Jean-Claude Michel

Effective properties of composite materials with periodic microstructure: a computational approach

Abstract This study reviews several problems which are specific of composites with periodic microstructure composed of linear or nonlinear constituents. The theoretical background of the method is recalled first. Two different families of numerical methods are considered to solve the problem. The first is based on the Finite Element Method. The concept of ‘macroscopic degrees of freedom’ is presented. The implementation of periodicity conditions is discussed. A general framework permitting either a strain or stress control is proposed. The second numerical method is based on Fast Fourier Transforms. It considers first the problem of a homogeneous linear reference material undergoing a nonhomogeneous periodic eigenstrain. The solution of this problem is based on the explicit form of the periodic Greens function of the reference medium. The relative merits of the two methods are compared and several examples are discussed. Both methods give very comparable results on test examples and their domains of applications appear to be complementary.

Second-order estimate of the macroscopic behavior of periodic hyperelastic composites: theory and experimental validation

Abstract This paper deals with some theoretical and experimental aspects of the behavior of periodic hyperelastic composites. We focus here on composites consisting of an elastomeric matrix periodically reinforced by long fibers. The paper is composed of three parts. The first part deals with the theoretical aspects of compressible behavior. The second-order theory of Ponte Castaneda (J. Mech. Phys. Solids 44 (1996) 827) is considered and extended to periodic microstructures. Comparisons with results obtained by the finite element method show that the composite behavior predicted by the present model is much more accurate for compressible than for incompressible materials. The second part deals with the extension of the method to incompressible behavior. A mixed formulation (displacement–pressure) is used which improves the accuracy of the estimate given by the model. The third part presents experimental results. The composite tested is made of a rubber matrix reinforced by steel wires. Firstly, the matrix behavior is identified with a tensile test and a shear test carried out on homogeneous samples. Secondly, the composite is tested under shearing. The experimentally measured homogenized stress is then compared with the predictions of the model.

Analysis of Inhomogeneous Materials at Large Strains using Fast Fourier Transforms

A computational technique based on fast Fourier transforms has been recently developed to analyze the response of inhomogeneous materials with complex microstructure, both at the global level (effective properties) and local level (stress and strain fields). Initially proposed for elastic constituents and subsequently extended to elastoplastic constituents in the approximation of infinitesimal strains, the method is extended here to large strains, using either a Lagrangian formulation in the reference configuration, or an Eulerian formulation in the deformed configuration. In the latter case, the algorithm inspired by “particle-in-cell” method, makes use of two grids, a computational grid and a material grid. Applications to hyperelastic materials reinforced by stiff inclusions (Lagrangian formulation) and to the motion of a rigid particle in a viscous fluid (Eulerian formulation) are given.

Effective potentials in nonlinear polycrystals and quadrature formulae

This study presents a family of estimates for effective potentials in nonlinear polycrystals. Noting that these potentials are given as averages, several quadrature formulae are investigated to express these integrals of nonlinear functions of local fields in terms of the moments of these fields. Two of these quadrature formulae reduce to known schemes, including a recent proposition (Ponte Castañeda 2015 Proc. R. Soc. A 471, 20150665 (doi:10.1098/rspa.2015.0665)) obtained by completely different means. Other formulae are also reviewed that make use of statistical information on the fields beyond their first and second moments. These quadrature formulae are applied to the estimation of effective potentials in polycrystals governed by two potentials, by means of a reduced-order model proposed by the authors (non-uniform transformation field analysis). It is shown how the quadrature formulae improve on the tangent second-order approximation in porous crystals at high stress triaxiality. It is found that, in order to retrieve a satisfactory accuracy for highly nonlinear porous crystals under high stress triaxiality, a quadrature formula of higher order is required.