A virtual element method for isotropic hyperelasticity

TL;DR

This paper introduces a virtual element method tailored for plane hyperelasticity problems, demonstrating robustness, convergence, and effectiveness across various material models and element geometries, including near-incompressible cases.

Contribution

The paper presents a novel stabilization parameter selection approach for virtual element methods applied to hyperelasticity, ensuring robustness and locking-free performance across diverse geometries.

Findings

Method is robust and locking-free.

Converges well for various strain energy functions.

Performs effectively with highly concave elements.

Abstract

This work considers the application of the virtual element method to plane hyperelasticity problems with a novel approach to the selection of stabilization parameters. The method is applied to a range of numerical examples and well known strain energy functions, including neo-Hookean, Mooney-Rivlin and Ogden material models. For each of the strain energy functions the performance of the method under varying degrees of compressibility, including near-incompressibility, is investigated. Through these examples the convergence behaviour of the virtual element method is demonstrated. Furthermore, the method is found to be robust and locking free for a variety of element geometries, including elements with a high degree of concavity.