We develop a continuum model for the dynamics of grain boundaries in three dimensions that incorporates the motion and reaction of the constituent dislocations. The continuum model is based on a simple representation of densities of curved dislocations on the grain boundary. Illposedness due to nonconvexity of the total energy is fixed by a numerical treatment based on a projection method that maintains the connectivity of the constituent dislocations. An efficient simulation method is developed, in which the critical but computationally expensive long-range interaction of dislocations is replaced by another projection formulation that maintains the constraint of equilibrium of the dislocation structure described by the Frank's formula. This continuum model is able to describe the grain boundary motion and grain rotation due to both coupling and sliding effects, to which the classical motion by mean curvature model does not apply. Comparisons with atomistic simulation results show that our continuum model is able to give excellent predictions of evolutions of low angle grain boundaries and their dislocation structures.

In order to relieve the misfitting elastic energy, the hetero-interfaces become semicoherent by forming networks of dislocations. These microscopic structures strongly influence the materials properties associated with the development of advanced materials. We develop a continuum model for the dislocation structures of semicoherent interfaces. The classical Frank-Bilby equation that governs the dislocation structures on semicoherent interfaces is not able to determine a unique solution. The available methods in the literature either use further information from atomistic simulations or consider only special cases (dislocations with no more than two Burgers vectors) where the Frank-Bilby equation has a unique solution. In our continuum model, the dislocation structure of a semicoherent interface is obtained by minimizing the energy of the equilibrium dislocation network with respect to all the possible Burgers vectors, subject to the constraint of the Frank-Bilby equation. The continuum model is validated by comparisons with atomistic simulation results.

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