A novel analytical-empirical Eulerian method for the hyperelastic layers under punch compression

Abstract

Accurately and efficiently solving the nonlinear mechanical fields in large deformation problems involving hyperelastic soft layers in contact with indenters has long been a challenge in mechanics. This paper aims to propose a new analytical method for incompressible hyperelastic layers under uniform compression in plane strain conditions based on Eulerian description. Utilizing this method, explicit analytical solutions for the mechanical fields of finite-thickness hyperelastic layers under uniform compression are derived. The principles for transforming the mechanical field forms when extending the neo-Hookean model to the Mooney-Rivlin model under plane strain are outlined in this study, providing a unified solution form applicable to incompressible Rivlin-type strain energy functions. During the model validation process, a mesh-to-mesh solution mapping method was implemented to effectively improve numerical accuracy. Building on the analytical solutions for uniformly compressed layers and finite element results, the method is extended to an analytical-empirical model for hyperelastic layers compressed by a flat punch with rounded edges. This model exhibits high congruence with finite element results and allows for the direct extraction of stresses and displacements from the deformed configuration without the need for coordinate transformations, significantly enhancing computational efficiency. This research provides new insights into solving contact mechanics problems of hyperelastic materials under large deformations and offers valuable references for further studies in related fields.