The action of a rigid stamp on a half-plane weakened by a regular system of cracks

Abstract

The action of a rigid stamp on an isotropic half-plane weakened by a regular system of rectilinear cracks is studied. The problem in question differs fundamentally from the wide class of problems solved in recent times for symmetric bodies in the non-periodic character of the boundary conditions at the edge of the half-plane. The unknown function of the contact pressures is sought in the form of an expansion in terms of Chebyshev polynomials. The coefficients of this expansion are found from the system of algebraic equations obtained by transforming the condition of compatibility of the vertical displacements of the stamp and its foundation. A series of problems arising in this connection and concerning the stress-strain state of the half-plane under the action of loads applied to its edge described in terms of Chebyshev polynomials, is solved using a general scheme which makes use of the symmetric properties of the medium. Results of a numerical analysis and the functions of contact pressures under the stamp and the stress intensity coefficients at the crack tips are given.