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    Evolution of Pulse and Periodic Elastic Waves in Media with Quadratically-Bimodular Nonlinearity

    2018, Vol. 14, no. 3, pp.  331-342

    Author(s): Nazarov V. E., Kiyashko S. B.

    On the basis of the elastic contact model of rough surfaces of solids, a quadraticallybimodular equation of state for micro-inhomogeneous media containing cracks is derived. A study is made of the propagation of elastic single unipolar pulse perturbations and bipolar periodic waves in such media. Exact analytical solutions that describe the evolution of initially triangular pulses and periodic sawtooth waves are obtained. A numerical and graphical analysis of the solutions is also carried out.
    Keywords: elastic contact, quadratically-bimodular nonlinearity, pulse perturbation, periodic waves
    Citation: Nazarov V. E., Kiyashko S. B., Evolution of Pulse and Periodic Elastic Waves in Media with Quadratically-Bimodular Nonlinearity, Rus. J. Nonlin. Dyn., 2018, Vol. 14, no. 3, pp.  331-342

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    [1] Landau, L. D. and Lifshitz, E. M., Course of Theoretical Physics: In 10 Vols., v. 6, Fluid Mechanics, 2nd ed., Butterworth-Heinemann, Oxford, 2003, 552 pp.  mathscinet
    [2] Rudenko, O. V. and Soluyan, S. I., Theoretical Foundations of Nonlinear Acoustics, Springer, New York, 2013, 274 pp.  mathscinet
    [3] Naugolnykh, K. A. and Ostrovsky, L. A., Nonlinear Wave Processes in Acoustics, Cambridge Texts Appl. Math., Cambridge Univ. Press, Cambridge, 1998, 312 pp.  mathscinet  zmath
    [4] Ryskin, N. M. and Trubezkov, D. I., Nonlinear Waves, Fizmatlit, Moscow, 2000, 272 pp. (Russian)
    [5] Isakovich, M. A., General Acoustics, Nauka, Moscow, 1973, 496 pp. (Russian)
    [6] Guyer, R. A. and Johnson, P. A., Nonlinear Mesoscopic Elasticity: Behaviour of Granular Media including Rocks and Soil, Wiley/VCH, Weinheim, 2009, 410 pp.
    [7] Nazarov, V. E. and Radostin, A. V., Nonlinear Acoustic Waves in Micro-Inhomogeneous Solids, Wiley, Chichester, 2015, 264 pp.
    [8] Ambartsumyan, S. A. and Khachatryan, A. A., “The Different-Modules Theory of Elasticity”, Mech. Solids, 1966, no. 1, 29–34  mathscinet; Inzh. Zh. Mekh. Tverd. Tela, 1966, no. 6, 64–67 (Russian)
    [9] Beresnev, I. A. and Nikolaev, A. V., “Experimental Investigations of Nonlinear Seismic Effects”, Phys. Earth Planet. Inter., 50:1 (1988), 83–87  crossref  adsnasa
    [10] Benveniste, Y., “One-Dimensional Wave Propagation in Materials with Different Moduli in Tension and Compression”, Int. J. Eng. Sci., 18:6 (1980), 815–827  crossref  mathscinet  zmath
    [11] Maslov, V. P. and Mosolov, P. P., “General Theory of the Solutions of the Equations of Motion of an Elastic Medium of Different Moduli”, J. Appl. Math. Mech., 49:3 (1985), 322–336  crossref  mathscinet  zmath; Prikl. Mat. Mekh., 49:3 (1985), 419–437  mathscinet  zmath
    [12] Nazarov, V. E. and Sutin, A. M., “Harmonic Generation in the Propagation of Elastic Waves in Nonlinear Solid Media”, Sov. Phys. Acoust., 35:4 (1989), 410–413; Akust. Zh., 35:4 (1989), 711–716
    [13] Nazarov, V. E. and Ostrovsky, L. A., “Elastic Waves in Media with Strong Acoustic Nonlinearity”, Sov. Phys. Acoust., 36:1 (1990), 57–60; Akust. Zh., 36:1 (1990), 106–110 (Russian)
    [14] Gavrilov, S. N. and Herman, G. C., “Wave Propagation in a Semi-Infinite Heteromodular Elastic Bar Subjected to a Harmonic Loading”, J. Sound Vibration, 331:20 (2012), 4464–4480  crossref  adsnasa
    [15] Radostin, A., Nazarov, V., and Kiyashko, S., “Propagation of Nonlinear Acoustic Waves in Bimodular Media with Linear Dissipation”, Wave Motion, 50:2 (2013), 191–196  crossref  mathscinet  zmath
    [16] Nazarov, V. E., Kiyashko, S. B., and Radostin, A. V., “Self-Similar Waves in Media with Bimodular Elastic Nonlinearity and Relaxation”, Nelin. Dinam., 11:2 (2015), 209–218 (Russian)  mathnet  crossref  zmath
    [17] Nazarov, V. E., Kiyashko, S. B., and Radostin, A. V., “The Wave Processes in Micro-Inhomogeneous Media with Different-Modulus Nonlinearity and Relaxation”, Radiophys. Quantum El., 59:3 (2016), 246–256  crossref  mathscinet; Izv. Vyssh. Uchebn. Zaved. Radiofizika, 59:3 (2016), 275–285 (Russian)
    [18] Rudenko, O. V., “Inhomogeneous Burgers Equation with Modular Nonlinearity: Excitation and Evolution of High-Intensity Waves”, Dokl. Math., 95:3 (2017), 291–294  crossref  mathscinet; Dokl. Akad. Nauk, 474:6 (2017), 671–674 (Russian)  zmath
    [19] Rudenko, O. V., “Modular Solitons”, Dokl. Math., 94:3 (2016), 708–711  crossref  mathscinet  zmath; Dokl. Akad. Nauk, 471:6 (2016), 651–654 (Russian)  zmath
    [20] Hedberg, C. M. and Rudenko, O. V., “Collisions, Mutual Losses and Annihilation of Pulses in a Modular Nonlinear Medium”, Nonlinear Dynam., 90:3 (2017), 2083–2091  crossref  mathscinet
    [21] Nazarov, V. E. and Sutin, A. M., “Nonlinear elastic constants of solids with cracks”, J. Acoust. Soc. Am., 102:6 (1997), 3349–3354  crossref  adsnasa
    [22] Johnson, K. L., Contact Mechanics, Cambridge Univ. Press, Cambridge, 1985, 468 pp.  zmath
    [23] Sneddon, I. N., Fourier Transformations, McGraw-Hill, New York, 1951, 542 pp.  mathscinet
    [24] Timoshenko, S. P. and Goodier, J. N., Theory of Elasticity, 3rd ed., McGraw-Hill, New York, 1970, 591 pp.  mathscinet  zmath
    [25] Landau, L. D. and Lifshitz, E. M., Course of Theoretical Physics: Vol. 7. Theory of Elasticity, 3rd ed., Butterworth/Heinemann, Oxford, 1986, 195 pp.  mathscinet
    [26] Mathews, J. and Walker, R. L., Mathematical Methods of Physics, 2nd ed., Addison-Wesley, Reading, Mass., 1971, 501 pp.  adsnasa

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