New theoretical modelling of thermo-elastic behaviour using the geometric average: theoretical and numerical investigations J. FAJOUI, S.

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1 0 New theoretical modelling of thermo-elastic behaviour using the geometric average: theoretical and numerical investigations J. FJOU, S. FREOUR nstitut de Recherche en énie Civil et Mécanique (UMR CNRS ) Université de Nantes - Centrale Nantes UT de Saint-Nazaire,, rue Michel nge, 00 Saint-Nazaire cedex, FRNCE Corresponding author: Fax number: + address: jamal.fajoui@univ-nantes.fr bstract For the first time, Eshelby-Kröner self-consistent model extended to thermo-elastic loading is developed using the geometric mean for performing the set average operations. new homogenization Eshelby-Kröner Self-Consistent approach is developed using a geometric average during thermo-mechanical loading. The description of the material thermo-elastic behaviour is carried out through new, appropriate, geometric polarisation tensors. n this paper, the classical Eshelby-Kröner thermo-elastic model based upon the arithmetic mean is used as a reference. discussion and comparison between both average types is also provided. n particular, the coefficient of thermal expansion of silicon carbide, identified in the range K - K from the knowledge of the thermal expansion of aluminium-silicon carbide metal matrix composites and pure aluminium samples do not depend on the average type chosen for achieving the computations. Keywords Thermo-elastic behaviour, Scale transition, Eshelby-Kröner Self-Consistent approach, geometric average, arithmetic average, identification, coefficients of thermal expansion. Page /

2 0. ntroduction Scale transition methods are widely employed in the context of materials science. Mechanics of materials is a field of choice for applying scale transition models. s an example, they enable to determine homogenized effective properties such as the elastic stiffness or the coefficients of thermal expansion of a polycrystalline sample []. Besides, inversing the homogenization relations provided by scale transition models enables to identify interesting local materials properties such as the singlecrystal elasticity constants as well as their coefficients of thermal expansion [, ]. nother application of this kind of approach consists in the prediction of multi-scale mechanical states, application that proved its usefulness in the context of multi-physics phenomena such as the diffusion of moisture in reinforced polymers []: the moisture diffusion coefficients as well as the maximum moisture absorption capacities of a polymer are actually dependent on the local mechanical strains it experiences [, ]. The statistical scale transition models are based on the calculation of macroscopic quantities from the corresponding mesoscopic ones through an adapted mathematical framework featuring sets averaging operations. n that context, the classical arithmetic mean is often used. However, some authors use various methods for performing set averaging operations: as an example, the harmonic mean or the geometric mean, according to references [-]. For instance, the effective macroscopic stiffness determined according to the approximation proposed by Reuss actually results from a harmonic mean achieved on the local stiffness of the constituents []. Usually, in the context of determining various properties of polycrystals, one uses the arithmetic mean for performing the computations. Nevertheless, the geometric mean is sometimes used as a rplacement to the arithmetic mean, instead, as shown in references [, ]. The geometric mean, historically introduced in particular cases by [] is based on the condition of the commutation of inversion and averaging operations. ccording to [], if for a given single crystal property E 0 0 0, the inverse property H E generally exists, - physically, the corresponding relation also holds for a polycrystalline sample, E H E. Unfortunately, the arithmetic mean does not provide this important relation. E a E a - may be 0 quite different but constitute the extreme limits of the experimental data lying in between. Nevertheless, the consequences of replacing arithmetic sets averages by their geometric counterparts were neither extensively discussed by the authors, nor the mathematical/physical considerations having driven such a modification of the traditional calculation framework. The consequences induced by the use of the geometric mean instead of the arithmetic mean in a given scale transition Page /

