An asymptotic solution of the hypersingular boundary integral equation simulating wave scattering by the interface strip-like crack

The ultrasonic non-destructive testing is widely used in different civil and engineering applications as one of the most effective and convenient method of structural health monitoring. It is necessary to have a reliable mathematical model simulating scattering caused by defects and inhomogeneities in order to apply effective ultrasonic methods. Modern composite materials used in manufacturing have a laminated structure; therefore it is important to detect damages occurrence located between two materials. Scattering caused by interface cracks can be investigated using the boundary integral equation (BIE), the method which is analytically oriented. The unknown function of the crack opening displacement in the BIE is expanded in terms of orthogonal polynomials. Then the integral equation is projected onto a set of polynomials. Regularization of the hypersingular BIE using the Bubnov-Galerkin scheme is obtained through a repeated integration on the crack faces. This paper uses the BIE method to derive an asymptotic solution describing the elastic wave diffraction by the strip-like crack located at the interface between two dissimilar elastic half-spaces. The Fourier transformation of Green's matrix is applied to obtain a scattered field. Asymptotic representations of the equation kernel around zero and at infinity are derived with the assumption that the crack size is much less than a wavelength of an incident wave. The Bubnov-Galerkin scheme is used to obtain the frequency dependent asymptotic solution of BIE which has a wider accuracy frequency range than the existing quasi-static solution. A good agreement of the derived asymptotic solution with the numerical solution is shown for different materials of the considered structure. The asymptotic solution allows increasing the BIE method potency by reducing the computational cost of integrals. It can also be used to describe dynamic damaged interfaces in the Bostrom-Wickham model's term.