Je Remercie

Sébastien Brisard

Yann Monerie

Stéphane Roux

Stéphane Berbenni

Djimédo Kondo

Renaud Masson

Pierre Suquet

Volker Schmidt

Pedro Ponte

Castañeda J'

Ai Formé

'ai l'honneur également de compter dans mon jury

The present manuscript is submitted in partial fulfillment of my application to the degree of Habilitation à diriger des recherches at Sorbonne University. Its main contribution is a study in theoretical mechanics devoted to homogenization problems in the context of degenerate (nonstrictly convex) local response of one of the phases, which can serve as idealized models for porous or rigidly-reinforced materials exhibiting perfectly-plastic behavior. In these situations plastic flow preferentially concentrates along shear bands; as a result the material effective response is governed by those regions within the material where the field localizes. A form of localization also occurs in linear problems governed by asymptotically hyperbolic partial derivative equations, where the strain field is found to develop banding patterns. The solutions are relevant to stronglyanisotropic elastic or thermoelastic media. Such a linear problem is studied in the present work in the context of a random microstructure, specifically a polycrystal containing cracks. A related topic is whether incipient localization may develop as a result of the microstructure itself. This question is investigated in the case of a Stokes flow occurring in a porous medium around a set of obstacles exhibiting unusual spatial distribution, characterized by long-range correlations. The different model problems are addressed by means of various techniques. Limit analysis bounds are combined with integral geometry to provide insight on the material behavior. Comparisons with rigorous bounds and estimates of homogenization theories, and full-field "Fourier-based" numerical results, allow us to interpret and assess the response of certain random microstructures. Although a complete solution is out-of-reach, the interplay between field localization and microstructure is elucidated in specific cases. v

I would like to briefly describe here the different activities I have had as a researcher, teacher and supervisor of PhD students at Mines ParisTech and École des Mines since 2010. But before I proceed and enumerate some of my activities, it is useful to point out that the Center for Mathematical Morphology where I am located is a research center common to Mines ParisTech and the research association Armines. Beside academic research, the purpose of the Armines and Mines ParisTech centers is the development of applied research and its transfer to industry. This transfer occurs thanks to collaborative projects, grants from French or European agencies, "direct contracts" with industry and partnership through chairs and collaborative projects. The same philosophy holds at the Center of Materials, which is a center common to CNRS, Mines and Armines, and in which I also work as a research associate, on a weekly basis.

After my PhD thesis at the Commissariat à l'Énergie atomique, in the Département de Physique Théorique et Appliqué and the Laboratoire de Mécanique des Solides of École Polytechnique, and after a subsequent post-doctoral position at University of Pennsylvania, I have, as soon as I became a tenure track at Mines ParisTech, contributed to and directly managed several national and international research projects. Some of the projects I contributed to are listed in Appendix (B). The project topics are quite diverse, ranging from the modeling of optical properties for paint layers in automotive industry (LIMA ANR project), microstructure modeling and optimization for the development of low-temperature fuel cells (EVOLVE European project), the study and modeling of damage in TATB polycrystals (CEA Le Ripault), microstructure optimization of the viscoelastic properties of tires (Yokohama Rubber Cie), the modeling of cosmetics at the microscopic scale (L'Oréal), transport properties of mesoporous alumina (IFPEN), the conducting properties of coldspray materials (US Air Force grant), the mechanical properties of concrete (EDF MAI grant), the prediction of piezoelectricity in woven (FUI), classification of defects in wielding (CEA Valduc) to the study of damage in reservoirs (ministry grant). Although many of these projects are actually collaborations with colleagues scientists, I have taken an active part in obtaining grants so that the research may be carried out.

Obviously, my role and activities have evolved with time, but a key aspect consists in keeping a good equilibrium between academic and applied research. On the one hand, state-of-the-art academic research is required for applied research to be useful to industry. On the other hand, many examples show that industrial problems are at the origin of breakthrough in mathematical morphology and in the wide domain of material and engineering science. An example of industrial problems which have motivated new theoretical developments are the modeling of anode layers and other components of fuel cells. The later have led to the developments of new methodologies for simulating n-phase random sets [START_REF] Abdallah | Morphological modeling of three-phase microstructures of anode layers using SEM images[END_REF], image analysis tools specific to FIB-SEM and interesting optimization problems [START_REF] Masson | Morphological modeling of a metal foam SOFC configuration[END_REF]. Today, these developments continue [START_REF] Cadiou | Numerical prediction of multiscale electronic conductivity of Lithium-ion battery positive electrodes[END_REF] and have led to a fruitful international collaboration [START_REF] Neumann | Stochastic 3D modeling of three-phase microstructures for the prediction of transport properties in solid oxide fuel cells[END_REF][START_REF] Neumann | Quantifying the influence of microstructure on effective conductivity and permeability: virtual materials testing[END_REF].

The main topics of my research activities cover the development of spectral, Fourier-based numerical methods in homogenization [START_REF] Azzimonti | Optical properties of deposit models for paints:full-fields FFT computations and representative volume element[END_REF][START_REF] Willot | Fourier-based schemes for computing the mechanical response of composites with accurate local fields[END_REF][START_REF] Willot | Fourier-based schemes with modified Green operator for computing the electrical response of heterogeneous media with accurate local fields[END_REF][START_REF] Figliuzzi | Eikonal-based models of random tesselations[END_REF] and the modeling of heterogeneous microstructures (see e.g. [START_REF] Escoda | Modélisation morphologique et micromécanique 3D de matériaux cimentaires[END_REF][START_REF] Abdallah | Analyse morphologique et modélisation pour l'optimisation structurelle d'électrodes[END_REF][START_REF] Wang | Modeling of mesoporous alumina microstructure by 3D random models of platelets[END_REF]). A summary of my recent activities in this domain is provided in Appendix (A).

Another aspect of my work as a research scientist is teaching. I organize a one-week course at École des Mines devoted to random sets microstructure modeling and homogenization. The course includes theoretical lectures as well as training sessions carried out with the Fourier code I develop for homogenization. The courses I give take the point of view of numerical analysis (PDEs, convex analysis, optimization etc.) rather than that of algebra. The same philosophy is followed in the one-week course I contribute to, which is part of the course "Ingénierie Digitale des Systèmes Complexes"1 in Mines ParisTech, organized by D. Ryckelynck and É. Hachem. The one-week course, organized together with H. Proudhon and B. Figliuzzi, aims at predicting and simulating the properties of materials based on images (microtomography etc.).

I wish to pursue my teaching activities and if possible, enlarge them. I would also like to not only teach, but organize courses, and invite scientists to deliver top-level lectures in topics that are rarely taught. One example is the course on nonlinear homogenization in the Athens week "Physics and Mechanics of Random Media"2 that PhD students as well as students from the international "Athens" University network attend. In that respect, I have been pleased to organize (with the invaluable help of Samuel Forest) the international workshop in honor of Dominique Jeulin "Physics and mechanics of random structures: from morphology to material properties" 3which was held in the Peninsula of Oléron in 2018. A book was published at the occasion [START_REF] Willot | Physics & Mechanics of Random Media: from Morphology to Material Properties. A tribute to Dominique Jeulin's contributions to science, research and teaching[END_REF] at Presses de l'École des Mines as well as a special topic in Image Analysis & Stereology [START_REF] Serra | Special topic on multiscale modeling of granular media: a tribute to Prof. Dominique Jeulin[END_REF] and a special issue of the International Journal of Solids and Structures4 (Elsevier).

To guarantee a global coherence in this HDR thesis manuscript, I have decided to focus on the homogenization of the mechanical response and transport properties of heterogeneous media, in the context of field localization. As a result of this choice, part of my research work, some of which is briefly summarized in Appendix (A) has been left out. Nevertheless, the present HDR study includes developments in integral geometry, which, while they are relevant to the homogenization problems at stake, are also of interest, I believe, to microstructure simulations and probabilistic models.

This HDR manuscript therefore does not constitute a program of the research activities I would like to pursue in the future. In particular, I am presently working on the implementation of Stokes flow solvers in Fourier-based methods, and, thanks to the ongoing PhD thesis of F. Rabette, on a FFT implementation [START_REF] Rabette | Modélisation multiéchelle par champ de phase de la microfissuration d'un polycristal organique de forte anisotropie cristalline par FFT[END_REF] of phase-field methods for the initiation and propagation of cracks in quasi-brittle materials, based on the variational theory of Francfort and Marigot [START_REF] Francfort | Revisiting brittle fracture as an energy minimization problem[END_REF] and by the works of Bourdin et al. [START_REF] Bourdin | Numerical experiments in revisited brittle fracture[END_REF] and Miehe et al. [START_REF] Miehe | A phase field model for rate-independent crack propagation: Robust algorithmic implementation based on operator splits[END_REF]. The PhD thesis of É. Kaeshammer, focusing on wave propagation in energetic materials is another important domain of research [START_REF] Kaeshammer | Étude expérimentale et numérique de la sensibilité de compositions énergétiques : influence de la microstructure et rôle de l'endommagement[END_REF] that I would definitely like to pursue. I hope to be able to develop these topics of research in the future and have new interesting collaborations with other researchers.

The following section gives a short introduction to the main topic of this HDR thesis.

An investigation of localization in random media

A broad range of physical phenomena in material science involve the emergence of surfaces or paths of minimal energy. Examples include domain walls in random Ising systems [START_REF] Huse | Pinning and roughening of domain walls in Ising systems due to random impurities[END_REF], current localization in varistors (or non-Newtonian fluid flow) [START_REF] Donev | Random manifolds in non-linear resistor networks: applications to varistors and superconductors[END_REF], the onset of voltage in polycrystals [START_REF] Haslinger | Theory of percolative conduction in polycrystalline hightemperature superconductors[END_REF], or ductile fracture [START_REF] De Arcangelis | Scaling laws in fracture[END_REF][START_REF] Bouchaud | Models of fractal cracks[END_REF]. Some of these problems have been addressed using idealized random network models (see [START_REF] Donev | Random manifolds in non-linear resistor networks: applications to varistors and superconductors[END_REF][START_REF] Duxbury | Critical manifolds in non-linear response of complex materials[END_REF] and references therein). In nonlinear networks relevant to varistors and superconductors, localized flow paths emerge at special points, which act as thresholds for the network's macroscopic response [START_REF] Donev | Random manifolds in non-linear resistor networks: applications to varistors and superconductors[END_REF]. As noted by Duxbury et al. [START_REF] Duxbury | Critical manifolds in non-linear response of complex materials[END_REF], these paths are the solutions of the "shortest-path" and "minimum-cut" problems. The equivalence between a network's effective response and an optimization problem on graphs is demonstrated by the min-cut/max-flow theorem [START_REF] Elias | A note on the maximum flow through a network[END_REF]. Interestingly, the optimal surfaces exhibit, in the presence of disorder, rich self-affine structures, characterized, notably, by non-standard scaling laws [START_REF] Bovier | Lines and domain walls in dilute ferromagnets[END_REF][START_REF] Alava | Disorder-induced roughening in the three-dimensional Ising model[END_REF]. In the continuum, the maximal flow of a vector field through a domain is likewise related to a minimal cut problem [START_REF] Strang | Maximal flow through a domain[END_REF].

In a numerical and theoretical study of an "analog electric network" that mimic perfectplasticity, Roux and Hansen [START_REF] Roux | Perfect plasticity in a random medium[END_REF] have used the connection between minimal surface problems and effective yield stress. In the continuum, limit analysis results obtained by Drucker [START_REF] Drucker | The continuum theory of plasticity on macroscale and microscale[END_REF] in a periodic problem in plane strain highlight a similar non-analytic dependence. Drucker's bounds demonstrate that the leading-order term to the effective flow stress y 0 of a periodic arrangement of pores of volume fraction f scales as " f 1{2 in the limit of vanishingly small porosity f :

y 0 " y " 1 ´af 1{2 `opf 1{2 q ‰ , f Ñ 0, (1.1)
with prefactor a depending on the geometry of the inclusions. This results holds for a rigid, perfectly-plastic matrix. The infinite slope of the flow stress (By 0 {Bf " 8) when f " 0 underlines the strong effect of the voids in this limit. The trial displacement fields used to derive the upperbound are piecewise-constant and present surface discontinuities in the matrix [START_REF] Francescato | Ductile failure of cylindrically porous materials. Part I: plane stress problem and experimental results[END_REF]. Optimal upperbounds are thus attained by minimal surfaces. In antiplane strain, these surfaces reduce to minimal paths, or in geometrical terms, geodesics. The paths are minimal in the sense that they have smallest cut in the matrix and span the array of pores in a direction that depends on the applied macroscopic loading. Examples relevant to a 2D square array of pores with prescribed "simple" and "pure shear" (45 ˝-rotated) strain loadings are given in [START_REF] Idiart | Infinite-contrast periodic composites with strongly nonlinear behavior: effective-medium theory versus full-field simulations[END_REF]. The strain field in porous materials tends to localize in bands seeking to pass through the pores in order to minimize the plastic dissipation (see example Fig. 1.1). In this case, the interaction between the pores is obviously quite strong and dependent on the geometry of the configuration as well as the relative orientation of the loading. As a consequence of the strong interaction between shear bands and pores, a slight increase of the porosity leads, in specific configurations, to a sudden rearrangement of the shear bands, as demonstrated by limit analysis and Fourier computations [START_REF] Idiart | Infinite-contrast periodic composites with strongly nonlinear behavior: effective-medium theory versus full-field simulations[END_REF]. Consequently, the response of the material is essentially driven by the geometry rather than details of the material law. In the rigid case, in periodic structures, the shear bands may avoid inclusions entirely while spanning the medium, highlighting a much lower influence of the rigid reinforcement. Despite the aforementioned works, the role of a random distribution of heterogeneities (typically pores) on the effective behavior of perfectly-plastic media, is not completely elucidated. Theoretical results suggest that the effective yield stress of 2D random porous media also exhibits power-laws with non-integer exponents, in the dilute porosity limit f Ñ 0. Different exponents however have been proposed, depending on the problem at hand. In a square lattice model made of broken bonds in proportion f , optimal surfaces are obtained by minimizing the sum of the capacity (analog to the yield stress) taken over each bond that crosses the surface. Thus, it is in principle possible to infer the behavior of the effective response of heterogeneous materials by solving minimal paths problems. In a two-phase resistor lattice with a dilute proportion of "broken" (insulating) bonds, the effective response exhibits interesting non-analytic behavior. The exponent characterizing the leading-order term correction in the dilute limit f Ñ 0 is found to be 1 for minimal paths oriented along the directions of the lattice, and 1{2 along the diagonal [START_REF] Roux | A simple model for ductile fracture of porous materials[END_REF]. Dilute limit singularities, of a different type, have been reported in the dielectric breakdown of square and cubic networks [START_REF] Duxbury | Breakdown of two-phase random resistor networks[END_REF] and in the continuum [START_REF] Duxbury | Capacitance and dielectric breakdown of metal loaded dielectrics[END_REF]. More recently, the non-analytical behavior of the effective response characterizing nonlinear ionic transport has been discussed by Sillamoni and Idiart [START_REF] Sillamoni | Nonlinear ionic transport through microstructured solid electrolytes: homogenization estimates[END_REF]. The authors obtain an exponent 2{3 as the result of their application of the nonlinear homogenization theory of P. Ponte Castañeda [START_REF] Castañeda | Second-order theory for nonlinear dielectric composites incorporating field fluctuations[END_REF].

The "second-order" nonlinear homogenization theory, which provides analytical estimates based on variational principles, indeed predicts a fractional exponent 2{3 in the dilute limit f Ñ 0, in the plane strain [START_REF] Castañeda | Second-order homogenization estimates for nonlinear composites incorporating field fluctuations. II -Applications[END_REF] and anti-plane [START_REF] Furer | A symmetric fully optimized second-order method for nonlinear homogenization[END_REF] problems. The result has been confirmed in plane strain by numerical Fourier computations on a pixel-based porous material [START_REF] Willot | Fast Fourier Transform computations and build-up of plastic deformation in 2D, elastic-perfectly plastic, pixelwise-disordered porous media[END_REF][START_REF] Willot | Contribution à l'étude théorique de la localisation plastique dans les poreux[END_REF]. As pointed out in [START_REF] Sillamoni | Nonlinear ionic transport through microstructured solid electrolytes: homogenization estimates[END_REF], this suggests a lower influence of the pores in a random configuration rather than in the periodic problem. Nevertheless, the exact role of the pores in a random medium and the onset of a non-analytical powerlaw has not been elucidated. The studies developed in the following chapters, devoted to various problems relevant to minimal paths and random structures, give partial answers, in the porous and rigid problems.

It is interesting to note that the second-order nonlinear homogenization theory is based on a linear comparison composite material and underlying linear homogenization theory. In the perfectly-plastic limit, the linear comparison composite makes use of phases that are in general anisotropic [START_REF] Idiart | Infinite-contrast periodic composites with strongly nonlinear behavior: effective-medium theory versus full-field simulations[END_REF]. Nonlinearity is therefore approached by a locally (elastic or thermoelastic) anisotropic matrix phase. In effect, in the limit of certain "infinitely-anisotropic" constitutive laws (where the local potential looses strict convexity), the strain field organizes through banding patterns [START_REF] Willot | Localization of elastic deformation in strongly anisotropic, porous, linear materials with periodic microstructures: exact solutions and dilute expansions[END_REF], which are of a different nature than plastic shear bands, but are nevertheless useful to study localization phenomena. This approach is followed in [START_REF] Willot | Effectivemedium theory for infinite-contrast two-dimensionally periodic linear composites with strongly anisotropic matrix behavior: dilute limit and crossover behavior[END_REF] in the case of a periodic arrangements of voids, and is pursued in the case of a random material in Chap. [START_REF] Abdallah | Stokes flow through a Boolean model of spheres: Representative volume element[END_REF]. The chapter is complemented by a study of elastic field distributions in random media (Chap. 3), which although not directly related to localization, is included because of the possible perspectives of this work. The two chapters, which form the first part of this manuscript is thus concerned by a linear behavior. The second part is devoted to nonlinear media for mechanical and transport properties, and the third part is devoted to Stokes flow.

Outline of the present dissertation

Some of the studies presented in this HDR thesis are reworked versions of previously-published journal articles. Chap. [START_REF] Abdallah | Stokes flow through a Boolean model of spheres: Representative volume element[END_REF], based on reference [START_REF] Willot | The thermoelastic response of cracked polycrystals with hexagonal symmetry[END_REF], deals with self-consistent estimates for polycrystals containing cracks. It incorporates a figure and results taken from [START_REF] Willot | Effectivemedium theory for infinite-contrast two-dimensionally periodic linear composites with strongly anisotropic matrix behavior: dilute limit and crossover behavior[END_REF], with additional discussion. The study of elastic field distributions (Chap. 3) is part of a manuscript accepted for publication [START_REF] Willot | Elastostatic field distributions in polycrystals and cracked media[END_REF]. Chap. ( 4), a study of the length of geodesics in Boolean and Cox-Boolean models, is a part of ref. [START_REF] Willot | The power laws of geodesics in some random sets with dilute concentration of inclusions[END_REF]. The link between minimal paths and effective yield stress is examined in Chap. [START_REF] Alava | Disorder-induced roughening in the three-dimensional Ising model[END_REF], which is part of a recently accepted manuscript [START_REF] Willot | The effective conductivity of strongly nonlinear media: the dilute limit[END_REF]. The effective yield stress occurring in multiscale random sets is the subject of Chap. [START_REF] Altendorf | Influence of the fiber geometry on the macroscopic elastic and thermal properties[END_REF]. Bounds related to strongly-nonlinear media with periodic microstructures are presented in Chap. [START_REF] Amar | On the Finsler metrics obtained as limits of chessboard structures[END_REF]. These two chapters have not been published. Chap. [START_REF] Ambos | Numerical modeling of the thermal expansion of an energetic material[END_REF] is concerned with the scaling properties of the covariance function, integral range and variance properties of a Boolean set of particles with high aspect ratio, namely oblate and prolate cylinders. This work, slightly amended in the present manuscript, is published in ref. [START_REF] Willot | Mean covariogram of cylinders and applications to Boolean random sets[END_REF]. Chap. [START_REF] Anderson | An introduction to multivariate statistical analysis[END_REF] investigates the effective permeability of the corresponding porous media, making use of analytical upper-bounds. This last work has been published as a part of ref. [START_REF] Willot | The permeability of Boolean sets of cylinders[END_REF], but also incorporates an additional section (9.3.3) which is new.

Part I Strongly anisotropic, linear-elastic and thermoelastic media

Chapter 2

Self-consistent estimates for cracked polycrystals

This chapter is devoted to the influence of a population of randomly-oriented cracks on the macroscopic thermal and linear-elastic response of a polycrystal. Emphasis is put on crystals exhibiting hexagonal symmetry with strong anisotropy. The problem is addressed using a self-consistent method. Coupling between micro-cracks and crystal anisotropy is taken into account through an effective medium where all inhomogeneities are embedded. In the absence of cracks, the self-consistent estimates reduce to that of [START_REF] Berryman | Bounds and self-consistent estimates for elastic constants of random polycrystals with hexagonal, trigonal, and tetragonal symmetries[END_REF], whereas the method is new in the case of a homogeneous cracked body. The accuracy of the present method is first assessed using numerical, Fourier-based computations. In the absence of crystal anisotropy, the estimates for the effective elastic properties are close to that obtained numerically for a homogeneous body containing a Boolean model of disk-shaped cracks. In the case, of homogeneous cracked bodies, the method is also compared to various other analytical estimates and bounds. The prediction of the self-consistent estimates are then investigated in the case of a strongly-anisotropic polycrystal in which the in-plane shear modulus of the crystal is much smaller than the elastic moduli related to other deformation modes. The cracks-density percolation threshold is found to diminish abruptly in this limit. This "advanced" percolation threshold is concomitant to the onset of large, weakly-loaded regions surrounding cracks.

Introduction

The present work focuses on a model problem of a polycrystal, involving open micro-cracks in a macroscopically isotropic material in its initial and damaged states as well, in a locally linear, thermoelastic framework. Initially isotropic materials should remain so if micro-cracking results from the application of a non-directional load, in particular by heating or cooling slowly enough for the temperature field to remain uniform. Such a situation may be encountered in materials made of strongly thermally and elastically anisotropic crystals, such as polycrystalline graphite [START_REF] Naum | Thermal expansion of polycrystalline graphite[END_REF][START_REF] Delannay | Predictions of inter-granular cracking and dimensional changes or irradiated polycrystalline graphite under plane strain[END_REF], or the energetic material 1,3,5-triamino-2,4,6-trinitrobenzene (TATB) [START_REF] Kolb | Growth of 1,3,5-Triamino-2,4,6-Trinitrobenzene (TATB). I. Anisotropic thermal expansion[END_REF][START_REF] Ambos | Numerical modeling of the thermal expansion of an energetic material[END_REF].

In the general case, elastic softening in quasi-brittle materials is accompanied by induced 11 anisotropy, known to reflect the non-uniform distribution of micro-cracks orientations, itself the result of either the strong initial heterogeneity and anisotropy of the material or the directionality of the load. Modeling this softening effect is not straightforward, and has generated a large amount of work, either by phenomenological means (see in particular the brief review of Cormey and Welemane [START_REF] Cormery | A critical review of some damage models with unilateral effect[END_REF] or more recently [START_REF] Bargellini | Modelling of quasi-brittle behaviour: a discrete approach coupling anisotropic damage growth and frictional sliding[END_REF][START_REF] Cormery | A stress-based macroscopic approach for microcracks unilateral effect[END_REF][START_REF] Challamel | A variationally based nonlocal damage model to predict diffuse microcracking evolution[END_REF]) or via micromechanical modeling (see [START_REF] Budiansky | Elastic moduli of a cracked solid[END_REF][START_REF] Horii | Overall moduli of solids with microcracks: load-induced anisotropy[END_REF][START_REF] Andrieux | Un modèle de matériau microfissuré pour les bétons et les roches[END_REF][START_REF] Krajcinovic | Damage mechanics[END_REF][START_REF] Kachanov | Effective elastic properties of cracked solids: critical review of some basic concepts[END_REF], and the more recent studies [START_REF] Pensée | Micromechanics of anisotropic brittle damage: comparative analysis between a stress based and a strain based formulation[END_REF][START_REF] Dormieux | Stress-based estimates and bounds of effective elastic properties: the case of cracked media with unilateral effects[END_REF][START_REF] Zhu | A micromechanics-based thermodynamic formulation of isotropic damage with unilateral and friction effects[END_REF][START_REF] Monchiet | A micromechanical approach of crack-induced damage in orthotropic media: application to a brittle matrix composite[END_REF][START_REF] Levasseur | A microcracks-induced damage model for initially anisotropic rocks accounting for microcracks closure[END_REF] among many others). Most of these works have considered the damaged material as a homogeneous matrix containing a distribution of micro-cracks. However, real materials are not homogeneous in their initial state, and this assumption may be seen as being too restrictive, as noted for example by Ortiz [START_REF] Ortiz | A constitutive theory for the inelastic behavior of concrete[END_REF], Pijaudier-Cabot and Bazant [START_REF] Pijaudier-Cabot | Nonlocal damage theory[END_REF], or Challamel [START_REF] Challamel | A variationally based nonlocal damage model to predict diffuse microcracking evolution[END_REF]. The problem however is potentially very complex, as it may involve open and closed cracks, strongly nonlinear unilateral effects, closed crack kinking, and friction. Therefore, the focus in this chapter will be restricted to isotropic polycrystals containing a population of open micro-cracks with isotropic distribution in space.

The following work is restricted to crystals with hexagonal symmetry (transverse isotropy). This choice is motivated by applications to polycrystalline graphite or TATB, and because hexagonal symmetry is amenable to a relatively simple analytical treatment. A previous attempt was made by Huang and Hu [START_REF] Huang | Elastic moduli of a microcracked composite with spherical inclusions of cubic anisotropy[END_REF], aiming at predicting the elastic behavior of micro-cracked polycrystals, in the case of crystals of cubic symmetry.

The model derived in this chapter uses the self-consistent approach. Exact expressions for the Hill tensor (or equivalently Eshelby's tensor) have been derived for cracks embedded in a transversely-isotropic [START_REF] Laws | A note on penny-shaped cracks in transversely isotropic materials[END_REF] or orthotropic case [START_REF] Gruescu | Eshelby tensor for a crack in an orthotropic elastic medium[END_REF], for particular crack shape and orientations. Closed-form explicit solutions of Eshelby's problem for an ellipsoidal crack in a medium with arbitrary anisotropy are not available, although some analytical results are available in two dimensions [START_REF] Masson | New explicit expressions of the Hill polarization tensor for general anisotropic elastic solids[END_REF]. Therefore we consider the grains, of any crystallographic orientation, and the randomly-oriented cracks as separate phases embedded in the homogeneous effective medium. Under this assumption, we do not take into account the exact interaction between an oriented crack and the crystallographic orientation of the surrounding medium.

The obtained formulation is first discussed in terms of its percolating behavior, and compared to previous models of microcracked materials. Quantitative model predictions are then confronted to numerical ones obtained by Fourier-based simulations, first on a homogeneous matrix in which (isotropic) populations of micro-cracks of various crack densities are embedded, then on polycrystalline microstructures representative of a TATB-based material, in which the same populations of micro-cracks as before are embedded. The confrontation uses the Bristow-Budiansky-O'Connell [START_REF] Bristow | Microcracks, and the static and dynamic elastic constants of annealed and heavily cold-worked metals[END_REF][START_REF] Budiansky | Elastic moduli of a cracked solid[END_REF] definition of micro-crack density, widely accepted to be a good quantifier (see for example [START_REF] Cormery | A stress-based macroscopic approach for microcracks unilateral effect[END_REF]), up to relatively large crack densities (up to around 0.7) representing several thousands of micro-cracks at most.

Thermoelasticity problem for cracked polycrystals

Consider a polycrystal in a domain Ω of volume V , containing a population of frictionless cracks along a set of surfaces Γ and subjected to mechanical and thermal loading. In the rest of this chapter, it is assumed that the applied load results in the opening of all micro-cracks, so that:

σpxq ¨n " 0, x P Γ (2.1)
where n is the normal at point x along Γ (oriented indifferently) and σ is the stress tensor. Outside of the cracks, the strain field ε is determined as a function of the stress by the linear thermoelastic law:

εpxq " Spxq : σpxq `αpxq∆T, σpxq " Cpxq : rεpxq ´αpxq∆T s , x P ΩzΓ, (

where ∆T " T ´T0 is the difference of temperature with respect to a reference temperature T 0 . The permanent regime is attained at all time during cooling or heating, so that ∆T is homogeneous in Ω and represents a macroscopic thermal loading. Small deformation ε ij " p1{2qpB i u j `Bj u i q, with u the displacement vector, and quasi-static equilibrium, i.e. divσ " 0 is assumed for all thermo-mechanical loadings. Hereafter, the compliance S and stiffness tensor C " S ´1 of the crystal in (2.2) have hexagonal (i.e. transversely-isotropic) symmetry. Equation (2.2) is rewritten as: pε 11 , ε 22 , ε 33 , 2ε 23 , 2ε 13 , 2ε 12 q t " S ¨pσ 11 , σ 22 , σ 33 , σ 23 , σ 13 , σ 12 q t `αt ∆T, (2.3) where S is expressed, in the above Voigt notation, as the following 6 ˆ6 symmetric matrix:

S " ¨1{E p ´νp {E p ´νpz {E p 0 0 0 1{E p ´νpz {E p 0 0 0 1{E z 0 0 0 1{G zp 0 0 sym 1{G zp 0 2p1 `νp q{E p ‹ ‹ ‹ ‹ ‹ ‹ ' , ( 2.4)
in an orthonormal basis (e 1 , e 2 , e 3 ) local to each grain where e 3 coincides with the crystal's axis of symmetry. The compliance tensor S depends on five elastic moduli: the in-plane (E p ) and out-of-plane (E z ) Young moduli, the out-of-plane shear modulus G zp , the in-plane (ν p ) and out-ofplane (ν pz ) Poisson ratios. In contrast to isotropic elasticity, Poisson effects depend on the loading direction, hence, the coefficients ν p and ν pz can exceed 1{2. They vary in the range:

´1 ď ν p ď 1 ´2ν 2 pz E z E p ď 1, |ν pz | ď d p1 ´νp qE p 2E z ď c E p E z . ( 2.5)
With the condition E z ě 0, E p ě 0, G zp ě 0, the above guarantees that S (or C) is positive semi-definite. The two Young moduli E z and E p vary in the range: where C is represented by the 6 ˆ6 matrix inverse of (2.4).

2ν 2 zp 1 ´νp E p ď E z ď 1 ´νp 2ν 2 pz E p , 2ν 2 pz 1 ´νp E z ď E p ď E z 1 ´νp 2ν 2 zp , ( 2
In the crystal, we assume that a purely-thermal loading induces transversely-isotropic dilation or contraction with the same axis of symmetry as that of S. In the same basis used in (2.4), where e 3 is the axis of symmetry of the crystal lattice, the thermal expansion tensor entering (2.3) reads, in vector notation:

αpxq " pα 11 , α 11 , α 33 , 0, 0, 0q t , (

i.e. α 11 " α 22 and α 12 " α 13 " α 23 " 0.

The polycrystal is a space-tessellation of Ω, in which all grains are made from the same, randomly-oriented crystal. In an orthornormal basis px I q attached to the laboratory: S IJ,KL pxq " R iI pxqR jJ pxqR kK pxqR lL pxqS ij,kl , α IJ pxq " R iI pxqR jJ pxqα ij , x P ΩzΓ, (2.9) where Rpxq are constant-per-grain rotation matrix and where S ij,kl and α ij are given by (2.3), (2.4) and (2.8).

The effective thermoelastic response of the polycrystal reads:

σ " xσpxqy " r C : pε ´r α∆T q , ( 2.10)
where ε " xεpxqy and σ are the macroscopic strain and stress fields, computed by averaging the fields εpxq and σpxq over the domain Ω. In the following, the rotation matrix R and cracks's orientations are uniformly-distributed random variables in the space of 3D matrix rotations and in the sphere, respectively, and the polycrystal is statistically isotropic. In particular, the orientations of two different crystals are uncorrelated. The effective stiffness tensor r C reduces to its bulk modulus r K and shear modulus r G or equivalently, its Young modulus r E and Poisson ratio r ν, and the effective thermal expansion tensor is diagonal with the unique eigenvalue r α 11 . The macroscopic volumetric expansion is r α V " trp r αq " 3r α 11 . In the analytical treatment that follows, we use Eshelby-type solutions for the cracks, in a manner that model them as infinitesimally thin ellispoidal voids. Under this hypothesis, the strain field in the cracks is singular and each crack contributes to the spatial mean xεpxqy by (see e.g. [START_REF] Kachanov | Effective elastic properties of cracked solids: critical review of some basic concepts[END_REF]):

1

V ż Γ i dS b ¨n `n ¨b 2 , ( 2.11)
where V is the volume of Ω, Γ i the surface along crack i, dS an elementary surface along Γ i and b is the displacement jump at each point of Γ i . In the above, denoting u ˘pxq the displacement vector field on each side of the lips of the cracks, b " u `´u ´where n is oriented positively from the ´to the `side.

In a homogeneous body containing cracks, the effective elastic moduli depend on a nondimensional cracks-density parameter η. This parameter arises naturally from Eshelby's singleinclusion problem in the "dilute limit" where cracks interaction are neglected [START_REF] Bristow | Microcracks, and the static and dynamic elastic constants of annealed and heavily cold-worked metals[END_REF] and has been subsequently used in most self-consistent schemes, e.g. [START_REF] Budiansky | Elastic moduli of a cracked solid[END_REF]. When the cracks embedded in the polycrystal are equisized disks of radius a, the cracks-density parameter reads:

η " n 0 a 3 V , (2.12)
where n 0 is the number of cracks in Ω and n 0 {V the number of cracks per unit volume. The cracks's locations are arbitrary, in particular, cracks may interpenetrate with one another. In the rest of this work, we assume that the spatial distribution of cracks in the polycrystal and their morphology can be described by the cracks-density parameter 0 ď η ď 8.

Self-consistent estimate for cracked polycrystals

We now derive estimates for the homogenized thermoelastic response of a cracked polycrystal using classical self-consistent assumptions. Elastic moduli are first considered. Our method is based on Eshelby's problem for: (i) an isolated penny-shaped crack in an isotropic medium and (ii) a spherical grain with transversely-isotropic elastic response embedded in an isotropic medium.

Assume for now that the grains and cracks are isolated from one another and that they are embedded in a homogeneous isotropic body of compliance tensor S 0 , subjected to an applied stress σ. The contributions of the cracks and grains to the material's macroscopic strain ε are summed up as: ε " r S : σ " S 0 : σ `f xHpS; S 0 q : σy C `ηxGpS 0 q : σy O , (2.13) where the second-term on the r.h.s. represents the strain change due to the presence of grains, of volume fraction 0 ď f ď 1, and the third-term on the r.h.s. the strain change due to the cracks. The contributions of individual grains (tensor H) are averaged over all crystallographic orientations C. Likewise, the contributions of the cracks (tensor G) are averaged over all cracks orientations O. The transversely-isotropic tensors H and G are obtained in terms of Eshelby's or Hill's tensor [START_REF] Hill | A self-consistent mechanics of composite materials[END_REF] for their respective problem. Tensor H reads (see e.g. [START_REF] Sevostianov | Effective elastic properties of matrix composites with transversely-isotropic phases[END_REF]):

HpS; S 0 q " " `S ´S0 ˘´1 `QpS 0 q ı ´1 , (2.14)
in which Q is the isotropic stiffness tensor defined by [176, Chap. 2]:

Q 11 " Q 22 " Q 33 " 16 15
G 0 1 ´ν0 , Q 12 " Q 23 " Q 13 " 2G 0 15 1 `5ν 0 1 ´ν0 , ( 2.15)
where we refer to the Voigt notation (2.7). Other components are obtained from the above, e.g.

Q 44 " Q 55 " Q 66 " pQ 11 ´Q12 q{2.
The variables G 0 refer to the shear modulus associated to S 0 and ν 0 to its Poisson ratio, i.e. G 0 " 1{S 0 44 and ν 0 " S 0 44 {p2S 0 11 q ´1. An equivalent form is given in [START_REF] Levin | The effective thermoelectroelastic properties of microinhomogeneous materials[END_REF] for the reciprocal tensor:

P " S 0 pI ´QS 0 q, (2.16) whereas a more general expression for Q is given in [START_REF] Sevostianov | Explicit cross-property correlations for anisotropic two-phase composite materials[END_REF] for an ellipsoidal inclusion with revolution symmetry.

Tensor G entering (2.13) is given in closed-form expression as [START_REF] Mura | Micromechanics of defects in solids[END_REF]Chap. 5]:

G 33 " 8p1 ´ν0 q 3G 0 , G 44 " G 55 " 16p1 ´ν0 q 3G 0 p2 ´ν0 q , G 11 " G 22 " G 12 " G 13 " G 66 " 0, (2.17)
in the Voigt notation (2.3) where e 3 is normal to the crack. Assume now that the embedding medium with compliance S 0 has the same elastic response as the heterogeneous medium, so that S 0 " r S. Accordingly, when the material is subjected to the macroscopic stress σ, the deformation induced by the cracks compensates that of the grains:

xHpS; r Sqy C `ηxGp r Sqy O " 0, (2.18)
where we took f " 1, so that the grains cover the entire domain and the cracks have no volume.

The assumption S 0 " r S can be thought of as the construction of a hierarchical material with infinitely many scales, in the following way. Consider a material made of a homogeneous matrix containing widely-separated grains and cracks. Replace the embedding matrix by a set of grains and cracks at a much smaller length-scale, and repeat this operation until the matrix is entirely covered by grains and cracks. Now, in a polycrystal where cracks and grains have the same typical size, one may assume that each crack or grain is surrounded by many different, randomly-oriented cracks and grains. The approximation S 0 « r S amounts to replace the surrounding medium around each inhomogeneity by a homogeneous effective material.

Rewrite now Equation (2.18) using (2.14), (2.15) and (2.17). We refer to [START_REF] Sevostianov | Explicit cross-property correlations for anisotropic two-phase composite materials[END_REF] for amenable formula for the product and inverse of transversely-isotropic tensors. The means xAy C, O of a transversely-isotropic stiffness tensor A with axis of symmetry parallel to e 3 is the isotropic tensor with bulk and shear moduli [START_REF] Levin | The effective thermoelectroelastic properties of microinhomogeneous materials[END_REF]: one obtains the expressions below for the inverse moduli:

K A " 2pA 11 `A12 `2A 13 q `A33 9 , G A " 3p2A 44 `A66 q `2pA 11 ´A13 q `A33 ´A12
K ´1 T " 2u 2 `u1 ´4u 3 u 1 u 2 ´2u 2 3 `8ηp1 ´r νq 3 r G , r ν " 3 r K ´2 r G 2p3 r K `r Gq , ( 2.21a
)

G ´1 T " 2u 2 `4pu 1 `2u 3 q 15pu 1 u 2 ´2u 2 3 q `6 15u 4 `6 15u 5 `32ηp5 ´6r ν `r ν 2 q 45 r Gp2 ´r νq , (2.21b)
where r ν is the effective Poisson ratio and the u i are a set of five independent components of the tensor U " H ´1 as follows:

u 1 " U 1111 `U1122 " " 18 r K r G ´2p r G `3 r KqE z ı E p E z ∆ u `2 r Gp9 `5r νq 15p1 ´r νq , (2.22a) u 2 " U 3333 " " 18 r K r Gp1 ´νp q ´p4 r G `3 r KqE p ı E 2 z ∆ u `16 r G 15p1 ´r νq , (2.22b) u 3 " U 1133 " " 18 r K r Gν zp `p2 r G ´3 r KqE z ı E p E z ∆ u `2 r Gp1 `5r νq 15p1 ´r νq , (2.22c) u 4 " U 1212 " r GE p 2 r Gp1 `νp q ´Ep `r Gp7 ´5r νq 15p1 ´r νq , (2.22d) u 5 " U 1313 " r GG zp r G ´Gzp `r Gp7 ´5r νq 15p1 ´r νq , (2.22e) ∆ u " r2pν p ´1qE z `p8ν zp r ν ´1qE p s p3 r K `r GqE z `18p1 ´νp q r K r GE z `3E p " E 2 z ´p12 r Kν 2 zp `Ez q r G ı , ( 2.22f)
again, expressed in a basis where e 3 is the axis of transverse symmetry. The bulk and shear moduli K T and G T are, accordingly, rational functions of the bulk and shear moduli r K and r G and of all parameters of the problem. Condition (2.18) therefore can be rewritten as two polynomial equations in r K and r G. The first one is of order 4 in r K and 5 in r G, whereas the second one is of order 9 in r K and 5 in r G. For an isotropic tensor S with bulk modulus K and shear modulus G, they reduce to a system of cubic equations:

r G r K " p9 `16ηq r K ´9K ı " 3 r G 2 pK ´r Kq ´12η r K 2 K, (2.23a) 27p75 ´32ηq r K 2 G " 3 r G r K " 27p25 `16ηq r K `p704η ´675qG ı `p450 `512ηq r G 3 `3 r G 2 " 9p75 `64ηq r K `2p128η ´75qG ı . (2.23b)
Numerical solutions of the above are determined using the software "Mathematica" [262]. We obtained a unique set of real solutions in the domain r K ą 0, r G ą 0, for all elastic moduli r K and r G and crack-density parameter η considered in the present work. For arbitrary anisotropy, numerical solutions are readily obtained using the following fixed-point algorithm:

Initialization: set r K ´1 " 9∆{r2E p `p1 ´νp `4ν pz qE z s, r G ´1 " 6∆{rE p `2p1 ´νp ´2ν pz qE z s, ∆ " 1 ´νp ´2ν pz ν zp .
(2.24a)

Iterations: set r K ´1 Ð r K ´1 `K´1 T p r K, r Gq, r G ´1 Ð r G ´1 `G´1 T p r K, r Gq,
until the absolute values of r K and r G change by less than 10 ´10 . (

In this algorithm, the effective moduli are initialized to the Voigt-Hill upper-bound given by xS ´1y C (step 2.24a) whereas iterations (2.24b) enforce infinite bulk and shear moduli K T " G T " 8, i.e. the compliance tensor T vanishes. We emphasize that the value of r K used to evaluate successively the functions K T and G T in step (2.24b) changes, as r K is updated before r G. No convergence is observed, except at small cracks-density values, when the same value of r K is used to evaluate K T and G T . With the above algorithm, nevertheless, we obtained fast convergence for arbitrary crack-density η and all sets of crystal anisotropy considered in the present study.

