JOURNAL OF THEORETICAL AND APPLIED MECHANICS 44, 3, pp. 553-583, Warsaw 2006 DAMAGE MODELLING FRAMEWORK FOR VISCOELASTIC PARTICULATE COMPOSITES VIA A SCALE TRANSITION APPROACH Carole Nadot Andre Dragon Laboratoire de Mécanique et de Physique des Matériaux, Ecole Nationale Supérieure de Mécanique et d’Aérotechnique/CNRS, Futuroscope-Chassenenil, Cedex, France; e-mail: carole.nadot@lmpm.ensma.fr; andre.dragon@lmpm.ensma.fr Herve Trumel Commissariat à l’Energie Atomique-Le Ripault, Monts, France; e-mail: herve.trumel@cea.fr Alain Fanget Centre d’Etudes de Gramat, Gramat, France; e-mail: alain.fanget@dga.defense.gouv.fr The aim of this paper is to pursue, in the wake of the work by Nadot-Martin et al. (2003), a non-classicalmicromechanical study and scale transition for highly filled particulate compositeswith viscoelasticmatrices. The present extension of a morphology-based approach due to Christoffersen (1983), carried forward to the viscoelastic small strain context byNadot-Martin et al. (2003), consists here in introducing a supplementarymechanism, namely damageby grain/matrixde- bonding. Displacement discontinuities (microcracks) on grain/matrix interfaces are first incorporated in a compatible way within geometric and kinematic hy- potheses regarding the grains-and-layers assembly of Christoffersen. Then, local field expressions as well as homogenized stresses are established and discussed for a given state of damage (i.e. for a given actual number of open and clo- sed microcracks) and using the hypothesis of no sliding on closed crack lips. A comparison with the results obtained for the sound viscoelastic composite by Nadot-Martin et al. (2003) allows to quantify the damage influence on local and global levels.At last, the basic formulation of themodel obtainedby scale transi- tion is completed by the second stage leading to a thermodynamically consistent formulation eliminating some superfluous damaged-induced strain-like variables related to open cracks. This second stage is presented here for a simplified sys- tem where delayed (viscoelastic) effects are (tentatively) neglected. It appears as a preliminary and crucial step for further generalization in viscoelasticity. Key words:micro-macro transition, heterogeneousmaterials,morphology, visco- elasticity, anisotropic damage, microcracking 554 C. Nadot et al. 1. Introduction This paper deals with a two step scale transition for modelling anisotropic damage behaviour of viscoelastic particulate composites, starting from the methodology initially proposed by Christoffersen (1983) for elastic bonded granulates. Thismethodology is built on geometric and kinematic hypotheses regarding a granular assembly with interconnecting layers constituting thus a consistent framework of a microstructural morphology pattern. The latter forms in fact an advantageous starting point for a micromechanical descrip- tion and further localization-homogenization procedure. The recent extension of the method, performed by Nadot-Martin et al. (2003) for composites invo- lving viscoelastic matrices, has confirmed its efficiency since it allows one to account for genuine viscoelastic interactions between constituents and for their macroscopic consequence – the ”long rangememory” effect. It is to be recalled that the presence of truly viscoelastic (i.e. viscous and elastic) coupled inte- ractions on themicroscale level and of associated global ”long rangememory” constitute two crucial criteria for relative evaluation of the pertinency of sca- le transition in the viscoelastic context (see e.g. Beurthey and Zaoui, 2000; Brenner et al., 2002). The present contribution attempts to further extend the technique in the presence of damage by grain-matrix debonding. It is to be emphasized that the resulting two step scale transition presented is done for a given diffuse distribution of open and closed interface microdefects (i.e. without coalescence). TheaimofSection 2 is to extend the techniquedue toChristoffersen (1983) – with its geometrical and kinematical ingredients, its averaging scheme and the relevant strategy of the approach of the local problem – in the presence of interfacial discontinuities. In such a way, Section 2 provides generalization, involving the damage mechanism mentioned, of the conceptual structure and relevant consistency requirements by Christoffersen. Section 3 deals with the solution to the localization-homogenization problem for composites with a vi- scoelastic matrix as it was done for the sound aggregate by Nadot-Martin et al. (2003), while here it is performed in the presence of interfacial damage. A discussion is put forward (Subsection 3.3) in order to quantify the coupling between damage and viscoelasticity regarding several aspects as e.g. local in- teractions and themacroscopic consecutive long rangememory effect, induced anisotropy, moduli recovery under crack closure. At this stage, local fields and global stresses involve a full set of internal relaxation variables (as for the so- undmaterial) and a new set of strain-like variables related to (discrete) sites of microcracking. In the same time, the reversible global moduli tensor lacks Damage modelling framework for viscoelastic... 555 crucial symmetries. The discussion at the end of Subsection 3.3 brings out the necessity of complementary analysis in order to express local open defects- related strains as functions of macroscopic state variables. Nevertheless, the simultaneous presence of viscoelastic variables anddamage related onesmakes the problem complex to deal with. This is why the above mentioned specific analysis, called the ’complementary localization-homogenization approach’ is conducted here (Section 4) for an elastic aggregate only (elastic grains and matrix + microcracks open/closed). This is a (necessary) crucial step, and the results obtained will constitute the basis for further genuine viscoelastic analysis. 2. Extension of Christoffersen’s method in presence of damage 2.1. Microstructure schematization Figure 1 shows a close-up schematic for grains separated bymatrix layers according to the scheme proposed byChristoffersen (1983) for a sound, i.e. an undamaged particulate composite. The grains are considered as polyhedral; any two of themare interconnected by a thinmaterial layer of a given uniform thickness (noted hα for the αth layer). The grain-layer interfaces are charac- terized by their orientation (nα for the αth layer). The spatial distribution of grains is accounted for through vectors linking grain centroids (dα for the αth layer). Moreover, no restriction is imposed on the grain size – the repre- sentation allows granulometric variations. As a result, such a schematization, givingmuch attention to the granular character, makes it possible to describe with sufficient accuracy the real initial microstructure geometry of an am- ple class of strongly charged particulate composites. Moreover, such a direct morphological description will allow one to introduce local interfacial defects (discontinuities) in a relatively direct and simplemanner (see Subsection 2.2). 2.2. Local problem approach 2.2.1. Kinematics The purpose consists here in introducing material discontinuities and re- lative displacement jumps in a compatible way with the original kinematical framework of Christoffersen (1983), and following step by step the strategy of this author. This implies more detailing of this framework. The latter is 556 C. Nadot et al. Fig. 1. Two neighbouring grains GA and GB with an interconnectingmaterial layer according to Christoffersen (1983) defined by four assumptions for the local displacement field that are recalled below: • The kinematics of grain centroid is characterized by the global (macro- scopic) displacement gradient ∇U =F. • Thegrains are supposedhomogeneously deformedandthe corresponding displacement gradient f0 is assumed to be common to all members of the Representative Volume Element (RVE). • Each interconnecting layer is subject to a homogeneous deformation, proper for the layer α under consideration. The corresponding displace- ment gradient is denoted fα for the αth layer. • Local disturbances at grain edges and corners are neglected on the basis of thinness of the layers (see surrounded zones in Fig.1). Following themethodology byChristoffersen, the first stage consists in in- terpreting the above hypotheses. The resulting three following equations cor- respond to relations (2.1)-(2.3) in the original paper by Christoffersen (1983). According to the first item, the centroid displacements uAi and u B i of two grains GA and GB separated by the layer α (see Fig.1) are given by uAi =u 0 i +Fijy A j u B i =u 0 i +Fijy B j (2.1) where u0 designates a global constant vector and yj (j = 1,2,3) represent local, cartesian coordinates in the RVE. Therefore and with the second as- sumption, the displacements of the grains GA and GB are uGAi (y)=u 0 i +(Fij −f 0 ij)y A j +f 0 ijyj (2.2) uGBi (y)=u 0 i +(Fij −f 0 ij)y B j +f 0 ijyj Damage modelling framework for viscoelastic... 