Mihail Garajeu, Pierre Suquet. To cite this version: HAL Id: hal

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1 On the inluence o local luctuations in volume raction o constituents on the eective properties o nonlinear composites. Application to porous materials Mihail Garajeu, Pierre Suquet To cite this version: Mihail Garajeu, Pierre Suquet. On the inluence o local luctuations in volume raction o constituents on the eective properties o nonlinear composites. Application to porous materials. Journal o the Mechanics and Physics o Solids, Elsevier, 007, 55 (4), pp < /j.jmps >. <hal > HAL Id: hal Submitted on 4 Jan 008 HAL is a multi-disciplinary open access archive or the deposit and dissemination o scientiic research documents, whether they are published or not. The documents may come rom teaching and research institutions in France or abroad, or rom public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diusion de documents scientiiques de niveau recherche, publiés ou non, émanant des établissements d enseignement et de recherche rançais ou étrangers, des laboratoires publics ou privés.

2 On the inluence o local luctuations in volume-raction o constituents on the eective properties o nonlinear composites. Application to porous materials M. Gărăjeu 1,, P. Suquet L.M.A./C.N.R.S., 31 Chemin Joseph Aiguier, 1340 Marseille Cedex 0, France Abstract Composite materials oten exhibit local luctuations in the volume raction o their individual constituents. This paper studies the inluence o such small luctuations on the eective properties o composites. A general asymptotic expansion o these properties in terms o powers o the amplitude o the luctuations is given irst. Then, this general result is applied to porous materials. As is well-known, the eective yield surace o ductile voided materials is accurately described by Gurson s criterion. Suitable extensions or viscoplastic solids have also been proposed. The question adressed in the present study pertains to nonuniorm distributions o voids in a typical volume element or in other words to the presence o matrix-rich and pore-rich zones in the material. It is shown numerically and analytically that such deviations rom a uniorm distribution result in a weakening o the macroscopic carrying capacity o the material. Key words: Inhomogeneous material, Viscoplastic material, Homogenization, Field luctuations, Porous material, Gurson criterion 1 Introduction The macroscopic properties o heterogeneous materials, such as composites or porous materials, depend in an essential manner on their microstructure. Corresponding author. addresses: mihai.garajeu@univ.u-3mrs.r (M. Gărăjeu), suquet@lma.cnrs-mrs.r (P. Suquet). 1 Present address: Aix-Marseille Univ, EA 596, Marseille, F-13397, France Preprint submitted to Journal o the Mechanics and Physics o Solids 9 July 007

3 One o the most important descriptor o this microstructure is the volume raction o the individual constituents. While the volume raction o the phases is macroscopically constant, its luctuations at a local level are essential in understanding a number o problems, including scattering by heterogeneities or racture o composite materials. For instance, random, but nonuniorm, distributions o particles arise during processing o metal-matrix composites. Clustering o particles may play a role in the stress concentrations which eventually lead to breakage o particles and racture o the composite. Clustering eects in particle-reinorced composites have been simulated numerically by Segurado et al. (003). They ound that the eect o reinorcement clustering is weak in the linear elastic regime but more pronounced, although limited, in the nonlinear plastic regime. They also ound a dramatic increase in the number o particles broken in the clusters. This increase is due to the modiication o the local ields due to clustering. Similar eects have been observed in voided materials. For instance, Rossoll et al. (00) observed in an experimental study on nuclear steels that clusters o precipitates constitute, ater debonding, initiation sites or void growth and coalescence. These clusters are characterized by a local volume-raction which is higher than the average volume-raction in the material. Motivated by this observation, Bilger et al. (005) investigated void clustering eects in ideally plastic materials. Their study conirms that while void clustering has almost no eect on the linear elastic properties o voided materials, it has a noticeable eect on their plastic properties. Considering three dierent types o void distributions, random distribution without clusters, distributions with disconnected clusters and distributions with connected clusters, they ound that the low surace o materials containing a random distribution o voids is a strict outer bound or the low surace o materials containing disconnected clusters o voids which itsel is an outer bound or materials in which the void clusters are connected. The degeneracy o the ideally plastic model explains probably why the inluence observed by Bilger et al. (005) is larger than that observed by Segurado et al. (003) whose study concerned a matrix with signiicant isotropic hardening. Bilger et al. noticed in addition a strong inluence o the stress state on the clustering eect. More speciically the eect o clusters was larger or hydrostatic loadings than or pure shear loadings. To conclude, there is a indisputable body o results, both experimental or computational, showing the role o local volume raction luctuations on the eective properties o heterogeneous materials. This eect, enhanced by the material nonlinearity, is probably more pronounced or porous materials than or reinorced composites and depends on the loading. The aim o the present study is to assess this problem by analytical means. Clustering eects can be seen as a local variation in the volume raction o the constituents inside the composite materials, the clusters o one phase being

4 zones o higher volume-raction o this particular phase. Thereore the question o clustering eects is transormed into the ollowing one: how do luctuations in the volume raction o constituents aect the eective properties o composites? Do they strengthen or weaken the material? What is the role played by the material nonlinearity? How does this inluence depend on the loading conditions? This paper is organized as ollows. Section presents the class o composites under consideration in this study. They are characterized by three separate length-scales, called the micro, meso and macro-scale. The volume raction o the constituents is uniorm at the macro-scale but has luctuations at the mesoscale. The irst theoretical result o the paper is given in section 3, where an asymptotic expansion o the macroscopic energy o the composite in powers o the volume-raction luctuations is given. It is shown in particular that the dierence between the energy with and without luctuations is, to second-order, proportional to the standard deviation o the luctuations. This expansion is applied in section 4 to porous nonlinear materials, or which the eect o clustering is expected to be important. The eective properties o porous materials containing an isotropic distribution o spherical pores with uniorm porosity is well described by Gurson s model (Gurson, 1977) (see section.3 or details). The present study will also make use o an extension o Gurson s model to viscous materials, based on the same choice o velocity ields as in the original work by Gurson. This extension, proposed by Leblond et al. (1994) is recalled in section.3. The main inding o section 4 is that, or materials whose eective properties are well described by Gurson s model or by its extension to viscous materials (roughly speaking, these materials have a microstructure close to an ideal assemblage o hollow spheres), local luctuations in porosity always result in a weakening o the material properties. This observation, irst made by means o numerical simulations, is conirmed by analytical results or most loading conditions. A very strong inluence o the nonlinearity is ound. In the ideally plastic limit the corrective term can even become ininite or deormation conditions which are investigated in detail in section 5. Finally section 6 investigates the eect o volume-raction luctuations on the low surace o porous materials, rather than on their eective potential. Throughout the text, vectors and second-order tensors will be denoted with boldace letters, whereas ourth-order tensors will be denoted by barred letters. In this connection, the various types o products will be denoted by dots (e.g., u.v = u i v i, (L : ε) ij = L ijkh ε kh ). 3

