The perturbation (small-parameter) method is used to analyze the propagation of a harmonic longitudinal plane wave in a quadratic nonlinear hyperelastic material described by the classical Murnaghan model. The three first approximations are obtained, and the contribution of each of them into the wave pattern is analyzed. It is shown that the third approximation somewhat improves the prediction of the evolution of the initial waveprofile: the tendency to generate the second harmonic goes over into the tendency to generate the fourth harmonic
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References
N. Blombergen, Nonlinear Optics, Benjamin, New York (1965).
I. A. Viktorov, “Second-order effects of wave propagation in solids,” Akust. Zh., 9, No. 3, 121–126 (1963).
M. B. Vinogradova, O. V. Rudenko, and A. P. Sukhorukov, Theory of Waves [in Russian], Nauka, Moscow (1990).
Z. A. Goldberg, “Interaction of longitudinal and transverse plane waves,” Akust. Zh., 6, No. 2, 307–310 (1960).
A. N. Guz, Elastic Waves in Prestressed Bodies [in Russian], Naukova Dumka, Kyiv (1986).
Kun Hsiu-Fen, L. K. Zarembo, and V. A. Krasil’nikov, “Experimental investigation of combination scattering of sound by sound in solids,” Sov. Phys. JETP, 21, 1073–1077 (1965).
V. I. Erofeev, Wave Processes in Solids with Microstructure [in Russian], Izd. Mosk. Univ., Moscow (1999).
L. K. Zarembo and V. A. Krasil’nikov, An Introduction to Nonlinear Acoustics [in Russian], Nauka, Moscow (1966).
V. V. Krylov and V. A. Krasil’nikov, An Introduction to Physical Acoustics [in Russian], Nauka, Moscow (1986).
G. N. Polozhii (ed.), Practical Mathematics [in Russian], Fizmatgiz, Moscow (1960).
J. J. Rushchitsky and S. I. Tsurpal, Waves in Microstructural Materials [in Ukrainian], Inst. Mekh. im. S. P. Timoshenka, Kyiv (1998).
C. Cattani and J. J. Rushchitsky, “The subharmonic resonance and second harmonic of a plane wave in nonlinearly elastic bodies,” Int. Appl. Mech., 39, No. 1, 93–98 (2003).
C. Cattani and J. J. Rushchitsky, “Cubically nonlinear elastic waves: Wave equations and methods of analysis,” Int. Appl. Mech., 39, No. 10, 1115–1145 (2003).
C. Cattani and J. J. Rushchitsky, “Cubically nonlinear versus quadratically nonlinear elastic waves: Main wave effects,” Int. Appl. Mech., 39, No. 12, 1361–1399 (2003).
C. Cattani and J. J. Rushchitsky, “Nonlinear cylindrical waves in Signorini’s hyperelastic material,” Int. Appl. Mech., 42, No. 7, 765–774 (2006).
C. Cattani and J. J. Rushchitsky, “Nonlinear plane waves in Signorini’s hyperelastic material,” Int. Appl. Mech., 42, No. 8, 895–903 (2006).
C. Cattani and J. J. Rushchitsky, “Similarities and differences between the Murnaghan and Signorini descriptions of the evolution of quadratically nonlinear hyperelastic plane waves,” Int. Appl. Mech., 42, No. 9, 997–1010 (2006).
A. N. Guz, J. J. Rushchitsky, and I. A. Guz, “Establishing fundamentals of the mechanics of nanocomposites,” Int. Appl. Mech., 43, No. 3, 247–271 (2007).
C. Cattani and J. J. Rushchitsky, Wavelet and Wave Analysis as Applied to Materials with Micro or Nanostructure, World Scientific, Singapore–London (2007).
F. R. Rollins, “Interaction of ultrasonic waves in solid media,” Appl. Phys. Let., 2, No. 1, 147–151 (1963).
J. J. Rushchitsky, “Interaction of waves in solid mixtures,” Appl. Mech. Rev., 52, No. 2, 35–74 (1999).
J. J. Rushchitsky, “Extension of the microstructural theory of two-phase mixtures to composite materials,” Int. Appl. Mech., 36, No. 5, 586–614 (2000).
J. J. Rushchitsky, “Quadratically nonlinear cylindrical hyperelastic waves: Primary analysis of evolution,” Int. Appl. Mech., 41, No. 7, 770–777 (2005).
J. J. Rushchitsky, “On universal deformations in analysis of Signorini’s nonlinear theory of hyperelastic media,” Int. Appl. Mech., 43, No. 12, 1347–1352 (2007).
J. J. Rushchitsky and C. Cattani, “Generation of the third harmonics by plane waves in Murnaghan materials,” Int. Appl. Mech., 38, No. 12, 1482–1487 (2002).
J. J. Rushchitsky, A. P. Kovalenko, and E. V. Savel’eva, “Self-excitation of transverse waves in hyperelastic materials (third approximation),” Int. Appl. Mech., 32, No. 5, 349–356 (1996).
J. J. Rushchitsky and E. V. Savel’eva, “Self-switching of a transverse plane wave propagating through a two-component elastic composite,” Int. Appl. Mech., 43, No. 7, 734–744 (2007).
J. J. Rushchitsky and Ya. V. Simchuk, “Higher-order approximations in the analysis of nonlinear cylindrical waves in a hyperelastic medium,” Int. Appl. Mech., 43, No. 4, 388–394 (2007).
J. J. Rushchitsky and Ya. V. Simchuk, “Modeling cylindrical waves in nonlinear elastic composites,” Int. Appl. Mech., 43, No. 6, 638–646 (2007).
J. J. Rushchitsky and Ya. V. Simchuk, “Quadratic nonlinear torsional hyperelastic waves in isotropic cylinders: Primary analysis of evolution,” Int. Appl. Mech., 44, No. 3, 304–312 (2008).
J. J. Rushchitsky and Ya. V. Simchuk, “Quadratically nonlinear torsional hyperelastic waves in a transversely isotropic cylinder: Primary analysis of evolution,” Int. Appl. Mech., 44, No. 5, 505–515 (2008).
N. S. Shiren, “Nonlinear acoustic interaction in MgO at 9 Gc/sec,” Phys. Rev. Let., 11, No. 3, 561–563 (1963).
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Translated from Prikladnaya Mekhanika, Vol. 45, No. 2, pp. 46–58, February 2009.
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Rushchitsky, J.J. Analysis of a quadratic nonlinear hyperelastic longitudinal plane wave. Int Appl Mech 45, 148–158 (2009). https://doi.org/10.1007/s10778-009-0169-8
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DOI: https://doi.org/10.1007/s10778-009-0169-8