see the 

MAC/GMC 4.0 Theory Manual Section 2.1.2. 

4) Loading: 

a) Mechanical (*MECH) [KM_4]: 

Loading option: 1 (LOP=1) 

Number of points: 2 (NPT=2) 

Time points: 0., 200. sec. (TI=0.,200.) 

Load magnitude: 0., 0.02 (MAG=0.,0.02) 

Loading mode: strain control (MODE=1) 

b) Thermal (*THERM) [KM_4]: 

Number of points: 2 (NPT=2) 

Time points: 0., 200. sec. (TI=0.,200.) 

Temperature points: 650., 650. °C (TEMP=650.,650.) 

c) Time integration (*SOLVER) [KM_4]: 

Time integration method: Forward Euler (METHOD=1) 

Number of points: 2 (NPT=2) 

Time points: 0., 200. sec. (TI=0.,200.) 

Time step sizes: 1. sec. (STP=1.) 

Order of Legendre polynomial: 5 (NLEG=5) 

No. integration points per subcell: 11 (NINTEG=11) 

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MAC/GMC 4.0 Example Problem Manual 

As mentioned above, unlike GMC the local fields within HFGMC vary within each subcell. Thus, a 

number of integration points (NINTEG) at which to track the field variables within each subcell must 

be specified. Further, the inelastic strain field is approximated using Legendre polynomials. The 

order of these polynomials (NLEG) must also be specified. While users may employ desired values 

for these terms, NLEG=5 and NINTEG=11 have been shown to yield good precision in the local 

results. It is recommended that these values be employed unless the user is confident in altering 

them. For more information, see the MAC/GMC 4.0 Keywords Manual Section 4. 

5) Damage and Failure: None 

6) Output: 

a) Output file print level (*PRINT) [KM_6]: 

Print level: 6 (NPL=6) 

b) Matlab output data (*MATLAB) [KM_6]: 

Number of Matlab output times: 1 (N=1) 

Matlab output times: 200. sec. (TIMES=200.) 

In order to display the local fields generated by the GMC and HFGMC models in the present 

example, the *MATLAB keyword has been utilized. This option generates data files containing the 

local fields within the composite that can be used to generate surface or ìfringeî plots using the 

MATLAB software product. The user specifies the number of times (N) during the MAC/GMC 4.0 

simulation that MATLAB output will be written, as well as the actual output times (TIMES) 

themselves. For more information on generating MATLAB fringe plots like those shown in this 

example, see the MAC/GMC Keywords Manual Section 6 and Example Problem 6c in this manual. 

c) x-y plots (*XYPLOT) [KM_6]: 

Frequency: 5 (FREQ=5) 

Number of macro plots: 1 (MACRO=1) 

Macro plot names: example_3f (NAME=example_3f) 

Macro plot x-y quantities: ε22, σ22 (X=2 Y=8) 

Number of micro plots: 0 (MICRO=0) 

7) End of file keyword: (*END) 

Results

Figure 3.12 shows that the global transverse tensile response predicted by HFGMC and GMC is very 

similar. However, the local stress component fields plotted in Figure 3.13 (at an applied load of 0.02 

transverse strain) show significant differences between the two models. First, since GMC lacks shear 

coupling and only normal global stresses and strains are imposed, the shear stress (σ23) within the 

composite is zero everywhere in Figure 3.13a. In contrast, Figure 3.13b shows that HFGMC predicts 

significant shear stress concentrations near the fiber-matrix interface. Second, GMC predicts in-plane 

normal stress fields (σ22 and σ33) that are not only uniform within each subcell, but also constant in rows 

(along the x3-axis for σ33) or columns (along the x2-axis for σ22) of subcells. This is another manifestation 

of GMCís lack of shear coupling. HFGMC, on the other hand, predicts in-plane stress fields that vary 

throughout the repeating unit cell with noticeable concentrations at various locations. 

NASA/TM—2002-212077/VOL3 88
Section 3: Analysis Type and Architecture Example 3f: High-Fidelity Generalized Method of Cells 

Figure 3.14 compares the inelastic strain component fields predicted by HFGMC and GMC at an applied 

load of 0.02 transverse strain. As in the stress fields, HFGMC predicts an in-plain shear concentration at 

the fiber-matrix interface while GMC predicts zero in-plane inelastic shear strain. Otherwise, the GMC 

appears to match the HFGMC inelastic strain results quite well, although HFGMC does predict higher 

magnitude in

22 ε and in

33 ε concentrations near the fiber-matrix interface. The improved local fields 

predicted by HFGMC will have the greatest effect on the predicted overall composite response when 

effects that depend strongly on the local fields (e.g., damage, debonding, failure) are included in the 

simulation. 

0

10

20

30

40

50

60

70

0.000 0.005 0.010 0.015 0.020 0.025

strain

stress (ksi)

GMC

HFGMC

Stress and 

inelastic strain 

fields plotted at 

this point

Figure 3.12 Example 3f: plot of the global transverse tensile stress-strain (σ22-ε22) response for a 0.25 

fiber volume fraction SiC/Ti-21S composite at 650 °C as represented by a 26×26 RUC 

using GMC and HFGMC.

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MAC/GMC 4.0 Example Problem Manual 

Figure 3.13 Example 3f: plots of the local stress component fields for a 0.25 fiber volume fraction 

SiC/Ti-21S composite at 650 °C as simulated by (a) GMC and (b) HFGMC at an applied 

global load of 0.02 transverse strain. 

(a) 

(b) 

NASA/TM—2002-212077/VOL3 90
Section 3: Analysis Type and Architecture Example 3f: High-Fidelity Generalized Method of Cells 

Figure 3.14 Example 3f: plots of the local inelastic strain component fields for a 0.25 fiber volume 

fraction SiC/Ti-21S composite at 650 °C as simulated by (a) GMC and (b) HFGMC at an 

applied global load of 0.02 transverse strain. 

(b) 

(a) 

NASA/TM—2002-212077/VOL3 91
MAC/GMC 4.0 Example Problem Manual 

Example 3g: Laminate Theory ñ Single Ply

This example problem employs the classical lamination theory capabilities of MAC/GMC 4.0 to simulate 

the response of a single ply SiC/Ti-21S laminate with several different fiber orientations. It is important 

to keep in mind that there are several major differences between analyzing a laminate and a composite 

material, even if both are composed of the same phases. First, while a composite material is a threedimensional continuum point without edges and boundary influences, a laminate has a finite thickness 

and thus top and bottom boundaries. Further, classical lamination theory is based on the assumption that 

the laminate is in a state of plane stress at every point. That is, (in MAC/GMCís laminate coordinate 

system, Figure 1.2) the out-of-plane normal stress (σzz or σ33) as well as the out-of-plane shear stresses σxz

and σyz or σ13 and σ23) are required to be zero throughout the laminate. Thus global loading that involves 

any out-of-plane stress (or strain) components is inadmissible. Finally, since the layers within the 

laminate are composed of composite materials, MAC/GMC 4.0 treats the laminate problem with an 

embedded multi-scale approach. The local behavior of the composite material in each layer is modeled 

using GMC, which is embedded within the lamination theory analysis that handles the structural 

(laminate) scale. For more information on the lamination theory and the codeís laminate analysis 

capabilities, see the MAC/GMC 4.0 Theory Manual Section 3. 

In the present example, a single 0.25 fiber volume fraction SiC/Ti-21S composite ply is considered in 

which the fiber orientation is varied between 0° (along the x loading direction) and 90° (transverse to the 

x loading direction). A simple 2×2 square fiber, square pack RUC architecture is employed, and a 

midplane strain of 0.02 is applied at a temperature of 650 °C. 

MAC/GMC Input File: example_3g.mac

MAC/GMC 4.0 Example 3g - Lamination theory, single ply, varying fiber angle

*CONSTITUENTS

NMATS=2

M=1 CMOD=6 MATID=E

M=2 CMOD=4 MATID=A

*LAMINATE

NLY=1

LY=1 MOD=2 THK=1. ANG=0 ARCHID=1 VF=0.25 F=1 M=2

# LY=1 MOD=2 THK=1. ANG=15 ARCHID=1 VF=0.25 F=1 M=2

# LY=1 MOD=2 THK=1. ANG=45 ARCHID=1 VF=0.25 F=1 M=2

# LY=1 MOD=2 THK=1. ANG=30 ARCHID=1 VF=0.25 F=1 M=2

# LY=1 MOD=2 THK=1. ANG=60 ARCHID=1 VF=0.25 F=1 M=2

# LY=1 MOD=2 THK=1. ANG=75 ARCHID=1 VF=0.25 F=1 M=2

# LY=1 MOD=2 THK=1. ANG=90 ARCHID=1 VF=0.25 F=1 M=2

*MECH

LOP=1

NPT=2 TI=0.,200. MAG=0.,0.02 MODE=1

*THERM

NPT=2 TI=0.,200. TEMP=650.,650.

*SOLVER

METHOD=1 NPT=2 TI=0.,200. STP=1.

*PRINT

NPL=6

*XYPLOT

NASA/TM—2002-212077/VOL3 92
Section 3: Analysis Type and Architecture Example 3g: Laminate Theory ñ Single Ply 

FREQ=5

LAMINATE=1

NAME=example_3g X=1 Y=10

MACRO=0

MICRO=0

*END

Annotated Input Data

1) Flags: None 

2) Constituent materials (*CONSTITUENTS) [KM_2]: 

Number of materials: 2 (NMATS=2) 

Materials: SiC fiber (MATID=E) 

Ti-21S (MATID=A) 

Constitutive models: SiC fiber: linearly elastic (CMOD=6) 

Ti-21S matrix: Isotropic GVIPS (CMOD=4) 

3) Analysis type (*LAMINATE) → Laminate Analysis [KM_3]: 

Number of layers: 1 (NLY=1) 

Layer analysis model: Doubly periodic GMC (MOD=2) 

Layer thickness: 1. (THK=1.) 

Layer fiber angle: 0° (ANG=0) 

 15° (ANG=15) 

 30° (ANG=30) 

 45° (ANG=45) 

 60° (ANG=60) 

 75° (ANG=75) 

 90° (ANG=90) 

Architecture: 2×2, square fiber, square pack (ARCHID=1) 

Material assignment: SiC fiber (F=1) 

Ti-21S matrix (M=2) 

The information for each layer is specified on a separate line of the input file. In the present example, 

the laminate contains only one layer and thus requires only one line. The information on the line is 

similar to that which is specified under *RUC for repeating unit cell analysis, with the addition of the 

layer thickness (THK) and the layer fiber orientation angle (ANG). Note that the thickness values are 

arbitrary, but the laminate force resultants are integrated through the thickness and thus scale with the 

overall laminate thickness. In order to execute the code for the seven different fiber orientations, the 

appropriate lines in the input file must be commented and uncommented. For more information on 

the laminate analysis input requirements, see the MAC/GMC 4.0 Keywords Manual Section 3. 

4) Loading: 

a) Mechanical (*MECH) [KM_4]: 

Loading option: 1 (loading in the laminate x-direction) (LOP=1) 

Number of points: 2 (NPT=2) 

Time points: 0., 200. sec. (TI=0.,200.) 

Load magnitude: 0., 0.02 (MAG=0.,0.02) 

Loading mode: midplane strain/curvature control (MODE=1) 

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MAC/GMC 4.0 Example Problem Manual 

Rather than applying loads in the form of stress and strain components as is done in repeating unit 

cell analysis, in laminate analysis, stress/moment resultants and midplane strains/curvatures are 

applied. In addition, as mentioned above, only in-plane loading is admissible so the loading options 

are numbered differently. Loading options 1 ñ 3 are associated with applied in-plane force resultants 

or midplane strains. Loading options 4 ñ 6 are associated with applied in-plane moment resultants or 

midplane curvatures. For more information on mechanical loading in laminate analysis, see the 

MAC/GMC 4.0 Keywords Manual Section 4. 

b) Thermal (*THERM) [KM_4]: 

Number of points: 2 (NPT=2) 

Time points: 0., 200. sec. (TI=0.,200.) 

Temperature points: 650., 650. °C (TEMP=650.,650.) 

c) Time integration (*SOLVER) [KM_4]: 

Time integration method: Forward Euler (METHOD=1) 

Number of points: 2 (NPT=2) 

Time points: 0., 200. sec. (TI=0.,200.) 

Time step sizes: 1. sec. (STP=1.) 

5) Damage and Failure: None 

6) Output: 

a) Output file print level (*PRINT) [KM_6]: 

Print level: 6 (NPL=6) 

b) x-y plots (*XYPLOT) [KM_6]: 

Frequency: 5 (FREQ=5) 

Number of laminate plots: 1 (LAMINATE=1) 

Laminate plot name: example_3g (NAME=example_3g) 

Laminate plot x-y quantities: 0

xx ε , Nxx (X=1 Y=10) 

Number of macro plots: 0 (MACRO=0) 

Number of micro plots: 0 (MICRO=0) 

Because this example problem involves a laminate, an additional level of x-y plot output is available 

above the macro and micro plot output available in repeating unit cell analysis. The number of these 

laminate scale plots is specified by LAMINATE=. ì_lam.dataî is appended to the laminate level 

plot names, thus, in this example, the x-y plot output is written to a file named 

ìexample_3g_lam.dataî. Note that, since different quantities are available for plotting on the 

scale of the laminate, the numbering scheme for X= and Y= is different than that used for the macro 

and micro plots. For more information on generating laminate scale x-y plot files, see the 

MAC/GMC Keywords Manual Section 6. 

 

7) End of file keyword: (*END) 

NASA/TM—2002-212077/VOL3 94
Section 3: Analysis Type and Architecture Example 3g: Laminate Theory ñ Single Ply 

Results

The following is taken from the MAC/GMC 4.0 output file for this example problem for the 0° ply: 

--------------------- EFFECTIVE PROPERTIES AT TEMPERATURE = 650.00 ----------------------------

* FOR LAYER NUMBER 1 *

CG - Effective/Macro Stiffness Matrix

0.3082E+05 0.1277E+05 0.1277E+05

0.1277E+05 0.2517E+05 0.1305E+05

0.1277E+05 0.1305E+05 0.2517E+05

0.5360E+04

0.5719E+04

0.5719E+04

CI - Effective/Macro Compliance Matrix

0.4488E-04 -0.1500E-04 -0.1500E-04

-0.1500E-04 0.5935E-04 -0.2317E-04

-0.1500E-04 -0.2317E-04 0.5935E-04

0.1866E-03

0.1749E-03

0.1749E-03

Effective Engineering Moduli

E11S= 0.2228E+05

N12S= 0.3342

E22S= 0.1685E+05

N23S= 0.3903

E33S= 0.1685E+05

G23S= 0.5360E+04

G13S= 0.5719E+04

G12S= 0.5719E+04

Effective Thermal Expansion Coefficients

0.7452E-05 0.1105E-04 0.1105E-04

Local Q Stiffness For Layer 1

2.434E+04 6.150E+03 0.000E+00

6.150E+03 1.840E+04 0.000E+00

0.000E+00 0.000E+00 5.719E+03

Global Q Stiffness For Layer 1

2.434E+04 6.150E+03 0.000E+00

6.150E+03 1.840E+04 0.000E+00

0.000E+00 0.000E+00 5.719E+03

------------------------------------------------------------------------------------------------

----------------------- LAMINATE RESULTS AT TEMPERATURE = 650.00 -------------------------------

Laminate Axial Stiffness Matrix [A]

2.434E+04 6.150E+03 0.000E+00

6.150E+03 1.840E+04 0.000E+00

0.000E+00 0.000E+00 5.719E+03

Laminate Coupling Stiffness Matrix [B]

0.000E+00 0.000E+00 0.000E+00

0.000E+00 0.000E+00 0.000E+00

0.000E+00 0.000E+00 0.000E+00

NASA/TM—2002-212077/VOL3 95
MAC/GMC 4.0 Example Problem Manual 

Laminate Bending Stiffness Matrix [D]

2.028E+03 5.125E+02 0.000E+00

5.125E+02 1.534E+03 0.000E+00

0.000E+00 0.000E+00 4.766E+02

Laminate Engineering Constants (only valid for symmetric laminates)

Exx= 2.228E+04

Nxy= 3.342E-01

Eyy= 1.685E+04

Gxy= 5.719E+03

================================================================================================

***** Section III: Time-Based Output *****

================================================================================================

1 TIME: 1.0000D+00 TEMP: 6.5000D+02 TSTEP: 1.0000D+00

-------------------------------------------------------------------------------------------------------

FORCE(N), MOMENT(M): 2.2280D+00 0.0000D+00 0.0000D+00 0.0000D+00 0.0000D+00 0.0000D+00

STRAIN, CURVATURE: 1.0000D-04 -3.3418D-05 0.0000D+00 0.0000D+00 0.0000D+00 0.0000D+00

INELASTIC N, M: 0.0000D+00 0.0000D+00 0.0000D+00 0.0000D+00 0.0000D+00 0.0000D+00

THERMAL N, M: 0.0000D+00 0.0000D+00 0.0000D+00 0.0000D+00 0.0000D+00 0.0000D+00

OUT-OF-PLANE STRAIN: -3.3418D-05 0.0000D+00 0.0000D+00

ABD MATRIX:

| 2.43355D+04 6.14987D+03 0.00000D+00 0.00000D+00 0.00000D+00 0.00000D+00 |

| 6.14987D+03 1.84029D+04 0.00000D+00 0.00000D+00 0.00000D+00 0.00000D+00 |

| 0.00000D+00 0.00000D+00 5.71872D+03 0.00000D+00 0.00000D+00 0.00000D+00 |

| 0.00000D+00 0.00000D+00 0.00000D+00 2.02796D+03 5.12489D+02 0.00000D+00 |

| 0.00000D+00 0.00000D+00 0.00000D+00 5.12489D+02 1.53357D+03 0.00000D+00 |

| 0.00000D+00 0.00000D+00 0.00000D+00 0.00000D+00 0.00000D+00 4.76560D+02 |

As shown above, for laminate analysis, the output to the MAC/GMC 4.0 output file is different than for 

repeating unit cell analysis. For each layer, the local (in principal material coordinates) and global (in 

laminate coordinates) reduced stiffness matrices (Q) are output. Further, the axial, coupling, and bending 

stiffness matrices (A, B, and D) for the laminate are output, as are the effective (or apparent) engineering 

properties for the laminate. The time-based output has changed as well. Instead of printing the six 

components of stress and strain at each time step (as is done for repeating unit cell analysis), for laminate 

analysis, the three components of force and moment resultant and the three components of midplane 

strain and curvature are printed. In place of the inelastic and thermal strains, inelastic and thermal force 

and moment resultants are output. Also, due to the plane stress assumption inherent within classical 

lamination theory, the out-of-plane stress components are zero throughout the laminate. However, the 

out-of-plane strain components may or may not (due to Poisson effects) be zero. Thus, these out-of-plane 

strain components are printed at each time step. Finally, rather than the composite stiffness, the laminate 

ABD matrix is output at each time step. Note that the above description pertains to a print level of 6 

(NPL=6). More or less data is written to the MAC/GMC 4.0 output file if this value is altered (see the 

Keywords Manual Section 6 for details on the codeís print levels). 

Figure 3.15 shows that the [0°] ply, with its fibers oriented along the loading direction, gives the stiffest 

response. In fact, because a ply thickness of 1. was employed and no bending occurs, the response of the 

[0°] ply shown in Figure 3.15 is identical to the longitudinal stress-strain response generated for the 

SiC/Ti-21S composite at 650 °C in Example 1d (Figure 1.5). As the orientation angle of the ply rises 

from 0°, the response rapidly becomes more compliant as one might expect. However, at the higher 

angles this trend reverses ñ the most compliant response is exhibited by the [60°] ply. Then, as the 

orientation angle continues to rise to 75° and 90°, the plyís response stiffens. This trend is somewhat 

counterintuitive as one might expect the response of a [90°] ply, with its fibers oriented completely 

NASA/TM—2002-212077/VOL3 96
Section 3: Analysis Type and Architecture Example 3g: Laminate Theory ñ Single Ply 

normal to the loading direction, to be the most compliant orientation. However, as stated by Jones 

(1974), ìthe extremum material properties [of a ply] do not necessarily occur in the principal material 

directions.î Indeed, consider the formula for the apparent stiffness of an off-axis ply (Jones, 1975), 

( ) () () ( )

22

4

22

11

12

1211

4 sin cossin 1cos1 2

xx EEGEE

θ θ θ θ ν

+ 















−+=

where Exx is the apparent stiffness in the x-direction, E11 and E22 are the longitudinal and transverse ply 

stiffness (i.e., principal material coordinate stiffnesses), G12 is the ply axial shear modulus, ν 12 is the ply 

axial Poisson ration, and θ is the fiber orientation angle. For the SiC/Ti-21S composite considered in the 

present example, the ply properties in principal material coordinates ( E11 , E22 ,G12 , and ν 12 ) can be 

obtained from the output file: 

Effective Engineering Moduli

E11S= 0.223E+05

N12S= 0.334E+00

E22S= 0.168E+05

N23S= 0.390E+00

E33S= 0.168E+05

G23S= 0.536E+04

G13S= 0.572E+04

G12S= 0.572E+04

Utilizing these values in conjunction with the formula for the apparent ply stiffness (given above) yields a 

plot of the apparent ply stiffness vs. fiber orientation angle for the 0.25 fiber volume fraction SiC/Ti-21S 

ply as shown in Figure 3.16. Clearly, the apparent stiffness of the ply first decreases, and then increases 

as the fiber orientation angle rises. The apparent modulus reaches a minimum at an angle of 55.6 °. Thus 

the trend shown in Figure 3.15, while somewhat counterintuitive, is correct. 

NASA/TM—2002-212077/VOL3 97
MAC/GMC 4.0 Example Problem Manual 

0

20

40

60

80

100

120

140

160

0.000 0.005 0.010 0.015 0.020 0.025

midplane strain εxx

force resultant Nxx (ksi

×unit thickness)

[0]

[15]

[30]

[45]

[60]

[75]

[90]

Figure 3.15 Example 3g: plot of the global tensile force resultant ñ midplane strain ( 0 N xxxx − ε ) 

response for a 0.25 fiber volume fraction single ply SiC/Ti-21S laminate at 650 °C with 

varying fiber orientation angle. 

0

5

10

15

20

25

0 15 30 45 60 75 90

Fiber Orientation Angle (°)

Apparent Stiffness Exx (Msi)

Figure 3.16 Example 3g: plot of the apparent stiffness of a 0.25 fiber volume fraction SiC/Ti-21S ply at 

650 °C as a function of fiber orientation angle.

NASA/TM—2002-212077/VOL3 98
Section 3: Analysis Type and Architecture Example 3h: Cross-Ply and Quasi-Isotropic Laminates 

Example 3h: Cross-Ply and Quasi-Isotropic Laminates

This example problem employs the classical lamination theory capabilities of MAC/GMC 4.0 to simulate 

the response of two SiC/Ti-21S laminates: cross-ply [90°/0°]s and quasi-isotropic [90°/45°/0°/-45°]s. As 

in Example 3g, a midplane strain of 0.02 is applied to the laminate, however, in the present problem a 

more refined RUC architecture is employed for each layer. For more information on the lamination 

theory and the codeís laminate analysis capabilities, see the MAC/GMC 4.0 Theory Manual Section 3. 

MAC/GMC Input File: example_3h.mac

MAC/GMC 4.0 Example 3h - Cross-ply and quasi-isotropic laminates

*CONSTITUENTS

NMATS=2

M=1 CMOD=6 MATID=E

M=2 CMOD=4 MATID=A

*LAMINATE

# -- Cross-ply

NLY=3

LY=1 MOD=2 THK=0.25 ANG=90 ARCHID=6 R=1. VF=0.25 F=1 M=2

LY=2 MOD=2 THK=0.50 ANG=0 ARCHID=6 R=1. VF=0.25 F=1 M=2

LY=3 MOD=2 THK=0.25 ANG=90 ARCHID=6 R=1. VF=0.25 F=1 M=2

# -- Quasi-isotropic

# NLY=7

# LY=1 MOD=2 THK=0.125 ANG=90 ARCHID=6 R=1. VF=0.25 F=1 M=2

# LY=2 MOD=2 THK=0.125 ANG=45 ARCHID=6 R=1. VF=0.25 F=1 M=2

# LY=3 MOD=2 THK=0.125 ANG=0 ARCHID=6 R=1. VF=0.25 F=1 M=2

# LY=4 MOD=2 THK=0.25 ANG=-45 ARCHID=6 R=1. VF=0.25 F=1 M=2

# LY=5 MOD=2 THK=0.125 ANG=0 ARCHID=6 R=1. VF=0.25 F=1 M=2

# LY=6 MOD=2 THK=0.125 ANG=45 ARCHID=6 R=1. VF=0.25 F=1 M=2

# LY=7 MOD=2 THK=0.125 ANG=90 ARCHID=6 R=1. VF=0.25 F=1 M=2

*MECH

LOP=1

NPT=2 TI=0.,200. MAG=0.,0.02 MODE=1

*THERM

NPT=2 TI=0.,200. TEMP=650.,650.

*SOLVER

METHOD=1 NPT=2 TI=0.,200. STP=1.

*PRINT

NPL=6

*XYPLOT

FREQ=5

LAMINATE=1

NAME=example_3h X=1 Y=10

MACRO=0

MICRO=0

*END

Annotated Input Data

1) Flags: None 

NASA/TM—2002-212077/VOL3 99
MAC/GMC 4.0 Example Problem Manual 

2) Constituent materials (*CONSTITUENTS) [KM_2]: 

Number of materials: 2 (NMATS=2) 

Materials: SiC fiber (MATID=E) 

Ti-21S (MATID=A) 

Constitutive models: SiC fiber: linearly elastic (CMOD=6) 

Ti-21S matrix: Isotropic GVIPS (CMOD=4) 

3) Analysis type (*LAMINATE) → Laminate Analysis [KM_3]: 

Cross-Ply Laminate: 

Number of layers: 3 (NLY=3) 

Layer Analysis 

Model 

Thickness Fiber 

Angle 

Architecture Aspect 

Ratio 

Volume 

fraction 

Fiber 

material 

Matrix 

material 

(LY=) (MOD) (THK) (ANG) (ARCHID) (R) (VF) (F) (M) 

1 GMC-2D 0.25 90° 7×7 circle, 

rect. pack 

1. 0.25 SiC Ti-21S 

2 GMC-2D 0.50 0° 7×7 circle, 

rect. pack 

1. 0.25 SiC Ti-21S 

3 GMC-2D 0.25 90° 7×7 circle, 

rect. pack 

1. 0.25 SiC Ti-21S 

Quasi-Isotropic Laminate: 

Number of layers: 7 (NLY=7) 

Layer Analysis 

Model 

Thickness Fiber 

Angle 

Architecture Aspect 

Ratio 

Volume 

fraction 

Fiber 

material 

Matrix 

material 

(LY=) (MOD=2) (THK=) (ANG=) (ARCHID=6) (R=1.) (VF=0.25) (F=1) (M=2) 

1 GMC-2D 0.125 90° 7×7 circle, 

rect. pack 

1. 0.25 SiC Ti-21S 

2 GMC-2D 0.125 45° 7×7 circle, 

rect. pack 

1. 0.25 SiC Ti-21S 

3 GMC-2D 0.125 0° 7×7 circle, 

rect. pack 

1. 0.25 SiC Ti-21S 

4 GMC-2D 0.25 -45° 7×7 circle, 

rect. pack 

1. 0.25 SiC Ti-21S 

5 GMC-2D 0.125 0° 7×7 circle, 

rect. pack 

1. 0.25 SiC Ti-21S 

6 GMC-2D 0.125 45° 7×7 circle, 

rect. pack 

1. 0.25 SiC Ti-21S 

7 GMC-2D 0.125 90° 7×7 circle, 

rect. pack 

1. 0.25 SiC Ti-21S 

! Note: In order to execute the code for the two different laminates, the appropriate lines in the input 

file must be commented and uncommented. For more information on the laminate analysis 

input requirements, see the MAC/GMC 4.0 Keywords Manual Section 3. 

NASA/TM—2002-212077/VOL3 100
Section 3: Analysis Type and Architecture Example 3h: Cross-Ply and Quasi-Isotropic Laminates 

! Note: The middle layers of both laminates are twice as thick as the other layers because these layers 

actually represent two individual layers that are adjacent, and thus can be combined in the 

analysis. 

4) Loading: 

a) Mechanical (*MECH) [KM_4]: 

Loading option: 1 (loading in the laminate x-direction) (LOP=1) 

Number of points: 2 (NPT=2) 

Time points: 0., 200. sec. (TI=0.,200.) 

Load magnitude: 0., 0.02 (MAG=0.,0.02) 

Loading mode: midplane strain/curvature control (MODE=1) 

b) Thermal (*THERM) [KM_4]: 

Number of points: 2 (NPT=2) 

Time points: 0., 200. sec. (TI=0.,200.) 

Temperature points: 650., 650. °C (TEMP=650.,650.) 

c) Time integration (*SOLVER) [KM_4]: 

Time integration method: Forward Euler (METHOD=1) 

Number of points: 2 (NPT=2) 

Time points: 0., 200. sec. (TI=0.,200.) 

Time step sizes: 1. sec. (STP=1.) 

5) Damage and Failure: None 

6) Output: 

a) Output file print level (*PRINT) [KM_6]: 

Print level: 6 (NPL=6) 

b) x-y plots (*XYPLOT) [KM_6]: 

Frequency: 5 (FREQ=5) 

Number of laminate plots: 1 (LAMINATE=1) 

Laminate plot name: example_3h (NAME=example_3h) 

Laminate plot x-y quantities: 0

xx ε , Nxx (X=1 Y=10) 

Number of macro plots: 0 (MACRO=0) 

Number of micro plots: 0 (MICRO=0) 

7) End of file keyword: (*END) 

Results

Table 3.1 and Figure 3.17 show that the cross-ply laminate exhibits a stiffer global response than does the 

quasi-isotropic laminate. The difference is especially clear in the post-yield response depicted in Figure 

3.17. In addition, from Table 3.1, it is clear that neither laminate exhibits global extensional normal-shear 

coupling as the terms A16 and A26 are zero (or nearly zero) in both cases. However, since the terms D16

and D26 are non-zero for the quasi-isotropic laminate, normal-shear bending coupling is present, whereas 

this coupling is absent in the cross-ply laminate. Also plotted in Figure 3.17 is the response of a 

NASA/TM—2002-212077/VOL3 101
MAC/GMC 4.0 Example Problem Manual 

0.25 fiber volume fraction SiC/Ti-21S composite whose geometry is represented by a cross-ply 

continuum architecture (see Figure 3.10) using triply periodic GMC from Example 3e. For the present 

case, the continuum and lamination theory simulations are in good agreement. Finally, examining the 

s

termed ìquasi-isotropicî, while the [90°/0°]s laminate is not. For isotropic materials, the relation among 

the elastic engineering constants is, 

( ) +ν = 12

E G

Substituting the effective laminate E and ν engineering constants for the quasi-isotropic laminate results 

in a calculated isotropic shear modulus value of G = .6650 ksi, which agrees with the value determined 

for the laminate given in Table 3.1. In contrast, this value calculated for the cross-ply laminate is 

G = .7596 ksi, which does not agree with the value determined for the laminate in Table 3.1. Thus, the 

[90°/45°/0°/-45°]s laminate exhibits ìquasi-isotropyî in its effective engineering constants, while the 

[90°/0°]s laminate does not. 

Table 3.1 Example 3h: effective stiffness and engineering constant results for the 0.25 volume 

fraction SiC/Ti-21S laminates analyzed taken from the MAC/GMC 4.0 output file. 

Cross-Ply Quasi-Isotropic 

Laminate Axial Stiffness Matrix [A]

2.130E+04 6.108E+03 7.791E-08

6.108E+03 2.130E+04 -6.978E-07

7.791E-08 -6.978E-07 5.704E+03

Laminate Coupling Stiffness Matrix [B]

0.000E+00 0.000E+00 0.000E+00

0.000E+00 0.000E+00 0.000E+00

0.000E+00 0.000E+00 0.000E+00

Laminate Bending Stiffness Matrix [D]

1.586E+03 5.090E+02 1.136E-08

5.090E+02 1.964E+03 -1.018E-07

1.136E-08 -1.018E-07 4.754E+02

Laminate Engineering Constants

E11= 1.955E+04

N12= 2.868E-01

E22= 1.955E+04

G12= 5.704E+03

Laminate Axial Stiffness Matrix [A]

2.035E+04 7.054E+03 3.896E-08

7.054E+03 2.035E+04 -3.489E-07

3.896E-08 -3.489E-07 6.650E+03

Laminate Coupling Stiffness Matrix [B]

0.000E+00 5.684E-14 -2.344E-15

5.684E-14 0.000E+00 1.556E-15

-2.344E-15 1.556E-15 -1.137E-13

Laminate Bending Stiffness Matrix [D]

1.608E+03 5.583E+02 3.542E+01

5.583E+02 1.844E+03 3.542E+01

3.542E+01 3.542E+01 5.246E+02

Laminate Engineering Constants

E11= 1.791E+04

N12= 3.466E-01

E22= 1.791E+04

G12= 6.650E+03

NASA/TM—2002-212077/VOL3 102

effective laminate engineering constants in Table 3.1, it is clear why the [90°/45°/0°/-45°] laminate is 
Section 3: Analysis Type and Architecture Example 3h: Cross-Ply and Quasi-Isotropic Laminates 

0

20

40

60

80

100

120

140

160

180

200

0.000 0.005 0.010 0.015 0.020 0.025

midplane strain εxx

force resultant Nxx (ksi

×unit thickness)

Cross-ply

Quasi-isotropic

Triply periodic GMC cross-ply simulation

Figure 3.17 Example 3h: plot of the global tensile force resultant ñ midplane strain ( 0 N xxxx − ε ) 

response for 0.25 fiber volume fraction cross-ply [90°/0°]s and quasi-isotropic [90°/45°/0°/-

45°]s SiC/Ti-21S laminates at 650 °C. Also plotted is a triply periodic GMC prediction 

using an RUC like that employed in Example 3e that simulates a cross-ply architecture. 

NASA/TM—2002-212077/VOL3 103
MAC/GMC 4.0 Example Problem Manual 

Example 3i: Asymmetric Laminate

This example problem employs the classical lamination theory capabilities of MAC/GMC 4.0 to simulate 

the response of an asymmetric SiC/Ti-21S laminate. This fictional laminate consists of four plies, each of 

which is different. In fact, one layer is monolithic, one layer employs doubly periodic GMC, and the final 

two layers employ triply periodic GMC. Note that the ability to include doubly and triply periodic layers 

within the same laminate is a new capability of MAC/GMC 4.0. The laminate is subjected to a midplane 

strain of 0.02 at 23 °C. The asymmetric lay up of the laminate plies gives rise to non-zero terms in the 

laminate coupling stiffness matrix [B], which induces bending when the laminate is subjected to the 

applied midplane strain. For more information on the lamination theory and the codeís laminate analysis 

capabilities, see the MAC/GMC 4.0 Theory Manual Section 3. 

MAC/GMC Input File: example_3i.mac

MAC/GMC 4.0 Example 3i - Asymmetric laminate

*CONSTITUENTS

NMATS=2

M=1 CMOD=6 MATID=E

M=2 CMOD=4 MATID=A

*LAMINATE

NLY=4

LY=1 MOD=1 THK=0.25 M=2

LY=2 MOD=2 THK=0.25 ANG=60 ARCHID=6 R=1. VF=0.25 F=1 M=2

LY=3 MOD=3 THK=0.25 ANG=0. ARCHID=3 ASP1=3. ASP2=2. DR=1. VF=0.25 F=1 M=2

LY=4 MOD=3 THK=0.25 ANG=0. ARCHID=1 VF=0.25 ASP=1. F=1 M=2

*MECH

LOP=1

NPT=2 TI=0.,200. MAG=0.,0.02 MODE=1

*THERM

NPT=2 TI=0.,200. TEMP=23.,23.

*SOLVER

METHOD=1 NPT=2 TI=0.,200. STP=0.5

*PRINT

NPL=6

*XYPLOT

FREQ=5

LAMINATE=13

NAME=example_3i_nxx X=1 Y=10

NAME=example_3i_nxxp X=1 Y=16

NAME=example_3i_nyyp X=1 Y=17

NAME=example_3i_nxyp X=1 Y=18

NAME=example_3i_mxxp X=1 Y=19

NAME=example_3i_myyp X=1 Y=20

NAME=example_3i_mxyp X=1 Y=21

NAME=example_3i_eyy X=1 Y=2

NAME=example_3i_ezz X=1 Y=3

NAME=example_3i_exy X=1 Y=6

NAME=example_3i_kxx X=1 Y=7

NAME=example_3i_kyy X=1 Y=8

NAME=example_3i_kxy X=1 Y=9

MACRO=0

MICRO=0

*END

NASA/TM—2002-212077/VOL3 104
Section 3: Analysis Type and Architecture Example 3i: Asymmetric Laminate 

Annotated Input Data

1) Flags: None 

2) Constituent materials (*CONSTITUENTS) [KM_2]: 

Number of materials: 2 (NMATS=2) 

Materials: SiC fiber (MATID=E) 

Ti-21S (MATID=A) 

Constitutive models: SiC fiber: linearly elastic (CMOD=6)

Ti-21S matrix: Isotropic GVIPS (CMOD=4) 

3) Analysis type (*LAMINATE) → Laminate Analysis [KM_3]: 

Number of layers: 4 (NLY=4) 

Layer Analysis 

Model 

Thickness Fiber 

Angle 

Architecture Volume 

fraction 

Aspect 

ratio 

Fiber 

material 

Matrix 

material 

(LY) (MOD) (THK) (ANG) (ARCHID) (VF) (R) (F) (M)

1 monolithic 0.25 ñ ñ ñ ñ ñ Ti-21S 

2 GMC-2D 0.25 60° 7×7 circle, 

rect. pack 

0.25 1. SiC Ti-21S 

3 GMC-3D 0.25 0° off-set fibers (3) 0.25 ñ SiC Ti-21S 

4 GMC-3D 0.25 0° short fiber (1) 0.25 ñ SiC Ti-21S 

Additional information for layers 3 and 4: 

Fiber aspect ratio 3. (ASP1=3.) 

Unit cell aspect ratio: 2. or 1. (ASP2=2. or ASP=1.) 

D ratio (ARCHID=3 only): 1. (DR=1.) 

The laminate in this example problem consists of a monolithic Ti-21S layer and three composite 

SiC/Ti-21S layers: a 60° continuous layer, a short fiber discontinuous layer, and a particulate 

discontinuous layer. For more information on the laminate analysis input requirements, see the 

MAC/GMC 4.0 Keywords Manual Section 3. 

4) Loading: 

a) Mechanical (*MECH) [KM_4]: 

Loading option: 1 (loading in the laminate x-direction) (LOP=1) 

Number of points: 2 (NPT=2) 

Time points: 0., 200. sec. (TI=0.,200.) 

Load magnitude: 0., 0.02 (MAG=0.,0.02) 

Loading mode: midplane strain/curvature control (MODE=1) 

b) Thermal (*THERM) [KM_4]: 

Number of points: 2 (NPT=2) 

Time points: 0., 200. sec. (TI=0.,200.) 

Temperature points: 23., 23. °C (TEMP=23.,23.) 

NASA/TM—2002-212077/VOL3 105
MAC/GMC 4.0 Example Problem Manual 

c) Time integration (*SOLVER) [KM_4]: 

Time integration method: Forward Euler (METHOD=1) 

Number of points: 2 (NPT=2) 

Time points: 0., 200. sec. (TI=0.,200.) 

Time step sizes: 0.5 sec. (STP=0.5) 

5) Damage and Failure: None 

6) Output: 

a) Output file print level (*PRINT) [KM_6]: 

Print level: 6 (NPL=6) 

b) x-y plots (*XYPLOT) [KM_6]: 

Frequency: 5 (FREQ=5) 

Number of laminate plots: 13 (LAMINATE=13) 

Laminate plot names: example_3i_nxx (NAME=example_3i_nxx) 

 example_3i_nxxp (NAME=example_3i_nxxp) 

 example_3i_nyyp (NAME=example_3i_nyyp) 

 example_3i_nxyp (NAME=example_3i_nxyp) 

 example_3i_mxxp (NAME=example_3i_mxxp) 

 example_3i_myyp (NAME=example_3i_myyp) 

 example_3i_mxyp (NAME=example_3i_mxyp) 

 example_3i_eyy (NAME=example_3i_eyy) 

 example_3i_ezz (NAME=example_3i_ezz) 

 example_3i_exy (NAME=example_3i_exy) 

 example_3i_kxx (NAME=example_3i_kxx) 

 example_3i_kyy (NAME=example_3i_kyy) 

 example_3i_kxy (NAME=example_3i_kxy) 

Laminate plot quantities: 0

xx ε , Nxx (X=1 Y=10) 

 0

xx ε , p Nxx (X=1 Y=16) 

 0

xx ε , p Nyy (X=1 Y=17) 

 0

xx ε , p Nxy (X=1 Y=18) 

 0

xx ε , p M xx (X=1 Y=19) 

 0

xx ε , p M yy (X=1 Y=20) 

 0

xx ε , p Mxy (X=1 Y=21) 

 0

xx ε , 0

yy ε (X=1 Y=2) 

 0

xx ε , 0xy ε (X=1 Y=3) 

 0

xx ε , zz ε (X=1 Y=6) 

 0

xx ε , 0 κ xx (X=1 Y=7) 

 0

xx ε , 0 κ yy (X=1 Y=8) 

 0

xx ε , 0 κ xy (X=1 Y=9) 

NASA/TM—2002-212077/VOL3 106
Section 3: Analysis Type and Architecture Example 3i: Asymmetric Laminate 

Number of macro plots: 0 (MACRO=0) 

Number of micro plots: 0 (MICRO=0) 

7) End of file keyword: (*END) 

Results

Table 3.2 shows that the asymmetric laminate ABD matrix is fully populated. Thus, due to the applied 

normal midplane strain, global shear strains (due to the presence of the 60° layer), as well as bending (due 

to the laminateís asymmetry) will result. 

Figure 3.18 shows plots of the laminate force resultant ( Nxx ) and laminate inelastic force resultants ( p Nij ) 

that arise due to the applied normal midplane strain. The point at which the laminate yields is clear in this 

figure as the inelastic force resultants begin to arise. Also, the inelastic shear force resultant is small 

relative to the normal inelastic force resultants. Figure 3.19 shows the laminate inelastic moment 

resultants ( p Mij ) that rise during the MAC/GMC 4.0 simulation (which are an indicator of inelastic 

laminate bending). Again, it is clear that the shear component is smaller than the normal components. 

The inelastic bending moment component associated with the loading direction ( p M xx ) appears to be the 

dominant component, and several reversals in the p M xx history occur during the simulation. These 

reversals are caused by the onset and progression of inelastic deformation in the layers of the laminate as 

it bends. Figure 3.20 shows the midplane strain and curvature components that arise in the laminate 

during the simulation. Again, the shear components are less significant than the normal components. The 

normal curvature histories exhibit reversals in slope, as do the normal inelastic moment results plotted in 

Figure 3.19. 

NASA/TM—2002-212077/VOL3 107
MAC/GMC 4.0 Example Problem Manual 

Table 3.2 Example 3i: effective stiffness and engineering constant results for the asymmetric SiC/Ti21S laminate analyzed taken from the MAC/GMC 4.0 output file. 

Asymmetric Laminate 

Laminate Axial Stiffness Matrix [A]

2.342E+04 8.364E+03 8.295E+01

8.364E+03 2.392E+04 4.308E+02

8.295E+01 4.308E+02 7.570E+03

Laminate Coupling Stiffness Matrix [B]

6.044E+02 1.239E+02 -1.037E+01

1.239E+02 5.176E+02 -5.385E+01

-1.037E+01 -5.385E+01 1.024E+02

Laminate Bending Stiffness Matrix [D]

1.888E+03 6.630E+02 1.728E+00

6.630E+02 1.898E+03 8.974E+00

1.728E+00 8.974E+00 5.857E+02

Laminate Engineering Constants (only valid for symmetric laminates)

E11= 2.029E+04

N12= 3.515E-01

E22= 2.074E+04

G12= 7.541E+03

-50

0

50

100

150

200

250

0.000 0.005 0.010 0.015 0.020 0.025

Applied Midplane Strain εxx

Force Resultant (ksi

× unit thickness)

Nxx

NPxx

NPyy

NPxy

Figure 3.18 Example 3i: plot of the global force resultant (N) and inelastic force resultant (NP) 

response for an asymmetric SiC/Ti-21S laminate at 23 °C to an applied midplane strain. 

NASA/TM—2002-212077/VOL3 108
Section 3: Analysis Type and Architecture Example 3i: Asymmetric Laminate 

-5

-4

-3

-2

-1

0

1

0.000 0.005 0.010 0.015 0.020 0.025

Applied Midplane Strain εxx

Moment Resultant (ksi × unit thickness^2)

MPxx

MPyy

MPxy

Figure 3.19 Example 3i: plot of the global inelastic moment resultant (MP) response for an asymmetric 

SiC/Ti-21S laminate at 23 °C to an applied midplane strain. 

-0.012

-0.01

-0.008

-0.006

-0.004

-0.002

0

0.002

0.004

0.006

0.000 0.005 0.010 0.015 0.020 0.025

Applied Midplane Strain εxx

Midplane Strain, Curvature (per unit thickness)

eyy

exy

ezz

kxx

kyy

kxy

Figure 3.20 Example 3i: plot of the global midplane strain (e) and midplane curvature (k) response for 

an asymmetric SiC/Ti-21S laminate at 23 °C to an applied midplane strain. 

NASA/TM—2002-212077/VOL3 109
MAC/GMC 4.0 Example Problem Manual 

Section 4 : Applied Loading 

This section presents example problems demonstrating various ways in which loading may be applied to 

composites and laminates in MAC/GMC 4.0. To understand the various loading options, it is useful to 

examine the global, or effective, constitutive equation for a composite material: 

[ ]









































−−

−−

−−

−−

−−

−−

=









































∗

Tp

Tp

Tp

Tp

Tp

Tp

121212

131313

232323

333333

222222

111111

12

13

23

33

22

11

γ γ γ

γ γ γ

γ γ γ

ε ε ε

ε ε ε

ε ε ε

σ

σ

σ

σ

σ

σ

C

where σ ij are the global (or average) stress components on the composite, ∗ C is the global (or effective) 

stiffness matrix, ij ε are the global (or average) normal strain components on the composite, ij ε are the 

global (or average) normal strain components on the composite, ij γ are the global (or average) 

engineering shear strain components on the composite, p

ij ε are the global (or average) normal inelastic 

strain components on the composite, p`

ij γ are the global (or average) engineering shear inelastic strain 

components on the composite, T

ij ε are the global (or average) normal thermal strain components on the 

composite, and T

ij γ are the global (or average) engineering shear thermal strain components on the 

composite. For repeating unit cell analysis, loading takes the form of an admissible time profile of global 

stress and strain components. Admissibility simply requires that, if a particular strain component is 

specified, the corresponding stress component cannot be specified (and vice versa). Thus, the user may 

specify a time history for each of the six components. Indeed, the arbitrary mechanical loading option 

within MAC/GMC 4.0 allows just this. However, there are several internal loading options within the 

code that simplify the application of loading for commonly used mechanical loading types. For instance, 

a simulated strain controlled tensile test on a composite entails applying a single global strain component 

(with a corresponding unknown global stress component) while all remaining stress components are kept 

at zero. MAC/GMCís internal loading options 1 ñ 6 allow this type of application of a single load 

component. Some additional internal loading options simplify biaxial mechanical loading. For details, 

see the MAC/GMC 4.0 Keywords Manual Section 4. 

The loading history that is applied to a laminate is different than that applied to a composite material. 

Globally, rather than dealing with average stresses and strains, the applicable quantities are stress and 

moment resultants along with midplane strains and curvatures. For laminate analysis, the codeís internal 

loading options are numbered differently. Loading options 1 ñ 3 correspond to application of force 

resultants or midplane strains in the x1-x2 plane, while loading options 4 ñ 6 correspond to application of 

moment resultants and curvatures. For additional information on the codeís loading options, see Section 

4 of the Keywords Manual. 

NASA/TM—2002-212077/VOL3 110
Section 4: Applied Loading Example 4a: MAC/GMC Internal Loading Options 

Example 4a: MAC/GMC Internal Loading Options

This example problem demonstrates all of the internal MAC/GMC 4.0 mechanical loading options that 

are applicable to repeating unit cell analysis. These internal loading options allow for the easy application 

of many common uniaxial and biaxial loading conditions. In the present example, doubly periodic GMC 

is employed in conjunction with loading options 1 through 10. For triply periodic GMC, loading options 

1 ñ 8 are identical to the coinciding doubly periodic GMC loading options. However, loading options 9 

and 10 are different, and triply periodic GMC allows two additional loading options, 11 and 12. For more 

information on the codeís internal mechanical loading options, see the MAC/GMC 4.0 Keywords Manual 

Section 4. 

MAC/GMC Input File: example_4a.mac

MAC/GMC 4.0 Example 4a - RUC analysis internal loading options

*CONSTITUENTS

NMATS=2

M=1 CMOD=6 MATID=E

M=2 CMOD=4 MATID=A

*RUC

MOD=2 ARCHID=13 VF=0.25 R=1.25 F=1 M=2

*MECH

LOP=1

# LOP=2

# LOP=3

# LOP=4

# LOP=5

# LOP=6

# LOP=7

# LOP=8

# LOP=9

# LOP=10

# -- Note: LOP > 6 are biaxial & require 2 load profiles

NPT=2 TI=0.,200. MAG=0.,0.02 MODE=1

# NPT=2 TI=0.,200. MAG=0.,0.02 MODE=1

*THERM

NPT=2 TI=0.,200. TEMP=650.,650.

*SOLVER

METHOD=1 NPT=2 TI=0.,200. STP=1.

*PRINT

NPL=6

*XYPLOT

FREQ=5

MACRO=6

NAME=example_4a X=1 Y=7

NAME=example_4a X=2 Y=8

NAME=example_4a X=3 Y=9

NAME=example_4a X=4 Y=10

NAME=example_4a X=5 Y=11

NAME=example_4a X=6 Y=12

MICRO=0

*END

NASA/TM—2002-212077/VOL3 111
MAC/GMC 4.0 Example Problem Manual 

Annotated Input Data

1) Flags: None 

2) Constituent materials (*CONSTITUENTS) [KM_2]: 

Number of materials: 2 (NMATS=2) 

Materials: SiC fiber (MATID=E) 

Ti-21S (MATID=A) 

Constitutive models: SiC fiber: linearly elastic (CMOD=6) 

Ti-21S matrix: Isotropic GVIPS (CMOD=4) 

3) Analysis type (*RUC) → Repeating Unit Cell Analysis [KM_3]: 

Analysis model: Doubly periodic GMC (MOD=2) 

RUC architecture: 26×26 circular fiber, rect. pack (ARCHID=13) 

Fiber volume fraction: 0.25 (VF=0.25) 

RUC aspect ratio: 1.25 (R=1.25) 

Material assignment: SiC fiber (F=1) 

Ti-21S matrix (M=2) 

4) Loading: 

a) Mechanical (*MECH) [KM_4]: 

Loading option: 11-component (LOP=1) 

 22-component (LOP=2) 

 33-component (LOP=3) 

 23-component (LOP=4) 

 13-component (LOP=5) 

 12-component (LOP=6) 

 11- & 22-components (LOP=7) 

 22- & 33-components (LOP=8) 

 11- & 23-components (LOP=9) 

 22- & 13-components (LOP=10) 

Number of points: 2 (NPT=2) 

Time points: 0., 200. sec. (TI=0.,200.) 

Load magnitude: 0., 0.02 (MAG=0.,0.02) 

Loading mode: strain control (MODE=1) 

In order to execute the code using each of the ten loading options, the appropriate lines in the input 

file must be commented and uncommented. For loading options 7 ñ 10, two load components are 

applied to the composite simultaneously. Thus, two sets of data are necessary to specify timemagnitude points and the loading mode (i.e., one set for each component). These sets of data are 

input on two separate lines in the MAC/GMC 4.0 input file. The identical loading history is applied 

for all loading options in this example. Note that loading options 4 ñ 6, 9, and 10 involve application 

of shear strains. MAC/GMC 4.0 employs engineering shear strains (often denoted by γ) as opposed 

to tensorial shear strains (often denoted by ε). See Section 4 of the Keywords Manual for additional 

information on the mechanical loading options. 

NASA/TM—2002-212077/VOL3 112
Section 4: Applied Loading Example 4a: MAC/GMC Internal Loading Options 

b) Thermal (*THERM) [KM_4]: 

Number of points: 2 (NPT=2) 

Time points: 0., 200. sec. (TI=0.,200.) 

Temperature points: 650., 650. °C (TEMP=650.,650.) 

c) Time integration (*SOLVER) [KM_4]: 

Time integration method: Forward Euler (METHOD=1) 

Number of points: 2 (NPT=2) 

Time points: 0., 200. sec. (TI=0.,200.) 

Time step sizes: 1. sec. (STP=1.) 

