multiscale damage modelling for composite materials: theory and computational framework

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERINGInt. J. Numer. Meth. Engng 2001; 52:161–191 (DOI: 10.1002/nme.276)

Multiscale damage modelling for composite materials:theory and computational framework

Jacob Fish∗;† and Qing Yu

Departments of Civil; Mechanical and Aeronautical Engineering; Rensselaer Polytechnic Institute;Troy; NY 12180; U.S.A.

SUMMARY

A non-local multiscale continuum damage model is developed for brittle composite materials. A triple-scale asymptotic analysis is generalized to account for the damage phenomena occurring at micro-,meso- and macroscales. A closed-form expressions relating microscopic, mesoscopic and overall strainsand damage is derived. The damage evolution is stated on the smallest scale of interest and non-localweighted phase average ;elds over micro- and mesophases are introduced to alleviate the spuriousmesh dependence. Numerical simulation conducted on a composite beam made of Blackglas=Nextel2-D weave is compared with the test data. Copyright ? 2001 John Wiley & Sons, Ltd.

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

1. INTRODUCTION

Damage phenomena in composite materials are very complex due to signi;cant heterogeneitiesand interactions between microconstituents. Typically, damage can be either discrete or con-tinuous and described on at least three di@erent scales: discrete for atomistic voids and latticedefects; and continuous for micromechanical and macromechanical scales, which describeeither distributed microvoids and microcracks or discrete cracks whose size is comparableto the structural component. Here, attention is restricted to continuum scales only. On themicromechanical scale, the representative volume element (RVE) is introduced to model theinitiation and growth of microscopic damage and their e@ects on the material behaviour.The RVE is de;ned to be small enough to distinguish microscopic heterogeneities, but suf-;ciently large to represent the overall behaviour of the heterogeneous medium. Most re-search in this area is focused on the two-scale micro–macro problems with homogeneous

∗Correspondence to: Jacob Fish, Department of Civil, Mechanical and Aeronautical Engineering, RensselaerPolytechnic Institute, Troy, NY 12180, U.S.A.

†E-mail: ;[email protected]

Contract=grant sponsor: Air Force OFce of Scienti;c Research; contract=grant number: F49620-97-1-0090Contract=grant sponsor: NSF; contract=grant number: CMS-9712227Contract=grant sponsor: Sandia National Laboratories; contract=grant number: AX-8516

Copyright ? 2001 John Wiley & Sons, Ltd.

162 J. FISH AND Q. YU

microconstituents (see, for example, [26; 32; 33; 35; 37; 41–43]). For certain compositematerials systems, such as woven composites [1], the two-scale model might be insuFcientdue to strong heterogeneities in one of the microphases. The question arises as to how toaccount for damage e@ects in these heterogeneous phases. Most commonly, macroscopic-likepoint of view [2–6] is adopted by idealizing the heterogeneous phases as anisotropic homo-geneous media. Anisotropic continuum damage theory is then employed to model damageevolution in each phase. As an alternative, which is explored here, is to de;ne smaller-scaleRVE(s) for the heterogeneous phases and then to carry out multiple scale damage analysiswith various RVEs at di@erent length scales.The objective of this paper is to extend the two-scale (macro–micro) non-local damage

theory developed in Reference [7] to three scales in attempt to account for evolution ofdamage in heterogeneous microphases. Throughout the manuscript we term the larger-scaleRVE(s) as mesoscopic while RVE(s) comprising heterogeneous mesophases as microscopic. InSection 1, the three-scale (macro-, meso- and micro-) damage theory within the framework ofthe mathematical homogenization theory is developed. The triple-scale asymptotic expansionsof damage and displacements lead to closed-form expressions relating local (microscopic andmesoscopic) ;elds to overall (macroscopic) strains and damage. In Section 3, the non-localphase ;elds for multi-phase composites are de;ned as weighted averages over each phasein the mesoscopic and microscopic characteristic volumes with piecewise constant weightfunctions. A more general case of weight functions is discussed in Reference [7]. In Section 4,a simpli;ed variant of the non-local damage model for the two-phase composite materials isdeveloped. The computational framework including stress update procedure and consistenttangent sti@ness are presented in Section 5. In Section 6, we ;rst study the axial loadingcapacity of the Blackglas=Nextel 2-D woven composite [1; 8]. Numerical results obtainedby the present three-scale formulation are compared with those obtained by the two-scalemodel [7]. We then consider a four-point bending test conducted on the composite beammade of Blackglas=Nextel 2-D woven composite and compare the simulation results withthe experiments data provided by Butler et al. [8]. Discussion and future research directionsconclude the manuscript.

2. MATHEMATICAL HOMOGENIZATION FOR DAMAGED COMPOSITES

As shown in Figure 1, the composite material is represented by two locally periodic RVEson the mesoscale (Y -periodic) and the microscale (Z-periodic), denoted by Oy and Oz, re-spectively. Let x be the macroscopic co-ordinate vector in the macrodomain P; y≡x=& bethe mesoscopic position vector in Oy and z≡ y=& be the microscopic position vector in Oz.Here, & denotes a very small positive number; y≡x=& and z≡ y=& are regarded as the stretchedlocal co-ordinate vectors. When a solid is subjected to some load and boundary conditions, theresulting deformation, stresses, and internal variables may vary from point to point within theRVE(s) due to a high level of heterogeneity. We assume that all quantities on the mesoscalehave two explicit dependences: one on the macroscale x and the other on the mesoscale y.For the quantities on the microscale, additional dependence on the microscale z is introduced.For any microscopically periodic response function f, we have f(x; y; z)=f(x; y+ky; z+kz)in which vectors y and z are the basic periods in the meso- and microstructures and k is a3× 3 diagonal matrix with integer components. Adopting the classical nomenclature [28], any

Copyright ? 2001 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2001; 52:161–191

MULTISCALE DAMAGE MODELLING FOR COMPOSITE MATERIALS 163

Figure 1. Three-scale composite materials.

locally periodic function can be represented as

f&(x)≡f(x; y(x); z(x; y)) (1)

where superscript & indicates that the corresponding function f is locally periodic and is afunction of macroscopic spatial variables. The indirect macroscopic spatial derivative of f& iscalculated by the chain rule as

f&; xi(x)=f;xi(x; y; z) +

1&f;yi(x; y; z) +

1&2f;zi(x; y; z) (2)

where the comma followed by a subscript variable denotes the partial derivative (i.e. f;xi ≡@f=@xi). Summation convention for repeated subscripts is employed, except for subscripts x; yand z.To model the isotropic damage process in meso- and microconstituents, we de;ne a scalar

damage variable !&=!(x; y; z). The constitutive equation can be derived from thestrain-based continuum damage theory based on the thermodynamics of irreversible pro-cesses and internal state variable theory. We assume that microconstituents possess homo-geneous properties and satisfy equilibrium, constitutive, kinematics and compatibilityequations as well as jump conditions at the interface. The corresponding boundary valueproblem on the smallest (micro) scale of interest is described by the following set ofequations:

&ij; xj + bi =0 in P (3)

&ij = (1−!&)L&ijkl�&kl in P (4)

�&ij = u&(i; xj) in P (5)

u&i = Tui on Uu (6)

&ijnj = t i on Ut (7)



where !& ∈ [0; 1) is a scalar damage variable governed by a strain history parameter (seeSection 4); &ij and �&ij are components of stress and strain tensors; L&ijkl represents componentsof elastic sti@ness satisfying symmetry

L&ijkl =L&jikl =L&ijlk =L&klij (8)

and positivity

∃C0¿0; L&ijkl�&ij�

&kl¿C0�&ij�

&ij ∀�&ij = �&ji (9)

conditions; bi is a body force assumed to be independent of local position vectors y and z;u&i denotes the components of the displacement vector; the subscript pairs with parenthesesdenote the symmetric gradients de;ned as

u&(i; xj) ≡ 12 (u

&i; xj + u&j; xi) (10)

P denotes the macroscopic domain of interest with boundary U; Uu and Ut are boundary por-tions where displacements Tui and tractions t i are prescribed, respectively, such that Uu ∩Ut = ∅and U=Uu ∩Ut; ni denotes the normal vector on U. We assume that the interface between thephases is perfectly bonded, i.e. [ &ij nj]= 0 and [u&i ]= 0 at the interface, Uint, where ni is thenormal vector to Uint and [•] is a jump operator.

