Homogenization of multi-coated inclusion-reinforced linear elastic composites with eigenstrains: application to the thermo-elastic behavior

Transcription

1 Homogenization of multi-coated inclusion-reinforced linear elastic composites with eigenstrains: application to the thermo-elastic behavior Stephane Berbenni Mohammed Cherkaoui To cite this version: Stephane Berbenni Mohammed Cherkaoui. Homogenization of multi-coated inclusion-reinforced linear elastic composites with eigenstrains: application to the thermo-elastic behavior. Philosophical Magazine Taylor Francis 00 0 ) pp.0-. <0.00/00>. <hal-00> HAL Id: hal-00 Submitted on Jun 0 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents whether they are published or not. The documents may come from teaching and research institutions in France or abroad or from public or private research centers. L archive ouverte pluridisciplinaire HAL est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche publiés ou non émanant des établissements d enseignement et de recherche français ou étrangers des laboratoires publics ou privés.

2 Homogenization of multi-coated inclusion-reinforced linear elastic composites with eigenstrains: application to the thermo-elastic behavior Journal: Manuscript ID: TPHM-0-Oct-00.R Journal Selection: Philosophical Magazine Date Submitted by the Author: 0-Feb-00 Complete List of Authors: Berbenni Stephane; LPMM FRE CNRS ENSAM Cherkaoui Mohammed; Georgia Tech Mechanical Engineering Keywords: micromechanics thermal expansion Keywords user supplied): interphase multi-coated inclusion eigenstrains Note: The following files were submitted by the author for peer review but cannot be converted to PDF. You must view these files e.g. movies) online. articlepr.tex

3 February 00 0:0 Philosophical Magazine articlepr Page of 0 0 Philosophical Magazine Vol. 00 No. 00 February 00 RESEARCH ARTICLE Homogenization of multi-coated inclusion-reinforced linear elastic composites with eigenstrains: application to the thermo-elastic behavior Stéphane Berbenni ab and Mohammed Cherkaoui bc a Laboratoire de Physique et Mécanique des Matériaux LPMM FRE CNRS ENSAM Technopole 0 Metz Cedex 0 France; b UMI Georgia Tech-CNRS Technopole 00 Metz France; c George W. Woodruff School of Mechanical Engineering Atlanta GA -0 USA v. released March 00) A new micromechanical approach for arbitrary multi-coated ellipsoidal elastic inclusions with general eigenstrains is developed. We start from the integral equation of the linear elastic medium with eigenstrains adopting the Green s function technique and we apply a n+)-phase model with a self-consistent condition to determine the homogenized behavior of multi-coated inclusion-reinforced composites. The effective elastic moduli and eigenstrains are obtained as well as the residual stresses through the local stress concentration equations. The effective eigenstrains are determined either with the concentration tensors here obtained by the present model or more classically with Levin s formula. In order to assess our micromechanical model some applications to the isotropic thermo-elastic behavior of composites with and without interphase are given. In particular -phase and -phase models are derived for isotropic homothetic spherical inclusion-reinforced materials and the results are successfully compared to exact analytical solutions regarding the effective elastic moduli and the effective thermal expansion. Keywords: Interphase; Multi-coated inclusion; Eigenstrains; Thermo-elastic Behavior; Micro-mechanics. Introduction Numerous composite materials contain an interphase between a matrix phase and reinforcements like fibers inclusions etc. This interphase may significantly change the mechanical properties of composites. Classical linear elastic properties were extensively studied by various authors in solid mechanics and in civil engineering applications to account for the role of interphase on the homogenized behavior of composites [ ]. Meanwhile the influence of other materials properties in the interphase giving rise to the so called eigenstrains [ 0] are less explored. These eigenstrains or stress-free strains) are inelastic strains which occur in many physical problems e.g. thermal strain lattice mismatch strain static configurations of dislocations chemical compositions magnetostrictive strain poroelasticity etc.). The purpose of the paper is twofold. The first issue is to extend a recently developed self-consistent model for linear elastic heterogeneous composites based on the multi-coated inclusion problem [] to the case of same types of composites Corresponding author. Stephane.Berbenni@metz.ensam.fr ISSN: - print/issn - online c 00 Taylor & Francis DOI: 0.00/YYxxxxxxxx