5) Damage and Failure: None 

6) Output: 

a) Output file print level (*PRINT) [KM_6]: 

Print level: 6 (NPL=6) 

b) x-y plots (*XYPLOT) [KM_6]: 

Frequency: 5 (FREQ=5) 

Number of macro plots: 6 (MACRO=6) 

Macro plot names: example_4a (NAME=example_4a) 

Macro plot x-y quantities: 11 ε , σ 11 (X=1 Y=7) 

 22 ε , σ 22 (X=2 Y=8) 

 33 ε , σ 33 (X=3 Y=9) 

 23 γ , σ 23 (X=4 Y=10) 

 13 γ , σ 13 (X=5 Y=11) 

 12 γ , σ 12 (X=6 Y=12) 

Number of micro plots: 0 (MICRO=0) 

7) End of file keyword: (*END) 

Results

Figure 4.1 shows that, among the plots associated with a single applied load component, the normal 

response of the repeating unit cell in the fiber direction is stiffest, followed by the normal response in the 

two transverse directions, with the three shear response curves exhibiting significantly more compliance. 

Figure 4.2 shows the composite response when two strain components are applied simultaneously. Most 

interesting are the LOP=8 results, which indicate that the composite response is significantly stiffer when 

22 ε and 33 ε are applied simultaneously compared to when each component is applied individually (see 

LOP=2 and LOP=3 results in Figure 4.1). This is caused by the constraining effect of the biaxial loading 

in addition to the higher hydrostatic stress state induced when applying the components simultaneously 

(which tends to suppress the inelastic deformation in the matrix). 

NASA/TM—2002-212077/VOL3 113
MAC/GMC 4.0 Example Problem Manual 

0

10

20

30

40

50

60

70

80

0.000 0.005 0.010 0.015 0.020 0.025

strain

stress (ksi)

LOP = 1

LOP = 2

LOP = 3

LOP = 4

LOP = 5

LOP = 6

Figure 4.1 Example 4a: plot of the simulated stress-strain response for a 0.25 fiber volume fraction 

SiC/Ti-21S composite at 650 °C where MAC/GMC 4.0 mechanical loading options 1 ñ 6 

have been employed. 

0

50

100

150

200

250

300

350

400

0.000 0.005 0.010 0.015 0.020 0.025

strain

stress (ksi)

LOP = 7, 11 response

LOP = 7, 22 response

LOP = 8, 22 response

LOP = 8, 33 response

LOP = 9, 11 response

LOP = 9, 23 response

LOP = 10, 22 response

LOP = 10, 13 response

Figure 4.2 Example 4a: plot of the simulated stress-strain response for a 0.25 fiber volume fraction 

SiC/Ti-21S composite at 650 °C where MAC/GMC 4.0 mechanical loading options 7 ñ 10 

have been employed.

NASA/TM—2002-212077/VOL3 114
Section 4: Applied Loading Example 4b: SiC/Ti-21S Composite with Residual Stresses 

Example 4b: SiC/Ti-21S Composite with Residual Stresses

This is an example of a MAC/GMC 4.0 simulation for a metal matrix composite in which residual 

stresses due to fabrication have been included. The composite configuration is identical to that of 

Example 1d. The changes lie in the loading section of the input file, wherein, prior to application of the 

simulated mechanical loading, a simulated stress-free cool-down from 900 °C is performed. Thus, when 

the simulated mechanical loading is applied, the composite contains internal local stresses within each 

constituent resulting from property mismatch of the composite phases and the simulated cool-down. The 

stress-strain curve that results from the subsequent applied mechanical loading can then be shifted such 

that the origin corresponds to the onset of the mechanical loading. This then simulates an actual 

composite tensile test in which it is assumed that the zero global strain condition corresponds to the start 

of the mechanical test. 

MAC/GMC Input File: example_4b.mac

MAC/GMC 4.0 Example 4b - SiC/Ti-21S composite w/ residual

*CONSTITUENTS

NMATS=2

M=1 CMOD=6 MATID=E

M=2 CMOD=4 MATID=A

*RUC

MOD=2 ARCHID=1 VF=0.25 F=1 M=2

*MECH

LOP=1 REFTIME=57600.

# LOP=2 REFTIME=57600.

NPT=3 TI=0.,57600.,57800. MAG=0.,0.,0.02 MODE=2,1

*THERM

NPT=3 TI=0.,57600.,57800 TEMP=900.,23.,23.

*SOLVER

METHOD=1 NPT=3 TI=0.,57600.,57800 STP=40.,1.

*PRINT

NPL=6

*XYPLOT

FREQ=5

MACRO=4

NAME=example_4b_11 X=1 Y=7

NAME=example_4b_22 X=2 Y=8

NAME=example_4b_TH1 X=100 Y=1

NAME=example_4b_TH2 X=100 Y=2

MICRO=0

*END

Annotated Input Data

1) Flags: None 

2) Constituent materials (*CONSTITUENTS) [KM_2]: 

Number of materials: 2 (NMATS=2) 

NASA/TM—2002-212077/VOL3 115
MAC/GMC 4.0 Example Problem Manual 

Materials: SiC fiber (MATID=E) 

Ti-21S (MATID=A) 

Constitutive models: SiC fiber: linearly elastic (CMOD=6) 

Ti-21S matrix: Isotropic GVIPS (CMOD=4) 

3) Analysis type (*RUC) → Repeating Unit Cell Analysis [KM_3]: 

Analysis model: Doubly periodic GMC (MOD=2) 

RUC architecture: square fiber, square pack (ARCHID=1) 

Fiber volume fraction: 0.25 (VF=0.25) 

Material assignment: SiC fiber (F=1) 

Ti-21S matrix (M=2) 

4) Loading: 

a) Mechanical (*MECH) [KM_4]: 

Loading option: 1 or 2 (LOP=1 or LOP=2) 

Strain reference time: 57600. sec. (REFTIME=57600.) 

Number of points: 3 (NPT=3) 

Time points: 0., 57600., 57800. sec. (TI=0.,57600.,57800.) 

Load magnitudes: 0., 0., 0.02 (MAG=0.,0.,0.02) 

Loading mode: stress control, strain control (MODE=2,1) 

The mechanical loading profile now contains three time-magnitude pairs. For the first segment of the 

mechanical loading, the magnitude is kept at zero, then, during the second segment, the mechanical 

loading progresses to a magnitude of 0.02. Since there are now three time-magnitude pairs, and thus 

two loading segments, two loading modes (one for each segment) must be specified. During the first 

segment, it is the stress that is held at zero, while during the second segment, it is the strain that is 

applied up to a level of 0.02. 

The strain reference time (REFTIME) is the time taken to be the zero strain point for subsequent 

specified strains. Thus, the strain that results at time = 57600. sec. from the cool-down is taken as the 

reference strain. Then, the subsequent applied 0.02 strain is taken with respect to this reference strain 

such that a total increment in strain of 0.02 is applied. This mimics a tensile stress-strain test on a 

specimen containing an unknown residual strain. The unknown residual strain is the reference 

(thought to be zero at the beginning of the tensile test) and the amount of strain applied during the test 

is on top of this reference strain. If it is not specified, the strain reference time defaults to zero. For 

additional information on the strain reference time, see the MAC/GMC 4.0 Keywords Manual 

Section 4. 

b) Thermal (*THERM) [KM_4]: 

Number of points: 3 (NPT=3) 

Time points: 0., 57600., 57800. sec. (TI=0.,57600.,57800.) 

Temperature points: 900., 23., 23. (TEMP=900.,23.,23.) 

Three time-temperature pairs are used to specify the two thermal loading segments ñ a cool-down 

from 900 °C to 23 °C followed by a hold at 23 °C. The second segment corresponds to the 

application of the non-zero simulated mechanical loading. 

NASA/TM—2002-212077/VOL3 116
Section 4: Applied Loading Example 4b: SiC/Ti-21S Composite with Residual Stresses 

c) Time integration (*SOLVER) [KM_4]: 

Time integration method: Forward Euler (METHOD=1) 

Number of time points: 3 (NPT=3) 

Time points: 0., 57600., 57800. sec. (TI=0.,57600.,57800.) 

Time step size: 40., 1. sec. (STP=40.,1.) 

5) Damage and Failure: None 

6) Output: 

a) Output file print level (*PRINT) [KM_6]: 

Print level: 6 (NPL=6) 

b) x-y plots (*XYPLOT) [KM_6]: 

Frequency: 5 (FREQ=5) 

Number of macro plots: 4 (MACRO=4) 

Macro plot names: example_4b_11 (NAME=example_4b_11) 

 example_4b_22 (NAME=example_4b_22) 

 example_4b_TH1 (NAME=example_4b_TH1) 

 example_4b_TH2 (NAME=example_4b_TH2) 

Macro plot x-y quantities: ε11, σ11 (X=1 Y=7) 

 ε22, σ22 (X=2 Y=8) 

 Temperature, ε11 (X=100 Y=1) 

 Temperature, ε22 (X=100 Y=2) 

Number of micro plots: 0 (MICRO=0) 

In this example, four macro (repeating unit cell level) plots are generated; two represent the 

mechanical response of the composite, two represent the thermal response of the composite 

7) End of file keyword: (*END) 

Results

Figure 4.3 shows the strain that results from the simulated stress-free cool-down applied to the SiC/Ti21S composite. It should be noted that, due to the fact the Ti-21S matrix undergoes inelastic deformation 

during the cool-down, the overall or global strain plotted in Figure 4.3 is not the global thermal strain. 

Rather, it is the sum of the global thermal strain and the global inelastic strain. Examining the output file 

for this example reveals the following global strain state at the conclusion of the simulated cool-down 

(but before application of the simulated mechanical loading): 

1440 TIME: 5.7600D+04 TEMP: 2.3000D+01 TSTEP: 4.0000D+01

-----------------------------------------------------------------------------------------------

STRESS: 0.0000D+00 0.0000D+00 0.0000D+00 0.0000D+00 0.0000D+00 0.0000D+00

STRAIN: -5.5186D-03 -8.7271D-03 -8.7271D-03 0.0000D+00 0.0000D+00 0.0000D+00

IN. STRAIN: 7.1719D-04 -4.2385D-04 -4.2385D-04 0.0000D+00 0.0000D+00 0.0000D+00

TH. STRAIN: -6.2358D-03 -8.3033D-03 -8.3033D-03 0.0000D+00 0.0000D+00 0.0000D+00

Clearly, there is an inelastic strain contribution to the total strain, in addition to the thermal strain. An 

interesting ramification of this observation is that, it would be erroneous to use Figure 4.3 to determine 

tangential coefficients of thermal expansion (CTEs) for the composite by taking slopes of the curves at 

NASA/TM—2002-212077/VOL3 117
MAC/GMC 4.0 Example Problem Manual 

different temperatures. This would give the slope of the total strain with respect to temperature, while the 

definition of tangent CTE is the slope of thermal strain with respect to temperature. Taking the slope of 

an experimental composite thermal expansion curve as a tangential CTE is similarly erroneous if the 

composite has undergone any inelastic deformation during the test, consequently micromechanics is 

needed to extract the true composite CTE. 

Figure 4.4 shows that the effect of residual stresses on the SiC/Ti-21S composite (at 23 °C) is greater in 

the longitudinal direction compared to the transverse direction. The composite yields sooner 

longitudinally for the case that includes residual stresses. In the transverse direction, the simulation with 

residual stresses exhibits yielding slightly sooner, but the differences in the predicted stress-strain curves 

is minor. Clearly, the presence of the continuous fibers causes greater residual effects in the longitudinal 

direction as these fibers have a significant restraining effect on the matrix during the cool down. 

It should also be noted that in Figure 4.4, the simulated stress-strain curves with residual stresses have 

been shifted from the strain value at the completion of the cool-down to a strain value of zero. That is, for 

example, the composite exhibited a longitudinal strain of -0.00552 upon completion of the cool-down. 

The longitudinal stress-strain curve with residual in Figure 4.4 was thus shifted to the right by a strain 

value of +0.00552. Therefore, as is the case in an experimental tensile stress-strain test on a composite, it 

appears that, at the beginning of the test, the composite starts at zero macro strain. In reality, of course, 

during manufacture, the composite has experienced some strain, but the as delivered, room-temperature 

composite state is commonly used as the zero strain reference. 

-0.01

-0.008

-0.006

-0.004

-0.002

0

0 200 400 600 800 1000

Temperature (C)

Strain

Longituginal

Transverse

Figure 4.3 Example 4b: plot of the simulated longitudinal and transverse thermal response of a 25% 

SiC/Ti-21S composite to a stress-free cool-down from 900 °C to 23 °C. 

NASA/TM—2002-212077/VOL3 118
Section 4: Applied Loading Example 4b: SiC/Ti-21S Composite with Residual Stresses 

0

50

100

150

200

250

300

350

400

450

0.000 0.005 0.010 0.015 0.020

Strain

Stress (ksi)

Longitudinal with residual

Longitudinal no residual

Transverse with residual

Transverse no residual

Figure 4.4 Example 4b: plots of the simulated longitudinal and transverse tensile response of a 25% 

SiC/Ti-21S composite with and without residual stresses included. The plots for the cases 

without residual stresses are the results from Example 1d. 

NASA/TM—2002-212077/VOL3 119
MAC/GMC 4.0 Example Problem Manual 

Example 4c: General Loading Option for RUC analysis

This example demonstrates how to apply a general mechanical loading history to a given repeating unit 

cell. This option, specified by LOP=99, allows the user to specify all six mechanical loading components 

applied to the composite. Of course, for each component only a single stress or strain may be specified, 

not both. This example simulates a complex thermomechanical loading history on a continuous fiber 

SiC/Ti-21S composite that involves application of stresses and strains in the x1-x2 plane. This loading 

profile might mimic a material point on the skin of an aerospace vehicle or engine housing. The 26×26 

circular fiber approximation RUC architecture is employed. For more information on the codeís general 

mechanical loading option, see the MAC/GMC 4.0 Keywords Manual Section 4. 

MAC/GMC Input File: example_4c.mac

MAC/GMC 4.0 Example 4c - RUC analysis general loading option

*CONSTITUENTS

NMATS=2

M=1 CMOD=6 MATID=E

M=2 CMOD=4 MATID=A

*RUC

MOD=2 ARCHID=13 VF=0.25 R=1. F=1 M=2

*MECH

LOP=99

NPT=6 TI=0.,300.,310.,2310.,2320.,2620. MAG=0.,0.,0.01,0.01,0.,0. &

MODE=1,1,1,2,2

NPT=6 TI=0.,300.,310.,2310.,2320.,2620. MAG=0.,0.,0.005,0.005,0.,0. &

MODE=1,1,1,2,2

NPT=2 TI=0.,2620. MAG=0.,0. MODE=2

NPT=2 TI=0.,2620. MAG=0.,0. MODE=2

NPT=2 TI=0.,2620. MAG=0.,0. MODE=2

NPT=6 TI=0.,300.,310.,2310.,2320.,2620. MAG=0.,0.,0.0075,0.0075,0.,0. &

MODE=1,1,1,2,2

*THERM

NPT=6 TI=0.,300.,310.,2310.,2320.,2620. TEMP=23.,650.,650.,650.,650.,23.

*SOLVER

METHOD=1 NPT=6 TI=0.,300.,310.,2310.,2320.,2620. STP=1.,0.2,2.,1.,1.

*PRINT

NPL=6

*XYPLOT

FREQ=1

MACRO=4

NAME=example_4c X=1 Y=7

NAME=example_4c X=2 Y=8

NAME=example_4c X=6 Y=12

NAME=example_4c X=100 Y=101

MICRO=0

*END

Annotated Input Data

1) Flags: None 

NASA/TM—2002-212077/VOL3 120
Section 4: Applied Loading Example 4c: General Loading Option for RUC analysis 

2) Constituent materials (*CONSTITUENTS) [KM_2]: 

Number of materials: 2 (NMATS=2) 

Materials: SiC fiber (MATID=E) 

Ti-21S (MATID=A) 

Constitutive models: SiC fiber: linearly elastic (CMOD=6) 

Ti-21S matrix: Isotropic GVIPS (CMOD=4) 

3) Analysis type (*RUC) → Repeating Unit Cell Analysis [KM_3]: 

Analysis model: Doubly periodic GMC (MOD=2) 

RUC architecture: 26×26 circular fiber, rect. pack (ARCHID=13) 

Fiber volume fraction: 0.25 (VF=0.25) 

RUC aspect ratio: 1. (square pack) (R=1.) 

Material assignment: SiC fiber (F=1) 

Ti-21S matrix (M=2) 

4) Loading: 

a) Mechanical (*MECH) [KM_4]: 

Loading option: general loading (LOP=99) 

Component #1 (ε11 or σ11) 

Number of points: 6 (NPT=6) 

Times (TI=) (sec.) 0. 300. 310. 2310. 2320. 2620. 

Magnitudes (MAG=) 0. 0 0.01 0.01 0. 0. 

Control (MODE=) strain strain strain stress stress 

Component #2 (ε22 or σ22) 

Number of points: 6 (NPT=6) 

Times (TI=) (sec.) 0. 300. 310. 2310. 2320. 2620. 

Magnitudes (MAG=) 0. 0 0.005 0.005 0. 0. 

Control (MODE=) strain strain strain stress stress 

Component #3 (ε33 or σ33) 

Number of points: 2 (NPT=2) 

Times (TI=) (sec.) 0. 2620. 

Magnitudes (MAG=) 0. 0. 

Control (MODE=) stress 

Component #4 (γ23 or σ23) 

Number of points: 2 (NPT=2) 

Times (TI=) (sec.) 0. 2620. 

Magnitudes (MAG=) 0. 0. 

Control (MODE=) stress 

NASA/TM—2002-212077/VOL3 121
MAC/GMC 4.0 Example Problem Manual 

Component #5 (γ13 or σ13) 

Number of points: 2 (NPT=2) 

Times (TI=) (sec.) 0. 2620. 

Magnitudes (MAG=) 0. 0. 

Control (MODE=) stress 

Component #6 (γ12 or σ12) 

Number of points: 6 (NPT=6) 

Times (TI=) (sec.) 0. 300. 310. 2310. 2320. 2620. 

Magnitudes (MAG=) 0. 0 0.0075 0.0075 0. 0. 

Control (MODE=) strain strain strain stress stress 

The simulated loading constitutes a state of plane stress in the x1-x2 plane, thus the stress associated 

with components 3 ñ 5 are kept at zero. The first loading segment involves the in-plane strain 

components being constrained (i.e., kept at zero) during which time a heat-up is specified in the 

thermal loading. Then, in-plane normal and shear strains are quickly applied and then held for 2000 

sec. The control mode next switches from strain to stress for all three in-plane components, and the 

mechanical loading is reduced to zero in stress control. Finally, the composite is kept globally stressfree as the temperature is reduced during the final segment. This simulates a (fictitious) mission 

profile that might be encountered at a material point on the skin of an aerospace vehicle or engine as 

the vehicle warms up, takes off, flies, lands, and cools. 

b) Thermal (*THERM) [KM_4]: 

Number of points: 6 (NPT=6) 

Time points: 0., 300., 310., 2310., 2320., 2620. sec. 

 (TI=0.,300.,310.,2310.,2320.,2620.) 

Temperature points: 23., 650., 650., 650., 650., 23. °C 

(TEMP=23.,650.,650.,650.,650.,23.) 

c) Time integration (*SOLVER) [KM_4]: 

Time integration method: Forward Euler (METHOD=1) 

Number of points: 6 (NPT=6) 

Time points: 0., 300., 310., 2310., 2320., 2620. sec. 

 (TI=0.,300.,310.,2310.,2320.,2620.) 

Time step sizes: 1., 0.2, 2., 1., 1. sec. (STP=1.,0.2,2.,1.,1.) 

5) Damage and Failure: None 

6) Output: 

a) Output file print level (*PRINT) [KM_4]: 

Print level: 6 (NPL=6) 

b) x-y plots (*XYPLOT) [KM_4]: 

Frequency: 5 (FREQ=5) 

Number of macro plots: 6 (MACRO=6) 

NASA/TM—2002-212077/VOL3 122
Section 4: Applied Loading Example 4c: General Loading Option for RUC analysis 

Macro plot names: example_4c (NAME=example_4c) 

Macro plot x-y quantities: 11 ε , σ 11 (X=1 Y=7) 

 22 ε , σ 22 (X=2 Y=8) 

 12 γ , σ 12 (X=6 Y=12) 

 temperature, time (X=100 Y=101) 

Number of micro plots: 0 (MICRO=0) 

7) End of file keyword: (*END) 

Results

Figure 4.5 and Figure 4.6 show that during the first loading segment (0 ñ 300 sec.), while the temperature 

is increasing from 23 ñ 650 °C, no strain results and the normal stress components (σ11 and σ22) become 

compressive. This is due to the fact that the composite has been constrained against its natural thermal 

expansion during this heat-up. The shear stress component remains at zero because the composite does 

not experience shear thermal expansion (i.e., it is orthotropic). From 300 ñ 310 sec., the three strain 

components are quickly applied, and the stress components rise rapidly in response. Next, these three 

strain components are held constant for 2000 sec., during which time the stress components are free to 

relax. From 2310 ñ 2320 sec., the mechanical load, which was applied in strain control, is unloaded 

under stress control to zero for all three stress components. Finally, the temperature is decreased from 

650 ñ 23 °C from 2320 ñ 2620 sec., during which time the stress components on the composite are held at 

zero. Therefore, the normal strain components decrease as the composite contracts while the shear strain 

remains constant. From Figure 4.6, it is clear that the composite experience permanent normal and shear 

strains at the completion of this thermomechanical loading history.

-150

-100

-50

0

50

100

150

200

0 500 1000 1500 2000 2500 3000

time (s)

stress (ksi)

σ11

σ22

σ12

Figure 4.5 Example 4c: plot of the simulated stress vs. time history for a 0.25 fiber volume fraction 

SiC/Ti-21S composite. 

NASA/TM—2002-212077/VOL3 123
MAC/GMC 4.0 Example Problem Manual 

0.000

0.002

0.004

0.006

0.008

0.010

0.012

0 500 1000 1500 2000 2500 3000

time (s)

strain

ε11

ε22

γ12

Figure 4.6 Example 4c: plot of the simulated strain vs. time history for a 0.25 fiber volume fraction 

SiC/Ti-21S composite. 

NASA/TM—2002-212077/VOL3 124
Section 4: Applied Loading Example 4d: Internal Loading Options for Laminate 

Example 4d: Internal Loading Options for Laminate

This example problem demonstrates how to invoke the various internal mechanical loading options 

available for a laminate in MAC/GMC 4.0. In particular, the laminate analyzed is the identical quasiisotropic 0.25 fiber volume fraction SiC/Ti-21S laminate presented in Example 3h. Because of the plane 

stress assumption inherent in lamination theory, only loading in the x-y plane of the laminate is 

permissible (where the z-direction is the through-thickness direction of the laminate, Figure 1.2). Further, 

loading in the form of midplane curvatures and moment resultants (in addition to midplane strains and 

force resultants) may be applied to the laminate using the internal loading option 4 ñ 6. Thus, the 

meaning of the various loading options is different in the case of a laminate as compared to the repeating 

unit cell analysis. For details, see Section 4 of the Keywords Manual. In the present example, the 

loading option is successively increased from 1 to 6 to apply the three midplane strain components and 

the three midplane curvatures in separate executions of the code. 

MAC/GMC Input File: example_4d.mac

Lamination theory - Internal loading options for laminates

*CONSTITUENTS

NMATS=2

M=1 CMOD=6 MATID=E

M=2 CMOD=4 MATID=A

*LAMINATE

NLY=7

LY=1 MOD=2 THK=0.125 ANG=90 ARCHID=6 R=1. VF=0.25 F=1 M=2

LY=2 MOD=2 THK=0.125 ANG=45 ARCHID=6 R=1. VF=0.25 F=1 M=2

LY=3 MOD=2 THK=0.125 ANG=0 ARCHID=6 R=1. VF=0.25 F=1 M=2

LY=4 MOD=2 THK=0.25 ANG=-45 ARCHID=6 R=1. VF=0.25 F=1 M=2

LY=5 MOD=2 THK=0.125 ANG=0 ARCHID=6 R=1. VF=0.25 F=1 M=2

LY=6 MOD=2 THK=0.125 ANG=45 ARCHID=6 R=1. VF=0.25 F=1 M=2

LY=7 MOD=2 THK=0.125 ANG=90 ARCHID=6 R=1. VF=0.25 F=1 M=2

*MECH

# -- Alter LOP from 1 - 6 to generate example problem results

LOP=1

NPT=2 TI=0.,200. MAG=0.,0.02 MODE=1

*THERM

NPT=2 TI=0.,200. TEMP=650.,650.

*SOLVER

METHOD=1 NPT=2 TI=0.,200. STP=1.

*PRINT

NPL=6

*XYPLOT

FREQ=5

LAMINATE=6

NAME=example_4d_exx X=1 Y=10

NAME=example_4d_eyy X=2 Y=11

NAME=example_4d_exy X=6 Y=12

NAME=example_4d_kxx X=7 Y=13

NAME=example_4d_kyy X=8 Y=14

NAME=example_4d_kxy X=9 Y=15

MACRO=0

MICRO=0

*END

NASA/TM—2002-212077/VOL3 125
MAC/GMC 4.0 Example Problem Manual 

Annotated Input Data

1) Flags: None 

2) Constituent materials (*CONSTITUENTS) [KM_2]: 

Number of materials: 2 (NMATS=2) 

Materials: SiC fiber (MATID=E) 

Ti-21S (MATID=A) 

Constitutive models: SiC fiber: linearly elastic (CMOD=6) 

Ti-21S matrix: Isotropic GVIPS (CMOD=4) 

3) Analysis type (*LAMINATE) → Laminate Analysis [KM_3]: 

Number of layers: 7 (NLY=7) 

Layer Analysis 

Model 

Thickness Fiber 

Angle 

Architecture Aspect 

Ratio 

Volume 

fraction 

Fiber 

material 

Matrix 

material 

(LY=) (MOD=2) (THK=) (ANG=) (ARCHID=6) (R=1.) (VF=0.25) (F=1) (M=2) 

1 GMC-2D 0.125 90° 7×7 circle, 

rect. pack 

1. 0.25 SiC Ti-21S 

2 GMC-2D 0.125 45° 7×7 circle, 

rect. pack 

1. 0.25 SiC Ti-21S 

3 GMC-2D 0.125 0° 7×7 circle, 

rect. pack 

1. 0.25 SiC Ti-21S 

4 GMC-2D 0.25 -45° 7×7 circle, 

rect. pack 

1. 0.25 SiC Ti-21S 

5 GMC-2D 0.125 0° 7×7 circle, 

rect. pack 

1. 0.25 SiC Ti-21S 

6 GMC-2D 0.125 45° 7×7 circle, 

rect. pack 

1. 0.25 SiC Ti-21S 

7 GMC-2D 0.125 90° 7×7 circle, 

rect. pack 

1. 0.25 SiC Ti-21S 

4) Loading: 

a) Mechanical (*MECH) [KM_4]: 

Loading option: 1 ñ 6 (LOP=1 through LOP=6) 

Number of points: 2 (NPT=2) 

Time points: 0., 200. sec. (TI=0.,200.) 

Load magnitude: 0., 0.02 (MAG=0.,0.02) 

Loading mode: midplane strain/curvature control (MODE=1) 

! Note: To generate the results presented in this example, the loading option (LOP) must be changed 

from 1 to 6, successively. 

b) Thermal (*THERM) [KM_4]: 

Number of points: 2 (NPT=2) 

Time points: 0., 200. sec. (TI=0.,200.) 

Temperature points: 650., 650. °C (TEMP=650.,650.) 

NASA/TM—2002-212077/VOL3 126
Section 4: Applied Loading Example 4d: Internal Loading Options for Laminate 

c) Time integration (*SOLVER) [KM_4]: 

Time integration method: Forward Euler (METHOD=1) 

Number of points: 2 (NPT=2) 

Time points: 0., 200. sec. (TI=0.,200.) 

Time step sizes: 1. sec. (STP=1.) 

5) Damage and Failure: None 

6) Output: 

a) Output file print level (*PRINT) [KM_6]: 

Print level: 6 (NPL=6) 

b) x-y plots (*XYPLOT) [KM_6]: 

Frequency: 5 (FREQ=5) 

Number of laminate plots: 6 (LAMINATE=6) 

Laminate plot name: example_4d_exx (NAME=example_4d_exx) 

 example_4d_eyy (NAME=example_4d_eyy) 

 example_4d_exy (NAME=example_4d_exy) 

 example_4d_kxx (NAME=example_4d_kxx) 

 example_4d_kyy (NAME=example_4d_kyy) 

 example_4d_kxy (NAME=example_4d_kxy) 

Laminate plot x-y quantities: 0

xx ε , Nxx (X=1 Y=10) 

 0

yy ε , Nyy (X=2 Y=11) 

 0

xy ε , Nxy (X=6 Y=12) 

 κ xx , M xx (X=7 Y=13) 

 κ yy , M xy (X=8 Y=14) 

 κ xy , M xy (X=9 Y=15) 

Number of macro plots: 0 (MACRO=0) 

Number of micro plots: 0 (MICRO=0) 

7) End of file keyword: (*END) 

Results

Figure 4.7 shows the response of the quasi-isotropic laminate to the three applied midplane strain 

components (LOP=1–3), whereas Figure 4.8 shows the laminate response to the three applied curvature 

components (LOP=4–6). It is clear from Figure 4.7 that the extensional behavior of the laminate is 

isotropic while Figure 4.8 shows that the bending behavior is anisotropic. The response of the laminate to 

the applied κ yy is stiffer than that to the applied κ xx because the outer plies of the laminate (which are 

farthest from the midplane and therefore contribute more to the bending stiffness) are oriented along the 

y-axis. It should also be noted that this laminate exhibits normal-shear coupling in its bending response 

(see Table 3.1). 

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MAC/GMC 4.0 Example Problem Manual 

0

20

40

60

80

100

120

140

160

0.000 0.005 0.010 0.015 0.020 0.025

Midplane Strain

Force Resultant (ksi 

× unit thickness)

xx response

yy response

xy response

Figure 4.7 Example 4d: 0 N ijij − ε response of a 0.25 fiber volume fraction quasi-isotropic SiC/Ti-21S 

laminate to applied midplane strains at 650 °C. 

0

5

10

15

20

25

0.000 0.005 0.010 0.015 0.020 0.025

Curvature (1/unit thickness)

Moment Resultant (ksi × unit thickness^2)

xx response

yy response

xy response

Figure 4.8 Example 4d: M −κ ijij response of a 0.25 fiber volume fraction quasi-isotropic SiC/Ti-21S 

laminate to applied curvatures at 650 °C. 

NASA/TM—2002-212077/VOL3 128
Section 4: Applied Loading Example 4e: General loading Option for Laminates 

Example 4e: General loading Option for Laminates

This problem exercises the general loading option for a laminate. As in the previous example problem, 

the loading associated with the laminate involves force and moment resultants along with midplane 

strains and curvatures. Now, however, all six components applied to the laminate are specified via the 

LOP=99 loading option. In particular, an asymmetric elastic graphite/epoxy laminate is considered with 

temperature-dependent user-defined constituent properties. The laminate is subjected to a stress-free 

cool-down from 150 °C, which is intended to simulate the cure temperature. During the cool-down, the 

asymmetric laminate experiences bending (non-zero curvatures), and the subsequent mechanical loading 

eliminates this bending by returning the curvatures to zero. Thus, at the completion of the applied 

loading, the code has determined the moment resultants required to flatten the laminate after the thermally 

induced curvature. 

MAC/GMC Input File: example_4e.mac

MAC/GMC 4.0 Example 4e - General loading option for laminates

*CONSTITUENTS

NMATS=2

# -- Graphite fiber

M=1 CMOD=6 MATID=U MATDB=1

NTP=2

TEM=23.,150.

EA=388.2E9,390.E9

ET=7.6E9,7.6E9

NUA=0.41,0.41

NUT=0.45,0.45

GA=14.9E9,15.1E9

ALPA=-0.68E-6,-0.45E-6

ALPT=9.74E-6,10.34E-6

# -- Epoxy matrix

M=2 CMOD=6 MATID=U MATDB=1

NTP=2

TEM=23.,150.

EA=3.45E9,3.10E9

ET=3.45E9,3.10E9

NUA=0.35,0.35

NUT=0.35,0.35

GA=1.278E9,1.148E9

ALPA=45.E-6,55.E-6

ALPT=45.E-6,55.E-6

*LAMINATE

NLY=4

LY=1 MOD=2 THK=0.25 ANG=90 ARCHID=6 R=1. VF=0.65 F=1 M=2

LY=2 MOD=2 THK=0.25 ANG=60 ARCHID=6 R=1. VF=0.65 F=1 M=2

LY=3 MOD=2 THK=0.25 ANG=-60 ARCHID=6 R=1. VF=0.65 F=1 M=2

LY=4 MOD=2 THK=0.25 ANG=0 ARCHID=6 R=1. VF=0.65 F=1 M=2

*MECH

LOP=99

NPT=3 TI=0.,1.,2. MAG=0.,0.,0. MODE=2,2

NPT=3 TI=0.,1.,2. MAG=0.,0.,0. MODE=2,2

NPT=3 TI=0.,1.,2. MAG=0.,0.,0. MODE=2,2

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MAC/GMC 4.0 Example Problem Manual 

NPT=3 TI=0.,1.,2. MAG=0.,0.,0. MODE=2,1

NPT=3 TI=0.,1.,2. MAG=0.,0.,0. MODE=2,1

NPT=3 TI=0.,1.,2. MAG=0.,0.,0. MODE=2,1

*THERM

NPT=3 TI=0.,1.,2. TEMP=150.,23.,23.

*SOLVER

METHOD=1 NPT=3 TI=0.,2. STP=1.

*PRINT

NPL=0

*END

Annotated Input Data

1) Flags: None 

2) Constituent materials (*CONSTITUENTS) [KM_2]: 

Number of materials: 2 (NMATS=2) 

Constitutive models: Elastic (CMOD=6) 

Materials: User-defined (Graphite) (MATID=U) 

 User-defined (Epoxy) (MATID=U) 

Material property source: Read from input file (MATDB=1) 

Material properties: See Table 4.1 

Table 4.1 Constituent material elastic properties for Example 4e. 

Temp 

(°C) 

EA

(GPa)

ET

(GPa)

νA νT GA

(GPa)

αA

(10-6/ °C)

αT

(10-6/ °C)

Graphite 23. 388.2 7.6 0.41 0.45 14.9 -0.68 9.74 

150. 390. 7.6 0.41 0.45 15.1 -0.45 10.34 

Epoxy 23. 3.45 3.45 0.35 0.35 1.278 45. 45. 

150. 3.10 3.10 0.35 0.35 1.148 55. 55. 

3) Analysis type (*LAMINATE) → Laminate Analysis [KM_3]: 

Number of layers: 4 (NLY=4) 

Layer Analysis 

Model 

Thickness Fiber 

Angle 

Architecture Volume 

fraction 

Aspect 

ratio 

Fiber 

material 

Matrix 

material 

(LY=) (MOD) (THK) (ANG) (ARCHID) (VF) (R) (F) (M) 

1 GMC-2D 0.25 90° 7×7 circle, 

rect. pack 

0.65 1. graphite epoxy 

2 GMC-2D 0.25 60° 7×7 circle, 

rect. pack 

0.65 1. graphite epoxy 

3 GMC-2D 0.25 -60° 7×7 circle, 

rect. pack 

0.65 1. graphite epoxy 

4 GMC-2D 0.25 0° 7×7 circle, 

rect. pack 

0.65 1. graphite epoxy 

NASA/TM—2002-212077/VOL3 130
Section 4: Applied Loading Example 4e: General loading Option for Laminates 

4) Loading: 

a) Mechanical (*MECH) [KM_4]: 

Loading option: general loading (LOP=99) 

Component #1 ( 0

xx ε or Nxx ) 

Number of points: 3 (NPT=3) 

Times (TI=) (sec.) 0. 1. 2. 

Magnitudes (MAG=) 0. 0. 0. 

Control (MODE=) force resultant force resultant 

Component #2 ( 0

yy ε or Nyy ) 

Number of points: 3 (NPT=3) 

Times (TI=) (sec.) 0. 1. 2. 

Magnitudes (MAG=) 0. 0. 0. 

Control (MODE=) force resultant force resultant 

Component #3 ( 0

xy γ or Nxy ) 

Number of points: 3 (NPT=3) 

Times (TI=) (sec.) 0. 1. 2. 

Magnitudes (MAG=) 0. 0. 0. 

Control (MODE=) force resultant force resultant 

Component #4 (κ xx or M xx ) 

Number of points: 3 (NPT=3) 

Times (TI=) (sec.) 0. 1. 2. 

Magnitudes (MAG=) 0. 0. 0. 

Control (MODE=) moment resultant curvature 

Component #5 (κ yy or M yy ) 

Number of points: 3 (NPT=3) 

Times (TI=) (sec.) 0. 1. 2. 

Magnitudes (MAG=) 0. 0. 0. 

Control (MODE=) moment resultant curvature 

Component #6 (κ xy or M xy ) 

Number of points: 3 (NPT=3) 

Times (TI=) (sec.) 0. 1. 2. 

Magnitudes (MAG=) 0. 0. 0. 

Control (MODE=) moment resultant curvature 

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MAC/GMC 4.0 Example Problem Manual 

b) Thermal (*THERM) [KM_4]: 

Number of points: 3 (NPT=3) 

Time points: 0., 1., 2. sec. (TI=0.,1.,2.) 

Temperature points: 150., 23., 23. °C (TEMP=150.,23.,23.) 

c) Time integration (*SOLVER) [KM_4]: 

Time integration method: Forward Euler (METHOD=1) 

Number of points: 3 (NPT=3) 

Time points: 0., 1., 2. sec. (TI=0.,1.,2.) 

Time step sizes: 1., 1. sec. (STP=1.,1.) 

! Note: Since this example problem is linearly elastic and the temperature-dependent constituent 

material properties are linear, the problem solution is independent of the number of time steps 

taken through the loading history. Therefore the minimum number of time steps (2) has been 

employed. 

5) Damage and Failure: None 

6) Output: 

a) Output file print level (*PRINT) [KM_6]: 

Print level: 6 (NPL=6) 

b) x-y plots (*XYPLOT): None 

7) End of file keyword: (*END) 

Results

The results below are taken from the MAC/GMC 4.0 output file for this example. 

1 TIME: 1.0000D+00 TEMP: 2.3000D+01 TSTEP: 1.0000D+00

-------------------------------------------------------------------------------------------------------

FORCE(N), MOMENT(M): 0.0000D+00 0.0000D+00 0.0000D+00 0.0000D+00 0.0000D+00 0.0000D+00

STRAIN, CURVATURE: -1.2843D-03 -5.4894D-05 1.3757D-04 3.3269D-03 -4.3171D-04 -3.6875D-03

INELASTIC N, M: 0.0000D+00 0.0000D+00 0.0000D+00 0.0000D+00 0.0000D+00 0.0000D+00

THERMAL N, M: -1.6159D+07 -1.0899D+07 -1.0789D-03 1.9726D+06 -1.9726D+06 -5.6944D+05

OUT-OF-PLANE STRAIN: -5.2503D-03 0.0000D+00 0.0000D+00

2 TIME: 2.0000D+00 TEMP: 2.3000D+01 TSTEP: 1.0000D+00

-------------------------------------------------------------------------------------------------------

FORCE(N), MOMENT(M): 0.0000D+00 0.0000D+00 0.0000D+00 -6.5415D+06 2.9887D+06 1.1301D+06

STRAIN, CURVATURE: -1.9617D-04 -4.3628D-05 -6.2573D-14 0.0000D+00 0.0000D+00 0.0000D+00

INELASTIC N, M: 0.0000D+00 0.0000D+00 0.0000D+00 0.0000D+00 0.0000D+00 0.0000D+00

THERMAL N, M: -1.6159D+07 -1.0899D+07 -1.0789D-03 1.9726D+06 -1.9726D+06 -5.6944D+05

OUT-OF-PLANE STRAIN: -5.7715D-03 0.0000D+00 0.0000D+00

Upon completion of the stress-free cool down of the laminate (at time = 1. sec.), the laminate is bent 

(as indicated by the non-zero curvatures). Then, upon completion of the simulated mechanical loading 

(at time = 2. sec.), the laminate is no longer bent (as indicated by the zero curvatures). The 

moment resultants required to eliminate the laminateís bending are: −= 54.6 M xx MPa × thickness2

, 

= 99.2 M yy MPa × thickness2, = 13.1 M xy MPa × thickness2, where thickness is the total 

thickness of the laminate (e.g., if the laminate were 1 mm thick, the required 

NASA/TM—2002-212077/VOL3 132
Section 4: Applied Loading Example 4e: General loading Option for Laminates 

6

2

N 6.54 10

m

M xx =− × ( )2× 0.001 m = −6.54 N ). Recall that a moment resultant is a 

moment (with units force × length) per unit width of the laminate, yielding units of force. 

NASA/TM—2002-212077/VOL3 133
MAC/GMC 4.0 Example Problem Manual 

Example 4f: RUC Yield Surface Analysis

This example problem demonstrates one of the more powerful capabilities of MAC/GMC 4.0, the 

generation of yield surfaces. A yield surface is defined as the locus of points in a stress space at which a 

specified ìyieldî condition is satisfied. For metals, a 0.1 % or 0.2 % permanent (inelastic) strain is often 

used as a yield condition, for example. However, this type of condition is based on uniaxial tension or 

compression, whereas, a yield surface, by definition is concerned with a multi-axial state of stress. A 

typical yield surface for a monolithic metal (aluminum, in this case) is shown in Figure 4.9. The ìstress 

spaceî for this yield surface is the σ11-σ22 plane defined by the axes. The plotted yield surface then 

represents σ11-σ22 points that cause the metal to yield. Of course, this implies that some definition of 

what constitutes yielding has been employed. In Figure 4.9, yielding was defined as an equivalent plastic 

strain (EPS) of 0.001 (0.1 %), where p

ij

p EPS ij ε ε

3

2 = and p

ij ε are the plastic strain components. Since 

EPS is a scalar value based on the multi-axial plastic strain state, it is suitable for use as a yield condition. 

The elliptical shape of the yield surface indicates that it is based on a von Mises type yield criterion. 

Indeed, the ìyield surfaceî in Figure 4.9 was generated using the Bodner-Partom viscoplastic constitutive 

model, which employs a von Mises (J2) type flow law. In viscoplasticity theory, no explicit yield 

condition exists that must be satisfied at all times. Therefore, a ìyieldî or ìthresholdî surface can be 

predicted for a viscoplastic material by specifying a particular criterion and checking for when it is 

satisfied during applied loading. This type of surface can then be generated for any material, regardless 

of the inelastic model formulation. However, some yield conditions are better suited than others for use 

in conjunction with particular inelastic constitutive models. For example, a surface generated from a ratebased criterion would not be appropriate for use in conjunction with a rate independent inelastic model 

such as incremental plasticity. It should be noted that, in its present form, the yield surface analysis 

implementation within MAC/GMC 4.0 is most relevant to composites and laminates that contain metals. 

However, the capabilities could also be applied to polymers and ceramics that exhibit inelastic 

deformation. For additional information on the codeís yield surface predictions, see Section 6 of the 

Theory Manual. 

In order to generate a yield surface like that shown in Figure 4.9, MAC/GMC 4.0 applies combined 

mechanical loading in the given stress space. That is, loading is applied along a specific angle in the σ11-

σ22 plane (for the present case) until the yield definition is encountered. Then, a new angle is chosen 

along which loading is applied until yield. This angle is referred to as the ìprobing angleî. When a 

sufficient number of angles have been probed, the locus of all the points at which yield occurred defines 

the yield surface. In MAC/GMC 4.0, the user specifies the increment to employ for the probing angle. 

The code then probes at angles from 0° to 360° using this angle increment. A small angle increment will 

generate a smoother yield surface, but requires more execution time as the code must execute for each 

probing angle. 

NASA/TM—2002-212077/VOL3 134
Section 4: Applied Loading Example 4f: RUC Yield Surface Analysis 

-120

-80

-40

0

40

80

120

-120 -80 -40 0 40 80 120

σ11 (MPa)

σ22 (MPa)

Figure 4.9 Yield surface for aluminum at room temperature based on EPS=0.001.

MAC/GMC Input File: example_4f.mac

MAC/GMC 4.0 Example 4f - RUC yield surface analysis

*CONSTITUENTS

NMATS=2

M=1 CMOD=6 MATID=E

M=2 CMOD=4 MATID=A

*RUC

MOD=2 ARCHID=1 VF=0.25 F=1 M=2

#*THERM

# NPT=3 TI=0.,57600.,64800. TEMP=900.,23.,650.

#*SOLVER

# METHOD=1 NPT=3 TI=0.,57600.,64800. STP=40.,40.

*SURF

TMAX=400. STP=0.01 MMAX=0.04 MODE=1 TREF=650.

OPTION=1,2,3,4

ISPX=1 ISPY=2 ANGINC=5.

EPS=0.0001 DR=0.00025 ISR=0.00002 IP=0.001

*PRINT

NPL=1

*END

Annotated Input Data

1) Flags: None 

2) Constituent materials (*CONSTITUENTS) [KM_2]: 

Number of materials: 2 (NMATS=2) 

Materials: SiC fiber (MATID=E) 

Ti-21S (MATID=A) 

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MAC/GMC 4.0 Example Problem Manual 

Constitutive models: SiC fiber: linearly elastic (CMOD=6) 

Ti-21S matrix: Isotropic GVIPS (CMOD=4) 

3) Analysis type (*RUC) → Repeating Unit Cell Analysis [KM_3]: 

Analysis model: Doubly periodic GMC (MOD=2) 

RUC architecture: Square fiber, square pack (ARCHID=1) 

Fiber volume fraction: 0.25 (VF=0.25) 

Material assignment: SiC fiber (F=1) 

Ti-21S matrix (M=2) 

4) Loading: 

a) Mechanical (*MECH): None 

The loading for yield surface probing is controlled under *SURF. Any specified mechanical or 

thermal loading is interpreted as ìpreloadingî, which is applied prior to beginning the yield surface 

probing. This allows the user to incorporate a certain stress/strain state into the composite prior to 

determination of the yield surface. 

b) Thermal (*THERM) [KM_4]: (case with residual stresses only) 

Number of points: 3 (NPT=3) 

Time points: 0., 57600., 64800. sec. (TI=0.,57600.,64800.) 

Temperature points: 900., 23., 650. (TEMP=900.,23.,650.) 

c) Time integration (*SOLVER) [KM_4]: (case with residual stresses only) 

Time integration method: Forward Euler (METHOD=1) 

Number of points: 3 (NPT=3) 

Time points: 0., 57600., 64800. sec. (TI=0.,57600.,64800.) 

Time step size: 40., 40. sec. (STP=40.,40.) 

! Note: As with the mechanical loading, the thermal loading in yield surface analysis is interpreted by 

the code as ìpreloadingî. Thus, the thermal loading specified allows residual stresses to be 

included in the composite prior to the yield surface probing. To execute this example problem 

without residual stresses, the *THERM and *SOLVER keyword data should be commented as 

shown above. To execute with residual stresses, these should be uncommented. 

d) Yield surface generation (*SURF) [KM_4]: 

Maximum probing time: 400. sec. (TMAX=400.) 

Probing time step size: 0.01 sec. (STP=0.01) 

Maximum probing load: 0.04 (MMAX=0.04) 

Probing mode: strain control (MODE=1) 

Reference temperature: 650. °C (TREF=650.) 

Yield surface options: Generate global data (1) (OPTION=1,2,3,4) 

 Generate 1st subcell data (2) 

 Generate all subcells data (3) 

 Generate local subcell data (4) 

NASA/TM—2002-212077/VOL3 136
Section 4: Applied Loading Example 4f: RUC Yield Surface Analysis 

Stress space x-axis: σ11 (ISPX=1) 

Stress space y-axis: σ22 (ISPY=2) 

Probe angle increment: 5.° (ANGINC=5) 

Quantities defining yield: Equivalent plastic strain = 0.0001 (EPS=0.0001) 

 Dissipation rate = 0.00025 (DR=0.00025) 

 Inelastic strain rate = 0.00002 (ISR=0.00002) 

 Inelastic power = 0.001 (IP=0.001) 

The mechanical loading for the yield surface probing is specified under *SURF in the form of a 

maximum time (TMAX), time step (STP), maximum load (MMAX), and mode (MODE). The yield surface 

mechanical loading is unique from that specified under *MECH in that only one segment may be 

employed. If, at a particular probing angle, the maximum time is reached before all yield criteria are 

satisfied, a warning is written to the output file, and the probing proceeds to the next angle. Thus a large 

maximum probing load and maximum probing time should be employed. In addition, the time step size 

needs to be reasonably small such that the exact point of yielding can be captured. The rate of the applied 

loading is determined from the maximum time and maximum load data. This rate is interpreted as a J2

(stress) or J2í (strain) quantity. This allows the loading rate to be consistent as the probing angle changes. 

If thermal preloading is not specified with *THERM, then TREF must be specified under *SURF in order 

to indicate the temperature at which the yield surface is determined. If thermal preloading is specified, 

this TREF is ignored and the ending temperature from the thermal preloading is employed. 

The yield surface option (*SURF) generates up to four ASCII files (for the case of RUC analysis) that 

contain the yield surface data. These are: surf_global.dat, surf_1st.dat, surf_all.dat, 

and surf_local.dat. surf_global.dat contains the global yield surface data for each type of 

yield surface (i.e., EPS, DR, ISR, and IP), surf_1st.dat contains the yield surface data based on first 

subcell yield for each type of yield surface, surf_all.dat contains the yield surface data based on the 

requirement that all subcells (that can yield) yield for each type of yield surface, and surf_local.dat

contains the yield surface data for each individual subcell for each type of yield surface. 

The stress space is defined by ISPX and ISPY. In this example, the σ11-σ22 stress space is specified, 

which indicates that loading will be applied in the x1-direction (along the fiber direction) and the x2-

direction (transverse to the fiber direction). Additional options are available using ISPX and ISPY, 

including applying additional loading components that are slaves to (i.e., equal to) the primary stress 

space loading components. These options are described in the MAC/GMC 4.0 Keywords Manual 

Section 4. 

The yield surface probing occurs along angles in the stress space determined by the specified probe angle 

increment (ANGINC). That is, for example, when probing at an angle of 5° in the σ11-σ22 stress space, the 

code applies loading such that σ11/σ22 = tan(5°). Upon fulfillment of all yield criteria (or upon reaching 

the end of the specified maximum time), the code proceeds to the next probing angle. If preloading is 

specified, the code begins each probing angle not at a zero stress-strain state, but rather at the stress-strain 

state that existed at the end of the preload. A smaller probe angle increment results in a smoother 

predicted yield surface, but will increase the execution time. For additional information on the 

MAC/GMC 4.0 yield surface capabilities and procedures, see the MAC/GMC 4.0 Keywords Manual 

Section 4. 

The particular yield values chosen in this example problem are arbitrary. It is possible to see when 

yielding occurs at a particular probe angle by generating an x-y plot and obtaining a particular yield point 

NASA/TM—2002-212077/VOL3 137
MAC/GMC 4.0 Example Problem Manual 

from the notification written to the output file. The yield values can the be adjusted as desired to make 

the yield prediction more or less conservative. 

5) Damage and Failure: None 

6) Output: 

a) Output file print level (*PRINT) [KM_6]: 

Print level: 0 (NPL=0) 

b) x-y plots (*XYPLOT): None 

7) End of file keyword: (*END) 

Results

Some of the yield surface data results from this example are plotted in Figure 4.10 ñ Figure 4.15. All 

yield surface data generated have not been plotted as this example generates data for six yield surfaces of 

each type (i.e., EPS, DR, ISR, and IP) for both the cases with and without residual. Figure 4.10 shows the 

global yield surfaces for the case without residual stresses. The term ìglobalî refers to the fact that the 

composite or RUC level stresses, strains, and rates have been employed to determine when yielding 

occurs. All four yield surfaces in Figure 4.10 are biased (i.e., elongated) along the σ11-axis due to the 

presence of the continuous fibers, which elevate the yield stress, in the x1-direction. Based on the yield 

values specified in the input files (which are somewhat arbitrary), the composite tends to yield first with 

respect to the dissipation rate (DR), followed by the inelastic power (IP), the inelastic strain rate (ISR), 

and finally by the equivalent plastic strain (EPS). The four yield surfaces are similar in shape and all are 

centered about the stress space origin, with the ISR surface being the most elongated. 

Figure 4.11 shows the same four yield surfaces for the case in which residual stresses have been 

incorporated. Qualitatively, these yield surfaces are similar to their counterparts from Figure 4.10. The 

major difference is that the yield surfaces are shifted to the left due to the compressive residual stresses in 

the matrix. 

Figure 4.12 presents all six equivalent plastic strain (EPS) yield surfaces from the case in which residual 

stresses were not included. The global yield surface is the same as that plotted in Figure 4.10 (labeled 

ìEPS Globalî). Also plotted are the yield surfaces associated with the three individual matrix subcells 

(β,γ = 1,2; 2,1; and 2,2), see Figure 4.16. Because the 2,2 subcell is farthest from the fiber, this subcell 

has a significantly larger yield surface than the 1,2 and 2,1 subcells. The 1,1 subcell is occupied by the 

elastic fiber and thus does not have a yield surface. The first subcell yield surface generated by 

MAC/GMC 4.0 traces the intersection of the individual subcell yield surfaces. Conversely the yield 

surface generated by requiring all subcells to yield traces the union of the individual subcell yield surfaces 

(which, in Figure 4.12, coincides with the subcell 2,2 yield surface). 