557 At last, bymeans of the thirdassumption, thedisplacement field of the layer α is uαi (y)=u α i (y AB)+fαij(yj −y AB j ) (2.3) where AB stands for an arbitrary point on Iα1 , the interface of the layer α and the grain GA. For a sound material, further developments by Christoffersen consist in expressing uα and finally the displacement gradient fα of any layer α as functions of F, themacroscopic displacement gradient, f0, the grain displace- ment gradient and of morphological parameters of the layer fαij = f 0 ij +(Fik −f 0 ik)d α k nαj hα (2.4) To this aim, Christoffersen employs the continuity of the displacement field successively on Iα1 and I α 2 , namely, makes use of what happens at the gra- in/layer interfaces. These developments are here revisited to take into account the presence of discontinuities. Following the spirit of the author, it leads to consideration of the jumps (discontinuities) as data of the local problem and to search for uα and fα as functions of these. It is stressed that this is also the option taken by some works regarding homogenization of microcracked solids (Andrieux et al., 1986; Kachanov, 1994; Basista and Gross, 1997) where the local problem is solved by considering the displacement jumps (corresponding to cracks) as the relevant data. So, consider the presence of a discontinuity on the first interface Iα1 of the layer α. According to the previous remark, the corresponding displacement discontinuity vector, denoted bα1i , is considered as a data of the local problem. Nevertheless, its form cannot be arbitrary. Indeed, the linearity (according to kinematical assumptions) of the displacement field leads to assignment of a linear form to bα1i , namely bα1i (y AB)= fαD1ij y AB j + c αD1 i (2.5) where the tensor fαD1 and the vector cαD1 are homogeneous and stand for data characterizing the crack. So, in the presence of a discontinuity on the first interface Iα1 , instead of researching u α i by means of the continuity condition uαi (y AB)=uGAi (y AB) for anypoint AB on Iα1 as itwasdonebyChristoffersen for the soundmaterial, one has to find it in such a way that uαi (y AB)=uGAi (y AB)+ bα1i (y AB) ∀yAB ∈ Iα1 (2.6) with bα1i given by (2.5). By reporting (2.6) where u GA i and b α1 i are expres- sed using (2.2)1 and (2.5) respectively, in (2.3), the displacement field in the 558 C. Nadot et al. layer α is obtained in the following form uαi (y)=u 0 i +(Fij −f 0 ij)y A j +(f 0 ij −f α ij)y AB j +f α ijyj +f αD1 ij y AB j + c αD1 i (2.7) As the above expression must be independent of the choice of the point AB, the condition (f0ij −f α ij +f αD1 ij )m α j =0 (2.8) must hold for any tangent mα to the grain-layer interface Iα1 . It follows that f α must have the form fαij = f 0 ij +f αD1 ij +g α i n α j (2.9) where gα denotes a homogeneous vector. Furthermore, expression (2.7) for uαi becomes independent of the choice of the point AB, and is finally given by uαi (y)=u 0 i +(Fij −f 0 ij)y A j +f 0 ijyj +g α i z AB +fαD1ij yj +c αD1 i (2.10) where zAB = nαi (yi − y AB i ) is the distance from the debonded interface I α 1 . Except for the starting point consisting in satisfying (2.6) – in the place of the displacement continuity for any point AB on Iα1 – the foregoing reasoning in the presence of damage constitutes a simple extension of that advanced by Christoffersen (1983) for the sound composite (see for comparison relations (2.3)-(2.6) in the reference quoted). For the soundmaterial, furtherdevelopments byChristoffersen concern the determination of the homogeneous vector gα regarding the continuity on the second interface Iα2 . In thepresence of a discontinuity on thefirst interface I α 1 , the (basic) hypothesis stipulating a homogenous displacement gradient for two grains separated by the layer α – making two opposite faces deform in the manner to stay parallel – makes necessary to introduce simultaneously a discontinuity on the second interface Iα2 , see Fig.2. Fig. 2. A layer with cracks at its boundaries Damage modelling framework for viscoelastic... 559 For the same reasons as for bα1i on the first interface, one assigns a linear form to the displacement discontinuity vector, noted bα2i , across I α 2 bα2i (y BA)= fαD2ij y BA j + c αD2 i (2.11) where the tensor fαD2 and the vector cαD2 are homogeneous and stand for new data of the local problem that could be a priori considered as different from fαD1 and cαD1. The vector gα is then searched in such a way that uαi (y BA)=uGBi (y BA)+ bα2i (y BA) ∀yBA ∈ Iα2 (2.12) be satisfied with bα2i given by (2.11) and u α i , u GB i expressed via (2.10) and (2.2)2, respectively. In this manner, one mathematically proves that f αD1 = fαD2. It is stressed that this relation is not a choice but a consequence of themethodology assumed. In the following, this displacement gradient will be denoted fαD. The finally obtained form of gα allows one to expressfα in the following manner fαij = f 0 ij +(Fik −f 0 ik)d α k nαj hα +fαDij +(c αD2 i − c αD1 i ) nαj hα (2.13) The supplementary terms in (2.13) compared to (2.4) represent a specific con- tributionof twomicrocracks located at theboundariesof thedebonded layer α considered. At this stage, all the ingredients of the methodology by Christoffersen in order to express fα have been exploited, and it appears necessary to recapitu- late different implications of the kinematical hypotheses.The latter, consisting in a piecewise linearization of the microscopic displacement field, impose first that the displacement discontinuity vectors across the debonded interfaces are necessarily affine functions of spatial coordinates. Since an affine function can not be equal to zero on a segment and different from zero elsewhere, the- re is either (total) decohesion almost everywhere or there is no decohesion. This means that the simplified (piece-wise linear) kinematics put forward by Christoffersen (1983) does not allow one to account for partial decohesion of grain/matrix interfaces. Moreover, the hypothesis stipulating the identical di- splacement gradient f0 for opposite grains separated by a given layer imposes either no decohesion, or simultaneous decohesion of its both interfaces. Phy- sically speaking, for grains of different size and, in particular, two opposite interfaces of different geometry and area, it is clear that a single crack along one of the interfaces (one-sided decohesion) would be more realistic than two simultaneous events. Unfortunately, the kinematics framework of theChristof- fersen pattern does not allow for such one-sided local decohesion. So, in order 560 C. Nadot et al. tomake thedouble decohesion acceptable, one completes the geometrical basis of the Christoffersen theory by adding the following assumption: • any two opposite interfaces are supposed tohave comparable geometrical properties (shape and area). Since two opposite interfaces remain parallel duringmotion, such an assump- tion regarding their geometry gives some physical justification to the fact that when the first one is debonded, the second is too. We are aware that the lat- ter assumption, by adding a supplementary constraint to the schematization, leads to restriction of the class of particulate composite microstructures that could bemodeledwithChristoffersen’s original geometrical scheme.Neverthe- less, it seems to be a necessary compromise