A nonlinear plane longitudinal elastic displacement wave is studied theoretically and numerically using the Murnaghan model for two forms of initial profile: harmonic and bell-shaped. A major novelty is that the evolution of waves is analyzed by approximate methods taking into account the first three approximations. The harmonic wave is analyzed only to compare with the new results for the bell-shaped wave. Some significant differences between the evolution of waves are shown. The initially symmetric profiles transform differently due to distortion: symmetrically (for the harmonic profile) and asymmetrically (for the bell-shaped profile). The third approximation introduces the fourth harmonic for the harmonic wave when this wave is analyzed by the method of successive approximations, while the bell-shaped wave is characterized in the third approximation differently when using the method of constraints on the displacement gradient. At relatively long distances from the beginning of the propagation, the one-hump bell-shaped wave transforms into a two-hump one. These humps adjoin each other halving their lengths. The third approximation allows us to observe new wave effects: the asymmetry of the left and right humps about their peaks and the asymmetry of the humps about each other; the lowering of the left hump and the rise of the right one. The results obtained are analyzed.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
J. J. Rushchitsky and S. I. Tsurpal, Waves in Microstructural Materials [in Ukrainian], Inst. Mekh. S. P. Timoshenka, Kyiv (1998).
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).
C. Cattani and J. Rushchitsky, Wavelet and Wave Analysis as Applied to Materials with Micro and Nanostructure,World Scientific, Singapore–London (2007).
I. S. Gradstein and I. M. Ryzhik, Table of Integrals, Series, and Products, 7th ed., Academic Press Inc., New York (2007).
A. N. Guz and J. J. Rushchitsky, “For the 100th anniversary of the S. P. Timoshenko Institute of Mechanics of the NASU: Books (monographs and textbooks) published by the institute,” Int. Appl. Mech., 54, No. 2, 121–142 (2018).
J. J. Rushchitsky, Elements of the Theory of Mixtures [in Russian], Naukova Dumka, Kyiv (1991).
J. J. Rushchitsky, “Certain class of nonlinear hyperelastic waves: classical and novel models, wave equations, wave effects,” Int. J. Appl. Math. Mech., 9, No. 12, 600–643 (2013).
J. J. Rushchitsky, Nonlinear Elastic Waves in Materials, Springer, Heidelberg (2014).
J. J. Rushchitsky, “On constraints for displacement gradients in elastic materials,” Int. Appl. Mech., 52, No. 2, 119–132 (2016).
J. J. Rushchitsky, “Plane nonlinear elastic waves: approximate approaches to the analysis of evolution-plenary lecture,” in: Abstracts of 19th Int. Conf. on Dynamical System Modeling and Stability Investigations – DSMSI 2019, Ukraine, Taras Shevchenko Kyiv National University, May 22–24 (2019), pp. 221–223.
J. J. Rushchitsky, “Plane nonlinear elastic waves: Approximate approaches to analysis of evolution,” Ch. 3 of the book W. A. Cooper (ed.), Understanding Plane Waves, Nova Science Publishers, London (2019), pp. 201–220.
J. J. Rushchitsky, C. Cattani, and S. V. Sinchilo, “Physical constants for one type of nonlinearly elastic fibrous micro and nanocomposites with hard and soft nonlinearities,” Int. Appl. Mech., 41, No. 12, 1368–1377 (2005).
J. J. Rushchitsky and V. N. Yurchuk, “An approximate method for analysis of solitary waves in nonlinear elastic materials,” Int. App. Mech., 52, No. 3, 282–290 (2016).
V. N. Yurchuk and J. J. Rushchitsky, “Numerical analysis of the evolution of the plane longitudinal nonlinear elastic waves with different initial profiles,” Int. App. Mech., 53, No. 1, 104–110 (2017).
J. J. Rushchitsky and V. M. Yurchuk, “Evolution of SV-wave with Gaussian profile,” Int. Appl. Mech., 53, No. 3, 300–304 (2017).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Prikladnaya Mekhanika, Vol. 56, No. 5, pp. 65–77, September–October 2020.
This study was sponsored by the budget program “Support for Priority Areas of Scientific Research” (KPKVK 6541230).
Rights and permissions
About this article
Cite this article
Rushchitsky, J.J., Yurchuk, V.N. Effect of the Third Approximation in the Analysis of the Evolution of a Nonlinear Elastic P-wave. Part 1*. Int Appl Mech 56, 581–589 (2020). https://doi.org/10.1007/s10778-020-01036-4
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10778-020-01036-4