A concise derivation of membrane theory from three-dimensional nonlinear elasticity

Membrane theory may be regarded as a special case of the Cosserat theory of elastic surfaces, or, alternatively, derived from three-dimensional elasticity theory via asymptotic or variational methods. Here we obtain membrane theory directly from the local equations and boundary conditions of the three-dimensional theory. 1. Three-dimensional prismatic bodies Let F, with detF > 0, be the gradient of a deformation function χ(X), where X is the position of a material point in a reference placement. Let P̂(F) be a smooth constitutive function for the Piola stress P in an elastic body, and let W (F) be a twice-differentiable function furnishing the strain energy stored in the body, per unit reference volume. Then, P̂(F) =WF, the derivative with respect to the deformation gradient. To simplify matters we suppose the material to be uniform in the sense that the functions P̂ and W do not depend explicitly on reference position. We consider deformations that are C in the interior of the body and we suppress body forces. If such a deformation is equilibrated, then the associated stress satisfies DivP = 0 (1) pointwise, where Div is the divergence with respect to position in the reference placement. We assume that equilibria satisfy the strong-ellipticity condition a⊗ b ·M(F)[a⊗ b] > 0 for all a⊗ b 6= 0, (2) where M(F) =WFF (3) is the tensor of elastic moduli. We note that the constitutive hypothesis of strong ellipticity is used in [1] to prove an existence theorem for C equilibria under restrictive conditions on the boundary data. Our assumption of strong ellipticity at equilibrium, without restrictions on the boundary data, is insufficient to meet the hypotheses of this theorem. It is nevertheless consistent with the degree of smoothness assumed for the equilibrium deformation, and so it is natural to restrict attention to deformations that are sufficiently smooth for (1) to be meaningful everywhere in the interior of the body. The relaxation of this restriction in the context of the theory of gamma convergence is known to yield a model that is energetically optimal