On finite element implementation of a selected class of hyperelastic-plastic constitutive equations

Abstract In this study the finite element implementation of the hyperelastic-plastic constitutive relationship with isotropic hardening as proposed by Simo and Hughes (Computational Inelasticity, Interdisciplinary Applied Mathematics Vol. 7, Springer-Verlag), Simo (Comput Meth in Appl Mech and Eng 66:199–219. https://doi.org/10.1016/0045-7825(88)90076-X , 1988), Simo (Comput Meth Appl Mech Eng 68:1–31. https://doi.org/10.1016/0045-7825(88)90104-1 , 1988) is discussed. The major novel contributions of this work focus around the development of a general purpose UMAT code. The code enables to implement a family of hyperelastic-plastic constitutive relationships into the non-commercial finite element (FE) program CalculiX. The built-in implementation of hyperelasto-plasticity in this program permits only to use piecewise-linear isotropic hardening curves and a single selected formulation of volumetric stored-energy function. The present implementation of the model via subroutine UMAT allows the usage of nonlinear hardening curves and various formulations of the volumetric elastic energy. Thus, a whole family of hyperelastic-plastic models can be implemented into CalculiX using the developed UMAT code. What is more, several questions regarding the derivation of the flow rule, the radial return mapping algorithm and the consistent tangent operator have been more extensively discussed than previously in the literature. The presented relations will be useful for implementing the hyperelastic-plastic constitutive model into other FE packages. Numerous benchmark test problems have been solved and compared with both analytical results and experimental measurements which validated the performance of the developed code. The UMAT subroutine is attached to this work as supplementary material.