Numerical solution of crystal plasticity problems by a mixed formulation

A mixed crystal plasticity formulation, where displacements and crystal slips are independent variables, is proposed here to simulate the behavior of single crystals. Higher order boundary conditions are also considered. An artificial viscous effect is introduced in a way that the methodology is able to simulate, in the limit, rate dependent and rate independent constitutive laws. The methodology is implemented within a finite element context. It is shown that, in this framework, pure rate independent relations tend to present uniqueness problems while rate dependent relations tend to introduce numerical locking. Attention is restricted to two-dimensional cases. Slip system rotations are neglected. Analyses of the indentation of a single crystal with up to three active slip systems are carried out considering rigid indenters. When higher order terms are taken into account, a size dependent hardness emerges. A comparison with experiments indicates the most appropriate higher order boundary condition for indentation.