computational damage mechanics for composite materials based on mathematical homogenization

*Correspondence to: Jacob Fish, Department of Civil Engineering, Rensselaer Polytechnic Institute, Troy, NY 12180,U.S.A. E-mail: "[email protected]

Contract/grant sponsor: Air Force O$ce of Scienti"c Research; Contract/grant number: F49620-97-1-0090Contract/grant sponsor: National Science Foundation; Contract/grant number: CMS 9712227Contract/grant sponsor: Sandia National Laboratories; Contract/grant number: AX 8516

CCC 0029-5981/99/231657}23$17.50 Received 7 August 1998Copyright ( 1999 John Wiley & Sons, Ltd. Revised 4 December 1998

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING

Int. J. Numer. Meth. Engng. 45, 1657}1679 (1999)

COMPUTATIONAL DAMAGE MECHANICSFOR COMPOSITE MATERIALS BASED ON

MATHEMATICAL HOMOGENIZATION

JACOB FISH*, QING YU AND KAMLUN SHEK

Departments of Civil, Mechanical and Aerospace Engineering, Rensselaer Polytechnic Institute, ¹roy, N> 12180, ;.S.A.

SUMMARY

This paper is aimed at developing a non-local theory for obtaining numerical approximation to a boundaryvalue problem describing damage phenomena in a brittle composite material. The mathematical homogen-ization method based on double-scale asymptotic expansion is generalized to account for damage e!ects inheterogeneous media. A closed-form expression relating local "elds to the overall strain and damage isderived. Non-local damage theory is developed by introducing the concept of non-local phase "elds (stress,strain, free energy density, damage release rate, etc.) in a manner analogous to that currently practiced inconcrete [1, 2], with the only exception being that the weight functions are taken to be C0 continuous overa single phase and zero elsewhere. Numerical results of our model were found to be in good agreement withexperimental data of 4-point bend test conducted on composite beam made of BlackglasTM/Nextel 5-harnesssatin weave. Copyright ( 1999 John Wiley & Sons, Ltd.

KEY WORDS: damage; composites; homogenization; non-local; asymptotic

1. INTRODUCTION

Damage in composite materials occurs through di!erent mechanisms that are complex andusually involve interaction between microconstituents. During the past two decades, a number ofmodels have been developed to simulate damage and failure process in composite materials,among which the damage mechanics approach is particularly attractive in the sense that itprovides a viable framework for the description of distributed damage including material sti!nessdegradation, initiation, growth and coalescence of microcracks and voids. Various damagemodels for brittle composites can be classi"ed into micromechanical and macromechanicalapproaches. In the macromechanical damage approach, composite material is idealized (orhomogenized) as an anisotropic homogeneous medium and damage is introduced via internalvariable whose tensorial nature depends on assumptions about crack orientation [3}9]. The

micromechanical damage approach, on the other hand, treats each microphase as a statisticallyhomogeneous medium. Local damage variables are de"ned to represent the state of damage ineach phase and phase e!ective material properties are de"ned thereafter. The overall response issubsequently obtained by homogenization [10}14].

From the mathematical formulation stand point, both approaches can be viewed as a two-stepprocedure. The main di!erence between the two approaches is in the chronological order inwhich the homogenization and evolution of damage are carried out. In the macromechanicalapproach, homogenization is performed "rst followed by application of damage mechanicsprinciples to homogenized anisotropic medium, while in the micromechanical approach, damagemechanics is applied to each phase followed by homogenization.

The primary objective of the present manuscript is to simultaneously carry out the two steps(homogenization and evolution of damage) by extending the framework of the classical math-ematical homogenization theory [15}17] to account for damage e!ects. This is accomplished byintroducing a double-scale asymptotic expansion of damage parameter (or damage tensor ingeneral). This leads to the derivation of the closed-form expression relating local "elds to overallstrains and damage (Section 2). The second salient feature of our approach is in developinga non-local theory by introducing the concept of non-local phase "elds (stress, strain, free energydensity, damage release rate, etc.) in Section 3. Non-local phase "elds are de"ned as weightedaverages over each phase in the characteristic volume in a manner analogous to that currentlypracticed in concrete [1, 2] with the only exception being that the weight functions are taken to beC0 continuous over a single phase and zero elsewhere. On the global (macro) level we limit the"nite element size to ensure a valid use of the mathematical homogenization theory and to limitlocalization. In Sections 4 and 5 we develop a mathematical and numerical model for the case ofpiecewise constant weight function, which is the simplest variant of the model presented inSection 3. The stress update procedure and the consistent tangent sti!ness matrix are thenderived. Section 6 compares the results of our numerical model to the experimental data. Weconsider a 4-point bend test conducted on the composite beam made to BlackglasTM/Nextel5-harness satin weave and compare our numerical simulations to experiments conducted atRutgers University [18].

2. MATHEMATICAL HOMOGENIZATION FOR DAMAGED COMPOSITES

In this section we extend the classical mathematical homogenization theory [15] for statisticallyhomogeneous composite media to account for damage e!ects. The strain-based continuumdamage theory is adopted for constructing constitutive relations at the level of microconstituents.Closed-form expressions of local strain and stress "elds in a multi-phase composite medium arederived. Attention is restricted to small deformations.

