THE QUASI-CONVEX ENVELOPE OF THE SAINT VE NANT-KIRCHHOFF STORED ENERGY FUNCTION

We give art explicit expression for the quasiconvex envelope of the Sa int Venant-Kirchhoff stored energy function. This envelope is zero on, and only on, the set of matrices whose singular values are less than 1. It coincides with the Saint Venant-Kirchhoff stored energy function on, and only on, the set of matrices whose singular values arranged i n increasing order are located outside of an ellipsoid Moreover, this envelope is also the convex and polyconvex envelope of the Saint Venan t-Kirchhoff stored energy function.