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Erschienen in: Continuum Mechanics and Thermodynamics 4/2023

09.01.2023 | Original Article

Structures of longitudinal-torsional shock waves and special discontinuities in nonlinearly viscoelastic media with dispersion

verfasst von: A. P. Chugainova, A. G. Kulikovskii

Erschienen in: Continuum Mechanics and Thermodynamics | Ausgabe 4/2023

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Abstract

Weakly nonlinear longitudinal-torsional waves in rods are considered. The structure of discontinuities in the solutions of the hyperbolic system of equations describing these waves is studied. Previously, the authors studied discontinuity structures under more special assumptions about dissipative processes in these structures. In the present study, no constraints are imposed on the matrix of dissipative coefficients except for positive definiteness. Conditions are formulated for the existence of special discontinuities, that is, discontinuities with additional boundary conditions that are independent of conservation laws.

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Literatur
1.
Zurück zum Zitat Giorgio, I., Della Corte, A.: Dynamics of 1D nonlinear pantographic continua. Nonlinear Dyn. 88(1), 21–31 (2017)CrossRef Giorgio, I., Della Corte, A.: Dynamics of 1D nonlinear pantographic continua. Nonlinear Dyn. 88(1), 21–31 (2017)CrossRef
2.
Zurück zum Zitat Misra, A., Nejadsadeghi, N.: Longitudinal and transverse elastic waves in 1D granular materials modeled as micromorphic continua. Wave Motion 90, 175–195 (2019)ADSMathSciNetCrossRefMATH Misra, A., Nejadsadeghi, N.: Longitudinal and transverse elastic waves in 1D granular materials modeled as micromorphic continua. Wave Motion 90, 175–195 (2019)ADSMathSciNetCrossRefMATH
3.
Zurück zum Zitat Eugster, S.R.: Numerical analysis of nonlinear wave propagation in a pantographic sheet. Math. Mech. Complex Syst. 9(3), 293–310 (2021)MathSciNetCrossRefMATH Eugster, S.R.: Numerical analysis of nonlinear wave propagation in a pantographic sheet. Math. Mech. Complex Syst. 9(3), 293–310 (2021)MathSciNetCrossRefMATH
4.
Zurück zum Zitat Ciallella, A., Giorgio, I., Eugster, S.R., Rizzi, N.L., dell’Isola, F.: Generalized beam model for the analysis of wave propagation with a symmetric pattern of deformation in planar pantographic sheets. Wave Motion 113, 102986 (2022)MathSciNetCrossRefMATH Ciallella, A., Giorgio, I., Eugster, S.R., Rizzi, N.L., dell’Isola, F.: Generalized beam model for the analysis of wave propagation with a symmetric pattern of deformation in planar pantographic sheets. Wave Motion 113, 102986 (2022)MathSciNetCrossRefMATH
5.
Zurück zum Zitat Turco, E., Barchiesi, E., dell’Isola, F.: A numerical investigation on impulse-induced nonlinear longitudinal waves in pantographic beams. Math. Mech. Solids 27(1), 22–48 (2022)MathSciNetCrossRefMATH Turco, E., Barchiesi, E., dell’Isola, F.: A numerical investigation on impulse-induced nonlinear longitudinal waves in pantographic beams. Math. Mech. Solids 27(1), 22–48 (2022)MathSciNetCrossRefMATH
6.
Zurück zum Zitat Barchiesi, E., Laudato, M., Di Cosmo, F.: Wave dispersion in non-linear pantographic beams. Mech. Res. Commun. 94, 128–132 (2018)CrossRef Barchiesi, E., Laudato, M., Di Cosmo, F.: Wave dispersion in non-linear pantographic beams. Mech. Res. Commun. 94, 128–132 (2018)CrossRef
