Gradient polyconvex material models and their numerical treatment

TL;DR

This paper introduces gradient polyconvex material models that allow for less smoothness in strains, expanding the class of admissible deformations, and demonstrates their implementation with practical examples.

Contribution

It provides a new geometric interpretation of gradient polyconvexity and details a numerical implementation for these advanced material models.

Findings

Successful implementation of gradient polyconvex elastic energies

Comparison with standard second-grade materials

Application to St.-Venant Kirchhoff and double well energy models

Abstract

Gradient polyconvex materials are nonsimple materials where we do not assume smoothness of the elastic strain but instead regularity of minors of the strain is required. This allows for a larger class of admissible deformations than in the case of second-grade materials. We describe a possible implementation of gradient polyconvex elastic energies. Besides, a new geometric interpretation of gradient-polyconvexity is given and it is compared with standard second-grade materials. Finally, we demonstrate application of the proposed approach using two different models, namely, a St.-Venant Kirchhoff material and a double well stored energy density.