The counterpart to Figure 4.12 is Figure 4.13, which contains the same six yield surfaces for the case 

including residual stresses. Now, the subcell 2,2 yield surface intersects the subcell 1,2 and 2,1 yield 

surfaces, giving rise to irregularly shaped first subcell and all subcells yield surfaces in the vicinity of the 

tensile σ11-axis. This irregularity of the all subcells EPS yield surface can be seen more clearly in Figure 

4.14, which directly compares EPS yield surfaces for the cases with and without residual stresses. 

Further, this figure shows that residual stresses have translated the yield surfaces slightly in the direction 

of the tensile σ22-axis in addition to the compressive σ11 translation mentioned previously. Finally, Figure 

4.15 compares inelastic strain rate (ISR) yield surfaces for the cases with and without residual stresses. 

NASA/TM—2002-212077/VOL3 138
Section 4: Applied Loading Example 4f: RUC Yield Surface Analysis 

-60

-40

-20

0

20

40

60

-60 -40 -20 0 20 40 60

σ11 (ksi)

σ22 (ksi)

EPS Global

DR Global

ISR Global

IP Global

Figure 4.10 Global (RUC) yield surfaces for 0.25 fiber volume fraction SiC/Ti-21S at 650 °C with no 

residual stresses. 

-60

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-20

0

20

40

60

-60 -40 -20 0 20 40 60

σ11 (ksi)

σ22 (ksi)

EPS Global (RUC)

EPS 1st subcell

EPS All subcells

EPS subcell (1,2)

EPS subcell (2,1)

EPS subcell (2,2)

Center of EPS Global (RUC)

Figure 4.11 Global (RUC) yield surfaces for 0.25 fiber volume fraction SiC/Ti-21S at 650 °C with 

residual stresses.

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0

20

40

60

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σ11 (ksi)

σ22 (ksi)

EPS Global (RUC)

EPS 1st subcell

EPS All subcells

EPS subcell (1,2)

EPS subcell (2,1)

EPS subcell (2,2)

Figure 4.12 Local and global equivalent plastic strain (EPS) yield surfaces for 0.25 fiber volume 

fraction SiC/Ti-21S at 650 °C with no residual stresses. 

-60

-40

-20

0

20

40

60

-60 -40 -20 0 20 40 60

σ11 (ksi)

σ22 (ksi)

EPS Global (RUC)

EPS 1st subcell

EPS All subcells

EPS subcell (1,2)

EPS subcell (2,1)

EPS subcell (2,2)

Center of EPS Global (RUC)

Figure 4.13 Local and global equivalent plastic strain (EPS) yield surfaces for 0.25 fiber volume 

fraction SiC/Ti-21S at 650 °C with residual stresses.

NASA/TM—2002-212077/VOL3 140
Section 4: Applied Loading Example 4f: RUC Yield Surface Analysis 

-60

-40

-20

0

20

40

60

-60 -40 -20 0 20 40 60

σ11 (ksi)

σ22 (ksi)

EPS Global (no residual)

EPS All subcells (no residual)

EPS Global (with residual)

EPS All subcells (with residual)

Figure 4.14 Comparison of equivalent plastic strain (EPS) yield surfaces for 0.25 fiber volume fraction 

SiC/Ti-21S at 650 °C with and without residual stresses. 

-60

-40

-20

0

20

40

60

-60 -40 -20 0 20 40 60

σ11 (ksi)

σ22 (ksi)

ISR Global (no residual)

ISR All subcells (no residual)

ISR Global (with residual)

ISR All subcells (with residual)

Figure 4.15 Comparison of inelastic strain rate (ISR) yield surfaces for 0.25 fiber volume fraction 

SiC/Ti-21S at 650 °C with and without residual stresses.

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MAC/GMC 4.0 Example Problem Manual 

Figure 4.16 2×2 RUC representing the 0.25 fiber volume fraction SiC/Ti-21S composite in Example 4f. 

The subcell indices (β,γ) are indicated. 

NASA/TM—2002-212077/VOL3 142
Section 4: Applied Loading Example 4g: Laminate Yield Surface Analysis 

Example 4g: Laminate Yield Surface Analysis

This example problem illustrates the laminate yield surface capabilities within MAC/GMC 4.0. In 

particular, a cross-ply 0.25 fiber volume fraction SiC/Ti-21S laminate, with each layer represented by the 

7×7 circular fiber cross-section approximation RUC architecture, is examined. For the RUC yield surface 

analyses presented in Example 4f, the stress space was defined by global stress quantities. For laminate 

yield surface analysis, the global loading quantities are force resultants, which define the force resultant 

space for the yield surface. In accordance with the plane stress formulation of lamination theory, only inplane yield surfaces may be generated for laminates. Yield surfaces that involve bending of the laminate 

may not be generated. Additional options are available for laminate yield surface analysis as yield 

surfaces can be generated on the level of the laminate, the ply, and the individual subcells that constitute 

each ply. 

MAC/GMC Input File: example_4g.mac

MAC/GMC 4.0 Example 4f - Laminate yield surface analysis

*CONSTITUENTS

NMATS=2

M=1 CMOD=6 MATID=E

M=2 CMOD=4 MATID=A

*LAMINATE

NLY=3

LY=1 MOD=2 THK=0.25 ANG=0 ARCHID=6 VF=0.25 R=1. F=1 M=2

LY=2 MOD=2 THK=0.50 ANG=90 ARCHID=6 VF=0.25 R=1. F=1 M=2

LY=3 MOD=2 THK=0.25 ANG=0 ARCHID=6 VF=0.25 R=1. F=1 M=2

*SURF

TMAX=400. STP=0.05 MMAX=0.04 MODE=1 TREF=650.

OPTION=1,2,3,4,5,6,7

ISPX=1 ISPY=2 ANGINC=5.

EPS=0.0001 DR=0.00025 ISR=0.00002 IP=0.001

*PRINT

NPL=0

*END

Annotated Input Data

1) Flags: None 

2) Constituent materials (*CONSTITUENTS) [KM_2]: 

Number of materials: 2 (NMATS=2) 

Materials: SiC fiber (MATID=E) 

Ti-21S (MATID=A) 

Constitutive models: SiC fiber: linearly elastic (CMOD=6) 

Ti-21S matrix: Isotropic GVIPS (CMOD=4) 

3) Analysis type (*LAMINATE) → Laminate Analysis [KM_3]: 

Number of layers: 3 (NLY=3) 

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MAC/GMC 4.0 Example Problem Manual 

Layer Analysis 

Model 

Thickness Fiber 

Angle 

Architecture Aspect 

Ratio 

Volume 

fraction 

Fiber 

material 

Matrix 

material 

(LY=) (MOD) (THK) (ANG) (ARCHID) (R) (VF) (F) (M) 

1 GMC-2D 0.25 90° 7×7 circle, 

rect. pack 

1. 0.25 SiC Ti-21S 

2 GMC-2D 0.50 0° 7×7 circle, 

rect. pack 

1. 0.25 SiC Ti-21S 

3 GMC-2D 0.25 90° 7×7 circle, 

rect. pack 

1. 0.25 SiC Ti-21S 

4) Loading: 

a) Mechanical (*MECH) [KM_4]: no mechanical preloading 

b) Thermal (*THERM) [KM_4]: no thermal preloading 

c) Time integration (*SOLVER) [KM_4]: no preloading 

d) Yield surface generation (*SURF) [KM_4]: 

Maximum probing time: 400. sec. (TMAX=400.) 

Probing time step size: 0.05 sec. (STP=0.05) 

Maximum probing load: 0.04 (MMAX=0.04) 

Probing mode: strain control (MODE=1) 

Reference temperature: 650. °C (TREF=650.) 

Yield surface options: Generate global (laminate) data (1) (OPTION=1,2,3,4,5,6,7) 

 Generate 1st ply data (2) 

 Generate all plies data (3) 

 Generate data for each ply (4) 

 Generate 1st subcell data (5) 

 Generate all subcells data (6) 

 Generate local subcell data (7) 

Force resultant space x-axis: Nxx (ISPX=1) 

Force resultant space y-axis: Nyy (ISPY=2) 

Probe angle increment: 5.° (ANGINC=5) 

Quantities defining yield: Equivalent plastic strain = 0.0001 (EPS=0.0001) 

 Dissipation rate = 0.00025 (DR=0.00025) 

 Inelastic strain rate = 0.00002 (ISR=0.00002) 

 Inelastic power = 0.001 (IP=0.001) 

For laminate analysis, the yield surface option (*SURF) generates up to seven ASCII files that 

contain the yield surface data. surf_global.dat contains the global (i.e., laminate level) yield 

surface data for each type of yield surface (i.e., EPS, DR, ISR, and IP), surf_1st_ply.dat

contains the yield surface data based on first ply yield for each type of yield surface, 

surf_all_plies.dat contains the yield surface data based on the requirement that all plies (that 

can yield) yield for each type of yield surface, surf_plies.dat contains the yield surface data for 

each individual ply for each type of yield surface, surf_1st.dat contains the yield surface data 

NASA/TM—2002-212077/VOL3 144
Section 4: Applied Loading Example 4g: Laminate Yield Surface Analysis 

based on first subcell yield for each type of yield surface, surf_all.dat contains the yield surface 

data based on the requirement that all subcells (that can yield) yield for each type of yield surface, 

and surf_local.dat contains the yield surface data for each individual subcell for each type of 

yield surface. Thus, laminate analysis adds an additional level in the yield surface data hierarchy. 

! Note: In general, the quantities necessary for determining global (laminate level) yield are not present 

in standard lamination theory. That is, for instance, lamination theory calculates laminate level 

inelastic force resultants rather than laminate level inelastic strains, which are needed for 

evaluation of yielding. Therefore, for use within the yield calculations, MAC/GMC 4.0 

determines effective laminate level quantities when appropriate. 

5) Damage and Failure: None 

6) Output: 

a) Output file print level (*PRINT) [KM_6]: 

Print level: 0 (NPL=0) 

b) x-y plots (*XYPLOT): None 

7) End of file keyword: (*END) 

Results

Figure 4.17 presents the four global (laminate level) yield surfaces generated in this example. Because a 

[0°/90°]s laminate is simulated and residual stresses are not included, a particular yield surface crosses 

each axis at the same force resultant magnitude. In addition, the yield surfaces are less elongated than 

those that correspond to the continuous SiC/Ti-21S composite in Figure 4.10. 

Figure 4.18 shows additional equivalent plastic strain (EPS) yield surfaces. The yield surfaces generated 

based on yield of the 0° ply and the 90° ply intersect (much like the subcell-based yield surfaces intersect 

in Figure 4.12). The yield surface based on first ply yield traces the intersection of the individual ply 

curves, while the yield surface based on requiring yield of all plies traces the union. Also plotted in 

Figure 4.18 is the EPS yield surface based on first subcell yield. This is the smallest surface plotted 

because it is based on yielding of any subcell in any of the plies. Conversely, the yield surface based on 

requiring yielding of all subcells in all plies tends to be large. At certain locations, however, the global 

laminate yield surface is larger than the all subcells yield surface because the global yield surface is based 

on volume-average quantities, which include the 0.25 volume fraction fiber. Thus, while the subcellbased yield surfaces relate to the matrix material occupying the particular subcell, the laminate (and ply) 

level yield surfaces have averaged in the effect of the elastic fiber (which contributes zero plastic strain). 

Figure 4.19 shows the yield surfaces associated with each of the individual subcells within each layer. 

The first subcell yield surface and the all subcells yield surface automatically select the intersection and 

union, respectively, of the individual subcell yield surfaces. 

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MAC/GMC 4.0 Example Problem Manual 

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0

10

20

30

40

50

-50 -40 -30 -20 -10 0 10 20 30 40 50

Nxx (ksi × thickness)

Nyy (ksi

× thick

ness)

EPS Global

DR Global

ISR Global

IP Global

Figure 4.17 Global (laminate) yield surfaces for a 0.25 fiber volume fraction [0°/90°]s SiC/Ti-21S 

laminate. 

-50

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-20

-10

0

10

20

30

40

50

-50 -40 -30 -20 -10 0 10 20 30 40 50

Nxx (ksi × thickness)

Nyy (ksi

× thick

ness)

EPS Global

EPS 1st ply

EPS All Plies

EPS 0 ply

EPS 90 ply

EPS 1st subcell

EPS All subcells

Figure 4.18 Equivalent plastic strain (EPS) yield surfaces for a 0.25 fiber volume fraction [0°/90°]s

SiC/Ti-21S laminate.

NASA/TM—2002-212077/VOL3 146
Section 4: Applied Loading Example 4g: Laminate Yield Surface Analysis 

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0

10

20

30

40

50

-50 -40 -30 -20 -10 0 10 20 30 40 50

Nxx (ksi × thickness)

N

yy (ksi 

× thickness)

EPS 1st subcell

EPS All subcells

EPS 0 ply subcells

EPS 90 ply subcells

Figure 4.19 Subcell-based equivalent plastic strain (EPS) yield surfaces for a 0.25 fiber volume fraction 

[0°/90°]s SiC/Ti-21S laminate.

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MAC/GMC 4.0 Example Problem Manual 

Example 4h: Predictor-Corrector Time Integration

This example problem demonstrates the use of the predictor-corrector time integration method within 

MAC/GMC 4.0, which employs a self-adaptive time stepping scheme. This method is based on a fourthorder Runge-Kutta starter, an Adams-Bashforth four-step predictor, and an Adams-Moulton four-step 

corrector. Instead of specifying an explicit time step size (as in the forward Euler time integration 

method), the user specifies an acceptable level of error between the predicted and corrected values (for 

each local and global variable). The predictor-corrector integration algorithm then automatically adjusts 

the time step size in order to take the largest possible time step while ensuring that the error remains 

below the error tolerance. For additional information see Section 7.2 of the Theory Manual. The present 

problem considers a unidirectional 0.35 fiber volume fraction SiC/Ti-21S composite as represented by the 

7×7 circular fiber cross-section approximation RUC architecture. A stress-free cool-down is first applied 

to incorporate residual stresses (see Example 4b), followed by cyclic mechanical loading. 

MAC/GMC Input File: example_4h.mac

MAC/GMC 4.0 Example 4h - Predictor-corrector time integration method

*CONSTITUENTS

NMATS=2

M=1 CMOD=6 MATID=E

M=2 CMOD=4 MATID=A

*RUC

MOD=2 ARCHID=6 VF=0.35 R=1. F=1 M=2

*MECH

LOP=2 REFTIME=57600.

NPT=8 TI=0.,57600.,57630.,57650.,57690.,57710.,57770.,57790. &

MAG=0.,0.,0.015,0.,0.02,0.,0.03,0. MODE=2,1,2,1,2,1,2

*THERM

NPT=3 TI=0.,57600.,57790. TEMP=900.,23.,23.

*SOLVER

METHOD=2 ISTM=0.001 ISTT=0.1 ERR=0.001 MINSTEP=0.00001

*PRINT

NPL=3

*XYPLOT

FREQ=5

MACRO=3

NAME=example_4h X=2 Y=8

NAME=example_4h_th1 X=100 Y=1

NAME=example_4h_th2 X=100 Y=2

MICRO=0

*END

Annotated Input Data

1) Flags: None 

2) Constituent materials (*CONSTITUENTS) [KM_2]: 

Number of materials: 2 (NMATS=2) 

NASA/TM—2002-212077/VOL3 148
Section 4: Applied Loading Example 4h: Predictor-Corrector Time Integration 

Materials: SiC fiber (MATID=E) 

Ti-21S (MATID=A) 

Constitutive models: SiC fiber: linearly elastic (CMOD=6) 

Ti-21S matrix: Isotropic GVIPS (CMOD=4) 

3) Analysis type (*RUC) → Repeating Unit Cell Analysis [KM_3]: 

Analysis model: Doubly periodic GMC (MOD=2) 

RUC architecture: 7×7 circle approx., rect. pack (ARCHID=6) 

Fiber volume fraction: 0.35 (VF=0.35) 

RUC aspect ratio: 1. (square pack) (R=1.) 

Material assignment: SiC fiber (F=1) 

Ti-21S matrix (M=2) 

4) Loading: 

a) Mechanical (*MECH) [KM_4]: 

Loading option: 2 (LOP=2) 

Strain reference time: 57600. sec. (REFTIME=57600.) 

Number of points: 8 (NPT=8) 

Times (TI=) (sec.) 0. 57600. 57630. 57650. 57690. 57710. 57770. 57790. 

Magnitudes (MAG=) 0. 0 0.015 0.0 0.02 0. 0.03 0. 

Control (MODE=) stress strain stress strain stress strain stress 

In this example, the mechanical loading is applied in strain control and unloaded in stress control. 

This allows the stress to be precisely controlled so that it returns to zero after each cycle. 

b) Thermal (*THERM) [KM_4]: 

Number of points: 3 (NPT=3) 

Time points: 0., 57600., 57790. sec. (TI=0.,57600.,57790.) 

Temperature points: 900., 23., 23. (TEMP=900.,23.,23.) 

c) Time integration (*SOLVER) [KM_4]: 

Time integration method: Predictor-corrector (METHOD=2) 

Initial mech. time step: 0.001 sec. (ISTM=0.001) 

Initial thermal time step: 0.1 sec. (ISTT=0.1) 

Error tolerance: 0.001 (ERR=0.001) 

Minimum time step size: 0.00001 sec. (MINSTEP=0.00001) 

The initial mechanical (ISTM) and thermal (ISTT) time steps are used to start the time integration 

with the fourth-order Runge-Kutta method. The error tolerance (ERR) is maximum allowable 

fractional difference between a predicted and corrected value for any local or global variable. The 

minimum time step size (MINSTEP) limits how small the time step can become, regardless of the 

predicted and corrected variable values. As the results of this example indicate, the above values for 

these terms work well in the present case, but they can be case dependent. Users are thus encouraged 

to employ values for the terms based on experience with their own particular problems. For 

additional information of the predictor-corrector, see the MAC/GMC 4.0 Keywords Manual Section 4 

and the Theory Manual Section 7.2. 

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MAC/GMC 4.0 Example Problem Manual 

5) Damage and Failure: None 

6) Output: 

a) Output file print level (*PRINT) [KM_6]: 

Print level: 3 (NPL=3) 

b) x-y plots (*XYPLOT) [KM_6]: 

Frequency: 5 (FREQ=5) 

Number of macro plots: 3 (MACRO=3) 

Macro plot name: example_4h (NAME=example_4h) 

 example_4h_th1 (NAME=example_4h_th1) 

 example_4h_th2 (NAME=example_4h_th2) 

Macro plot x-y quantities: ε22, σ22 (X=2 Y=8) 

 temperature, ε11 (X=100 Y=1) 

 temperature, ε22 (X=100 Y=2) 

Number of micro plots: 0 (MICRO=0) 

7) End of file keyword: (*END) 

Results

Figure 4.20 shows the longitudinal and transverse strain induced in the SiC/Ti-21S composite as it is 

cooled from 900 °C to room temperature. The predictor corrector begins with a small initial time step, 

which soon increases as the temperature is dropping. At approximately 700 °C, the composite begins to 

experience inelastic deformation, causing the time step to become very small. At lower temperatures, the 

inelastic strain rate becomes smaller, and the time step is increased. 

Figure 4.21 shows the subsequent transverse cyclic tensile response of the composite. Again, as the 

mechanical loading begins, the time step is small. The composite behaves as initially elastic, thus the 

time step increases rapidly. As the composite begins to yield, the time step decreases, and subsequently 

increases several times. This is due to yielding of the individual subcells within the composite. Once a 

given subcell yields and begins to flow at a predictable rate, the predictor-corrector can increase the time 

step until another subcell starts to yield. The unloading and reloading behavior of the composite is 

mainly elastic in nature, so the time step increases for these segments of the stress-strain response. When 

the composite begins to yield for each cycle, the time step decreases once more. 

NASA/TM—2002-212077/VOL3 150
Section 4: Applied Loading Example 4h: Predictor-Corrector Time Integration 

-0.009

-0.008

-0.007

-0.006

-0.005

-0.004

-0.003

-0.002

-0.001

0.000

0 200 400 600 800 1000

Temperature (°C)

Strain

ε11

ε22

Figure 4.20 Example 4h: Stress-free cool-down response of 0.35 fiber volume fraction SiC/Ti-21S 

generated using the predictor-corrector time integration scheme.

0

50

100

150

200

250

300

0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035

ε22

σ22 (ksi)

Figure 4.21 Example 4h: Cyclic transverse tensile response of 0.35 fiber volume fraction SiC/Ti-21S at 

room temperature generated using the predictor-corrector time integration scheme. 

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MAC/GMC 4.0 Example Problem Manual 

Section 5 : Damage and Failure 

In this section, example problems are presented that invoke the failure and damage analysis capabilities 

resident in MAC/GMC 4.0. The code is capable of performing an array of failure and damage 

simulations with varying levels of complexity. The six keywords used to specify the failure and damage 

analysis input are: 

*ALLOWABLES

*FAILURE_SUBCELL

*FAILURE_CELL

*DAMAGE

*DEBOND

*CURTIN

The simplest (non-conservative) type of failure analysis is estimation of composite elastic allowable stress 

and strain values from stress and strain allowable data of the constituent materials. Calculation of these 

estimated allowables is efficient since they do not require application of any load history and are based 

solely on the linear elastic properties of the composite constituents. Static failure analysis, on the other 

hand, is inherently linked to applied loading histories in that the analysis constantly checks user specified 

failure criteria, on either the RUC or subcell level, during application of the simulated loading. Thus, if 

inelastic deformation or thermal strains are present in the composite, they will affect the static failure 

analysis results. 

Continuum fatigue damage analysis is distinct from allowables estimation and static failure analysis in 

that it involves simulation of the effects of cyclic loading. MAC/GMC 4.0 includes two fatigue damage 

models, one that involves stiffness degradation due to the cyclic loading (suitable for simulating the 

matrix phase), and one that involves strength degradation due to the cyclic loading (suitable for 

simulating the fiber phase). Fiber-matrix debonding and longitudinal fiber breakage are two additional 

failure/damage phenomena that can be modeled using MAC/GMC 4.0. 

All of the failure and damage capabilities of MAC/GMC 4.0 can be attributed to the ability of the 

generalized method of cells to localize the multi-axial state of stress and strain on a composite or laminate 

to stresses and strains on the material occupying each subcell. This local state of stress and strain can 

then be employed within a particular failure or damage model to determine the state of local failure or 

damage. Then, GMC homogenizes the local results so that the effect of the local failure or damage of a 

given constituent on the overall composite or laminate behavior can be determined. 

NASA/TM—2002-212077/VOL3 152
Section 5: Damage and Failure Example 5a: Composite Allowables Calculation 

Example 5a: Composite Allowables Calculation

MAC/GMC 4.0 includes the ability to determine elastic stress and strain allowables for a composite based 

on the stress and strain allowables for the constituents. The calculations are based on pure elastic 

concentration analyses in which, through GMC, the global stress or strain is determined that leads to the 

stress or strain allowable being reached locally. Because the calculated allowables are based on linear 

elastic calculations, include no residual stresses or damage, and have not been validated experimentally, 

they should be thought of as estimated values only. The user is strongly cautioned that the estimated 

allowables should not be interpreted as ìdesign allowablesî that are, by definition, certified via extensive 

comparison with experimental data. The estimated allowables calculated by the code should also be 

considered non-conservative. 

This problem considers a continuous 0.60 fiber volume fraction graphite/epoxy composite represented by 

the refined 26×26 circular fiber cross-section approximation RUC architecture. No thermal or mechanical 

loading is applied as the problem is only concerned with the determination of the allowables. 

MAC/GMC Input File: example_5a.mac

MAC/GMC 4.0 Example 5a - RUC allowables calculation

*CONSTITUENTS

NMATS=2

# -- Graphite fiber

M=1 CMOD=6 MATID=U MATDB=1

NTP=2

TEM=23.,150.

EA=388.2E3,390.E3

ET=7.6E3,7.6E3

NUA=0.41,0.41

NUT=0.45,0.45

GA=14.9E3,15.1E3

ALPA=-0.68E-6,-0.45E-6

ALPT=9.74E-6,10.34E-6

# -- Epoxy matrix

M=2 CMOD=6 MATID=U MATDB=1

NTP=2

TEM=23.,150.

EA=3.45E3,3.10E3

ET=3.45E3,3.10E3

NUA=0.35,0.35

NUT=0.35,0.35

GA=1.278E3,1.148E3

ALPA=45.E-6,55.E-6

ALPT=45.E-6,55.E-6

*RUC

MOD=2 ARCHID=13 VF=0.60 R=1. F=1 M=2

*ALLOWABLES

NMAT=2

MAT=1

S11=3500. S22=91.2 S33=91.2 S23=31.4 S13=134. S12=134. COMPR=SAM

E11=0.009 E22=0.012 E33=0.012 E23=0.012 E13=0.009 E12=0.009 COMPR=SAM

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MAC/GMC 4.0 Example Problem Manual 

MAT=2

S11=80. S22=80. S33=80. S23=40. S13=40. S12=40. COMPR=SAM

E11=0.023 E22=0.023 E33=0.023 E23=0.031 E13=0.031 E12=0.031 COMPR=SAM

*PRINT

NPL=-1

*END

Annotated Input Data

1) Flags: None 

2) Constituent materials (*CONSTITUENTS) [KM_2]: 

Number of materials: 2 (NMATS=2) 

Constitutive models: Elastic (CMOD=6) 

Materials: User-defined (Graphite) (MATID=U) 

 User-defined (Epoxy) (MATID=U) 

Material property source: Read from input file (MATDB=1) 

Material properties: See Table 4.1 

3) Analysis type (*RUC) → Repeating Unit Cell Analysis [KM_3]: 

Analysis model: Doubly periodic GMC (MOD=2) 

RUC architecture: 26×26 circular fiber approx., rect. pack (ARCHID=13) 

Fiber volume fraction: 0.60 (VF=0.60) 

Unit cell aspect ratio: 1.0 (square pack) (R=1.0) 

Material assignment: graphite fiber (F=1) 

epoxy matrix (M=2) 

4) Loading: None 

5) Damage and Failure: 

a) Stress and strain allowables estimation (*ALLOWABLES) [KM_5]: 

Number of materials: 2 (NMAT=2) 

Fiber stress allowables: σ11 = 3500. MPa (S11=3500.) 

 (MAT=1) σ22 = 91.2 MPa (S22=91.2) 

 σ33 = 91.2 MPa (S33=91.2) 

 σ23 = 31.4 MPa (S23=31.4) 

 σ13 = 134. MPa (S13=134.) 

 σ12 = 134. MPa (S12=134.) 

Fiber strain allowables: ε11 = 0.009 (E11=0.009) 

 ε22 = 0.012 (E22=0.012) 

 ε33 = 0.012 (E33=0.012) 

 γ23 = 0.012 (E23=0.012) 

 γ13 = 0.009 (E13=0.009) 

 γ12 = 0.009 (E12=0.009) 

Fiber compression flag: Compression same as tension (COMPR=SAM) 

Matrix stress allowables: σ11 = 80. MPa (S11=80.) 

 (MAT=2) σ22 = 80. MPa (S22=80.) 

NASA/TM—2002-212077/VOL3 154
Section 5: Damage and Failure Example 5a: Composite Allowables Calculation 

 σ33 = 80. MPa (S33=80.) 

 σ23 = 40. MPa (S23=40.) 

 σ13 = 40. MPa (S13=40.) 

 σ12 = 40. MPa (S12=40.) 

Matrix strain allowables: ε11 = 0.023 (E11=0.023) 

 ε22 = 0.023 (E22=0.023) 

 ε33 = 0.023 (E33=0.023) 

 γ23 = 0.031 (E23=0.031) 

 γ13 = 0.031 (E13=0.031) 

 γ12 = 0.031 (E12=0.031) 

Matrix compression flag: Compression same as tension (COMPR=SAM) 

As shown, stress and strain allowables may be specified for one or more materials within the 

composites. If allowables are specified for some, but not all constituent materials, the materials for 

which allowables were not specified will be skipped (i.e., not considered in the allowables 

calculation). Each of the six stress and strain allowable components may be specified for each 

material. If a particular component is not specified for a particular material, the calculation will skip 

that component for that material. A compression flag (COMPR) must also be specified for each 

material. This indicates how compression is handled for that material. COMPR=SAM indicates that 

the normal compressive allowables are of the same magnitude as the normal tensile allowables, 

COMPR=OFF indicates that code should not perform compressive allowables calculations, and 

COMPR=DIF indicates that the normal compressive allowables are of different magnitude than the 

normal tensile allowables. For this latter case, three compressive allowables may be specified as: 

SC11= SC22= SC33= and EC11= EC22= EC33= . The actual constituent 

allowable values employed in this problem are fictitious. For more information on the estimated 

allowables calculations, see the MAC/GMC 4.0 Keywords Manual Section 5. 

6) Output: 

a) Output file print level (*PRINT) [KM_6]: 

Print level: -1 (effective properties only) (NPL=-1) 

b) x-y plots (*XYPLOT): None 

7) End of file keyword: (*END) 

Results

Results from the estimated allowables calculations are written to the MAC/GMC 4.0 output file: 

* ESTIMATED ELASTIC STRESS ALLOWABLES *

*************************************************************

** CAUTION - These allowables have not been validated and **

** are based only on linear elastic calculations **

*************************************************************

--- 1st Subcell --- --- Avg Material --- - All Subcells -

ALLOWABLE CRITICAL ALLOWABLE CRITICAL ALLOWABLE

COMP STRESS SUBCELL STRESS MATERIAL STRESS

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MAC/GMC 4.0 Example Problem Manual 

---- ---------- -------- ---------- -------- ----------

1 0.2112E+04 325 0.2112E+04 1 0.5599E+04

2 0.6808E+02 13 0.8416E+02 1 0.1199E+03

3 0.6808E+02 339 0.8416E+02 1 0.1199E+03

4 0.3140E+02 39 0.3140E+02 1 0.4000E+02

5 0.2489E+02 313 0.5764E+02 2 0.3179E+03

6 0.2489E+02 13 0.5764E+02 2 0.3179E+03

---- ---------- --------- ---------- -------- ----------

1 -0.2112E+04 325 -0.2112E+04 1 -0.5599E+04

2 -0.6808E+02 13 -0.8416E+02 1 -0.1199E+03

3 -0.6808E+02 339 -0.8416E+02 1 -0.1199E+03

* ESTIMATED ELASTIC STRAIN ALLOWABLES *

*************************************************************

** CAUTION - These allowables have not been validated and **

** are based only on linear elastic calculations **

*************************************************************

--- 1st Subcell --- --- Avg Material --- - All Subcells -

ALLOWABLE CRITICAL ALLOWABLE CRITICAL ALLOWABLE

COMP STRAIN SUBCELL STRAIN MATERIAL STRAIN

---- ---------- -------- ---------- -------- ----------

1 0.9000E-02 39 0.9000E-02 1 0.2300E-01

2 0.1283E-01 325 0.1417E-01 1 0.2770E-01

3 0.1283E-01 351 0.1417E-01 1 0.2770E-01

4 0.1704E-01 39 0.1704E-01 1 0.2147E-01

5 0.6549E-02 313 0.1517E-01 2 0.8454E-01

6 0.6549E-02 13 0.1517E-01 2 0.8454E-01

---- ---------- --------- ---------- -------- ----------

1 -0.9000E-02 39 -0.9000E-02 1 -0.2300E-01

2 -0.1283E-01 325 -0.1417E-01 1 -0.2770E-01

3 -0.1283E-01 351 -0.1417E-01 1 -0.2770E-01

NOTE: 1st subcell --> based on 1st subcell reaching specified constituent allowable

Avg material --> based on 1st material whose avg stress/strain reaches allowable

All subcells --> based on all subcells reaching specified constituent allowable

Three sets of results are generated for the estimated composite stress and strain allowables. ì1st Subcellî 

allowables are determined based on any subcell within the RUC reaching the specified allowable value 

for the material within the subcell. This is the most conservative estimated allowable as, due to stress 

concentrations, certain subcells may locally reach the allowable at a low global stress or strain level. An 

intermediate ìAvg Materialî allowable is determined by finding the global stress or strain at which the 

average stress or strain for each constituent material reaches the specified allowable for that material. 

Finally, the most liberal (non-conservative) allowable estimate (ìAll Subcellsî) is determined by 

requiring every subcell within the RUC to reach its allowable stress or strain. For the present example, 

the composite compressive results are identical (in magnitude) to the tensile results as, for both 

constituent materials, the compressive allowables are the same as the tensile allowables. 

The results from this example problem are presented in graphical form in Figure 5.1 and Figure 5.2. The 

independent estimated allowable components have been normalized by the corresponding specified fiber 

allowable values. For comparison, the normalized values specified for the matrix material have been 

plotted as well. Figure 5.1 and Figure 5.2 clearly show that the average material allowables are bounded 

by the 1st subcell and all subcells allowables. Further, in all cases, the composite average material 

allowable falls in between the specified matrix and fiber allowables. This is not the case for the 1st

subcell allowable, which falls below both constituent allowables for σ22, σ12, and γ12. Likewise, the all 

subcells allowable is greater than the allowable for both constituents for σ11, σ22, σ12, ε22, and γ12. Clearly, 

the all subcells stress allowables are unrealistically high as they require every point in the composite to 

reach the local allowable stress. Obviously, global failure occurs much sooner in real-world structures 

after only a few local failures initiate. 

NASA/TM—2002-212077/VOL3 156
Section 5: Damage and Failure Example 5a: Composite Allowables Calculation 

0.0

0.5

1.0

1.5

2.0

2.5

σ11 σ22 σ23 σ12

Component

N

ormalized Stress Allo

wable

Pure Matrix

Composite - 1st Subcell

Composite - Avg Material

Composite - All Subcells

Figure 5.1 Example 5a: Specified matrix and calculated composite stress allowables normalized by the 

fiber stress allowables for 60% unidirectional graphite/epoxy. 

0

1

2

3

4

5

6

7

8

9

10

ε11 ε22 γ23 γ12

Component

N

ormalized Strain Allowable

Pure Matrix

Composite - 1st Subcell

Composite - Avg Material

Composite - All Subcells

Figure 5.2 Example 5a: Specified matrix and calculated composite strain allowables normalized by the 

fiber strain allowables for 60% unidirectional graphite/epoxy.

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MAC/GMC 4.0 Example Problem Manual 

Example 5b: Subcell Static Failure Analysis 

In MAC/GMC 4.0, static failure analysis is completely different from the allowables estimation presented 

in Example 5a. Since the composite allowables are estimated based on a quick elastic concentration 

analysis, composite static failure analysis is associated with an entire simulated loading history. During 

each increment of the applied loading, the specified RUC or subcell failure criteria are checked to 

determine if failure has occurred. Thus, inelasticity, residual stresses, and even damage can be 

incorporated within a static failure simulation, giving such a simulation the potential to be considerably 

more accurate than an allowables estimation. Unlike the estimated allowables, however, a failure stress 

determined via a MAC/GMC 4.0 simulation is not a property-like quantity for a particular composite. 

The value can change dramatically based on the loading history applied. Note that the identical 

graphite/epoxy composite employed in Example 5a is employed in the present example. Now, however, 

simulated thermal and mechanical loading is applied. 

MAC/GMC Input File: example_5b.mac

MAC/GMC 4.0 Example 5b - Subcell static failure analysis

*CONSTITUENTS

NMATS=2

# -- Graphite fiber

M=1 CMOD=6 MATID=U MATDB=1

NTP=2

TEM=23.,150.

EA=388.2E3,390.E3

ET=7.6E3,7.6E3

NUA=0.41,0.41

NUT=0.45,0.45

GA=14.9E3,15.1E3

ALPA=-0.68E-6,-0.45E-6

ALPT=9.74E-6,10.34E-6

# -- Epoxy matrix

M=2 CMOD=6 MATID=U MATDB=1

NTP=2

TEM=23.,150.

EA=3.45E3,3.10E3

ET=3.45E3,3.10E3

NUA=0.35,0.35

NUT=0.35,0.35

GA=1.278E3,1.148E3

ALPA=45.E-6,55.E-6

ALPT=45.E-6,55.E-6

*RUC

MOD=2 ARCHID=13 VF=0.60 R=1. F=1 M=2

*MECH

LOP=2

NPT=2 TI=0.,100. MAG=0.,0.02 MODE=1

# LOP=2 REFTIME=100.

# NPT=3 TI=0.,100.,200. MAG=0.,0.,0.02 MODE=2,1

*THERM

NPT=2 TI=0.,100. TEMP=23.,23.

# NPT=3 TI=0.,100.,200. TEMP=150.,23.,23.

NASA/TM—2002-212077/VOL3 158
Section 5: Damage and Failure Example 5b: Subcell Static Failure Analysis 

*SOLVER

METHOD=1 NPT=2 TI=0.,100. STP=0.1

# METHOD=1 NPT=3 TI=0.,100.,200. STP=5.,0.1

*ALLOWABLES

NMAT=2

MAT=1

S11=3500. S22=91.2 S33=91.2 S23=31.4 S13=134. S12=134. COMPR=SAM

E11=0.009 E22=0.012 E33=0.012 E23=0.012 E13=0.009 E12=0.009 COMPR=SAM

MAT=2

S11=80. S22=80. S33=80. S23=40. S13=40. S12=40. COMPR=SAM

E11=0.023 E22=0.023 E33=0.023 E23=0.031 E13=0.031 E12=0.031 COMPR=SAM

*FAILURE_SUBCELL

NMAT=2

MAT=1 NCRIT=2

CRIT=1 X11=3500. X22=91.2 X33=91.2 X23=31.4 X13=134. X12=134. &

COMPR=OFF ACTION=1

CRIT=2 X11=0.009 X22=0.012 X33=0.012 X23=0.012 X13=0.009 X12=0.009 &

COMPR=OFF ACTION=1

MAT=2 NCRIT=2

CRIT=1 X11=80. X22=80. X33=80. X23=40. X13=40. X12=40. &

COMPR=OFF ACTION=1

CRIT=2 X11=0.023 X22=0.023 X33=0.023 X23=0.031 X13=0.031 X12=0.031 &

COMPR=OFF ACTION=1

*PRINT

NPL=6

*XYPLOT

FREQ=1

MACRO=1

NAME=example_5b X=2 Y=8

MICRO=0

*END

Annotated Input Data

1) Flags: None 

2) Constituent materials (*CONSTITUENTS) [KM_2]: 

Number of materials: 2 (NMATS=2) 

Constitutive models: Elastic (CMOD=6) 

Materials: User-defined (Graphite) (MATID=U) 

 User-defined (Epoxy) (MATID=U) 

Material property source: Read from input file (MATDB=1) 

Material properties: See Table 4.1 

3) Analysis type (*RUC) → Repeating Unit Cell Analysis [KM_3]: 

Analysis model: Doubly periodic GMC (MOD=2) 

RUC architecture: 26×26 circular fiber approx., rect. pack (ARCHID=13) 

Fiber volume fraction: 0.60 (VF=0.60) 

Unit cell aspect ratio: 1.0 (square pack) (R=1.0) 

Material assignment: graphite fiber (F=1) 

epoxy matrix (M=2) 

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MAC/GMC 4.0 Example Problem Manual 

4) Loading: 

a) Mechanical (*MECH) [KM_4]: 

Loading option: 2 (LOP=2) 

Without residual

Number of points: 2 (NPT=2) 

Time points: 0., 100. sec. (TI=0.,100.) 

Load magnitudes: 0., 0.02 (MAG=0.,0.02) 

Loading mode: strain control (MODE=1) 

With residual

Reference strain time: 100. sec. (REFTIME=100.) 

Number of points: 3 (NPT=3) 

Time points: 0., 100., 200. sec. (TI=0.,100.,200.) 

Load magnitudes: 0., 0.02 (MAG=0.,0.,0.02) 

Loading mode: stress control, strain control (MODE=2,1) 

! Note: The appropriate lines must be commented and uncommented to execute with and without 

residual. 

b) Thermal (*THERM) [KM_4]: 

Without residual

Number of points: 2 (NPT=2) 

Time points: 0., 100. sec. (TI=0.,100.) 

Temperature points: 23., 23. (TEMP=23.,23.) 

With residual

Number of points: 3 (NPT=3) 

Time points: 0., 100., 200. sec. (TI=0.,100.,200.) 

Temperature points: 150., 23., 23. (TEMP=150.,23.,23.) 

! Note: The appropriate lines must be commented and uncommented to execute with and without 

residual. 

c) Time integration (*SOLVER) [KM_4]: 

Time integration method: Forward Euler (METHOD=1) 

Without residual

Number of points: 2 (NPT=2) 

Time points: 0., 100. sec. (TI=0.,100.) 

Time step size: 0.1 sec. (STP=0.1) 

With residual

Number of points: 3 (NPT=3) 

Time points: 0., 100., 200. sec. (TI=0.,100.,200.) 

Time step sizes: 5., 0.1 sec. (STP=5.,0.1) 

NASA/TM—2002-212077/VOL3 160
Section 5: Damage and Failure Example 5b: Subcell Static Failure Analysis 

! Note: The small step size is employed during the mechanical loading despite the elastic response of 

the composite is employed in order to detect failure as soon as it occurs. The appropriate lines 

must be commented and uncommented to execute with and without residual. 

5) Damage and Failure: 

a) Stress and strain allowables estimation (*ALLOWABLES) [KM_5]: 

Number of materials: 2 (NMAT=2) 

Fiber stress allowables: σ11 = 3500. MPa (S11=3500.) 

 (MAT=1) σ22 = 91.2 MPa (S22=91.2) 

 σ33 = 91.2 MPa (S33=91.2) 

 σ23 = 31.4 MPa (S23=31.4) 

 σ13 = 134. MPa (S13=134.) 

 σ12 = 134. MPa (S12=134.) 

Fiber strain allowables: ε11 = 0.009 (E11=0.009) 

 ε22 = 0.012 (E22=0.012) 

 ε33 = 0.012 (E33=0.012) 

 γ23 = 0.012 (E23=0.012) 

 γ13 = 0.009 (E13=0.009) 

 γ12 = 0.009 (E12=0.009) 

Fiber compression flag: Compression same as tension (COMPR=SAM) 

Matrix stress allowables: σ11 = 80. MPa (S11=80.) 

 (MAT=2) σ22 = 80. MPa (S22=80.) 

 σ33 = 80. MPa (S33=80.) 

 σ23 = 40. MPa (S23=40.) 

 σ13 = 40. MPa (S13=40.) 

 σ12 = 40. MPa (S12=40.) 

Matrix strain allowables: ε11 = 0.023 (E11=0.023) 

 ε22 = 0.023 (E22=0.023) 

 ε33 = 0.023 (E33=0.023) 

 γ23 = 0.031 (E23=0.031) 

 γ13 = 0.031 (E13=0.031) 

 γ12 = 0.031 (E12=0.031) 

Matrix compression flag: Compression same as tension (COMPR=SAM) 

b) Subcell static failure analysis (*FAILURE_SUBCELL) [KM_5]: 

Number of materials: 2 (NMAT=2) 

Material #1 (MAT=1) 

Number of criteria: 2 (NCRIT=2) 

Criterion #1: Maximum stress criterion (CRIT=1) 

Failure stresses: σ11 = 3500. MPa (X11=3500.) 

 σ22 = 91.2 MPa (X22=91.2) 

 σ33 = 91.2 MPa (X33=91.2) 

 σ23 = 31.4 MPa (X23=31.4) 

NASA/TM—2002-212077/VOL3 161
MAC/GMC 4.0 Example Problem Manual 

 σ13 = 134. MPa (X13=134.) 

 σ12 = 134. MPa (X12=134.) 

Criterion #2: Maximum strain criterion (CRIT=2) 

Failure strains: ε11 = 0.009 (X11=0.009) 

 ε22 = 0.012 (X22=0.012) 

 ε33 = 0.012 (X33=0.012) 

 γ23 = 0.012 (X23=0.012) 

 γ13 = 0.009 (X13=0.009) 

 γ12 = 0.009 (X12=0.009) 

Compression flag: Do not check for compressive failure (COMPR=OFF) 

Action to take upon failure: Zero subcell stiffness and continue (ACTION=1) 

Material #2 (MAT=2) 

Number of criteria: 2 (NCRIT=2) 

Criterion #1: Maximum stress criterion (CRIT=1) 

Failure stresses: σ11 = 80. MPa (X11=80.) 

 σ22 = 80. MPa (X22=80.) 

 σ33 = 80. MPa (X33=80.) 

 σ23 = 40. MPa (X23=40.) 

 σ13 = 40. MPa (X13=40.) 

 σ12 = 40. MPa (X12=40.) 

Criterion #2: Maximum strain criterion (CRIT=2) 

Failure strains: ε11 = 0.023 (X11=0.023) 

 ε22 = 0.023 (X22=0.023) 

 ε33 = 0.023 (X33=0.023) 

 γ23 = 0.031 (X23=0.031) 

 γ13 = 0.031 (X13=0.031) 

 γ12 = 0.031 (X12=0.031) 

Compression flag: Do not check for compressive failure (COMPR=OFF) 

Action to take upon failure: Zero subcell stiffness and continue (ACTION=1) 

MAC/GMC 4.0 includes three failure criteria: maximum stress (CRIT=1), maximum strain 

(CRIT=2), and Tsai-Hill (CRIT=3). For all three criteria, the input takes the same form where the 

critical failure values for each component are specified as Xij=. As in the allowables calculation, 

the subcell static failure analysis requires a flag to indicate how to handle compression (COMPR=) 

(this should be omitted in the case of the Tsai-Hill Criterion). In addition, subcell static failure 

analysis requires specification of the action to take upon detection of a subcell failure (ACTION=). A 

value of -1 indicates that the code should stop the execution upon 1st failure, a value of 0 indicates 

that the code should only write notification of the failure to the output file and continue, and a value 

of 1 indicates that the code should treat the subcell as completely damage (i.e., zero the subcellís 

stiffness) and continue the execution. For more information on subcell static failure analysis, see the 

MAC/GMC 4.0 Theory Manual Section 5.2 and the MAC/GMC 4.0 Keywords Manual Section 5. 

6) Output: 

a) Output file print level (*PRINT) [KM_6]: 

Print level: 6 (NPL=6) 

NASA/TM—2002-212077/VOL3 162
Section 5: Damage and Failure Example 5b: Subcell Static Failure Analysis 

b) x-y plots (*XYPLOT): 

Frequency: 1 (FREQ=1) 

Number of macro plots: 1 (MACRO=1) 

Macro plot names: example_5b (NAME=example_5b) 

Macro plot x-y quantities: ε22, σ22 (X=2 Y=8) 

Number of micro plots: 0 (MICRO=0) 

7) End of file keyword: (*END) 

Results

The results for Example Problem 5b are shown in Figure 5.3 in the form of the transverse stress-strain 

response of the composite with and without residual stresses included. Without residual stresses, the 

failure of the first matrix subcell occurs at 68.1 MPa (due to the maximum stress criterion), which 

corresponds to the 1st subcell transverse allowable stress estimated for the composite in Example 5a. 

Because ACTION was set to 1, the subcell(s) that failed at this point have their stiffnesses set to zero, 

which causes the dramatic drop in the stress-strain response evident in Figure 5.3. Further, the following 

notification of subcell failure is written to the output file: 

******* SUBCELL FAILURE ******

SUBCELL # 13 MATERIAL # 2

FAILED DUE TO THE MAX STRESS CRITERION

COMPONENT = 2

STRESS = 80.0299221493631

ALLOWABLE STRESS = 80.0000000000000

ACTION = 1 ----> ZERO SUBCELL AND CONTINUE

******* SUBCELL FAILURE ******

SUBCELL # 14 MATERIAL # 2

FAILED DUE TO THE MAX STRESS CRITERION

COMPONENT = 2

STRESS = 80.0299221493631

ALLOWABLE STRESS = 80.0000000000000

ACTION = 1 ----> ZERO SUBCELL AND CONTINUE

******* SUBCELL FAILURE ******

SUBCELL # 663 MATERIAL # 2

FAILED DUE TO THE MAX STRESS CRITERION

COMPONENT = 2

STRESS = 80.0299221493631

ALLOWABLE STRESS = 80.0000000000000

ACTION = 1 ----> ZERO SUBCELL AND CONTINUE

******* SUBCELL FAILURE ******

SUBCELL # 664 MATERIAL # 2

FAILED DUE TO THE MAX STRESS CRITERION

COMPONENT = 2

STRESS = 80.0299221493631

ALLOWABLE STRESS = 80.0000000000000

ACTION = 1 ----> ZERO SUBCELL AND CONTINUE

So clearly, four subcells have failed due to the maximum stress criterion. Because the simulated 

transverse loading is applied under strain control, it can continue despite the drop in the stress-strain 

response. These drops are due to the instantaneous ìzeroingî of the failed subcells stiffness properties. 

Less dramatic drops would be present were this problem analyzed using HFGMC. As the loading 

continues in Figure 5.3, additional subcell failures occur, resulting in more drops in the composite 

transverse response. Clearly, the predicted response without residual stresses reaches its maximum stress 

just prior to the 1st subcell failure, and as such, this is the predicted composite transverse UTS. 

NASA/TM—2002-212077/VOL3 163
MAC/GMC 4.0 Example Problem Manual 

Also plotted in Figure 5.3 is the predicted transverse response of the graphite/epoxy composite in which 

residual stresses have been included via a simulated cool down from 150 °C. This is intended to simulate 

the cure temperature of the composite. Now, even though all subcell failure parameters are identical to 

the case without residual, 1st subcell failure is delayed until the composite reaches a global stress of 77.4 

MPa. This is because during the simulated cool down, the matrix is placed in a state of residual 

transverse compression. Then, upon application of the simulated transverse mechanical loading, this 

transverse residual compression must first be overcome in addition to loading the matrix subcells to their 

transverse tensile strength of 80 MPa. Note that, even when residual stresses are included, the estimated 

allowables for the composite are identical to those presented in Example 5a as the allowables are 

calculated as a first step in the codeís execution and are not linked to the applied loading. In this 

example, however, it is clear that the MAC/GMC 4.0 subcell static failure analysis is linked to the applied 

loading. For identical failure parameters, the simulations with and without residual stresses differ by 9.3 

MPa (13.7%) in their transverse UTS predictions for the graphite/epoxy composite. It should be noted 

that, without micromechanics, this effect could not be predicted. 

0

10

20

30

40

50

60

70

80

90

0.000 0.005 0.010 0.015 0.020 0.025

ε22

σ22 (M

Pa)

With Residual

Without Residual

Predicted transverse 

UTS with residual 

= 77.4 MPa

Predicted transverse 

UTS without residual 

= 68.1 MPa

Figure 5.3 Example 5b: Simulated transverse static failure response of unidirection 60% 

graphite/epoxy at room temperature with and without residual stresses included. 

NASA/TM—2002-212077/VOL3 164
Section 5: Damage and Failure Example 5c: Laminate Static Failure Analysis 

Example 5c: Laminate Static Failure Analysis

In this example problem, a 60% fiber volume fraction graphite/epoxy composite is again considered, as in 

Examples 5a and 5b. However, now a quasi-isotropic [0°/45°/-45°/90°]s laminate is the subject of static 

failure analysis utilizing both the subcell and RUC static failure analysis options. Biaxial loading (NxxNyy) is applied to the laminate, and by altering the load magnitudes applied in the two directions, a plot of 

the predicted laminate ìfailure surfaceî is generated. This ìfailure surfaceî is the locus of points in the 

plane of the applied biaxial stress loading at which failure occurs. Unlike the MAC/GMC 4.0 yield 

surface capabilities, this failure surface analysis procedure is not automated at this time. 

MAC/GMC Input File: example_5c.mac

MAC/GMC 4.0 Example 5c - Laminate static failure analysis

*CONSTITUENTS

NMATS=2

# -- Graphite fiber

M=1 CMOD=6 MATID=U MATDB=1

NTP=2

TEM=23.,150.

EA=388.2E3,390.E3

ET=7.6E3,7.6E3

NUA=0.41,0.41

NUT=0.45,0.45

GA=14.9E3,15.1E3

ALPA=-0.68E-6,-0.45E-6

ALPT=9.74E-6,10.34E-6

# -- Epoxy matrix

M=2 CMOD=6 MATID=U MATDB=1

NTP=2

TEM=23.,150.

EA=3.45E3,3.10E3

ET=3.45E3,3.10E3

NUA=0.35,0.35

NUT=0.35,0.35

GA=1.278E3,1.148E3

ALPA=45.E-6,55.E-6

ALPT=45.E-6,55.E-6

*LAMINATE

NLY=7

LY=1 THK=0.125 ANG=0 MOD=2 ARCHID=13 VF=0.60 R=1. F=1 M=2

LY=2 THK=0.125 ANG=45 MOD=2 ARCHID=13 VF=0.60 R=1. F=1 M=2

LY=3 THK=0.125 ANG=-45 MOD=2 ARCHID=13 VF=0.60 R=1. F=1 M=2

LY=4 THK=0.250 ANG=90 MOD=2 ARCHID=13 VF=0.60 R=1. F=1 M=2

LY=5 THK=0.125 ANG=-45 MOD=2 ARCHID=13 VF=0.60 R=1. F=1 M=2

LY=6 THK=0.125 ANG=45 MOD=2 ARCHID=13 VF=0.60 R=1. F=1 M=2

LY=7 THK=0.125 ANG=0 MOD=2 ARCHID=13 VF=0.60 R=1. F=1 M=2

*MECH

LOP=99

NPT=2 TI=0.,100. MAG=0.,2000 MODE=2

NPT=2 TI=0.,100. MAG=0.,2000 MODE=2

NPT=2 TI=0.,100. MAG=0.,0. MODE=2

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MAC/GMC 4.0 Example Problem Manual 

NPT=2 TI=0.,100. MAG=0.,0. MODE=2

NPT=2 TI=0.,100. MAG=0.,0. MODE=2

NPT=2 TI=0.,100. MAG=0.,0. MODE=2

*THERM

NPT=2 TI=0.,100. TEMP=23.,23.

