First basic elasticity theory problem in a half-space with several parallel round cylindrical cavities

Abstract

When designing different kinds of structures and forecasting the strength of mine workings in rock and geotechnical mechanics, there occur problems in which it is necessary to know the stress-strain state of a half-space with cylindrical cavities and take into account the mutual influence of the cavities and the half-space boundaries. The article gives an analytical and numerical solution to the first main spatial problem of the theory of elasticity (stresses are specified on boundaries) for a homogeneous half-space with several circular cylindrical cavities parallel to each other and the boundary of the half-space. The specified stresses are assumed to rapidly decay to zero at great distances from the origin of coordinates, on the boundaries of the cavities at coordinates z and on the boundary of the half-space at coordinates z and x. To solve the problem, the generalized Fourier method is used in relation to a system of the Lame equations in the cylindrical coordinates associated with the cylinders, and the Cartesian coordinates related to the half-space. For transition between the basic solutions of the Lame equation, special formulas for transition between local cylindrical coordinate systems and between the Cartesian and cylindrical coordinate systems were used. Infinite systems of linear algebraic equations for which the problem is reduced is solved by the truncation method. As a result, displacements and stresses were found in an elastic body. As an example, a detailed numerical analysis of the stress-strain state for two parallel cylindrical cavities in a half-space at various values of the geometric parameters of the problem is given. The graphs given show a picture of the distribution of stresses in a body in the most interesting zones and illustrate the mutual influence of cylindrical cavities as well as the mutual influence of a half-space and cylindrical cavities, depending on the geometric parameters of the problem.