A general result states that the thermal expansion tensor of a homogeneous body is unchanged in the presence of pores [START_REF] Rosen | Effective thermal expansion coefficients and specific heats of composite materials[END_REF], including cracks. Nevertheless, if the body is a polycrystal, or is in general heterogeneous, cracks do have an effect on the effective thermal expansion. This effect may be captured by homogenization techniques, such as the "generalized" self-consistent schemes relevant to multi-coated inclusion-reinforced composites [START_REF] Berbenni | Homogenization of multicoated inclusion-reinforced linear elastic composites with eigenstrains: Application to thermoelastic behavior[END_REF]. In the present work, we rely on the exact relationship [START_REF] Benveniste | Universal Relations in Piezoelectric Composites With Eigenstress and Polarization Fields, Part 2: Multiphase Media-Effective Behavior[END_REF]:

r α : xσpxqy " xαpxq : σpxqy, (2.25)
where σpxq is the local stress field resulting from the macroscopic stress loading xσpxqy " σ, with no thermal loading applied (∆T " 0). Since the stress is distributed over all grains, the l.h.s. and r.h.s. may be rewritten as a mean over all crystallographic orientations i:

r α : xσ i y i " xα i : σ i y i , (2.26)
where σ i is the constant stress field occurring in an isolated spherical inclusion of compliance tensor S i and thermal expansion tensor α i , oriented along direction i, and embedded in a medium with compliance tensor S 0 . Now, the macroscopic strain ε in a large domain containing one single inclusion can be decomposed as:

ε " S 0 : pσ ´fi σ i q `fi S i : σ i ,
where f i is the inclusion volume fraction. Comparing the above to (2.13), one obtains σ i " Γ : σ where the tensor Γ " rI `Q : pS ´S0 qs ´1 allows one to express the uniform field inside the inhomogeneity in terms of the prescribed field (see e.g. [START_REF] Sevostianov | Explicit cross-property correlations for anisotropic two-phase composite materials[END_REF]). Equation (2.26) now reduces to:

r α : xΓy C : σ " xα : Γy C : σ, (2.27)
and, taking σ ij " δ ij :

r α V " 6α 11 pΓ 1111 `Γ1122 `Γ1133 q `3α 33 pΓ 3333 `2Γ 3311 q Γ 3333 `2pΓ 1111 `Γ1122 `Γ1133 `Γ3311 q , ( 2.28)
where the components of Γ are given in a basis with axis of symmetry e 3 . Making use of (2.15) the effective thermal expansion coefficient reads:

r α V 3 " α 11 γ z `α33 γ p γ z `γp , ∆ γ " r ν 2 r Gp1 `r νq ´νzp E z , (2.29a) γ z " p2u 3 ´u1 q∆ γ `pu 2 ´u3 q « 1 E z ´1 2 r Gp1 `r νq ff , (2.29b) γ p " pu 3 ´u2 q∆ γ `pu 1 ´2u 3 q « 1 ´νp 2E p ´1 ´r ν 4 r Gp1 `r νq ff , ( 2.29c)
where the u i are given in Equation (2.22), and the effective bulk and shear moduli r K and r G are the solutions of (2.18).

In what follows, the self-consistent estimates (2.18) and (2.29) are denoted (SC 1 ). When the polycrystal contains no crack (η " 0), the system (2.18) reduces to two polynomial equations, one of degree 2 in r K and the other of degree 6 in r G (not shown here). These two polynomial equations are equivalent to the self-consistent estimate of Berryman [START_REF] Berryman | Bounds and self-consistent estimates for elastic constants of random polycrystals with hexagonal, trigonal, and tetragonal symmetries[END_REF] or Willis [START_REF] Willis | Bounds and self-consistent estimates for the overall properties of anisotropic composites[END_REF] which is well suited to describe the overall behavior of cell materials [START_REF] Suquet | Mathematics of multiscale materials[END_REF] and polycrystalline media with moderate contrast [START_REF] Lebensohn | On the accuracy of the selfconsistent approximation for polycrystals: comparison with full-field numerical simulations[END_REF][START_REF] Lebensohn | Study of the antiplane deformation of linear 2D polycrystals with different microstructure[END_REF][START_REF] Brenner | Elastic anisotropy and yield surface estimates of polycrystals[END_REF]. It is useful also to consider the exact cross-property relation, due to Hashin [START_REF] Hashin | Thermal expansion of polycrystalline aggregates: I. Exact analysis[END_REF], which relates the bulk and volumetric thermal expansion coefficient of isotropic polycrystalline aggregates containing hexagonal crystals such that S and α share a common axis of symmetry:

r α V " r α 11 `r α 22 `r α 33 " α V `pα 33 ´α11 q 1{ r K ´Siikk S ii33 ´Sii11 . (2.30)
where e 3 is the axis of symmetry of the crystal and repeated indices are to be summed up. Hashin's result applies to the present problem when η " 0, and as expected estimates (2.18) and (2.29) satisfy (2.30) in that case. When η ‰ 0 instead, (2.30) is not verified.

Percolating behavior

In porous and rigidly-reinforced materials, self-consistent estimates typically predict a percolation threshold. The results of these methods should be taken with care, especially with respect to percolating behavior, where the theory's predictions are sometimes unrealistic compared to real materials [START_REF] Hashin | Analysis of composite materials-a survey[END_REF]. In linear elasticity, for instance, different percolation thresholds may be obtained for the bulk and shear moduli -whereas in real materials, the mechanical and geometrical percolation thresholds are expected to coincide (see [START_REF] Milton | The Theory of Composites[END_REF]Chap. 10] for a discussion). This is because self-consistent estimates are realizable by very peculiar, hierarchical microstructures made of infinitely-many scales [START_REF] Avellaneda | Iterated homogenization, differential effective medium theory and applications[END_REF] and with no finite correlation length [START_REF] Kröner | Bounds for effective elastic moduli of disordered materials[END_REF]. That being said, it is nevertheless a merit of the self-consistent theory that, in contrast with most other approximations and bounds, it does predict a percolation threshold (see e.g. [START_REF] James G Berryman | Long-wavelength propagation in composite elastic media II. Ellipsoidal inclusions[END_REF] for a discussion).

For cracked media, the percolation thresholds is often expressed in terms of a critical crackdensity threshold η c , as, for instance, in the context of conductivity [START_REF] Saevik | Anisotropic effective conductivity in fractured rocks by explicit effective medium methods[END_REF], permeability [START_REF] Saevik | A 3D computational study of effective medium methods applied to fractured media[END_REF] or elastic wave propagation [START_REF] Saenger | Effective elastic properties of randomly fractured soils: 3D numerical experiments[END_REF]. In the present problem of quasi-static elasticity, the percolation threshold is associated to the loss of connectivity of the matrix [START_REF] Rosenzweig | Solid matrix partition by fracture networks[END_REF], at which point the effective elastic moduli vanish. For a Boolean set of uniformly-oriented disks, the geometrical percolation threshold of the complementary set of the disks has been estimated numerically in [START_REF] Yi | Computational measurement of void percolation thresholds of oblate particles and thin plate composites[END_REF]. The authors give the threshold p4{3qπη c « 22.9 i.e. η c « 5.5 for the critical crack-density.

The zeros of the Euler-Poincaré characteristic (or connectivity number) has also been used to estimate the percolation thresholds of binary media [START_REF] Bretheau | Caractéristiques morphologiques des constituants et comportement à la limite élastique d'un matériau biphasé Fe/Ag[END_REF]. Its expression reads, for a Boolean model of volume fraction 1 ´q, intensity θ and primary grain A:

qθ ˆ1 ´θM A S A 4π `πθ 2 S 3 A 384 ˙, (2.31)
where S A is the surface area of A and M A the integral of the mean curvature along the boundary of A. For a disk of radius a, S A " 2πa 2 , M A " π 2 a so that, setting the above expression to zero provides us with the approximation η c " θa 3 " p4{π 2 q ˆp3 ˘?6q. The first value η c « 0.2231 is close to the numerical estimate η c « 0.2230 [START_REF] Yi | Geometric percolation thresholds of interpenetrating plates in three-dimensional space[END_REF] for the percolation threshold of the disks. However the prediction using the connectivity number for the second threshold, η c « 2.21, related to the loss of connectivity of the matrix, is significantly smaller than the estimate given in [START_REF] Yi | Computational measurement of void percolation thresholds of oblate particles and thin plate composites[END_REF]. For linear elasticity, the self-consistent scheme of Budiansky and O'Conell [START_REF] Budiansky | Elastic moduli of a cracked solid[END_REF] predicts the percolation threshold η c " 9{16 « 0.56 for "dry cracks" embedded in an isotropic homogeneous body, whereas a different percolation threshold η c " 45{32 « 1.41 is obtained for "wet" (fluid-saturated) cracks.

For the self-consistent estimates proposed in this work, the percolating behavior is obtained by expanding K ´1 T and G ´1 T (Equation 2.21) as a Taylor series of order 2 near the point η Ñ η c where η c is unknown. The equations K ´1

T " 0 and G ´1 T " 0 provide a set of conditions on r K, r G and η c . We look for solutions of the form r K " K 1 pη c ´ηq, r G " G 1 pη c ´ηq where the prefactors K 1 and G 1 are independent of η. These asymptotic forms will be confirmed later on in this study, using numerical results. We obtain the following condition on η c : 135p225 `1424η c q " p64η c q 2 p64η c ´28q, (2.32) which provides the percolation threshold η c « 1.15752. Furthermore, in the limit η Ñ η c , the self-consistent estimates predicts constant thermal expansion coefficient and Poisson ratio, independently of the crystal's elastic moduli:

r α V Ñ α V " 2α 11 `α33 , r ν Ñ 8704η 2 c ´4608η c ´6075 512η 2 c `1800η c « 0.09144. (2.33)
The limiting value for r α V implies, in light of ( where σpxq is the local stress field occurring in the polycrystalline material subjected to stress loading, with no thermal loading. The above equality can be interpreted as a lack of correlation between the crystal's orientations and the stress field, as the percolation threshold is approached.

The considerations leading to (2.32) do not hold when some of the eigenvalues of C vanish. We first investigate the case G zp " 0 with the moduli E z and E p strictly positive. Equation (2.21) yields the lower percolation threshold η 1 c solution of:

4 `2048η 12 c `2448η 1 c `675 ˘" p2176η 1 c `1125q a 16η 1 c `9 , ( 2.35)
so that η 1 c « 0.373126. Second, when E z " 0, keeping G zp ą 0, E p ą 0, a third percolation threshold η 2 c , in-between η 1 c and η c is obtained:

η 2 c " 32

Isotropic body containing cracks

In the following, we compare the predictions of SC 1 with three homogenization schemes developed in the literature: the self-consistent scheme of Budiansky and O'Connell [START_REF] Budiansky | Elastic moduli of a cracked solid[END_REF] (denoted SC), the Mori-Tanaka method of Benveniste [START_REF] Benveniste | A new approach to the application of Mori-Tanaka's theory in composite materials[END_REF] (denoted MT) and the Hashin-Shtrikman-type bounds of Ponte Castañeda and Willis [START_REF] Castañeda | The effect of spatial distribution on the effective behavior of composite materials and cracked media[END_REF] (denoted HS). Scheme (MT) amounts to using an asymptotic expansion truncated to first-order in η ! 1 for the inverse 1{ r E, and is equivalent to the "noninteracting approximation" presented in [START_REF] Sevostianov | Effective elastic properties of matrix composites with transversely-isotropic phases[END_REF]. Method (HS) provides rigorous upper-bounds for classes of materials containing cracks embedded in uncracked regions ("security spheres"). In this construction, the cracks do not interpenetrate and the cracks-density parameter is constrained in the range η ď 3{p4πq « 0.239.

We also carry out Fourier-based numerical computations using a Poisson-Voronoï tessellation as polycrystalline aggregate. This tesselation is parametrized by the density of germs, equal to n g {V , the number of grains per unit volume. Assuming the volume Ω is a cube of dimension L, we choose to parametrize the tessellation by the grains typical size " r3{p4π{n g qs 1{3 L, defined so that the mean grain volume equals that of a sphere of radius . We model the cracks by a Boolean model of equisized disks of radius a with homogeneous Poisson point process. The intensity of the Poisson point process is ηa 3 . The Boolean model of disks is "superimposed" on the tessellation, so that the orientations of the grains and cracks, and their position compared to one another, are uncorrelated. In method SC 1 , no scale-separation hypothesis is made between the typical length scales of the cracks and that of the grains. Hereafter we choose a " so that the cracks and grains have similar size. A schematic view of the model is presented in Fig. (2.2). The Voronoï-Boolean model is not representative of real materials, but is useful nevertheless to draw comparisons between analytical estimates and exact numerical results. Numerical Fourier-based computations are carried out on the random cracked-polycrystal model, using the backward-and-forward difference scheme [START_REF] Willot | Fast Fourier Transform computations and build-up of plastic deformation in 2D, elastic-perfectly plastic, pixelwise-disordered porous media[END_REF][START_REF] Willot | Fourier-based schemes for computing the mechanical response of composites with accurate local fields[END_REF]. The domain Ω is discretized on grids of L 3 " 128 3 , 256 3 and 512 3 voxels, the cylindrical cracks have a diameter of D " 40 or D " 80 voxels and the thickness is set to w " 1.25, 1.4, 1.5, 2, 3, 4 and 5 voxels. The diameter of the cracks in each configuration is adjusted so that the crack-density parameter stay the same. As described in another study [START_REF] Gasnier | Thermoelastic properties of microcracked polycrystals. Part I: Adequacy of Fourier-based methods for cracked elastic bodies[END_REF], the effective elastic moduli are overestimated when w ď 1.4, whereas a linear behavior is observed with respect to w in the region w ě 1.5. Results for all data sets are extrapolated in the limit w Ñ 0. The mean absolute difference between the apparent moduli computed for the various data sets, i.e. different values of L and D, are used to estimate error bars. Data detailed in [START_REF] Gasnier | Thermoelastic properties of microcracked polycrystals. Part I: Adequacy of Fourier-based methods for cracked elastic bodies[END_REF] show that the error on the bulk modulus is not more than 1% when η « 0.38. Nevertheless, the precision of the method deteriorates when the number of cracks increases, and hereafter we restrict ourselves to η ď 0.8.

FFT predictions for the normalized bulk modulus r K{K and normalized shear modulus r G{G are shown in Fig. (2.3), and compared to the analytical estimates SC, SC 1 , MT and HS. The four methods SC, SC 1 , MT and HS are equivalent up to the first-order correction in η in the limit η Ñ 0. The stiffest estimate is provided by scheme MT which percolates at η " 8, whereas method SC is the most compliant. This behavior is a consequence of the self-consistent assumption employed in method SC, which reads [START_REF] Horii | Overall moduli of solids with microcracks: load-induced anisotropy[END_REF]:

r S " S `ηxGp r Sqy O .
This is in contrast with the present method, where both the matrix phase and the cracks are embedded in the equivalent medium. Similar observations hold when the Poisson ratio is ν " 0 in the matrix (Fig. At non-dilute crack density, the analytical estimates differ significantly. Numerical results for the Boolean microstructure considered here are very close to the predictions of methods SC 1 , up to about η « 0.6. At higher crack density (η ě 0.6), the estimates provided by method SC 1 are more compliant than numerical results. This behavior is not surprising since the percolation threshold for the loss of connectivity of the matrix, equal to 5.46 according to Yi and Esmail [START_REF] Yi | Computational measurement of void percolation thresholds of oblate particles and thin plate composites[END_REF], is much higher than threshold η c « 1.16 predicted by SC 1 (Eq. 2.32). Note that this percolation threshold is independent of the elastic moduli, except in degenerate cases, as will be seen later on.

E ~Ep = 1 0 -2 E p = 0 .1 E p =7 ν zp =0.2, E z =1 E p =5 E p =2 E p = 1 E p =9.999 E p = 0 . 5 E p =4 0 0.2 0.4 0.6 0.8 1 η -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 ν ~Ep =10 -2 E p =0.1 E p =7 ν zp =0.2, E z =1 E p =5 E p =10 -3 E p =1 E p =9.999 E p =0.5 E p =4 E p =2 0 0.2 0.4 0.6 0.8 1 η 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 α ~Ep =10 -2 E p =0.1 ν zp =0.2, E z =1 E p =10 -3 E p =1 E p =9 .99 9 E p =0.5 (a) (b) (c)

Cracked polycrystalline aggregates

Method SC 1 is now applied to a polycrystal containing cracks. To illustrate the predictions of the method, we fix E z " 1, G zp " 5{12, ν zp " ν p " 0.2 and let E p vary between 0 to its maximum value 10 (see Equation 2.6). Isotropic crystals are recovered when E p " E z " 1 (black solid line). The effective thermoelastic properties are represented as a function of the cracks density η in Fig. (2.4) values of G zp monitored by the parameter γ " G zp {p5{12q " 0.50, 0.1, 0.01, 10 ´3 and 0. At G zp " 0, SC 1 predicts vanishing bulk and shear moduli when η ě η 1 c « 0.37. Likewise, in the limit E z Ñ 0 a percolation threshold η 2 c « 0.75 (Equation 2.36) is obtained (Fig. 2.5c). Additionally, we have carried out numerical FFT computations for varying values of G zp ! 1 (Fig. 2.5a). They lie very close to the predictions of SC 1 when the anisotropy is moderate (G zp ą 0.1) or when η is smaller than η 1 c . This is not the case when the crack density is high (η ą η 1 c ) and the out-of-plane shear modulus is very small (G zp ď 0.01). On the one hand, FFT results indicate a finite limit for the effective elastic moduli of the cracked polycrystals, even when G zp " 0. This is confirmed in Fig. (2.5b) which represents the effective bulk modulus as a function of G zp , in semi-log plot, when η " 0.3815 is fixed. In this plot, the dotted line is the FFT prediction when G zp is strictly zero. On the other hand, although no advanced percolation threshold is observed according to FFT results, a strong decrease of the effective moduli with respect to η is observed up to about 0.4, followed by a slower decrease in the range η ą 0.4.

0.5 1 1.5 E ~Ez =10 -4 E z =0.01 E z =3 ν p =ν pz =0.2 E z =1 E z =0.1 E p =1, G zp =5/12 E z =9.9 E z =0.3 (a) (b) (c)
These results suggest a mechanism is responsible for the strong weakening of the moduli at moderate crack density, visible on the local stress fields. The stress component σ 33 occurring in a crystal is represented in Fig. (2.6), with cracks shown in white. The crystal has been chosen so that its axis of symmetry, e 3 , lies almost exactly in the same plane as the 2D cut where the fields are represented. Accordingly, σ 33 corresponds to the traction component in the out-of-plane direction, and is oriented about 45 degrees from the horizontal in Fig. (2.6). Maps (a) to (e) show the fields at decreasing values of G zp . When the shear modulus G zp approaches zero, the material is weakened by large regions where the stress field σ 33 is close to zero (Fig. 2.6e, regions colored in blue). These regions organize through large bands parallel to direction e 3 . Indeed, G zp " 0 implies in the crystal σ 31 " σ 32 " 0. Accordingly, using stress equilibrium, B 3 σ 33 " 0. Along cracks embedded in the crystal, however, σ ¨n " 0 implies σ 33 n 3 " 0. These two results explain the development of weakening bands parallel to direction e 3 , enclosing cracks. The weakening zones however do not cross the grain boundary (not shown). When the crystals are not covered by such bands, an increase of η has a strong weakening effect as each crack will weaken large regions in the surrounding crystals. Conversely, when most of the material is covered by weakening bands, an increase of η has a smaller weakening effect on the macroscopic properties.

This behavior may be compared to the effect of cracks on tension-field elastic sheets which are stiff in traction but very soft in compression. The sheets sustain tension only if a set of regions surrounding the cracks, free of stress, do not form a percolating phase connecting boundaries where a load is applied [START_REF] Mahmood | Cracks in Tension-Field Elastic Sheets[END_REF]. Weakening bands have also been observed in periodic structures subjected to plane strain and made of pores embedded in a strongly-anisotropic elastic matrix [START_REF] Willot | Localization of elastic deformation in strongly anisotropic, porous, linear materials with periodic microstructures: exact solutions and dilute expansions[END_REF]. Specifically, the authors of [START_REF] Willot | Effectivemedium theory for infinite-contrast two-dimensionally periodic linear composites with strongly anisotropic matrix behavior: dilute limit and crossover behavior[END_REF] consider a transversely-isotropic, incompressible crystal with, referring to (2.4), ν p " ν pz " ν zp " 1{2, E p " E z and G zp " kE p where k ! 1 monitors the anisotropy. The material is subjected to strain shear loading, resulting in a macroscopic strain ε 11 " ´ε33 ą 0 and ε 13 " 0. The crystal contains a cylindrical void with axis of symmetry parallel to e 2 . The authors have observed the presence of localized shear bands that develop from the void as the porosity increases. The length of the band scales as ξ " a p { ? k where a p is the pore radius. In the limit k Ñ 0 where the stiffness tensor is not positive and the governing partial derivative equations become hyperbolic, two regimes have been identified. The "dilute regime" (porosity f ! f r ) occurs when the length of the localized shear bands are smaller than the size of the elementary cell, whereas a "strongly anisotropic regime" (porosity f " f r ) holds when the shear bands span the periodic array of pores (see Fig. (see text). Porosities from left to right: f " f r {10 (a, regular "dilute" regime); f " f r (b, crossover regime); f " 10f r (c, "strongly anisotropic" regime), where f r " k{π is the crossover porosity. Incompressible matrix. Black represents the lowest field values, highest in white.

f r " k{π. In the present problem, one may expect the development of similar shear bands as G zp Ñ 0, with the size of the bands increasing as G zp decreases. However, recall that the bands surrounding cracks are, according to FFT maps, constrained to lie within the grains the crack is embedded in, and have therefore a limited impact. This explains why, according to the predictions of SC 1 , the polycrystal sustains macroscopic loads in the domain η ă η 1 c , even when G zp is strictly zero. K at increasing values of the out-of-plane shear modulus G zp , monitored by the anisotropy ratio γ " G zp {G 0 (G 0 " 5{12). Solid lines: cracks and grains of similar size. Dotted lines: cracks much larger than the grains. A different gray color is used for each value of γ.

The effect of scale separation is illustrated in Fig. (2.8) of cracked polycrystals as a function of the crack density η, for various values of the anisotropy ratio γ " G zp {G 0 . The value γ " 1 corresponds to an isotropic tensor C with Young moulus 1 and Poisson ratio 0.2. Two configurations are considered. In the first one, modeled using SC 1 , the cracks and grains have similar size, as previously considered (solid lines). In the second one, the size of the cracks is much larger than that of the grains (dotted lines). This latter case is modeled using method SC 1 two times. The matrix surrounding the crack is first replaced by an effective medium by applying SC 1 to an uncracked polycrystal (in this case the method is equivalent to Berryman's). Second, the cracks are accounted for using method SC 1 specialized to a homogeneous body containing cracks (Eq. 2.23a). The two estimates coincide only when the crystals are isotropic (γ " 1, black line). In all other cases, the elastic response is stiffer when the cracks are larger than the grains. If scale separation is assumed, no advanced percolation threshold develops when γ Ñ 0. In that case percolation occurs when η « 1.158 independently of γ. Accordingly, a homogenization approach where grains and cracks are treated separately, using e.g. Berryman's method for the polycrystal and an homogenization scheme for the cracks (e.g. MT, HS or SC), would give incorrect results, in the case where the cracks and grains have similar size and the grains have strong anisotropy. Such situation occurs in TATB polycrystals, as investigated in e.g. [START_REF] Gasnier | Thermoelastic properties of microcracked polycrystals. Part II: The case of jointed polycrystalline TATB[END_REF].

To illustrate the predictions of SC 1 when G zp " 0, two effective elastic moduli and the thermal expansion coefficient are represented in Fig. (2.9) for various values of E z , with G zp " 0, E p " 1, ν p " ν zp " 0.2, α 11 " 1 and α 33 " 0. At η " η 1 c , method SC 1 predicts r ν « 0.31 independently of E z whereas r α " 2{3 as when G zp ‰ 0 (Fig. 2.4).

Conclusion

In the present chapter, we have derived explicit formula that approximate the effective thermal and elastic, response of a polycrystal with hexagonal symmetry containing randomly-oriented cracks. This model is not an extension of that developed by Huang and Hu [START_REF] Huang | Elastic moduli of a microcracked composite with spherical inclusions of cubic anisotropy[END_REF] for cubic symmetry.

In particular, the coupling between crystal anisotropy and cracks is much stronger in the present case.

The main results are the following. First, in the absence of crystal anisotropy, i.e. for a homogeneous cracked body, our self-consistent approach yields estimates that are very close to that of a Boolean model of disk-shaped cracks. Good agreement is observed up to a crack-density parameter η « 0.7. In this domain, most cracks intersect another crack, and the matrix's Young modulus is weakened by as much as 75% for a Poisson ratio in the embedding medium of 0.2. For general transversely-isotropic crystals, the self-consistent estimates predict a percolation threshold equal to about η « 1.16 for the crack density parameter.

Second, "advanced" percolation thresholds at η " 0.37 and 0.75 are predicted for crystals with vanishing out-of-plane shear or Young modulus. The low percolation threshold predicted by the self-consistent estimate in the former case can be related to the onset of "weakening" regions that surround the cracks and lie inside the grains. The existence of such regions has been confirmed by Fourier-based numerical computations carried out on a Voronoï-Boolean model. However, in contrast to the predictions of the self-consistent estimates, Fourier computations do not indicate a percolation threshold in highly-anisotropic crystals at low-crack density, but rather a strong weakening of the elastic moduli. The analytical estimates derived in the present approach fail to predict this mechanism.

Although the methodology presented herein has focused on cracked polycrystals, it is worth mentioning that such approach can be extended, in theory, to hexagonal polycrystals containing other types of inhomogeneities, including pores or rigid inclusions.

Chapter 3

Elastostatic field distributions: self-consistent estimates & Van Hove singularities

This chapter addresses the problem of the reconstruction of the local fields distribution occurring in heterogeneous linear elastic solids. The constitutive heterogeneities are crystals and cracks. Through comparisons with FFT computations, it is shown that self-consistent estimates together with an assumption of normal distribution at the phase scale provide an accurate description of the elastostatic field histograms in polycrystals without cracks. In the case of inter and transgranular cracks, full-field FFT simulations indicate that the field histograms present van Hove singularities. Their natures are determined analytically in the low-density regime, in the case of an homogeneous medium containing cracks.

Introduction

The probability distribution functions (or field histograms) of the local stress and strain fields in heterogeneous media are useful to gain insight on a material's mechanical response. On the one hand, distribution functions embody the material's macroscopic (homogenized) response, as well as statistics on the stress or strain extreme values (distribution tails). On the other hand, the information contained in probability distribution functions is highly condensed compared to the medium's full-field microscopic response. This property may be regarded as an advantage, since one-dimensional functions are easier to characterize and model than a material's full-field response, which requires an explicit description of the geometry.

Various works in micromechanics and in homogenization have investigated the distribution function of mechanical fields in heterogeneous media. Sevostianov [START_REF] Sevostianov | Effect of pore distribution on the statistics of peak stress and overall properties of porous material[END_REF] has modeled the distribution of the peak stresses occurring along the interface of porous inclusions in plane stress, using a Gumbel (double exponential) law. Idiart et al. [START_REF] Idiart | Macroscopic behavior and field fluctuations in viscoplastic composites: second-order estimates versus full-field simulations[END_REF] considered the fluctuations of mechanical fields in particulate viscoplastic composites, predicted by homogenization theories. Using FFT computations, they showed that the field distributions in the matrix are well approximated by Gaussian distributions in the linear case but strongly deviate from the latter when dealing with strong nonlinearities such as perfect plasticity. The distribution of elastic fields in polycrystals is considered in [START_REF] Kowalski | Finite element modeling of periodic polycrystalline aggregates with intergranular cracks[END_REF].

Probability distribution functions present, in general, Van Hove singularities, named after the work of Van Hove on the density of states of phonons in crystals [START_REF] Van Hove | The occurrence of singularities in the elastic frequency distribution of a crystal[END_REF]. Along Van Hove singularities, a distribution exhibits discontinuities, sharp peaks (blow-up and cusps) and non-differentiable points. These singularities occur whenever the field's spatial gradient vanishes [START_REF] Aa Abrikosov | Experimentally observed extended saddle point singularity in the energy spectrum of YBa 2 Cu 3 O 6.9 and YBa 2 Cu 4 O 8 and some of the consequences[END_REF][START_REF] Meunier | Van Hove singularities in probability density functions of scalars[END_REF], typically extrema or saddle-points. Conversely, regions where the fields blow-up control the behavior of the distributions at infinity (distribution tails). Van Hove singularities occur in the probability distribution of the elastic fields in periodic media (see e.g. [START_REF] Willot | Localization of elastic deformation in strongly anisotropic, porous, linear materials with periodic microstructures: exact solutions and dilute expansions[END_REF]), as well as in their electrical response [START_REF] Cule | Electric-field distribution in composite media[END_REF]. Giordano [START_REF] Giordano | Electrical behaviour of a single crack in a conductor and exponential laws for conductivity in micro cracked solids[END_REF] determined Van Hove singularities for the electric field surrounding an insulating crack. The local elastic response is dealt with in [START_REF] Giordano | Local elastic fields around cracks and their stress density of states[END_REF]. The effect of a crack on the field distributions has also been considered in [START_REF] Duxbury | Breakdown of two-phase random resistor networks[END_REF] for dielectric breakdown. The case of a random dispersion of inclusions has been investigated by Cule & Torquato [START_REF] Cule | Electric-field distribution in composite media[END_REF]. Comparing deterministic and random microstructures, the authors argue that, in random media, Van Hove singularities are smoothed out by disorder. Stokes flow in random porous media has also been investigated in [START_REF] Abdallah | Stokes flow through a Boolean model of spheres: Representative volume element[END_REF].

The electrical (or dielectric) response has been considered in other works. Barthelemy and Orland [START_REF] Barthélémy | Local field probability distribution in random media[END_REF] computed the local field distribution in random dielectric networks. The method makes use of the Lipmann-Schwinger equation and generalizes the classical self-consistent analytical treatment to field distributions. The method proposed by the authors show that the field distributions are well captured when taking into account first-neighbor interactions between bonds, and treating long-range interactions by an effective medium. Pellegrini [START_REF] Pellegrini | Field distributions and effective-medium approximation for weakly nonlinear media[END_REF] has modeled the field distributions in random dielectrics using Gaussian distributions to predict the response of "weakly nonlinear" media.

The present study aims at investigating up to which extent one may use homogenization theories to predict not only the mean (and second-order moments) of the fields but also the entire probability distribution functions of such fields, in linear elastic solids. The effect of a random dispersion of heterogeneities, made of crystals or cracks are examined. Self-consistent estimates and analytical results for the field surrounding an isolated crack are employed to model the elastic field distributions. Fourier-based computations are carried out to assess the validity of our approach, and to determine the presence of Van Hove singularities in random media. The asymptotics of the distribution tail for certain stress components, are determined as well in specific cases.

This chapter is organized as follows. The probability distribution functions of elastic fields occurring in 3D sound polycrystals are considered in Sec. (3.2). Sec. (3.3) deals with a homogeneous body containing an isolated crack, or a population of parallel, or randomly-oriented cracks, in plane strain, and investigates Van Hove singularities in the field distributions. Numerical and selfconsistent results pertaining to the field histograms of cracked polycrystals in 3D are presented in Sec. (3.4).

Field histograms in sound polycrystals

Multivariate distributions of the elastostatic fields

A polycrystal made of elastically anisotropic grains can be considered as a composite material occupying a domain Ω whose constitutive phase prq represents a given crystalline orientation.

Linear homogenization theories provide estimates for the intraphase first (i.e. mean) and secondorder moments [START_REF] Bobeth | Field fluctuations in multicomponent mixtures[END_REF][START_REF] Kreher | Residual stresses and stored elastic energy of composites and polycrystals[END_REF][START_REF] Castañeda | Nonlinear composites[END_REF] The overall distribution of the strain field P ε in domain Ω reads

P ε ptq " 1 |Ω| ż Ω δpεpxq ´tq dx " N ÿ r"1 f r P r ε ptq, t P S 2 , ( 3.4)
with f r " |Ω r |{|Ω| the volume fraction of phase prq. Similar definitions hold for the intraphase stress moments M r σ and Σ r σ as well as the stress probability distributions P r σ and P σ . In order to use estimates on M r ε and Σ r ε provided by mean-field homogenization models, it is natural to consider multivariate normal (i.e. Gaussian) distributions for random tensor-valued variables. However, a symmetric second-order tensor in R 3 can also be seen as a vector in R 6 . Therefore, the well-known theory of vector-valued normal distributions can advantageously be used [START_REF] Anderson | An introduction to multivariate statistical analysis[END_REF][START_REF] Basser | A normal distribution for tensor-valued random variables: applications to diffusion tensor MRI[END_REF][START_REF] Johnson | Multivariate statistical simulation: A guide to selecting and generating continuous multivariate distributions[END_REF]. The probability distribution function for the strain field εpxq then reads [START_REF] Basser | A normal distribution for tensor-valued random variables: applications to diffusion tensor MRI[END_REF] P r ε ptq "

1 a |A r ε |p2πq 6 exp ˆ´1 2 pt ´Mr ε q T ¨pA r ε q ´1 ¨pt ´Mr ε q ˙, t P V. (3.5)
with V the vector space in R 6 . The covariance matrix A r ε is defined by

A r ε " xpε ´Mr ε q b pε ´Mr ε qy prq " Σ r ε ´Mr ε b M r ε (3.6)
and |A r ε | " detpA r ε q. The inverse of the covariance matrix, which enters (3.5), is known as the precision matrix. Theorems available for vector-valued normal distributions [START_REF] Anderson | An introduction to multivariate statistical analysis[END_REF] allow one to derive useful properties for the strain field distribution. In particular, any linear combination of the components of the strain field εpxq follows a univariate Gaussian intraphase distribution. Since the constitutive law is linear elastic, it follows that each component of the stress field σpxq is also normally-distributed in phase prq.

Let us diagonalize the covariance matrix A r ε as:

A r ε " B r ε ¨Dr ε ¨pB r ε q T . ( 3.7)
The orthogonal matrix B r ε is composed by the eigenvectors b r i (i " 1, . . . , 6) whereas the diagonal matrix D r ε give the eigenvalues λ i of the covariance matrix. Since the field ε follows a normal intraphase distribution, the eigenvectors b r i are independent normal random fields. This result allows one to obtain statistical realizations of the intraphase strain field by generating vectors ŷ whose components ŷi (i " 1, . . . , 6q are independent normal random variables with mean µ " 0 and variance σ 2 " 1 (ŷ i " N p0, 1q). A particular realization p ε r of the strain field in phase prq reads (see e.g. [START_REF] Johnson | Multivariate statistical simulation: A guide to selecting and generating continuous multivariate distributions[END_REF]Chap. 4]):

p ε r " M r ε `Br ε ¨aD r ε ¨ŷ. (3.8)
Using the above, one may compute numerically the distribution of the equivalent von Mises strain ε eq " a p2{3q e : e with e the strain deviatoric tensor. More generally, Eqs. (3.7) and (3.8) allows one to compute the distribution of any function of the strain field components.

Self-consistent estimates

To get a statistical description of the local fields through mean-field homogenization, we adopt the self-consistent (SC) model (2.18) specialized to a sound polycrystal (η " 0). It can be written [START_REF] Willis | Bounds and self-consistent estimates for the overall properties of anisotropic composites[END_REF]:

C " " Cpxq ´r C ı ´1 `Pp r Cq * ´1G " 0 (3.9)
with C (resp. r C) the local (resp. effective) elastic tensors and P the Hill microstructural tensor (2.16) which depends on r C and on the spatial distribution of the phases. The elastic properties being uniform per phase (i.e. crystalline orientation), that is:

Cpxq " N ÿ r"1 χ r pxq C r , @x P Ω, (3.10)
with χ r pxq the characteristic function of phase prq, relation (3.9) can be written

N ÿ r"1 f r " pC r ´r Cq ´1 `Pp r Cq ı ´1 " 0. (3.11)
At the local scale, the self-consistent scheme provides information about the average fields for each crystalline orientation. For the strain field, for instance, the mean reads:

M r ε " xεy prq " pC r `C˚q´1 : ´r C `C˚¯: ε. (3.12)
Besides, due to the quadratic dependence of the elastic energy on the strain field, the intraphase second moment of the strain field distribution can be obtained from the partial derivatives of the overall elastic energy r ω with respect to the local elastic tensors C r [START_REF] Bobeth | Field fluctuations in multicomponent mixtures[END_REF][START_REF] Kreher | Residual stresses and stored elastic energy of composites and polycrystals[END_REF][START_REF] Castañeda | Nonlinear composites[END_REF]]

Σ r ε " xε b εy prq " 2 f r Br ω BC r with r ω " 1 2 ε : r C : ε. (3.13)
In a general context of anisotropy, the SC estimate of the intraphase second-moment is solution of a linear system which requires the numerical integration of the Hill microstructural tensor P and its derivative BP{B r C [START_REF] Brenner | Mechanical field fluctuations in polycrystals estimated by homogenization techniques[END_REF][START_REF] Brenner | Elastic anisotropy and yield surface estimates of polycrystals[END_REF].

FFT-based computations

The distribution of the elastostatic fields in polycrystals can be addressed by considering unitcell computations with different microstructure models. We consider the classical Poisson-Voronoi tessellation [START_REF] Meijering | Interface area, edge length, and number of vertices in crystal aggregates with random nucleation[END_REF][START_REF] Miles | The random division of space[END_REF] and, for comparison, a modified Poisson-Johnson-Mehl partition of the 3D space, with anisotropic distance functions [START_REF] Gasnier | A Fourier-based numerical homogenization tool for an explosive material[END_REF] (see Fig. 3.1a). The latter has been optimized to model a TATB (triamino-trinitrobenzene, a molecular triclinic crystal) polycrystal material [START_REF] Ambos | Numerical modeling of the thermal expansion of an energetic material[END_REF] and in particular contains much more grains with elongated and non-convex shapes than in the Voronoi tessellation. The TATB crystals are strongly anisotropic. Their elastic behavior displays a general triclinic symmetry [START_REF] Bedrov | A molecular dynamics simulation study of crystalline 1,3,5-triamino-2,4,6-trinitrobenzene as a function of pressure and temperature[END_REF] which may be approached by transverse isotropy. In this study, we adopt the approximation of the crystal elastic tensor used in [START_REF] Gasnier | Thermoelastic properties of microcracked polycrystals. Part II: The case of jointed polycrystalline TATB[END_REF]. For simplicity, the crystallographic orientations in the grains are independent and uniformly-distributed on the sphere, leading to an isotropic effective response. Numerical computations are carried out using a spectral Fourier method [START_REF] Moulinec | A fast numerical method for computing the linear and non linear mechanical properties of the composites[END_REF][START_REF] Willot | Fourier-based schemes for computing the mechanical response of composites with accurate local fields[END_REF] on voxel grids containing 1024 3 voxels and about 10 4 grains for the Voronoi model, and 256 3 voxels and about 200 grains for the Johnson-Mehl model. As an example, we apply a macroscopic hydrostatic strain, normalized so that xε m y " 1. The field distributions P εxx are computed for three independent realizations of the Johnson-Mehl and Voronoi models (Fig. 3.1b). The overall distribution is nearly the same for all three realizations of the Voronoi model, except for small variations near the maximum ε xx « 0.6 of the distribution (Fig. 3.1b, gray lines). As expected, the Johnson-Mehl model displays much higher variations with respect to the realizations (Fig. 3.1, inset). When averaged, however, the distributions for the Voronoi and Johnson-Mehl models nearly coincide (Fig. 3.1, black and gray solid lines). This result suggests that the histograms for the elastic fields are not very sensitive to the grains shape, at least for moderate anisotropy. In the following, all computations are carried out using the Johnson-Mehl tessellation, discretized on a volume of 512 3 voxels and containing about 1, 250 grains. In the sequel, the full-field distributions are compared with the mean-field approximation based on the self-consistent model together with an assumption of intraphase normal distribution.

P σ [GPa -1 ] ε xy = 0.5 σ xy σ xx σ m σ eq (c) (d) -1 0 1 2 ε 0 1 2 3 4 P ε ε xx = 1 ε xy ε eq ε m ε xx -20 0
P σ [GPa -1 ] ε xx = 1 σ xy σ xx σ m σ eq (e) (f)

Elastic fields distribution: self-consistent estimates vs. FFT results

We evaluate the reconstruction of the fields distribution which is obtained by assuming a multivariate normal probability distribution within each phase prq (r " 1, . . . , N ). Each distribution is characterized by the intraphase mean and variance which are estimated with the SC model. These "mean-field" distributions are compared with reference FFT results for isotropic TATB-like polycrystals made of grains with hexagonal symmetry (transversely isotropic elastic behavior). The implicit SC equation (3.11) is solved by considering an isotropic distribution of the phases (i.e. spherical inclusion for the ancillary Eshelby inclusion problem) and a set of crystalline orientations. To approximate an isotropic crystallographic texture (i.e. uniform orientation distribution function), the space of orientations has been discretized by considering a division of the standard triangle, for hexagonal crystal symmetry, in equal area domains [START_REF] Ruer | Méthode vectorielle d'analyse de la texture[END_REF]. The isotropic texture is represented by a set of 2160 crystalline orientations (i.e phases) with equal volume fractions. Our numerical results on the overall bulk and shear moduli agree with the analytical result of the SC estimate for isotropic polycrystals made of hexagonal grains [START_REF] Berryman | Bounds and self-consistent estimates for elastic constants of random polycrystals with hexagonal, trigonal, and tetragonal symmetries[END_REF].

The probability distribution functions P ε and P σ , for field components and invariant quantities, have been computed for macroscopic hydrostatic xε m y " 1, shear xε xy y " 0.5 and uniaxial xε xx y " 1 strain loadings. The comparisons with the FFT computations show an overall good agreement with a correct description of the distributions asymmetry (Fig. 3.2). These results on linear elastic polycrystalline aggregates are consistent with those reported for linear viscous two-phase particulate composites [START_REF] Idiart | Macroscopic behavior and field fluctuations in viscoplastic composites: second-order estimates versus full-field simulations[END_REF].

Field distributions in cracked media: homogeneous body under plane strain

This section focuses on the elastic field distributions in a homogeneous body containing cracks and subjected to plane strain. In Subsection 3.3.1, a model problem made of an isolated crack or a population of randomly-oriented non-interacting cracks is investigated analytically. Emphasis is put on Van Hove singularities and a methodology is proposed to reconstruct the field distributions in the case of a population of interacting cracks. In subsection 3.3.2, the method's predictions are compared to FFT-based computations.

