NASA/TM--1998-208805

Flow/Damage Surfaces for

Fiber-Reinforced Metals Having

Different Periodic Microstructures

Cliff J. Lissenden

Pennsylvania State University, University Park, Pennsylvania

Steven M. Arnold

Lewis Research Center, Cleveland, Ohio

Saiganesh K. Iyer

Pennsylvania State University, University Park, Pennsylvania

November 1998

https://ntrs.nasa.gov/search.jsp?R=19990008648 2018-05-11T16:56:14+00:00Z
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NASA/TM--1998-208805

Flow/Damage Surfaces for

Fiber-Reinforced Metals Having

Different Periodic Microstructures

Cliff J. Lissenden

Pennsylvania State University, University Park, Pennsylvania

Steven M. Arnold

Lewis Research Center, Cleveland, Ohio

Saiganesh K. Iyer

Pennsylvania State University, University Park, Pennsylvania

National Aeronautics and

Space Administration

Lewis Research Center

November 1998
Acknowledgments

We would like to thank Dr. T.E. Wilt for his assistance in extending MAC/GMC

so as to conduct flow surface probing. The first author gratefully acknowledges the support

of the NASA Lewis Research Center, grant NCC3-481.

NASA Center for Aerospace Information

7121 Standard Drive

Hanover, MD 21076

Price Code: A03

Available from

National Technical Information Service

5285 Port Royal Road

Springfield, VA 22100

Price Code: A03
FLOW/DAMAGE SURFACES FOR FIBER-REINFORCED METALS

HAVING DIFFERENT PERIODIC MICROSTRUCTURES

Cliff J. Lissenden, and Saiganesh K. Iyer

Pennsylvania State University

University Park, Pennsylvania 16802

Phone: (814) 863-5754, FAX: (814) 863-6031

email: cjlesm@psu.edu

and

Steven M. Arnold,

National Aeronautics and Space Administration

Lewis Research Center

Cleveland, Ohio 44135

SUMMARY

Flow/damage surfaces can be defined in terms of stress, inelastic strain rate, and internal variables using a thermodynamics framework. A macroscale definition relevant to thermodynamics and usable in an experimental program is employed to map out surfaces of constant inelastic power in various stress planes. The inelastic flow of a

model silicon carbide/titanium composite system having rectangular, hexagonal, and square diagonal fiber packing

arrays subjected to biaxial stresses is quantified by flow/damage surfaces that are determined numerically from

micromechanics, using both finite element analysis and the generalized method of cells. Residual stresses from processing are explicitly included and damage in the form of fiber-matrix debonding under transverse tensile and/or

shear loading is represented by a simple interface model. The influence of microstructural architecture is largest

whenever fiber-matrix debonding is not an issue; for example in the presence of transverse compressive stresses.

Additionally, as the fiber volume fraction increases, so does the effect of microstructural architecture. With regard to

the micromechanics analysis, the overall inelastic flow predicted by the generalized method of cells is in excellent

agreement with that predicted using a large number of displacement-based finite elements.

I. INTRODUCTION

The advent of man-made fiber-reinforced composite materials some forty years ago enabled the design of more

efficient structures and greatly expanded the domain of engineered materials. Recent developments in the processing

of unidirectional metal matrix composites (MMCs) provide new opportunities for engineers and materials scientists

to tailor microstructural architecture for specific applications. For example, placing individual fibers in photo-etched

grooves in foils of matrix material results in a very uniform microstructure. Since the grooves hold the fiber in place

during consolidation, the microstructure can be engineered by simply specifying the foil thickness and groove pattern. Consequently, this paper addresses the effect that various microstructures have on the overall inelastic material

response in the presence of multiaxial stress states.

This research merges and builds upon two recent publications on unidirectional composites; one dealing with

different microstructural architectures subjected to uniaxial loadings (Arnold et al., 1996a) and the other with

macroscale flow/damage surfaces (Lissenden and Arnold, 1997a) given a fixed architecture. Hence, a brief summary

of each follows, after which the objectives of the current work are enumerated.

Arnold et al. (1996a) begins with an extensive survey of the literature dealing with the effects of fiber shape and

distribution on the response of composites. The generalized method of cells (Paley and Aboudi, 1992) was summarized and then used to predict the response of a silicon carbide/titanium system (SiC/Ti). The results presented demonstrate the relative effects of fiber arrangement and shape on the axial and transverse stress-strain and creep

responses for both strong and weak fiber/matrix interracial bonds. Fiber arrangements considered include rectangular, square, square diagonal, and hexagonal periodic arrays as well as a random arrangement. Circular, square, and

NASA/TM--1998-208805 1
cross-shaped fibercross-sections werestudiedasweretheeffectsoffibervolumefractionandstrainrateonthe

stress-strain response. Additionally, thestress-strainndcreepresponses of"hybrid"composites(thosewithdifferentsizefibershavingdifferentarrangements andbondstrengths) wereinvestigatedtodeterminethefeasibilityof

usingthisapproachtoenhancethetransversetoughnessandcreepresistance.

LissendenandArnold(1997a)addressedtheoretical andexperimental issuesregardingconstitutivemodel

development foranisotropicheterogeneous materialsusingmicromechanics inlieuofactualexperimental data.The

generalizedmethodofcellswithfoursubcells, arrangedtorepresent asquarearrayoffiberswasusedtopredictthe

multiaxial responseofMMCsintermsofoverallfow/damagesurfaces(i.e.,thoseincorporatingmatrixviscoplasticityandfiber/matrixdebonding) inthreedifferentstressplanes.Theflow/damagesurfaceswereshowntohave

their centers offset from the origin by residual stresses and their shape altered by debonding. The normality condition was shown to be reasonably well satisfied for initial surfaces of constant inelastic power (or in this case dissipation) in the presence of fiber-matrix debonding. Results indicated which types of flow/damage surfaces should be

characterized and what loading histories applied to obtain the most meaningful experimental data for guiding theoretical model development and verification. This work differed from the extensive studies of plasticity in MMCs

conducted by Dvorak and coworkers (summarized by Dvorak, 1991), in that different definitions of rate-dependent

flow and the effects of fiber-matrix debonding were addressed by Lissenden and Arnold (1997a).

Herein we will demonstrate, using micromechanics, the influence that the continuously reinforced periodic

microstructures shown in figure 1 (namely rectangular, hexagonal, and square diagonal fiber packing arrays) have

on overall flow/damage surfaces in the axial-transverse (or11- _22 )' transverse-transverse (_22- _33 ), axial-shear

(CYl1- CYl2), and transverse-shear (cY22- CYl2) stress planes. Both a displacement-based finite element analysis (FEA)

and the generalized method of cells (GMC) micromechanics approach are employed. Since the material response is

in general viscoplastic, we will consider flow surfaces (typically important at elevated temperature), which are geometrically analogous to yield surfaces (typically important at room temperature). Furthermore, silicon carbide/

titanium (SiC/Ti) composites exhibit a weak bond between the constituents and this damage mode, when active, can

greatly affect the overall material response. The effects of damage are included in the definition of flow surfaces,

hence the name flow/damage surfaces. We will pay particular attention to definitions which indicate a prescribed

level of inelasticity or damage, in an attempt to balance theoretical and experimental considerations. Sometimes

results for the strong bond case (which is fictitious for current SiC/Ti systems) are shown in order to demonstrate

how fiber/matrix debonding influences the predicted overall inelasticity.

