A line-integral representation for the stresses due to an arbitrary dislocation in an isotropic half-space

Abstract

A representation is developed that allows the stresses due to an arbitrary dislocation in an isotropic, homogeneous half-space to be expressed as a line integral around that dislocation. For the special case of a dislocation half-line the integral is evaluated analytically to yield closed form solutions for the stresses. The given formulae allow the study of three-dimensional dislocation configurations to be carried through with much greater computational simplicity than has hitherto been possible. As an example, two approximations are assessed that are commonly used in the literature to infer the energies of dislocation loops in a half-space from infinite-body results: the energy of a buried loop in a half-space is commonly taken to be the same as the energy of the corresponding loop in a whole-space; the energy of a surface half-loop is commonly taken to be half the energy of the completed loop in a whole-space. It is demonstrated that simple correlation factors may be evaluated, which modify the approximate formulae, yielding exact results. The case is considered of formation of a dislocation with Burgers vector 1 2 [101̄] on a (111) glide plane in a strained layer with (100) surface normal; such a loop would lead to the formation of a misfit dislocation of the commonly observed 60 type at the substrate-layer interface. Simple correction factors are presented for the energies of a square loop buried in the layer, and of rectangular and semicircular half-loops forming at the free surface. It is found that the approximation for the energy of a buried loop yields accurate activation energies for loop formation unless the layer thickness is small, corresponding to the early stages of layer growth, in which case the approximation may over-estimate the activation energy by a factor of one and a half. Moreover, in some cases the approximation yields as the critical loop dimension a side length for which the loop would no longer be contained within the layer. In such a situation it is not clear what significance can be attached to the obtained activation energy. By contrast, the results presented here allow the critical loop dimension and activation energy to vary with layer thickness, yielding physically reasonable results in all cases. The approximation for the energy of a surface half-loop is found to be more seriously in error, yielding over-estimates by as much as factors of two and three of, respectively, the activation energies of rectangular and semicircular surface half-loops.