7.
Zurück zum Zitat Malkhanov, A.O., Erofeev, V.I., Leontieva, A.V.: Nonlinear travelling strain waves in a gradient-elastic medium. Contin. Mech. Thermodyn. 31, 1931–1940 (2019)ADSMathSciNetCrossRef Malkhanov, A.O., Erofeev, V.I., Leontieva, A.V.: Nonlinear travelling strain waves in a gradient-elastic medium. Contin. Mech. Thermodyn. 31, 1931–1940 (2019)ADSMathSciNetCrossRef
8.
Zurück zum Zitat Erofeev, V.I., Leontieva, A.V., Malkhanov, A.O.: A longitudinal magnetoelastic wave in a rod with account of the damage of its material. Contin. Mech. Thermodyn. 32, 1271–1285 (2020)ADSMathSciNetCrossRef Erofeev, V.I., Leontieva, A.V., Malkhanov, A.O.: A longitudinal magnetoelastic wave in a rod with account of the damage of its material. Contin. Mech. Thermodyn. 32, 1271–1285 (2020)ADSMathSciNetCrossRef
9.
Zurück zum Zitat Porubov, A.V., Krivtsov, A.M.: Dispersive propagation of localized waves in a mass-in-mass metamaterial lattice. Contin. Mech. Thermodyn. 34, 1475–1483 (2022)ADSMathSciNetCrossRef Porubov, A.V., Krivtsov, A.M.: Dispersive propagation of localized waves in a mass-in-mass metamaterial lattice. Contin. Mech. Thermodyn. 34, 1475–1483 (2022)ADSMathSciNetCrossRef
10.
Zurück zum Zitat Chugainova, A.P., Kulikovskii, A.G.: Longitudinal and torsional shock waves in anisotropic elastic cylinders. Z. Angew. Math. Phys. 71(1), 17 (2020)MathSciNetCrossRefMATH Chugainova, A.P., Kulikovskii, A.G.: Longitudinal and torsional shock waves in anisotropic elastic cylinders. Z. Angew. Math. Phys. 71(1), 17 (2020)MathSciNetCrossRefMATH
11.
Zurück zum Zitat Kulikovskii, A.G., Chugainova, A.P.: On the structures of nonclassical discontinuities in solutions of hyperbolic systems of equations. Russ. Math. Surv. 77(1), 47–79 (2022)CrossRefMATH Kulikovskii, A.G., Chugainova, A.P.: On the structures of nonclassical discontinuities in solutions of hyperbolic systems of equations. Russ. Math. Surv. 77(1), 47–79 (2022)CrossRefMATH
12.
Zurück zum Zitat Gel’fand, I.M.: Some problems in the theory of quasilinear equations. Transl. Ser. 2. Am. Math. Soc. 29, 295–381 (1963) Gel’fand, I.M.: Some problems in the theory of quasilinear equations. Transl. Ser. 2. Am. Math. Soc. 29, 295–381 (1963)
13.
Zurück zum Zitat Oleinik, O.A.: Construction of a generalized solution of the cauchy problem for a quasi-linear equation of first order by the introduction of vanishing viscosity. Am. Math. Soc. Transl. II. Ser. 33, 277–283 (1963)CrossRefMATH Oleinik, O.A.: Construction of a generalized solution of the cauchy problem for a quasi-linear equation of first order by the introduction of vanishing viscosity. Am. Math. Soc. Transl. II. Ser. 33, 277–283 (1963)CrossRefMATH
14.
Zurück zum Zitat Galin, G Ya.: Shock waves in media with arbitrary equations of state. Sov. Phys. Dokl. 119(3), 244–247 (1958) Galin, G Ya.: Shock waves in media with arbitrary equations of state. Sov. Phys. Dokl. 119(3), 244–247 (1958)
15.
Zurück zum Zitat Galin, G Ya.: A theory of shock waves. Sov. Phys. Dokl. 4, 757–760 (1960) Galin, G Ya.: A theory of shock waves. Sov. Phys. Dokl. 4, 757–760 (1960)
16.