MMCs are anticipated to be typically used in elevated temperature environments. Unless stated otherwise, the

results presented are for a TIMETAL-21S titanium matrix reinforced by a 35 percent fiber volume content of continuous SCS-6 fibers. This unidirectional SiC/Ti system is taken as our model material system (see Arnold and

Castelli, 1995). Two fairly extreme temperatures are considered; 23 and 650 °C (where time-dependent effects

dominate the titanium matrix). The fiber response is taken to be linear elastic and temperature-independent, while

the matrix response is elastic-viscoplastic (Arnold et al., 1996b and c) and highly temperature-dependent as illustrated in figure 2. Note that the protective fiber coatings used to reduce fiber strength degradation (associated with

chemical reactions with the matrix during processing) result in a weak bond to the matrix. A discussion of modeling

this weak bond and the viscoplastic matrix response is postponed until Section III. Finally, as we are interested in

surfaces in prescribed stress planes, all loadings are stress controlled. An equivalent stress rate,,JCslC51 , of

2 MPa/sec is used in all cases, where cyI is the contracted stress tensor and I = 1,2 ..... 6.

In the next section (II) we review the theoretical framework and basic concepts applicable to inelastic flow in

MMCs. Flow surface definitions and factors that influence flow are discussed and an appendix (A) provides a continuum example. The FEA and GMC micromechanics models are introduced along with the constituent constitutive

models in Section III. The stress-strain response and inelastic power-time response for different microstructural architectures are presented in Section IV. A related appendix (B) discusses the accuracy of the FEA and GMC models.

Finally, Section V presents the effects of microstructure and fiber-matrix debonding on macroscale flow/damage

surfaces in four different stress planes at 23 and 650 °C. Section VI provides a closure for the paper.

II. INELASTIC FLOW

Our subject is the inelastic multiaxial response of MMCs for applications in elevated temperature environments.

Hence, the viscoplasticity of anisotropic composite materials susceptible to internal damage must be addressed.

From the viewpoint of a structural analyst investigating complex components, a macroscale continuum model is

NASA/TM--1998-208805 2
preferredduetoitsnumerical efficiencyrelativetoalternativemicromechanics analyses. However, suchacontinuummodelisnotcurrentlyavailable; noraretheexperimental resultsthatarenecessarytodevelopsuchamodel.

Thus,inthepresentstudywewillutilizeatheoretical frameworkappropriateforcontinuumodelingtoguideus

andemploytwomicromechanics modelstosimulatetherequiredexperiments numerically, soastounderstandthe

theoretical andexperimental implicationsand/orassumptions necessary ineitheramacro(continuum) or

micromechanics baedapproach.

II.1.Theoretical Framework

Thetheoretical considerations usedinthispaperarebasedonanenergybalance. Theprimaryvariablesatany

pointaretheCauchystresstensor,(YiJ' theinternalstresstensor,c_ij,andtemperature, T. Other internal state variables could also be defined and used if we were so motivated. The current values of these variables can be used to

define the Gibbs thermodynamic potential, G = G((yij, czij,Dij,T), where Dij is the preferred direction tensor formed by

the self product of the unit vector denoting the fiber direction. Conjugate to these variables are the total strain tensor,

eij, the internal strain tensor, Aij, and the entropy, S,

0G o_G o_G

eij- O_ij Aij Oaij S o3F (1)

Our basis is that the total work performed on the system must be equal to the sum of the stored energy and the energy dissipated, where the stored energy includes an elastic component as well as an inelastic component associated

...... • / •

with the internal state. Thus, the dissipation potential, _ = _((y!j,czij,Dij,T), can be defined to be (YijSij - o_ijAij.

The associated flow law (for the inelastic strain rate) and evolution equations (for the internal strain rate) are

given by normality,

_i._ O_ /4ij_ O_

O(YiJ , Oc_ij (2)

and

• m I •

c_ij = Qijl;1A1;1,

where

and Qijkl is called the internal compliance operator (Arnold and Saleeb, 1994). Thus, once the functional dependencies of the Gibbs and dissipation potentials have been determined, all of the variables are known by simple

differentiation. Let us now assume that the dissipation potential can be written in terms of two scalar functions,

= _(F,H), where F depends on the deviatoric effective stress, Zij, and H depends only on the internal stress

(Robinson and Ellis, 1986). The deviatoric effective stress is the difference between the deviatoric Cauchy stress and

the deviatoric internal stress. Now the flow law can be written,

_/. 0f2 OF

- OF OcYij" (3)

Thus, the direction of the inelastic strain rate vector is normal to surfaces having F = constant. However, if it is

true that the normality condition is not satisfied in MMCs, as indicated by Nigam et al. (1994a,b) for boron/aluminum, then it becomes necessary to develop a nonassociated flow law and evolution equations. We return to this issue

at the end of Appendix A.

NASA/TM--1998-208805 3
11.2.SurfaceDefinitions

Theconcept ofayieldsurfaceiswellknowninrate-independent plasticity,evenif noonedefinitionofyielding

hasbeenuniversallyadopted. Themostcommondefinitionsemployedaretheproportional limit,asmall(usually

5to20jam/m)offsetstrain,aback-extrapolation, andalarge(usually2000jam/m)offsetstrain.Forrate-dependent

plasticity(viscoplasticity), theconceptofastrictyieldsurfacebreaksdownasstressstatesoutsidetheyieldsurface

areaccessible(sincenoconsistency conditionapplies). Thustheneedforgeometricallyanalogous, thermodynamicallybased,flowsurfacedefinitions. Atleasttwodifferentrate-dependent definitionshavebeenproposedformonolithicmaterials(ClinardandLacombe, 1988);

(1) surfacesofconstant dissipationrate(SCDRs), definedby (Yij_q.r_OcijAi j and

•/ ./

(2) surfaces of constant inelastic strain rate (SCISRs), defined by alCTiC_i.

These definitions can also be applied to composite materials. However, one needs to be careful when using

micromechanics as the inelastic strain is an eigenstrain and consequently;

eijeij _ j_ _ dV

(4)

where an over-bar indicates a macroscale quantity and V is the total volume of the representative volume element.

The difference is due to the existence of residual stresses, at least some of which are associated with stored elastic

energy. Being an additive quantity, the overall dissipation potential can be taken to be the volume average of the

local dissipation potentials (Suquet, 1987), and from a theoretical standpoint that is what we would like to quantify.

However, it is indeterminate experimentally because the local (internal) variables are unknown and not measurable.

This issue is dealt with more completely by Lissenden et al. (1998). Here let us define another type of flow surface

I

using experimentally measurable macroscopic quantities, -_ijEij, that are available from both micromechanics and

macroscale models. We will call them surfaces of constant inelastic power (SCIPs). This is in fact the nomenclature

that should have been used by Lissenden and Arnold (1997a) instead of the term initial SCDRs. The actual difference between SCIPs and SCDRs depends on many factors, but as shown in figure 3 for transverse tensile loading, it

can be substantial.

Lissenden and Arnold (1997a) demonstrated, using micromechanics, that the direction of the overall inelastic

strain rate vector can differ significantly from the outward normal of a SCISR. Whereas, for the stress planes considered, the direction of the overall inelastic strain rate vector was reasonably close to the outward normal of the

SCDRs (actually SCIPs) 1 considered. SCISRs, however, are certainly more amenable to experimental methods than

are SCDRs, and may be more amenable than SCIPs given that stress quantities are not included in the definition. For

macroscale continuum theories it is not possible to distinguish between SCDRs and SCIPs. A macroscale continuum

example is given in Appendix A.

II.3. Factors Influencing Flow

Many factors influence inelastic flow in metallic materials. Certainly, temperature and loading rate as well as

the past loading history can be important in many metals. Additionally, microstructural architecture, degree of

1Notethatfor a near vh'gin state, SCDRs and SCIPs are nearly the same given o:ij= 0, but as incleastic flow occurs (o:ij;_0) they will diverge.

NASA/TM--1998-208805 4
anisotropy, fiber-matrixbondstrength, anddamageinfluenceflowincomposites. Otherfactors,suchasthestress

plane,definition,andtargetvalueinfluencehowflowisrepresented. LissendenandArnold(1997b)illustratedthe

effectsofmanyofthesefactors.Thecurrentpaperfocusesontheinfluenceofmicrostructural architectureandfibermatrixbondstrength. Weconsiderrepeatingmicrostructures, specificallyoneshavingrectangular, hexagonal,nd

squarediagonal fiberpackingarraysasshowninfigure1.Therectangular arrayhasanaspectratio,R = a/b. For the

special case of a square array, R = 1. Additionally, we consider strong and weak fiber-matrix bonding. Our

definition of strong is that there is no discontinuity in the displacement field at the interface. Likewise, weak means

that the interface can transmit a finite traction before debonding causes the fiber and matrix to separate.

