Automatic differentiation of strain‐energy functions in the context of user‐defined materials for the FEM

In order to implement material models as user‐defined material subroutines into implicit finite element method (FEM) codes, a stress tensor and a fourth‐order tangent tensor are required. In the context of hyperelasticity, these tensors can be obtained as the first and second derivative of the strain‐energy function. Manual formulation and implementation of these derivatives can be tedious and error‐prone. In this contribution, we discuss automatic differentiation (AD) as an easy‐to‐use alternative to express derivatives for the FEM [1] in an efficient way and without extensive manual implementation [2]. AD is applied to automatically obtain a user‐defined material subroutine for an implicit analysis based on source code of the strain‐energy. The approach is validated by utilizing a hyperelastic anisotropic model of arterial layers [3] in ABAQUS. Additionally, a pseudo‐elastic model of the Mullins effect in filled rubber [4] is implemented to evaluate the applicability of the discussed approach for inelastic behaviors which require internal variables.