Surveys in Mathematics and its Applications

ISSN 1842-6298 (electronic), 1843-7265 (print)
 Volume 19 (2024), 217 -- 232

Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License.

THE DIFFRACTION OF ELASTIC WAVES BY SPHERICAL DEFECTS

Oleg Nazarenko, Angela Stekhun, Anatoly Yarovyi

Abstract. Generalization of the method of discontinuous solutions [9, 11] to the case of spherical defects (cracks or thin rigid spherical inclusions). A method for constructing a discontinuous solution of the wave equation for a spherical coordinate system is proposed.

2020 Mathematics Subject Classification: 74BXX; 74HXX
Keywords: wave equation; elasticity theory; defect; inclusion; crack; discontinuous solution; jump; spherical coordinates; stress; displacement.

Full text

References

  1. I. M. Gel'fand and G. Ye. Shilov, Generalized functions and operations on them, Fizmatgiz, M. (1958), 439. MR0097715. Zbl 0091.11102.

  2. A. G. Gorshkov and D. V. Tarlakovskii, The Unsteady Aerohydroelasticity of Spherical Solids, Nauka, M. (1990), 264.

  3. I. S. Gradshtein and I. M. Ryzhik Tables of Integrals, Sums, Series and Products, Nauka, M. (1971), 1108. MR0987590. Zbl 0697.46035.

  4. A. N. Guz, V. D. Kubenko & M. A. Cherevko, The Diffraction of Elastic Waves, Kiev, Naukova Dumka (1978), 307.

  5. L. V. Kantorovich and G. P. Akilov, Functional Analysis, Nauka, M. (1977), 742. MR0511615.

  6. V. D. Kupradze and T. V. Burchuladze, The dynamical problems of the theory of elasticity and thermoelasticity, Journal of Soviet Mathematics 7, (1977), 415–500. MR0987590. Zbl 0697.46035.

  7. L. F. Lependin, Akustika [Acoustics], High school publ. (1978), 448.

  8. G. Ya. Popov, The use of Jacobi polynomials to solve integral equations, Izv. Vuzov. Matematika, 4, (1966), 77-85. MR0199666.

  9. G. Ya. Popov, The Concentration of Elastic Stresses Near Punches, Cuts, Thin Inclusions and Supports, Nauka, M. (1982), 342.

  10. G. Ya. Popov, A new approach to problems of elastic-stress concentration near crack, J. Appl. Math. Mech. 55, No. 1, 120-128 (1991); translation from Prikl. Mat. Mekh. 55, 1, (1991), 148-156. MR1107510. Zbl 0757.73004.

  11. G. Ya. Popov, Problems of stress concentration in the neighbourhood of a spherical defect, Advances in Mechanics, No 15, 1-2 (1992), 71-110. MR1220257.

  12. A. P. Prudnicov, Yu. A. Brychkov and O. I. Marichev, Integrals and Series. Spetsialnye funktsii, Nauka Publ., M. (1983), 758.

  13. E. C. Titchmarshe, Expansion in Eigen functions Related to Second-order Differential Equations, Pt 1. Izd. Inost. Lit., M. (1960), 278.

  14. F. Tricomi, Integral Equations, Izd. Inost. Lit., M. (1960), 299. MR0123160. Zbl 0092.10803.





Oleg Nazarenko
Candidate of physical and mathematical science,
Associate professor of the Department of Higher Mathematics,
Odesa State Academy of Civil Engineering and Architecture,
Didrichson street, 4, Odesa, 65029, Ukraine
e-mail: gelo.fabric@gmail.com
orcid: 0000-0003-0405-6522



Angela Stekhun
Candidate of physical and mathematical science,
Associate Professor of the Department of Optimal Management and Economic Cybernetics,
Faculty of Mathematics, Physics and Information Technologies,
Odesa I.I. Mechnikov National University,
Dvoryanskaya street, 2, Odesa, 65023, Ukraine
e-mail: angela.stehun@gmail.com
orcid: 0000-0003-3140-2689



Anatoly Yarovyi
Candidate of physical and mathematical science,
Associate Professor of the Department of Optimal Management and Economic Cybernetics,
Faculty of Mathematics, Physics and Information Technologies,
Odesa I.I. Mechnikov National University,
Dvoryanskaya street, 2, Odesa, 65023, Ukraine
e-mail: jarovyiat@gmail.com
orcid: 0009-0000-7111-3609


http://www.utgjiu.ro/math/sma