Prediction of Springback in Unconstrained Bending by a Model for Evolving Elastic and Plastic Anisotropy

Abstract

Sheet metals exhibit anisotropic plastic behavior due to the large plastic deformations that occur during the rolling of the sheet and which induce texture and are responsible for the initial anisotropy. There exist various possibilities to introduce plastic anisotropy into the finite element modelling of sheet metal forming. The initial yield anisotropy can be incorporated either through an anisotropic yield surface or directly by means of a crystallographic texture model. Here, one basically differentiates between empirical and phenomenological anisotropic yield function equations, where the anisotropy coefficients can be obtained from mechanical tests, and texture-based models the coefficients of which are directly determined based on experimentally obtained orientation distributions. Another type of anisotropy that can be usually found in anisotropic materials is the elastic anisotropy. In metal plasticity one often considers the effect of elastic anisotropy significantly smaller than the effect of plastic anisotropy. Consequently, elastic isotropic expressions are often used for elastic stored energy functions with anisotropic yield criteria. However, the influence of elastic anisotropy in the elastoplastic behavior can be very important especially during elastic recovery processes during unloading after forming and springback. This research focuses, therefore, on the study of the influence of elastic anisotropy on the amount of springback in bending processes such as e.g. unconstrained bending. We discuss a finite strain material model for evolving elastic and plastic anisotropy combining nonlinear isotropic and kinematic hardening. The evolution of elastic anisotropy is described by representing the Helmholtz free energy as a function of a family of evolving structure tensors. In addition, plastic anisotropy is modelled via the dependence of the yield surface on the same family of structure tensors. Exploiting the dissipation inequality leads to the interesting result that all tensor-valued internal variables are symmetric. Thus, the integration of the evolution equations can be efficiently performed by means of an algorithm that automatically retains the symmetry of the internal variables in every time step. The material model has been implemented as a user material subroutine UMAT into the commercial finite element software ABAQUS/Standard and has been applied to the simulation of springback of unconstrained bending.