THE QUASI-CONVEX ENVELOPE OF THE SAINT-VENANT-KIRCHHOFF STORED ENERGYFUNCTION

We give an explicit expression for the quasiconvex envelope of the Sai nt Venant-Kirchhoff stored energy function in terms of the singular va lues. This envelope is also the convex, polyconvex, and rank 1 convex envelope of the Saint Venant-Kirchhoff stored energy function. Moreove r, it coincides with the Saint Venant-Kirchhoff stored energy function itself on, and only on, the set of matrices whose singular values arr anged in increasing order are located outside an ellipsoid. It vanishe d on, and only on, the set of matrices whose singular values are less than 1. Consequently, a Saint Venant-Kirchhoff material can be compres sed under zero external loading.