III. MICROMECHANICS

Biaxial experiments on unidirectional continuous-fiber reinforced SiC/Ti in the axial-transverse (t311 -- 1322)'

transverse-transverse (1322- 1333), axial-shear (1311- 1312), and transverse-shear (1322- 1312)stress planes were simulated numerically using micromechanics. Initial overall (macroscale) flow/damage surfaces were mapped out by a

sequence of stress-controlled proportional loading probes at different angles in a prescribed stress plane as shown in

figure 4. The two nonzero stress rate components were calculated to be the cosine and sine of the equivalent stress

rate (2 MPa/sec). Each probe started at the origin and continued until the inelastic power target value had been

reached, after which the material was returned to its virgin state, the probe angle increased (usually by 5°), and the

next probe conducted; until the entire stress plane (0 --_ 360 °) had been probed. For surface determinations at room

temperature thermal residual stresses were accounted for by cooling the composite from a stress-free temperature of

815 °C in 2 hr. These simulations were conducted using both FEA and GMC, with the specifics described below.

III. 1. Finite Element Analysis

The commercial FEA program ABAQUS (HKS,1995) was used to determine the response of the microstructures of interest (fig. 1). Each microstructure can be represented by a repeating unit cell that has two planes of symmetry. Thus, it is only necessary to analyze a quarter of each unit cell. Coarse and fine discretizations of one quarter

of the repeating unit cell for both square and square diagonal arrays are shown in figure 5. Only strongly bonded

SiC/Ti was considered using FEA for determining SCIPs. Generalized plane strain triangular elements formulated

by Lissenden and Herakovich (1995), and implemented into ABAQUS through a UEL subroutine, were used for

modeling the repeating unit cell. These triangular elements have an extra node that is common to all elements in the

mesh. The degree of freedom in the 1-direction (axial) at this common node is what makes this a generalized plane

strain element. Furthermore, warping of the 23-plane permits axial-shear loading to be simulated. Overall inelastic

strain components were calculated as the difference between the total strain components, found by volumetric averaging, and the elastic strain, found from the overall stress (volumetric average) and elastic properties.

The boundary conditions applied to each FEA mesh are illustrated schematically in figure 6. To maintain compatibility with the adjacent unit cells, multipoint constraints (MPCs) were applied on the edges that are in contact

with adjacent unit cells. For example, MPCs were applied along the right edge such that the edge remains straight.

Stress control loading was simulated by applying concentrated forces at the nodes along an appropriate edge; except

for axial loading, where a concentrated force was applied to the common node, which is shown in figure 6. Details

of the applied loadings are summarized in table I.

111.2. Generalized Method of Cells

The generalized method of cells (GMC) (Paley and Aboudi, 1992; Aboudi, 1995) is an approximate analytical

micromechanics model that extends the original method of cells (Aboudi, 1991) to an arbitrary number of subcells,

permitting the study of different microstructures. The reader is referred to Aboudi (1995) for the derivation of the

detailed equations of GMC. The primary equations of GMC relate to standard micromechanics equations as follows.

NASA/TM--1998-208805 5
] N_ NI_ N_,

-- [3--1 y=l

(_=1 [3=1 y=l

(6)

(7)

Cijkl• = ((Cijrs (x)Arskl (x) )), c

1 Na N3 N r

dhl E E E dahfll7 C(aflT)A(aflT)

a=l ]3=1 7=1

(8)

N a N fl N r

ah, 1 (c*)-lE E E - (9)

a=l 3=1 7=1

N a N fl N r

-_T-ldhl (C*)-I E E EdahfllTC(aflT)(D(aflT)8Ts-8T(aflT)) (10)

a=l fl=l 7=1

In equations (5) to (10) an overbar indicates a macroscale variable; (x) explicitly indicates that the local variable is a

1 I.T" C*]k l denotes the overall effective elastic stiff- function of position; ({'))='_ov dVis the volumeaverageoperator;

ness tensor; the superscript (c_y) refers to subcell (c_y) where cq _, and y are indices in the three coordinate directions; d, h, and ! are the overall dimensions of the repeating unit cell and the same variables with subscripts denote

dimensions of subcells; Ai]kl, Bijkl, and Dijkl are the elastic strain, stress, and nonelastic strain localization tensors;

and EsI and Er are arrays containing the subcell inelastic and thermal strain vectors.

GMC has been implemented into the recently developed micromechanics analysis code (MAC/GMC) which

has many user friendly features and significant flexibility (Wilt and Arnold, 1996). The GMC unit cell representations utilized herein for square, hexagonal, and square diagonal arrays are shown in figure 7.

III.3. Constituent Models

As mentioned previously, the fiber response is assumed to be linear elastic and temperature independent

(E = 400 GPa, v = 0.2, CTE = 3.5x10 4 °C 1). The elastic-viscoplastic behavior of the matrix is represented using a

generalized viscoplastic potential structure (GVIPS) model (Arnold et al., 1996b,c). This model is a fully associative, multiaxial, nonisothermal, nonlinear kinematic hardening viscoplastic model for use with initially isotropic

metallic materials. A unique aspect of this model is the inclusion of nonlinear hardening through the use of a compliance operator Qi/kl in the evolution law for the back stress. This nonlinear tensorial operator is significant in that it

allows both the flow and evolutionary laws to be fully associative and greatly influences the multiaxial response

under nonproportional loading paths so prevalent within composite materials. The primary equations are summarized by,

NASA/TM--1998-208805 6
_/=ZEij if /_>0

AiJ = QijklCklpqbpq bij ifif aij_,ij aijZij <>_00

=

flow law

evolution law

internal constitutive rate law

(16)

where

•I .I 1

Z = ,I 'J '' , J2 = ZijZij

2J2 2

Lijkl = Q,711 = Kl( Iijkl + K2aijakl )

bi j = _I _ K3ai j

O(jK4a(j

E = Sij -

(J aij

S(j = c_(j - 3 c_kkf(J

a(j = aij - 1 akkf(j

where the Macauley brackets are defined by

(x-a)= {x-a0 x>a. x<a

The constants K 1 - K 4 contain material parameters and internal stress invariants and can be inferred from Arnold

et al. (1996b).

Weak bonding between the fiber and matrix is modeled by assuming that a jump in the displacement field may

occur under certain conditions, while the traction vector remains continuous. In this model debonding initiates when

the normal traction exceeds a critical value or when the tangential traction exceeds a critical value, with no interaction between the two,

l_n = RnTn 1 ' > Tt crit

l_t = RtTt j if Tn >_TCnt or T t

(17)

where u denotes interfacial displacement, T the interfacial traction, and R the flexibility of the failed interface. The

subscripts n and t denote the directions normal and tangent to the interface respectively. This simple model combines elements of interface models developed by Jones and Whittier (1967) and Achenbach and Zhu (1990). Once

debonding has initiated, the interfacial displacement rate is made proportional to the stress rate, simulating a perfectly plastic-like behavior. In the results for SiC/Ti with a weak bond the critical normal and tangential tractions

have been taken to be 103 and 41.4 MPa, respectively, and the flexibility of the failed interface has been taken to be

0.271 mm/MPa.

IV. STRESS-STRAIN RESPONSE

Before determining flow surfaces we validated our FEA models by performing a convergence study and comparing stress-strain responses with those published by Brockenbrough et al. (1991) for boron/aluminum. In figure 8,

the stress-strain and inelastic power-time responses are shown for axial ((Y11), transverse (_22), and shear ((Y12) loadings at 650 °C. Here the effect of microstructure, i.e., square, hexagonal, and square diagonal repeating arrays of

NASA/TM--1998-208805 7
fibers, is demonstrated using GMC and the unit cells shown in figure 7. Both strong and weak fiber-matrix bonds are

considered, but as expected for axial loading the interface does not affect the response. Also as expected, microstructure has no affect on the axial response (fig. 8(a)). Hexagonal and square diagonal array responses for transverse and shear loadings (figs. 8(b) and (c)) for strongly bonded composites are very similar and diverge from the

response of a square array for transverse loading, but agree closely with the response of a square array for shear

loading. It is clear that the transverse response is strongly affected by microstructural architecture, while the axial

and shear responses are not affected and very mildly affected, respectively. However, for weakly bonded composites

the square and hexagonal array transverse and shear responses are similar, while the square diagonal array exhibits a

more compliant response (figs. 8(b) and (c)).

