A numerical method for solving three-dimensional elasticity problems for bodies with flat cracks and flattened cavities with due respect of contact of their surfaces

A numerical method is developed for solving three-dimensional elasticity problems for bodies with cracks and flattened cavities with allowance for the possible formation of contact regions where their surfaces interact without friction. In accordance with the approach proposed here, the initial boundary value problem is replaced by an equivalent variational problem, and the transition from the continuous variational problem to a discrete one is performed using a variational method proposed in an earlier paper (Key et al., 1978). The discrete variational problem is then solved using the gradient projection method. The method is demonstrated for problems of normal crack extension and of a flattened cavity of arbitrary shape in an elastic space. In particular, the method has been found to be effective in solving the problem of hanging the roofs of mining excavations of complicated shapes.