Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

New results for the effects of higher-order moduli and material properties on strain waves propagating through elastic circular cylindrical rods

  • 41 Accesses

Abstract

The equation that describes the propagation of macrostrain waves is studied using a unified approach quoted earlier by one of the authors. New exact traveling wave solutions, mainly solitary, soliton, periodic and elliptic waves are obtained. It is believed that the family of obtained solutions covers all possible explicit solutions that could be found by all the other methods. Here, attention is focused on the case when the speed of the moving frame is equal to the wave speed. It is found that the nonlinearity produced by higher order of elasticity moduli acts as an impeding factor to the wave intensity. While the double dispersion relative to properties of the material leads to the existence of critical values for some nonlinearity and double dispersion parameters for which the wave amplitude reduces to zero. Such critical values do not seem to have been reported earlier. Finally, the stability of solution is analyzed and stability criteria are obtained.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

References

  1. 1.

    G.A. Maugin, Solitons in elastic solids (1938–2010). Mech. Res. Commun. 38, 341–349 (2011)

  2. 2.

    A.V. Porubov, E.L. Aero, G.A. Maugin, Two approaches to study essentially nonlinear and dispersive properties of the internal structure of materials. Phys. Rev. E 79(4), 046608 (2009)

  3. 3.

    YuI Meshcheryakov, N.I. Zhigacheva, B.K. Barakhtin, G.V. Stepanov, V.I. Zubov, A.N. Olisov, V.A. Fedorchuk, J.R. Klepaczko, Studies on localized strain in sheet metals under impact tension and shear. Strength Mater. 33(4), 325–332 (2001)

  4. 4.

    V.I. Erofeev, A.O. Malkhanov, Localized strain waves in a nonlinearly elastic conducting medium interacting with a magnetic field. Mech. Solids 52(2), 224–231 (2017)

  5. 5.

    R. Hirota, M. Ito, Resonance of solitons in one dimension. J. Phys. Soc. Japan 52(3), 744–748 (1983)

  6. 6.

    R. Hirota, The Direct Method in Soliton Theory (Cambridge University Press, Cambridge, 2004)

  7. 7.

    C. Gu, Soliton Theory and Its Applications (Zhejiang Science and Technology Publishing House, Zhejiang, 1990)

  8. 8.

    M.J. Ablowitz, P.A. Clarkson, Soliton, Nonlinear Evolution Equations and Inverse Scattering (Cambridge University Press, New York, 1991)

  9. 9.

    S.Y. Lou, J.Z. Lu, Special solutions from the variable separation approach: Davey–Stewartson equation. J. Phys. A Math. Gen. 29, 4209–4215 (1996)

  10. 10.

    V.B. Matveev, M.A. Salle, Darboux transformation and solitons, Springer Series in Nonlinear Dynamics (Springer, Berlin, 1991)

  11. 11.

    H.D. Wahlquist, F.B. Estabrook, Bäcklund transformation for solutions of the Korteweg–de Vries equation. Phys. Rev. Lett. 31(23), 1386–1390 (1973)

  12. 12.

    V. Kuznetsov, P. Vanhaecke, Bäcklund transformations for finite-dimensional integrable systems: a geometric approach. J. Geom. Phys. 44(1), 1–40 (2002)

  13. 13.

    F. Caruello, M. Tabor, Painlevé expansions for nonintegrable evolution equations. Phys. D 39, 77–94 (1989)

  14. 14.

    C.T. Yan, A simple transformation to nonlinear waves. Phys. Lett. A 224, 77–84 (1996)

  15. 15.

    Z.Y. Yan, H.Q. Zhang, E.G. Fan, New explicit and travelling wave solutions for a class of nonlinear evolution equations. Acta Phys. Sin. 48, 1–5 (1999)

  16. 16.

    E.J. Parkes, B.R. Duy, Travelling solitary wave solutions to a compound KdV Burgers equation. Phys. Lett. A 229, 217–220 (1997)

  17. 17.

    E. Fan, Extended tanh-function method and its applications to nonlinear equations. Phys. Lett. A 277, 212–218 (2000)

  18. 18.

    Z.Y. Yan, New explicit travelling wave solutions for two new integrable coupled nonlinear evolution equations. Phys. Lett. A 292, 100–106 (2001)

  19. 19.

    Y. Chen, Y. Zheng, Generalized extended tanh-function method to construct new explicit exact solutions for the approximate equations for long water waves. Int. J. Mod. Phys. C 4, 1–14 (2003)

  20. 20.

    J.H. He, An elementary introduction to recently developed asymptotic methods and nanomechanics in textile engineering. Int. J. Mod. Phys. B 22, 3487–3578 (2008)

  21. 21.

    S.K. Liu, Z.K. Fu, S.D. Liu, Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations. Phys. Lett. A 289, 69–74 (2001)

  22. 22.

    F. Xie, Y. Zhang, Z. Lü, Symbolic computation in non-linear evolution equation: application to (3+1)-dimensional Kadomtsev–Petviashvili equation. Chaos Solitons Fractals 24(1), 257–263 (2005)

  23. 23.

    M.L. Wang, Application of a homogeneous balance method to exact solutions of nonlinear equations in mathematical physics. Phys. Lett. A 216, 67–75 (1996)

  24. 24.

    J.F. Alzaidy, Extended mapping method and its applications to nonlinear evolution equations. J. Appl. Math. 2012, 1–5 (2012)

  25. 25.

    R. Castro López, G.H. Sun, O. Camacho-Nieto, C. Yáñez-Márquez, S.H. Dong, Analytical traveling-wave solutions to a generalized Gross–Pitaevskii equation with some new time and space varying nonlinearity coefficients and external fields. Phys. Lett. A 381, 2978–2985 (2017)

  26. 26.

    H.I. Abdel-Gawad, Towards a unified method for exact solutions of evolution equations. An application to reaction diffusion equations with finite memory transport. J. Stat. Phys. 147(3), 506–518 (2012)

  27. 27.

    H.I. Abdel-Gawad, M. Tantawy, Exact solutions of the Shamel–Korteweg–de Vries equation with time dependent coefficients. Inf. Sci. Lett. 3(3), 103–109 (2014)

  28. 28.

    L.H. Zhang, Travelling wave solutions for the generalized Zakharov–Kuznetsov equation with higher-order nonlinear terms. Appl. Math. Comput. 208(1), 144–155 (2009)

  29. 29.

    Z. Fu, S. Liu, On some classes of breather lattice solutions to the sinh–Gordon equation. Z. Naturforsch. 62, 555–563 (2007)

  30. 30.

    J.G. Liu, M.-X. You, L. Zhou, G.-P. Ai, The solitary wave, rogue wave and periodic solutions for the (3\(+\)1)-dimensional soliton equation. Z. Angew. Math. Phys. 70(4), 1–11 (2019)

Download references

Author information

Correspondence to M. Tantawy.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Tantawy, M., Abdel-Gawad, H.I. & Ghaleb, A.F. New results for the effects of higher-order moduli and material properties on strain waves propagating through elastic circular cylindrical rods. Eur. Phys. J. Plus 135, 7 (2020). https://doi.org/10.1140/epjp/s13360-019-00014-1

Download citation