to legitimate the Christoffersen kinematical framework in the presence of damage. Having supposed the above simplification and considering the parallelism of interfaces in the course of deformation, it is reasonable to consider that the mean displacement discontinuity vectors across the interfaces Iα1 and I α 2 of the debonded layer α defined by 〈bα1i 〉Iα1 = f αD ij y B1 j + c αD1 i = b α1 i (y B1) (2.14) 〈bα2i 〉Iα2 = f αD ij y B2 j + c αD2 i = b α2 i (y B2) are opposite. B1 and B2 are the centres of the interfaces I α 1 and I α 2 , respec- tively. In the following, one attempts to simplify expression (2.13) obtained for f α involving for instance the terms fαD, cαD1 and cαD2 considered as data characterizing microcracks at the boundaries of the debonded layer α consi- dered. The relevant motivation is the advantage of reducing the number of entities that will characterize the effects of microcracks inside the RVE in the expressions further obtained for local fields and the homogenized stress-strain relation. To this aim, one introduces now the following assumption concerning the vectors cαD1 and cαD2, forwhich – it shouldbe emphasized –no condition has been imposed by the Christoffersenmethodology: • The contribution of constant vectors cαD1 and cαD2 in displacement jumps bα1i (y) = f αD ij yj + c αD1 i and b α2 i (y) = f αD ij yj + c αD2 i across I α 1 and Iα2 , respectively, are considered negligible (i.e. null). The latter hypothesis consists in fact in choosing particular, simple and linear forms of displacement jumps considered as data of the local problem. Damage modelling framework for viscoelastic... 561 General forms (2.5) and (2.11) are thus replaced (withmoreover fαD ≡ fαD1 = f αD2) by bα1i (y)= f αD ij yj b α2 i (y)= f αD ij yj (2.15) In this way, the displacement gradient fα for a debonded layer α takes the simplified expression fαij = f 0 ij +(Fik −f 0 ik)d α k nαj hα +fαDij (2.16) Moreover, the following simple relationship exists now between 〈bα1〉Iα 1 = −〈bα2〉Iα 2 and the unique term fαD representing the two microcracks effect on fα by subtracting (2.14)2 from (2.14)1 and suppressing the contribution of cαD1 and cαD2) 〈bα1i 〉Iα1 =−〈b α2 i 〉Iα2 =− 1 2 fαDij c α j c α j = y B2 j −y B1 j (2.17) In (2.17), cα designates the vector connecting the centres B1 and B2 of two opposite interfaces (see Fig.2). The displacement gradient fα for any layer α whose both interfaces are cohesive, obtained by using the continuity of displacements on the grain/layer interfaces according to theChristoffersenmethodology, remains given by (2.4) fαij = f 0 ij +(Fik −f 0 ik)d α k nαj hα (2.18) In view of (2.16) for a debonded layer and (2.18) for a cohesive one, the strain as well as rotation is controlled by F, the macroscopic displacement gradient, f0, the grain displacement gradient, but also by geometrical features of the layer α under consideration. Onemay emphasize the physical relevance of such a dependence on local morphological parameters: it allows one to ac- count for themicrostructure effect on deformationmechanisms of thematrix. In this way, theChristoffersen kinematical framework offers away to take into account some strain heterogeneity in the matrix phase in the homogenized behaviour estimation. It is stressed that taking into account field fluctuations in phases represents actually a crucial challenge in micromechanics especially for non linear and/or time-dependent behaviour (see e.g. Ponte Castañeda, 2002; Moulinec et Suquet, 2003). It is to be noted that the strain heteroge- neity in the matrix is also influenced by damage via the dependence of fα on f αD (for debonded layers). At last, due to the assumption neglecting the de- scription of complex effects in interlayer zones (see surroundedzones inFig.1), each layer is in theChristoffersen framework subjected to loading uniquely via 562 C. Nadot et al. its adjacent grains. In this way, there is no direct interaction between layers; the transmission through the grains-and-layers assembly strongly involves the grains as expressed through the presence of f0 in (2.16) and (2.18). 2.2.2. Micro-macro relations The focus ishereonestablishingmicro–macro relationships essential for the ultimate solution to the local problem, i.e. for determination of the unknown f 0 according to the procedure outlined further. In the same spirit as in the kinematic description, one follows step-by-step – in the presence of damage – the correspondingmethod by Christoffersen (1983) for the soundmaterial. In order to ensure compatibility between local motion in accordance with the above kinematical description and global motion characterized by F, the following average relation, the counterpart of relation (2.13) in Christoffersen (1983), including now the contribution of material discontinuities, is imposed Fij =(1− c)f 0 ij + 1 V ∑ α fαijA αhα+ 1 V ∑ k (∫ Ik1 bk1i n k j da− ∫ Ik2 bk2i n k j da ) (2.19) where V represents the volume of grains and layers, Aα is the projected area of the αth layer and c = V−1 ∑ αA αhα is the ratio of the layer volume to the volume V . The subscripts α, k under summation symbols designate sum- mations over all layers contained in the RVE and over layers with debonded interfaces, respectively. After somemanipulations using (2.18), and (2.16), for f α for the layers α whose both interfaces are cohesive, respectively debon- ded, and (2.15) to express bk1 and bk2, one may prove that the geometrical condition established by Christoffersen for the soundmaterial, namely 1 V ∑ α dαin α jA α = δij (2.20) remains necessary in the presence of damage to ensure the compatibility be- tween local andglobalmotions, i.e. relationship (2.19). In (2.20) δij is theKro- necker’s symbol. In the work by Christoffersen (1983), geometrical condition (2.20) related to the composite morphology may be seen as a discriminating criterion of applicability for the Christoffersen-type approach. Thus, it seems coherent to retrieve such a condition in the presence of interfacial damage (cracks). The principle of macro-homogeneity for the RVE subjected to uniform tractions is given by Christoffersen (1983) – see Eq. (3.1) in the reference qu- oted.The corresponding expression extendedhere andaccounting for interface discontinuities takes the following form Damage modelling framework for viscoelastic... 563 ΣijFji =(1− c)σ 0 ijf 0 ji+ 1 V ∑ α σαijf α jiA αhα+ (2.21) + 1 V ∑ k (∫ Ik 1 σijn k jb k1 i da− ∫ Ik 2 σijn k jb k2 i da ) for anyarbitrary Fand f0 andany stressfield σ, statically admissiblewith the macroscopic stress Σ. σ0 and σα represent average stresses in the grains and in the αth layer, respectively.After somemanipulationsusing (2.16) and (2.18) to express fα, (2.15) for bk1 and bk2, and taking successively two particular values for f0, namely f0 =F and f0 =0 as it was done for the soundmaterial, it can be shown from (2.21) that the system established by Christoffersen          Σij = 〈σij〉V =(1−c)σ 0 ij + 1 V ∑ α σαijA αhα Σij = 1 V ∑ α tαi d α j = 1 V ∑ α tαjd α i t α j =σ α kjn α kA α (2.22) remainsvalid in thepresence of damage. In (2.22), tα represents the total force transmitted through the interfacial layer. Note that, although the first avera- ging is ”classically” exploited in the micromechanics, the second one remains specific to the Christoffersen-type approach: stresses are seen from a granular viewpoint as forces transmitted fromgrain to grainby layers acting as contacts zones. For the debonded layer α, two cases must be considered.When cracks located at its boundaries are open (i.e 〈bα1i 〉Iα1n α i = −〈b α2 i 〉Iα2 n α i > 0) then tα = 0. When they are closed ( i.e 〈bα1i 〉Iα1 n α i = −〈b α2 i 〉Iα2n α i = 0) and in the framework of this exploratory study, it is supposed that no sliding is allowed, so that tα is integrally transmitted. For a cohesive layer α, tα is considered as fully conveyed as it was in the case of all layers in the absence of damage. According to the Christoffersen methodology, the following consists in se- arching f0 in such a way that the real stress field, namely this associated to the strain field by local constitutive laws, satisfies system (2.22). 