The Cosserat Vector In Membrane Theory : A Variational Approach

In a previous article (see [BFM]) the authors studied a model of nonlinear membrane where the external surface loading induces a density of bending moment. Due to the special form of the applied surface forces, the emerging Cosserat vector, result of the 3D-2D dimension reduction, was restricted to a class of two dimensional functions. In this paper we analyze the more general case where the Cosserat vector depends also on the thickness variable. In order to detail our main result, relating it with the one in [BFM], we will use the same notations. Let ω be an open bounded subset of R and let I be the interval (−1/2, 1/2). Define Ω := ω × I, Σ± := ω × {±1/2}, Γ := ∂ω × I and, for each ε > 0, Ωε := ω × εI, Σε := ω × {±ε/2}, Γε := ∂ω × εI. In what follows LN stands for the N -dimensional Lebesgue measure in R , N = 2, 3, and H2 denotes the 2-dimensional Hausdorff measure in R. Greek indexes will be used to distinguish the first two components of a tensor, for instance (xα) and (xα, x3), designates (x1, x2) and (x1, x2, x3), respectively. We write R3×2 to denote the vector space of 3× 2 real-valued matrices, and for F ∈ R3×2 and b ∈ R, let (F |b) denote the 3× 3 matrix whose first two columns are those of F and the last one is b. Consider the rescaled total energy of a deformation U : x̃ ∈ Ωε 7→ U(x̃) ∈ R,