Quasiconvex functions incorporating volumetric constraints are rank-one convex

AbstractWe prove that a quasiconvex function W:Mn×n→[0,∞] which is finite on the set Σ={F:detF=1} is rank-one convex, and hence continuous, on Σ; and the same for constraints on minors. This implies that the rank-one convex envelope gives an upper bound on the quasiconvex envelope of any energy density modeling an incompressible material. Our result is based on the construction of an appropriate piecewise affine function u such that ∇u∈Σ almost everywhere