Zurück zum Zitat Kulikovskii, A.G., Chugainova, A.P.: Classical and non-classical discontinuities in solutions of equations of non-linear elasticity theory. Russ. Math. Surv. 63(2), 283–350 (2008)CrossRefMATH Kulikovskii, A.G., Chugainova, A.P.: Classical and non-classical discontinuities in solutions of equations of non-linear elasticity theory. Russ. Math. Surv. 63(2), 283–350 (2008)CrossRefMATH
17.
Zurück zum Zitat LeFloch, P.G.: Hyperbolic systems of conservation laws: The theory of classical and nonclassical shock waves. Lectures in Mathematics, Birkhauser, ETH Zurich (2002)CrossRefMATH LeFloch, P.G.: Hyperbolic systems of conservation laws: The theory of classical and nonclassical shock waves. Lectures in Mathematics, Birkhauser, ETH Zurich (2002)CrossRefMATH
18.
Zurück zum Zitat Bedjaoui, N., LeFloch, P.: Diffusive-dispersive travelling waves and kinetic relations V. Singular diffusion and nonlinear dispersion. Proc. R. Soc. Edinb. Sect. A Math. 134(5), 815–843 (2004) Bedjaoui, N., LeFloch, P.: Diffusive-dispersive travelling waves and kinetic relations V. Singular diffusion and nonlinear dispersion. Proc. R. Soc. Edinb. Sect. A Math. 134(5), 815–843 (2004)
19.
Zurück zum Zitat Kulikovskii, A.G., Pogorelov, N.V., Semenov, A.Y.: Mathematical Aspects of Numerical Solution of Hyperbolic Systems. Chapman and Hall/CRC, Boca Raton (2001)MATH Kulikovskii, A.G., Pogorelov, N.V., Semenov, A.Y.: Mathematical Aspects of Numerical Solution of Hyperbolic Systems. Chapman and Hall/CRC, Boca Raton (2001)MATH
20.
Zurück zum Zitat Kulikovskii, A.G., Chugainova, A.P.: Classical and nonclassical discontinuities and their structures in nonlinear elastic media with dispersion and dissipation. Proc. Steklov Inst. Math. 276(Suppl. 2), 1–68 (2012)MathSciNetCrossRefMATH Kulikovskii, A.G., Chugainova, A.P.: Classical and nonclassical discontinuities and their structures in nonlinear elastic media with dispersion and dissipation. Proc. Steklov Inst. Math. 276(Suppl. 2), 1–68 (2012)MathSciNetCrossRefMATH
21.
Zurück zum Zitat El, G.A., Hoefer, M.A., Shearer, M.: Dispersive and diffusive-dispersive shock waves for non-convex conservation laws. SIAM Rev. 59, 3–61 (2015)CrossRefMATH El, G.A., Hoefer, M.A., Shearer, M.: Dispersive and diffusive-dispersive shock waves for non-convex conservation laws. SIAM Rev. 59, 3–61 (2015)CrossRefMATH
22.
Zurück zum Zitat Jacobs, D., McKinney, B., Shearer, M.: Travelling wave solutions of the modified Korteweg-de Vries-Burgers equation. J. Differ. Equ. 116, 448–467 (1995)ADSMathSciNetCrossRefMATH Jacobs, D., McKinney, B., Shearer, M.: Travelling wave solutions of the modified Korteweg-de Vries-Burgers equation. J. Differ. Equ. 116, 448–467 (1995)ADSMathSciNetCrossRefMATH
23.
24.
Zurück zum Zitat Hayes, B., Shearer, M.: Undercompressive shocks and Riemann problems for scalar conservation laws with non-convex fluxes. Proc. R. Soc. Edinb. A. 129, 733–754 (1999)MathSciNetCrossRefMATH Hayes, B., Shearer, M.: Undercompressive shocks and Riemann problems for scalar conservation laws with non-convex fluxes. Proc. R. Soc. Edinb. A. 129, 733–754 (1999)MathSciNetCrossRefMATH
25.