The microstructure of a composite material is assumed to be locally periodic (>-periodic) witha period de"ned by a Statistically Homogeneous Volume Element (SHVE), denoted by #, asshown in Figure 1. Let x be a macroscopic coordinate vector in macro domain ) and y,x/1 bea microscopic position vector in #. Here, 1 denotes a very small positive number compared withthe dimension of ), and y,x/1 is regarded as a stretched co-ordinate vector in the microscopicdomain. When a solid is subjected to some load and boundary conditions, the resultingdeformation, stresses, and internal variables may vary from point to point within the SHVE dueto the high level of heterogeneity. We assume that all quantities have two explicit dependencies:

1658 J. FISH, Q. YU AND K. SHEK

Copyright ( 1999 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng. 45, 1657}1679 (1999)

Figure 1. Macroscopic and microscopic structures

one on the macroscopic level x, and the other one on the level of microconstituents y,x/1. Forany >-periodic response function f, we have f (x, y)"f (x, y#ky; ) in which vector y; is the basicperiod of the microstructure and k is a 3]3 diagonal matrix with integer components. Adoptingthe classical nomenclature, any >-periodic function f can be represented as

f 1 (x),f (x, y (x)) (1)

where superscript 1 denotes a >-periodic function f. The indirect macroscopic spatial derivativesfor f 1 can be calculated by the chain rule as

f 1

,xi(x)"f

,xi(x, y)#

1

1f,yi

(x, y) (2)

where the comma followed by a subscript variable xidenotes a partial derivative with respect to

the subscript variable (i.e. f,xi,Lf/Lx

i). Summation convention for repeated subscripts is em-

ployed, except for subscripts x and y.The constitutive equation on the microscale is derived from continuum damage theory based

on the thermodynamics of irreversible processes and internal state variable theory. To model theisotropic damage process, we de"ne a scalar damage parameter u1 as a function of microscopicand macroscopic position vectors, i.e., u1

"u(x, y).Based on the strain-based continuum damage theory, the free energy density has the form of

( (u1, e1

ij)"(1!u1 )(

%(e1

ij) (3)

where u13[0, 1) is the damage parameter. For small deformations, elastic free energy density is

given as (%(e1

ij)"1

2¸ijkl

e1

ije1

kl. The constitutive equation, thermodynamic force (also known as

a damage energy release rate) and dissipative inequality follow from (3).

p1

ij"

L((u1, e1

ij)

Le1

ij

"(1!u1 )¸ijkl

e1

kl(4)

>"L((u1, e1

ij)

Lu1 "(%(e1 ) (5)

>uR 1*0 (6)

With this brief glimpse into the constitutive theory, we proceed to outlining the strong form of thegoverning di!erential equations on the "ne scale*the scale of microconstituents. Further detailson the evolution of damage are given in Section 4.

COMPUTATIONAL DAMAGE MECHANICS FOR COMPOSITE MATERIALS 1659


We assume that microconstituents possess homogeneous properties and satisfy equilibrium,constitutive, kinematics and compatibility equations. The corresponding boundary value prob-lem is governed by the following set of equations:

p1

ij ; xj#b

i"0 in ) (7)

p1

ij"(1!u1 )¸

ijkle1

klin ) (8)

e1

ij"u 1

(i,xj)in ) (9)

u1

i"uN

ion !

u(10)

p1

ijnj"tN

ion !

i(11)

where u1 is a scalar damage parameter; p1

ijand e1

ijare components of stress and strain tensors:

¸ijkl

represents components of elastic sti!ness satisfying conditions of symmetry

¸ijkl

"¸jikl

"¸ijlk

"¸klij

(12)

and positivity

&C0'0, ¸

ijklm1

ijm1

kl*C

0m1

ijm1

ij∀m1

ij"m 1

ji(13)

biis a body force assumed to be independent of y; u1

idenotes the components of the displacement

vector; the subscript pairs with parentheses denote the symmetric gradients de"ned as

u1

(i,xj),1

2(u1

(i,xj)#u1

j,xi) (14)

) denotes the macroscopic domain of interest with boundary !; !uand !

tare boundary portions

where displacements uNiand tractions tN

iare prescribed, respectively, such that !

uW!

t"0 and

!"!uX!

t; n

idenotes the normal vector on !. We assume that the interface between the phases is

perfectly bonded, i.e. [p1

ijnLj]"0 and [u1

i]"0 at the interface, !

*/5, where nL

iis the normal vector

to !*/5

and [f] is a jump operator.Clearly, a brute force approach attempting discretization of the entire macrodomain with

a grid spacing comparable to that of the microscale features is not computationally feasible. Thus,a mathematical homogenization method based on the double-scale asymptotic expansion isemployed to account for microstructural e!ects on the macroscopic response without explicitlyrepresenting the details of the microstructure in the global analysis. As a starting point, weapproximate the displacement "eld, u1

i(x)"u

i(x, y), and the damage parameter, u1(x)"u(x, y),

in terms of double-scale asymptotic expansions on )]#:

ui(x, y)+u0

i(x, y)#1u1

i(x, y)#. . . (15)

u(x, y)+u0(x, y)#1u1(x, y)#. . . (16)

