Likely cavitation in stochastic elasticity

TL;DR

This paper investigates cavitation in stochastic elasticity, revealing probabilistic outcomes of cavitation at critical loads and introducing the concept of 'likely cavitation' due to material uncertainties.

Contribution

It extends classical cavitation analysis to stochastic materials, deriving probability distributions of deformation and demonstrating the coexistence of stable and unstable cavitation.

Findings

Probabilistic cavitation occurs at a range of loads due to material variability.

Unstable snap cavitation can occur in certain isotropic materials.

Cavitation likelihood depends on material parameter distributions.

Abstract

We revisit the classic problem of elastic cavitation within the framework of stochastic elasticity. For the deterministic elastic problem, involving homogeneous isotropic incompressible hyperelastic spheres under radially symmetric tension, there is a critical dead-load traction at which cavitation can occur for some materials. In addition to the well-known case of stable cavitation post-bifurcation at the critical dead load, we show the existence of unstable snap cavitation for some isotropic materials satisfying Baker-Ericksen inequalities. For the stochastic problem, we derive the probability distribution of the deformations after bifurcation. In this case, we find that, due to the probabilistic nature of the material parameters, there is always a competition between the stable and unstable states. Therefore, at a critical load, stable or unstable cavitation occurs with a given probability, and there is also a probability that the cavity may form under smaller or greater loads than the expected critical value. We refer to these phenomena as `likely cavitation'. Moreover, we provide examples of homogeneous isotropic incompressible materials exhibiting stable or unstable cavitation together with their stochastic equivalent.