Power-logarithmic singularities of solution for a class of singular integral equations arising in two-dimensional elasticity

The study is concerned with a class of one-dimensional singular integral equations (SIE) with generalized kernels and the complex conjugate unknown function that describes the elasticity problems in two-dimensional domains with singular points. Within the theory of complex variable functions and based on the formalism of the theory of special functions, a method for determination of the power-logarithmic type singularity in the solution of the integral equation is developed. By means of an asymptotic analysis for the characteristic part of a SIE, the problem of determining a solution singularity exponent at the end of the integration interval is reduced to a group of independent transcendental equations for this exponent. The analysis of the obtained equations for complex and real exponents is carried out, and a comparison with the previous results for classical power-type solution asymptotics is performed. It is shown that the power-logarithmic singularity with a complex exponent can only take place when the boundary value problem is not divided into normal and shear subproblems, and for the real exponent the logarithmic intensification of singularity is not realized in the general case. Numerical results for the complex power-logarithmic singularity exponent are presented for the two-dimensional elasticity problem of a crack approaching the interface at arbitrary angle.