3 0 0 model were only recently envisaged through a series of articles focused on Eshelby-Kröner Self- Consistent model [, ]. The geometric mean is relevant for calculating averages of products of terms, whereas it is considered that the arithmetic mean better suits averaging operations performed on sums of terms. n statistics, achieving sets averages using either the geometric or the arithmetic mean often yields (see [] for an extensive discussion) close outcome, if the terms do not have extremely different values. n illustration of such a closeness can be found in [] where the results obtained for the elastic properties of polycrystals, through the geometric mixture law, were compared to others scale transition approaches involving the arithmetic mean. n the context of the present work, this statement suggests comparing the writing of a single scale transition model, from an arithmetic, sum based approach, or a geometric, product based formalism. n both references [, ], the fundamental equations of Eshelby-Kröner Self-Consistent elastic model have been investigated from the standpoint of either the historical, classical arithmetic framework or, for the first time, the productbased, geometric, deviation of the mechanical states experienced by a single crystallite from the corresponding macroscopic quantities. n reference [], it was demonstrated that the geometric polarization tensors are proportional to either the strain localization tensor or the stress concentration tensor. This is interesting because many scale transitions models are based on these two quantities mainly because of their strong physical meaning. The extensive numerical investigation achieved in [] did show the closeness of the quantities (macroscopic stiffness, lattice strains,...) predicted by Eshelby-Kröner Self-Consistent model in the case that single-phase polycrystals, possibly presenting a strong morphologic texture, was considered, whatever the chosen type of set average: the geometric or the arithmetic mean. However, the geometric, product-based Kröner-Eshelby Self-Consistent model developed in [, ] is restricted to the case of purely elastic loads. For the first time, in the present work, this very model will be extended to thermo-elastic loads, through the introduction of some appropriate, original, product-based thermo-elastic polarization tensors. The fundamental definitions introduced by Kröner [], especially those of the so-called polarization tensors, that linearly relate the thermo-elastic strain (or, respectively, the stress) experienced by a given heterogeneous inclusion to the macroscopic mechanical stress (or, respectively the strain) experienced by the effective medium in which it is embedded, are carefully examined from the standpoint of the mathematical method envisaged for performing sets average operations. On the basis of the practical independence of physical properties from the mathematical method applied in order to proceed to their determination, analytical forms are determined for the so-called geometric polarization tensors. The obtained expressions are Page /

4 0 compared to their counterparts, satisfied according to the traditional, arithmetic average version of the model. Eventually, the homogenization relations enabling to predict the effective thermo-elastic properties of a polycrystal (i.e. the elastic stiffness and the coefficients of thermal expansion) according to the product-based definition of the model will be established. The consistency of the obtained expressions with their counterparts corresponding to the classical sum-based version of the model will be checked. Numerical investigations were performed to simulate the coefficient of thermal expansion in a twophase material: l-sic (aluminum-silicon carbide) by Eshelby-Kröner Self-Consistent model depending on the set average type chosen for realizing the computations: either the geometric or the arithmetic mean. The CTE back calculated for SiC from the knowledge of the macroscopic CTE of the composite as well as those of the pure aluminum (according to reference [0]) will be compared to an experimental value found in the literature []. The CTE back calculated for SiC from the comparison between the numerical results predicted for the effective CTE of the composite (knowing the CTE of the l matrix) and the temperature dependent CTE of the metal matrix composite determined in [0] agrees with the experimental value (known at one temperature from []).. Homogenization approach The scale transition approach enables to predict the macroscopic behaviour from the mesoscopic quantities. n the present paper, it is intended to reproduce and explain the deformation mechanisms at different scales during the thermo-elastic loading. n this paper, Eshelby-Kröner Self-Consistent model was used. This model is based on the following assumptions: o Each basic volume is considered as an ellipsoidal inclusion embedded in a Homogeneous Effective Medium (HEM) having the average properties of the polycrystalline aggregate. o The overall response of the material (X) is determined from averaging the mesoscopic terms (x) over a representative set of crystallographic orientations (): X xdv x volume. o The mesoscopic quantities (properties as well as the mechanical states) are uniform in a given inclusion. Page /

5 . Presentation of arithmetic and geometric averages brief description of the arithmetic and geometric averages is presented in this section. For more details, the readers are referred [,, ], and in []. n statistics, given a set of data, X = {x, x,..., xi,..., xn} and corresponding weights, W = {w, w,..., wi,..., wn}, the weighted geometric (respectively, arithmetic) mean α X α,,..., i,..., n (respectively, α X α,,..., i,...,n ) is calculated as: α X α,,..., i,..., n n w α n w α α α x α () α,,..., i,..., n n n α α α α X w x w () α α. Eshelby-Kröner Self-Consistent model with arithmetic average When the polycrystal simultaneously experiences a uniform temperature increment ΔT and a macroscopic stress, the resulting total strain ( ) in each grain is the sum of the elastic strain ( el ) and the thermal strain ( ) as follows: th el th L : ΔT () Where L is the elastic stiffness tensor of the polycrystal and, the thermal expansion tensor. :B denotes the double scalar product ijklbkltu using the Einstein summation convention. From (), one can deduce the total stress: L : ΔT () t the mesoscopic scale, the relation between the stress and the strain in a given inclusion can be written as follows: 0 L : ΔT () Page /

