Plane stress finite element modelling of arbitrary compressible hyperelastic materials

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.

Figures (8)

• Figure 1: The Cook's cantilever under transverse loading: (a) Schematic representation. (b) Vertical displacement of the tip versus applying traction $f$.
• Figure 2: The deformed Cook's cantilever with $32 \times 32$ mesh size, $p_o=2$ for different models under $f=40 \, \text{N}/\text{mm}^2$.
• Figure 3: The effect of the thickness of the Cook's cantilever in the three-dimensional model on the vertical displacement of the tip.
• Figure 4: The block under inhomogeneous compression; the top surface is constrained in $x_1$ direction.
• Figure 5: Maximum compression of the block: (a) for different mesh sizes and applied traction of $f=6000 \, \text{N}/\text{mm}^2$ (b) versus applied traction $f$.