A calibration method for estimating critical cavitation loads from below in 3D nonlinear elasticity

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 $I(u)={\int }_{\mathrm{\Omega }}W(\mathrm{\nabla }u)dx$ 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 $\mathbf{1}$$,where$$\mathbf{1}$$isthe$3 \times 3$ 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 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 $\lambda_1(\nabla u)$ofacompetingdeformation$u$,thatisnecessaryfortheinequality$I(u) < I(u_{\lambda})$, and which, in particular, does not exclude the possibility of cavitation.