# Plane stress finite element modelling of arbitrary compressible hyperelastic materials

## TL;DR

This paper introduces a robust method for modeling large deformations of compressible and nearly incompressible hyperelastic materials under plane stress conditions using finite element analysis.

## Contribution

A general procedure and open-source code for modeling compressible hyperelastic materials under plane stress conditions are presented.

## Key findings

- The proposed method uses an isochoric/volumetric decomposition for nearly incompressible materials.
- A Newton–Raphson procedure at the quadrature point level solves the out-of-plane deformation gradient equation.
- Benchmark simulations and challenging numerical examples validate the model's performance and accuracy.

## 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.