An accuracy comparison between these GMC unit cell representations and their FEA cotmter-parts was also

undertaken and is described in Appendix B. A summary of our observations based on stress-strain and inelastic

power-time responses, relative to the appropriate micromechanics model (FEA and GMC) and microstructural architecture (square, hexagonal, and square diagonal arrays) is as follows:

• GMC predictions agree well with FEA predictions if a fine FEA mesh is employed;

• A fine FEA mesh is required to accurately predict gross inelastic flow in the transverse direction, but for axial

and shear loadings a coarse FEA mesh is adequate, however, a coarse mesh appears adequate for any of the

loadings considered, provided the inelastic power is small;

• Microstructural architecture is very significant for transverse loading, but has much less effect for axial

(none) and shear loading (only significant for weakly bonded composites). Note that we only considered axial

shear ((Y12)loading, not transverse shear ((Y23)loading.

V. MACROSCALE FLOW/DAMAGE SURFACES

Our focus now turns to macroscale flow/damage surfaces in order to obtain information on the initial overall

1

dissipation potential. Surfaces of constant inelastic power (SCIPs), defined simply by criie;i, will be determined in

order to give this work relevance to experimental and macroscale continuum approaches. Here the inelastic strain

tensor includes matrix inelasticity as well as nonlinear effects associated with fiber-matrix debonding. First, we

exercise the FEA models for a strongly bonded composite, then compare FEA and GMC predicted flow surfaces,

and then finally consider a weakly bonded composite using GMC.

Consider first the effect that the target value has on SCIPs in the axial-transverse stress plane at 23 °C. Figure 9

shows 1, 5, and 10 kPa/sec SCIPs for both a square array and a square diagonal array as predicted by the fine and

coarse FEA meshes shown in figure 5 for strong bonding. The first observation that we make is that the surfaces for

different target values are not concentric for either array and that they are offset in the axial compression direction

due to thermal residual stresses. While the surfaces for square and square diagonal arrays are approximately the

same size for the same target value, their shapes are different, particularly the 5 kPa/sec SCIPs in the first quadrant

(i.e., tension-tension). The 5 kPa/sec SCIP for a square array has a 'nose' at approximately 35 °, while the 'nose' on

the same SCIP for a square diagonal array is at approximately 15°. The differences in shapes of the overall flow

surfaces are apparently related to the different local stress and strain fields.

Local J2 (1/2 S(jS(j, where S(j is the deviatoric Cauchy stress) contours in the matrix are plotted in figure 10 for

both arrays given their fine mesh idealizations and all three target values for loading at 35 ° from the axial stress

axis. The smallest values (darkest) occur in the matrix adjacent to the fiber near the bottom edge (and top edge for

the square diagonal array). This is interesting because J2 is small where the transverse stress is large and large

where the transverse stress is small. Another way to see the differences in the local fields for square and square

diagonal arrays is to compare the local stress invariants 11 (the sum of the normal stress components) and J2' In

figure 11, the stress invariants 11 and J2 for each matrix element have been sorted and are plotted in descending

order. Note that while the two arrays each have a fiber volume fraction of 0.35, the square array has 227 matrix

elements in the fine mesh, while the square diagonal array has 281. The square diagonal array has a relatively

small number of elements with large hydrostatic stress, while the square array has a more uniform distribution of

hydrostatic stress (fig. 11(a)). For example, the difference between the square and square diagonal arrays for

5 kPa/sec SCIPs is related to the different J2 distributions shown in figure 1l(b). The square array has fewer elements with high values of J2 and therefore requires a higher overall stress to achieve the same flow target value as

the square diagonal array.

NASA/TM--1998-208805 8
Comparing the SCIPs from the fine and coarse mesh FEAs shown in figure 9 indicates that the coarse mesh for

a square array of fibers provides reasonably good results in the axial-transverse stress plane. However, in the case of

a square diagonal array of fibers agreement between the fine and coarse mesh results is not as good, implying that

too few elements were used in the coarse square diagonal array mesh.

Clearly, it is highly preferable to use a coarse mesh in lieu of a fine mesh, provided the accuracy of the results

can be maintained, as the execution time is greatly reduced.

Nimmer et al. (1991) used FEA to study the effect that the aspect ratio of a rectangular array has on the transverse tensile response of a weakly bonded SCS-6/Ti-6-4 composite system. Here we expand the spirit of Nimmer's

study to the biaxial loading behavior of a strongly bonded S CS-6/TIMETAL 21S composite system. A wide range

of aspect ratios (R = a/b in fig. 1), 0.5 < R < 2.0, are considered and compared with results for a square array

(R = 1.0). Figure 12 shows 5 kPa/sec SCIPs at 23 °C in four stress planes for aspect ratios of 0.5, 1.0, and 2.0. As

demonstrated previously, the microstructural architecture, defined by the aspect ratio here, has the largest effect by

far on the transverse response.

One application for the kind of information that flow surfaces provide is the design of components subjected to

loads resulting in deterministic multiaxial stress states. The microstructure can then be engineered to best resist the

loading. For example, a ring mounted on a shaft in a jet engine will rotate in service. Thus, the radial and circumferential stress components will be tensile. Suppose the ring is to be fabricated from hoop-wound, strongly bonded SiC/

Ti, and that the goal is to delay the onset of inelastic flow as long as possible. According to the first quadrant of the

axial-transverse stress plane in figure 12, the largest possible aspect ratio should be used. However, for a weakly

bonded composite, once the interface fails the stress distribution is much different because the transverse stress in

the fiber gets redistributed to the matrix. In this case, a small aspect ratio is preferred because it provides a long ligament of matrix material between fibers for the transverse stress to flow through, while a large aspect ratio is associated with a short ligament of matrix between fibers and more localized flow (Lissenden and Herakovich, 1996).

In figure 13, FEA (coarse mesh) and GMC predicted 1 and 10 kPa/sec SCIPs at 23 °C are compared in all four stress

planes for composites with a strong fiber-matrix bond. The SCIPs predicted by FEA and GMC are in good agreement. This is in line with the excellent agreement between yield surfaces defined by local yielding (Mises stress)

that Pindera and Aboudi (1988) reported for the method of cells (square packing) and FEA. In figure 13 overall

SCIPs are determined based on overall inelastic strain rates and stresses, as might be done in an experiment. These

results are in contrast to the FEA/GMC comparisons made in Lissenden and Arnold (1997c), where GMC and FEApredicted flow surfaces were obtained at 650 °C using strain-controlled and stress-controlled loading, respectively.

While initial results indicated that flow surfaces in the axial-transverse stress plane were not very sensitive to

whether stress or strain control probing was used (Lissenden et al., 1998); in the transverse-transverse stress plane

the control mode is important. Thus, the larger flow surfaces obtained by GMC in strain-control (fig. 8 in Lissenden

and Arnold, 1997c) are easily explained. Inelastic deformation is strain rate dependent, thus more deformation occurs for a slower overall strain rate than a faster one. Consequently, in strain control the overall strain rate is fixed, it

is the stress rate that decreases as inelastic deformation occurs. However, in stress control the overall strain rate must

decrease as inelastic deformation occurs, resulting in more inelastic flow relative to strain control.

Figure 14 shows excellent agreement between GMC and FEA (coarse mesh) 10 kPa/sec SCIPs at 650 °C in all

four stress planes. The sharp points apparent on some of the surfaces are artifacts due to the large step in probe angle

(5 °) used to make the analysis less time consuming. These SCIPs are actually smooth curves. SCIPs from finer

GMC discretizations having 16 subcells (a cross shaped fiber) and 49 subcells (a roughly circular fiber) were also

examined in the axial-transverse stress plane. As was found for FEA (fig. B 1), the axial response was independent of

the discretization, but the transverse stress was 7 percent less for the 49 subcell model than for the 4 subcell model

shown in figure 14. This difference is almost imperceptible at the scale it is drawn. Local stress and strain fields,

which ultimately determine the strength or cyclic life of a material, are not reported here, but are the topic of a companion paper (Lissenden et al., 1998).

