Advances in the numerical treatment of grain-boundary migration: Coupling with mass transport and mechanics

摘要

This work is based upon a coupled, lattice-based continuum formulation that was previously applied to problems involving strong coupling between mechanics and mass transport; e.g. diffusional creep and electromigration [K. Garikipati, L.C. Bassman, M.D. Deal, A lattice-based micromechanical continuum formulation for stress-driven mass transport in polycrystalline solids, J. Mech. Phys. Solids 49(6) (2001) 1209-1237; K. Garikipati, L.C. Bassman, Atomically-based field formulations for coupled problems of composition and mechanics, in: L.P. Kubin, J.L. Bassani, K. Cho, H. Gao, R.L.B. Selinger (Eds.), MRS symposium proceedings, Multiscale Modeling of Materials—2000, vol. 653, Materials Research Society, Warrendale, 2001, pp. Z9.6.1-Z9.6.6]. Here we discuss an enhancement of this formulation to account for migrating grain boundaries. The level set method is used to model grain-boundary migration in an Eulerian framework where a grain boundary is represented as the zero level set of an evolving higher-dimensional function. This approach can easily be generalized to model other problems involving migrating interfaces; e.g. void evolution and free-surface morphology evolution. The level-set equation is recast in a remarkably simple form which obviates the need for spatial stabilization techniques. This simplified level-set formulation makes use of velocity extension and field re-initialization techniques. In addition, a least-squares smoothing technique is used to compute the local curvature of a grain boundary directly from the level-set field without resorting to higher-order interpolation. A notable feature is that the coupling between mass transport, mechanics and grain-boundary migration is fully accounted for. The complexities associated with this coupling are highlighted and the operator-split algorithm used to solve the coupled equations is described.