Strain expansions on the composite domain )]# can be obtained by substituting (15) into (9)with consideration of the indirect di!erentiation rule (2)

eij(x, y)+

1

1e~1ij

(x, y)#e0ij(x, y)#1e1

ij(x, y)#. . . (17)

where strain components for various orders of 1 are given as

e~1ij

"eyij

(u0), esij"e

xij(us)#e

yij(us`1), s"0, 1, . . . (18)



and

exij

(us)"us(i,xj)

, eyij

(us)"us(i,yj)

(19)

Stresses and strains for di!erent orders of 1 are related by the constitutive equation (8)

p~1ij

"(1!u0)¸ijkl

e~1kl

(20)

psij"(1!u0)¸

ijkleskl#

s+r/0

us~r`1¸ijkl

er~1kl

, s"0, 1, . . . (21)

The resulting asymptotic expansion of stress is given as

pij(x, y)+

1

1p~1ij

(x, y)#p0ij(x, y)#1p~1

ij(x, y)#. . . (22)

Inserting the stress expansion (22) into equilibrium equation (7) and making use of equation (2)yield the following equilibrium equations for various orders:

O (1~2): p~1ij,yj

"0 (23)

O(1~1): p~1ij,xj

#p0ij,yj

"0 (24)

O(10) : p0ij,xj

#p1ij,yj

#bi"0 (25)

O(1 s) : psij,xj

#ps`1ij,yj

"0, s"1, 2, . . . (26)

We consider of O(1~2) equilibrium equation (23) "rst. Pre-multiplying it by u0i

and integratingover # yields

P#u0ip~1ij,yj

d#"0 (27)

and subsequently integrating by parts gives

P!#u0ip~1ij

njd!#!P!#

(1!u0)u0(i,yj)

¸ijkl

u0(k,yl)

d#"0 (28)

where !# denotes the boundary of #. The boundary integral term in (28) vanishes due to>-periodicity on !# , and hence, with the positivity of ¸

ijkland the assumption of u03[0, 1) (see

Section 3), we have

eyij

(u0)"u0(i,yj)

"0 N u0i"u0

i(x) (29)

and

p~1ij

(x, y)"e~1ij

(x, y)"0 (30)

We proceed to the O(1~1) equilibrium equation (24). From (18) and (20) follow

M(1!u0)¸ijkl

(exkl

(u0)#eykl

(u1))N,yj"0 on # (31)

To solve for (31) up to a constant we introduce the following separation of variables:

u1i(x, y)"H

ikl(y)Me

xkl(u0)#du

kl(x)N (32)



where Hikl

is a >-periodic function. We assume that dukl(x) is macroscopic damage-induced strain

driven by the macroscopic strain eNkl,e

xkl(u0). More speci"cally we can state that if eN

kl"0, then

dukl(x)"0 and u0(x, y)"0. Note that vice versa is not true, i.e. if du

kl(x)"0 or u0 (x, y)"0, the

macroscopic strain eNkl

may not be necessarily zero. In (32) both Hikl

and dukl

are symmetric withrespect to indices k and l.

Based on the decomposition given in (32), the O (1~1) equilibrium takes the following form:

M(1!u0)¸ijkl

[(Iklmn

#Gklmn

)exmn

(u0)#Gklmn

d0mn

(x)]N,yj"0 in # (33)

where

Iklmn

"12

(dmk

dnl#d

nkdml

), Gklmn

(y)"H(k,yl)mn

(y) (34)

and dmk

is the Kronecker delta, while Gklmn

is known as a polarization function. It can be shownthat the integrals of the polarization functions in # vanish due to periodicity conditions. Sinceequation (33) should be valid for arbitrary macroscopic "elds, we may "rst consider the case ofdukl(x)"0 and (u0"0) but eN

klO0, which yields the following equation in #:

M¸ijkl

(Iklmn

#H(k,yl)mn

)N,yj"0 (35)

Equation (35) together with the >-periodic boundary conditions is a linear boundary valueproblem in #. By exploiting the symmetry with respect to the indexes (m, n), the weak form of (35)is solved for three right-hand side vectors in 2-D and six right-hand side vectors in 3-D (see forexample [17, 19]).

In an absence of damage, the asymptotic expansion of strain (17) can be expressed in terms ofthe macroscopic strain eN

ijas follows:

eij"A

ijkleNkl#O(1) (36)

where Aijkl

is termed as the elastic strain concentration function de"ned as

Aijkl

"Iijkl

#Gijkl

(37)

The elastic homogenized sti!ness 1̧ijkl

follows from the O (10) equilibrium equation [20]:

1̧ijkl

,

1

D#D P#¸ijmn

Amnkl

d#"

1

D#D P#A

mnij¸

mnstA

stkld# (38)

where D#D is the volume of a SHVE.After solving (35) for H

imn, we proceed to "nd du

mnfrom (33). Premultiplying it by H

istand

integrating it by parts with consideration of >-periodic boundary conditions yields

P#(1!u0)G

ijst¸ijkl

(Aklmn

exmn

(u0)#Gklmn

dumn

(x)) d#"0 (39)

from where the expression of the macroscopic damaged induced strain can be shown to be

dumn

(x)"!GP# (1!u0)Gijst

¸ijkl

Gklmn

d#H~1

GP# (1!u0)Gijst

¸ijkl

Aklmn

d#HeNmn

(40)