4 February 00 0:0 Philosophical Magazine articlepr Taylor & Francis and I.T. Consultant Page of 0 0 undergoing eigenstrains in addition to elastic strains. This means that the problem needs to be solved from the integral equation of the linear elastic microheterogeneous solid with eigenstrains. Thus inclusions interphases and matrix will have different elastic properties and different eigenstrains i.e. piecewise uniform). The extensions of the technique based on the so called interfacial operators [ ] to this multi-coated inclusion problem is here highlighted. The second objective is to apply the present model to the thermo-elastic behavior of multiple coated inclusion-reinforced composites. Actually this problem can be solved analytically using exact results derived by Hervé and Zaoui [] for the effective elastic properties who extended the -phase model of Christensen and Lo []. Later Hervé [] derived the effective thermal expansion using the results of Hervé and Zaoui [] and the Levin s formula [ ]. Mean field approximations based on the multi-inclusion model derived by Nemat-Nasser and Hori [] were also applied to this particular case by Li []. Variational principles to derive sharp bounds on the effective thermal expansion coefficients of multiphase composites were given by Willis [] Gibiansky and Torquato [] and Stolz [ ]. In particular the exact solutions for the -phase model are retrieved [ ]. Results of the present model are given in the particular case of a spherical inclusion-reinforced material i.e. containing an interphase and compared to exact solutions [ ]. In contrast to [ ] the presented methodology can be applied to more complex anisotropic problems and to ellipsoidal inclusions of arbitrary shapes. The paper is organized as follows. In section the micromechanical approach for arbitrary multi-coated ellipsoidal elastic inclusions with general eigenstrains is detailed. We start from the integral equation of the problem adopting the Green s function technique and we use interfacial relations for perfect bonded interfaces with an averaging procedure which gives solutions in the form of recurrence relations. Then we apply a generalized self-consistent scheme or n+)-phase model [ ] to determine the homogenized behavior of multi-coated inclusion-reinforced composites. As a result we obtain the effective elastic properties and the effective eigenstrains as well as the residual stresses due to eigenstrains. It is highlighted that the effective eigenstrains can be determined by two different ways: the one detailed in the present paper using concentration tensors and the one derived from Levin s formula [ ]. In section we apply the present micromechanical approach -phase and -phase versions) to layered homothetic spherical inclusion-reinforced composite materials with isotropic elastic properties and thermal properties in each phase. In particular we compare the results of our -phase and -phase approaches respectively to the exact analytical results of Hervé [] and Stolz [ ] regarding the effective thermal expansion. Section is devoted to conclusions and perspectives of this work.. Micromechanical approach.. Field Equations and Integral Equation On the boundary V of V a prescribed displacement u d Dirichlet conditions) is considered: where E is a uniform imposed strain on V. The other field equations are constituted of: u d = E x on V )

5 February 00 0:0 Philosophical Magazine articlepr Page of Philosophical Magazine the stress equilibrium condition for the symmetric Cauchy stress tensor σ : div σx) = 0 in V ) - the compatibility relation for total strain ǫ where u is the displacement field: ǫx) = ux) + t ux) ) ) - the total strain in the small perturbation hypothesis which writes as the sum of an elastic strain ǫ e and an eigenstrain ǫ : ǫx) = ǫ e x) + ǫ x) ) - the constitutive equation for linear elasticity Hooke s law) with the presence of eigenstrains: σx) = Cx) : ǫ e x) = Cx) : ǫx) ǫ x)) = Cx) : ǫx) + λx) ) where Cx) denotes the elastic moduli. In this problem the unknown fields are the displacement u from which the total strain ǫ and the Cauchy stress σ are derived. In the following we consider λx) = Cx) : ǫ x) as the eigenstress associated to the eigenstrain ǫ x). Then first order spatial variations of elastic properties and eigenstresses are respectively denoted δcx) and δλx) so that: Cx) = C 0 + δcx) λx) = λ 0 + δλx) where C 0 and λ 0 = C 0 : ǫ 0 denote respectively the homogeneous elastic moduli and eigenstresses of the infinite reference medium 0) described in Fig.. By introducing these fluctuations in the set of field equations eqs.) to )) we obtain the so-called integral equation [ ] of the problem as follows: ǫ ij x) = E ij Γ 0 ijkl x x ) δc klmn x )ǫ mn x ) + δλ kl x ) ) dv ) V where Γ 0 x x ) is the modified Green s tensor associated to C 0. According to [] this one classically writes: Γ 0 ijkl x x ) = ) G 0 ikjl x x ) + G 0 jkil x x ) ) ) with G 0 being the Green s function of the infinite homogeneous medium C 0... Case of a n-phase multi-coated composite inclusion... Concentration equations We first apply the integral equation eq.)) to the case of a n-phase multi-coated composite inclusion of volume V I embedded in an infinite reference medium denoted

6 February 00 0:0 Philosophical Magazine articlepr Taylor & Francis and I.T. Consultant Page of 0 0 Figure. Multi-coated inclusion problem with eigenstrains considering n phases. The inclusion is embedded in an infinite reference medium 0). In this decription Phase is called the interphase when n = for instance. 0 see Fig.) and containing a set of phases k with k [..n] characterized by their volume V k and the characteristic function θ k x) defined by: θ k x) = { if x Vk 0 if x / V k Then the first order variations of elastic moduli and eigenstresses follow a piecewise uniform decomposition in the form of: δcx) = δλx) = n C k/0) θ k x) with C k/0) = C k C 0 k= n λ k/0) θ k x) with λ k/0) = λ k λ 0. k= In the following and according to eq.0) C k and λ k denote piecewise constant values per phase k respectively for Cx) and λx). By construction the volume V I of the composite inhomogeneity is constituted of the first inhomogeneity ) and n other coatings. Thus: V I = ) 0) n V k ) k=

