A Calibration Method for Estimating Critical Cavitation Loads from Below in Three Dimensional Nonlinear Elasticity

Abstract

Abstract. In this paper we give an explicit sufficient condition for the affine map u λ (x) := λx to be the global energy minimizer of a general class of elastic stored-energy functionals I(u) = Ω W (∇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 [4], 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(u λ ), and which, in particular, does not exclude the possibility of cavitation.