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International audience; The electronic conductivity, at the multiscale, of lithium-ion positive composite electrodes based on LiNi_{1/3}Mn_{1/3}Co_{1/3}O_2 and/or carbon-coated LiFePO_4, carbon black and poly(vinylidene fluoride) mixture is modeled. The electrode microstructures are acquired numerically in 3D by X-ray tomography and FIB/SEM nanotomography and numerically segmented to perform electrostatic simulations using Fast Fourier Transform (FFT) method. Such simulations are easy and quick to perform because they are directly computed on the grid represented by the voxels in the 3D volumes. Numerical results are compared with experimental measurements of the multiscale electronic conductivity by broadband dielectric spectroscopy (BDS). A good prediction is realized for the bulk conductivities of the C/LiFePO_4 phase and the CB/PVdF mixture. The combination of X-ray and FIB/SEM tomography, FFT simulation method, and BDS is thus well adapted to understand the influence of the electrode composition and microstructure on its electronic conductivity.

International audience; The thermoelastic responses of a cracked TATB polycrystal is modeled using Fourier-based numerical computations and linear homogenization estimates. The numerical method is based on a Fourier scheme previously investigated (Gasnier et al., 2018) and the homogenization schemes include a new self-consistent estimate recently developed for cracked polycrystals. The effect of various populations of intergranular and transgranular microcracks are compared, with special attention to thermal behavior in the percolating regime. A model of micro-cracks is proposed for the TATB material in its initial state and the variations of the volumetric expansion coefficient of the TATB polycrystal under thermal cycles, during cooling and heating, are interpreted.

International audience; The influence of a population of randomly-oriented cracks on the macroscopic thermal and linear-elastic response of a hexagonal polycrystal is addressed using a self-consistent method. Coupling between micro-cracks and crystal anisotropy is taken into account through the effective medium where all inhomogeneities are embedded. In the absence of cracks, the proposed approach reduces to the self-consistent estimate of Berryman (2005). 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 disk-shaped cracks, with Boolean spatial dispersion. Various other analytical estimates and bounds, that are available for homogeneous cracked bodies, are also considered and compared to the present approach. Second, the combined role of crystal anisotropy and micro-cracks is investigated analytically, specifically when the in-plane shear modulus of the crystal becomes zero. The cracks-density perco-lation 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 in strongly-anisotropic crystals.

International audience; The granular structures of energetic materials made of hexogene particles embedded in a matrix are characterized using a combination of flotation, light-scattering measurements and micro-computed tomography images. The complementary nature of the three characterization techniques, when employed with this type of material, allows one to derive accurate estimates for the grain size distribution and the particle bulk density distribution. Three types of granular formulations, with the same weight fraction of particles but markedly different grains morphology and shock-sensitivity properties, are addressed. The particles granulometry and intra-granular porosities differ from one material to another. The grains spatial dispersion in the three formulations, characterized by the scale-dependent local density, also varies from one formulation to another. Specifically, we show that samples containing elongated grains display “abnormal” variance scalings for the grains local density, as a result of long-range correlations in the granular packing structure. The use of virtual microstructures to predict the materials mechanical properties is addressed and discussed.

Preface to a special issue of the International Journal of Solids and Structures. This is not a research article.; International audience; This volume gathers contributions presented at the International Workshop on Physics and Mechanics of Random Structures: From Morphology to Material Properties held on the Ile d’Oléron (Atlantic coast, France), June 17–22, 2018 (Willot and Forest, 2018; Oleron, 2019). This exceptional workshop was organized on the occasion of the (official) retirement of Prof. Dominique Jeulin from his position at Mines ParisTech, where he has been working for more than 30 years to the development of mathematical morphology and physics and mechanics of random media.