3. Application to viscoelastic composite materials The class of heterogeneous materials considered is that of particulate compo- site materials which can be considered as composed of isotropic linear-elastic grains embedded in a viscoelastic matrix (see Nadot-Martin et al., 2003). At 564 C. Nadot et al. first,mean features of thematrix viscoelastic law are recalled. This constitutes a preliminary step before going on to find f0 and to establish the full set of localization relations, as it was done for the soundmaterial by Nadot-Martin et al. (2003), but here is done in the presence of damage by grain/matrix de- bonding. Then, the macroscopic homogenized stress is derived from (2.22)1. Finally, a discussion is presented in order to quantify the damage influence on the local and global scale levels. 3.1. Viscoelastic law for the matrix Thematrix occupying each elementary layer α is considered as viscoelastic and isotropic according to the thermodynamically consistent internal variable representation given byNadot-Martin et al. (2003). The dissipative process re- lated to viscoelastic relaxation is accounted for via the symmetric, strain-like, tensorial internal variable γ. The free energy per unit volume and correspon- dingly the total stress are decomposed into two terms, a reversible function of the total strain ε, and a viscous function of γ. The reversible and viscous stresses are obtained by partial derivation of the free energy with respect to ε and γ. The evolution of γ which can be interpreted as inelastic-viscous or otherwise as ’delayed elastic’ strain is given by law (3.3)1 employing, for simplicity, a single relaxation time τ w(ε,γ)= 1 2 ε :L(e)` : ε+ 1 2 γ :L(v) :γ (3.1) σ=σ(r)+σ(v) =L(e)` : ε+L(v) :γ (3.2) γ̇+ 1 τ γ = ε̇ γ(t=0)=0 (3.3) d(v) =σ(v) : (ε̇− γ̇)= 1 τ γ :L(v) :γ ­ 0 L (e)` and L(v) are fourth-order tensors of the elastic and viscous moduli for the matrix. 3.2. Solution to the local problem and expression of homogenized stress The purpose is to resolve system (2.22) in order to get f0 by considering grains as isotropic linear-elastic and thematrix layers as viscoelastic according to the model presented in Subsection 3.1. All the grains have here identical moduli denoted by L0.Mechanical properties of thematrix (moduli L(e)`,L(v) and relaxation characteristic τ) are considered as homogeneous, namely the Damage modelling framework for viscoelastic... 565 same for all layers. Consequently, as εα = Symfα is uniform over the αth layer (see kinematical assumption 3 in Subsection 2.2.1), the corresponding viscoelastic relaxation γ introduced by (3.3)1 is also uniform for a given re- laxation state; it is denoted by γα. It is also the case for all thermodynamic quantities involved in the matrix model. Fromamethodological viewpoint, calculations to determine f0 from (2.22) in the presence of damage are similar to those required for the soundmaterial (see for comparisonSubsection 3.2 inNadot-Martin et al., 2003). Nevertheless, it is to be recalled that the summation in (2.22)2 is here to be considered over layers either cohesive or with closed cracks. One begins by inserting the microscopic laws formulated in terms of displacement gradients rather than in terms of strain in system (2.22). Then, (2.18) is substituted for fα for the layers α whose both interfaces are cohesive, while (2.16) is put for the layers debonded. Finally, by using geometrical condition (2.20) and eliminating Σij between both equations of (2.22), one obtains the form of f0 relevant to the local problem in the presence of damage as follows f0ij =(Id 1−B′ −1 :A′)ijklFlk ︸ ︷︷ ︸ f 0(r) ij + −B′ −1 ijuvL (v) mukl ( 1 V ∑ α′ Πα ′ vmγ α′ lkA α′hα ′ + δvm 1 V ∑ β γ β lk Aβhβ ) ︸ ︷︷ ︸ f 0(v) ij + (3.4) −B′ −1 ijuvL (e)` mukl ( 1 V ∑ f Πfvmε fD lk Afhf + δvm 1 V ∑ β ε βD lk Aβhβ ) ︸ ︷︷ ︸ f 0(d) ij with, for any layer α, Πα = δ−dα ⊗nα/hα and where the tensors A′, B′ degraded by the presence of damage are defined as follows A′ijkl = 〈L (e) ijkl 〉V −L (e)` mjkl (δim−Dim) Aijkl = 〈L (e) ijkl 〉V −L (e)` ijkl (3.5) B′ijkl =Aijkl−L (e)` mjkl (δim−Dim)+L (e)` mjnl (T imkn−Dimkn) (3.6) T ijkl = 1 V ∑ α dαin α jd α kn α l Aα hα (3.7) Dij = 1 V ∑ β d β in β jA β Dijkl = 1 V ∑ β d β in β jd β k n β l Aβ hβ (3.8) 566 C. Nadot et al. In the above relations, the subscripts α, α′, β and f under summation sym- bols denote summations over all layers, layers either cohesive or with closed cracks, layers with open cracks only and layers with closed cracks only. In (3.4), εβD = SymfβD, εfD = SymffD and one has assumed invertibility of B ′ with respect to the identity tensor Id1 defined by Id1ijkl = δilδjk. The form of (3.4) represents a remarkable decomposition into a reversible term f0(r), depending linearly on the macroscopic gradient F, a viscous one f0(v), func- tion of variables γα for α = 1, . . . ,N – with N being the total number of layers inside the RVE – and a damage-induced one f0(d) involving the full set {εkD}= {εfD}∪{εβD} related to the effect of any kind of cracks (closed and open) inside the RVE. These three contributions depend on the damage state through the tensors D and D (see A′,B′). The same can be done for fα after employing (2.18) and (2.16) for a cohesive and debonded layer, respectively . At last, the local strain fieldwith respect to y in the grains andmatrix layers is obtained in the following additive form ε(y)=C(y) :E+ε(v)(y)+ε(d)(y)+ { εαD for y∈ debonded layer α 0 elsewhere (3.9) Cijkl(y)=        C0ijkl(D,D)= (Id− Id :B ′−1 :A′)ijkl for y∈ grains Cαijkl(D,D)= Idijkl+ −Idijuv(B ′−1 :A′)vmklΠ α mu for y∈ layer α, ∀α (3.10) ε (v) ij (y)=    ε 0(v) ij ({γ α},D,D)= Idijklf 0(v) lk for y∈ grains ε α(v) ij ({γ α},D,D)= Idijuvf 0(v) vm Π α mu for y∈ layer α, ∀α (3.11) ε (d) ij (y)=    ε 0(d) ij ({ε kD},D,D)= Idijklf 0(d) lk for y∈ grains ε α(d) ij ({ε kD},D,D)= Idijuvf 0(d) vm Π α mu for y∈ layer α, ∀α (3.12) with E= SymF. As expected, the degraded elastic strain concentration ten- sor satisfies 〈C〉V = Id – with Id being the classical fourth-order identity tensor defined by Idijkl = (δikδjl + δilδjk)/2 – and the fields ε (v) and ε(d) the properties 〈ε(v)〉V = 0 and 〈ε (d)〉V = 0, respectively. At last, the overall (average) stress is derived from (2.22)1 Σ=L(D,D) :E+Σ(v)({γα},D,D)+Σ(d)({εkD},D,D) (3.13) L(D,D)= 〈L(e)〉V −A :B ′−1 :A′ (3.14) Damage modelling framework for viscoelastic... 567 Σ(v) =A : f0(v)({γα},D,D)+L(v) : 1 V ∑ α γαAαhα (3.15) Σ(d) =A : f0(d)({εkD},D,D)+L(e)` : 1 V ∑ k εkDAkhk (3.16) 3.3. Discussion In order to discuss the forms of results on micro and macro levels in the presenceofdamage, itmaybeconvenient to compare themwith thoseobtained by Nadot-Martin et al. (2003) for the soundmaterial. Table 1.Localization results and expression of the homogenized stress for the soundmaterial (Nadot-Martin et al., 2003) Local strain field: ε(y)=C(y) :E+ε(v)(y) Cijkl(y)= { C0ijkl =(Id− Id :B −1 :A)ijkl for y∈ grains Cαijkl = Idijkl− Idijuv(B −1 :A)vmklΠ α mu for y∈ layer α,∀α ε (v) ij (y)=    ε 0(v) ij ({γ α}) = Idijklf 0(v) lk for y∈ grains ε α(v) ij ({γ α}) = Idijuvf 0(v) vm Π α mu for y∈ layer α, ∀α Homogenized stress: Σ=L :E+Σ(v)({γα}) L= 〈L(e)〉V −A :B −1 :A Σ(v) =A : f0(v)({γα})+L(v) : 1 V ∑ α γαAαhα . