Variational convergence for nonlinear shell models with directors and related semicontinuity and relaxation results

We use a variational convergence method to study the consistency of various Cosserat hypotheses in shell theory with the limit nonlinear membrane mode l derived from three-dimensional elasticity. In the course of the analysis, we introduce a generalization of quasiconvexity that is suitable for probl ems of the calculus of variations with two vectorial unknowns, one of which appears through its gradient, the other one through its value, in a weak W -1,W-p x L-p framework.