Abstract
The 3D problem of elastic-wave scattering by cracks in a homogeneous isotropic medium is considered. Using the method of Green’s function and specially introduced auxiliary functions (potentials), it is shown how, in the general case, for plane cracks, the boundary conditions can be divided into two independent parts, one of which is a system of two differential equations, the solution of which leads to Rayleigh waves on the crack surfaces, and the second of which includes one differential equation, reduced to a similar equation for acoustic-wave scattering by an absolutely rigid inclusion. Using as an example scattering by a disk-shaped crack, it is shown that the expressions for the scattered fields can be reduced to quadratures, which is important, e.g., for the study of crack detection by ultrasonic methods of flaw detection.
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Notes
In the region z > 0, this parametrization coincides with the potentials introduced by the relationship u = \(\nabla f\, + \,{\text{curl}}([\nabla h,{{{\mathbf{z}}}^{0}}]~\) + χz0), where z0 is the unit vector of the z axis. On the opposite surface of the crack, the selected parameterization differs simply by the substitution \({{{\mathbf{z}}}^{0}} \to - ~{{{\mathbf{z}}}^{0}}.\)
REFERENCES
E. Ginzel, Ultrasonic Time of Flight Diffraction (Eclipse Scientific, Waterloo, Ontario, Canada, 2013).
N. P. Aleshin, N. V. Krysko, D. M. Kozlov, and A. G. Kusyy, Russ. J. Nondestr. Test. 57, 13 (2021).
N. P. Aleshin, N. V. Krys’ko, N. A. Shchipakov, and L. Yu. Mogil’ner, Nauka Tekhnol. Truboprov. Transp. Nefti Nefteprod. 10 (6), 352 (2020).
A. W. Maue, Zeitschr. Angew. Math. Mech. 33, 1 (1953).
J. Miklowitz, The Theory of Elastic Waves and Waveguides (North-Holland, Amsterdam, 1978).
P. A. Martin and G. R. Wickham, Proc. R. Soc. London, Ser. A 390, 91 (1983).
S. Kanaun, Wave Motion 51, 360 (2014).
V. Zernov, L. Fradkin, and M. Darmon, Ultrasonics 52, 830 (2012).
A. K. Djakou, M. Darmon, L. Fradkin, and C. Potel, J. Acoust. Soc. Am. 138, 3272 (2015).
H. Hönl, A. W. Maue, and K. Westpfahl, in Handbuch der Physik (Springer, Berlin, 1961).
N. P. Aleshin, V. S. Kamenskii, D. V. Kamenskii, and L. Yu. Mogil’ner, Dokl. Akad. Nauk SSSR 302, 777 (1988).
E. P. Savelova, Gen. Relat. Grav. 48, 85 (2016).
I. A. Viktorov, Sound Surface Waves in Solids (Nauka, Moscow, 1981) [in Russian].
L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics, Vol. 7: Theory of Elasticity (Nauka, Moscow, 1982; Pergamon, New York, 1986).
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Translated by E. Chernokozhin
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Aleshin, N.P., Kirillov, A.A., Mogilner, L.Y. et al. A General Solution of the Problem of Elastic-Wave Scattering by a Plane Crack. Dokl. Phys. 66, 202–208 (2021). https://doi.org/10.1134/S1028335821070016
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DOI: https://doi.org/10.1134/S1028335821070016