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . with: A= 〈L(e)〉V −L (e)` Bijkl =Aijkl−L (e)` ijkl +L (e)` mjnl T imkn T ijkl = 1 V ∑ α dαin α jd α kn α l Aα hα f 0(v) ij ({γ α})=−B−1ijuvL (v) mukl 1 V ∑ α Παvmγ α lkA αhα At the local level, onemay observe that the degraded elastic concentration tensor given by (3.10) has the same form as C for the soundmaterial with A′ 568 C. Nadot et al. and B′ replacing A and B. Moreover, the strain field (3.9) for any point in grains or layers depends through the term ε(v) on the full set of relaxations {γα}. The internal variable γα representing memory of the αth layer clearly indicates viscoelastic interactions between the full set ofmatrix layers and the set of grains in the RVE. In the same manner as for the sound material, this dependence directly results from the term f0(v) which, bymeans of (2.16) and (2.18) appears in the expression of fα and, therefore, in that of the strain field, see (3.11). Nevertheless, onemaynote that themore complex structure of f0(v) (see (3.4)) in the presence of damage shows that the damage tends to enhance the complexity of viscoelastic interferences taken into account. Moreover, the strain field (3.9) for any point in grains or layers (cohesive or not) depends on damage through the tensors D and D (appearing in A′ and B′) but also on the term ε(d) depending on the full set {εkD}= {εβD}∪{εfD} related to the effect of any kind of cracks (open or closed) inside theRVE.The latter depen- dence results from the term f0(d) which, for the same reasons as f0(v), appears in the expression of the strain field, see (3.12). This is not surprising when reported to comments formulated at the end of Subsection 2.2.1 concerning the transmission inside the aggregate. In particular, for the debonded layer α, onemay distinguish two kinds of contribution of damage to the corresponding ”overall” strain in the layer: a ”local” one, εαD, related to microcracks loca- ted at its own boundaries (its ”own” defects) and a ”non-local” one, εα(d), involving the effect (via f0(d)) of the whole set of microcracks inside the RVE, in other words the effect of microcracks at the interfaces of other layers in addition to the influence of those at its own boundaries. At the global level, the overall stress givenby (3.13) is split into a reversible part and a viscous one influenced by damage through D and D and comple- ted (when compared to that for the sound material) by the damage-induced stress Σ(d). Note that the forms of viscous stress for the sound and damaged materials are the same, the difference is in the detailed expression of f0(v) relevant to local viscoelastic interactions depending here on the damage state in addition to the set {γα} acquiring the status of macroscopic viscoelastic internal variables. The first terms of (3.15) and (3.16), namely A : f0(v) and A : f0(d), correspond respectively to themacroscopic consequences of viscoela- stic interactions and ”non-local” damage effects. It can be seen that A′, B′ and therefore C and L(D,D), are degraded only by open cracks via D and D, see (3.8). This is due to the assumption of no sliding on closed crack lips (infinite friction coefficient). Being tensorial by nature, D and D allow one to account for the damage induced anisotropy. By dependingon the vectors dβ andnot only on the crack normal vectors nβ, the damage tensors D and D emerging fromthepresentmorphology-basedmodel- Damage modelling framework for viscoelastic... 569 ling take into account the granular character of the composite microstructure considered. Moreover, since D is not symmetric, the damage induced aniso- tropy may be very complex. It is stressed that the scale transition at stake accounts also for the initial morphology and internal organization of consti- tuents through the presence of the fourth-order structural tensor T given by (3.7) in the local and homogenized expressions (via B′ for the damagedmate- rial and B for the sound one). The reader may refer to Christoffersen (1983), where it is shown that T reflects material texture and irregularities in the grain shape and in the layer thickness. In this way, the Christoffersen-type approach extension in the presence of damage, applied here to a viscoelastic composite, allows one to take into account, in a general 3D context, coupling effects between the primary anisotropy, if any, (via T) and the secondary, damage-induced one (via D and D). At last, when the number of open cracks is equal to zero, i.e. for exclusively closed cracks, the reversible part of the local strain field and, furthermore, the homogenized reversible moduli become equal to those of the sound material. Theviscouspart ε(v) of the local strainfieldbecomes equal to that of the sound material as well as Σ(v) at the macroscopic scale. The term ε(d), depending only on {εfD}, accounts for the blockage effect of closed cracks inside the RVE as the corresponding macroscopic damage-induced stress Σ(d). Thus, themodelling is potentially capable of describingunilateral effects. In the limit case where there is no crack, the local and global responses are identical to those obtained for the soundmaterial. This principal backwards confrontation shows that themethodological coherence is beingpreservedbetween the sound and damaged composites. As macroscopic state variables, one has already mentioned the whole set {γα} accounting for the relaxation state of the composite. Homogenized stress (3.13) conveys also a full set {εkD} = {εβD}∪{εfD}. Let examine now the status of {εkD}. Remarking, according to (3.4), that f 0(d)({εkD},D,D)=−(B′ −1 :L(e)`)ijkl 1 V ∑ β ε βD lk Aβhβ ︸ ︷︷ ︸ f 0(d)1({εβD},D,D) + (3.17) −B′ −1 ijuv :L (e)` mukl 1 V ∑ f Πfvmε fD lk Afhf ︸ ︷︷ ︸ f 0(d)2({εfD},D,D) onemaydiscern that the respective contributions in thedamage-induced stress Σ(d) of open and closed cracks are clearly additive. Indeed, when detailing 570 C. Nadot et al. somewhat (3.16) on the basis of partition (3.17), one obtains Σ(d) =A : f0(d)1+L(e)` : 1 V ∑ β εβDAβhβ ︸ ︷︷ ︸ Σ (d)1({εβD},D,D) +A : f0(d)2+L(e)` : 1 V ∑ f εfDAfhf ︸ ︷︷ ︸ Σ (d)2({εfD},D,D) (3.18) In (3.18), the set {εfD} acquires the status of macroscopic internal variables accounting for the distortion due to the blockage of closed cracks inside the RVE, and Σ(d)2 appears as the corresponding residual stress.At themicrosco- pic level, εβD represents for a layer β the ”local” contribution of open cracks located at its own boundaries to its total strain. It seems natural to think that the crack opening depends on the macroscopic strain E and therefore εβD as well. So, each εβD cannot a priori be considered as a macroscopic variable independent of E. This is confirmedwhen noting that L(D,D), given by (3.14), does not exhibit all symmetries required for the effective reversi- ble moduli tensor suggesting that Σ(d)1 must depend, through {εβD}, on E. This remark shows that further analysis is necessary to explicit the depen- dence of each εβD on E, that – via the term Σ(d)1 in the expression of Σ – will complete the linear part L(D,D) : E of Σ and, therefore the form of reversible moduli. This is the aim of the next section where a complementary localization-homogenization procedure is advanced in order to express the lo- cal strain induced in a layer β by open cracks at its interfaces as a function of E,D,D and local geometrical features of the layer concerned. In the spirit of a gradual, step-by-step approach to difficulties, this procedure is developed hereafter in the context of pure elasticity. It is a necessary and preliminary stage for further generalization in viscoelasticity. 4. A complementary localization-homogenization procedure for an elastic aggregate 4.1. Preliminaries Thepurpose of this Section is to express εβD for an arbitrary layer β with open cracks at its ownboundaries as a function ofmacroscopic state variables, the global strain E inparticular. In the framework of the exploratory character of the approach advanced in this paper, the developments put forward below are performed in the elastic context, namely by considering the grains and the matrix (i.e. the set of layers) as linear elastic and isotropic. Damage modelling framework for viscoelastic... 571 The first step consists in the determination of the overall free-energy for the elastic heterogeneous material as the volume average of the local energy. After some calculations using the localization relations (see (3.9) to (3.12) where the viscous field ε(v) is suppressed), employing geometrical statement (2.20) and themajor symmetry of B, the overall free energy is obtained in the following additive form W = 〈1 2 ε :L(e) : ε 〉 =W1(E,D,D)+W2(E,{εβD},D,D)+ +W3(E,{εfD},D,D)+W4({εβD},D,D)+ (4.1) +W5({εβD},{εfD},D,D)+W6({εfD},D,D) where W1(E,D,D)= 1 2 E : 〈tC :L(e) :C〉V :E (4.2) 〈tC :L(e) :C〉V = 〈L (e)〉V + t G :B :G− [A :G+ tG :A] (4.3) G=B′ −1 :A′ andwith W i for i=2, . . . ,6 given inAppendixA. W2 and W3 are explicitly linear in E and linearwith respect to each εβD and εfD. W5 depends linearly on each εβD. The terms W i for i=4,5,6 do not depend explicitly on E. A quick comparison between the homogenized free energy and the expres- sion of the global stress given by (3.13), where the viscous stress Σ(v) is suppressed, shows immediately that the explicitly quadratic term in E of W , i.e. W1, cannot give by derivation the linear term L(D,D) : E of the stress except for only closed cracks inside the RVE. This provides a new confirma- tion of the dependence of each εβD on E. A non-trivial problem consists then in quantifying the relationship between each εβD (for an arbitrary layer β) and E. Note that the expression for εβD is not a priori postulated so that the strategy proposedcanbeviewedas a complementary ”localization” procedure. To this aim, the thermodynamic framework is used as a guide. From a thermodynamic viewpoint, the macroscopic stress must derive from the overall free energy with respect to E. Consequently, the relation εβD = εβD(E) has to be searched in such a way that the global elasticity law Σ= ∂W ∂E (4.4) be explicitly verified with Σ = 〈σ〉V and W = 〈w〉V given by (3.13), with Σ(v) suppressed, and by (4.1). 