Consider now, the case of a weak fiber-matrix bond using GMC, which is more realistic for the current SiC/Ti

system. In figure 15, 1 kPa/sec SCIPs at 23 °C for a square array are shown for weak and strong bonds. The primary

effect of a weak bond is to significantly reduce the tensile stress at which deviation from proportional response

begins. Since compressive interfacial tractions are not detrimental to the integrity of the interface, debonding does

not occur for transverse compressive loading (unless it is due to Poisson expansion, see Lissenden and Arnold

(1997a)). Additionally, axial loading is not observed to cause debonding.

The effect that microstructural architecture has on 1 kPa/sec SCIPs in the axial-transverse and transverse-shear

stress planes for weakly bonded SiC/Ti at 650 °C is shown in figure 16(a). Microstructural architecture has the most

NASA/TM--1998-208805 9
influence when a compressive transverse stress is present. As was observed for the stress-strain response (fig. 8), the

square array exhibits less inelastic flow than the hexagonal and square diagonal arrays.

Brockenbrough et al. (1991) and Arnold et al. (1996a) demonstrated that as the fiber volume fraction increases,

the effect of microstructural architecture increases for uniaxial loadings. Figure 16(b) shows 1 kPa/sec SCIPs in the

axial-transverse stress plane at 650 °C for square, hexagonal, and square diagonal packings and a fiber volume fraction of 0.50. The primary effect of increasing the fiber volume fraction is to enlarge the SCIP. Additionally, larger

differences in the SCIPs for the three microstructures are observed for the higher fiber volume fraction, especially

where the transverse stress is compressive (and the interface has not debonded).

VI. CLOSURE

Micromechanics has been used to generate overall flow/damage surfaces for unidirectional MMCs having different microstructures. The generalized method of cells results compared very well with those from the computationally more demanding finite element analysis approach, wherein many elements are required to obtain the same

level of accuracy. Except for the special case of uniaxial loading in the direction of the fibers, microstructure influences the shape and location of the macroscale flow/damage surface. The magnitude of the effect that microstructure

has on flow/damage surfaces depends strongly on fiber/matrix bond strength, fiber volume fraction, and the type of

loading, among other secondary factors. A weak fiber/matrix bond significantly influences the shape of the flow/

damage surface. A dramatic flattening of the surface is observed for transverse tensile loading. Debonding also

occurs under shear loadings, but its effect is less drastic.

Flow/damage surfaces are important because multiaxial stress states are encountered in the analysis and design

of most structural components due to complex loadings and geometries, as well as due to stress concentrations. The

response of MMCs to multiaxial stress states can not be inferred from uniaxial test data. It is necessary to conduct

multiaxial tests to guide and later validate theoretical models. To implement the thermodynamically based framework discussed in Section II for a macroscale continuum model, the functional form for the Gibbs and dissipation

potentials must be known. Neither of these potentials can be directly determined experimentally. However, one possible approach for continuum model development is to experimentally determine SCIPs in various stress planes and

use a micromechanics model to interpret the results. The functional form of the Gibbs and dissipation potentials

could then be inferred from the micromechanics model having an inelastic power that correlates well with the

experimental results over a wide range of conditions. This work has shown that in addition to the dependence of

these potential functions on stress invariants, internal state invariants, and temperature, as for isotropic monolithic

materials, they may also depend on directionality, microstructure, and bond strength for MMCs. This adds yet another level of complexity to the macro-based continuum approach for modeling multiaxial inelastic response of

MMCs.

NASA/TM--1998-208805 10
APPENDIX A

A MACROSCALE CONTINUUM EXAMPLE

Consider, a unidirectional MMC and smear the microstructure such that we have a homogeneous anisotropic

material. Thus, all variables in this appendix refer to composite quantities, there are no local (micro) variables. The

effect of hydrostatic stress on the flow of most metals is quite small and usually neglected, as is the case here, even

though composites can be dependent upon hydrostatic stresses (e.g., Aboudi, 1991 and Jansson, 1995). Assume that

the dissipation potential can be written in terms of two scalar functions, _ = _(F,H), where H is a function of the

deviatoric internal stress, aq, and the reinforcement direction, defined by Dq, and

F= IE'+ kT2 1 +--12-I 4(4O)291_ 1) 13 -1 (A1)

where

1

12 = I - 13 J2 = _ ZijZji

13 = I2 J3 = 3 ZqZjkZki

I o = Dij Zji

and kT is the transverse shear strength; also, q and m denote threshold stress ratios for shear and normal loading,

respectively. The constant c scales the effect of J3 on inelastic flow and must be determined experimentally. This

form ofF is for transversely isotropic materials and is due to Robinson and Duffy (1990), except here J2 has been

replaced with Drucker's (1949) J2/J3 form (c = -9/4). In this theory, the inelastic strain rate vector is normal to surfaces having constant F in stress space (see eq. (3)). Thus, normality only applies to flow surfaces proportional to

surfaces having F = constant. It can be shown that

• 1 3&_

cyije/= 2(F + ) _--_, (A2)

in the neighborhood of the virgin state, which are proportional to F = constant surfaces. On the other hand, SCISRs

turn out to be

_= 2"I'I (1/__T2.._)].I O_[j3+cj2)--4/312J52+4cj2j2(l+__7c) 1

2(4 1)(J 3 +cJ3 )2x-2/3F | 2 q/ ) 2 ( )8

+/0/1 + 41/2 q- 4o)2_9 1 )[(j3 +cj2)-2/3(ioj2 _4cJ2J3+ 2cJ3DijY_jkY_k i)+ I+-13 (Dqy_jkDki +DqDjkY_ki -210 )1

+{+-l)2[(DkiXjk+DjkXki)(DliXjl+DjlXli)+413(Io-2)l

2 )1/2

+ 13 (41 - 7 q 40)29)(- 1" DijDij - 1)t . (A3)

NASA/TM--1998-208805 11
For an isotropic material (that is, T1= CO= 1), SCISRs reduce to

,.,., - 1+ (A4) k3(F+I) 0F (F+I)_ k6(F+l)2 -_

and if the J3 contribution to inelastic flow is negligible (that is, c = 0) we get

__•I • I 2_+1)0_ (A5)

k OF

Thus, SCISRs are only proportional to surfaces having F = constant for the special case of isotropic J2 materials,

and therefore the normality criterion only applies to SCISRs for this special class of materials. Also, Lissenden and

[2II

Arnold (1997a) extrapolated these results to rate-independent plasticity by considering _ije I and _-_ eij eij, where

the latter is the usual equivalent inelastic strain definition often used to define a yield surface, and found results

analogous to those for SCDRs and SCISRs. Hence, the Nigam et al. (1994a,b) results on boron/aluminum that the

plastic strain rate vector was not normal to the yield surface could be merely dependent on their definition of yield.

NASA/TM--1998-208805 12
APPENDIXB

A MESHREFINEMENTSTUDY

Hereacomparisonfthestress-strain responsesforuniaxialandbiaxialloadingsusingbothFEAandGMCis

conducted. Additionally, sinceweareinterestedininelasticpower,itsincreaseasafunctionoftimewillalsobe

plottedfortheseloadings. InfigureB1thestress-strainndinelasticpower-timersponses areshownforaxial

(cY11), transverse (_22), and shear (cY12)loadings at 650 °C. A square array of fibers is discretized using the coarse

and fine FEA meshes (fig. 5) and the four subcell GMC model (fig. 7). For axial loading the three predictions are

nearly identical for both stress-strain and inelastic power (fig. B l(a)), which is not surprising since the material

response is dominated by the fiber as evidenced by the high stresses and small inelastic power relative to the other

two types of loading. On the other hand, transverse and shear loadings are matrix dominated and differences between analysis techniques are anticipated.