Van Hove singularities

As previously stated, Van Hove singularities are non-differentiable points in the probability distribution function P ptq of a spatial field. These singularities are contributions of regions where the spatial gradient of the field vanishes. The type of singularity depends on the eigenvalues of the Hessian (second-order spatial derivatives) matrix being of the same or opposite signs [START_REF] Meunier | Van Hove singularities in probability density functions of scalars[END_REF]. For a field in 2D space, a saddle-point generates a logarithmic singularity of the type P ptq " ´log |t ´t0 | where t 0 is the value of the field at the saddle-point. In 2D again, a local extremum generates a finite jump in P ptq at t " t 0 , where t 0 is the field extremal value. In 3D, saddle-points and local extrema induce powerlaw singularities with exponent 1{2, therefore the probability distribution P ptq remains finite while its derivative is unbounded. "Extended" Van Hove singularities [START_REF] Aa Abrikosov | Experimentally observed extended saddle point singularity in the energy spectrum of YBa 2 Cu 3 O 6.9 and YBa 2 Cu 4 O 8 and some of the consequences[END_REF] may in addition occur if the Hessian matrix has one or more zero eigenvalues. In 2D, blow-ups of the probability distribution function as a powerlaw of exponent ´1{2 [START_REF] Aa Abrikosov | Experimentally observed extended saddle point singularity in the energy spectrum of YBa 2 Cu 3 O 6.9 and YBa 2 Cu 4 O 8 and some of the consequences[END_REF] or ´1{3 [START_REF] Willot | Localization of elastic deformation in strongly anisotropic, porous, linear materials with periodic microstructures: exact solutions and dilute expansions[END_REF] have been reported.

θ 1 θ θ 2

Isolated crack

We now examine histograms for the stress field surrounding an isolated crack.

Vector and tensor components refer to a Cartesian coordinates system (e x , e y , e z ). The crack is modeled as an infinitely-thin cylinder of equation |x| ă a, y " 0, that is, its length is 2a and its axis is parallel to e z (see Fig. 3.3). The crack is subjected to plane strain in the (e x , e y ) plane (i.e. ε iz " 0, i " x, y, z) and to remote biaxial stress loading, resulting in σ xx " σ yy " σ k ą 0, σ xy " 0 where σ " xσy Ω is the mean of the stress field over a very large domain Ω surrounding the crack.

In the vicinity of the crack tips, the stress components σ ij become singular [START_REF] Williams | On the stress distribution at the base of a stationary crack[END_REF]. In the three polar coordinate systems (r, θ), (r 1 , θ 1 ) and (r 2 , θ 2 ) the stress around an isolated crack is expressed as [229]:

σxx yy " σ k r ? r 1 r 2 " cos ˆθ ´θ1 `θ2 2
˙¯a 2 r 1 r 2 sin θ sin 3 2 pθ 1 `θ2 q  , (3.14a) σ xy " σ k ra 2 pr 1 r 2 q 3{2 sin θ cos 3 2 pθ 1 `θ2 q. (3.14b)
Variables r 1,2 and r denote the distances to the crack tips and to the mid-point along the crack, respectively, and θ 1,2 , θ are resp. the angles formed between the x-axis and the line joining the crack tips and the mid-point along the crack (see Fig. 3.3). The stress intensity factor "seen" by the crack is σ k { ? 2πa [229]. Maps of the three stress components are shown in Figs. (3.4a-c). Consider now the probability distribution P ij pσ k ; tq of σ ij " t, computed within a region Ω k . We assume that Ω k is a circular domain with radius D and surface fraction f " πD 2 {S and seek for the behavior of P ij pσ k ; tq when t " 0, or t " ˘8. Set θ 1 " π ´dθ 1 and θ 2 " dθ 2 and expand (3.14) in the limit dθ 1 Ñ 0, dθ 2 Ñ 0. One obtains:

σ yy " σ k 1 `4px{aq 2 r1 ´px{aq 2 s 7{2 ´y a ¯3 `Opy 5 q. (3.15)
When t is small, the region where 0 ă σ yy ă t is accordingly delimited by the curves of equation:

y " ˘a r1 ´px{aq 2 s 7{6 r1 `4px{aq 2 s 1{3 t 1{3 σ 1{3 k , ´a ă x ă a (3.16) (region A, Fig. 3.4b).
Integrating the above function leads to the extended Van Hove singularity:

P tσ yy pxq ă t; x P Ω k u " 4a 2 t 1{3 σ 1{3 k |Ω i | ż 1 0 dup1 ´u2 q 7{6 p1 `4u 2 q 1{3 `Optq « 0.701 a 2 t 1{3 D 2 σ 1{3 k , ( 3.17a
)

P yy pσ k ; tq « 0.234 a 2 D 2 σ 1{3 k t 2{3 hptq, t Ñ 0, (3.17b)
where hp'q " maxpsignp'q, 0q denotes the Heaviside function.

Component σ xx vanishes along the crack lips (region A, Fig. 3.4a), and its probability distribution function P xx presents a jump at t " 0:

σ xx " 2σ k r1 ´px{aq 2 s 3{2 y a `Opy 2 q, P xx pσ k ; tq " 3a 2 8D 2 σ k hptq, t Ñ 0. (3.18)
Component σ xy is zero along the two lines y " 0, x " 0 (regions A 1 and A 3 , Fig. 3.4c) and along four curves starting from the crack tip at angles θ 1 " ˘π{3, θ 2 " ˘2π{3 (region A 2 , Fig. 3.4c).

The gradient of the field σ xy is zero along the segment y " 0, ´a ă x ă a (region A 1 ), and is non-zero in regions A 2 and A 3 . The contributions to P xy ptq (t ! 1) of regions A 2 and A 3 are accordingly finite, so that only the contribution of region A 1 matters. A Taylor expansion provides in the right-upper quadrant x ą 0, y ą 0:

σ xy " ´3σ k x{apy{aq 2 r1 ´px{aq 2 s 5{2 , P xy " ´a2 Γp´3{4qΓp9{4q 2 ? 3D 2 π 3{2 a σ k |t| « 0.2839a 2 D 2 a σ k |t| , t Ñ 0, (3.19)
where Γp'q is the extended factorial function, or Gamma function.

The tails of the distributions are now derived using the asymptotic near-tip expansions [229] (see regions B and C in Figs. 3.4a-c):

σxx yy " σ k ? a ? 2r 1 cos θ 1 2 ˆ1 ¯sin θ 1 2 sin 3θ 1 2 ˙, σ xy " σ k ? a ? 2r 1 sin θ 1 2 cos θ 1 2 cos 3θ 1 2 . ( 3.20)
We solve the three equations σ ij " t for t Ñ ˘8 and obtain r 1 as a function of θ 1 and t. The probability distributions P xx pσ k ; tq and P yy pσ k ; tq are obtained by integration of ´p4{πD 2 qr 1 B t r 1 over θ 1 in the range r´π; πs. The distribution P xy pσ k ; tq is derived by integration of the same quantity over the domain r´π; ´π{3s Y r0; π{3s (when t Ñ 8, see region B, Fig. 3.4c) and r´π{3; 0s Y rπ{3; πs (when t Ñ ´8, see region C, Fig. 3.4c). This yields, to leading order-term in t:

P xx pσ k ; tq " 177a 2 σ 4 k 512D 2 t 5 , t Ñ `8, (3.21a
)

P yy pσ k ; tq " 1089a 2 σ 4 k 512D 2 t 5 , t Ñ `8, (3.21b
)

P xy pσ k ; tq " 9a 2 σ 4 k 1024D 2 |t| 5 , t Ñ ˘8. (3.21c)
The various probability distribution functions P ij ptq " P ij pσ k " 1; tq, computed numerically from (3.14) on grids of 5000 2 voxels, are shown in Fig. (3.5a). They present Van Hove singularities not only at t " 0, ˘8 but also at various finite values of t. For reasons that will be clear later on, we do not investigate the later.

Randomly-oriented non-interacting cracks

Assume now that the material is made up of widely-separated cracks with uniformly-distributed orientations in the pe x , e y ) plane. Apply as previously a biaxial remote stress σ xx " σ yy " σ k ą 0, σ xy " 0 in plane strain. Consider a crack oriented along a direction 0 ď β ď π with respect to the x-axis. The stress components σ β ij are obtained by a rotation of angle β in the plane (e x , e y ) of the stress tensor given in (3.14). For instance σ β xx reads: The stress component σ β xx vanishes along the crack tips and a Taylor expansion provides:

σ β xx " σ xx cos 2 β `σyy sin 2 β ´σxy sinp2βq. (3.22) -0.5 0 0.5 1 σ ij [GPa] 0 1 2 3 4 5 6 P ij _ (t) [GPa -1 ] P xx _ D/a=2.98 P xy _ P yy _ -0.5 0 0.5 1 σ ij [GPa] 0 1 2 3 4 5 6 P ij _ * (t) [GPa
σ β xx " 2 cos 2 βσ k py{aq{r1 ´px{aq 2 s 3{2 , y Ñ 0, ´a ă x ă a. (3.23)
Accordingly, the corresponding distribution presents a jump and:

P β xx pσ k ; t " 0q « 3a 2 8D 2 σ k cos 2 β hptq, t Ñ 0. (3.24)
The above equation reduces to (3.18) in the special case β " 0. The jump in (3.24) also blows up when β " π{2, at which point P β xx develops the singularity given in (3.17b). Consider now the p.d.f. P xx pσ k ; tq " xP β xx pσ k ; tqy β of the stress fields surrounding cracks with stress factor σ k , averaged over all orientations β. The smallest values of σ k are concentrated in a region near the crack tips. Furthermore, the size of this region is much more important for cracks oriented with an angle β « π{2. Let us first examine the contribution of the values located along the line x " 0. Set r " y, θ " π{2, θ 1 `θ2 " π, r 1 " r 2 in (3.14) and (3.22). We obtain:

σ β xx p0, yq " σ k y pa 2 cosp2βq `a2 `y2 q pa 2 `y2 q 3{2 . (3.25)
We solve σ β xx " t for β and determine the interval β P rβ 1 py, tq; π{2s that contributes to values of σ β xx in the range r0; ts. This interval is not empty whenever y ă

y max " at 1{3 { b σ 2{3 k
´t2{3 . The region of interest, where the r.h.s. of (3.24) blows up, is y « y max (as also confirmed by Eq. 3.16), therefore we make use of the variable change y " y 1 t 1{3 in the following integral and obtain to leading-order in t:

ż ymaxptq 0 dy rπ{2 ´β1 py, tqs 9 at 2{3 σ 2{3 k , t Ñ 0 `. (3.26)
Accordingly, the contribution of the line x " 0 to P xx scales as " t ´1{3 . The analytical computation of the contribution of the entire domain Ω proves cumbersome and we resort to numerical computations. The distribution P β xx is computed by discretization of the field σ β xx on a grid of 10, 000 2 pixels, and P xx is computed by averaging over 100 regularly-spaced values of the angle β. A fit of the numerical data, gives, with excellent agreement over two decades (see Fig. 3.6a):

P xx pσ k ; tq " 0.44 a 2 σ 1{3 k D 2 t ´0.33 , t Ñ 0 `, (3.27)
which suggests an extended Van Hove singularity P xx " P ẙy " t ´1{3 , as hinted by (3.26). This is a less singular behavior at the origin than that of the probability distribution P yy obtained for parallel cracks (Eq. 3.17b), which blows up as " t ´2{3 . Regarding component σ xy , numerical computations also provide the following fit:

P xy pσ k ; tq « ´0.91 a 2 D 2 log |t| σ k , t Ñ 0, (3.28)
with good agreement ranging over two and a half decades (see Fig. 3.6b). A powerlaw with small exponent P xy " t ´0.1 instead of a logarithm can not be ruled out (Fig. 3.6b, inset) but appears less consistent with the data available. The logarithmic blow-up (3.28) is a weaker singularity than that obtained for the traction components σ xx or σ yy (Eq. 3.27). Finally, the behavior at t " ˘8 is computed analytically in the same manner as in Sec. (3.3.1). We obtain, the x-and y-axis being statistically equivalent directions:

P xx yy pσ k ; tq " 585a 2 σ 4 k 512D 2 t 5 , t Ñ `8, (3.29a
)

P xy pσ k ; tq " 9a 2 σ 4 k 1024D 2 |t| 5 , t Ñ ˘8. (3.29b)
Observe that the prefactor for P xx (or P ẙy ) lies in-between that of P xx and P yy , while it is unchanged for the shear component xy (Eq. 3.21c).

The probability distribution functions P ij ptq, are shown in Fig. (3.5b). They are computed numerically from (3.14) and (3.22) on grids of 5000 2 voxels. The parameter β is discretized along 200 values. Note that, as for the distribution functions P ij , they present Van Hove singularities at finite values of t, not investigated in the present work.

Parallel and randomly-oriented interacting cracks Consider now a population of interacting identical cracks, either parallel to one another or randomly-oriented. To take into account crack interactions, we assume that the overall distribution of the stress field around the cracks is the same as that of a set of isolated cracks subjected to random stress intensity factors, with a probability law for the stress intensity factors to be determined. As in the previous sections, the material is subjected to plane strain and in-plane biaxial stress loading. The cracks density is monitored by the non-dimensional parameter η " N c {Sa 2 where S " |Ω| is the surface area of the domain Ω and N c the number of cracks within the domain.

Consider a particular component σ ij in a domain Ω k containing a crack with remote stress σ k , and with field distribution P ij . Clearly:

P ij pσ k ; tq " 1 |σ k | P ij ˆt σ k ˙, (3.30)
where P ij ptq " P ij p1; tq is the distribution obtained when taking σ k " 1. For a continuous distribution qpsq of stress intensity factors in domains Ω k (k " 1, ..., N c ) with N c very large, the probability distribution of σ ij over Ω, is given by the convolution product:

r P ij ptq " ż `8 ´8 ds qpsq |s| P ij ˆt s ˙" ż `8
´8 du

P ij puq |u| q ˆt u ˙, ż 8
´8 ds qpsq " 1.

(3.31)

We also denote r P ij ptq the orientation-averaged distribution r P ij ptq, equal to:

r P ij ptq " ż `8 ´8 ds q ˚psq |s| P ij ˆt s ˙, ż 8 ´8 ds q ˚psq " 1, (3.32)
where P ij ptq " P ij pσ k " 1; tq. The fields moments (n " 1, 2, ...) read:

xt n y r P ij " xt n y P ij xt n y q , xt n y r P ij " xt n y P ij xt n y q ˚. (3.33)
In principle, these formula can be used to reconstruct q knowing r P ij and P ij , under some conditions [START_REF] Shohat | The problem of moments[END_REF]. In practice, the problem is ill-posed and efficient numerical algorithms must be implemented [START_REF] John | Techniques for the reconstruction of a distribution from a finite number of its moments[END_REF]. In the present situation, our knowledge of the two moments in (3.33) leads to the natural assumption that q and q ˚are Gaussian kernels:

qpsq " 1 ? 2πv q exp ˆ´ps ´µq q 2 2v q ˙, q ˚psq " 1 a 2πv q exp ˆ´ps ´µq q 2 2v q ˙, (3.34)
where we assume µ q ą 0, µ q ą 0. In doing so, we smooth out singularities that may be present in distributions P ij and P ij except possibly at t " 0 and t " `8, which are fixed under the transformation t Þ Ñ t{s. We emphasize that other choices are possible. Kushch et al. [START_REF] Vi Kushch | SIF statistics in micro cracked solid: effect of crack density, orientation and clustering[END_REF], notably, modeled the distribution of stress intensity factors in similar heterogeneous media using a Gumbel (double exponential) law. ). We finally consider distribution r P xx which takes a finite value when t " 0 `and is zero when t ă 0 (3.18). The integral in (3.32) is splitted into two intervals r0; t{t 0 s and rt{t 0 ; 8s. Assuming 0 ă t ! t 0 ! 1:

ż t{t 0 0 ds s q ˆt s ˙P xx psq « ´qpt 0 qP xx p0q logpt 0 q, (3.38a) ż 8 t{t 0 ds s q ˆt s ˙P xx psq « ´qpt 0 qP xx p0q log t t 0 . ( 3

.38b)

A similar behavior is obtained when t Ñ 0 ´so that:

r P xx ptq " ´qp0qP xx p0 `q log |t|, t Ñ 0 ˘. (3.39)
This singularity, a logarithmic blow-up, is generated by regions around cracks subjected to nearly zero remote stress, consistently with (3.32). Notice that P xx displays a finite jump at t " 0 (Eq. 3.18), unlike r P xx . A similar treatment, carried out on distribution P xy (Eq. 3.28), leads to: 3.1: Van Hove singularities of the probability distribution functions of the stress components σ ij in the non-interacting model of parallel (P ij ) or randomly-oriented (P ij ) cracks, and in the respective interacting models ( r P ij and r P ij ). Prefactors are omitted for clarity. The term h ' ptq designates a function with a finite jump at t " 0, of the type 1 `bhptq with b ą 0.

r P xy ptq " qp0q 2 log 2 |t|, t Ñ 0 ˘. (3.40)
σ ij P ij r P ij P ij r P ij t " 0 ˘σxx hptq ´log |t| hptq|t| ´0.33 h ' ptq|t| ´0.33 σ yy hptq|t| ´2{3 h ' ptq|t| ´2{3 σ xy |t| ´1{2 |t| ´1{2 ´log |t| log 2 |t| t " ˘8 σ xx , σ yy hptq|t| ´5 h ' ptq|t| ´5 hptq|t| ´5 h ' ptq|t| ´5 σ xy |t| ´5 Table
Again, the nature of the singularity is different from the logarithmic blow-up found for P xy in (3.28).

The Van Hove singularities at t " 0 and t " ˘8 of the distribution functions P ij , r P ij , P ij and r P ij are summarized in Tab. (3.1). As previously noted, the singularities of r P ij are different from that of P ij . For instance r P yy develops powerlaw singularities when t Ñ 0 ´or t Ñ ´8, contrarily to P yy which has support on r0; 8[. Likewise, r P xx develops a logarithmic singularity when t Ñ 0

´, contrarily to P xx . Also, as will be seen later on, distributions r P ij ptq and r P ij ptq do not present singularities at finite (non-zero) values of t.

FFT-based computations

Reconstruction of the overall stress distribution vs. FFT data We compute numerically the stress distributions occurring in the solid phase of a 2D Boolean set [START_REF] Serra | The Boolean model and random sets[END_REF] of cracks subjected to plane strain. The cracks are parallel to one another. Again, we use the FFT scheme with "backward-forward" finite difference [START_REF] Willot | Fast Fourier Transform computations and build-up of plastic deformation in 2D, elastic-perfectly plastic, pixelwise-disordered porous media[END_REF], appropriate to the presence of discretized cracks [START_REF] Gasnier | Thermoelastic properties of microcracked polycrystals. Part I: Adequacy of Fourier-based methods for cracked elastic bodies[END_REF], and computations are carried out on two independent realizations of the Boolean model, discretized on a 2048 2 -pixels grid. We follow [START_REF] Gasnier | Thermoelastic properties of microcracked polycrystals. Part I: Adequacy of Fourier-based methods for cracked elastic bodies[END_REF] and discretize the cracks as thin rectangles of width w " 1.5 voxel. In all computations hereafter, the overall applied stress is by convention equal to 1 GPa so that xσ xx y " xσ yy y " 1. The cracks have a finite thickness, therefore the mean of the stress field in the solid phase is slightly above 1 GPa.

The probability distribution function r P xx ptq is compared with FFT results in Fig. (3.7a). The first and second moments of the distribution of the stress intensity factor q, used to compute r P xx ptq, are adjusted so that the moments of r P xx ptq fit the corresponding FFT data (see 3.33). The values of a and D are chosen so that the crack density parameter η is the same in the FFT computations and in each domain Ω k , i.e.

η " N c a 2 S " a 2 πD 2 , ( 3.41)
where N c " 60 is the number of cracks in FFT computations, S " 2048 2 is the surface of the unit cell, a " 25 pixels, and so η « 0.035, D « 75 pixels. It is emphasized that FFT numerical results have been used, in the present work, to tailor the first and second-order moments of r P xx . In principle, however, such moments could be determined using self-consistent estimates. For a cracked body in plane strain, these estimates, which include the case of an anisotropic embedding A similar procedure is used in the random case. Consider the low crack-density regime η ! 1. Eq. (3.33) gives the first and second-moments of q ˚, given that of r P ij . The latter are provided by FFT computations, carried out on a 8192 2 pixels grid. We use a " 25 pixels, N c " 20, D " 129 and η " 0.011. The mean and second-order moments of r P ẙy , given by FFT computations, are 1.0007 and 1.0279 resp. and the mean and variances of q are 1.0103 and 3 10 ´4 resp. Results are shown in Fig. (3.7b).

The method however breaks down when η is larger than about 0.05, at which point (3.33) predicts a negative variance for q, so that this constraint can not be satisfied anymore. Indeed, our method assumes that all cracks are loaded in mode-I. In random media, in reality, the regions in which cracks are embedded are subjected to varying loadings, combining several modes. Furthermore, in the present method, the distribution q entering (3.33) does not depend on the stress component considered, hence, one can not simultaneously satisfy all set of constraints on the first and second moments of the stress components. Taking into account other modes would, in theory, provide additional degrees of freedom, at the expense of a more analytically-involved treatment.

Van Hove singularities

We now focus on the singularities of the probability distribution functions P ˚,F F T ij , predicted by numerical FFT computations. All computations are carried out on two independent realizations of a Boolean model of cracks, discretized on a 8192 2 pixel grid containing N c " 200 cracks. To highlight possible Van Hove singularities, we choose a crack density equal to η " 0.12 with a " 25 pixels. The following powerlaw behaviors are observed. When t Ñ ˘8:

P ˚,F F T xx,yy « |t| ´5 ˆ" 1.8, t Ñ `8, 0.03, t Ñ ´8, (3.42a
) 33 , t Ñ 0

P ˚,F F T xy « 0.02|t| ´5, t Ñ ˘8 (3.42b)
P ˚,F F T xx,yy « " 0.19t ´0.18 , t Ñ 0 `, 0.074|t| ´0.
´. for the stress components σ yy in media with randomly-oriented cracks, as predicted by FFT computations. Dots: FFT data for η " 0.01, 0.12 and 0.6. Dashed lines: powerlaws " t ´1{3 (see text). (b) Comparison between the probability distribution functions P F F T yy , P ˚,F F T yy , predicted by FFT (dots) and models r P yy and r P ẙy (solid black lines), in log-log plot.

density parameter η. For η " 0.01, the data is consistent with the prediction r P ẙy ptq " |t| ´0.33 (Tab. 3.1) in the region t Ç 0 (see dashed lines with slope ´1{3).

Field distributions in cracked polycrystals

This section is concerned with various models of cracked polycrystals based on the Johnson-Mehl model and introduced in [START_REF] Gasnier | Thermoelastic properties of microcracked polycrystals. Part II: The case of jointed polycrystalline TATB[END_REF]. The first one is a polycrystal with intergranular cracks, denoted JM-I and represented in Fig. (3.10a). Three transgranular cracks models are also considered. In the first one, denoted (JM-T), crystals and cracks orientations are uncorrelated and uniformly-distributed on the sphere (Fig. 3.10b). In the other two transgranular crack models, the cracks orientation are correlated to the crystal directions. In the "weak-plane" transgranular model, denoted (JM-TW), cracks are oriented parallel to the crystal basal plane (Fig. 3.10c). In the stiff-plane model, denoted (JM-TS), the cracks are oriented perpendicular to the basal plane. Finally, for comparison purposes, we also consider a homogeneous body containing cracks, denoted (JM-IB), obtained by replacing the grains in model (JM-I) by a homogeneous isotropic medium with the same elastic properties as those of the sound polycrystal. The density of cracks is fixed to η " 0.24 in all models.

The distributions of the equivalent strain ε eq , mean strain ε m and shear strain ε xy are shown in Figs. (3.11a-f), for hydrostatic strain loading xε m y " 1 and shear strain xε xy y " 0.5. Solid lines .12: Tail of the distributions P εxx ptq (t " 1) of the strain field ε xx in media subjected to hydrostatic strain loading: comparison between sound Voronoi (V) and Johnson-Mehl (JM) polycrystals and cracked models JM-I (intergranular cracks) and JM-T (uncorrelated transgranular cracks), represented in log-log plot. Black dashed lines: powerlaw fits P εxx ptq " 0.5t ´4. 7 and P εxx ptq " 10t ´4.7 . Gray dashed line marked SC: self-consistent estimate for a sound polycrystal.

represent polycrystals with transgranular cracks. Dashed lines refer to the integranular model (JM-I) and homogeneous cracked medium (JM-IB). Overall, the field distributions differ significantly depending on the microstructure. In models with transgranular cracks, the correlation between the cracks orientation and the grains basal plane has a strong effect on the field distribution ε m , for hydrostatic strain loading (Figs. 3.11c). All field distributions exhibit a singularity, or accumulation point at ε " 0. For the von Mises equivalent deviatoric part ε eq , a local peak appears near ε " 0, in all models, whereas the distributions for ε eq and ε xy blow up at ε " 0. The peak appears to be more important for model with intergranular cracks than for models with transgranular cracks.

The decay of the distribution tails for the strain component ε xx is shown in Fig. (3.12), for various models subjected to hydrostatic loading. The transgranular model (JM-T) and intergranular model (JM-I) as well as a sound polycrystal model, with Johnson-Mehl (JM) or Voronoi (V) tessellation are considered. Models (JM-T) and (JM-I) exhibit powerlaw decays P εxx ptq " t ´ν as t Ñ 8 with ν « 4.7 « 5. This exponent is close to that obtained in plane strain (Sec. 3.3). As shown by Fig. (3.12), the distribution of the field ε xx decays at a much faster rate in sound polycrystal models (light gray curves) than in cracked polycrystal models (dark gray curves). For Jonsohn-Mehl and Voronoi sound polycrystals, the rate of decay for the distribution P εxx ptq as t Ñ 8 is close to that of a powerlaw. The exponent of the powerlaw is about ´8. It varies greatly with the realization of the model (not shown) but is nevertheless much lower than ´5. In any case, such powerlaw decay may result from the blow-up of the elastic fields at multimaterial corners, which has been studied, notably, in plane strain and plane stress problems [START_REF] Barroso | Computing stress singularities in transversely isotropic multimaterial corners by means of explicit expressions of the orthonormalized Stroh-eigenvectors[END_REF][START_REF] Barroso | Singularity analysis of anisotropic multimaterial corners[END_REF][START_REF] Mantič | Singular elastic solutions in anisotropic multimaterial corners: applications to composites[END_REF][START_REF] Barroso | Determination of Generalized Fracture Toughness in composite multimaterial closed corners with two singular terms-Part I: Test proposal and numerical analysis[END_REF][START_REF] Mantič | Stress singularities in 2D orthotropic corners[END_REF]. The powerlaw decay is not predicted by the self-consistent method described in Sec. (3.2), which assumes a Gaussian intraphase field distribution (dashed line, Fig. 3.12).

We close this section by further considerations regarding a particular model, the one containing weak-plane transgranular cracks (JM-TW). We make use of a self-consistent estimate, proposed in [START_REF] Gasnier | Thermoelastic properties of microcracked polycrystals. Part II: The case of jointed polycrystalline TATB[END_REF] for this model, which assumes separation of scales between the cracks and grains. In this method, the elastic moduli of each grain is weakened by the presence of micro-cracks. Thus, the same methodology as proposed in Sec. (3.2) may be applied to model (JM-TW), provided that the grains are replaced by weakened grains. Our results for the distribution of the stress components σ eq , σ xx and σ xy are shown in Fig. (3.13) and compared to FFT predictions. As previously, hydrostatic or shear loading is applied, and the FFT histograms are computed in the matrix only, as for the strain. In FFT computations, Van Hove singularities appear at σ " 0. As expected, the reconstruction making use of the self-consistent estimates are unable to reproduce such singularities.

Conclusion

To investigate how one may use homogenization theories to predict the field probability distribution functions in linear elastic solids, the present contribution has examined two model problems in mechanics, that of a polycrystal and that of cracked media. Two different methods have been followed, based on an ansatz for the field distributions in grains or in regions surrounding a crack. Our main results are as follows.

The probability distribution function of the local elastic fields in sound polycrystals, with highly anisotropic grains and Johnson-Mehl or Voronoi tessellation microstructures, is accurately predicted by the self-consistent model and an hypothesis of multivariate Gaussian distribution of the intraphase strain and stress fields, except for the distribution tails. The method is not sufficient, however, when dealing with media containing cracks. The elastic fields in this case exhibit extended Van Hove singularities at ε " 0 or σ " 0, even when the cracks are randomly-distributed and randomly-oriented.

For the model problem of a homogeneous body containing a population of cracks and subjected to plane strain, in particular, the probability distribution for the stress field exhibits a powerlaw decay " |σ ij | ´5 as σ ij Ñ ˘8, resulting from the singularity of the elastic field at the crack tip. A method has been proposed to reconstruct the probability distribution function of the stress fields, based on the assumption that the field surrounding each crack is governed by a stress intensity factor which follows a Gaussian probability distribution. The method gives accurate predictions for the probability distribution when the density of cracks is small, except for negative field values. Finally, numerical Fourier-based computations carried out in cracked media in 2D and 3D suggest a powerlaw decay " |σ ij | ´5. In 3D cracked polycrystals, FFT data confirm the existence of Van Hove singularities at σ ij " 0.

Chapter 4

Minimal paths in random media

In this chapter, we develop a method for computing upper-bounds on the length of geodesics spanning random sets in two, three (or more) dimensions, with emphasis on Boolean models containing a vanishingly small surface or volume fraction of inclusions f ! 1. The distance function is zero inside the grains and equal to the Euclidean distance outside of them, and the geodesics are shortest paths connecting two points far from each other. The asymptotic behavior of the upper-bounds is derived in the limit f Ñ 0. The scalings involve powerlaws with fractional exponents " f 2{3 for Boolean sets of disks and " f 1{2 for the Boolean set of spheres. These results are extended to models of hyperspheres in arbitrary dimension and, in two and three dimensions, to a more general problem where the distance function is non-zero in the inclusions. Finally, other fractional exponents are derived for geodesics spanning multiscale (Cox) Boolean sets, based on inhomogeneous Poisson point processes, in two and three dimensions.

Introduction

The present chapter deals with various types of random sets representing microstructures, which are made of particles embedded in a matrix. We focus on Boolean sets [START_REF] Matheron | Image analysis and mathematical morphology[END_REF], studied in G. Matheron's seminal works [START_REF] Matheron | Eléments pour une théorie des milieux poreux[END_REF][START_REF] Matheron | Random sets theory and its applications to stereology[END_REF], for both theoretical and practical reasons. On the theoretical side, a key property of these models is that of "infinite divisibility" [START_REF] Serra | The Boolean model and random sets[END_REF]. Boolean models are obtained as the limit of unions of non-Boolean models such as tessellations [START_REF] Serra | The Boolean model and random sets[END_REF]. This property is regarded as analogous, in the context of random sets, to that of the central limit theorem [START_REF] Serra | The Boolean model and random sets[END_REF][START_REF] Cressie | Random set theory and problems of modeling[END_REF] and suggest that, from a theoretical point of view, Boolean sets play the same role with respect to unions as the Gaussian distribution with respect to summation. On the practical side, Boolean models are straightforward to simulate, and are arguably the most commonly-used for representing random structures of the matrix-inclusion type (for a few examples among many, see e.g. [START_REF] Greco | The use of the texture analyser to study sinter structure: application to the morphology of calcium ferrites encountered in basic sinters of rich iron ores[END_REF][START_REF] Wang | Modeling of mesoporous alumina microstructure by 3D random models of platelets[END_REF][START_REF] Bortolussi | Morphological modeling of cold spray coatings[END_REF]).

Boolean set of disks in two dimensions

This section details how to construct an upper-bound for the length of minimal paths spanning random media in the continuum. Out attention is restricted to particulate materials with dilute concentration of heterogeneities. We compare the bound with numerical results for the length of 53 geodesics in finite-size systems. We also investigate the "rugosity" of the path used to construct the bound and that of the minimal path.

Hereafter, a Boolean set [START_REF] Matheron | Random sets and integral geometry[END_REF][START_REF] Serra | The Boolean model and random sets[END_REF] of disks in R 2 , of surface fraction 0 ď f ď 1 is considered. The disks have constant diameter D ą 0 and may interpenetrate. Their centers follow a homogeneous Poisson point process. Disks are crossed at no cost whereas the embedding medium is crossed at a unit cost. The distance between two points A and B therefore reads:

dpA, Bq " inf pPK ż 1 0 dt χppptqq ||B t pptq|| , (4.1) χpM q " " 0 if M lies inside a disk, 1 otherwise,
where χ is the indicator function of the embedding medium, || ¨|| is the Euclidean norm and:

K " p P C `r0; 1s, R 2 ˘, pp0q " A, pp1q " B ( (4.2)
is the set of continuous curves from A to B. Some immediate properties of the function dp¨, ¨q follow from (4.1). For all points A, B and C:

dpA, Aq " 0, dpA, Bq " dpB, Aq ě 0, (4.3a) dpA, Bq ď dpA, Cq `dpB, Cq, dpA, Bq ď ||B ´A||, (4.3b)
so that d is a pseudo-distance. Also from (4.1), it is clear that any continuous portion p 1 of the path p is a minimal path between its extremal points. Accordingly, if the path p 1 lies entirely in the embedding medium (χpp 1 ptqq " 1), it is necessarily straight. Therefore, minimal paths are unions of segments joining disk centers. Assume, for convenience, that the end points of the line segments are pA; C 1 ; ...; C N ; Bq (N ě 0) where the C i are disk centers of coordinates pC i 1 ; C i 2 q. We now focus on the limiting behavior of the normalized distance:

ξ " dpA, Bq L , L " ||A ´B|| Ñ 8. (4.4)
Without loss of generality, we assume that A is the center of a disk at the origin of a Cartesian coordinate system pe 1 ; e 2 q and that the line joining A and B is parallel to e 1 . When A is fixed and L Ñ 8, the computation of ξ amounts to study the limit shape of the set:

S t " " 1 t B; dpA, Bq ď t * , 1 t B " ˆB1 t ; B 2 t ˙, (4.5)
as t Ñ 8. The above can be regarded as a growth process in the continuum [START_REF] Howard | Euclidean models of first-passage percolation[END_REF][START_REF] Deijfen | Asymptotic shape in a continuum growth model[END_REF]. In the present work, an isotropic Boolean set of disks is considered, and so the limiting shape S 8 of S t is a disk with radius lim LÑ8 p1{ξq.

In the rest of this work, we study the behavior of lim LÑ8 ξ in the dilute limit f Ñ 0, and more exactly its leading-order correction in f . In effect, this amount to take the double limit:

lim f Ñ0 lim LÑ8 ξ. (4.6)
The two limits are not interchangeable, for if A and B are fixed and f Ñ 0, the minimal path between A and B is almost surely a straight line and ξ " 1. Therefore, in the rest of this work f ! 1 is fixed and L " D is chosen sufficiently large that the minimal path between A and B passes through a very large number of disks (N " 1).

Three discs

Consider three non-intersecting discs, identified by their centers U , V and W . We denote by V 1 the projection of V onto the line pU W q and set " ||U ´V 1 ||, m " ||V ´V 1 || and r " ||U ´W || (see Fig. 4.1). We are interested in the condition under which a path of minimal length joining U to W must pass through V . Inequality:

||U ´V || ´D `||V ´W || ´D ď ||U ´W || ´D yields: m ď a DpD `2rqpD `2r ´2 qpD `2 q 2pD `rq . ( 4.7)
In the dilute limit f Ñ 0, taking r " D and " D, (4.7) reduces to:

m ď a 2D p1 ´ {rq. (4.8)
Assuming that U and W are fixed, equation (4.7) defines a domain of interest where disks may be looked for, in order to construct paths with small length. The surface of domain (4.8), scales as:

S " 2 ż r 0 a 2D p1 ´ {rqd " πr ? Dr ? 8 . (4.9)
The width of this domain grows as " ? r, which suggests a power-law behavior for the length of minimal paths in the dilute limit, in the continuum, following the argument of Roux & François [START_REF] Roux | A simple model for ductile fracture of porous materials[END_REF]. This is detailed hereafter by the derivation of an upper-bound on ξ.

Upper-bound in the dilute limit

We identify U with the center C i of the i th disk on a path pC 1 ; ...; C N q and W as the "first" disk encountered in the direction e 1 starting from U , if the path was straight. The choice C i`1 " W amounts to follow closely direction e 1 and is not advantageous, unless there exists no disk V satisfying (4.7), an event that has a low probability. Hence, we consider instead the disk V satisfying (4.8) with minimal value of . Clearly, ! r and condition (4.8) becomes:

m ď ? 2D . (4.10)
The above property suggests the following iterative procedure for constructing a path pC 0 ; C 1 ; ...; C N q starting with the point C 0 . Knowing C i , C i`1 is the disk in the domain delimited by two

C i`1 " arginf CPK 1 C 1 , (4.11) K 1 " " C a disk center; C 1 ą C i 1 , |C 2 ´Ci 2 | ď b 2D|C 1 ´Ci 1 | * .
This procedure may be compared to that used by Lee [START_REF] Lee | The powerlaws of M and N in greedy lattice animals[END_REF] who derived a bound on the minimal length of self-avoiding paths in random lattices by selecting a series of points which move "as directly as possible" from one vertex to another. In the present method, however, we consider a continuum medium and the geodesics must follow a privileged direction. Let us first replace condition (4.11) by:

C i`1 " arginf CPK 2 C 1 , (4.12) K 2 " " C a disk center; C 1 ą C i 1 `D, |C 2 ´Ci 2 | ď α b D|C 1 ´Ci 1 | * ,
where α ą 0 is a constant to be optimized on. We also request that C i 1 ą C 1 `D so that the disks do not overlap. This technical assumption simplifies the analytical treatment, and has no effect in the dilute limit where disks are "almost always" far from each other. Starting from C 0 " A, we construct the path pA; C 1 ; ...; C N ; Bq where the number N is chosen so that C N `1 is the first disk center with coordinate along e 1 larger than B 1 , i.e. C N `1

1 ą B 1 and C N 1 ď B 1 . As in Sec. 4.2.1, we set i " |C i 1 ´Ci´1 1 |, m i " C i 2 ´Ci´1 2 (i ě 1)
. The path (4.12) provides the following upper bound on ξ " dpA, Bq{L:

ξ ď ř N i"1 ´a 2 i `m2 i ´D¯`Z ř N i"1 i , ( 4.13)
where Z " ||C N ´B|| is the Euclidean distance from C N to B (see Fig. 4.2b). Rewrite (4.13) as:

1 ´ξ ě N ř i"1 ´D ` i ´a 2 i `m2 i ¯´Z ř N i"1 i « N ř i"1 rD ´m2 i {p2 i qs ´Z ř N i"1 i . (4.14)
To evaluate the above, we determine the mean of the i and of m 2 i { i . The m i are uniform random variables in the interval r´α ? i D; α ? i Ds. The probability P t i ą u is the probability that the domain delimited by the two curves in Fig. (4.2a) and enclosed by the lines x 1 " C i´1 1 `D and x 1 " C i´1 1 ` i contains no disk center. This probability is given by the Choquet capacity of a Poisson point process [START_REF] Matheron | Random sets theory and its applications to stereology[END_REF] as expp´θV q where V is the size of the domain and θ " ´4 logp1 ´f q{pπD 2 q is the intensity of the Poisson point process (i.e. the disks centers) used to build the Boolean set. We compute V and obtain:

P t i ď u " 1 ´p1 ´f q 16α{p3πqrp {Dq 3{2 ´1s , ( 4.15)
as the cumulative probability function of the random variable i in rD; 8q. The above yields, for the average of the i , using the variable change η " pD{ q 3{2 : 1

DN N ÿ i"1 i « 1 D ż ěD P t ď i ď `d u (4.16) " 1 `2 3 p1 ´f q ´16α 3π E 1{3 ˆ´16α logp1 ´f q 3π ˙" ˆπ 4α ? 6f ˙2{3 Γ ˆ2 3 ˙`op1q,
where E 1{3 ptq " ş 1 0 dηe ´t{η η ´5{3 is the exponential integral function of parameter 1{3. Furthermore:

1 N N ÿ i"1 m 2 i i « ż ěD ż α ? D m"0 m 2 P t ď i ď `d u dm α ? D " Dα 2 3 . ( 4.17)
At lowest order in f :

ξ ď 1 ´2α 2{3 p6 ´α2 q 3Γ `5 3 ˘ˆ2 3π ˙2{3 f 2{3 `Opf 4{3 q `Z ř N i"1 i , ( 4.18)
where Γ is the Gamma (or extended factorial) function. Furthermore, (4.16) entails m i " f ´1{3 and:

Z " ˇˇˇˇN ÿ i"1 m i ˇˇˇˇ" ? N f ´1{3 " ? L. (4.19)
Accordingly the term Z{

ř N i"1 i " 1{ ? L in (4.18) is negligible when L is large. The choice α "
a 3{2 in (4.18) then yields:

ξ ď 1 ´3 Γ `2 3 ˘ˆ3f 2π ˙2{3 `Opf 4{3 q « 1 ´1.3534f 2{3 . (4.20)

Comparison between the length of the minimal path and its upper-bound

Hereafter, we compare the dilute limit expansion (4.20) to numerical results, obtained using the algorithm in Appendix (4.A). The paths used to construct bound (4.20) and the minimal path are represented in Fig. (4.5). The two paths join two opposite corners of a realization of a Boolean set in a square domain. Our numerical computations are carried out on random configurations containing on average 10, 000 disks at increasing surface fractions f " 10 ´7, ..., 10 ´1. For each value of f , 20 random configurations are averaged. We also compute numerically the bound (4.13).

