Necessary and Sufficient Conditions for Polyconvexity of Isotropic Functions

Isotropic energy densities in finite-strain elasticity can be written in terms of the singular values of the deformation gradient, which is assumed to have nonnegative determinant. It is of importance to characterize convexity conditions like rank-one convexity, polyconvexity and full convexity in terms of the singular-value representation. Necessary and sufficient conditions for rank-one convexity (strict ellipticity) were found in 1977 and 1983. Polyconvexity is a more difficult notion since it is a nonlocal condition. The case of space dimension 2 was solved by Rosakis 1998 and Silhavy 1999. This work is based on a different approach and leads to new necessary conditions in every space dimension. For dimensions 2 and 3 the conditions are also shown to be sufficient. The result in 3D relies on the fact that the extremal points of the set of all Schur products of two rotation matrices is given by 24 signed permutations.