Evan J. Pineda and Brett A. Bednarcyk

Glenn Research Center, Cleveland, Ohio

Anthony M. Waas

University of Michigan, Ann Arbor, Michigan

Steven M. Arnold

Glenn Research Center, Cleveland, Ohio

Implementation of a Smeared Crack Band Model

in a Micromechanics Framework

NASA/TM—2012-217603

June 2012
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Evan J. Pineda and Brett A. Bednarcyk

Glenn Research Center, Cleveland, Ohio

Anthony M. Waas

University of Michigan, Ann Arbor, Michigan

Steven M. Arnold

Glenn Research Center, Cleveland, Ohio

Implementation of a Smeared Crack Band Model

in a Micromechanics Framework

NASA/TM—2012-217603

June 2012

National Aeronautics and

Space Administration

Glenn Research Center

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Level of Review: This material has been technically reviewed by technical management. 
Implementation of a Smeared Crack Band Model in 

a Micromechanics Framework 

Evan J. Pineda and Brett A. Bednarcyk 

National Aeronautics and Space Administration 

Glenn Research Center 

Cleveland, Ohio 44135 

Anthony M. Waas 

University of Michigan 

Ann Arbor, Michigan 48109 

Steven M. Arnold 

National Aeronautics and Space Administration 

Glenn Research Center 

Cleveland, Ohio 44135 

NASA/TM—2012-217603 1
1 Introduction

Micromechanics techniques can be employed to model the individual constituents

within the composite. Typically, a repeating unit cell (RUC) in the composite microstructure is identified and analysis is performed on that RUC assuming periodic

boundary conditions. The response of a point in a continuum is determined assuming an infinite array of the RUCs. However, representative volume element (RVE)

methodologies exist which incorporate applying non-periodic boundary conditions to

a subvolume that accurately represents the composite microstructure. The RVE is

meant to represent the actual microstructure of the continuum, and the size of the

features of the microstructure is preserved. Micromechanics can be utilized to provide

the homogenized composite stiffness, or they can be used to model damage and failure within the constituents. If utilized for the latter, the global mechanisms can arise

through the natural evolution and interaction of the mechanisms in the constituents

in the micromechanics model. Numerous micromechanical frameworks exist that encompass analytical, semi-analytical, and numerical techniques. An expansive review

of many micromechanics theories is given in Ref. [Aboudi et al. (2012)]. In this work

a discretization insensitive continuum damage model (CDM) is implemented within

the high-fidelity generalized method of cells (HFGMC) to model transverse cracking

and compressive shear banding in a unidirectional fiber-reinforced polymer matrix

composite (PMC).

The method of cells (MOC) developed by Aboudi (1991) discretized a rectangular composite RUC into four subvolumes, called subcells. One of the subcells was

occupied by the fiber material and the rest were occupied by the matrix. Linear

displacement fields were assumed in each of the subcells. Displacement and traction

continuity conditions were enforced, in an average integral sense, at the subcell interfaces, along with periodic boundary conditions at the RUC boundaries to derive

a set of equations that would yield a strain concentration matrix that could, in turn,

NASA/TM—2012-217603 2
be used to obtain the local subcell strains from the applied fields. Following determination of the subcell strains, the subcell stress are readily calculated using the

local constitutive laws, and volume averaging can be used to obtain the homogenized

thermomechanical properties of the composite. MOC was later extended to the generalized method of cells (GMC) by Paley and Aboudi (1992) which accommodated

any number of subcells and constituents in two-periodic directions. Aboudi (1995)

adapted the formulation to accommodate triply-periodic materials. Finally, Aboudi

et al. (2001) developed HFGMC which utilized second order displacement field approximations in the subcells, rather than linear. Haj-Ali and Aboudi (2009) showed

the local elastic field accuracy produced by HFGMC corresponded very well to FEM;

whereas, Aboudi et al. (2003) compared the accuracy of the fields in inelastic phases

to composite cylinder assemblage (CCA). Bednarcyk et al. (2004) utilized HFGMC to

model fiber-matrix debonding in metal matrix composites (MMCs), and Bednarcyk

et al. (2010) implemented a multi-axial damage model in HFGMC. Reformulations, of

GMC and HFGMC, which reduced the total number of unknowns in the problem were

introduced by Pindera and Bednarcyk (1997) and Aboudi et al. (2012), respectively.

The generality of the GMC and HFGMC formulations admit any constitutive

behavior at the subcell level. However if the response of the subcell material exhibits

post-peak strain softening the tangent stiffness tensor of the subcell loses positive

definiteness. This leads to pathologically mesh dependent behavior.

Baˇzant and Oh (1983) developed the smeared crack band approach which introduced a characteristic element length into the post-peak softening damage evolution

formulation. The tangent slope of the softening stress-strain curve was scaled by

the characteristic length to ensure that total strain energy release rate (SERR) upon

complete failure (i.e. zero stress) is always equal to the prescribed fracture toughness, regardless of the element size. In the original formulation, the band was always

oriented perpendicular to the direction of maximum principal stress; thus, the crack

NASA/TM—2012-217603 3
band always advanced under pure mode I. de Borst and Nauta (1985); Rots and

de Borst (1987) later reformulated the model to incorporate a fixed crack band that

evolved under mixed-mode conditions. Both formulations employ linear degradation

schemes. Later, Camanho et al. (2007) incorporated more sophisticated initiation

criteria to predict the onset of mixed-mode crack bands. All of these smeared crack

formulations assume linear elastic behavior up to the initiation of the crack band,

followed by immediate post-peak strain softening. However, Spencer (2002) coupled

pre-peak plasticity with crack band post-peak strain softening in modeling failure of

concrete.

A variation of the crack band model developed by Baˇzant and Oh (1983) for concrete structures is implemented here within the HFGMC micromechanics framework

to model mesh objective failure of continuous fiber-reinforced polymer matrix composites. The focus is restricted to the microscale to evaluate the capabilities of the

smeared crack band model to predict progressive failure evolution within a composite

microstructure. Thus, a detailed, multiple-fiber RUC is chosen demonstrate these

capabilities. HFGMC is utilized as the micromechanics platform because the level of

fidelity required to accurately model failure evolution in a complex microstructure is

not provided by GMC.

In Section 2 the formulation for HFGMC is given, followed by the implementation

of the smeared crack band model in Section 3. Two scenarios are considered to determine the mode in which the cracks within the crack band grow. If the principal stress

that has the highest magnitude is tensile, it is assumed that it is more energetically

favorable for the crack band to form perpendicular to the maximum principal stress

and for the cracks within the band to advance under mode I conditions. Conversely, if

the the magnitude of a compressive principal stress is higher than the other principal

stresses, the cracks within the crack band evolve under mode II conditions (due to

internal, Mohr-Coulomb friction) and are oriented with the plane of maximum shear

NASA/TM—2012-217603 4
stress. In Section 4 the objectivity of the post-peak softening behavior of the model

with respect to the subcell mesh is shown. An example is presented in Section 5

that consists of an RUC composed of 13, randomly placed fibers that is subjected to

transverse tension and compression. The model is verified against experimental data

in Section 6.

The motivation behind the following developments is to arrive at a failure methodology that is capable of accurately capturing localization in a multiscale model without

any spurious dependencies. This paper is intended to present results that verify a

model for such an application. In Section 7 a multiscale methodology is introduced

that ensures mesh objectivity and addresses the issues presented by Baˇzant (2007).

2 The High-Fidelity Generalized Method of Cells

HFGMC, first introduce by Aboudi et al. (1999) resolves some the shortcomings of

the original GMC. The biggest of which is the lack of normal-shear coupling in GMC

which results from the enforcement of displacement and traction continuity in an

average, integral sense and a linear approximation of the displacement fields in each

subcell. HFGMC employs quadratic displacement approximations. However, this

requires more equations than can be produced using displacement and traction continuity. Thus, the zeroth, first, and second moments of equilibrium are used to solve

the problem.

It is assumed that a composite microstructure can be represented as a collection of

triply periodic RUCs containing a general number of constituents, as shown in Figure

1. The RUC is then discretized into Nα x Nβ x Nγ rectangular, prismatic subcells, as

exhibited in Figure 2. Each of these subcells are occupied by one of the constituents in

the composite. The number of subcells and the materials occupying each subcell are

completely general. For a two-phase fibrous composite any desired micro-architecture

NASA/TM—2012-217603 5
can be represented by occupying each subcell with either a matrix or fiber constituent.

