Anisotropic additive plasticity in the logarithmic strain space: modular kinematic formulation and implementation based on incremental minimization principles for standard materials Original Research Article

The paper presents a modular formulation and computational implementation of a class of anisotropic plasticity models at finite strains based on incremental minimization principles. The modular kinematic setting consists of a constitutive model in the logarithmic strain space that is framed by a purely geometric pre- and postprocessing. On the theoretical side, the point of departure is an a priori six-dimensional approach to finite plasticity based on the notion of a plastic metric. In a first step, a geometric preprocessor defines a total and a plastic logarithmic strain measure obtained from the current and the plastic metrics, respectively. In a second step, these strains enter in an additive format a constitutive model of anisotropic plasticity that may have a structure identical to the geometrically linear theory. The model defines the stresses and consistent tangents work-conjugate to the logarithmic strain measure. In a third step these objects of the logarithmic space are then mapped back to nominal, Lagrangian or Eulerian objects by a geometric postprocessor. This geometric three-step-approach defines a broad class of anisotropic models of finite plasticity directly related to counterparts of the geometrically linear theory. It is specified to a model problem of anisotropic metal plasticity. On the computational side we develop an incremental variational formulation of the above outlined constitutive structure where a quasi-hyperelastic stress potential is obtained from a local constitutive minimization problem with respect to the internal variables. It is shown that this minimization problem is exclusively restricted to the logarithmic strain space in a structure identical to the small-strain theory. The minimization problem determines the internal state of the material for finite increments of time. We develop a discrete formulation in terms of just one scalar parameter for the amount of incremental flow. The existence of the incremental stress potential provides a natural basis for the definition of the geometric postprocessor based on function evaluations. Furthermore, the global initial-boundary-value-problem of the elastic–plastic solid appears in the incremental setting as an energy minimization problem. Numerical examples show that the results obtained are surprisingly close to those obtained by a reference framework of multiplicative plasticity.