*SOLVER

METHOD=1 NPT=2 TI=0.,100. STP=0.1

*FAILURE_SUBCELL

NMAT=2

MAT=1 NCRIT=1

CRIT=3 X11=3500. X22=91.2 X33=91.2 X23=31.4 X13=134. X12=134. &

ACTION=-1

MAT=2 NCRIT=1

CRIT=3 X11=80. X22=80. X33=80. X23=40. X13=40. X12=40. &

ACTION=-1

#*FAILURE_CELL

# NCRIT=1

# -- Data from 1st subcell allowables (Example 5a)

# CRIT=3 X11=2112. X22=68.08 X33=68.08 X23=31.4 X13=24.89 X12=24.89 &

# ACTION=-1

# -- Data from avg material allowables (Example 5a)

# CRIT=3 X11=2112. X22=84.16 X33=84.16 X23=31.4 X13=57.64 X12=57.64 &

# ACTION=-1

*PRINT

NPL=6

*END

Annotated Input Data

1) Flags: None 

2) Constituent materials (*CONSTITUENTS) [KM_2]: 

Number of materials: 2 (NMATS=2) 

Constitutive models: Elastic (CMOD=6) 

Materials: User-defined (Graphite) (MATID=U) 

 User-defined (Epoxy) (MATID=U) 

Material property source: Read from input file (MATDB=1) 

Material properties: See Table 4.1 

3) Analysis type (*LAMINATE) → Laminate Analysis [KM_3]: 

Number of layers: 7 (NLY=7) 

Layer Analysis 

Model 

Thickness Fiber 

Angle 

Architecture Volume 

fraction 

Aspect 

ratio 

Fiber 

material 

Matrix 

material 

(LY=) (MOD) (THK) (ANG) (ARCHID) (VF) (R) (F) (M) 

1 GMC-2D 0.125 0° 26×26 circle 

rect. pack 

0.60 1. graphite epoxy 

2 GMC-2D 0.125 45° 26×26 circle 

rect. pack 

0.60 1. graphite epoxy 

3 GMC-2D 0.125 -45° 26×26 circle 

rect. pack 

0.60 1. graphite epoxy 

NASA/TM—2002-212077/VOL3 166
Section 5: Damage and Failure Example 5c: Laminate Static Failure Analysis 

4 GMC-2D 0.25 90° 26×26 circle 

rect. pack 

0.60 1. graphite epoxy 

5 GMC-2D 0.125 -45° 26×26 circle 

rect. pack 

0.60 1. graphite epoxy 

6 GMC-2D 0.125 45° 26×26 circle 

rect. pack 

0.60 1. graphite epoxy 

7 GMC-2D 0.125 0° 26×26 circle 

rect. pack 

0.60 1. graphite epoxy 

4) Loading: 

a) Mechanical (*MECH) [KM_4]: 

Loading option: general loading (LOP=99) 

Component #1 ( 0

xx ε or Nxx ) 

Number of points: 2 (NPT=2) 

Times (TI=) (sec.) 0. 100. 

Magnitudes (MAG=) 0. 2000. MPa × thickness 

Control (MODE=) force resultant 

Component #2 ( 0

yy ε or Nyy ) 

Number of points: 2 (NPT=2) 

Times (TI=) (sec.) 0. 100. 

Magnitudes (MAG=) 0. 2000. MPa × thickness 

Control (MODE=) 

Component #3 ( 0

xy γ or Nxy ) 

Number of points: 2 (NPT=2) 

Times (TI=) (sec.) 0. 100. 

Magnitudes (MAG=) 0. 0. 

Control (MODE=) force resultant 

Component #4 (κ xx or M xx ) 

Number of points: 2 (NPT=2) 

Times (TI=) (sec.) 0. 100. 

Magnitudes (MAG=) 0. 0. 

Control (MODE=) moment resultant 

Component #5 (κ yy or M yy ) 

Number of points: 2 (NPT=2) 

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 force resultant 
MAC/GMC 4.0 Example Problem Manual 

Times (TI=) (sec.) 0. 100. 

Magnitudes (MAG=) 0. 0. 

Control (MODE=) moment resultant 

Component #6 (κ xy or M xy ) 

Number of points: 2 (NPT=2) 

Times (TI=) (sec.) 0. 100. 

Magnitudes (MAG=) 0. 0. 

Control (MODE=) moment resultant 

! Note: To generate the ìfailure surfaceî shown in the results for this example problem, the magnitudes 

of the component #1 and #2 applied loads must be adjusted accordingly. For example, 

applying Nxx=0 and Nyy=2000 MPa × thickness will allow the determination of the failure points 

on the positive Nyy -axis in Figure 5.4. 

b) Thermal (*THERM) [KM_4]: 

Number of points: 2 (NPT=2) 

Time points: 0., 100. sec. (TI=0.,100.) 

Temperature points: 23., 23. (TEMP=23.,23.) 

c) Time integration (*SOLVER) [KM_4]: 

Time integration method: Forward Euler (METHOD=1) 

Number of points: 2 (NPT=2) 

Time points: 0., 100. sec. (TI=0.,100.) 

Time step size: 0.1 sec. (STP=0.1) 

5) Damage and Failure: 

a) Subcell static failure analysis (*FAILURE_SUBCELL) [KM_5]: 

Number of materials: 2 (NMAT=2) 

Material #1 (MAT=1) 

Number of criteria: 1 (NCRIT=1) 

Criterion #1: Tsai-Hill criterion (CRIT=3) 

Failure stresses: σ11 = 3500. MPa (X11=3500.) 

 σ22 = 91.2 MPa (X22=91.2) 

 σ33 = 91.2 MPa (X33=91.2) 

 σ23 = 31.4 MPa (X23=31.4) 

 σ13 = 134. MPa (X13=134.) 

 σ12 = 134. MPa (X12=134.) 

Action to take upon failure: Stop execution (ACTION=-1) 

Material #2 (MAT=2) 

Number of criteria: 1 (NCRIT=1) 

Criterion #1: Tsai-Hill criterion (CRIT=3) 

Failure stresses: σ11 = 80. MPa (X11=80.) 

 σ22 = 80. MPa (X22=80.) 

NASA/TM—2002-212077/VOL3 168
Section 5: Damage and Failure Example 5c: Laminate Static Failure Analysis 

 σ33 = 80. MPa (X33=80.) 

 σ23 = 40. MPa (X23=40.) 

 σ13 = 40. MPa (X13=40.) 

 σ12 = 40. MPa (X12=40.) 

Action to take upon failure: Stop execution (ACTION=-1) 

b) RUC static failure analysis (*FAILURE_CELL) [KM_5]: 

Number of criteria: 1 (NCRIT=1) 

Criterion #1: Tsai-Hill criterion (CRIT=3) 

Data From 1st Subcell Allowables for Composite

Failure stresses: σ11 = 2112. MPa (X11=2112.) 

 σ22 = 68.08 MPa (X22=68.08) 

 σ33 = 68.08 MPa (X33=68.08) 

 σ23 = 31.4 MPa (X23=31.4) 

 σ13 = 24.89 MPa (X13=24.89) 

 σ12 = 24.89 MPa (X12=24.89) 

Data From Avg. Material Allowables for Composite

Failure stresses: σ11 = 2112. MPa (X11=2112.) 

 σ22 = 84.16 MPa (X22=84.16) 

 σ33 = 84.16 MPa (X33=84.16) 

 σ23 = 31.4 MPa (X23=31.4) 

 σ13 = 57.64 MPa (X13=57.64) 

 σ12 = 57.64 MPa (X12=57.64) 

Action to take upon failure: Stop execution (ACTION=-1) 

The RUC level failure analysis input data is similar to the input data for subcell static failure analysis. 

As with the subcell static failure capabilities, MAC/GMC 4.0 includes the maximum stress and strain 

criteria, as well as the Tsai-Hill criterion within its RUC static failure capabilities. In this example 

problem, two sets of RUC static failure parameters are included within the input file (both of which 

are commented in the input file above). These two sets of data were taken from the allowables 

estimated for the unidirectional graphite/epoxy composite (which occupies each of the seven plies in 

the present example) in Example Problem 5a. One set corresponds to the 1st subcell failure allowable 

stresses, while the other corresponds to the average material failure allowable stresses. 

In order to execute the code to generate all of the results shown below, the appropriate lines in the 

input file must be commented and uncommented such that only one RUC static failure criterion is 

active with the subcell failure criteria inactive or such that only the subcell failure criteria are active. 

6) Output: 

a) Output file print level (*PRINT) [KM_6]: 

Print level: 6 (NPL=6) 

b) x-y plots (*XYPLOT): None 

7) End of file keyword: (*END) 

NASA/TM—2002-212077/VOL3 169
MAC/GMC 4.0 Example Problem Manual 

Results

For the case that utilizes the subcell static failure criterion and identical σ11 and σ22 mechanical loading 

rates (as shown in the input file above), the laminate reaches 1st subcell failure at an applied force 

resultant level of Nxx = Nyy = 774 MPa × thickness. At this point the code stops execution in accordance 

with the specified ACTION=-1, and the following is written to the output file: 

******* SUBCELL FAILURE ******

LAYER # 1 INTEGRATION POINT #1

SUBCELL # 325 MATERIAL # 1

FAILED DUE TO THE TSAI-HILL CRITERION

STRESSES: 0.2470E+04 0.6041E+02 -0.4911E+01

0.0000E+00 0.0000E+00 0.1427E-07

TSAI-HILL FUNCTION = 1.00000697203627

ALLOWABLE FUNCTION = 1.00000000000000

ACTION = -1 ----> STOP EXECUTION

As stated above, by altering the ratio of the applied Nxx and Nyy loading, the ìfailure surfacesî shown in 

Figure 5.4 can be generated. To generate the three distinct failure surfaces, the failure simulation is 

switched between the two sets of RUC failure parameters (1st Cell Failure surfaces) and the subcell failure 

criteria (1st Subcell Failure surfaces). Figure 5.4 shows that localizing all the way to the level of the 

subcells within each ply of the laminate in order to generate the 1st Subcell Failure surface for the 

laminate gives very similar results to the surface generated by localizing only to the level of the RUC 

within each ply and using the previously generated 1st subcell allowables for the RUC failure stresses. 

Employing the average material allowables for the RUC failure stresses, however, produces a much larger 

(i.e., less conservative) laminate failure surface. 

-1000

-750

-500

-250

0

250

500

750

1000

-1000 -750 -500 -250 0 250 500 750 1000

Nxx (MPa × thickness)

Nyy (MPa 

× thickness)

1st Cell Failure (data from

1st subcell allowables)

1st Subcell Failure

1st Cell Failure (data from

avg material allowables)

Figure 5.4 Example 5c: Predicted failure surfaces for a 60% fiber volume fraction quasi-isotropic 

graphite/epoxy laminate at room temperature. Note that these failure surfaces were 

generated by manually adjusting the ratio of the applied Nxx and Nyy.

NASA/TM—2002-212077/VOL3 170
Section 5: Damage and Failure Example 5d: Fatigue Damage Analysis 

Example 5d: Fatigue Damage Analysis 

MAC/GMC 4.0 includes two fatigue damage models that function on the local subcell level, both of 

which are employed in this example problem. The first involves degradation of the subcell material 

stiffness properties due to applied cyclic loading (suitable for modeling the composite matrix behavior), 

while the second involves degradation of the subcell material strength properties (suitable for modeling 

the composite fiber behavior). The same 60% fiber volume fraction graphite/epoxy composite is again 

considered as both a unidirectional composite and a quasi-isotropic laminate. 

The first step in this example problem involves characterizing the epoxy and graphite phases in terms of 

the fatigue damage model parameters. First, considering the epoxy matrix, the stiffness degradation 

fatigue damage model will be employed. The transversely isotropic form of this damage model present 

within MAC/GMC 4.0 is multi-axial, isothermal, and it employs a single scalar internal damage 

parameter, D (Arnold and Kruch, 1994). This damage parameter begins at zero for an undamaged 

material and grows (for a particular subcell) as damage occurs due to cyclic loading. The stiffness of the 

material in a damaged subcell is reduced by a factor of (1 ñ D) to account for the damage. A value of D = 

1 corresponds to the completely damaged (zero stiffness) case. When employed (as in this example) to 

model the fatigue behavior of the isotropic material present within a particular subcell, this model reduces 

to the NonLinear Cumulative Damage Rule (NLCDR) developed at ONERA (Wilt et al., 1997). For 

details on this fatigue damage model, see the MAC/GMC 4.0 Theory Manual Section 5.3. 

For an isotropic material, the damage parameters that must be selected reduce to M , β and aà , and the 

pertinent equation relating the fatigue life of the isotropic material to the cyclic stress state is, 

( )

( )( ) 0 à 1 max

max

max

>

−−+

















− −

= F

fl

u

F Nfor

a

M

N

β σ σ σ

σ σ

σ σ

β

where σ u is the material ultimate strength, σ fl is the material fatigue limit (stress below which damage 

does not occur), σ max is the maximum stress during a loading cycle, σ is the mean stress during a 

loading cycle, and NF is the number of cycles to failure. Note that, in the terminology of Arnold and 

Kruch (1994), 

fl

u aa

σ

σ à = . Utilizing the above equation, the damage model parameters M , β and aà can 

be selected for an isotropic material based on the materialís S-N curve (stress level vs. cycles to failure). 

A suggested characterization procedure for this damage model can be found in Arnold and Kruch (1994). 

An S-N curve for epoxy was obtained from Plastics Design Library (1995), and the fatigue damage model 

parameters were selected as M =150 MPa, β = 9 , and aà = 05.0 , with = 80 σ u MPa, and = 27 σ fl

MPa. A plot showing the fatigue model characterization is given in Figure 5.5. 

The second damage model within MAC/GMC 4.0 is much simpler and involves degradation of a 

materialís strength due to cyclic loading. As shown by Wilt et al. (1997), this type of damage model can 

be used to simulate the fatigue behavior of fibers that occurs in-situ during fatigue of a composite. 

NASA/TM—2002-212077/VOL3 171
MAC/GMC 4.0 Example Problem Manual 

The model assumes a logarithmic relation between the materialís strength and the number of cycles 

within a certain range such that: 

( ) ( )

( )

1 1

12 1

1 12

2 1

2 2

0

log

log

u u

u u

u u

u u

N N

N N

N N N

N N

N N

σ σ

σ σ

σ σ

σ σ

= ≤≤

− = − ≤≤

= ≤

This strength degradation model was employed in the present example to model the longitudinal fatigue 

behavior of the graphite fiber. The necessary parameters for the model are 121 σ σ uu ,, N , and N2 . The 

values of these parameters chosen for the graphite fiber are shown in Figure 5.6. Note that these data 

were not correlated with experiment, but rather chosen based on the expected trend. 

Given these required parameters for the fatigue damage models for each phase in the graphite/epoxy 

composite, this example problem, which predicts the fatigue life of a composite and a laminate, can be 

executed. 

0

10

20

30

40

50

60

70

80

90

1.E-01 1.E+01 1.E+03 1.E+05 1.E+07 1.E+09

Nf - Cycles to Failure

Maximum Stress (MPa)

Experimental Data

Fatigue Model Fit

Figure 5.5 Example 5d: Characterization of the stiffness reduction fatigue damage model parameters 

for the epoxy matrix. Experimental data are from Plastics Design Library (1995).

NASA/TM—2002-212077/VOL3 172
Section 5: Damage and Failure Example 5d: Fatigue Damage Analysis 

0

500

1000

1500

2000

2500

3000

3500

4000

1.E+02 1.E+03 1.E+04 1.E+05 1.E+06 1.E+07 1.E+08 1.E+09

N - Cycles

Fiber Ultimate Stren

gth (MPa)

N1 = 1000 N2 = 3.E+08

SU1 = 3500 MPa

SU2 = 2000 MPa

Figure 5.6 Example 5d: Strength reduction fatigue model parameters assumed for the graphite fiber. 

MAC/GMC Input File: example_5d.mac

MAC/GMC 4.0 Example 5d - Fatigue damage analysis

*CONSTITUENTS

NMATS=2

# -- Graphite fiber

M=1 CMOD=6 MATID=U MATDB=1

NTP=2

TEM=23.,150.

EA=388.2E3,390.E3

ET=7.6E3,7.6E3

NUA=0.41,0.41

NUT=0.45,0.45

GA=14.9E3,15.1E3

ALPA=-0.68E-6,-0.45E-6

ALPT=9.74E-6,10.34E-6

# -- Epoxy matrix

M=2 CMOD=6 MATID=U MATDB=1

NTP=2

TEM=23.,150.

EA=3.45E3,3.10E3

ET=3.45E3,3.10E3

NUA=0.35,0.35

NUT=0.35,0.35

GA=1.278E3,1.148E3

ALPA=45.E-6,55.E-6

ALPT=45.E-6,55.E-6

*RUC

MOD=2 ARCHID=6 VF=0.60 R=1. F=1 M=2

#*LAMINATE

# NLY=7

# LY=1 THK=0.125 ANG=0 MOD=2 ARCHID=6 R=1. VF=0.60 F=1 M=2

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# LY=2 THK=0.125 ANG=45 MOD=2 ARCHID=6 R=1. VF=0.60 F=1 M=2

# LY=3 THK=0.125 ANG=-45 MOD=2 ARCHID=6 R=1. VF=0.60 F=1 M=2

# LY=4 THK=0.250 ANG=90 MOD=2 ARCHID=6 R=1. VF=0.60 F=1 M=2

# LY=5 THK=0.125 ANG=-45 MOD=2 ARCHID=6 R=1. VF=0.60 F=1 M=2

# LY=6 THK=0.125 ANG=45 MOD=2 ARCHID=6 R=1. VF=0.60 F=1 M=2

# LY=7 THK=0.125 ANG=0 MOD=2 ARCHID=6 R=1. VF=0.60 F=1 M=2

*MECH

LOP=2

# LOP=1

NPT=4 TI=0.,50.,150.,200. MAG=0.,40.,-40.,0. MODE=2,2,2

*THERM

NPT=4 TI=0.,50.,150.,200. TEMP=23.,23.,23.,23.

*SOLVER

METHOD=1 NPT=4 TI=0.,50.,150.,200. STP=10.,10.,10.

*DAMAGE

MAXNB=100 DINC=0.2 DMAX=1.0 BLOCK=0.,200.

NDMAT=2

MAT=1 MOD=2 SU1=3500,91.2,91.2,31.4,134.,134 &

SU2=2000.,91.2,91.2,31.4,134.,134. &

N1=1000,1000,1000,1000,1000,1000 &

N2=300000000,300000000,300000000,300000000,300000000,300000000

MAT=2 MOD=1 ANG=0. BN=0.0 BP=0.0 OMU=1. OMFL=1. OMM=1. ETU=1. &

ETFL=1. ETM=1. BE=9. A=0.05 SFL=27. XML=150. &

SU=80.

*FAILURE_SUBCELL

NMAT=2

MAT=1 NCRIT=1

CRIT=1 X11=3500. X22=91.2 X33=91.2 X23=31.4 X13=134. X12=134. &

COMPR=SAM

MAT=2 NCRIT=1

CRIT=1 X11=80. X22=80. X33=80. X23=40. X13=40. X12=40. &

COMPR=SAM

*FAILURE_CELL

NCRIT=1

CRIT=2 X11=0.05 X22=0.05 X33=0.05 X23=0.05 X13=0.05 X12=0.05 &

COMPR=SAM

*PRINT

NPL=3

*XYPLOT

FREQ=1

# LAMINATE=1

# NAME=example_5d X=1 Y=10

MACRO=2

NAME=example_5d_11 LYR=1 X=1 Y=7

NAME=example_5d_22 LYR=1 X=2 Y=8

MICRO=0

*END

Annotated Input Data

1) Flags: None 

NASA/TM—2002-212077/VOL3 174
Section 5: Damage and Failure Example 5d: Fatigue Damage Analysis 

2) Constituent materials (*CONSTITUENTS) [KM_2]: 

Number of materials: 2 (NMATS=2) 

Constitutive models: Elastic (CMOD=6) 

Materials: User-defined (Graphite) (MATID=U) 

 User-defined (Epoxy) (MATID=U) 

Material property source: Read from input file (MATDB=1) 

Material properties: See Table 4.1 

3) Analysis type:

a) Unidirectional Composite Case (*RUC) → Repeating Unit Cell Analysis [KM_3]: 

Analysis model: Doubly periodic GMC (MOD=2) 

RUC architecture: 7×7 circular fiber approx., rect. pack (ARCHID=6) 

Fiber volume fraction: 0.60 (VF=0.60) 

Unit cell aspect ratio: 1.0 (square pack) (R=1.0) 

Material assignment: graphite fiber (F=1) 

epoxy matrix (M=2) 

b) Quasi-Isotropic Laminate Case (*LAMINATE) → Laminate Analysis [KM_3]: 

Number of layers: 7 (NLY=7) 

Layer Analysis 

Model 

Thickness Fiber 

Angle 

Architecture Volume 

fraction 

Aspect 

ratio 

Fiber 

material 

Matrix 

material 

(LY=) (MOD) (THK) (ANG) (ARCHID) (VF) (R) (F) (M) 

1 GMC-2D 0.125 0° 7×7 circle 

rect. pack 

0.60 1. graphite epoxy 

2 GMC-2D 0.125 45° 7×7 circle 

rect. pack 

0.60 1. graphite epoxy 

3 GMC-2D 0.125 -45° 7×7 circle 

rect. pack 

0.60 1. graphite epoxy 

4 GMC-2D 0.25 90° 7×7 circle 

rect. pack 

0.60 1. graphite epoxy 

5 GMC-2D 0.125 -45° 7×7 circle 

rect. pack 

0.60 1. graphite epoxy 

6 GMC-2D 0.125 45° 7×7 circle 

rect. pack 

0.60 1. graphite epoxy 

7 GMC-2D 0.125 0° 7×7 circle 

rect. pack 

0.60 1. graphite epoxy 

! Note: To generate the data in the Results for both the unidirectional composite and the quasi-isotropic 

laminate, the appropriate lines in the input file must be commented and uncommented. 

4) Loading: 

a) Mechanical (*MECH) [KM_4]: 

Loading option: 2 or 1 (LOP=2) or (LOP=1) 

Number of points: 4 (NPT=4) 

Time points: 0., 50., 150., 200. sec. (TI=0.,50.,150.,200.) 

Load magnitudes: 0., 40., -40., 0. MPa (MAG=0.,40.,-40.,0.) 

Loading mode: stress control (MODE=2,2,2) 

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For fatigue loading, the loading cycle must be defined in *MECH. This cycle must start and end at the 

same magnitudes. Since this example involves a fully reversed fatigue simulation, the cycle starts at 

a stress of 0, rises to 40 MPa, then decreases to -40 MPa, and finally returns to 0. 

! Note: To execute the code with different maximum (and minimum) stress levels (i.e., to generate an 

S-N curve), the magnitudes of the applied loading must be altered and the code executed 

repeatedly. 

! Note: In the case of the laminate, the loading is actual force resultant control. Since the laminate 

thickness is 1., equivalency exists between the stress magnitudes for the unidirectional 

composites and the force resultant magnitudes for the laminate. 

b) Thermal (*THERM) [KM_4]: 

Number of points: 4 (NPT=4) 

Time points: 0., 50., 150., 200. sec. (TI=0.,50.,150.,200.) 

Temperature points: 23., 23., 23., 23. (TEMP=23.,23.,23.,23.) 

As with the mechanical loading, the thermal loading in fatigue damage analysis defines the cycle and 

must start and end at the same temperature. In this case, the loading cycle does not involve a 

temperature change. 

c) Time integration (*SOLVER) [KM_4]: 

Time integration method: Forward Euler (METHOD=1) 

Number of points: 4 (NPT=4) 

Time points: 0., 50., 150., 200. sec. (TI=0.,50.,150.,200.) 

Time step sizes: 10., 10., 10. sec. (STP=10.,10.,10.) 

5) Damage and Failure: 

a) Fatigue Damage Analysis (*DAMAGE) [KM_5]: 

MAXNB=100 DINC=0.2 DMAX=1.0 BLOCK=0.,200.

Max. no. of load blocks: 100 (MAXNB=100) 

Damage increment: 0.2 (DINC=0.2) 

Max. damage value: 1.0 (DMAX=1.0) 

Load block times: 0., 200. sec. (BLOCK=0.,200.) 

The load block associated with one cycle is specified using BLOCK= to indicate the start and end 

times of the load block. This allows additional loading to occur before or after the actual cyclic load 

block if desired (e.g., to incorporate residual stresses). The MAC/GMC 4.0 fatigue analysis applies 

this load block and then determines the number of cycles of this load block required to cause the local 

damage increment specified as DINC=. This is a local increment of the damage parameter, D, which 

pertains to a single subcell. Since there can be many subcells, each with its own value of D, the code 

selects a controlling subcell that reaches the damage increment first. The number of cycles (of the 

specified load block) required to cause the local damage increment is then imposed upon the 

composite, resulting in a state of damage throughout the composite. Then, the simulated load block 

is applied again, and a new number of cycles required to increment the local damage (by DINC) is 

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Section 5: Damage and Failure Example 5d: Fatigue Damage Analysis 

calculated. This process is repeated until the maximum number of load blocks (MAXNB) has been 

applied or complete failure has occurred. By applying load blocks to increment the damage in this 

fashion, the stress state in the composite is permitted to redistribute based on the evolving state of 

damage. In general, a smaller damage increment will cause longer execution times, but also allows a 

greater degree of load redistribution. 

NDMAT=2

MAT=1 MOD=2 SU1=3500,91.2,91.2,31.4,134.,134 &

SU2=2000.,91.2,91.2,31.4,134.,134. &

N1=1000,1000,1000,1000,1000,1000 &

N2=300000000,300000000,300000000,300000000,300000000,300000000

No. damaging materials: 2 (NDMAT=2) 

For Material #1 (MAT=1) 

Fatigue model: Strength reduction model (MOD=2) 

Ultimate stress point 1: σ11 = 3500. MPa (SU1=3500.,91.2,...) 

 σ22 = 91.2 MPa 

 σ33 = 91.2 MPa 

 σ23 = 31.4 MPa 

 σ13 = 134. MPa 

 σ12 = 134. MPa 

Ultimate stress point 2: σ11 =2000. MPa (SU1=2000.,91.2,...) 

 σ22 = 91.2 MPa 

 σ33 = 91.2 MPa 

 σ23 = 31.4 MPa 

 σ13 = 134. MPa 

 σ12 = 134. MPa 

Number of cycles point 1: 1000 (N1=1000,1000,...) 

Number of cycles point 2: 300,000,000 (N2=300000000,300000000,...) 

The ultimate stress points and number of cycles points are listed for the six stress components. In the 

present example, only the σ11 ultimate stress component is changing with number of cycles. 

MAT=2 MOD=1 ANG=0. BN=0.0 BP=0.0 OMU=1. OMFL=1. OMM=1. ETU=1. &

ETFL=1. ETM=1. BE=9. A=0.05 SFL=27. XML=150. &

SU=80.

For Material #2 (MAT=2) 

Fatigue model: Stiffness reduction model (MOD=1) 

θ: 0. (ANG=0.) 

b: 0.0 (BN=0.0) 

bí: 0.0 (BP=0.0) 

ωu: 1. (OMU=1.) 

ωfl: 1. (OMFL=1.) 

ωm: 1. (OMM=1.) 

ηu: 1. (ETU=1.) 

ηfl: 1. (ETFL=1.) 

ηm: 1. (ETM=1.) 

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β: 9. (BE=9.) 

aà : 0.05 (A=9.) 

σfl: 27. MPa (SFL=27.) 

M: 150. MPa (XML=150.) 

σu: 80. MPa (SU=80.) 

For the meaning of all stiffness reduction fatigue damage model parameters, see the MAC/GMC 4.0 

Theory Manual Section 5.3. For an isotropic material such as epoxy, the last 5 parameters listed are 

all that must be selected. 

b) Subcell static failure analysis (*FAILURE_SUBCELL) [KM_5]: 

Number of materials: 2 (NMAT=1) 

Material #1 (MAT=1) 

Number of criteria: 1 (NCRIT=1) 

Criterion #1: Maximum stress criterion (CRIT=1) 

Failure stresses: σ11 = 3500. MPa (X11=3500.) 

 σ22 = 91.2 MPa (X22=91.2) 

 σ33 = 91.2 MPa (X33=91.2) 

 σ23 = 31.4 MPa (X23=31.4) 

 σ13 = 134. MPa (X13=134.) 

 σ12 = 134. MPa (X12=134.) 

Compression flag: Compressive strengths same as tensile (COMPR=SAM) 

Material #2 (MAT=2) 

Number of criteria: 1 (NCRIT=1) 

Criterion #1: Maximum stress criterion (CRIT=1) 

Failure stresses: σ11 = 80. MPa (X11=80.) 

 σ22 = 80. MPa (X22=80.) 

 σ33 = 80. MPa (X33=80.) 

 σ23 = 40. MPa (X23=40.) 

 σ13 = 40. MPa (X13=40.) 

 σ12 = 40. MPa (X12=40.) 

Compression flag: Compressive strengths same as tensile (COMPR=SAM) 

! Note: In the case of fatigue damage analysis, ACTION is not needed because the fatigue analysis must 

continue until overall failure or the maximum number of load blocks is applied. If ACTION is 

specified in this case, it will be ignored by the code. 

c) RUC static failure analysis (*FAILURE_CELL) [KM_5]: 

Number of criteria: 1 (NCRIT=1) 

Criterion #1: Maximum strain criterion (CRIT=2) 

Failure strains: ε11 = 0.05 (X11=0.05) 

 ε22 = 0.05 (X22=0.05) 

 ε33 = 0.05 (X33=0.05) 

 ε23 = 0.05 (X23=0.05) 

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Section 5: Damage and Failure Example 5d: Fatigue Damage Analysis 

 ε13 = 0.05 (X13=0.05) 

 ε12 = 0.05 (X12=0.05) 

Compression flag: Compressive strains same as tensile (COMPR=SAM) 

In the present example, the RUC static failure analysis option is employed to limit the amount of 

strain permitted for an RUC. This allows the code to treat the RUC as failed if damage has occurred 

such that the stiffness of the RUC is very low and a large strain results. Limiting the strain in this 

matter can also prevent numerical overflow when the loading is in stress control, as in the present 

example. The RUC level static failure capabilities function similarly to the subcell level static failure 

capabilities. The code simply employs the global quantities rather than the local quantities when 

evaluating the appropriate failure criteria. 

6) Output: 

a) Output file print level (*PRINT) [KM_6]: 

Print level: 3 (NPL=3) 

b) x-y plots (*XYPLOT): 

Frequency: 1 (FREQ=1) 

Number of laminate plots: 1 (LAMINATE=1) 

Laminate plot names: example_5d (NAME=example_5d) 

Laminate x-y quantities: Nxxxx , 0 ε (X=1 Y=10) 

Number of macro plots: 2 (MACRO=2) 

Macro plot names: example_5d_11 (NAME=example_5d_11) 

 example_5d_22 (NAME=example_5d_22) 

Macro plot x-y quantities: ε11, σ11 (X=1 Y=7) 

ε22, σ22 (X=2 Y=8) 

Number of micro plots: 0 (MICRO=0) 

! Note: In this example, the lines from the input file associated with the laminate plots should be 

commented for the RUC analyses. 

7) End of file keyword: (*END) 

Results

Results for the fatigue damage analysis are written not only to the MAC/GMC 4.0 output file, but also to 

a damage file. This damage file is given the same name as the output file, with _dam.data appended. 

Thus, in the present example, the damage file is named ìexample_5d_dam.dataî. This file includes 

a summary of the damage calculations and state of damage after each applied load block, as well as the 

total number of cycles to failure. A portion of this file is shown below: 

Completed Applied Load Block Number 3 (Loading Increment Number 60 )

----------------------------------------------------------------------------

A) Cycles Required to Incur a Damage Increment of DINC = 0.2000

Controlling subcell ---> 46

Number of Cycles => 1460.389

Number of Cycles => 1460.000 (rounded)

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---------------------------------------------------

B) Current TOTAL number of cycles ---> 7530637

(after applied load cycle 3, D = 0.5999 )

---------------------------------------------------

C) Current Damage in each subcell (after 7530637 cycles):

Subcell NF* D

------- ------- -------

1 - inf - 0.0000

2 - inf - 0.0000

3 0.2876E+09 0.0000

4 1486.0947 0.5993

5 0.2876E+09 0.0000

6 - inf - 0.0000

7 - inf - 0.0000

8 - inf - 0.0000

.

. → Lines Omitted .

*NOTE: NF = Remaining life assuming no further stress redistribution

.

. → Lines Omitted .

********************************************

* ALL SUBCELLS HAVE FAILED *

* TOTAL NUMBER OF CYCLES = 7530637.0 *

* *

********************************************

Results from the x-y plot file are shown in Figure 5.7 for a transverse fatigue analysis of the 

unidirectional 60% graphite/epoxy composite. In this case, it was determined that 7,517,751 cycles of 

load block #1 were required to achieve a damage level of 0.2 in a subcell. Then, load block #2 was 

applied, and, as shown in Figure 5.7, the transverse composite response becomes more compliant due to 

the damage state caused by the 7,517,751 cycles of load block #1. Only 11,426 cycles of load block #2 

are required to cause a local damage increment of 0.2, resulting in a total number of cycles of 7,529,177. 

When load block #3 is applied, the transverse composite response is even more compliant due to the 

higher level of damage throughout the composite. Only 1,460 cycles of load block #3 are required to 

cause an additional local damage increment of 0.2. During applied load block #4 local failures occur that 

cause the non-linearity evident in Figure 5.7. Then, when the code attempts to apply load block #5, 

additional failures occur that lead to the complete failure of the composite. That is, after application of 

load block #4, the code cannot withstand any additional cycles. The predicted life of the composite is 

7,530,637 cycles. 

By altering the applied stress magnitudes in *MECH, a transverse S-N curve can be predicted for the 

graphite/epoxy composite. Similarly, by altering the loading option, a longitudinal S-N curve for the 

composite can be predicted. Finally, switching the simulation to a laminate fatigue analysis (commenting 

*RUC and uncommenting *LAMINATE in the input file), an S-N curve for the quasi-isotropic laminate 

can be predicted. All three of these predicted S-N curves are plotted in Figure 5.8, whereas, for additional 

detail, each S-N curve is plotted separately in Figure 5.9, Figure 5.10, and Figure 5.11. As one would 

expect, the S-N curve for the longitudinal composite is highest due to the strong and stiff fibers oriented 

along the loading direction. The transverse S-N curve is lowest since the compositeís transverse response 

is matrix dominated, and the laminate S-N curve is intermediate. 

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Section 5: Damage and Failure Example 5d: Fatigue Damage Analysis 

-40

-30

-20

-10

0

10

20

30

40

-0.010 -0.005 0.000 0.005 0.010 0.015

ε22

σ22 (MPa)

Load Blocks→ 1 2 3 4

Figure 5.7 Example 5d: Transverse stress-strain response for a 60% graphite/epoxy composite at room 

temperature for the four applied load blocks prior to failure. 

0

500

1000

1500

2000

2500

1.E+03 1.E+04 1.E+05 1.E+06 1.E+07 1.E+08 1.E+09

Nf - Cycles to Failure 

Max. 

σ (MPa) or N (MPa × thickness)

Longitudinal [0]

Transverse [90]

[0/45/-45/90]s laminate

Figure 5.8: Example 5d: Predicted S-N curves for 60% graphite/epoxy at room temperature.

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MAC/GMC 4.0 Example Problem Manual 

25

30

35

40

45

50

55

60

65

70

75

1.E+03 1.E+04 1.E+05 1.E+06 1.E+07 1.E+08 1.E+09

Nf - Cycles to Failure 

Maximum Stress (

MPa)

Figure 5.9: Example 5d: Predicted transverse S-N curve for 60% graphite/epoxy at room temperature.

1800

1850

1900

1950

2000

2050

2100

2150

1.E+07 1.E+08 1.E+09

Nf - Cycles to Failure 

Maximum Stress (

MPa)

Figure 5.10 Example 5d: Predicted longitudinal S-N curve for 60% graphite/epoxy at room 

temperature.

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Section 5: Damage and Failure Example 5d: Fatigue Damage Analysis 

300

350

400

450

500

550

600

650

700

750

800

1.E+03 1.E+04 1.E+05 1.E+06 1.E+07 1.E+08 1.E+09

Nf - Cycles to Failure 

Max. Force Resultant (

MPa 

× thickness)

Figure 5.11 Example 5d: Predicted S-N curve for a quasi-isotropic [0°/45°/-45°/90°]s 60% 

graphite/epoxy laminate. 

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MAC/GMC 4.0 Example Problem Manual 

Example 5e: Fiber-Matrix Debonding

This example problem involves the simulation of transverse fiber-matrix debonding in a SiC/Ti-21S 

composite. This type of failure has proven to be a major obstacle in the utilization of titanium matrix 

composites. MAC/GMC 4.0 incorporates two distinct fiber-matrix debonding models, each of which is 

based on the same overall concept of imposing a discontinuity or ìjumpî in displacement at a particular 

interface. This displacement jump is modeled as proportional to the stress at the interface and is not 

activated until the interfacial stress reaches a critical debond stress, 

[ ] I

DB

I II

Ru σ σ ≥= σ

where [u] is the resulting displacement component at interface I, σ is a particular stress component at 

interface I, σDB is the debond stress of interface I, and R is a debonding parameter for interface I. This 

interfacial representation has been employed by Jones and Whittier (1967), Aboudi (1987), Achenbach 

and Zhu (1989), and Wilt and Arnold (1996). In MAC/GMC 4.0, as was done by Bednarcyk and Arnold 

(2002), the debonding parameter R is permitted to evolve with time. The form of this time evolution 

distinguishes between the two debonding models within MAC/GMC 4.0. 

First, the constant compliant interface (CCI) model employs, 

( ) ( )

Β

Λ=

∂

∂ t

t

tR σ

at a particular interface. The evolving compliant interface (ECI) model, on the other hand, employs, 

( ) ( ) ( ) 0 exp ≥−

Β

− Λ=

∂

∂

DB

DB tt

tt

t

tR

at a particular interface. Λ and Β are additional parameters that characterize the behavior of a particular 

interface and tDB is the time of debonding for the interface. Thus, the time dependence in the CCI model 

is implicitly based on the time dependence of the interfacial stress while the ECI model incorporates 

explicit time dependence in its evolution equation. The dependence of the CCI model on the interfacial 

stress causes this model to saturate to a steady state condition (when the parameters Λ and Β are chosen 

properly). This characteristic is due to the fact that, as the parameter R rises with the rising interfacial 

stress, the interface becomes more and more compliant, causing the interfacial stress to level off. Once 

the interfacial stress stops increasing, the debonding parameter R can no longer increase, and a steady 

state condition is reached. Conversely, in the ECI model, with its explicit exponential time dependence, 

the debonding parameter R can continue to rise. As shown in this example problem, this ability allows 

the interfacial stress to unload in composite simulations involving the ECI model. For additional 

information on the CCI and ECI models, see the MAC/GMC 4.0 Theory Manual Section 5.4. 

MAC/GMC Input File: example_5e.mac

MAC/GMC 4.0 Example 5e - Fiber-matrix transverse debonding

*CONSTITUENTS

NMATS=2

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Section 5: Damage and Failure Example 5e: Fiber-Matrix Debonding 

M=1 CMOD=6 MATID=E

M=2 CMOD=4 MATID=A

*RUC

MOD=2 ARCHID=11 VF=0.35 R=0.9 F=1 M=2

*MECH

LOP=3 REFTIME=64800.

NPT=5 TI=0.,24000.,57600.,64800.,64908. MAG=0.,0.,0.,0.,0.018 MODE=2,2,2,1

*THERM

NPT=5 TI=0.,24000.,57600.,64800.,64908. TEMP=900.,534.583,23.,650.,650.

*SOLVER

METHOD=1 NPT=5 TI=0.,24000.,57600.,64800.,64908. STP=250.,40.,40.,0.2

*DEBOND

NII=1

# DBCH=1 NBI=1 NGI=1 FACE=2 BDN=7. LN=0.0001 BN=60. TOLN=0. &

# BDS=40 LS=0.1 BS=100. DELAY=64800.

DBCH=2 NBI=1 NGI=1 FACE=2 BDN=7. LN=0.0001 BN=8. TOLN=0. &

BDS=40 LS=0.1 BS=100 DELAY=64800.

*PRINT

NPL=6

*XYPLOT

FREQ=1

MACRO=1

NAME=example_5e X=3 Y=9

MICRO=1

NAME=example_5e IB=1 IG=2 X=3 Y=9

*END

Annotated Input Data

1) Flags: None 

2) Constituent materials (*CONSTITUENTS) [KM_2]: 

Number of materials: 2 (NMATS=2) 

Materials: SiC fiber (MATID=E) 

Ti-21S (MATID=A) 

Constitutive models: SiC fiber: linearly elastic (CMOD=6) 

Ti-21S matrix: Isotropic GVIPS (CMOD=4) 

3) Analysis type (*RUC) → Repeating Unit Cell Analysis [KM_3]: 

Analysis model: Doubly periodic GMC (MOD=2) 

RUC architecture: Square fiber, rectangular pack (ARCHID=11) 

Unit cell aspect ratio: 0.9 (R=0.9) 

Fiber volume fraction: 0.35 (VF=0.35) 

Material assignment: SiC fiber (F=1) 

Ti-21S matrix (M=2) 

4) Loading: 

a) Mechanical (*MECH) [KM_4]: 

Loading option: 3 (LOP=3) 

Strain reference time: 57600. sec. (REFTIME=57600.) 

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Number of points: 5 (NPT=5) 

Time points: 0., 24000., 57600., 64800., 64908. sec. (TI=0.,24000.,...) 

Load magnitude: 0., 0., 0., 0., 0.018 (MAG=0.,0.,...,0.018) 

Loading mode: stress/strain control (MODE=2,2,2,1) 

b) Thermal (*THERM) [KM_4]: 

Number of points: 5 (NPT=5) 

Time points: 0., 24000., 57600., 64800., 64908. sec. (TI=0.,24000.,...) 

Temperature points: 900., 534.583, 23., 650., 650. °C (TEMP=900.,...,650.) 

! Note: The second temperature (534.583 °C) is chosen in order to preserve the rate of change of the 

temperature. 

c) Time integration (*SOLVER) [KM_4]: 

Time integration method: Forward Euler (METHOD=1) 

Number of points: 5 (NPT=5) 

Time points: 0., 24000., 57600., 64800., 64908. sec. (TI=0.,24000.,...) 

Time step sizes: 250., 40., 40., 0.2 sec. (STP=250.,40.,40.,0.2) 

5) Damage and Failure: 

a) Fiber-matrix debonding (*DEBOND) [KM_5]: 

NII=1

# DBCH=1 NBI=1 NGI=1 FACE=2 BDN=7. LN=0.0001 BN=60. TOLN=0. &

# BDS=40 LS=0.1 BS=100. DELAY=64800.

DBCH=2 NBI=1 NGI=1 FACE=2 BDN=7. LN=0.0001 BN=8. TOLN=0. &

BDS=40 LS=0.1 BS=100 DELAY=64800.

No. debonding interfaces: 1 (NII=1) 

CCI Model (DBCH=1) 

Interface subcell indices: 1, 1 (NBI=1 NGI=1) 

Interface identifier: x3-interface (FACE=2) 

Normal debond stress: 7. ksi (BDN=7.) 

Normal Λ parameter: 0.0001 /ksi (LN=0.0001) 

Normal Β parameter: 60. s (BN=60.) 

Load reversal tolerance: 0. ksi (TOLN=0.) 

Shear debond stress: 40. ksi (BDS=40.) 

Shear Λ parameter: 0.1 /ksi (LS=0.1) 

Shear Β parameter: 100. s (BS=100.) 

Debond time delay 64800. sec. (DELAY=64800.) 

ECI Model (DBCH=2) 

Interface subcell indices: 1, 1 (NBI=1 NGI=1) 

Interface identifier: x3-interface (FACE=2) 

Normal debond stress: 7. ksi (BDN=7.) 

Normal Λ parameter: 0.0001 /ksi (LN=0.0001) 

Normal Β parameter: 60. s (BN=60.) 

NASA/TM—2002-212077/VOL3 186
Section 5: Damage and Failure Example 5e: Fiber-Matrix Debonding 

Load reversal tolerance: 0. ksi (TOLN=0.) 

Shear debond stress: 40. ksi (BDS=40.) 

Shear Λ parameter: 0.1 /ksi (LS=0.1) 

Shear Β parameter: 100. s (BS=100.) 

Debond time delay 64800. sec. (DELAY=64800.) 

The input format for debonding shown above involves first specifying the number of interfaces that 

could possibly debond, then specifying the required data for each of these interfaces. MAC/GMC 4.0 

treats the normal and shear debonding at a particular interface independently. Thus, independent 

normal and shear values for the debond stress (σDB), Λ parameter, and Β parameter are required for 

each interface that is permitted to debond. In addition, a normal load reversal tolerance (TOLN) and a 

debond time delay (DELAY) must be specified for each interface. The normal load reversal tolerance 

is the stress below which the model treats the normal interfacial stress as zero, thus allowing the 

interface to close and support compressive stress. This can become important in cyclic load cases. 

The debond time delay is the time at which the code begins to consider the possibility of debonding. 

Before this time, the interface is treated as perfectly bonded independently of the other specified 

parameters. For more information on the debonding models and the associated input requirements, 

see the MAC/GMC 4.0 Theory Manual Section 5.4 and the MAC/GMC 4.0 Keywords Manual 

Section 5. 

! Note: In order to execute the three cases presented in the results for this example, the appropriate 

lines under *DEBOND must be commented and uncommented. 

6) Output: 

a) Output file print level (*PRINT) [KM_6]: 

Print level: 6 (NPL=6) 

b) x-y plots (*XYPLOT) [KM_6]: 

Frequency: 1 (FREQ=1) 

Number of macro plots: 1 (MACRO=1) 

Macro plot names: example_5e (NAME=example_5e) 

Macro plot x-y quantities: ε33, σ33 (X=3 Y=9) 

Number of micro plots: 1 (MICRO=1) 

Micro plot names: example_5e (NAME=example_5e) 

Micro plot subcell indices: 1, 2 (IB=1 IG=2) 

Micro plot x-y quantities: ε33, σ33 (X=3 Y=9) 

7) End of file keyword: (*END) 

Results

Figure 5.12 shows the predicted transverse tensile response of the composite at 650 °C for the three 

different cases (perfect bonding, debonding represented by the CCI model, and debonding represented by 

the ECI model). Both the global (composite) stress vs. strain response and the local interface stress vs. 

global strain response are plotted for each case. In the case of perfect bonding, the interfacial stress is 

higher than the composite stress at a given global strain level. Including fiber-matrix debonding in the 

simulation via the CCI model has a major effect on the transverse response. The obvious ìkneeî in the 

NASA/TM—2002-212077/VOL3 187
MAC/GMC 4.0 Example Problem Manual 

composite curve corresponds to the onset of fiber-matrix debonding. The interface response predicted by 

the CCI model is identical to the perfectly bonded interface behavior until debonding occurs. Then the 

interfacial stress quickly saturates to a nearly constant value. Since the interface can support no 

additional stress, the intact region of the composite is placed under greater stress and flows to a greater 

extent. Finally, in the case of the ECI model, after the interface debonds, the interfacial stress unloads. 

This is thought to be the more accurate mechanism as a debonded interface cannot support stress. Now 

the intact region of the composite must not only support the additional applied load, but also the stress 

that is unloaded from the interface. This causes even more flow in the intact region, and a composite 

stress strain curve that is considerably more compliant than that predicted using the CCI model. Note that 

residual stresses were incorporated in the present example, which is why the interfacial stresses plotted in 

Figure 5.12 are compressive at zero applied global strain. 

-10

0

10

20

30

40

50

60

0.000 0.005 0.010 0.015 0.020

Applied Global Strain

Stress (ksi)

Composite: Perfect Bonding

Composite: DBCH=1 (CCI)

Composite: DBCH=2 (ECI)

Interface: Perfect Bonding

Interface: DBCH=1 (CCI)

Interface: DBCH=2 (ECI)

Figure 5.12 Example 5e: Predicted local and global transverse stress-strain response of 35% SiC/Ti-21S 

at 650 °C with fiber-matrix debonding. 

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Section 5: Damage and Failure Example 5f: Fiber Breakage 

Example 5f: Fiber Breakage

This example problem demonstrates the two methods incorporated within MAC/GMC 4.0 that can be 

used to model longitudinal fiber breakage in composite materials. The first is the evolving compliant 

interface (ECI) model, which was applied to transverse fiber-matrix debonding in the previous example 

problem. Now, instead of applying the ECI model to a fiber-matrix interface, it is applied to an internal 

fiber interface within a triply periodic RUC. The second model is the Curtin effective fiber breakage 

model (Curtin, 1991, 1993). This model combines a shear-lag analysis with fiber strength statistics to 

degrade the stiffness of an effective fiber that represents all fibers within a composite. Both models, as 

implemented within MAC/GMC 4.0, are capable of predicting the longitudinal strength of continuous 

fiber composites. 

MAC/GMC Input File: example_5f.mac

MAC/GMC 4.0 Example 5f - Fiber breakage

*CONSTITUENTS

NMATS=2

M=1 CMOD=6 MATID=E

M=2 CMOD=4 MATID=A

*RUC

MOD=2 ARCHID=1 VF=0.35 F=1 M=2

# MOD=3 ARCHID=99

# NA=1 NB=2 NG=2

# D=1.

# H=0.5916,0.4084

# L=0.5916,0.4084

# SM=1,2

# SM=2,2

*MECH

LOP=1 REFTIME=64800.

NPT=5 TI=0.,24000.,57600.,64800.,64908. MAG=0.,0.,0.,0.,0.018 MODE=2,2,2,1

*THERM

NPT=5 TI=0.,24000.,57600.,64800.,64908. TEMP=900.,534.583,23.,650.,650.

*SOLVER

METHOD=1 NPT=5 TI=0.,24000.,57600.,64800.,64908. STP=250.,40.,40.,0.2

#*DEBOND

# NII=1

# DBCH=2 NAI=1 NBI=1 NGI=1 FACE=1 BDN=311. LN=0.00000001 BN=3. TOLN=0. &

# BDS=400 LS=0.1 BS=100 DELAY=64800.

*CURTIN

NCURT=1

NBI=1 NGI=1 D=142.E-6 L0=0.0127 SIG0=508. TAU0=2.03 M=17.0 &

DELAY=64800. ACTION=0

*PRINT

NPL=6

*XYPLOT

FREQ=1

MACRO=1

NAME=example_5f X=1 Y=7

MICRO=1

NAME=example_5f IA=1 IB=1 IG=1 X=1 Y=7

*END

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MAC/GMC 4.0 Example Problem Manual 

Annotated Input Data

1) Flags: None 

2) Constituent materials (*CONSTITUENTS) [KM_2]: 

Number of materials: 2 (NMATS=2) 

Materials: SiC fiber (MATID=E) 

Ti-21S (MATID=A) 

Constitutive models: SiC fiber: linearly elastic (CMOD=6) 

Ti-21S matrix: Isotropic GVIPS (CMOD=4) 

3) Analysis type (*RUC) → Repeating Unit Cell Analysis [KM_3]: 

Perfect Fiber and Curtin Model Simulations

Analysis model: Doubly periodic GMC (MOD=2) 

RUC architecture: square fiber, square pack (ARCHID=1) 

Fiber volume fraction: 0.35 (VF=0.35) 

Material assignment: SiC fiber (F=1) 

Ti-21S matrix (M=2) 

ECI Model Simulation

Analysis model: Triply periodic GMC (MOD=3) 

Architecture: User-defined (ARCHID=99) 

No. subcells in x1-dir.: 1 (NA=1) 

No. subcells in x2-dir.: 2 (NB=2) 

No. subcells in x3-dir.: 2 (NG=2) 

Subcell depths: 1. (D=1.0) 

Subcell heights: 0.5961,0.4084 (H=0.5961,0.4084) 

Subcell lengths: 0.5961,0.4084 (L=0.5961,0.4084) 

Material assignment: square fiber, square pack (SM=1,2 / SM=2,2) 

! Note: To generate the results for all three cases presented in the results, the appropriate lines in the 

input file must be commented and uncommented. 

4) Loading: 

a) Mechanical (*MECH) [KM_4]: 

Loading option: 3 (LOP=3) 

Strain reference time: 57600. sec. (REFTIME=57600.) 

Number of points: 5 (NPT=5) 

Time points: 0., 24000., 57600., 64800., 64908. sec. (TI=0.,24000.,...) 

Load magnitude: 0., 0., 0., 0., 0.018 (MAG=0.,0.,...,0.018) 

Loading mode: stress/strain control (MODE=2,2,2,1) 

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Section 5: Damage and Failure Example 5f: Fiber Breakage 

b) Thermal (*THERM) [KM_4]: 

Number of points: 5 (NPT=5) 

Time points: 0., 24000., 57600., 64800., 64908. sec. (TI=0.,24000.,...) 