572 C. Nadot et al. Since W6 is independent of E, ∂W/∂E = ∑5 i=1∂W i/∂E. Assuming, for each layer β with open cracks at its boundaries, linearity of the relation εβD = εβD(E) and its independence of the set {εfD}, one can discern that: 1) W2, W4 and Σ(d)1 do not depend on {εfD}; 2) W3 and W5, depending on the set {εfD}, are linear functions of E (explicitly for W3, implicitly thro- ugh each εβD for W5). Thus, for such a strategy (∂W3/∂E) + (∂W5/∂E) must correspond to the residual stress due to the blockage of closed crack lips. Precisely, one must have explicitly ∂W3 ∂E + ∂W5 ∂E =Σ(d)2 (4.5) ∂W1 ∂E + ∂W2 ∂E + ∂W4 ∂E =L(D,D) :E+Σ(d)1 In fact, while searching a relation between εβD and E directly (in order to assure that Σ = ∂W/∂E = ∑5 i=1∂W i/∂E be satisfied) appears too complex when examining the detailed expressions of W i for i = 1, . . . ,5, it is easier to search it by satisfying (4.5)1. It is stressed that ensuring (4.5)1 is sufficient to ensure simultaneously (4.5)2 and more generally (4.4). But the converse is not true. Indeed, searching a solution to (4.5)2 would not be sufficient to ensure simultaneously (4.4) since the solutionwould not take into account the contribution of the set {εfD}. 4.2. Solution The aim is to find a linear relation εβD = εβD(E) for each layer β with open cracks at its boundaries, satisfying equation (4.5)1 with W 3, W5 and Σ(d)2 given by (A.2), (A.4) and (3.18). From (A.2)-(A.4) and due to symme- tries of A,L(e)` and A′, the expression for (∂W3/∂E)+(∂W5/∂E) is obtained as follows ∂W3 ∂Eij + ∂W5 ∂Eij =Aijklf 0(d)2 lk +L (e)` ijkl 1 V ∑ f ε fD lk Afhf + (4.6) + {[ tGijuv − t (∂f 0(d)1 uv ∂Eij )] (B′−B)vukl+L (e)` vunl 1 V ∑ β Π β kn ∂εβDuv ∂Eij Aβhβ } f 0(d)2 lk where one recognizes Σ(d)2 (see the first line) given by (3.18). So, (4.5)1 is satisfied for any damage configuration, in particular for any f0(d)2, only if the term between braces in (4.6) is null. Moreover, by using the expressions for Damage modelling framework for viscoelastic... 573 B and B′ (see Table 1 and (3.6), respectively) in order to develop B′−B, it follows L (e)` vunl 1 V ∑ β Π β kn ∂εβDuv ∂Eij Aβhβ = (4.7) =− [ tGijuv − t (∂f 0(d)1 uv ∂Eij )] L (e)` munl 1 V ∑ β dβv nβm hβ Π β kn Aβhβ with (see (3.17)) ∂f 0(d)1 uv ∂Eij =−B′ −1 uvabL (e)` bats 1 V ∑ β ∂ε βD st ∂Eij Aβhβ (4.8) Considering complex forms (4.7) and (4.8), it appears useful to exploit the consequences of the linearity of the relation εβD = εβD(E) supposed for any layer βwith open cracks at its boundaries. It implies the linearity of f0(d)1, i.e. theexistence of amacroscopic fourth-order tensor K′ suchthat ∂f 0(d)1 uv /∂Eij = K′uvij. Note that the ”non local” effects of open cracks in the heterogeneous mediumrepresentedby f0(d)1 will bedescribedbya linear functionof E. Thus, the final form of (4.7) to satisfy is L (e)` vunl 1 V ∑ β Π β kn ∂εβDuv ∂Eij Aβhβ =−t(G−K′)ijuvL (e)` munl 1 V ∑ β dβv nβm hβ Π β kn Aβhβ (4.9) After calculations (see Appendix B for details), one obtains a linear rela- tionship between εβD and E, i.e. solution to (4.9) ε βD ij =−Idijmud β v nβm hβ (G−K′)uvklElk+r βD ij (4.10) The constant tensor with respect to E, rβD, represents a residual strain indu- ced in the layer β by residual opening of the cracks at its boundaries when E=0. It is reasonable to think that it is a function of D and D. In view of (B.5), if (4.10) is satisfied for every βth layer with open cracks at its boundaries, then (4.9) is verified and furthermore (4.5)1 also, but the converse is not true. In other words, from amathematical viewpoint, the solu- tion is not unique. Nevertheless, the strategy employed in Appendix B, based on the assumption that the relations εβD = εβD(E) have the same form for every βth-layer, is in accordance with the Christoffersen framework. More- over, the result seems pertinent since the ”local” strain induced in the layer β 574 C. Nadot et al. by open cracks at its interfaces depends on damage through D and D appe- aring explicitly in A′ and B′, but also on the geometrical features of the layer considered. It remains now to determine the expression for K′. The latter has to satisfy ∂f 0(d)1 uv /∂Eij = K ′ uvij with ∂f 0(d)1 uv /∂Eij given by (4.8) and with ε βD being represented by (4.10). After some manipulations, it follows K ′ = [B′+ t(A′−A)]−1 : t(A′−A) :G (4.11) where one has assumed the invertibility of B′ + t(A′ −A) with respect to the identity tensor Id1. Once relation (4.10) obtained for εβD is accepted as pertinent, the resulting form (4.11) of K′ is unique. Furthermore, the ”non- local” effects of open cracks inside theRVEare now represented by (see (3.17) with (4.10)) f 0(d)1(E,D,D)=K′ :E−B′−1 :L(e)` : 1 V ∑ β r βDAβhβ (4.12) The second term in (4.12) characterizes the specific contribution of the re- sidual opening of cracks when E = 0. It will be denoted by f0(d)1Res. One may emphasise the complex structure of K′ involving elastic moduli of both constituents, the tensor T (via B′) related to thematerial initial morphology and internal organization and D, D. This is in perfect accordance with the role of f0(d)1. Remark: In addition to its linearity, one has supposed the independence of the relation εβD = εβD(E) on the set {εfD}. Such an assumption seems reasonable when recalling that εβD represents the local strain induced in a layer β by open cracks at its own interfaces and that the influence of the distorsion of closed cracks on the strain of this layer is already taken into account throughthenon local term f0(d)2.Note thatwithout suchan assumption, the disconnection in (4.5) would be no longer valid, so that feasibility of obtaining an analytical solution would remain questionable considering the complexity of various couplings involved. 4.3. Macroscopic stress-strain relation With (4.10)-(4.11), onemay formulate thewhole elasticmodel giving local and global responses of the elastic damaged composite in terms of macrosco- pic variables E, {εfD} and damage tensors D, D. For simplicity, one just reports below themacroscopic stress-strain relation obtained by derivation of Damage modelling framework for viscoelastic... 575 the overall free energy given by (4.1)-(4.2) and (A.1) to (A.5), in which (4.10) is substituted for εβD Σ= ∂W ∂E = L̃(D,D) :E+Σ(d)({εfD},D,D) (4.13) Σ(d) =A : f0(d)2+L(e)` : 1 V ∑ f εfDAfhf ︸ ︷︷ ︸ Σ (d)2({εfD},D,D) + (4.14) +A : f0(d)1Res+L(e)` : 1 V ∑ β r βDAβhβ ︸ ︷︷ ︸ Σ (d)1(D,D) L̃(D,D)= 〈L(e)〉V + t(G−K′) : [H− t(A′−A)− (A′−A)] : (G−K′) (4.15) Hijkl =L (e)` mjnl Dimkn−Bijkl (4.16) In (4.14), the expression for Σ(d)2 representing themacroscopic residual stress induced by the blockage of closed cracks remains unchanged, and Σ(d)1 cor- responds now to the residual stress induced by the residual opening of (open) cracks. Direct calculation of Σ using (3.13) with Σ(v) suppressed, (3.14) and (3.18) in which (4.10) is substituted for εβD, leads to Σ= 〈σ〉V = [〈L (e)〉V − t A ′ : (G−K′)] :E+Σ(d)({εfD},D,D) (4.17) with Σ(d) given by (4.14). With (4.11), one may fortunately prove that L̃(D,D) = 〈L(e)〉V − t A ′ : (G−K′) so that (4.17) and (4.13) are equiva- lent. This equivalence shows that solution (4.10)-(4.11) satisfies explicitly, as expected, (4.4), i.e. 