Rank-One-Convex and Quasiconvex Envelopes for Functions Depending on Quadratic Forms

In this paper we are interested in functions defined, on a set of matrices, by the mean of quadratic forms and we compute the rank-one-convex, quasiconvex, polyconvex and convex envelopes of these functions. For that, and for a given quadratic form, we prove, in a first part, some general decomposition results for matrices, with a rank-one-compatibility condition. We also study the James-Ericksen stored energy function.