The nonlinear membrane energy : Variational derivation under the constraint “ detu 0 ”

where ε > 0 is very small and Σ ⊂ R is Lipschitz, open and bounded. A point of Σε is denoted by (x, x3) with x ∈ Σ and x3 ∈]− ε 2 , ε 2 [. Let W : M 3×3 → [0,+∞] be the stored-energy function supposed to be continuous and coercive, i.e., W (F ) ≥ C|F | for all F ∈ M and some C > 0. In order to take into account the important physical properties that the interpenetration of matter does not occur and that an infinite amount of energy is required to compress a finite volume into zero volume, i.e., W (F ) → +∞ as detF → 0, where detF denotes the determinant of the 3× 3 matrix F , we assume that: (1) W (F ) = +∞ if and only if detF ≤ 0;