5 Problem setting.1 Clusters, volume-raction luctuations and three-scale composites For deiniteness, the composites considered in this study are two-phase materials. For simplicity, they are also assumed to be composed o isotropic phases arranged isotropically (both assumptions can be removed). The existence o clusters o one phase is modelled as luctuations in the local volume raction o one o the phases, say phase 1, about a mean value 0. a L d Ω Fig. 1. Three-scale composite. Let: representative volume element Ω seen at the meso-scale showing the microstructure. is volume raction o the white phase. Right: representative volume element V seen at the macro-scale showing luctuations in the local volume raction (x). To put this inormation in a more rigorous setting, these composite are considered as three-scale materials exhibiting heterogeneities at dierent scales hereater called the micro, meso and macro-scales. These scales are characterized by dierent lengths d, a and L (see igure 1) corresponding respectively to the size o a single inhomogeneity (d), to the size o clusters o inhomogeneities (a) and to the size o a very large representative volume element (L). At the smallest scale (micro) the composite is highly heterogeneous, as shown in igure 1a. At the intermediate scale (meso) the sharp micro-inhomogeneities are smeared out as shown in igure 1b and the composite appears now as an intermediate composite, or meso-composite, comprised o dierent constituents with a mesostructure representing the dierent regions, with dierent volume raction o constituents, in the actual composite. Finally, at the largest scale all inhomogeneities are smeared out and the composite is seen as an homogeneous medium, the properties o which being, by deinition, the macroscopic or eective properties o the composite. V 4

6 Consider or instance a two-phase composite composed o identical spherical inclusions distributed in a surrounding matrix in an almost uniorm manner. By almost uniorm, it is meant that the local volume raction o inclusions may vary slightly within a large volume element. In other words, the volume raction o the inclusions is 0 at the level o the volume element. But at a smaller (meso) scale it can luctuate slightly about this mean value and is described by a variable volume raction (x). In the present study it is assumed that the microstructures o the dierent regions are o the same type (inclusion/matrix or instance), so that the volume raction o one phase is the leading parameter in the microstructure description and that the volumeraction luctuations at the meso-scale are small : (x) 0 0. Obviously a three-scale composite is also a two-scale composite, the two scales being the micro and the macro scale. Thereore the eective properties o the composite can be obtained by an homogenization scheme operating directly between the micro and the macro scales under the sole assumption that d L. Depending on the type o microstructure under consideration, examples o such homogenization schemes are given by the Hashin-Shtrikman estimates, the sel-consistent scheme or any other appropriate scheme. However, when the three scales are well separated, which means that the three lengths urther satisy the string o inequalities d a L, the eective properties o the composite can be obtained by splitting the homogenization procedure into two successive steps. The eective properties at the meso-scale (in the dierent regions) are determined in a irst step. It is assumed here that this step o the procedure can been completed and that the eective meso-properties or the dierent regions are known under the orm o an energy unction w(, ε) depending on the volume raction o phase 1. For the sake o deiniteness examples o unctions w are given in section.3 or voided materials.. Microscopic, mesoscopic and macroscopic potentials The constitutive behavior o each individual constituent (α) is governed at the microscopic scale by a convex potential or strain-energy unction w (α) (e) in such a way that the ininitesimal strain (or strain-rate) e and the stress s 5

7 at the microscopic scale are related by s = w(α) (e) in phase e (α). A volume element Ω, whose size a is large with respect to the size d o the heterogeneities, is subjected to an average deormation ε. As is classical in homogenization theories, the eective energy o the composite at the mesoscale is given by the variational principle : 1 w(ε) = In w(e(u ))dω, (.1) u Ω Ω where the trial displacement ields u are such that the average over Ω o the microscopic strain ield e(u ) is equal to ε. The eective energy w is a convex unction o ε. In a second step a macroscopically representative volume element V is chosen, whose size L is much larger that the size a o the meso-representative volume element Ω. The volume raction (x) o phase 1 depends on the position x at the meso-scale. The average volume raction at the macroscopic scale is 0. The eective energy is given by (as is classical) : w ( 0, ε) = In u K(ε) w((x), ε(u)), (.) where K(ε) = {v = ε.x+v, v periodic on V }. For deiniteness, periodicity conditions have been chosen on V but the type o boundary conditions is not important as long as the volume element V is much larger than the typical length o the luctuations o in V. In the rest o the paper it will be assumed that the irst step o homogenization has been achieved and that w(, ε) is known, either exactly or through an accurate estimate. Our aim is to examine the dierence between w and w which stems rom the deviations o (x) rom 0..3 Mesoscopic energy unctions or voided materials The main theoretical result obtained in this paper in section 3 will be applied to voided materials in section 4. At the microscopic scale, voids (phase 1) Dierent notations, e, ε and ε are used in the paper to denote the ininitesimal strain at the micro, meso and macro-scale respectively. Similarly s, σ and σ denote the Cauchy stress at the three dierent scales. The three quantities are related by classical average relations. 6

8 are distributed in a matrix (phase ) which is a power-law incompressible material characterized by the strain-energy (or dissipation potential when e is interpreted as a strain-rate) : w(e) = σ 0ε 0 m + 1 ( eeq ε 0 ) m+1 when tr(e) = 0, + otherwise. (.3) The particular case m = 1 corresponds to a linear-elastic material with shear modulus µ = σ 0 /3ε 0. As is well-known the eective energy o linear elastic voided materials are bounded rom above by the upper Hashin-Shtrikman bound : w(, ε) σ 0(1 ) ε 0 ( 4 ε m + 1 ) 1 + ε eq. (.4) 3 In addition the right hand-side o (.4) is an accurate estimate o the eective energy at the meso-scale when the microstructure o the material is o the composite sphere assemblage type. The other extreme case m = 0 corresponds to a rigid-plastic matrix obeying the von Mises criterion with low stress σ 0. Bounds and estimates or the eective energy o ideally-plastic voided materials have been obtained by means o various variational methods (Ponte Castañeda (1991, 199), Willis (1991), Suquet (199, 1993)). However the strain-energy unctions obtained by these methods are known to be inaccurate especially at high stress triaxialities (although they are rigorous upper bounds). The most widely used model or porous ideally-plastic materials is due to Gurson (1977) and based on the approximate analysis o a single hollow sphere. The yield unction proposed by this author to describe the low surace o voided ideally-plastic materials reads as : F(, σ) = σ ( ) eq 3 σ m + cosh 1 0. (.5) σ 0 σ 0 The associated energy (or dissipation) unction at the meso-scale can be calculated as : w(, ε) = Sup σ, F(σ) σ : ε. 0 Straightorward algebra shows that the energy w associated with Gurson s criterion (.5) reads as : w(, ε) = σ 0 1 ( 4ε m y + ε eq ) 1/ dy. (.6) Following the work o Gurson (1977), Perrin (199) proved that (.6) is an upper bound or the eective potential o a hollow sphere subjected to uniorm displacements on its boundary. The potential (.6) remains a rigorous upper bound or the eective potential o any composite assemblage o hollow spheres with porosity. Numerical studies (see Tvergaard, 1990, or a review) show that this upper bound is also an accurate estimate or spherical voids which are 7

