A variational formulation for finite deformation wrinkling analysis of inelastic membranes

Abstract

This paper is concerned with a novel, fully variational formulation for finite deformation analysis of inelastic membranes with wrinkling. In contrast to conventional approaches, every aspect of the physical problem derives from minimization of suitable energy functionals. A variational formulation of finite strain plasticity theory, which leads to a minimization problem for the constitutive updates, serves as the starting point for the derivations. In order to take into account the kinematics induced by wrinkles and slacks, a relaxed version of the finite strain functional is postulated. In effect, the local incremental stress–strain relations are established via differentiation of the relaxed energy functional with respect to the strains. Hence, the presented formulation is fully analogous to that of hyperelasticity with the sole exception that the aforementioned functional depends on history variables and, accordingly, it is path dependent. The advantages associated with the developed variational method are manifold. From a practical point of view, the possibility of applying standard optimization algorithms to solve the minimization problem describing inelastic membranes is remunerative. From a mathematical point of view, on the other hand, the energy of the system induces some sort of natural metric representing an essential requirement for error estimation and thus, for adaptive finite element methods. The presented derivation of the model allows to consider possible material symmetries in the elastic as well as plastic response of the material. As a prototype, a von Mises-type model is implemented. The efficiency and performance of the resulting algorithm are demonstrated by means of numerical examples.