Quasiconvex envelopes in nonlinear elasticity

We give several examples of modeling in nonlinear elasticity where a quasiconvexification procedure is needed. We first recall that the three-dimensional Saint Venant-Kirchhoff energy fails to be quasiconvex and that its quasiconvex envelope can be obtained by means of careful computations. Second, we turn to the mathematical derivation of slender structure models: an asymptotic procedure using Γ-convergence tools leads to models whose energy is quasiconvex by construction. Third, we construct an homogenized quasiconvex energy for square lattices. 1 The Saint Venant-Kirchhoff stored energy function 1.1 Non quasiconvexity of the Saint Venant-Kirchhoff stored energy function This section is based on Raoult (1986) from which it is immediately derived that the Saint Venant-Kirchhoff stored energy function is not rankone convex, and as a consequence not polyconvex, nor quasiconvex. The internal energy of an elastic material reads J(φ) = ∫ Ω W (∇φ(x))dx where Ω ⊂ R is a reference configuration (here assumed to be homogeneous), W : M3×3 7→ R is the stored energy function that is most of the time assumed to be continuous and the deformation φ : Ω 7→ R is sufficiently regular. This is the energy due to the deformation φ. Actually, the domain of W should be restricted to the set M3×3 of matrices with positive determinant and φ should satisfy in some sense det∇φ(x) > 0 in order to express that orientation is preserved by realistic deformations and to prevent matter interpenetration. This restriction leads to mathematical difficulties and is quite often left aside. The total energy is the sum of the internal energy and of the external energy which takes into account the action of external loads (body forces such as gravity, surface forces such as