Since the microstructure does not vary along the axial, fiber-direction, a unidirectional

composite can be modeled using a doubly-periodic RUC (Figure 3), where the x2-x3

plane is discretized. Herein, doubly-periodic formulations of HFGMC [Aboudi et al.

(1999)] will be employed; the reader is referred to Ref. [Aboudi et al. (2012)] for the

fully 3D formulations, as well as a more efficient reformulation.

2.1 Governing Equations

The 3D equilibrium equations for subcell βγ (see Figure 4) in a doubly-periodic RUC

(Figure 3) are given by

∂1σ

(βγ)

1i + ∂2σ

(βγ)

2i + ∂3σ

(βγ)

3i = 0 (1)

Since there is no variation in the x1-direction for a doubly-periodic material, ∂1 =

∂/∂x1 = 0. Therefore Equation (1) becomes

∂2σ

(βγ)

2i + ∂3σ

(βγ)

3i = 0 (2)

Averaging Equation (2) over the volume of the subcell yields

J

(βγ)

2i(00) + K

(βγ)

3i(00) = 0 (3)

where

J

(βγ)

2i(00) =

1

hβ

h

(2)t

+(βγ)

i −

(2) t

−(βγ)

i

i

(4)

and

K

(βγ)

3i(00) =

1

lγ

h

(3)t

+(βγ)

i −

(3) t

−(βγ)

i

i

(5)

NASA/TM—2012-217603 6
The surface tractions are given by

(2)t

±(βγ)

i =

1

lγ

Z lγ/2

−lγ/2

σ

(βγ)

2i



x ̄

(β)

2 = ±

hβ

2



dx ̄

(γ)

3

(6)

(3)t

±(βγ)

i =

1

hβ

Z hβ/2

−hβ/2

σ

(βγ)

3i



x ̄

(γ)

3 = ±

lγ

2



dx ̄

(β)

2

(7)

where “(2)” and “(3)” are superscripts that indicate the applicable normal direction

to the subcell surface and “±” indicates the sign of the normal. Similarly, the first

moment of equilibrium can also be averaged over the subcell volume resulting in

J

(βγ)

2i(10) − S

(βγ)

2i(00) = 0 (8)

and

K

(βγ)

3i(01) − S

(βγ)

3i(00) = 0 (9)

where S

(βγ)

ij(mn)

is an average stress quantity, and

J

(βγ)

2i(10) =

1

2

h

(2)t

+(βγ)

i +

(2) t

−(βγ)

i

i

(10)

K

(βγ)

3i(01) =

1

2

h

(3)t

+(βγ)

i +

(3) t

−(βγ)

i

i

(11)

S

(βγ)

ij(mn) =

1

hβlγ

Z hβ/2

−hβ/2

Z lγ/2

−lγ/2

σ

(βγ)

ij ( ̄x

(β)

2

)

m( ̄x

(γ)

3

)

n

dx ̄

(β)

2 dx ̄

(γ)

3

(12)

Finally, the second moment of equilibrium can be averaged over the subcell volume,

then utilizing integration by parts leads to

h

2

β

4

J

(βγ)

2i(00) +

h

2

β

12

K

(βγ)

3i(00) − 2S

(βγ)

2i(10) = 0 (13)

and

l

2

γ

12

J

(βγ)

2i(00) +

l

2

γ

4

K

(βγ)

3i(00) − 2S

(βγ)

3i(01) = 0 (14)

NASA/TM—2012-217603 7
Substituting Equations (13) and (14) into Equation (3) gives

J

(βγ)

2i(00) =

12

h

2

β

S

(βγ)

2i(10) (15)

K

(βγ)

3i(00) =

12

l

2

γ

S

(βγ)

3i(01) (16)

Utilizing Equations (15) and (16) in Equation (14):

1

h

2

β

S

(βγ)

2i(10) +

1

l

2

γ

S

(βγ)

3i(01) = 0 (17)

Equation (17) represents the average form, in HFGMC, of the equilibrium equations,

Equation (2), of subcell βγ within a doubly-periodic composite RUC.

2.2 Second Order Displacement Approximation

Contrary to GMC, HFGMC employs a quadratic expansion of the subcell displacements

u

(βγ)

i =  ̄ijxj + W

(βγ)

i(00) +  ̄x

(β)

2 W

(βγ)

i(10) +  ̄x

(γ)

3 W

(βγ)

i(01)+

1

2



3 ̄x

(β)

2

2 −

h

2

β

4



W

(βγ)

i(20) +

1

2



3 ̄x

(γ)

2

3 −

l

2

γ

4



W

(βγ)

i(02)

(18)

where  ̄ij are, as in GMC, the average global strain components, and W

(βγ)

i(mn)

are

microvariables describing the spatial variation of the displacement field and are determined by enforcing interfacial and periodic conditions.

2.3 Constitutive Relations

The local strains in subcell βγ are related to the global strains and local displacement

fields through



(βγ)

ij =  ̄ij +

1

2



∂iu

(βγ)

j + ∂ju

(βγ)

i



(19)

NASA/TM—2012-217603 8
Employing the local constitutive law,

σ

(βγ)

ij = C

(βγ)

ijkl 

(βγ)

kl (20)

ignoring plastic and thermal effects (for HFGMC formulation with plasticity and

thermal strains please see Ref. [Aboudi et al. (2003)]) in Equation (12) along with

Equations (18) and (19), the relationship between zeroth order stress components

and the microvariables is determined to be:

S

(βγ)

11(00) = C

(βγ)

11  ̄11 + C

(βγ)

12 

W

(βγ)

2(10) +  ̄22

+ C

(βγ)

13 

W

(βγ)

3(01) +  ̄33

+ C

(βγ)

14 

W

(βγ)

2(01) + W

(βγ)

3(10) + 2 ̄23

+ C

(βγ)

15 

W

(βγ)

1(01) + 2 ̄13

+ C

(βγ)

16 

W

(βγ)

1(10) + 2 ̄12

(21)

S

(βγ)

22(00) = C

(βγ)

12  ̄11 + C

(βγ)

22 

W

(βγ)

2(10) +  ̄22

+ C

(βγ)

23 

W

(βγ)

3(01) +  ̄33

+ C

(βγ)

24 

W

(βγ)

2(01) + W

(βγ)

3(10) + 2 ̄23

+ C

(βγ)

25 

W

(βγ)

1(01) + 2 ̄13

+ C

(βγ)

26 

W

(βγ)

1(10) + 2 ̄12

(22)

S

(βγ)

33(00) = C

(βγ)

13  ̄11 + C

(βγ)

23 

W

(βγ)

2(10) +  ̄22

+ C

(βγ)

33 

W

(βγ)

3(01) +  ̄33

+ C

(βγ)

34 

W

(βγ)

2(01) + W

(βγ)

3(10) + 2 ̄23

+ C

(βγ)

35 

W

(βγ)

1(01) + 2 ̄13

+ C

(βγ)

36 

W

(βγ)

1(10) + 2 ̄12

(23)

S

(βγ)

44(00) = C

(βγ)

14  ̄11 + C

(βγ)

24 

W

(βγ)

2(10) +  ̄22

+ C

(βγ)

34 

W

(βγ)

3(01) +  ̄33

+ C

(βγ)

44 

W

(βγ)

2(01) + W

(βγ)

3(10) + 2 ̄23

+ C

(βγ)

45 

W

(βγ)

1(01) + 2 ̄13

+ C

(βγ)

46 

W

(βγ)

1(10) + 2 ̄12

(24)

NASA/TM—2012-217603 9
S

(βγ)

55(00) = C

(βγ)

15  ̄11 + C

(βγ)

25 

W

(βγ)

2(10) +  ̄22

+ C

(βγ)

35 

W

(βγ)

3(01) +  ̄33

+ C

(βγ)

45 

W

(βγ)

2(01) + W

(βγ)

3(10) + 2 ̄23

+ C

(βγ)

55 

W

(βγ)

1(01) + 2 ̄13

+ C

(βγ)

56 

W

(βγ)

1(10) + 2 ̄12

(25)

S

(βγ)

66(00) = C

(βγ)

16  ̄11 + C

(βγ)

26 

W

(βγ)

2(10) +  ̄22

+ C

(βγ)

36 

W

(βγ)

3(01) +  ̄33

+ C

(βγ)

46 

W

(βγ)

2(01) + W

(βγ)

3(10) + 2 ̄23

+ C

(βγ)

56 

W

(βγ)

1(01) + 2 ̄13

+ C

(βγ)

66 

W

(βγ)

1(10) + 2 ̄12

(26)

where the contracted notation for the stiffness tensor components C

(βγ)

ij is used. Similarly the first stress moments are

S

(βγ)