Temperature points: 900., 534.583, 23., 650., 650. °C (TEMP=900.,...,650.) 

! Note: The second temperature (534.583 °C) is chosen in order to preserve the rate of change of the 

temperature. 

c) Time integration (*SOLVER) [KM_4]: 

Time integration method: Forward Euler (METHOD=1) 

Number of points: 5 (NPT=5) 

Time points: 0., 24000., 57600., 64800., 64908. sec. (TI=0.,24000.,...) 

Time step sizes: 250., 40., 40., 0.2 sec. (STP=250.,40.,40.,0.2) 

5) Damage and Failure: 

a) Fiber-matrix debonding (*DEBOND) [KM_5]: 

No. debonding interfaces: 1 (NII=1) 

Interface subcell indices: 1, 1, 1 (NAI=1 NBI=1 NGI=1) 

Interface identifier: x1-interface (FACE=1) 

Normal debond stress: 311. ksi (BDN=311.) 

Normal Λ parameter: 0.00000001 /ksi (LN=0.00000001) 

Normal Β parameter: 3. s (BN=3.) 

Load reversal tolerance: 0. ksi (TOLN=0.) 

Shear debond stress: 400. ksi (BDS=400.) 

Shear Λ parameter: 0.1 /ksi (LS=0.1) 

Shear Β parameter: 100. s (BS=100.) 

Debond time delay 64800. sec. (DELAY=64800.) 

b) Curtin effective fiber breakage model (*CURTIN) [KM_5]: 

*CURTIN

NCURT=1

NBI=1 NGI=1 D=142.E-6 L0=0.0127 SIG0=508. TAU0=2.03 M=17.0 &

DELAY=64800. ACTION=0

No. Curtin model fibers: 1 (NCURT=1) 

Fiber subcell indices: 1, 1 (NBI=1 NGI=1) 

Fiber diameter: 142. μm (D=142.E-6) 

Fiber gauge length: 12.7 mm (L0=0.0127) 

Fiber mean strength: 508. ksi (SIG0=508.) 

Fiber-matrix shear friction: 2.03 ksi (TAU0=2.03) 

Fiber Weibull modulus: 17.0 (M=17.0) 

Curtin time delay 64800. sec. (DELAY=64800.) 

Action to take upon failure: Only write notification and continue (ACTION=0) 

The format for specifying the Curtin model data is similar to that employed in the debond model data 

specification. Most of the Curtin model parameters are physical or statistical in nature and relatively 

easily obtained. The exception is the fiber-matrix frictional sliding shear stress (TAU0). Attempts 

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MAC/GMC 4.0 Example Problem Manual 

have been made to extract this value from fiber push out or pull out tests on the composite, but this 

can be problematic since the value will then depend on the residual stress state in the particular 

composite. Hence, the fiber-matrix frictional sliding shear stress may alternately be thought of as an 

internal parameter. As in the fiber-matrix debonding data, a time delay must be specified in the 

Curtin model data. Finally, as in the subcell failure data, an action to take upon Curtin model failure 

of the fiber must be specified. 

! Note: In order to execute the three cases presented in the results for this example, the appropriate 

lines under *DEBOND must be commented and uncommented. 

6) Output: 

a) Output file print level (*PRINT) [KM_6]: 

Print level: 6 (NPL=6) 

b) x-y plots (*XYPLOT) [KM_6]: 

Frequency: 1 (FREQ=1) 

Number of macro plots: 1 (MACRO=1) 

Macro plot names: example_5f (NAME=example_5f) 

Macro plot x-y quantities: ε11, σ11 (X=1 Y=7) 

Number of micro plots: 1 (MICRO=1) 

Micro plot names: example_5e (NAME=example_5e) 

Micro plot subcell indices: 1, 1 (IA=1 IB=1 IG=1) 

Micro plot x-y quantities: ε11, σ11 (X=1 Y=7) 

7) End of file keyword: (*END) 

Results

Figure 5.13 shows the predicted longitudinal tensile response of the composite at 650 °C for the three 

different cases (perfect fiber, ECI model for the fiber, and Curtin model for the fiber). Both the global 

(composite) stress vs. strain response and the local fiber stress vs. global strain response are plotted for 

each case. In the present case, the ECI and Curtin models give similar results both locally and globally. 

Both models cause the fiber response to diverge from that of the perfect fiber case, reach a maximum, and 

then decrease. Because the present case involves the longitudinal behavior of the continuous fiber 

composite, the fiber response has a dominant influence on the global composite response. Both the ECI 

model and Curtin model composite curves reach a maximum at 173 ksi, which may be considered the 

predicted UTS of the composite. In fact, the following is written to the output file: 

CURTIN FAILURE:

> STRESS = 418.189691475676 X = 0.125681408909315

at the time corresponding to the maximum in the Curtin model composite prediction. Were the ACTION

specifier under *CURTIN set to -1, execution of the code would have stopped at this point. When 

employing the ECI model to simulate fiber breakage, it is not possible to stop execution when this 

maximum in the stress-strain response is reached. After the maximum composite stress is reached, the 

ECI and Curtin model predictions diverge in what is a non-physical domain of the predictions. 

NASA/TM—2002-212077/VOL3 192
Section 5: Damage and Failure Example 5f: Fiber Breakage 

While the ECI and Curtin models give similar results for the present case, the Curtin model is often 

preferable for modeling the longitudinal fiber breakage behavior of composites. This is because most of 

the Curtin model parameters have physical interpretation, whereas the ECI model parameters Λ and Β are 

internal variables with no real physical meaning. As shown by Bednarcyk and Arnold (2001), the ECI 

model debond stress can be assigned based on fiber strength statistics in an RUC containing multiple 

fibers, but the internal parameters Λ and Β still remain to be chosen. The advantage of the ECI model is, 

that by controlling the parameters for each fiber individual, the model can provide some additional 

flexibility. 

-100

0

100

200

300

400

500

0.000 0.005 0.010 0.015 0.020

Applied Global Strain

Stress (ksi)

Composite - Perfect

Composite - ECI

Composite - Curtin

Fiber - Perfect

Fiber - ECI

Fiber - Curtin

Predicted Composite 

UTS = 173 ksi

Figure 5.13 Example 5f: Predicted local and global longitudinal stress-strain response of 35% SiC/Ti21S at 650 °C with fiber breakage.

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MAC/GMC 4.0 Example Problem Manual 

Section 6 : Output and Data Visualization 

The output data output in MAC/GMC 4.0 involves several levels of complexity. The simplest level of 

output is the data written to the MAC/GMC 4.0 output file. As illustrated in several example problems 

(Example 1a and 1b, for instance), this output file contains the effective properties of the composite or 

laminate. Detailed global field results can also be written to the output file at each time step, depending 

on the print level specified. The next level of complexity comes through the use of the *XYPLOT

keyword. This enables generation of laminate, macro, and micro level x-y scatter plot data files 

containing the desired field variable components printed at each time step. These data files can then be 

imported or copied into plotting software (such as Microsoft Excel) to generate the desired x-y scatter 

plots. Many such plots (e.g., stress-strain curves) have been presented in this manual. The final (and 

most complex) level of results data output involves data for generating fringe plots. A fringe plot is a 

color-coded plot of a particular scalar field component over the composite geometry. They thus enable 

visualization of the spatial variation of the field component within the composite. MAC/GMC 4.0 

enables output of fringe plot data through the *PATRAN and *MATLAB keywords. These keywords cause 

the code to write large data files containing the local fields within all of the RUC subcells at various 

(specified) time steps. In addition, data files containing information about the RUC geometry are 

generated. *PATRAN generates data files intended for post-processing with the MSC/PATRAN software 

package, in conjunction with the MACPOST software add-on. Note that the MACPOST software files 

are distributed with MAC/GMC 4.0, and that the MACPOST software has its own User Manual 

(Goldberg et al., 1999). *MATLAB generates data files for post-processing with the MATLAB software 

package. Three MATLAB source files (stress.m, strain.m, and epsp.m) are distributed with 

MAC/GMC 4.0 that generate fringe plots from the MATLAB data files. 

This section presents three example problems dealing with the MAC/GMC 4.0 data output and 

visualization. Example 6a illustrates the use of the code for constructing micro (local) x-y scatter plots. 

Example 6b demonstrates the used of MAC/GMC 4.0 in conjunction with the MSC/PATRAN software 

and the MACPOST add-on to generate fringe plots. Finally, Example 6c shows how MAC/GMC 4.0 

results can be used with the MATLAB software to generate fringe plots. Note that Example Problem 3f 

also constructed MATLAB fringe plots for both GMC and HFGMC. 

NASA/TM—2002-212077/VOL3 194
Section 6: Output and Data Visualization Example 6a: Local x-y Plots 

Example 6a: Local x-y Plots

As illustrated in a number of the previous example problems, access is given to the MAC/GMC 4.0 local 

field quantities through generation of ASCII files representing data of particular subcells. This example 

problem exercises this micro scale x-y plot capability to show how certain micro and macro field 

quantities are related. The simple 2×2 square fiber, square pack RUC architecture is employed to 

represent a continuous 0.25 fiber volume fraction SiC/Ti-21S composite at 650 °C subjected to applied 

transverse strain. 

MAC/GMC Input File: example_6a.mac

MAC/GMC 4.0 Example 6a - Micro x-y plots

*CONSTITUENTS

NMATS=2

M=1 CMOD=6 MATID=E

M=2 CMOD=4 MATID=A

*RUC

MOD=2 ARCHID=1 VF=0.25 F=1 M=2

*MECH

LOP=2

NPT=2 TI=0.,200. MAG=0.,0.02 MODE=1

*THERM

NPT=2 TI=0.,200. TEMP=650.,650.

*SOLVER

METHOD=1 NPT=2 TI=0.,200. STP=1.

*PRINT

NPL=6

*XYPLOT

FREQ=5

MACRO=2

NAME=example_6a X=2 Y=8

NAME=example_6ai X=2 Y=14

MICRO=8

NAME=example_6a_11 IB=1 IG=1 X=2 Y=8

NAME=example_6a_11i IB=1 IG=1 X=2 Y=14

NAME=example_6a_12 IB=1 IG=2 X=2 Y=8

NAME=example_6a_12i IB=1 IG=2 X=2 Y=14

NAME=example_6a_21 IB=2 IG=1 X=2 Y=8

NAME=example_6a_21i IB=2 IG=1 X=2 Y=14

NAME=example_6a_22 IB=2 IG=2 X=2 Y=8

NAME=example_6a_22i IB=2 IG=2 X=2 Y=14

*END

Annotated Input Data

1) Flags: None 

2) Constituent materials (*CONSTITUENTS) [KM_2]: 

Number of materials: 2 (NMATS=2) 

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MAC/GMC 4.0 Example Problem Manual 

Materials: SiC fiber (MATID=E) 

Ti-21S (MATID=A) 

Constitutive models: SiC fiber: linearly elastic (CMOD=6) 

Ti-21S matrix: Isotropic GVIPS (CMOD=4) 

3) Analysis type (*RUC) → Repeating Unit Cell Analysis [KM_3]: 

Analysis model: Doubly periodic GMC (MOD=2) 

RUC architecture: square fiber, square pack (ARCHID=1) 

Fiber volume fraction: 0.25 (VF=0.25) 

Material assignment: SiC fiber (F=1) 

Ti-21S matrix (M=2) 

4) Loading: 

a) Mechanical (*MECH) [KM_4]: 

Loading option: 2 (LOP=2) 

Number of points: 2 (NPT=2) 

Time points: 0., 200. sec. (TI=0.,200.) 

Load magnitude: 0., 0.02 (MAG=0.,0.02) 

Loading mode: strain control (MODE=1) 

b) Thermal (*THERM) [KM_4]: 

Number of points: 2 (NPT=2) 

Time points: 0., 200. sec. (TI=0.,200.) 

Temperature points: 650., 650. °C (TEMP=650.,650.) 

c) Time integration (*SOLVER) [KM_4]: 

Time integration method: Forward Euler (METHOD=1) 

Number of points: 2 (NPT=2) 

Time points: 0., 200. sec. (TI=0.,200.) 

Time step sizes: 1. sec. (STP=1.) 

5) Damage and Failure: None 

6) Output: 

a) Output file print level (*PRINT) [KM_6]: 

Print level: 6 (NPL=6) 

b) x-y plots (*XYPLOT) [KM_6]: 

*XYPLOT

FREQ=5

MACRO=2

NAME=example_6a X=2 Y=8

NAME=example_6ai X=2 Y=14

MICRO=8

NAME=example_6a_11 IB=1 IG=1 X=2 Y=8

NAME=example_6a_11i IB=1 IG=1 X=2 Y=14

NAME=example_6a_12 IB=1 IG=2 X=2 Y=8

NAME=example_6a_12i IB=1 IG=2 X=2 Y=14

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Section 6: Output and Data Visualization Example 6a: Local x-y Plots 

NAME=example_6a_21 IB=2 IG=1 X=2 Y=8

NAME=example_6a_21i IB=2 IG=1 X=2 Y=14

NAME=example_6a_22 IB=2 IG=2 X=2 Y=8

NAME=example_6a_22i IB=2 IG=2 X=2 Y=14

Frequency: 5 (FREQ=5) 

Number of macro plots: 2 (MACRO=2) 

Macro plot names: example_6a (NAME=example_6a) 

 example_6ai (NAME=example_6ai) 

Macro plot x-y quantities: ε22, σ22 (X=2 Y=8) 

 ε22, i

22 ε (X=2 Y=14) 

Number of micro plots: 8 (MICRO=8) 

Micro plot names: example_6a_11 (NAME=example_6a_11) 

 example_6a_11i (NAME=example_6a_11i) 

 example_6a_12 (NAME=example_6a_12) 

 example_6a_12i (NAME=example_6a_12i) 

 example_6a_21 (NAME=example_6a_21) 

 example_6a_21i (NAME=example_6a_21i) 

 example_6a_22 (NAME=example_6a_22) 

 example_6a_22i (NAME=example_6a_22i) 

Micro plot subcell indices: 1, 1 (IB=1 IG=1) 

 1, 2 (IB=1 IG=2) 

 2, 1 (IB=2 IG=1) 

 2, 2 (IB=2 IG=2) 

Micro plot x-y quantities: ε22, σ22 (X=2 Y=8) 

 ε22, i

22 ε (X=2 Y=14) 

The major difference in the data specification between micro and macro plots is, with micro plots, the 

subcell for which the data are to be plotted must be specified. In the present doubly periodic GMC 

example, the βγ indices are specified. In the case of triply periodic RUC analysis, the three αβγ

indices are specified. Finally, in the case of laminate analysis, the layer from which the subcell is 

taken must be specified. In addition, there are some micro scale quantities that may be specified for 

x-y plot file generation that cannot be specified for macro scale x-y plots (e.g., constitutive model 

state variables, debonding parameters). For more information on micro x-y plots, see the MAC/GMC 

4.0 Keywords Manual Section 6. 

7) End of file keyword: (*END) 

Results

Figure 6.1 shows the RUC employed in the present example plus the βγ indices of each subcell and the 

applied loading. Figure 6.2 is a plot of the local and global σ22-ε22 response of the SiC/Ti-21S composite. 

Examining the four subcell curves, the Subcell 11 curve, which is associated with the fiber, is stiff, 

elastic, and experiences a large local σ22 stress. The subcell that is in series with the fiber along the 

loading direction (Subcell 12) experiences the same σ22 stress as the fiber subcell (at a given point in 

time), but, since the material associated with the subcell is Ti-21S, the subcell response is more compliant 

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MAC/GMC 4.0 Example Problem Manual 

and exhibits inelastic deformation. The remaining two subcells (21 and 22), which are in parallel with the 

fiber (with respect to the loading direction) experience far less σ22 stress than do subcells 11 and 12. 

While the σ22 stress is the same for subcells 21 and 22 (at a given point in time), the ε22 strain experienced 

by the subcell adjacent to the fiber (Subcell 21) is significantly higher. 

A final point illustrated by Figure 6.2 involves the volume averaging of the subcell stresses and strains. 

As shown, if the four subcell σ22 stress and ε22 strain values are averaged in a volume-weighted sense at 

each point, the resulting averaged σ22-ε22 curve is identical to the macro (composite) σ22-ε22 curve plotted 

directly from the macro MAC/GMC 4.0 x-y plot file. The fact that the volume weighted micro stress and 

total strain components sum to the macro stress and total strain components is an important attribute of a 

micromechanics model and is often considered a consistency condition. In fact, the formulation of GMC 

enforces this consistency condition. 

In contrast, Figure 6.3 plots the local subcell inelastic strain i

22 ε vs. the local total strain ε22. The fiber 

subcell (11) exhibits no inelastic strain as the fiber material is elastic. The subcells adjacent to the fiber 

(12 and 21) exhibit a great deal of inelastic strain, while Subcell 22 exhibits less. However, when the 

volume weighted average of the subcell inelastic strain i

22 ε values is taken at each point, the resulting 

curve does not correspond to the macro (composite) level curve plotted directly from the MAC/GMC 4.0 

macro x-y plot file. Thus, while a consistency condition is applicable to the total strain components, such 

a condition does not exist for the inelastic strain components. Similarly, since the total strain is the sum 

of the elastic, inelastic, and thermal strains, the elastic and thermal strains do not obey a consistency 

condition either. 

Figure 6.1 MAC/GMC 4.0 2×2 doubly periodic repeating unit cell with subcell indices identified.

NASA/TM—2002-212077/VOL3 198
Section 6: Output and Data Visualization Example 6a: Local x-y Plots 

0

10

20

30

40

50

60

70

80

90

0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040

ε22

σ22 (ksi)

Macro

Volume Averaged

Subcell 11

Subcell 12

Subcell 21

Subcell 22

Figure 6.2 Example 6a: Macro (composite) and micro (subcell) stress-strain curves for 25% SiC/Ti21S at 650 °C.

0.000

0.002

0.004

0.006

0.008

0.010

0.012

0.014

0.016

0.018

0.020

0.000 0.005 0.010 0.015 0.020 0.025

Total ε22

Inelastic ε22

Macro

Volume Averaged

Subcell 11

Subcell 12

Subcell 21

Subcell 22

Figure 6.3 Example 6a: Macro (composite) and micro (subcell) inelastic strain vs. total strain plots for 

25% SiC/Ti-21S at 650 °C.

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MAC/GMC 4.0 Example Problem Manual 

Example 6b: PATRAN Post-Processing 

This example problem demonstrates how to obtain full access to all micro and macro field data that can 

be produced by MAC/GMC 4.0 for use in conjunction with the MSC/PATRAN software package. This 

enables the generation of fringe plots, which display the spatial variation of the predicted field 

components over the composite geometry. The MSC/PATRAN software is available from 

http://www.mscsoftware.com/. In order to generate fringe plots from the MAC/GMC 4.0 software using 

MSC/PATRAN, an add-on to the software, known as MACPOST, is needed. MACPOST is distributed 

with MAC/GMC 4.0 and has its own User Manual (Goldberg et al., 1999). As indicated in the 

MACPOST User Manual, the add-on is capable of generating local and global x-y plots in addition to the 

fringe plot capabilities illustrated here. An advantage of using PATRAN to generate fringe plots is that a 

point can be chosen from the macro response at which to examine the micro response via fringe plots. 

This linkage can be used to gain some additional insight into the local mechanisms that affect the global 

composite response. The present example problem generates fringe plots for a triply periodic 0.25 

volume fraction short fiber SiC/Ti-21S composite subjected to a stress-free cool-down followed by 

applied transverse strain. The evolution of the J2 stress invariant and an inelastic strain component are 

displayed in the fringe plots during application of simulated transverse tensile loading. 

MAC/GMC Input File: example_6b.mac

MAC/GMC 4.0 Example 6b - PATRAN post-processing

*CONSTITUENTS

NMATS=2

M=1 CMOD=6 MATID=E

M=2 CMOD=4 MATID=A

*RUC

MOD=3 ARCHID=3 VF=0.25 ASP1=2. ASP2=1. DR=1. F=1 M=2

*MECH

LOP=2 REFTIME=57600.

NPT=3 TI=0.,57600.,57800. MAG=0.,0.,0.02 MODE=2,1

*THERM

NPT=3 TI=0.,57600.,57800. TEMP=900.,23.,23.

*SOLVER

METHOD=1 NPT=3 TI=0.,57600.,57800. STP=40.,1.

*PRINT

NPL=6

*PATRAN

TPRE=57600. STP=20

*XYPLOT

FREQ=5

MACRO=1

NAME=example_6b X=2 Y=8

MICRO=0

*END

Annotated Input Data

1) Flags: None 

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Section 6: Output and Data Visualization Example 6b: PATRAN Post-Processing 

2) Constituent materials (*CONSTITUENTS) [KM_2]: 

Number of materials: 2 (NMATS=2) 

Materials: SiC fiber (MATID=E) 

Ti-21S (MATID=A) 

Constitutive models: SiC fiber: linearly elastic (CMOD=6) 

Ti-21S matrix: Isotropic GVIPS (CMOD=4) 

3) Analysis type (*RUC) → Repeating Unit Cell Analysis [KM_3]: 

Analysis model: Triply periodic GMC (MOD=3) 

RUC architecture: Off-set short fibers, diagonal array (ARCHID=3) 

Fiber volume fraction: 0.25 (VF=0.25) 

Fiber aspect ratio: 2. (ASP1=2.) 

Unit cell aspect ratio: 1. (ASP2=1.) 

D ratio: 1. (DR=1.) 

Material assignment: SiC fiber (F=1) 

Ti-21S matrix (M=2) 

4) Loading: 

a) Mechanical (*MECH) [KM_4]: 

Loading option: 2 (LOP=2) 

Strain reference time: 57600. sec. (REFTIME=57600.) 

Number of points: 3 (NPT=3) 

Time points: 0., 57600., 57800. sec. (TI=0.,57600,57800.) 

Load magnitude: 0., 0., 0.02 (MAG=0.,0.,0.02) 

Loading mode: stress/strain control (MODE=2,1) 

b) Thermal (*THERM) [KM_4]: 

Number of points: 3 (NPT=3) 

Time points: 0., 57600., 57800. sec. (TI=0.,57600.,57800.) 

Temperature points: 900., 23., 23. °C (TEMP=900.,23.,23.) 

c) Time integration (*SOLVER) [KM_4]: 

Time integration method: Forward Euler (METHOD=1) 

Number of points: 3 (NPT=3) 

Time points: 0., 57600., 57800. sec. (TI=0.,57600.,57800.) 

Time step sizes: 40., 1. sec. (STP=40.,1.) 

5) Damage and Failure: None 

6) Output: 

a) Output file print level (*PRINT) [KM_6]: 

Print level: 6 (NPL=6) 

b) PATRAN output (*PATRAN) [KM_6]: 

PATRAN preloading time: 57600. sec. (TPRE=57600.) 

Output step frequency: 20 (STP=20) 

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MAC/GMC 4.0 Example Problem Manual 

Two specifiers control the PATRAN output. A preloading time (TPRE) defines the time at which 

writing of PATRAN output begins. The output time step frequency (STP) specifies how often the 

PATRAN output is written. In the present example, the code writes PATRAN output every 20 time 

steps starting at time = 57600 sec. Thus, during the 200 time steps occurring during application of 

the mechanical loading, PATRAN output is written a total of 11 times. 

MAC/GMC 4.0 generates 14 PATRAN output ASCII files. These 14 files are: 

outfile.macgeo RUC geometry data 

outfile.total_pat.data Number of time steps and subcells 

outfile.macro1_pat.data RUC level strain data at each output time 

outfile.macro2_pat.data RUC level stress data at each output time 

outfile.macro3_pat.data RUC level inelastic strain data at each output time 

outfile.macro4_pat.data RUC level thermal strain, creep time, temperature, and stress 

invariant data at each output time 

outfile.micro1_pat.data Subcell level strain data at each output time 

outfile.micro2_pat.data Subcell level stress data at each output time 

outfile.micro3_pat.data Subcell level inelastic strain data at each output time 

outfile.micro4_pat.data Subcell level thermal strain, creep time, temperature, and 

stress invariant data at each output time 

outfile.micro1_pat.contour Subcell level strain data for fringe plots 

outfile.micro2_pat.contour Subcell level stress data for fringe plots 

outfile.micro3_pat.contour Subcell level inelastic strain data for fringe plots 

outfile.micro4_pat.contour Subcell level thermal strain, creep time, temperature, and 

stress invariant data for fringe plots 

where outfile is the name of the MAC/GMC 4.0 output file. In order to employ these ASCII files 

within PATRAN via MACPOST, all files but the first (outfile.macgeo) must be renamed. The 

leading ìoutfile.î must be stripped from the filename. For more information on the MAC/GMC 

4.0 PATRAN output capabilities, see the MAC/GMC 4.0 Keywords Manual Section 6. For more 

information on using MACPOST, see the MACPOST User Guide (Goldberg et al., 1999). 

c) x-y plots (*XYPLOT) [KM_6]: 

Frequency: 5 (FREQ=5) 

Number of macro plots: 1 (MACRO=1) 

Macro plot names: example_6b (NAME=example_6b) 

Macro plot x-y quantities: ε22, σ22 (X=2 Y=8) 

Number of micro plots: 0 (MICRO=0) 

7) End of file keyword: (*END) 

Results

Plotted in Figure 6.4 is the overall (global) transverse stress-strain response of the discontinuous SiC/Ti21S composite considered in this example problem. Note that this plot is generated from the macro x-y 

plot file data. The arrows in this figure indicate the six points at which PATRAN fringe plots have been 

generated. Note that, although only six points have been selected, the PATRAN output files actually 

contain data for 11 points (i.e., fringe plots have only been generated for every other PATRAN output 

time). 

NASA/TM—2002-212077/VOL3 202
Section 6: Output and Data Visualization Example 6b: PATRAN Post-Processing 

The fringe plots generated using PATRAN with MACPOST are shown in Figure 6.5 and Figure 6.6. 

Figure 6.5 shows the evolution of the inelastic strain component corresponding with the direction of the 

applied loading ( in

22 ε ). Each of the six fringe plots shown in this figure were generated individually using 

the MSC/PATRAN software. Note that these fringe plots were stored as bitmap images, and some 

manipulation was performed beyond that of the raw PATRAN fringe plot display. Below each fringe 

plot, the corresponding applied global (composite level) transverse strain at which the inelastic strain 

component field is plotted is indicated. 0 ε 22 = corresponds to the residual inelastic strain component 

field in the composite after the applied cool-down but before the application of any simulated mechanical 

loading. Note that, in this first fringe plot, the subcells that are occupied by the SiC fiber material are 

outlined in red. This ARCHID=3 geometry can also be seen in Figure 3.8. 

Figure 6.5 indicates that during the cool-down, only the matrix subcells that are between the fibers exhibit 

noticeable yielding. It is not until the applied global stain reaches 0.008 that the fringe plot shows a 

noticeable change in the inelastic strain component field. In Figure 6.4, it is clear that this is near the 

point at which global yielding of the composite is occurring. As the global loading continues to increase, 

additional subcells yield and the inelastic strain component in the yielded subcells continues to rise. By 

the end of the applied loading ( 020.0 ε 22 = ), all matrix subcell exhibit noticeable in

22 ε values, with the 

inelastic strain component magnitudes being highest in the matrix subcells in between the fibers along the 

loading direction. 

Figure 6.6 shows the evolution of the J2 stress invariant during the applied global transverse mechanical 

loading. The J2 stress invariant is a von Mises type stress and is thus related to the inelastic deformation 

in the composite. Further, it is a scalar invariant quantity that takes into account all stress components. 

As the simulated mechanical loading increases, the J2 stress increases in the composite. The highest 

magnitude J2 occurs in the fibers and the matrix in between the fibers. As observed in Figure 6.5, this 

matrix located in between the fibers exhibits the highest in

22 ε magnitude as well. 

0

50

100

150

200

250

0.000 0.005 0.010 0.015 0.020 0.025

strain

stress (ksi)

Fields plotted at these points

Figure 6.4 Example 6b: Simulated transverse tensile stress-strain response of 25% fiber volume 

fraction discontinuous SiC/Ti-21S at 23 °C. The arrows indicate the points at which the 

fields are plotted using the MSC/PATRAN software package in Figure 6.5 and Figure 6.6. 

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MAC/GMC 4.0 Example Problem Manual 

0 ε 22 = 004.0 ε 22 =

008.0 ε 22 = 012.0 ε 22 =

016.0 ε 22 = 020.0 ε 22 =

Figure 6.5 Example 6b: Inelastic strain component in

22 ε at six applied global strain levels for the 25% 

fiber volume fraction discontinuous SiC/Ti-21S composite at 23 °C as plotted using the 

MSC/PATRAN software package in conjunction with MACPOST. The fiber subcells are 

outlined in red in the figure for the lowest applied strain level.

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Section 6: Output and Data Visualization Example 6b: PATRAN Post-Processing 

0 ε 22 = 004.0 ε 22 =

008.0 ε 22 = 012.0 ε 22 =

016.0 ε 22 = 020.0 ε 22 =

Figure 6.6 Example 6b: ijij SSJ

2

3

2 = stress invariant at six applied global strain levels for the 25% 

fiber volume fraction discontinuous SiC/Ti-21S composite at 23 °C as plotted using the 

MSC/PATRAN software package in conjunction with MACPOST. The fiber subcells are 

outlined in red in the figure for the lowest applied strain level.

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MAC/GMC 4.0 Example Problem Manual 

Example 6c: MATLAB Post-Processing 

This example problem demonstrates the use of MAC/GMC 4.0ís capability to generate ASCII data files 

that can be used to generate local fringe plots using the MATLAB software package. MATLAB is a 

commonly-used commercial engineering and science oriented software package (see 

http://www.mathworks.com/) and must be independently licensed. Three MATLAB source files are 

shipped with the MAC/GMC 4.0 software. These are: stress.m, strain.m, and epsp.m. These files 

can be used within the MATLAB software to generate fringe plots from the MAC/GMC 4.0 MATLAB 

ASCII files. The user is free to customize these files in order to generate the plots desired. 

The present example considers a 0.35 fiber volume fraction SiC/Ti-21S composite as both a cross-ply 

laminate and a unidirectional composite. The 7×7 circular fiber cross-section approximation RUC 

architecture is employed in both cases, and loading takes the form of a stress-free cool-down. The local 

residual fields for the unidirectional composite and the laminate are then compared using the MATLAB 

software fringe plots. 

MAC/GMC Input File: example_6c.mac

MAC/GMC 4.0 Example 6c - MATLAB post-processing

*CONSTITUENTS

NMATS=2

M=1 CMOD=6 MATID=E

M=2 CMOD=4 MATID=A

*LAMINATE

NLY=3

LY=1 THK=0.25 ANG=0. MOD=2 ARCHID=6 VF=0.35 R=1. F=1 M=2

LY=2 THK=0.50 ANG=90. MOD=2 ARCHID=6 VF=0.35 R=1. F=1 M=2

LY=3 THK=0.25 ANG=0. MOD=2 ARCHID=6 VF=0.35 R=1. F=1 M=2

#*RUC

# MOD=2 ARCHID=6 VF=0.35 R=1. F=1 M=2

*THERM

NPT=2 TI=0.,57600. TEMP=900.,23.

*SOLVER

METHOD=1 NPT=2 TI=0.,57600. STP=40.

*PRINT

NPL=6

*MATLAB

N=1 TIMES=57600.

*XYPLOT

FREQ=40

LAMINATE=1

NAME=example_6c X=100 Y=1

MACRO=0

# MACRO=2

# NAME=example_6c_1 X=100 Y=1

# NAME=example_6c_2 X=100 Y=2

MICRO=0

*END

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Section 6: Output and Data Visualization Example 6c: MATLAB Post-Processing 

Annotated Input Data

1) Flags: None 

2) Constituent materials (*CONSTITUENTS) [KM_2]: 

Number of materials: 2 (NMATS=2) 

Materials: SiC fiber (MATID=E) 

Ti-21S (MATID=A) 

Constitutive models: SiC fiber: linearly elastic (CMOD=6) 

Ti-21S matrix: Isotropic GVIPS (CMOD=4) 

3) Analysis type: 

Cross Ply Laminate (*LAMINATE) → Laminate Analysis [KM_3]: 

Number of layers: 3 (NLY=3) 

Layer Analysis 

Model 

Thickness Fiber 

Angle 

Architecture Aspect 

Ratio 

Volume 

fraction 

Fiber 

material 

Matrix 

material 

(LY=) (MOD) (THK) (ANG) (ARCHID) (R) (VF) (F) (M) 

1 GMC-2D 0.25 0° 7×7 circle 

rect. pack 

1. 0.35 SiC Ti-21S 

2 GMC-2D 0.50 90° 7×7 circle 

rect. pack 

1. 0.35 SiC Ti-21S 

3 GMC-2D 0.25 0° 7×7 circle 

rect. pack 

1. 0.35 SiC Ti-21S 

Unidirectional Composite (*RUC) → Repeating Unit Cell Analysis [KM_3]: 

Analysis model: Doubly periodic GMC (MOD=2) 

RUC architecture: 7×7 circle approx., rect. pack (ARCHID=6) 

Fiber volume fraction: 0.35 (VF=0.35) 

RUC aspect ratio: 1. (square pack) (R=1.) 

Material assignment: SiC fiber (F=1) 

Ti-21S matrix (M=2) 

! Note: In order to execute the code for the laminate and the unidirectional composite, the appropriate 

lines in the input file must be commented and uncommented. 

4) Loading: 

a) Mechanical (*MECH): None 

b) Thermal (*THERM) [KM_4]: 

Number of points: 2 (NPT=2) 

Time points: 0., 57600. sec. (TI=0.,57600.) 

Temperature points: 900., 23. °C (TEMP=900.,23.) 

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MAC/GMC 4.0 Example Problem Manual 

c) Time integration (*SOLVER) [KM_4]: 

Time integration method: Forward Euler (METHOD=1) 

Number of points: 2 (NPT=2) 

Time points: 0., 57600. sec. (TI=0.,57600.) 

Time step sizes: 40. sec. (STP=40.) 

5) Damage and Failure: None 

6) Output: 

a) Output file print level (*PRINT) [KM_6]: 

Print level: 6 (NPL=6) 

b) MATLAB output (*MATLAB) [KM_6]: 

Number of MATLAB output times: 1 (N=1) 

MATLAB output times: 57600. sec. (TIMES=57600.) 

As shown above, the number of times desired for writing MATLAB output, and the times 

themselves, must be specified. MAC/GMC 4.0 generates five MATLAB output ASCII files for RUC 

analysis. In the case of laminate analysis, five separate files are output for each layer. These five 

files are: 

outfile_x2.dat → contains x2 direction (β) grid coordinates

outfile_x3.dat → contains x3 direction (γ) grid coordinates

outfile_sig.dat → contains local stresses

outfile_eps.dat → contains local strains

outfile_epsp.dat → contains local inelastic strains (plus material numbers) 

where outfile is the name of the MAC/GMC 4.0 output file. In the case of laminate analysis, for 

layer numbers greater than 1, the layer number is appended to the file name root (e.g., 

outfile_sig_2.dat). In the case of RUC analysis, the local data for all specified TIMES is 

written to the same ASCII file, with the time value written prior to the data for each time. In the case 

of laminate analysis, the integration point number is also written on the same line as the time. The 

three MATLAB source files distributed with MAC/GMC 4.0 (stress.m, strain.m, and epsp.m) 

do not read these five ASCII files directly, rather they read the files: x2.dat, x3.dat, sig.dat, 

eps.dat, and epsp.dat. Thus, the ASCII files generated must be renamed. Further, the 

MATLAB source files only read the local data for a single time. Thus, the files sig.dat, 

eps.dat, and epsp.dat should contain only the local data for a single time (and integration 

point); the line on which the time (and integration point number) are written should not be present. 

For more information on the MAC/GMC 4.0 MATLAB output capabilities, see the MAC/GMC 4.0 

Keywords Manual Section 6. 

c) x-y plots (*XYPLOT) [KM_6]: 

Frequency: 40 (FREQ=40) 

Number of laminate plots: 1 (LAMINATE=1) 

Laminate plot name: example_6c (NAME=example_6c) 

Laminate plot x-y quantities: temperature, 0

xx ε (X=100 Y=1) 

Number of macro plots: 2 (MACRO=2) 

Macro plot names: example_6c_1 (NAME=example_6c_1) 

 example_6c_2 (NAME=example_6c_2) 

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Section 6: Output and Data Visualization Example 6c: MATLAB Post-Processing 

Macro plot x-y quantities: temperature, ε11 (X=100 Y=1) 

 temperature, ε22 (X=100 Y=2) 

Number of micro plots: 0 (MICRO=0) 

! Note: In the case of laminate analysis, the lines of the input file associated with the macro c-y plots 

must be commented. Likewise, in the case of RUC analysis, the input file lines associated with 

the laminate level x-y plots must be commented. 

7) End of file keyword: (*END) 

Results

Figure 6.7 shows the predicted global strain response of the laminate and the unidirectional composite to 

the imposed thermal loading. The unidirectional composite exhibits significantly more strain in the 

transverse (x2) direction compared to the longitudinal (x1) direction. The response of the cross-ply 

laminate is identical in both the x and y directions, and the associated curve in Figure 6.7 falls in between 

the two curves that represent the unidirectional composite response. 

In this example problem, MATLAB data was written at the end of the cool-down, at 23 °C. The stress 

and inelastic strain component fields, which represent residual fields in the composite and laminate due to 

the imposed stress-free thermal cool-down, are plotted in Figure 6.8 and Figure 6.9. It should be noted 

that the x3-direction corresponds to the through-thickness (z) direction of the laminate (see Figure 1.2). 

From Figure 6.8 it is apparent that the stresses are noticeably higher in the cross-ply laminate as compared 

to the unidirectional composite. The presence of the stiff fibers along the x and y directions in the 

laminate act to restrain the thermal expansion of the composite, giving rise to the larger stresses. Figure 

6.9 shows that, transverse to the fiber direction (x2 and x3 directions), the higher stresses present in the 

laminate give rise to higher tensile and compressive inelastic strain concentrations in certain subcells. In 

the unidirectional composite, the transverse inelastic strain fields present in the x2 and x3 directions are 

identical (but rotated 90°). In the fiber (x1) direction, on the other hand, the restraint effect of the 

orthogonal layers of the laminate limits the matrix inelastic strain. Thus, the overall magnitude of in

11 ε is 

lower in the laminate compared to the unidirectional composite. 

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MAC/GMC 4.0 Example Problem Manual 

-0.009

-0.008

-0.007

-0.006

-0.005

-0.004

-0.003

-0.002

-0.001

0

0 100 200 300 400 500 600 700 800 900 1000

Temperature (°C)

Strain

Unidirectional Composite - EPS11

Unidirectional Composite - EPS22

Cross-Ply Laminate

Local fields 

plotted at 23 °C

Figure 6.7 Example 6c: Predicted global response of unidirectional and cross-ply 35% SiC/Ti-21S 

during a stress-free cool-down.

NASA/TM—2002-212077/VOL3 210
Section 6: Output and Data Visualization Example 6c: MATLAB Post-Processing 

Figure 6.8 Predicted residual stress fields (plotted from MATLAB output data) for a unidirectional 

35% SiC/Ti-21S composite and each ply within a cross-ply 35% SiC/Ti-21S laminate. 

Note that the fields in each ply of the laminate are identical in its local coordinate system.

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MAC/GMC 4.0 Example Problem Manual 

Figure 6.9 Predicted residual inelastic fields (plotted from MATLAB output data) for a unidirectional 

35% SiC/Ti-21S composite and each ply within a cross-ply 35% SiC/Ti-21S laminate. 

Note that the fields in each ply of the laminate are identical in its local coordinate system.

NASA/TM—2002-212077/VOL3 212
Section 7: Advanced Topics 

Section 7 : Advanced Topics 

In this Section, example problems are presented that deal with advanced topics within MAC/GMC 4.0. 

These advanced topics have been designated as such because they are newer to the code and thus less well 

established than many of the other topics dealt with in this manual. Example 7a presents the use of a new 

multimechanism GVIPS constitutive model. This model includes not only material viscoplastic effects, 

but also material viscoelastic effects. Because it employs a local tangent stiffness matrix (while globally, 

MAC/GMC 4.0 does not) along with a local implicit integration scheme (which is unconditionally stable), 

the multimechanism GVIPS model requires global equilibrium iterations (specified under the *SOLVER

keyword). Non- U.S. government users of MAC/GMC 4.0 will need to obtain an additional software 

license in order to use the multimechanism GVIPS constitutive model. 

Examples 7b and 7c present the use of the new MAC/GMC 4.0 electromagnetic analysis capabilities to 

analyze a smart composite material and a smart composite laminate. A good deal of additional material 

input data is required when the constituent materials are electromagnetically active. The final two 

Example Problems in this section, 7d and 7e, illustrate the analysis of woven composites using 

MAC/GMC 4.0. Example 7d shows a standard one step approach to the woven composite simulation, 

while Example 7e presents a new and more accurate two step approach. 

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MAC/GMC 4.0 Example Problem Manual 

Example 7a: Implicitly Integrated Multimechanism GVIPS

This example problem illustrates the use of a new multimechanism GVIPS constitutive model (see Saleeb 

et al., 2001) that has been implemented within MAC/GMC 4.0. This constitutive model incorporates an 

arbitrary number of viscoelastic and viscoplastic mechanisms, giving it a wide range of applicability. 

While this model can be characterized for any material, currently, only material parameters for Ti-21S 

have been implemented within MAC/GMC 4.0. The multimechanism GVIPS model has an additional 

distinction from the standard isotropic GVIPS model available within MAC/GMC 4.0 in that it is 

implicitly integrated on the local level. That is, the model, on the scale of the subcell, returns a new local 

stress and inelastic strain state rather than the usual local inelastic strain increments. In the standard 

isotropic GVIPS model available within MAC/GMC 4.0, the local inelastic strain increments returned by 

the constitutive model are, in contrast, explicitly integrated (to obtain inelastic strains) outside of the 

model itself. As discussed earlier, global equilibrium iterations are necessary at each increment of the 

applied simulated loading (as with the incremental plasticity model, see Example 2c). 

A major benefit associated with the implicitly integrated multimechanism GVIPS model is that it is 

unconditionally stable, regardless of the step size employed to apply the simulated loading. Thus, unlike 

the previous version of GVIPS available within MAC/GMC 4.0, a very large time step size can be 

employed in cases that utilize this new constitutive model. This example problem illustrates this point by 

comparing simulations of SiC/Ti-21S composites using both GVIPS constitutive models for the Ti-21S 

matrix. 

! Note: In order for non- U.S. government users to access the multimechanism GVIPS capabilities, an 

additional software license is required. 

MAC/GMC Input File: example_7a.mac

MAC/GMC 4.0 Example 7a - Implicitly Integrated Multimechanism GVIPS

*CONSTITUENTS

NMATS=2

M=1 CMOD=6 MATID=E

# -- GVIPS

# M=2 CMOD=4 MATID=A

# -- Multimechanism GVIPS

M=2 CMOD=22 MATID=A

*RUC

MOD=2 ARCHID=6 VF=0.25 R=1. F=1 M=2

*MECH

LOP=2 REFTIME=57600.

NPT=3 TI=0.,57600.,57800. MAG=0.,0.,0.02 MODE=2,1

*THERM

NPT=3 TI=0.,57600.,57800. TEMP=900.,23.,23.

*SOLVER

# -- Multimechanism GVIPS Cases

# METHOD=1 NPT=3 TI=0.,57600.,57800. STP=5760.,1. ERR=0.001 ITMAX=100

# METHOD=1 NPT=3 TI=0.,57600.,57800. STP=5760.,2.5. ERR=0.001 ITMAX=100

# METHOD=1 NPT=3 TI=0.,57600.,57800. STP=5760.,4. ERR=0.001 ITMAX=100

# METHOD=1 NPT=3 TI=0.,57600.,57800. STP=5760.,8. ERR=0.001 ITMAX=100

METHOD=1 NPT=3 TI=0.,57600.,57800. STP=5760.,20. ERR=0.001 ITMAX=100

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Section 7: Advanced Topics Example 7a: Implicitly Integrated Multimechanism GVIPS 

# METHOD=1 NPT=3 TI=0.,57600.,57800. STP=5760.,50. ERR=0.001 ITMAX=100

# METHOD=1 NPT=3 TI=0.,57600.,57800. STP=5760.,100. ERR=0.001 ITMAX=100

# -- GVIPS Cases

# METHOD=1 NPT=3 TI=0.,57600.,57800. STP=40.,1. ERR=0.001 ITMAX=100

# METHOD=1 NPT=3 TI=0.,57600.,57800. STP=40.,2.5 ERR=0.001 ITMAX=100

# METHOD=1 NPT=3 TI=0.,57600.,57800. STP=40.,4. ERR=0.001 ITMAX=100

# METHOD=1 NPT=3 TI=0.,57600.,57800. STP=40.,5. ERR=0.001 ITMAX=100

# METHOD=1 NPT=3 TI=0.,57600.,57800. STP=40.,8. ERR=0.001 ITMAX=100

*PRINT

NPL=6

*XYPLOT

FREQ=1

MACRO=2

NAME=example_7a X=2 Y=8

NAME=example_7a_th X=100 Y=1

MICRO=0

*END

Annotated Input Data

1) Flags: None 

2) Constituent materials (*CONSTITUENTS) [KM_2]: 

Number of materials: 2 (NMATS=2) 

Materials: SiC fiber (MATID=E) 

Ti-21S (MATID=A) 

Constitutive models: SiC fiber: linearly elastic (CMOD=6) 

Ti-21S matrix: Isotropic GVIPS (CMOD=4) 

Multimechanism GVIPS (CMOD=22) 

! Note: To generate all results in this example problem, the appropriate lines in the input file must be 

commented and uncommented. 

3) Analysis type (*RUC) → Repeating Unit Cell Analysis [KM_3]: 

Analysis model: Doubly periodic GMC (MOD=2) 

RUC architecture: 7×7 circle approx. (ARCHID=6) 

Fiber volume fraction: 0.25 (VF=0.25) 

RUC aspect ratio: 1. (square pack) (R=1.) 

Material assignment: SiC fiber (F=1) 

Ti-21S matrix (M=2) 

4) Loading: 

a) Mechanical (*MECH) [KM_4]: 

Loading option: 2 (LOP=2) 

Strain reference time: 57600. sec. (REFTIME=57600.) 

Number of points: 3 (NPT=3) 

Time points: 0., 57600., 57800. sec. (TI=0.,57600,57800.) 

Load magnitude: 0., 0.02 (MAG=0.,0.02) 

Loading mode: stress/strain control (MODE=2,1) 

NASA/TM—2002-212077/VOL3 215
MAC/GMC 4.0 Example Problem Manual 

b) Thermal (*THERM) [KM_4]: 

Number of points: 3 (NPT=3) 

Time points: 0., 57600., 57800. sec. (TI=0.,57600.,57800.) 

Temperature points: 900., 23., 23. (TEMP=900.,23.,23.) 

c) Time integration (*SOLVER) [KM_4]: 

Time integration method: Forward Euler (METHOD=1) 

Number of time points: 3 (NPT=3) 

Time points: 0., 57600., 57800. sec. (TI=0.,57600.,57800.) 

Time step sizes: variable (STP=*) 

Error tolerance for global iteration: 0.001 (ERR=0.001) 

Maximum number of iterations: 100 (ITMAX=100) 

The error tolerance (ERR) is the maximum allowable fractional difference between a global (cell or 

laminate level) energy increment value between iterations. Note, to generate all results presented in 

this example problem, the appropriate lines in the input file must be commented and uncommented. 

For more information on the implicitly integrated GVIPS constitutive model and its time integration 

control, see the MAC/GMC 4.0 Keywords Manual Sections 2 and 4 and the MAC/GMC 4.0 Theory 

Manual Section 4.4.3. 

5) Damage and Failure: None 

6) Output: 

a) Output file print level (*PRINT) [KM_6]: 

Print level: 6 (NPL=6) 

b) x-y plots (*XYPLOT) [KM_6]: 

Frequency: 1 (FREQ=1) 

Number of macro plots: 2 (MACRO=2) 

Macro plot name: example_7a (NAME=example_7a) 

 example_4h_th (NAME=example_7a_th) 

Macro plot x-y quantities: ε22, σ22 (X=2 Y=8) 

 temperature, ε11 (X=100 Y=1) 

Number of micro plots: 0 (MICRO=0) 

7) End of file keyword: (*END) 

Results

Figure 7.1 shows the simulated transverse tensile response of the SiC/Ti-21S composite wherein the 

standard isotropic GVIPS constitutive model (CMOD=4) has been employed to represent the matrix. As 

this model is explicitly integrated in time, a small time step is required in order to achieve convergence. 

Figure 7.1 illustrates the effect of using a successively coarser time step to apply the 0.02 global 

transverse strain over 200 seconds. Convergence is achieved for time step sizes of 1 and 2.5 seconds, but 

for a time step of 4 seconds, some oscillations are present in the simulated stress-strain curve. 

NASA/TM—2002-212077/VOL3 216
Section 7: Advanced Topics Example 7a: Implicitly Integrated Multimechanism GVIPS 

These oscillations become more apparent for a time step of 5 seconds. Finally, for a time step of 

8 seconds, the solution diverges (causing the code to fail). 

Figure 7.2 shows the same simulations for the SiC/Ti-21S composite, but now the new implicitly 

integrated multimechanism GVIPS constitutive model (CMOD=22) has been employed for the matrix. 

Employing a time step up to and including 20 seconds allows the simulated stress-strain curve to be well 

represented without divergence or oscillations. A time step size of 50 seconds still does a reasonable job 

of reproducing the stress-strain curve predicted using much finer time step sizes. Finally, using a time 

step size of 100 seconds (which applies the entire simulated load in just 2 steps) the integration still 

achieves convergence. Note that even though the intermediate point on this simulated stress-strain curve 

does fall significantly outside the previous predictions, the ending stress (at an applied strain of 0.02) is 

still within 3.5 % of the previously predicted stress levels. Also, it is important to remember that the 

global equilibrium iterations required in the case of the multimechanism GVIPS model adds additional 

overhead to the codeís execution. Thus, to preserve the codeís efficiency, a relatively large time step 

should be employed when using the multimechanism GVIPS model. 

0

50

100

150

200

250

300

350

400

0.000 0.005 0.010 0.015 0.020 0.025

Strain

Stress (ksi)

GVIPS, STP = 1. s

GVIPS, STP = 2.5 s

GVIPS, STP = 4. s

GVIPS, STP = 5. s

GVIPS, STP = 8. s

Figure 7.1 Example 7a: Simulated transverse tensile response of 25% SiC/Ti-21S at room 

temperature wherein the Ti-21S matrix response is simulated using the standard isotropic 

GVIPS constitutive model. Effect of the loading time step size is highlighted.

NASA/TM—2002-212077/VOL3 217
MAC/GMC 4.0 Example Problem Manual 

0

50

100

150

200

250

0.000 0.005 0.010 0.015 0.020 0.025

Strain

Stress (ksi)

Multimechanism, STP = 1. s

Multimechanism, STP = 2.5 s

Multimechanism, STP = 4. s

Multimechanism, STP = 8. s

Multimechanism, STP = 20. s

Multimechanism, STP = 50. s

Multimechanism, STP = 100. s

Figure 7.2 Example 7a: Simulated transverse tensile response of 25% SiC/Ti-21S at room 

temperature wherein the Ti-21S matrix response is simulated using the new implicitly 

integrated multimechanism GVIPS constitutive model. Effect of the loading time step size 

is highlighted. 