6 L : ΔT () Where L () is the elastic stiffness tensor and () the coefficients of thermal expansion, at mesoscopic scale. The Eshelby-Kröner approach enables to link the components of the mesoscopic stresses, and strains tensors, of a crystallite to the macroscopic strains or stresses, respectively, through the following scale transition relations: q : u ΔT () p : v ΔT () where p, u, q and These tensors are linked together through: q p L : p v are the arithmetic ( subscripted) polarization tensors. : L () L :q : L () L : v q : L u () : L : u q : L : v () 0 n order to simulate the mechanical behaviour of a polycrystal ( ) from that of the single crystal ( () ()) using equations () and (), an explicit expression of the polarization tensors is required. Within Eshelby-Kröner model, these tensors are traditionally written as sums of two terms as follows: p L r () L t q () u n () v L : m () Page /

7 The fourth-order tensors r and t stand for the arithmetic deviation of the elastic stiffness or compliance (respectively) of a single crystallite from the corresponding macroscopic quantity. n is the arithmetic deviation of the thermal expansion of the crystallite from its macroscopic counterpart. Eventually, the product T v represents the deviation between the macroscopic and the mesoscopic stresses induced by the temperature change By introducing the polarisation tensors ()-() in () and (), one eventually obtains: L : L : ΔT r : m ΔT () L : ΔT t : n ΔT () Where L : (respectively L : ) and L Δrespectively Δ stand for the mechanical stress (respectively elastic strain) and thermal stress (respectively thermal strain). ccording to [Kröner, ], r represents the arithmetic deviation between the macroscopic mechanical stress state : and that experienced by the considered crystallite. t : is the arithmetic deviation between the mesoscopic and macroscopic mechanical strains. By a similar line of reasoning, we can explain the physical meaning of the additional terms appearing in (-): i) m ΔT is the arithmetic deviation between the macroscopic thermal stress and the mesoscopic one; whereas, ii) n ΔT the arithmetic deviation between the thermal strain and its mesoscopic counterpart. is The volume averages of the local stress and strain tensors have to equal the overall strain and stress, respectively. s a consequence, the so-called Hill s average principles should be fulfilled []: () 0 (0) pplying these volume weighted set averages to equations () and (), one obtains: L : ΔT t : n ΔT () L : L : ΔT r : m ΔT () By comparing together relations () and (), the following equations should be satisfied: Page /

8 t 0 () n 0 () Similarly, from the comparison between relations () and (), we find: r 0 () m 0 () The equations ()-() are the fundamental constitutive relations of Eshelby-Kröner Self-Consistent thermo-elastic model. ndeed, these equations ensure the consistency between global and local mechanical states. The consistency of strain (respectively of stress) is satisfied within this model: the average over the mesoscopic strains (respectively stresses) is equal to the corresponding macroscopic quantity. Thus, the global response of the material to a thermo-mechanical load is determined from averaging the mesoscopic terms as described in []. n order to achieve the extensive description of Eshelby-Kröner thermo-elastic model based upon the use of the classical arithmetic average, it is necessary to find the expressions of r, t, m and n. The following of the paper will be devoted to give explicit forms for these tensors. The literature provides the following relation between the stresses and strains []: esh esh L : S :S : () 0 Where S esh stands for the classical Eshelby tensor. This tensor takes into account the effect of inclusion morphology upon the average crystal behaviour. n the case that a purely elastic load is considered (i.e. T = 0 C), and starting from relation (), one eventually obtains the following expression for can refer to []): r (for an extensive demonstration, the reader r esh esh : L : S :S : L esh esh L : S :S : L ntroducing () and () in (), one finds: : L () t : L L L : r () Page /