11(10) =

h

2

β

4

h

C

(βγ)

12 W

(βγ)

2(20) + C

(βγ)

14 W

(βγ)

3(20) + C

(βγ)

16 W

(βγ)

1(20)i

(27)

S

(βγ)

11(01) =

l

2

γ

4

h

C

(βγ)

13 W

(βγ)

3(02) + C

(βγ)

14 W

(βγ)

2(02) + C

(βγ)

15 W

(βγ)

1(02)i

(28)

S

(βγ)

22(10) =

h

2

β

4

h

C

(βγ)

22 W

(βγ)

2(20) + C

(βγ)

24 W

(βγ)

3(20) + C

(βγ)

26 W

(βγ)

1(20)i

(29)

S

(βγ)

22(01) =

l

2

γ

4

h

C

(βγ)

23 W

(βγ)

3(02) + C

(βγ)

24 W

(βγ)

2(02) + C

(βγ)

25 W

(βγ)

1(02)i

(30)

S

(βγ)

33(10) =

h

2

β

4

h

C

(βγ)

23 W

(βγ)

2(20) + C

(βγ)

34 W

(βγ)

3(20) + C

(βγ)

36 W

(βγ)

1(20)i

(31)

S

(βγ)

33(01) =

l

2

γ

4

h

C

(βγ)

33 W

(βγ)

3(02) + C

(βγ)

34 W

(βγ)

2(02) + C

(βγ)

35 W

(βγ)

1(02)i

(32)

S

(βγ)

44(10) =

h

2

β

4

h

C

(βγ)

24 W

(βγ)

2(20) + C

(βγ)

44 W

(βγ)

3(20) + C

(βγ)

46 W

(βγ)

1(20)i

(33)

S

(βγ)

55(01) =

l

2

γ

4

h

C

(βγ)

35 W

(βγ)

3(02) + C

(βγ)

45 W

(βγ)

2(02) + C

(βγ)

55 W

(βγ)

1(02)i

(34)

S

(βγ)

66(10) =

h

2

β

4

h

C

(βγ)

26 W

(βγ)

2(20) + C

(βγ)

46 W

(βγ)

3(20) + C

(βγ)

66 W

(βγ)

1(20)i

(35)

S

(βγ)

66(01) =

l

2

γ

4

h

C

(βγ)

36 W

(βγ)

3(02) + C

(βγ)

46 W

(βγ)

2(02) + C

(βγ)

56 W

(βγ)

1(02)i

(36)

NASA/TM—2012-217603 10
2.4 Displacement Continuity

The unknown microvariables (the volume-averaged displacement vector W

(βγ)

i(00) and the

higher order terms W

(βγ)

i(mn)

) can be calculated from the governing equation, Equation

(17), subcell interfacial continuity conditions, and periodicity conditions.

The periodic boundary conditions for an RUC are given by

ui|x2=0 = ui|x2=H(37)

σ2i|x2=0 = σ2i|x2=H(38)

ui|x3=0 = ui|x3=L(39)

σ3i|x3=0 = σ3i|x3=L(40)

The displacement continuity conditions are enforced, as in GMC, in an average (integral) sense, yielding:

Z lγ/2

−lγ/2



u

(1γ)

i







x ̄

(1)

2 =−h1/2



dx ̄

(γ)

3 =

Z lγ/2

−lγ/2



u

(Nβγ)

i







x ̄

(Nβ)

2 =hNβ

/2



dx ̄

(γ)

3

, γ = 1, . . . , Nγ

(41)

Z hβ/2

−hβ/2



u

(β1)

i







x ̄

(1)

3 =−l1/2



dx ̄

(β)

2 =

Z hβ/2

−hβ/2



u

(βNγ)

i







x ̄

(Nγ)

3 =lNγ

/2



dx ̄

(β)

2

, β = 1, . . . , Nβ

(42)

Using the definition of the subcell displacements, Equation (18), in the average displacement continuity conditions, Equations (41) and (42), gives:

W

(1γ)

i(00) −

h1

2

W

(1γ)

i(10) −

h

2

1

4

W

(1γ)

i(20) = W

(Nβγ)

i(00) −

hNβ

2

W

(Nβγ)

i(10) −

h

2

Nβ

4

W

(Nβγ)

i(20) , γ = 1, . . . , Nγ

(43)

W

(β1)

i(00) −

l1

2

W

(β1)

i(01) −

l

2

1

4

W

(β1)

i(02) = W

(βNγ)

i(00) −

lNγ

2

W

(βNγ)

i(01) −

l

2

Nγ

4

W

(βNγ)

i(02) , β = 1, . . . , Nβ

(44)

Equations (43) and (44) represent the displacement continuity conditions necessary

NASA/TM—2012-217603 11
to enforce periodicity of the RUC. Similar relations can be derived to enforce displacement continuity across adjacent subcell boundaries,

W

(βγ)

i(00) +

hβ

2

W

(βγ)

i(10) +

h

2

β

4

W

(βγ)

i(20) = W

(β+1γ)

i(00) +

hβ+1

2

W

(β+1γ)

i(10) +

h

2

β+1

4

W

(β+1γ)

i(20) ,

β = 1, . . . , Nβ − 1, γ = 1, . . . , Nγ

(45)

W

(βγ)

i(00) +

lγ

2

W

(βγ)

i(01) +

l

2

γ

4

W

(βγ)

i(02) = W

(βγ+1)

i(00) +

lγ+1

2

W

(βγ+1)

i(01) +

l

2

γ+1

4

W

(βγ+1)

i(02) ,

β = 1, . . . , Nβ, γ = 1, . . . , Nγ − 1

(46)

Note that Equations (43)-(44) can be relaxed to incorporate the effects of local,

interfacial debonding [Bednarcyk et al. (2004)].

2.5 Traction Continuity

Similarly using Equations (6) and (7), the average implementation of the stress periodicity conditions (Equations (38) and (40)) is

(2)t

+(1γ)

i =(2) t

+(Nβγ)

i

, γ = 1, . . . , Nγ

(47)

(3)t

+(β1)

i =(3) t

+(βNγ)

i

, β = 1, . . . , Nβ

(48)

Using Equations (4) and (5) with Equations (10) and (11) gives

(2)t

±(βγ)

i = J

(βγ)

2i(10) ±

hβ

2

J

(βγ)

2i(00), β = 1, . . . , Nβ, γ = 1, . . . , Nγ

(49)

(3)t

±(βγ)

i = K

(βγ)

3i(01) ±

lγ

2

K

(βγ)

3i(00), β = 1, . . . , Nβ, γ = 1, . . . , Nγ

(50)

Substituting Equations (8), (9), (15), and (16)

(2)t

±(βγ)

i = S

(βγ)

2i(00) ±

6

hβ

S

(βγ)

2i(10), β = 1, . . . , Nβ, γ = 1, . . . , Nγ (51)

NASA/TM—2012-217603 12
(3)t

±(βγ)

i = S

(βγ)

3i(00) ±

6

lγ

S

(βγ)

3i(01), β = 1, . . . , Nβ, γ = 1, . . . , Nγ (52)

Thus, the average stress periodicity conditions, Equations (47) and (48), become

S

(1γ)

2i(00) −

6

h1

S

(1γ)

2i(10) = S

(Nβγ)

2i(00) −

6

hNβ

S

(Nβγ)

2i(10) , γ = 1, . . . , Nγ (53)

S

(β1)

3i(00) −

6

l1

S

(β1)

3i(01) = S

(βNγ)

3i(00) −

6

lNγ

S

(βNγ)

3i(01) , β = 1, . . . , Nβ (54)

Equations (53) and (54) represent the stress continuity conditions at the boundaries

of the RUC necessary to enforce periodicity. Similarly, traction continuity between

adjacent subcells is given by:

S

(βγ)

2i(00) −

6

hβ

S

(βγ)

2i(10) = S

(β+1γ)

2i(00) −

6

hβ+1

S

(β+1γ)

2i(10) , β = 1, . . . , Nβ − 1, γ = 1, . . . , Nγ

(55)

S

(βγ)

3i(00) −

6

lγ

S

(βγ)

3i(01) = S

(βγ+1)

3i(00) −

6

lγ+1

S

(βγ+1)

3i(01) , β = 1, . . . , Nβ, γ = 1, . . . , Nγ − 1

(56)

2.6 Summary of Global Equations

The number of unknown microvariables in the displacement expansion, Equation (18),

is 15NβNγ. The governing equations, Equation (17) provide 3NβNγ relations for the

unknown microvariables while Equations (43) and (44) provide 3(Nβ + Nγ) relations,

Equations (45) and (46) provide 3[(Nβ − 1)Nγ + Nβ(Nγ − 1)] relations, Equations

(53) and (54) provide 3(Nβ + Nγ) relations, and Equations (55) and (56) provide

3[(Nβ −1)Nγ +Nβ(Nγ −1)] relations, which yields a total of 15NβNγ linear equations

for the unknown microvariables.

