Abstract : We explicit the closure for the Mosco-convergence in of the set of elasticity functionals. We prove that this closure coincides with the set of all non-negative lower-semicontinuous quadratic functionals which are objective, i.e. which vanish for rigid motions. The result is still valid if we consider only the set of elasticity functionals which have a prescribed Poisson coefficient. This shows that a very large family of materials can be reached when homogenizing a composite material with highly contrasted rigidity coefficients.

Keywords : Homogenization, Mosco-convergence, Gamma-convergence, Composite materials.

Introduction : Finding all possible materials which may, a priori, be the limit of a given sequence of elastic materials has a great mathematical and practical interest. Indeed, once the form of the limit energy is known it is enough to consider a few typical situations (and to make a few experiments) to identify the parameters which determine this limit. This is generally the way followed by those who make experiments in order to describe the behavior of composite materials.

This paper is devoted to the case of linear elasticity in the three-dimensional physical space. Then, the considered functionals are non-negative quadratic forms defined on a functional space, namely on the Sobolev space H1(W,R³)and take the form

Here W is the domain (a regular bounded open set of R³) occupied by the material. The function f(x,.) is a non-negative quadratic form on the space of 3x3-matrices.

Rigid motions, in the linear theory, are those functions u which take the form u(x):= c+ w \wedge x for some given vectors c and w of R³ .
A physical law (called objectivity) leads to consider only functions f which vanish for rigid motions. It is well known that this implies that f(x,.) depends only on the strain tensor, i.e. the symmetric part e(u) of the gradient of u (2 e(u):=Du + D^t u). The particular case of isotropic elastic materials is of great interest. In that case the energy density f is determined by two scalar quantities a and b (the Lam\'e coefficients) :

Here ||e(u)|| denotes the Euclidian norm of the matrix e(u) and tr(e(u)) its trace. We consider a set S of functionals of type (1)-(2) and we search the set of all possible limits of sequences contained in this set. In other words we search the sequential closure of the considered set of functionals.

This problem has been extensively studied under assumptions which ensure that the limit functionals are still elasticity functionals. Suitable assumptions can be  the existence of positive constants c, C such that, for any F in S, the energy density f satisfies \

This is the case, for instance, of a composite material made of two isotropic materials with comparable rigidity. Then f has the form (2) with

where Wⁿ is the part of the domain occupied by one of the two components. In these cases, the so-called bounds theory gives estimations for the limit functional. However the precise determination of the closure of the considered set of functionals has not already been completed.

But in many composite materials, the different components have very different Lame coefficients. Then assumptions (3) are no more suitable and one must consider, for instance, sequences of functionals in which the ratio a₁/ a₂  tends to zero or infinity. It is known that, in this case, the limit functional can be very different from the initial ones. Examples have been given in which the limit functional is non-local or in which the limit functional involves some second derivative of u. Many questions arise : Can any dependence of the second gradient be reached? In particular can we obtain the following functional :

where x₁ and u₁ denote the components of x and u in a same direction. Such an energy would lead to a very unusual behavior of the material.

Can the limit functional depend on higher derivatives of u? For instance, can we obtain the following functional :

In an other range, can a sequence of functionals of type (1)-(2) with positive coefficients b lead to a limit of the same type but with a negative coefficient b? Using the definition of the Poisson coefficient nu in terms of Lame coefficients :

this question can be reformulate in a more classical way : can a material with negative Poisson coefficient be the limit of materials with positive Poisson coefficients? In particular, under the same restrictions, can we obtain the degenerated functional (when nu=-1) :

A positive answer to this last question has been given by Milton by the explicit and intricate construction of a sequence of suitable composite materials. Of course, this list of questions is not exhaustive and constructing explicit sequences of functionals in order to answer to all these questions seems beyond reach. In this paper we give a positive answer to all these questions using an indirect way. We prove that any objective, non-negative quadratic functional can be obtained (provided it is lower semi-continuous for the considered topology). This result must be compared with the scalar case. We recently determined the closure of the set of diffusion functionals. We proved that this closure coincides with the set of all Dirichlet forms which vanish on constant fields. This scalar result can be easily extended to a vectorial case. Indeed, any functional of type (1) with

is the sum of three independent diffusion functionals acting on the different components of u. The limit of any sequence of such functionals will still be the sum of three diffusion functionals. Then the Deny-Beurling formula provides a representation for this limit as

in which, for all k in {1,2,3}, nu^k and mu^k are non-negative Radon measures respectively on W and WxW, while eta^k is a Radon measure on W taking values in the set of non-negative symmetric matrices. Hence, any limit of a sequence of functionals of type (1)-(9) is the sum of a diffusion term, a non-local two-points interaction and a killing (or strange) term, but, in no way, functionals like (5) or (6) can appear. In the present paper we are concerned by elasticity functionals (1)-(2) and our result is very different : the closure we find is much larger than the set of functionals of type (10). In our opinion, this phenomenon is due to the particular structure of the kernel of elasticity functionals. Note that, as the set of elasticity functionals contains only objective functionals, we cannot obtain killing terms (i.e. terms of type \int_W (u^k(x))² nu^k(dx) ) in a limit functional. For lack of space, we do not study here the case of elasticity functionals submitted to some Dirichlet boundary condition. In that case, we could prove that the closure coincides with the whole set of all non-negative quadratic functionals (which are lower semi-continuous for the considered topology).