Let (3 ,Mt(g)(y)Nn1

be a set of C~1 continuous functions, then the damage parameter u0 (x, y) isassumed to have the following decomposition:

u0(x, y)"n+g/1

t(g) (y)u(g)(x) (41)

where t(g)(y) is a damage distribution function on the microscale. Rewriting (40) in terms of strainconcentration function A

ijkland manipulating it with (38) and (41) yield

dumn

(x)"Dklmn

(x)eNmn

(42)

where

Dklmn

(x)"AIklst!n+g/1

B(g)klst

u(g)(x)B~1

An+g/1

C(g)stmn

u(g)(x)B (43)

B(g)ijkl

"

1

D#D( 3̧

ijmn! 1̧

ijmn)~1 P#

t(g)Gstmn

¸stpq

Gpqkl

d# (44)

C(g)ijkl

"

1

D#D( 3̧

ijmn! 1̧

ijmn)~1 P#

t(g)Gstmn

¸stpq

Apqkl

d# (45)

3̧ijmn

"

1

D#D P#¸ijmn

d# (46)

In conjunction with (32) and (42), the asymptotic expansion of strain "eld (17) can be "nallycast as

eij(x, y)"A

ijmn(y)eN

mn(x)#G

ijkl(y)D

klmn(x)eN

mn(x)#O(1) (47)

where Gijkl

(y) can be interpreted as a damage strain in#uence function. Note that the asymptoticexpansion of the strain "eld is given as a sum of mechanical "elds induced by the macroscopicstrain via elastic strain concentration function and thermodynamical "elds governed by damage-induced strain, du

kl(x)"D

klmn(x)eN

mn(x), through the damage strain in#uence function.

Finally, we integrate the O(10) equilibrium equation (25) over #. The :#p1ij,yj

d# term vanishesdue to periodicity and we obtain

A1

D#D P#p0ij

d#B,xj

#bi"0 in ) (48)

Substituting the constitutive relation (20) and the asymptotic expansion of the strain "eld (47) into(48) yields the macroscopic equilibrium equation

A1

D#D P#(1!u0)¸

ijkl(A

klmneNmn#G

klmndumn

) d#B,xj

#bi"0 (49)

If we de"ne the macroscopic stress pNij

as

pNij,

1

D#D P#p0ij

d# (50)

then the equilibrium equations (48) and (49) can be recast into more familiar form

pNij,xj

#bi"0 and (L

ijmneNmn

),xj

#bi"0 (51)



where Lijmn

is an instantaneous secant sti!ness given as

Lijmn

"A 1̧ijkl

#

n+g/1

u(g)D#D P#

t(g)¸ijst

Astkl

d#B ) (Iklmn#D

klmn)

!A 3̧ijkl

#

n+g/1

u(g)D#D P#

t(g)¸ijkl

d#B ) Dklmn(52)

3. NON-LOCAL DAMAGE MODEL FOR MULTI-PHASE MATERIALS

Accumulation of damage leads to strain softening and loss of ellipticity. The local approach,stating that in the absence of thermal e!ects, stresses in a material at a point are completelydetermined by the deformation and the deformation history at that point, may result ina physically unacceptable localization of the deformation [21]. The principal fault of the localapproach, as indicated in [2, 21, 22], is that the energy dissipation at failure is incorrectlypredicted to be zero and the corresponding "nite element solution converges to this spurioussolution as the mesh is re"ned. To remedy the situation, a number of approaches have beendevised to limit strain localization and to circumvent mesh sensitivity associated with strainsoftening [23]. One of these approaches is based on the nonlocal damage theory [2, 22], theessence of which is to smear solution variables causing strain softening over the characteristicvolume of the material. For other forms of localization limiters including introduction of higherorder gradients, arti"cal (or real) viscosity and elements with embedded localization zones werefer to [22, 24}34].

Following [21, 2], the non-local damage parameter uN (x) is de"ned as

uN (x)"1

D#CD P#C

u (y)u0(x, y) d# (53)

where u (y) is a weight function, #Cis the characteristic volume, and l

Cis the characteristic length,

de"ned (for example) as a radius of the largest inscribed sphere in #C. The characteristic length

lC

is related to the size of the material inhomogeneity [2], whereas lH*the radius of the largest

inscribed sphere in #*primarily depends on the distribution and interaction of inclusions anddiscrete defects [35, 36]. Several guidelines for determining the value of characteristic length havebeen provided in [1] and [33]. l

C, as indicated in [2], is usually smaller than l

Hin particular for

random microstructures. In the present manuscript, we de"ne the Representative VolumeElement (RVE) as the maximum between the statistically homogeneous volume element, forwhich the local periodicity assumption is valid, and the characteristic volume. Schematically, thiscan be expressed as

lRVE

"maxMlH, lCN (54)

where lRVE

denotes the radius of the largest inscribed sphere in #RVE

.We further assume that the microscopic damage distribution function t(g)(y) introduced in (41)

is a piecewise function, i.e., it is continuous within the domain of microphase, #(g)L#CL#, but

vanishes elsewhere, i.e.

t(g)(y)"Gg(g)(y)

0

if y3#(g)otherwise

(55)



Figure 2. Selection of the representative volume element

where Zni/1

#(g)"# and #(j)W#(g)"0 for jOg, g"1, 2, . . . , ; n is the product of the numberof di!erent microphases and the number of characteristic volumes in RVE; t(g)(y) is a distributionfunction; g(g)(y) is a C0 continuous function in #(g) ; and u(g) (x) is a macroscopically variableamplitude. Figure 2 illustrate two possibilities for construction of RVE in a two-phase medium:one for random microstructure where RVE typically coincides with SHVE, and the other one forperiodic microstructure, where l

Cand l

Hare of the same order of magnitude.