This system of equations can be expressed as

KU = f (57)

NASA/TM—2012-217603 13
where K, the structural stiffness matrix, contains geometric and mechanical constitutive information from each of the subcells in the RUC. The displacement vector U

contains the unknown displacement microvariables for each subcell.

U = [U(11), . . . , U(NβNγ)] (58)

and

U(βγ) = [W(βγ)

(00)

,W(βγ)

(10)

,W(βγ)

(01)

,W(βγ)

(20)

,W(βγ)

(02)

] (59)

where the vectors W(βγ)

(mn)

contain the three components of that displacement microvariable for that particular subcell

W(βγ)

(mn) =

h

W

(βγ)

1(mn)

, W(βγ)

2(mn)

, W(βγ)

3(mn)

i

(60)

The force vector f contains details of the applied average strains  ̄ij . To prevent

rigid body motion, the displacement of the corners of the RUC are pinned and those

equations are eliminated from Equation (57).

Solving Equation (57) yields a strain concentration matrix that relates the local,

average subcell strains to the global, applied, average strains.



(βγ) = A

(βγ)

HF  ̄ (61)

Once the local strains are obtained, the local stresses are readily determined through

the local constitutive laws, Equation (20). The average global stresses are simply the

volume average of the local stresses.

σ ̄ =

1

HL

X

Nβ

β=1

X

Nγ

γ=1

hβlγσ ̄

(βγ)

(62)

where σ ̄

(βγ) are the average subcell stresses.

NASA/TM—2012-217603 14
The global, composite constitutive relationship is taken to be

σ ̄ = C

∗

 ̄ (63)

Thus, the effective elastic stiffness matrix for the composite is given by substituting

Equations (61) and (62) into Equation (63).

C

∗ =

1

HL

X

Nβ

β=1

X

Nγ

γ=1

hβlγC

(βγ)A

(βγ)

HF (64)

It should be noted that HFGMC was reformulated using the displacement continuity

conditions to reduce the overall number of unknown microvariables, thus reducing

computer memory requirements and computational cost. The reader is referred to

[Bansal and Pindera (2004); Aboudi et al. (2012)] for details on this reformulation.

3 Modeling Constituent-Level Post-Peak Strain

Softening with the Smeared Crack Band Approach

HFGMC is an efficient (relative to fully numerical methods), useful tool for modeling

details of the microstructure of a composite material. Additionally, it is readily

amenable for implementation into a multiscale framework. Although, physics-based,

discretization objective, progressive failure constitutive models must be in place for

the constituents of the composite to accurately predict the response of a structure

that is a damaging. For pre-peak loading (i.e. positive-definite tangent stiffness

tensor), there are a multitude of non-linear elasticity, plasticity, continuum damage

mechanics, and visoelastic/plastic theories available that can predict the evolution of

the appropriate mechanisms in the composite. However when the local fields enter

NASA/TM—2012-217603 15
the post-peak regime of the stress-strain laws, most of these theories breakdown in

a numerical setting and display pathological mesh dependence [Baˇzant and Cedolin

(1979); Pietruszczak and Mroz (1981)].

The lack of positive definiteness of the elastic, or inelastic, tangent stiffness tensor

leads to imaginary wave speeds in the material. The longitudinal wave speed in an

isotropic material is given by

cL =

s

E(1 − ν)

ρ(1 + ν)(1 − 2ν)

(65)

where cL is the wave velocity, E is the Young’s modulus of the material, ν is the Poisson’s ratio, and ρ is the material density. A one-dimensional approximation yields

v =

qE

ρ

. The existence of an imaginary wave speed results in a boundary value

problem that is ill-posed Baˇzant and Cedolin (1979); Pietruszczak and Mroz (1981);

deBorst (1987); Baˇzant and Cedolin (1991). Physically, a material must posses a

positive-definite tangent stiffness tensor, and in fact, at the micro-scale the material

tangent stiffness tensor always remains positive-definite. However for practical purposes, engineers must model structures at scales much larger than the flaws in the

material, and the homogenized continuum representation of a material with nucleation

and propagation of discontinuities, such as cracks or voids, exhibits post-peak strain

softening in the macroscopic, homogenized, stress-strain response. This homogenized

response is assumed to govern over a suitable volume of the material, appropriate

to the microstructure of the material. Loss of positive-definiteness of the tangent

stiffness tensor leads to a material instability which manifests as a localization of

damage into the smallest length scale in the continuum problem[Baˇzant and Cedolin

(1991)]. In GMC or HFGMC this is a single subcell. Thus, the post-peak softening

strain energy is dissipated over the volume of the subcell that the damage localizes

to. Since a stress-strain relationship prescribes the energy density dissipated during

NASA/TM—2012-217603 16
the failure process, the total amount of energy dissipated in the subcell decreases as

the size of the element is reduced, and in the limit zero energy is required to fail the

structure.

A simple way to remedy this non-physical behavior in a numerical setting is to

judiciously scale the post-peak softening slope of the stress-strain constitutive law.

Then, the failure energy density dissipated becomes a function of the characteristic

length of discretized continuum. Baˇzant (1982); Baˇzant and Oh (1983) first proposed a crack band model in which post-peak softening damage (herein referred to

as failure) in the material was assumed to occur within a band. The post-peak slope

of the material constitutive law was scaled by the characteristic length of the finite

element exhibiting failure; such that, the total strain energy release rate in the element, upon reaching a state of zero stress, and the material fracture toughness were

coincident. In this reference, equivalence between this smeared crack approach and a

line crack approach is presented. Subsequently, Baˇzant and Cedolin (1983) exhibited

propagation of a crack band not aligned with the mesh bias. In this work, the crack

band model is implemented within the HFGMC framework, in the MAC/GMC suite

of micromechanics codes developed at the NASA Glenn Research Center [Bednarcyk

and Arnold (2002a,b)], and used to analyze crack band growth in composite RUCs.

The following subsections provide theoretical details on the crack band model.

3.1 Physical Behavior of Crack band

The smeared crack band model is meant to capture the behavior of a region of a

material wherein numerous microcracks have initiated and they coalesce to form a

larger crack. Figure 6 displays a crack band of width wc embedded in a continuum.

The domain of the crack band is denoted as Ω0 and the remaining continuum as Ω.

The crack band is oriented within the continuum such that, for a given point within

the crack band, the unit vector normal to the crack band is n.

NASA/TM—2012-217603 17
The total energy dissipated during the failure process is dissipated over Ω0

, and the

size wc of Ω0is a material property directly related to the material fracture toughness

[Baˇzant and Oh (1983)].

wc =

2GC

σ

2

C



1

E

−

1

ET

−1

(66)

where σC is the critical stress for initiation of the post-peak regime in the 1D material

stress-strain law (given in Figure 7a), and ET is the negative tangent slope in that

regime. The fracture toughness GC, or critical strain energy release rate, of the

material is given by the area under the 1D traction-separation law (Figure 7b) which

governs the cohesive response of the separation of crack faces as a crack propagates

in the material. The energy density dissipated during failure WF is related to the the

material fracture toughness by the characteristic length in the material.

GC = wcWF (67)

3.2 Numerical Implementation of Crack Band Model in

GMC/HFGMC

The crack band model is implemented in the HFGMC micromechanical framework.

The local subcell fields are used to govern crack brand evolution in the constituents of

the composite. Figure 8 shows the discretization of the continuum displayed in Figure

6. A magnified view of the crack band embedded in a single subcell is also displayed

in Figure 8. Since the all of the energy dissipated in the crack band is smeared over

the subcell volume, the subcell must be large enough to contain the crack band of

width wC. Note that Figure 8 shows a 2D geometry for illustrative purposes, but the

crack bands can also evolve in a general 3D space.

