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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?

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The plane-stress model is physically more meaningful for modelling thin structures. This is the reason for its popularity. It's rare to encounter problems with hyperelastic material models where the plane-strain condition is appropriate.

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.

Almost all benchmark 2D examples of hyperelasticity are with the plane-strain model only. Please refer to my paper and references therein for the details. I suggest this textbook for comprehensive fundamental details.

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  • $\begingroup$ Thank you for the detailed answer, the formulation in your paper is also nicely detailed. So, the plane strain examples in your paper were solved using 2D formulations, correct? I wonder how you handled the non-zero normal stress in the third dimension? It's because of this non-zero stress component that I am unable to correctly condense it to 2D only. $\endgroup$ Nov 21, 2022 at 14:26
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    $\begingroup$ 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. $\endgroup$
    – Chenna K
    Nov 25, 2022 at 15:33
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    $\begingroup$ Thank you very much, I figured it out. The 2nd Piola Kirchhoff component in the third direction being non-zero threw me off. $\endgroup$ Nov 26, 2022 at 5:42

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