A general method is proposed to model 3D microstructures representative of three-phases anode layers used in fuel cells. The models are based on SEM images of cells with varying morphologies. The materials are first characterized using three morphological measurements: (cross-)covariances, granulometry and linear erosion. They are measured on segmented SEM images, for each of the three phases. Second, a generic model for three-phases materials is proposed. The model is based on two independent underlying random sets which are otherwise arbitrary. The validity of this model is verified using the cross-covariance functions of the various phases. In a third step, several types of Boolean random sets and plurigaussian models are considered for the unknown underlying random sets. Overall, good agreement is found between the SEM images and three-phases models based on plurigaussian random sets, for all morphological measurements considered in the present work: covariances, granulometry and linear erosion. The spatial distribution and shapes of the phases produced by the plurigaussian model are visually very close to the real material. Furthermore, the proposed models require no numerical optimization and are straightforward to generate using the covariance functions measured on the SEM images.

International audience; The mechanical role of the shape of the aggregates, and their spatial distribution in concrete materials is examined. The effect on the macroscopic mechanical response as well as on the local stress fields are investigated by numerical means, making use of a recently developed Poisson-polyhedra model in which the aggregates have polyhedral shapes. Comparison is made with previous microstructure models of spheres. Full-field computations are carried out on large volume size using Fourier methods. The field maps are used to determine zones of highest stress. Furthermore, the scale separation hypothesis for the computation of the elastic properties is investigated using numerical calculations on various multiscale microstructures. Finally, numerical predictions for the elastic properties are compared to experimental measurements, and the results discussed.

This work focuses on a mesoporous material made of nanometric alumina "platelets" of unknown shape. We develop a 3D random microstructure to model the porous material , based on 2D Transmission Electron Microscopy (TEM) images, without prior knowledge on the spatial distribution of alumina inside the material. The TEM images , acquired on samples with thickness 300 nm, a scale much larger than the platelets's size, are too blurry and noisy to allow one to distinguish platelets or platelets aggregates individually. In a first step, the TEM images correlation function and integral range are estimated. The presence of long-range fluctuations, due to the TEM inhomogeneous detection , is detected and corrected by filtering. The corrected correlation function is used as a morphological descriptor for the model. After testing a Boolean model of platelets, a two-scales model of microstructure is introduced to replicate the statistical dispersion of platelets observed on TEM images. Accordingly a set of two-scales Boolean models with varying physically-admissible platelets shapes is proposed. Upon optimization, the model takes into account the dispersion of platelets in the microstructure as observed on TEM images. Comparing it to X-ray diffraction and nitrogen porosimetry data, the model is found to be in good agreement with the material in terms of specific surface area.

International audience; The Stokes flow is numerically computed in porous media based on 3D Boolean random sets of spheres. Two configurations are investigated in which the fluid flows inside the spheres or in the complementary set of the spheres. Full-field computations are carried out using the Fourier method of Wiegmann (2007). The latter is applied to large system sizes representative of the microstructure. The overall permeability of the two models as well as the representative volume element are estimated as a function of the pore volume fraction. We give numerical estimates for the asymptotic behavior of the permeability in the dilute limit for the solid phase, and close to the percolation threshold of the pores. FFT maps of the velocity field are presented, for increasing values of the pore volume fraction. The patterns of the local velocity field is analyzed using various morphological criteria. The tortuosity of the streamlines is found to be much higher than the geometrical tortuosity, for both models. The histograms of the velocity field are computed at increasing pore volume fraction. The covariance of orientation is used to characterize the spatial correlation of the velocity field.

This paper aims at developing a random morphological model for concrete microstructures. A 3D image of concrete is obtained by microtomography and is used in conjunction with the concrete formulation to build and validate the model through morphological measurements. The morphological model is made up of two phases, corresponding to the matrix, or cement paste and to the aggregates. The set of aggregates in the sample is modelled as a combination of Poisson polyhedra of different scales. An algorithm is introduced to generate polyhedra packings in the continuum space. The latter is validated with morphological measurements.

This is a preface to a special issue of IA&S, not a research article; International audience; A few words on the present special topic, devoted to the multiscale modeling of granular media, and published in honor of Prof. Dominique Jeulin's enduring contribution to the wide field of image analysis, random structures and material science.

