Energetically Consistent Invertible Elasticity

Abstract

We provide a smooth extension of arbitrary isotropic hyperelastic energy density functions to inverted configurations. This extension is designed to improve robustness for elasticity simulations with ex- tremely large deformations and is analogous to the extension given to the first Piola-Kirchoff stress in [ITF04]. We show that our energy-based approach is significantly more robust to large deformations than the first Piola-Kirchoff fix. Furthermore, we show that the robustness and stability of a hyper- elastic model can be predicted from a characteristic contour, which we call its primary contour. The extension to inverted configurations is defined via extrapolation from a convex threshold surface that lies in the uninverted portion of the principal stretches space. The extended hyperelastic energy den- sity yields continuous stress and unambiguous stress derivatives in all inverted configurations, unlike in [TSIF05]. We show that our invertible energy-density-based approach outperforms the popular hy- perelastic corotated model, and we also show how to use the primary contour methodology to improve the robustness of this model to large deformations.