Results, indicated in Fig. (4.3a), show an excellent agreement between bound (4.13) and expansion (4.20) (solid lines and black dots). A power-law fit on the numerical data for the exact minimal path length provides ξ " 1 ´1.85f 0.67 . The upper-bound (4.13) is generalized for non-dilute surface fraction of grains as:

ξ ď lim N Ñ8 ř N i"1 max ! 0; a 2 i `m2 i ´D) ř N i"1 i , ( 4.21)
The above bound is now computed for the full range of porosity 0 ď f ď 1 using (4.16) and:

1 DN N ÿ i"1 max ! 0; a 2 i `m2 i ´D) « ż xě1 2dx π p1 ´f q 16αx 3{2 3π logp1 ´f q ˆ"4α ? x ´2αx ? α 2 `x ´x2 log ˆ1 `2α x ´α `?α 2 `x¯˙ `ż 1 x" b 1`α 4 4 ´α2 2 ´4dx π p1 ´f q 16αx 3{2 3π logp1 ´f q ˆ"αx ? α 2 `x `?1 ´x2 ´2α ? x `x2 log ˆ?x α `?α 2 `x 1 `?1 ´x2 ˙ . ( 4.22)
The two integrals in the above are computed numerically. Numerical experiments indicate that α " a 3{2 is optimal, i.e. produces the sharpest bounds, for all values of f . The bound is compared to numerical estimates of ξ in Fig. ) is significantly higher than the exact result when the surface fraction f of the disks is not small. For small values of f , however, the bound becomes a good estimate of the geodesics. As expected, the upperbound percolates at f " 1, a value larger than the actual percolation threshold, which is about f " f 2D p « 0.68 [START_REF] Quintanilla | Efficient measurement of the percolation threshold for fully penetrable discs[END_REF].

Rugosity exponent

We close this section by a discussion of the rugosity of the minimal path and that of the path used to derive bound (4.20). The m i are independent variables with zero mean and finite variance:

xm 2 i y " ˆπ 12f ˙2{3 Γ ˆ2 3 ˙D2 2 " x i y D 2 " LD 2N , ( 4.23)
hence, by the Central Limit theorem, p1{N q ř i m i is Gaussian for N large. Accordingly, Z follows the probability distribution function:

P tz ă Z ă z `dzu 1 dz " 2 ?
LDπ e ´z2 {pLDq . We have simulated the probability distribution function for Z on 40, 000 configurations each containing 15, 000 disks, with f " 0.01 fixed. The value of Z is estimated at various values of L{D and compared with that obtained for the shortest path (Fig. 4.4). For the later, Z is defined to be the deviation along the direction transverse to propagation, i.e. it is equal to |x 2 | where x lies on the shortest path such that x 1 " L, the origin of the shortest path being 0 by convention. The data points for the numerical estimates of Z are in excellent agreement with (4.24). We also observe that the distribution of the variable Z for the exact shortest path, at given value of L{D, is much wider than that of the bound (triangles in Fig. 4.4a). We study the width of the distribution of Z hereafter.

Consider the path transverse fluctuations w " pxZ 2 y´xZy 2 q 1{2 as a function of L, for fixed value of f " 10 ´2, 10 ´3 and 10 ´4. It has been shown [START_REF] Huse | Pinning and roughening of domain walls in Ising systems due to random impurities[END_REF][START_REF] Kardar | Scaling of directed polymers in random media[END_REF] that the minimal path is very rough with scaling law w " L ´2{3 as L becomes large (see numerical computations, Fig. 4.4b). As expected, this is not so for the path used to derive bound (4.20) which exhibits a much more common scaling law " L ´1{2 (black line, Fig. 4.4b). Thus, the exponent 2{3 obtained in bound (4.20) is not directly linked to the "roughness exponent" related to the scaling law for w.

Other Boolean and Cox-Boolean sets in two dimensions

In this section, we extend result (4.20) obtained for a Boolean set of discs to other random sets. The discs are replaced by an arbitrary compact, non-random, fully connected grain in Sec.

Boolean set of compact grains

Let us first suppose that the inclusions are aligned squares of side D and that points A and B are aligned with one of the direction of the squares. The path pA; C 1 ; ...; C N ; Bq is defined by:

|C i`1 1 ´Ci 1 | " inf |C 1 ´Ci 1 |; C a square center; C 1 ą C i 1 `D, |C 2 ´Ci 2 | ď α b D|C 1 ´Ci 1 | * . ( 4.25)
Note that compared to Eq. (4.12), the C i are now square centers and we have added the condition

C 1 ą C i 1 `D. Like in Sec. (4.2.2), we set i " C i 1 ´Ci´1 1 , m i " C i 2 ´Ci´1 2 with i ě D, |m i | ď α ? i D.
Again, α is a constant to be optimized on and we denote f ! 1 the surface fraction of the squares. The distance function d sq is defined as in (4.1) with χ replaced by the indicator function of the Boolean set of squares. Note that:

d sq pC i´1 , C i q ď a p i ´Dq 2 `pm i ´Dq 2 ,
which yields for the normalized geodesic distance:

ξ sq " d sq pA, Bq L ď Z `řN i"1 a p i ´Dq 2 `pm i ´Dq 2 ř N i"1 i , ( 4.26)
with Z " |C N B|, L " |AB|. For squares, Eq. (4.15) now takes the form:

P t i ď u " 1 ´p1 ´f q 4α 3 rp {Dq 3{2 ´1s ,
and the average of the i reads:

1

N N ÿ i"1 i D « 1 `2 3 p1 ´f q ´4α 3 E 1{3 ˆ´4α logp1 ´f q 3 ˙(4.27) " Γ `2 3 p? 6αf q 2{3 `Opf 1{3 q, f Ñ 0. (4.28)
The mean of the term a p i ´Dq 2 `pm i ´Dq 2 occurring in (4.26) is expressed as a double integral like in Sec. (4.2.2). We develop it for i " D and integrate over i and m i :

1 DN N ÿ i"1 a p i ´Dq 2 `pm i ´Dq 2 " Γ `2 3 ?6αf ˘2{3 `ˆα 2 6 ´1˙`O pf 1{3 q. (4.29)
We neglect Z and choose α " a 3{2:

ξ sq ď 1 ´35{3 4Γ `2 3 ˘f 2{3 `opf 2{3 q « 1 ´1.1521f 2{3 . ( 4.30)
This correction is smaller than that derived for the disks model. We emphasize that this model is anisotropic and that the geodesics are directed parallel to the sides of the squares. More generally, it is clear that, in a Boolean model with non-random connected compact grain, the minimal path used in the present bound depends, to leading-order term in the dilute limit, solely on the width of the grain, in the direction of propagation of the geodesics. The "width" of a shape is the distance between parallel lines (transverse to the direction of propagation) bounding it. The mean width, taken over all orientations, is the perimeter of the convex hull of the grain divided by π [START_REF] Schneider | Convex bodies: the Brunn-Minkowski theory[END_REF].

For instance, consider a Boolean model made of grains of surface A g and width w g in the direction of propagation of the geodesics. The width w g is the highest difference between the coordinates of two points in the grain, measured along the direction of propagation. The upperbound in the dilute limit reads:

ξ c ď 1 ´35{3 4Γ `2 3 ˘ˆw 2 g f A g ˙2{3 `opf 2{3 q, (4.31)
where f {A g is the density of grains (number of grains per unit surface). This result holds for arbitrary compact, non-random grain. Note that since the convex hull of a shape has the same width in all directions, bound (4.31) is unchanged when a grain is replaced by its convex hull. In the case of a set of parallel segments (modeling, e.g., cracks), (4.31) reduces to 1 ´ξc ě g 0 η 2{3 where η is the crack density parameter (2.12) and g 0 depends on the orientation of the crack with respect to the direction of propagation of the geodesics.

The homogenized metric in bound (4.31) is isotropic for particles of constant width [96, Chap. 6.7], such as a Reuleaux triangle. By Barbier's theorem [START_REF] Ew Weisstein | CRC concise encyclopedia of mathematics[END_REF], the bound then reduces to:

ξ c ď 1 ´35{3 4Γ `2 3 ˘ˆp 2 g f A g π 2

˙2{3

`opf 2{3 q, (4.32)

where p g is the perimeter of the grain. The shape of given constant width with smallest area is the Reuleaux triangle [START_REF] Chakerian | Convex Bodies of Constant Width[END_REF]. Therefore, for a given surface fraction f , the Reuleaux triangle is also the shape minimizing bound (4.32) among particles of constant width. The shape of constant width maximizing the same bound is the disk, for fixed surface fraction f .

Boolean set of disks with non-zero distance function inside the disks

In this section, we consider a Boolean model of disks with the following modified distance function:

d p pA, Bq " inf pPK ż 1 0 dt χ p ppptqq ||B t pptq|| , ( 4.33
)

χ p pM q " " p if M lies inside a disk, 1 otherwise,
where 0 ď p ă 1 is the cost associated to the distance in the disks. The distance function d in (4.1) is recovered when p " 0. We consider a similar path pA; C 1 ; ...; C N ; Bq as in Sec. (4.2.2), defined by:

|C i`1 1 ´Ci 1 | " inf |C 1 ´Ci 1 |; C a disk center; C 1 ą C i 1 `D, |C 2 ´Ci 2 | ď α b D|C 1 ´Ci 1 | * . ( 4
ξ p " d p pA, Bq L ď ř N i"1 " a 2 i `m2 i ´p1 ´pqD ı ř N i"1 i , ( 4.35)
with

Z " |C N B|, i " C i 1 ´Ci´1 1 , m i " C i 2 ´Ci´1 2 , i ě D, |m i | ď α ? i D
and α ą 0. Note that the path pA; C 1 ; ...; C N ; Bq used to derive bound (4.35) consists in a set of segments joining the disks centers. Most geodesics will not pass through disk centers when p ą 0. Nevertheless, in the dilute regime considered here we expect |m i | ! i so that bound (4.35) should be a very good estimate of the length of the path pA; C 1 ; ...; C N ; Bq . The asymptotic expansions for the means of the i and of the quantity m 2 i { i are the same as in (4.16) and (4.17). This yields:

ξ p ď 1 ´ˆ2α 3π ˙2{3 6p1 ´pq ´α2 Γ `2 3 ˘f 2{3 `opf 2{3 q, (4.36
)

" 1 ´3p1 ´pq 4{3 Γ `2 3 ˘ˆ3 2π ˙2{3 f 2{3 `opf 2{3 q « 1.3534p1 ´pq 4{3 f 2{3 , ( 4.37)
with α " a 3p1 ´pq{2. The upper-bound above is sharper than the trivial bound ξ p ď 1 ´p1 ´pqf in the domain f ! 1 ´p.

Multiscale Boolean set of disks

Consider first two Boolean sets of disks with constant diameter, denoted M 1 and M 2 . The disks of set M 1 have constant diameter D 1 and that of set M 2 have diameter D 2 ! D 1 . The centers of the disks in the Boolean sets M 1 and M 2 follow a homogeneous Poisson point process. We denote by f 1 and f 2 the surface fractions of sets M 1 and M 2 respectively and assume

f 1 ! 1, f 2 ! 1.
Hereafter we consider the intersection of the two sets M " M 1 X M 2 which is a two-scales random set with surface fraction f " f 1 f 2 . The model is assumed "symmetric" so that f 1 " f 2 " ? f and the distance function defined as in (4.1).

Accordingly to (4.20), the distance dpA 1 , B 1 q between two points A 1 and B 1 that lie in a disk contained in M 1 admits the following upper bound:

dpA 1 , B 1 q L 1 ď 1 ´3 Γ `2 3 ˘ˆ3 2π ˙2{3 f 2{3 2 `opf 2{3 2 q, when L 1 " |A 1 B 1 | " D 2 .
ξ ms " dpA, Bq L ď 1 ´335{9 p2πq 14{9 Γ `2 3 ˘7{3 f 7{9 `opf 7{9 q « 1 ´2.0261f 7{9 . ( 4.38)
This correction is smaller than that derived in the one-scale model (4.20) and indicates that clustering tend to "constraint" the shortest paths and increase their lengths. Similarly, the 2D periodic model, which has a very homogeneous spatial distribution of voids, has an exponent 1{2 and its geodesics are shorter than in the one-scale Boolean set of disks.

Boolean sets in three dimensions or more 4.4.1 Boolean set of spheres

In this section, we study a Boolean set of spheres of volume fraction f . The distance function d 3D is defined in 3D as in (4.1):

d 3D pA, Bq " inf pPK ż 1 0 dt χ 3D ppptqq ||B t pptq|| , ( 4.39)
with 1 ´χ3D is the indicator of the Boolean set of spheres and K is given by (4.2). We define the path pC 0 " A; C 1 ; ...; C N ; C N `1 " Bq by:

|C i`1 1 ´Ci 1 | " inf |C 1 ´Ci 1 |; C a sphere center; C 1 ą C i 1 `D, |C ´C1 | ď α b D|C 1 ´Ci 1 | * , ( 4.40)
where C 1 is the orthogonal projection of C onto the line pC i ; C i `e1 q and α is a constant to be optimized on. As in Sec. (4.2), the axis e 1 is aligned with the line passing by A and B. We set

i " C i 1 ´Ci´1 1 and m i " |C i C i 1 | so that |C i´1 C i | 2 " 2 i `m2 i .
The path (4.40) leads to the bound:

ξ 3D " d 3D pA, Bq L ď ř N i"1 ´a 2 i `m2 i ´D¯`Z ř N i"1 i , ( 4.41)
with Z " |C N B|. Using the Choquet capacity [START_REF] Matheron | Random sets theory and its applications to stereology[END_REF], the variables i P rD; 8q follow the cumulative probability function according t:

P t i ď u " 1 ´p1 ´f q 3α 2 rp {Dq 2 ´1s . (4.42)
and so, for the mean of the i :

1 DN N ÿ i"1 i « 1 `p1 ´f q ´3α 2 2α c π ´3 logp1 ´f q erfc ´αa ´3 logp1 ´f q " 1 2α c π 3f `Op a f q, f Ñ 0, (4.43)
where erfcpzq " 2{ ? π ş 8 z dt e ´t2 is the complementary error function. The mean of the quantity a 2 i

`m2

i reads:

1

N N ÿ i"1 a 2 i `m2 i D « ż 8 "D ż α ? D m"0 ? 2 `m2 P t ď i ď `d u 2mdm α 2 D 2 " ´ż 8
x"1

dx 4x 3{2 logp1 ´f qp1 ´f q 3α 2 px 2 ´1q " px `α2 q 3{2 ´x3{2 ‰ " 1 2α c π 3f `α2 4 `Op a f q, (4.44)
where the expression behind the integral has been expanded for x Ñ 8. With α " 2{ ? 3 and Z ! 1:

ξ 3D ď 1 ´8 3 ? π a f `op a f q « 1 ´1.5045 a f , f Ñ 0. (4.45)
The number of "possible choices" for picking C i`1 knowing C i is greater in 3D than in 2D and results in a lower exponent in 3D.

Non-zero distance function in spheres

The reasoning above extends to a distance function d 3D p which is non-zero in the spheres. Define the distance d 3D p as in (4.39) with χ 3D replaced by:

χ 3D p pM q " " p if M lies inside a sphere, 1 otherwise, (4.46)
where 0 ď p ă 1 is a parameter. The following upper-bound on the normalized shortest paths is derived:

ξ 3D p " d 3D p pA, Bq |AB| ď 1 ´8p1 ´pq 3{2 3 ? π a f `op a f q, f Ñ 0. (4.47)
Again, this bound is non-trivial when f ! 1 ´p.

Multiscale Boolean set of spheres

The two-scales "symmetric" Boolean set of spheres is constructed similarly as in Sec. (4.3.3). The latter is the intersection of two Boolean models of spheres with scale separation. The indicator function χ 3D ms of the two-scales Boolean model is used to define the distance d 3D ms as in (4.39). The normalized shortest path is expanded in the dilute limit as:

ξ 3D ms " d 3D ms pA, Bq |AB| ď 1 ´ˆ8 3 ? π ˙5{2 f 5{8 `opf 5{8 q « 1 ´2.7764f 5{8 , f Ñ 0. (4.48)
Our conclusions are the same as that given in 2D. At fixed volume fraction of pores, geodesics are higher in the two-scales symmetric model than in the one-scale Boolean model. The shortest geodesics are found for the 3D periodic model, in which the spatial distribution of voids is very homogeneous and the voids well-separated.

Boolean model of hyperspheres

In this section we consider a Boolean model of hyperspheres in dimension d ě 2. The geodesic distance d dD and path pA; C 1 ; ...; C N ; Bq are defined as in (4.39) and (4.40) with spheres replaced by hyperspheres. A bound for the geodesic is given by the path from

A to B. With i " C i 1 ´Ci´1 1 , m i " |C i C i 1 |, Z " |C N B|
and C 1 the orthogonal projection of C onto the line pC i ; C i `e1 q, we have:

ξ dD " d dD pA, Bq L ď ř N i"1 ´a 2 i `m2 i ´D¯`Z ř N i"1 i « 1 ´´Z `řN i"1 ´D ´m2 i 2 i řN i"1 i . ( 4.49)
Denote by π d pDq the volume of the hyperdimensional ball of diameter D and V that of the domain:

! C; D ď C 1 ď , 0 ď |C 1 | ď α ? D
) .

The probability law for the i P rD, 8p depends on V by:

P t i ď u " 1 ´p1 ´f q V π d pDq " 1 ´p1 ´f q 2 d`1 α d´1 Γ `1 `d 2 pd `1q ? πΓ `1`d 2 ˘«ˆ D ˙d`1 2 ´1ff , ( 4.50)
which provides the sum:

1 DN N ÿ i"1 m 2 i i « ż ěD, mďα ? D P t ď i ď `d u pd ´1qm d dm α d´1 pD q d`1 2 " α 2 d ´1 d `1 . (4.51)
The mean of the i , approximated by ş ěD P t ď i ď `d u, is determined using the symbolic solver Mathematica [262]. The expression involves the function Γ as well as incomplete Γ functions (not shown). Carrying out a Taylor expansion of the latter and optimizing on α yield α " a p1 `dq{d. Finally:

ξ dD ď 1 ´pd `1q 1`3d 1`d Γ `2 1`d ˘" Γ `1 `d 2 2? πd d Γ `3`d 2 ˘ 2 1`d f 2 1`d `opf 2 1`d q, f Ñ 0. (4.52)
The above generalizes (4.20) and (4.45). Taking successively the limits f Ñ 0 and d Ñ 8:

ξ dD ď 1 ´2 " 1 ´logp2dπq ´2p1 `γq d  f 2 1`d " 1 ´˜f a f p ¸2{d (4.53)
where γ « 0.5772 is Euler's constant and f p " 2 ´d is the asymptotic percolation threshold in dimension d " 1 [START_REF] Torquato | Effect of dimensionality on the continuum percolation of overlapping hyperspheres and hypercubes[END_REF].

Conclusion

Powerlaws with fractional exponents 2{3 and 1{2 have been derived for the lowest-order corrections to the lengths of geodesics in 2D and 3D Boolean models of discs and spheres, respectively. The method is general and provides an upper-bound with lowest-order correction " f

2 d´1 in dimension d ě 2.
The bounds obtained for multiscale models, which scale as " f 7{9 in 2D and " f 5{8 indicate lower variations of the geodesics near the point f " 0. These results underline that the the singularities for the geodesics are small for highly-heterogeneous dispersion of particles, and high when the dispersion is homogeneous. This phenomenon will be investigated in more details in Chap. (6).

4.A Algorithm for computing minimal paths in porous media

Our numerical algorithm for computing shortest paths in a model of disks is described here. Assume domain Ω contains N disks of centers C i and radii R i (i " 1, ...., N ). Set d ij " max p0; |C i ´Cj | ´Ri ´Rj q where | ' | is the Euclidean distance. The distances d i " dpA, C i q between a given point A and any disk is computed by the following algorithm, described hereafter.

(i) Set d i :" max p0; |C i ´A| ´Ri q.

(ii) For all i, j, set d i :" min pd i , d j `dij q.

(iii) If any value of d i has been changed in step (ii) GOTO (ii), otherwise STOP.

Note that distances d ij are computed on the fly and need not be saved in memory, and that only the distances d i are saved. This algorithm nevertheless requires OpN 2 q computations at each step and becomes impractical for certain Boolean sets with inhomogeneous Poisson point process, such as multiscale random sets. To increase its speed, disk centers are sorted beforehand with respect to the distance of propagation |C i ´A|, and the distance d i is updated by checking disks C j in an interval i ´N0 ď j ď i `N0 , while we let N 0 increases, until N 0 " N . The justification rests on the fact that the distance function usually increases with |C i ´A| and that d i is not usually updated when a disk far from C i is checked. The new algorithm reads:

(i) Label each disk with a number so that |C i ´A| increases with the index i.

(ii) Set d i :" max p0; |C i ´A| ´Ri q and N 0 " 5 ? N .

(iii) For i " 1, ..., N , and for j " maxp1, i´N 0 q, ..., minpN, i`N 0 q, set d i :" min pd i , d j `dij q.

(iv) If any value of d i has been changed in step (iii) GOTO (iii), otherwise continue to next step.

(v) if N 0 ă N set N 0 :" 2M and GOTO (iii), otherwise STOP.

The initial value of N 0 is so that, for a homogeneous distribution of inclusions, on average the 5 disks closest to each disk are checked. The results given by the algorithm is illustrated Fig.

Chapter 5

Effective behavior and geodesics: lattice vs. continuum

In this chapter, we investigate the effective conductivity of strongly nonlinear media in two dimensions. The nonlinear behavior is characterized by a threshold value for the maximal absolute current. Our main focus is on random media containing an infinitesimal proportion f ! 1 of insulating phase. We first consider a random conducting network on a square grid and establish a relationship between the length of minimal paths spanning the network and the network's effective response. In the dilute limit f ! 1, the network's effective conductivity scales, to leading-order correction in f , as " f ν with ν " 1 or ν " 1{2, depending on the direction of the applied field with respect to the grid. Second, we introduce coupling between local bonds, and observe an exponent ν « 2{3, in agreement with the upper-bound for the length of geodesics spanning random media in the continuum. We argue that ν " 2{3 for random composites in the continuum with homogeneously-distributed, monodisperse particles, in two dimensions.

Outline The present chapter is articulated around three main sections. A model problem in a random square lattice is formulated in Sec. (5.1). In Sec. (5.2), a scaling law is established for the effective conductivity with respect to f , the proportion of broken bonds, in the limiting case f Ñ 0. In Sec. (5.3), we introduce local coupling between the bonds and draw comparisons with the problem in the continuum. We conclude in Sec. 5.4.

Nonlinear resistor network

Consider a resistor network on a square grid (Fig. 5.1b) with nodes x " px 1 , x 2 q P Z 2 . Each node is linked to its four closest neighbours by bonds oriented along e 1 and e 2 (see Fig. 5.1a). At each node x lies a potential φpxq, and along each bond (x, x `ei q lies component J i pxq of the current and component E i pxq of the electric field. Following Kirchhoff's law:

2 ÿ i"1
rJ i pxq ´Ji px ´ei qs " 0, E i pxq " ´rφpx `ei q ´φpxqs , (

for all x and i " 1, 2. Each bond in the resistor network is either insulating, in which case J i pxq " 0, or conducting. The conducting bonds follow the nonlinear constitutive law:

J i pxq " " σE i pxq if σ|E i | ď J 0 , J 0 signpE i pxqq if σ|E i | ą J 0 , (5.2)
where signp'q " '{| ' | is the sign function, J 0 ą 0 is the yield current and σ ą 0 is a parameter used for numerical regularization (see Fig. 5.1c). In the limit σ Ñ 8, Eq. (5.2) reads:

J i " J 0 if E i ą 0, (5.3a) J i " ´J0 if E i ă 0, (5.3b
)

|J i | ă J 0 if E i " 0. (5.3c)
This constitutive law is obtained as the limiting behavior of "power-law materials". This class of materials, extensively studied in the context of nonlinear homogenization theories [START_REF] Castañeda | Nonlinear composites[END_REF], is characterized by local potentials which are power-law of the electric or current fields. In the present case, referred to as "strongly nonlinear", the potential corresponding to the constitutive law (5.3) is not strictly convex, and fields may localize along critical paths [START_REF] Duxbury | Critical manifolds in non-linear response of complex materials[END_REF]. For numerical purposes, we assume that the medium is periodic in the two directions and is given by its elementary cell Ω " r0; M ´1s 2 made of M 2 bonds. A random configuration of Ω is depicted in Fig. 5.1b for M " 16. In domain Ω, the network is made up of two phases, the phase label following a Bernouilli distribution, that is, a bond has constitutive law (5.2) with probability 1 ´f and is insulating with probability f . The vector fields E and J are Ω-periodic whereas the potential is the sum of a periodic and linear function:

φ(x) φ(x+e 2 ) E 1 (x) E 2 (x) φ(x+e 1 ) J 1 (x) J 2 (x) E 1 (x+e 2 ) J 1 (x+e 2 ) E 1 (x+e 1 ) J 1 (x+e 1 ) 0 E i (x) -J 0 0 J 0 J i (x) J 0 -J 0 (a) (b) (c)
φpxq " ´E ¨x `φ˚p xq, φ ˚#, (5.4)
where E is a remote field applied at infinity and # denotes doubly-periodic fields. Problem (5.1)-(5.2)-(5.4) is solved exactly on the finite domain Ω using the Fourier-based method with "discrete"

Green operator appropriate to resistor networks [START_REF] Willot | Fourier-based schemes with modified Green operator for computing the electrical response of heterogeneous media with accurate local fields[END_REF]. As for the iterative algorithm, we make use of the so-called "augmented Lagrangian" scheme [START_REF] Michel | A computational scheme for linear and non-linear composites with arbitrary phase contrast[END_REF]. When applied to materials with nonlinear threshold law (5.2), this scheme requires one to solve the equation in E:

σ 0 E `J pEq " P , (

for arbitrary P . Making use of (5.2), the solution to this problem is straightforward. In the above, σ 0 is the reference conductivity of the homogeneous medium, which we choose equal to 1. We also choose J 0 " 1 and restrict ourselves to |E| " 1, so that the problem depends on σ, f , M and the direction of the applied field E. Convergence is obtained when current conservation is observed at all nodes (Eq. 5.1). Our convergence criterion is the L 2 -norm of the divergence of the current field:

|divJ | " ˜ÿ xPΩ 2 ÿ i"1 |J i pxq ´Ji px ´ei q| 2 ¸1{2 ă η, ( 5.6)
with η " 10 ´10 . Convergence is somehow erratic but nevertheless obtained (Fig. 5.2a). Define the effective conductivity tensor as:

J " xJ pxqy Ω " χ ¨E, (5.7)
where x'y Ω denotes a spatial mean over Ω. We also define the scalar effective conductivity, that depends on the orientation of the applied field:

χ 0 " J ¨E |E| 2 .
(5.8)

We seek for high values of the slope at origin σ so that χ 0 represents the effective yield current.

In the rest of this work, we set σ " 10, so that the effective behavior has reached its yield value when xEy " E (Fig. 5.2b, with E " e 1 ).

Fourier-base numerical results

Using σ " 10, the effective conductivity is computed on grids of increasing size along each dimension M " 64, 128 and 256 and insulator fraction f in the range r0; f c s where f c " 1{2 is the bond-percolation threshold for square lattices (Fig. 5.3a). We observe that the effective conductivity χ 0 for all FFT data point on this graph is of the form k{M where 0 ď k ď M is an integer, up to an error of less than 10 ´7 (Fig. 5.3b). A connection may be established between this property and the onset of a minimal path (of integer length) along which the electric field localizes. This is illustrated in Figs. (5.4) which shows component E 1 of the electric field, represented as an image (Fig. 5.4a), as well as the grid itself (Fig. 5.4b). On the dual graph, there exists two paths of minimal length (Fig. 5.4b) that span the medium in the direction e 2 transverse to the applied field (horizontal on the image). The length of a path is defined by counting a unit cost for crossing a conducting bond and 0 for an insulating bond, thereby minimizing the number of conducting bonds crossed by the path. Using this definition, the two minimal paths shown in Fig. (5.4b) have normalized length ξ " 17{M and we notice that χ 0 " ξJ 0 . It is obvious that ξJ 0 is an upper-bound of χ 0 . Conversely, the existence of a divergence-free current field with mean ξJ 0 is a consequence of the "max-flow min-cut" theorem [START_REF] Dantzig | On the max flow min cut theorem of networks[END_REF]. Hence, ξ " χ 0 {J 0 . The non-unicity of the electric field is a consequence, in general, of the non-unicity of minimal paths. Assume that there is a closed loop in the set of minimal paths on the dual graph such as, for instance, two minimal paths joining at infinity. Along the bonds crossing the loop, the current field is J 0 and the electric field is larger than, or equal to, J 0 {σ. Therefore, if the electric component along any of these bonds is larger than J 0 {σ `δE with δE ą 0, a solution of the problem may be defined by adding a quantity smaller than δE to the potential φpxq at each node x located in the interior of the loop. The electric fields E should be different in each solution, while the current field J , and the overall energy p1{2q

ř Ω E ¨J , is unique. The length of minimal paths in random lattices is closely related to the "time constant" in firstpassage percolation theory [START_REF] Kesten | Percolation theory and first-passage percolation[END_REF]. In this problem, domain that grows with respect to a discrete time t is considered. Growth at each time step is controlled by the value along each bond at the frontier of the domain. After a long time (t Ñ 8), the region asymptotically defines a limiting shape [START_REF] Hoffman | Geodesics in first passage percolation[END_REF] which in general, is not a ball [START_REF] Durrett | The shape of the limit set in Richardson's growth model[END_REF], due to the anisotropy of the lattice. Thus, if spx, x 1 q is the minimal length between two points x and x 1 , we expect spx, x 1 q{|x ´x1 | to converge to a finite value when |x ´x1 | Ñ 8, which depends on the direction of x ´x1 . Equivalently, χ 0 depends on the loading direction E in the square lattice, as expected. We now perform FFT computations of the effective conductivity χ 0 for small values of f , equal to 10 ´4, 10 ´3, 10 ´2 and 10 ´1 (Fig. 5.5). A fit of the data provides χ 0 « 1 ´?f , with nearly unit prefactor, when E is oriented along direction e 1 `e2 (light gray symbols). FFT results obtained for a macroscopic field oriented along e 1 , are less conclusive (black symbols) but point nevertheless to a linear correction χ 0 « 1 ´2.7f . As pointed out by Roux & François [START_REF] Roux | A simple model for ductile fracture of porous materials[END_REF], the scaling law correction " f 1{2 in the p1, 1q direction is related to the rugosity (width in the transverse direction) of minimal paths. Suppose that a path of minimal length passes through two points x and x 1 far from each other with x ´x1 parallel to e 1 `e2 (Fig. 5.6). In the absence of broken bonds, the length of the two gray paths are the same. Therefore, if any broken bond appears inside the square rx 1 ; x 1 1 s ˆrx 2 ; x 1 2 s the length of the path will be reduced by 1. Accordingly, the length of the minimal path normalized by |x 1 ´x| will be reduced by 1{p ? 2|x 1 1 ´x1 |q whereas f increases by about 1{L 2 , thus the normalized length of the minimal path change by a factor " ? f . In the p1, 0q direction, minimal paths are mostly flat [START_REF] Derrida | Interface energy in random systems[END_REF][START_REF] Roux | A simple model for ductile fracture of porous materials[END_REF], therefore, the leading-order correction to χ 0 is expected to be linear (" f ), as would be obtained for a straight line.

1-χ 0 M=512 M=1024 1-χ 0 =2.7f 1-χ 0 =f 1/2

Network with local coupling between bonds

Hereafter we consider a resistor network that may be used as a model for a material in the continuum. The elementary cell Ω is now made of a set of M ˆM pixels, rather than nodes. The electric and current field Epxq and J pxq, and the potential field φpxq are defined in each pixel. We keep the conservation and admissibility equations (5.1) and (5.4), i.e., we use finite differences, but consider instead of (5.2) the constitutive law:

J i pxq " " σE i pxq if σ|Epxq| ď J 0 , J 0 E i pxq{|Epxq| if σ|Epxq| ą J 0 , (5.9)
where |Epxq| 2 " E 1 pxq 2 `E2 pxq 2 . Problem (5.1)-(5.4)-(5.9) is that of a random resistor network with local coupling between bonds pointing from the same node. As previously we solve this problem numerically using FFT computations, carried out using σ " 10, this time on grids of 2048 2 and 4096 2 voxels. The augmented Lagrangian scheme is found to be somehow slow to deal with this problem, and we turn to the "accelerated scheme" [START_REF] Eyre | A fast numerical scheme for computing the response of composites using grid refinement[END_REF] with "discrete" Green operator [START_REF] Willot | Fourier-based schemes with modified Green operator for computing the electrical response of heterogeneous media with accurate local fields[END_REF].

The accelerated scheme was originally devised for linear behavior. However, it is straightforward to extend it to nonlinear behavior, provided one is able to invert an equation of the type (5.5). We choose σ 0 " 0.01 for the conductivity of the reference medium. The effective conductivity χ 0 is again defined by (5.8). We consider the limiting case of a dilute concentration of insulators. FFT results for the effective behavior are represented in Fig. log-log plot. The leading-order term in f exhibits a power-law scaling with exponent « 2{3. This scaling law is consistent with the interpretation of the effective conductivity in terms of minimal path, and the bound (4.20) obtained in the continuum.

Assume now that the constitutive law in each phase is of type (5.9) with yield current J 0 in the matrix and J 1 in inclusions, i.e.:

J " minpσ|E|, J c q E |E| , ( 5.10)
with J c " 1 in the matrix and J c " J 1 (0 ď J 1 ď 1) in inclusions. The effective conductivity of such material may be compared to bound (4.37) appropriate to a Boolean model in the continuum with finite contrast, i.e.:

ξ ď 1 ´3p1 ´J1 q 4{3 Γ `2 3 ˘ˆ3f 2π ˙2{3 `Opf 4{3 q « 1 ´1.3534p1 ´J1 q 4{3 f 2{3 , ( 5.11)
in the dilute limit f Ñ 0. We carry out FFT computations of the effective conductivity of a random network containing two nonlinear phases obeying (5.9). Results, shown in Fig. (5.8), are compared to the analytical bound (5.11). The estimate ξ « 1 ´1.6p1 ´J1 q 4{3 f 2{3 is also represented as a dashed line in Fig. (5.8). The bound (5.11) appears meaningful as it displays the same concavity as the FFT data, although FFT results are quite sensitive to the size of the elementary cell Ω.

Conclusion

The present chapter examined how one may characterize the effective behavior of a binary mixture made of an insulating phase embedded in a conducting nonlinear phase characterized by a yield current. The problem is addressed in the context of a random resistor network in two dimensions.

χ 0 « 1 ´1.6p1 ´J1 q 4{3 f 2{3 (see text).
We observe that the network's effective conductivity is given by the length of the minimal path defined on the dual lattice. The path spans the network in the direction transverse to the applied field and its length is weighted by the yield current along each bond. For a dilute concentration of insulating phase, the effective conductivity exhibits singular powerlaw behavior when the applied macroscopic field is oriented transverse to the directions of the lattice, or if local coupling, that mimic an isotropic law, is introduced between the nodes and the applied field is parallel to the lattice directions. In these two cases, the responses are characterized by a correction " f ν with exponent ν equal to 1{2 and 2{3 respectively. Furthermore, we derive an upper-bound for the length of geodesics spanning a composite in the continuum, a Boolean set of disks. This bound exhibits an exponent 2{3 in the dilute limit, consistently with numerical data. These results suggest a leading-order correction " f 2{3 for the effective conductivity of nonlinear composite materials containing a dilute concentration f of homogeneously-distributed, monodisperse, insulating particles. Other problems of interest include hierarchical microgeometries, some of which are exactly solvable [START_REF] Idiart | Estimates for two-phase nonlinear conductors via iterated homogenization[END_REF].

In the dual problem of an ideally-plastic media reinforced by rigid particles, strong connections have also been reported between geodesics and shear bands [START_REF] Jeulin | On the geodesic property of strain field patterns in elastoplastic composites[END_REF], highlighting the links between effective flow stress, geodesics and the localization patterns of the shear bands [START_REF] Poliakov | Fractal plastic shear bands[END_REF].

Chapter 6

Homogenized metrics in planar, multiscale random sets

In this chapter, we investigate homogenized metrics in planar, multiscale random sets. The length of geodesics (minimal paths) spanning a stationary random sets of grains is considered. The pseudometric is zero in "pores" and infinite in "rigid obstacles", and equals the Euclidean distance in the embedding matrix. Accordingly, minimal paths tend to cross porous grains while avoiding rigid particles. Multiscale random structures based on Boolean sets of disk-shaped grains are considered, as well as random sequential adsorption models of squares, and we focus on the asymptotic limit of an infinitesimal density of grains. Our approach relies on numerical computations on the one hand, and on upper-bounds on the other hand. This work is motivated by the connection between geodesics and effective conducting behavior, outlined in the previous chapter for nonlinear resistor networks. Our results, of a purely-geometric nature, are compared to the predictions of nonlinear homogenization theories for the effective yield stress of particulate composites subjected to anti-plane shear, and to rigorous bounds established for the same problem.

Outline This chapter is organized as follows. In Sec. (6.1) we define various "porous" random models of disks in 2D, with emphasis on multiscale structures. In Sec. (6.2), the minimal path problem is defined, and numerical results are obtained for the porous case. Sec. (6.3) deals specifically with the dilute porosity limit. Sec. (6.4) is concerned by the dual problem of minimal paths spanning a medium containing a population of "rigid" grains. Analytical results related to nonlinear homogenization problems are discussed in (6.4.4). We conclude in Sec. (6.5).

Multiscale random structures

Our starting point is a Boolean model of equisized disks, denoted B E , with homogeneous Poisson intensity in the plane [START_REF] Stoyan | Stochastic geometry and its applications[END_REF][START_REF] Molchanov | Theory of random sets[END_REF]. A realization of B E is the union of translations G i of a primary grain G implanted along a set of germs g i which are Poisson-distributed in the plane [START_REF] Miles | On the homogeneous planar Poisson point process[END_REF]:

B E " 8 ď i"1 G i , G i " tx `gi ; x P Gu, (6.1)
where, for simplicity, we denote B E both the probabilistic model and one of its realization. The random Poisson point process pg i q ią0 is defined by its intensity θpxq, also known as a a regionalized density, which depends on the space variable x P R 2 . The quantity θpxqdS is the expectation (mean over realizations) of the number of germs g i contained in a domain of infinitesimal surface area dS centered around x. Additionally, the number of germs in two disjoint domains are independent random variables. These two properties completely define a Poisson point process [START_REF] Stoyan | Stochastic geometry and its applications[END_REF]. Furthermore, the Poisson point process is called "homogeneous" if θpxq is uniform in space (θpxq " cst).

In the present work, the primary grain G is a disk of radius R. The mean surface fraction 0 ď f ď 1 of the disks, averaged over the plane, is given by:

f " 1 ´e´θπR 2 . (6.2)
The above is a consequence of a more general result, known as the Choquet-Matheron-Kendall theorem [START_REF] Stoyan | Stochastic geometry and its applications[END_REF] which gives an explicit form of the capacity functional, i.e. the probability that a compact set K hits B E . The theorem, which involves morphological transforms of the primary grain with structuring element K (e.g. dilation, erosion), reduces to (6.2) when the convex K is a point, and is the basis of many other analytical results related to Boolean media. However the simple formula (6.2) may also be derived by elementary means (see "Avrami equation" [START_REF] Avrami | Kinetics of phase change. I General theory[END_REF]). Consider now a square domain Ω of surface area S containing a realization of the set B E X Ω, and denote n the number of germs (or disk centers) lying in Ω. The number n is a random variable which follows the discrete Poisson distribution law of mean θS: n " P pθSq.

(6.3)

In the dilute limit θ ! 1 (or f ! 1), Eq. (6.2) entails the obvious result f « θπR 2 « nπR 2 {S. The "representativity" of realizations of the model in Ω for estimating a statistical quantity may be defined by the asymptotic behavior of the variance of that quantity as S Ñ 8. For instance, the surface fraction of disks in domain Ω scales asymptotically as " 1{S when S Ñ 8 [START_REF] Is Molchanov | A limit theorem for scaled vacancies of the Boolean model[END_REF][START_REF] Lantuéjoul | Ergodicity and integral range[END_REF].

A realization of B E in a domain Ω is represented in Fig. (6.1a). The model parameters θ and R have been chosen so as to enforce, on average, n " 500 and f " 30%, referring to (6.2) and (6.3).

The Boolean model B E pθpxq, Rq made of disks of constant radius R and based on a homogeneous or inhomogeneous intensity θpxq serves as an ingredient to define the following "two-scale" models:

B P " B 1 Y B 2 , B C " B E pθ 2 1 B 1 pxq, R 2 q with B 1 " B E pθ 1 , R 1 q, B 2 " B E pθ 2 , R 2 q, (6.4)
where θ 1 , θ 2 ą 0 and R 1 ą R 2 ą 0 are parameters, 1 B 1 pxq is the characteristic function of the Boolean set B 1 , and B i " B E pθ i , R i q (i " 1, 2) designates independent random realizations of Boolean models of equisized discs.

The "polydisperse" model B P is according to (6.4) the union of two Boolean models of disks of radii R 1 and R 2 . It is also a Boolean model of randomly-sized grains with intensity θ 1 `θ2 (see Fig. 6.1b for an example). The "Cox-Boolean" model B C is built in two steps. In the first step, a realization of a Boolean set B 1 of disks of radius R 1 and homogeneous intensity θ 1 is simulated. In the second step, smaller-sized disks of radius R 2 ă R 1 are homogeneously implanted in domain B 1 . At the end of the process, the largest disks, of radius R 2 , are discarded. The point process for the disk centers in this later model is known as a Cox point process [START_REF] Møller | Structured spatio-temporal shot-noise Cox point process models, with a view to modelling forest fires[END_REF]. This procedure allows one to generate a set of grains aggregated into clusters (Fig. 6.1c, clusters shown in the embedded image). When R 2 ! R 1 , i.e. for widely-separated disk radii, models B P and B C can be described as two-scale random structures. The length scales R 1 and R 2 are related to the granulometry of the grains (model B P ) or to the typical sizes of grains and of grains aggregates (model B C ). The two models nevertheless depend on four parameters R 1 , R 2 , θ 1 and θ 2 . In the following, we specialize and parametrize them using quantities relevant to homogenization theories, in particular the mean disk surface fraction. Denote f 1 and f 2 the surface fractions of the sets B 1 and B 2 . The surface fractions f P and f C of models B P and B C read:

f P " f 1 `f2 ´f1 f 2 , f C « f 1 f 2 . (6.5)
In the above, the second equation is only an approximation. Equality between f C and f 1 f 2 would hold if B C had been defined as the intersection B C " B 1 X B 2 . This choice has not been made here because B 1 X B 2 is not a union of disks, and minimal paths in this set are much more complex to study analytically and numerically. Nevertheless, relation f C " f 1 f 2 is asymptotically exact in the limiting case of scale separation R 1 {R 2 Ñ 8.

