Modeling of polycrystals with gradient crystal plasticity: A comparison of strategies

Abstract

This paper treats the computational modeling of size dependence in microstructure models of metals. Different gradient crystal plasticity strategies are analyzed and compared. For the numerical implementation, a dual-mixed finite element formulation which is suitable for parallelization is suggested. The paper ends with a representative numerical example for polycrystals.

Keywords:

• crystal plasticity
• gradient hardening
• grain modeling
• dual mixed finite element
• polycrystalline material
• constitutive behavior

Acknowledgements

This work was supported by the Swedish Research Council, which is gratefully acknowledged. This research was carried out while Swantje Bargmann was at Chalmers University, Sweden, as a post-doctoral fellow. Partial financial support (B.S.) for this work, provided by the German Science Foundation (DFG) under contract Sv 8/8-1 is greatly acknowledged. The authors are very grateful to the scientific editor, Prof. Samuel Forest, and to the anonymous referees for their helpful comments and suggestions leading to the improvement of this contribution.

Notes

Notes

1. In a follow-up paper 20, it is shown that this model can be recast in a true work-conjugated formulation.

2. The model introduced in 14 is formulated for edge and screw dislocation densities. However, the numerical example presented in 14 is restricted to self-hardening and double slip where only edge dislocations are considered.

3. Different stress terms and their relations are compared in 16, with the main focus on linking the phenomenological Cosserat theory, the statistical theory of dislocations and strain gradient crystal plasticity.

4. As Macaulay brackets have been used in the formulation of the flow rule for model II and III, the microforces are determinate because we assume microclamped respectively microfree grain boundary conditions (cf. Equation (31)). In case of an elastically deforming grain with a plastically deforming neighboring grain, the microforces at its grain boundary might be indeterminate, e.g. if continuity conditions are assumed at the grain boundary.

5. Both microclamped and microfree boundary conditions rule out the possibility of interaction of slip-systems across a grain boundary and are therefore quite restrictive. Nevertheless, they are quite popular and have been used by, e.g. 3–6,8,26. There also exist contributions introducing boundary conditions that may account for grain interactions, see, e.g. 25. In this contribution, we limit our analysis to the particular case described above, as the influence of different grain boundary conditions is part of future research and a topic of its own.

6. Lele and Anand implemented the theory by writing a nine-node quadratic user-element subroutine for the commercial finite element package ABAQUS/Standard.