9 periodically distributed in an ideally-plastic matrix. However it is not known i it remains an upper bound or more general microstructures. For instance, it is not known whether the Gurson s potential remains an upper bound, or an accurate estimate, when the voids shapes are not spherical. For intermediate values o m, Leblond et al. (1994) and Gărăjeu et al. (000), generalizing the analysis o Gurson with the same velocity ields, have proposed a strain-energy unction which reads : w(, ε) = σ 0 (m + 1)ε m 0 1 ( ) m+1 4ε m y + ε eq dy. (.7) When m = 0 the strain-energy (.7) reduces to the Gurson s energy (.6). When m = 1 it coincides with the upper Hashin-Shtrikman bound (.4) or hydrostatic loadings but exceeds slightly this bound or deviatoric loadings. Section 4 will deal with potentials in the orm (.7). a L d 1 (a) Fig.. Two uniorm volume ractions. (a) Actual microstructure, (b) mesostructure. The question addressed in this study, when specialized to voided materials, can be ormulated in the ollowing way. Assume that the microstructure o the voided materials under study can be reasonably considered as a generalized composite sphere assemblage (isolated voids in a matrix). Void clusters can be seen as regions in the r.v.e. where the volume raction o the composite spheres is above the average porosity and regions where it is below, as schematically depicted in igure a (or clarity the dierence in the local volume ractions 1 and has been much exaggerated in this igure). The corresponding microstructure is not strictly speaking a composite sphere assemblage, but is a generalized composite sphere assemblage in the sense that it results rom the piling-up o two dierent classes o sel-similar spheres with two dierent porosities 1 and. How does the eective potential o such a generalized asemblage dier rom that o the corresponding microstructure (b) 8

10 where all the sel-similar spheres have the same porosity? A numerical approach to the question would be to consider the mesostructure shown in igure b. Then considering that each region is governed by the Gurson criterion (.5), one could compute numerically the (macroscopic) eective yield surace o the material. This is not the direction that we have pursued here. Instead we have developed an analytical approach, based on the assumption that the volume ractions 1 and do not dier too much rom. This restriction is counter-balanced by the act that our analytical results do not depend on the orm o the potential and gives qualitative results which shed light on the role, positive or negative, o volume-raction luctuations. 3 A general result or small luctuations in the local volume raction o constituents 3.1 Asymptotic expansion As recalled in the introduction, the homogenization procedure or three-scale composites involves a irst step by which the mesoscopic eective energy w(, ε) is determined. Once the irst step o homogenization is completed, in exact or approximate orm, it remains to perorm the second step o homogenization, i.e. to achieve the meso-macro transition. At the meso-scale the local volume raction o phase 1 exhibits luctuations about its average: t (x) = 0 + tδ(x), with δ = 0. (3.1) The parameter t serves to measure the amplitude o the luctuations about the mean value 0. The luctuations are assumed to be small so that t is a small parameter. Then, the macroscopic eective energy o the composite, which depends on the small parameter t, is deined as w (t, ε) = In u K(ε) w( t, ε(u)) = w( t, ε(u t )). (3.) where K(ε) = {v = ε.x + v, v periodic}. The ields u t and ε(u t ) are the local displacement and associated strain ields at the meso-scale induced by appropriate boundary conditions generating an average strain ε in V. It will be assumed that w (., t) and u t are continuously dierentiable unctions o t. Since t is small, it is appropriate to look or a perturbation series expansion o w about t = 0. 9

11 The main result result o this section is the ollowing expansion w (t, ε) = w( 0, ε)+ t ( δ w ( 0, ε) w ε (ε) : H : ) w ε (ε) + O(t 3 ), where H is a ourth-order tensor speciied by (3.1). (3.3) Comments: 1. In general, luctuations in the volume raction have no irst-order eect on the eective potential but have a second-order eect. This is true as long as the Taylor expansion (3.3) is legitimate. In particular, when the tensor H becomes singular, the expansion (3.3) is no more valid.. Fluctuations have a irst-order eect on the local ields, relected by the irst-order terms u 0 and σ 0 (solutions o the problem (3.10). 3. The luctuations enter this second-order expansion through their covariance δ and through the ourth-order tensor H which depends on the geometrical arrangement o the domains where the perturbations around the mean value 0 take place. 4. Fluctuations in volume raction can have, at second-order, a beneicial or deleterious eect on the eective potential w, depending on the sign o the second-order term. 5. It should be kept in mind or urther consideration that the Taylor expansion(3.3) is valid or small values o the volume raction luctuation (t) and can break down i the second-order term blows up (this will the case or rigid-plastic materials under speciic loading conditions as studied in detail in section 5. In the same spirit, it should be noted that, although w and w are convex unctions o ε and ε respectively, nothing guarantees that the Taylor expansion truncated at order as given by (3.3), is a convex unction with respect to ε when t is not small enough. Derivation o (3.3): The expansion o w in powers o t is ormally given by its Taylor expansion at t = 0. The irst derivatives in this expansion may be computed by successive derivation o (3.); this procedure will naturally involve derivatives o u t. The problem to be solved or u t is given by σ t = w ε ( t, ε(u t )), div(σ t ) = 0, u t K(ε). (3.4) By dierentiation o (3.4), it is seen that u t (the dot denoting the derivative 10