In the following, the contracted (matrix) notation is adopted. The indices of the contractednotation follow the de;nition 11≡ 1, 22≡ 2, 33≡ 3, 12≡ 4, 13≡ 5 and 23≡ 6. The scalarquantities and tensor components are denoted by lightface letters. XT and X−1 denote thetranspose and inverse of matrix X (or vector), respectively; the subscripts in Xmic, Xmeso

and Xmac represent the scale on which the quantities are measured, i.e. Xmic ≡X(x; y; z),Xmeso ≡X(x; y) and Xmac ≡X(x).

Clearly, the straightforward approach attempting at discretization of the entire macrodomainwith a grid spacing comparable to that of the microconstituents or even mesoscale featuresis not computationally feasible. Instead, a mathematical homogenization method based onthe triple-scale asymptotic expansion is employed to account for meso- and micromechanicale@ects without explicitly representing the details of local structures in the global analysis. Asa starting point, we approximate the microscopic displacement ;eld, u&(x)≡ u(x; y; z), andthe damage variable, !&(x)≡!(x; y; z), in terms of the triple-scale asymptotic expansions onP×Oy ×Oz:

umic ≈ u0(x) + &u1(x; y) + &2u2(x; y; z) (11)

!mic ≈!0(x; y; z) + &!1(x; y; z) + &2!2(x; y; z) (12)

where the superscripts on u and ! denote the length scale of each term in the asymptoticexpansions. The strain expansion on P×Oy ×Oz can be obtained by substituting (11) into(5) with consideration of the indirect di@erentiation rule (2):

�mic ≡ �(x; y; z)= �0(x; y; z) + &�1(x; y; z) + &2�2(x; y; z) (13)

where the strain components for various orders of & are given as

�0 = u0; x + u1; y + u2; z (14)



�1 = u1; x + u2; y (15)

�2 = u2; x (16)

We further de;ne scale-average strains by integrating (13) in Oz and Oy ×Oz with consider-ation of Y - and Z-periodic conditions, respectively:

�meso(x; y)≡ 1|Oz|

∫Oz

�0(x; y; z) dO= u0; x + u1; y (17)

�mac(x)≡ 1|Oy|

1|Oz|

∫Oy

∫Oz

�0(x; y; z) dOdO =1

|Oy|∫Oy

�meso dO= u0; x (18)

Following (17) and (18), the mesoscopic strain obeys the decomposition

�meso = �mac + u1; y(x; y) (19)

where the macroscopic strain �mac(x) in P, represents the average strain on mesoscale andu1; y(x; y) is the local oscillatory part of mesoscopic strain in Oy. The asymptotic expansion ofthe microscopic strain (13) can be expressed in a similar fashion as

�mic = �meso + u2; z(x; y; z) +O(&) (20)

where the mesoscopic strain in Oy; �meso(x; y), represents the average strain on the microscale,whereas u2; z(x; y; z) is the local oscillatory part of the microscopic strain in Oz.Stresses and strains for di@erent orders of & are related by the constitutive equation (4) and

the expansion of the damage variable (12):

0 = (1−!0)Lmic�0 (21)

1 = (1−!0)Lmic�1 −!1Lmic�0 (22)

2 = (1−!0)Lmic�2 −!1Lmic�1 −!2Lmic�0 (23)

The resulting asymptotic expansion of stress is given as

mic ≡ (x; y; z)= 0(x; y; z) + &1 1(x; y; z) + &2 2(x; y; z) +O(&3) (24)

Once again, we de;ne the scale-average stresses by integrating (24) in Oz and Oy ×Oz:

meso(x; y)≡ 1|Oz|

∫Oz

0(x; y; z) dO (25)

mac(x)≡ 1|Oy|

1|Oz|

∫Oy

∫Oz

0(x; y; z) dOdO=1

|Oy|∫Oy

meso dO (26)

Inserting the stress expansion (24) into the equilibrium equation (3) and making use ofindirect di@erentiation rule (2) yields the following equilibrium equations for various



orders:

O(&−2): 0; z =0 (27)

O(&−1): 0; y + 1

; z =0 (28)

O(&0): 0; x + 1

; y + 2; z + b=0 (29)

Consider the O(&−2) equilibrium equation (27) ;rst. From (13), (14), (17) and (21) follow

{Lmic�mic}; z = {Lmic(�meso + u2; z)}; z =0 in Oz (30)

Lmic = {1−!0(x; y; z)}Lmic (31)

where Lmic is a history-dependent microscopic [email protected] solve for (30) we introduce the following decomposition:

u2(x; y; z)=H(z){�meso(x; y) + dmeso(x; y)} (32)

where the third-order tensor H(z)≡Hikl(z) is symmetric with respect to indices k and l [7; 9]and Z-periodic in Oz. We assume that dmeso(x; y) is mesoscopic damage-induced strain drivenby mesoscopic strain �meso(x; y). More speci;cally, we can state that if �meso = 0, then dmeso = 0and !0 = 0. However, vice versa is not true, i.e. if dmeso = 0 or !0 = 0, the mesoscopic strain�meso may not be necessarily zero.Based on the decomposition given in (32), O(&−2) equilibrium equation takes the following

form:

{Lmic{(I+Gz)�meso +Gzdmeso}}; z =0 in Oz (33)

where I is identity matrix and

Gz(z)=H; z(z) ≡ H(i; zj)kl(z) (34)

is a microscale polarization function. The integrals of the polarization function in Oz vanishdue to periodicity conditions. Since Equation (33) should be valid for arbitrary mesoscopic;elds, we may ;rst consider the case of dmeso = 0 (and !0 = 0) but �meso �=0, which yields

{Lmic(I+Gz)}; z =0 (35)

Equation (35) together with the Z-periodic boundary conditions comprise a linear boundaryvalue problem for H in Oz. The weak form of (35) is solved for three right-hand side vectorsin 2-D and six in 3-D (see for example References [10; 11]). In absence of damage, the strainasymptotic expansions (13) and (20) can be expressed in terms of the mesoscopic strain �meso

as

�mic =Az�meso +O(&) (36)

where Az is elastic strain concentration function in Oz de;ned as

Az = I+Gz (37)



After solving (35) for H, we proceed to ;nding dmeso from (33). Premultiplying it by Hand integrating it by parts in Oz with consideration of Z-periodic boundary conditions yields∫

Oz

GTz Lmic(Az�meso +Gzdmeso) dO=0 (38)

and the expression for the mesoscopic damage-induced strain becomes

dmeso = −{∫

Oz

GTz LmicGz dO

}−1{∫Oz

GTz LmicAz dO

}�meso (39)

Let f ≡{f(�)(y); �=1; 2; : : :} and g≡{g(��)(z); �; �=1; 2; : : :} be the two sets of C−1 con-tinuous functions, then the damage variable !0(x; y; z) is assumed to have the followingdecomposition:

!0(x; y; z)=∑�

∑�!(x)f(�)(y)g(��)(z) (40)

where !(x) is the macroscopic damage variable, f(�)(y) is a damage distribution function onthe mesoscale RVE and g(��)(z) represents the damage distribution function in the microscaleRVE � corresponding to phase � in the mesoscale RVE. Rewriting (39) in terms of Amic and!0(x; y; z) yields

dmeso =Dmeso�meso (41)

where

Dmeso =

(I −∑

�

∑�!(x)f(�)(y)B(��)

