The edge dislocation problem in a layered elastic medium

Abstract

Analytic solutions are provided for the displacement and stress fields induced by an edge dislocation in a layered medium composed of two welded half-spaces endowed with different elastic parameters. A plane strain configuration is considered, with the dislocation surface perpendicular to the interface between the two half-spaces. Two cases are considered: in case I the dislocation surface is entirely embedded in one half-spaces, in case II the dislocation surface cuts across the interface. From these elementary solutions, a closed Volterra dislocation (with constant slip) can be easily obtained: in model A the dislocation opens and closes within the same medium, in model B it opens in one half-space and closes in the other. These elementary solutions also provide the singular kernel of the integral equation governing the equilibrium configuration of a mode-II crack (in which the stress drop is assigned). The stress field induced by closed Volterra dislocations (models A and B) or by a crack with constant stress drop (model C) is computed and compared with the solutions in a homogeneous medium. Important differences appear in the overall pattern of the stress maps. Lower values are generally found for the vertical normal component and for the shear component near and within the softer half-space, which might be anticipated on intuitive grounds as due to the different rigidity of the two media in welded contact. However, unexpected differences are found in the normal stress component parallel to the interface (which needs not be continuous at the interface): this component shows wide regions of high stress on the hard side of the interface which, in models B and C, give rise to clear maxima and minima within two narrow lobes elongated along the interface. These interface stress concentrations extend up to distances comparable to the vertical extension of the dislocation surface and their amplitude is comparable to the stress change induced over the dislocation surface. The physical motivation for the presence of these maxima are discussed and their dependence is studied from the rigidity contrast between the two media and from the position of the dislocation surface relative to the interface.