Likely equilibria of the stochastic Rivlin cube

TL;DR

This paper investigates the stability and likely equilibria of a stochastic hyperelastic cube model, accounting for randomness in material parameters, and derives the probability distribution of deformations under uncertainty.

Contribution

It introduces a stochastic version of the Rivlin cube problem, analyzing how parameter variability affects equilibrium stability and deformation probabilities.

Findings

Identifies likely equilibria considering parameter randomness.

Analyzes stability dependence on material constitutive laws.

Derives probability distributions of deformations under stochastic parameters.

Abstract

The problem of the Rivlin cube is to determine the stability of all homogeneous equilibria of an isotropic incompressible hyperelastic body under equitriaxial dead loads. Here, we consider the stochastic version of this problem where the elastic parameters are random variables following standard probability laws. Uncertainties in these parameters may arise, for example, from inherent data variation between different batches of homogeneous samples, or from different experimental tests. As for the deterministic elastic problem, we consider the following questions: what are the likely equilibria and how does their stability depend on the material constitutive law? In addition, for the stochastic model, the problem is to derive the probability distribution of deformations, given the variability of the parameters.