On the Boundary Conditions for Scattering of Elastic Waves on Plane Cracks

Abstract

In ultrasonic flaw detection of materials and welds, the use of fine effects associated with the diffraction of elastic waves by cracks of various orientations is expanding. Therefore, the article notes the relevance of a detailed study of the problems of surface wave scattering during propagation along cracks. These problems are also relevant for studying the conditions of crack growth. In this regard, the article considers the issue of elastic surface wave scattering during propagation along a family of angular half-planes connected by a cylindrical sector. The conditions for stitching the corresponding solutions in various areas are analyzed. It is shown that no reflection of the Rayleigh wave occurs on such an edge; the wave undergoes only a phase shift. At the same time, on the cylindrical sector the wave generates sources that radiate into the volume. The boundary conditions and the dispersion relation for Rayleigh waves are considered. The possibility of a derivative jump at the boundary of two surfaces with different external curvatures is analyzed. The general question of setting boundary conditions for plane cracks is also considered. It is argued that for the reflection of surface waves on the edge of a crack, the presence of internal curvature of its surface or the presence of inhomogeneities near the surface of the crack is necessary

Please cite this article in English as:

Kirillov A.A., Mogilner L.Yu., Savelova E.P. On the boundary conditions for scattering of elastic waves on plane cracks. Herald of the Bauman Moscow State Technical University, Series Natural Sciences, 2025, no. 3 (120), pp. 62--79 (in Russ.). EDN: NARUPY

References

[1] Ginzel E. Ultrasonic time of flight diffraction. Eclipse Scientific, 2013.

[2] Kretov E.F. Ultrazvukovaya defektoskopiya v energomashinostroenii [Ultrasonic flaw detection in power machine construction]. St. Petersburg, Sven Publ., 2014.

[3] Neganov D.A. Basics of deterministic normative methods of pipeline strength substantiation. Nauka i tekhnologii truboprovodnogo transporta nefti i nefteproduktov [Science and Technologies: Oil and Oil Products Pipeline Transportation], 2018, vol. 8, no. 6, pp. 608--617 (in Russ.). EDN: YRNHBJ

[4] Von Honl H., Maue A.W., Westpfahl K. Theorie der Beugung. Springer, 1961.

[5] Kostrov B.V. Diffraction of a plane wave by a rigid wedge inserted without friction into an infinite elastic medium. Prikladnaya matematika i mekhanika, 1966, vol. 30, no. 1, pp. 198--202 (in Russ.).

[6] Poruchikov V.B. Metody dinamicheskoy teorii uprugosti [Methods of dynamics elasticity theory]. Moscow, Nauka Publ., 1986.

[7] Guz A.N., Kubenko V.D., Cherevko M.A. Difraktsiya uprugikh voln [Diffraction of elastic waves]. Kiev, Naukova dumka Publ., 1978.

[8] Aleshin N.P., Kamenskiy V.S., Kamenskiy D.V., et al. Diffraction of an elastic wave on a stress-free disk. Doklady AN SSSR, 1988, vol. 302, no. 4, pp. 777--780 (in Russ.). EDN: YAGSJN

[9] Aleshin N.P., Mogilner L.Yu. Elastic-wave scattering by a plane crack: application of flaw detection. Dokl. Phys., 2023, vol. 68, no. 3, pp. 97--105. DOI: https://doi.org/10.1134/S1028335823030011

[10] Mogilner L.Yu., Krysko N.V. Elastic wave 3d scattering on a crack edge in the weld joint shve. Izvestiya vysshikh uchebnykh zavedeniy. Mashinostroenie [BMSTU Journal of Mechanical Engineering], 2024, no. 3, pp. 42--55 (in Russ.).EDN: EUNZDH

[11] Aleshin N.P., Mogilner L.Yu., Shchipakov N.A., et al. On use of slots in modelling cracks in ultrasonic testing. Russ. J. Nondestruct. Test., 2022, vol. 58, no. 2, pp. 71--80. DOI: https://doi.org/10.1134/S1061830922020036

[12] Dymkin G.Ya., Maksimov A.V. Study of reflection of Rayleigh waves from subsurface defects. Defektoskopiya, 1988, no. 3, pp. 93--95 (in Russ.).

[13] Razin A.V. Scattering of a Rayleigh surface acoustic wave by a small-size inhomogeneity in a solid half-space. Radiophys. Quantum El., 2010, vol. 53, no. 7, pp. 417--431. DOI: https://doi.org/10.1007/s11141-010-9239-3

[14] Yavorskaya I.M. Diffraction of plane stationary elastic waves on smooth convex cylinders. Prikladnaya matematika i mekhanika, 1965, vol. 29, no. 3, pp. 493--508 (in Russ.).

[15] Miklowitz J. The theory of elastic waves and waveguides. North-Holland, 1978.

[16] Mogilner L.Yu. Use of a cylindrical reflector for adjusting sensitivity in ultrasonic testing. Defektoskopiya, 2018, no. 7, pp. 27--36 (in Russ.). EDN: YLFBZZ. DOI: https://doi.org/10.1134/S0130308218070047

[17] Sheriff R.E., Geldart L.P. Exploration seismology. Cambridge Univ. Press, 1995.

[18] Aki K., Richards P.G. Quantitative seismology. University Science Books, 2002.

[19] Viktorov I.A. Zvukovye poverkhnostnye volny v tverdykh telakh [Sound surface waves in solids]. Moscow, Nauka Publ., 1981.

[20] Biryukov S.V., Gulyaev Yu.V., Krylov V.V., et al. Poverkhnostnye akusticheskie volny v neodnorodnykh sredakh [Surface acoustic waves in inhomogeneous media]. Moscow, Nauka, 1991.

[21] Aleshin N.P., Kirillov A.A., Mogilner L.Yu., et al. A general solution of the problem of elastic-wave scattering by a plane crack. Dokl. Phys., 2021, vol. 66, no. 7, pp. 202--208. DOI: https://doi.org/10.1134/S102833582107001622

[22] Lokhov V.P. Investigation of Rayleigh wave diffraction at the crack edge. Defektoskopiya, 1989, no. 3, pp. 39--47 (in Russ.).

[23] Budaev B.V. Elastic wave diffraction by a wedge-shaped inclusion. J. Math. Sci., 1995, vol. 73, no. 3, pp. 330--341. DOI: https://doi.org/10.1007/BF02362817