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Longitudinal and torsional shock waves in anisotropic elastic cylinders. (English) Zbl 1430.74067
Summary: Discontinuities in solutions of one-dimensional hyperbolic equation system describing nonlinear longitudinal and torsional waves propagating in elastic rods are studied. The discontinuity amplitude is assumed to be small so that in the equations all nonlinear terms, except for the quadratic ones, are neglected. The shock adiabat form and evolutionary conditions are analyzed depending on the model parameters. The results of this study can be applied not only to waves in rods, but also to shock waves in anisotropic elastic media. shock waves in rods, elasticity, shock adiabat, evolutionary conditions.
MSC:
74J30 Nonlinear waves in solid mechanics
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
74B20 Nonlinear elasticity
74M20 Impact in solid mechanics
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[1] Rakhmatulin, Kh. A., Adylov, K.A.: Normal transverse impact against spiral wire cables. Moscow Univ. Mech. Bull. 31(5-6), 60-64 (1976) · Zbl 0364.73074
[2] Ergashov, M., A study of the propagation of elastic waves in wound structures taking into account their rotation under extension, J. Appl. Math. Mech., 56, 1, 117-124 (1992)
[3] Erofeev, Vi; Klyueva, Nv, Propagation of nonlinear torsional waves in a beam made of a different modulus material, Mech. Solids., 38, 5, 122-126 (2003)
[4] Erofeev, Vi, Nonlinear flexural and torsional waves in rods and rod systems, Vestn. Nauchn. Tekh. Razvitiya. No., 4, 46-50 (2009)
[5] Malashin, Aa, Longitudinal, transverse, and torsion waves and oscillations in musical strings, Dokl. Phys., 54, 1, 43-46 (2009) · Zbl 1255.70020
[6] Malashin, Aa, Waves and vibrations in strings, J. Appl. Math. Mech., 75, 1, 69-72 (2011) · Zbl 1272.74290
[7] Sugimoto, N.; Yamane, Y.; Kakutani, T., Oscillatory structured shock waves in a nonlinear elastic rod with weak viscoelasticity, J. Appl. Mech., 51, 4, 766-772 (1984) · Zbl 0543.73023
[8] Zhang, S.; Liu, Z., Three kinds of nonlinear dispersive waves in elastic rods with finite deformation, Appl. Math. Mech., 29, 7, 909-917 (2008) · Zbl 1231.74278
[9] Liu, Y.; Khajeh, E.; Lissenden, Cj; Rose, Jl, Interaction of torsional and longitudinal guided waves in weakly nonlinear circular cylinders, J. Acoust. Soc. Am., 133, 5, 2541-2553 (2013)
[10] Kulikovskii, Ag; Chugainova, Ap, Long nonlinear waves in anisotropic cylinders, Comput. Math. Math. Phys., 57, 7, 1194-1200 (2017) · Zbl 1457.74103
[11] Kulikovskii, Ag; Chugainova, Ap, Shock waves in anisotropic cylinders, Proc. Steklov Inst. Math., 300, 100-113 (2018) · Zbl 1452.74058
[12] Landau, Ld; Lifshits, Em, Course of Theoretical Physics, Fluid Mechanics (1987), Oxford: Pergamon, Oxford
[13] Lax, Pd, Hyperbolic systems of conservation laws, Commun. Pure Appl. Math., 10, 537-566 (1957) · Zbl 0081.08803
[14] Kulikovskii, Ag; Chugainova, Ap, Simulation of the influence of small-scale dispersion processes in a continuum on the formation of large-scale phenomena, Comput. Math. Math. Phys., 44, 6, 1062-1068 (2004)
[15] Kulikovskii, Ag; Chugainova, Ap, Classical and non-classical discontinuities in solutions of equations of non-linear elasticity theory, Russ. Math. Surv., 63, 2, 283-350 (2008) · Zbl 1155.74019
[16] Kulikovskii, Ag, Surfaces of discontinuity separating two perfect media of different properties: recombination waves in magnetohydrodynamics, J. Appl. Math. Mech., 32, 1145-1151 (1968)
[17] Chugainova, Ap, Asymptotic behavior of nonlinear waves in elastic media with dispersion and dissipation, Theor. Math. Phys., 147, 2, 646-659 (2006) · Zbl 1177.74207
[18] Chugainova, Ap, Self-similar asymptotics of wave problems and the structures of non-classical discontinuities in non-linearly elastic media with dispersion and dissipation, J. Appl. Math. Mech., 71, 5, 701-711 (2007)
[19] Kulikovskii, Ag; Chugainova, Ap, On the steady-state structure of shock waves in elastic media and dielectrics, J. Exp. Theor. Phys., 110, 5, 851-862 (2010)
[20] Chugainova, Ap; Shargatov, Va, Stability of discontinuity structures described by a generalized KdV-Burgers equation, Comput. Math. Math. Phys., 56, 2, 263-277 (2016) · Zbl 1346.35178
[21] Chugainova, Ap, Special discontinuities in nonlinearly elastic media, Comput. Math. Math. Phys., 57, 6, 1013-1021 (2017) · Zbl 1457.74102
[22] Kulikovskii, Ag; Chugainova, Ap; Shargatov, Va, Uniqueness of self-similar solutions to the Riemann problem for the Hopf equation with complex nonlinearity, Comput. Math. Math. Phys., 56, 7, 1355-1362 (2016) · Zbl 1362.35069
[23] Chugainova, Ap; Il’Ichev, At; Kulikovskii, Ag; Shargatov, Va, Problem of arbitrary discontinuity disintegration for the generalized Hopf equation: selection conditions for a unique solution, J. Appl. Math., 82, 3, 496-525 (2017) · Zbl 1404.35082
[24] Sedov, Li, Mechanics of Continuous Media (1997), River Edge: World Sci, River Edge
[25] Berdichevsky, Vl, Variational Principles of Continuum Mechanics. I: Fundamentals. II: Applications (2009), Berlin: Springer, Berlin · Zbl 1183.49002
[26] Rabotnov, Yn, Mechanics of a Deformable Solid (1988), Moscow: Nauka, Moscow
[27] Kulikovskii, Ag; Sveshnikova, Ei, On shock wave propagation in stressed isotropic nonlinearly elastic media, J. Appl. Math. Mech., 44, 3, 367-374 (1980) · Zbl 0484.73013
[28] Kulikovskii, Ag; Sveshnikova, Ei, Investigation of the shock adiabat of quasitransverse shock waves in a prestressed elastic medium, J. Appl. Math. Mech., 46, 5, 667-673 (1982) · Zbl 0542.73024
[29] Kulikovskii, Ag; Sveshnikova, Ei, Nonlinear Waves in Elastic Media (1995), Boca Raton: CRC, Boca Raton
[30] Kulikovskii, Ag; Pogorelov, Nv; Semenov, Ay, Mathematical Aspects of Numerical Solution of Hyperbolic Systems (2001), Boca Raton: Chapman & Hall/CRC, Boca Raton
[31] Hanyga, A., On the Solution to the Riemann Problem for Arbitrary Hyperbolic System of Conservation Laws (1976), Warszawa: Panstw. Wydawn. Nauk, Warszawa · Zbl 0353.73007
[32] Kulikovskii, Ag, Properties of shock adiabats in the neighborhood of Jouguet points, Fluid Dyn., 14, 317-320 (1979)
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