Abstract
In this paper we give an explicit sufficient condition for the affine map uλ(x):=λx to be the global energy minimizer of a general class of elastic stored-energy functionals I(u)=∫ΩW(∇u)dx in three space dimensions, where W is a polyconvex function of 3×3 matrices. The function space setting is such that cavitating (i.e., discontinuous) deformations are admissible. In the language of the calculus of variations, the condition ensures the quasiconvexity of I(⋅) at λ1, where 1 is the 3×3 identity matrix. Our approach relies on arguments involving null Lagrangians (in this case, affine combinations of the minors of 3×3 matrices), on the previous work Bevan & Zeppieri, 2015, and on a careful numerical treatment to make the calculation of certain constants tractable. We also derive a new condition, which seems to depend heavily on the smallest singular value λ1(∇u) of a competing deformation u, that is necessary for the inequality I(u)<I(uλ), and which, in particular, does not exclude the possibility of cavitation.