12 with respect to t) is, or all t, the solution o the systems o equations: where σ t = L t : ε( u t ) + τ t, div( σ t ) = 0, u t K(0), (3.5) L t = w ε ε ( t, ε(u t )), τ t = δ(x) w ε ( t, ε(u t )). L t is the tensor o instantaneous or tangent moduli. It ollows rom (3.) and rom Hill s lemma that w w (t, ε)= t ε ( t, ε(u t )) : ε( u t ) + δ(x) w ( t, ε(u t )) = δ(x) w ( t, ε(u t )). (3.6) Dierentiating again yields: w (t, ε) = (δ(x)) w t ( t, ε(u t )) + δ(x) w ε ( t, ε(u t )) : ε( u t ). (3.7) By means o the system o equations (3.5) satisied by u t, one obtains inally that : w w (t, ε) = (δ(x)) t ( t, ε(u t )) ε( u t ) : L t : ε( u t ). (3.8) The relations (3.6) and (3.8), which hold or arbitrary values o t, give the irst two derivatives o w at t = 0. The material is homogeneous at the meso-scale at t = 0. Thereore u 0 = ε.x and : w (0, ε) = w( 0, ε), w w (0, ε) = δ t ( 0, ε) = 0 (since δ = 0), w t (0, ε) = (δ(x)) w ( 0, ε) ε( u 0 ) : L : ε( u 0 ), where L = w ε ε ( 0, ε). (3.9) In these relations, u 0 is the solution o the linear thermoelasticity problem: σ 0 = L : ε( u 0 ) + τ, div( σ 0 ) = 0, u 0 K(0), where τ(x) = δ(x) w ε ( (3.10) 0, ε). Given that the modulus tensor L is constant, problem (3.10) is a standard linear thermoelasticity problem or a homogeneous material with a distribution 11

13 o body orces determined by the polarization ield τ. Its solution can be expressed as ε( u 0 ) = IΓ τ, where IΓ is the Green operator associated with L. When the luctuations are piecewise uniorm, i.e. uniorm in N regions V r with characteristic unction χ (r), the polarization ield is piecewise constant with : N τ(x) = χ (r) (x)τ (r), τ (r) = w ε ( 0, ε)δ (r). r=1 Then, taking advantage o Hill s lemma, one obtains that : N N ε( u 0 ) : L : ε( u 0 ) = τ : ε( u 0 ) = τ : IΓ τ = τ (r) : IΓ (rs) : τ (s), r=1 s=1 (3.11) where IΓ (rs) = χ (r) IΓ χ (s). In conclusion, when luctuations have zero average around the mean value 0, the expansion (3.3) is obtained with : N N H = δ (r) : IΓ (rs) : δ (s). (3.1) r=1 s=1 3. Inclusion-rich zones and inclusion-poor zones Assume that the luctuations are piecewise uniorm on only two subdomains (N = ) (see igure ), whose distribution is compatible with the overall isotropy o the composite. It ollows rom the act that the luctuations around the mean value 0 have zero average that δ (1) and δ () should have the ollowing orm : δ (1) = c, (1) δ() = c, () where c (1) and c () denote the volume ractions o the domains where the perturbations take place. Then δ(x) = c (1) c () ( χ1 (x) c (1)) and δ = ( ) c (1) c (). The tensor H is computed rom (3.1) with δ (1) and δ () given above and, since in the case o a two-phases composite the microstructural tensors IΓ (rs) have particular properties :: IΓ (1) = IΓ (1) = IΓ (11) = IΓ ( ), 1

14 it ollows : H = ( ) 1 c (1) c ()P, with P = χ (1) IΓ χ (1) (3.13) c (1) c () and thereore the second order derivative o the eective potential is : w ( ( ) w ( ε, 0) = t c (1) c () ( 0, ε) w ε ( ε) : P : ) w ε ( ε). (3.14) The two domains on which the perturbation take place can be chosen arbitrarily to a certain extent (with the constraint that their distribution should be such that the composite is macroscopically isotropic). The inluence o this choice on the eective potential w is measured by the product c (1) c () and the largest eect is obtained when c (1) = c () = 1/. 3.3 Materials with mesoscopic isotropy Assume that the composites under consideration are isotropic at the mesoscopic scale. Their mesoscopic energy w depends only on the irst three invariants o ε. Assume or urther simplicity that w depends only on the irst two invariants : w(, ε) = w(, ε m, ε eq ). (3.15) I, additionally, the geometrical distribution o the two luctuation domains is statistically uniorm and isotropic (i.e. the two-point correlation unctions depend only on the distance between any two points in the composite), then ollowing Willis (1981), the tensor P deined in (3.13) is related to the instantaneous elastic tensor L (3.9) by : P = 1 ξ N(ξ) ξ sym ds(ξ). (3.16) 4π ξ =1 with N(ξ) = (K(ξ)) 1 is the inverse o the acoustic tensor K(ξ) = ξ.l.ξ. The tensor L is the second derivative o the potential w(ε) with respect to ε evaluated at ε. The irst derivative reads 3 : w ε (ε) = 1 w (ε)i + w (ε) e, 3 ε m 3 ε eq ε eq 3 For simplicity we shall drop, in what ollows, the dependence on which is just a parameter. 13

15 where i is the second-order tensor identity and e is the deviator o ε. The second derivative reads w ε (ε) = 1 w J + 3 εm(ε) w (ε) E + 1 w (ε) F + w (ε) G 3 ε eq 3 ε eq ε eq 9 ε m ε eq where J = 1 3 i i, E = ê, F = I J E, G = i ê + ê i, 3ê with ê = e ε eq. In the above relations I is the ourth-order tensor representing the identity between symmetric ourth-order tensors, J is the projector over spherical tensors and thereore K = I J is the projector over deviatoric tensors, E is the projector over ê and E + F = K. e materials. The ourthorder tensors E and F were introduced by Ponte Castañeda (1996) to express the tangent moduli in incompressible nonlinear materials. Another tensor G is necessary here to account or compressibility eects. Set k = 1 w λ = 9 εm( ε), 1 3 Then Let η = ê.ξ. Since w ( ε), µ = 1 ε eq 3 1 w ε eq ( ε), γ = 1 w ( ε). ε eq 9 ε m ε eq (3.17) L = 3kJ + λe + µf + γg. (3.18) ξ.j.ξ = 1 3 ξ ξ, ξ.e.ξ = 3 η η, ξ.f.ξ = 1 (i ξ ξ 4 ) 3 η η, ξ.g.ξ = η ξ + ξ η, the acoustic tensor K(ξ) reads as : K(ξ) = ξ.l.ξ = µ and can be put in the orm : [ i + ( k µ )ξ ξ + γ µ (η ξ + ξ η) K = i + λ 1 ξ ξ + λ (ξ η + η ξ) + λ 3 η η, ( ) ] λ µ 1 η η with λ 1 = k µ + 1 3, λ = γ µ, λ 3 = 4 ( ) λ 3 µ 1. (3.19) According to lemma 8 o appendix A the inverse o K can be expressed as : N(ξ) = 1 µ { i 1 } [Λ 1ξ ξ + Λ (ξ η + η ξ) + Λ 3 η η], 14

