# Modeling of Scattering by Cracks in Anisotropic Solids - Application to Ultrasonic Detection Doctoral thesis, 1996

Elastodynamic wave scattering in anisotropic solids with application to ultrasonic nondestructive testing (NDT) is treated and mathematically modeled in this thesis. The scatterers are infinite strip-like cracks and the solids are homogeneous and linearly elastic with negligible damping. The anisotropy is restricted to the case where the planes perpendicular to the crack axes are elastic symmetry planes. First a problem dealing with the scattering of 2D-SH time harmonic planar waves is considered. The anisotropic solid consists of two joined dissimilar anisotropic half-spaces. The crack is situated at the interface between the half-spaces. An integral equation technique based on Fourier representations for the scattered fields and crack boundary conditions leads to an integral equation of the second kind (though not explicitly so) for the crack opening displacement (COD). This equation is solved by expanding the COD in modified Chebyshev functions. These functions have a square root behaviour at the crack edges which also is the displacement field behaviour at these edges. Numerical results for the far field amplitude and the integrated scattered energy is calculated. A superposition of the above incoming planar 2D-SH waves is performed to model the wave fields from a 2D-SH wave transducer. The scattered fields due to these incoming fields are obtained as superpositions of the scattered fields above. The transducer signal response, i.e. the voltage change in a receiving transducer due to the scattering by the strip-like crack, is calculated. A solution for a surface breaking crack is also obtained, as a special case, by the method of imaging. Numerical examples for the scattered displacement field and the transducer signal response are given in frequency and time domain. The same integral equation technique is used to solve the scattering problem with 2D in-plane time harmonic planar waves. These waves are scattered by an infinite strip-like crack in an anisotropic full-space. The integral equation consists of two coupled scalar integral equations for the two vector components of the COD. Numerical examples in the frequency and time domain for the scattered displacement field are given. The above integral equation technique is also applied to model the scattering of 3D transducer generated wave fields. Two transducers, transmitting and receiving, are attached to an anisotropic half-space with a strip-like crack. The mathematical derivations are performed for general 3D waves. However, the dimensionality is reduced to two, under certain circumstances, in the expression for the transducer signal response. Numerical examples for the transducer signal response are given in the time domain. Finally, a rotated back surface is added to the above 3D problem. The crack is surface breaking or almost surface breaking at this back surface. A vector integral equation for the COD is derived from an integral representation of the total displacement field. The representation contains an anisotropic half-space Green's tensor. The integral equation is solved by expanding the COD in modified Chebyshev functions as above. An expression for the transducer signal response is then derived with the same reduction of coordinate dependence as above. The signal response is presented in the time domain.

elastic wave

ultrasonics

crack

integral equations

transducer

anisotropy

scattering

nondestructive testing

## Author

### Johan Mattsson

Department of Theoretical and Applied Mechanics

### Subject Categories

Mechanical Engineering