)−1(∑�

∑�!(x)f(�)(y)C(��)

)(42)

B(��) =1

|Oz| (Lmeso − TLmeso)−1∫Oz

g(��)(z)GTz LmicGz dO (43)

C(��) =1

|Oz| (Lmeso − TLmeso)−1∫Oz

g(��)(z)GTz LmicAz dO (44)

Lmeso =1

|Oz|∫Oz

Lmic dO (45)

TLmeso =1

|Oz|∫Oz

LmicAz dO=1

|Oz|∫Oz

ATzLmicAz dO (46)

where Lmeso is the overall sti@ness on Oz; TLmeso is the elastic homogenized sti@ness [7; 9];|Oz| denotes the volume of the microscale RVE. Note that the integrals in B(��) and C(��) arehistory-independent and thus can be precomputed. This provides one of the main motivationsfor the decomposition given in (40).Based on (32) and (41), the asymptotic expansion of the strain ;eld (13) can be ;nally

cast as

�mic = (Az +GzDmeso)�meso +O(&) (47)



where Gz is a local damage strain distribution function in Oz. Note that the asymptoticexpansion of the strain ;eld is given as a sum of mechanical ;elds induced by the mesoscopicstrain via elastic strain concentration function Az and thermodynamical ;elds governed by thedamage-induced strain dmeso =Dmeso�meso through the distribution function Gz.We now consider the O(&−1) equilibrium equation (28). By integrating (28) over Oz,

making use of the Z-periodicity condition and the de;nition of the mesoscopic stress in (25),we get

( meso); y =(

1|Oz|

∫Oz

0(x; y; z) dO); y

=0 in Oy (48)

which represents the mesomechanical equilibrium equation in Oy. Based on the asymptoticexpansion of the strain ;eld in (13) and (19), the constitutive equation in (21) and thedecomposition (47) we can rewrite (48) as

{Lmeso�meso}; y = {Lmeso(�mac + u1; y)}; y =0 in Oy (49)

Lmeso =1

|Oz|∫Oz

Lmic(Az +GzDmeso) dO (50)

where Lmeso is a history-dependent mesoscopic sti@ness matrix.To solve for the mesomechanical equilibrium equation, we ;rst note that (49) is similar to

its microscopic counterpart in (30). Thus, a similar procedure can be employed by introducingthe decomposition:

u1(x; y)= H(y){�mac(x) + dmac(x)} (51)

where H(y)≡ Hikl(y) is a Y -periodic third order tensor on the mesoscale, symmetric withrespect to indices k and l; dmac(x) is the macroscopic damage-induced strain driven by themacroscopic strain �mac(x). Based on this decomposition, (49) becomes

{Lmeso{(I+Gy)�mac +Gydmac}}; y =0 in Oy (52)

where

Gy(y)= H; y(y)≡ H(i; yj)kl(y) (53)

is a polarization function on the mesoscale whose integral in Oy vanishes due to Y -periodicityconditions. Once again, since (53) is valid for arbitrary macroscopic ;elds, we ;rst considerthe case of damage-free, i.e. dmac =0 and !0 = 0 but �mac �=0, which yields

{TLmeso(I+Gy)}; y =0 (54)

Equation (54) comprise a linear boundary value problem for H in the mesoscale domainOy subjected to Y -periodic boundary conditions. Based on the decomposition in (51) and inabsence of damage, the mesoscopic strain in (19) can be expressed in terms of the macroscopicstrain �mac as

�meso =Ay�mac (55)



where Ay is the mesoscopic elastic strain concentration function de;ned as

Ay = I+Gy (56)

After obtaining H; dmac can be obtained from (52) by premultiplying it with H and integratingit by parts in Oy with consideration of Y -periodic boundary conditions, which yields

dmac =Dmac�mac (57)

Dmac =−{∫

Oy

GTyLmesoGy dO

}−1{∫Oy

GTyLmesoAy dO

}(58)

where Lmeso in (50) is a history-dependent sti@ness matrix. Based on the decomposition ofthe damage variable in (40), Lmeso is given as

Lmeso =

(TLmeso −

∑�

∑�!(x)f(�)(y)TL(��)

meso

)(I+Dmeso)

−(Lmeso −

∑�

∑�!(x)f(�)(y)L(��)

meso

)Dmeso (59)

where

L(��)meso =

1|Oz|

∫Oz

g(��)(z)Lmic dO (60)

TL(��)meso =

1|Oz|

∫Oz

g(��)(z)LmicAz dO (61)

To this end, the mesoscopic strain (19) and the asymptotic expansion of the microscopicstrain ;eld (13) can be directly linked to the macroscopic strain as

�meso = (Ay +GyDmac)�mac (62)

�mic = (Az +GzDmeso)(Ay +GyDmac)�mac +O(&) (63)

where Gy, the counterpart of Gz in Oz, is a local distribution function of the damage-inducedstrain in Oy.

Finally, we integrate the O(&0) equilibrium equation (29) over Oy×Oz. The∫Oy 1; y dO and∫

Oz 2; z dO terms in the integral vanish due to periodicity and we obtain(

1|Oy|

1|Oz|

∫Oy

∫Oz

0(x; y; z) dOdO

); x

+ b=0 in P (64)



Substituting the constitutive relation (21), the asymptotic expansion of the strain ;eld (63),the microscopic and mesoscopic instantaneous sti@nesses in (31) and (50) into (64) yieldsthe macroscopic equilibrium equation

( mac); x + b=0 and (Lmac�mac); x + b=0 (65)

where

Lmac =1

|Oy|∫Oy

Lmeso(Ay +GyDmac) dO (66)

is a macroscopic instantaneous secant sti@ness.

3. NON-LOCAL PIECEWISE CONSTANT DAMAGE MODELFOR MULTI-PHASE MATERIALS

Accumulation of damage leads to strain softening and loss of ellipticity in quasi-static prob-lems. The local approach, stating that in absence of thermal e@ects, stresses at a materialpoint are completely determined by the deformation and the deformation history at that point,may result in a physically unacceptable localization of the deformation [12–17]. A numberof regularization techniques have been developed to limit strain localization and to alleviatemesh sensitivity associated with strain softening [12–16; 18–22; 29; 34; 40]. One of these ap-proaches is based on smearing solution variables causing strain softening over the characteristicvolume of the material [12; 14]. Following [14] and [7], the non-local damage variable T!(x) isde;ned as

T!(x)=1

|OCy |1

|OCz |∫OCy

∫OCz

’y(y)’z(z)!0(x; y; z) dOdO (67)

where ’z(y) and ’y(y) are weight functions on micro- and mesoscale, respectively and OCz

and OCy are the characteristic volumes on the micro- and mesoscales with characteristic lengthlCz and lCy , respectively. The characteristic length is de;ned (for example) as a radius of thelargest inscribed sphere in a characteristic volume, which is related to the size of the materialinhomogeneties [14]. lHz and lHy are the radii of the largest inscribed spheres in the statisti-cally homogeneous volumes (SHV), which is the smallest volume for which the correspondinglocal periodicity assumptions are valid. Several guidelines for determining the value of char-acteristic length have been provided in References [13; 17]. The characteristic lengths, lCz andlCy , as indicated in Reference [14], are usually smaller than the corresponding lHz and lHy inparticular for random local structures. Following the two-scale non-local damage model in Ref-erence [7], we rede;ne the representative volume element (RVE) as the maximum betweenstatistically homogeneous volume and the characteristic volume. Schematically, this can beexpressed as

lRVEmic = max{lHz ; lCz} (68)

lRVEmeso = max{lHy ; lCy} (69)