16 where the detailed expressions o and o the Λ i s are given in appendix A. The polarization ield τ specializes to τ = w ε ( 0, ε) = τ m i+ 3 τeqê, τ m = 1 w ( 0, ε), τ eq = w ( 0, ε). 3 ε m ε eq (3.0) To determine the second-order expansion (3.3) o the eective potential is suicient to evaluate τ : P : τ in (3.14) rather than computing P itsel : τ : P : τ = 1 4π = 1 4π ξ =1 ξ =1 τ : (ξ N(ξ) ξ) : τds(ξ) (τ.ξ).n(ξ).(τ.ξ)ds(ξ). Note that τ.ξ = τ m ξ + 3 τ eqη and thereore the integrand reads (τ.ξ).n(ξ).(τ.ξ) = τ m ξ.n(ξ).ξ τ mτ eq ξ.n(ξ).η τ eq η.n(ξ).η. Straightorward algebra gives : ξ.n(ξ).ξ = 1 ( 1 Λ 1 + Λ p + Λ 3 p ) µ ξ.n(ξ).η = 1 µ η.n(ξ).η = 1 µ [ ( p Λ 1p + Λ (b + p ) + Λ 3 pb b Λ 1p + Λ pb + Λ 3 b 4 = 1 [ 1 + λ3 (b p ) ], µ ) ] = 1 [ p λ (b p ) ], µ = 1 [ b + λ 1 (b p ) ]. µ with the notations o appendix A (b = η, p = ξ η). Finally the integral reads τ : P : τ = 1 τm + 4τ 3 mτ eq p τ eq b + ( τm λ 3 4τ 3 mτ eq λ τ eq λ 1) (b p ) ds(ξ) 4π ξ =1 µ (3.1) This integral cannot be computed explicitly in general. As remarked in other circumstances by Ponte Castañeda and Suquet (1998), it depends strongly on the third invariant o the strain tensor ε. Indeed, ε appears in the integral (3.1) through p = ξ.ê.ξ and b = (ξ.ê.ê.ξ) 1/ which are independent o the reerence rame. Relative to its principal axes, ê may be represented (Kachanov, 1971) in terms o a single parameter ω, 0 ω < π/3, the principal values o ê taking the orm ( ê 1 = cos ω π ) (, ê = cos ω + π ), ê 3 = cosω. (3.)

17 The angle ω is related to the third invariant o the macroscopic strain ε. Indeed, the determinant o ê, which is closely related to this third invariant, can be expressed in terms o ω as det(ê) = cos(3ω)/4. The values ω = 0 and ω = π/6 correspond to an axisymmetric strain (deviatoric part o a uniaxial deormation) and simple shear, respectively. Comments: Our results show that the perturbed potential w depends on the third invariant o the strain even though the unperturbed potential w (3.15) has no such dependence. This raises the question o the validity o assumption (3.15). The inluence o the third invariant o stress (or strain) on the eective properties o porous materials has been the subject o contradictory observations. On the one hand Duva and Hutchinson (1984) and Richelsen and Tvergaard (1994) have observed a rather weak eect o this invariant. On the other hand theoretical results (or weakly contrasted materials) by Ponte Castañeda and Suquet (1998) have shown potentially strong eects. Since this is still an open question (to the best o the authors knowledge) and by lack o an accurate model including third invariant eects, the assumption (3.15) was made. 4 Nonlinear porous materials 4.1 The main result This section is devoted to the inluence o isotropic porosity luctuations on the eective behaviour o nonlinear porous materials. Our main observation is the ollowing one: Conjecture 1: For any porous material whose eective behaviour is described by the Gurson-type potential (.6) or (.7), the second-order perturbation due to luctuations in porosity is always negative. In other words, luctuations in porosity always weaken porous materials with eective properties described by (.7). We have a complete analytical proo o conjecture 1 in three dierent cases : irst when m = 1, second or arbitrary m when the macroscopic strain is either a pure dilatation or a simple shear (see section 4.3), and third when m = 0 and when the macroscopic strain triaxiality T = ε m /ε eq is larger than 1. In the remaining cases, numerical simulations presented in section 4.5 strongly support the conjecture s validity. 16

18 The perturbation o the eective potential w (ε, t) induced by porosity luctuations can be measured by the normalized quantity: δ(, ε) = ( w (, ε) w ε (, ε) : P : )/ w (, ε) ε w(, ε) (4.1) This quantity does not depend on the amplitude o the luctuations t but depends, through the tensor P, on the geometrical arrangement o the domains where the luctuations take place. Conjecture 1 can be alternatively ormulated by saying that δ is negative or all m s and all macroscopic strains ε. It should be noted that since w is a positively homogeneous unction o ε o degree m + 1, L and P are positively homogeneous unctions o ε with degree m 1 and 1 m respectively. It ollows that δ depends on the macroscopic strain ε only through the strain triaxiality-ratio deined as T = ε m ε eq, (4.) and through the angle ω deined by (3.) and related to the third invariant o the strain. Thereore δ = δ(, T, ω). 4. Linear materials In the linear case m = 1, the expression o δ is quite simple. The potential w(, ε) takes the orm : w(, ε) = σ ( ) 0 4ε (1 ) m ε 0 + ε eq The relations (3.17) give γ = 0 and λ = µ and thereore the elastic tensor L (3.18) and the corresponding P tensor (3.16) whose expression can be ound in Willis (1981) are isotropic : L = [ ] σ 0 (1 ) 3 ε 0 J + K, P = 1 σ 0 3 ε 0 (1 ) [ J ] K, 5 where K is the projector over deviatoric tensors (to avoid conusion, the second order acoustic tensor is always denoted in the paper by K(ξ)). The polarization ield τ and the term τ : P : τ reduce to τ = ( ) σ 0 εm 3 ε 0 i + ε eqê, τ : P : τ = σ ( 0 1 4ε m ε ) ε 3 eq. 5 17

