Likely equilibria of stochastic hyperelastic spherical shells and tubes

TL;DR

This paper investigates the probability of stable inflation in stochastic hyperelastic spherical shells and tubes, revealing that inherent material variability always introduces a chance of instability, unlike deterministic models.

Contribution

It introduces a probabilistic framework for analyzing inflation stability in hyperelastic shells, accounting for material variability and deriving associated probability distributions.

Findings

Inherent variability leads to always-present competition between stable and unstable inflation.

Derived probability distributions for inflation responses based on experimental rubber data.

Showed that stochastic parameters influence the occurrence of inflation instability.

Abstract

In large deformations, internally pressurised elastic spherical shells and tubes may undergo a limit-point, or inflation, instability manifested by a rapid transition in which their radii suddenly increase. The possible existence of such an instability depends on the material constitutive model. Here, we revisit this problem in the context of stochastic incompressible hyperelastic materials, and ask the question: what is the probability distribution of stable radially symmetric inflation, such that the internal pressure always increases as the radial stretch increases? For the classic elastic problem, involving isotropic incompressible materials, there is a critical parameter value that strictly separates the cases where inflation instability can occur or not. By contrast, for the stochastic problem, we show that the inherent variability of the probabilistic parameters implies that there is always competition between the two cases. To illustrate this, we draw on published experimental data for rubber, and derive the probability distribution of the corresponding random shear modulus to predict the inflation responses for a spherical shell and a cylindrical tube made of a material characterised by this parameter.