Figure 2. Selection of the representative volume element.

where lRVEmic and lRVEmeso denote the radii of the largest inscribed spheres in the mesoscopicand macroscopic RVEs, respectively. Figure 2 illustrates the two possibilities for constructionof the RVE on the mesoscale in a two-phase medium: one for random microstructure whereRVE typically coincides with SHV, and another one for periodic microstructure, where RVEand SHV are of the same order of magnitude. Figure 2 is also applicable to the de;nition ofthe microscopic RVE.In particular, we assume that the microscopic and mesoscopic damage distribution functions,

g(��)(z) and f(�)(y) in (40), are both piecewise constant; f(�)(y) is assumed to be unitywithin the domain of the mesoscopic phase O(�)

y which satis;es O(�)y ⊂OCy ⊂Oy, and to vanish

elsewhere, i.e.

f(�)(y) =

{1 if y∈O(�)

y

0 otherwise(70)

where⋃k�

�=1O(�)y =Oy and O(i)

y ∩O(j)y = ∅ for i �= j and i; j=1; 2; : : : ; k�; k� is a product of the

number of di@erent mesoscopic phases and the number of mesoscopic characteristic volumesin the mesoscopic RVE Oy; g(��)(z) is unity within the microphase O(��)

z , such that O(��)z ⊂

O(�)Cz

⊂O(�)z , but vanish elsewhere, i.e.

g(��)(z) =

{1 if z∈O(��)

z

0 otherwise(71)

where⋃ k��

�=1O(��)z =O(�)

z and O(i)y ∩O( j)

y = ∅ for i �= j and i; j=1; 2; : : : ; k��; k�� is the productof the number of di@erent microphases and the number of microscopic characteristic volumesin O(�)

z .



We further de;ne the weight function in (67) as

’y(y)= (�)y f(�)(y) and ’z(z)= (��)

z g(��)(z) (72)

where the constants (�)y and (��)

z are determined by the orthogonality conditions

(�)y

|OCy |∫OCy

f(�)(y)f(�)(y) dO= !��; �; �=1; 2; : : : ; k� (73)

(��)z

|O(�)Cz|

∫O(�)

Cz

g(��)(z)g(��)(z) dO= !��; �; �=1; 2; : : : ; k�� (74)

!�� and !�� are the Kronecker deltas. Substituting (40) and (70)–(74) into (67) shows that!(��) coincides with the non-local phase average damage variable:

T!(x) = (�)y

|OCy | (��)z

|O(�)Cz|

∫OCy

∫O(�)

Cz

!(x){f(�)(y)g(��)(z)}2 dOdO≡!(��)(x) (75)

The average strains in each microphase, termed as microscopic phase average strain, areobtained by integrating (63) over O(��)

z

�(��)meso =1

|O(��)z |

∫O(��)z

�mic dO=( TA(��)z + TG(��)

z D(�)meso)(A

(�)y +G(�)

y Dmac)�mac (76)

where O(��)z represents the domain of phase � in O(�)

z

G(�)y ≡Gy(y∈O(�)

y ) and A(�)y = I+G(�)

y (77)

TG(��)z =

1

|O(��)z |

∫O(��)z

Gz(z) dO and TA(��)z = I+ TG(��)

z (78)

D(�)meso ≡Dmeso(x; y∈O(�)

y ) =(I −∑

�!(��)B(��)

)−1(∑

�!(��)C(��)

)(79)

and following (43)–(46) and (71) yields

B(��) =1

|O(�)z |

(L(�)meso − TL(�)

meso)−1∫O(��)z

GTz LmicGz dO (80)

C(��) =1

|O(�)z |

(L(�)meso − TL(�)

meso)−1∫O(��)z

GTz LmicAz dO (81)



L(�)meso =

1

|O(�)z |

∫O(�)z

Lmic dO (82)

TL(�)meso =

1

|O(�)z |

∫O(�)z

LmicAz dO=1

|O(�)z |

∫O(�)z

ATzLmicAz dO (83)

We denote the volume fractions for phase � in O(�)z by "(��) ≡ |O(��)

z |=|O(�)z | such that∑

� "(��) = 1. It can be readily seen that

�(�)meso ≡ �meso(x; y∈O(�)y )=

∑�"(��)�(��)meso (84)

Similarly, the phase average strain in the mesoscopic RVE Oy can be obtained by integrating(62) over O(�)

y , which yields

�(�)mac =1

|O(�)y |

∫O(�)y

�meso dO=( TA(�)y + TG(�)

y Dmac)�mac (85)

where the mesoscopic phase average strain concentration function TA(�)y is de;ned as

TG(�)y =

1

|O(�)y |

∫O(�)y

Gy(y) dO and TA(�)y = I+ TG(�)

y (86)

Also, we have the relation

�mac =∑�"(�)�(�)mac (87)

where "(�) ≡ |O(�)y |=|Oy| is the volume fraction of phase � in Oy satisfying

∑� "

(�) = 1.To construct the non-local constitutive relation between phase averages, we de;ne the local

averages stress in O(�)z as

(��)meso ≡

1

|O(��)z |

∫O(��)z

0(x; y; z) dO (88)

By combining (88) with microscopic constitutive equation (21), the asymptotic expansion ofstrain in (13) and (47) and the piecewise constant damage variable de;ned in (40), (70) and(71), we get

(��)meso = (1−!(��))L(��)

meso�(��)meso (89)

where L(��)meso ≡Lmic(x; y∈O(�)

y ; z∈O(��)z ). Here, we assume that each phase in the microscale

RVE is homogeneous, i.e. L(��)meso is assumed to be independent of the position vector z within

each phase O(��)z .



The phase average stress in the mesoscopic RVE Oy is de;ned in a similar fashion to(88) as

(�)mac ≡

1

|O(�)y |

∫O(�)y

meso dO (90)

With the de;nition of meso in (25), the microscopic constitutive equation (21), Equations(31) and (50), the de;nition in (90) can be restated as

(�)mac =L(�)

mac�mac (91)

where the macroscopic phase average instantaneous secant sti@ness L(�)mac in O(�)

y is given as

L(�)mac =

1

|O(�)y |

∫O(�)y

L(�)meso(Ay +GyDmac) dO (92)

and

mac =∑�"(�) (�)

mac and Lmac =∑�"(�)L(�)

mac (93)

which follows from (26) and (91). The mesoscopic instantaneous sti@ness L(�)meso can be

obtained from (59) in conjunction with the de;nition of the piecewise constant damage sothat

L(�)meso ≡Lmeso(x; y∈O(�)

y )

=

(TLmeso −

∑�"(��)!(��) TL(��)

meso

)(I+D(�)

meso)−(Lmeso −

∑�"(��)!(��)L(��)

meso

)D(�)

meso

(94)

where the phase average sti@ness L(��)meso and the homogenized sti@ness TL(��)

meso are obtained from(60) and (61):

L(��)meso =

1

|O(��)z |

∫O(��)z

Lmic dO and TL(��)meso =

1

|O(��)z |

∫O(��)z

LmicAz dO (95)

The constitutive equations (89) and (91) have a non-local character in the sense that theyrelate between phase averages in the microscopic and mesoscopic RVEs, respectively. Theresponse characteristics between the phases are not smeared as the damage evolution lawand thermomechanical properties of phases might be signi;cantly di@erent, in particular whendamage occurs in a single phase. In the next section we focus on the multiscale damagemodeling in woven composites.