Standard relations (Eq. 6.2) entail:

θ 1 " ´logp1 ´f1 q πR 2 1 " n 1 S , θ 2 " ´log p1 ´f2 q πR 2 2 " n 2 S , (6.6)
where n 1 and n 2 are the number of germs implanted in domain Ω for models B 1 and for B 2 . Consider the polydisperse model B P . A proportion of disks of radius R 2 which are entirely contained in B 1 are not observable from a realization of B P , once the largest disks are implanted. When f 1 and f 2 are small and R 1 ! R 2 , the total number of observable disks n is about:

n " n 1 1 `n1 2 , n 1 1 « n 1 , n 1 2 « n 2 p1 ´f1 q, (6.7)
where n 1 1 and n 1 2 are the number of observable disks of radius R 1 and R 2 . In the following, we choose the parameters θ 1 , θ 2 , R 1 and R 2 for model B P so as to impose f 1 « f 2 and n 1 1 « ? n.

The first condition aims to enforce equal effect of the two population of disks in terms of surface fraction of disks, which is the most important parameter in homogenization theories. The second condition is a compromise between two opposite goals: (i) achieving representativity of realizations of B 1 in Ω, and (ii) scale separation R 2 ! R 1 . Accordingly, we set: Model B C is now parametrized by three variables which represent the surface fraction of the disks (f C ), the surface fraction of clusters (f 1 ą f C ) and the mean number of disks (n). We also introduce a "symmetric" Cox-Boolean model, denoted B S , corresponding to model B C (Eq. 6.9) with the choice f 1 " f 2 " ? f C . This model depends on two parameters, 0 ď f C ď 1 and n. A realization of model B S is represented in Fig. (6.1c) with f C " 30%, n " 500.

R 2 1 " ´logp1 ´fP q 2π ? n S, R 2 2 "
The critical disk surface fraction at the percolation threshold is, for the Boolean model B E , f c « 0.6763 [START_REF] Quintanilla | Asymmetry in the percolation thresholds of fully penetrable disks with two different radii[END_REF]. Regarding model B P , it is clear that the disk percolate whenever maxpf 1 , f 2 q ě f c . The inverse proposition holds at scale separation, assuming R 2 is infinitesimal. Indeed, if f 2 ă f c , no connected path of length " R 1 exists in the complementary set of B 1 , hence percolation of the disks imply percolation of the largest disks. Accordingly, the percolation threshold for B P must be f P c " 2f c ´f 2 c « 0.8952. Similarly, disk percolation in model B C occurs if and only if minpf 1 , f 2 q ě f c , assuming scale separation, hence, f S c " f 2 c « 0.4574 for the disk percolation threshold in model B S .

Geodesics and homogenized distance

In the following, the length of a path is defined as the Euclidean length of the parts of the path located in the complementary set of the union of disks. The distance dpA, Bq between two points A and B is the minimal length of a path joining the two points, and we call geodesics such paths of minimal length.

The distance function in a stationary random (or periodic) medium may be homogenized, in the sense that it behaves, at sufficiently large scales, as the distance function of a homogeneous medium [START_REF] Oberman | Homogenization of metric Hamilton-Jacobi equations[END_REF]. If the medium is isotropic, as is the case for the models defined in Sec. (6.1), the homogenized distance function is necessarily isotropic. We denote ξ the limit distance rate lim tÑ8 dpA, A `tvq{t where A is any fixed point and v a unit vector. ξ More generally, we also make use of the homogenized distance function ξ 1 of models with "finite contrast p". As previously, the length of a path contained in the complementary set of the union of disks is its Euclidean length. However, when the path is inside disks, its length is p times its Euclidean length, instead of 0. When p " 0, ξ 1 " ξ, and, trivially, ξ 1 " 1 when p " 1. For any stationary (translation invariant) random material, a straight line provides the bound:

f c f c S f c P B S B C (f 1 =0.2) B P B E
ξ 1 ď 1 `pp ´1qf.
(6.10)

In the rest of this work, we denote ξ E , ξ P , ξ C and ξ S the homogenized distance for models B E , B P , B C and B S , respectively. The homogenized distance function relevant to model B E with "cost" p ą 0 in the disk is denoted ξ 1 E . We compute numerically distance ξ for various models of disks. Our algorithm, summarized in Appendix (4.A), does not rely on the discretization of sets of grains on a regular grid, as in fast marching [START_REF] Sethian | A fast marching level set method for monotonically advancing fronts[END_REF] or fast sweeping [START_REF] Zhao | A fast sweeping method for eikonal equations[END_REF] methods. Instead, the geodesics and the distance function are computed exactly, for a given realization, using the disk centers and radii as input. The geodesics shown as gray lines in Fig. (6.1) have been obtained using this method.

Remark that as a consequence of scale separation R 1 {R 2 " 8, distance ξ P is obtained from ξ E as: .11) where 1 ´?1 ´f represents the surface fractions f 1 " f 2 in Eq. (6.5).

ξ P pf q " " ξ E ´1 ´a1 ´f ¯ı2 , ( 6
Estimates of ξ vs. f , the overall disk surface fraction, are shown in Fig. (6.2) for models B E , B P , B C (with f 2 " 20%) and B S . The data has been obtained using two random realizations made of n " 10 5 disks. This value was deemed sufficient for a rough estimate, indeed estimates for n " 10 4 disks are similar while displaying larger fluctuations (not shown). Error bars, corresponding to standard deviation, are shown for all models except for B E , for clarity. The variances of ξ were found to be very small for this model and need not be represented. The percolation thresholds f c , f P c and f S c are indicated in the figure as dotted lines.

Distance ξ P (model B P ) is similar to that of the one-scale model B E (Fig. 6.2, left triangles). However, ξ P ą ξ for f ą 0.3, consistently with a higher percolation threshold f P c ą f c , whereas ξ P ă ξ at small surface fraction f ă 0.1. The numerical data for ξ P can as well be computed using the data for ξ E together with Eq. (6.11). For the sake of clarity, this second estimate has not been represented in Fig. (6.2), although it is probably more accurate. Nevertheless, the two sets of data are similar.

The distance function ξ S in model B S (Fig. 6.2, down triangles), is significantly lower than that of the one-scale model B E , for all values of f simulated. This is consistent with a lower percolation threshold f S c ă f c . As expected, the distance function ξ C (Fig. 6.2, diamonds), defined in the range 0 ď f ď f 1 (f 1 " 0.2), reduces to ξ E when f " f 2 . Although not visible in the figure, the curves for ξ C and ξ S also cross each other when f " f 2 1 " 0.04.

Porous media in the dilute limit 6.3.1 Two-scale random microstructures

In the limit of a vanishing surface fraction of disks f Ñ 0, the distance function ξ E in the one-scale model decays at a rate at least as high as " f 2{3 . Indeed, according to (4.20):

ξ E ď 1 ´αf 2{3 , α « 1.3534. (6.
12)

The relevance of this bound is confirmed by numerical results in Sec. (4.2.3) which indicate a powerlaw behavior with a similar exponent and slightly higher prefactor: At the largest scale, minimal paths in B C join clusters to one another. Assuming separation of length scales, we identify these minimal paths as that occurring in a one-scale Boolean model B E of equisized disks (the clusters) with finite contrast p (Sec. 6.2) where p is a positive "cost" associated to paths crossing the disks. The bound (4.37) extending (6.12) for such a model reads:

ξ E " 1 ´γf ν `opf ν q, γ « 1.85, ν « 0.
ξ 1 E ď 1 ´αp1 ´pq 4{3 f 2{3 1 , (6.14)
where 0 ď p ď 1. Bound (6.14) as well as (6.12) has been obtained by constructing a path and computing its length explicitly (see Chap. 4). These paths are represented by white lines in Figs. (6.1a,c). Observe that, depending on the value of p, bound (6.14) may be higher or lower than that in (6.10). Therefore, combining the two bounds, we obtain:

ξ 1 E ď 1 ´max ! p1 ´pqf 1 ; αp1 ´pq 4{3 f 2{3 1
) . (6.15)

Accordingly, bound (6.10) improves on bound (6.14) in the low-contrast case α 3 p1 ´pq ď f 1 . This is because bound (6.14) assumes that the minimal path is made of segment of lines joining disk centers. This condition becomes too restrictive in the low contrast case. To show this, let us consider the simple configuration of We now make use of (6.15) to compute a bound for minimal paths spanning B C . Assume separation of length scales and replace p with the r.h.s. expression in (6.12):

ξ C ď 1 ´max ! αf 2{3 2 f 1 ; α 7{3 f 8{9 2 f 2{3 1
) . (6.16)

When specialized to f 1 " cste ! 1, one obtains, replacing f 2 with f {f 1 : (6.17) with α ´6 « 0.16 (γ ´6 « 0.025). The above holds when the surface fraction of pores in clusters decreases at a much faster rate than the surface fraction of clusters. The scaling law (6.17a) is obtained when the minimal path is flat at the largest scale and "rough" inside clusters (Fig. 6.4a). In contrast, bound (6.17b) is obtained when the minimal path is rough both at the largest scale and inside clusters (Fig. 6.4b).

ξ C ď # 1 ´αf 1{3 1 f 2{3 if f ď f 5{2 1 α ´6, 1 ´α7{3 f ´2{9 1 f 8{9 if f ě f 5{2 1 α ´6,
When specialized to model B S , bound (6.10) becomes irrelevant, since f 1 " ? f is always smaller than α 3 p1 ´pq " α 4 f 1{3 . We thus obtain:

ξ S ď 1 ´α7{3 f 7{9 , (6.18)
a result identical to (4.38).

The numerical data obtained in Sec. (4.2.3) suggests the same scaling law as in (6.14), with prefactor γ instead of α:, i.e.

ξ 1 E « 1 ´γp1 ´pq 4{3 f 2{3 , f Ñ 0. (6.19)
This entails the following expressions for the dilute limit expansions of ξ C and ξ S , similar to (6.17

ξ C « # 1 ´γf 1{3 1 f 2{3 if f ď f 5{2 1 γ ´6 1 ´γ7{3 f ´2{9 1 f 8{9 if f ě f 5{2 1 γ ´6 (6.20a) ξ S « 1 ´γ7{3 f 7{9 , (6.20c)
with f Ñ 0. Finally, for the polydisperse model B P , Eqs. (6.11) and (6.13) require that:

ξ P « 1 ´21{3 γf 2{3 , f Ñ 0. (6.21)
The lowest-order correction in this asymptotic expansion is higher than that in (6.13). This is consistent with the data in Fig. 6.2 (Sec. 6.2) which suggests that ξ P ă ξ E for f ă 0.1.

In the following, we carry out numerical computations of the lengths ξ P , ξ C and ξ S of minimal paths spanning models B P , B C and B S . We focus on small surface fractions of disks, ranging from f " 10 ´6 to f " 10 ´1. The number of inclusion is set to n " 10 6 , and use is made of two random realizations for each data point. Our results are represented in Figs. [START_REF] Altendorf | Influence of the fiber geometry on the macroscopic elastic and thermal properties[END_REF].5a (polydisperse model, ξ P ), 6.5b (symmetric Cox-Boolean model, ξ S ) and 6.6 (Cox-Boolean model, ξ C ).

The numerical data for ξ P is in good agreement with the scaling law predicted by expansion (6.21) (see Fig. 6.5a, triangles). The data for ξ S , related to the cluster model B S is close to a powerlaw (Fig. 6.5b, triangles) with slope clearly in-between 2{3 and 1. A fit of the data points in the region f P r10 ´6; 10 ´2s provides ξ S « 4.16f 0.79 (Fig. 6.5a, solid black line) in excellent agreement with expansion (6.20c), shown as a light-gray solid line. The numerical data relevant to ξ C is represented in Fig. (6.6). The surface fraction of the clusters take on values f 1 " 0.01 (`signs) and f 1 " 0.02 (diamond symbols). The predictions of Eq. (6.20a) are shown as solid lines. Numerical data points exhibit a powerlaw behavior with exponent close to 8{9, as predicted by (6.20a 2 ), in the range f ě f 5{2 1 {γ 6 . In the highly-dilute regime f ! f 5{2 1 {γ 6 , numerical computations have been carried out using two configurations of n " 3 10 6 disks, for each value of f " 10 ´7 and f " 10 ´8. Results are less conclusive, because, the representativity of the simulated realizations, is insufficient in this case. Nevertheless, we observe that the correction 1 ´ξC is higher when f 1 " 0.02 (diamonds) compared to that obtained when f 1 " 0.01 (plus signs). The opposite tendency holds when f ď 10 ´6.

"Second-order" nonlinear homogenization theory

In this section, the results of the bounds on geodesics are compared to the predictions of the "non symmetric fully-optimized second-order" method, with strain potential, denoted SO here, and reported in [START_REF] Furer | A symmetric fully optimized second-order method for nonlinear homogenization[END_REF]Annex B]. Closed-form expressions are given, in this reference, in the ideally plastic limit. It is straightforward to expand these in the dilute limit f Ñ 0. We obtain, for a 2D medium:

ξ 1 pSOq E « 1 ´max α 1 p1 ´pq 4{3 f 2{3 ; p1 ´pqf ( , α 1 " 3 ˆ2´5{3 « 0.94, (6.22)
as f Ñ 0, where ξ

1 pSOq E ă 1 is the effective yield stress, normalized by the yield stress in the inclusions, and p is the yield stress in the inclusions divided by that in the matrix. Formula (6.22) is the same as (6.15), save for a prefactor. The second-order theory accordingly entails the same (f 1 " 0.01 for `signs, f " 0.02 for diamonds). Black and gray solid lines: scaling law (6.20a). Dashed lines, for comparison: scaling law (6.12) and linear correction " f . expressions as in (6.17) and (6.18), with α replaced by α 1 :

1-ξ C 1-ξ C (f 1 =0.01) 1-ξ C (f 1 =0.02) max(γf 1 1 _ 3 f 2 _ 3 ;γ 7 _ 3 f 1 -2 _ 9 f 8 _ 9 ) (f 1 =0.01) max(γf 1 1 _ 3 f 2 _ 3 ;γ 7 _ 3 f 1 -2 _ 9 f 8 _ 9 ) (f 1 =0.02) 17f γ f 2/3
ξ pSOq C « # 1 ´α1 f 1{3 1 f 2{3 if f ď f 5{2 1 α 1 ´6 1 ´α1 7{3 f ´2{9 1 f 8{9 if f ě f 5{2 1 α 1 ´6 (6.23a) ξ pSOq S « 1 ´α1 7{3 f 7{9 , ( 6.23c)
with f , f 1 Ñ 0. The predictions of the second-order theory, computed numerically for model ξ C in the dilute limit, are shown in Fig. (6.7) as symbols. The data is very close to the asymptotic formula (6.23a) (not shown), and also to bound (6.17), shown as solid lines in Fig. (6.7). The lower envelope of the curves, i.e. the set of points where the regime change occurs (see 6.17), is the curve of equation ξ C " α 9{5 f 4{5 , represented as a dotted grey line in Fig. (6.7). Along this curve, the effect of the pores is minimal, for a given porosity f .

Multiscale random microstructures

Eq. ( 6.20a) relevant to model B C assumes that f 1 is constant but very small. This limit is obtained by letting f 1 decrease with respect to f in a much slower way than f 2 , for instance by enforcing f 1 " f β , f 2 " f 1´β with β ! 1. Said otherwise, the dilute limit is obtained when, as f goes to 0, pores are replaced by the solid phase uniformly in each cluster. This is in contrast with the dilute limit of model B S , where both pores and entire clusters are removed as f Ñ 0, according to f 1 " f 2 " ? f . Other dilute limits arise when, more generally, f 1 " f β and f 2 " f 1´β , with 0 ď β ď 1. Eq. (6.16) then leads to:

1 ´ξC " f ν , ν " min " 2 `β 3 , 8 ´2β 9 * , ( 6.24)
where constant prefactors on the r.h.s. have been omitted for simplicity. When ν " p2 `βq{3, the geodesics are "flat" at the scale of the clusters, and display rugosity when ν " p8 ´2βq{9. The Solid gray lines: bound (6.17). Each gray-level refers to the same value of f 1 .

maximum value of ν is obtained when the two expressions for ν coincide, that is β " 2{5, ν " 4{5. Also observe that ν ě 2{3 with equality when β " 0, 1.

Let us now examine the prediction of the upper-bounds on geodesics for multiscale microstructures with more than two scales. Consider a N -scale Boolean microstructure analogous to B C . For instance, when N " 3, the microstructure is made of grains, aggregated into clusters, and the clusters themselves are aggregated into super-clusters. We denote f 1 the surface fraction of the clusters at the largest scale, f 2 at the next-to-largest scale etc. and f n the porosity inside the smallest clusters. Assume further that f 1 " f β 1 , ..., f N " f β N , with f " f 1 ...f N the overall porosity and β 1 `... `βN " 1. Making use of (6.15) one obtains the recurrence relation:

1 ´ξpNSq C " f ν 1 , ν N " 2β N {3, ν i " ν i`1 `2 3 β i `1 3 min tβ i ; ν i`1 u , 1 ď i ă N. (6.25)
for the leading-order correction to the geodesic length ξ pN Sq C

. We have that:

ν 1 " 2 3 `minpβ 1 , ν 2 q `... `minpβ N ´1, ν N q,
hence ν 1 ě 2{3, this value being attained when β 1 "

1 ´or β N " 1 ´. It is also clear that ν 1 ă 1.
The maximum value is obtained by taking β i " ν i`1 (i ă N ) and choosing β N so that ř i β i " 1. We find that β N " 3{p1 `2N q and ν 1 " 1{p1 `2´N q, a value that approaches 1 as N Ñ 8. For the "symmetric" model corresponding to β 1 " ... " β N " 1{N , it is straightforward to show that ν 1 " 1 ´4{p9N q, which again tends to 1 as N Ñ 8. A linear correction is accordingly recovered for these various media with infinitely-many scales.

Rigid grains in the dilute and highly-packed limit

In this section, we consider the dual problem of geodesics spanning a medium containing a set of "rigid" obstacles. The distance function is now infinite inside particles and geodesics are entirely contained in the complementary set of the inclusions. To simplify the analysis and numerical computations, we assume that the inclusions are a set of non-overlapping squares. Indeed, in such model, the minimal path in-between two points outside the squares consists in a continuous line made of a set of segments joining square corners, which are known a priori (Fig. 6.10a). We emphasize that the squares are aligned and not randomly-oriented, and so the model is anisotropic. Furthermore, the direction of propagation of the minimal paths are parallel to one of the axis of symmetry of the squares. The results in this section are therefore restricted to this particular direction. We mention nevertheless, that a study of a similar medium, the random checkerboard, suggests that the metric of that set is in fact isotropic [START_REF] Oberman | Homogenization of metric Hamilton-Jacobi equations[END_REF].

Random checkerboard

Square-shaped obstacles

Consider an infinite checkerboard, made of randomly-occupied cells, as depicted in Fig. (6.8a). Each cell is occupied, with probability f , by a square-shaped grain with side length D made of two closed and two open boundaries. More precisely, the grain are translations of the set s0; Drˆr0; Ds.

We define a path spanning the complementary set of the squares as shown in Fig. (6.8). In the figure, the frontiers of the squares are open along the vertical sides and closed along the horizontal ones. We seek for a minimal path oriented along the horizontal direction.

The "visibility" i from the corner of a square may be defined as the length of the longest ray starting from that point and oriented in the horizontal direction, which does not intersect a square or the frontier of a square. At its end, the ray necessarily hits two adjacent cells (right of Fig. 6.8). Around the end of the ray, we define a region of maximal size 1 i containing no occupied cells and made of two rows of cells on each side of the ray (Fig. 6.8a). Furthermore, we denote m i the length of the path, oriented in the vertical direction, which follows the frontier of the obstacle at the end of the ray. This length is the minimal in the up and down direction.

Let us now define a path that we will use to construct a bound. The path follows the ray along a distance i ´ 1 i , then follows an oblique direction along a distance a 1 2 i `D2 , and finally circumvent the obstacle following a path of length m i ´D. The square open boundaries in the transverse direction allow for configurations where i " D or 1 i " 0 and guarantee that the path is still in the complementary set of the squares.

The path normalized length reads:

ξ " ř N k"1 ´ i ´ 1 i `a 1 i 2 `D2 `mi ´Dř N k"1 i " 1 `Aa 1 i 2 `D2 ´ 1 i E i `xm i y i ´D x i y i . (6.26)
Since the cells are occupied independently of each other, one has:

P t i " kDu " f 2 p1 ´f 2 q k´1 ,
x i y i " D{f 2 , (6.27a) P tm i " kDu " f 2pk´1q p1 ´f 2 q, xm i y i " D{p1 ´f 2 q, (6.27b)

for k ě 1. For 1 i , one obtains the conditional probability: P t 1 i " jD| i " kDu " " 2f p1 ´f q 2j`1 p1 ´f 2 q ´j´1 , j " 0, ..., k ´2, p1 ´f q 2pk´1q p1 ´f 2 q 1´k , j " k ´1. (6.28) and so, making use of (6.27a):

P j " P t 1 i " jDu " f 2 p1 ´f q 2j `2f p1 ´f q 2j`1 , j ě 0, (6.29)

1 D B b 1 i 2 `D2 ´ 1 i F i " ÿ jě0 P j ´aj 2 `1 ´j¯" P 0 `p? 2 ´1qP 1 `ÿ jě2 P j ´aj 2 `1 ´j¯. (6.30)
Let us replace the term ? j 2 `1 in the r.h.s. by its Taylor expansion as j Ñ 8. The symbolic calculator [262] provides an exact summation of each term of the Taylor expansion up to order 20, involving "polylogarithmic" special functions [START_REF] Weisstein | Polylogarithm. From MathWorld-A Wolfram Web Resource[END_REF] (not explicited here). The resulting expressions for the r.h.s. of (6.30) admit, in the dilute limit f Ñ 0, the following expansion:

pC 0 ´log f qf `Op´f 2 log f q, (6.31)
where C 0 is a constant approaching C 0 « 1.0888 as the number of terms in the Taylor expansion increases. Thus:

ξ " 1 `pC 0 ´log f qf 3 `Op´f 4 log f q. (6.32)
To make sure that the path considered here may be used as a bound, we compute its transverse deviation. Denote n i " ˘mi the transverse distance, counted positive if the "next" point is "up" (as in Fig. 6.8) and negative otherwise. The transverse deviation of the path is ř N i"0 n i . By the central limit theorem: .33) where N denotes a normal distribution. The normalized deviation reads:

1 ? N N ÿ i"0 n i " N `0, xm 2 i y i ˘, xm 2 i y i " D 2 p1 `f 2 qp1 ´f 2 q ´2. ( 6
| ř N i"0 n i | ř N i"0 i " f 2 c 2 N π ? 1 `f 2 1 ´f 2 Ñ 0, N Ñ 8. (6.34)
Thus ξ ď ξ where ξ admits the asymptotic expansion (6.32).

Multiscale random checkerboards

Consider a two-scale random checkerboard which appears as in Fig. ( 6.8a) at the largest scale. Each occupied cell is itself a random checkerboard at much smaller scale. The occupied cells at the largest scale are therefore clusters, of surface fraction f 1 , whereas the surface fraction of cells at the smallest scale in each cluster is f 2 . The total surface fraction of the grains is therefore f 1 f 2 . Applying the above construction at the largest scale, one may consider a path that avoids completely all clusters, and therefore:

ξ ď 1 `p´log f 1 qf 3 1 , f 1 Ñ 0. (6.35)
Now, one may consider a nearly straight path at the largest scale (bound 6.10), and make use of bound (6.32) within each cluster:

ξ ď 1 `p´log f 2 qf 3 2 f 1 , f 2 Ñ 0. (6.36)
Assume that the surface fractions f 1 and f 2 tends to 0 as f 1 " f β , f 2 " f 1´β . Making use of the two previous bounds, one obtains:

ξ ´1 ď cstp´log f qf ν , ν " maxp3β, 3 ´2βq, f 1 , f 2 Ñ 0. (6.37)
The exponent ν varies between 9{5 and 1. Bound (6.35) is lower than (6.36) when 3{5 ă β ă 1.

Conversely, (6.36) gives a lower bound when β ă 3{5. Similar regime changes have been observed in the periodic checkerboard for which exact analytical results are available [START_REF] Amar | On the Finsler metrics obtained as limits of chessboard structures[END_REF].

For a medium with N iterated scales, and surface fraction f "

f 1 ...f N , with f i " f β i (0 ă β i ă 1, ř i β i " 1)
, one obtains, more generally:

ξ ´1 ď cstp´log f qf ν , ν " β 1 `maxp2β 1 , β 2 `maxp2β 2 , β 3 `... `maxp2β N ´1, 3β N q...q, (6.38)
as f Ñ 0. The exponent ν is less than 3, this value being attained when β 1 " 1 ´or β N " 1 ´. The minimal value of the exponent ν is obtained when 3β N " 2β N ´1, 3β N ´1 " 2β N ´2, ...., 2β 1 " 3β 2 , i.e. β i " p2{3q i´1 β 1 . Since β 1 `... `βN " 1, this entails ν " r1 ´p2{3q N s ´1 which approaches 1 (i.e. a quasi-linear correction) as N Ñ 8, for such a medium containing infinitely-many scales.

Rectangular obstacles

Consider, as previously, a 2D checkerboard made of a square-shaped cells. Each bond parallel to the horizontal direction is either occupied with probability η or free with probability 1 ´η. Furthermore, we assume that the state of a bond (occupied or free) is independent from the state of other horizontal bonds. We define the rigid obstacles in the plane by the set of rectangles made of the two adjacent cells that are either "above" or "below" each occupied bond (see Fig. 6.8b, hashed areas). Again, in order to derive rigorous bound, we assume that the rectangles are open sets, i.e. the matrix-rectangles boundaries are in the matrix. Accordingly, vertical lines in the lattice are free of obstacles, whereas in the horizontal direction, straight geodesics are prohibited by the presence of occupied bonds (Fig. 6.8b).

As previously, we construct a path that spans the checkerboard in the horizontal direction while avoiding the set of rectangular obstacles. The method is defined in much the same way as in Sec. (6.4.1). The path is broken into a set of subpaths comprising three segments, one horizontal of length i ´ 1 i , one oblique, of length 1 i in the horizontal direction and D in the vertical direction, and the third vertical of length m i ´D (see gray line, Fig. 6.8b). As previously, the variable i denotes the visibility in the horizontal direction from the initial point. As a consequence, the length of such a path is given by Eq. (6.26). However, the variables i , m i and 1 i now follows different probability laws. We obtain, for i and m i (k ě 1):

P t i " kDu " ηp1 ´ηq k´1 ,
x i y i " D{η, (6.39a) P tm i " kDu " η 2pk´1q p1 ´η2 q, xm i y i " D{p1 ´η2 q. (6.39b)

The conditional probability of 1 i reads:

P t 1 i " jD| i " kDu " " ηp2 ´ηqp1 ´ηq 2j , j " 0, ..., k ´2, p1 ´ηq 2pk´1q , j " k ´1, (6.40)
which entails, making use of (6.39a):

P j " P t 1 i " jDu " ηp1 ´ηq 3j " η 2 `3p1 ´ηq ‰ , j ě 0. (6.41)
Inserting this expression in (6.30) and expanding the term ? j 2 `1 as j Ñ 8:

1 D B b 1 i 2 `D2 ´ 1 i F i « ˆC1 0 ´3 2 log η ˙η `Op´η 2 log ηq, ( 6.42)
where the constant C 1 0 « 1.025 is given by the symbolic calculator [262] as an expression involving the Riemann Zeta function. Again, the number of terms in the expression increases with the order of the Taylor expansion. The first ten digits however are unchanged when ? j 2 `1 is expanded to order 20 or more in j.

Inserting expressions (6.42) and (6.39a) into (6.26), one obtains:

ξ " 1 `ˆC 1 0 ´3 2 log η ˙η2 `Op´η 3 log ηq, η Ñ 0. ( 6.43)
The surface fraction of the rectangles is f " ηp2 ´ηq and so, in terms of f :

ξ " 1 `ˆC 1 1 ´3 8 log f ˙f 2 `Op´f 3 log f q, C 1 1 « 0.513, f Ñ 0. ( 6.44)
As in the previous model, the expression for ξ is an upper-bound of the length of the geodesic. This bound exhibits a power 2 of f and is significantly higher that that in (6.32). Indeed, in the square model relevant to (6.32), the presence of an obstacle at the end of a ray is much less likely than in the model of rectangles. For a medium with N iterated scales of surface fraction f i "

f β i (0 ă β i ă 1, ř i β i " 1)
, and total surface fraction f " f 1 ...f N , with f 1 the concentration of clusters at the largest scale and f N the concentration of obstacles at the smallest scale, we derive, more generally:

ξ ´1 ď cstp´log f qf ν , ν " β 1 `maxpβ 1 , β 2 `maxpβ 2 , β 3 `... `maxpβ N ´1, 2β N q...q, (6.45)
as f Ñ 0. This may be interpreted in a similar way as the model of squares. The exponent ν is less than 2, this value being attained when β 1 " 1 ´or β N " 1 ´. The minimal value for ν is obtained when 2β N " β N ´1, 2β N ´1 " β N ´2, ...., β 1 " 2β 2 , i.e. β i " 2 1´i β 1 . Since β 1 `... `βN " 1, this entails ν " r1 ´2´N s ´1 which approaches 1 as N Ñ 8.

High-packing limit

Let us extend the results of Sec. (6.4.1) and construct an upper-bound for the length of the minimal path in the checkerboard with square-shaped obstacles appropriate to the high-packing limit f Ñ 1. The same microstructure as in Sec. (6.4.1) is considered. Furthermore, we assume that the cost for crossing an occupied cell is equal to pD with p ě 1. Compared to Sec. (6.4.1), a slightly simpler path is considered. The path is made of N subpaths each of them consisting of a segment of length m i D oriented along direction e 2 (transverse to the macroscopic direction of the minimal path) followed by a segment of line of length D, oriented along direction e 1 . If p " 8, we follow the same procedure as in Sec. (6.4.1), so that m i is chosen to be the minimal length of segments oriented along e 2 that reach a non-occupied cell (see Fig. 6.9). If the segments of minimal length in the upward and downward directions have the same length, one of them is chosen at random. When p is finite, the same value of m i is picked, except if the cost pm i `1qD is larger than pD, in which case we set m i " 0. In the latter case only, an occupied cell is crossed with cost pD at the end of the subpath.

Let us denote m 1 i the value of m i when p " 8. We obtain:

P tm 1 i " kDu " f 2k p1 ´f 2 q, (6.46a
)

P tm i " kDu " $ & % 0 if k ě tpu, P tm 1 i " kDu if 1 ď k ď tpu ´1, P tm 1 i " 0u `řjětpu P tm 1 i " jDu if k " 0, (6.46b)
where k ě 0 and t¨u is the integer part (or "floor") function. The mean length of the subpath is then:

D tpu´1 ÿ k"0 pk `1qP tm 1 i " kDu `pD 8 ÿ k"tpu P tm 1 i " kDu, ( 6.47)
which provides the bound:

ξ ď ξpp, f q " 1 ´f 2tpu 1 ´f 2 `pp ´tpuq f 2tpu . ( 6.48)
Consider now a n-scales microstructure consisting of a sequential iteration of n random checkerboards with widely-separated length scales. At the smallest scale, the microgeometry is made of a (one-scale) random checkerboard, characterized by a concentration f 1 of obstacles and a contrast p. The i th scale is obtained by replacing each occupied cell in a random checkerboard with concentration of occupied cells f i by the microgeometry at the pi ´1q th scale. A bound for such n-scale iterated material read:

ξ n pf q " ξpξ n´1 pf {f n q, f n q, (

where ξ i is a bound for the iterated microstructure at scale i. Suppose that occupied cells are highly-packed at each scale (f i « 1) and that f 1 " ... " f n . The overall concentration of obstacles is f " f 1 ...f n with f i " f 1{n . We have, when n Ñ 8:

ξ n´1 pf {f n q « ξ n´1 pf ´pf {nq log f q " ξ n´1 pf q ´ξ1 n´1 pf q f n log f. ( 6.50)
Insert the above into (6.49) and make use of (6.48). For simplicity, we assume that p is an integer. After some algebra:

1 ´p1 ´f 2{n qξ n pf q « 1 `2 n " ξ n´1 pf q ´ξ1 n´1 pf q f n log f  log f `2 n 2 ξ 2 n´1 log 2 f. (6.51)
Simplifying this expression further, all terms cancel up to order Op1{nq whereas setting to zero the term of order Op1{n 2 q provides us with the ordinary differential equation:

ξ n ´ξ2 n `f ξ 1 n " 0, n Ñ 8. (6.52)
Its solution reads:

ξ n " 1 1 ´p1 ´1{pqf , n Ñ 8. (6.53)
In the high-packing limit f Ñ 1 Eq. ( 6.53) coincides with (6.48) up to Op1 ´f q:

ξ n " ξ " p ´ppp ´1qp1 ´f q `Op1 ´f q 2 , n " 8, p ă 8, f Ñ 1. (

This is not so when p " 8, in which case:

ξ n " p1 ´f q ´1, ξ " 1 2 p1 ´f q ´1 `Op1q, n " 8, p " 8, f Ñ 1. (6.55)
Accordingly, if bounds are to be interpreted as approximations of the exact behavior, the sequential microstructure is stiffer than the one-scale checkerboard in the high-packing limit.

m i + 1 i m m i + 2 Figure 6
.9: Path used to compute bound (6.48) in the random checkerboard.

Random sequential adsorption model of rigid squares

Hereafter, we consider the "random sequential adsorption" model [231] made of squares in the plane. In this model, trial grains are added sequentially in a 2D domain at random position, with uniform regionalized density. When a grain overlaps a previously inserted grain, it is rejected, otherwise, it is implanted. Random sequential adsorption is one among many models of nonoverlapping grains [START_REF] Månsson | Random patterns of nonoverlapping convex grains[END_REF]. For dilute surface fraction of grains, the position of the grains in random sequential adsorption models is asymptotically equivalent to that of a Poisson point process [START_REF] Månsson | Random patterns of nonoverlapping convex grains[END_REF].

Realizations of the model are generated in a domain Ω of surface S. The model is parametrized by the surface fraction of squares f " nb 2 {S where n is the number of disks in Ω and b 2 the surface of a square.

Inspired by the algorithm relevant to a union of disks (Appendix 4.A), we devise an algorithm for computing the minimal path between two points in a model of rigid squares, again without discretizing the medium. The task is significantly more demanding, because of the important computational cost involved in determining whether a given segment intersects a square. The algorithm is described in Sec. (6.A). We now compute the minimal path joining two mid-points on the "right" and "left" sides of Ω, as shown in Fig. (6.10a). In this figure, the direction of propagation is horizontal. We let f " 0.1, 0.03 and 0.01. For each value of f , the number of disks takes on values n " 400, 800, 1600, ..., 102400. In the dilute regime f Ñ 0, we expect very "flat" minimal paths. Hence, we consider rectangular domains Ω of dimension 1 (a.u.) along the direction of propagation and h ! 1 along the transverse direction. We let h " 100{n. With this choice, all minimal paths are constrained in a narrow strip of width at most 24% of the domain. Note that if the value chosen for h is too small, the minimal path predicted by the algorithm is close to the border of the domain and "avoids" the entire set of squares. This can be detected a posteriori. In that case, a new computation would be carried out with a larger value of h.

Numerical results for the quantity n 4{3 pξ ´1q are represented in Figs. (6.10b) as a function of n 1{3 ? f in log-log plot. This graph shows the existence of two regimes, one where ξ ´1 « p5{nq ? f , when n ď n c " 125f ´3{2 , and the other in which:

ξ « 1 `α0 f 2 , α 0 « 0.04, (6.56)
which holds when n ą n c (Fig. 6.10b). Behavior (6.56) may be interpreted as follows. Let us write the length of the minimal path as:

ξ " ř N i"1 a 2 i `m2 i ř N i"1 i " x a m 2 i ` 2 i y i x i y i , ( 6.57)
where the square root term is the length of the i-th segment in the path, i and m i are the length of the projection of the segment along the direction of propagation and transverse to it, and N " 1 is the number of segments. Averages x'y i are to be taken over all segments in the minimal path. Now, most segments are expected to be nearly horizontal in the dilute limit so that m i ! i . A Taylor expansion of (6.57) in {m Ñ 8 gives, neglecting the contribution of small values of the i :

ξ « 1 `xm 2 i { i y i 2x i y i . ( 6.58)
As an approximation, one may identify i as the length of the "visibility" in the direction of propagation, which is the longest distance from a given point in that direction that does not intersect a grain (see [START_REF] Calka | Asymptotics of the visibility function in the Boolean model[END_REF] from some exact results of related quantities in Boolean models). We expect i to be of order " b{f , since the fraction of a line intersected by the squares is, on average, f , and m i " b{4. One accordingly expects ξ ´1 " f 2 {32 « 0.03f 2 . We emphasize that, unlike in Sec. (6.3), this reasoning is not rigorous, because no explicit (nor admissible) path has been considered.

Comparison with non-linear homogenization theories

The problem of a rigidly-reinforced, perfectly plastic material in anti-plane shear has been considered in [START_REF] Goldsztein | Bound on the yield set of fiber reinforced composites subjected to antiplane shear[END_REF] and [START_REF] Goldsztein | Effects of the Fibers' Shape and Volume Fraction on the Strength of Ideally Plastic Fiber Reinforced Composites[END_REF]. The author derives an upper-bound for the effective yield stress Y of such a medium. The inclusions are much stiffer than the matrix, and the bound depends on the perimeter and area of the inclusions. The latter reads, for a dilute concentration of squares:

Y ď Y m ˆ1 `7 2 f ˙, (6.59)
where Y m is the yield stress in the matrix. This result is consistent with (6.56), yet in the dilute limit, the bound's prediction, i.e. a leading-order term " f , is significantly higher than our numerical result, which is of order " f 2 (6.56). Note however that bound (6.59) holds for an arbitrary distribution of squares. Specifically, consider a N -scale arrangement of square-shaped particles similar to the iterated model B C , with particle surface fraction f " f 1 ...f N where f 1 is the concentration of clusters at the largest scale and f N that of particles in the smallest clusters. Making use of (6.56) and following a similar treatment as in Sec. (6.4.1) we obtain:

ξ ď 1 `α0 min `f 2 1 ; f 1 f 2 2 ; f 1 f 2 f 2 3 ; ...; f 1 ...f N ´1f 2 N ˘. (6.60)
To maximize this quantity, we choose:

f 1 " f p1{2qr1´2 ´N s ´1 , f 2 " a f 1 , ..., f N " a f N ´1, (6.61)
so that f 2 1 " f 1 f 2 2 " ... " f 1 ...f N ´1f 2 N . Taking now N Ñ 8: ξ ď 1 `α0 f. (6.62)
The latter expansion explains the scaling law " f in (6.59), assuming the r.h.s. of (6.62) is a good approximation of the length of the exact minimal path in a RSA model of squares with infinitely-many separated scales following (6.61).

In this section, we also consider, as in the porous case, the "non symmetric fully-optimized second-order" (SO) method [83, Annex B]. In the dilute limit f Ñ 0, the (SO) theory predicts, asymptotically:

ξ 1 pSOq E « 1 `min " f 2 8 ; pp ´1qf * , (6.63)
where ξ

1 pSOq E ą 1 is the effective yield stress, normalized by the yield stress in the inclusions, and p is the yield stress in the inclusions divided by that in the matrix. When specialized to p " 8, this formula is close to (6.56). Since bound (6.56) also holds for arbitrary value of p, and given (6.10), it is clear that:

ξ 1 E ď 1 `min α 0 f 2 ; pp ´1qf ( , f Ñ 0, (6.64)
which is very similar to (6.63). We remark that the (SO) theory also predicts that, when p ě p ˚(where p ˚is a function depending on f only), then ξ " a 1 `f 2 {4 becomes independent of p. This prediction may be interpreted by the fact that, when inclusions are sufficiently stiff, shear bands span the matrix only. Thus, the effective response is unchanged when the inclusions yield stress is increased.

Conclusion

In this chapter, we have examined the effect of a multiscale distribution of rigid obstacles on the length of minimal paths spanning the media. The dual problem of a multiscale distribution of porous inclusions has also been considered. In the porous case, we have extended bounds on the minimal paths spanning Boolean random media to multiscale materials. In Cox-Boolean models, obtained as the intersection of Boolean models with widely-separated length scales, the length of minimal paths increase with the number of scales, at small surface fraction of particles f . In the limit of infinitely-many scales, the bound predict a linear correction in f at leading-order term. In the rigid case, a small effect of the particles is observed in the dilute limit, resulting in a leadingorder correction " f 2 , according to numerical data. Bounds obtained for a random checkerboard geometry confirm a behavior which is essentially quadratic in f . A multiscale distribution of obstacles, where particles are organized as clusters, increase the minimal path length, according to the bounds. For infinitely-many scales, a linear correction in f is predicted. The numerical results obtained in the present chapter, as well as the predictions of nonlinear homogenization theories for the effective yield stress of media in antiplane shear suggest that the present bounds are sharp.

Acknowledgements I thank P. Suquet for pointing to me Refs. [START_REF] Goldsztein | Bound on the yield set of fiber reinforced composites subjected to antiplane shear[END_REF][START_REF] Goldsztein | Effects of the Fibers' Shape and Volume Fraction on the Strength of Ideally Plastic Fiber Reinforced Composites[END_REF].

6.A Algorithm for computing shortest paths in a model of squares

In general, geodesics bypassing an obstacle are made of two straight-line segments and of a segment of a curve inbetween, that follows the frontier of the obstacle's convex hull [START_REF] Vladimir | Catastrophe theory[END_REF]Chap. 13]. Furthermore, the two straight-line segments are tangent to the frontier of the obstacle. Accordingly, in the case of polyhedral obstacles, geodesics are made of segments of lines joining the corners of polyhedra. Lozano-Pérez & Wesley have proposed a numerical algorithm to compute geodesics in such systems of polyhedra [START_REF] Lozano-Pérez | An algorithm for planning collision-free paths among polyhedral obstacles[END_REF]. Indeed, for polyhedral shapes, the problem in the continuum can be reformulated as that of a minimal path problem in the "visibility graph", which can be solved using Djikstra's efficient algorithm for graphs [START_REF] Ew Dijkstra | A note on two problems in connexion with graphs[END_REF]. The nodes in the visibility graph are the vertices of the obstacles, and the bonds between nodes are segments joining two of the obstacles vertices, that do not intersect an obstacle. The size of the visibility graph, and so the number of steps to compute the shortest path between two points, however increases rapidly with the number of obstacles [START_REF] Ch Papadimitriou | Shortest-path motion[END_REF]. Faster algorithms have been proposed for L 1 -metrics [START_REF] Mitchell | Shortest rectilinear paths among obstacles[END_REF] of for special geometries, requiring only a subgraph of the visibility graph, such as a set of parallel barriers [START_REF] Lee | Euclidean shortest paths in the presence of rectilinear barriers[END_REF]. The latter can not be used in the present work and we follow the visibility graph approach. Our algorithm is summarized hereafter. Consider a model of non-interpenetrable squares in two dimensions with infinite distance function inside the squares, i.e. the geodesics are located in the complementary set of the union of squares. Since the geodesics are locally straight lines, they reduce to a set of segments joined by square corners. Two square corners are said to be "visible" from one another if the segment joining the two corners is included in the matrix (if it lies exactly along the matrix-square frontier, the two corners are visible to one another). Thus, for instance a square corner is visible by two of the other other corners in the same square, but not by the third.

