Hervé Le Dret

The quasiconvex envelope of the Saint Venant-Kirchhoff stored energy function

We give an explicit expression for the quasiconvex envelope of the Saint Venant–Kirchhoff stored energy function in terms of the singular values. This envelope is also the convex, polyconvex and rank 1 convex envelope of the Saint Venant–Kirchhoff stored energy function. Moreover, it coincides with the Saint Venant–Kirchhoff stored energy function itself on, and only on, the set of matrices whose singular values arranged in increasing order are located outside an ellipsoid. It vanishes on, and only on, the set of matrices whose singular values are less than 1. Consequently, a Saint Venant–Kirchhoff material can be compressed under zero external loading.

Convergence of displacements and stresses in linearly elastic slender rods as the thickness goes to zero

367 Le Dret, H., Convergence of displacements and stresses in linearly elastic slender rods as the thickness goes to zero, Asymptotic Analysis 10 (1995) 367-402. We show the convergence in an appropriate sense of displacements and stresses in a linearly elastic slender rod when its thickness tends to zero. The limit displacements and stresses are determined by well-posed variational problems.

Modeling heterogeneous martensitic wires

We give a direct derivation of a theory of martensitic heterogeneous wires in the zero thickness and homogenization limit via a convergence result. We start from three-dimensional nonlinear hyperelasticity theory augmented by a term of interfacial energy of the van der Waals type. The derivation involves no a priori choice of asymptotic expansion or Ansatz. It yields a wire theory with two Cosserat vector fields. A formal derivation is given of higher theories for homogeneous wires, which yields one corrector for the deformation of the central line and two correctors for the Cosserat vector fields. Finally, we present a few numerical results.

An Up-to-the Boundary Version of Friedrichs's Lemma and Applications to the Linear Koiter Shell Model

In this work, we introduce a variant of the standard mollifier technique that is valid up to the boundary of a Lipschitz domain in

Two Finite Element Approximations of Naghdi's Shell Model in Cartesian Coordinates

\mathbb{R}^n

Homogenization of hexagonal lattices

. A version of Friedrichss lemma is derived that gives an estimate up to the boundary for the commutator of the multiplication by a Lipschitz function and the modified mollification. We use this version of Friedrichss lemma to prove the density of smooth functions in the new function space introduced in our earlierwork concerning the linear Koiter shell model for shells with little regularity. The density of smooth functions is in turn used to prove continuous dependence of the solution of Koiters model on the midsurface. This provides a complete justification of our new formulation of the Koiter model.

Heterogeneous wires made of martensitic materials

We present a penalized version of Naghdis model and a mixed formulation of the same model, in Cartesian coordinates for linearly elastic shells with little regularity, and finite element approximations thereof. Numerical tests are given that validate and illustrate our approach.

Partial Differential Equations: Modeling, Analysis and Numerical Approximation

We characterize the macroscopic effective mechanical behavior of a graphene sheet modeled by a hexagonal lattice of elastic bars, using

The Finite Difference Method for the Heat Equation

\Gamma

The Variational Formulation of Elliptic PDEs

-convergence.