9 f L is a symmetric tensor, introducing () as a replacement rule in () yields: t esh esh : S :S : L esh esh L L L : L : S :S : L (0) n the case that a purely thermal load is considered (i.e. = 0, = T) and by using relation (), one obtains: : L esh esh L : S :S : ΔT Taking into account (), () and (), the following relation can be written: esh esh L L : S :S : L n : () () Eventually, from the equations (), (), () and (), one obtains: 0 m L : L : n r : () n this section, we summarized the Self-Consistent thermo-elastic model with arithmetic polarization tensors as it is presented in recent papers [, ]. With this average, by applying the Eshelby-Kröner model through localization tensors and / or concentration tensors, we can study the thermomechanical behaviour of materials taking into account the effects related to the heterogeneities existing at mesoscopic scale. n the next section, the Eshelby-Kröner model will be rewritten through product based polarization tensors, compatible with sets averages involving the geometric mean instead of the classical arithmetic mean. The benefit of this average will be discussed.. Eshelby-Kröner self-consistent thermo-elastic model featuring product based polarization tensors.. Defining the proper product-based polarizations tensors n a recent work, Fréour et al. [] introduced product based polarization tensors in the case that a purely elastic load was considered within Eshelby-Kröner self-consistent model. These productbased, polarization tensors are compatible with the geometric average. Thus they are denoted by the subscript in the following of this work. These polarization tensors are defined by: Page /

10 p q r : L () t : L () n the present context of extending the previously published approach to the case of a thermo-elastic load, additional thermo-elastic polarization tensors u and written as the following products of factors: m : L u : v are necessary. They can be () n v : () q : u T The strain (respectively stress) experienced by the material at the mesoscopic scale is linked together with the macroscopic stress (respectively strain) through the geometric polarization tensors as follows: 0 q : u T () p : v T () Whatever the set average intended to be used, one should obtain the same mechanical behaviour for a given crystallographically oriented grain. s a consequence, the geometric polarization tensors must numerically be identical to their arithmetic counterparts: p p p (0) q q q () u u u () v v v () By considering () and () in () and (0), respectively, one obtains: Page /

11 L p L : L q v r u m t n : L : L : : L : () Relations () yield the following interesting average forms over the geometric polarization tensors: r () m i () t () n i () 0 Where is the th -order identity tensor, whereas i is the nd -order identity tensor. The mathematical results ()-() make it possible to coincide with the physical reality. These four expressions are actually necessary to fulfil the historical set average principle, introduced by Hill [] in the case that the geometric average is used instead of the classical arithmetic mean. s a result of the conditions () to (), the geometric average of the mesoscopic strains or stresses is respectively identical to its macroscopic counterparts. Using the relations (), () and (), and after some calculations, one obtains the following expression of the thermo-elastic mesoscopic strain experienced by a crystallographic orientation : esh esh esh :S : L L : S :S : L : esh L L : S L : () n the case that the studied material is subjected to thermal variations (denoted by the subscript th in the following) in absence of a macroscopic mechanical load ( = 0, = ΔT), the relation () becomes: v T n : T n ; with v n th : : ΔT (0) Page /

12 ccording to (), while its arithmetic historical counterpart physical meaning, or practical use, in the field of micro-macro modelling, n does not have a very strong n provides the scale transition relation linking the macroscopic strains to those experienced by an crystallographically oriented inclusions family. Thus n actually corresponds to the so-called (strain) localization tensor in the case that the polycrystal experiences a pure thermal macroscopic load. n order to achieve the development of Eshelby-Kröner Self-Consistent model it is necessary to find the relations existing between the geometric and / or the arithmetic polarizations tensors. ctually, the expression satisfied by n esh L L : S n can be deduced from relations () and (): esh esh esh :S : L : S :S L : () : The relations (), (), (), (0), () and () enable finding the following expressions relating the geometric polarization tensors to their arithmetic counterparts: 0 n i n : () i m : L m : r t () r : L () t : L ().. The macroscopic stiffness and coefficients of thermal expansion n order to achieve this theoretical study, it is necessary to determine the global behaviour of the material, subjected to thermal variations (ΔT) as well as to an external mechanical stress ( ). s a result, the present section will be focused on both the macroscopic elastic stiffness tensor L and the macroscopic coefficients of thermal expansion tensor. The equation () should be satisfied for any set of numerical values of the quantities and ΔT (provided that the material is kept in the elastic deformation domain). Let us first consider a specific condition ΔT=0 in equation (). The volume weighted set averages of the local stress and strain tensors must coincide with the overall strain and stress experienced by the polycrystal. These conditions provide the relation between the macroscopic stiffness tensor L as Page /

