Dimension reduction
Calculus of variations
Overview
Polyconvexity – convexity of an energy density in all minors of its matrix argument – is a central structural condition in nonlinear elasticity, as it guarantees the existence of minimisers. In general, however, it is hard to verify, because the convex representative lives in a high-dimensional space.
For isotropic energy densities, we show that it suffices to consider diagonal matrices. The convex representative then only has to depend on the diagonal entries and their products. In three dimensions, this reduces the dimension of the convex representative from \(\mathbb{R}^{19}\) to \(\mathbb{R}^{7}\), which considerably simplifies both the analytical and the numerical verification of polyconvexity. In addition, the work provides a new polyconvexity criterion for functions expressed in terms of principal invariants of the left or right stretch tensor.