When Rank-One Convexity Meets Polyconvexity: An Algebraic Approach to Elastic Binodal

In the variational problems involving non-convex integral functionals, finding the binodal, the boundary of validity of the quasiconvexity of the energy density, is of central importance. We develop a systematic methodology for identifying a part of the binodal corresponding to simple laminates by showing that in this case the supporting null-Lagrangians, establishing polyconvexity, can be constructed explicitly. We present a nontrivial example from nonlinear elasticity where this approach allows one to obtain the entire quasiconvex envelope.

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