19 Finally, the second-order perturbation δ deined in (4.1) reads : δ = σ ( 0 1 4ε m ε Conjecture 1 is thereore proved or linear materials. ε eq ) / w(, ε) 0. Motivated by this result and by the numerical simulations presented in section 4.5, a second conjecture can be ormulated : Conjecture : For any porous materials whose eective behaviour is described by the Gurson-type potential (.7), the second-order perturbation δ is an increasing unction o m when 0 m 1 (δ(0) δ(m) δ(1)). Conjecture 1 is a consequence o conjecture and o the result established in the linear case since, i conjecture holds true, one has or ixed porosity and ixed applied strain ε : δ(m) δ(1) < Proo o conjectures 1 and or speciic loading conditions The evaluation o δ or speciic loading conditions requires detailed expressions or the dierent terms in (4.1) which are given in appendix B Pure dilatation When the macroscopic strain is a pure dilatation (ε eq = 0, T = ), the potential w(, ε) reduces to (appendix B.1): w(, ε) = σ 0 ε m 0 ( εm ) m+1 (1 m ) m(m + 1) and its second derivative with respect to reads: w (, ε) = σ 0 ε m 0 1 ( εm ) m+1 Finally, using results o the appendix B, δ = δ T= can be expressed in closed orm : δ (, m) = m + 1 m (m 1) m (4.3) 1 m 1 + (1 m)(1 m ) 18

20 δ depends on and m only. As the product o three increasing unctions m m + 1, m m 1 m, m (m 1) m 1 + (1 m)(1 m ), the irst two being positive, δ is an increasing unction o m. Conjecture and consequently conjecture 1 are thereore proved or purely dilatational deormations. 0-0 δ m=1 m=0.35 m=0.05 m=0.01 m= , 0,4 0,6 0,8 1 Fig. 3. Normalized second-order perturbation δ as a unction o or dierent values o m The normalized second-order perturbation δ is plotted in Figure 3 as a unction o or dierent values o m. For completeness, the igure shows δ when ranges rom 0 to 1. It it is thereore implicitely assumed that Gurson s model remains accurate in this range o volume raction. In other words, it is assumed that the porosity remains closed even or large porosities. I this is not the case, and in particular i the porosity becomes open over a certain percolation threshold, the asymptotic expansion (3.3) remains valid upon use o a proper expression or w accounting or this change in microstructure. The singularity observed in these curves in the vicinity o = 0 is due to the act that the material becomes incompressible in the limit as goes to 0. The potential w, its second derivative with respect to and the dierent terms entering the expression o the second-order perturbation become ininite in this limit. Similarly, when approaches 1 the potential w tends to 0 (the material loses its carrying capacity) and the normalization by w(, ε) creates a singularity or δ. 19

21 4.3. Simple shear When the overall deormation is a simple shear (ε m = 0, i.e. T = 0) the potential w is an aine unction o, and thereore w (, ε) = 0. w(, ε) = σ 0ε m+1 eq (m + 1) ε m 0 (1 ) Then, using appendix B. one obtains the expression o the perturbation δ 0 = δ T=0 : δ 0 = 1 3π m + 1 (1 ) ξ =1 ê.ξ 4+ 4(1+) (ξ.ê.ξ) (m 1) ( ê.ξ 4+ 4(1+) (ξ.ê.ξ) )ds(ξ) (4.4) which is a unction o and ω. Note that or all values o and ω, δ 0 is an increasing unction on m. Indeed, m m+1 is an increasing and positive (1 ) unction and, since the unction ê.ξ m 1 3π ξ =1 is an increasing unction (1 + ) (ξ.ê.ξ) ê.ξ (ξ.ê.ξ) 0, ê.ξ 4+ 4(1+) (ξ.ê.ξ) (m 1) ( ê.ξ 4+ 4(1+) (ξ.ê.ξ) ) Conjecture and consequently conjecture 1 are thereore proved or simple shear. The normalized second-order perturbation δ 0 or ω = 0 is plotted in Figure 4a or dierent values o m. Again, the large values o δ observed when approaches 1 are due to the act that the normalization actor w(, ε) tends to 0 in this limit. The plots in Figure 4b show δ 0 as a unction o when m = 0 and or dierent values o ω, except or ω = π where the integral in 6 (4.4) is divergent or any (as proved in section 5.1). Compared to the order o magnitude o δ shown in igure 3, the second order perturbation δ 0 is small or most s and ω s which means that the Gurson potential is almost insensitive to small perturbations in the void volume raction when the overall strain is a pure shear. 0

22 δ m=1 m=0.35 m=0.05 m= ,1 0, 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 (a) δ ω = 0 ω = π /30 ω = π /15 ω = π /10 ω = π / ,1 0, 0,3 0,4 0,5 0,6 0,7 0,8 0,9 Fig. 4. Normalized second-order perturbation δ 0 : (a) when ω = 0, as a unction o and dierent values o m; (b) when m = 0, as a unction o and dierent values o ω 4.4 Proo o conjecture 1 or rigid-plastic materials and T 1. (b) In this section we prove that δ is negative when m = 0 and T 1. The proo requires in particular an estimate o (3.1). Note irst that both the numerator and the denominator o the integrand in (3.1) are positive since, according to section 5.1 the acoustic tensor K(ξ) and its inverse N(ξ) are positive deinite when T 1/. The idea o the proo is to ind a uniorm upper bound max or (ξ) which is independent o ξ. Such an inequality or will lead to a lower bound or τ : P : τ : τ : P : τ 1 1 4π µ max ξ =1 τ m +4 3 τ mτ eq p τ eq b + ( τ m λ τ mτ eq λ τ eq λ 1) (b p )ds(ξ). The integral on the right-hand-side o the inequality can be computed in closed orm : τ : P : τ 1 [ ( 4 1 µ max ) 10 λ 1 τeq ( 5 λ τ eq τ m ) ] 10 λ 3 τm. Then using (B.4) and (B.) with m = 0 (note that in the case m = 0, a(t, ) simpliies to a(t, ) = 4T + ) the above inequality reads τ : P : τ 1 [ 4σ 0 ε eq a(t, ) 3 + max 15 (a(t, 1) a(t, )) a(t, 1) + ( ] 1) 3 a(t, ) The normalized second perturbation δ has the same sign as w (, ε) τ : P : τ. We get rom the above inequality : w (, ε) τ : P : τ 4σ { 0ε eq T 1 a(t, ) max B [1 + (1 5(1 B) 3 B + 1 )] } B 1