In theory, algorithm (4.A) may be used to compute a solution provided one determines the pairs of square corners that are visible to one another. A brute-force approach requires one to determine, for every pair of corners, if a square intersects the segment joining the two corners. This has complexity OpN 3 q where N is the number of squares and is prohibitively expensive. Instead, we rely on a different approach, described below.

Assume the domain is divided into subdomains of rectangular shapes whose dimensions are larger than the squares. We are interested in a square corner C located in a subdomain. We first determine the square corners in the same subdomain that are visible from C, using a "brute-force" technique. We then determine the set of "forbidden" angular sectors which are obstructed by the neighboring squares, located in the same subdomain. We then consider a larger set of 8 (or less) subdomains around the square and determine visible square corners in that domain. If a corner is located in a direction included in the forbidden angular sectors, it is discarded, otherwise we determine if it is obstructed by another square. We update the set of obstructed angular sectors, and consider an even larger set of subdomains. This process continues until obstructed angular sectors entirely cover the rest of the (unchecked) domain.

Chapter 7

Cross-over behavior in periodic composites under plane strain and anti-plane shear

The previous chapters have investigated how certain parameters of the microstructure may influence the effective response of materials using a combination of geometrical bounds for the homogenized distance function and FFT numerical results. The results are relevant to nonlinear conductivity with a non-strictly convex potential and to the yield stress of materials subjected to anti-plane shear. This study aims, in contrast with the previous ones, to understand how localization bands may develop, in a setting where the matrix phase follow a strictly convex, powerlaw potential. The problem is investigated in simple, periodic configurations in two dimensions, using limit analysis.

Conductivity problem

Consider a periodic medium in two dimensions made of a square-shaped elementary cell Ω containing a single disc of radius a (Fig. 7.1a). We assume that the size L of the domain Ω is the length unit, i.e. L " 1. The quantity a may alternatively be considered as a non-dimensional parameter that monitors the disc surface fraction f , with a " a f {π. Points in Ω are referred to in a Cartesian coordinates system (e 1 , e 2 ) so that Ω " r´1{2; 1{2s 2 and the disc center is at the origin.

We are looking for the solution to the problem:

W pEq " r χ E n`1 n `1 " inf EPK " 1 Ω ż Ω d xχpxq |Epxq| n`1 n `1 * , ( 7.1)
K " E, Dφ : E " ´∇φ, xEy " E ( , where 0 ď n ď 1 is the nonlinearity exponent, E the electric field, E the mean applied field, φ the electric potential, χpxq the conductivity at point x, W pEq the macroscopic energy density and r χ the effective conductivity. In the following, we assume the inclusions is insulating and embedded in a conducting matrix, accordingly χpxq " 0 in the disc and χpxq " χ m ą 0 outside of the disc. The applied field is oriented along the x 1 -axis, so that E " |E|e 1 .

99

Consider a trial electrical field E ψ with components of the form E ψ 2 pxq " 0, E ψ 1 pxq " ψpx¨e 1 q where the function ψ is to be optimized on, under the constraint ş 1 0 ψ " 1.

(a) (b)

W pEq ď W ψ " 2χ m n `1 « ż 1{2 a dx|ψpxq| n`1 `ż a 0 dx|ψpxq| n`1 p1 ´2? a 2 ´x2 q ff . ( 7.2)
Set BW ψ {Bψpzq " 0 where W ψ " W ψ `θ ´1 ´ş1 0 ψ ¯and solve for ψ:

E ψ pxq " E χ ´1{n ψ ˆ# 1 if |x 1 | ą a, ´1 ´2a a 2 ´x2 1 ¯´1{n if |x 1 | ď a, (7.3a) r χ χ m ď χ ψ " " 1 ´2a `2a ż 1 0 du ´1 ´2a ? 1 ´u2 ¯´1{n  ´n . ( 7.3b)
A Taylor expansion of the integrand in (7.3b) in the limit a Ñ 0 (n fixed) provides, upon integration:

χ ψ " 1 ´πa 2 ´8pn `1q 3n a 3 `Opa 4 q. ( 7.4)
The previous expansion is a good approximation of χ ψ when the integrand in (7.3b) stays close to 1 `2a{n ? 1 ´u2 , hence when a ! a c " n{2 or equivalently n " n c " 2a. However, when n Ñ 0 (a fixed), the integral in (7.3b) is governed by the values of its integrand around u « 0. Let us expand the term ? 1 ´u2 to second-order in u in (7.3b). Using the variable change u Ñ u ? n and integrating, we obtain:

ż 1 0 du ´1 ´2a ? 1 ´u2 ¯´1{n " p1 ´2aq ´1{n « c nπp1 ´2aq 4a `opnq ff , ( 7.5)
so that:

χ ψ " p1 ´2aq " 1 ´n 2 logpnπap1 ´2aqq `Opn logpnqq ı . ( 7.6)
We expect this expansion to hold when the logarithmic term in the r.h.s. is much smaller than 2a. Solving for 2a " ´pn{2q logpπnaq entails a " a 1 c with a 1 c " pn{4qW p4{pπn 2 qq, where W p'q is the special Lambert function, inverse of W Þ Ñ W e W . As n Ñ 0, a 1 c is equivalent to:

a 1 c " ´n 2 log n. ( 7.7)
We also define n 1 c so that a 1 c pn 1 c q " n 1 c which is close to

n 1 c " 2a ´log a , ( 7.8)
as a ! 1. Notice that expansions (7.4) and (7.6) do not hold in the interval a P ra c ; a 1 c s, and in particular when 1 ! 2a{n ! ´log n. The behavior of χ ψ in this intermediate regime is obtained by approximating the integrand in χ ψ as follows:

´1 ´2a ? 1 ´u2
¯´1{n « e 2a{n e ´au 2 {n .

Integrating the above yields:

χ ψ « " 1 ´2a `?πan erf ˆc a n ˙e2a{n  ´n (7.9)
where "erf" is the "error function" erfpzq " p2{ ? πq ş z 0 expp´t 2 qdt. When t Ñ 0, erfptq « 2t{ ? π and one obtains:

χ ψ « 1 `3an ´e2a{n ? πan 3{2 `n2 , ( 7.10)
which is a good approximation of χ ψ in the region a c ! a ! a 1 c (Fig. 7.2a). The shape of the local electrical field in the band surrounding the inclusion, given by (7.3a), is plotted in Fig. (7.2b), after normalization between 0 and 1. Various values of the exponent n are used. The profile E ψ 1 px 1 , 0q for the trial electric field changes in the interval n P rn 1 c ; n c s. A concave shape at n " n c is replaced by a curve exhibiting an inflection point when n " n 1 c , highlighting the electric field localization. The coordinate of the inflection point is obtained as the root of a polynomial equation of degree 3, given by the second derivative of E ψ 1 px 1 , 0q with respect to coordinate x 1 (not shown). The typical width b of the localization band then reads:

b " c 2n a `n 2a `opnq, n Ñ 0. (7.11)
It is clear that when n " 0, the electric field is localized along a one-dimensional path that passes through the disc center, with minimum length 1 ´2a. When n is small but not zero, the field is localized over a band of finite width.

Observe that the present analysis would hold if the disc had been cut along lines parallel to e 1 and e 2 into pieces and each piece moved at a different location, in such a way that points located along a line x 1 " cst stay along a line x 1 " cst (Fig. 7.1b). Obviously, the distribution of the pore fractions along lines parallel to axis e 2 is the same in the original microstructure and in the elementary cell with "scrambled disk". In such a configuration, the localization band are spread into many vertical paths, of suboptimal length, smaller than 1 ´2a `n{p4a 2 q. Such solution bears similarity to localization bands of varying intensity that span elastic, perfectly-plastic media under plane strain (see e.g. [START_REF] Willot | Fast Fourier Transform computations and build-up of plastic deformation in 2D, elastic-perfectly plastic, pixelwise-disordered porous media[END_REF]).

The effective behavior is represented in Fig. (7.3a). In the region a ! a c , the effective conductivity scales, to leading-order term, as " a 2 , i.e. a linear correction in term of the volume fraction f . The field are not localized in this domain, corresponding to dilute regime (Fig. 7.3b). The electric field in the matrix, in particular, is close to the macroscopic field E. In the intermediate regime a c ! a ! a 1 c , the electric field (and local energy density) in the matrix diminishes as a increases, and increases in the band surrounding the inclusion. The increase is maximum along the line of minimal path (Fig. 7.4). In the region a " a 1 c (Fig. 7.3b), characterized as a "localized regime", the electric field takes on very high values within a thin band. The effective response χ ψ scales as " 1 ´2a or equivalently, as " 1 ´2a f {π.

Nonlinear elasticity 7.2.1 Local and effective behavior

This section is devoted to the analysis of the plane strain, mechanical response of a composite material with the same periodic geometry as considered in the previous section. In the matrix, the local infinitesimal deformation and stress tensors are related by:

σ " Bw Bε , wpεq " y 0 ε 0 1 `n ˆεe ε 0 ˙1`n `p8qtrpεq, (7.12)
where y 0 and ε 0 are the flow stress and reference strain and w " wpε e q is an isotropic strain potential depending on the Von-Mises equivalent strain ε e " a p2{3qε 1 : ε 1 , in which ε 1 denotes the Figure 7.3: Solid lines: leading-order correction to the effective conductivity upper-bound χ ψ in the dilute limit a Ñ 0, for various values of the nonlinearity exponent n, in log-log plot. Dashed lines: "highly-dilute" and "localized" regimes described by Eqs. (7.4) and (7.6).

strain deviatoric part. The matrix is incompressible, the potential ω being infinite when trpεq ‰ 0.

The constitutive law's nonlinearity is monitored by the exponent 0 ď n ď 1. The limiting values n " 1 and n " 0 accordingly characterize a linear and rigid-ideally plastic behavior, in the context of deformation theory.

In the following, periodic boundary conditions are applied along the frontiers BΩ of the elementary cell Ω. The medium is accordingly subjected to the macroscopic strain field ε " xεy Ω , where x'y Ω denotes a mean over Ω. The "effective" behavior is described by the effective potential: r wpεq " inf εPKpεq xwpx, εqy Ω , (7.13) where Kpεq is the set of kinematically admissible strain fields:

Kpεq " tε; Du : ε " pgraduq sym , u " ε ¨x `u˚, u ˚#, trpεq " 0u . (7.14) Let us focus on the case of a 'simple shear' loading where the principal axes of ε are aligned with e 1 ˘e2 , referring to the Cartesian axis of Fig. (7.1a). Accordingly, ε 12 " ε 21 is the only non-zero component of ε. The effective potential r w is then characterized by an effective flow stress r y 0 which may be rewritten as:

r wpεq " r y 0 ε 0 1 `n ˆεe ε 0 ˙1`n , ( 7.15)
from which the macroscopic stress xσy Ω may be obtained. In both (7.12) and (7.15), due to the isochoric loading direction considered and the material incompressibility, the Von Mises strains read:

ε e " 2 ? 3 c ε 2 12 `´ε 11 ´ε22 2
¯2, ε e " 2|ε 12 | ? 3 . (7.16)
In the rest of this chapter, the variational formulation (7.13) defining the homogenized behavior of the composite is used to derive bounds for the effective flow stress r y 0 . Two cases of interest are considered, namely, a material containing an array of cylindrical pores and a composite reinforced by an array of rigid, cylindrical inclusions. To assess the domains of validity of the bounds, use is made of the nonlinear second-order homogenization theory, relevant to such periodic microstructures [START_REF] Idiart | Infinite-contrast periodic composites with strongly nonlinear behavior: effective-medium theory versus full-field simulations[END_REF], and of Fourier numerical computations. The latter computations are carried out on grids of 1024 2 voxels, using the "augmented Lagrangian" method [START_REF] Michel | A computational method based on Augmented Lagrangians and Fast Fourier Transforms for composites with high contrast[END_REF] with 'discrete' Green operator [START_REF] Willot | Fast Fourier Transform computations and build-up of plastic deformation in 2D, elastic-perfectly plastic, pixelwise-disordered porous media[END_REF]. A regularized form of the constitutive law (7.12) is implemented in Fourier computations. Namely, the behavior is linear-elastic at small strain and the medium is compressible with bulk modulus κ:

σ 1 " " 2µε 1
if 3µε e ă y 0 ε n e , p2{3qy 0 ε n´1 e ε 1 otherwise, trpσq " 3κtrpεq, (7.17) so that the Von Mises equivalent stress σ e " a p3{2qσ 1 : σ 1 reads:

σ e "
" 3µε e if 3µε e ă y 0 ε n e , y 0 ε n e otherwise. (7.18) where σ 1 the stress deviatoric part. In the FFT computations, we set y 0 " 1 (a.u.), κ " 1000 (a.u.), µ " 1 (a.u.) and apply a strain loading ε 12 " 4, using steps of ∆ε 12 " 0.1. These values allow one to approach an incompressible behavior, and to recover the effective nonlinear response σ e « r y 0 ε n e (see [START_REF] Idiart | Infinite-contrast periodic composites with strongly nonlinear behavior: effective-medium theory versus full-field simulations[END_REF]). Note that the augmented Lagrangian method requires one to invert the equation L 0 : ε `σ " τ where τ is a given symmetric second-order tensor and σ is provided by (7.17). This inversion is carried out numerically by an iterative method, at each grid point. Newton's method fails when very small exponents are considered, therefore, we use the bisection method. More involved FFT schemes, such as the ones discussed in [START_REF] Schneider | On polarization-based schemes for the FFTbased computational homogenization of inelastic materials[END_REF], have not been considered in this work.

In the porous case, we set σ " 0 in the cylindrical void. For the rigid inclusion, we use σ " L : ε with isotropic stiffness tensor L with elastic moduli µ 2 " 1000 and κ 2 " κ. For the porous medium, the fastest convergence rate offered by the Fourier algorithm, in term of the L 1 -norm of the divergence of the stress field, is obtained with quite small values of the reference moduli, about 10 ´4. The convergence rate is less sensitive to the reference medium for quasi-rigid inclusions. In that case, we use reference elastic moduli slightly stiffer than κ and µ.

In the rest of this chapter, we focus on the dilute-limit behavior for the effective flow stress, that is, we assume the cylindrical inclusion has a very small volume fraction f ! 1. In this limit, the second-order nonlinear homogenization theory predicts the isotropic macroscopic behavior: r y 0 y 0 " " 1 ´p1 `?nq 2 {p2 ? nqf `Opf 2 q (porous inclusion), 1 `1 2 ? np1 `?nq 2 f `Opf 2 q (rigid reinforcement), (7.19) as f Ñ 0. Note that scaling law (7.19 1 ) above has been obtained in the case of incompressible pores.

In the following, we focus on the 'strongly-nonlinear' case n ! 1. Limit analysis bounds provide that, when n " 0 [START_REF] Drucker | The continuum theory of plasticity on macroscale and microscale[END_REF][START_REF] Idiart | Infinite-contrast periodic composites with strongly nonlinear behavior: effective-medium theory versus full-field simulations[END_REF]:

r y 0 " y 0 `1 ´α0 f 1{2 ˘, 2{ ? π ď α 0 ď 2 a 2{π, (7.20)
in the porous case, to leading-order term in f . In effect, numerical results indicate [START_REF] Willot | Contribution à l'étude théorique de la localisation plastique dans les poreux[END_REF]:

r y 0 « y 0 `1 ´1.185f 1{2 ˘. (7.21)
In the rigid case, the upper-bound provides the exact result: r y 0 " y 0 , (7.22) in the interval f ă π{4. The purpose of the next sections is twofold. First, we wish to assess the validity of the predictions (7.19). Our second goal is to clarify the regime change that (presumably) occurs between (7.19) and (7.20)-(7.22).

Figure 7.5: Field pattern for the limit analysis trial field in the unit cell.

Porous material

One can attempt to estimate the behavior of the crossover threshold f c between the two regimes by equaling expressions (7.21) and (7.20 1 ) to find:

f c " 4α 2 0 n. (7.23)
We will show that the actual regime change occurs at a much lower porosity. Inspired by the bounds obtained in conductivity, we approach the porous problem by means of a continuous trial field:

u 1 px 1 , 2 q " ε 12 " rx 2 , |x 2 | ă b, rbsignpx 2 q `p1 ´2rbqp1 ´2bq ´1 rx 2 ´signpx 2 qbs, |x 2 | ă b, ( 7.24)
and u 2 px 1 , x 2 q " u 1 px 2 , x 1 q, with ´1{2 ď x 1,2 ď 1{2. The parameter b monitors the unknown half-width of a band passing through the pore (see Fig. 7.5) and the parameter r the field intensity along that band. The two parameters are in the intervals 0 ă b ă a and 0 ă r ă 1{p2bq.

For convenience, we restrict ourselves to b ă a{ ? 2, so that the square intersection area of the bands remains within the pore. Denote f B pa, bq the area of the band that lie outside the void. Straightforward geometrical considerations lead to:

f B pa, bq " 4 ˆb ´a2 sin ´1 b a ´ab a 1 ´b2 {a 2 ˙. (7.25)
We insert the trial field ε corresponding to the above displacement field into (7.13) which provides us with:

r y 0 y 0 ď ż matrix d 2 x ˆε12 ε 12 ˙n`1 " f B 2 n`1 ˆr `1 ´2rb 1 ´2b ˙n`1 `p1 ´f ´fB q ˆ1 ´2rb 1 ´2b ˙n`1 . ( 7

.26)

A simplified bound is obtained by taking r " 1{p2bq, i.e. ε 12 " 0 outside the bands, in which case:

r y 0 y 0 ď inf bďa{ ? 2 f B pa, bqp4bq ´pn`1q . (7.27)
This choice is accurate in the regime where the strain field localizes, i.e. when n is very small. An optimal band width b ˚is now sought for to minimize (7. In practice, curves drawn with this expression are indistinguishable from those obtained by a numerical minimization of (7.27). When n Ñ 0, bound (7.29) reduces to the upper-bound (7.20). It should be noted that bound (7.29) becomes useless when a Ñ 0 with n fixed, as the bound will then be greater than 1. In this regime, the field pattern can not be approached by a straight, localized band that join pores. We find numerically with good precision that this occurs when ´2a{ log a À n. Thus, to account for the dilute regime a ! n, we take b " a{ ?

2 in (7.26) and obtain the bound: The behavior of this upper-bound is similar to results obtained in conductivity (Sec. 7.1), see Fig. (7.6). The curves in Fig. (7.6) are not smooth at a single point where the minimum for the trial field switches from bound (7.30) to (7.29). The behavior in the regime a ! n is, as expected, inconsistent with (7.19 1 ). Although both analytical results predict a linear correction in f , the prefactor of that correction depends on n, contrary to the upper-bound (7.31). This is because, in the dilute limit, the simple trial field corresponding to our bound does not adequately describe the fields. Those fields are necessarily homogeneous except in the region surrounding the inclusion, unlike the trial field which always display a straight band. Nevertheless, bound (7.31) may capture some information regarding the regime change between the 'localized' and 'dilute' regime. Since bound (7.30) is not meaningful, we restrict our attention to bound (7.29). This is compared to the dilute estimates (7.19) and to FFT computations in Fig. (7.7). The quantity 1 ´r y 0 {y 0 is represented in log-log scale as a function of a, for various values of the exponent n " 10 ´1, ..., 10 ´4. Bound (7.30) ceases to be meaningful when 2a " ´n log a, hence: Figure 7.7: Normalized effective yield stress r y 0 {y 0 vs. disk radius a in a periodic medium, in log-log plot, for various values of the nonlinear exponent n. Symbols: FFT computations. Solid and dotted lines: asymptotic dilute estimates (7.19) predicted by the nonlinear "second-order" homogenization theory [START_REF] Idiart | Infinite-contrast periodic composites with strongly nonlinear behavior: effective-medium theory versus full-field simulations[END_REF]. Gray dashed lines: upper-bound (7.29). Black dashed line: expansion (7.21).

r y 0 y 0 ď ζpa,
f c « ´pπ{4qn 2 log 2 n. ( 7
The scaling (7.21), corresponding to a localized strain field, is confirmed by FFT computations in the domain n À ´2a{ log a. The dilute estimates (7.19) are also confirmed by FFT computations when n " 10 ´1. For smaller values of n, FFT data is not conclusive, owing the difficulty of obtaining numerical results in highly-dilute, strongly-nonlinear cases. Nevertheless, the FFT data confirms that the scaling law r y 0 {y 0 " 1 ´α0 a (Eq. 7.21) ceases to be valid at much smaller values than that predicted by (7.23) and in fact suggests a cross-over consistent with the much smaller threshold (7.32) provided by bound (7.29), although the numerical data can not conclusively confirm it.

Rigid inclusion

As in the porous case, the bound (7.22) is extended as a bound for rigidly-reinforced, powerlaw materials of rate-sensitivity exponent n, by considering a continuous regularization of the trial displacement field. Again, we limit ourselves to simple shear loading and strain fields. The situation is simpler than in the porous case, since we already know from FFT results that the relevant band width is 2a (see FFT maps in). A continuous trial field consistent with the rectangular patterns of [START_REF] Idiart | Infinite-contrast periodic composites with strongly nonlinear behavior: effective-medium theory versus full-field simulations[END_REF]Tab. 2] on the unit cell ´1{2 ď x, y ď 1{2 is:

u 1 px 1 , 2 q " ε 12 2 " 0, |x 2 | ă a, signpx 2 qp|x 2 | ´aqp1 ´2aq ´1, |x 2 | ą a, ( 7.33)
and u 2 px 1 , x 2 q " u 1 px 2 , x 1 q. Accordingly, ε 12 is constant within each of a set of 9 rectangles, and vanishes in the square circumscribed to the rigid inclusion disk. The following upper bound is Being valid for any a ď 1{2 and always greater than one for 0 ď n ď 1, it does not suffer the limitations encountered in the porous case. It moreover reduces to (7.22) when n " 0, and constitutes a good estimate as long as n is small enough for the rectangular field pattern to hold.

In the dilute limit, Eq. (7.34) reads:

r y 0 y 0 " 1 `2 `n `2´n ´1˘a `Opa 2 q, (7.35)
so that comparing to Eq. (7.19) the latter provides a crossover at a r c pnq " p4{πqp1 ´log 2q

? n. Both regimes can be encoded in the form:

r y 0 y 0 " 1 `n3{2 G pa{a r c pnqq , ( 7.36)
where the scaling function is such that Gptq " t 2 (resp. Gptq " t) when t ! 1 (resp. t " 1). To check these results, composite curves built from the infimum of expressions (7.19) and (7.34) are compared to FFT data in Fig. (7.8), where the inset shows the scaling function G. The agreement is excellent, and confirms that Eq. (7.34) and (7.19 2 ) hold when a ą a r c and a ă a r c , respectively. In terms of the volume fraction, the regime change occurs when:

f " f r c pnq "
16 π p1 ´log 2q 2 n « 2.49n. (7.37) It would be useful to study this problem in the context of a random distribution of particles, such as the one investigated in Sec. (6.4.3).

Conclusion

This chapter presented an investigation of the effective response of powerlaw materials with decreasing nonlinearity (or strain-rate sensitivity) exponent n and dilute porosity f , or in the case of rigid reinforcements, small concentration of particles. Our attention has been restricted to periodic structures. Numerical FFT results, limit analysis bounds and the predictions of nonlinear homogenization theories have been used to investigate the material response. One of the main finding of this work is the observation of a crossover-type behavior for the porous case in the dilute concentration and small rate-sensitivity regime. As f goes to 0, the effective flow stress tends to the flow stress in the matrix with a correction proportional to f , consistently with a set of non-interacting inhomogeneities (Eq. 7.19 1 ). However, the prefactor blows up as n ´1{2 as n Ñ 0, suggesting that the range of validity of this expansion tends to zero in the the ideally plastic limit. In contrast to this result, a regime consistent with a collective-type behavior induced by strongly-interacting voids is obtained when taking the limit n Ñ 0, in which case the effective flow stress scales as f 1{2 as leading-order correction in f . The limit analysis study presented in this chapter suggests a cross-over in between the two regimes, at f " n. The bound obtained by limit analysis and FFT results also suggest a much more abrupt transition between the weakly and strongly-interacting regimes than in the corresponding linear problem with strongly anisotropic matrix [START_REF] Willot | Effectivemedium theory for infinite-contrast two-dimensionally periodic linear composites with strongly anisotropic matrix behavior: dilute limit and crossover behavior[END_REF], also discussed in Sec. (2.6).

For the rigidly-reinforced materials, the two asymptotic regimes were found to be consistent up to the first correction and a weaker crossover-type behavior was found, which could be completely elucidated. In this case, the strongly dilute regime is found to be linear in f (cf. 7.19 2 ) with a prefactor that tends to zero as m Ñ 0, while the strongly nonlinear regime has a vanishing correction for sufficiently small values of f (cf. 7.22). This result has implications also for random material, such as the "RSA" model of squares (Sec. 6.4.3).

Part III

Stokes flow in porous media

Chapter 8

Covariance of a Boolean set of anisotropic grains

This chapter focuses on the variance properties of isotropic Boolean random sets containing randomly-oriented cylinders with circular cross-section. Emphasis is put on cylinders with large aspect ratios, of the oblate and prolate types. A link is established between the powerlaw decay of the covariance function and the variance of the estimates of the volume fraction of cylinders. The covariance and integral range of the Boolean mixtures are expressed in terms of the orientation-averaged covariogram of cylinders, for which exact analytical formulas and approximate expressions are provided.

Introduction

The covariogram, originally introduced by G. Matheron [START_REF] Matheron | Les variables régionalisées et leur estimation: une application de la théorie des fonctions aléatoires aux sciences de la nature[END_REF], gives the volume of the intersection of a (convex) body with a translation of itself. The function is closely related to the distribution of the length of the chords of a body, and also a key ingredient in the theory of Boolean stochastic models based on Poisson point processes (see [START_REF] Matheron | Eléments pour une théorie des milieux poreux[END_REF][START_REF] Matheron | Random sets theory and its applications to stereology[END_REF][START_REF] Matheron | Random sets and integral geometry[END_REF][START_REF] Serra | The Boolean model and random sets[END_REF][START_REF] Stoyan | Stochastic geometry and its applications[END_REF] and references therein). A key-theorem relates the covariance function of stationary Boolean sets in the Euclidean space to the Poisson intensity of the point process and to the covariogram of the primary grain. In turn, the covariance function itself governs basic features of the model, such as the specific surface area and the integral range. The integral range is linked to the estimates on finite-size volumes of the random set volume fraction.

Another property of interest concerns the probability that a segment is entirely contained in the complementary set of a Boolean model. This probability takes a simple form for convex primary grains, which depends on the derivative of the covariogram at 0 [START_REF] Serra | The Boolean model and random sets[END_REF]. Linear erosion allows one to compute this probability numerically which is especially useful for model identification [START_REF] Peyrega | 3D morphological modelling of a random fibrous network[END_REF][START_REF] Jeulin | Gypsum morphological analysis and modeling[END_REF].

An important subclass of Boolean models, commonly used in material science for modelling heterogeneous materials, concern isotropic random sets. In dimension 3, this model requires one to average the covariogram over all directions uniformly on the sphere. In the rest of this chapter, the orientation-averaged covariogram, also denoted "isotropized covariogram" in the literature, will be referred to as "mean covariogram". The mean covariogram is a special case of so-called "kinematic" integrals (see [START_REF] Schneider | Stochastic and integral geometry[END_REF], Chap. 5) of the form (in dimension d):

ż G d V d pM X gAq µpdgq (8.1)
where a convex body A is moved into gA and the volume of the intersection of gA with a convex set M is averaged with respect to the measure µ over all transformations g in the motion group G d . The measure µ is invariant in G d . The mean covariogram required for describing isotropic Boolean models is obtained by taking translations at fixed distance uniformly distributed on the sphere for G d and µ and M " A. Unfortunately, such kinematic covariogram is usually not known explicitly, especially in 3D. Some notable exceptions include the sphere, parallelepiped, cylinder [START_REF] Gille | The intercept length distribution density of a cylinder of revolution[END_REF][START_REF] Gille | Particle and particle systems characterization: small-angle scattering (SAS) applications[END_REF] and Poisson polyhedra [START_REF] Matheron | Polyédres poissoniens et ensembles semi-markoviens[END_REF]. The mean covariogram for cylinders is useful for identifying stochastic fibrous models from experimental 2D (e.g. SEM) or 3D (e.g. tomography) images of fibrous materials [START_REF] Robb | Fiber orientation estimation from 3D image data: Practical algorithms, visualization, and interpretation[END_REF][START_REF] Peyrega | 3D morphological modelling of a random fibrous network[END_REF][START_REF] Redenbach | Statistical analysis and stochastic modelling of fibre composites[END_REF]. More generally, heterogeneous microstructures studied for various industrial applications can be approached by random models of cylinders. Examples include flakes in optics [START_REF] Couka | A mixed Boolean and deposit model for the modeling of metal pigments in paint layers[END_REF] or platelets in mesoporous materials [START_REF] Wang | Modeling of mesoporous alumina microstructure by 3D random models of platelets[END_REF]. This work focuses on the mean covariogram of 3D cylinders with circular cross-section and its applications to Boolean models. It is organized as follows: the covariogram of a cylinder is recalled in Sec. (8.2). The mean covariogram is derived in Sec. (8.3). Boolean models of cylinders are considered in Sec. (8.4). The integral range is given in Sec. (8.4.1). Variance properties are discussed in Secs. (8.4.2) and (8.4.3). We conclude in Sec. (8.5).

Covariogram of a cylinder

Consider a cylinder C of height h and circular cross-section of radius r, and a Cartesian coordinate system with origin O and axes e x , e y and e z . We assume that O is at the center of one of the bases of C and that e z is parallel to the cylinder main axis. Consider now the translation C 1 of the cylinder C by a vector v. We parametrize v by its norm t " |v| and two angles φ P r0; 2πs and θ P r´π{2; π{2s in spherical coordinates. The azimuthal angle φ is the angle between e x and the projection of v onto the plane p0; e x , e y q. The variable θ denotes the angle between v and the plane p0; e x , e y q so that θ " π{2 when v is parallel to e z and θ " 0 when v is contained in the plane p0; e x , e y q. Note that, using this convention, θ is the complementary of the polar angle.

The (oriented) covariogram of a cylinder Kpθ, tq is defined as the volume of the intersection of

C with C 1 : Kpθ, tq " L 3 pC X C 1 q , ( 8.2)
where L 3 denotes the Lebesgue measures in R 3 . The covariogram K depends on r, h, t and θ, but not on φ. For conciseness, the dependence on r and h is omited in the notation for K and in the notations for other variables hereafter. Let us introduce the ratios:

x " t 2r , y " h t . (8.
3)

The two variables x and y, not to be confounded with Cartesian coordinates, will be used preferentially to t, h and r. Furthermore, the variable t as argument of a function will be replaced indifferently by x or y. For instance Kpθ, tq is also denoted Kpθ, xq.

The expression for the covariogram K is derived from the formula of a covariogram of a disk in the plane and reads [START_REF] Harutyunyan | Covariogram of a cylinder[END_REF] where the mean is taken over all directions on the sphere, assuming the distribution of orientations is uniform on the latter. Here, the covariogram K is normalized by the cylinder volume and surface area of the unit sphere so that the function kptq " 1 when t " 0 and kptq " 0 when t " 8. Like K, the quantity k depends on r ě 0 and h ě 0.

Mean covariogram of a cylinder

In this section, the isotropized covariogram of a cylinder with circular cross-section is given. We refer to [START_REF] Gille | The intercept length distribution density of a cylinder of revolution[END_REF][START_REF] Gille | The chord length distributions of selected infinitely long geometric figuresconnections to the field of small-angle scattering[END_REF][START_REF] Gille | Particle and particle systems characterization: small-angle scattering (SAS) applications[END_REF][START_REF] Hs Sukiasian | Relation between the chord length distribution of an infinitely long cylinder and that of its base[END_REF] where this problem has been studied in details.

Prolate cylinders

In this section, we compute the limit k 8 ptq of kptq when h Ñ 8 with r and t fixed. The condition t ď t max (Eq. 8.5) reduces to x cos θ ď 1 which is satisfied for all θ when x ă 1 and for θ ą cos ´1p1{xq when x ą 1. Therefore the term K can be replaced by r K in (8.6) provided the integration is carried out in the intervals r0; π{2s (x ă 1) and rcos ´1p1{xq; π{2s (x ą 1):

k 8 pxq " # lim hÑ8 1 hπr 2 ş π{2 0 dθ r Kpθ, xq cos θ, if x ă 1, lim hÑ8 1 hπr 2 ş π{2 cos ´1p1{xq dθ r Kpθ, xq cos θ, if x ą 1. (8.7)
Observe however that, when x ą 1, r Kpθ, tq is purely imaginary in the domain 0 ă θ ă cos ´1p1{xq. Accordingly, it is sufficient to integrate in the interval r0; π{2s instead of rcos ´1p1{xq; π{2s, provided the imaginary part of the integral is discarded:

k 8 pxq " lim hÑ8 1 hπr 2 Re # ż π{2 0 dθ r Kpθ, xq cos θ + , ( 8.8)
where Re stands for the real part of complex numbers. Replacing K with its expression (8.4) and expanding the integrand at first order in 1{h yields:

k 8 pxq " 2 π Re # ż π{2 0 dθ cos θ " cos ´1 px cos θq ´x cos θ ? 1 ´x2 cos 2 θ ı + . ( 8.9)
As seen above, k 8 depends on t and r through x " t{p2rq only. The integral in (8.9) is readily computed in the complex domain using a symbolic calculator [262]. The expression depends on the complete elliptic integrals of the first and second kind, denoted F and E respectively and defined by:

F pzq " ż π{2 0 du ? 1 ´z sin 2 u , Epzq " ż π{2 0 du a 1 ´z sin 2 u. (8.10)
The above functions F and E are real-valued when z ď 1 and complex with non-zero real and imaginary parts when z ą 1. We refer to the online resources [START_REF] Weisstein | Complete Elliptic Integral of the First Kind[END_REF] (and references therein) for an overview of their properties. Note that the elliptic functions are usually defined by z Ñ F p ? zq, Ep ? zq. In this work we follow the notation used in [262].

The elliptic functions F and E appear in the integration of both the square root and inverse cosine terms in (8.9). For instance, when x ă 1, the cos ´1 term is integrated by parts as follows:

ż π{2 0 dθ cos ´1 px cos θq cos θ " ż π{2 0 x sin 2 θdθ ? 1 ´x2 cos 2 θ " ˆx ´1 x ˙F px 2 q `Epx 2 q x " ? 1 ´x2 x Im " E ˆ1 1 ´x2 ˙* , ( 8.11)
where Im denotes the imaginary part of complex numbers. The final formula for kpxq reads, after simplification in the regions x ă 1 and x ą 1:

k 8 pxq " $ & % 1 ´2? 1´x 2 πx ! Im " E `1 1´x 2 ˘‰ `2x 2 ´1 3 E ´x2 x 2 ´1 ¯`1 3 F ´x2 x 2 ´1 ¯), if x ă 1, 1 ´2? x 2 ´1 πx ! E `1 1´x 2 ˘`Im " 2x 2 ´1 3 E ´x2 x 2 ´1 ¯`1 3 F ´x2 x 2 ´1 ¯ı) , if x ą 1.
(8.12)

The function k 8 is represented in Fig. 8.1a (top curve).

Cylinders of arbitrary aspect ratio

This section is devoted to the general case, i.e. finite h and r. The intervals of integration for θ are required for computing (8.6). Take first h ă 2r and examine the condition t ď t max in (8.5 As in Sec. (8.3.1), r K is purely-imaginary when θ P r0; cos ´1p1{xqs, so one may carry out the integration in (8.6) along intervals r0; as with a " π{2 or a " sin ´1pyq depending on conditions precised in (8.13) and (8.14). We are left with the evaluation of the following integral: where H 1 is the step function:

H 1 pxq " 1 r1;`8s " " 1 if x ă 1, 0 if x ą 1. (8.18)
As expected, the quantity between square brackets in (8.17) is real when x ă 1 and so is kpx, yq for all x.

We now consider the case y ă 1 (t ą h) and the integral in (8.15) with a " sin ´1 y. The symbolic calculator [262] provides a lengthy formula for the solution. After simplification, the expression takes different forms for x ă 1 and x ą 1. For x ą 1 and y ă 1, the covariogram k reads:

k yă1 xą1 px, yq " 4 ? x 2 ´1 3πx Im " F ˆsin ´1 y ˇˇx 2 x 2 ´1 ˙´p1 `x2 qE ˆsin ´1 y ˇˇx 2 x 2 ´1 ˙ `ˆy π `4x 2 ´1 4πyx 2 ˙cos ´1 ´xa 1 ´y2 1 2π ˆxy 3 `1 2xy `x y ˙a1 ´y2 a 1 ´x2 p1 ´y2 q. (8.19)
The functions of two variables F pφ|zq and Epφ|zq are the incomplete elliptic integrals of the first and second kind, respectively, defined by [START_REF] Weisstein | Elliptic Integral of the Second Kind[END_REF]:

E `φˇˇz˘" ż φ 0 du a 1 ´z sin 2 u, F `φˇˇz˘" ż φ 0 du ? 1 ´z sin 2 u . ( 8

.20)

The complete elliptic integrals in (8.10) are special cases: The incomplete elliptic integrals are real-valued if z sin 2 φ ă 1. Note that this is never the case for the arguments of E and F in (8. [START_REF] Barthélémy | Local field probability distribution in random media[END_REF], so that the two functions have complex values. Note also that, owing to t ď ? h 2 `4r 2 :

Epzq " E ´π 2 ˇˇz ¯, F pzq " F ´π 2 ˇˇz ¯. ( 8
x a 1 ´y2 ă 1, 1 ´x2 p1 ´y2 q ă 1, (8.22)
and so k yă1 xą1 px, yq in (8.19) is real. Now, when x, y ă 1, one finds for kpx, yq:

k yă1 xă1 px, yq " 4 ? 1 ´x2 3πx " F ˆsin ´1 y ˇˇx 2 x 2 ´1 ˙´p1 `x2 qE ˆsin ´1 y ˇˇx 2 x 2 ´1˙ `ˆy π `5x 2 ´2 4πyx 2 ˙cos ´1 ´xa 1 ´y2 ¯`2 ´5x 2 4πyx 2 cos ´1 x ´1 2π ˆxy 3 `1 2xy `x y ˙a1 ´y2 a 1 ´x2 p1 ´y2 q `1 `2x 2 4πyx ? 1 ´x2 `1 ´x2 4πyx 2 cos ´1 ´x2 a 1 ´y2 `?1 ´x2 a 1 ´x2 p1 ´y2 q ¯. (8.23)
Note the similarities with the formula for x ą 1 in (8.19). Also, all three expressions inside cos ´1 in the above are comprised between 0 and 1 and so kpx, yq is real.

To summarize, the exact expression for kpx, yq is given by (8.17) and (8.12) when y ą 1, by (8.19) when y ă 1 ă x, and by (8.23) when x ă 1 and y ă 1. The covariogram is plotted as a function of t in Fig. (8.1a) and (8.1b) for various values of r and h (solid lines), and compared with that of a sphere (dotted lines).

An asymptotic expansion of kptq when t Ñ 0 is now carried out. Use the expansion (m Ñ 0):

F `φˇˇm˘" φ `sinp2ψq ´2ψ 8 m `Opm 2 q, E `φˇˇm˘" φ `sinp2ψq ´2ψ 8 m `Opm 2 q, (8.24)
to find:

k 8 ptq " 1 ´t 2r `Opt 3 q, kptq " 1 ´1 2 ˆ1 r `1 h ˙t `2 3πhr t 2 `Opt 3 q, t Ñ 0. (8.25)
As expected the derivative of kptq at t " 0 is equal (up to the sign) to the surface/volume ratio of the cylinder.

The second-order derivative of kptq becomes infinite at the point t " h [START_REF] Gille | The intercept length distribution density of a cylinder of revolution[END_REF]. Connections between the second-order derivative of the covariogram of a compact set and and its singular points have been previously noticed in the plane [START_REF] Emery | Geometric covariograms, indicator variograms and boundaries of planar closed sets[END_REF]. We now consider the domain h " r and compute an approximation of kptq valid in the region h " r. This task is more difficult than in the oblate case h ! r because the exact result for h " 8 involves elliptic functions. In the domain 2r ă t ă h, we let h and t Ñ 8 with r and the ratio h{t fixed and expand (8.17) to Oph ´4q. We use the same expression in the domain t ą h, as long as the latter is positive and 0 otherwise. In the domain t ă 2r, we expand (8.19) when h, t Ñ 8 with r and the ratio h{t fixed, to Oph ´4q. One finds:

k(t) k r≫h (t) k r≪h (t) r = 1, h = 2.9 r = h = 1 k(t)
kptq « k r!h ptq " $ ' ' ' & ' ' ' % 1 ´t 2r « 1 ´1 2 ˆt 4r ˙2 ´1 4 ˆt 4r ˙4ff ´t 2h `2t 2 3πhr ˆ1 ´t2 40r 2 ˙, t ă 2r, max " 0; r 2 2t 2 ˆ1 ´t h `r2 2t 2 ˙*, t ą 2r.
(8.28) We emphasize that, contrary to (8.27), the expressions above are not asymptotically correct in the limit h Ñ 8. Also, approximation k r!h ptq is discontinuous at t " 2r. The maximal error sup t |kptq ´kr!h ptq| is attained at t Ç 2r, is constant and equal to about 1.1% for h ą 2r. In the domain 1.6r ă h ă 2r, the maximal error is less than 1.4% (Fig. 8.2).

To summarize, a good approximation of the covariogram is given by (8.27) when h ă 1.6r and by (8.28) when h ą 1.6r with a maximal absolute error of 1.8%.

