Quasiconvexity and Rank-One Convexity Conditions in the Nonlinear Theory of Elastic Shells

Abstract We consider the general theory of 6-parameter shells, in which material points on the midsurface are endowed with 3 translational and 3 rotational degrees of freedom. In this framework, we derive quasiconvexity conditions and rank-one convexity conditions. These inequalities represent necessary conditions for energy minimizers; they are the two-dimensional counterparts of the well-known relaxed convexity conditions in three-dimensional finite elasticity. As a specific feature, the quasiconvexity inequality for shells contains the gradients in the tangent plane of the variation fields associated to deformation and microrotation. Finally, we also deduce the Legendre-Hadamard condition for shells, as a consequence of the rank-one convexity inequality.