On the approximate analysis of the evolution of a plane longitudinal hyperelastic wave

Rushchitsky, JJ
Dopov. Nac. akad. nauk Ukr. 2019, 8:34-45
Section: Mechanics
Language: Ukrainian

Three approaches (methods) are used to analyze the evolution of a plane longitudinal wave that propagates in a nonlinear hyperelastic medium — method of successive approximations, method of slowly varying amplitudes, and method of restriction on the displacement gradient. The evolution is understood as changing the initial wave profile during the propagation of a wave in the nonlinear elastic medium. Ten variants (known and new) of an approximate analysis of the evolution of a plane longitudinal wave propagating in an hyperelastic medium are described and commented. It is shown that each variant gives answer on some aspect in studying the wave evolution. An attention is drawn to the similarity and the difference in the results of analysis.

Keywords: distortion of a wave initial profile, longitudinal hyperelastic wave, variants of approximate analysis of the wave evolution

1. Rushchitsky, J. J. (2014). Nonlinear Elastic Waves in Materials. Series: Foundations of engineering mechanics. Heidelberg: Springer. doi: https://doi.org/10.1007/978-3-319-00464-8
2. Rabinovich, M. I. & Trubetskov, D. I. (1989). Oscillations and Waves in Linear Systems. New York: Kluwer Academic Publishers. doi: https://doi.org/10.1007/978-94-009-1033-1
3. Rushchitsky, J. J. (2012). Theory of waves in materials. Copenhagen: Ventus Publishing ApS.
4. Yurchuk, V. N. & Rushchitsky, J. J. (2017). Numerical Analysis of Evolution of the Plane Longitudinal Nonlinear Elastic Waves with Different Initial Profiles. Int. App. Mech., 53, No. 1, pp. 104110. doi: https://doi.org/10.1007/s10778-017-0794-6
5. Kamke, E. (1977). Differentialgleichungen. Lösungmethoden und Lösungen. Wiesbaden: Vieweg+Teubner, Springer Fachmedien Wiesbaden GmbH. doi: https://doi.org/10.1007/978-3-663-05925-7
6. Rushchitsky, J. J., Cattani, C. & Sinchilo, S. V. (2005). Physical constants for one type of nonlinearly elastic fib rous microand nanocomposites with hard and soft nonlinearities. Int. Appl. Mech., 41, No. 12, pp. 13681377. doi: https://doi.org/10.1007/s10778-006-0044-9