NASA/TM—2012-217603 18
3.2.1 Mode I Crack Band Growth Under Tensile Principal Stress

The orientation of the crack band in subcell βγ is given by the vector n

(βγ)

1

(see Figure 8) and is determined from the local principal stress state ( ̄σ

(βγ)

1

, σ ̄

(βγ)

2

, σ ̄

(βγ)

3

). In a

monolithic material, cracks orient such that the crack tips are always subjected to pure

mode I (opening mode) conditions unless there are constraints that limit the crack

orientation. In a micromechanical analysis the composite material is composed of separate, monolithic constituents; thus, in the matrix, the crack band runs perpendicular

to  ̄σ

(βγ)

1

, the principal stress with the largest magnitude, |σ ̄

(βγ)

1

| > |σ ̄

(βγ)

2

| > |σ ̄

(βγ)

3

|, if

σ ̄

(βγ)

1 ≥ 0 (tensile). Under these conditions, a crack oriented as such, is subjected to

pure mode I loading, locally. Although, the resulting global behavior may appear to

be mixed mode because of the influence of the fibers on the matrix crack band path.

Crack orientation and evolution is determined differently if  ̄σ

(βγ)

1 < 0 as described in

3.2.2. The characteristic length of the subcell l

(βγ)

C

is determined as the dimension

of the subcell running parallel to n

(βγ)

1

. Crack band initiation is determined using a

very simple, but physical, maximum stress criterion.

σ ̄

(βγ)

1

σ

(βγ)

C

= 1, σ ̄

(βγ)

1 ≥ 0 (68)

where σ

(βγ)

C

is the cohesive strength of the crack band. Once the crack band has

initiated, the crack band orientation is fixed as time evolves.

Once the crack band orientation has been calculated, the subcell compliance is

rotated into the principal frame using the transformation matrix.

T = [n

(βγ)

1 n

(βγ)

2 n

(βγ)

3

][e1e2e3] (69)

where n

(βγ)

1

, n

(βγ)

2

, and n

(βγ)

3 are the principal stress directions, and e1, e2, and e3

are the unit basis vectors. All material degradation due to crack band evolution is

NASA/TM—2012-217603 19
imposed on the rotated compliance S ̄(βγ)

, the components of which are given by:

S ̄

(βγ)

ijkl = TpiTqjS

(βγ)

pqrsTkrTls (70)

The strain energy released during the formation of new surfaces corresponding

to the growth of cracks within the crack band is assumed to be dissipated over the

entire subcell volume. Therefore, the post-peak softening slope E

(βγ)

IT , and the strain

at which a the principal stress state is zero, is calculated using the characteristic

length of the subcell l

(βγ)

C

and the material fracture toughness G

(βγ)

IC (see Figure 7a).



(βγ)

F =

2G

(βγ)

IC

σ

(βγ)

C

l

(βγ)

C

(71)

E

(βγ)

IT =

 

1

E ̄

110

−



(βγ)

F

σ

(βγ)

C

!−1

(72)

where E ̄

110 is the undamaged, axial Young’s modulus in the principal frame. It should

be noted that E

(βγ)

IT must be less than zero; therefore, by Equations (71) and (72), a

restriction is placed on the maximum allowable subcell size.

l

(βγ)

C <

2G

(βγ)

IC E ̄

110

σ

(βγ)

C

(73)

The local, rotated, subcell strain state  ̄

(βγ)

i







 ̄

(βγ)

1

 ̄

(βγ)

2

 ̄

(βγ)

3







=







T

(βγ)

1i

 ̄

(βγ)

ij T

(βγ)

1j

T

(βγ)

2i

 ̄

(βγ)

ij T

(βγ)

2j

T

(βγ)

3i

 ̄

(βγ)

ij T

(βγ)

3j







(74)

is used to degrade the rotated compliance components. The scalar damage factor

NASA/TM—2012-217603 20
D(βγ)

is calculated using the rotated strain corresponding to  ̄σ

(βγ)

1

.

D

(βγ) = 1 +

E

(βγ)

IT 



(βγ)

C −  ̄

(βγ)

1



E ̄

110 ̄

(βγ)

1

(75)

where 

(βγ)

C

is the value of  ̄

(βγ)

1 when the initiation criterion, Equation (68), is satisfied.

If D(βγ)is less than zero, no damage occurs, and a maximum damage level of one

corresponds to a zero stress state on the softening stress-strain curve. Also, damage

healing is inadmissable.

D ̇

(βγ) ≥ 0 (76)

Components of the rotated compliance matrix are degraded with the damage factor.

S ̄(βγ) =







































































S ̄0

(βγ)

1111

(1 − D(βγ))

S ̄

(βγ)

1122 S ̄

(βγ)

1133 0 0 0

S ̄

(βγ)

1122 S ̄

(βγ)

2222 S ̄

(βγ)

2233 0 0 0

S ̄

(βγ)

1133 S ̄

(βγ)

2233 S ̄

(βγ)

3333 0 0 0

0 0 0 S ̄

(βγ)

2323 0 0

0 0 0 0

S ̄0

(βγ)

1313

(1 − D(βγ))

0

0 0 0 0 0

S ̄0

(βγ)

1212

(1 − D(βγ))







































































(77)

Since the crack band orientation is fixed upon initiation, the S ̄

(βγ)

1313 and S ̄

(βγ)

1212 shear

compliances in the rotated frame are degraded, as well as the S ̄

(βγ)

1111 compliance, so

that the crack band faces normal to n

(βγ)

1 are free of normal and shear tractions when

NASA/TM—2012-217603 21
all of the crack band energy has been dissipated (i.e. l

(βγ)

C W

(βγ)

F = G

(βγ)

IC ). Once the

compliance in the rotated frame is degraded, the compliance is transformed back to

the global frame to yield the new subcell compliance.

S

(βγ)

ijkl = T

−1

pi T

−1

qj S ̄(βγ)

pqrsT

−1

kr T

−1

ls (78)

Note that, damage introduced in the principal frame, through Equation (77), can

induce normal-shear coupling in the global frame.

3.2.2 Mode II Crack Band Growth Under Compressive Principal Stress

Crack band growth under pure mode I conditions are energetically favorable; however,

conditions arise, under which, mode I crack growth is not possible. For instance,

the crack cannot grow under in mode I if the normal traction at the crack tip is

compressive. This occurs when the principal stress with the maximum magnitude is

compressive ( ̄σ

(βγ)

1 < 0). However, experimental data shows that monolithic materials

subjected to global compression will eventually fracture.

In brittle and quasi-brittle materials, it has been shown that local, internal friction

results in shear (mode II) fracture when the monolithic material is subjected to compressive stresses [Hoek and Bieniawski (1965); Horii and Nemat-Nasser (1986); Ashby

and Sammis (1990); Chen and Ravichandran (2000)]. Consequently, if  ̄σ

(βγ)

1 < 0 it

is assumed that the crack band is aligned with plane of maximum shear stress  ̄τ

(βγ)

in the matrix subcell and a Mohr-Coulomb failure criterion will be used to indicate

crack band initiation under maximum principal compression.

τ ̄

(βγ)

E

τ

(βγ)

C

= 1, σ ̄

(βγ)

1 < 0 (79)

where τ

(βγ)

C

is the cohesive shear strength of the matrix, and  ̄τ

(βγ)

E

is an effective shear

NASA/TM—2012-217603 22
stress that includes the influence of the traction normal to the crack band  ̄σn

(βγ)

.

τ ̄

(βγ)

E = |τ ̄

(βγ)

| + μiσ ̄n

(βγ)

(80)

where μiis the internal friction coefficient and must lie between 0 < μi ≤ 1.5 to obtain physically reasonable surface friction coefficient [Chen and Ravichandran (2000)].

Since the principal stress with the largest magnitude is compressive, the traction normal to the maximum shear stress plane  ̄σn

(βγ) must also be compressive. Hence, in

Equation (80), an increase in the normal compressive stress will yield a reduction in

the effective shear stress and an increase in apparent shear strength. Taliercio and

Sagramoso (1995) derived relationships between internal friction coefficient, shear

strength, the tensile strength and compressive strength of the material, yielding an

expression for μiin terms of σ

(βγ)

C

and τ

(βγ)

C

.

μi = tan







sin−1









2τ

(βγ)

C

2

− σ

(βγ)

C

2

(2 ̄τ

(βγ)

)

2 + σ

(βγ)

C

2













(81)

Similar criteria are used for shear failure under compressive loading in homogenized composite materials [Puck and Sch ̈urmann (1998, 2002); Pinho et al. (2005)].

In these theories, the orientation of the crack is not aligned with the plane of maximum shear stress because of the influence of the fibers on the crack path, but rather,

orientation is an input to the theories obtained from experimental data. Since the

influence of the fibers are explicitly accounted for in the present methodology, the

crack may remain oriented with the maximum shear stress plane and the local stress

in the matrix will drive the crack path.

