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Teoreticheskaya i Matematicheskaya Fizika, 2020, Volume 202, Number 3, Pages 364–381
DOI: https://doi.org/10.4213/tmf9803 (Mi tmf9803) 		 
This article is cited in 11 scientific papers (total in 11 papers)

Longitudinal bulk strain solitons in a hyperelastic rod with quadratic and cubic nonlinearities

F. E. Garbuzova, Y. M. Beltukova, K. R. Khusnutdinovab

a Ioffe Institute, St. Petersburg, Russia
b Department of Mathematical Sciences, Loughborough University, Loughborough, United Kingdom
Full-text PDF (729 kB) Citations (11)
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DOI: https://doi.org/10.4213/tmf9803
Abstract: We study long nonlinear longitudinal bulk strain waves in a hyperelastic rod of circular cross section in the framework of general weakly nonlinear elasticity leading to a model with quadratic and cubic nonlinearities. We systematically derive extended equations of the Boussinesq and Korteweg–de Vries types and construct a family of approximate weakly nonlinear soliton solutions using near-identity transformations. We compare these solutions with the results of direct numerical simulations of the original nonlinear problem formulation, showing excellent agreement in the range of their asymptotic validity (waves of small amplitude) and extending their relevance beyond it (to waves of moderate amplitude) as a very good initial condition. In particular, we can observe a stably propagating “table-top” soliton.
Keywords: hyperelastic rod, Korteweg–de Vries-type equation, near-identity transformation, soliton.
Funding agency Grant number
Russian Science Foundation 17-72-20201
The research of F. E. Garbuzov and Y. M. Beltukov was supported by a grant from the Russian Science Foundation (Project No. 17-72-20201).
Received: 30.08.2019
Revised: 30.08.2019
English version:
Theoretical and Mathematical Physics, 2020, Volume 202, Issue 3, Pages 319–333
DOI: https://doi.org/10.1134/S0040577920030046
Bibliographic databases:
Document Type: Article
PACS: 62.30, 43.25
MSC: 35Q51, 35Q53
Language: Russian
Citation: F. E. Garbuzov, Y. M. Beltukov, K. R. Khusnutdinova, “Longitudinal bulk strain solitons in a hyperelastic rod with quadratic and cubic nonlinearities”, TMF, 202:3 (2020), 364–381; Theoret. and Math. Phys., 202:3 (2020), 319–333
Citation in format AMSBIB
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    • This publication is cited in the following 11 articles:
    1. F.E. Garbuzov, Y.M. Beltukov, “Slowly decaying strain solitons in nonlinear viscoelastic waveguides”, International Journal of Non-Linear Mechanics, 2025, 105043  crossref
    2. Nerijus Sidorovas, Dmitri Tseluiko, Wooyoung Choi, Karima Khusnutdinova, “Nonlinear concentric water waves of moderate amplitude”, Wave Motion, 128 (2024), 103295  crossref  mathscinet
    3. Andrea Nobili, “A weakly nonlinear Love hypothesis for longitudinal waves in elastic rods”, International Journal of Non-Linear Mechanics, 163 (2024), 104737  crossref
    4. L. Ostrovsky, E. Pelinovsky, V. Shrira, Y. Stepanyants, “Localized wave structures: Solitons and beyond”, Chaos: An Interdisciplinary Journal of Nonlinear Science, 34:6 (2024)  crossref
    5. F. E. Garbuzov, Y. M. Beltukov, “Viscoelastic relaxation of nonlinear strain waves in polymeric bars”, International conference of numerical analysis and applied mathematics ICNAAM 2021, AIP Conf. Proc., 2849, no. 1, 2023, 450024  crossref
    6. C. G. Hooper, K. R. Khusnutdinova, J. M. Huntley, P. D. Ruiz, “Theoretical estimates of the parameters of longitudinal undular bores in polymethylmethacrylate bars based on their measured initial speeds”, Proc. R. Soc. A, 478:2266 (2022)  crossref
    7. A. A. Zhikhoreva, A. V. Belashov, I. V. Semenova, Y. M. Beltukov, Ch. Zhou, L. Cao, T.-C. Poon, H. Yoshikawa, “Analysis of longitudinal strain wave evolution in polystyrene waveguides using digital holography and spectral decomposition”, Holography, Diffractive Optics, and Applications XII, 2022, 94  crossref
    8. K. R. Khusnutdinova, M. R. Tranter, “Periodic solutions of coupled Boussinesq equations and Ostrovsky-type models free from zero-mass contradiction”, Chaos: An Interdisciplinary Journal of Nonlinear Science, 32:11 (2022)  crossref  mathscinet
    9. C. G. Hooper, P. D. Ruiz, J. M. Huntley, K. R. Khusnutdinova, “Undular bores generated by fracture”, Phys. Rev. E, 104:4 (2021), 044207  crossref  isi
    10. A. V. Belashov, A. A. Zhikhoreva, Ya. M. Beltukov, I. V. Semenova, “Combined data from digital and classical holographic recording provides insight on early stages of strain soliton formation”, Optical Measurement Systems For Industrial Inspection XII, Proceedings of Spie, 11782, eds. P. Lehmann, W. Osten, A. Goncalves, Spie-Int Soc Optical Engineering, 2021, 117821Q  crossref  isi
    11. F. E. Garbuzov, I. V. Semenova, A. V. Belashov, Y. M. Beltukov, 2021 Days on Diffraction (DD), 2021, 58  crossref
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    		Теоретическая и математическая физика Theoretical and Mathematical Physics
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