NASA/TM—2002-212077/VOL3 218
Section 7: Advanced Topics Example 7b: Electromagnetic RUC Analysis 

Example 7b: Electromagnetic RUC Analysis

This example problem involves the use of the new electromagnetic micromechanics capabilities within 

MAC/GMC 4.0. The basic difference between standard and electromagnetic micromechanics involves 

the form of the material constitutive equation employed. In the standard micromechanics constitutive 

equations, stresses and strains are related by the stiffness matrix. In electromagnetic micromechanics, the 

constitutive equations are expanded, 

( )

11 11 12 13 14 15 16 11 21 31 11 21 31

22 12 22 23 24 25 26 12 22 32 12 22 32

33 13 23 33 34 35 36 13 23 33 13 23 33

23 14 24 34 44 45 46 1

13

12

1

2

3

1

2

3

CCCCCC e e e q q q

CCCCCC e e e q q q

CCCCCC e e e q q q

CCCCCC e

D

D

D

B

B

B

αβγ

σ

σ

σ

σ

σ

σ

 

 

 

 

 

 

 

 

  =  

 

 

 

 

 

 

 

   

4 24 34 14 24 34

15 25 35 45 55 56 15 25 35 15 25 35

16 26 36 46 56 66 16 26 36 16 26 36

11 12 13 14 15 16 11 12 13 11 12 13

21 22 23 24 25 26 12 22 23 21 22 23

31 32 33 34 35 36 13 23

eeqqq

CCCCCC e e e q q q

CCCCCC e e e q q q

eeeeee a a a

eeeeee a a a

eeeeee

κ κ κ

κ κ κ

κ κ κ

−−−−−−

−−−−−−

−−−

( )

11 11 11

22 2

33 31 32 33

11 12 13 14 15 16 11 21 31 11 12 13

21 22 23 24 25 26 12 22 32 12 22 23

31 32 33 34 35 36 13 23 33 13 23 33

I T

aaa

qqqqqq a a a

qqqqqq a a a

qqqqqq a a a

αβγ

εεε

ε ε

μ μ μ

μ μ μ

μ μ μ

  − −   −

−−−

−−−−−−

−−−−−−

  −−−−−−

( )

2 22

33 33 33

23 23 23

13 13 13

12 12 12

1 1

2 2

3 3

1 1

2 2

3 3

222

222

222

I T

I T

I T

I T

I T

T

T

T

T

T

T

E E

E E

E E

H H

H H

H H

αβγ

ε

εεε

εεε

ε ε ε

εεε

 

  −  

  − −   − −  

  − −  

  − −

  − −  

  − −

  − −  

  − −

  − −  

    − −

where ( )

ij

αβγ σ are the stress components, ( ) Dkαβγ are the electric displacement components, ( ) Bkαβγ are the 

magnetic flux density components, ( )

ij

αβγ ε are the total strain components, I( )

ij

αβγ ε are the inelastic strain 

components, T ( )

ij

αβγ ε are the thermal strain components, ( ) Ekαβγ are the electric field components, T ( ) Ekαβγ

are the thermo-electric field components, ( ) Hk

αβγ are the magnetic field components, T ( ) Hkαβγ are the 

thermo-magnetic field components, ( ) Cij

αβγ are the material stiffness components, ( )

kj e αβγ are the material 

piezoelectric components, ( )

kj q αβγ are the material piezomagnetic components, ( )ij

αβγ κ are the material 

dielectric components, ( )

ij a αβγ are the material magnetoelectric components, and ( )ij

αβγ μ are the material 

magnetic permeability components of a given subcell (denoted by the indices αβγ ). The thermal strain 

and thermal field components are related to a change in temperature from a given reference temperature 

(i.e., ∆T ) by, 

( )

[ ]( )

11 22 33 23 13 12 1 2 3 1 2 3

11 22 33 23 13 12 1 2 3 1 2 3

222 TT T T T T T T T T T T EEEHHH

T

αβγ

αβγ

ε ε ε ε ε ε

α α α α α α ζ ζ ζ ψ ψ ψ

  =  

∆

where ( )

ij

αβγ α are the subcell material coefficients of thermal expansion (CTEs), ( )

k

αβγ ζ are the subcell 

material pyroelectric constants, and ( )

k

αβγ ψ are the subcell material pyromagnetic constants. 

NASA/TM—2002-212077/VOL3 219
MAC/GMC 4.0 Example Problem Manual 

For the truly anisotropic case described by the above constitutive equation, there are indeed a great 

number of additional terms that must be known to characterize a materialís electromagnetic response. 

Fortunately, in practice, a great number of the coefficients in the stiffness/electromagnetic coefficient 

matrix are zero. MAC/GMC 4.0 admits electromagnetic materials of class C6v with an arbitrary poling 

direction. These types of materials are in many ways analogous to transversely isotropic materials. 

Assuming an x1 poling direction, the above equations reduce to, 

( )

11 11 12 12 11 11

22 12 22 23 12 12

33 12 23 22 12 12

23 44

13 55 26 26

12 66 26 26

1 11 12 12

2

3

1

2

3

000 0 0 0 0

000 0 0 0 0

000 0 0 0 0

000 00 0 0 0 0 0 0

0000 0 0 0 0 0

00000 0 0 0 0

000

CCC e q

CCC e q

CCC e q

C

C eq

Ce q

D eee

D

D

B

B

B

αβγ

σ

σ

σ

σ

σ

σ

 

 

 

 

 

 

 

 

  =  

 

 

 

 

 

 

 

   

( )

11 11 11

22 22 22

33 33 33

23

11 11

26 22 22

26 22 22

11 12 12 11 11

26 22 22

26 22 22

2

00 00

00000 0 0 0 0

0000 0 0 0 0 0

000 0 0 0 0

00000 0 0 0 0

0000 0 0 0 0 0

I T

I T

I T

a

e a

e a

qqq a

q a

q a

αβγ

εεε

εεε

εεε

ε

κ

κ

κ

μ

μ

μ

  − −   − −

− −

− −

− −

− −

− −

− −

  − −

( )

23 23

13 13 13

12 12 12

1 1

2 2

3 3

1 1

2 2

3 3

2 2

222

222

I T

I T

I T

T

T

T

T

T

T

E E

E E

E E

H H

H H

H H

αβγ

ε ε

εεε

εεε

 

 

 

 

  − −  

  − −  

  − −

  − −  

  − −

  − −  

  − −

  − −  

    − −

( )

[ ]( )

11 22 33 23 13 12 1 2 3 1 2 3

11 22 22 1 2 2 1 2 2

222

000

TT T T T T T T T T T T EEEHHH

T

αβγ

αβγ

ε ε ε ε ε ε

α α α ζ ζ ζ ψ ψ ψ

  =  

∆

Thus, in addition to the standard thermo-mechanical material coefficients, for an electromagnetic 

material, the additional material parameters are: three piezoelectric parameters (e11, e12, e26), three 

piezomagnetic parameters (q11, q12, q26), two dielectric parameters (κ11, κ22), two magnetoelectric 

parameters (a11, a12), two magnetic permeability parameters (μ11, μ22), two pyroelectric parameters (ζ1, 

ζ2), and two pyromagnetic parameters (ψ1, ψ2), for a total of sixteen additional material parameters. In 

MAC/GMC 4.0, these parameters are read directly from the input file for an assumed x1 poling direction. 

However, thanks to MAC/GMC 4.0ís transversely isotropic elastic constitutive model that allows an 

arbitrary direction of transverse isotropy, the actual poling direction of the material is arbitrary. 

The present example problem considers a particulate composite consisting of a BaTiO3 (barium titanate) 

inclusion and a CoFe2O4 (cobalt ferrite) matrix. The BaTiO3 is a piezoelectric material (non-zero eij

parameters), while the CoFe2O4 is a piezomagnetic material (non-zero qij terms). When combined to form 

a composite, the resulting BaTiO3/CoFe2O4 material is electromagnetic. Composites with one or more 

piezoelectric or electromagnetic phases are often referred to as ìsmart compositesî. The smart composite 

is subjected to strain-controlled loading at 24 °C. 

MAC/GMC Input File: example_7b.mac

MAC/GMC 4.0 Example 7b - Electromagnetic RUC analysis

*ELECTROMAG

*CONSTITUENTS

NMATS=2

NASA/TM—2002-212077/VOL3 220
Section 7: Advanced Topics Example 7b: Electromagnetic RUC Analysis 

# -- BaTiO3 (Barium Titanate)

M=1 CMOD=9 MATID=U MATDB=1 EM=1

NTP=2

TEM=24.,600.

EA=111.93E9,111.93E9

ET=116.33E9,116.33E9

NUA=0.321,0.321

NUT=0.307,0.307

GA=43.0E9,43.0E9

ALPA=1.99E-6,1.99E-6

ALPT=8.53E-6,8.53E-6

D=0.,0.,1.

ES11=18.6,18.6

ES12=-4.4,-4.4

ES26=11.6,11.6

QS11=0.0,0.0

QS12=0.0,0.0

QS26=0.0,0.0

KS11=12.6E-9,12.6E-9

KS22=11.2E-9,11.2E-9

AS11=0.0,0.0

AS22=0.0,0.0

MS11=10.0E-6,10.0E-6

MS22=5.0E-6,5.0E-6

PELS1=0.13E5,0.13E5

PELS2=0.13E5,0.13E5

PMGS1=0.0,0.0

PMGS2=0.0,0.0

# -- CoFe2O4 (Cobalt Ferrite)

M=2 CMOD=9 MATID=U MATDB=1 EM=1 &

EL=143.57E9,154.57E9,0.37,0.368,45.3E9,0.00E-6,0.00E-6 &

D=0.,0.,1.

ES=0.0,0.0,0.0

QS=699.7,580.3,550.

KS=0.93E-10,0.08E-9

AS=0.0,0.0

MS=157.E-6,-590.E-6

PELS=0.0,0.0

PMGS=0.0,0.0

*RUC

MOD=3 ARCHID=1 VF=0.25 ASP=1. F=1 M=2

*MECH

LOP=99

NPT=2 TI=0.,200. MAG=0.,0. MODE=2

NPT=2 TI=0.,200. MAG=0.,0. MODE=2

NPT=2 TI=0.,200. MAG=0.,0.02 MODE=1

NPT=2 TI=0.,200. MAG=0.,0. MODE=2

NPT=2 TI=0.,200. MAG=0.,0. MODE=2

NPT=2 TI=0.,200. MAG=0.,0. MODE=2

NPT=2 TI=0.,200. MAG=0.,0. MODE=2

NPT=2 TI=0.,200. MAG=0.,0. MODE=2

NPT=2 TI=0.,200. MAG=0.,0. MODE=2

NPT=2 TI=0.,200. MAG=0.,0. MODE=2

NPT=2 TI=0.,200. MAG=0.,0. MODE=2

NPT=2 TI=0.,200. MAG=0.,0. MODE=2

*THERM

NPT=2 TI=0.,200. TEMP=24.,24.

NASA/TM—2002-212077/VOL3 221
MAC/GMC 4.0 Example Problem Manual 

*SOLVER

METHOD=1 NPT=2 TI=0.,200. STP=100.

*PRINT

NPL=8

*XYPLOT

FREQ=1

MACRO=2

NAME=example_7b_se X=3 Y=9

NAME=example_7b_De X=3 Y=46

NAME=example_7b_Be X=3 Y=49

MICRO=0

*END

Annotated Input Data

1) Flags: 

a) Perform electromagnetic analysis (*ELECTROMAG) [KM_1]: 

2) Constituent materials (*CONSTITUENTS) [KM_2]: 

*CONSTITUENTS

NMATS=2

# -- BaTiO3 (Barium Titanate)

M=1 CMOD=9 MATID=U MATDB=1 EM=1

NTP=2

TEM=24.,600.

EA=111.93E9,111.93E9

ET=116.33E9,116.33E9

NUA=0.321,0.321

NUT=0.307,0.307

GA=43.0E9,43.0E9

ALPA=1.99E-6,1.99E-6

ALPT=8.53E-6,8.53E-6

D=0.,0.,1.

ES11=18.6,18.6

ES12=-4.4,-4.4

ES26=11.6,11.6

QS11=0.0,0.0

QS12=0.0,0.0

QS26=0.0,0.0

KS11=12.6E-9,12.6E-9

KS22=11.2E-9,11.2E-9

AS11=0.0,0.0

AS22=0.0,0.0

MS11=10.0E-6,10.0E-6

MS22=5.0E-6,5.0E-6

PELS1=0.13E5,0.13E5

PELS2=0.13E5,0.13E5

PMGS1=0.0,0.0

PMGS2=0.0,0.0

Number of materials: 2 (NMATS=2) 

Constitutive model: Arbitrary transversely isotropic (CMOD=9) 

Materials: User-Defined (MATID=U) 

NASA/TM—2002-212077/VOL3 222
Section 7: Advanced Topics Example 7b: Electromagnetic RUC Analysis 

Material property source: Read from input file (MATDB=1) 

Electromagnetic specifier: Material is electromagnetic (EM=1) 

! Note: Electromagnetic analysis requires that the material properties be specified by the user in the 

MAC/GMC 4.0 input file. The electromagnetic specifier indicates whether or not the particular 

material has electromagnetic properties (EM=0 indicated that the material does not have 

electromagnetic properties). Further, the arbitrary transversely isotropic elastic constitutive 

model (CMOD=9) must be employed. 

For illustrative purposes, the material properties for the barium titanate material have been input as 

temperature-dependent (although the same material properties are employed for each input 

temperature). As indicated, the sixteen additional material parameters are specified for the two input 

temperatures. ES11, ES12, and ES26 are the three piezoelectric parameters, QS11, QS12, and 

QS26 are the three piezomagnetic parameters, KS11 and KS22 are the two magnetoelectric 

parameters, AS11 and AS22 are the two magnetoelectric parameters, and MS11 and MS22 are the 

two magnetic permeability parameters. PELS1 and PELS2 correspond to the two pyroelectric 

parameters (ζ1, ζ2), while PMGS1 and PMGS2 correspond to the two pyromagnetic parameters (ψ1, ψ2). 

In addition, a direction vector (D=0.,0.,1.) has been specified. This direction vector is standard 

input associated with the arbitrary transversely isotropic elastic constitutive model (CMOD=9). It 

specifies the direction of transverse isotropy with respect to the RUC coordinate axes (i.e., it specifies 

the vector that is normal to the plane of transverse isotropy) (see the MAC/GMC 4.0 Keywords 

Manual Section 2). When employed in conjunction with electromagnetic analysis, the direction 

vector also specifies the poling direction for the electromagnetic material. Thus, in the present case, 

an x3 poling direction has been specified for the materials. 

# -- CoFe2O4 (Cobalt Ferrite)

M=2 CMOD=9 MATID=U MATDB=1 EM=1 &

EL=143.57E9,154.57E9,0.37,0.368,45.3E9,0.00E-6,0.00E-6 &

D=0.,0.,1.

ES=0.0,0.0,0.0

QS=699.7,580.3,550.

KS=0.93E-10,0.08E-9

AS=0.0,0.0

MS=157.E-6,-590.E-6

PELS=0.0,0.0

PMGS=0.0,0.0

For the cobalt ferrite material, temperature-independent material properties have been input. The 

ordering for the electromagnetic material parameter input is intuitive. For more information on the 

electromagnetic material parameter specification, see the MAC/GMC Keywords Manual Section 2. 

3) Analysis type (*RUC) → Repeating Unit Cell Analysis [KM_3]: 

Analysis model: Triply periodic GMC (MOD=3) 

RUC architecture: short fiber, square array (ARCHID=1) 

Fiber volume fractions: 0.25 (VF=0.25) 

Fiber aspect ratio 1. (ASP=1.) 

Material assignment: BaTiO3 (barium titanate) fiber (F=1) 

CoFe2O4 (cobalt ferrite) matrix (M=2) 

! Note: Electromagnetic analysis requires use of a triply periodic RUC (MOD=3). 

NASA/TM—2002-212077/VOL3 223
MAC/GMC 4.0 Example Problem Manual 

4) Loading: 

a) Mechanical (*MECH) [KM_4]: 

Loading option: general loading (LOP=99) 

Component #1 (ε11 or σ11) 

Number of points: 2 (NPT=2) 

Times (TI=) (sec.) 0. 200. 

Magnitudes (MAG=) 0. 0. 

Control (MODE=) stress 

Component #2 (ε22 or σ22) 

Number of points: 2 (NPT=2) 

Times (TI=) (sec.) 0. 200. 

Magnitudes (MAG=) 0. 0. 

Control (MODE=) stress 

Component #3 (ε33 or σ33) 

Number of points: 2 (NPT=2) 

Times (TI=) (sec.) 0. 200. 

Magnitudes (MAG=) 0. 0.02 

Control (MODE=) strain 

Component #4 (γ23 or σ23) 

Number of points: 2 (NPT=2) 

Times (TI=) (sec.) 0. 200. 

Magnitudes (MAG=) 0. 0. 

Control (MODE=) stress 

Component #5 (γ13 or σ13) 

Number of points: 2 (NPT=2) 

Times (TI=) (sec.) 0. 200. 

Magnitudes (MAG=) 0. 0. 

Control (MODE=) stress 

Component #6 (γ12 or σ12) 

Number of points: 2 (NPT=2) 

Times (TI=) (sec.) 0. 200. 

Magnitudes (MAG=) 0. 0. 

Control (MODE=) stress 

NASA/TM—2002-212077/VOL3 224
Section 7: Advanced Topics Example 7b: Electromagnetic RUC Analysis 

Component #7 (E1 or D1) 

Number of points: 2 (NPT=2) 

Times (TI=) (sec.) 0. 200. 

Magnitudes (MAG=) 0. 0. 

Control (MODE=) electric displacement (D1) 

Component #8 (E2 or D2) 

Number of points: 2 (NPT=2) 

Times (TI=) (sec.) 0. 200. 

Magnitudes (MAG=) 0. 0. 

Control (MODE=) electric displacement (D2) 

Component #9 (E3 or D3) 

Number of points: 2 (NPT=2) 

Times (TI=) (sec.) 0. 200. 

Magnitudes (MAG=) 0. 0. 

Control (MODE=) electric displacement (D3) 

Component #10 (H1 or B1) 

Number of points: 2 (NPT=2) 

Times (TI=) (sec.) 0. 200. 

Magnitudes (MAG=) 0. 0. 

Control (MODE=) magnetic flux density (B1) 

Component #11 (H2 or B2) 

Number of points: 2 (NPT=2) 

Times (TI=) (sec.) 0. 200. 

Magnitudes (MAG=) 0. 0. 

Control (MODE=) magnetic flux density (B2) 

Component #12 (H3 or B3) 

Number of points: 2 (NPT=2) 

Times (TI=) (sec.) 0. 200. 

Magnitudes (MAG=) 0. 0. 

Control (MODE=) magnetic flux density (B3) 

For electromagnetic analysis, the general loading option must be selected. Further, there are now six 

additional load components that may be applied to the composite: three components of electric 

displacement or electric field and three components of magnetic flux density or magnetic field. The 

present case simulates application of strain in the x3-direction to a composite that is otherwise free of 

loading. As usual, these ìfreeî loading conditions take the form of zero applied global stresses for all 

components other than the normal 33 component. The non-loaded strain components are permitted to 

NASA/TM—2002-212077/VOL3 225
MAC/GMC 4.0 Example Problem Manual 

arise naturally. In terms of electromagnetic effects, ìfreeî loading conditions correspond to zero 

applied electrical displacement and magnetic flux density (rather than zero applied electric and 

magnetic field components). The electric and magnetic field components are permitted to arise 

naturally. For more information on loading specification for electromagnetic analysis, see the 

MAC/GMC Keywords Manual Section 4. 

b) Thermal (*THERM) [KM_4]: 

Number of points: 2 (NPT=2) 

Time points: 0., 200. sec. (TI=0.,200.) 

Temperature points: 24., 24. (TEMP=24.,24.) 

c) Time integration (*SOLVER) [KM_4]: 

Time integration method: Forward Euler (METHOD=1) 

Number of points: 2 (NPT=2) 

Time points: 0., 200. sec. (TI=0.,200.) 

Time step size: 100. sec. (STP=100) 

! Note: Since this problem is linear electro-magneto-elastic, the time step size is unimportant in terms 

of convergence. 

5) Damage and Failure: None 

6) Output: 

a) Output file print level (*PRINT) [KM_6]: 

Print level: 8 (NPL=8) 

b) x-y plots (*XYPLOT) [KM_6]: 

Frequency: 1 (FREQ=1) 

Number of macro plots: 3 (MACRO=3) 

Macro plot name: example_7b_se (NAME=example_7b_se) 

 example_7b_De (NAME=example_7b_De) 

 example_7b_Be (NAME=example_7b_Be) 

Macro plot x-y quantities: ε33, σ33 (X=3 Y=9) 

 ε33, D3 (X=3 Y=46) 

 ε33, B3 (X=3 Y=49) 

Number of micro plots: 0 (MICRO=0) 

7) End of file keyword: (*END) 

Results

Since the particulate BaTiO3/CoFe2O4 electromagnetic (smart) composite considered in this example 

problem consists of phases that are both linear electro-magneto-elastic, results in the form of effective 

properties (from the MAC/GMC 4.0 output file) will be examined. First, consider the effective (global) 

electro-magneto-elastic matrix for the smart composite: 

NASA/TM—2002-212077/VOL3 226
Section 7: Advanced Topics Example 7b: Electromagnetic RUC Analysis 

ZG - Effective/Macro Stiffness/Electromagnetic Coefficient Matrix

| 2.449D+11 1.382D+11 1.353D+11 0.000D+00 0.000D+00 0.000D+00 0.000D+00 0.000D+00 -1.565D-02 0.000D+00 0.000D+00 3.078D+02 |

| 1.382D+11 2.449D+11 1.353D+11 0.000D+00 0.000D+00 0.000D+00 0.000D+00 0.000D+00 -1.565D-02 0.000D+00 0.000D+00 3.078D+02 |

| 1.353D+11 1.353D+11 2.404D+11 0.000D+00 0.000D+00 0.000D+00 0.000D+00 0.000D+00 9.518D-02 0.000D+00 0.000D+00 3.861D+02 |

| 0.000D+00 0.000D+00 0.000D+00 4.733D+10 0.000D+00 0.000D+00 0.000D+00 4.891D-02 0.000D+00 0.000D+00 3.397D+02 0.000D+00 |

| 0.000D+00 0.000D+00 0.000D+00 0.000D+00 4.733D+10 0.000D+00 4.891D-02 0.000D+00 0.000D+00 3.397D+02 0.000D+00 0.000D+00 |

| 0.000D+00 0.000D+00 0.000D+00 0.000D+00 0.000D+00 5.306D+10 0.000D+00 0.000D+00 0.000D+00 0.000D+00 0.000D+00 0.000D+00 |

| 0.000D+00 0.000D+00 0.000D+00 0.000D+00 4.891D-02 0.000D+00 -1.332D-10 0.000D+00 0.000D+00 2.179D-10 0.000D+00 0.000D+00 |

| 0.000D+00 0.000D+00 0.000D+00 4.891D-02 0.000D+00 0.000D+00 0.000D+00 -1.332D-10 0.000D+00 0.000D+00 2.179D-10 0.000D+00 |

| -1.565D-02 -1.565D-02 9.518D-02 0.000D+00 0.000D+00 0.000D+00 0.000D+00 0.000D+00 -1.547D-10 0.000D+00 0.000D+00 -1.210D-11 |

| 0.000D+00 0.000D+00 0.000D+00 0.000D+00 3.397D+02 0.000D+00 2.179D-10 0.000D+00 0.000D+00 3.517D-04 0.000D+00 0.000D+00 |

| 0.000D+00 0.000D+00 0.000D+00 3.397D+02 0.000D+00 0.000D+00 0.000D+00 2.179D-10 0.000D+00 0.000D+00 3.517D-04 0.000D+00 |

| 3.078D+02 3.078D+02 3.861D+02 0.000D+00 0.000D+00 0.000D+00 0.000D+00 0.000D+00 -1.210D-11 0.000D+00 0.000D+00 -1.011D-04 |

This matrix relates the global stress/electric displacement/magnetic flux density vector to the global 

strain/electric field/magnetic field vector, and is of identical form to that of each phase as shown in the 

local constitutive equation given earlier in this example problem. The composite, like the barium titanate 

inclusion, has non-zero piezoelectric parameters (eij). In addition, like the cobalt ferrite matrix, the 

composite has non-zero piezomagnetic parameters (qij). Most interesting, however, is the fact that the 

smart composite has non-zero magnetoelectric parameters (aij), whereas the aij of each constituent is zero. 

These magnetoelectric parameters provide coupling between the electric and magnetic effects in the 

composite. That is, for instance, when this coupling is present, electric displacement components can 

arise due to an applied magnetic field. It is noteworthy that the smart composite exhibits this electromagnetic coupling when neither constituent does. 

Next, the effective engineering properties of the smart composite, along with the thermal coefficients, are 

output: 

Effective Engineering Moduli

E11S= 0.1459E+12

N12S= 0.3679

E22S= 0.1459E+12

N23S= 0.3557

E33S= 0.1449E+12

G23S= 0.4733E+11

G13S= 0.4733E+11

G12S= 0.5306E+11

Effective Thermal Expansion Coefficients

NOTE: Shear CTEs are "engineering" shear CTEs

0.1927E-05 0.1927E-05 0.3256E-06

0.3074E-08 0.3074E-08 0.0000E+00

Effective Pyroelectric Vector

0.5223E+04 0.5223E+04 0.6427E+04

Effective Pyromagnetic Vector

-0.1131E-02 -0.1131E-02 0.2688E+01

Here again, the electromagnetic coupling gives rise to interesting effects in the smart composite that are 

absent in both of the constituents. First of all, shear coefficients of thermal expansion arise in the 

composite. This indicates that in response to an applied temperature change, global shear strain will 

NASA/TM—2002-212077/VOL3 227
MAC/GMC 4.0 Example Problem Manual 

result in the composite. Further, while the pyromagnetic parameters for each constituent are zero, the 

composite has non-zero pyromagnetic parameters. Thus, thermo-magnetic field components ( T Hk ) will 

arise in the smart composite due to applied thermal loading even though this would not occur in either of 

the monolithic constituents. 

Finally, the present example problem involved application of a global normal strain to the smart 

composite. The time-based output written to the MAC/GMC 4.0 output file includes: 

2 TIME: 2.0000D+02 TEMP: 2.4000D+01 TSTEP: 1.0000D+02

-----------------------------------------------------------------------------------------------

STRESS: 0.0000D+00 0.0000D+00 2.9041D+09 0.0000D+00 0.0000D+00 0.0000D+00

E-M FLUX: 0.0000D+00 0.0000D+00 0.0000D+00 0.0000D+00 0.0000D+00 0.0000D+00

STRAIN: -7.0886D-03 -7.0886D-03 2.0000D-02 0.0000D+00 0.0000D+00 0.0000D+00

E-M FIELD: 0.0000D+00 0.0000D+00 -1.3739D+07 0.0000D+00 0.0000D+00 -3.3223D+04

TH. STRAIN: 0.0000D+00 0.0000D+00 0.0000D+00 0.0000D+00 0.0000D+00 0.0000D+00

PYRO E-M FIELD: 0.0000D+00 0.0000D+00 0.0000D+00 0.0000D+00 0.0000D+00 0.0000D+00

TANGENT CTE: 1.9274D-06 1.9274D-06 3.2555D-07 3.0735D-09 3.0735D-09 0.0000D+00

SECANT CTE: 1.9274D-06 1.9274D-06 3.2555D-07 3.0735D-09 3.0735D-09 0.0000D+00

TAN PYRO E-M VECT: 5.2231D+03 5.2231D+03 6.4268D+03 -1.1308D-03 -1.1308D-03 2.6880D+00

SEC PYRO E-M VECT: 5.2231D+03 5.2231D+03 6.4268D+03 -1.1308D-03 -1.1308D-03 2.6880D+00

NOTE: TREF = 24.000

Thus, in response to the applied normal strain loading (ε33), the composite experiences the standard 

mechanical response (non-zero σ33, ε11, and ε22), in addition to a non-zero electric field component (E3 = -

1.3739×107

 V/m) and a non-zero magnetic field component (H3 = -3.3223×104 A/m). The S.I. units for 

the electric field (Ek) are volts per meter (V/m) or Newtons per Coulomb (N/C) and for magnetic field 

(Hk), the S.I. units are Amperes per meter (A/m). The S.I units for electric displacement (Dk) are 

Coulombs per square meter (C/m2

) and for magnetic flux density (Bk), the S.I. units are Newtons per 

Ampere-meter (N/Am). For additional information on electromagnetic analysis with MAC/GMC 4.0, see 

the MAC/GMC 4.0 Keywords Manual Section 2. 

Finally, the local electric and magnetic field components that are induced in the BaTiO3/CoFe2O4

composite are shown in Figure 7.3. A relatively constant electric field is induced while the induce 

magnetic field varies widely in the composite. 

 Figure 7.3 Example 7b: Local electric field component E3 and magnetic field component H3 induced 

in the smart BaTiO3/CoFe2O4 composite by an applied global strain of 02.0 ε 33 = . 

NASA/TM—2002-212077/VOL3 228
Section 7: Advanced Topics Example 7c: Electromagnetic Laminate Analysis 

Example 7c: Electromagnetic Laminate Analysis

This example problem considers an electromagnetic (smart) composite laminate. In particular, a hybrid 

smart/metal matrix composite laminate is analyzed consisting of a unidirectional B/Al layer sandwiched 

between two BaTiO3/CoFe2O4 layers (see Figure 7.4). By reversing the electromagnetic polarity of the 

bottom smart composite layer, a completely different laminate response can be obtained. The applied 

loading involves a through-thickness magnetic field component. 

z

y

0.25

0.25

0.50

Top and Bottom Plies:

Variable Vf continous [0] BaTiO3/CoFe2O4 Smart Composite

Vf = 0.25 continous [90] B/Al MMC

Middle Ply:

Figure 7.4 Hybrid smart/MMC laminate analyzed in MAC/GMC 4.0 Example Problem 7c. 

 

MAC/GMC Input File: example_7c.mac

MAC/GMC 4.0 Example 7c - Electromagnetic laminate analysis

*ELECTROMAG

*CONSTITUENTS

NMATS=5

# -- BaTiO3 (Barium Titanate)

M=1 CMOD=9 MATID=U MATDB=1 EM=1 &

EL=111.93E9,116.33E9,0.321,0.307,43.0E9,1.99E-6,8.53E-6 &

D=1.,0.,0.

ES=18.6,-4.4,11.6

QS=0.0,0.0,0.0

KS=12.6E-9,11.2E-9

AS=0.0,0.0

MS=10.0E-6,5.0E-6

PELS=0.13E5,0.13E5

PMGS=0.0,0.0

NASA/TM—2002-212077/VOL3 229
MAC/GMC 4.0 Example Problem Manual 

# -- CoFe2O4 (Cobalt Ferrite) - positive poling direction

M=2 CMOD=9 MATID=U MATDB=1 EM=1 &

EL=143.57E9,154.57E9,0.37,0.368,45.3E9,0.00E-6,0.00E-6 &

D=0.,0.,1.

ES=0.0,0.0,0.0

QS=699.7,580.3,550.

KS=0.93E-10,0.08E-9

AS=0.0,0.0

MS=157.E-6,-590.E-6

PELS=0.0,0.0

PMGS=0.0,0.0

# -- CoFe2O4 (Cobalt Ferrite) - negative poling direction

M=3 CMOD=9 MATID=U MATDB=1 EM=1 &

EL=143.57E9,154.57E9,0.37,0.368,45.3E9,0.00E-6,0.00E-6 &

D=0.,0.,-1.

ES=0.0,0.0,0.0

QS=699.7,580.3,550.

KS=0.93E-10,0.08E-9

AS=0.0,0.0

MS=157.E-6,-590.E-6

PELS=0.0,0.0

PMGS=0.0,0.0

# -- Boron

M=4 CMOD=6 MATID=U MATDB=1 EM=0 &

EL=400.E9,400.E9,0.20,0.20,166.6667E9,8.3E-6,8.3E-6

# -- Aluminum

M=5 CMOD=1 MATID=U MATDB=1 EM=0 &

EL=72.6E9,72.6E9,0.33,0.33,72.283E9,22.5E-6,22.5E-6 &

VI=1.E4,65.E6,150.E6,50.,10.,1.

*LAMINATE

NLY=3

LY=1 THK=0.25 ANG=0. MOD=3 ARCHID=99 EM=1

NA=1 NB=2 NG=2

D=1.

H=0.5,0.5

L=0.5,0.5

SM=1,2

SM=2,2

LY=2 THK=0.5 ANG=90. MOD=2 ARCHID=1 VF=0.25 F=4 M=5 EM=0

LY=3 THK=0.25 ANG=0. MOD=3 ARCHID=99 EM=1

NA=1 NB=2 NG=2

D=1.

H=0.5,0.5

L=0.5,0.5

SM=1,2

SM=2,2

# SM=1,3

# SM=3,3

*MECH

LOP=99

NPT=2 TI=0.,600. MAG=0.,0. MODE=2

NPT=2 TI=0.,600. MAG=0.,0. MODE=2

NPT=2 TI=0.,600. MAG=0.,0. MODE=2

NPT=2 TI=0.,600. MAG=0.,0. MODE=2

NPT=2 TI=0.,600. MAG=0.,0. MODE=2

NPT=2 TI=0.,600. MAG=0.,0. MODE=2

NPT=2 TI=0.,600. MAG=0.,0. MODE=1

NASA/TM—2002-212077/VOL3 230
Section 7: Advanced Topics Example 7c: Electromagnetic Laminate Analysis 

NPT=2 TI=0.,600. MAG=0.,0. MODE=1

NPT=2 TI=0.,600. MAG=0.,0. MODE=2

NPT=2 TI=0.,600. MAG=0.,0. MODE=1

NPT=2 TI=0.,600. MAG=0.,0. MODE=1

NPT=2 TI=0.,600. MAG=0.,6.E6 MODE=1

*THERM

NPT=2 TI=0.,600. TEMP=24.,24.

*SOLVER

METHOD=1 NPT=2 TI=0.,600. STP=0.25

*PRINT

NPL=8

*XYPLOT

FREQ=40

LAMINATE=5

NAME=example_7c_ex X=34 Y=1

NAME=example_7c_ey X=34 Y=2

NAME=example_7c_ez X=34 Y=3

NAME=example_7c_kx X=34 Y=7

NAME=example_7c_ky X=34 Y=8

MACRO=0

MICRO=0

*END

Annotated Input Data

1) Flags: 

a) Perform electromagnetic analysis (*ELECTROMAG) [KM_1]: 

2) Constituent materials (*CONSTITUENTS) [KM_2]: 

Number of materials: 5 (NMATS=5) 

Materials: User-Defined (MATID=U) 

Constitutive models: Arbitrary transversely isotropic (CMOD=9) 

 Linearly elastic (CMOD=6) 

 Bodner-Partom (CMOD=1) 

Material property source: Read from input file (MATDB=1) 

Electromagnetic specifier: Material is electromagnetic (EM=1) 

 Material is not electromagnetic (EM=0) 

The boron and aluminum materials are specified with EM=0 to indicate that they do not have 

electromagnetic properties associated with them. The cobalt ferrite material properties are specified 

twice ñ each time with a different direction vector. Since the direction vector is a material property, 

the same material with only a change in direction vector constitutes a new material. In the present 

case, the sign of the direction vector of the cobalt ferrite is reversed. While this has no effect on the 

mechanical properties, it causes a sign reversal in the materialís electromagnetic properties. 

3) Analysis type (*LAMINATE) → Laminate Analysis [KM_3]: 

Number of layers: 3 (NLY=3) 

NASA/TM—2002-212077/VOL3 231
MAC/GMC 4.0 Example Problem Manual 

Layer Analysis 

Model 

Thickness Fiber 

Angle 

Architecture Volume 

fraction 

Fiber 

material 

Matrix 

material 

E-M 

Specifier 

(LY=) (MOD) (THK) (ANG) (ARCHID) (VF) (F) (M) (EM) 

1 GMC-3D 0.25 90° short fiber 

square array 

0.25 BaTiO3 CoFe2O4 electromagnetic 

2 GMC-2D 0.50 0° short fiber 

square array 

0.25 boron aluminum not electromagnetic 

3 GMC-3D 0.25 90° short fiber 

square array 

0.25 BaTiO3 CoFe2O4 Electromagnetic 

For electromagnetic laminate analysis, there is an additional piece of information required for each 

layer that specifies whether or not the layer is electromagnetic. EM=1 indicates that the layer is 

electromagnetic, while EM=0 indicates that the layer is not electromagnetic. 

! Note: To generate the results for the magnetically asymmetric laminate, the appropriate lines in the 

input file must be commented and uncommented. 

4) Loading: 

a) Mechanical (*MECH) [KM_4]: 

Loading option: general loading (LOP=99) 

Component #1 ( 0

xx ε or Nxx ) 

Number of points: 2 (NPT=2) 

Times (TI=) (sec.) 0. 600. 

Magnitudes (MAG=) 0. 0. 

Control (MODE=) force resultant 

Component #2 ( 0

yy ε or Nyy ) 

Number of points: 2 (NPT=2) 

Times (TI=) (sec.) 0. 600. 

Magnitudes (MAG=) 0. 0. 

Control (MODE=) force resultant 

Component #3 ( 0

xy γ or Nxy ) 

Number of points: 2 (NPT=2) 

Times (TI=) (sec.) 0. 600. 

Magnitudes (MAG=) 0. 0.02 

Control (MODE=) force resultant 

Component #4 (κ xx or M xx ) 

Number of points: 2 (NPT=2) 

NASA/TM—2002-212077/VOL3 232
Section 7: Advanced Topics Example 7c: Electromagnetic Laminate Analysis 

Times (TI=) (sec.) 0. 600. 

Magnitudes (MAG=) 0. 0. 

Control (MODE=) moment resultant 

Component #5 (κ yy or M yy ) 

Number of points: 2 (NPT=2) 

Times (TI=) (sec.) 0. 600. 

Magnitudes (MAG=) 0. 0. 

Control (MODE=) moment resultant 

Component #6 (κ xy or M xy ) 

Number of points: 2 (NPT=2) 

Times (TI=) (sec.) 0. 600. 

Magnitudes (MAG=) 0. 0. 

Control (MODE=) moment resultant 

Component #7 (E1) 

Number of points: 2 (NPT=2) 

Times (TI=) (sec.) 0. 600. 

Magnitudes (MAG=) 0. 0. 

Control (MODE=) electric field (E1) 

Component #8 (E2) 

Number of points: 2 (NPT=2) 

Times (TI=) (sec.) 0. 600. 

Magnitudes (MAG=) 0. 0. 

Control (MODE=) electric field (E2) 

Component #9 (E3) 

Number of points: 2 (NPT=2) 

Times (TI=) (sec.) 0. 600. 

Magnitudes (MAG=) 0. 0. 

Control (MODE=) electric field (E3) 

Component #10 (H1) 

Number of points: 2 (NPT=2) 

Times (TI=) (sec.) 0. 600. 

Magnitudes (MAG=) 0. 0. 

Control (MODE=) magnetic field (H1) 

NASA/TM—2002-212077/VOL3 233
MAC/GMC 4.0 Example Problem Manual 

Component #11 (H2) 

Number of points: 2 (NPT=2) 

Times (TI=) (sec.) 0. 600. 

Magnitudes (MAG=) 0. 0. 

Control (MODE=) magnetic field (H2) 

Component #12 (H3) 

Number of points: 2 (NPT=2) 

Times (TI=) (sec.) 0. 600. 

Magnitudes (MAG=) 0. 6,000,000 A/m 

Control (MODE=) magnetic field (H3) 

! Note: Currently, in the case of an electromagnetic laminate, only electromagnetic field components 

(Ek and Hk) may be applied. The electromagnetic lamination theory does not admit application 

of laminate level electric displacements (Dk) or magnetic flux density components (Bk), 

although average laminate electric displacements and magnetic flux densities can arise. For 

more information on loading specification for electromagnetic analysis, see the MAC/GMC 

Keywords Manual Section 4. 

b) Thermal (*THERM) [KM_4]: 

Number of points: 2 (NPT=2) 

Time points: 0., 600. sec. (TI=0.,600.) 

Temperature points: 24., 24. (TEMP=24.,24.) 

c) Time integration (*SOLVER) [KM_4]: 

Time integration method: Forward Euler (METHOD=1) 

Number of points: 2 (NPT=2) 

Time points: 0., 600. sec. (TI=0.,600.) 

Time step size: 0.25 sec. (STP=0.25) 

! Note: Unlike Example Problem 7b, this example problem contains an inelastic material (aluminum). 

Thus, a small time step must be employed to achieve convergence. 

5) Damage and Failure: None 

6) Output: 

a) Output file print level (*PRINT) [KM_6]: 

Print level: 8 (NPL=8) 

b) x-y plots (*XYPLOT) [KM_6]: 

Frequency: 40 (FREQ=40) 

Number of laminate plots: 5 (LAMINATE=5) 

Laminate plot name: example_7c_ex (NAME=example_7c_ex) 

 example_7c_ey (NAME=example_7c_ey) 

 example_7c_ez (NAME=example_7c_ez) 

 example_7c_kx (NAME=example_7c_kx) 

 example_7c_ky (NAME=example_7c_ky) 

NASA/TM—2002-212077/VOL3 234
Section 7: Advanced Topics Example 7c: Electromagnetic Laminate Analysis 

Laminate plot x-y quantities: Hz, 0

xx ε (X=34 Y=1) 

 Hz, 0

yy ε (X=34 Y=2) 

 Hz, zz ε (X=34 Y=3) 

 Hz, κ xx (X=34 Y=7) 

 Hz, κ yy (X=34 Y=8) 

Number of macro plots: 0 (MACRO=0) 

Number of micro plots: 0 (MICRO=0) 

7) End of file keyword: (*END) 

Results

This example problem highlights the effect of reversing the magnetic polarity of the CoFe2O4 in one of 

the BaTiO3/CoFe2O4 layers in the hybrid smart/MMC laminate. First, for the case in which the poling 

direction of both 0° BaTiO3/CoFe2O4 layers is the x3-direction (laminate z-direction), the laminate 

responds to the applied through-thickness magnetic field with extension (midplane strains), but no 

curvature. The midplane strain response of the magnetically symmetric laminate is plotted in Figure 7.5. 

Also plotted is the average through-thickness strain in the laminate. The laminate exhibits more midplane 

strain in the x-direction than the y-direction because the continuous and stiff boron fibers of the middle 

90° layer are oriented in the y-direction. The out-of-plane strain is of the largest magnitude because of 

the large effect piezomagnetic term q33 of the smart plies. It is clear from the knees in the curves plotted 

in Figure 7.5 the point at which yielding commences in the B/Al ply. The MAC/GMC 4.0 output file 

confirms that, in addition to non-zero magnetic force resultants, non-zero inelastic force resultants arise in 

the laminate. It should be noted that, in addition to the strains, the laminate experiences an average 

electric displacement component, Dx, and an average magnetic flux component Bz. 

When the poling direction of the CoFe2O4 in the bottom 0° BaTiO3/CoFe2O4 layer is reversed, the 

laminate responds to the applied through-thickness magnetic field in a completely different way. As 

shown in Figure 7.6, this magnetically asymmetric laminate responds with curvature but no midplane 

extension. The reversal in poling direction causes a reversal of the sign of the effective qij and aij terms 

for the bottom smart composite layer; the signs are now opposite of those in the top smart composite 

layer, causing the magnetic asymmetry. As a result, magnetic moment resultants, rather than force 

resultants, arise in the composite due to the applied magnetic field. An identical average magnetic flux 

component, Bz arises in the magnetically asymmetric laminate, but the average electric displacement 

component, Dx, that was present in the magnetically symmetric laminate, is now zero. Note that, 

mechanically, the two laminates are identical, with identical ABD matrices. A slight knee is present in the 

curves of Figure 7.6, but the effect of the B/Al ply inelasticity is much smaller than in the magnetically 

symmetric laminate. The magnitude of the curvature is smaller in the y-direction than the x-direction due 

to the stiff boron fibers oriented along the y-direction. 

NASA/TM—2002-212077/VOL3 235
MAC/GMC 4.0 Example Problem Manual 

0.0000

0.0005

0.0010

0.0015

0.0020

0.0025

0 1,000,000 2,000,000 3,000,000 4,000,000 5,000,000 6,000,000 7,000,000

Applied Magnetic Field, Hz (A/m)

Strain

Midplane epsxx

Midplane epsyy

Average laminate epszz

Figure 7.5 Example 7c: Strain response of a symmetric [0°/90°]s hybrid smart/MMC laminate to an 

applied through-thickness magnetic field.

-0.006

-0.005

-0.004

-0.003

-0.002

-0.001

0

0 1,000,000 2,000,000 3,000,000 4,000,000 5,000,000 6,000,000 7,000,000

Applied Magnetic Field, Hz (A/m)

Midplane Curvature (1/m)

kxx

kyy

Figure 7.6 Example 7c: Curvature response of a magnetically asymmetric [0°/90°]s hybrid 

smart/MMC laminate to an applied through-thickness magnetic field. 

NASA/TM—2002-212077/VOL3 236
Section 7: Advanced Topics Example 7d: Woven Composite Analysis ñ Single Step 

Example 7d: Woven Composite Analysis ñ Single Step

This example problem and the next illustrate MAC/GMC 4.0ís ability to analyze composites with woven 

(or braided) reinforcements. The present example problem performs the analysis using a one step 

homogenization procedure. The next example problem uses a two step procedure that gives improved 

results. Triply periodic GMC has the ability to represent the inherently three-dimensional architecture of 

the reinforcement in woven composites. Currently, woven composites may be treated within MAC/GMC 

4.0 by employing the internal constitutive models that allow specification of an arbitrary direction of 

transverse isotropy (CMOD= 3, 7, or 9). Then, by choosing the direction vector appropriately for the 

material occupying each subcell, the woven reinforcement architecture can be represented. 

Woven polymer matrix composites (PMCs), such as graphite/epoxy, are a common type of woven 

composite. The polymer matrix in these composites is often treated as linearly elastic. An important 

class of woven PMC is considered in this example problem: plain weave reinforced PMCs. The plain 

weave architecture involves fibers, or more commonly tows of fibers, in a repeating over-one, under-one 

pattern as shown in Figure 7.7. As indicated in the figure, an RUC can be identified from the architecture 

that remains unchanged when the plain weave reinforcement is infiltrated with a polymer matrix to form a 

composite. The triply periodic GMC RUC representation of the plain weave composite is shown in 

Figure 7.8, with an exploded view given in Figure 7.9. As shown, the subcells within the triply periodic 

RUC are themselves occupied by composite materials that represent the fiber tows of the weave. In 

addition, some subcells are occupied by the pure matrix material. Figure 7.9 clearly shows the threedimensional nature of the woven reinforcement as the fiber tows undulate in and out of each other. 

In this example problem, the composite material within subcells is represented using the transversely 

isotropic elastic model with arbitrary plane of isotropy (CMOD = 9). The ability to employ the arbitrary 

direction of transverse isotropy allows the representation of the subcells containing inclined fibers in 

Figure 7.9. The material system considered is graphite/epoxy. The effective properties of the composite 

subcells have been taken from Example 1a (see Table 1.1), which determined the effective properties of a 

0.65 fiber volume fraction unidirectional graphite/epoxy composite. Thus, in the present example 

problem, it is assumed that the infiltrated fiber tows that occupy the subcells have a fiber volume fraction 

of 0.65. Due to the chosen dimension (a, g, and H in Figure 7.8) this results in an overall woven 

composite fiber volume fraction of 0.325. 

Figure 7.7 Top view of the plain weave architecture with the repeating unit cell identified. 

NASA/TM—2002-212077/VOL3 237
MAC/GMC 4.0 Example Problem Manual 

x

a g a g

a

a

g

g

H

x

x

1

3

2

Figure 7.8 MAC/GMC 4.0 triply periodic RUC that represents a plain weave reinforced composite. 

 

Figure 7.9 MAC/GMC 4.0 triply periodic RUC that represents a plain weave reinforced composite ñ 

exploded view.

MAC/GMC Input File: example_7d.mac

MAC/GMC 4.0 Example 7d - graphite/epoxy plain weave reinforced composite

*CONSTITUENTS

NMATS=7

M=1 CMOD=9 MATID=U MATDB=1 &

EL=253.5E9,6.05E9,0.3901,0.4682,4.167E9,-0.4724E-6,26.63E-6 D=0.,0.,1.

M=2 CMOD=9 MATID=U MATDB=1 &

EL=253.5E9,6.05E9,0.3901,0.4682,4.167E9,-0.4724E-6,26.63E-6 D=0.,1.,0.

NASA/TM—2002-212077/VOL3 238
Section 7: Advanced Topics Example 7d: Woven Composite Analysis ñ Single Step 

M=3 CMOD=9 MATID=U MATDB=1 &

EL=253.5E9,6.05E9,0.3901,0.4682,4.167E9,-0.4724E-6,26.63E-6 D=0.25,1.,0.

M=4 CMOD=9 MATID=U MATDB=1 &

EL=253.5E9,6.05E9,0.3901,0.4682,4.167E9,-0.4724E-6,26.63E-6 D=-0.25,1.,0.

M=5 CMOD=6 MATID=U MATDB=1 &

EL=3.45E9,3.45E9,0.35,0.35,1.278E9,45.E-6,45.E-6

M=6 CMOD=9 MATID=U MATDB=1 &

EL=253.5E9,6.05E9,0.3901,0.4682,4.167E9,-0.4724E-6,26.63E-6 D=0.25,0.,1.

M=7 CMOD=9 MATID=U MATDB=1 &

EL=253.5E9,6.05E9,0.3901,0.4682,4.167E9,-0.4724E-6,26.63E-6 D=-0.25,0.,1.

*RUC

MOD=3 ARCHID=99

NA=4 NB=4 NG=4

D=0.25,0.25,0.25,0.25

H=1.,1.,1.,1.

L=1.,1.,1.,1.

# -- gamma = 1

SM=1,5,2,5

SM=1,3,2,4

SM=2,3,1,4

SM=2,5,1,5

# -- gamma = 2

SM=5,5,5,5

SM=7,5,6,5

SM=7,5,6,5

SM=5,5,5,5

# -- gamma = 3

SM=2,5,1,5

SM=2,4,1,3

SM=1,4,2,3

SM=1,5,2,5

# -- gamma = 4

SM=5,5,5,5

SM=6,5,7,5

SM=6,5,7,5

SM=5,5,5,5

*PRINT

NPL=-1

*END

Annotated Input Data

1) Flags: None 

2) Constituent materials (*CONSTITUENTS) [KM_2]: 

Number of materials: 7 (NMATS=7) 

Constitutive models: Arbitrary transversely isotropic (CMOD=9) 

 Elastic (CMOD=6) 

Materials: User-defined (MATID=U) 

Material property source: Read from input file (MATDB=1) 

Material properties: See Table .1 (EL=...) 

NASA/TM—2002-212077/VOL3 239

7
MAC/GMC 4.0 Example Problem Manual 

Direction of trans. Isotropy: Material #1: (0, 0, 1) (D=0.,0.,1.) 

 Material #2: (0, 1, 0) (D=0.,1.,0.) 

 Material #3: (0.25, 1, 0) (D=0.25,1.,0.) 

 Material #4: (-0.25, 1, 0) (D=-0.25,1.,0.) 

 Material #6: (0.25, 0, 1) (D=0.25,0.,1.) 

 Material #7: (-0.25, 0, 1) (D=-0.25,0.,1.) 

Table 7.1 Constituent material properties for example 7d.

EA

(GPa)

ET

(GPa)

νA νT GA

(GPa)

αA

(10-6/ °C)

αT

(10-6/ °C)

Composite 253.5 6.05 0.3901 0.4682 4.167 -0.4724 6.63 

Epoxy 3.45 3.45 0.35 0.35 1.278 45. 45. 

In this example problem, the material occupying many of the subcells represents a unidirectional 

composite material (see Figure 7.8 and Figure 7.9) intended to model the infiltrated graphite/epoxy 

fiber tow of which the woven composite reinforcement is composed. Thus, transversely isotropic 

effective properties for the unidirectional composite material, taken from the results of Example 1a, 

are specified (for materials #1 ñ 4 and 6 ñ 7). Material #5 is the pure isotropic epoxy matrix material. 

In order to account for the directionality of the fibers in each subcell, the direction vector (D=) is used 

to specify the direction of transverse isotropy for each of the six transversely isotropic constituent 

materials. As indicated, a slope of o (D=0.25,...) has been employed for the inclined fibers in the 

appropriate subcells. When placed in the correct arrangement (under *RUC), these constituent 

materials will represent the plain weave reinforced composite shown in Figure 7.8. For more 

information on constitutive material properties, see the MAC/GMC 4.0 Keywords Manual Section 2. 

3) Analysis type (*RUC) → Repeating Unit Cell Analysis [KM_3]: 

Analysis model: Triply periodic GMC (MOD=3) 

RUC architecture: User-defined (ARCHID=99) 

No. subcells in x1-dir.: 4 (NA=4) 

No. subcells in x2-dir.: 4 (NB=4) 

No. subcells in x3-dir.: 4 (NG=4) 

Subcell depths: 0.25, 0.25, 0.25, 0.25 (D=0.25,0.25,0.25,0.25) 

Subcell heights: 1., 1., 1., 1. (H=1.,1.,1.,1.) 

Subcell lengths: 1., 1., 1., 1. (L=1.,1.,1.,1.) 

Material assignment: see input file (SM=...) 

4) Loading: None 

5) Damage and Failure: None 

6) Output: 

a) Output file print level (*PRINT) [KM_6]: 

Print level: -1 (effective properties only) (NPL=-1) 

b) x-y plots (*XYPLOT): None 

7) End of file keyword: (*END) 

NASA/TM—2002-212077/VOL3 240
Section 7: Advanced Topics Example 7d: Woven Composite Analysis ñ Single Step 

Results

The results from this example problem are the predicted effective properties of the woven composite 

printed to the MAC/GMC 4.0 output file: 

CG - Effective/Macro Stiffness Matrix

NOTE: Stiffness relates stresses to normal and ENGINEERING SHEAR STRAINS

0.6650E+10 0.3432E+10 0.3432E+10 0.0000E+00 0.3260E-08 0.0000E+00

0.3432E+10 0.1061E+11 0.2873E+10 0.0000E+00 -0.1010E-26 -0.9413E-08

0.3432E+10 0.2873E+10 0.1061E+11 0.0000E+00 -0.4891E-09 -0.9458E-09

0.0000E+00 0.0000E+00 0.0000E+00 0.2079E+10 0.0000E+00 0.0000E+00

0.1281E-08 0.1164E-09 -0.3193E-09 0.0000E+00 0.1859E+10 0.0000E+00

-0.4657E-09 -0.1791E-09 -0.1979E-08 0.0000E+00 0.5404E-28 0.1859E+10

Effective Engineering Moduli

Use with caution when global stiffness matrix is anisotropic

E11S= 0.4902E+10

N12S= 0.2545

E22S= 0.8703E+10

N23S= 0.1245

E33S= 0.8703E+10

G23S= 0.2079E+10

G13S= 0.1859E+10

G12S= 0.1859E+10

Effective Thermal Expansion Coefficients

NOTE: Shear CTEs are "engineering" shear CTEs

0.4267E-04 0.2045E-04 0.2045E-04

0.0000E+00 -0.2846E-22 0.3251E-22

The effective stiffness matrix results show that, even through materials #3, #4, #6, and #7 are monoclinic 

(in the global coordinates of the woven composite), the woven composite is orthotropic. This is because 

for every subcell that contains inclined fibers, there is a subcell containing fibers with the incline reversed 

(see Figure 7.9). The anisotropic terms thus add to zero during the GMC triply periodic homogenization 

procedure. The same is true of the ìshearî CTE terms. 

