Numerical implementation of orthotropic plasticity for sheet-metal forming analysis

Abstract

Due to rolling and crystallographic texture development, thin sheet-metal for stamping exhibits an initial orthotropic plastic behaviour. The objective of the present work is to define a stress-computation algorithm taking into account this anisotropic plastic behaviour and the update of the orthotropic frame directions within a finite-element simulation of sheet-metal forming. Following Mandel's approach, a macroscopic behaviour model is considered which is based on the use of a rotational objective derivative defined from the initial micro-structure rotation governing the orthotropic direction update. The expression of the constitutive law in a frame rotated by this rotation permits a numerical scheme including an elastic prediction and a plastic correction for the stress calculation. The elastic prediction ensures the incremental objectivity and a second-order level of accuracy. The correction stage is based on an implicit Euler-backward scheme and leads to the solution of a non-linear equation using a local Newton method. In the proposed simulation approach, C 0 shell finite elements are used to model the blank. The proposed stress-calculation algorithm takes into account the mixed interpolation used in the element formulation in order to avoid shear locking. A set of numerical results is presented in cases of the simulation of deep-drawing processes. The accuracy of the solutions, and especially the good agreement with experimental results, shows the efficiency of the proposed approach.