We further de"ne the weight function in (53) as

u(y)"k(g)t(g)(y) (56)

where the constant k(g) is determined by the orthogonality condition

k(g)D#

CD P#C

g(j) (y)g(g)(y) d#"djg , j, g"1, 2, . . . , n (57)

and djg is Kronecker delta. Substituting (41) and (55)}(57) into (53) yields

uN (x)"k(g)D#

CD P#C

(g(g) (y))2u(g) (x) d#"u(g)(x) (58)

which provides the motivation for the speci"c choice of the weight function. It can be seen thatu (g) has a meaning of the non-local phase damage parameter.

The average strains in each subdomain in RVE are obtained by integrating (47) over #(g) :

e(g)ij"

1

D#(g) D P#(g)eij

d#"A(g)ijkl

eNkl#G(g)

ijklD

klmneNmn#O(1 ) (59)

where

A(g)ijkl

"

1

D#(g) D P#(g)A

ijkld# (60)

G(g)ijkl

"

1

D#(g) D P#(g)G

ijkld# (61)



To construct the non-local constitutive relation between the phase averages we de"ne the localaverage stress in #(g) as

p(g)ij"

1

D#(g) D P#(g)p0ij

d# (62)

By combining (21), (41), (55), (59)}(61) we get

p(g)ij"(I

klmn!u(g)N (g)

klmn)¸(g)

ijkle(g)mn

(63)

where

N(g)klmn

"(AM (g)klst

#GM (g)klpq

Dpqst

) ) (A(g)mnst

#G(g)mnij

Dijst

)~1 (64)

AM (g)ijkl

"

1

D#(g) D P#(g)g(g)A

ijkld# (65)

GM (g)ijkl

"

1

D#(g) D P#(g)g(g)G

ijkld# (66)

The constitutive equation (63) has a non-local character in the sense that it represents the relationbetween phase averages. The response characteristics between the phases are not smeared as thedamage evolution law and thermomechanical properties of phases might be considerably di!er-ent, in particular when damage occurs in a single phase.

For the isotropic strain-based damage model adopted in this paper, the phase free energydensity corresponding to the non-local constitutive equation (63) is given as

((g)(u(g), e(g)ij

)"12

(Iklmn

!u(g)N(g)klmn

)¸(g)ijkl

e(g)mn

e(g)ij

(67)

and the corresponding non-local phase damage energy release rate can be expressed as

>(g)"L((g)Lu(g)

"

1

2N(g)

klmn¸(g)

ijkle(g)mn

e(g)ij

(68)

4. NON-LOCAL PIECEWISE CONSTANT DAMAGE MODELFOR TWO-PHASE MATERIALS

As a special case we consider a composite material consisting of two phases, matrix andreinforcement, denoted by #(m) and #(f) such that #"#(m)X#(f). Superscripts m and f representmatrix and reinforcement phases, respectively. For simplicity, we assume that damage occursin the matrix phase only, i.e. u(f),0. The volume fractions for matrix and reinforcement aredenoted as l(m) and l(f), respectively, such that l(m)#l(f)"1. The overall elastic properties aregiven as in [37]

¸ijkl

"l(m)¸(m)ijmn

A(m)mnkl

#l(f)¸(f)ijmn

A(f)mnkl

(69)

To further simplify the matters, we de"ne the microscopic damage distribution function t(g) (y)(41) as a piecewise constant function

t(g) (y)"G1

0

if y3#(g)otherwise

(70)



The corresponding weight function becomes piecewise constant function with k(g)"D#(g) D/ D#CD.

A piecewise constant approximation of damage distribution has been also considered in [38].Since damage in the reinforcement phase is neglected, the average strains in the matrix and

reinforcement can be written as

e(g)ij"A(g)

ijmneNmn#G(g)

ijklD(m)

klmneNmn#O(1) , g"m, f (71)

where

D(m)klmn

"MIklpq

!B(m)klpq

u(m)N~1C(m)pqmn

u(m) (72)

B(g)ijkl

"

1

D#(g) D( 3̧

ijmn! 1̧

ijmn)~1 P#(g)

Gstmn

¸stpq

Gpqkl

d# (73)

C(g)ijkl

"

1

D#(g) D( 3̧

ijmn! 1̧

ijmn)~1 P#(g)

Gstmn

¸stpq

(Ipqkl

#Gpqkl

) d# (74)

The corresponding non-local phase stresses (63) are given as

p(g)ij"(1!u(g))¸(g)

ijkle(g)kl#O(1), g"m, f (75)

and the overall stresses de"ned in (50) reduce to

pNij"l(m)p(m)

ij#l(f)p(f)

ij(76)

The non-local energy release rate and the energy dissipation inequality in (67) and (68) become

