# Finite Element Modelling of Hyperelastic Material under 2D Plane Strain Conditions

I am currently working on writing a MATLAB code for running a finite element simulation of a hyperelastic material in 2D. Since I am building this simulation as a part of a fluid-structure interaction analysis, the 2D assumption is usually carried out using plane strain conditions. I started from Kim's textbook since he included examples written in MATLAB on simulating hyperelasticity but only in 3D, while the 2D assumptions for hyperelasticity were not discussed.

My two questions are:

1. Most of the literature I found on finite element modelling of hyperelasticity in 2D addresses the plane stress conditions. I understand that in real-life 2D plane strain rubber might not be that common, but I wanted to inquire if there is a more fundamental reason behind this popularity of plane stress over plane strain in numerical simulations of hyperelasticity?

2. I am working on condensing the 3D model into the 2D plane strain model. Is there any resources on finite element modelling on hyperelasticity under plane strain conditions?

However, the plane-strain model for hyperelasticity is relatively easier to implement than the plane-stress model. The plane-strain model is nothing but the 3D model with zero-strain in the third direction, and this can be realised by setting $$F_{33}=1$$ in the deformation gradient. But, the incorporation of the zero-stress condition in the third direction makes the plane-stress model quite complicated for hyperelasticity.
• Yes, that is correct! We don't have to worry about the stress components in the Z direction in the solution phase as we don't have any constraints on stresses. Set $F_{33}=1$, calculate the elasticity tensor and Cauchy stress and retain only the relevant components to compute the element stiffness matrix and residual. Nov 25, 2022 at 15:33