Subsequent to mode II crack band initiation via Equation (79), the compliance

tensor of the subcell is rotated into the maximum shear stress frame using Equations

(69) and (70), where n

(βγ)

1

is a unit-vector perpendicular to the plane of maximum

NASA/TM—2012-217603 23
shear stress. Features of the post-peak softening shear stress-shear strain curve, such

as the endpoint and post-peak tangent stiffness, can be calculated from the mode II

fracture parameters.

γ

(βγ)

F =

2G

(βγ)

IC

τ

(βγ)

C

l

(βγ)

C

(82)

E

(βγ)

IIT =

 

1

G ̄

120

−

γ

(βγ)

F

τ

(βγ)

C

!−1

(83)

where G ̄

120 is the undamaged, axial shear modulus in the maximum shear stress

frame. Again the tangent stiffness E

(βγ)

IIT must be less than zero; therefore, Equations

(82) and (83) place a restriction is on the maximum allowable subcell size.

l

(βγ)

C <

2G

(βγ)

IC G ̄

120

τ

(βγ)

C

(84)

The shear strain corresponding to the maximum shear stress  ̄γ

(βγ)

is obtained by

rotating the strain tensor.

γ ̄

(βγ) = T

(βγ)

1i

 ̄

(βγ)

ij T

(βγ)

2j

(85)

An effective shear strain  ̄γ

(βγ)

E

can be defined that is work conjugate with the effective

shear stress tEbg.

τ ̄

(βγ)

E

dγ ̄

(βγ)

E =  ̄τ

(βγ)

dγ ̄

(βγ) +  ̄σn(βγ)

d ̄n

(βγ)

(86)

where  ̄τ

(βγ) and  ̄σn(βγ) are the shear and normal tractions acting on the crack faces

oriented parallel to the plane of maximum shear stress, and  ̄γ

(βγ) and  ̄n(βγ) are the

apparent shear and normal strains with respect to the maximum shear orientation

in the subcell, including the effects of crack tip opening displacement. Taking the

derivative of Equation (86) with respect to  ̄τ

(βγ) yields

∂τ ̄

(βγ)

E

∂τ ̄

(βγ)

dγ ̄

(βγ)

E = dγ ̄

(βγ)

(87)

NASA/TM—2012-217603 24
and along with the derivative of Equation (80) with respect to  ̄τ

(βγ)

∂τ ̄

(βγ)

E

∂τ ̄

(βγ)

= 1 (88)

finally, after integrating, gives:

γ ̄

(βγ)

E =  ̄γ

(βγ)

(89)

which states that the shear strain in the rotated, maximum shear stress frame and

the effective shear strain that is work conjugate to the effective, Mohr-Coulomb shear

stress are equivalent.

Thus,  ̄γ

(βγ)

is used to degrade the rotated compliance components. The damage

factor D(βγ)is calculated using  ̄γ.

D

(βγ) = 1 +

E

(βγ)

IIT 

γ

(βγ)

C − γ ̄

(βγ)



G ̄

120γ ̄

(βγ)

(90)

where γ

(βγ)

C

is the value of  ̄γ

(βγ) when the initiation criterion, Equation (79), is satisfied. In the case of mode II fracture, only the shear moduli are degraded. The normal

NASA/TM—2012-217603 25
direction is subjected to compression, and thus, retains its stiffness.

S ̄(βγ) =





































































S ̄

(βγ)

1111 S ̄

(βγ)

1122 S ̄

(βγ)

1133 0 0 0

S ̄

(βγ)

1122 S ̄

(βγ)

2222 S ̄

(βγ)

2233 0 0 0

S ̄

(βγ)

1133 S ̄

(βγ)

2233 S ̄

(βγ)

3333 0 0 0

0 0 0 S ̄

(βγ)

2323 0 0

0 0 0 0

S ̄0

(βγ)

1313

(1 − D(βγ))

0

0 0 0 0 0

S ̄0

(βγ)

1212

(1 − D(βγ))





































































(91)

Then, the compliance tensor is rotated back to the global frame using Equation (78).

4 Subcell Mesh Objectivity

The key advantage of using the smeared crack band approach is that it has been verified to provide mesh objective results within an FEM setting; this can also be said for

implementation of this model within HFGMC. To illustrate this, a uniform, uniaxial,

tensile displacement u was applied to monolithic, doubly-periodic, unit square RUCs

in the x3-direction. As shown in Figure 9, the levels of subcell refinement ranged

from: 35 subcells x 35 subcells, 45 subcells x 45 subcells, 55 subcells x 55 subcells, to

65 subcells x 65 subcells. The elastic properties were chosen to match data for in-situ

MY750/HY917/DY063 epoxy matrix [Hinton et al. (2004)]. The fracture properties

were chosen such that the problem would demonstrate significant post-peak energy

dissipation. All properties are given in Table 1. A single subcell, colored red in Figure

NASA/TM—2012-217603 26
9, was given a 10% lower initiation strain then the rest of the domain. Additionally,

only the subcells that lie along a horizontal line running through the weaker subcell

were allowed to fail because as the damaged region grows, the opposite ends of the

damaged region would interact with each other (due to periodicity), as a result of

the periodic boundary conditions, altering the damage path and resulting in damage

growth that is not self-similar. Restricting the damage path ensures that only subcell

dimensions would influence the results from case to case.

The resulting load P in the x3-direction is plotted in Figure 10 against the applied

displacement for the four different levels of subcell refinement. It is clear that the total

strain energy release rate dissipated in the system, and the ultimate load, is insensitive

to the dimension of the subcell size. Of course, in more complicated RUCs, a certain

level of subcell refinement is necessary to capture the local fields accurately; however,

that behavior is not pathological subcell mesh dependence because the solution is

bounded.

5 Details of Numerical Models

The smeared crack band model, presented in Section 3.2, is implemented within

HFGMC, and it is used to investigated the progression of post-peak softening damage within a unidirectional fiber-reinforced composite RUC. The main objective was

to capture the failure evolution of a composite RUC under transverse tension and compression. In tension, the lamina is susceptible to transverse cracking, as evidenced by

the scanning electron microscope (SEM) image of a unidirectional composite failed

under tension in Figure 11. The behavior of this mechanism is observed to be quite

brittle, as is evidenced by tension tests on 90◦coupon laminates. However, in compression a network of shear bands develop, accompanied by some matrix cracking

and fiber-matrix debonding (see the SEM image in Figure 12) and the stress-strain

NASA/TM—2012-217603 27
response if much more ductile. To emulate these failure modes, a doubly-periodic

HFGMC RUC of the 2-3 plane of the composite was created (see Figure 13). This

RUC was subjected to global tensile and compressive, transverse, uniaxial strains:

± ̄22. The constitutive behavior of the matrix subcells follows linear elasticity, coupled with the crack band model for post-peak strain softening formulated earlier.

This model is verified against experimental data in Section 6.

A representation of the 2-3 plane of a typical composite lamina (where the 1-axis

runs in the longitudinal direction of the fibers and the 2- and 3-axes span the plane of

transverse isotropy) was produced to examine the evolution of transverse cracks within

the composite. The blueprint for the discrete model is created by randomly placing

thirteen circular fibers with a diameter of 5 μm in a square box. The domain was

then discretized into a number of subcells. The considered formulation of HFGMC

only admits rectangular subcells in doubly-periodic models (parallelepiped in triplyperiodic models); so, some of the fiber domains would overlap or lie directly adjacent

to one another. An isoparametric formulation of HFGMC has been recently developed

by Haj-Ali and Aboudi (2010). Mesh objective, subcell interfacial debonding has been

formulated for GMC (see Pineda et al. (2010a,b)) but has not been fully developed

within the HFGMC framework. As a result, thin matrix channels were inserted

between any adjacent fibers to avoid arrest of the crack band for non-physical reasons

such as the inability of two adjacent fiber to separate. The final architecture and

discretization is shown in Figure 13, which contains 81 subcells x 85 subcells. The blue

subcells indicate fiber material and the green represent the matrix. The dimensions

of the RUC (21.25 μm x 21.25 μm) were chosen such that the final fiber volume

fraction in Figure 13 would be preserved at 58%, corresponding to the experimental

data provided in Hinton et al. (2004).

The elastic properties for a Silenka E-glass/MY750/HY917/DY063 lamina were

taken from data from the world wide failure exercise WWFE [Hinton et al. (2004)].

