Nonlinear plane longitudinal elastic waves with different profiles are studied using the Murnaghan model. The novelty is that the waves are analyzed using the same approximate method and the solutions of the nonlinear wave equations are similar in form. The distortion of the initial wave profile described by cosinusoidal, Gaussian, and Whittaker functions is described theoretically and numerically. About 80 variants of initial parameters are studied numerically: three analytical representations of the initial profile, three materials (aluminum, copper, steel), three wave lengths, three initial maximum amplitudes. For each variant, four (cosine) and five (Gauss, Whittaker) two-dimensional graphs of wave shape versus traveled distance are plotted to demonstrate the distortion of the wave profile
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
J. J. Rushchitsky, “On constraints for displacement gradients in elastic materials,” Int. Appl. Mech., 52, No. 2, 119–132 (2016).
J. J. Rushchitsky and V. N. Yurchuk, “An approximate method for analysis of solitary waves in nonlinear elastic materials,” Int. Appl. Mech., 52, No. 3, 282–289 (2016).
J. J. Rushchitsky and S. I. Tsurpal, Waves in Microstructural Materials [in Ukrainian], Inst. Mekh. S. P. Timoshenka, Kyiv (1998).
C. Cattani and J. Rushchitsky, Wavelet and Wave Analysis as applied to Materials with Micro and Nanostructure, World Scientific, Singapore–London (2007).
V. I. Erofeev, Wave Processes in Solids with Microstructure, World Scientific, Singapore–London (2003).
I. S. Gradstein and I. M. Ryzhik, Table of Integrals, Series, and Products, Academic Press, New York (2007).
I. A. Guz, A. A. Rodger, A. N. Guz, and J. J. Rushchitsky, “Predicting the properties of micro and nanocomposites: from the micro whiskers to bristled nanocentipedes,” Phil. Trans., Royal Soc. A: Ìathematical, Physical and Engineering Sciences, 365, No. 1860, 3233–3239 (2008).
I. A. Guz and J. J. Rushchitsky, “Theoretical description of a delamination mechanism in fibrous micro- and nanocomposites,” Int. Appl. Mech., 40, No. 10, 1129–1136 (2004).
M. I. Hussein and R. Khayehtourian, “Nonlinear elastic waves in solids: Deriving simplicity from complexity,” Bull. Amer. Phys. Soc., 60, No. 1 (2015), http://meetings.aps.org/link/BAPS.2015,MAR_Q8.10
J. Janno and A. Seletski, “Reconstruction of coefficients of higher order nonlinear wave equation by measuring solitary waves,” Wave Motion, 52, 15–25 (2015).
G. A. Maugin, Nonlinear Waves in Elastic Crystals, Oxford University Press, Oxford (1999).
K. Narahara, “Asymmetric solitary waves in coupled nonlinear transmissions lines,” Wave Motion, 58, 13–21 (2015).
A. B. Olde Daalhuis, “Whittaker functions,” Secs. 13.14–13.26 of Ch. 13 Confluent Hypergeometric Functions, in: F. W. J. Olver, D. W. Lozier, R. F. Bousvert, and C. W. Clark (eds.), NIST (National Institute of Standards and Technology) Handbook of Mathematical Functions, Cambridge University Press, Cambridge (2010), pp. 383–402.
A. V. Porubov, Amplification of Nonlinear Strain Waves in Solids, World Scientific, Singapore-London (2003).
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 three facts of reticences in the classical mathematical modeling of elastic materials,” Math. Model. Comp. (MMC), 1, No. 2, 245–255 (2014).
V. Hauk (ed.), Structural and Residual Stress Analysis, Elsevier Science, Amsterdam (1997) (e-variant 2006).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Prikladnaya Mekhanika, Vol. 53, No. 1, pp. 121–130, January–February, 2017.
Rights and permissions
About this article
Cite this article
Yurchuk, V.N., Rushchitsky, J.J. Numerical Analysis of the Evolution of Plane Longitudinal Nonlinear Elastic Waves with Different Initial Profiles. Int Appl Mech 53, 104–110 (2017). https://doi.org/10.1007/s10778-017-0794-6
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10778-017-0794-6