7 February 00 0:0 Philosophical Magazine articlepr Page of Philosophical Magazine 0 0 and the volume fraction φ k of phase k in V I is defined by: The average strain ǫ I over V I is defined by: φ k = V k V I for k [..n]. ) ǫ I = V I V I ǫx)dv ) After simple manipulations using eq.) and eq.0) we obtain: where: and ǫ I = E T I C 0 ) : n φ k C k/0) : ǫ k + λ k/0)) ) k= T I C 0 ) = V I From eq.) and eq.) we can also write: V I V I Γ 0 x x )dv dv ) ǫ k = V k V k ǫx)dv. ) ǫ I = n φ k ǫ k. ) k= Thus by comparing eq.) and eq.) we find out: E = n φ k [ǫ k + T I C 0 ) : k= C k/0) : ǫ k + λ k/0))]. ) The concentrations tensors A I and b I for the composite formed by volume V I and the concentrations tensors α k and β k for each phase k can be introduced [ ] so that : with from eq.): ǫ I = A I : E + b I ǫ k = α k : ǫ I + β k n φ k α k = I k= n φ k β k = 0 k= ) 0)

8 February 00 0:0 Philosophical Magazine articlepr Taylor & Francis and I.T. Consultant Page of 0 0 where 0 and I are respectively null and unit fourth order tensors. From eqs.) we can also write: where: Using eq. ) it comes: ǫ k = A k : E + b k ) A k = α k : A I b k = α k : b I + β k. n )) A I = I + T I C 0 ) : φ k C k/0) : α k k= n b I = A I : T I C 0 ) : φ k [ C k/0) : β k + λ k/0)]).... Interfacial relations k= We consider a perfect bonding at the interface k/k + Fig. ) denoted k) for which essential properties at any point x of the interface are the continuity of displacement vector [u] k) i x) u k+ i x) u k i x) = 0 and the continuity of the interfacial traction vector [σ] k) ij N k) j σij k+ N k) j σij k Nk) j = 0 where N k) i is the outward unit normal on the interface k). As detailed elsewhere [ ] the strains and the stresses are discontinuous and their jumps can be related to the so-called interfacial operators introduced by Walpole [] or Hill []. The strain jump is obtained from the compatibility relation at any point of the interface and taking the symmetric part of the displacement gradient jump: ij x) ǫ k+ ij x) ǫ k ijx) = [ǫ] k) ν k) i N k) j + ν k) j N k) i where ν k) i is a vector describing the direction and the magnitude of the strain jump on the interface k). Starting from eq.) in k and k + respectively: σ k+ ij = C k+ ijkl ǫk+ ij kl + λ k+ σ k ij = C k ijkl ǫk kl + λk ij and using the continuity of the interfacial traction vector at the interface we obtain: C k+ ijkl [ǫ]k) kl N k) j + [C] k) ijkl ǫk kl Nk) j and [λ]k) ij = λ k+ ij where [C] k) ijkl = Ck+ ijkl Cijkl k by its expression given by eq.). It gives: + [λ] k) ij N k) ) ) ) ) ) j = 0 ) λ k ij. In eq.) we replace [ǫ]k) kl h k+ ik νk) k + [C] k) ijkl ǫk kl Nk) j + [λ] k) ij N k) j = 0 ) where h k+ ik is the Christoffel matrix associated with k + and defined by h k+ ik = N k). The Christophel matrix is found symmetric if the classic symmetry C k+ ijkl Nk) j l

9 February 00 0:0 Philosophical Magazine articlepr Page of Philosophical Magazine 0 0 for the elastic moduli are assumed i.e. C k+ ijkl = C k+ jikl = C k+ ijlk ). Following Walpole [] and Hill [] the interfacial operator P k+ ijkl only depends on the elastic moduli C k+ of phase k + and on the unit normal N of the interface k): P k+ ijkl = ) h k+ ) ik N jn l + h k+ ) jk N in l + h k+ ) il N j N k + h k+ ) jl N in k ) From eq.) assuming the determinant of h k+ ik is non zero) and eq.) it comes directly : ) ǫ k+ ij x) = I ijmn + P k+ ijkl Ck klmn Ck+ klmn ) ǫ k k+ mn x) + Pijkl λk kl λk+ kl ) ) where λ k C k and λ k+ C k+ are respectively uniform in phases k and k Approximation by an averaging procedure and solutions Figure. Multi-coated inclusion problem with eigenstrains considering n phases. Approximation using the average strain fields over Ω k = V... V k the volume of the composite formed by the phases to k. For each level k) Fig. ) we denote by Ω k = V... V k the volume of the composite formed by the phases to k. Then in order to solve the problem we adopt the following assumption which avoids complex full field calculations: ǫ k x) is substituted by the averaged value of ǫx) over Ω k denoted ǫ Ωk. Thus by performing the average strain over the coating of volume V k+ denoted ǫ k+ we obtain the following recurrence relation at each level k) from eq.): ǫ ij k+ = ) I ijmn + T k+ ijkl Ck+ )C Ωk klmn Ck+ klmn ) Ω ǫ k mn +T k+ ijkl Ck+ )λ Ωk kl λ k+ kl ) )

