HyperSym: An Educational MATLAB Code for Hyperelasticity

Abstract

Engineering students may find two challenges while studying finite element-based structural analysis: the transition from linear to nonlinear analysis theory and implementing finite element algorithms. Unlike damage and plasticity, which require often complex return mapping algorithms to update internal variables, introducing material nonlinearity with hyperelasticity is simpler as the stress tensor therein is computed explicitly from a deformation measure. To simplify the derivation process, we present HyperSym, an educational MATLAB-based tool that leverages symbolic differentiation to derive hyperelastic tensors from the strain energy density functional and automatically generate ready-to-use functions. We integrate these functions into the educational open-source finite element software  NLFEA to illustrate the connection between user-defined subroutines and a finite element framework often found in commercial packages. This paper outlines  HyperSym's core features and demonstrates its educational potential through numerical examples applicable to lecture and homework settings. Lastly, we explore potential extensions and customizations to  HyperSym for further academic projects or research. The complete version of MATLAB implementation of  HyperSym is available in a public repository, and some extensions and modifications are provided as Supporting Information.