Boolean model of cylinders

In this section, we consider a Boolean model [START_REF] Matheron | Random sets and integral geometry[END_REF] of cylinders with radius r and height h. The Boolean model is defined by an homogeneous Poisson point process of intensity ψ (average number of points per unit area). A cylinder C oriented in a random direction, uniformly distributed on the sphere, is implanted on each Poisson point. The cylinder volume fraction, denoted p, is linked to the Poisson intensity through [START_REF] Matheron | Random sets theory and its applications to stereology[END_REF]: logppq " ´ψπhr 2 .

(8.29)

Hereafter, we examine the integral range of Boolean random sets made of cylinders. The Boolean model, denoted B is the union of all cylinders C, implanted at Poisson points so that cylinders may interpenetrate. Its characteristic function is denoted χ B .

Integral range

The covariance Cptq of the Boolean model of cylinders B reads:

Cptq " tz P B, z `t P B, |t| " tu " 2p ´1 `p1 ´pq 2´kptq . (8.30)

Its integral range is defined by [START_REF] Matheron | Random sets and integral geometry[END_REF]:

A 3 " 1 pp1 ´pq ż 8 0 dt 4πt 2 " Cptq ´p2 ‰ " 4πp1 ´pq p ż 8 0 dt t 2 " e ´kptq logp1´pq ´1‰ , ( 8.31)
where the integrand Cptq ´p2 is the centered covariance. For cylinders the centered covariance is identically zero for t ě t c where t c " ? 4r 2 `h2 is the maximum length of a chord in the cylinder. The integral range is useful to quantify the representative volume element for the set B. Using the exact expression derived in Sec. (8.3.2), we compute the normalized covariogram:

r A 3 " A 3 πhr 2 , ( 8.32)
for r " 1{2 (Fig. 8.3). The latter is equal to 1 when p " 0. Remark that the integral range is very close, but not equal, to that of a Boolean model of sphere of diameter 1:

r A sphere 3 " 6 πp ż 1 0 dt " p1 ´pq 3t{2´t 3 {2 ´1 `pı . ( 8.33)
When p " 1{2, the maximum value of A 3 is found to occur for h very slightly smaller than 1 (Fig. 8.4). This value is still smaller than r A sphere

3

. Furthermore, numerical computations show that the normalized integral range A 3 is almost unchanged when h is changed to 1{h (Fig. 8.3).

Variances and representative volume element: prolate and oblate cylinders

Assume p W is the measure of the volume fraction of cylinders over a domain W of volume V : .34) Denote p W the mean of the estimates p W i taken over N independent realizations W " W 1 , ..., W N of volume V . The variance D 2 B pV q of the estimates p W i is given by: Taking N large and p W « p in the above:

p W " L 3 pB X V q V " ż W du χ B puq. ( 8
D 2 B pV q " 1 N N ÿ i"1 " 1 V ż W i du χ B puq ´pW  2 " 1 N N ÿ i"1 « ˆ1 V ż W i du χ B puq ˙2 ´p2 W ff . ( 8
D 2 B pV q " 1 N V 2 N ÿ i"1 żż u,vPW i dudv " χ B puqχ B pvq ´p2 ‰ . ( 8.36)
Take W i Ñ R 3 and make use of the variable change t " v ´u. The double integral amounts to an integral of the centered covariance function and so:

D 2 B pV q " 1 V ż R 3 dt " Cptq ´p2 ‰ " pp1 ´pq A 3 V , ( 8.37)
a result first derived by Matheron [START_REF] Matheron | Random sets and integral geometry[END_REF]. The asymptotic expansion (8.37) is actually valid for volumes V much larger than A 3 and so requires that A 3 (and t c ) is finite. The latter shows that, when V " A 3 , the volume W acts as n independent domains of volume A 3 with n " W {A 3 . Thus the volume A 3 is said to be "representative" of the Boolean model B. Note however that the variance D 2 B pV q also depends on the point variance pp1 ´pq. When A 3 is infinite, a scaling law different from (8.37) is expected [START_REF] Lantuéjoul | Ergodicity and integral range[END_REF]. For prolate and oblate cylinders, the theory respectively predicts [START_REF] Jeulin | Power laws variance scaling of Boolean random varieties[END_REF]:

D 2 B pW q " α V 2{3 , D 2 B pW q " α 1 V 1{3 , V " A 3 , ( 8
k 8 ptq " r 2 2t 2 `O ˆ1 t 4 ˙, pif h " `8q, k 0 ptq " h 2t `O ˆ1 t 2 ˙, pif r " `8q. (8.40)
For prolate and oblate cylinders, the integral of the centered covariance on a spherical domain of radius diverges as:

A 3 p q " ż tď p4πt 2 qdt " Cptq ´p2 ‰ " # ´2πp1´pq p r 2 logp1 ´pq , if h Ñ 8, ´πp1´pq p h logp1 ´pq 2 , if r Ñ 8. (8.41)
We remark that A 3 p q{ 3 behave as " V ´2{3 and " V ´1{3 respectively for prolate and oblate cylinders, where V is the sphere of radius . This qualitatively explains the expansions (8.38). B pV q of the estimate of the volume fraction measured in a spherical domain W vs. volume size V of the domain for oblate cylinders (r " `8). The cylinders height is fixed to h " 1. Top to bottom: volume fraction of cylinders p " 0.5, 0.4, 0.6, 0.3, 0.7, 0.2, 0.8, 0.1, 0.9.

However, the variable change t " v ´u leading to (8.37) can not be directly carried out for finite domains W i and infinite integral range. We derive it here for spherical domains W i of radius . One needs to compute the probability dP pt, q that two points A and B in a sphere are separated from a distance in the interval rt; t `dts. We first fix A, a point at a distance 0 ă a ă of the sphere center. The volume of points at a distance in the interval rt; t `dts from A is given by: dV

D 2 B pV q " 1 N N ÿ i"1 1 V 2 żż u,vPW i dudv " χ B puqχ B pvq ´p2 ‰ « ż tď2 dP pt, q " Cptq ´p2 ‰ . (8.44)
The integral above is analytically solvable as a closed-form expression, for oblate cylinders (r " 8).

When 2 ą h, the solution involves the exponential integral function E i pzq " ´ş8 ´z e ´s{sds. The exact solution is a lengthy expression, which, for conciseness, is not given here. It is provided by the software Mathematica [262]. The variance D 2 B pV q is plotted as a function of V for various volume fraction of cylinders in Fig. (8.5).

Taking the limit Ñ 8 and using the expansion (8.40) yields:

D 2 B pV q "
´9p1 ´pq 2 r 2 logp1 ´pq 8 2 `Op1{ q 3 , ph " `8q, (8.45a)

D 2 B pV q "
´3hp1 ´pq 2 logp1 ´pq 5

`9h 2 p1 ´pq 2 rlogp1 ´pqs 2 32 2 `Op1{ q 3 , pr " `8q. (8.45b) when Ñ 8, with V " p4{3qπ 3 . Note that the results above are identical up to a constant factor to those obtained by replacing A 3 in (8.37) by A 3 p q from (8.41).

Variances and representative volume element: cylinders with finite height and radius

This section is concerned with the behavior of D 2 B pV q for r or h large but not infinite. We first examine h large. The integral in Eq. (8.44) is computed numerically for increasing values of h " 10, 10 2 , ..., 10 4 with r " 1{2 fixed (Fig. 8.6a). The data is compared to h " 8 (solid line, top) and to the expansion (8.45) (dashed line, top). The variance D 2 B pV q asymptotically scales as " 1{V for very large V , as long as h is finite. However, for h ě 10, an intermediate regime appears where D 2 B pV q " 1{V 2{3 . This scaling law occurs for V ! h 3 , i.e. ! h. It is very close to the asymptotic limit (8.45). The change between the two regimes takes place, as expected, when is of the same order as h. Numerical data indicates « 3h. A similar behavior happens for r " h with h " 1 fixed (Fig. 8.6b): when " r, the scaling law D 2 B pV q " 1{V holds, whereas D 2 B pV q " 1{V 1{3 when ! r. The change between the two regimes occurs when r « . Pluging " 3h and " r into (8.45) yields:

D 2 B pV q " ´27p1 ´pq 2 logp1 ´pqhr 2 8 3 , p " h " rq, ( 8.46a

Conclusion

In this chapter, the covariance and integral range of the Boolean model of cylinders have been computed using the geometrical covariogram of cylinders. As expected, the integral range takes the form of a divergent integral for flat (oblate) or highly-elongated (prolate) cylinders. This results in peculiar scaling laws of the variance of the measurement of the volume fraction of cylinders over subdomains of volume V . The latter variance scales as " V ´1{3 for the oblate and " V ´2{3 for the prolate type when V Ñ 8. The lowest-order correction in the asymptotic expansion for the variance has been derived for domains of volume V Ñ 8 with a spherical shape. For cylinders with finite (but large) aspect ratio, the scaling law " V ´1{3 or " V ´2{3 occurs in an intermediate region where L " V 1{3 is comprised between the lowest and highest dimensions of the cylinders. The classical scaling law " 1{V is recovered when L is much larger than both cylinders dimensions. Such scaling laws should in turn affect the size of the representative volume element relative to the apparent thermal and elastic response of architectured materials [START_REF] Dirrenberger | Computational Homogenization of Architectured Materials[END_REF][START_REF] Dirrenberger | Towards gigantic RVE sizes for 3D stochastic fibrous networks[END_REF], fiber-reinforced [START_REF] Altendorf | Influence of the fiber geometry on the macroscopic elastic and thermal properties[END_REF] or quasi-brittle composites [START_REF] Pélissou | Determination of the size of the representative volume element for random quasi-brittle composites[END_REF].

numerical methods, such as the Lattice Boltzmann method, and comparisons with analytical formula for regular arrays have been used. In one of the first such study [START_REF] Jackson | The permeability of fibrous porous media[END_REF], an analytical estimate, based on numerical predictions, is proposed for disordered fibrous media. The estimate has been found to be accurate for a porosity less than about 25% [START_REF] Clague | Hydraulic permeability of (un) bounded fibrous media using the lattice Boltzmann method[END_REF]. A fiber-web material, with a structure similar to that found in fibrous sheets or paper, has been investigated numerically in [START_REF] Koponen | Permeability of three-dimensional random fiber webs[END_REF]. Extensive numerical computations have been performed in [START_REF] Claeys | Suspensions of prolate spheroids in Stokes flow. Part 2. Statistically homogeneous dispersions[END_REF] on models of prolate spheroids with varying aspect ratio. More recently, the efficient Lattice Boltzmann method has been used [START_REF] Thomas | Prediction of the Permeability of Fibrous Porous Media Using the Lattice Boltzmann Method in Conjuction with Coarse Numerical Lattices[END_REF] to compute the permeability fibrous media, where the fluid flows outside of the cylinders. In one such study [START_REF] Nabovati | A general model for the permeability of fibrous porous media based on fluid flow simulations using the lattice Boltzmann method[END_REF], several models made of random cylinders are investigated. An empirical law is proposed for the permeability, based on the theoretical analysis of the permeability of regular arrays of cylinders [START_REF] Gebart | Permeability of unidirectional reinforcements for RTM[END_REF]. The effect of the cylinders curvature is found to be small compared to straight cylinders. Computations have also been carried out with varying cylinders aspect ratios.

The combined numerical and experimental study in [START_REF] Rief | Computer simulation of Air Filtration including electric surface charges in three-dimensional fibrous micro structures[END_REF] considers fibrous media. The authors investigate in particular the domain of validity of the relevant analytical estimates. In another experimental investigation, Darcy and Darcy-Forchheimer laws have been identified on packings of non-spherical particles, including prisms and cylinders [START_REF] Clavier | Experimental investigation on single-phase pressure losses in nuclear debris beds: Identification of flow regimes and effective diameter[END_REF].

Much less work has been devoted to the "reverse" model of fluid flow through fibers, despite some early works [START_REF] Matheron | L'émergence de la loi de Darcy[END_REF]. Systems of random flat cylinders have been investigated in hydro-geological literature on fractured rocks. The permeability of 3D random models of fracture networks has been investigated numerically and theoretically in various works [START_REF] Koudina | Permeability of threedimensional fracture networks[END_REF][START_REF] Lang | Permeability tensor of three-dimensional fractured porous rock and a comparison to trace map predictions[END_REF]. Among important geometrical factors that influence the permeability, authors have highlighted the role of contact regions [START_REF] Berkowitz | Characterizing flow and transport in fractured geological media: A review[END_REF] or that of a wide size distribution in fractures [START_REF] Mourzenko | Macroscopic permeability of threedimensional fracture networks with power-law size distribution[END_REF].

This chapter is organized as follows. The micrsotructure model is introduced in Sec. (9.2), the equations to solve for Stokes flow are given in Sec. (9.2). Analytical upper-bounds are derived in Sec. (9.3) for flows in Boolean models of cylinders in the dilute limit. Concluding remarks are given in Sec. (9.4).

Stokes flow in a Boolean model of cylinders

The microstructures considered hereafter consist in Boolean sets of cylinders. Boolean models [START_REF] Matheron | Random sets theory and its applications to stereology[END_REF] are a classical type of random models which depend on a primary grain and on a Poisson point process [START_REF] Miles | On the homogeneous planar Poisson point process[END_REF]. In the present work, the primary grain is defined by a distribution of cylinders, and the homogeneous Poisson point process by a spatially-constant intensity n, the mean number of points per unit volume. A cylinder is attached to each point of the Poisson point process, and the Boolean set is the union of cylinders. In the following, we choose to focus on cylinders with high aspect ratio that significantly depart from sphere packings. For such ideal materials, we expect in particular a strong influence of the microstructure on the permeability.

The cylinders have circular cross-section of radius r and height h but their main axis is randomly-oriented. The direction of the axis follows a uniform distribution on the sphere, resulting in (macroscopically) isotropic Boolean sets. Cylinder intersections are allowed so that the volume fraction of cylinder p varies from 0 to 1 and is given by [START_REF] Matheron | Random sets theory and its applications to stereology[END_REF]:

p " 1 ´q " 1 ´exp p´nV 0 q , (9.1) where V 0 " πr 2 h is the volume of a cylinder. The Boolean sets accordingly depend on n, h and r, or, equivalently q, h and r. In the following, cylinders with very small aspect ratio h{r are considered, with p fixed, as detailed below. All microstructures accordingly depend on two parameters, the cylinders volume fraction q and a characteristic sizes for the cylinder, either r or h.

Let h ! r and consider oblate cylinders. The configuration of interest is that of "porous cylinders" where the fluid flows inside cylinders. In the case h ! r, the cylinders have asymptotically the same shape as sections enclosed by two (randomly-oriented) parallel planes. The percolation threshold f C c again tends to 0 (we refer to [START_REF] Huseby | Geometry and topology of fracture systems[END_REF] for a study on the connectivity of such systems). Accordingly, f is defined in the entire interval rf C c ; 1s « r0; 1s. Note that, for r " 8, the "reverse" model where fluids flow outside the cylinders presents no interest as the complementary set does not percolate. A section of the microstructure considered is shown in Fig. (9.1).

We consider hereafter a viscous fluid of velocity u satisfying the Stokes equation in the pores:

µ∆u i pxq " B i ppxq, B i u i pxq " 0, (9.2)
where ppxq is the pressure at point x and µ the fluid viscosity. The fluid velocity is zero (u " 0) along the interface with the solid. Periodic boundary conditions are applied on the computational domain, of the form [START_REF] Ene | Equations et phénomenes de surface pour l'écoulement dans un modele de milieu poreux[END_REF]:

u #, x∇py " p∆P qe 1 , p ˚" p ´∆P x ¨e1 #, (9.3) here # denotes periodicity and ∆P is the macroscopic pressure drop, oriented along the first axis e 1 of a Cartesian coordinate system. The permeability k is a scalar for isotropic media. It is given by Darcy's law:

Upper-bound for the permeability of Boolean sets of oblate cylinders

The Doi [START_REF] Doi | A new variational approach to the diffusion and the flow problem in porous media[END_REF] upper-bound for permeability reads: where s is the specific surface area of the void-solid interface, and F vv , F sv and F ss are the "volumevolume", "surface-volume" and "surface-surface" correlation functions:

k ď 2
F vv ptq " xZpxqZpx `tuqy u , (9.6) F sv ptq " x|∇Zpxq|Zpx `tuqy u , (9.7) F ss ptq " x|∇Zpxq||∇Zpx `tuq|y u .

(9.8)

In the above, Z is the indicator function for the complementary B c of set B (the obstacles), ∇ is the gradient operator and u is a unit vector (|u| " 1). The quantity F vv is given by the probability that two points separated by a distance t lie in B c . The quantities F sv and F ss can also be interpreted in terms of probability, replacing the interface of B by an interphase of vanishingly small width δ.

The quantity F sv is proportional to the probability (divided by δ) that one point lies in B c and the other in the interphase. The quantity F ss is related to the probability that the two points lie in the interphase. Formula (9.5) is a rigorous bound which provides the exact lowest-order correction to the permeability for dilute concentration of spherical obstacles q Ñ 0 [START_REF] Rubinstein | Flow in random porous media: mathematical formulation, variational principles, and rigorous bounds[END_REF].

The correlation function F vv of a Boolean model of cylinders is given by [START_REF] Matheron | Random sets and integral geometry[END_REF]:

F vv ptq " P tz R B, z `t R B, |t| " tu " q 2 ´Kpt; h, rq , (9.9)

where Kpt; h, rq (not to be confounded with k) is the normalized mean cylinder covariogram. The latter is defined as the volume intersected by two cylinders as follows:

Kpt; h, rq " E " |C X C ´t| |C| * , ( 9.10)
where |C| is the volume of the cylinder C and C ´t is C translated by a vector of length t. The quantity is averaged over orientations uniformly distributed on the sphere and normalized by the cylinder volume, so that kpt " 0; h, rq " 1.

Case r " 8

We first consider the case of infinitely flat cylinders with r " 8 (Chap. 8). Note that h and q are kept finite in this limit. Accordingly, the Poisson intensity θ (mean number of cylinders per unit of volume) should tend to 0 in a manner inversely proportional to the cylinder volume. Indeed θ is related to q by [START_REF] Matheron | Random sets and integral geometry[END_REF]: q " exp `´θπr 2 h ˘, (9.11) and so θ " 1{r 2 Ñ 0. Accordingly, the microstructure should be understood as a set of randomlyoriented, very large cylinders, with a very low density (in number per unit volume), so that the cylinders volume fraction remains finite. This limit microstructure could be usefully compared to Poisson tessellations (or dilated versions thereof). We refer to [START_REF] Jeulin | Modèles de fonctions aléatoires multivariables[END_REF][START_REF] Jeulin | Modèles Morphologiques de Structures Aléatoires et de Changement d'Échelle[END_REF] for theoretical results on the latter. The covariogram reads (Sec. 8.2):

Kpt; h, rq " " 1 ´t{p2hq, t ă h, h{p2tq, t ą h. (9.12)

The formula for F sv and F ss in Boolean media are similar to that of F vv [START_REF] Torquato | Microstructure characterization and bulk properties of disordered twophase media[END_REF][START_REF] Rubinstein | Diffusion-controlled reactions: Mathematical formulation, variational principles, and rigorous bounds[END_REF] but quite more complex. The formula involve first and second-order derivatives at h 1 " h of the following functional:

Kpt; h, h The three expressions for the correlation functions allow one to compute the integrand in (9.5): ˆq2 s 2 ˙Fss ptq ´ˆ2q s ˙Fsv ptq `Fvv ptq "

q 2 h 4t , t Ñ 8, (9.15)
and so the integral (9.5) diverges and the upper-bound is infinite. The bound accordingly provides no information on the permeability of materials with "infinitely" flat-cylinders. This negative result will nevertheless prove useful for interpreting the results of Sec. (9.3.2).

Case r finite

We now seek for the computation of upper-bounds on the permeability for cylinders of the oblate type but with finite radius r. The isotropized covariogram K of general cylinders with r and h finite has been studied in Chap. [START_REF] Ambos | Numerical modeling of the thermal expansion of an energetic material[END_REF]. The exact form provided in is a rather complicate analytical expression involving incomplete Elliptic functions. Therefore, we use instead the approximate expression taken from (8.27): The previous upper-bound may be meaningful in the dilute limit (q ! 1) only as shown in [START_REF] Willot | The permeability of Boolean sets of cylinders[END_REF], where FFT computations have been carried out for this model. Therefore we focus on the dilute regime q Ñ 0. Fix r and expand the above to zero-order correction Op1q in the dilute limit q Ñ 0 and let r Ñ 8 afterwards. We obtain:

Kptq « K app ptq
k ď 8h 2 3qplog qq 2 " 1 `ˆ1 2 log q ´1˙? q  `Op1q, q Ñ 0. (9.19)
At the lowest-order correction, one recovers an asymptotic regime " 1{rqplog qq 2 s. We remark that this constitutes a higher correction than that predicted by the James-Jackson model [START_REF] Jackson | The permeability of fibrous porous media[END_REF] which provides:

k " ´3a 2 log q 20q , q Ñ 0, (9.20) for the permeability of fibrous media of radius a and volume fraction q. Estimate (9.20) is itself higher than the dilute expansion for the permeability of a model of hard spheres of volume fraction q, which scales as " 1{q when q Ñ 0. Interestingly, the second leading-order term in (9.19) scales as 1{p ? q log qq. A similar nonanalytic dependence appears in the dilute limit expansion of the permeability of a bed of totally impenetrable spheres, obtained by self-consistent methods [START_REF] Childress | Viscous flow past a random array of spheres[END_REF]:

k " 2a 2 9q
" 1 ´3 ? 2 ? q `Opq log qq  , q Ñ 0, (9.21) where a is the sphere radius. The ? q term is related to hydrodynamic screening effects [START_REF] Brady | Stokesian dynamics[END_REF][START_REF] Tam | The drag on a cloud of spherical particles in low Reynolds number flow[END_REF]. For a dilute packing of spheres, as shown in [START_REF] Rubinstein | Flow in random porous media: mathematical formulation, variational principles, and rigorous bounds[END_REF], no screening term is predicted by the bound (9.17) or by Doi's bound [START_REF] Doi | A new variational approach to the diffusion and the flow problem in porous media[END_REF]. As shown in expansion (9.19), a screening term is indeed predicted by the variational bound, for the particular highly-elongated shapes considered here. Note that in the dilute limit considered here, the obstacles are not cylinders, but have a variety of shapes with very large surface/volume ratio.

Although the bound (9.17) is in general less accurate than Doi's bound which uses the surfacesurface correlation information [START_REF] Rubinstein | Flow in random porous media: mathematical formulation, variational principles, and rigorous bounds[END_REF], both are relevant in the dilute limit. Doi's bound predicts the correct leading-order term in the dilute limit for Boolean spherical obstacles. The bound (9.17) predicts the correct scaling law " a 2 {q, with a slightly overestimated prefactor 4{15 « 0.27 instead of 2{9 « 0.22 for the Boolean model of spheres.

Compute now the limit r Ñ 8 and afterwards q Ñ 0 in (9.18): k ď ´8hr 9π log q `Op1q, r Ñ 8. (9.22)

The expansion above depends on r unlike (9.21) and tends to `8 which explains the result obtained previously for Doi's bound with r " 8. When taking the limits r Ñ 8 and q Ñ 0, 9.17) for the permeability k of a Boolean model of flat cylinders with height h " 1 and radius r " 10 2 , 10 3 , 10 4 vs. concentration of obstacles q, in loglog plot (black). Dark-gray solid lines: r Ñ 8 expansion (9.22). Light-gray solid line: dilute limit expansion q Ñ 0 (9.21). (b) Permeability k vs. volume fraction of obstacles q (x-axis) in log-log plot, for a Boolean model of flat cylinders of height h " 1 and radius r " 10 4 (a.u.). Comparison between the Berryman-Milton upper-bound (9.17, solid blue line), its two contributions over t ă h and t ą h (orange and green solid lines), and expansions (9.19, blue dashed line) and (9.22, orange dashed line).

different regimes are obtained depending on the order one takes the limits. The two regimes appear when the permeability is plotted as a function of q in log-log plot (Fig. 9.2a). The regime change occurs at some points r « r c pqq, q « q c prq which are obtained by equaling the two highest-order corrections (9.19) and (9.22): r c " 3πh q c p´log q c q 3 . (9.23)

When r " r c pqq, or equivalently q ! q c prq, the permeability follows expansion (9.22). Expansion (9.21) holds in the domain r ! r c pqq or q " q c prq.

Comparison with FFT data would be needed to confirm the existence of the two regimes, however, accurate computations in the dilute limit requires high computational costs [START_REF] Abdallah | Stokes flow through a Boolean model of spheres: Representative volume element[END_REF]. This is left to future work.

Dilute and "correlated" regimes

To interpret the two regimes identified above, we decompose the Berryman-Milton upper-bound into two contributions, corresponding to near and far-field terms. The integral in (9.17) is split into domains r0; hs and rh; 8s. (Fig. 9.2b, orange and green solid lines). The former and latter contributions match expansions (9.19) and (9.22) rexpectively. As expected, the dilute behavior q ! q c corresponds to the covariance's main contributions in the range r0; hs, whereas the non-dilute regime q " q c is a controlled by the behavior of the covariance in the interval t " h. These observations allow us to interpret the two regimes predicted by the covariance bound. On the one hand, when q ! q c , the medium is made of uncorrelated, isolated obstacles of size 9h (Fig. 9.3a), and the permeability scales as k " h 2 (9.19), i.e. the permeability in this regime does not depend on r, at first-order. When q " q c , on the other hand, the fluid flow is driven through large-scales structures, which constrain the fluid path into the flat cylinders. This is illustrated in Fig. (9.3b) which shows areas free of obstacles in-between clusters of particles. Some particles are clearly aligned to one another. Accordingly, the permeability in this regime scales as k " hr (9.22), highlighting the role of the large-scale structures (of size " r). Also, the bound predicts that the permeability increases much more with respect to porosity in the highly-dilute regime q ! q c . This is consistent with our intuition, which suggests that, in the non-dilute regime q " q c , the fluid flow inside clusters of particles is much smaller than in zones that are free of particles. Hence, removing obstacles has a lesser effect on the macoscopic flow than in regime q ! q c .

Conclusion

In this chapter, the permeability of a Boolean model of very flat cylinders in the case where the fluid flows inside the cylinders. The flat cylinders are tantamount to models of planes of finite width. Because of the cylinder overlap, the resulting microstructure, which consist, in the high porosity limit, of isolated obstacles, displays markedly anisotropic structures and long-range correlations. It should be noted that strongly-anisotropic arrangements are not uncommon in materials, and in fact appear in a wide array of applications [START_REF] Wang | Modeling of mesoporous alumina microstructure by 3D random models of platelets[END_REF][START_REF] Brisard | FFT-based methods for the mechanics of composites: A general variational framework[END_REF].

In this chapter we have investigated the behavior of the Berryman-Milton upper-bound in the dilute limit of such a Boolean set, making use of an approximate formula for the covariogram of cylinders. The upper-bound predicts two distinct regimes depending on the diameter of the cylinders relative to the volume fraction of obstacles. In the strongly-dilute regime, a scaling law " 1{rqplog qq 2 s is predicted, whereas a much weaker regime " log q arises when r " h{q.

Chapter 10 Conclusion

The main problem tackled in this manuscript concerns a nonlinear medium in the continuum, in two dimensions, weakened or reinforced by a population of voids or rigid particles. The matrix is rigid perfectly-plastic (in deformation theory) and subjected to anti-plane shear. In the equivalent conducting problem, the particles are either insulating or perfectly-conducting, and the matrix exhibits a threshold-type nonlinear constitutive law. The macroscopic behavior is characterized by an effective plastic yield stress or effective current threshold.

Bounds (4.20) and (4.37), computed in Chap. ( 4), together with the predictions of nonlinear homogenization theories, have allowed us to gain insights on the effective behavior of such nonlinear medium in the porous case. Namely, in the dilute porosity limit, the effective yield stress relative to the yield stress in the matrix tends to unity with a correction that scales as the porosity to the power 2{3. Although this scaling law holds for Boolean media with homogeneous Poisson point process intensity, it is in fact much more general. The location of particles in random sequential adsorption models, for instance, is asymptotically equivalent to a Poisson point process in the dilute limit, so these models are also identical to Boolean random sets for a dilute concentration of particles [START_REF] Månsson | Random patterns of nonoverlapping convex grains[END_REF]. A key for deriving bound (4.20) is the property that the presence of the center of a pore in a given domain is independent of the presence of another pore center in a disjoint domain. As a consequence, the correlation function of the "one-scale" Boolean model B E (Chap. 6) vanishes at finite distance. That is, the probability that a point x is in a pore is independent of the probability that point x `h is in a pore, for a finite value of h. This property holds for any media with finite integral range, the integral range being the integral of the correlation function [START_REF] Matheron | Estimating and choosing: an essay on probability in practice[END_REF]. It would be interesting to investigate if microstructures with finite integral range exhibit a "dilute exponent" equal to 2{3.

The properties of a multiscale dispersion of equisized pores has been studied in Chap. ( 6), with emphasis on the dilute limit. By tailoring the variation of the volume fraction of clusters with respect to the total porosity in the medium, it has been shown that a non-analytic behavior may be obtained, different from that observed for a homogeneous distribution of pores. This non-analyticity is characterized by a powerlaw with exponent greater than 2{3 and strictly lower than 1, for a two-scale medium. The highest exponent, and consequently the lowest effect of the pores, is attained when the shear bands are flat at the scale of the clusters, but would in fact display rugosity at the largest scale if the porosity was only very slightly increased. By increasing the number of scales, analytical bounds predict that any exponent inbetween 2{3 and 1 may be obtained. For infinitely-many scales, an exponent 1 is obtained for most arrangements (among the 137 ones considered in Sec. 6.3.3).

As predicted by Duxbury et al. [START_REF] Duxbury | Critical manifolds in non-linear response of complex materials[END_REF], the non-analyticity of the effective behavior with respect to the volume fraction is a signature of the localization of the strain field and of the strong interaction of pores along shear bands. Clearly, as shown by the multiscale structures studied in Chap. ( 6), the exponent depends in general on the tortuosity at the different scales of the minimal path.

Furthermore, analytical bounds obtained in the dilute limit in Chap. (4), in particular (4.31), suggest that the effective behavior depends on the shape of the pores through the width of the inclusion, measured in the direction parallel to the applied load or to the macroscopic direction of the geodesics. As a consequence, according to bounds, elongated inclusions (tantamount to segment of lines, or open cracks) have the most important weakening effect on the effective yield stress at fixed porosity -an unsurprising result. In contrast to this, disk-shaped inclusions have, in the dilute limit, the least important weakening effect on the effective yield stress, in the space of shapes of constant width resulting in an isotropic macroscopic behavior.

Although the bounds on geodesics extend to multidimensional spaces (Sec. 4.4.4), the latter can not be used to predict the behavior of minimal surfaces spanning media in three dimensions. Therefore, they can not be used at the moment to assess the validity of the predictions of nonlinear homogenization theories in three dimensions. To achieve this goal, a description of "sufficiently general" surfaces spanning a volume would be required. Another difficulty concerns the extension of the bounds in Chap. (4) to polydisperse inclusions, for instance a population of pores of two different sizes. Such a bound is not available at the moment. Nevertheless, results obtained in Chap. [START_REF] Altendorf | Influence of the fiber geometry on the macroscopic elastic and thermal properties[END_REF] for model B P suggest that the exponent 2{3 is unchanged in this case, although the prefactor is not.

The dual problem related to the response of a nonlinear medium containing a population of rigid particles, could not be investigated by the same integral geometry techniques. This is because particles interpenetration is a major difficulty that prevents us from deriving rigorous bounds equivalent to the ones in Chap. [START_REF] Aa Abrikosov | Experimentally observed extended saddle point singularity in the energy spectrum of YBa 2 Cu 3 O 6.9 and YBa 2 Cu 4 O 8 and some of the consequences[END_REF]. The random sequential adsorption model is a random set in which particles interpenetration are forbidden, very few analytical result is known about this model, save in one dimension [START_REF] Torquato | Nearest-neighbor statistics for packings of hard spheres and disks[END_REF], which prevents us to derive bounds.

To alleviate this difficulty, we have turned to random checkerboard materials where cells do not interpenetrate. A rigorous bound has been obtained for a random checkerboard of square-shaped cells. In the dilute limit of rigid particles volume fraction f Ñ 0, the bound's leading-order term in f scales as " | log f |f 3 . A markedly different behavior " | log f |f 2 was obtained for a random checkerboard of rectangles which are not aligned to each other. We argue that the much lower bound obtained in the former case is an effect of particle alignment and long-range correlations, although long-range correlations are also present in the rectangular case. In any case, the scaling " f 2 has been numerically confirmed for the RSA model of squares, and is qualitatively close to the predictions of the bound obtained for the random checkerboard of rectangles. Finally, note that the same scaling law " f 2 is predicted by the "second-order" nonlinear homogenization theory [START_REF] Furer | A symmetric fully optimized second-order method for nonlinear homogenization[END_REF], a result which supports our conclusion that a " f 2 correction holds for spatial distributions without long-range correlations.

Extending our results for multiscale dispersions of rigid particles, it has been found that the exponent in the dilute limit also varies when the number of scales increases. Specifically, for multiscale random materials containing rigid particles, we have shown that any exponent between 1 and 2 may be obtained. In the limit of a material with infinitely-many scales, made of particles arranged into clusters, themselves arranged into superclusters (etc.) an exponent 1 (i.e. a linear correction in f ) is recovered. Different regimes are predicted which depend on the fact that shear bands may avoid entire clusters, or pass through them while avoiding rigid particles inside each cluster they go through. The rigid particles have the highest strengthening effect when the length of the minimal path is the same for a path that avoids clusters entirely or for a path that goes through them. This linear correction in the limit of infinitely-many scales explains the behavior of certain rigorous upper-bounds [START_REF] Goldsztein | Bound on the yield set of fiber reinforced composites subjected to antiplane shear[END_REF]. It would be useful to extend these results to random microstructures in three-dimensions, such as the ones considered in [START_REF] Fritzen | Computational homogenization of elasto-plastic porous metals[END_REF] and [START_REF] Fritzen | Computational homogenization of porous materials of Green type[END_REF].

The results obtained in the present work also highlight the limits -or the extent -of the domain of validity of linear and nonlinear homogenization theories, as the nonlinearity or heterogeneity contrast becomes large. Chap. [START_REF] Amar | On the Finsler metrics obtained as limits of chessboard structures[END_REF] has investigated how shear bands may develop from a strictly convex powerlaw potential, in the context of periodic structures. Different regime changes have been identified, in the porous and rigid case, including in plane strain. The study may be usefully compared to [START_REF] Willot | Effectivemedium theory for infinite-contrast two-dimensionally periodic linear composites with strongly anisotropic matrix behavior: dilute limit and crossover behavior[END_REF] and Chap. [START_REF] Abdallah | Stokes flow through a Boolean model of spheres: Representative volume element[END_REF] where a regime change occurring in linear, stongly-anisotropic media is reported. In particular, it would be interesting to generalize these results to randomlydistributed particles.

The two closing chapters (Chaps. 8 and 9) present a study of the behavior of a Stokes flow passing through a medium with statistically-aligned obstacles. In the dilute limit, the fluid avoids a set of obstacles. While the latter are isolated from one another, their positions are correlated to each other so that they form organized patterns corresponding to long-range alignments. In this regime, the Berryman-Milton upper-bound predicts markedly different behavior of the permeability than when the obstacles are isolated and their positions are uncorrelated. Namely, the increase of the permeability with respect to porosity is much smaller in the former case.

Part IV

Addenda

T his work is a study in theoretical mechanics devoted to homogenization problems in the con- text of degenerate (non-strictly convex) local response of one of the phases, which can serve as idealized models for porous or rigidly-reinforced materials exhibiting perfectly-plastic behavior. In these situations plastic flow preferentially concentrates along shear bands; as a result the material effective response is governed by those regions within the material where the field localizes. A form of localization also occurs in linear problems governed by asymptotically hyperbolic partial derivative equations, where the strain field is found to develop banding patterns. The solutions are relevant to strongly-anisotropic elastic or thermoelastic media. Such a linear problem is studied in the present work in the context of a random microstructure, specifically a polycrystal containing cracks. A related topic is whether incipient localization may develop as a result of the microstructure itself. This question is investigated in the case of a Stokes flow occurring in a porous medium around a set of obstacles exhibiting unusual spatial distribution, characterized by long-range correlations. The different model problems are addressed by means of various techniques. Limit analysis bounds are combined with integral geometry to provide insight on the material behavior. Comparisons with rigorous bounds and estimates of homogenization theories, and full-field "Fourier-based" numerical results, allow us to interpret and assess the response of certain random microstructures. Although a complete solution is out-of-reach, the interplay between field localization and microstructure is elucidated in specific cases.

Figure 1 . 1 :

 11 Figure 1.1: A shear band going through two pores in plane strain. Perfectly-plastic medium. Numerical computation.

Figure 2 . 2 :Figure 2 . 3 :

 2223 Figure 2.2: Random model of polycrystal containing a population of cracks: Poisson-Voronoï tessellation superimposed on a Boolean model of penny-shaped cracks (2D cut).

Figure 2 . 4 :

 24 Figure 2.4: Self-consistent estimate SC 1 for the effective response of a cracked polycrystal vs. crack density η, for varying in-plane Young modulus E p ď 10 with ν zp " 0.2, E z " 1, α 11 " 1 and α 33 " 0. (a) Young modulus r E, (b) Poisson ratio r ν, (c) bulk modulus r K, (d) shear modulus r G, (e) thermal expansion coefficient r α " r α V {3.

Figure 2 . 5 :Figure 2 . 6 :

 2526 Figure 2.5: (a) Bulk modulus r K vs. crack density η as the out-of-plane shear modulus G zp tends to 0 (G zp " γG 0 , G 0 " 5{12). Solid lines: SC 1 . Symbols: FFT results. (b) Bulk modulus r K vs. G zp when η " 0.3815, in lin-log plot. Symbols and dashed line: FFT results. (c) Effective Young modulus r E vs. crack-density η as E z Ñ 0.

Figure 2 . 7 :

 27 Figure 2.7: FFT map of the strain shear component ε 11 ´ε33 , parallel to the applied load, in the plane (e 1 ; e 3 ), in the elementary cell of a periodic material with strong anisotropy ratio k " 10´3

  z =1 ν p =ν zp =0.2

Figure 2 . 8 :

 28 Figure 2.8: Effect of scale separation on the elastic response of a cracked polycrystal as a function of η: effective bulk modulus rK at increasing values of the out-of-plane shear modulus G zp , monitored by the anisotropy ratio γ " G zp {G 0 (G 0 " 5{12). Solid lines: cracks and grains of similar size. Dotted lines: cracks much larger than the grains. A different gray color is used for each value of γ.

Figure 3 . 1 :

 31 Figure 3.1: (a) Poisson-Johnson-Mehl tesselation of space made of elongated grains (2D cut). (b) Mean over three realizations of the distribution P εxx of the strain component ε xx in Poisson-Voronoi (dashed gray line) and Poisson-Johnson-Mehl (black) polycrystals. Inset (bottom-right): individual distributions, computed in each one of the three realizations.

Figure 3 . 2 :

 32 Figure 3.2: (a,c,e) Probability distribution function P ε for the mean and equivalent strains (ε m , ε eq ) and strain components (ε xx , ε xy ), (b,d,f) Probability distribution function P σ for the mean and equivalent stress (σ m , σ eq ) and stress components (σ xx , σ xy ). Gray solid line: Selfconsistent estimate; Black solid line: FFT computation. (a,b) Overall hydrostatic loading with xε m y " 1. (c,d) Shear loading with xε xy y " 0.5 (e,f) Uniaxial strain loading with xε xx y " 1.

Figure 3 . 3 :

 33 Figure 3.3: Coordinate systems around a crack (hashed area), in plane strain.

Figure 3 . 4 :

 34 Figure 3.4: Maps of the stress components σ ij around an isolated crack with σ k " 1 GPa (Eqs. 3.14), in a domain 2a ą x ą 0, y ą 0 (see Fig. 3.3). Annotated regions enclosed by contour lines determine the behavior of P ij ptq as t Ñ 0, ˘8. Regions A, A 1 , A 2 , A 3 : t Ñ 0. Region B: t Ñ 8. Region C: t Ñ ´8. To highlight the field patterns, values lower than a minimum and higher than a maximum are thresholded out (see legend, units are GPa). Lowest values are shown in blue and highest values in yellow. Intermediate values in green and orange.

Figure 3 . 5 :

 35 Figure 3.5: (a) Probability distribution functions P ij ptq for the stress component σ ij around an isolated crack of length a, in a disk-shaped domain of radius D centered around the crack. (b) Probability distribution functions P ij ptq for the stress component σ ij around a set of isolated cracks of length a with uniform orientations, in disk-shaped domains of radius D centered around each crack.

Figure 3 . 6 :

 36 Figure 3.6: Numerical fit of the probability distribution functions P xx , in log-log plot (a) and P xy , in semi-log plot (b) for components σ xx and σ xy , in media with randomly-oriented cracks. Solid line in (a): fit (3.27); in (b): fit (3.28), up to a constant term. Inset in (b): log-log plot of the data with fit P xy " 5.5pa{Dq 2 pt{σ k q ´0.1 .

Figure 3 . 7 :

 37 Figure 3.7: Probability distribution functions r P yy ptq and r P ẙy ptq (solid black lines) for the stress component σ yy , in a matrix containing a set of (a) parallel and (b) randomly-oriented cracks, resp., under plane strain. Gray solid lines: FFT results obtained on two random realizations. Solid line marked G: Gaussian distribution. Inset: enlargement in the region r P ẙy ptq ă 0.5.

Figure 3 . 8 :

 38 Figure 3.8: Probability distribution functions P ˚,F F T yy

(3. 43 )Fits ( 3 .Figure 3 . 9 :

 43339 Figure 3.9: (a) Log-log plot of the probability distribution function P ˚,F F T yy

Fig. ( 3 Figure 3 .

 33 Figure 3.10: (a) Polycrystal model JM-I with integranular cracks. (b) Model JM-T with transgranular cracks. Cracks orientation uncorrelated to the crystal symmetry axis. (c) Model JM-TW with "weak-plane" transgranular cracks. Cracks oriented parallel to the basal plane in each crystal.

Figure 3 . 11 :

 311 Figure 3.11: Histograms of the strain field components ε eq (a,b), ε m (c,d), ε xy (e,f) in various cracked media, subjected to hydrostatic strain loading (a,c,e) or shear (b,d,f). JM-I: polycrystal with intergranular cracks. JM-T: transgranular cracks. JM-TW: "weak-plane" transgranular cracks. JM-TS: "stiff-plane" transgranular cracks. JM-IB: homogeneous cracked body. Embedded graphs: strain distribution in the region ε « 0.

