Comparison of two integration algorithms for finite plasticity combined with nonlinear kinematic and isotropic hardening at the example of springback

Abstract

The constitutive modelling of combined nonlinear kinematic and isotropic hardening in the finite strain regime is still a very active area of research. Nowadays, hyperelasticity-based formulations using the multiplicative decomposition of the deformation gradient into elastic and plastic parts are widely accepted. The most characteristic phenomena of the hardening behaviour of metals are the Bauschinger effect and ratcheting. The first one is related to the fact that straining in one direction reduces the yield stress in the opposite direction, whilst the latter denotes the progressive increase of the mean strain as a result of unsymmetric stress cycles. A main focus of the present work is the comparison of two algorithms used for the numerical integration of the evolution equations. Due to the deviatoric character of the flow rule, an integration scheme which preserves the plastic volume is required. Dettmer & Reese [2004] proposed a new type of the exponential map algorithm which fulfills plastic incompressibility. Here, we develop this scheme further and use the spectral decomposition to evaluate the exponential tensor functions in closed form. This integration algorithm has the advantage that the symmetry of the internal variables is automatically preserved. In addition, we discuss a modified backward Euler schemes that preserve plastic incompressibility. The results show higher accuracy and improved performance when using large time steps in favour of the exponential algorithm. In the finite strain regime the use of the classical multiplicative decomposition F = Fe Fp of the deformation gradient into elastic and plastic parts is well established. In analogy to the additive split of the plastic strain into two parts modelling the deformation in the hardening spring and the dashpot, we carry out the multiplicative split Fp = Fpe Fpi . The Helmholtz free energy represents the elastic energy storage. Based on the principle of material objectivity, the free energy depends on the deformation only through the elastic right Cauchy-Green tensors Ce and Cpe . Thus, for the ArmstrongFrederick model of kinematic hardening the Helmholtz free energy per unit volume decomposes additively into the three parts