Superposition of power-logarithmic and power singular solutions in two-dimensional elasticity problems

A comparison of the results obtained recently for power and power-logarithmic singular asymptotics of solution associated with a class of singular integral equations of the two-dimensional elasticity is performed. It is noted that leading parts of the integral equation contain similar terms for these singular solutions. In this connection, transcendental equations in regard to singularity exponents for additive form (superposition) of power and power-logarithmic solution asymptotics were constructed. It was established that superposition of the mentioned singular solutions has the singularity exponent which is known for the classical power asymptotics of elastic stress. The general nature of the obtained results is discussed that is related to the description of numerous boundary value problems of the two-dimensional elasticity by means of systems of singular integral equations belonging to the class under consideration. Based on theory of the Kolosov-Muskhelishvili complex potentials power-logarithmic singular solution of a boundary value problem is constructed. This solution represents the obtained results from point of view of direct asymptotic analysis of the boundary value problems. The parametric approach for equations on real singularity exponent is suggested to extend domain where non-oscillatory asymptotic is implemented. Numerical results on leading power-logarithmic singularity exponent for the two-dimensional problem of the elasticity theory for the crack terminating an interface are presented. The efficiency of the developed parametric approach for examined crack problem is demonstrated.