10 February 00 0:0 Philosophical Magazine articlepr Taylor & Francis and I.T. Consultant Page of 0 0 where: and T k+ C k+ ) = P k+ dv ) V k+ V k+ ǫ Ωk = k i= V i Ω k ǫ i = k φ i ǫ i i=. ) k φ i C Ωk and λ Ωk are respectively the elastic moduli and eigenstresses of the composite inclusion formed by volume Ω k. It is noteworthy that C Ωk and λ Ωk will be naturally eliminated in the following equations by recurrence relations starting from the basic configuration described by n=. Indeed for n = we have ǫ Ω = ǫ from eq.) C Ω = C and λ Ω = λ because Ω = V Fig. ). Thus eq.) reduces to: ǫ ij = I ijmn + Tijkl C )Cklmn C klmn )) ǫ mn + Tijkl C )λ kl λ kl ). ) Following [] we can demonstrate in the general case of non homothetic ellipsoidal inclusions that the expression of T k+ C k+ ) takes the form of: with: T k+ C k+ ) = T Ωk C k+ ) k i= φ i i= ) T Ωk+ C k+ ) T Ωk C k+ ) ) φ i+ T Ωk C k+ ) = Ω k Ω k ΓC k+ )dv T Ωk+ C k+ ) = Ω k+ Ω k+ ΓC k+ )dv. Furthermore eq.) can be written in the following form: where: ǫ k+ = ǫ Ωk T k+ C k+ ) : ) C k+/ωk) : ǫ Ωk + λ k+/ωk)) ) C k+/ωk) = C k+ C Ωk λ k+/ωk) = λ k+ λ Ωk. Using simple manipulations on the averaged values with the Hooke s law of each )

11 February 00 0:0 Philosophical Magazine articlepr Page of Philosophical Magazine 0 0 phase from to k we find out: C k+/ωk) : ǫ Ωk + λ k+/ωk) = k i= φ i C k+/i) : ǫ i + λ k+/i)). ) k φ i For n = both sides of eq.) give C /) : ǫ + λ /) because ǫ Ω = ǫ C Ω = C and λ Ω = λ. By using eqs.))) the expressions of α k+ and β k+ in eq.) are obtained: where: α k+ = β k+ = k φ i w k+/i) : α i i= k φ i i= i= k φ i w k+/i) : β i v k+/i)) i= k φ i i= w k+/i) = I T k+ C k+ ) : C k+/i) v k+/i) = T k+ C k+ ) : λ k+/i). Then by recurrence we can transform eqs.) into the following equations: α k+ = X k+ : α β k+ = X k+ : β + Y k+ with the recurrence relations for X k+ and Y k+ : X k+ = Y k+ = k φ i w k+/i) : X i i= k i= k φ i i= φ i w k+/i) : Y i v k+/i)). k φ i Thus it is sufficient to derive α and β to completely solve the problem. This is i= ) ) ) )

12 February 00 0:0 Philosophical Magazine articlepr 0 Taylor & Francis and I.T. Consultant Page 0 of 0 0 done by applying eqs.0) such that: n ) α = φ k X k k= β = α : n φ k Y k. Let us note that the purely linear elastic solution for the n-phase composite inclusion described by Fig. recently obtained by Lipinski et al. [] is retrieved in eq.) by setting λ k = 0 and β k = 0 for all k in eq.) and eq.). This can be considered as a particular case of the general formalism developed in the present paper. k=.. n + )-phase self-consistent model In this section we adopt the same methodology as the Generalized Self-Consistent Scheme GSCS) introduced by Christensen and Lo [] the so-called -phase model) for composite spheres or cylinders) and extended by Hervé and Zaoui [ ] to a n + )-phase model with a self-consistent condition. A Homogeneous Equivalent Medium HEM) is introduced in Fig. in addition to the n phases including the matrix phase denoted 0). In the following the effective properties associated with the HEM are denoted eff. We consider an imposed strain E Fig. ) such that u d = E x on the boundary of the Representative Volume Element RVE) of total volume V T. As highlighted by Hervé and Zaoui [] and by Zaoui [0] the Christensen and Lo s self-consistent energy condition [] coincides with the following average strain condition: E = ǫ VT = V T ) n n ǫx)dv = f q ǫ q = f 0 ǫ 0 + f q ǫ q ) V T where f 0 = V 0 V for the matrix phase 0) and f q = V q T V for the other phases q) T from to n such that: q=0 q= n f q =. ) q=0 Thus the homogenized behavior or effective behavior associated with the HEM) writes: Σ = C eff : E + λ eff ) where C eff and λ eff are unknown effective elastic moduli and eigenstresses. In eq.) Σ is the volume stress average in the RVE: n Σ = f 0 σ 0 + f q σ q. ) q=