NASA/TM—2012-217603 28
Elastic properties for Silenka E-glass fiber and MY750/HY917/DY063 epoxy matrix

used are given in Table 2. Note that the matrix properties were correlated to represent

the in-situ properties of the matrix, which differ significantly from those of the neat

material [Ng et al. (2010)], and match the global properties of the composite. As a

result of this calibration, the isotropic relationship between the shear modulus and the

Young’s modulus and Poisson’s ratio of the matrix is not maintained. The fracture

properties (also Table 2) were calibrated to produce and ultimate transverse tensile

stress of 40 MPa, and the compressive properties were calibrated to match transverse

compression data, reported in Hinton et al. (2004).

6 Results

6.1 Transverse Tension

A comparison between the tensile results obtained from the HFGMC model and the

experiment is provided in Figure 14. Only one data point is given in Hinton et al.

(2004), implying that the tensile response is linear until ultimate failure. The first

peak in the  ̄σ22- ̄22 obtained from HFGMC at  ̄σ22 = 39.76 MPa and  ̄22 = 0.00244

was calibrated to match the experimental data, and thus, correlates well. This is

expected since the failure parameters were calibrated to match the experimental data.

Subsequent to the first peak, the model exhibits a sudden drop in the global stress;

however, the RUC then continues to reload in a nonlinear manner. A second peak is

achieved at an applied transverse strain of 0.00308 and a transverse stress of 37.97

MPa, which is slightly less than the stress at the first peak. After the second peak,

the load carrying capability of the RUC is severely diminished.

The evolution of the tensile crack band in the RUC is presented in Figure 15. At

σ ̄22 = 37.84 MPa a crack band initiates near the top of the RUC between two fibers

(please refer back to Figure 13 for the fiber-matrix architecture). At the ultimate

NASA/TM—2012-217603 29
stress 39.76 MPa, the initial crack band has grown significantly into a fully developed

crack band that propagated across the periodic boundary into the bottom of the

RUC. The tips of the initial crack band are arrested in lower stress, matrix rich

regions. At  ̄σ22 = 35.09 MPa, after the ultimate stress has been achieved and the

RUC is reloading, A new crack band initiates between two adjacent fibers. When the

second peak is reached, the subcells composing the second crack band have degraded

significantly, and the original crack band has extended further. As the stress drops,

Figure 15e, a third crack band initiates. The newest crack band develops rapidly, and

the second crack band becomes stagnant. When the RUC has completely failed and

can no longer sustain any load, the first and third crack band have bridged to form a

nearly continuous crack that has maneuvered around the fiber inclusions and spans

the entire height of the RUC. This failure path closely resembles the SEM image of a

transverse crack in the glass/epoxy composite shown in Figure 11b, and although the

model does not account for fiber-matrix debonding, the model exhibits appreciable

matrix failure in matrix subcells adjacent to fiber subcells.

6.2 Transverse Compression

The same HFGMC RUC is loaded in transverse compression and the stress-strain

response is compared to the experimental data in Figure 16. The nonlinear behavior

of the model and experiment correlate well until the model exhibits catastrophic

failure at -104.8 MPa, which is well below the reported ultimate compressive strength

of 145 MPa.

Investigations into the failure progression within the RUC reveal the cause of

the premature ultimate failure. Figure 17 shows the progression of mode II failure

(DC), resulting from maximum, compressive principal stresses (left column), mode I

failure (DT ), resulting from maximum, tensile principal stress (middle column), and

the superposition of mode I and mode II failure (DT + DC, right column) as the

NASA/TM—2012-217603 30
RUC is loaded in global, transverse compression. The first failure initiation occurs

between two fibers near the top of the RUC under mode II conditions at a global

stress of -65.68 MPa. As the applied strain increases, many mode II crack bands

form in angular matrix regions between closely packed fibers. This is exhibited in

Figures 17b and 17c. When the global stress reaches -92.88 MPa, a mode I crack

band initiates in the top-right corner of the RUC. At the ultimate stress, Figure

17f both the mode I and mode II crack band have evolved. In Figure 17g, shortly

after the ultimate stress was reached, a network of angular mode II crack bands has

formed. These mode II crack bands are bridged by a horizontal, mode I crack band

forming one continuous crack band that has progressed throughout the RUC. When

the global stress has dropped severely, in Figure 17h, there is substantial matrix

degradation resulting from mode II crack band propagation. Furthermore, one fiber

is completely surrounded by crack bands. Comparing Figure 17h to an SEM image

of a compressively failed carbon fiber/epoxy laminate in Figure 12b, displays closely

resembling failure patterns, indicating that the qualitative failure mode was captured

accurately, despite the quantitative discrepancy.

The development of mode I crack bands corresponded to a drop in global stress

shortly thereafter. Thus, to determine if the mode I crack bands were responsible

for the premature failure of the RUC, the simulation was re-executed but mode I

crack band growth was prohibited. The global  ̄σ22- ̄22 response is presented in Figure

16, along with the previous results and experimental data. It can be observed that

ultimate failure is delayed considerably until the  ̄σ22 = -146.4 MPa, which corresponds

well to the strength reported in Ref. [Hinton et al. (2004)] although the ultimate

failure strain 16% higher. Moreover, the majority of the stress-strain response of the

RUC matches the experimental data. This indicates that, in the model, the mode II

crack bands are responsible for the observed nonlinearity; whereas, the mode I crack

bands are responsible for the early onset of ultimate failure, as the mode I fracture

NASA/TM—2012-217603 31
toughness is an order of magnitude lower than mode II.

Extending this observation to the behavior of actual composite would suggest

that its non-linearity is controlled by shear band evolution, and the ultimate failure

is due to progression of tensile cracks and fiber-matrix debonding. It may be that

the size of the RUC (i.e. number of fibers included in the RUC) was too small to

get an accurate representation of the quantitative response of the composite under

compression, and the growth of the tensile crack band made an unrealistically large

impact on the response of the RUC. Further studies evaluating the influence of fibermatrix architecture on the compressive response of composites needs to be performed.

Additionally, adjacent constraining plies with different local architectures, present in

the experiment, may have helped to diminish the effect of formation of tensile cracks,

which is an effect not captured in this analysis.

The mode II crack band progression, produced by the simulation in which mode

I crack band growth was disabled, is displayed in Figure 18. Prior to the initiation

of the mode I crack band in Figure 17, Figures 17 and 18 are identical. However as

the stress continues beyond the ultimate stress observed in the original simulation,

more angular, distributed, mode II crack bands arise, as shown in Figure 18i at the

ultimate stress of the simulation with mode I crack band preclusion. Upon ultimate

failure, the expansive array of mode II crack bands is almost entirely adjoined and

nearly all of the matrix subcells in the RUC have failed. The SEM image, Figure 12b,

does not display such extensive failure. This further supports that the failure mode

predicted with both mode I and mode II crack bands is more physically correct, and

other factors are influencing the discrepancy between the response of the model and

the experiment.

NASA/TM—2012-217603 32
7 Conclusions

A mesh objective, smeared crack band model was implemented at the matrix subcell

level within the HFGMC micromechanics framework. Mode I crack bands were allowed to propagate normal to the maximum principal stress when the principal stress

component with the maximum magnitude was tensile, and the associated strain exceeded a critical value. If the principal stress component with the maximum magnitude was compressive, however, it was assumed that mode I cracks within the crack

band could not evolve, and instead, they grew in mode II as a result of Mohr-Coulomb

friction, upon satisfaction of a Mohr-Coulomb failure criterion. The mode II crack

bands were aligned with the plane of maximum shear stress. Although, the damage

model and framework are verified using a polymer matrix composite as an example,

the model is applicable to any brittle or quasi-brittle material including ceramics or

concrete.

An RUC containing 13, randomly distributed glass fibers in epoxy matrix was

simulated under global transverse tension and compression. The results for both

cases were compared to experimental data. The tensile response correlated extremely

well with the test results. Although, the failure parameters were calibrated to match

the experimental data, the successful correlation verifies that the smeared crack band

model, utilized within HFGMC, can capture the evolution of brittle transverse cracks.

Furthermore, the predicted failure mode mirrored SEM images of composites failed in

tension. Under compression, the micromechanics model predicted failure well below

the reported compressive strength. This was attributed to the formation of tensile,

mode I crack bands in the matrix which exhibit very low fracture toughness. A

simulation wherein mode I crack band evolution was restricted, provided quantitative

results that more closely agreed with the experiment. However, the failure mode

exhibited by the simulation allowing for both mode I and mode II crack bands more

closely resembled an SEM image of a compressively failed composite compared to

NASA/TM—2012-217603 33
the latter simulation. This indicates that a sensitivity study on the size and level of

refinement of the RUC should be enacted. Future studies will determine the number

of fibers in the RUC required to achieve convergence of the stiffness and the failure

progression.