13 a function of the individual elastic behaviour of the crystallites. The interested reader can refer to [], where an extensive demonstration is given: esh esh esh L L : S :S : L L : S esh :S : L L () Equation () is an implicit equation since the unknown L is featured in both its right-hand and lefthand sides. This equation should consequently be solved by an iterative method. The above presented fundamental equation () is actually compatible with either the historical, classical arithmetic framework or, as demonstrated for the first time in [], with a product-based, geometric rewriting of the polarization tensors. This yields a privileged (but not exclusive) link between each formulation and the corresponding averaging operation-type. t was demonstrated in [] that the geometric polarization tensors are proportional to either the strain localization tensor or the stress concentration tensor. For practical applications, many scale transition models are based upon the strain localization tensor and the stress concentration tensor, mainly because of their strong physical meaning. On the contrary, the classical arithmetic polarization tensors did hold very little appeal on the scientific community working on this field of research. The last step of the development of the Eshelby-Kröner model with the geometric average, is to find the macroscopic coefficients of thermal expansion using the geometric mean. Therefore, we are going to determine the expression of this tensor using the arithmetic average. The volume weighted average of the local strain tensor has to coincide with the overall strain. s a result, equation () becomes: 0 esh esh L L : S :S : esh esh L L : S :S : L : L : ΔT () From this relation (), we deduce []: L esh esh L : S : S : L : L : 0 () Thus, we can write the expression of macroscopic coefficients of thermal expansion as follows []: Page /

14 L esh esh esh esh L : S : S : L : L L : S : S : L : (0) Using this relation (0), one can estimate the macroscopic coefficients of thermal expansion provided that the macroscopic elastic stiffness tensor (relation ) is known. Thereafter, the global thermomechanical response of the polycrystal can be described. By introducing the set average operator in relation (), we find: n i n : () ccording to (), the term n i n is equal to zero. Thus, we find: () From this relation (), we can write: n : i () : esh esh esh esh L L : S :S : L : S :S : esh esh L L : S :S : L : By comparing between the two expressions of, (0) and (), we have to demonstrate that: () esh esh esh esh esh L L : S :S : L : S :S L L : S :S esh : L () n fact, Kocks et al. [] provide the demonstration of the following identity: esh esh esh L L : S :S : L : S esh :S () This relation can be developed as: Page /

15 esh esh esh L L : S :S : L L : S esh :S () esh esh esh L L : S :S : L L L : S esh esh :S : L : S :S () esh esh esh L L : S :S : L L L : S esh esh :S : L : S :S () esh esh Finally, we find the researched relation (): esh esh esh esh esh L L : S :S : L : S :S L L : S :S esh : L () 0 The development presented between () and () enable us to find the same expression for the macroscopic coefficients of thermal expansion using either the geometric mean (product-based) or the arithmetic average (sum-based). f we consider only the mathematical point of view, we observe that the geometric average provides another alternative in addition to the arithmetic one. n fact, the mathematical development, with the two averages, leads to the same thermo-elastic macroscopic behavior: consistent expressions were found for the macroscopic stiffness tensor and the macroscopic coefficients of thermal expansion. But, the originality of this work is the physical interpretation of the mathematical equations obtained with the geometric mean. n the case that purely mechanical load is considered, we shown [] that the geometric average is used to link directly the global behavior with the local one through the new geometric polarizations tensors. These tensors are proportional to either the strain localization tensor or the stress concentration tensor, two quantities on which many scale transition models are based upon, for practical applications. n the case that the material is subjected to thermal variations without external mechanical loading, we find the same important physical discussion. ndeed, the equation (0) shows the scale transition between the macroscopic and the microscopic behavior through the new geometric tensor n. Usually, this connection is made by the localization tensor. Thus, this new expression will allow physicists to establish the homogenization approach using this tensor. Page /

