Shear waves in a nonlinear elastic cylindrical shell

A. I. Zemlyanukhin ; A. V. Bochkarev ; N. A. Artamonov

Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 24 (2024) no. 4, pp. 578-586.

Résumé

Asymptotic integration methods have been used to model the propagation of a shear wave beam along a nonlinear-elastic cylindrical shell of the Sanders – Koiter model. The shell is assumed to be made of a material characterized by a cubic dependence between stress and strain intensities, and the dimensionless parameters of thinness and physical nonlinearity are considered to have the same order of smallness. The multiscale expansion method is used, which makes it possible to determine the wave propagation speed from the equations of the linear approximation, and in the first essentially nonlinear approximation, to obtain a nonlinear quasi-hyperbolic equation for the main term of the expansion of the shear displacement component. The derived equation is a cubically nonlinear modification of the Lin – Reisner – Tsien equation modeling unsteady near-sonic gas flow and can be transformed into the modified Khokhlov – Zabolotskaya equation used to describe narrow beams in acoustics. The solution of the derived equation is found in the form of a single harmonic with slowly changing complex amplitude, since in deformable media with cubic nonlinearity the effect of self-induced wave essentially prevails over the effect of generation of higher harmonics. As a result, a perturbed nonlinear Schrödinger equation of defocusing type is obtained for the complex amplitude, for which there is no possibility of modulation instability development. In terms of the elliptic Jacobi function, an exact physically consistent solution, periodic along the dimensionless circumferential coordinate, is constructed.

Bibliographie

[1] Rudenko O., Sarvazyan A., “Wave biomechanics of skeletal muscle”, Journal of the Acoustical Society of America, 120 (2006), 3270 | DOI

[2] Rudenko O. V., “Nonlinear waves: Some biomedical applications”, Physics-Uspekhi, 50:4 (2007), 359 | DOI

[3] Sarvazyan A. P., Rudenko O. V., Swanson S. D., Fowlkes J. B., Emelianov S. Y., “Shear wave elasticity imaging: A new ultrasonic technology of medical diagnostics”, Ultrasound in Medicine and Biology, 24:9 (1998), 1419–1435 | DOI

[4] Rénier M., Gennisson J.-L., Tanter M., Catheline S., Barrière C., Royer D., Fink M., “7B-2 nonlinear shear elastic moduli in quasi-incompressible soft solids”, 2007 IEEE Ultrasonics Symposium Proceedings (New York, NY, 2007), 554–557 | DOI

[5] Gennisson J.-L., “New parameters in shear wave elastography in vivo”, Mécanique pour le vivant. Identification et modélisation du comportement des tissus biologiques humains et animaux. Avancées et perspectives, Colloque National (du 18 au 22 janvier, 2016) (accessed May 10, 2024) https://mecamat.ensma.fr/Aussois/2016/DOCUMENT/TexteGenisson.pdf

[6] Andreev V. G., Dmitriev V. N., Pishchal'nikov Yu. A., Rudenko O. V., Sapozhnikov O. A., Sarvazyan A. P., “Observation of shear waves excited by focused ultrasound in a rubber-like medium”, Acoustical Physics, 43:2 (1997), 123–128

[7] Cormack J. M., Hamilton M. F., “Plane nonlinear shear waves in relaxing media”, The Journal of the Acoustical Society of America, 143:2 (2018), 1035–1048 | DOI

[8] Lindley B. S., Linear and nonlinear shear wave propagation in viscoelastic media, University of North Carolina, Chapel Hill, 2008, 78 pp. (accessed May 10, 2024) https://cdr.lib.unc.edu/downloads/z029p543p

[9] Rajagopal K. R., Saccomandi G., “Shear waves in a class of nonlinear viscoelastic solids”, The Quarterly Journal of Mechanics and Applied Mathematics, 56:2 (2003), 311–326 | DOI

[10] Wochner M. S., Hamilton M. F., Ilinskii Y. A., Zabolotskaya E. A., “Cubic nonlinearity in shear wave beams with different polarizations”, Journal of the Acoustical Society of America, 123:5 (2008), 2488–2495 | DOI

[11] Destrade M., Saccomandi G., “Solitary and compact-like shear waves in the bulk of solids”, Physical Review E, 73:6 (2006), 065604 | DOI

[12] Doronin A. M., Erofeev V. I., “A generation of second harmonic of shear wave in elastoplastic media”, Letters on Materials, 6:2 (2016), 102–104 (in Russian) | DOI

[13] Erofeev V. I., “Propagation of nonlinear shear waves in a solid with microstructure”, International Applied Mechanics, 29 (1993), 262–266 | DOI

[14] Raskin I. G., Erofeyev V. I., “Propagation of sound shear waves in the nonlinear elastic solids”, Soviet Applied Mechanics, 27:1 (1991), 127–129 (in Russian)