4. DAMAGE EVOLUTION FOR TWO-PHASE COMPOSITES

As a special case depicted in Figure 1, we consider a three-scale composite whose mesoscopicstructure is composed of reinforcement phase (�=F) and matrix phase (�=M) such thatOy =O(F)

y ∪O(M)y and the volume fractions satisfy "(F) + "(M) =1. We assume that the matrix

phase O(M)y in the mesoscopic RVE Oy is homogeneous and isotropic, i.e. the sti@ness of

matrix is independent of any position vectors such that

TL(M)meso ≡ L(M)

meso ≡L(M)mac = constant (96)

Since the matrix phase in Oy is homogeneous, there are no microscopic structures for O(M)y ;

and therefore Gz ≡ 0 and Az ≡ I in O(M)y . From (79)–(83), follows that

D(M)meso ≡ 0 (97)

For the reinforcement phase O(F)y , we assume that it consists of a two-phase composite (re-

inforcement �=f and matrix �=m) characterized by the microscopic RVE O(F)z , such that

O(F)z =O(Ff)

z ∪O(Fm)z and "(Ff ) + "(Fm) =1.

For simplicity, we further assume that damage occurs in the matrix only. The expres-sion of the damage variable (40), (70) and (71) can be further simpli;ed for this specialcase as

!0(x; y; z)=

!(M)(x); y∈O(M)

y

!(Fm)(x); y∈O(F)y and z∈O(Fm)

z

0; otherwise

(98)

Accordingly, the mesoscopic instantaneous sti@ness for O(F)y and O(M)

y in (94) becomes

L(F)meso = {TL(F)

meso − "(Fm)!(Fm) TL(Fm)meso}(I+D(F)

meso)− {L(F)meso − "(Fm)!(Fm)L(Fm)

meso}D(F)meso (99)

L(M)meso = (1−!(M))L(M)

mac (100)

where

D(F)meso = (I −!(Fm)B(Fm))−1!(Fm)C (Fm) (101)

The macroscopic phase average instantaneous sti@ness L(�)mac (�=F;M) given in (92) takes

the following form:

L(F)mac =

1

|O(F)y |

∫O(F)y

L(F)meso(Ay +GyDmac) dO (102)

L(M)mac =L(M)

meso( TA(M)y + TG(M)

y Dmac) (103)



The expression of Dmac can be further modi;ed by substituting (99) and (100)

Dmac =−{∫

O(F)y

GTyL

(F)mesoGy dO+

∫O(M)y

GTyL

(M)mesoGy dO

}−1

{∫O(F)y

GTyL

(F)mesoAy dO+

∫O(M)y

GTyL

(M)mesoAy dO

}(104)

The damage variable !(#) (#=M;Fm) is assumed to be a monotonically increasing functionof non-local phase deformation history parameter $(#) (see, for example, [2; 7; 17; 23–25])which characterizes the maximum deformation experienced throughout the loading history. Ingeneral, the evolution of phase damage at time t can be expressed as

!(#)(x; t)=f(〈$(#)(x; t)− T#(#)ini 〉+) and

@f($(#)(x; t))@$(#) ¿0 (105)

where #=M;Fm; the operator 〈〉+ denotes the positive part, i.e. 〈 • 〉+ = sup{0; •}; the phasedeformation history parameter $(#) is determined by the evolution of the nonlocal phase dam-age equivalent strain, denoted by T#(#)

$(#)(x; t)= max{T#(#)(x; &) | (&6t); T#(#)

ini

}(106)

where T#(#)ini represents the threshold value of the damage equivalent strain prior to the initiation

of phase damage. Based on the strain-based damage theory [25] we de;ned the damageequivalent strain T#(#) as

T#(#) =√

12 (F

(#) �(#)mac)TL(M)mac(F(#) �(#)mac); #=M;FM (107)

where �(#)mac represents the principal non-local phase strain vector, i.e. �(#)mac = [�(#)1 ; �(#)2 ; �(#)3 ]Tmac

and L(M)mac is the mapping of L(M)

mac in the corresponding principal strain directions; F(#) denotesthe weighting matrix aimed at accounting for di@erent damage accumulation in tension andcompression

F(#) =

h(#)1 0 0

0 h(#)2 0

0 0 h(#)3

; #=M;FM (108)

h(#)� ≡ h(�(#)� )=12+

1(atan[c(#)1 (�(#)� − c(#)2 )]; �=1; 2; 3 (109)

where c(#)1 and c(#)2 are constants selected to represent the contributions of each componentof the non-local principal phase strain to phase damage equivalent strain T#(#). Figure 3 illus-trates the inXuences of both constants. As an extreme case, when c(#)1 →∞ and c(#)2 = 0, the



Figure 3. Weight functions for principal phase average strains.

weight function reduces to h(�(#)� )= 〈�(#)� 〉+=�(#)� so that the compressive principal phase straincomponents have no contribution to the phase damage equivalent strain.The phase average strain in (107) is de;ned as

�(��)mac =1

|O(�)y |

1

|O(��)z |

∫O(�)y

∫O(��)z

�mic dOdO (110)

When damage occurs in the matrix phase only, ��= #=M;Fm, Equation (110) is applied tomatrix phases on the two scales. Exploiting (76) and (85), we get

�(Fm)mac =

1

O(F)y

∫O(F)y

( TA(Fm)z + TG(Fm)

zTD(F)meso)(Ay +GyDmac) dO�mac (111)

�(M)mac = ( TA(M)

y + TG(M)y Dmac)�mac (112)

The arctangent form evolution law [7]

Y(#) ≡ atan{a(#)(〈 T#(#) − T#(#)ini 〉+= T#(#)

0 )− b(#)}+ atan(b(#))(=2 + atan(b(#))

−!(#) = 0; #=M; fm (113)

is adopted, in which a(#); b(#) are material parameters; T#(#)0 denotes the critical value of the

strain history parameter beyond which the damage will develop very quickly.Based on the de;nition of the phase damage equivalent strain T#(#) in (107), the damage

evolution conditions can be expressed as

if T#(#) − $(#) = 0; $(#)¿0 ⇒ damage process: !(#)¿0 (114)



if T#(#) − $(#)¡0

or

if T#(#) − $(#) = 0; $(#) = 0

⇒ elastic process: !(#) = 0 (115)

where !(#) denotes the rate of damage.

5. COMPUTATIONAL ISSUES

In this section, we describe the computational aspects of the non-local piecewise constantdamage model for the two-phase material developed in Section 4. Due to the non-linearcharacter of the problem an incremental analysis is employed. Prior to the non-linear analysiselastic strain concentration factors, Gy(y) and Gz(z), are computed in the mesoscopic andmicroscopic RVEs, respectively, using the ;nite element method. Subsequently, the phaseaverage elastic strain concentration factor TA(M)

y and the damage distribution factor TG(M)y are

precomputed using (86), (78) and subsequently TA(Fm)z and TG(Fm)

z are evaluated.

