Top

Continuum Mechanics and Thermodynamics

Published in:

09-01-2023 | Original Article

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

Authors: A. P. Chugainova, A. G. Kulikovskii

Published in: Continuum Mechanics and Thermodynamics | Issue 4/2023

Activate our intelligent search to find suitable subject content or patents.

search-config

AI-assisted search

Off

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.

previous article A new block-based approach for the analysis of damage in masonries undergoing large deformations

next article Right-hand side method for the numerical solution of nonlinear partial differential equations

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 390 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Maschinenbau + Werkstoffe

Jetzt Wissensvorsprung sichern!

inform now

Giorgio, I., Della Corte, A.: Dynamics of 1D nonlinear pantographic continua. Nonlinear Dyn. 88(1), 21–31 (2017)CrossRef

Misra, A., Nejadsadeghi, N.: Longitudinal and transverse elastic waves in 1D granular materials modeled as micromorphic continua. Wave Motion 90, 175–195 (2019)ADSMathSciNetCrossRefMATH

Eugster, S.R.: Numerical analysis of nonlinear wave propagation in a pantographic sheet. Math. Mech. Complex Syst. 9(3), 293–310 (2021)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

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

Barchiesi, E., Laudato, M., Di Cosmo, F.: Wave dispersion in non-linear pantographic beams. Mech. Res. Commun. 94, 128–132 (2018)CrossRef

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

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

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.

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.

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.

Gel’fand, I.M.: Some problems in the theory of quasilinear equations. Transl. Ser. 2. Am. Math. Soc. 29, 295–381 (1963)

13.

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.

Galin, G Ya.: Shock waves in media with arbitrary equations of state. Sov. Phys. Dokl. 119(3), 244–247 (1958)

15.

Galin, G Ya.: A theory of shock waves. Sov. Phys. Dokl. 4, 757–760 (1960)

16.

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.

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.

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.

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.

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.

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.

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.

Bertozzi, A.L., Munch, A., Shearer, M.: Undercompressive shocks in thin film flows. Phys. D 134(2), 431–464 (1999)MathSciNetCrossRefMATH

24.

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.

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.

Bakhvalov, N.S., Eglit, M.E.: Effective dispersive equations for wave propogation in periodic media Dokl. Math. 61(1), 1–4 (2000)

27.

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.

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.

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.

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.

Landau, L.D., Lifshits, E.M.: Course of Theoretical Physics, Fluid Mechanics, vol. 6. Pergamon, Oxford (1987)MATH

32.

Lax, P.D.: Hyperbolic systems of conservation laws. Comm. Pure Appl. Math. 10, 537–566 (1957)MathSciNetCrossRefMATH

33.

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)

Title: Structures of longitudinal-torsional shock waves and special discontinuities in nonlinearly viscoelastic media with dispersion
Authors: A. P. Chugainova
A. G. Kulikovskii
Publication date: 09-01-2023
Publisher: Springer Berlin Heidelberg
Published in: Continuum Mechanics and Thermodynamics / Issue 4/2023
Print ISSN: 0935-1175
Electronic ISSN: 1432-0959
DOI: https://doi.org/10.1007/s00161-022-01182-9

Springer Professional

Abstract

Please log in to get access to your license.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Technik"

Other articles of this Issue 4/2023

How material and geometrical nonlinearity influences diastolic function of an idealized aortic valve

Torsion of a multilayer elastic cylinder with sequential attachment of layers with multiple superposition of large deformations

Multi-scale constitutive model of human trabecular bone

Modeling and numerical investigation of damage behavior in pantographic layers using a hemivariational formulation adapted for a Hencky-type discrete model

A new block-based approach for the analysis of damage in masonries undergoing large deformations

Theoretical and experimental validation of the variable-thickness topology optimization approach for the rib-stiffened panels

Premium Partners