Zurück zum Zitat Chugainova, A.P., Shargatov, V.A.: Traveling waves and undercompressive shocks in solutions of the generalized Korteweg-de Vries-Burgers equation with a time-dependent dissipation coefficient distribution. Eur. Phys. J. Plus. 135(8), 1–18 (2020)CrossRef Chugainova, A.P., Shargatov, V.A.: Traveling waves and undercompressive shocks in solutions of the generalized Korteweg-de Vries-Burgers equation with a time-dependent dissipation coefficient distribution. Eur. Phys. J. Plus. 135(8), 1–18 (2020)CrossRef
26.
Zurück zum Zitat Bakhvalov, N.S., Eglit, M.E.: Effective dispersive equations for wave propogation in periodic media Dokl. Math. 61(1), 1–4 (2000) Bakhvalov, N.S., Eglit, M.E.: Effective dispersive equations for wave propogation in periodic media Dokl. Math. 61(1), 1–4 (2000)
27.
Zurück zum Zitat Kulikovskii, A.G., Chugainova, A.P.: Modeling the influence of small-scale dispersion processes in a continuum on the formation of large-scale phenomena. Comput. Math. Math. Phys. 44(6), 1062–106 (2004)MathSciNetMATH Kulikovskii, A.G., Chugainova, A.P.: Modeling the influence of small-scale dispersion processes in a continuum on the formation of large-scale phenomena. Comput. Math. Math. Phys. 44(6), 1062–106 (2004)MathSciNetMATH
28.
Zurück zum Zitat Kulikovskii, A.G., Chugainova, A.P., Shargatov, V.A.: 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)MathSciNetCrossRefMATH Kulikovskii, A.G., Chugainova, A.P., Shargatov, V.A.: 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)MathSciNetCrossRefMATH
29.
Zurück zum Zitat Chugainova, A.P., Il’ichev, A.T., Kulikovskii, A.G., Shargatov, V.A.: Problem of arbitrary discontinuity disintegration for the generalized Hopf equation: selection conditions for a unique solution. J. Appl. Math. 82(3), 496–525 (2017)MathSciNetMATH Chugainova, A.P., Il’ichev, A.T., Kulikovskii, A.G., Shargatov, V.A.: Problem of arbitrary discontinuity disintegration for the generalized Hopf equation: selection conditions for a unique solution. J. Appl. Math. 82(3), 496–525 (2017)MathSciNetMATH
30.
Zurück zum Zitat Chugainova, A.P., Il’ichev, A.T., Shargatov, V.A.: Stability of shock wave structures in nonlinear elastic media. Math. Mech. Solids 24(II), 3456–3471 (2019) Chugainova, A.P., Il’ichev, A.T., Shargatov, V.A.: Stability of shock wave structures in nonlinear elastic media. Math. Mech. Solids 24(II), 3456–3471 (2019)
31.
Zurück zum Zitat Landau, L.D., Lifshits, E.M.: Course of Theoretical Physics, Fluid Mechanics, vol. 6. Pergamon, Oxford (1987)MATH Landau, L.D., Lifshits, E.M.: Course of Theoretical Physics, Fluid Mechanics, vol. 6. Pergamon, Oxford (1987)MATH
33.
Zurück zum Zitat The stability of shock waves in magnetohydrodynamics: Akhiezer, A.I., Liubarskii, G Ia., Polovin, R.V. J. Exptl. Theor. Phys. (USSR) 35, 731–737 (1959) The stability of shock waves in magnetohydrodynamics: Akhiezer, A.I., Liubarskii, G Ia., Polovin, R.V. J. Exptl. Theor. Phys. (USSR) 35, 731–737 (1959)
Metadaten
Titel
Structures of longitudinal-torsional shock waves and special discontinuities in nonlinearly viscoelastic media with dispersion
verfasst von
A. P. Chugainova
A. G. Kulikovskii
Publikationsdatum
09.01.2023
Verlag
Springer Berlin Heidelberg
Erschienen in
Continuum Mechanics and Thermodynamics / Ausgabe 4/2023
Print ISSN: 0935-1175
Elektronische ISSN: 1432-0959
DOI
https://doi.org/10.1007/s00161-022-01182-9

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