For a woven composite such as the plain weave graphite/epoxy modeled here, the in-plane properties (i.e., 

x2-x3 plane) properties are the most important. This is because woven composites are often made in thin 

plate form. Thus, the in-plane properties for the woven composite predicted by this single step GMC 

homogenization procedure are: 

E = 8.703 GPa, ν = 0.1245, G = 2.079 GPa, and α = 20.45×10-6 /°C 

Thus, these predictions indicate that the in-plane elastic modulus of 8.703 GPa is quite small, on the order 

of the transverse elastic modulus of the fiber (7.6 GPa) and the elastic modulus of the epoxy (3.45 GPa). 

The extreme longitudinal stiffness of the fiber (388.2 GPa) appears not to impart much stiffness to the 

woven composite in this single step homogenization procedure. For an alternative two step procedure, 

which has been shown to be significantly more accurate (particularly in the case of large constituent 

property mismatch), see Example 7e. 

NASA/TM—2002-212077/VOL3 241
MAC/GMC 4.0 Example Problem Manual 

Example 7e: Woven Composite Analysis ñ Two Step

This example considers the previous plain weave graphite/epoxy composite considered in Example 7d. In 

this example, however, a two step approach is employed to homogenize the triply periodic RUC in order 

to arrive at the effective properties of the composite. This two step homogenization procedure has been 

shown to yield significantly more accurate effective properties for the composite than the one step 

approach demonstrated in Example 7d (Bednarcyk, 2000). 

Considering the plain weave reinforced composite shown in Figure 7.8, the six unique through-thickness 

subcell groups shown in Figure 7.10 can be identified. Note that reversing the stacking sequence of 

Group 1 does not result in a unique through-thickness subcell group. In step one of the two step 

homogenization procedure, the six unique subcell groups are individually homogenized in the throughthickness direction. The result is a set of effective thermo-elastic properties for each of the groups. Note 

that, since Group 5 consists only of matrix material, its effective properties are that of the matrix. Further, 

Groups 2, 3, 4, and 6 contain subcells with inclined fibers. Thus, the effective properties of these Groups 

will be monoclinic. 

Once the effective properties of each through-thickness subcell group are determined, an RUC, with only 

one through-thickness subcell, can be assembled that represents the woven composite. This is shown in 

Figure 7.11, where the effective properties of the appropriate through-thickness group are used to 

represent the subcell materials. Then, as step two of the procedure, this two-dimensional RUC is 

homogenized to determine the effective properties of the woven composites. As will be shown, the 

results of this two step homogenization procedure can be quite different from those of the corresponding 

one step homogenization procedure. 

Group 1 Group 2 Group 3

Group 4 Group 5 Group 6

 

Figure 7.10 Unique through-thickness groups in the plain weave reinforced composite shown in Figure 

7.8. These through-thickness subcell groups are homogenized using MAC/GMC 4.0 to 

determine their effective stiffness matrices in Step 1 of the two step homogenization 

procedure.

NASA/TM—2002-212077/VOL3 242
Section 7: Advanced Topics Example 7e: Woven Composite Analysis ñ Two Step 

Figure 7.11 Representation of the plain-weave reinforced composite after though-thickness 

homogenization in Step 1. Step 2 involves homogenizing this RUC to determine the 

effective properties of the plain weave reinforced composite. 

MAC/GMC Input Files: example_7e_1.mac, example_7e_2.mac

MAC/GMC 4.0 Example 7e_1 - Two step plain weave composite analysis (step 1)

*CONSTITUENTS

NMATS=7

M=1 CMOD=9 MATID=U MATDB=1 &

EL=253.5E9,6.05E9,0.3901,0.4682,4.167E9,-0.4724E-6,26.63E-6 D=0.,0.,1.

M=2 CMOD=9 MATID=U MATDB=1 &

EL=253.5E9,6.05E9,0.3901,0.4682,4.167E9,-0.4724E-6,26.63E-6 D=0.,1.,0.

M=3 CMOD=9 MATID=U MATDB=1 &

EL=253.5E9,6.05E9,0.3901,0.4682,4.167E9,-0.4724E-6,26.63E-6 D=0.25,1.,0.

M=4 CMOD=9 MATID=U MATDB=1 &

EL=253.5E9,6.05E9,0.3901,0.4682,4.167E9,-0.4724E-6,26.63E-6 D=-0.25,1.,0.

M=5 CMOD=6 MATID=U MATDB=1 &

EL=3.45E9,3.45E9,0.35,0.35,1.278E9,45.E-6,45.E-6

M=6 CMOD=9 MATID=U MATDB=1 &

EL=253.5E9,6.05E9,0.3901,0.4682,4.167E9,-0.4724E-6,26.63E-6 D=0.25,0.,1.

M=7 CMOD=9 MATID=U MATDB=1 &

EL=253.5E9,6.05E9,0.3901,0.4682,4.167E9,-0.4724E-6,26.63E-6 D=-0.25,0.,1.

*RUC

MOD=3 ARCHID=99

NA=4 NB=1 NG=1

D=0.25,0.25,0.25,0.25

H=1.

L=1.

# -- Group 1

SM=1

SM=1

SM=2

SM=2

# -- Group 2

# SM=5

# SM=3

# SM=3

# SM=5

# -- Group 3

# SM=5

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MAC/GMC 4.0 Example Problem Manual 

# SM=4

# SM=4

# SM=5

# -- Group 4

# SM=5

# SM=7

# SM=7

# SM=5

# -- Group 6

# SM=5

# SM=6

# SM=6

# SM=5

*PRINT

NPL=-1

*END

MAC/GMC 4.0 Example 7e_2 - Two step plain weave composite analysis (step 2)

*CONSTITUENTS

NMATS=6

M=1 CMOD=15 MATID=U MATDB=1 &

EL=7.827E9, 4.103E9, 4.103E9, 0., 0., 0., &

132.4E9, 4.519E9, 0., 0., 0., &

132.4E9, 0., 0., 0., &

4.167E9, 0., 0., &

2.757E9, 0., &

2.757E9 &

38.56E-6, 0.3901E-6, 0.3901E-6, 0., 0., 0.

M=2 CMOD=15 MATID=U MATDB=1 &

EL=6.677E9, 4.843E9, 3.274E9, 0., 0., 0.1887E9, &

55.80E9, 3.384E9, 0., 0., 6.969E9, &

6.662E9, 0., 0., -0.006711E9, &

2.625E9, 0.1830E9, 0., &

1.613E9, 0., &

2.269E9 &

43.74E-6, 4.043E-6, 39.04E-6, 0., 0., -12.19E-6

M=3 CMOD=15 MATID=U MATDB=1 &

EL=6.677E9, 4.843E9, 3.274E9, 0., 0., -0.1887E9, &

55.80E9, 3.384E9, 0., 0., -6.969E9, &

6.662E9, 0., 0., 0.006711E9, &

2.625E9,-0.1830E9, 0., &

1.613E9, 0., &

2.269E9 &

43.74E-6, 4.043E-6, 39.04E-6, 0., 0., 12.19E-6

M=4 CMOD=15 MATID=U MATDB=1 &

EL=6.677E9, 3.274E9, 4.843E9, 0., -0.1887E9, 0., &

6.662E9, 3.384E9, 0., 0.006711E9, 0., &

55.80E9, 0., -6.969E9, 0., &

2.625E9, 0., -0.1830E9, &

2.269E9, 0., &

1.613E9 &

43.74E-6, 39.04E-6, 4.043E-6, 0., 12.19E-6, 0.

M=5 CMOD=6 MATID=U MATDB=1 &

EL=3.45E9,3.45E9,0.35,0.35,1.278E9,45.E-6,45.E-6

M=6 CMOD=15 MATID=U MATDB=1

NTP=2

NASA/TM—2002-212077/VOL3 244
Section 7: Advanced Topics Example 7e: Woven Composite Analysis ñ Two Step 

TEM=23.,150.

C11=6.677E9,6.677E9

C12=3.274E9,3.274E9

C13=4.843E9,4.843E9

C14=0.,0.

C15=0.1887E9,0.1887E9

C16=0.,0.

C22=6.662E9,6.662E9

C23=3.384E9,3.384E9

C24=0.,0.

C25=-0.006711E9,-0.006711E9

C26=0.,0.

C33=55.80E9,55.80E9

C34=0.,0.

C35=6.969E9,6.969E9

C36=0.,0.

C44=2.625E9,2.625E9

C45=0.,0.

C46=0.1830E9,0.1830E9

C55=2.269E9,2.269E9

C56=0.,0.

C66=1.613E9,1.613E9

ALF1=43.74E-6,43.74E-6

ALF2=39.04E-6,39.04E-6

ALF3=4.043E-6,4.043E-6

ALF4=0.,0.

ALF5=-12.19E-6,-12.19E-6

ALF6=0.,0.

*RUC

MOD=3 ARCHID=99

NA=1 NB=4 NG=4

D=1.

H=1.,1.,1.,1.

L=1.,1.,1.,1.

SM=1,2,1,3

SM=4,5,6,5

SM=1,3,1,2

SM=6,5,4,5

*PRINT

NPL=-1

*END

Annotated Input Data

Step 1: Homogenization of the through-thickness subcell groups → example_7e_1.mac

1) Flags: None 

2) Constituent materials (*CONSTITUENTS) [KM_2]: 

Number of materials: 7 (NMATS=7) 

Constitutive models: Arbitrary transversely isotropic (CMOD=9) 

 Elastic (CMOD=6) 

NASA/TM—2002-212077/VOL3 245
MAC/GMC 4.0 Example Problem Manual 

Materials: User-defined (MATID=U) 

Material property source: Read from input file (MATDB=1) 

Direction of trans. Isotropy: Material #1: (0, 0, 1) (D=0.,0.,1.) 

 Material #2: (0, 1, 0) (D=0.,1.,0.) 

 Material #3: (0.25, 1, 0) (D=0.25,1.,0.) 

 Material #4: (-0.25, 1, 0) (D=-0.25,1.,0.) 

 Material #6: (0.25, 0, 1) (D=0.25,0.,1.) 

 Material #7: (-0.25, 0, 1) (D=-0.25,0.,1.) 

In this example problem, the constituent materials are the same as those employed in Example 7d. 

3) Analysis type (*RUC) → Repeating Unit Cell Analysis [KM_3]: 

Analysis model: Triply periodic GMC (MOD=3) 

RUC architecture: User-defined (ARCHID=99) 

No. subcells in x1-dir.: 4 (NA=4) 

No. subcells in x2-dir.: 1 (NB=1) 

No. subcells in x3-dir.: 1 (NG=1) 

Subcell depths: 0.25, 0.25, 0.25, 0.25 (D=0.25,0.25,0.25,0.25) 

Subcell height: 1. (H=1.) 

Subcell length: 1. (L=1.) 

Material assignment: see input file (SM=...) 

In this example problem, effective elastic properties must be generated for each of five subcell 

groups. Recall that Group 5 contains only the matrix material and thus does not need to be 

homogenized. The appropriate lines in the input file must be commented and uncommented in order 

to generate all of these effective properties. 

4) Loading: None 

5) Damage and Failure: None 

6) Output: 

a) Output file print level (*PRINT) [KM_6]: 

Print level: -1 (effective properties only) (NPL=-1) 

b) x-y plots (*XYPLOT): None 

7) End of file keyword: (*END) 

Step 2: Homogenization of the 2D RUC → example_7e_2.mac

1) Flags: None 

2) Constituent materials (*CONSTITUENTS) [KM_2]: 

Number of materials: 6 (NMATS=6) 

Constitutive models: Anisotropic elastic (CMOD=15) 

 Elastic (CMOD=6) 

NASA/TM—2002-212077/VOL3 246

Material properties: See Table 7.1 (EL=...) 
Section 7: Advanced Topics Example 7e: Woven Composite Analysis ñ Two Step 

Materials: User-defined (MATID=U) 

Material property source: Read from input file (MATDB=1) 

Material properties: See input file (EL=...) 

In step two of this example problem, the material occupying many of the subcells represents 

homogenized subcell groups that contain subcells with inclined fibers (see Figure 7.10). These 

materials are monoclinic and thus require a more general elastic constitutive model than presented 

this far in this Example Manual. For this reason, a completely anisotropic elastic model (CMOD=15) 

has been incorporated within MAC/GMC 4.0. This model allows the user to specify all 21 

components of the constituent material stiffness matrix, as well as six CTE components. For material 

#6, the use of this model in conjunction with temperature-dependent material properties is illustrated. 

For more information on constituent materials, see the MAC/GMC 4.0 Keywords Manual Section 2. 

3) Analysis type (*RUC) → Repeating Unit Cell Analysis [KM_3]: 

Analysis model: Triply periodic GMC (MOD=3) 

RUC architecture: User-defined (ARCHID=99) 

No. subcells in x1-dir.: 1 (NA=1) 

No. subcells in x2-dir.: 4 (NB=4) 

No. subcells in x3-dir.: 4 (NG=4) 

Subcell depth: 1. (D=1.) 

Subcell heights: 1., 1., 1., 1. (H=1.,1.,1.,1.) 

Subcell lengths: 1., 1., 1., 1. (L=1.,1.,1.,1.) 

Material assignment: see input file (SM=...) 

The anisotropic materials that represent the homogenized subcell groups are arranged as shown in 

Figure 7.11 to represent the plain weave reinforced composite. Note that the anisotropic constitutive 

model can only be used with triply-periodic GMC (i.e., the doubly periodic GMC implementation 

assumes the subcell materials to be at most orthotropic in the RUC coordinate system). Thus, even 

though the RUC in this example problem is two-dimensional, triply periodic GMC (with one x1-

direction subcell) has been employed. For more information on the MAC/GMC 4.0 RUC 

capabilities, see the MAC/GMC 4.0 Keywords Manual Section 3. 

4) Loading: None 

5) Damage and Failure: None 

6) Output: 

a) Output file print level (*PRINT) [KM_6]: 

Print level: -1 (effective properties only) (NPL=-1) 

b) x-y plots (*XYPLOT): None 

7) End of file keyword: (*END) 

NASA/TM—2002-212077/VOL3 247
MAC/GMC 4.0 Example Problem Manual 

Results

Results for this example problem take the form of the effective thermo-elastic properties of the subcell 

groups (in step 1) and of the plain weave reinforced graphite/epoxy composite (step 2). The effective 

stiffness matrix, elastic moduli, and CTEs of each through-thickness subcell group (determined in step 1) 

are given below. Because Group 1 contains no subcells with inclined fibers, the groupís effective thermoelastic properties are orthotropic. The remaining subcell groups (with the exception of Group 5, which 

contains only the matrix material and thus does not need to be homogenized in step 1) do contain subcells 

with inclined fibers. The effective elastic behavior of these homogenized subcell groups is monoclinic 

(i.e., 13 independent constants, see Jones (1975)). The non-zero C16 term for Group 2 indicates that, for 

example, a non-zero shear stress σ12 component would arise were a normal strain component ε11 applied. 

In addition, each monoclinic subcell group has a non-zero shear CTE. This indicates that, were a 

temperature change applied to the subcell group, a shear strain component would arise. The effective 

stiffness matrix of each group, as well as the effective CTEs, are employed as constituent material 

properties in step 2 of the homogenization procedure. 

GROUP 1

CG - Effective/Macro Stiffness Matrix

NOTE: Stiffness relates stresses to normal and ENGINEERING SHEAR STRAINS

0.7827E+10 0.4103E+10 0.4103E+10 0.0000E+00 0.0000E+00 0.0000E+00

0.4103E+10 0.1324E+12 0.4519E+10 0.0000E+00 0.0000E+00 0.0000E+00

0.4103E+10 0.4519E+10 0.1324E+12 0.0000E+00 0.0000E+00 0.0000E+00

0.0000E+00 0.0000E+00 0.0000E+00 0.4167E+10 0.0000E+00 0.0000E+00

0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.2757E+10 0.0000E+00

0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.2757E+10

Effective Engineering Moduli

Use with caution when global stiffness matrix is anisotropic

E11S= 0.7581E+10

N12S= 0.0300

E22S= 0.1302E+12

N23S= 0.0182

E33S= 0.1302E+12

G23S= 0.4167E+10

G13S= 0.2757E+10

G12S= 0.2757E+10

Effective Thermal Expansion Coefficients

NOTE: Shear CTEs are "engineering" shear CTEs

0.3856E-04 0.3901E-06 0.3901E-06

0.0000E+00 0.0000E+00 0.0000E+00

GROUP 2

CG - Effective/Macro Stiffness Matrix

NOTE: Stiffness relates stresses to normal and ENGINEERING SHEAR STRAINS

0.6677E+10 0.4843E+10 0.3274E+10 0.0000E+00 0.0000E+00 0.1887E+09

0.4843E+10 0.5580E+11 0.3384E+10 0.0000E+00 0.0000E+00 0.6969E+10

0.3274E+10 0.3384E+10 0.6662E+10 0.0000E+00 0.0000E+00 -0.6711E+07

0.0000E+00 0.0000E+00 0.0000E+00 0.2625E+10 0.1830E+09 0.0000E+00

0.0000E+00 0.0000E+00 0.0000E+00 0.1830E+09 0.1613E+10 0.0000E+00

0.3775E+09 0.5575E+10 -0.5369E+07 0.0000E+00 0.0000E+00 0.2269E+10

NASA/TM—2002-212077/VOL3 248
Section 7: Advanced Topics Example 7e: Woven Composite Analysis ñ Two Step 

Effective Engineering Moduli

Use with caution when global stiffness matrix is anisotropic

E11S= 0.4889E+10

N12S= 0.0733

E22S= 0.3586E+11

N23S= 0.3140

E33S= 0.4994E+10

G23S= 0.2604E+10

G13S= 0.1600E+10

G12S= 0.1552E+10

Effective Thermal Expansion Coefficients

NOTE: Shear CTEs are "engineering" shear CTEs

0.4374E-04 0.4043E-05 0.3904E-04

0.0000E+00 0.0000E+00 -0.1219E-04

GROUP 3

CG - Effective/Macro Stiffness Matrix

NOTE: Stiffness relates stresses to normal and ENGINEERING SHEAR STRAINS

0.6677E+10 0.4843E+10 0.3274E+10 0.0000E+00 0.0000E+00 -0.1887E+09

0.4843E+10 0.5580E+11 0.3384E+10 0.0000E+00 0.0000E+00 -0.6969E+10

0.3274E+10 0.3384E+10 0.6662E+10 0.0000E+00 0.0000E+00 0.6711E+07

0.0000E+00 0.0000E+00 0.0000E+00 0.2625E+10 -0.1830E+09 0.0000E+00

0.0000E+00 0.0000E+00 0.0000E+00 -0.1830E+09 0.1613E+10 0.0000E+00

-0.3775E+09 -0.5575E+10 0.5369E+07 0.0000E+00 0.0000E+00 0.2269E+10

Effective Engineering Moduli

Use with caution when global stiffness matrix is anisotropic

E11S= 0.4889E+10

N12S= 0.0733

E22S= 0.3586E+11

N23S= 0.3140

E33S= 0.4994E+10

G23S= 0.2604E+10

G13S= 0.1600E+10

G12S= 0.1552E+10

Effective Thermal Expansion Coefficients

NOTE: Shear CTEs are "engineering" shear CTEs

0.4374E-04 0.4043E-05 0.3904E-04

0.0000E+00 0.0000E+00 0.1219E-04

GROUP 4

CG - Effective/Macro Stiffness Matrix

NOTE: Stiffness relates stresses to normal and ENGINEERING SHEAR STRAINS

0.6677E+10 0.3274E+10 0.4843E+10 0.0000E+00 -0.1887E+09 0.0000E+00

0.3274E+10 0.6662E+10 0.3384E+10 0.0000E+00 0.6711E+07 0.0000E+00

0.4843E+10 0.3384E+10 0.5580E+11 0.0000E+00 -0.6969E+10 0.0000E+00

0.0000E+00 0.0000E+00 0.0000E+00 0.2625E+10 0.0000E+00 -0.1830E+09

-0.3775E+09 0.5369E+07 -0.5575E+10 0.0000E+00 0.2269E+10 0.0000E+00

0.0000E+00 0.0000E+00 0.0000E+00 -0.1830E+09 0.0000E+00 0.1613E+10

NASA/TM—2002-212077/VOL3 249
MAC/GMC 4.0 Example Problem Manual 

Effective Engineering Moduli

Use with caution when global stiffness matrix is anisotropic

E11S= 0.4889E+10

N12S= 0.4541

E22S= 0.4994E+10

N23S= 0.0343

E33S= 0.3586E+11

G23S= 0.2604E+10

G13S= 0.1552E+10

G12S= 0.1600E+10

Effective Thermal Expansion Coefficients

NOTE: Shear CTEs are "engineering" shear CTEs

0.4374E-04 0.3904E-04 0.4043E-05

0.0000E+00 0.1219E-04 0.0000E+00

GROUP 6

CG - Effective/Macro Stiffness Matrix

NOTE: Stiffness relates stresses to normal and ENGINEERING SHEAR STRAINS

0.6677E+10 0.3274E+10 0.4843E+10 0.0000E+00 0.1887E+09 0.0000E+00

0.3274E+10 0.6662E+10 0.3384E+10 0.0000E+00 -0.6711E+07 0.0000E+00

0.4843E+10 0.3384E+10 0.5580E+11 0.0000E+00 0.6969E+10 0.0000E+00

0.0000E+00 0.0000E+00 0.0000E+00 0.2625E+10 0.0000E+00 0.1830E+09

0.3775E+09 -0.5369E+07 0.5575E+10 0.0000E+00 0.2269E+10 0.0000E+00

0.0000E+00 0.0000E+00 0.0000E+00 0.1830E+09 0.0000E+00 0.1613E+10

Effective Engineering Moduli

Use with caution when global stiffness matrix is anisotropic

E11S= 0.4889E+10

N12S= 0.4541

E22S= 0.4994E+10

N23S= 0.0343

E33S= 0.3586E+11

G23S= 0.2604E+10

G13S= 0.1552E+10

G12S= 0.1600E+10

Effective Thermal Expansion Coefficients

NOTE: Shear CTEs are "engineering" shear CTEs

0.4374E-04 0.3904E-04 0.4043E-05

0.0000E+00 -0.1219E-04 0.0000E+00

STEP 2 RESULTS (Effective Properties of the Woven Composite)

The results of step 2 in the two step homogenization procedure for the plain weave reinforced 

graphite/epoxy composite are the predicted effective thermo-elastic properties of the woven composite. 

As in Example 7d, the effective stiffness matrix results show that, even through Groups 2, 3, 4, and 6 are 

monoclinic (in the global coordinates of the woven composite), the woven composite is orthotropic. This 

is because for every subcell that contains inclined fibers, there is a subcell containing fibers with the 

incline reversed (see Figure 7.9). The anisotropic terms thus add to zero during the GMC triply periodic 

homogenization procedure. The same is true of the ìshearî CTE terms. 

NASA/TM—2002-212077/VOL3 250
Section 7: Advanced Topics Example 7e: Woven Composite Analysis ñ Two Step 

Table 7.2 provides a comparison of the MAC/GMC 4.0 prediction for the in-plane (i.e., plane of the 

woven reinforcement) thermo-elastic properties of the woven composite. The effect of utilizing the 2 step 

approach rather than the 1 step approach is stunning. The predicted in plane elastic modulus is 3.2 times 

higher, the predicted in-plane Poisson ratio is 2.3 times lower, and the predicted in-plane CTE is 2.9 times 

lower. The effect on the predicted in-plane shear modulus is less significant; it is only 7.5% higher when 

using the 2 step approach. The reason for this dramatic effect on the effective properties is the fact that 

the triply periodic RUC that represents the woven composite (see Figure 7.8 and Figure 7.9) has no 

continuous fibers spanning the x2- and x3-directions. Following any row or column of subcells in the x2- 

and x3-directions (at a constant x1 position), a subcell containing either the pure matrix material or 

transversely oriented fibers will be encountered. These subcells serve as weak links and cause the 1 step 

homogenization approach to predict a low effective in-plane stiffness (along with a high in-plane Poisson 

ratio and CTE). In the 2 step homogenization procedure, the woven composite properties are 

homogenized, or ìsmearedî, in the through-thickness (x1) direction in step 1. This, in effect, removes the 

extreme weak links from the RUC. In Figure 7.11, two rows of subcells in the x2-direction exists that 

contain only homogenized Groups 1, 2, and 3, while in the x3-direction, two columns of subcells exist that 

contain only homogenized Groups 1, 4, and 6 (the remaining two rows or columns in the two directions 

do contain extreme weak link subcells). Examining Figure 7.10, it is clear that, while Groups 2, 3, 4, and 

6 are less stiff than Group 1, they still contain subcells with fibers mainly aligned in the appropriate inplane direction. This fact can also be seen in the effective properties of the Groups given above. 

Therefore, the RUC depicted in Figure 7.11 has two rows or columns of subcells in each of the in-plane 

directions that do not contain extreme weak link subcells. The result is the stiffer predicted composite 

properties obtained in the 2 step approach. Note that the large constituent property mismatch between the 

graphite fiber and epoxy matrix in the present example magnifies the discrepancy between the 1 step and 

2 step approaches. A significant improvement in the 1 step procedure will result from the use of HFGMC 

rather than GMC to model the woven composite. However, in order to model the architecture of the 

woven reinforcement, triply periodic HFGMC is required. This theory is currently under development. 

Finally, Bednarcyk (2000) showed that the 2 step MAC/GMC homogenization procedure provides much 

more accurate predictions of woven composite properties than does the 1 step approach. The two step 

approach is somewhat cumbersome compared to most of the other features within MAC/GMC 4.0. 

Future versions of MAC/GMC will automate the woven composite analysis procedure in order to relieve 

this burden from the user. 

 

CG - Effective/Macro Stiffness Matrix

NOTE: Stiffness relates stresses to normal and ENGINEERING SHEAR STRAINS

0.6669E+10 0.3662E+10 0.3662E+10 0.0000E+00 0.5215E-07 0.2980E-07

0.3662E+10 0.3022E+11 0.3562E+10 0.0000E+00 0.2068E-24 0.0000E+00

0.3662E+10 0.3562E+10 0.3022E+11 0.0000E+00 0.5679E-08 0.0000E+00

0.0000E+00 0.0000E+00 0.0000E+00 0.2234E+10 0.0000E+00 0.0000E+00

-0.4657E-08 0.0000E+00 0.3950E-09 0.0000E+00 0.1626E+10 0.0000E+00

-0.1397E-08 0.4740E-08 0.3725E-08 0.0000E+00 0.0000E+00 0.1626E+10

Effective Engineering Moduli

Use with caution when global stiffness matrix is anisotropic

E11S= 0.5875E+10

N12S= 0.1084

E22S= 0.2813E+11

N23S= 0.0550

E33S= 0.2813E+11

G23S= 0.2234E+10

G13S= 0.1626E+10

G12S= 0.1626E+10

NASA/TM—2002-212077/VOL3 251
MAC/GMC 4.0 Example Problem Manual 

Effective Thermal Expansion Coefficients

NOTE: Shear CTEs are "engineering" shear CTEs

0.5595E-04 0.7048E-05 0.7048E-05

0.0000E+00 -0.1206E-21 0.1138E-22

Table 7.2 Predicted effective in-plane thermo-elastic properties for 32.5% plain weave reinforced 

graphite/epoxy composite using the MAC/GMC 4.0 1 step and 2 step homogenization 

procedures. 

 E (GPa) ν G (GPa) α (×10-6 /°C) 

1 Step Approach 8.703 0.1245 2.079 20.45 

2 Step Approach 28.13 0.0550 2.234 7.048 

NASA/TM—2002-212077/VOL3 252
References 

References 

Aboudi, J. (1987) ìDamage in Composites ñ Modeling of Imperfect Interfacesî Composites Science and 

Technology 28, 103-128. 

Achenbach, J.D. and Zhu, H. (1989) ìEffect of Interfacial Zone on Mechanical Behavior and Failure of 

Fiber-Reinforced Compositesî Journal of the Mechanics and Physics of Solids 37(3), 381-393. 

Arnold, S.M. and Kruch, S. (1994) ìDifferential Continuum Damage Mechanics Models for Creep and 

Fatigue of Unidirectional Metal Matrix Compositesî International Journal of Damage Mechanics 3(2), 

170-191. 

Bednarcyk, B.A. (2000) ìModeling Woven Polymer Matrix Composites With MAC/GMCî NASA/CRñ

2000-210370. 

Bednarcyk, B.A. and Arnold, S.M. (2001) ìMicromechanics-Based Deformation and Failure Prediction 

for Longitudinally Reinforced Titanium Compositesî Composites Science and Technology 61, 705-729. 

Bednarcyk, B.A. and Arnold, S.M. (2002) ìTransverse Tensile and Creep Modeling of Continuously 

Reinforced Titanium Composites with Local Debondingî International Journal of Solids and Structures

39, 1987-2017. 

Curtin, W.A. (1991) ìTheory of Mechanical Properties of Ceramic-Matrix Compositesî Journal of the 

American Ceramics Society 74(11), 2837-2845. 

Curtin, W.A. (1993) ìUltimate Strengths of Fibre-Reinforced Ceramics and Metalsî Composites 24(2), 

98-102. 

Goldberg, R.K., Comiskey, M.D., and Bednarcyk, B.A. (1999) ìMicromechanics Analysis Code PostProcessing (MACPOST) User Guide Version 1.0î NASA/TMñ1999-209062. 

Jones, J.P. and Whittier, J.S. (1967) ìWaves at Flexibly Bonded Interfacesî Journal of Applied 

Mechanics 34, 905-909. 

Jones, R.M. (1974) ìStiffness of Orthotropic Materials and Laminated Fiber-Reinforced Compositesî 

AIAA Journal, January, pp. 112-114. 

Jones, R.M. (1975) Mechanics of Composite Materials, Hemisphere Publishing Corp., New York. 

Pahr, D.H. and Arnold, S.M. (2002) ìThe Applicability of the Generalized Method of Cells for Analyzing 

Discontinuously Reinforced Compositesî Composites: Part B 33, 153-170. 

Paley, M. and Aboudi, J. (1992) ìMicromechanical Analysis of Composites by the Generalized Cells 

Modelî Mechanics of Materials 14, 127-139. 

Plastics Design Library (1995) Fatigue and Tribological Properties of Plastics and Elastomers. Plastics 

Design Library, William Andrew, Inc., Norwich, NY. 

NASA/TM—2002-212077/VOL3 253
MAC/GMC 4.0 Example Problem Manual 

Saleeb, A.F., Arnold, S.M., Castelli, M.G., Wilt, T.E., and Graf, W. (2001) ìA General Hereditary 

Multimechanism-Based Deformation Model with Application to the Viscoelastoplastic Response of 

Titanium Alloysî International Journal of Plasticity 17, 1305-1350. 

Wilt, T.E. and Arnold, S.M. (1996) ìMicromechanics Analysis Code (MAC) User Guide: Version 2.0î 

NASA-TM-107290. 

Wilt, T.E., Arnold, S.M., and Saleeb, A.F. (1997) ìA Coupled/Uncoupled Computational Scheme for 

Deformation and Fatigue Damage Analysis of Unidirectional Metal Matrix Compositesî in Applications 

of Continuum Damage Mechanics to Fatigue and Fracture, D.L. McDowell (ed.), ASTM STP 1315, 

American Society for Testing and Materials, 65-82. 

NASA/TM—2002-212077/VOL3 254
Appendix 

Appendix 

This Appendix contains the following: 

• Complete MAC/GMC 4.0 output file for Example Problem 4.0 

• Complete distribution (default) version of the usrfun.F90 subroutine 

• Complete distribution (default) version of the usrmat.F90 subroutine 

• Complete distribution (default) version of the usrformde.F90 subroutine 

 

NASA/TM—2002-212077/VOL3 255
MAC/GMC 4.0 Example Problem Manual 

Complete output file for Example 1a 

__ __ ____ _______

/ |/ | / | / _____/

/ | / /| | / /

/ /|_/| | / /_| | / /

/ / | | / ___ | / /_____

/_/ |_|icromechanics /_/ |_|nalysis \______/ode

with

_______ __ __ _______

/ _____/ / |/ | / _____/

/ / ____ / | / /

/ / /_ / / /|_/| | / /

/ /___/ / / / | | / /_____

\______/eneralized /_/ |_|ethod of \______/ells

Version 4.0

NASA Glenn Research Center

Ohio Aerospace Institute

Point of Contact:

Dr. Steven M. Arnold

NASA Glenn Research Center

http://www.grc.nasa.gov/WWW/LPB/mac/index.html

___________________________________________________________________________

Disclaimer: This software package is considered to be a research code

and therefore no expressed or implied warranties are granted

pertaining to the results produced from the use of this code

___________________________________________________________________________

*** WARNING *** This code is export controlled - unauthorized transmission

to non-U.S. citizens may be a violation of federal law

___________________________________________________________________________

================================================================================================

Execution Initiated ---> 15:38:50 11-22-2002

================================================================================================

Input file ---> C:\mac\mac119\example_1a.mac

Output file --> C:\mac\mac119\test.out

******* PROBLEM TITLE ** PROBLEM TITLE ** PROBLEM TITLE ** PROBLEM TITLE ** PROBLEM TITLE *******

MAC/GMC 4.0 Example 1a - graphite/epoxy effective properties

*************************************************************************************************

================================================================================================

***** Section I: Problem Input Data *****

================================================================================================

NASA/TM—2002-212077/VOL3 256
Appendix Complete output file for Example 1a 

------------------------------------------------------------------------------------------------

** A: Execution Flags **

------------------------------------------------------------------------------------------------

Thermal Conductivities Requested

At the following temperatures:

0.2100E+02

No Electro-Magnetic Properties Requested

------------------------------------------------------------------------------------------------

** B: Constituent Material Information **

------------------------------------------------------------------------------------------------

Database material data (if selected) will be read from *internal* database

Number of constituent materials (nmats) = 2

COMMENT: # -- GRAPHITE FIBER

---------------------------------------------------------

MATERIAL # 1

---------------------------------------------------------

Constitutive Model Chosen:

Standard Elastic Model (RUC X2-X3 Plane of Transverse Isotropy)

(CH1) = U

User input material properties

Material properties read from input file

Thermal Conductivities:

Axial = 0.500E+03

Transverse = 0.100E+02

Elastic Properties:

1) E Longitudinal = 0.3882D+12

2) E Transverse = 0.7600D+10

3) Nu Longitudinal = 0.4100D+00

4) Nu Transverse = 0.4500D+00

5) G = 0.1490D+11

6) Alpha Longitudinal = -.6800D-06

7) Alpha Transverse = 0.9740D-05

COMMENT: # -- EPOXY MATRIX

---------------------------------------------------------

MATERIAL # 2

---------------------------------------------------------

Constitutive Model Chosen:

Standard Elastic Model (RUC X2-X3 Plane of Transverse Isotropy)

(CH1) = U

User input material properties

NASA/TM—2002-212077/VOL3 257
MAC/GMC 4.0 Example Problem Manual 

Material properties read from input file

Thermal Conductivities:

Axial = 0.190E+00

Transverse = 0.190E+00

Elastic Properties:

1) E Longitudinal = 0.3450D+10

2) E Transverse = 0.3450D+10

3) Nu Longitudinal = 0.3500D+00

4) Nu Transverse = 0.3500D+00

5) G = 0.1278D+10

6) Alpha Longitudinal = 0.4500D-04

7) Alpha Transverse = 0.4500D-04

------------------------------------------------------------------------------------------------

** C: Analysis Type & Architecture Information **

------------------------------------------------------------------------------------------------

--------------------------------------------------

ANALYSIS APPROACH INFORMATION

--------------------------------------------------

Selected Analysis Model: 2

1) Monolithic Material Analysis

2) Double Periodic GMC Micromechanics Model

3) Triple Periodic GMC Micromechanics Model

12) Double Periodic HFGMC Micromechanics Model

--------------------------------------------------

RUC INFORMATION

--------------------------------------------------

Selected RVE = 1

Monolithic Layer = 0

Square Pack With 4 Subcells = 1

Triangular pack = 2

Square Diagonal Pack = 3

Cross Shape Fiber = 4

(Square Pack)

---------------------------

Circular Fiber Approximation = 6

(49 Cell)

Circular Fiber Approximation = 7

(196 Cell)

Square Pack = 9

(Two Fibers of Different Size)

Rectangular Pack = 11

26x26 circular x-section fiber = 13

8x8 circular fiber approx. = 20

10x10 circular fiber approx. = 21

User Defined RVE = 99

Interface IS NOT present

Fiber Volume Ratio = 0.650E+00

Fiber Material ID Number: 1

Matrix Material ID Number: 2

Material Arrangement:

2 2

1 2

Subcell Dimensions:

H = 0.8062 0.1938

L = 0.8062 0.1938

NASA/TM—2002-212077/VOL3 258
Appendix Complete output file for Example 1a 

------------------------------------------------------------------------------------------------

** D: Applied Loading & Time Integration Information **

------------------------------------------------------------------------------------------------

No Yield Surface Analysis Requested

------------------------------------------------------------------------------------------------

** E: Damage & Failure Information **

------------------------------------------------------------------------------------------------

No Fatigue Damage Analysis Requested

No Elastic Allowable Estimates Requested

No Subcell Failure Analysis Requested

No RUC Failure Analysis Requested

No Debonding Analysis Requested

No Curtin Effective Fiber Breakage Model Analysis Requested

------------------------------------------------------------------------------------------------

** F: Results Output Information **

------------------------------------------------------------------------------------------------

Print Level (nplvl) = -1

No PATRAN Output Requested

No MATLAB Output Requested

================================================================================================

***** Section II: Effective Property Results *****

================================================================================================

------------------- SUBCELL IDENTIFICATION, VOLUME, AND MATERIAL ARRANGEMENT -------------------

(BETA , GAMMA) SUBCELL # SUBCELL MATERIAL SUBCELL VOLUME

1 , 1 1 1 0.650E+00

1 , 2 2 2 0.156E+00

2 , 1 3 2 0.156E+00

2 , 2 4 2 0.375E-01

TOTAL VOLUME = 0.1000E+01

MATERIAL NO.= 1 VOLUME RATIO= 0.6500E+00

MATERIAL NO.= 2 VOLUME RATIO= 0.3500E+00

------------------------------------------------------------------------------------------------

NASA/TM—2002-212077/VOL3 259
MAC/GMC 4.0 Example Problem Manual 

-------------------------------------------------

Effective Thermal Conductivities

At Temperature = 21.0

K11 = 325.0665

K22 = 0.7694

K33 = 0.7694

-------------------------------------------------

--------------------- EFFECTIVE PROPERTIES AT TEMPERATURE = 21.00 ----------------------------

CG - Effective/Macro Stiffness Matrix

0.2571E+12 0.4500E+10 0.4500E+10

0.4500E+10 0.7828E+10 0.3707E+10

0.4500E+10 0.3707E+10 0.7828E+10

0.1916E+10

0.4166E+10

0.4166E+10

CI - Effective/Macro Compliance Matrix

0.3944E-11 -0.1539E-11 -0.1539E-11

-0.1539E-11 0.1653E-09 -0.7738E-10

-0.1539E-11 -0.7738E-10 0.1653E-09

0.5219E-09

0.2400E-09

0.2400E-09

Effective Engineering Moduli

E11S= 0.2535E+12

N12S= 0.3901

E22S= 0.6050E+10

N23S= 0.4682

E33S= 0.6050E+10

G23S= 0.1916E+10

G13S= 0.4166E+10

G12S= 0.4166E+10

Effective Thermal Expansion Coefficients

-0.4724E-06 0.2663E-04 0.2663E-04

------------------------------------------------------------------------------------------------

STOPPING: NPLVL == -1

ENDENDENDENDENDENDENDENDENDENDENDENDENDENDENDEND

_______ ______

/ ____/ | / / __ \\

/ __/ / |/ / / / /

/ /___/ /| / /_/ /

/_____/_/ |_/_____/

ENDENDENDENDENDENDENDENDENDENDENDENDENDENDENDEND

Execution Time: 0.234 second(s)

NASA/TM—2002-212077/VOL3 260
Appendix usrfun.F90: User-defined material property function file 

usrfun.F90: User-defined material property function file 

!***********************************************************************

! BEGIN USER EDITS *

!***********************************************************************

! -- Consider placing your own message here to ensure that your code is

! being executed

IF (TIME .EQ. 0.D0) THEN

WRITE(NIO, *) '*** NOTE: Default USRFUN routine being executed ***'

WRITE(NIO, *) ' material number', MN

WRITE(*, *) '*** NOTE: Default USRFUN routine being executed ***'

WRITE(*, *) ' Material number', MN

ENDIF

!-----------------------------------------------------------------------

! MATERIAL # 1: Elastic, E = function of strain & previous E

!-----------------------------------------------------------------------

IF (MN .EQ. 1) THEN

IF (SA(1) .GT. 0) THEN

EAOLD = PEM(1, MN)

EA = 700.E9 - EAOLD * SA(1) * 80.0

ELSE

EA = 700.E9

EAOLD = 700.E9

ENDIF

IF (EA .GT. EAOLD) THEN

WRITE(NIO, *) 'NON-PHYSICAL REGIME ENTERED'

STOP

ENDIF

EA = EA

ET = EA

FNA = 0.41

FNT = 0.41

GA = EA / (2.0 * (1.0 + FNA))

ALPA(MN) = 4.5E-6

ALPT(MN) = 4.5E-6

NE = 5

PEM(1, MN) = EA

PEM(2, MN) = ET

PEM(3, MN) = FNA

PEM(4, MN) = FNT

PEM(5, MN) = GA

EA = EA

ET = EA

FNA = 0.41

FNT = 0.41

GA = EA / (2.0 * (1.0 + FNA))

ALPA(MN) = 4.5E-6

ALPT(MN) = 4.5E-6

NE = 5

PEM(1, MN) = EA

PEM(2, MN) = ET

PEM(3, MN) = FNA

PEM(4, MN) = FNT

PEM(5, MN) = GA

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!-----------------------------------------------------------------------

! MATERIAL # 2: Incremental plasticity with strain rate dependence

!-----------------------------------------------------------------------

ELSEIF (MN .EQ. 2) THEN

EA = 13161.

ET = EA

FNA = 0.26

FNT = 0.26

GA = EA / (2.0 * (1.0 + FNA))

ALPA(MN) = 9.00E-6

ALPT(MN) = 9.00E-6

NE = 5

PEM(1, MN) = EA

PEM(2, MN) = ET

PEM(3, MN) = FNA

PEM(4, MN) = FNT

PEM(5, MN) = GA

NP = 3

NV = 2 * NP + 1

STRAIN_RATE = 0.4714 * (DSQRT((DSA(1) - DSA(2))**2 + (DSA(1) - DSA(3))**2 + &

(DSA(2) - DSA(3))**2 + 1.5 * DSA(4)**2 + &

1.5 * DSA(5)**2 + 1.5 * DSA(6)**2)) / TSTEP

IF (STRAIN_RATE .GT. 1.E-2) STRAIN_RATE = 1.E-2

IF (STRAIN_RATE .LT. 1.E-7) THEN

SY = 70.

S1 = 80.

S2 = 84.

S3 = 87.

ELSE

SY = 70. + 3. * LOG(1.E7*STRAIN_RATE)

S1 = SY + 10. * SY / 70.

S2 = SY + 14. * SY / 70.

S3 = SY + 17. * SY / 70.

ENDIF

E1=0.01

E2=0.015

E3=0.2

PVM(1, MN) = SY

PVM(2, MN) = S1

PVM(3, MN) = S2

PVM(4, MN) = S3

PVM(5, MN) = E1

PVM(6, MN) = E2

PVM(7, MN) = E3

!-----------------------------------------------------------------------

! MATERIAL # 3: properties = linear function of temperature

! for use with user constitutive model (actually b-p)

!-----------------------------------------------------------------------

ELSEIF (MN .EQ. 3) THEN

IF (CTEMP .LT. 21.0) CTEMP = 21.0

IF (CTEMP .GT. 400.) CTEMP = 400.

! -- Elastic

E = 72.4E9 - 81.53E6 * (CTEMP - 21.)

FN = 0.33 + 7.916E-5 * (CTEMP - 21.)

ALP = 22.5E-6 + 3.958E-9 * (CTEMP - 21.)

NE = 5

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PEM(1, MN) = E

PEM(2, MN) = E

PEM(3, MN) = FN

PEM(4, MN) = FN

PEM(5, MN) = E / (2. * (1. + FN))

ALPA(MN) = ALP

ALPT(MN) = ALP

! -- Viscoplastic (Bodner-Partom)

D0 = 1.E4

Z0 = 340.E6

Z1 = 435.E6

BM = 300.0

AN = 10.0 - 0.02493 * (CTEMP - 21.)

Q = 1.0

NV = 6

PVM(1, MN) = D0

PVM(2, MN) = Z0

PVM(3, MN) = Z1

PVM(4, MN) = BM

PVM(5, MN) = AN

PVM(6, MN) = Q

!-----------------------------------------------------------------------

! MATERIAL # 4: Fiber (NCMD = 6) with local failure = funct of temp

!-----------------------------------------------------------------------

ELSEIF (MN .EQ. 4) THEN

PEM(1, MN) = 58.E3

PEM(2, MN) = 58.E3

PEM(3, MN) = 0.25

PEM(4, MN) = 0.25

PEM(5, MN) = 23.2E3

ALPA(MN) = 3.5E-6

ALPT(MN) = 3.5E-6

! -- Note: *SUBCELL_FAILURE must be specified in the input file to

! enable local failure checking. Any failure properties

! specified here will replace those specified in the input file

XFAILT(MN, 1, 1) = 3500. - 1.8 * (CTEMP - 21.)

!-----------------------------------------------------------------------

! MATERIAL #5: Matrix (NCMD = 6) with damage

!-----------------------------------------------------------------------

ELSEIF (MN .EQ. 5) THEN

PEM(1, MN) = 12770.

PEM(2, MN) = 12770.

PEM(3, MN) = 0.32

PEM(4, MN) = 0.32

PEM(5, MN) = PEM(1, MN)/(2. * (1.+ PEM(3, MN)))

ALPA(MN) = 21.06E-6

ALPT(MN) = 21.06E-6

! -- Note: *DAMAGE must be specified in the input file to enable fatigue

! damage analysis. Any damage properties specified here will

! replace those specified in the input file

ZDAM(MN, 1) = 0.

ZDAM(MN, 2) = 0.

ZDAM(MN, 3) = 0.

ZDAM(MN, 4) = 1.

ZDAM(MN, 5) = 1.

ZDAM(MN, 6) = 1.

ZDAM(MN, 7) = 1.

ZDAM(MN, 8) = 1.

ZDAM(MN, 9) = 1.

ZDAM(MN, 10)= 2.27

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MAC/GMC 4.0 Example Problem Manual 

ZDAM(MN, 11) = 0

ZDAM(MN, 12) = 0.2302

ZDAM(MN, 13) = 20.3

ZDAM(MN, 14) = 900.

ZDAM(MN, 15) = 128.

NZDAM(MN) = 0

ENDIF

!***********************************************************************

! END USER EDITS *

!***********************************************************************

NASA/TM—2002-212077/VOL3 264
Appendix usrmat.F90: User-defined material constitutive model file 

usrmat.F90: User-defined material constitutive model file 

!***********************************************************************

! BEGIN USER EDITS *

!***********************************************************************

! -- Consider placing your own message here to ensure that your code is

! being executed

IF (TIME .EQ. 0.D0) THEN

WRITE(NIO, *) '*** NOTE: Default USRMAT routine being executed ***'

WRITE(*, *) '*** NOTE: Default USRMAT routine being executed ***'

ENDIF

!-----------------------------------------------------------------------

! MATERIAL #1 -- Power Law Creep

!-----------------------------------------------------------------------

IF (MN .EQ. 1) THEN

! -- Creep material properties

CREEPCOEF = PV(1)

POWER = PV(2)

! -- Deviatoric stress [S] in the subcell

TRACE3 = (SA(7) + SA(8) + SA(9)) / 3.

S(1) = SA(7) - TRACE3

S(2) = SA(8) - TRACE3

S(3) = SA(9) - TRACE3

S(4) = SA(10)

S(5) = SA(11)

S(6) = SA(12)

! -- Effective Stress

SEFF = SQRT((SA(7) - SA(8))**2 + (SA(8) - SA(9))**2 + &

(SA(7) - SA(9))**2 + &

+ 6 * (SA(10)**2 + SA(11)**2 + SA(12)**2)) / SQRT(2.)

