A LINE-INTEGRAL REPRESENTATION FOR THE STRESSES DUE TO AN ARBITRARY DISLOCATION IN AN ISOTROPIC HALF-SPACE

A representation is developed that allows the stresses due to an arbit rary dislocation in an isotropic, homogeneous half-space to be express ed as a fine integral around that dislocation. For the special case of a dislocation half-line the integral is evaluated analytically to yie ld closed form solutions for the stresses. The given formulae allow th e study of three-dimensional dislocation configurations to be carried through with much greater computational simplicity than has hitherto b een possible. As an example, two approximations are assessed that are commonly used in the literature to infer the energies of dislocation l oops in a half-space from infinite-body results: the energy of a burie d 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 ha lf-loop is commonly taken to be half the energy of the completed loop in a whole-space. It is demonstrated that simple correlation factors m ay be evaluated, which modify the approximate formulae, yielding exact results. The case is considered of formation of a dislocation with Bu rgers vector 1/2[101BAR] on a (111) glide plane in a strained layer wi th (100) surface normal; such a loop would lead to the formation of a misfit dislocation of the commonly observed 60-degrees type at the sub strate-layer interface. Simple correction factors are presented for th e energies of a square loop buried in the layer, and of rectangular an d semicircular haff-loops forming at the free surface. It is found tha t the approximation for the energy of a buried loop yields accurate ac tivation energies for loop formation unless the layer thickness is sma ll, corresponding to the early stages of layer growth, in which case t he approximation may over-estimate the activation energy by a factor o f 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 cl ear what significance can be attached to the obtained activation energ y. By contrast, the results presented here allow the critical loop dim ension and activation energy to vary with layer thickness, yielding ph ysically reasonable results in all cases. The approximation for the en ergy of a surface half-loop is found to be more seriously in error, yi elding over-estimates by as much as factors of two and three of, respe ctively, the activation energies of rectangular and semicircular surfa ce half-loops.