Bonded contact and general crack problems in generalized materials and their relationships

Abstract

The term of generalized material is introduced here as the material, whose state is described by the second order homogeneous differential equations with constant coefficients. The generalized point sources are described by homogeneous expressions, containing the first order derivatives with constant coefficients. The Green’s function for a half-space, made of generalized material, subjected to the action of generalized point sources, is derived in the form of a single integral over a circle. Some of the components of the surface Green’s function are presented in finite form, no computation of any integral is needed. The bonded contact problem is described as mixed–mixed boundary value problem for a half-space, with normal and tangential displacements prescribed inside domain S and vanishing tractions outside this domain on the plane $$x_{3}=0$$ . A set of governing integral equations for bonded contact problem was derived, with some of the kernels defined in finite form. The general crack problem is defined as that of a flat crack of arbitrary shape S in the plane $$x_{3}=0$$ , with normal tractions $$\sigma _{33}$$ applied symmetrically to the crack faces and tangential tractions $$\sigma _{31}$$ and $$\sigma _{23}$$ applied anti-symmetrically to the crack faces. A set of 3 governing integral equations is derived, with some of the kernels presented in the finite form. The relationship between the integrands in Fourier transform of the kernels of the sets of the integral equations of both problems is established: they are related in such a way, as if they were the coefficients in the schematic sets of algebraic equations. This relationship can also be presented as a general relationship between matrices of arbitrary rank n, with their components being determinants of certain matrices of rank q, with $$q\ge n$$ .