Additionally, the response of the RUC to applied intralaminar shear strain  ̄γ12 was

not enacted. When subjected to in-plane shear loading numerous microcracks form

between fibers along the fiber direction in the composite laminae [Ng et al. (2010)].

This 3-D effect results in the composite exhibiting a very ductile response in shear.

To capture these 3-D, geometric effects with the proposed model, a very complex,

3-D RUC is required.

The RUC examples that were provided are far too computationally expensive for

a multiscale analysis. However, insight gained from the micromechanics analysis can

be utilized to formulate traction-separation laws, or to postulate fracture toughnesses

that include the geometric effects and can be implemented into simpler, more computationally feasible RUCs. To preserve the stress-strain response a consistent strain

energy density must exist across the scales. However, the mesh objectivity gained by

utilizing the smeared crack band model at the microscale is lost if the SERR is not

also preserved across the scales. To accomplish this, a consistent length must be used

at both scales. Thus, the volume of the microscale RUC and the volume associated

with the corresponding macroscale integration point must be equivalent.

NASA/TM—2012-217603 34
Property Value

E (GPa) 3.7

ν 0.35

C 0.0135

C (weak) 0.01215

GC (N/mm) 750

Table 1: Elastic and fracture properties used in mesh objectivity study.

NASA/TM—2012-217603 35
Property Value

Ef (GPa) 74.000

νf 0.200

Em (GPa) 4.65

νm 0.350

Gm 1.4



(βγ)

C

0.0167

γ

(βγ)

C

0.0256

G

(βγ)

IC (N/mm) 0.00076

G

(βγ)

IC (N/mm) 0.00435

Table 2: Elastic and fracture properties used in RUC simulation of 2-3 plane of a

Silenka E-Glass/MY750/HY917/DY063 composite lamina.

NASA/TM—2012-217603 36
Figure 1: Representation of triply-periodic microstructure of a composite material

Bednarcyk et al. (2010).

Figure 2: Discretization of a 3D, triply-periodic RUC Bednarcyk et al. (2010).

Figure 3: Discretization of a 2D, doubly-periodic RUC.

NASA/TM—2012-217603 37
Figure 4: 2D GMC/HFGMC subcell with local coordinate frame.

Figure 5: Effects of mesh refinement on overall load-displacement response for a

material exhibiting post-peak softening. Dashed lines indicate non-physical snapback.

NASA/TM—2012-217603 38
Figure 6: Crack band domain Ω0 of width wc oriented normal to vector n within a

continuum Ω.

(a) (b)

Figure 7: Stress-strain (a) and traction-separation (b) laws governing material behavior. Failure energy density WF is related to fracture toughness GC through the

characteristic length.

NASA/TM—2012-217603 39
Figure 8: Crack band embedded in discretized continuum. Magnified subcell displays

crack band orientation within subcell as well as characteristic length of subcell.

(a) 35 subcells x 35 subcells.(b) 45 subcells x 45 subcells.

(c) 55 subcells x 55 subcells.(d) 65 subcells x 65 subcells.

Figure 9: Four subcell meshes used in mesh objectivity study. Red subcell was given

a lower crack band initiation strain than others.

NASA/TM—2012-217603 40
Figure 10: Load versus displacement for four different subcell meshes. Total strain

energy release rate and ultimate load are unaffected by mesh refinement.

(a) Multiple transverse cracks. [Roberts

(2000)].

(b) Magnification of a single transverse

crack. [Gamstedt and Sj ̈ogren (1999)].

Figure 11: Transverse cracks in glass/epoxy laminates.

NASA/TM—2012-217603 41
(a) Macroscopic failure [Aragon ́es (2007)]. (b) Localized shear bands and cracks

[Gonzal ́ez and Llorca (2007)].

Figure 12: Transverse compressive failure of carbon fiber/epoxy laminates. Macroscopic failure path observed at an angle. Magnification illustrates shear banding,

cracking and fiber-matrix debonding.

Figure 13: HFGMC mesh used in simulations of RUC in 2-3 plane of lamina containing 13 fibers and a fiber volume fraction of 58%. Subcells/elements occupied by

fibers are colored blue, and subcells/elements occupied by matrix are colored green.

NASA/TM—2012-217603 42
Figure 14: Transverse tensile stress versus transverse strain of E-glass/Epoxy composite from experiment compared to HFGMC micromechanics model.

NASA/TM—2012-217603 43
(a) 22 = 0.00232, σ22 = 37.84 MPa. (b) 22 = 0.00248, σ22 = 39.76 MPa (First peak

stress).

(c) 22 = 0.00268, σ22 = 35.09 MPa. (d) 22 = 0.00300, σ22 = 38.35 MPa (Second peak

stress).

(e) 22 = 0.00308, σ22 = 37.88 MPa. (f) 22 = 0.00316, σ22 = 32.68 MPa.

Figure 15: Evolution of mode I crack band in RUC subjected to applied transverse

tensile strain.

NASA/TM—2012-217603 44
(g) 22 = 0.00336, σ22 = 4.11 MPa.

Figure 15: Evolution of mode I crack band in RUC subjected to applied transverse

tensile strain.

Figure 16: Compressive Transverse stress versus transverse strain of E-glass/Epoxy

composite from experiment compared to HFGMC micromechanics model.

NASA/TM—2012-217603 45
(a) 22 = -0.00403, σ22 = -65.68 MPa.

(b) 22 = -0.00490, σ22 = -79.66 MPa.

(c) 22 = -0.00543, σ22 = -87.71 MPa.

(d) 22 = -0.00578, σ22 = -92.88 MPa.

Figure 17: Evolution of mode I (left), mode II (middle), and superposition of mode I

and mode II (right) crack band in RUC subjected to applied transverse compressive

strain.

NASA/TM—2012-217603 46
(e) 22 = -0.00630, σ22 = -99.56 MPa.

(f) 22 = -0.00683, σ22 = -104.78 MPa (ultimate stress).

(g) 22 = -0.00770, σ22 = -96.37 MPa.

(h) 22 = -0.01750, σ22 = -16.06 MPa.

Figure 17: Evolution of mode I (left), mode II (middle), and superposition of mode I

and mode II (right) crack band in RUC subjected to applied transverse compressive

strain.

NASA/TM—2012-217603 47
(a) 22 = -0.00403, σ22 = -65.68 MPa. (b) 22 = -0.00420, σ22 = -68.53 MPa.

(c) 22 = -0.00490, σ22 = -79.66 MPa. (d) 22 = -0.00543, σ22 = -87.71 MPa.

(e) 22 = -0.00578, σ22 = -92.88 MPa. (f) 22 = -0.00630, σ22 = -100.15 MPa.

Figure 18: Evolution of mode II crack band in RUC subjected to applied transverse

compressive strain if mode I crack band evolution is supressed.

NASA/TM—2012-217603 48
(g) 22 = -0.00683, σ22 = -106.75 MPa. (h) 22 = -0.00770, σ22 = -116.36 MPa.

(i) 22 = -0.01420, σ22 = -146.43 MPa (ultimate

stress).

(j) 22 = -0.01750, σ22 = -82.87 MPa.

Figure 18: Evolution of mode II crack band in RUC subjected to applied transverse

compressive strain if mode I crack band evolution is supressed.

NASA/TM—2012-217603 49
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NASA/TM—2012-217603 52
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4. TITLE AND SUBTITLE

Implementation of a Smeared Crack Band Model in a Micromechanics Framework 

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Pineda, Evan, J.; Bednarcyk, Brett, A.; Waas, Anthony, M.; Arnold, Steven, M. 

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14. ABSTRACT

The smeared crack band theory is implemented within the generalized method of cells and high-fidelity generalized method of cells 

micromechanics models to capture progressive failure within the constituents of a composite material while retaining objectivity with 

respect to the size of the discretization elements used in the model. An repeating unit cell containing 13 randomly arranged fibers is modeled 

and subjected to a combination of transverse tension/compression and transverse shear loading. The implementation is verified against 

experimental data (where available), and an equivalent finite element model utilizing the same implementation of the crack band theory. To 

evaluate the performance of the crack band theory within a repeating unit cell that is more amenable to a multiscale implementation, a single 

fiber is modeled with generalized method of cells and high-fidelity generalized method of cells using a relatively coarse subcell mesh which 

is subjected to the same loading scenarios as the multiple fiber repeating unit cell. The generalized method of cells and high-fidelity 

generalized method of cells models are validated against a very refined finite element model.

15. SUBJECT TERMS

Fiber composites; Polymer matrix composites; Carbon fiber reinforced plastics; Computational mechanics; Damage; Fracture 

mechanics; Micromechanics 

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