〈σ〉V = ∂〈w〉V/∂E.Moreover, the degraded elasticmoduli tensor L̃(D,D) has now all symmetries (notably themajor symmetry) contra- rily to L(D,D) given by (3.14). This result concerning the indicial symmetries of the effective moduli obtained bymeans of the complementary localization- homogenization procedure stands as the proof for its efficiency. From a the- oretical viewpoint, equation (4.15) for the effective moduli seems to be more appropriate than the one in (4.17) since it clearly shows the major symme- try. In the limit cases, where there is no open crack inside the RVE, i.e. for only closed cracks or for the sound material, one may observe that the effec- tive moduli L̃(D,D) correspond, as expected, to those of the soundmaterial: L = 〈L(e)〉V −A : B −1 : A. This result constitutes a new confirmation of the coherence of the complementary localization-homogenization approach. 576 C. Nadot et al. At last, a comment should be made regarding practical determination of the variables {εfD} accounting for the frictional locking effect of closed cracks. Considering thehypothesis of no slidingon closed crack lips, amicrocrack is, in thepresent framework,necessarily openbeforebeing closed.Moreover, εfD for a layerwith closed cracks at its boundariesdoesnot evolve as longas the cracks remain closed.Therefore, the components of εfD maybecalculated fromthose of εβD given by (4.10) at the crack closure, precisely when the layer under consideration – initiallywith open cracks – becomes a layerwith closed cracks. The crucial problem is to simultaneously ensure the homogenized energy and stress-response continuity in spite of discontinuity of effective moduli, see e.g. Dragon andHalm (2004). The respective conditions formulated in the context of the scale transition at stake should also give tools to express the tensors rβD in function of D,D and geometrical features of the layer β. Such a strategy is necessarily associatedwith the formulation of rigorous criteria of unilaterality. This is the aim of present investigations concerning the unilateral effect (i.e. opening/closure transition modelling). 5. Conclusion A non-classical homogenization method that constitutes an extension of the Christoffersen approach for both viscoelasticity and damage by grain/matrix debonding is presented. The discontinuities have been first introduced in a compatible waywith theChristoffersen framework (geometry and kinematics) and following the strategy of this author. It is shown that the direct patter- ning of the material microstructure and associated local kinematics due to Christoffersen can accommodate the grain/layer discontinuities with just one additional assumption regarding geometry of opposite interfaces of a given layer, introduced in order to make acceptable their simultaneous decohesion (if any) resulting from the hypothesis of the identical displacement gradient f 0 for grains. Moreover, the Christoffersen’s morphology and kinematics fra- mework, extended in the presence of damage, offers an advantage to take into account some strain heterogeneity in thematrix phase in estimation of homo- genized behaviour (see in (2.16)-(2.18) the dependence of fα onmorphological features of the layer α and on fαD if it is debonded). The solution to the localization-homogenization problem obtained in Sec- tion 3 for composites with a viscoelastic matrix, in the presence of interfacial damage, allows one to discern several crucial features. First, the scale transi- tion leads to natural emergence of two macroscopic damage tensors involving granular aspects – a second-order one and a fourth-order one. They describe Damage modelling framework for viscoelastic... 577 damage-induceddegradation effects and inducedanisotropy.These two tensors – in addition to the textural tensor T related to the initial morphology and internal organisation of constituents – allow one to account, in a general 3D context, for coupling the primary anisotropy with the damage-induced one. Local viscoelastic interactions and the macroscopic consecutive long range memory effect are clearly shown to be affected by microcracking. Other re- markable features as recovery of some properties of the soundmaterial under microcrack closure may also be discerned through the comparison with local and global relations obtained for the undamaged material by Nadot-Martin et al. (2003). In particular, in the absence of discontinuities, the correspon- ding expressions for micro- and macro-scale levels reduce to the ones for the sound composite, confirming thus that the methodological coherence is being preserved between both cases (sound and damaged composite) and endorsing specific hypotheses regarding the damaged aggregate and relevant generaliza- tion.At last, the advanced scale transition doesnotmake use of thehypothesis of non-interacting cracks (each microcrack is not considered as isolated in an infinitemedium) so that some ”non-local” damage effectsmay be identified at both scales. One should realize in the same time that this does notmean that defects interact in the sense pointed out by e.g. Kachanov (1994). Indeed, on can note in particular that the influence of ”non-local” damage effects within the RVE, embodied by the term f0(d), on the strain of any layer is just pon- dered bymorphological features of the layer considered, and does not involve any distance separating this layer from ”remote” defects. Some superfluous damage-induced strain-like variables related to open cracks (i.e. {εβD}) are still explicitly present at this stage. Their status aswell as someother properties of homogenized expressions indicate that further ana- lysis is needed to obtain a net and thermodynamically consistent formulation. The latter has been achieved via complementary localization-homogenization analysis under notable simplification regarding behaviour of constituents: only elastic-damaged systemhas been considered inSection 4.The local strain εβD induced in any layer β with open cracks at its boundaries is thus expressed as a function of the macroscopic variable E, damage tensors D, D and geo- metrical features of the layer at stake in such a way that the homogenized stress derives explicitly from the global free energy. By giving access to the effective moduli in a direct and thermodynamically consistent manner, such a preliminary analysis performed in the elastic context will stand as a reference for furtherviscoelasticity-damage complementary localization-homogenization approach. The latter will also include replacement of the set of relaxation in- ternal variables {γα} by a single variable as it was done in Nadot-Martin et al. (2003) for the soundmaterial. 578 C. Nadot et al. Further work will include – apart from fully viscoelasticity-damage com- plementary study – a detailed treatment of unilateral phenomena and damage evolution. It is to be noted that the strategy regarding modelling of the uni- lateral effect proposed in the elastic context at the end of Section 4, remains valid for viscoelastic constituents. Acknowledgement The authors gratefully acknowledge the financial support from the French Mini- stry of Defence. A. Appendix This appendix presents detailed expressions of the terms W i for i=2, . . . ,6 figuring in homogenized free energy (4.1) obtained for the damaged elastic aggregate in Subsection 4.1. W2(E,{εβD},D,D)=Euv [ (A− tG :B−B′)vuklf 0(d)1 lk + (A.1) −tGvurmL (e)` srkl 1 V ∑ β ∓βmsε βD lk Aβhβ ] W3(E,{εfD},D,D)= (A.2) =Euv { [A− tG : (B−B′)]vuklf 0(d)2 lk +L (e)` vukl 1 V ∑ f ε fD lk Afhf } W4({εβD},D,D)= (1 2 f0(d)1uv Bvukl+L (e)` mlvu 1 V ∑ β Π β km εβDuv A βhβ ) f 0(d)1 lk + (A.3) +L (e)` vukl 1 2V ∑ β εβDuv ε βD lk Aβhβ W5({εβD},{εfD},D,D)= (A.4) = [ f0(d)1uv (B−B ′)vukl+L (e)` mlvu 1 V ∑ β Π β km εβDuv A βhβ ] f 0(d)2 lk W6({εfD},D,D)= (A.5) = f0(d)2uv (1 2 B−B′ ) vukl f 0(d)2 lk +L (e)` vukl 1 2V ∑ f εfDuv ε fD lk Afhf Damage modelling framework for viscoelastic... 