Integral equations for a thin inclusion in a homogeneous elastic medium

A study is made of equilibrium in a homogeneous elastic medium containing a thin inclusion whose elastic moduli differ substantially from those of the medium. The solution depends on two non-dimensional parameters: the ratio δ1 of the characteristic linear dimensions of the inclusion and the ratio δ2 of the elastic moduli of the inclusion and the medium. While δ1 is always small, δ2 may be either small or large. The problem of constructing the principal asymptotic terms of the elastic fields in the neighbourhood of a thin inhomogeneity based on these parameters has been reduced [1] to the solution of integral (pseudodifferential) equations on the middle surface of the inclusion. Similar equations are obtained with two-dimensional models of thin inclusions[2-5]. Some properties of the solutions of these equations will be discussed below. A method is proposed for the numerical solution of the equations, based on introducing a special class of approximating functions, thanks to which the problem can be reduced to a system of linear algebraic equations whose matrix can be calculated by analytical means. The idea of the method is due to V. G. Maz'ya.