A modified Green operator is proposed as an improvement of Fourier-based numerical schemes commonly used for computing the electrical or thermal response of heterogeneous media. Contrary to other methods, the number of iterations necessary to achieve convergence tends to a finite value when the contrast of properties between the phases becomes infinite. Furthermore, it is shown that the method produces much more accurate local fields inside highly-conducting and quasi-insulating phases, as well as in the vicinity of the phases interfaces. These good properties stem from the discretization of Green's function, which is consistent with the pixel grid while retaining the local nature of the operator that acts on the polarization field. Finally, a fast implementation of the "direct scheme" of Moulinec et al. (1994) that allows for parcimonious memory use is proposed.

This work is a combined analytical and numerical study of the extreme values of the thermal and elastic fields occurring in a propellant composite material, a granular medium containing dense self-assembled spheroidal grains embedded in a matrix. First, a 3D microtomography image of a representative sample is segmented using morphological operators. Second, the local temperature and heat flux in the quasi-static thermal response is computed numerically, making use of the segmented microstructure and of experimental values of the heat conductivity in each phase. Such fields are readily derived by means of a Fast Fourier Transform method. Emphasis is put on the maximum values of the local fields: even at low contrast of properties between the grain and the matrix, the heat flux patterns is made of hot-spot zones located in-between grains that are close to each other with preferential directions. Third, the local extrema of the fields are investigated in the context of linear elasticity. Finally, analytical approximations are examined at low and high contrast of properties, on a simple hard-core model of cylinders, which is tantamount to a 2D granular microstructure.

International audience; This study concerns the prediction of the elastic properties of a 3D mortar image, obtained by micro-tomography, using a combined image segmentation and numerical homogenization approach. The microstructure is obtained by segmentation of the 3D image into aggregates, voids and cement paste. Full-fields computations of the elastic response of mortar are undertaken using the Fast Fourier Transform method. Emphasis is made on highly-contrasted properties between aggregates and matrix, to anticipate needs for creep or damage computation. The Representative Volume Element, i.e. the volume size necessary to compute the effective properties with a prescribed accuracy, is given. Overall, the volumes used in this work were sufficient to estimate the effective response of mortar with a precision of 5%, 6% and 10% for contrasts ratio of 100, 1000 and 10000, respectively. Finally, a statistical and local characterization of the component of the stress field parallel to the applied loading is carried out.

International audience; The overall mechanical response as well as strain and stress field statistics of an heterogeneous material made of two randomly distributed, linear elastic phases, are investigated numerically. The Boolean model of spheres is used to generate microstructures consisting of either porous or rigid inclusions, at any volume fraction of the phases. Stress and strain field integral ranges, or equivalently the representative volume element, are computed and linked to features of the field statistics, and to the microstructure geometry.

International audience; Images of a hematite-based epoxy coating are obtained by scanning electron microscopy (SEM). At the scale of a few micrometers, they show aggregates of hematite nano-particles organized along thin curved channels. We first segment the images and analyze them using mathematical morphology. The heterogeneous dispersion of particles is quantified using the correlation function and the granulometry of the embedding (epoxy) phase. Second, a two-scales, 3D random microstructure model with exclusion zones is proposed to simulate the spatial distribution of particles. This simple model is parametrized by four geometrical parameters related to the exclusion zones solely. The microstructure is numerically optimized, in the space of morphological parameters, on the granulometry of the embedding epoxy phase and on the microstructure correlation function, by standard gradient-descent methods. Excellent agreement is found between the SEM images and our optimized model. Finally, the size of the representative volume element associated to the optimized microstructure model is compared with that of the SEM images.

A general method is proposed to model 3D microstructures representative of three-phases anode layers used in fuel cells. The models are based on SEM images of cells with varying morphologies. The materials are first characterized using three morphological measurements: (cross-)covariances, granulometry and linear erosion. They are measured on segmented SEM images, for each of the three phases. Second, a generic model for three-phases materials is proposed. The model is based on two independent underlying random sets which are otherwise arbitrary. The validity of this model is verified using the cross-covariance functions of the various phases. In a third step, several types of Boolean random sets and plurigaussian models are considered for the unknown underlying random sets. Overall, good agreement is found between the SEM images and three-phases models based on plurigaussian random sets, for all morphological measurements considered in the present work: covariances, granulometry and linear erosion. The spatial distribution and shapes of the phases produced by the plurigaussian model are visually very close to the real material. Furthermore, the proposed models require no numerical optimization and are straightforward to generate using the covariance functions measured on the SEM images.