5.1. Stress update (integration) procedure

Given: Displacement vector tumac; overall strain t T�mac; strain history parameters t$(M) andt$(Fm); phase damage variables t!(M) and t!(Fm); and displacement increment Zumac calcu-lated from the ;nite element analysis for the global problem. The left subscript denotes theincrement step, i.e. t+Zt is the variables in the current increment, whereas t is a convergedvariable from the last increment. For simplicity, we will omit the left subscript for the currentincrement, i.e. ≡ t+Zt , and use superscript # (=M;Fm) to denote the matrix phases in bothmesoscopic and microscopic RVEs.Find: Overall strain �mac; non-local strain history parameters $(M) and $(Fm); non-local phase

damage variables !(M) and !(Fm); overall stress mac and non-local phase stresses (F)mac and

(M)mac .The stress update procedure consists of the following steps:

Step (i): Calculate the macroscopic strain increment, ZT�mac =Zu; x, and update macroscopicstrains through T�mac = t T�mac + ZT�mac.Step (ii): Compute the damage equivalent strain T#(#) de;ned by (111) and (112) in terms

of t!(#) and T�mac.Step (iii): Check the damage evolution conditions (114) and (115). Note that $(#) is de;ned

by (106) and $(#) is integrated as Z$(#) =$(#) − t$(#). The procedure for strain and damagevariable updates is given as

IF: elastic process, T#(M)6 t$(M) and T#(Fm)6 t$(Fm), THEN

set !(#)= t!(#) and $(#)= t$(#) with #≡M;Fm

ELSE: damage process,

update for T#(#) and !(#) by solving the system of non-linear equations (113);

set $(#) = T#(#)



IF: elastic process in mesoscopic matrix phase; i.e. T#(M)¡t$(M), THEN

set !(M) = t!(M) and $(M) = t$(M);

update for T#(Fm) and !(Fm) by solving (113) with #=Fm only;

set $(Fm) = T#(Fm)

ENDIF

IF: elastic process in microscopic matrix phase, i.e. T#(Fm)¡t$(Fm), THEN

set !(Fm) = t!(Fm) and $(Fm) = t$(Fm);

update for T#(M) and !(M) by solving (113) with #=M only;

set $(M) = T#(M)

ENDIF

ENDIF

Since T#(#) is determined by the current phase average strains in the meso- and microscalematrix phases, which in turn depend on the current damage variable, it follows that the damageevolution laws in (113) comprise a set of non-linear equations for !(#). Using the Newton’smethod we construct an iterative process for the damage variables on both scales:

[k+1!(M)

k+1!(Fm)

]=

[k!(M)

k!(Fm)

]−

@Y(M)

@!(M)

@Y(M)

@!(Fm)

@Y(Fm)

@!(M)

@Y(Fm)

@!(Fm)

−1

k!(#)

[Y(M)

Y(Fm)

]k!(#)

(116)

The derivatives with respect to !(M) and !(Fm) in (116) can be evaluated by di@erentiating(113) with $(#) = T#(#) (#=M;Fm) provided that the damage processes occur on both scales,such that

@Y(M)

@!(M) = ,(M) @ T#(M)

@!(M) − 1;@Y(M)

!(Fm) = ,(M) @ T#(M)

@!(Fm) (117)

@Y(Fm)

@!(M) = ,(Fm) @ T#(Fm)

@!(M) ;@Y(Fm)

@!(Fm) = ,(Fm) @ T#(Fm)

@!(Fm) − 1 (118)

where ,(M) and ,(Fm) are given as

,(#) ={a(#)$(#)

0 }={(=2 + atan(b(#))}($(#)

0 )2 + {a(#)( T#(#) − T#(#)ini )− b(#)$(#)

0 }2; #=M;Fm (119)

The derivatives of the nonlocal damage equivalent strain T#(M) and T#(Fm) with respect to thedamage variables are presented in the appendix.



When damage process occurs only on one scale, either micro- or mesoscale, the set ofnon-linear equations (113) reduces to a single non-linear function and the Newton’s iterativemethod gives

k+1!(#) = k!(#) −(@Y(#)

@!(#)

)−1

Y(#)

∣∣∣∣∣k!(#)

; #=M or Fm (120)

where @Y(#)=@!(#) is given in (117) for #=M and in (118) for #=Fm.Step (iv): Update the non-local phase stresses, (F)

mac and (M)mac , using (91) with macroscopic

instantaneous sti@ness L(F)mac and L

(M)mac de;ned in (102) and (103), respectively. The macro-

scopic stress T mac can be ;nally obtained from (93).

5.2. Consistent tangent sti3ness

To this end we focus on the computation of the consistent tangent sti@ness matrix on themacro level. We start by taking material derivative of the incremental form of the constitutiveequation (91)

(F)mac = L(F)

mac�mac +L(F)mac�mac; (M)

mac = L(M)mac �mac +L(M)

mac �mac (121)

where L(�)mac can be obtained by taking time derivative of (102) and (103), which yields

L(F)mac =

1

|O(F)y |

∫O(F)y

{L(F)meso(Ay +GyDmac) +L(F)

mesoGyDmac} dO (122)

L(M)mac = L

(M)meso( TA

(M)y + TG(M)

y Dmac) +L(M)meso

TG(M)y + Dmac (123)

Following (99) and (100), the rate form of the mesoscopic sti@ness is given as

L(F)meso = L(F)

meso!(Fm); L

(M)meso = L(M)

meso!(M) (124)

where

L(F)meso = {(TL(F)

meso − L(F)meso)− "(Fm)!(Fm)(TL(Fm)

meso − L(Fm)meso)}D(F)

meso

− "(Fm){TL(Fm)meso + (TL(Fm)

meso − L(Fm)meso)D

(F)meso} (125)

L(M)meso =−L(M)

mac (126)

and D(F)meso in (125) is obtained by taking time derivative of D(F)

meso (101)

D(F)meso ≡ D(F)

meso!(Fm) = (I −!(Fm)B(Fm))−2C (Fm)!(Fm) (127)

From (124)–(127), the time derivative of Dmac de;ned in (104) can be expressed as

Dmac = D(M)mac!

(M) + D(Fm)mac !

(Fm) (128)



where

D(M)mac =−

{∫O(F)y

GTyL

(F)mesoGy dO+

∫O(M)y

GTyL

(M)mesoGy dO

}−1

{∫O(M)y

GTy L

(M)mesoGy dODmac +

∫O(M)y

GTy L

(M)mesoAy dO

}(129)

D(Fm)mac =−

{∫O(F)y

GTyL

(F)mesoGy dO+

∫O(M)y

GTyL

(M)mesoGy dO

}−1

{∫O(F)y

GTy L

(F)mesoGy dODmac +

∫O(F)y

GTy L

(F)mesoAy dO

}(130)

Substituting (124) and (128) into (122) and (123), the rate form of the macroscopic phasesti@ness becomes

L(F)mac = L(FM)

mac !(M) + L(FF)mac !

(Fm) (131)

L(M)mac = L(MM)

mac !(M) + L(MF)mac !(Fm) (132)

where

L(FM)mac =

1

|O(F)y |

∫O(F)y

L(F)mesoGy dOD(M)

mac (133)

L(FF)mac =

1

|O(F)y |

{∫O(F)y

L(F)meso(Ay +GyDmac) dO +

∫O(F)y

L(F)mesoGy dOD(F)

mac

}(134)

L(MM)mac = L(M)

meso( TA(M)y + TG(M)

y Dmac) +L(M)meso

TG(M)y D(M)

mac (135)

L(MF)mac =L(M)

mesoTG(M)y D(F)

mac (136)

To this end, the rate form of the constitutive equation (121) takes the following form:

(F)mac = L(FM)

mac �mac!(M) + L(FF)mac �mac!(Fm) +L(F)

mac�mac (137)

(M)mac = L(MM)

mac �mac!(M) + L(MF)mac �mac!(Fm) +L(M)

mac �mac (138)

In order to obtain !(M) and !(Fm), we make use of damage cumulative law (113) withconsideration of damage=elastic processes as described in Section 5.1. In the case of elasticprocess, !(M) =0 and=or !(Fm) =0. For damage process, !(M) and !(Fm) are obtained by taking



time derivatives of (113). The derivation is detailed in the appendix and the ;nal expressionscan be summarized as

!(M) = (w(M))T�mac; !(Fm) = (w(Fm))T�mac (139)

Substituting (139) into (137) and (138), we get the following relations between the rate ofthe overall strain and the non-local phase average stresses in the matrix phases:

(F)mac =˝(F)

mac�mac; (M)mac =˝(M)

mac �mac (140)

where

˝(F)mac = L(FM)

mac �mac(w(M))T + L(FF)mac �mac(w(Fm))T +L(F)

mac (141)

˝(M)mac = L(MM)

mac �mac(w(M))T + L(MF)mac �mac(w(Fm))T +L(M)

mac (142)

The overall consistent tangent sti@ness is constructed by substituting (140) into the rate formof the overall stress–strain relation (93)

mac =˝mac�mac (143)

˝mac = "(F)˝(F)mac + "(M)˝(M)

mac (144)

Finally, we remark that the integrals in the mesoscopic RVE, (129) and (130), have tobe evaluated at each Gaussian point in the global ;nite element mesh and for every loadincrement. This may lead to enormous computational complexity, especially when the ;nermesh is used for mesoscopic RVE. A close look at the above formulations reveals that thesource of the computational complexity stems from the history-dependent form of D(F)

meso. Toremedy the situation, we make use of Taylor expansion in (101) with respect to !(Fm) so thatthe history-dependent variable can be moved out of the integrals.