>(m)"12¸(m)

ijkle(m)ij

e(m)kl

(77)

>(m)uR (m)*0 (78)

The non-local isotropic damage state variable u(m) is assumed to be a monotonically increasingfunction of nonlocal phase deformation history parameter i(m) [3}6, 33], which characterizes theultimate deformation experienced throughout the loading history. In general, the evolution ofmatrix damage at time t can be expressed as

u(m)(x, t)"f (i(m) (x, t)) (79)

The non-local phase deformation history parameter i(m) is determined by the evolution ofnonlocal phase damage equivalent strain, denoted by 0M (m), as follows:

i(m)(x, t)"maxM0M (m)(x, q) D(q)t),i(m)i

N (80)

where the threshold value for damage initiation in the matrix, i(m)i

, represents the extreme value ofthe equivalent strain prior to the initiation of damage. Equation (80) can be also expressed by theKuhn}Tucker relations

iR (m)*0, 0M (m)!i(m))0, iR (m) (0M (m)!i(m))"0 (81)

In the present manuscript the non-local phase damage equivalent strain, 0M (m), is de"ned as squareroot of the non-local phase damage energy release rate [6]

0M (m),J>(m)"J12¸(m)

ijkle(m)ij

e(m)kl

(82)

Since ¸(m)ijkl

is a positive-de"nite-fourth-order tensor, it follows that >(m)*0. Consequently, thenon-local phase energy dissipation inequality (78), together with the de"nition of damage



evolution (79), yield

uR (m)"iR (m)Lf (i(m) (x, t))

Li(m)*0 (83)

Combining this inequality with Kuhn}Tucker relations, we arrive at the following two con-clusions: (1) the damage evolution law f (i(m)(x, t) ) is an increasing function of i(m)3[i(m)

i, i(m)

u]

since (Lf (i(m)(x, t)))/Li(m)'0, where i(m)u

is the ultimate equivalent strains at rupture; and (2) thedamage evolution condition can be expressed as

if 0M (m)!i(m)"0, iR (m)'0 N damage process: uR (m)'0 (84)

if 0M (m)!i(m)(0 or if 0M (m)!i(m)"0, iR (m)"0 N elastic process: uR (m)"0 (85)

In accordance with the above thermodynamic considerations, it is possible to construct anappropriate damage evolution law. An extensive review of a variety of damage evolution law hasbeen reported in [33]. In the present manuscript we propose an arctangent form of evolution lawto ensure regularity of the tangent sti!ness matrices in almost completely damaged state

'(m)(a, b,u(m),i(m), i(m)0

)"u(m)!atan (a(i(m)/i(m)

0)!b)#atan(b)

(n/2)#atan(b)"0 (86)

where a, b are material parameters; and i(m)0

denotes the threshold of the strain history parameterbeyond which the damage will develop very quickly. For simplicity, we set i(m)

i"0. From (86), it

can be seen that u(m)3[0, 1) ensures (29) to be the necessary and su$cient conditions for (28).Furthermore, this evolution law accounts for initial microcracks which are often present inceramic composites.

5. COMPUTATIONAL ISSUES

In this section, we describe computational aspects of the non-local piecewise constant damagemodel for two-phase materials developed in Section 4. Due to the non-linear character of theproblem an incremental analysis is employed. Prior to non-linear analysis elastic strain concen-tration factors, A

ijkl(y), are computed using (35), (37) by either "nite element method or if possible

by analytically solving an inclusion problem. Subsequently, non-local phase elastic strain concen-tration factors A(g)

ijkl(g"m, f ) and damage strain concentration factors G(g)

ijklare precomputed

using (60) and (61), respectively.The stress update (integration) problem can be stated as follows:

Given: displacement vectortum; overall strain

teNmn

; strain history parameterti(m) ; damage

parametertu(m) ; and displacement increment *u

mcalculated from the "nite element analysis of

the macro problem. Here left subscript denotes the increment step, i.e.t`*th is the variable in the

current increment, whereasth is a converged variable from the previous increment. For simpli-

city, we will omit the left subscript for the current increment, i.e. h,t`*th.

Find: displacement vector um,

t`*tum ; overall strain eNmn

; nonlocal phase strains e(m)mn

and e(f)mn

;non-local strain history parameter i(m) ; non-local phase damage parameter u(m) ; overall stresspNmn

and non-local phase stresses p(m)mn

and p(f)mn

.



The stress update procedure consists of the following steps:

(i) Calculate macroscopic strain increment, *eNmn"*u

(m,xn), and then update macroscopic

strains through eNmn"

teNmn#*eN

mn.

(ii) Compute the damage equivalent strain 0M (m) de"ned by (82) in terms oftu(m) and eN

mn.

(iii) Check the damage evolution conditions (84) and (85). Note that i(m) is de"ned by (80) andiR (m) is integrated as *i(m)"i(m)!

ti(m).

If damage process, i.e. 0M (m)'ti(m), then i(m)"0M (m) and update for u(m).

