A Variational Approximation Scheme for Radial Polyconvex Elasticity That Preserves the Positivity of Jacobians

and is also a sufficient condition for avoiding interpenetration of matter. The constitutive properties of hyperelastic materials are completely determined by the stored energy function W (F ) :M + → [0,∞), which — due to frame indifference — has to be invariant under rotations. For isotropic elastic materials W (F )=Φ(v1,v2,v2), where Φ is a symmetric function of the principal stretches v1,v2,v3 of F ; see [14]. Convexity of the stored energy is, in general, incompatible with certain physical requirements and is not a natural assumption. For instance, in order to avoid interpenetration of matter the stored energy should increase without bound as detF →0+ so that compression of a finite volume down to a point would cost infinite energy. This behavior is inconsistent with simultaneously requiring convexity and invariance of the stored energy under rotations. As an alternative, the assumption of polyconvexity [1] is often employed, which postulates that