# Likely oscillatory motions of stochastic hyperelastic solids

## TL;DR

This paper investigates how uncertainties in material properties affect the oscillatory behavior of stochastic hyperelastic solids, revealing probabilistic distributions of oscillation characteristics and conditions for oscillatory versus non-oscillatory motions.

## Contribution

It introduces a probabilistic framework for analyzing oscillations in stochastic hyperelastic solids, including new insights into the distribution of oscillation amplitudes and periods.

## Key findings

- Amplitude and period follow identifiable probability distributions.
- Existence of parameter intervals with competing oscillatory and non-oscillatory motions.
- Probabilistic characterization of likely oscillatory motions.

## Abstract

Stochastic homogeneous hyperelastic solids are characterised by strain-energy densities where the parameters are random variables defined by probability density functions. These models allow for the propagation of uncertainties from input data to output quantities of interest. To investigate the effect of probabilistic parameters on predicted mechanical responses, we study radial oscillations of cylindrical and spherical shells of stochastic incompressible isotropic hyperelastic material, formulated as quasi-equilibrated motions where the system is in equilibrium at every time instant. Additionally, we study finite shear oscillations of a cuboid, which are not quasi-equilibrated. We find that, for hyperelastic bodies of stochastic neo-Hookean or Mooney-Rivlin material, the amplitude and period of the oscillations follow probability distributions that can be characterised. Further, for cylindrical tubes and spherical shells, when an impulse surface traction is applied, there is a parameter interval where the oscillatory and non-oscillatory motions compete, in the sense that both have a chance to occur with a given probability. We refer to the dynamic evolution of these elastic systems, which exhibit inherent uncertainties due to the material properties, as `likely oscillatory motions'.