Weak Solutions, Elliptic Problems and Sobolev Spaces

Abstract

We study the structure and energy of (1 1 1) low-angle twist boundaries in face-centered cubic Al, Cu and Ni, using a generalized Peierls–Nabarro model incorporating the full disregistry vector in the slip plane and the associated stacking fault energy. It is found that dislocation network structures on these twist boundaries can be determined by a single dimensionless parameter, with two extreme cases of a hexagonal network of perfect dislocations and triangular network of partial dislocations enclosing stacking faults. We construct a simple model of these networks based upon straight partial dislocation segments. Based on this structural model, we derive an analytical expression for the twist boundary energy as a function of the twist angle θ, intrinsic stacking fault energy and parameters describing the isolated dislocation core. In addition to the θ and θ log θ terms [3] and the θ2 term [17] in the boundary energy, our new energy expression also includes additional terms in θ that represent the partial dissociation of the dislocation nodes and the effect of the stacking fault energy. The analytical predictions of boundary structure and energy are shown to be in excellent agreement with the Peierls–Nabarro model simulations.