Since 0M (m) is governed by the current average strains in the matrix phase, which in turndepend on the current damage parameter, it follows that the damage evolution law (86) isa non-linear function of u(m). Using Newton's method, we construct an iterative process forthe damage parameter

k`1u(m)"ku(m)!AL'(m)

Lu(m)B~1

'(m) Kku(m)

(87)

The derivative in (87) can be evaluated by (71), (82), (86) as

L'(m)

Lu(m)"1!

ai(m)0

) (L0M (m)/Lu(m))

(n/2#atan(b) ) ) M(i(m)0

)2#(a0M (m)!bi(m)0

)2N(88)

where

L0M (m)

Lu(m)"

1

20M (m)) e(m)

ij¸(m)

ijklG(m)

klstR(m)

stmneNmn

(89)

with

R(m)stmn

"(Istpq

!B(m)stpq

u(m))~2C(m)pqmn

(90)

Otherwise for elastic process: u(m)"tu(m).

(iv) Update the non-local strains e(m)kl

and e(f)kl

using (71) and update the non-local strain historyparameter i(m) in (80).

(v) Update macroscopic stresses pNij

de"ned by (76) and calculate non-local phase stressesp(m)ij

and p(f)ij

using (75).

To this end we focus on the computation of a consistent tangent sti!ness matrix needed for theNewton method on the macro level. We start by substituting (71) into (75) and then taking thematerial derivative of the incremental form of (75) in the matrix domain, i.e. g"m

pR (m)ij

"P(m)ijmn

eNRmn#Q(m)

ijmneNmn

uR (m) (91)

where

P(m)ijmn

"(1!u(m))¸(m)ijkl

(A(m)klmn

#G(m)klst

D(m)stmn

) (92)

Q(m)ijmn

"(1!u(m))¸(m)ijkl

G(m)klst

R(m)stmn

!¸(m)ijkl

(A(m)klmn

#G(m)klst

D(m)stmn

) (93)

In order to obtain uR (m), we take the material derivative of damage evolution law (86), '0 (m)"0,and make use of (75), (82), and (88), which yields

uR (m)"!&(m)kl

pR (m)kl

(94)

where

&(m)kl

"ce(m)kl

(95)



and c is a scalar given as

c"M!(1!u(m)) (n/2#atan(b))M(i(m)0

)2#(a0M (m)!bi(m)0

)2N#a0M (m)i(m)0

N~1ai(m)

020M (m)

(96)

Substituting (94) into (91) and manipulating the indices, we get the following relation between therate of overall strain and non-local phase stresses in the matrix domain:

pR (m)ij

")(m)ijmn

eN Qmn

(97)

where

)(m)ijmn

"(dikdjl#&(m)

ijQ(m)

klsteNst)~1P(m)

klmn(98)

By using Sherman}Morrison formula, (98) reduces to

)(m)ijmn

"Adikdjl!&(m)

ijQ(m)

klsteNst

1#&(m)ij

Q(m)ijst

eNstBP(m)

klmn(99)

A similar result relating the rate of the non-local reinforcement stress and the overall strain rate,can be obtained by substituting (71) into (75) and then taking the material derivative of (75) in thereinforcement domain

pR (f)ij

")(f)ijmn

eN Qmn

(100)

where

)(f)ijmn

"P(f)ijmn

!Q(f)ijst

eNst&(m)

kl)(m)

klmn(101)

and

P(f)ijmn

"¸(f)ijkl

(A(f)klmn

#G(f)klst

D(m)stmn

) (102)

Q(f)ijst

"¸(f)ijkl

G(f)klmn

R(m)mnst

(103)

Finally, the overall consistent tangent sti!ness is constructed by substituting (97) and (100) intothe rate form of the overall stress}strain relation (76)

pNQij")

ijmneNQmn

(104)

)ijmn

"l(m))(m)ijmn

#l(f))(f)ijmn

(105)

6. NUMERICAL EXAMPLES

6.1. Qualitative examples for two-phase ,brous composites under uniaxial loading

The "rst numerical example is aimed at qualitative study of the behaviour of the proposednon-local piecewise constant damage model for two-phase materials. We consider a macro-domain in the shape of a block discretized with a single brick element and a periodic "brousmicrostructures as shown in Figure 3. The block is subjected to the state of constant macrostrain"eld in the axial (parallel to the "bres) and transverse (normal to the "bres) directions. The axial



Figure 3. Finite element mesh of the RVE for "brous microstructure

direction is aligned along the Z axis whereas the two transverse directions coincide with theX- and >-axis. The phase properties of microconstituents are as follows:

Matrix: Volume fraction"0)733; Young's modulus"69 GPa; Poisson's ratio"0)33.

Fibre: Volume fraction"0)267; Young's modulus"379 GPa; Poisson's ratio"0)21.

The parameters of the damage evolution law are chosen as a"8)2, b"10)2 and i(m)0

"0)05. Thecorresponding damage evolution law is depicted in Figure 4.

The uniaxial stress}strain curves for the axial and transverse tension problems are illustrated inFigures 4 and 6, respectively. Figure 5 shows a rapid loss of sti!ness as the damage in the matrixphase accumulates and in the limit as the matrix material is completely damaged the axial loadingcapacity of composite is provided by the "ber only. Our numerical model is in good agreementwith the limit solution which gives limu(m)?1>0

p33"l(f)E(f)e

33. Results of the transverse tension

problem are shown in Figure 6. It can be seen that when the matrix is totally damaged, it fails totransfer the load into the "bre and consequently, the entire load carrying capacity of the "brouscomposite is lost in the transverse direction, i.e. limu(m)?1>0

p11"0. In both "gures, we also

demonstrate the evolution of the damage parameter in the matrix phase. Referring back to thedamage evolution curve shown in Figure 4, it can be seen that the sudden drop in load-carryingcapacity in both axial and transverse directions occurs when the damage parameter reachesu(m)+0)1 beyond which the damage parameter grows sharply.

