ELASTIC TRANSVERSAL-ISOTROPIC SPACE WITH TWO UNIAXIAL PARALLEL CIRCULAR CRACKS AND ASSOCIATED PROBLEMS OF BASICITY

Abstract

In the paper a problem of the stress state in a transversely isotropic space with two parallel circular cracks, the centers of which are located on the axis of anisotropy of the space, is investigated. A constant normal load acts on the crack planes. The problem is solved by the generalized Fourier method. For this purpose, systems of compressed spheroidal coordinates are introduced, the origins of which are connected to the centers of cracks. The general solution of the problem is constructed in the form of series based on axisymmetric variants of the general vector solutions of the system of equations of equilibrium of a transversely isotropic body in spheroidal coordinates, which were previously constructed by one of the authors of the paper. To implement the method, it is further generalized to compressed spheroidal coordinate systems with origins shifted along the axis. For this purpose, new addition theorems for basic vector displacements of transversally isotropic bodies in the above-mentioned coordinate systems are obtained. After applying the generalized Fourier method, the problem is reduced to an infinite system of linear algebraic equations. It is proved that under certain geometrical and mechanical conditions the operator of the system is a Fredholm operator. The reduction method is used for the numerical solution of the system. Graphs of normal stresses in the plane of one of the cracks outside its boundaries, as well as values of stress intensity factor at the top of the crack for different geometric parameters of the cracks, are obtained. The obtained results agree with the known value of the stress intensity factor in the problem with one crack. The practical convergence of the reduction method is studied.
 As an important related problem, the problem of proving the basicity of a general axisymmetric set of external solutions of the system of equilibrium equations of a transversally isotropic body whose boundary is described in compressed spheroidal coordinates is considered. The key problem here is obtaining subtle estimates from below of the modulus of the determinant of the first boundary value problem. As a corollary of the obtained result, several important estimates from the theory of special functions are derived, in which Legendre functions of the second kind from a purely imaginary argument appear.