16 . Numerical investigation n this part, we present an example of numerical comparisons between geometric and arithmetic averaging. We simulated the multi-scale Coefficients of Thermal Expansion (CTE) in a two-phases material: l-sic (aluminium-silicon carbide). These simulations aim to validate the self-consistent (SC) model using the geometric average () and to compare it with the classical SC model relying on the arithmetic average (). The input texture consists of the same set of 000 equally-weighted crystallographic orientations representing a random texture (for both calculations). The grains were assumed to be equiaxed. The average elastic behaviour of the phases is considered isotropic for l and for SiC. Table sums up the anisotropic single crystal elastic constants for both phases. The volume fraction of each phase is taken equal to 0. in the composite. Our simulations requires the introduction of the temperature dependent coefficient of thermal expansion (CTE) determined by [0] for pure aluminium and l-sic metal matrix composite in the temperature range raising from 0 C up to 00 C (Table ). Then, the SC model (with either the arithmetic or the geometric average) will enable us to identify the evolution of the average CTE of the SiC particulates. The Single Crystal Coefficients of Thermal Expansion (m α i (T)) are assumed to satisfy the following form as a function of the temperature: m i 0 T m m T m T m T (0) i i i i 0 0 where m α(k) i are the interpolation coefficient (k=0-). Their values are indicated in table. Figure gives the evolution of the coefficient thermal expansion (CTE) identified for the SiC reinforcements using the self-consistent approach according to either the arithmetic or the geometric average. n this figure we present also the experimental value determined at K in the previously published literature []:. - K - (dashed line). t can be observed that the model captures this value (figure.b). Figure.c presents the comparison between the CTE predicted for SiC obtained either from the model using arithmetic average or from the geometric average. The relative difference between the two sets of results is drawn on figure.c. t was noted that the deviation remains between 0. % and 0. %: it increases with the temperature. The difference is positive; i.e. values determined using the arithmetic average are always larger than those obtained by using the geometric average. However, the Page /

17 discrepancy between the two numerical methods is weak. Thus, materials properties identified from a statistical scale transition model, by using either the geometric or the arithmetic average are consistent.. Conclusions and perspectives n the present work, for the first time, the Eshelby-Kröner self-consistent model extended to thermoelastic load was expressed through product based geometric polarization tensors. The new formulations were defined and developed for these tensors using the geometric mean which corresponds to a product of factors. This approach enabled us to find new expressions for the effective macroscopic thermo-elastic behavior of polycrystals, compatible to those obtained with the arithmetic mean. The physical meaning of the geometric polarization tensors introduced for the purpose of establishing the homogenization relations was discussed also. We applied the self-consistent model, using the two averaging strategies, in order to identify the coefficient of thermal expansion of SiC from data obtained on pure aluminum on the one hand and l-sic composite (with 0 % volume fraction of SiC) on the other hand. good agreement is observed between the simulations results using either one or the other average type. Besides the predicted back calculated CTE of the SiC agrees with experimental value obtained at K []. s a result, identification of materials properties owing to a statistical scale transition model can be achieved as reliably by using either the arithmetic or the geometric average. Page /

18 0 0 References. Kocks UF (000) Texture and nisotropy: Preferred Orientations in Polycrystals and Their Effect on Materials Properties. Cambridge University Press. Matthies S, Merkel S, Wenk HR, et al. (00) Effects of texture on the determination of elasticity of polycrystalline ϵ-iron from diffraction measurements. Earth Planet Sci Lett :0. doi:./s00-x(0)00-. Freour S, loaguen D, Francois M, uillen R (00) pplication of inverse models and XRD analysis to the determination of Ti- β-phase coefficients of thermal expansion. Scr Mater :.. Youssef, Fréour S, Jacquemin F (00) Stress-dependent Moisture Diffusion in Composite Materials. J Compos Mater :. doi:./000. utran M, Pauliard R, autier L, et al. (00) nfluence of mechanical stresses on the hydrolytic aging of standard and low styrene unsaturated polyester composites. J ppl Polym Sci :.. Wan YZ, Wang YL, Huang Y, et al. (00) Hygrothermal aging behaviour of VRTMed threedimensional braided carbon-epoxy composites under external stresses. Compos Part ppl Sci Manuf :.. Lichtenecker K () Die Dielektrizitätskonstante künstlicher und natürlicher Mischkörper. Phys Z :.. Bruggeman D a. () Berechnung verschiedener physikalischer Konstanten von heterogenen Substanzen.. Dielektrizitätskonstanten und Leitfähigkeiten der Mischkörper aus isotropen Substanzen. nn Phys :. doi:.0/andp.0. Reuss () Berechnung der Fließgrenze von Mischkristallen auf rund der Plastizitätsbedingung für Einkristalle. ZMM - J ppl Math Mech Z Für ngew Math Mech :. doi:.0/zamm.000. Morawiec () Calculation of Polycrystal Elastic Constants from Single-Crystal Data. Phys Status Solidi B :. doi:.0/pssb.. Matthies S, Humbert M () The Realization of the Concept of a eometric Mean for Calculating Physical Constants of Polycrystalline Materials. Phys Status Solidi B :K K0. doi:.0/pssb.0. Baczmanski, Wierzbanowski K, Haije W, et al. () Diffraction elastic constants for textured materials : different methods of calculation. Cryst Res Technol :.. Matthies S, Humbert M, Schuman C () On the Use of the eometric Mean pproximation in Residual Stress nalysis. Phys Status Solidi B :K K. doi:.0/pssb.0. Matthies S, Humbert M () On the Principle of a eometric Mean of Even-Rank Symmetric Tensors for Textured Polycrystals. J ppl Crystallogr :. doi:./s0000 Page /