23 where B = a(t, )/a(t, 1) satisies the identity ( 1)/a(T, ) = 1 1/B. It remains to speciy the upper bound max in this inequality. For this purpose the ollowing technical lemma is proved in Appendix C : Lemma 1 When T 1 then max = 4 3 a(t,1) a(t,) (1 + T ). Then, upon substitution o max in the above inequality, we obtain that : w 4σ (, ε) τ : P : τ 0 ε eq a(t, )(1 + T ) D where : D = T (1 + T ) 3 a(t, ) [1 + (1 0(1 B) 3 B + 1 )]. B It remains to prove that D is negative. Another technical result is used : Lemma D is a decreasing unction o. Lemma is proved in the Appendix D. According to this lemma the maximum o D is reached or = 0. Note that or = 0, a(t, 0) = T, B = T /a(t, 1) and 1 = a(t, 1)(a(T, 1) + T ). 1 B It ollows that: D T (1 + T ) 3T ( B) 5(1 B) [ = T (1 + T ) a(t, 1)(a(T, 1) + T ) 3 + B ] 5 = T ( T 5 T 1 + 4T ) 5 (or T 1) T ( ) T T 5 5 T (11 5 5) 5 5 < 0 This completes the proo o the inequality δ 0 or m = 0 and T Numerical evaluation o the second-order perturbation The above sections prove conjectures 1 and in a certain range o macroscopic strains ε and o rate-sensitivity exponents m. To investigate the remaining cases the normalized second-order perturbation δ is computed numerically or porous materials governed by a Gurson-type potential using the relations (3.1) and (B.4).

24 δ δ m=1 m=0.35 m= m=0.01 m= ,5 0,5 0,75 1 1,5 1,5 1,75 T a) m=1 m=0.35 m=0.05 m=0.01 m=0 0 0,5 0,5 0,75 1 1,5 1,5 1,75 T Fig. 5. Second-order perturbation δ or 0 = 0.1, as a unction o T or dierent values o m: (a) ω = 0; (b) ω = π/6 The variations o δ as a unction o the strain rate triaxiality T are shown in Figure 5 or dierent values o the exponent m. The average porosity is 0 = 0.1. Figure 5a corresponds to uniaxial tension (ω = 0), whereas Figure 5b corresponds to simple slip (ω = π/6). The ollowing points are worth noticing : - First it is observed that δ is always negative. This supports conjecture 1. It is also noted that δ is an increasing unction o m. This observation supports conjecture. - Second, when m = 0 the plots indicate that δ becomes singular or speciic values o the strain triaxiality-ratio depending on ω. When ω = 0 the singularity is observed or T = 1/ and when ω = π/6, the singularity is observed or T = 0. - Another aspect o the singularity o δ in the rigid-plastic case is reported in igure 6 where δ is plotted as a unction o ω or a given strain triaxiality T (m = 0 or all plots). When the macroscopic strain triaxiality T is less than 1/ (igure 6a), there is a speciic value o ω or which δ becomes singular. In contrast, when T is larger than 1/ no singularity o ω is observed. Most o these observations will be conirmed by analytical results in section 5. b) 5 Singularity o the second-order perturbation or rigid-plastic materials It was observed in the numerical investigations reported in section 4.5 that the second-order term in the expansion (3.3) can be singular or rigid-plastic materials. When w is a twice-dierentiable unction o (which is usually the case), this singularity can only come rom the P tensor in (3.14) and thereore rom a degeneracy o the acoustic tensor K(ξ) = ξ.l(ε).ξ. The acoustic tensor 3

25 δ T=0-600 T=0.1 T=0. T= T=0.4 T= ,15 0,3 0,45 0,6 0,75 0,9 ω (a) δ T=0.75 T=1 T=1.5 T= T=3 0 0,15 0,3 0,45 0,6 0,75 0,9 1,05 ω (b) Fig. 6. Second-order perturbation δ or 0 = 0.1, and m = 0 as a unction o ω or dierent values o T: (a) 0 T 0.5; (b) 0.5 < T < 10 is always positive (by convexity o w) but can become degenerate when its determinant vanishes. This possibility is investigated here both or rigid-plastic and viscous materials. Considering an isotropic convex energy w in the orm (3.15), we prove the ollowing results : - When m = 0, the acoustic tensor K(ξ) can only be singular when ε is a rank-one tensor. - When m > 0, the acoustic tensor K(ξ) is always positive deinite. In conclusion, the second-order perturbation δ can only be singular or rigidplastic materials (m = 0) and only or rank-one macroscopic deormations. 5.1 Rigid-plastic materials m = 0 The energy unction w is positively homogeneous o degree m+1 and thereore satisies : w(λε) = λ m+1 w(ε) or all positive λ s. (5.5) Dierentiating (5.5) twice with respect to λ and taking the resulting relation or λ = 1 gives : ε : L(ε) : ε = m(m + 1) w(ε), (5.6) where L(ε) is the tensor o tangent moduli (second derivative o w) deined by (3.9). It results rom the convexity o w that the ourth-order symmetric tensor L(ε) is positive (in the sense o quadratic orms). But when m = 0 the relation (5.6) shows that L(ε) is not positive deinite, since (5.6) reduces in this case to : ε : L(ε) : ε = 0, or all second-order symmetric ε. (5.7) The tangent moduli L being singular, the associated acoustic tensor is likely to be singular as well, leading to = det(k(ξ)) = 0 and explaining the 4

26 singularity o δ observed in plots 6. This intuitive guess is in act not always true, but is true or speciic strains as rigorously shown by the main result o this section. Theorem 1 Let w be an isotropic energy unction in the orm (3.15) which is convex and positively homogeneous o degree 1. Let L(ε) be its tensor o tangent moduli or a given strain ε and let K(ξ) be the acoustic tensor associated with L(ε) through the relation K(ξ) = ξ.l(ε).ξ. Then, there exists a unit vector ξ such that det (K(ξ)) = 0 i and only i ε is a rank-one tensor. Recall that ε is a rank-one symmetric second-order tensor i there exist two vectors a and b in R 3 such that ε = a s b. A more thorough characterization o symmetric rank-one tensors is needed beore proceeding to the proo o theorem 1. Lemma 3 Let ε be an element o R 3 3 s, space o symmetric second-order tensors, with eigenvalues ε 1 ε ε 3. Then a) ε is rank-one i and only i ε = 0. b) Let T = ε m be the triaxiality-ratio o ε. Then, or all rank-one tensors ε, ε eq T 1. c) Conversely, or every T 1, there exists a unique (up to a rotation and a multiplicative actor) rank-one tensor whose triaxiality-ratio is T. Proo 4 o a): First we prove that or all rank-one tensors ε, the intermediate eigenvalue ε is equal to 0, rom what it ollows that the two other eigenvalues are o opposite sign. Let ε = a s b. Whenever a and b are parallel, two o the eigenvalues o ε are equal to 0 and the assertion is proved. When a and b Result 5.3are not collinear, a basis o R 3 can be ormed by considering the three vectors: e 1 = a a, e = a b a b, e 3 = e 1 e. In this basis, a, b and ε write respectively a b 1 a b a b 3 a = 0, b = 0, [ε] = b 3 a b An alternative proo o this result can be ound in Bhattacharya (003, Result 5.3). 5

