Analytical Solution of the Problem for a Set of Shear Fractures with Coulomb Friction

摘要

A simplified solution of a two-dimensional problem on a set of interaction shear fractures with Coulomb friction along their wings is proposed. In the classical statement of this problem, the equality between shear stresses and friction stresses must necessarily hold in a new equilibrium state at each point along the fracture plane. The solution of the problem is reduced to the solution of a set of singular integral equations with respect to unknown functions of a shear jump on fractures. In reality, our knowledge of fracture conditions is rather approximate. In addition, in the problems of seismology and tectonophysics it is sufficient to approximately estimate dynamic (taking into account the inertia forces) and static perturbations from a fracture in the form of the first terms of a true solution series. With this aim in view, point models of an earthquake source or continual representation of discontinuous deformations are used. Within the framework of the requirements of such approaches, it is assumed that the condition of equality between the sum of shear stresses and friction stresses on the shear fractures in a new equilibrium state is met. In addition, the complex potential function is calculated for each fracture based on the function of the jump of its wings obtained in the problem for a solitary fracture. Such statement of the problem of a set of neighboring and even intersecting fractures makes it possible to reduce its solution to a set of linear algebraic equations with respect to shear stresses relieved at each fracture and the average along its length, they being the unknown values.