Figure 3

 3 Figure3.12: Tail of the distributions P εxx ptq (t " 1) of the strain field ε xx in media subjected to hydrostatic strain loading: comparison between sound Voronoi (V) and Johnson-Mehl (JM) polycrystals and cracked models JM-I (intergranular cracks) and JM-T (uncorrelated transgranular cracks), represented in log-log plot. Black dashed lines: powerlaw fits P εxx ptq " 0.5t ´4.7 and P εxx ptq " 10t ´4.7 . Gray dashed line marked SC: self-consistent estimate for a sound polycrystal.

1 ] 1 ]Figure 3 . 13 :

 11313 Figure 3.13: Distribution of stress components σ xx , σ eq and σ xy in a polycrystal with transgranular cracks oriented along basal planes (model "TW"). Solid lines: self-consistent estimates; dashed lines: FFT computations. (a): Hydrostatic strain loading. (b) Shear strain loading.

Figure 4 . 1 :

 41 Figure 4.1: Three disks with centers U , V and W .

Figure 4 . 2 :

 42 Figure 4.2: (a) Method for choosing the disk C i`1 , knowing C i . (b) Path pA; C 1 ; ...; C N ; Bq defined by (4.12) and used to deliver an upper-bound on the length of geodesics.

  (4.3b). Results are represented in Fig. (4.3b), with error bars that indicate statistical fluctuations. As expected, the upper-bound (4.21

Figure 4 . 3 :

 43 Figure 4.3: (a) Leading-order correction to the length of geodesics ξ vs. disk concentration f , in log-log plot. Black dots: numerical computation of bound (4.13); black solid line: analytical estimate (4.20); triangles: numerical computation of ξ using the exact shortest path on finite-size systems; gray solid line: power-law fit of the latter; dotted lines: power-law corrections " f and " f 1{2 . (b) Upper-bound (4.21) (solid line) vs. numerical estimates for ξ (circles with error bars) for increasing values of the disks surface fraction f . Dashed line: asymptotic expansion (4.20). Vertical solid line: percolation threshold f c « 0.68.

Figure 4 . 4 :

 44 Figure 4.4: (a) Probability distribution function for the random variable Z, representing the deviation transverse to the direction of propagation, computed numerically (symbols) and analytically (solid lines). Black dots and triangles: path used in bound (4.20). Gray dots: shortest path fitted with a folded Gaussian distribution (4.24). (b) Transverse fluctuations w as a function of the path length L in log-log plot, for various values of f . Gray symbols: shortest path. Black symbols: path used in bound (4.20). Solid lines: power-law fit of the data.

. 34 )

 34 with the extra condition C 1 ą C i 1 `D. It provides a bound on d p pA, Bq:

  (4.5).

Figure 4 . 5 :

 45 Figure 4.5: Boolean set of discs with surface fraction f " 10% and 10 3 disks. Gray line: minimal path joining two opposite corners. White line: path used to derive bound (4.13).

Figure 5 . 1 :

 51 Figure 5.1: (a) Two bonds originating from a node at point x and joining nodes at x `e1,2 (solid lines); other nearby bonds are represented as dotted lines. (b) 16 ˆ16-nodes random conducting network. Conducting bonds shown in black, insulating bonds omitted. (c) Nonlinear constitutive law (5.2) satisfied along each conducting bond.

Figure 5 . 2 :

 52 Figure 5.2: Random network in a 64 ˆ64 grid. (a) Convergence criterion (5.6) vs. number of iterations. (b) Effective conductivity χ 0 vs. slope σ, with E " e 1 .

Figure 5 . 3 :

 53 Figure 5.3: (a) Effective scalar conductivity χ 0 vs. insulator fraction f , for various grid sizes, with applied electric E " e 1 . (b) Fractional part of M χ 0 showing that M χ 0 is an integer.

Figure 5 . 4 :

 54 Figure 5.4: FFT results obtained on a grid Ω of 32 ˆ32 nodes, with f " 0.2 and E " e 1 where e 1 is oriented top to bottom. (a) Component E 1 of the electric field, each bond being represented by a voxel. (b) Square grid. Missing bonds are insulating, the gray-level color in other bonds represents the value of the electric field: highest values in black, lowest in light gray.

Figure 5 . 5 :

 55 Figure 5.5: FFT results for the effective conductivity in the random resistor network. Gray symbols: loading in the p1, 1q direction E " e 1 `e2 . Black symbols: loading in the p1, 0q direction E " e 1 .

Figure 5 . 6 :

 56 Figure 5.6: Two minimal paths (in light gray) joining two broken bonds (top-left and bottomright) in the square lattice.

Figure 5 . 7 :

 57 Figure 5.7: Log-log plot of 1 ´χ0 vs. fraction of insulating bonds f , in the dilute limit f Ñ 0. Data points: FFT results. Black dashed line: power-law fit with 2{3 exponent of FFT data points. Gray dashed lines: power-law f and f 1{2 .

Figure 5 . 8 :

 58 Figure 5.8: Effective conductivity χ 0 of a binary resistor network with yield currents J 0 " 1 and 0 ď J 1 ď 1, as a function of J 1 . Black solid line: bound(5.11). Dashed line: estimate χ 0 « 1 ´1.6p1 ´J1 q 4{3 f 2{3 (see text).

Figure 6 . 1 :

 61 Figure 6.1: Random realizations of three Boolean random sets containing about 500 disks with surface fraction f « 30%. (a) Equisized disks (Boolean model B E ). (b) Polydisperse model B P . (c) Symmetric Cox-Boolean model B S (top-right: model B 1 used to generate the structures). Gray lines: path of minimal length joining two opposite corners. White line: paths used to compute bounds (6.12) and (6.18) for the distance between the two corners.

Figure 6 . 2 :

 62 Figure 6.2: Length of minimal paths in the Boolean model B E (black circles) and in various two-scale Boolean models: B P (left triangles), B C (diamonds) and B S (bottom triangles).

67 . ( 6 . 13 )

 67613 Consider now the two-scale model B C containing clusters of pores with total porosity f " f 1 f 2 , f 1 being the volume fraction of clusters and f 2 the volume fraction of pores relative to clusters.

Figure 6 . 3 :

 63 Figure 6.3: Path (solid line) crossing two disks of centers A and B.

  )

Figure 6 . 4 :

 64 Figure 6.4: Minimal path (black line) in two realizations of model B C containing n " 5000 disks, with the same surface fraction of clusters f 1 " 10%. The surface fraction of disks inside is clusters f 2 " 7 10 ´4 (a) and f 2 " 5 10 ´2 (b). The disks displayed in (a) and (b) have been enlarged so that they are visible. In the real microstructures, their radii is 70 times and 8 times smaller, for (a) and (b) resp.

Figure 6 . 5 :

 65 Figure 6.5: (a) Normalized distance ξ P in the polydisperse model B P . Solid gray line: scaling law (6.21). Dashed lines: scaling law (6.13) and linear correction " f . (b): Distance ξ S in the symmetric Cox-Boolean model B S . Triangles: numerical results. Gray solid line: scaling law (6.20c). Black solid line: powerlaw fit of the numerical data. Dashed lines, for comparison: scaling law (6.13) and linear correction " f .

Figure 6 . 6 :

 66 Figure 6.6: Normalized distance ξ C in the Cox-Boolean model B C . Symbols: numerical results(f 1 " 0.01 for `signs, f " 0.02 for diamonds). Black and gray solid lines: scaling law (6.20a). Dashed lines, for comparison: scaling law (6.12) and linear correction " f .

12 Figure 6 . 7 :

 1267 Figure 6.7: Normalized effective yield stress in the two-scale model B C as a function of the porosity f , for various values of cluster volume fraction f 1 . Symbols: prediction of the second-order theory.Solid gray lines: bound (6.17). Each gray-level refers to the same value of f 1 .

Figure 6 . 8 :

 68 Figure 6.8: Checkerboard containing randomly-occupied cells. (a) Square model. (b) Model of rectangles. Gray solid lines: path used to obtain a bound on geodesics.

Figure 6 . 10 :

 610 Figure 6.10: (a) Model of non-interpenetrable squares (in dark gray). Light gray lines: geodesics joining the mid-point on the left side of the domain with all square corners, forming a tree. Black solid line: minimal path joining the two mid-points on the left and right sides. (b) Rigid squares in the RSA model. Numerical data for the quantity n 4{3 pξ ´1q vs. n 1{3 ?f , in log-log plot.

Figure 7 . 1 :

 71 Figure 7.1: (a) Elementary cell Ω of a periodic microstructure, containing a single disk. (b): "scrambled disk".

Figure 7 . 2 :

 72 Figure 7.2: (a) Quantity 1 ´χψ vs. exponent n in log-log scale, with a " 10 ´5 fixed (data points). Solid and dashed lines (left to right): expansions (7.6), (7.10) and (7.4), respectively. Gray vertical lines: exponents n c and n 1 c . (b) Normalized profile of the local electrical field E ψ 1 within the band surrounding the inclusion, for various values of the nonlinearity exponent n, corresponding to the light gray points in (a).

Figure 7 . 4 :

 74 Figure 7.4: Local trial electric field E ψ 1 px 1 , 0q at various points of coordinate 0 ď x 1 ď a as a function of the nonlinearity exponent n for fixed a " 10 ´5, in log-log plot. Dotted line: scaling law E ψ 1 " 1{ ? nπa.

  27). The scaling law for b ˚turns out to be b ˚9? an for n ! 1. This is shown by replacing b by k ? an (k constant) in (7.27), expanding the latter as n Ñ 0 and optimizing on k. The optimal value of b ˚ensues: b ˚" min ´ap3{2qp1 ´2aqan; a{ ? 2 ¯, (7.28) where the restriction over b has been accounted for. The analytical upper-bound now reads: r y 0 y 0 ď f B pa, b ˚qp4b ˚q´pn`1q . (7.29)

Figure 7 . 6 :

 76 Figure 7.6: Upper-bound (7.31) for the normalized effective yield stress r y 0 {y 0 vs. disk radius a in a periodic medium, in log-log plot, for various values of the nonlinear exponent n.

Figure 7 . 8 :

 78 Figure 7.8: Normalized effective yield stress ry 0 {y 0 ´1 vs. disk radius a in a periodic medium, in log-log plot, for various values of the nonlinear exponent n. Comparisons between limit analysis bounds (7.34) (solid lines), second-order dilute predictions (7.19 2 ) (solid lines) and FFT estimates (symbols).

. 21 )Figure 8 . 1 :

 2181 Figure 8.1: Normalized mean covariogram kptq as a function of t: (a) with r " 1{2 fixed and increasing height h " 0.01, 0.05, 0.2, 0.5, 1, 2, `8 (bottom to top, solide lines); (b) with h " 1 fixed and increasing radius r " 0.02, 0.1, 0.3, 0.5, 1, 2, `8 (bottom to top, solide lines). Dashed lines: covariogram of a sphere of diameter 1.

Figure 8 . 5 :

 85 Figure 8.5: Variance D 2B pV q of the estimate of the volume fraction measured in a spherical domain W vs. volume size V of the domain for oblate cylinders (r " `8). The cylinders height is fixed to h " 1. Top to bottom: volume fraction of cylinders p " 0.5, 0.4, 0.6, 0.3, 0.7, 0.2, 0.8, 0.1, 0.9.

Figure 8 . 6 :

 86 Figure 8.6: Variance D 2B pV q of the estimate of the volume fraction measured in a spherical domain W vs. volume size V of the domain, in log-log plot. The cylinders volume fraction is fixed to p " 1{2. Grey solid lines, bottom to top : (a) cylinders height h " 1, 10, 100, 10 3 , 10 4 , 8 with radius fixed to r " 1{2; (b) cylinders radius r " 1, 10, 100, 10 3 , 8 with height fixed to h " 1. Dotted lines in black: expansions(8.45).

) D 2 B pV q " ´3p1 ´pq 2 logp1 ´pqhr 2 5 3

 23 , p " r " hq.(8.46b)The two expansions above are in good agreement with the numerical data in Fig.(8.6).

Figure 9 . 1 :

 91 Figure 9.1: 2D sections of a Boolean models of oblate cylinders (white). The fluid flows in the black phase. (a) and (b): Different volume fractions of cylinders.

xu 1

 1 pxqy " ´k µ xB 1 ppxqy.(9.4)

Figure 9 . 2 :

 92 Figure 9.2: (a) Berryman-Milton upper-bound(9.17) for the permeability k of a Boolean model of flat cylinders with height h " 1 and radius r " 10 2 , 10 3 , 10 4 vs. concentration of obstacles q, in loglog plot (black). Dark-gray solid lines: r Ñ 8 expansion (9.22). Light-gray solid line: dilute limit expansion q Ñ 0 (9.21). (b) Permeability k vs. volume fraction of obstacles q (x-axis) in log-log plot, for a Boolean model of flat cylinders of height h " 1 and radius r " 10 4 (a.u.). Comparison between the Berryman-Milton upper-bound (9.17, solid blue line), its two contributions over t ă h and t ą h (orange and green solid lines), and expansions (9.19, blue dashed line) and (9.22, orange dashed line).

Figure 9 . 3 :

 93 Figure 9.3: Section (1000 ˆ1000 voxels) of a Boolean set of cylinders of height 4 voxels and radius 200 voxels with q c « 0.04%. The fluid flow occurs inside the black phase, obstacles are shown in white. (a): q « 0.009% « 0.2q c . (c): q « 1.2% « 30q c .

  .[START_REF] Altendorf | Influence of the fiber geometry on the macroscopic elastic and thermal properties[END_REF] where we have introduced ν zp " ν pz E z {E p , inequalities (2.5) and (2.6) being equivalent. When needed, we refer to the following Voigt notation for stiffness tensors, equivalent to (2.3): pσ 11 , σ 22 , σ 33 , σ 23 , σ 13 , σ 12 q t " C ¨"pε 11 , ε 22 , ε 33 , 2ε 23 , 2ε 13 , 2ε 12 q t ´αt ∆T

	‰	,	(2.7)

  The predictions of estimates (2.23a) are represented in Figs.(2.1). The limiting value for the effective Poisson ratio at percolation (Equation 2.33, method SC 1 ) is confirmed in Fig.(2.1b)

	`13 160 ? 46	« 0.751064.	(2.36)

  , which shows the effective bulk modulus

	0.3 0.4 0.5 ν ~νp =ν zp =0.2 E z =0.101 E p =1, G zp =0 E z =10 7	0.4 0.6 0.8 K ~Ez =1 E z =0.101 E z =10 7			ν p =ν zp =0.2 E p =1, G zp =0	E z =1 ~νp =ν zp =0.2 0.7 0.75 α E z =0.101 E p =1, G zp =0 E z =0.15 E z =0.3 α 11 =1, α 33 =0
	0.1 0.2						0.2	E z =0.3 E z =0.15				0.65	E z =2 E z =10 7
	0 0	0.2	0.4	0.6	0.8	1 η	0 0	0.2	0.4	0.6	0.8	1 η	0 0.6	0.2	0.4	0.6	0.8	1 η
				(a)					(b)						(c)
	Figure 2.9: Effective response of a polycrystalline aggregate with out-of-plane shear modulus
	G zp " 0 as a function of the crack density parameter η, for various out-of-plane Young modulus
	E z . (a) Poisson coefficient r ν, (b) bulk modulus r K, (c) thermal expansion coefficient r α " r α V {3.

  of the strain field εpxq. For each phase prq, they read

		M r ε " xεy prq and Σ r ε " xε b εy prq , @r " 1, . . . , N.	(3.1)
	The multivariate probability distribution function of the tensorial field εpxq in phase prq is defined
	as	P r ε ptq " lim ζÑ0	1 ζ	P t||ε ´t|| ă ζu "	1 |Ω r |	ż Ωr	δpεpxq ´tq dx, t P S 2 ,	(3.2)
	with δ the Dirac delta function, |Ω r | the volume of phase r and S 2 the space of symmetric second-order tensors in R 3 equipped with norm || ' || " ? ' : '. The intraphase first and second order
	moments of the strain field can thus be written
			ż			ż
		M r ε "		S 2	t P r ε ptq dt and Σ r ε "	S 2	pt b tq P r ε ptq dt.	(3.3)

  Let us examine the possible Van Hove singularities of r P ij and r P ij as t Ñ ˘8. Using Eqs. (3.21) we obtain: ij pt{σ k q ´5 is the asymptotic behavior of P ij pσ k ; tq as t Ñ 8 provided by Eqs. (3.21). The same relations are obtained for r P ij with prefactors p ij given by (3.29a) instead. Note that r P xx , r P yy , r P xx and r P ẙy possess distribution tails as t Ñ ´8, unlike distributions P xx and P yy . The tails are asymmetric: the prefactor at t " ´8 is lower than that obtained in the t " `8 limit.

	r Pxx yy ptq " pxx yy |t|	0 ´5 ż 8	ds s 4 qpsignptqsq,	t Ñ ˘8,	(3.35a)
	r P xy ptq " p xy |t|	´5 ż 8	ds s 4 rqpsq `qp´sqs ,	t Ñ ˘8,	(3.35b)
			0
	where p

The behavior of r P yy as t Ñ 0 is determined as: r P yy ptq " p 1 yy |t| ´ν ż 8 0 ds s ν´1 qpsignptqsq, t Ñ 0 ˘, (3.36) where p 1 yy t ´ν (ν " 2{3) is the asymptotic behavior of P k yy pσ k ; tq as t Ñ 0 `(see Eq. 3.26). The behavior of r P xx " r P ẙy as t Ñ 0 is obtained by taking ν " 1{3 and by replacing prefactor p 1 yy by that provided by Eq. (3.27). The behavior of r P xy as t Ñ 0 is obtained as: r P xy ptq " p 1 xy |t| ´1{2 ż 8 0 ds s ´1{2 rqpsq `qp´sqs , t Ñ 0. (3.37) where p 1 xy |t| ´1{2 is the asymptotic behavior of P k xy given in (3.19

  In the limit D 2 ! D 1 the distance function in M is well approximated by that considered in Sec. (4.3.2) with p " 1 ´"3{Γ `2 3 ˘‰ r3f 2 {p2πqs 2{3 . Eq. (4.37) then provides the following bound, for two points A and B sufficiently far away from each other:

  P is now parametrized by the two variables f P and n, which approximate the surface fraction and mean number of observable disks in the domain, respectively. Scale separation R 1 {R 2 " 1 is achieved when n Ñ 8. The realization of model B P shown in Fig.(6.1b) has been generated by setting f P " 30% and n " 500 and using(6.8).Consider now model B C , which contains disks of radius R 2 . When f 2 Ñ 0 andR 2 ! R 1 , the number of disks in Ω is n « f 1 n 2 .To simulate representative configurations of the set of aggregates B 1 in Ω while approaching scale separation (R 1 {R 2 " 1), we choose, as for model B P , n 1 " ? n. The parameters θ 1 , θ 2 , R 1 and R 2 are accordingly set to:

				R 2 1	? 1 ´fP ? n ´1 , θ 1 " P	ˆ?n S	˙, θ 2 " P	ˆ?nR 2 1 2 SR 2	˙.	(6.8)
	Model B R 2 1 "	´logp1 ´f1 q π ? n	S, R 2 2 "	´logp1 ´fC {f 1 q πn	f 1 S, θ 1 " P	ˆ?n S	˙, θ 2 " P	ˆn f 1 S	˙. (6.9)

  nq " min

	ar	? 2ă1	#	f B pa, a{ ? 2q 2 n`1	ˆr	`1 ´ra 1 ´a? ? 2 2	˙n`1 `"1 ´f ´fB ˆa,	a ? 2	˙ ˆ1 ´ra 1 ´a? ? 2 2	˙n`1 +	,
												(7.30)
	which is numerically estimated, by optimizing on r. Our upper-bound on the entire domain
	0 ă n ă 1, 0 ă a ă 1{2 then reads:
				r y 0 y 0	ď min ζpa, nq, f B pa, b ˚qp4b ˚q´pn`1q (	.			(7.31)

  :

	Kpθ, tq " r Kpθ, xqHpθ, tq,					(8.4a)
	r Kpθ, xq " 4r 3	ˆh 2r	´x sin θ ˙"cos ´1 px cos θq ´x cos θ	? 1 ´x2 cos 2 θ	ı	,	(8.4b)
	where Hpθ, tq " 1 when θ and t are such that the two cylinders C and C 1 intersect, and 0 otherwise.
	More precisely:
	Hpθ, tq "	"	1 if t ď t max , 0 otherwise,	t max "	" 2r cos θ h sin θ	if θ P r0; tan ´1 h 2r s, if θ P rtan ´1 h 2r ; π 2 s.	(8.5)
	The term cos ´1 in Eq. (8.4) refers to the inverse cosine function, also denoted arccos and tan	´1
	in (8.5) denotes the inverse tangent function. Likewise, sin ´1 hereafter is used to denote the inverse
	sine function.
	The following section is concerned by the normalized mean covariogram:
	kptq "	1 4π	ż 2π 0	dφ	ż π{2 ´π{2	dθ	Kpθ, tq hπr 2 cos θ "	ż π{2 0	dθ	Kpθ, tq hπr 2 cos θ,	(8.6)

  ). The quantity K in (8.4) is non-zero in the following intervals for θ (recall that y " h{t):

	$ &	r0; π{2s	if t ă h,
	θ P	r0, sin ´1pyqs	if h ă t ă 2r,	(8.13)
	%	rcos ´1p1{xq, sin ´1pyqs	if 2r ă t.
	Whereas, when h ą 2r, K is non-zero if:
	$ &	r0; π{2s	if t ă 2r,
	θ P	rcos ´1p1{xq, π{2s	if 2r ă t ă h,	(8.14)
	%	rcos ´1p1{xq, sin ´1pyqs	if h ă t.

  where we have identified k 8 pxq, the covariogram for h " 8 given in (8.12). The integral in (8.[START_REF] Barroso | Determination of Generalized Fracture Toughness in composite multimaterial closed corners with two singular terms-Part I: Test proposal and numerical analysis[END_REF]) is computed using a software for symbolic computations, and, after rearranging the terms:

				kpx, yq " Re	# ż a 0 dθ	r Kpθ, xq cos θ hπr 2	+	.	(8.15)
	Now according to (8.13) and (8.14), a " π{2 when y ą 1 (t ă h) and a " sin ´1 y when y ă 1
	(t ą h). We first examine the case a " π{2, i.e. y ą 1. Equation (8.15) takes the form:
	kpx, yq " k yą1 px, yq " k 8 pxq					(8.16)
	´2 πy	Re	#	0 ż π{2 dθ cospθq sinpθq	" cos ´1 px cos θq ´x cos θ	? 1 ´x2 cos 2 θ	+ ı	,
	k yą1 px, yq " k 8 pxq	´1 8x 2 y	`H1 pxq 2yπ	"ˆ1 2x	`x˙? 1 ´x2 `ˆ1 2x 2	´2˙c os ´1 x		,	(8.17)

  , k r≫h (t), k r≪h (t)

	1
	0.8
	0.6
	0.4
	0.2
	0 0	0.5 1	1.5 2	2.5 3	t

Figure 8.2: Mean normalized covariogram k of a cylinder: comparison between the exact result (solid line) and approximations (8.

27

) and (8.28) (dotted lines) when r " h " 1 and r " 1, h " 2.9, respectively. respectively. Approximation (8.27) is good in the region h{2 ă r ă h. The errors are about sup t |k ´kr"h | " 2.6% and x|k ´kr"h |y t " 0.6% when r " h{2.

  Figure 8.3: Normalized integral range r A 3 ppq in a Boolean model of cylinders with radius r " 1{2 as a function of the volume fraction p of cylinders, with varying cylinder heights h " 3, 10, 10 2 (solid grey lines, top to bottom), 1{3, 10 ´1, 10 ´2, (dotted black lines, top to bottom) and h " 1 (solid black line). Dark-grey solid line on top: normalized integral range for a Boolean model of spheres of diameter 1.

	1 A 3 (p)
	0.8
	0.2 0.4 0.6	spheres h = 1 h = 0.3, 3 h = 10 -1 , 10 h = 10 -2 , 10 2
		0.2	0.4	0.6	0.8	1	p
							.35)

  .38) where α, α 1 are prefactors. The scaling laws above indicate a slower decrease of the variance with respect to the volume than in(8.37), due to infinite correlation lengths.For prolate and oblate cylinders, the behavior of D 2 B pW q should be linked to that of the integral of the covariance Cptq when t Ñ 8. Consider first the asymptotic behavior of kptq when t Ñ 8 in the cases h " 8 and r " 8. Making use of the expansions for φ Ñ 0: Figure 8.4: Normalized integral range r A 3 ppq in a Boolean model of cylinders with radius r " 1{2 and volume fraction p " 1{2 as a function of the cylinders height h (solid line). Dashed line: normalized integral range for a Boolean model of spheres of diameter 1.

		0.78 A 3 (p)
		0.76
		0.74
		0.72
		0.7		5	10	15	20	h
	one finds, when t Ñ 8:
	F	`φˇˇm˘"	φ	`mφ 3 6	`mp9m ´4qφ 5 120	`Opφ 7 q,	(8.39a)
	E	`φˇˇm˘"	φ	´mφ 3 6	`mp4 ´3mqφ 5 120	`Opφ 7 q,	(8.39b)

  Replace now the term χ B puqχ B pvq in (8.36) by Cptq and integrate over t:

	#	4πt 2 dt, 2πtdt ´t ´a 2 `	2 ´t2 2a ¯,		if t ă ´a, if ´a ă t ă `a.	(8.42)
	Integrating over a, one finds the required probability:
	dP pt, q "	3dt t 2 2 ˆ1	´t 2	˙2 ˆ1	`t 4	˙,	0 ď t ď 2 .	(8.43)

t "

  C 1 is a thick plane of width h 1 . Straightforward computations lead to:

			1 q " E	"	|C X C 1 ´t| |C|	*	,	(9.13)
	where, Kpt; h, h 1 q "	"	1 t rh{2 ´h1 `ph 1 `tq 2 {p2hqs, t rh `h12 {p2hq `pt{h ´1qph `h1 qs, t ą h. t ă h, 1	(9.14)

  The derivation Doi's bounds for general cylinders requires one to compute the volume of the intersection of two cylinders of different heights and radius averaged over uniformly-distributed directions, which we do not have. Accordingly, hereafter we use the simpler Berryman-Milton bound[START_REF] Berryman | Normalization constraint for variational bounds on fluid permeability[END_REF][START_REF] Rubinstein | Flow in random porous media: mathematical formulation, variational principles, and rigorous bounds[END_REF] which reads:For cylinders with r ą h, replacing F vv with (9.9) and using (9.16) yields, after integrating:

	k ď k BM "		h log q 3π	" 2h cos ´1 ˆh 2r	3 ˙´1	2r 2 ˆh2	`4˙? 4r 2 ´h2	
		`h2 3	"	8	`4? q plog q ´2q qplog qq 2	´1	.	(9.18)
	" 1	´pr `hqt 2rh	`2t 2 3πrh	, if t ă h,	(9.16a)
	"	ˆh2 6t 2	´1˙h 2πr	c 1	´t2 4r 2 `h πt	cos ´1 ˆt 2r	˙, if t ą h.	(9.16b)
			k ď	2 3q 2	ż 8 0	dt t	"	F vv ptq ´q2 ‰	.	(9.17)

http://mms2.ensmp.fr/option_IDSysCo/accueil_option.php

http://www.cmm.mines-paristech.fr/ESPRM/

http://www.cmm.mines-paristech.fr/~willot/PMRM/

https://www.sciencedirect.com/journal/international-journal-of-solids-and-structures/speci al-issue/103QGG5T8J0

CHAPTER 2. CRACKED POLYCRYSTALS

Remerciements

Acknowledgements I am indebted to R. Brenner for the reconstruction of the fields distribution in Fig. 3.2 and for the text in Sec. 3.2.2.

Part II

Perfectly-plastic media

Oblate cylinders

Consider now the limit r Ñ 8 with h fixed so that x Ñ 0 and y remains finite. We take the limit in the formula (8.23) when y ă 1 and (8.17) when y ą 1. The covariogram k depends of y only and reads:

Note that, in the limit r Ñ 8 the cylinders amount to infinite layers enclosed between two planes. The same asymptotic limit would accordingly be recovered when considering cylinders with other non-circular cross-section. This is in contrast to the prolate case (Sec. 8.3.1), where the covariogram depends on the shape of the cylinders cross-section.

Approximate formulas

We now give approximate expressions of the exact solutions which do not involve Elliptic functions. Such simpler expressions are useful for material applications, in particular for identifying random models, when a high accuracy is not required. We first consider the domain r " h. The exact solution is given by (8.26) when r " 8 and by (8.17) and (8.23) when r ă 8. We first let r Ñ 8 and t Ñ 8 with h and the ratio t{r fixed, and expand (8.23) to order Opr ´3q. In the region t ă h, i.e. y ą 1, we let t Ñ 0 and compute an expansion to Opt 3 q with r, h fixed. In the region t ą 2r, we set k « 0. One obtains: The expansion above is continuous except at points t " h, and is exact in the limit r " 8 only.

It turns out, however, that formula (8.27) is a very good approximation of the exact covariogram for r ą h. The maximal error sup t |kptq ´kr"h ptq| between (8.27) and the exact result is attained when t AE h whenever r ą h. As expected, this error decreases and tends to 0 when r{h Ñ 8.

It is about 0.5% for r " h and 0.1% for r " 4h (Fig. 8.2). The error is much smaller at points t ff h: for instance, the mean error x|kptq ´kr"h ptq|y t is 0.1% and 0.004% when r " h and r " 4h

Chapter 9

Permeability of a Boolean model of cylinders

A rigorous bound on the permeability of Boolean models of randomly-oriented cylinders with circular cross-section is reported. The bound is examined in detail in the case where the fluid flows inside strongly-oblate (flat) cylinders, in the limit of a vanishing volume fraction of obstacles. In such microstructure, the fluid flows around a set of quasi-isolated obstacles. Two regimes are identified for the permeability of such a medium, depending on the aspect ratio of the underlining cylinders and porosity. In one of the regime, the obstacles have a tendency to align with obstacles far from them, resulting in an unusual dilute expansion, which is explicited in this work.

Introduction

In the recent years, several so-called "Fourier-based" algorithms have been introduced to solve the problem of Stokes flow in heterogeneous media [START_REF] Wiegman | Computation of the permeability of porous materials from their microstructure by FFF-Stokes[END_REF][START_REF] Monchiet | A FFT-based method to compute the permeability induced by a Stokes slip flow through a porous medium[END_REF][START_REF] Bignonnet | FFT-based bounds on the permeability of complex microstructures[END_REF][START_REF] Willot | Fourier-based schemes for computing the permeability of porous media[END_REF]] some of which are based on a variational (Hashin-Shtrikman) framework [START_REF] Brisard | FFT-based methods for the mechanics of composites: A general variational framework[END_REF][START_REF] Brisard | Combining Galerkin approximation techniques with the principle of Hashin and Shtrikman to derive a new FFT-based numerical method for the homogenization of composites[END_REF]. These methods, which rely on digitized images of the microstructures and do not require meshing, are especially useful for predicting the permeability of materials based on microstructure models or segmented microtomography images.

They have been applied to compute the permeability of textiles [START_REF] Rief | Computer simulation of Air Filtration including electric surface charges in three-dimensional fibrous micro structures[END_REF][START_REF] Gervais | Permeability modeling of fibrous media with bimodal fiber size distribution[END_REF], ceramic foams [START_REF] Redenbach | Modeling a ceramic foam for filtration simulation[END_REF] or anode layers in fuel cells [START_REF] Masson | Morphological modeling of a metal foam SOFC configuration[END_REF], making use of 3D models. Numerical computations on large-scale 3D microstructures are also useful for investigating theoretical and empirical homogenization estimates, like the classical Carman-Kozeny [START_REF] Carman | Fluid flow through granular beds[END_REF][START_REF] Kozeny | Ueber kapillare leitung des wassers im boden[END_REF] formula, Doi's upper-bound [START_REF] Doi | A new variational approach to the diffusion and the flow problem in porous media[END_REF][START_REF] Torquato | Rigorous bounds on the fluid permeability: effect of polydispersivity in grain size[END_REF] based on correlation functions or dilute limit expansions [START_REF] Rubinstein | Flow in random porous media: mathematical formulation, variational principles, and rigorous bounds[END_REF]. The Boolean model of spheres has been considered in the literature [START_REF] Martys | Universal scaling of fluid permeability for sphere packings[END_REF] and, as expected, follows the Carman-Kozeny approximation [START_REF] Abdallah | Stokes flow through a Boolean model of spheres: Representative volume element[END_REF], whereas the upper-bound of Doi is valid in the dilute limit only, i.e. for a small volume fraction of obstacles.

The permeability of ordered arrays of cylinders has received considerable interest in experimental, theoretical and numerical works (see e.g. [START_REF] Martin | Pressure drop through stacked spheres. Effect of orientation[END_REF][START_REF] Richardson | The sedimentation of a suspension of uniform spheres under conditions of viscous flow[END_REF][START_REF] Tamayol | Analytical determination of viscous permeability of fibrous porous media[END_REF][START_REF] Yazdchi | Microstructural effects on the permeability of periodic fibrous porous media[END_REF][START_REF] Shahmardan | A New Mathematical Model for Permeability of Composites[END_REF]). Attention has also been devoted in the literature to various random fibrous media, where fluid flows around cylindrical obstacles. Many of the theoretical homogenization methods used for regular arrays, however, such as the construction of trial fields, cannot be employed for random media. To study random media,

Appendix A

Summary of recent activities (2016-2019)

This Appendix contains a brief description of my main achievements conducted in three topics of research: micromechanical modeling of polycrystals, probabilistic models of microstructures and Fourier-based methods.

Micromechanical modeling of polycristals

Several studies are currently being conducted regarding the modeling of a 1,3,5-triamino-2,4,6trinitrobenzen (TATB) polycrystal containing an amorphous thermoplastic polymer used as binder, and to its thermomechanical response, of a quasi-brittle nature [J11]. The material's response is strongly dependent on damage, irreversible deformation, viscoelastic mechanisms and thermal dilation (internship works of A. Ambos, and thesis of J.-B. Gasnier). The material is well characterized at the macroscopic scale, however its behavior is not well understood, although the main mechanisms have been qualitatively identified: inter and transgranular microcracking, viscoelastcicity due to the binder, and inelastic deformation of the TATB. The material is strongly dependent on temperature and stress triaxiality.

A numerical Fourier-based method has been devised [J10] making use of virtual microstructures [J27] representative of the main morphological features of the material (grains size distribution and aspect ratio, lack of geometrical convexity). In the linear thermoelastic case [J11], the important effect of crystal elastic anisotropy of the main phase, the TATB, has been highlighted. Transgranular micro-cracking along the graphitic plane of the monocrystal has been confirmed making use of SEM images, and explains the decrease of the thermal expansion coefficient during cooling-heating thermal cycles. Intergranular cracking, on the other hand, must be present in the material in its initial state [J11].

This work is currently being extended to the modeling of damage, crack initiation and propagation in polycrystals, making use of a numerical approach that combines phase-field and Fourierbased methods [P6] (thesis of F. Rabette). The role of damage and its effect on the macroscopic response of the polycrystal is investigated, in particular during thermal cycles. Image analysis tools have also been developed to assess how intergranular and transgranular micro-cracks propagate through heterogeneous structures.

Modeling of heterogeneous structures

Probabilistic models have recently been developed to represent a variety of heterogeneous microstructures. This includes N -phases random materials [J16] in the context of a European project devoted to fuel cell applications [P15] (thesis of B. Abdallah). The method proposed, which makes used of underlying independent random sets, has been compared to techniques developed elsewhere for the same type of materials, and based on graph-induced random sets [J2, J3]. We compared, in particular, the ability of the virtual microstructures to predict a variety of behavior such as conductivity or permeability, based on SEM images.

A method based on a mixed Boolean-tesselation random structure has been developed to model cold-spray coatings used in materials made for the aeronautical industry [J9] (thesis of V. Bortolussi), with funding from US Air Force. In this study as in others [J13, J18, C1, C2], use is made of automatic images segmentation methods.

Mesoporous structures for catalysis applications have been studied, making use of transmission electron microscopy and porosimetry curves to characterize the multiscale arrangement of alumina platelets (thesis of H. Wang). The two experimental techniques, transmission electron microscopy and porosimetry, have been modeled numerically to infer the material's structure [J14, J12].

Spectral methods

Fourier-based numerical methods have been proposed to predict the viscoelastic behavior of composites in the time-harmonic regime, in heterogeneous media [J18]. Fourier techniques that improve on the field discretization, using specific Green operators specifically adapted to the voxel grid, have been devised [J29]. The Fourier Fortran code has been used at the TSUBAME Japanese cluster to compute the viscoelastic response of virtual models of rubber at a very large scale, to carry out microstructure optimization (about 4 000 problems containing " one billion degrees of freedom) [P11]. Learning methods and data mining have been used to infer the main microstructural characteristics. Similar approaches are presently conducted in the context of L. Lacourt

Scientific responsibilities

• 2019-on: responsible of the ECMI (European Consortium of Mechanics and Industry) Special Interest Group "Shape and size in medecine and material science" together with J. Angulo & L. Bonilla. URL: https://ecmiindmath.org/special-interest-groups/shape-andsize-in-medicine-biotechnology-and-materials-science/.

• Responsible for the "Mécamat" (scientific association) working group "Probabilistic approaches in Mechanics of Materials". Workshops organized in Reims, 2012; S t -Martin d'Hères, 2013; Paris, May 2015; Paris, Dec. 2015; Aussois, 2018.

• Member of the organization comitte of the Mécamat/SF2M (Société française de Mécanique) "Matériaux numériques" from 2016.

• Responsible of the topic "microstructure modeling and analytical methods" for the GDR (CNRS working group) on fibrous media "3MF" (2013-2015) and "MECAFIB" (2018-on).

• • Reviewer of research projects for ANR (French national research agency) and for the Czech National Foundation.

• Examinateur in PhD jurys for B. Osman-Hoch & D. Komlan Sénam's thesis.

• Invited speaker at EUROMECH 559 (multiscale computational methods, Eindhoven, 2015).

• Editor together with S. Forest of the book "Physics & Mechanics of Random Media: from Morphology to Material Properties. A tribute to Dominique Jeulin's contributions to science, research and teaching", Presses des Mines, Paris, 2018.

• 2008-2012: co-animation with D. Jeulin of the working group "Nanomatériaux" at École des Mines (collaboration for grant applications and teaching).

Involvement in research projects

As main contributor:

• Project in collaboration with CEA Gramat (2018)

• Project in collaboration with CEA Le Ripault (2018)

• Project in collaboration with L'Oréal (2016, 2017)

• Project in collaboration with US Air Force (2016)

• Project in collaboration with CEA Le Ripault (2014)

• Project in collaboration with CERIB (2013)

Participation to various other projects, built with the Centre of Materials, and notably the CEA Valduc (2017), CERN (2016). In collaboration with the Process and surfaces team at centre of materials (A. Thorel, M. jeandin, A. Chesnaud, F. Delloro): european projects on fuel cells (2014) and ANR on colors in paintings (2014), cold-spray modeling (2017).

Teaching

• 2013-2019: organization of the one-week course in Mines ParisTech, in English "Physics and Mechanics of Random Media", for PhD students and to students from the "Athens" european university network. The courses include training session using Fourier methods, linear and nonlinear homogenization theory, and models of random structures.

• 2018-2019: one-week course "option ingénierie et dynamique des systèmes complexes" (organized by D. Ryckelynck and É. Hachem). Participation on a one-week course at École des Mines de Paris (together with H. Proudhon and B. Figliuzzi).

• 2017-2019: PSL master courses "Énergies" www.psl.eu/formation/master-energie

• 2014-2019: specialized courses on nanomaterials (organized by J.-F. Hochepied). Course on homogeneisation.

• 2016-2019: PSL master courses "Dynamique et Mécanique des Structures" (organized by F. Azzouz).

• 2011-2017: ESIEE engineering school. Courses on MPI code parallelization.

• 2015: doctoral courses at École des Ponts. Probabilistic modeling in mechanics.

• • Victor-Manuel Monclus (defense planed in 2019): Fabrication et étude de fibres piézoélectriques pour textiles auto-énergétiques (provisional title). Thesis directors: J. Renard, J.-F. Hochepied.

• Lucie Baudin (started in 2017): Structuration de surfaces par technologie Laser pour applications ultra-vide dans l'environnement des accélérateurs de particules (provisional title).

Thesis director: M. Jeandin.

Collaborations with other PhD students (no thesis supervision)

• François Cadiou: "Étude de l'impact de la microstructure sur les propriétés électriques des batteries Lithium-Ion" (provisional title). Thesis director: É. Maire.

• Matthias Neumann (ongoing). Thesis director: V. Schmidt.

• Grace Ham (2015-2018): "Propriétés physico-chimiques des oxydes de surface et analyse des données de l'imagerie hyperspectrale" (defense held in 2018). Thesis directors: C. Carteret & J. Angulo. É.D. Université de Lorraine.

Supervision of master students in the context of internships All master students have published their work in scientific journals (see Sec. C), except current internships.

• Siyuan Zhang (2019, in progress), collaboration with L'Oréal (T. Baldeweck).

• Jean-Baptiste Gasnier (2014), collaboration with CEA le Ripault (H. Trumel).

• Andrea Pietrasanta (2013), collaboration with IFPEN (M. Moreaud).

• Andrey Ambos (2013), collaboration with CEA le Ripault (H. Trumel).

• Amaury Vannier-Moreau (2013), collaboration with CERIB.

• Dario Azzimonti (2012), collaboration with PSA (ANR project).

Various

• Administration of the high-performance computing cluster at the Center for Mathematical Morphology in Mines Paristech, made of 9 racks each with 32 to 128 GB RAM, and inbetween 16 and 32 cpu cores.

• Webmaster for http://cmm.ensmp.fr

• Articles in anthology of comics (Glénat).

• Excellent knowledge of Fortran, latex PHP, matlab, bash, SQL. Sole contributor of FFT Fortran code (" 25K lines of code) used in training session and by PhDs students, and author of a PHP interface for a website (roughly 250, 000 visits per day, " 40K lines of code).

Appendix C

List of publications

When I have supervised their work, the name of PhD and master students is underlined. Google scholar: https://scholar.google.fr/citations?user=Gjv6430AAAAJ&hl=en, Scopus: https://www.scopus.com/authid/detail.uri?authorId=23981490600.

Journals