[15] Potapov A. I., Soldatov I. N., “Quasioptical approximation for a bundle of shear waves in a nonlinear hereditary medium”, Prikladnaya Mekhanika i Tekhnicheskaya Fizika, 27:1 (1986), 144–147 (in Russian)

[16] Kivshar Yu. S., Syrkin E. S., “Shear solitons in an elastic plate”, Akusticheskiy Zhurnal, 37:1 (1991), 104–109 (in Russian)

[17] Erofeev V. I., Sheshenina O. A., “Nonlinear longitudinal and shear stationary deformation waves in a gradient-elastic medium”, Mathematical Modeling of Systems and Processes, 2007, no. 15, 15–27 (in Russian)

[18] Erofeev V. I., Kolesov D. A., Sandalov V. M., “Demodulation of a shear wave in a nonlinear plate resting on an elastic foundation with the parameters changing following the running wave law”, Problems of Strength and Plasticity, 75:4 (2013), 268–272 (in Russian) | DOI

[19] Bogdanov A. N., Skvortsov A. T., “Nonlinear shear waves in a granular medium”, Akusticheskiy Zhurnal, 38:3 (1992), 408–412 (in Russian)

[20] Bykov V. G., “Solitary shear waves in a granular medium”, Acoustical Physics, 45:2 (1999), 138–142

[21] Erofeyev V. I., Sharabanova A. V., “Riemann shift waves in material with properties depending on the stress state type”, Problemy Mashinostroeniya i Nadezhnosti Mashin, 1 (2004), 20–23 (in Russian)

[22] Erofeyev V. I., Kajaev V. V., Semerikova N. P., Waves in rods. Dispersion. Dissipation. Nonlinearity, Fizmatlit, M., 2002, 208 pp. (in Russian)

[23] Kaplunov J. D., Kossovich L. Yu., Nolde E. V., Dynamics of thin walled elastic bodies, Academic Press, San Diego, 1998, 226 pp. | DOI

[24] Zemlyanukhin A. I., Andrianov I. V., Bochkarev A. V., Mogilevich L. I., “The generalized Schamel equation in nonlinear wave dynamics of cylindrical shells”, Nonlinear Dynamics, 98 (2019), 185–194 | DOI

[25] Zemlyanukhin A. I., Bochkarev A. V., Andrianov I. V., Erofeev V. I., “The Schamel – Ostrovsky equation in nonlinear wave dynamics of cylindrical shells”, Journal of Sound and Vibration, 491 (2021), 115752 | DOI

[26] Zemlyanukhin A. I., Bochkarev A. V., Artamonov N. A., “Physically admissible and inadmissible exact localized solutions in problems of nonlinear wave dynamics of cylindrical shells”, Russian Journal of Nonlinear Dynamics, 20:2 (2024), 219–229 | DOI

[27] Zemlyanukhin A. I., Bochkarev A. V., Ratushny A. V., Chernenko A. V., “Generalized model of nonlinear elastic foundation and longitudinal waves in cylindrical shells”, Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 22:2 (2022), 196–204 | DOI

[28] Zemlyanukhin A. I., Bochkarev A. V., “Axisymmetric nonlinear modulated waves in a cylindrical shell”, Acoustical Physics, 64 (2018), 408–414 | DOI

[29] Yamaki N., Simitses G. J., “Elastic stability of circular cylindrical shells”, Journal of Applied Mechanics, 52:2 (1985), 501–502 | DOI

[30] Amabili M., “A comparison of shell theories for large-amplitude vibrations of circular cylindrical shells: Lagrangian approach”, Journal of Sound and Vibration, 264:5 (2003), 1091–1125 | DOI

[31] Lukash P. A., Fundamentals of nonlinear structural mechanics, Stroyizdat, M., 1978, 208 pp. (in Russian)

[32] Volmir A., The nonlinear dynamics of plates and shells, Foreign Tech. Div., Wright-Patterson AFB, Ohio, 1974, 450 pp.

[33] Rudenko O. V., “The 40th anniversary of the Khokhlov – Zabolotskaya equation”, Acoustical Physics, 56 (2010), 457–466 | DOI

[34] Zarembo L. K., Krasil'nikov V. A., Introduction to nonlinear acoustics: High intensity sound and ultrasonic waves, Nauka, M., 1966, 520 pp. (in Russian)

[35] Zarembo L. K., Krasil'nikov V. A., “Nonlinear phenomena in the propagation of elastic waves in solids”, Soviet Physics Uspekhi, 13:6 (1971), 778–797 | DOI

[36] Ryskin N. M., Trubetskov D. I., Nonlinear waves, Lenand, M., 2017, 312 pp. (in Russian)