The disparity in the coarse and fine mesh FEA-predicted transverse stress-strain and inelastic power-time

responses shown in figure B 1(b) indicates that the coarse mesh is not adequate for predicting gross inelastic flow,

that is, the coarse mesh has not yet converged. Further refinement of the fine mesh indicated that it had converged

for the range of stresses obtained in the surface determinations presented in this paper, but for larger stresses an even

more refined mesh would be necessary. The GMC-predicted responses agree very well with the fine mesh FEA predictions for the range of stresses obtained in surface determinations. However, for larger stresses a more accurate

geometric representation of the fiber within the repeating unit cell would provide more accurate results, see figure 1

of Arnold et al. 1996a.

The shear responses (fig. B 1(c)) from the two FEA meshes are identical, indicating that the coarse mesh has

converged for shear loading. The GMC predicted response exhibits more inelastic flow and is in reasonably good

agreement with the FEA results. The exponential form of the inelastic power accumulation with time (fig. B 1(c)) is

a result of the constant stress rate used, which causes the inelastic strain rate to approach infinity as the slope of the

stress-strain curve approaches zero.

Stress-strain and inelastic power-time responses calculated utilizing FEA and GMC are shown in figure B2 for

proportional biaxial loading of a square array of fibers at 650 °C. The axial-transverse (cy11 = _22) response shown

in figure B2(a) indicates that each micromechanics approach is equally good for predicting axial response, but that

differences are present in the predicted transverse response, which lead to differences in the inelastic power. As for

uniaxial transverse loading, the coarse FEA mesh is inadequate for predicting gross inelastic flow while GMC provides a very reasonable approximation of the inelastic flow. Each model is equally good for predicting the transverse-transverse (_22 = _33) response, as shown in figure B2(b) and reasonable agreement is also observed for the

axial-shear ((Yll = (Y12) response shown in figure B2(c).

Note that the finely meshed FEA takes a relatively long time to execute compared to that of the coarse FEA

mesh (a clock-time ratio of approximately 4:1), and even more so relative to GMC. This lengthy execution time is

greatly exacerbated when performing flow surface determinations as at least 72 directions for one surface determination are required. Fortunately, the coarse FEA mesh is reasonably accurate for small values of inelastic power,

approximately 10 kPa/sec and below. Thus, it appears acceptable to use the coarse mesh for surface determinations

using transverse stresses provided the target value is small. Additionally, GMC, which is numerically more efficient

than either FEA model, agrees well with the fine meshed FEA results.

NASA/TM-- 1998-208805 13
REFERENCES

Aboudi,J.(1991),Mechanics of Composite Materials': a unified micromechanical approach, Elsevier, Amsterdam.

Aboudi, J. (1995), Micromechanical analysis of thermo-inelastic multiphase short-fiber composites. Composites

Eng. 5, 839.

Achenbach, J.D. and Zhu, H. (1990), Effect of interphases on micro and macromechanical behavior of hexagonalarray fiber composites. J. Appl. Mech. 57, 956.

Arnold, S.M. and Saleeb, A.F. (1994), On the thermodynamic framework of generalized coupled thermoelasticviscoplastic-damage modeling. Int. J. Plasticity 10, 263.

Arnold, S.M. and Castelli, M.G. (1995), What constitutes a model material? HITEMP Review 1995, Volume II:

Compressor/Turbine Materials, NASA CP- 10178, pp. 35A: 1-18

Arnold, S.M., Pindera, M-J., and Wilt, T.E. (1996a), Influence of fiber architecture on the inelastic response of

metal matrix composites. Int. J. Plasticity 12, 507.

Arnold, S.M., Saleeb, A.F., and Castelli, M.G. (1996b) ,A fully associative, nonisothermal, nonlinear kinematic,

unified viscoplastic model for titanium based matrices. In Thermomechanical Fatigue Behavior of Materials:

Second Volume, eds. M.J. Verrilli and M.G. Castelli, pp. 146-173. ASTM-STP 1263, American Society for Testing and Materials, Philadelphia.

Arnold, S.M., Saleeb, A.F., and Castelli, M.G. (1996c), A fully associative, nonlinear kinematic, unified viscoplastic

model for titanium based matrices. In Life Prediction Methodology for Titanium Matrix Composites, eds.

W.S. Johnson, J.M. Larsen, and B.N. Cox, pp. 231-256. ASTM-STP 1253, American Society for Testing and

Materials, Philadelphia.

Brockenbrough, J.R., Suresh, S. and Wienecke, H.A. (1991), Deformation of metal-matrix composites with continuous fibers: geometrical effects of fiber distribution and shape. Acta Metall. Mater. 39, 735.

Clinard, J.A. and Lacombe, C. (1988), Determination of multiaxial flow surfaces at elevated temperatures using the

concept of dissipation potential. ORNL TM-10787, Oak Ridge National Laboratory, Oak Ridge.

Drucker, D.C. (1949), Relation of experiments to mathematical theories of plasticity. J. Appl. Mech., 16, A349.

Dvorak, G.J. (1991), Plasticity theories for fibrous composite materials. In Metal Matrix Composites: Mechanisms

and Properties, eds. R.K. Everett and R.J. Arsenault, pp. 1-77. Academic Press, San Diego.

HKS (1995), ABAQUS/Standard User's Manual Version 5.5. Hibbitt, Karlsson, and Sorensen, Inc., Pawtucket.

Jansson, S. (1995), Non-linear constitutive equations for strongly bonded fibre-reinforced metal matrix composites.

Composites 26, 415.

Jones, J.P. and Whittier, J.S. (1967), Waves at a flexibly bonded interface. J. Appl. Mech. 34, 905.

Lissenden, C.J. and Arnold, S.M. (1997a), Theoretical and experimental considerations in representing macroscale

flow/damage surfaces for metal matrix composites. Int. J. Plasticity 13, 327.

Lissenden, C.J. and Arnold, S.M. (1997b), Factors influencing inelastic flow under multiaxial stress states: a perspective of their importance to composites. In HITEMP Review 1997 Advanced High Temperature Engine

Materials" Technology Program, NASA CP 10192, NASA Lewis Research Center, Cleveland, Paper No. 38.

Lissenden, C.J. and Arnold S.M. (1997c), Effect of microstructural architecture on flow/damage surfaces for metal

matrix composites. In Damage Mechanics in Engineering Materials, eds. G.Z. Voyadjis, D.H. Allen, and

J.W. Ju, Elsevier, in print.

Lissenden, C.J. and Herakovich, C.T. (1995), Numerical modelling of damage development and viscoplasticity in

metal matrix composites. Comput. Methods" Appl. Mech. Engng. 126, 289.

Lissenden, C.J. and Herakovich, C.T. (1996), Interracial debonding in laminated titanium matrix composites. Mech

of Materials 22, 279.

Lissenden, C.J., Iyer. S.K., and Arnold, S.M. (1998), Local and overall flow surfaces for composites predicted by

micromechanics. In preparation.

Nigam, H., Dvorak, G.J. and Bahei-E1-Din, Y.A. (1994b), An experimental investigation of elastic-plastic behavior

of a fibrous boron-aluminum composite: I. matrix-dominated mode. Int. J. Plasticity, 10, 23.

Nigam, H., Dvorak, G.J. and Bahei-E1-Din, Y.A. (1994b), An experimental investigation of elastic-plastic behavior

of a fibrous boron-aluminum composite: II. fiber-dominated mode. Int. J. Plasticity, 10, 49.

Nimmer, R.P., Bankert, R.J., Russel, E.S., Smith, G.A., and Wright, P.K. (1991), Micromechanical modeling of

fiber/matrix interface effects in transversely loaded SiC/Ti-6-4 metal matrix composites. J. Comp. Tech. Res.

13, 3.

NASA/TM--1998-208805 14
Paley, M. and Aboudi, J. (1992), Micromechanical analysis of composites by the generalized cells model. Mech.

Mater. 14, 127.

Pindera, M-J. and Aboudi, J. (1988), Micromechanical analysis of yielding of metal matrix composites. Int. J. Plasticity 4, 195.