6. NUMERICAL EXAMPLES

We consider a woven composite material made of Blackglas=Nextel 2-D 5-harness satin weaveas shown in Figure 4. The fabric is made of 600 denier bundles of Nextel 312 ;bres, spacedat 46 threads per inch, and surrounded by Blackglas 493C matrix material. We will refer tothis material system as AF10. The micrograph in Figure 4 was produced at Northrop-Gruman.In this set of numerical examples, the mesoscopic RVE is de;ned as a two-phase material

(bundle=Blackglas 493C), while bundles consist of unidirectional ;brous composite (micro-scopic RVE). The phase properties of RVEs on both scales are summarized below:

Microscopic (bundle) RVE:

Blackglas 493C matrix: volume fraction "(Fm) =0:733; Young’s modulus E(Fm) =82:7GPa;Poisson’s ratio 0.26.NextelTM 312 ;bre: volume fraction "(Ff ) = 0:267; Young’s modulus E(Ff ) = 151:7 GPa;Poisson’s ratio 0.24.



Figure 4. BlackglasTM=Nextel ;ve-harnesssatin weave.

Figure 5. Damage evolution law on micro-and mesoscales.

Mesoscopic RVE (AF10 woven composite architecture, Figure 4):

Blackglas 493C matrix (with reduce sti@ness): volume fraction "(M) =0:548; Young’smodulus E(M) =26:2 Gpa; Poisson’s ratio=0:26.Bundle: volume fraction "(F) = 0:452; properties determined by the homogenized sti@nessof the microscopic RVE.

Note that the matrix phase in both RVEs is made of the same material. The sti@ness reductionof matrix phase in the mesoscopic RVE is due to initial inter-bundle cracks. The parameters ofthe damage evolution laws are chosen as a(#) = 7:2, b(#) = 16:3 and T#(#)

0 = 0:22 with #=M;Fm.For the matrix phase in the mesoscopic RVE, we choose T#(M)

ini = 0. For the matrix phase in themicroscopic RVE we de;ne T#(Fm)

ini = (√E(Fm)=E(M) − 1) T#(M)

0 =0:17. We assume that the twomatrix phases reach the critical value T#(#)

0 at the same time under the equal uniaxial strains,i.e. T#(Fm)− T#(Fm)

ini = T#(M)− T#(M)ini = T#(M)

0 with h(�(Fm)� )�(Fm)

� = h(�(M)� )�(M)

� where �=1; 2; or 3. Thedamage evolution laws are depicted in Figure 5.We further assume that the compressive principal strain components do not contribute to

the damage evolution. Thus, the parameters in (109) are chosen as c(#)1 = 105 and c(#)2 = 0 (#=M;Fm). The microscopic RVE is discretized with 351 tetrahedral elements as shown inFigure 6.The ;nite element mesh of the bundle phase (mesoscopic RVE) is depicted in Figure 7. The

;nite element mesh of the mesoscopic RVE contains 6857 tetrahedral elements. We utilizeour model to predict the ultimate strength under uniaxial tension and the four-point bendingtests.The stress–strain curve for the uniaxial tension in the weave’s plane is illustrated in Figure 8.To illustrate the importance of the three-scale model, we compare the results with the two-

scale model developed in Reference [7], where the damage occurs in the matrix phase whilethe bundle is assumed to remain elastic throughout the loading. It can be seen from Figure 8that the post-ultimate loading curve obtained with the two-scale method has a signi;cant



Figure 6. Microscopic RVE for the bundle. Figure 7. Mesoscopic RVE of AF10 woven com-posites (only bundle displayed).

Figure 8. Strain–stress curves for uniaxial tension in weave plane.

loading capacity due to bending of elastic bundles. On the other hand, the stress–strain curveof the three-scale model shows two sharp drops due to successive failures of the matrix phasesin the two scales. The numerical simulation using the three-scale model gives u=148 MPaat �u=3:1× 10−3 compared with the ultimate experimental stress=strain values in the uniaxialtension test of u=150± 7 MPa at �u=2:5× 10−3 ± 0:3× 10−3.The con;guration of the composite beam used for four-point bending test is shown in

Figure 9, where the loading direction (normal to the plane of the weave) is aligned along



Figure 9. Con;guration and FE mesh of composite beam.

Figure 10. Four-point bending Xexure strain-stress curves.

the Y -axis. The ;nite element model of the composite beam (global structure) is composedof 1856 brick elements. Numerical simulation results as well as the test data for four-pointbending problem are shown in Figure 10. Experiments have been conducted on ;ve identicalbeams [8] and the scattered experimental data of the applied load versus the displacement atthe point of load application in the beam are shown by the gray area in Figure 10. It canbe seen that the numerical simulation results are in good agreement with the experimentaldata in terms of predicting the overall behaviour and the dominant failure mode. Both thenumerical simulation and the experimental data predict that the dominant failure mode is inbending. Plates 1 and 2 illustrate the distribution of the phase damage in the composite beamcorresponding to the ultimate point in the load–displacement curve in Figure 10.



7. SUMMARY AND FUTURE RESEARCH DIRECTIONS

A multiscale non-local damage theory for brittle composite materials has been developed basedon the triple-scale asymptotic expansions of damage and displacement ;elds. The closed-formexpressions relating microscopic, mesoscopic and overall strains and damage have been de-rived. The damage evolution is stated on the smallest scale of interest and the non-localityis taken into account to alleviate the spurious mesh dependence by introducing the weightedphase average ;elds over the micro- and mesophases. Numerical results revealed the supe-rior performance of the three-scale method over the two-scale damage model [7] for wovencomposites.The present work represents only the ;rst step towards developing a robust simulation

framework for prediction of complex damage processes in composite materials. Several im-portant issues, such as interfacial debonding, coupled plasticity-damage e@ects [23; 35; 41–43],have not been accounted for in our model. Moreover, the assumptions of local periodicity anduniformity of macroscopic ;elds, which are embedded in our formulation, may yield inaccu-rate solutions in the vicinity of free edges or in the case of the non-periodic microstructures[10; 27; 30; 31; 38; 39]. These issues will be studied in our future work.

APPENDIX

In this section we present detailed derivations for @ T#(1)=@!(#) 1; #=M, Fm in (117) and (118),and !(#) in (139). We start with the ;rst derivative by di@erentiating (107) with respect to!(1)

@ T#(#)

@!(1) = (e(#))T@(F(#)�(#)mac)

@!(1) ; 1; #=M;Fm (A1)

where the vector e(#) ≡ [e(#)1 ; e(#)2 ; e(#)3 ]T takes following form

(e(#))T =1

2 T#(m)(F(#)�(#)mac)

TL(#)mac (A2)

With the de;nition of F(#) in (108), the derivative in (A1) can be expressed as

@(F(#) �(#)mac)@!(1) =

[@(h(#)1 �(#)1 )@!(1)

@(h(#)2 �(#)2 )@!(1)

@(h(#)3 �(#)3 )@!(1)

]T(A3)

Since the three components of the vector in (A3) have a similar form, we simply denote themby @(h(#)� �(#)� )=(@!(1)) with � = 1; 2; 3; and then by using (109) we have

@(h(#)� �(#)� )@!(1) =

{@h(#)�

@�(#)�

�(#)� + h(#)�

}@�(#)�

@!(1) ; � = 1; 2; 3 (A4)

where

@h(#)�

@�(#)�

=c(#)1 =(

1 + {c(#)1 (�(#)� − c(#)2 )}2(A5)



To this end we need to compute the derivative of each component of principal strain �(#)macwith respect to the damage variable !(1). The principal components of a second order tensorsatisfy Hamilton’s Theorem, i.e.

