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On the Evolution of a Plane Harmonic Wave in a Nonlinear Elastic Composite Material Modeled by a Two-Phase Mixture*

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A nonlinear plane longitudinal elastic displacement wave is studied theoretically within the framework of the Murnaghan model for a harmonic initial profile. The main novelty is that the evolution of waves is analyzed by the developed approximate method of constraints on the displacement gradient within the framework of two-phase mixture theory. It is shown that the initially excited wave is divided into four modes, all modes are dispersed and distorted, and each mode wave is represented as a superposition of the first and second harmonics. The amplitude of the second harmonic depends on many parameters, including the direct dependence on time. This dependence means that the second harmonic will dominate over time.

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References

  1. V. M. Babich and A. P. Kiselev, Elastic Waves: High-Frequency Theory [in Russian], BKhV-Peterburg, St-Petersburg (2014).

  2. M. B. Vinogradova, O. V. Rudenko, and A. P. Sukhorukov, Wave Theory [in Russian], Nauka, Moscow (1990).

  3. 3. A. N. Guz, Propagation Laws, Vol. 2 of the two-volume series Elastic Waves in Prestressed Bodies [in Russian], Naukova Dumka, Kyiv (1986).

  4. L. K. Zarembo and V. A. Krasil’nikov, Introduction to Nonlinear Acoustics [in Russian], Nauka, Moscow (1966).

  5. V. V. Krylov and V. A. Krasil’nikov, Introduction to Physical Acoustics [in Russian], Nauka, Moscow (1986).

  6. M. K. Rabinovich and D. I. Trubetskov, Introduction to the Theory of Oscillations and Waves [in Russian], Nauka, Moscow (1984).

  7. J. J. Rushchitsky, Elements of Mixture Theory [in Russian], Naukova Dumka, Kyiv (1991).

  8. J. J. Rushchitsky, “Approximate analysis of the evolution of a longitudinal wave propagating in an elastic medium,” Dop. NAN Ukrainy, No. 8, 46–58 (2019).

  9. J. J. Rushchitsky and S. I. Tsurpal, Waves in Microstructural Materials [in Ukrainian], Inst. Mekh. S. P. Timoshenka, Kyiv (1998).

  10. V. M. Yurchuk, The Theory of Solitary Waves in Nonlinearly Elastic Materials [in Ukrainian], PhD Thesis, Inst. Mekh. S. P. Timoshenka, Kyiv (2019).

  11. C. Cattani and J. Rushchitsky, Wavelet and Wave Analysis as Applied to Materials with Micro and Nanostructure, World Scientific, Singapore–London (2007).

    Book  Google Scholar 

  12. F. Murnaghan, Finite Deformation in an Elastic Solid. 2nd ed., John Wiley, New York (1967).

  13. J. J. Rushchitsky, Nonlinear Elastic Waves in Materials, Springer, Heidelberg (2014).

    Book  Google Scholar 

  14. J. D. Achenbach, Wave Propagation in Elastic Solids, North-Holland, Amsterdam (1973).

    MATH  Google Scholar 

  15. A. Bedford and D. S. Drumheller, “Theories of immiscible and structured mixtures,” Int. J. Eng. Sci., 21, No. 8, 863–960 (1983).

    Article  MathSciNet  Google Scholar 

  16. A. Bedford and D. S. Drumheller, Introduction to Elastic Wave Propagation, John Wiley, Chichester (1994).

    Google Scholar 

  17. Y. Engelbrecht, Questions About Elastic Waves, Springer, Berlin (2015).

    Book  Google Scholar 

  18. B. M. Lempriere, Ultrasound and Elastic Waves: Frequently Asked Questions, Academic Press (2002).

  19. J. J. Rushchitsky, “Interaction of waves in solid mixtures,” Appl. Mech. Rev., 52, No. 2, 35–74 (1999).

    Article  Google Scholar 

  20. J. J. Rushchitsky, Nonlinear Elastic Waves in Materials, Springer, Heidelberg (2014).

    Book  Google Scholar 

  21. J. J. Rushchitsky, “Plane nonlinear elastic waves: approximate approaches to analysis of evolution-plenary lecture,” in: Abstracts of 19th Int. Conf. on Dynamical System Modeling and Stability Investigations (DSMSI 2019), Taras Shevchenko Kyiv National University, May 22–24 (2019), pp. 221–223.

  22. J. J. Rushchitsky, “Plane nonlinear elastic waves: approximate approaches to analysis of evolution,” in: William A. Cooper (ed.), Understanding Plane Waves, Nova Science Publishers, New York (2020), pp. 147–203.

  23. J. J. Rushchitsky, Theory of Waves in Materials, Ventus Publishing ApS, Copenhagen (2011).

    Google Scholar 

  24. J. J. Rushchitsky and V. M. Yurchuk, “An approximate method for analysis of solitary waves in nonlinear elastic materials,” Int. Appl. Mech., 51, No. 3, 282–290 (2016).

    Article  MathSciNet  Google Scholar 

  25. J. J. Rushchitsky and V. M. Yurchuk, “Numerical analysis of the evolution of plane longitudinal nonlinear elastic waves with different initial profiles,” Int. Appl. Mech., 52, No. 1, 104–110 (2017).

    MathSciNet  Google Scholar 

  26. J. J. Rushchitsky and V. M. Yurchuk, “Effect of the third approximation in the analysis of the evolution of a nonlinear elastic P-wave. Part 1,” Int. Appl. Mech., 56, No. 5, 581–589 (2020).

    Article  MathSciNet  Google Scholar 

  27. Y. Y. Rushchitskii and I. A. Guz, “Comparison of mechanical properties and effects in micro-and nanocomposites with carbon fillers (carbon microfibers, graphite microwhiskers, and carbon nanotubes),” Mech. Comp. Mat., 40, No. 3, 179–190 (2004).

    Article  Google Scholar 

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Authors and Affiliations

  1. S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, 3 Nesterova St., Kyiv, 03057, Ukraine

    J. K. Rushchitsky & V. M. Yurchuk

Authors
  1. J. K. Rushchitsky
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  2. V. M. Yurchuk
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Correspondence to J. K. Rushchitsky.

Additional information

Translated from Prikladnaya Mekhanika, Vol. 57, No. 2, pp. 58–69, March–April 2021.

* This study was sponsored by the budget program “Support for Priority Areas of Scientific Research” (KPKVK 6541230).

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Rushchitsky, J.K., Yurchuk, V.M. On the Evolution of a Plane Harmonic Wave in a Nonlinear Elastic Composite Material Modeled by a Two-Phase Mixture*. Int Appl Mech 57, 172–183 (2021). https://doi.org/10.1007/s10778-021-01071-9

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  • DOI: https://doi.org/10.1007/s10778-021-01071-9

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