Robinson, D.N. and Duffy, S.F. (1990), Continuum deformation theory for high-temperature metallic composites. J.

Eng. Mech. 116, 832.

Robinson, D.N. and Ellis, J.R. (1986), A multiaxial theory of viscoplasticity for isotropic materials. In Proc. Turbine

Engine Hot Section Technology, NASA CP_444, NASA Lewis Research Center, Cleveland, pp. 283-292.

Suquet, P. (1987), Elements of homogenization for inelastic solid mechanics. In Homogenization techniques for

Composite Materials', eds. E. Sanchez-Palencia and A. Zaoui, pp. 193-278. Springer-Verlag, Berlin.

Wilt, T.E. and Arnold, S.M. (1996) Micromechanics analysis code (MAC) User Guide: version 2.0.NASA

TM-107290, NASA Lewis Research Center, Cleveland.

TABLE I._OCATION AND DIRECTION OF APPLIED

NODAL FORCES

Axial, cy11

Transverse, (Y22

Yl'ansverse, (2_33

Shear, _12

Top edge

3 direction

Right edge

2 direction

1 direction

Conllllon node

1 direction

@ @ @ ® ®

@ @ @ ® @ @ @ @ ® ® ®

Rectangular Array Hexagonal Array Square Diagonal Array

Figure 1 .--Periodic microstrctures; (a) rectangular, (b) hexagonal, and

(c) square diagonal array.

1500

"_" 1000

500

B

/ SiC Ti

I o

IIIIIII /'___-/

650° C

0 i , J , I i , i , I , ,

0.00 0.01 0.02

Strain

Figure 2.--Predicted constituent response for SiC

fiber and Ti matrix.

NASA/TM--1998-208805 15
13-

or"

I..U

SiC/Ti at 650°C

Inelastic power, (TijE _.

/ _\

vol. avg. of local inelastic power, f(Tij_jl

.......... Dissipation, @Tij_l--o_ij;_ij}

0.04 -

/ j

0.03 Effect of residual

stress

Effect of internal ,,,-"'"'

/

0,02 stress //.L--'"

O.Ol y/..-

2" .'"

0.00

0.000 0.005 0.010

Transverse Strain (m/m)

Figure &--Inelastic power and dissipation for transverse

tensile loading.

(Y22 A

4

Figure 4.--Flow surface probes.

NASA/TM--1998-208805 16
Square array (coarse mesh)

N

Square diagonal array (coarse mesh)

Square array (fine mesh) Square diagonal array (fine mesh)

Figure 5.--Coarse and fine mesh FEA discretizations of the rectanglar, hexagonal,

and square diagonal microstructures.

NASA/TM-- 1998-208805 17
line ofT x3

symmetrvl u3 = constant

-'1

ul=O

ux =0

Discretized

domain

(1/4 of unit cell)

QCommon

node

J

Xl U3 = 0

n Ul = constant

symmetry

Figure 6.--Boundary conditions for FEA.

Figure 7.--GMC discretizations of the rectanglar, hexagonal, and square diagonal microstructures.

NASA/TM--1998-208805 1g
3OOO

2000

1000

0

0.00

- 0.02 -

O

D-

_= 0.01 o

13..

._

I I I I 0.00

0.01 0.02 0

Strain

Square

.......... Hexagonal

Square Diagonal

500 1000 1500

Time (sec)

13_

CD

CD

oO

800

400

0

0.00

- 0.40

(b) Transverse G"

13_

A

_= 0.20

o

13_

, I , I 0.00

0.01 0.02 0

Strain

i /,"

t /;

,_,' t Weak /,"

Bond ,/," ;// ,/ /

I ,' /,,"

I ,,' d,

200 400

Time (sec)

v

(D

(D

Of)

4OO

200

0 0.00 I

0.00 0 200

- (c) Shear 0.40

G"

_= 0.20

o

D_

o_

p- '--Weak Bond

' , I , I

0.01 0.02

Strain

V /

I i

___/_Weak'7'

J ! /,

I.."7 _j>-Strong

/// I 'end

,/t/

100

Time (sec)

Figure 8.--Uniaxial stress-strain and inelastic power-time responses at 650 °C for square, hexagonal, and

square diagonal arrays from GMC.

NASA/TM-- 1998-208805 19
13-

v

09

oO

G)

03

¢--

I3000 !4

-3000 "-r ,

-3000

i i i i ,,_

Square

Packing

I I I ]"

0 3000

Axial Stress (MPa)

13_

v

03

o_

09

Q)

09

¢-

3000 4 ' '

Square

. Diagonal

Packing °i<

-3000 _ .

-3000

, , 4-

_get Value,

(kPa/sec)

........ 1

5

i , 101"

0 3000

Axial Stress (MPa)

Figure 9.--Effect of target value on SCIPs at 23 °C in the axial-transverse stress plane for square and

square diagonal arrays discretized with fine (dark lines and o) and coarse (light lines and rn) FEA meshes.

1 kPa/se c

hi = 1.38E 6

Io = 0.232E 6

5 kPa/se c

hi = 1.53E 6

Io = 0.297E 6

10 kPa/se c

hi = 1.68E 6

Io = 0.482E 6

1 kPa/se c

hi = 1.48E 6

Io = 0.263E 6

5 kPa/se c

hi = 1.59E 6

Io = 0.322E 6

10 kPa/se

hi = 1.66E 6

Io = 0.386E 6

J2C°nt°urs iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii

(MPa)2 200,000 500,000 800,000 1,100,000 1,400,000 1,700,000

Figure 10.--Matrix J2 contours for a probe angle of 35° in the axial-transverse stress plane for square and

square diagonal arrays and SCIP target values of 1, 5, and 10 kPa/sec.

NASA/TM--1998-208805 20
13_

6000

4000

2000

(a) Target Value, ,,

"!, . kPa/sec i,.'

Square Array Square Diagonal Array

,

", .......... 10

""'---.. 1

I,:

Matrix Elements

2.0E+6

D_

1.0E+6

O.OE+O

(b)

,..S..quare Array ', Square Diagonal Array

\ : -, iI

" - X" ",k"

Matrix Elements

Figure 11.--Sorted distribution (maximum to minimum) of (a) 11and (b) J2 in the matrix for a probe

angle of 35° in the axial-transverse stress plane for square and square diagonal arrays and SCIP

target values of 1, 5, and 10 kPa/sec.

NASA/TM-- 1998-208805 21
_000F---------_ 2000_ ' 't

d

oi / J" 4._ ok ."" ._' - t e e_

u V/ ,// I _ / :' .4,

_-3000 -3000 _-t I , I l '

-3000 0 3000 -3000Transverse Stress,0 _MP 13000

Axial Stress, all (MPa) (_22 ,..._a,

Aspect Ratio, R

0.5

1.0

.......... 2.0

3000 -_ ' ' t ' ' _ 3000 -_ ' ' _ ' , 4-

J

a_ } TM

v

d 0 - 0 (

0.) ..,...,

•.-. o,)

m

-3000 t I I I I I -3000 i I I i i

-3000 0 3000 -3000TransverseStress,0 _MP _3000

Axial Stress, oll (MPa) (_22 ,..._a,

Figure 12.--Effect of aspect ratio on 5 kPa/sec SCIPs predicted by FEA (fine mesh) at 23 °C for a

rectangular array.

NASA/TM--1998-208805 22
3000

0

E

-3000

-3000

I , ' I ' ' J,: GMC

FEA (coarse mesh)

kPa/sec

1 kPa/se¢

t i I I i i 1

0 3000

Axial Stress (MPa)

3OOO

v

L..

I--

-3000

.j,. , /i IS _ _ 1

I_ J

-3000 0 3000

Transverse Stress (MPa)

3000 I

0

-3000

, ' I ' ' 4-

GMC

FEA (coarse mesh)

kPa/sec

1 kPa/sec

_" I I I I I

3000

0

I--

-3000

' ' I ' ' 4-

.#

t_."