27 The eigenvalues o ε solve the characteristic equation : λ [ λ(λ a b 1 ) a b 3 4 It ollows rom this expression that one o the eigenvalues o ε is 0 (the associated eigenvector is e ), and that the other two eigenvalues are o opposite sign since their product is negative ( a b 3 0). Thereore ε 4 1 0, ε = 0 and ε 3 0. Conversely, assume that the eigenvalues o ε satisy the string o inequalities ε 1 ε = 0 ε 3. A direct calculation shows that ε = a s b where the components o a and b in the principal basis o ε are: a = 1 ε3 ε 1 ] = 0 ε1 0, b = ε 1 ε 3 ε 1 0. ε3 ε3 This completes the proo o point a). Proo o b): Let ε be a rank-one tensor (such that ε 1 ε = 0 ε 3 according to part a) o the lemma). Its irst two invariants and its triaxiality-ratio read as : ε m = ε 1 + ε 3, ε eq = ε 1 + ε 1 + x 3 ε 1 ε 3, T = x x, x = ε 1 0. ε 3 (5.8) The unction x T(x) is a strictly increasing unction on ], 0] since its derivative T 3 3x = is strictly positive on this interval. Thereore the (1+x x) 3/ minimum and maximum o T are attained at the boundary o the interval, when x = 0 and x : This completes the proo o point b). lim T = 1 x T(x) T(0) = 1. Proo o c): Note that or given T 1 and unknown x, the equation (5.8) has a unique solution x T on the interval ], 0] because o the strict monotonicity o T(x) on this interval. Choose ε = 0 and pick-up arbitrarily ε 3 > 0 (i T = 1 then x T, ε 3 = 0 and ε 1 < 0 is chosen arbitrarily). Then the problem is reduced to inding a rank-one tensor ε with eigenvalues (ε 1 = ε 3 x T, 0, ε 3 ). According to point a) such a rank-one tensor exists. Corollary 1 There exists a unique axisymmetric rank-one tensor (up to a 6

28 rotation and a multiplicative actor): ε = ε n n, where n is an arbitrary unit vector and ε an arbitrary scalar. Indeed, let ε be an axisymmetric second-order rank-one tensor and let ε 1 = ε and ε 3 denote its eigenvalues. Then, either ε 1 ε 3 or ε 1 ε 3. In both cases, since ε is rank one, it ollows, rom the lemma 3 a), that ε 1 = ε = 0. Thereore two eigenvalues o ε vanish and the corollary is proved. Note that the triaxiality-ratio T o this tensor is T = 1. Lemma 4 Let w and L(ε) be as in Theorem 1 and such that the coeicients k, λ, µ and γ deined by (3.17) satisy µ 0 and k, γ and λ are not all equal to 0. Then the set L min = {θ Rs 3 3 such that θ : L : θ = 0} (L min is the set where the quadratic orm deined by L attains its minimum) is a one-dimensional space spanned by ε. Proo o lemma 4: We irst show that L min coincides with the kernel o L : L min = Ker(L) = {θ L : θ = 0}. The inclusion Ker(L) L min is straightorward. Conversely, let θ be such that θ : L : θ = 0. Then θ is a minimum point or the unction θ 1θ : L : θ since L is positive (rom the convexity o w). As such it satisies the stationarity condition ( ) 1 θ θ : L : θ = 0. Thereore L : θ = 0. The next step is to prove that i θ is such that L : θ = 0, its deviator is parallel to ê. Using the standard decomposition, θ = θ m i + θ d, one gets : L : θ = [ 3kθ m + γ(θ d : ê) ] i + µθ d + 6γθ m ê (λ µ)(θd : ê)ê. The equation L : θ = 0 reduces to two equations (hydrostatic and deviatoric part respectively) : 3kθ m + γ(θ d : ê) = 0, µθ d + 6γθ m ê (λ µ)(θd : ê)ê = 0. Since µ 0, the second equation shows that θ d is parallel to ê. Thereore θ is in the orm θ = θ m i + αê and the above system reduces to two scalar equations : kθ m + γα = 0, 3γθ m + λα = 0. (5.9) 7

29 This is a homogeneous system o equations or the two unknowns (θ m, α). The linear space o solutions o this homogeneous system cannot be o dimension since the coeicients k, γ and λ are not all equal to 0. It cannot be o dimension 0 since, rom the act that w is positively homogeneous o degree 1 it ollows ε : L : ε = 0, and thereore (ε m, ε eq ) is a solution o this system. The only remaining option is that the space o solutions o (5.9) is o dimension 1. As a consequence, there exists Λ R such that θ m = Λ ε m et α = Λ ε eq, or in other words θ = Λ ε. Thereore every element θ in L min is in the orm θ = Λ ε. This completes the proo o lemma 4. Proo o Theorem 1 : Let us irst prove that i ε is rank-one, there exists ξ such that det(k(ξ)) = 0. Since ε = a s b with two non-vanishing vectors a and b and since ε is in L min, one has, L(ε) : ε = L(ε) : a s b = 0. Given the symmetries o L(ε), this relation can be alternatively written as : L(ε).a.b = 0, and upon multiplication by a it becomes : a.l(ε).a.b = 0. Since b 0, one has det(a.l(ε).a) = 0. The acoustic tensor K evaluated at ξ = a/ a is singular. Conversely, let ξ be a vector or which det(ξ.l(ε).ξ) = 0. There exists a non-vanishing vector a such that ξ.l(ε).ξ.a = 0 and consequently a.ξ.l(ε).ξ.a = 0. By virtue o the symmetries exhibited by L(ε), this relation can be re-written as : ξ s a : L(ε) : ξ s a = 0. Thereore ξ s a belongs to L min. According to lemma 4, ξ s a is proportional to ε and there exists Λ such that ε = Λξ s a. This proves that ε is a rank-one symmetric tensor. Remark: According to theorem 1, all strains ε = ε eq (Ti + ê) or which there exists ξ such that det(ξ.l(ε).ξ) = 0 are rank-one tensors. According to lemma 3 they are such that T 1/ and their eigenvalues, which can be written as ε 1 = ε eq (T + ê 1 ), ε = ε eq (T + ê ), ε 3 = ε eq (T + ê 3 ), satisy ε 1 ε = 0 ε 3, (or ω [ 0, π 3] ). The condition ε = 0 gives : ( T + ê = T cos ω + π ) =