A second gradient material resulting from the homogenization of an heterogeneous linear elastic medium

eaire AppliquUniversitde Toulon et du Var, BP 132 - 83957 La Garde Cedex, France Homogenization may change fundamentally the constitutive laws of materials. We show how a heterogeneous Cauchy continuum may lead to a non Cauchy continuum. We study the effective properties of a linear elastic medium reinforced periodically with thin parallel fibers made up of a much stronger linear elastic medium and we prove that, when the Lam´ e coefficients in the fibers and the radius of the fibers have appropriate order of magnitude, the effective material is a second gradient material, i.e. a material whose energy depends on the second gradient of the displacement. Continuum mechanics is usually understood as a homogenized description of materials which are heteroge- neous at the microscopic level. Then, it is natural to expect from any general theory of continuum mechanics to be stable by homogenization procedures. We prove in this paper that the class of Cauchy continua does not enjoy this stability property. Indeed, we show that the effective properties of some periodic elastic material have to be described by a second gradient theory. We consider a composite material made up of an elastic matrix reinforced with elastic fibers. Both materials are isotropic linear elastic materials, the Lam ´ e coefficients in the fibers being larger than in the matrix. The structure is periodic: we assume that the fibers are parallel cylinders with the circular section arranged along a square lattice (see Fig. 1). Homogenization procedure consists in studying the limit behaviour of the material when the period of the structure tends to zero. What is the behaviour of the other physical quantities as the period tends to zero? The effective properties of the material strongly depend on them: when the elasticity coefficients in the fibers are of the same order of magnitude as in the matrix and when the radius of the fibers is of the same order of magnitude as the period, the problem is a classic one in homogenization theory: the effective material is still a linear elastic material whose coefficients can be expressed in terms of the geometry and of the elasticity coefficients of the matrix and the fibers (18). We study a different case: we want to describe a composite medium reinforced by very thin and very rigid fibers. Then, it is natural to assume that the radius of the fibers tends to zero faster than the period and that the elasticity coefficients in the fibers tend to infinity. Let us now fix some notations: by convention, we choose the characteristic length of the domain as the unit length. The period of the lattice is denoted by ". We study the limit " ! 0 and every quantity which is not assumed to be constant as " tends to zero, is indiced by ". For instance, the radius of the fibers is denoted by r", the Lam´ e coefficients in the fibers are denoted by " and " while the Lam´