(�(#)� )3 − I1(�(#)� )2 + I2�

(#)� − I3 = 0 (A6)

where I1; I2 and I3 are three invariants of �(#)mac (or �(#)mac) which can be expressed as

I1 = �(#)ii = �(#)1 + �(#)2 + �(#)3 (A7)

I2 = 12(�

(#)ii �

(#)jj − �(#)ij �

(#)ji ) = �(#)1 �(#)2 + �(#)2 �(#)3 + �(#)3 �(#)1 (A8)

I3 = 16(2�

(#)ij �

(#)jk �

(#)ki − 3�(#)ij �

(#)ji �

(#)kk + �(#)ii �

(#)jj �

(#)kk ) = �(#)1 �(#)2 �(#)3 (A9)

where the tensorial notations are adopted for �(#)mac, i.e. �(#)mac ≡ �(#)ij . Di@erentiating (A6) with

respect to !(1) gives

@�(#)�

@!(1) = {3(�(#)� )2 − 2I1 �(#)� + I2}−1

{@I1@!(1) (�

(#)� )2 − @I2

@!(1) �(#)� +

@I3@!(1)

}(A10)

where the derivative of the invariants with respect to !(1) can be obtained by using (A7)–(A9)such that

@I1@!(1) ≡ (p(#)1 )T

@�(#)mac

@!(1) = !jk!ik@�(#)ij

@!(1) (A11)

@I2@!(1) ≡ (p(#)2 )T

@�(#)mac

@!(1) = (�(#)mm!jk!ik − �(#)ji )@�(#)ij

@!(1) (A12)

@I3@!(1) ≡ (p(#)3 )T

@�(#)mac

@!(1)

=(�(#)jk �

(#)ki − �(#)mm�

(#)ji − 1

2�(#)mn�

(#)nm!jk!ik +

12�(#)mm�

(#)nn !jk!ik

) @�(#)ij

@!(1) (A13)

Substituting (A10)–(A13) into (A4), gives

@(h(#)� �(#)� )@!(1) = (q(#)

� )T@�(#)mac

@!(1) ; � = 1; 2; 3 (A14)

where q(#)� is given as

q(#)� =

{@h(#)�

@�(#)�

�(#)� + h(#)�

}{3(�(#)� )2 − 2I1 �

(#)� + I2}−1{p(#)1 (�(#)� )2 − p(#)2 �(#)� + p(#)3 } (A15)



Finally, @ T#(#)=@!(1) in (A1) can be written in a concise form by using (A3) and (A14), whichyields

@ T#(#)

@!(1) = (ℵ(#))T(@�(#)mac

@!(1)

); 1; # = M;Fm (A16)

where

ℵ(#) =3∑

�=1e(#)� q(#)� ; # = M;Fm (A17)

and the derivative on the right-hand side of (A16) can be evaluated by di@erentiating (111)and (112) such that

d�(M)mac = s(M)

1 d!(M) + s(M)2 d!(Fm) + S(M) d�mac (A18)

d�(Fm)mac = s(F)1 d!(M) + s(F)2 d!(Fm) + S(F) d�mac (A19)

where for (A18)

@�(M)mac

@!(M) ≡ s(M)1 = TG(M)

y D(M)mac�mac (A20)

@�(M)mac

@!(Fm) ≡ s(M)2 = TG(M)

y D(F)mac�mac (A21)

@�(M)mac

@�mac≡ S(M) = TA(M)

y + TG(M)y Dmac (A22)

and for (A19)

@�(Fm)mac

@!(M) ≡ s(Fm)1 =

1

|O(F)y |

∫O(F)y

( TA(Fm)z + TG(Fm)

z D(F)meso)GyD(M)

mac dO�mac (A23)

@�(Fm)mac

@!(Fm) ≡ s(Fm)2 =

1

|O(F)y |

∫O(F)y

{ TG(Fm)z D(F)

meso(Ay +GyDmac)

+ ( TA(Fm)z + TG(Fm)

z D(F)meso)GyD(F)

mac} dO�mac (A24)

@�(Fm)mac

@�mac≡ S(Fm) =

1

|O(F)y |

∫O(F)y

( TA(Fm)z + TG(Fm)

z D(F)meso)(Ay +GyDmac) dO (A25)

The time derivatives of the phase damage variable, !(M) and !(Fm), can be obtained by takingtime derivatives of damage evolution law (113) and making use of (A18) and (A19). From(113) and assuming that damage processes occur on both scales, i.e. $(#) = T#(#)(# = M;Fm),



we have

!(M) = ,(M) T#(M) (A26)

!(Fm) = ,(Fm) T#(Fm) (A27)

where ,(M) and ,(Fm) are given in (119); T#(M) and T#(Fm) can be derived in a similar way asfor @ T#(#)=@!(1), which yields

T#(#) = (ℵ(#))T�(#)mac; # = M;Fm (A28)

where �(#)mac can be evaluated by using (A18) and (A19). From (A26) and (A18), we can get

n(M)1 !(M) + n(M)

2 !(Fm) + (r(M))T�mac = 0 (A29)

where n(M)1 ; n(M)

2 and r(M) are given as

n(M)1 = (ℵ(M))Ts(M)

1 − 1=,(M) (A30)

n(M)2 = (ℵ(M))Ts(M)

2 (A31)

(r(M))T = (ℵ(M))TS(M) (A32)

Similarly, from (A28) and (A19), we have

n(Fm)1 !(M) + n(Fm)

2 !(Fm) + (r(Fm))T�mac = 0 (A33)

where n(Fm)1 ; n(Fm)

2 and r(Fm) are given as

n(Fm)1 = (ℵ(Fm))Ts(Fm)

1 (A34)

n(Fm)2 = (ℵ(Fm))Ts(Fm)

2 − 1=,(Fm) (A35)

(s(Fm))T = (ℵ(Fm))TS(Fm) (A36)

Finally, solving for (A29) and (A33) yields

!(M) = (w(M))T�mac; !(Fm) = (w(Fm))T�mac (A37)

where

w(M) =n(M)2 r(Fm) − n(Fm)

2 r(M)

n(M)1 n(Fm)

2 − n(M)2 n(Fm)

1

(A38)

w(Fm) =n(Fm)1 r(M) − n(M)

1 r(Fm)

n(M)1 n(Fm)

2 − n(M)2 n(Fm)

1

(A39)



The Jacobian matrix (116)–(118) follows from (A16), (A20)–(A24), (A30), (A31), (A34)and (A35):

J =

[,(M)n(M)

1 ,(M)n(M)2

,(Fm)n(Fm)1 ,(Fm)n(Fm)

2

](A40)

ACKNOWLEDGEMENTS

This work was supported in part by the Air Force OFce of Scienti;c Research under grant numberF49620-97-1-0090, the National Science Foundation under grant number CMS-9712227, and SandiaNational Laboratories under grant number AX-8516.

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Plate 1. Distribution of !(M) at ultimate point.

Plate 2. Distribution of !(FM) at ultimate point.



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multiscale damage modelling for composite materials: theory and computational framework

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