Finite strain discrete dislocation plasticity in a total Lagrangian setting

We present two total Lagrangian formulations for finite strain discrete dislocation plasticity wherein the discrete dislocations are presumed to be adequately represented by singular linear elastic fields thereby extending the superposition method of Van der Giessen and Needleman (1995) to finite strains. The finite deformation effects accounted for are (i) finite lattice rotations and (ii) shape changes due to slip. The two formulations presented differ in the fact that in the “smeared-slip” formulation the discontinuous displacement field is smeared using finite element shape functions while in the “discrete-slip” formulation the weak form of the equilibrium statement is written to account for the slip displacement discontinuity. Both these total Lagrangian formulations use a hyper-elastic constitutive model for lattice elasticity. This overcomes the issues of using singular dislocation fields in a hypo-elastic constitutive relation as encountered in the updated Lagrangian formulation of Deshpande et al. (2003). Predictions of these formulations are presented for the relatively simple problems of tension and compression of single crystals oriented for single slip. These results show that unlike in small-strain discrete dislocation plasticity, finite strain effects result in a size dependent tension/compression asymmetry. Moreover, both formulations give nearly identical predictions and thus we expect that the “smeared-slip” formulation is likely to be preferred due to its relative computational efficiency and simplicity.