579 B. Appendix This appendix deals with the determination of a linear relation between εβD and E that satisfies differential equation (4.9) established in Subsection 4.2. Since there is only one equation for M unknown functions, where M denotes thenumber of layerswith open cracks at their boundaries, it ismathematically impossible to find these functions in a unique manner. One proposes here a reasonable way based on the assumption that the above mentioned relations have the same structure for every βth layer. Equation (4.9) is satisfied if L (e)` vunl Π β kn ∂εβDuv ∂Eij =−t(G−K′)ijuvL (e)` munl dβv nβm hβ Π β kn (B.1) for every βth layer with open cracks at their boundaries. Consider a single βth layer. This particular layer verifies (B.1) if L (e)` vukl ∂εβDuv ∂Eij =−t(G−K′)ijuvL (e)` mukl dβv nβm hβ (B.2) Using the invertibility of L(e)`, (B.2) becomes equivalent to ∂εβDrs ∂Eij =−Idrsmud β v nβm hβ (G−K′)uvij (B.3) Finally, one obtains εβD in terms of E by solving (B.3) ε βD ij =−Idijmud β v nβm hβ (G−K′)uvklElk+r βD ij (B.4) where rβD is a constant tensor with respect to E. This simple calculation provides a linear form for εβD in function of E that, when satisfied for every layer β with open cracks at their boundaries, leads to (4.9) (B.4) ∀β⇔ (B.2) ∀β⇒ (B.1) ∀β⇒ (4.9) (B.5) Significant symbols Morphological parameters and tensors hα – thickness of αth layer nα – unit normal vector defining orientation of αth layer 580 C. Nadot et al. dα – vector linking centroids of grains interconnected by αth layer Aα – projected area of αth layer cα – vector connecting centres of two opposite boundaries of αth layer c – ratio of layer volume to volume V of grains and layers Πα – second-order tensor accounting for geometry of αth layer T – fourth-order structural tensor accounting formorphology and inter- nal organization of constituents inside the Representative Volume Element (RVE) Kinematical quantities F – global (macroscopic) displacement gradient f 0 – displacement gradient of grains f 0(r) – reversible part of f0 f 0(v) – viscous part of f0 accounting for viscoelastic interactions between constituents f 0(d) – damage-inducedpart of f0 accounting for ”non local” effects of whole set of defects inside RVE f 0(d)1 – ”non local” effects of open defects inside RVE f 0(d)2 – ”non local” effects of closed defects inside RVE f α – displacement gradient of αth layer f αD – contribution of defects located at boundaries of a debonded layer α to its displacement gradient fα uGA,uGB – displacement field of grains GA and GB, respectively uα – displacement field of αth layer bα1,bα2 – displacement discontinuity vectors across interfaces Iα1 and Iα2 of a debonded layer α Strain-like quantities E – global (macroscopic) strain tensor γα – viscoelastic internal second-order tensorial variable accoun- ting for relaxation state of αth layer ε – local strain tensor field ε(v),ε(d) – respectively viscous and damage-induced parts of local stra- in field ε0(v),ε0(d) – respectively viscous anddamage-inducedparts of strain ten- sor for grains εα(v),εα(d) – respectively viscous anddamage-inducedparts of strain ten- sor for αth layer εα = Symfα Damage modelling framework for viscoelastic... 581 εαD = SymfαD – for a debonded layer α, ”local” contribution of its own defects to its strain εα = Symfα εβD = SymfβD – for a layer β with open defects at its boundaries, ”local” contribution of its own defects to its strain εβ = Symfβ εfD = SymffD – for a layer f with closed defects at its boundaries, ”local” contribution of its own defects to its strain εf = Symff. Internal variable accounting for distor- sion due to blockage of corresponding closed defects r βD – residual strain induced ina layer β byresidualopening of defects at its boundaries when E=0 Stresses Σ – global (homogenized) stress tensor Σ(v),Σ(d) – respectively viscous and damage-induced parts of homoge- nized stress tensor Σ Σ(d)1 – contribution of open defects to the damage-induced stress tensor Σ(d) Σ(d)2 – contribution of closed defects to the damage-induced stress tensor Σ(d), macroscopic residual stress tensor correspon- ding to blockage of closed defects inside RVE σ0,σα – average stress tensors ingrains and in αth layer, respectively tα – total force transmitted through interfacial αth layer Local and global moduli and essential tensors involved L (e)`,L(v) – fourth-order tensors of elastic and viscous moduli for matrix L 0 – fourth-order tensor of elastic moduli for grains C – elastic concentration tensor field C 0,Cα – elastic concentration tensor for grains and αth layer, respec- tively L – reversible global moduli tensor for sound material (without damage) L(D,D) – ”incomplete” reversible global moduli tensor in presence of damage L̃(D,D) – reversible global moduli tensor (after complementary analy- sis) A,B – fourth-order tensors involved in local and global response expressions 582 C. Nadot et al. A ′,B′ – damage degraded forms (via D and D) of the tensors A and B D,D – second-order and fourth-order damage tensors Identity tensors and particular operators δ – symmetric second-order identity tensor Id – classical fourth-order identity tensor defined by Idijkl =(δikδjl+δilδjk)/2 Id 1 – fourth-order identity tensor defined by Id1ijkl = δilδjk 〈·〉V – volume average : – tensorial double contraction defined by: Cijkl =AijmnBnmkl if A,B and C are fourth-order tensors, Cij =AijklBlk if A is fourth-order tensor and B second-order one References 1. Andrieux S., Bamberger Y., Marigo J.J., 1986, Un modèle de matériau microfissuré pour les bétons et les roches, J. Méca. Théor. Appl., 5, 3 471-513 2. Basista M., Gross D., 1997, Internal variable representation of microcrack induced inelasticity in brittle materials, Int. J. Damage Mechanics, 6, 300-316 3. Beurthey S., Zaoui A., 2000, Structuralmorphology and relaxation spectra of viscoelastic heterogeneousmaterials,Eur. J. Mech. A/Solids, 19, 1-16 4. BrennerR.,MassonR., CastelnauO., ZaouiA., 2002,A”quasi-elastic” affine formulation for the homogenized behaviour of nonlinear viscoelastic po- lycrystals and composites,Eur. J. Mech. A/Solids, 21, 943-960 5. Christoffersen J., 1983,Bondedgranulates,J.Mech. Phys. Solids,31, 55-83 6. DragonA.,HalmD., 2004,DamageMechanics. SomeModelling Challenges, Centre of Excellence AMAS,Warsaw 7. Kachanov M., 1994, Elastic solids with many cracks and related problems, Advances in Applied Mechanics, 30, 259-445 8. MoulinecH., SuquetP., 2003, Intraphase strain heterogeneity in non linear composites: a computational approach,Eur. J. Mech A/Solids, 22, 751-770 9. Nadot-Martin C., Trumel H., Dragon A., 2003, Morphology-based ho- mogenization method for viscoelastic particulate composites, Part I: Viscoela- sticity sole,Eur. J. Mech. A/Solids, 22, 89-106 Damage modelling framework for viscoelastic... 583 10. Ponte Castañeda P., 2002, Second-order homogenization estimates for non linear composites incorporating field fluctuations: I – Theory., J. Mech. Phys. Solids, 50, 737-757 Modelowanie uszkodzenia w granulowanych kompozytach lepkosprężystych przy pomocy podejścia wieloskalowego Streszczenie Celem tej publikacji jest sformułowanie wieloskalowegomodelu mikromechanicz- nego dla granulowanychkompozytówowysokim stopniu upakowania inkluzji w osno- wie lepkosprężystej. Przedstawionymodel, będący rozwinięciemmorfologicznego po- dejściaChristoffersena (1983) iNadot-Martin i in. (2003)w zakresiemałychodkształ- ceń lepkosprężystych, polega na wprowadzeniu do analizy dodatkowegomechanizmu uszkodzenia – mikropękania na granicy inkluzji i osnowy. Mikroszczeliny na granicy inkluzji i osnowy uwzględniono w hipotezie geometrycznej i kinematycznej metody Christoffersena. Następnie, wyznaczono lokalne oraz uśrednione pola naprężenia dla zadanego stanu uszkodzenia (tzn. dla zadanej liczby otwartych i zamkniętych mi- kroszczelinprzy pominięciu poślizgównapowierzchniachmikroszczelin zamkniętych). Porównanie zwynikami uzyskanymi przezNadot-Martin i in. (2003) dla nieuszkodzo- nego kompozytu lepkosprężystego pozwoliło na określenie wpływu uszkodzenia na poziomie lokalnym i globalnym. Na koniec, podstawowy model wieloskalowy uzu- pełniono o drugą część sformułowania, która pozwoliła usunąć pewne nadmiarowe odkształcenia związane z mikroszczelinami otwartymi, czyniąc cały model termody- namicznie spójnym.Ta druga częśćmodelu wieloskalowego jest przeprowadzonaprzy założeniu upraszczającym, polegającym na (tymczasowym) pominięciu efektów lep- kosprężystych. Manuscript received December 22, 2005; accepted for print April 4, 2006