13 February 00 0:0 Philosophical Magazine articlepr Page of Philosophical Magazine 0 0 The respective constitutive behaviors for the matrix 0) and the other phases q) are: σ 0 = C 0 : ǫ 0 + λ 0 σ q = C q : ǫ q + λ q. The strain concentration equation for each phase q) including the matrix phase) writes: with: ) ǫ q = A q : E + b q ) A q = α q : A b q = α q : b + β q where the concentration tensors α q β q were introduced in section. through eqs.) ) ) and A b are adapted from eqs.) for the n + )-phase model: n A = I + T I C eff ) : f q C q/eff) : α q q=0 n [ b = A : T I C eff ) : f q C q/eff) : β q + λ q/eff)]. q=0 Applying the strain average condition eq.)) yields: n f q A q = A VT = I q=0 n f q b q = b VT = 0. q=0 The stress concentration equations yield: where from eqs.) and ): B q = C q : A q : C eff ) ) ) ) σ q = B q : Σ + d q ) d q = C q : A q : C eff ) : λ eff + C q : b q + λ q. )

14 February 00 0:0 Philosophical Magazine articlepr Taylor & Francis and I.T. Consultant Page of 0 0 d q denote the residual stresses in each phase q) i.e. for Σ = 0). From the stress average condition eq.)) we have: n f q B q = B VT = I q=0 n f q d q = d VT = 0. q=0 The last equation d VT = 0) means that the residual stresses are self-equilibrated over the RVE. Thus using the previous concentration equations in eq.) we find out the expressions for C eff and λ eff as: n C eff = C 0 + f q C q/0) : A q q= n λ eff = λ 0 + f q C q/0) : b q + λ q/0)) q= After a few manipulations and algebra using eq.) eq.) and eq.) we can also write C eff and λ eff as: n C eff = f q C q : α q : A q=0 n λ eff = f q λ q + C q : β q ). q=0 It is worth noticing that another way to obtain the expression of λ eff is to use directly the Levin s formula [ ] applied to the composite depicted in Fig.. From the knowledge of A q eq.)) we deduce from the Levin s formula: ) ) ) n λ eff = f q A qt : λ q ) q=0 where A qt is the transpose of A q such that A qt ijkl = Aq klij. To conclude this part two significant results of the present model can be reached through the last equations: the local mechanical fields like the residual stresses respectively residual strains) and the homogenized elastic properties and the effective eigenstresses respectively the effective eigenstrains). The Levin s formula can only achieve the latter because this does not need a relevant description of the microstructure.. Application to isotropic thermo-elastic materials with homothetic spherical inclusions In the case of layered homothetic spherical inclusion-reinforced n-phase composite with isotropic properties in each phase Hervé and Zaoui [] and Hervé [] re-

15 February 00 0:0 Philosophical Magazine articlepr Page of Philosophical Magazine 0 0 Figure. Schematic principle of the n + )-phase model self-consistent condition). spectively found the exact expressions for the effective elastic properties and the effective thermal expansion. In the following we adapt the equations of our model to this peculiar case in order to assess the good quality of the estimation of effective properties resulting from the n + )-phase self-consistent model described previously. In particular -phase and -phase models are respectively derived in the cases of -phase and -phase composites... -phase model As a first application let us apply the concept of interphase as described here in a -phase model. Thus we consider a heterogeneous elastic material with eigenstrains globally isotropic with isotropic phases and the peculiar case of a layered homothetic spherical inclusion-reinforced material is examined. Thus the -phase composite material is formed by a matrix phase denoted 0) inclusions reinforcements denoted ) and interphases between inclusions and matrix denoted ). Thus the RVE of this material is reported in Fig.. For the description of fourth order isotropic tensors we use the orthogonal projection tensors [ ] denoted J and K such that the unit tensor I decomposes as: I = J + K )

16 February 00 0:0 Philosophical Magazine articlepr Taylor & Francis and I.T. Consultant Page of 0 0 Figure. Schematic principle of the -phase scheme self-consistent condition). The inclusion phase phase )) is coated by the interphase phase )). Phases ) and ) are coated inclusions embedded in the matrix phase 0). In this decription we estimate the effective behavior of the Homogenized Equivalent Medium denoted HEM). with: I ijkl = δ ikδ jl + δ il δ jk ) J ijkl = δ ijδ kl K ijkl = δ ik δ jl + δ il δ jk ) δ ijδ kl where δ ij is the Kronecker operator. Furthermore J and K have the following properties: J : J = J K : K = K J : K = K : J = 0 J : I = I : J = I K : I = I : K = 0 where I is the second order unit tensor i.e. I ) ij = δ ij. For each phase q = 0 of the composite the elastic moduli are supposed ) )

