Plane stress finite element modelling of arbitrary compressible hyperelastic materials

TL;DR

This paper presents a comprehensive finite element modeling approach for large deformations of hyperelastic materials under plane stress, accommodating arbitrary compressibility and nearly incompressible behaviors with a robust open-source implementation.

Contribution

It introduces a general procedure for plane stress modeling of compressible hyperelastic materials, including a robust finite element formulation and solution method for out-of-plane deformation.

Findings

Accurate simulation of benchmark problems demonstrated.

Effective modeling of reinforced composite materials.

Robustness in nearly incompressible material cases.

Abstract

Modelling the large deformation of hyperelastic solids under plane stress conditions for arbitrary compressible and nearly incompressible material models is challenging. This is in contrast to the case of full incompressibility where the out-of-plane deformation can be entirely characterised by the in-plane components. A rigorous general procedure for the incorporation of the plane stress condition for the compressible case (including the nearly incompressible case) is provided here, accompanied by a robust and open source finite element code. An isochoric/volumetric decomposition is adopted for nearly incompressible materials yielding a robust single-field finite element formulation. The nonlinear equation for the out-of-plane component of the deformation gradient is solved using a Newton-Raphson procedure nested at the quadrature point level. The model's performance and accuracy are made clear via a series of simulations of benchmark problems. Additional challenging numerical examples of composites reinforced with particles and fibres further demonstrate the capability of this general computational framework.