I I I I I

-3000 0 3000 -3000 0 3000

Axial Stress (MPa) Transverse Stress (MPa)

Figure 13.--Comparison of I and 10 kPa/sec SCIPs predicted by GMC and FEA(coarse mesh) at

23 °C for a square array.

NASA/TM--1998-208805 23
3000 4 '

0

C

-3000 t i I I

-3000 0

AxialStress(MPa)

' I ' ' 4-

GMC

FEA (coarse mesh)

i 1

3000

3000 4

0

-3000

-3000

, , I ' i

I I I I I

0

Transverse Stress (MPa)

4-

1

3000

3000 4

-3000 r I

-3000

_3_

v

gO

0

O0

O0

' ' I ' ' 4.-

I I I I "

0 3000

Axial Stress (MPa)

3000 4 ' ' i ' '

-3000 t I I I I I

-3000 0

Transverse Stress(MPa)

3000

Figure 14.--Comparison of 10 kPa/sec SCIPs predicted by GMC and FEA (coarse mesh) at 650 °C

for a square array.

NASA/TM--1998-208805 24
2000

c_

0

t.--

_--2000

t } 'l , * I s , _ I

I

m ._,2000

r_

(gJ

0

t'--

N-.

-2000

i i !

s s t I

t

i •

/

o/O,

J

Weak Bond

.......... Strong Bond

I I t I I , , , I I , , , I , , , 1

-2000 0 2000 -2000 0 2000

Axial Stress (MPa) Transverse Stress (MPa)

20OO

13_

v

_O

0

u')

¢-

-2000

I ' ' ' I ' ' ' I

2000

c]._

v

(/)

0

o_

tot_

-2000

I ' ' ' I ' ' ' I

aj

I , , I I , , , I I i , , I , I , I

-2000 0 2000 -2000 0 2000

Axial Stress (MPa) Transverse Stress (MPa)

Figure 15.--Effect of debonding on 1 kPa/sec SCIPs predicted by GMC at 23 °C for a square array.

NASA/TM--1998-208805 25
O_

=E

u) 0

(D

(D

(D

F--

-400

400 11

Microstructure

Square

......... Square Diagonal

.... Hexagonal

-I , , , I , , , 1_

-400 0 400

Axial Stress (MPa)

(a) 35% Fiber

8OO

C)_

=Ev

_, 0

i i i I i i i

G.)

CO

I--

-800

-800

150

(3_

0

OO

Q

3=

O9

-150

' ' t ' ' I t

I , _ I , , I

-150 0 150

Transverse Stress (MPa)

2OO

(3-

v

ct_

0

O3

' -200

, h I , , , I , , , I , , a I

0 800 -200 0 200

Axial Stress (MPa) Transverse Stress (MPa)

(b) 50% Fiber

Figure 16.--Effect of microstructure and fiber volume fractron on 1 kPa/sec SCIPs

predicted by GMC at 650 °C given weak fiber-marix interface.

NASA/TM--1998-208805 26
3OOO

2ooo

1000

0

0.00

800 -

%-

4o0

El.-

ra9

(a) Axial

"G" 0.02

G_

i 0.01

o

i I i I 0.00

0.01 0.02

Strain

(b) Transverse 0.20

0.10

o

13-

._

¢9

C

0 I I I I 0.00

0.00 0.01 0.02

Strain

(c) Shear

G"

o

a..

.o_.

400 -

200

0.20

0.10

GMC

.......... FEA (Coarse)

FEA (Fine)

500 1000 1500

Time (sec)

0

0 I 0.00

0.00 0.01 0.02 0

Strain

..°

/j oo"

I

200 400

Time (sec)

/

/

_J" t I i

100 200

Time (sec)

Figure Bl.--Uniaxial stress-strain and inelastic power-time responses at 650 °C for a

square array using a coarse mesh FEA, a fine mesh FEA, and GMC.

NASA/TM--1998-208805 27
8OO

%-

(3-

400

r._

4000

D_

2000

CO

n

v

O9

400

- ! (a) Axial-Transverse ......

/ 0"11 = 0-22 ; "

ca

O

t_

ct_

0

0.00 0.01 0.02

Strain

(b) Transverse-Transverse,

G22 = 0"33 _"

g_

g_

o

0

0.00 0.01 0.02

Strain

I / (c) 0"11Axial-shear, = 0"12 _"

200 "'_""" 0.10

O

0 , I , I

0.00 0.01 0.02

Strain

0.20

0.10

0.00

0.02

0.01

0.00

0.20

0.00

GMC

FEA (Coarse)

FEA (Fine)

,//

25O 500

Time (sec)

1200 2400

Time (sec)

/

#

,,k

100 200

Time (sec)

Figure B2.--Biaxial stress-strain and inelastic power-time responses at 650 °C for a

square array using a coarse mesh FEA, a fine mesh FEA, and GMC.

NASA/TM--1998-208805 28
Form Approved REPORT DOCUMENTATION PAGE

OMB No. 0704-0188

Public reporting burden for this collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources,

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1. AGENCY USE ONLY (Leave blank) 2. REPORT DATE 3. REPORT TYPE AND DATES COVERED

November 1998 Technical Memorandum

4. TITLE AND SUBTITLE 5. FUNDING NUMBERS

Flow/Damage Surfaces for Fiber-Reinforced Metals Having Different

Periodic Microstructures

6. AUTHOR(S)

Cliff J. Lissenden, Steven M. Arnold, and Saiganesh K. Iyer

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)

National Aeronautics and Space Administration

Lewis Research Center

Cleveland, Ohio 44135- 3191

9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES)

National Aeronautics and Space Administration

Washington, DC 20546- 0001

WU-523-21-13-00

8. PERFORMING ORGANIZATION

REPORTNUMBER

E-11389

10. SPONSORING/MONITORING

AGENCY REPORT NUMBER

NASA TM--1998-208805

11. SUPPLEMENTARY NOTES

CliffJ. Lissenden and Saiganesh K. Iyer, Pennsylvania State University, Unive_ity Park, Pennsylvania 16802.

StevenM. Amold, NASA Lewis Research Center. Responsible person, StevenM. Arnold, organization code 5920,

(216) 433-3334.

12a. DISTRIBUTION/AVAILABILITY STATEMENT

Unclassified - Unlimited

Subject Categories: 24 and 39 Distribution: Nonstandard

This publication is available from the NASA Center for AeroSpace Information, (301) 6214)390.

12b. DISTRIBUTION CODE

13. ABSTRACT (Maximum 200 words)

Flow/damage surfaces can be defined in terms of stress, inelastic strain rate, and internal variables using a thermodynamics framework. A macroscale definition relevant to thermodynamics and usable in an experimental program is employed

to map out surfaces of constant inelastic power in various stress planes. The inelastic flow of a model silicon carbide/

titanium composite system having rectangular, hexagonal, and square diagonal fiber packing arrays subjected to biaxial

stresses is quantified by flow/damage surfaces that are determined numerically from micromechanics, using both finite

element analysis and the generalized method of cells. Residual stresses from processing are explicitly included and

damage in the form of fiber-matrix debonding under transverse tensile and/or shear loading is represented by a simple

interface model. The influence of microstructural architecture is largest whenever fiber-matrix debonding is not an issue;

for example in the presence of transverse compressive stresses. Additionally, as the fiber volume fraction increases, so

does the effect of microstrnctural architecture. With regard to the micromechanics analysis, the overall inelastic flow

predicted by the generalized method of cells is in excellent agreement with that predicted using a large number of

displacement-based finite elements.

14. SUBJECT TERMS

Metal matrix composites; Inelastic; Yield surfaces; Microstructure; Micromechanics

17. SECURITY CLASSIFICATION

OF REPORT

Unclassified

NSN 7540-01-280-5500

15. NUMBER OF PAGES

35

16. PRICE CODE

A03

18. SECURITY CLASSIFICATION 19. SECURITY CLASSIFICATION 20. LIMITATION OF ABSTRACT

OF THIS PAGE OF ABSTRACT

Unclassified Unclassified

Standard Form 298 (Rev. 2-89)

Prescribed by ANSI Std. Z39-1B

298-102