Relaxation of variational problems in two-dimensional nonlinear elasticity

Abstract

Consider a plate occupying in a reference configuration a bounded open set ⊂ R 2 , and let W : M 3×2 → [0, +∞] be its stored-energy function. In this paper we are concerned with relaxation of variational problems of type:= (x, 0) on ∂ } with p > 1, ·, · is the scalar product in R 3 and f ∈ L q ( ; R 3 ), with 1/ p + 1/q = 1, is the external loading per unit surface. We take into account the fact that an infinite amount of energy is required to compress a finite surface of the plate into zero surface, i.e., W ξ 1 | ξ 2 → +∞ as ξ 1 ∧ ξ 2 → 0.

Relaxation theorems in nonlinear elasticity

“…Other related results can be found in [7,5] where we refer the reader. The present work improves our previous one [1] (see also [2,3]). The main new contribution of the present paper is the treatment of the case N = 3.…”

The nonlinear membrane energy: Variational derivation under the constraint “det∇u≠0”

“…In [9], which is the paper corresponding to the note [8], the statement [8, Theorem 1] is partly proved (however, a more detailled proof, but not fully complete, can be found in his thesis [7]). Moreover, for Ben Belgacem W mem = QRW 0 where RW 0 denotes the rank one convex envelope of W 0 (in fact, as we proved in [3,4], QRW 0 = QW 0 ). Nevertheless, Ben Belgacem's thesis highlighted the role of approximation theorems for Sobolev functions by smooth immersions in the studying of the passage 3D-2D in presence of ( 1) and (2).…”

Homogenization of nonconvex unbounded singular integrals

“…This completes the proof of Theorem 2.33. for all n ≥ 1 and all k ≥ 1. Using (A.1) we deduce thatlim n→∞ lim k→∞ A L(ξ + ∇ϕ n,k (x)) dx = ZL(ξ) ,(A 3). and the result follows from (A.2) and (A.3) by diagonalization.Remark A.6.…”