CREEP_F = CREEPCOEF * SEFF**POWER

IF (SEFF .GT. 0.D0) THEN

COEF = 3 * CREEP_F / (2 * SEFF)

ELSE

COEF = 0.D0

ENDIF

! -- Inelastic strain increments

DSA(13) = COEF * S(1) * TSTEP

DSA(14) = COEF * S(2) * TSTEP

DSA(15) = COEF * S(3) * TSTEP

DSA(16) = COEF * S(4) * TSTEP

DSA(17) = COEF * S(5) * TSTEP

DSA(18) = COEF * S(6) * TSTEP

!-----------------------------------------------------------------------

! MATERIAL #2 --> Bodner-Partom

!-----------------------------------------------------------------------

ELSEIF (MN .EQ. 2) THEN

! -- Copy appropriate viscoplastic material constants

IF( NV .LT. 6 ) THEN

CALL FATALERROR(NIO)

WRITE(NIO, *) ' NOT ENOUGH PV SPACE: NV =', NV

STOP

ENDIF

NASA/TM—2002-212077/VOL3 265
MAC/GMC 4.0 Example Problem Manual 

D0 = PV(1)

Z0 = PV(2)

Z1 = PV(3)

BM = PV(4)

AN = PV(5)

Q = PV(6)

! -- Copy stress from [SA] to [SS]

SS(1) = SA(7)

SS(2) = SA(8)

SS(3) = SA(9)

SS(4) = SA(10)

SS(5) = SA(11)

SS(6) = SA(12)

! -- Compute the deviatoric stress [S] in the subcell

TEMP = (SS(1) + SS(2) + SS(3)) / 3.0

S(1) = SS(1) - TEMP

S(2) = SS(2) - TEMP

S(3) = SS(3) - TEMP

S(4) = SS(4)

S(5) = SS(5)

S(6) = SS(6)

AJ2 = 0.5 * (S(1)**2 + S(2)**2 + S(3)**2) + S(4)**2 + S(5)**2 + &

S(6)**2

SQ3AJ = DSQRT( SS(1)**2 + SS(2)**2 + SS(3)**2+2* (SS(4)**2 + &

SS(5)**2 + SS(6)**2) )

SQ2 = 1.414215

IF (SQ3AJ .EQ. 0.0) THEN

DO I = 1, 6

R(I) = 0

END DO

ELSE

R(1) = SS(1) / SQ3AJ

R(2) = SS(2) / SQ3AJ

R(3) = SS(3) / SQ3AJ

R(4) = SQ2 * SS(4) / SQ3AJ

R(5) = SQ2 * SS(5) / SQ3AJ

R(6) = SQ2 * SS(6) / SQ3AJ

ENDIF

! -- If D0=0 then assume elastic and zero-out [DSA(13-30)] (inelastic

! strain increments and internal variable increments), then return

IF (D0 .EQ. 0) THEN

DO JJ = 13, 30

DSA(JJ) = 0.0

END DO

RETURN

! -- Inelastic

ELSE

ZEF = Z0 + Q * SA(20) + (1 - Q) * (R(1) * SA(21) + R(2) * &

SA(22) + R(3) * SA(23) + R(4) * SA(24) + R(5) &

* SA(25) + R(6) * SA(26))

IF (AJ2 .EQ. 0.0) THEN

AL = 0.0

ELSE

ARG1 = ZEF**2.0 / (3.0 * AJ2)

IF (ARG1 .GT. 1E6) ARG1 = 1E6

CON = .5 * (AN + 1.0) / AN

ARG = CON * (ARG1)**AN

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Appendix usrmat.F90: User-defined material constitutive model file 

IF (ARG .GT. 50.0) ARG = 50.0

AL = D0 / (DEXP(ARG) * DSQRT(AJ2))

ENDIF

! -- Inelastic strain increments

DSA(13) = AL * S(1) * TSTEP

DSA(14) = AL * S(2) * TSTEP

DSA(15) = AL * S(3) * TSTEP

DSA(16) = 2 * AL * S(4) * TSTEP

DSA(17) = 2 * AL * S(5) * TSTEP

DSA(18) = 2 * AL * S(6) * TSTEP

! -- Plastic work increment

WPD = S(1) * DSA(13) + S(2) * DSA(14) + S(3) * DSA(15) + S(4) * &

DSA(16) + S(5) * DSA(17) + S(6) * DSA(18)

! -- State variable increments

DSA(19) = WPD

Z0M = BM / Z0

ZD = Z0M * (Z1 - ZEF) * WPD

DSA(20) = ZD

DSA(21) = ZD * R(1)

DSA(22) = ZD * R(2)

DSA(23) = ZD * R(3)

DSA(24) = ZD * R(4)

DSA(25) = ZD * R(5)

DSA(26) = ZD * R(6)

ENDIF

ENDIF

!***********************************************************************

! END USER EDITS *

!***********************************************************************

NASA/TM—2002-212077/VOL3 267
MAC/GMC 4.0 Example Problem Manual 

usrformde.F90: User-defined stiffness matrix calculation file 

!***********************************************************************

!* BEGIN USER EDITS *

!***********************************************************************

! -- Consider placing your own message here to ensure that your code is

! being executed

IF (TIME .EQ. 0.D0) THEN

WRITE(NIO, *) '*** NOTE: Default USRFORMDE routine being executed ***'

WRITE(NIO, *) ' Material number', MN

WRITE(*, *) '*** NOTE: Default USRFORMDE routine being executed ***'

WRITE(*, *) ' Material number', MN

ENDIF

! -- Obtain material properties from storage

E = PEM(1, MN)

FN = PEM(2, MN)

GA = E / (2. * (1. + FN))

! -- Zero the stiffness matrix

DO I = 1, 6

DO J = 1, 6

DNEW(I, J) = 0

END DO

END DO

! -- Determine the stiffness components

GT = 0.5 * E / (1 + FN)

FK = 0.25 * E / (0.5 * (1 - FN) - FN**2)

DNEW(1, 1) = E + 4.0 * FK * FN**2

DNEW(2, 1) = 2.0 * FK * FN

DNEW(3, 1) = 2.0 * FK * FN

DNEW(1, 2) = 2.0 * FK * FN

DNEW(2, 2) = FK + GT

DNEW(3, 2) = FK - GT

DNEW(1, 3) = 2.0 * FK * FN

DNEW(2, 3) = FK - GT

DNEW(3, 3) = FK + GT

DNEW(4, 4) = GT

DNEW(5, 5) = GA

DNEW(6, 6) = GA

!***********************************************************************

! END USER EDITS *

!***********************************************************************

NASA/TM—2002-212077/VOL3 268
Index 

Index 

A

A=, 174, 177, 178 

ABD matrix, 96, 107, 235 

accuracy, 3, 64, 81, 86, 158, 188, 213, 241, 242, 251 

ACTION=, 159, 162, 166, 168, 169, 170, 189, 191 

Adams-Bashforth, 148 

admissible, 94, 110 

ALF1=, 245 

ALF2=, 245 

ALF3=, 245 

ALF4=, 245 

ALF5=, 245 

ALF6=, 245 

all subcells, 37, 136, 137, 138, 144, 145, 156 

allowables, 152, 153, 154, 155, 156, 157, 158, 161, 162, 

164, 166, 169, 170 

ALPA=, 41, 42, 43, 44, 45, 46, 61, 129, 153, 158, 165, 173, 

221, 222 

ALPT=, 41, 42, 43, 44, 45, 46, 61, 129, 153, 158, 165, 173, 

221, 222 

aluminum, 29, 30, 60, 61, 62, 63, 75, 76, 77, 134, 135, 229, 

231, 232, 234, 235 

ANG=, 13, 92, 93, 99, 100, 104, 125, 126, 129, 143, 165, 

173, 174, 177, 206, 230 

ANGINC=, 135, 137, 143, 144 

angle increment, 134, 137, 144 

anisotropic, 7, 127, 220, 241, 247, 248, 249, 250, 251 

apparent stiffness, 97, 98 

ARCHID=, 7, 9, 10, 13, 23, 24, 65, 67, 68, 70, 72, 73, 75, 

76, 78, 79, 80, 82, 83, 86, 87, 92, 93, 99, 100, 104, 105, 

111, 112, 115, 116, 120, 121, 125, 126, 129, 135, 136, 

143, 148, 149, 153, 154, 158, 159, 165, 173, 174, 175, 

185, 189, 190, 195, 196, 200, 201, 203, 206, 207, 214, 

215, 221, 223, 230, 239, 240, 243, 245, 246, 247 

architecture, 7, 9, 10, 24, 64, 66, 67, 69, 71, 72, 73, 74, 75, 

76, 77, 78, 79, 80, 81, 82, 84, 85, 87, 92, 99, 102, 103, 

112, 116, 120, 121, 136, 143, 148, 149, 153, 154, 159, 

175, 185, 190, 195, 196, 201, 206, 207, 215, 223, 237, 

240, 246, 247, 251 

architecture library, 10, 66, 69, 74, 78, 81 

AS=, 221, 223, 229, 230 

ASCII files, 4, 11, 28, 60, 137, 144, 195, 202, 206, 208 

ASP=, 78, 79, 104, 105, 221, 223 

ASP1=, 79, 104, 105, 200, 201 

ASP2=, 79, 104, 105, 200, 201 

aspect ratio, 67, 68, 72, 73, 79, 87, 105, 112, 121, 149, 154, 

159, 175, 185, 201, 207, 215, 223 

asymmetric, 104, 107, 108, 109, 129, 235 

average material, 156, 169, 170 

axial stiffness matrix (A), 14 

B

barium titanate, 220, 221, 222, 223, 226, 228, 229, 232, 

235 

BaTiO3, 220, 223, 227 

BaTiO3/CoFe2O4, 220, 226, 228, 229, 235 

BDN=, 185, 186, 189, 191 

BDS=, 185, 186, 187, 189, 191 

BE=, 174, 177, 178 

bending, 14, 96, 101, 104, 107, 127, 129, 132, 143 

bending stiffness (D), 14, 96, 127 

biaxial, 110, 113, 165 

BLOCK=, 174, 176 

BN=, 174, 177, 185, 186, 189, 191 

Bodner-Partom, 1, 29, 30, 31, 32, 33, 34, 41, 44, 45, 50, 51, 

54, 55, 58, 61, 62, 76, 134, 231, 263, 265 

boron, 231, 232, 235 

BP=, 174, 177 

BS=, 185, 186, 187, 189, 191 

C

C11=, 245 

C12=, 245 

C13=, 245 

C14=, 245 

C15=, 245 

C16=, 245 

C22=, 155, 245 

C23=, 245 

C24=, 245 

C25=, 245 

C26=, 245 

C33=, 155, 245 

C34=, 245 

C35=, 245 

C36=, 245 

C44=, 245 

C45=, 245 

C46=, 245 

C55=, 245 

C56=, 245 

C66=, 245 

characterization, 171, 184, 220 

circular fiber, 67, 68, 72, 86, 87, 112, 120, 121, 143, 148, 

153, 154, 159, 175, 206, 258 

class C6v electromagnetic materials, 220 

CMOD=, 7, 8, 9, 13, 18, 19, 23, 24, 28, 29, 30, 32, 35, 36, 

38, 41, 42, 43, 44, 45, 46, 50, 51, 55, 56, 60, 61, 62, 65, 

67, 70, 72, 75, 76, 78, 79, 82, 83, 86, 87, 92, 93, 99, 100, 

104, 105, 111, 112, 115, 116, 120, 121, 125, 126, 129, 

130, 135, 136, 143, 148, 149, 153, 154, 158, 159, 165, 

166, 173, 175, 185, 189, 190, 195, 196, 200, 201, 206, 

NASA/TM—2002-212077/VOL3 269
MAC/GMC 4.0 Example Problem Manual 

207, 214, 215, 216, 217, 221, 222, 223, 229, 230, 231, 

237, 238, 239, 243, 244, 245, 246, 247 

cobalt ferrite, 220, 221, 223, 227, 229, 230, 231, 232, 235 

coefficient of thermal expansion, 9, 11, 28, 117, 228, 241, 

247, 248, 251 

CoFe2O4, 220, 221, 223, 227, 229, 230, 231, 232, 235 

Compaq Visual Fortran, 50 

compile, 50 

COMPR=, 153, 154, 155, 159, 161, 162, 174, 178, 179 

concentrations, 89, 153, 158 

conductivity, 8, 9, 11 

consistency, 198 

constant compliant interface (CCI) model, 184, 186, 187 

constitutive equation, 28, 110, 219, 220, 227 

constitutive model, 2, 3, 9, 28, 29, 30, 31, 32, 34, 35, 37, 

38, 39, 40, 41, 44, 45, 46, 48, 50, 55, 56, 58, 59, 60, 61, 

64, 72, 134, 197, 213, 214, 216, 217, 218, 220, 223, 237, 

247, 262, 265 

controlling subcell, 176 

convergence, 30, 36, 68, 216, 217, 226, 234 

coordinate system, 12, 14, 92, 211, 212, 247 

copper, 29, 30, 36, 37, 60, 61, 62, 171 

coupling stiffness matrix (B), 14, 104 

creep, 55, 57, 58, 59, 202 

creep time, 202 

cross shaped fiber, 67 

cross-ply, 12, 14, 99, 101, 102, 103, 143, 209, 210, 211, 

212 

Curtin model, 189, 190, 191, 192, 253, 259 

cyclic, 148, 150, 152, 171, 176, 187 

D

D=, 79, 82, 83, 189, 190, 191, 221, 222, 223, 229, 230, 238, 

239, 240, 243, 245, 246, 247 

damage, 3, 64, 89, 152, 153, 158, 162, 171, 172, 173, 176, 

178, 179, 180, 263 

damage increment, 176, 180 

damage parameter, 171, 172, 176, 178 

data output, 194 

DBCH=, 185, 186, 189 

debonding, 77, 89, 152, 184, 186, 187, 188, 189, 191, 193, 

197 

debonding parameter, 184, 197 

DELAY=, 185, 186, 187, 189, 191 

design allowables, 153 

diagonal array, 79, 201 

DINC=, 174, 176 

discontinuous, 64, 105, 202, 203, 204, 205 

displacement jump, 184 

dissipation rate, 138 

diverge, 192 

DR=, 79, 104, 105, 135, 137, 143, 144, 200, 201 

E

E1=, 61, 262 

E11=, 102, 108, 153, 154, 155, 159, 161 

E12=, 153, 154, 155, 159, 161 

E13=, 153, 154, 155, 159, 161 

E2=, 61, 262 

E22=, 102, 108, 153, 154, 155, 159, 161 

E23=, 153, 154, 155, 159, 161 

E33=, 153, 154, 155, 159, 161 

EA=, 41, 42, 43, 44, 45, 46, 61, 129, 153, 158, 165, 173, 

221, 222 

EC11=, 155 

EC22=, 155 

EC33=, 155 

effective compliance matrix, 11, 14 

effective properties, 6, 7, 9, 10, 11, 12, 14, 21, 155, 194, 

226, 237, 240, 241, 242, 243, 246, 247, 251, 256 

effective stiffness matrix, 11, 14, 241, 248, 250 

EL=, 7, 8, 13, 41, 42, 43, 44, 45, 46, 55, 56, 61, 70, 221, 

223, 229, 230, 238, 239, 243, 244, 246, 247 

elastic modulus, 8, 241, 251 

electric displacement, 219, 225, 227, 228, 234, 235 

electric field, 219, 225, 227, 228, 233 

electromagnetic, 3, 213, 219, 220, 222, 223, 225, 226, 227, 

228, 229, 231, 232, 234 

electromagnetic specifier, 223 

electromagnetically active, 213 

ellipsoidal, 80 

EM=, 221, 222, 223, 229, 230, 231, 232 

embedded, 64, 92 

end of file, 10 

ending time, 52 

engineering shear strain, 28, 110, 112 

EPS=, 135, 137, 143, 144 

equivalent plastic strain, 134, 138, 140, 141, 145, 147 

ERR=, 35, 36, 43, 47, 51, 52, 148, 149, 214, 215, 216 

error tolerance, 148, 149, 216 

errors, 19, 36, 47, 52, 60, 87, 148, 149, 216 

ES=, 221, 223, 229, 230 

ES11=, 221, 222 

ES12=, 221, 222 

ES26=, 221, 222 

ET=, 41, 42, 43, 44, 45, 46, 61, 129, 153, 158, 165, 173, 

221, 222 

ETFL=, 174, 177 

ETM=, 174, 177 

ETU=, 174, 177 

evolving compliant interface (ECI) model, 184, 186, 187, 

189, 190, 192 

Example Problem Manual, 1, 5, 9 

execution, 4, 30, 41, 50, 61, 87, 134, 137, 162, 164, 168, 

169, 170, 177, 192, 217 

external material database, 28, 60, 63 

EXTMAT=, 60, 62 

F

F=, 7, 9, 10, 13, 23, 24, 65, 67, 70, 72, 78, 79, 86, 87, 92, 

93, 99, 100, 104, 111, 112, 115, 116, 120, 121, 125, 126, 

129, 135, 136, 143, 148, 149, 153, 154, 158, 159, 165, 

173, 174, 175, 185, 189, 190, 195, 196, 200, 201, 206, 

207, 214, 215, 221, 223, 230 

FACE=, 185, 186, 189, 191 

failure, 3, 64, 89, 152, 157, 158, 161, 162, 163, 164, 165, 

168, 169, 170, 171, 177, 178, 179, 180, 181, 184, 191, 

192, 263 

NASA/TM—2002-212077/VOL3 270
Index 

failure criteria, 152, 158, 162, 169, 170, 179 

failure parameters, 164, 169, 170 

failure surface, 165, 168, 170 

fatigue, 152, 171, 172, 173, 176, 178, 179, 180, 263 

fatigue limit, 171 

fiber breakage, 152, 189, 191, 192, 193 

fiber material, 28, 198, 203 

fiber strength, 189, 193 

fiber volume fraction, 9, 11, 23, 65, 67, 68, 69, 73, 74, 77, 

78, 81, 85, 89, 90, 91, 92, 97, 98, 102, 103, 114, 123, 

124, 125, 128, 139, 140, 141, 142, 143, 146, 147, 148, 

151, 153, 165, 170, 171, 195, 203, 204, 205, 206, 237 

field variables, 50, 86, 88 

first subcell, 137, 138, 145 

flag-type keywords, 8, 154, 155, 161, 162, 178, 179 

flow law, 134 

force resultant, 93, 94, 98, 103, 107, 108, 110, 125, 131, 

143, 145, 167, 170, 176, 232, 235 

forward Euler, 20, 30, 148 

FREQ=, 18, 21, 23, 25, 29, 31, 32, 33, 35, 37, 38, 39, 43, 

47, 51, 52, 56, 57, 60, 62, 65, 68, 70, 73, 76, 77, 79, 80, 

83, 85, 86, 88, 93, 94, 99, 101, 104, 106, 111, 113, 115, 

117, 120, 122, 125, 127, 148, 150, 159, 163, 174, 179, 

185, 187, 189, 192, 195, 196, 197, 200, 202, 206, 208, 

215, 216, 222, 226, 231, 234 

frequency, 21, 201, 202 

fringe plots, 88, 194, 200, 202, 203, 206 

G

GA=, 41, 42, 43, 44, 45, 46, 61, 129, 153, 158, 165, 173, 

221, 222 

general loading, 120, 121, 129, 131, 167, 224, 225, 232 

general loading option, 120, 121, 129, 131, 167, 224, 225, 

232 

global equilibrium, 213, 214, 217 

graphite, 7, 9, 10, 11, 12, 13, 14, 28, 129, 130, 153, 154, 

157, 158, 159, 164, 165, 166, 167, 169, 170, 171, 172, 

173, 175, 180, 181, 182, 183, 237, 238, 240, 241, 242, 

248, 250, 251, 252, 256 

graphite/epoxy, 7, 10, 11, 12, 13, 14, 129, 157, 164, 165, 

169, 170, 171, 172, 180, 181, 182, 183, 237, 238, 240, 

241, 242, 248, 250, 252 

GVIPS, 2, 18, 24, 28, 30, 32, 33, 34, 41, 42, 45, 67, 72, 79, 

83, 87, 93, 100, 105, 112, 116, 121, 126, 136, 143, 149, 

185, 190, 196, 201, 207, 213, 214, 215, 216, 217, 218 

H

H=, 75, 76, 79, 83, 189, 190, 230, 239, 240, 243, 245, 246, 

247 

hexagonal pack, 67, 72 

HFGMC, 3, 64, 77, 86, 87, 88, 89, 90, 91, 163, 194, 251, 

258 

homogenization, 64, 237, 241, 242, 243, 248, 250, 251, 252 

hybrid, 69, 74, 229, 235, 236 

I

I=, 70, 72 

IA=, 189, 192 

IB=, 185, 187, 189, 192, 195, 196, 197 

IG=, 185, 187, 189, 192, 195, 196, 197 

inclined fibers, 237, 240, 241, 242, 247, 248, 250 

incremental plasticity model, 35, 36, 37, 41, 46, 47, 49, 50, 

52, 53, 64, 134, 214 

indices, 142, 186, 187, 191, 192, 197, 198, 219 

inelastic power, 138 

inelastic strain rate, 138, 139, 141, 150 

inelasticity, 64, 158, 235 

initial time step, 150 

input file, 4, 5, 7, 8, 9, 10, 12, 13, 18, 19, 21, 22, 25, 29, 31, 

41, 43, 44, 47, 52, 57, 60, 64, 67, 72, 73, 75, 76, 80, 83, 

84, 93, 100, 112, 115, 130, 138, 154, 159, 166, 169, 170, 

175, 179, 180, 190, 207, 209, 215, 216, 220, 223, 231, 

232, 239, 240, 246, 247, 257, 263 

integration, 20, 25, 30, 33, 36, 39, 47, 52, 57, 62, 68, 73, 

76, 80, 84, 86, 87, 88, 94, 101, 106, 113, 116, 122, 127, 

132, 136, 144, 148, 149, 151, 160, 168, 176, 186, 191, 

196, 201, 208, 213, 216, 217, 226, 234 

integration points, 86, 87, 88 

interface, 28, 65, 70, 71, 72, 73, 77, 88, 89, 184, 186, 187, 

189, 191 

interface material, 72 

internal material database, 19, 23, 24, 29, 31, 32, 35 

IP=, 135, 137, 143, 144 

ISPX=, 135, 137, 143, 144 

ISPY=, 135, 137, 143, 144 

ISR=, 135, 137, 143, 144 

ISTM=, 148, 149 

ISTT=, 148, 149 

iteration, 36, 64, 216 

ITMAX=, 35, 36, 43, 47, 51, 52, 214, 215, 216 

K

K=, 7, 8, 9, 13 

KS=, 221, 223, 229, 230 

KS11=, 221, 222 

KS22=, 221, 222 

ksi, 102, 186, 187, 191, 192 

L

L=, 75, 76, 83, 84, 189, 190, 230, 239, 240, 243, 245, 246, 

247 

L0=, 189, 191 

laminate, 3, 6, 12, 13, 14, 64, 82, 92, 93, 94, 96, 98, 99, 

100, 101, 102, 104, 105, 106, 107, 108, 109, 110, 125, 

127, 128, 129, 132, 143, 144, 145, 146, 147, 152, 165, 

170, 171, 172, 175, 176, 179, 180, 183, 194, 197, 206, 

207, 208, 209, 211, 212, 213, 216, 229, 232, 234, 235, 

236 

laminate analysis, 3, 14, 64, 92, 93, 94, 96, 99, 100, 104, 

105, 110, 144, 197, 208, 209, 229, 232 

laminate level plot, 94 

LAMINATE=, 93, 94, 99, 101, 104, 106, 125, 127, 174, 

179, 206, 208, 231, 234 

lamination theory, 3, 12, 14, 64, 82, 92, 96, 99, 102, 104, 

125, 143, 145, 234 

NASA/TM—2002-212077/VOL3 271
MAC/GMC 4.0 Example Problem Manual 

layer, 3, 12, 14, 64, 92, 93, 95, 96, 97, 98, 99, 101, 102, 

103, 104, 105, 107, 127, 143, 144, 145, 169, 170, 197, 

206, 208, 209, 210, 211, 212, 229, 232, 235 

Legendre polynomial, 87, 88 

library file, 50, 52, 55 

linear elastic, 28, 152, 153, 155, 156 

link, 50, 251 

LL=, 79 

LN=, 185, 186, 189, 191 

load block, 176, 178, 179, 180, 181 

load magnitude, 165 

loading, 1, 3, 6, 10, 12, 14, 18, 19, 20, 21, 23, 24, 25, 26, 

33, 35, 39, 57, 64, 65, 70, 78, 84, 92, 93, 94, 96, 101, 

105, 107, 110, 111, 112, 113, 114, 115, 116, 117, 120, 

121, 122, 123, 125, 126, 129, 131, 132, 134, 136, 137, 

143, 149, 150, 152, 158, 161, 163, 164, 165, 167, 170, 

171, 176, 179, 180, 197, 200, 202, 203, 206, 209, 214, 

217, 218, 220, 224, 225, 228, 229, 232, 234 

loading components, 120, 137 

loading cycle, 171, 176 

loading option, 19, 24, 84, 94, 110, 111, 112, 114, 120, 

125, 126, 129, 180, 225 

localization, 64 

LOP=, 18, 19, 23, 24, 25, 29, 30, 32, 33, 35, 36, 38, 39, 43, 

47, 50, 51, 55, 56, 60, 62, 65, 68, 70, 73, 75, 76, 79, 80, 

83, 84, 86, 87, 92, 93, 99, 101, 104, 105, 111, 112, 113, 

115, 116, 120, 121, 125, 126, 127, 129, 131, 148, 149, 

158, 160, 165, 167, 174, 175, 185, 189, 190, 195, 196, 

200, 201, 214, 215, 221, 224, 230, 232 

LS=, 185, 186, 187, 189, 191 

LY=, 13, 92, 99, 100, 104, 125, 126, 129, 130, 143, 144, 

165, 166, 173, 174, 175, 206, 207, 230, 232 

LYR=, 174 

M

M=, 7, 8, 9, 10, 13, 18, 19, 23, 24, 29, 30, 32, 33, 35, 36, 38, 

41, 42, 43, 44, 45, 46, 47, 50, 51, 55, 56, 60, 62, 65, 67, 

70, 72, 75, 78, 79, 82, 86, 87, 92, 93, 99, 100, 104, 111, 

112, 115, 116, 120, 121, 125, 126, 129, 135, 136, 143, 

148, 149, 153, 154, 158, 159, 165, 173, 174, 175, 185, 

189, 190, 191, 195, 196, 200, 201, 206, 207, 214, 215, 

221, 222, 223, 229, 230, 238, 239, 243, 244 

MACPOST, 194, 200, 202, 203, 204, 205, 253 

MAG=, 18, 19, 23, 24, 25, 29, 30, 32, 33, 35, 36, 38, 39, 43, 

47, 51, 55, 56, 60, 62, 65, 68, 70, 73, 75, 76, 79, 80, 83, 

84, 86, 87, 92, 93, 99, 101, 104, 105, 111, 112, 115, 116, 

120, 121, 122, 125, 126, 129, 130, 131, 148, 149, 158, 

160, 165, 166, 167, 168, 174, 175, 185, 186, 189, 190, 

195, 196, 200, 201, 214, 215, 221, 224, 225, 230, 231, 

232, 233, 234 

magnetic field, 219, 225, 227, 228, 229, 233, 234, 235, 236 

magnetic flux density, 219, 225, 227, 228, 234 

magnetic permeability, 219, 220, 223 

magnetically asymmetric, 232, 235, 236 

magnetoelectric, 219, 220, 223, 227 

MAT=, 60, 61, 153, 154, 159, 161, 162, 166, 168, 174, 177, 

178 

MATDB=, 7, 8, 13, 41, 42, 43, 44, 45, 46, 50, 51, 55, 56, 60, 

62, 70, 72, 129, 130, 153, 154, 158, 159, 165, 166, 173, 

175, 221, 222, 223, 229, 230, 231, 238, 239, 243, 244, 

246, 247 

material number, 41, 48, 50, 76, 84, 208, 261 

material point, 120, 122 

material properties, 7, 8, 9, 13, 18, 19, 20, 21, 23, 24, 28, 

41, 44, 45, 46, 48, 50, 54, 55, 60, 70, 72, 97, 132, 223, 

231, 240, 247, 248, 257, 265, 268 

material property database, 9, 18 

MATID=, 7, 8, 13, 18, 19, 23, 24, 29, 30, 32, 33, 35, 36, 38, 

41, 42, 43, 44, 45, 46, 50, 51, 55, 56, 60, 62, 65, 67, 70, 

72, 75, 76, 78, 79, 82, 83, 86, 87, 92, 93, 99, 100, 104, 

105, 111, 112, 115, 116, 120, 121, 125, 126, 129, 130, 

135, 136, 143, 148, 149, 153, 154, 158, 159, 165, 166, 

173, 175, 185, 189, 190, 195, 196, 200, 201, 206, 207, 

214, 215, 221, 222, 223, 229, 230, 231, 238, 239, 243, 

244, 246, 247 

MATLAB, 1, 4, 86, 88, 194, 206, 208, 209, 211, 212, 259 

matrix material, 6, 28, 145, 156, 237, 240, 242, 246, 248, 

251 

maximum strain, 162 

maximum stress, 162, 163, 169, 171 

MAXNB=, 174, 176 

mean stress, 171 

mechanical loading, 3, 10, 18, 19, 24, 25, 33, 38, 39, 50, 

55, 57, 65, 94, 110, 111, 112, 114, 115, 116, 117, 120, 

122, 125, 129, 132, 134, 136, 137, 148, 149, 150, 153, 

158, 161, 164, 170, 176, 202, 203 

METHOD=, 18, 20, 23, 25, 29, 30, 32, 33, 35, 36, 38, 39, 43, 

47, 51, 52, 55, 57, 60, 62, 65, 68, 70, 73, 75, 76, 79, 80, 

83, 84, 86, 87, 92, 94, 99, 101, 104, 106, 111, 113, 115, 

116, 120, 122, 125, 127, 130, 132, 135, 136, 148, 149, 

159, 160, 166, 168, 174, 176, 185, 186, 189, 191, 195, 

196, 200, 201, 206, 208, 214, 215, 216, 222, 226, 231, 

234 

micro level plot, 21 

micromechanics, 3, 28, 64, 118, 164, 198, 219 

midplane curvature, 94, 109, 125 

midplane strain, 92, 94, 96, 98, 99, 101, 103, 104, 105, 107, 

108, 109, 110, 125, 126, 127, 128, 129, 235 

minimal output, 31 

minimum time step, 149 

MINSTEP=, 148, 149 

mission profile, 122 

MMAX=, 135, 136, 143, 144 

MOD=, 7, 9, 13, 18, 19, 23, 24, 29, 30, 32, 33, 35, 36, 38, 

43, 47, 50, 51, 55, 56, 60, 62, 64, 65, 67, 70, 72, 75, 76, 

78, 79, 82, 83, 86, 87, 92, 93, 99, 100, 104, 111, 112, 

115, 116, 120, 121, 125, 126, 129, 135, 136, 143, 148, 

149, 153, 154, 158, 159, 165, 173, 174, 175, 177, 185, 

189, 190, 195, 196, 200, 201, 206, 207, 214, 215, 221, 

223, 230, 239, 240, 243, 245, 246, 247 

mode, 19, 24, 30, 33, 36, 39, 47, 51, 57, 62, 68, 73, 76, 80, 

84, 87, 94, 101, 105, 112, 116, 122, 126, 136, 137, 144, 

160, 175, 186, 190, 196, 201, 215 

MODE=, 18, 19, 23, 24, 25, 29, 30, 32, 33, 35, 36, 38, 39, 

43, 47, 51, 55, 56, 57, 60, 62, 65, 68, 70, 73, 75, 76, 79, 

80, 83, 84, 86, 87, 92, 94, 99, 101, 104, 105, 111, 112, 

115, 116, 120, 121, 122, 125, 126, 129, 130, 131, 135, 

136, 143, 144, 148, 149, 158, 160, 165, 166, 167, 168, 

174, 175, 185, 186, 189, 190, 195, 196, 200, 201, 214, 

215, 221, 224, 225, 230, 231, 232, 233, 234 

modified Bodner-Partom, 32, 33, 34 

NASA/TM—2002-212077/VOL3 272
Index 

moment resultant, 14, 94, 96, 107, 109, 110, 125, 129, 131, 

132, 167, 168, 233, 235 

monoclinic, 241, 242, 247, 248, 250 

monolithic, 6, 18, 19, 29, 30, 33, 36, 64, 105, 134, 228 

monolithic material, 19, 21, 30, 33, 35, 36, 50, 64 

MPa, 56, 132, 154, 155, 161, 162, 163, 164, 167, 168, 169, 

170, 171, 175, 176, 177, 178 

MS=, 221, 223, 229, 230 

MS11=, 221, 222 

MS22=, 221, 222 

multimechanism, 213, 214, 217, 218 

multi-scale, 92 

N 

N=, 86, 88, 206, 208 

N1=, 174, 177 

N2=, 174, 177 

NAI=, 189, 191 

NAME=, 18, 21, 23, 25, 29, 31, 32, 34, 35, 37, 38, 39, 43, 

47, 51, 52, 56, 57, 60, 61, 62, 65, 68, 70, 73, 76, 77, 79, 

80, 83, 85, 86, 88, 93, 94, 99, 101, 104, 106, 111, 113, 

115, 117, 120, 123, 125, 127, 148, 150, 159, 163, 174, 

179, 185, 187, 189, 192, 195, 196, 197, 200, 202, 206, 

208, 209, 215, 216, 222, 226, 231, 234 

NASA, 3, 253, 254, 256 

NBI=, 185, 186, 189, 191 

NDMAT=, 174, 177 

NGI=, 185, 186, 189, 191 

NII=, 185, 186, 189, 191 

NINTEG=, 86, 87, 88 

NiTi, 38, 39, 40 

NLEG=, 86, 87, 88 

NLY=, 13, 92, 93, 99, 100, 104, 105, 125, 126, 129, 130, 

143, 144, 165, 166, 173, 175, 206, 207, 230, 231 

NMAT=, 153, 154, 159, 161, 166, 168, 174, 178 

NMATS=, 7, 8, 13, 18, 23, 29, 30, 32, 35, 36, 38, 41, 43, 50, 

51, 55, 56, 60, 62, 65, 67, 70, 72, 75, 76, 78, 79, 82, 83, 

86, 87, 92, 93, 99, 100, 104, 105, 111, 112, 115, 120, 

121, 125, 126, 129, 130, 135, 143, 148, 153, 154, 158, 

159, 165, 166, 173, 175, 184, 185, 189, 190, 195, 200, 

201, 206, 207, 214, 215, 220, 222, 229, 231, 238, 239, 

243, 244, 245, 246 

NonLinear Cumulative Damage Rule, 171 

NP=, 42, 43, 45, 46 

NPL=, 7, 10, 13, 14, 18, 20, 23, 25, 29, 31, 32, 33, 35, 37, 

38, 39, 43, 47, 51, 52, 55, 57, 60, 62, 65, 68, 70, 73, 76, 

77, 79, 80, 83, 84, 86, 88, 92, 94, 96, 99, 101, 104, 106, 

111, 113, 115, 117, 120, 122, 125, 127, 130, 132, 135, 

138, 143, 145, 148, 150, 154, 155, 159, 162, 166, 169, 

174, 179, 185, 187, 189, 192, 195, 196, 200, 201, 206, 

208, 215, 216, 222, 226, 231, 234, 239, 240, 244, 245, 

246, 247 

NPT=, 18, 19, 20, 23, 24, 25, 29, 30, 32, 33, 35, 36, 38, 39, 

43, 47, 51, 52, 55, 56, 57, 60, 62, 65, 68, 70, 73, 75, 76, 

79, 80, 83, 84, 86, 87, 92, 93, 94, 99, 101, 104, 105, 106, 

111, 112, 113, 115, 116, 117, 120, 121, 122, 125, 126, 

127, 129, 130, 131, 132, 135, 136, 148, 149, 158, 159, 

160, 165, 166, 167, 168, 174, 175, 176, 185, 186, 189, 

190, 191, 195, 196, 200, 201, 206, 207, 208, 214, 215, 

216, 221, 222, 224, 225, 226, 230, 231, 232, 233, 234 

NPV=, 55, 56, 61 

NTEMP=, 7, 8 

NTP=, 41, 42, 43, 44, 45, 46, 61, 129, 153, 158, 165, 173, 

221, 222, 244 

NUA=, 41, 42, 43, 44, 45, 46, 61, 129, 153, 158, 165, 173, 

221, 222 

number of subcells, 87 

NUT=, 41, 42, 43, 44, 45, 46, 61, 129, 153, 158, 165, 173, 

221, 222 

O

OMFL=, 174, 177 

OMM=, 174, 177 

OMU=, 174, 177 

ONERA, 171 

open cell, 80 

OPT=, 79 

OPTION=, 135, 136, 143, 144 

order, 9, 35, 39, 41, 44, 45, 46, 50, 55, 84, 86, 88, 93, 100, 

112, 134, 137, 148, 149, 161, 169, 170, 186, 187, 191, 

192, 200, 202, 206, 207, 213, 214, 216, 240, 241, 242, 

246, 251 

orthotropic, 123, 241, 247, 248, 250 

out-of-plane, 7, 64, 85, 92, 96, 235 

output, 2, 3, 10, 11, 14, 21, 31, 36, 88, 94, 95, 96, 97, 102, 

108, 117, 132, 137, 138, 155, 162, 163, 170, 179, 192, 

194, 201, 202, 208, 211, 212, 226, 227, 228, 235, 241, 

255, 256 

output file, 2, 10, 11, 14, 21, 31, 36, 95, 96, 97, 102, 108, 

117, 132, 137, 138, 155, 162, 163, 170, 179, 192, 194, 

202, 208, 226, 228, 235, 241, 255, 256 

P

packing, 67, 68, 73, 80 

particulate, 64, 105, 220, 226 

PATRAN, 1, 4, 194, 200, 201, 202, 203, 204, 205, 259 

PELS=, 221, 223, 229, 230 

PELS1=, 221, 222 

PELS2=, 221, 222 

perfectly bonded, 187, 188 

piezoelectric, 219, 220, 223, 227 

piezomagnetic, 219, 220, 223, 227, 235 

plain weave, 237, 238, 240, 241, 242, 243, 244, 247, 248, 

250, 252 

plane of transverse isotropy, 223 

ply, 3, 12, 14, 64, 92, 93, 95, 96, 97, 98, 99, 101, 102, 103, 

104, 105, 107, 127, 143, 144, 145, 169, 170, 197, 206, 

208, 209, 210, 211, 212, 229, 232, 235 

PMGS=, 221, 223, 229, 230 

PMGS1=, 221, 222 

PMGS2=, 221, 222 

Poisson effect, 26, 96 

Poisson ratio, 8, 97, 251 

poling direction, 220, 223, 229, 230, 235 

post-processing, 194, 200, 206 

power law creep, 55, 58 

NASA/TM—2002-212077/VOL3 273
MAC/GMC 4.0 Example Problem Manual 

preload, 137 

preloading, 136, 137, 144, 201, 202 

principal material coordinates, 96, 97 

print level, 10, 14, 20, 21, 25, 30, 31, 33, 37, 39, 47, 52, 57, 

62, 68, 73, 77, 80, 84, 88, 94, 96, 101, 106, 113, 117, 

122, 127, 132, 138, 145, 150, 155, 162, 169, 179, 187, 

192, 194, 196, 201, 208, 216, 226, 234, 240, 246, 247 

probing, 134, 136, 137, 144 

pyroelectric, 219, 220, 223 

pyromagnetic, 219, 220, 223, 228 

Q

QS=, 221, 223, 229, 230 

QS11=, 221, 222 

QS12=, 221, 222 

QS26=, 221, 222 

quasi-isotropic, 99, 101, 102, 103, 125, 127, 128, 165, 170, 

171, 175, 180, 183 

R

R=, 65, 67, 68, 70, 72, 86, 87, 99, 100, 104, 111, 112, 120, 

121, 125, 126, 129, 143, 148, 149, 153, 154, 158, 159, 

165, 173, 174, 175, 185, 206, 207, 214, 215 

RA=, 79 

RAD=, 70, 72 

RAD1=, 65, 67, 70, 72 

RAD2=, 65, 67, 70, 72 

RC=, 79 

rectangular pack, 67, 68, 72, 87, 100, 105, 112, 121, 126, 

130, 144, 149, 154, 159, 166, 167, 175, 185, 207 

redistribution, 177, 180 

reduced stiffness matrices (Q), 96 

reference temperature, 19, 219 

REFTIME=, 115, 116, 148, 149, 158, 160, 185, 189, 190, 

200, 201, 214, 215 

repeating unit cell analysis, 1, 9, 12, 14, 19, 64, 93, 94, 96, 

110, 111, 120, 125, 137, 197, 208, 209, 220 

repeating unit cell dimensions, 79 

repeating unit cell failure, 170 

residual stress, 115, 118, 119, 136, 138, 139, 140, 141, 145, 

153, 158, 163, 164, 176, 188, 192, 211 

residual stresses, 115, 116, 118, 119, 136, 138, 139, 140, 

141, 145, 148, 153, 158, 160, 161, 163, 164, 176, 188, 

192, 203, 206, 209, 211, 212 

results, 3, 5, 9, 10, 11, 14, 21, 22, 23, 26, 31, 34, 35, 37, 41, 

47, 48, 52, 58, 63, 68, 77, 86, 88, 89, 102, 107, 108, 113, 

115, 116, 117, 119, 123, 125, 126, 132, 137, 138, 149, 

152, 156, 163, 168, 169, 170, 179, 187, 190, 192, 194, 

215, 216, 226, 232, 237, 240, 241, 242, 250, 256 

reversal tolerance, 186, 187, 191 

RITFR=, 70 

RITFR1=, 70 

RITFR2=, 70 

Runge-Kutta, 148, 149 

S

S11=, 153, 154, 159, 161 

S12=, 153, 154, 155, 159, 161 

S13=, 153, 154, 155, 159, 161 

S22=, 153, 154, 159, 161 

S23=, 153, 154, 155, 159, 161 

S33=, 153, 154, 155, 159, 161 

SC11=, 155 

SC22=, 155 

SC33=, 155 

scatter, 194 

SCS-6, 60, 61, 62, 125 

self-adaptive, 20, 148 

SFL=, 174, 177, 178 

shape memory alloy, 1, 38, 39, 40 

shear coupling, 77, 86, 88, 101, 127 

shear CTEs, 227, 241, 248, 249, 250, 252 

shear modulus, 8, 97, 102, 251 

short fiber, 79, 80, 105, 200, 201, 223, 232 

SiC, 1, 23, 24, 26, 27, 28, 67, 68, 69, 72, 73, 74, 79, 81, 83, 

85, 86, 87, 89, 90, 91, 92, 93, 96, 97, 98, 99, 100, 102, 

103, 104, 105, 108, 109, 112, 114, 115, 116, 117, 118, 

119, 120, 121, 123, 124, 125, 126, 128, 135, 136, 139, 

140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 

151, 184, 185, 188, 190, 193, 195, 196, 197, 199, 200, 

201, 202, 203, 204, 205, 206, 207, 210, 211, 212, 214, 

215, 216, 217, 218 

SiC/Ti-21S, 1, 23, 26, 27, 68, 69, 74, 81, 85, 89, 90, 91, 92, 

96, 97, 98, 99, 102, 103, 104, 105, 108, 109, 114, 115, 

117, 118, 119, 123, 124, 128, 139, 140, 141, 142, 143, 

145, 146, 147, 150, 151, 184, 188, 193, 197, 199, 202, 

203, 204, 205, 210, 211, 212, 214, 216, 217, 218 

SIG0=, 189, 191 

SM=, 75, 76, 83, 84, 189, 190, 230, 239, 240, 243, 244, 

245, 246, 247 

smart, 3, 213, 220, 226, 227, 228, 229, 235, 236 

S-N curve, 171, 176, 180, 181, 182, 183 

source, 8, 13, 43, 50, 51, 52, 56, 62, 130, 154, 159, 166, 

175, 194, 206, 208, 223, 231, 239, 246, 247 

special case, 10, 35, 46, 67, 73 

square array, 79, 223, 232 

square diagonal pack, 67, 72 

square fiber, 9, 13, 14, 24, 67, 68, 72, 92, 93, 116, 190, 195, 

196 

square pack, 9, 13, 14, 24, 67, 72, 87, 92, 93, 116, 121, 136, 

149, 154, 159, 175, 190, 195, 196, 207, 215 

state variables, 197 

static failure, 152, 158, 161, 162, 164, 165, 168, 169, 170, 

178, 179 

step size, 20, 25, 30, 33, 36, 37, 39, 47, 52, 57, 62, 68, 73, 

77, 80, 84, 87, 94, 101, 106, 113, 117, 122, 127, 132, 

136, 137, 144, 148, 149, 160, 161, 168, 176, 186, 191, 

196, 201, 208, 214, 216, 217, 218, 226, 234 

stiffness matrix, 2, 11, 14, 21, 28, 55, 104, 110, 213, 219, 

241, 247, 248, 249, 250, 251, 268 

stiffness reduction, 172, 178 

stop execution, 192 

STP=, 18, 20, 23, 25, 29, 30, 32, 33, 35, 36, 38, 39, 43, 47, 

51, 52, 55, 57, 60, 62, 65, 68, 70, 73, 75, 77, 79, 80, 83, 

84, 86, 87, 92, 94, 99, 101, 104, 106, 111, 113, 115, 117, 

120, 122, 125, 127, 130, 132, 135, 136, 143, 144, 159, 

160, 166, 168, 174, 176, 185, 186, 189, 191, 195, 196, 

200, 201, 206, 208, 214, 215, 216, 222, 226, 231, 234 

NASA/TM—2002-212077/VOL3 274
Index 

strain allowables, 153, 154, 155, 156, 157, 161 

strain rate, 20, 22, 32, 33, 34, 35, 50, 51, 53, 58, 137, 144, 

261 

strain reference time, 116, 149, 185, 190, 201, 215 

stress allowables, 154, 156, 157, 161 

stress resultant, 93, 107, 108, 110, 131, 143, 145, 232, 235 

stress-strain curve, 34, 35, 39, 52, 63, 115, 118, 194, 199, 

216, 217 

stress-strain curves, 21, 22, 25, 26, 29, 31, 34, 35, 37, 39, 

40, 46, 48, 49, 52, 53, 54, 57, 58, 63, 69, 74, 77, 81, 85, 

89, 96, 98, 103, 114, 115, 116, 118, 137, 150, 163, 181, 

188, 192, 193, 194, 199, 202, 203, 216, 217 

SU=, 174, 177, 178 

SU1=, 174, 177 

SU2=, 174, 177 

subcell failure, 158, 162, 163, 164, 169, 170, 192 

surf_1st.dat, 137, 145 

surf_1st_ply.dat, 144 

surf_all.dat, 137, 145 

surf_all_plies.dat, 144 

surf_global.dat, 137, 144 

surf_local.dat, 137, 145 

surf_plies.dat, 144 

T

TAU0=, 189, 191 

TEM=, 41, 42, 43, 44, 45, 46, 61, 129, 153, 158, 165, 173, 

221, 222, 245 

TEMP=, 7, 8, 23, 24, 25, 43, 47, 51, 52, 55, 57, 60, 62, 65, 

68, 70, 73, 79, 80, 83, 84, 86, 87, 92, 94, 99, 101, 104, 

105, 111, 113, 115, 116, 120, 122, 125, 127, 130, 132, 

135, 136, 148, 149, 158, 160, 166, 168, 174, 176, 185, 

186, 189, 191, 195, 196, 200, 201, 206, 207, 214, 216, 

221, 226, 231, 234 

temperature, 7, 8, 9, 19, 20, 21, 24, 25, 26, 27, 28, 29, 31, 

32, 35, 37, 41, 44, 46, 48, 50, 51, 52, 54, 92, 116, 117, 

118, 122, 123, 129, 132, 135, 136, 137, 144, 150, 151, 

164, 170, 176, 181, 182, 186, 191, 202, 208, 209, 216, 

217, 218, 219, 223, 228, 247, 248, 262 

temperature unit, 19 

temperature-dependent, 8, 9, 41, 44, 46, 48, 129, 132, 223, 

247 

temperature-independent, 7, 8, 41, 44, 48, 223 

tensorial shear strains, 112 

Theory Manual, 3, 5, 14, 64, 86, 87, 92, 99, 104, 134, 148, 

149, 162, 171, 178, 184, 187, 216 

thermal loading, 6, 19, 20, 21, 23, 24, 25, 26, 116, 122, 

136, 176, 209, 228 

thermal strain, 28, 96, 110, 117, 152, 198, 202, 219 

THK=, 13, 92, 93, 99, 100, 104, 125, 126, 129, 143, 165, 

173, 174, 206, 230 

through-thickness, 64, 125, 209, 229, 235, 236, 242, 245, 

248, 251 

TI=, 18, 19, 20, 23, 24, 25, 29, 30, 32, 33, 35, 36, 38, 39, 

43, 47, 51, 52, 55, 56, 57, 60, 62, 65, 68, 70, 73, 75, 76, 

77, 79, 80, 83, 84, 86, 87, 92, 93, 94, 99, 101, 104, 105, 

106, 111, 112, 113, 115, 116, 117, 120, 121, 122, 125, 

126, 127, 129, 130, 131, 132, 135, 136, 148, 149, 158, 

159, 160, 165, 166, 167, 168, 174, 175, 176, 185, 186, 

189, 190, 191, 195, 196, 200, 201, 206, 207, 208, 214, 

215, 216, 221, 222, 224, 225, 226, 230, 231, 232, 233, 

234 

Ti-15-3, 35, 36, 37 

Ti-21S, 1, 18, 19, 21, 22, 23, 24, 26, 27, 28, 32, 33, 34, 67, 

68, 69, 72, 74, 79, 81, 83, 85, 86, 87, 89, 90, 91, 92, 93, 

96, 97, 98, 99, 100, 102, 103, 104, 105, 108, 109, 112, 

114, 115, 116, 117, 118, 119, 120, 121, 123, 124, 125, 

126, 128, 136, 139, 140, 141, 142, 143, 144, 145, 146, 

147, 148, 149, 150, 151, 184, 185, 188, 190, 193, 195, 

196, 197, 199, 200, 201, 202, 203, 204, 205, 206, 207, 

210, 211, 212, 214, 215, 216, 217, 218 

Ti-24-11, 35, 36, 37 

time delay, 186, 187, 191, 192 

time integration, 20, 30, 148, 149, 151, 216 

time-based output, 96, 228 

TIMES=, 86, 88, 206, 208 

titanium, 1, 18, 19, 21, 22, 23, 24, 26, 27, 28, 29, 30, 32, 

33, 34, 35, 36, 37, 67, 68, 69, 72, 74, 79, 81, 83, 85, 86, 

87, 89, 90, 91, 92, 93, 96, 97, 98, 99, 100, 102, 103, 104, 

105, 108, 109, 112, 114, 115, 116, 117, 118, 119, 120, 

121, 123, 124, 125, 126, 128, 136, 139, 140, 141, 142, 

143, 144, 145, 146, 147, 148, 149, 150, 151, 184, 185, 

188, 190, 193, 195, 196, 197, 199, 200, 201, 202, 203, 

204, 205, 206, 207, 210, 211, 212, 214, 215, 216, 217, 

218 

TMAX=, 135, 136, 143, 144 

TOLN=, 185, 186, 187, 189, 191 

tows, 237 

TPRE=, 200, 201 

transversely isotropic, 9, 171, 220, 222, 223, 231, 237, 239, 

240, 245 

TREF=, 18, 19, 29, 30, 32, 33, 35, 36, 38, 75, 76, 135, 136, 

143, 144 

Tsai-Hill, 162, 168, 169 

U

ultimate strength, 171 

unconditionally stable, 213, 214 

units, 9, 19, 133, 228 

unloading, 39, 150 

user-defined, 8, 9, 13, 28, 41, 44, 45, 46, 48, 49, 50, 55, 56, 

58, 59, 72, 75, 76, 77, 82, 84, 85, 129 

usrformde, 2, 55, 58, 63, 255, 268 

usrfun, 2, 9, 50, 51, 52, 255, 261 

usrmat, 2, 55, 58, 63, 255, 265 

V

V1=, 42, 43, 44, 45, 46, 61 

V2=, 42, 43, 44, 45, 46, 61 

VF=, 7, 9, 10, 13, 23, 24, 65, 67, 70, 72, 78, 79, 86, 87, 92, 

99, 100, 104, 111, 112, 115, 116, 120, 121, 125, 126, 

129, 135, 136, 143, 148, 149, 153, 154, 158, 159, 165, 

173, 174, 175, 185, 189, 190, 195, 196, 200, 201, 206, 

207, 214, 215, 221, 223, 230 

VF1=, 65, 67, 70, 72 

VI=, 41, 42, 44, 45, 46, 55, 56, 61, 70, 230 

volume fraction, 9, 11, 23, 24, 65, 67, 68, 69, 72, 73, 74, 

77, 78, 79, 81, 85, 87, 89, 90, 91, 97, 98, 102, 103, 112, 

114, 116, 121, 123, 124, 125, 128, 136, 139, 140, 141, 

NASA/TM—2002-212077/VOL3 275
MAC/GMC 4.0 Example Problem Manual 

142, 145, 146, 147, 149, 151, 154, 159, 165, 170, 171, 

175, 185, 190, 196, 200, 201, 203, 204, 205, 207, 215, 

223, 237 

W

warnings, 21 

Weibull modulus, 191 

Windows, 50 

woven composite, 213, 237, 240, 241, 242, 250, 251 

X

X=, 18, 21, 23, 25, 29, 31, 32, 34, 35, 37, 38, 39, 43, 47, 51, 

52, 56, 57, 60, 62, 65, 68, 70, 73, 76, 77, 79, 80, 83, 85, 

86, 88, 93, 94, 99, 101, 104, 106, 111, 113, 115, 117, 

120, 123, 125, 127, 148, 150, 159, 163, 174, 179, 185, 

187, 189, 192, 195, 196, 197, 200, 202, 206, 208, 209, 

215, 216, 222, 226, 231, 235 

X11=, 159, 161, 162, 166, 168, 169, 174, 178 

X12=, 159, 162, 166, 168, 169, 174, 178, 179 

X13=, 159, 162, 166, 168, 169, 174, 178, 179 

X22=, 159, 161, 162, 166, 168, 169, 174, 178 

X23=, 159, 161, 162, 166, 168, 169, 174, 178 

X33=, 159, 161, 162, 166, 168, 169, 174, 178 

XA=, 65, 67 

XML=, 174, 177, 178 

x-y plot, 10, 14, 21, 25, 31, 33, 37, 39, 47, 52, 57, 62, 68, 

73, 77, 80, 85, 88, 94, 101, 106, 113, 117, 122, 127, 132, 

137, 138, 145, 150, 155, 163, 169, 179, 180, 187, 192, 

195, 196, 197, 198, 200, 202, 208, 209, 216, 226, 234, 

240, 246, 247 

Y

Y=, 18, 21, 23, 25, 29, 31, 32, 34, 35, 37, 38, 39, 43, 47, 51, 

52, 56, 57, 60, 62, 65, 68, 70, 73, 76, 77, 79, 80, 83, 85, 

86, 88, 93, 94, 99, 101, 104, 106, 111, 113, 115, 117, 

120, 123, 125, 127, 148, 150, 159, 163, 174, 179, 185, 

187, 189, 192, 195, 196, 197, 200, 202, 206, 208, 209, 

215, 216, 222, 226, 231, 235 

yield criteria, 134, 137 

yield surface, 35, 46, 50, 53, 58, 88, 101, 134, 135, 136, 

137, 138, 139, 140, 141, 143, 144, 145, 146, 147, 150, 

165, 203, 242 

Z

zero stiffness, 171 

NASA/TM—2002-212077/VOL3 276

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282

MAC/GMC 4.0 User's Manual—Example Problem Manual

Brett A. Bednarcyk and Steven M. Arnold

Software manual; Micromechanics; Elastic; Plastic; Lamination; Composite

Unclassified -Unlimited

Subject Categories: 24 and 39 Distribution: Nonstandard

Funding for this latest version was provided by the following three programs: 1) Revolutionary Aeropropulsion Concepts

(RAC) program, 2) Next Generation Launch Technology (NGLT)–PR&T program, and 3) Headquarters, Advanced Cross

Enterprise Technology Development (CETDP), NRA program. Brett A. Bednarcyk, Ohio Aerospace Institute, Brook Park,

Ohio 44142; Steven M. Arnold, NASA Glenn Research Center. Responsible person, Steven M. Arnold, organization code

5920, 216–433–3334.

This document is the third volume in the three volume set of User’s Manuals for the Micromechanics Analysis Code with Generalized

Method of Cells Version 4.0 (MAC/GMC 4.0). Volume 1 is the Theory Manual, Volume 2 is the Keywords Manual, and this document is the Example Problems Manual. MAC/GMC 4.0 is a composite material and laminate analysis software program developed at

the NASA Glenn Research Center. It is based on the generalized method of cells (GMC) micromechanics theory, which provides

access to the local stress and strain fields in the composite material. This access grants GMC the ability to accommodate arbitrary

local models for inelastic material behavior and various types of damage and failure analysis. MAC/GMC 4.0 has been built around

GMC to provide the theory with a user-friendly framework, along with a library of local inelastic, damage, and failure models.

Further, application of simulated thermo-mechanical loading, generation of output results, and selection of architectures to represent

the composite material, have been automated in MAC/GMC 4.0. Finally, classical lamination theory has been implemented within

MAC/GMC 4.0 wherein GMC is used to model the composite material response of each ply. Consequently, the full range of GMC

composite material capabilities is available for analysis of arbitrary laminate configurations as well. This volume provides in-depth

descriptions of 43 example problems, which were specially designed to highlight many of the most important capabilities of the code.

The actual input files associated with each example problem are distributed with the MAC/GMC 4.0 software; thus providing the user

with a convenient starting point for their own specialized problems of interest.