A method for computing upper-bounds on the length of geodesics spanning random sets in 2D and 3D is proposed, 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 → 0. The scalings involve powerlaws with fractional exponents ~f 2/3 for Boolean sets of disks or aligned squares and ~f 1/2 for the Boolean set of spheres. These results are extended to models of hyperspheres in arbitrary dimension and, in 2D and 3D, to a more general problem where the distance function is non-zero in the inclusions. Finally, other fractional exponents are derived for the geodesics spanning multiscale Boolean sets, based on inhomogeneous Poisson point processes, in 2D and 3D.

The overall behavior of a 2D lattice of voids embedded in an anisotropic matrix is investigated in the limit of vanishing porosity f. An effective-medium model (of the Hashin-Shtrikman type) which accounts for elastic interactions between neighboring voids, is compared to Fast Fourier Transform numerical solutions and, in the limits of infinite anisotropy, to exact results. A cross-over between regular and singular dilute regimes is found, driven by a characteristic length which depends on f and on the anisotropy strength. The singular regime, where the leading dilute correction to the elastic moduli is an O(f^{1/2}), is related to strain localization and to change in character - from elliptic to hyperbolic - of the governing equations.

The EVOLVE cell, named after a running FP7 project, is an innovative SOFC concept that integrates advanced materials, providing multiple functionalities (electrochemical activity, ion or/and electron conduction, gas diffusion…), and an inventive anode current collector made of a NiCrAl preoxidized foam impregnated with a conductive perovskite. Therefore, this anode combines the beneficial characteristics of MSC and ASC technologies, and strong improvements in terms of reliability and durability (inter-diffusion, Ni coarsening and agglomeration, sulphur tolerance, mechanical robustness and chemical stability under redox cycling in temperature) are expected. In order to enhance the electrode efficiency, the 3D microstructure needs to be precisely described at the microscopic scale by a morphology model. An optimized microstructure (controlled pores size, morphology and connectivity, grains size distribution, 3D tortuosity and 3D percolation of phases, volume fraction of phases, final thickness of components …) is closely related to the starting parameters (nature and quantity of starting powder, binder, shaping aids…) and to the control of the shaping process parameters (sintering treatment and atmosphere). It should meet the best compromise between a good electrocatalytic activity and a low ohmic resistivity while ensuring a stisfactory long term thermomechanical stability on the life time of the device. Based on a mathematical morphology approach applied on symmetrical LST/CGO anode layers, the present work aims at showing how a morphology model can be establised and 3D microstructural data relevant for shaping and performances can be derived from back-scattered electrons (BSE) SEM observations. For each sample, a series of images were used as input information for the microstructural modelling. All the original images were first filtered to remove the noise, thresholded and finally a series of morphological operators (openings, surface openings, closings and reconstruction with markers) were applied to remove artefacts. A 3D pluri-Gaussian model has been generated from 2D images, and then computed to predict volume fraction of the 3 phases, 3D tortuosity, gas permeation and ionic conductivities. These results were used for the optimization of the shaping process that includes the integration of the metal foam.

Exact solutions are derived for the problem of a two-dimensional, infinitely anisotropic, linear-elastic medium containing a periodic lattice of voids. The matrix material possesses either one infinitely soft, or one infinitely hard loading direction, which induces localized (singular) field configurations. The effective elastic moduli are computed as functions of the porosity in each case. Their dilute expansions feature half-integer powers of the porosity, which can be correlated to the localized field patterns. Statistical characterizations of the fields, such as their first moments and their histograms are provided, with particular emphasis on the singularities of the latter. The behavior of the system near the void close packing fraction is also investigated. The results of this work shed light on corresponding results for strongly nonlinear porous media, which have been obtained recently by means of the ``second-order'' homogenization method, and where the dilute estimates also exhibit fractional powers of the porosity., Comment: 22 pages, 10 figures

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