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

Abstract

In this paper we give an explicit sufficient condition for the affine map $u_\lambda(x):=\lambda x$ to be the global energy minimizer of a general class of elastic stored-energy functionals $I(u)=\int_{\Omega} W(\nabla u)\,dx$ in three space dimensions, where $W$ is a polyconvex function of $3 \times 3$ 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 $I(\cdot)$ at $\lambda {\bf 1}$, where \bf 1 is the $3 \times 3$ identity matrix. Our approach relies on arguments involving null Lagrangians (in this case, affine combinations of the minors of $3 \times 3$ matrices), on the previous work [J. Bevan and C. Zeppieri, Calc. Var. Partial Differential Equations, 55 (2015), pp. 1--25], 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 $\lambda_1(\nabla u)$ of a competing deformation $u$ that is necessary for the inequality $I(u) < I(u_{\lambda})$, and which, in particular, does not exclude the possibility of cavitation.