19 . Matthies S () Moment Pole Figures in Residual Stress nalysis. Textures Microstruct :. doi:./tsm... leksandrov KS, izenberg L () Method for computing the physical constants of polycrystalline materials. Dokl kad Nauk SSSR :.. Freour S, Lacoste E, Fajoui J, Jacquemin F (0) On the meaning of the chosen set-averaging method within Eshelby-Kröner self-consistent scale transition model: the geometric mean versus the classical arithmetic average. Z Für ngew Math Mech :.. Fréour S, Fajoui J (0) Eshelby-Kröner self-consistent elastic model: the geometric mean versus the arithmetic mean numerical investigation. ZMM - J ppl Math Mech Z Für ngew Math Mech :. doi:.0/zamm.00. Kröner E () Berechnung der elastischen Konstanten des Vielkristalls aus den Konstanten des Einkristalls. Z Für Phys :0. 0. Kim B, Dong SL, Park SD (00) Effects of thermal processing on thermal expansion coefficient of a 0 vol.% SiCp/l composite. Mater Chem Phys :.. Lonsdale K Thermal expansion in relation to structure. nt Tables Fir X-Ray Crystallogr chapitre.:. 0. Hill R () The Elastic Behaviour of a Crystalline ggregate. Proc Phys Soc Sect :. doi:./00-///0. Hill R () The essential structure of constitutive laws for metal composites and polycrystals. J Mech Phys Solids :.. Fajoui J, loaguen D, Courant B, uillén R (00) Micromechanical modelling of the elastoplastic behaviour of metallic material under strain-path changes. Comput Mech :. doi:.0/s loaguen D, Fajoui J, irard E, uillén R (0) X-ray measurement of residual stresses and texture development during a rolling sequence of zirconium alloy cladding tubes influence of plastic anisotropy on mechanical behaviour. J ppl Crystallogr :0. doi:./s000 Page /

20 Tables captions: Table : Elastic constant of the single crystal Table : Polynomial coefficients of equation (0) [0] Page 0/

21 Figures captions: Figure : a) Evolution, as a function of the temperature, of the CTE at macroscopic and mesoscopic scale in a l0%-sic0% metal matrix composite; b) Comparison between geometric and arithmetic average; c) Relative deviation between geometric and arithmetic average Page /

22 c (MPa) c (MPa) c (MPa) l... SiC... Table : Elastic constant of the single crystal Coefficient m α(0) i [K - ] m α() i [K - ] m α() i [K - ] m α() i [K - ] l l-sic Table : Polynomial coefficients of equation (0) [0] Page /

23 M [- K-] Deviation - (%) M [ - K - ] (a) 0 SiC CTE (SC model) : rithmetic. SiC CTE (SC model) : eometric. 0 l CTE (measured) Two-phase MMC CTE (measured) SiC CTE (measured at K) 0 Temperature [K] (b) 0, (c) SiC CTE (SC model) : rithmetic. SiC CTE (measured at K) 0, 0, Deviation rithmetic - eometric SiC CTE (SC model) : eometric. 0 Temperature [K] 0,0 Temperature [K] Figure : a) Evolution, as a function of the temperature, of the CTE at macroscopic and mesoscopic scale in a l0%-sic0% metal matrix composite; b) Comparison between geometric and arithmetic average; c) Relative deviation between geometric and arithmetic average Page /