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The presence of macropores and fractures significantly affects the effective transport properties of porous solids such as concrete and rocks. The dimensions of the fractures are generally large behind that of the initial porosity, so that the problem contains two porosities. The influence of the macroporosity is studied in the homogenization framework by solving at the intermediate scale, that of the macropores, a coupled Darcy/Stokes problem with the Beavers–Joseph–Saffman (BJS) interface condition. We derive analytic expressions of the macroscopic permeability in the case of an isotropic permeable matrix containing spheroidal-shaped macropores. To this aim, we consider a representative volume element (RVE) on which uniform boundary conditions are considered for the velocity and pressure fields. The local problem is written as minimum principles; kinematic and static approaches are developed to derive rigorous bounds for the macroscopic permeability. Closed-form expressions of the longitudinal permeability (along the revolution axe of the spheroid) are determined by considering a simplified RVE constituted of two confocal spheroids. They depend on the volume fraction and the eccentricity of the spheroidal macropores, the scale factor between the two porosities and the slip coefficient of the BJS model. Illustrations show the influence of these parameters.

International audience; The paper presents the study of elastic composites acting as resonant metamaterials. The composites are made of long bers of elliptic cross section. The condition for obtaining resonant metamaterials is presented in the context of homogenization by using asymptotic expansion. It is shown that inner resonance occurs in the case of soft inclusions embedded within a stier matrix, both components having mass densities of the same order of magnitude. The localization problem coming from the asymptotic expansion is solved in the case of bers with elliptic cross sections, leading to the dynamic density of the material. The dynamic density is characterized by resonance frequencies and the related eigenfunctions of the inclusions, that are obtained by using an expansion along Mathieu functions. The method allows us to obtain the (orthotropic) dynamic densityrelated to various cases of elliptic cross section, recovering the extreme cases of circularcross-sections and nearly plate inclusions.

International audience

International audience; This paper gives an update on recent results obtained in the field of prediction of the transport properties of fibrous media from the characterization of their microstructures. (1) The first stage of this work deals with the development of a procedure for the reconstruction of the three-dimensional geometry of the random fibrous medium and its experimental validation [J. Acoust. Soc. Am. 141, 4768 (2017)]. (2) In a second step, the proposed model is used to calculate the associated transport parameters over a wide range of variation of the geometric parameters. Correlations are proposed to approximate these properties when the geometry of the fibrous medium is known [Acta Acust united Ac. 103, 1050 (2017)]. (3) In the third step, the possibility of substituting a polydisperse distribution of fiber diameters by an equivalent monodisperse one is examined numerically [J. Acoust. Soc. Am. 141, EL96 (2017)]. (4) Finally, a fully stochastic methodology is proposed enabling to handle for complex fibrous media presenting both a wide distribution of fiber diameters and fiber lengths.

Colloque avec actes et comité de lecture. Internationale.; International audience; We consider the problem of finding the effective stiffness tensor \mathbb{C}^e$ of periodic heterogeneous martrix-inclusion materials. Given the distribution of the constituents, the cell problem must be solved first and the linear relation between average stress and strain is then established. Estimates can be obtained by making relevant approximation to the ingredients constituting the effective tensor. Although the present contribution concerns the theory of optimally estimating \mathbb{C}^e$ from the microstructure, it is closely related to FFT numerical homogenization methods. By introducing a reference material \mathbb{C}^0$, the heterogeneity effect can be viewed as a distribution of eigenstrains within an homogeneous material. Using the related Green tensor, our problem can be formulated as a Lippmann-Schwinger equation for eigenstrain. The integral equation is the origin of resolution methods based on iteration and Fast Fourier Transform (FFT) techniques (Michel et al., 1999; Bhattacharya and Suquet, 2005). Significant progresses have been made regarding the improvement of convergence rate (Michel et al., 1999; Eyre and Milton, 1999; Milton, 2002; Monchiet and Bonnet, 2012; Brisard and Dormieux, 2010). The study of convergence rate in those works will be extended in the present contribution in the case of new integral equations. The iteration scheme used to solve the Lippmann-Schwinger equation corresponds to the Neumann series summation. The latter can be used to derive exact theoretical relations and estimates. In this paper, we propose a new estimate based on series expansion that works at all contrast ratio, while using the matrix as a reference material. Additionally, we can control and optimize the convergence rate so that the series converges in the quickest way, and therefore produces the best estimates when using a finite sum in the series expansion. A class of integral equations for eigenstrain depending on two parameters \alpha,\beta$ is first derived. The spectral radius and norm of the corresponding operators are bounded by analytical expressions. Different optimization methods are proposed to find the fastest series convergence and the associated estimates. Similarly to the estimations of the effective elasticity tensor using correlation functions, the new method presented in this paper allows to estimate the effective elasticity tensor using the n-$ order structure factors, which represent the counterpart in Fourier space of correlation functions. As an example, a direct connection of the effective elasticity tensor to n-$th order structure factors is given in the case of randomly distributed spheres. Numerical applications for cubic arrays and random distribution of spheres yield very good results in comparison with FFT based methods and other results from the literature.

International audience; Angular orientation is a state of fibers which can be intentionally changed to modify the properties of the corresponding material. Here, the angular orientation tensor of a fibrous material is identified from scanning electron microscope images, in order to construct a model of the microgeometry, in which the partial differential equations governing the physics at the upper scale can be solved. Permeability and impedance tube measurements show that the corresponding model is a good candidate to simulate the transport properties of fibrous materials. It is also shown that the transport properties of the samples are highly affected and tunable through the angular orientation of the fibers. The proposed numerical approach can be applied to evaluate the effective transport and sound absorbing properties of general fibrous materials just from the geometrical information embodied in the microstructural parameters of single fibers, the fibers orientation distribution and the bulk porosity.

International audience

National audience; L'orientation angulaire désigne un état des fibres pouvant être intentionnellement changé pour modifier les propriétés du matériau correspondant. Dans cette communication, nous identifions le tenseur d'orientation angulaire d'un matériau fibreux, constitué par l'agencement de fibres végétales et de fibres bi-composants utilisées comme liant, par images MEB. On reconstruit alors un volume élémentaire représentatif (VER) de la micro-géométrie aléatoire à partir des distributions d'orientations angulaires précédemment mesurées. Pour plusieurs tirages de VER reconstruits, les équations locales (équations de Stokes, écoulement potentiel, conduction thermique) sont ensuite résolues par la méthode des éléments finis pour identifier les coefficients du modèle macroscopique. Des mesures de perméabilité et au tube d'impédance réalisées sur les échantillons réels montrent que le modèle correspondant est un bon candidat pour simuler l'ensemble des propriétés de transport de matériaux fibreux désordonnés à partir de la connaissance des principales caractéristiques de la géométrie (diamètres et proportions de fibres, distributions angulaires, porosité). Il est aussi montré par simulations que les propriétés de transport, en particulier la perméabilité, sont très affectées et donc modulables par l'orientation angulaire des fibres, ce qui en fait un levier privilégié pour l'amélioration ultérieure de leurs propriétés acoustiques.