17 February 00 0:0 Philosophical Magazine articlepr Page of Philosophical Magazine 0 0 isotropic: C q = k q J + µ q K ) where k q and µ q denote the bulk modulus and the shear modulus respectively. The thermal stresses λ q are also supposed isotropic: λ q = k q α q I Θ ) where α q is the thermal expansion of phase q) and Θ is the temperature rise relative to a reference temperature which is arbitrarily chosen as zero. In the isotropic case the effective elastic moduli C eff and the effective thermal stresses λ eff are of the form: C eff = k eff J + µ eff K λ eff = k eff α eff I Θ. In Eq.) k eff and µ eff are respectively the effective elastic bulk and shear moduli and α eff is the effective thermal expansion coefficient. In the particular case of concentric homothetic ellipsoidal inclusions eq.) reduces to see e.g. []): ) T q+ C q+ ) = T I C q+ ) = T Ω C q+ ) =... = T Ωq C q+ ) = T Ωq+ C q+ ). ) Furthermore for spherical inclusions T I C q ) reads []: T I C q J ) = + k q + µ q )K k q + µ q µ q k q + µ q ) Let us apply the general equations obtained in the section for the homogenized behavior of the composite described in Fig.. In this case by applying eq.) C eff reads: and λ eff reads: ) C eff = f 0 C 0 : α 0 + f C : α + f C : α ) : A ) λ eff = f 0 λ 0 + C 0 : β 0) + f λ + C : β ) + f λ + C : β ). ) where the concentration tensors A α 0 α α β 0 β β can write in the isotropic symmetry using eqs.)))))): A = MJ + DK α 0 = m 0 J + d 0 K α = m J + d K α = m J + d K β 0 = n 0 I Θ β = n I Θ β = n I Θ. )

18 February 00 0:0 Philosophical Magazine articlepr Taylor & Francis and I.T. Consultant Page of 0 0 In the last equation the expressions of M D m 0 d 0 m d m d n 0 n n are determined in the Appendix. Let us note that the expressions of n 0 n n represent new contributions due to thermal effects in comparison with the work of Lipinski et al. []. M and D depend on k eff and µ eff then from eq.) k eff and µ eff are the solutions of the following system of non linear equations: k eff k 0 f k k 0 )m M f k k 0 )m M = 0 µ eff µ 0 f µ µ 0 )d D f µ µ 0 )d D = 0. To solve this system of equations a standard Levenberg-Marquardt procedure [] is chosen with a starting guess at the solutions corresponding to the volume averages of elastic bulk and shear moduli over the RVE i.e. a Voigt approximation). To show the importance of the interphase elastic properties on the effective elastic moduli of the composite we used same materials parameters as the ones used by Hervé and Zaoui [] their phase ) corresponds to our phase 0)) i.e. identical Poisson s ratios for all the phases: ν = ν = ν 0 = 0. µ /µ 0 = and f + f = 0.. The results concerning the normalized effective shear moduli µ eff /µ 0 are reported in Fig. and Fig.. These results are compared with the ones obtained by Hervé and Zaoui [] from their equations numbered ) and ). Fig. shows the evolution of µ eff /µ 0 as a function of the interphase volume fraction f for different mechanical contrasts between the interphase and the matrix phase characterized by the ratio β = µ /µ 0. When f tends to 0. it is noteworthy that no influence of interphase occurs and the -phase model s solution [] is retrieved for all values of β or when β = i.e. µ = µ for all values of f ). In Fig. the interphase volume fraction f is fixed to 0.0 and the evolution of µ eff /µ 0 is plotted as a function of β = µ /µ 0. Two different regimes are observed: a first strong increase of the effective shear moduli when the values of β are lower than followed by a saturation for the values of β larger than. As already noticed by Lipinski et al. [] for the elastic regime only the present approach give same results as the exact solution of Hervé and Zaoui []. Once the effective bulk modulus k eff is obtained the effective thermal expansion coefficient α eff can be computed using eq.) and eq.) as: 0) α eff = k eff f n k f n k f 0 n 0 k 0 + f k α + f k α + f 0 k 0 α 0 ). ) The obtained effective thermal expansion coefficient α eff is reported in Fig. in the case of a composite material made of coated inclusions as described in Fig.. Here the volume fraction of the interphase f is fixed to 0.0 and the one of the inclusions f is fixed to 0.. The strong influence of both interphase properties α and k ) on α eff /α is here demonstrated. The results are found to be coherent with Hervé s result [] reported in Fig. using the equation numbered ) from her paper. Furthermore she used the effective elastic properties obtained by [] see Fig. and Fig. ) and Levin s formula [ ] to derive α eff. In order to double check the results reported in Fig. we also applied the Levin s formula [ ] according to eq.) applied to the composite described in Fig.. In this case isotropic configuration) α eff can be derived by the following formula: α eff = M k eff f m k α + f m k α + f 0 m 0 k 0 α 0 ) ) where M m m m 0 are given by eq.) and are detailed in the appendix. By

