On tensile cracks close to and across the interface between two welded elastic half-spaces

Tensile cracks are often employed to model magma migration in rift zones or within volcanic edifices through lateral or feeding dykes. In a crack model, the overpressure of magma with respect to the horizontal stress in the host rock is assumed to be responsible for dyke opening and propagation. In this paper the most simple heterogeneous medium is considered, made up of two welded half-spaces, characterized by different elastic parameters. The analytical solutions available for the elementary dislocation problem in such a medium are employed to set up an integral equation with generalized Cauchy kernel, representing the condition for static equilibrium. The unknown in such a problem is the dislocation density distribution, whose singular behaviour is studied near the crack tips and near the intersection with the interface between the two media. The problem for a crack entirely embedded in one of the two media is considered first (case A). The resulting Cauchy integral equation is solved by employing a truncated expansion in Chebyshev polynomials, and semi-analytical solutions are given for the displacement and stress fields provided by a constant overpressure within the crack. The problem of a crack terminating at the interface between the two half-spaces is then considered (case B) and a generalized Cauchy equation is obtained in which additional singularities, localized at the crack tip next to the interface, appear in the integral kernel. The singular stress behaviour in the proximity of the crack tip is affected considerably by the presence of the interface. Within case B, after assuming a vanishing rigidity for the upper medium, a crack open at the free surface of a half-space is obtained (case B1). The singular behaviour of the solution is found to change abruptly in this case (with respect to the case of small, non-vanishing rigidity in the upper medium). Finally, a crack crossing the interface between the two half-spaces is considered (case C). The crack is split into two interacting sections, each embedded in a homogeneous half-space, both open at the interface. A system of generalized Cauchy equations is obtained, which is responsible for the appearance of internal singularities in the dislocation density distribution, localized at the intersection between the crack plane and the interface. The order of this singularity depends only upon the elastic parameters of the media in welded contact. Crack opening, interface displacement and stress components are computed employing the method described for case A, assuming a known overpressure within the crack. It appears that layering can be responsible for stress changes, localized along the interface, which may be considerably higher than the overpressure within the dyke. These results may provide useful hints for the interpretation of induced seismicity in rift zones and in volcanic areas.