6.2. Four-point bending problem for woven composite

We next consider a 4-point bending problem carried out on a composite beam made ofBlackglasTM/Nextel 5-harness satin weave as shown in Figure 7. The fabric designs used 600denier bundles of NextelTM 312 "bres, spaced at 46 threads per inch, and surrounded byBlackglasTM matrix material. The bundle is assumed to be linear elastic throughout the analysis.The average transversely isotropic elastic properties were computed by the Mori}Tanakamethod. We will refer to this material system as AF10. The micrograph in Figure 7 was produced



Figure 4. Damage evolution law for the titanium matrix

Figure 5. Loading capacity in the axis (Z-axis) direction



Figure 6. Loading capacity in the transverse (X-axis) direction

Figure 7. BlackglasTM/Nextel 5-harness satin weave



Figure 8. Microstructure of AF10 woven composites

at Northrop}Grumman [18]. In this set of numerical examples, the non-local piecewise constantdamage model is employed and we assume that l

RVE"l

H"l

C. The phase properties of RVE are

summarized below:

BlackglasTM Matrix: volume fraction"0)548; Young's modulus"9)653 GPa;Poisson's ratio"0)244.

NextelTM 312 "bre: volume fraction"0)452; Young's modulus"151)7 GPa;Poisson's ratio"0)26

The microstructure of RVE is discretized with 6857 elements totaling 10608 degrees of freedomas shown in Figure 8. The issues of the automatic extraction, construction and linking ofthe geometry and attributes, automatic construction of matched meshes have been describedin [39]. The con"guration of the composite beam is shown in Figure 9 where the loadingdirection (normal to the plane of the weave) is aligned along the >-axis. The "nite elementmodel of the beam (macrostructure) is composed of 1856 brick elements totaling 7227 degreesof freedom. Figure 10 depicts the damage evolution law for BlackglasTM matrix witha"7)1, b"10)1 and i(m)

0"0)22, which are calibrated to the tensile and shear test data.

Comparison between tensile test data and the numerical simulation for the uniaxial tension isshown in Figure 11. It can be seen that the ultimate experimental stress/strain values in theuniaxial tension test are p

6"150$7 MPa and e

6"2)5]10~3$3)0]10~3, while the numer-

ical simulation gives p6"152 MPa at e

6"3)2]10~3.

Numerical simulation results as well as the test data for 4-point bending problem are shown inFigures 12 and 13. Experiments have been conducted on "ve identical beams and the scatteredexperimental data of force versus the displacement at the point of load application in the beamare shown by the gray area in Figure 12. It can be seen that the numerical simulation results are ingood agreement with the experimental data in terms of predicting the overall behaviour.



Figure 9. Con"guration and FE mesh of 4-point bending problem

Figure 10. Damage evolution law for the blackglasTM matrix

(Figure 12) and the dominant failure mode. Both numerical simulation and experimental datapredict that the dominant failure mode is tension/compression (so-called bending inducedfailure). Figure 13 illustrates the distribution of the damage parameter in the composite beam atthe peak load (Point A in Figure 12).



Figure 11. Strain}stress curves for the uniaxial tension problem

Figure 12. Force versus displacement for the 4-point bending problem



Figure 13. Damage distribution at the peak load in the 4-point bending problem

To this end we note that since bundles have been modelled as linear elastic spurious increase inload carrying capacity of the weave in the in-plane tension/compression eventually takes place.Remedies are discussed in Section 7.

7. SUMMARY AND FUTURE RESEARCH DIRECTIONS

A non-local damage theory for brittle composite materials based on double-scale asymptoticexpansion of damage has been developed. A closed-form expression relating local "elds tothe overall strains and damage has been derived. The concept of non-local phase "elds(stress, strain, free-energy density, damage release rate, etc.) has been introduced via weightingfunctions de"ned over the microphase. Numerical results revealed an excellent performance of themethod.

The present work by no means represents a complete account of all theoretical and numericalissues related to damage in composites and we apologize if some important works have beenomitted. We note that the assumptions of periodicity and uniformity of macroscopic "elds, whichare embedded in our formulation, may yield inaccurate solutions in the vicinity of boundarylayers. The remedies to this phenomenon range from changing the RVE size [40] to carryingout an iterative global}local analysis [31, 32, 41}43]. Moreover, various failure modes otherthan matrix cracking, such as damage at the interface and in the bundle domain,coupled plasticity-damage e!ects, di!erent responses in tension and compression have not beenaccounted for the present manuscript. These are just few of the issues that will be investigated inour future work.



ACKNOWLEDGEMENTS

This work was supported in part by the Air Force O$ce of Scienti"c Research under grantnumber F49620-97-1-0090, the National Science Foundation under grant number CMS-9712227,and Sandia National Laboratories under grant number AX-8516.

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