19 February 00 0:0 Philosophical Magazine articlepr Page of Philosophical Magazine 0 0 µ eff / µ f Figure. Estimation from the present model of normalized effective shear modulus µ eff /µ 0 lines) and comparison with the exact solutions given by Hervé and Zaoui [] points) for a composite material constituted of coated inclusions versus the volume fraction f of the interphase phase ) phase 0 denotes the matrix and phase denotes the inclusions). The results are plotted for different values of β = µ /µ 0 with ν = ν = ν 0 = 0. Poisson s ratios) µ /µ 0 = and f + f = 0.. µ eff / µ µ / µ 0 β=0.0 β=0. β= β= β=0 β=00 β=000 Present model Hervé and Zaoui Figure. Estimation from the present model of normalized effective shear modulus µ eff /µ 0 lines) and comparison with the exact solutions given by Hervé and Zaoui []dashed lines) for a composite material constituted of coated inclusions versus the normalized shear modulus µ /µ 0 of the interphase phase ) phase 0 denotes the matrix and phase denotes the inclusions). The results are plotted for ν = ν = ν 0 = 0. µ /µ 0 = f = 0. and f = 0.0.

20 February 00 0:0 Philosophical Magazine articlepr Taylor & Francis and I.T. Consultant Page of 0 0 α eff /α k /µ α /α Figure. Normalized effective thermal expansion coefficient α eff /α of a composite material made of coated inclusions as described in Fig. : result from the present model eq.)) with f = 0. f = 0.0. The elastic properties of the different phases are characterized by: k /µ 0 = µ /µ 0 = k 0 /µ 0 = µ /µ 0 = and α 0 /α = 0. applying this last formula we find the same numerical results as previously Fig. )... -phase model In the case of a -phase model Fig. 0) and using the same conventions and notations as in section. k eff µ eff and α eff are obtained from eqs.0) ) or ) by setting f = 0 no interphase). We set f = f and f 0 = f for the -phase composite 0) being the matrix phase and ) the inclusions). Then both eqs.0) and ) of the present modeling reduce to: k eff k 0 f k k 0 ) m M = 0 µ eff µ 0 f µ µ 0 )d D = 0 with f = 0 f = f and f 0 = f in m M d and D given in the Appendix and 0 ) α eff = k eff fn k f)n 0 k 0 + fk α + f)k 0 α 0 ) ) with f = 0 f = f and f 0 = f in n and n 0 given in the Appendix. In the case of thermo-elastic heterogeneous materials Stolz [] found rigorous bounds for different morphological assemblages starting from the free energy and developing a variational procedure. In the case of the -phase model [] the effective bulk modulus is exactly determined i.e. the bounds give the same value) and corresponds to the analytical solution given by the Composite Sphere Assemblage 00

21 February 00 0:0 Philosophical Magazine articlepr Page of Philosophical Magazine 0 0 α eff /α k /µ α /α Figure. Normalized effective thermal expansion coefficient α eff /α of a composite material made of coated inclusions as described in Fig. : result from [] using the effective elastic properties obtained by [] and Levin s formula [ ] with f = 0. f = 0.0. The elastic properties of the different phases are characterized by: k /µ 0 = µ /µ 0 = k 0 /µ 0 = µ /µ 0 = and α 0 /α = 0. model []: k eff = k 0 + and α eff reads using the Levin s formula: where: f k k 0 )k 0 + µ 0 ) k 0 + µ 0 + f)k k 0 ) ) α eff = α VT + /k eff /k VT /k 0 /k α 0 α ) ) α VT = fα + f)α 0 /k VT = f/k + f)/k 0. According to Fig. the results of the present -phase model in terms of effective thermal expansion given by eqs.) and ) are found in excellent agreement with the ones given by eqs.) and ) [] for -phase materials with different volume fractions f of inclusions and various ratios γ = µ /µ 0 with ν = ν 0 = 0. and α 0 /α = ).. Concluding remarks In this paper we investigated the effective behavior of multi-coated inclusionreinforced composites containing interphases with a linear elastic behavior with eigenstrains. These eigenstrains may be encountered in many physical situations of 0 ) 00

22 February 00 0:0 Philosophical Magazine articlepr 0 Taylor & Francis and I.T. Consultant Page 0 of 0 0 α eff /α k /µ α /α Figure. Normalized effective thermal expansion coefficient α eff /α of a composite material made of coated inclusions as described in Fig. : result from the present model using Levin s formula eq.)) with f = 0. f = 0.0. The elastic properties of the different phases are characterized by: k /µ 0 = µ /µ 0 = k 0 /µ 0 = µ /µ 0 = and α 0 /α = 0. Figure 0. Schematic principle of the -phase scheme self-consistent condition). The inclusions phase )) are embedded in the matrix phase 0). In this decription we estimate the effective behavior of the Homogenized Equivalent Medium denoted HEM). 0 00

