Keywords: Second gradient theory, Cosserat media, Homogenization, Gamma-convergence.

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 circular section arranged along a square lattice (see figure 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. We study a different case: we want to describe a composite medium reinforced by very thin and ved 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 e. We study the limit e tends to 0 and every quantity which is not assumed to be constant as e tends to zero, is indiced by e. For instance, the radius of the fibers is denoted by r_e, the Lamé coefficients in the fibers are denoted by l_e and mu_e while the Lamé coefficients in the matrix are denoted by l_0 and mu_0. Then our assumptions read r_e/ e tends to0, l_e tends to infty, mu_e tends to infty.

This situation has already been studied by D. Caillerie who, setting l_e=(r_e /e)^-t, mu_e=r_e /e)^-t, considered in two cases the limit (e,r_e /e) tending to (0,0): (r_e / e tends to 0 then e tends to 0) and (e tends to 0 then r_e /e tends to 0). He found that both cases lead to an elastic material but that the homogenized elasticity coefficients depend on the limit procedure: the two limits e tends to 0 and (r_e /e) tends to 0 do not commute. Here we let r_e/e, mu_e^-1 and l_e^-1 tend to zero together and assume that: e tends to 0, r_e/e tends to 0 , e^2 log (r_e) tends to 0 , mu_e r_e^4 /e^2 tends to m_1 >0 , l_e /mu_e} tends to a finite value. Th to a finite value. This particular scaling leads to a very different limit behaviour. We prove that the energy of the effective material depends not only on the strain tensor (as a classical elastic material) but also on the second gradient of the displacement.

Materials whose energy depends on the second gradient of the displacement cannot be considered as Cauchy continua otherwise one would be led to a thermodynamic paradox. This paradox can be removed by extending the thermodynamical framework but the fundamental point is that the Cauchy stress tensor is not sufficient to describe internal force. External forces concentrate along any edge of the boundary and the Cauchy theorem defining the Cauchy stress tensor cannot be applied. Moreover, a supplementary boundary condition is needed to write well-posed problems, which is unusual and not intuitive. The simplest way to describe these media is to use the second gradient theory or to consider them as Cosserat media. Our result gives a new example of such a material together with a ``microscopic" interpretation of its special features.

We emphasize that, going to the limit, the differential order of the energy changes (as does the system of partial differential equations associated with equilibrium). Such a change is not usual in homogenization theory. It arises in rod or plate theories but seems then to be connected with a change of dimension. Our result shows that Our result shows that this is not necessary. Notice also that such a change in the differential order of the energy can not arise when considering scalar problems (like thermal conductivity problems). Indeed, consider a sequence of energies which are quadratic functions of the gradient of a scalar quantity u; these energies decrease when truncating u and this property is preserved when going to the limit. Then, a representation theorem for Dirichlet forms assures that the limit energy can be represented as the sum of a term depending on u, the gradient of u and a non-local term. In other words, we can expect non-local effects but no increase of the differential order. Our result shows that this argument cannot be extended to elasticity problems.

Non-local effects actually arise for some scalar singular perturbation problems and we should probably have obtained non-local effects if assuming that e^2 |log (r_e)| converges to a finite positive value instead of zero. We do not have non-local effects under our assumptions: the second gradient part of the limit energy cannot be interpreted, as it is often done, as the limit of non-local interactions whose range is very short.

Our study is variational. We identify the Gamma-limit E_0 of the energy E_e of our composite material. The notion of Gamma-convergence corresponds to the intuitive notion of convergence of models: the result is obtained without considering external forceidering external forces, it remains valid in presence of body forces.

The limit energy is made explicit in section 2 where we state precisely our result. Section 3 is devoted to the more difficult part of the proof: considering a sequence of displacement fields u_e converging to some u, we have to express the lower bound for the energy E_e(u_e) in terms of u. This needs an accurate description of the asymptotic behaviour of u_e. Especially u_e has to be described at the scale r_e inside the fibers: we need a multiscale notion of convergence. However, we do not expect any periodicity with period r_e; the classical notions of multiscale convergence are not convenient. In subsection 3.1, we develop an adapted notion of double-scale convergence which describes the asymptotic behaviour of u_e in the fibers, that is in a set of scale r_e but with periodicity e. Section 4 is devoted to the end the proof: for any admissible displacement field u we have to construct an approximating sequence u_e whose limit energy is not larger than E_0(u). Such an approximation is obtained by choosing u_e=u in the main part of the matrix, a rod-like displacement field in the fibers and a suitable interpolation in transition layers around each fiber.