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

TL;DR

This paper provides an explicit criterion for when a simple affine deformation minimizes elastic energy in 3D nonlinear elasticity, accounting for cavitation and using advanced calculus of variations techniques.

Contribution

It introduces a new sufficient condition for energy minimization that explicitly handles cavitating deformations in 3D nonlinear elasticity.

Findings

Derived a sufficient condition for affine maps to be global minimizers.

Established a necessary condition involving the smallest singular value of deformations.

Utilized null Lagrangians and numerical methods to compute key constants.

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 $\lambda \mathbf{1}$, where $\mathbf{1}$ 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 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 $\lambda_1(\nabla u)$ 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.