23 February 00 0:0 Philosophical Magazine articlepr Page of Philosophical Magazine 0 0 α eff / α γ=0.0 γ=0. γ=0 γ= f Figure. Estimation from the present model eqs.) and )) lines) of the normalized effective thermal expansion coefficient α eff /α 0 of a composite material made of spherical inclusions phase ) in a matrix phase phase 0) see Fig. 0) as a function of the volume fraction f of inclusions and comparison with the exact solution given by the Levin s formula using the Composite Sphere Assemblage model for k eff [ ] eqs.) and )) points). The results are plotted for different values of γ = µ /µ 0 with ν = ν 0 = 0. Poisson s ratios) and α 0 /α =. importance in functional materials. By assuming perfectly bonded interfaces the effective elastic properties and the effective eigenstrains of multi-coated inclusionreinforced composites are retrieved with a n+)-phase self-consistent procedure. Even though the present micromechanical model is not exact due to averaging procedures introduced in section. this one can be applied to any anisotropic behaviors and reinforced composites with non-homothetic multi-coated ellipsoidal inclusions. Here we illustrated the efficiency of the model to study the influences of the interphase thermal expansion and elastic bulk modulus on the effective thermal expansion with our -phase model. Furthermore local and overall thermo-elastic behaviors are assumed isotropic. In this case the results of the present formulation match the exact results reported by Hervé [] for the -phase model and the ones reported by Stolz [ ] for the -phase model. Such framework is scheduled to be extended to the case of composite elastic materials with eigenstrains and with imperfect interfaces [ ] and to functional nanocomposites [ ] involving ellipsoidal nano-inhomogeneities.

24 February 00 0:0 Philosophical Magazine articlepr Taylor & Francis and I.T. Consultant Page of 0 0 Appendix: Expressions of M D m 0 d 0 m d m d n 0 n n used in section In this appendix we give the details for the complete expressions of M D m 0 d 0 m d m d n 0 n n present in eqs.) to derive the effective elastic moduli k eff and µ eff eqs.0)) and the effective thermal expansion coefficient α eff eq.)). By using simple algebra using eq.) and the properties of the orthogonal projection tensors [ ] denoted J and K introduced in section see also the developments in []) the following expressions of M and D eq.)) are obtained: M = D = + f k k eff ) m + f k k eff ) k eff + µ eff ) m 0 k eff + µ eff + f µ µ eff )k eff + µ eff ) µ eff k eff + µ eff ) f µ µ eff )k eff + µ eff ) µ eff k eff + µ eff ) k eff + µ eff m + f 0k 0 k eff ) d + d + f ) 0µ 0 µ eff )k eff + µ eff ) d 0 µ eff k eff + µ eff ) where m d m d m 0 d 0 are involved in the concentration tensors α α α 0 respectively see eqs.)). These ones cas be deduced from eqs.))))) as: m = f + f m + f ) 0 f m 0 + f m 0 m ) f + f d = f + f d + f ) 0 f d 0 + f d 0 d ) f + f m = f m ) + f + f 0 f m 0 m ) + f m 0) ) f + f d = f d ) + f + f 0 f d 0 d ) + f d 0) ) f + f [ m 0 f = f 0 + f m 0 + f ) m 0 m f + f m 0 m ) + f ) ] m 0 f + f f + f f + f f + f [ d 0 f = f 0 + f d 0 + f ) d 0 d f + f d 0 d ) + f ) ] d 0 f + f f + f f + f f + f with: m = µ + k and d = k µ + µ ) + µ µ + µ ) µ + k µ k + µ ) m 0 = µ 0 + k and d 0 = k 0µ + µ 0 ) + µ 0 µ + µ 0 ) µ 0 + k 0 µ 0 k 0 + µ 0 ) m 0 = µ 0 + k and d 0 = k 0µ + µ 0 ) + µ 0 µ + µ 0 ) µ 0 + k 0 µ 0 k 0 + µ 0 ) such that according to eq.) w /) = m J + d K w 0/) = m 0 J + d 0 K ) ) )

25 February 00 0:0 Philosophical Magazine articlepr Page of Philosophical Magazine 0 0 and w 0/) = m 0 J + d 0 K. Using again eq.) applied to eqs.))))) allows us to find the following expressions of n n n 0. These ones represent the thermal contributions present in eqs.) to determine the concentration tensors β β β 0. Their expressions are found to be: with: n = m f e + f ) 0 f e 0 + f m 0 e + e 0)) f + f n = m n e n 0 f = m 0 n e 0) + f ) m 0 n e 0) f + f f + f e = k α k α ) k + µ e 0 = k 0α 0 k α ) k 0 + µ 0 e 0 = k 0α 0 k α ) k 0 + µ 0 where v /) = e I Θ v 0/) = e 0 I Θ and v 0/) = e 0 I Θ following eq.). ) )

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