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Dissipative Nonlinear Strain Waves in Solids

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Selected Topics in Nonlinear Wave Mechanics

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Abstract

The study of strain localized waves of permanent form (solitary waves) is of theoretical and experimental interest because these waves may propagate and transfer energy over long distances along homogeneous free lateral surface elastic wave-guides. However, geometrical inhomogeneities of the waveguide, influence of an external medium or microstructure of the wave-guide material may result in an amplification of the strain wave causing the appearance of plasticity zones or micro cracks and eventually the breakdown of a wave-guide. This is of importance for an assessment of durability of elastic materials and structures, methods of nondestructive testing. Among the elastic wave-guides the cylindrical elastic rod is chosen since this real-life wave-guide admits an analytical description in a closed form. The theory is developed to account for longitudinal nonlinear strain waves in a rod with varying cross-sections, in a rod surrounded by a dissipative (active) external medium and in an elastic medium with microstructure. The governing equations obtained turn out nonintegrable, however, some exact traveling wave solutions are found. The asymptotic analysis reveals a possibility of a solitary wave selection when the initial KdV-like solitary wave transforms into the dissipative solitary wave with the amplitude and the velocity prescribed by the equation coefficients. Moreover, it was found that dissipation may be in balance with nonlinearity rather than with dispersion. As a result the kink-shaped strain wave propagates. Numerical simulations on the evolution of an initial arbitrary localized pulse confirm the predictions about solitary-wave selection done on the basis of the asymptotic solution. The asymptotic description of the amplification of the solitary wave in a narrowing rod is proven also in experiments.

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References

  1. A. N. Harker, Elastic Waves in Solids with application to Nondestructive Testing of Pipelines, Adam Hilger, Bristol and Philadelphia in association with British Gas plc 1988.

    Google Scholar 

  2. A. I. Lurie, Nonlinear Theory of elasticity, Elsevier, Amsterdam, 1990.

    MATH  Google Scholar 

  3. P. Pleus and M. Sayir, A second order theory for large deflections of slender beams, J. Appl. Math and Phys. (ZAMP), 34 (1983), 192-217.

    Article  MATH  Google Scholar 

  4. H. D. McNiven and J. J. McCoy, Vibrations and wave propagation in rods, in: G. Herrmann and R. Mindlin, eds., Applied Mechanics, Pergamon, New York, 1974, pp. 197-226.

    Google Scholar 

  5. A. E. H. Love, A Treatise on the Mathematical Theory of Elasticity, University Press, Cambridge 1927.

    Google Scholar 

  6. R. T. Shield, Equilibrium solutions for finite elasticity, Trans. ASME, J. Appl Math., 50 (1983), 1171-1180.

    Article  MATH  Google Scholar 

  7. J. Engelbrecht. Nonlinear wave processes of deformation in solids, Pitman, Boston 1983.

    MATH  Google Scholar 

  8. L. A. Ostrovsky and A.I. Potapov, Modulated waves in linear and nonlinear media, The John Hopkins University Press, Baltimore 1999.

    Google Scholar 

  9. A. M. Samsonov, Nonlinear strain waves in elastic waveguides, in: A. Jeffrey and J. Engelbrecht eds., Nonlinear Waves in Solids, Springer-Verlag, New York, 1994.

    Google Scholar 

  10. A. M. Samsonov, How to predict, generate and observe strain solitons in solids, Acta Tech. CSAV, 42 (1997), 93-113.

    MathSciNet  Google Scholar 

  11. A.V. Porubov and A.M. Samsonov, Refinement of the model for the propagation of longitudinal strain waves in a rod with nonlinear elasticity, Tech. Phys. Lett, 19 (1993), 365-366.

    Google Scholar 

  12. G. V. Dreiden, A. V. Porubov, A. M. Samsonov, I. V. Semenova, and E.V. Sokurinskaya, Experiments in the propagation of longitudinal strain solitons in a nonlinearly elastic rod, Tech. Phys. Lett, 21 (1995), 415-417.

    Google Scholar 

  13. G. V. Dreiden, A. V. Porubov, A. M. Samsonov, I. V. Semenova, Reflection of a longitudinal strain solitary wave from the end face of a nonlinearly elastic rod, Tech. Phys., 46:5 (2001), 505-511.

    Article  MathSciNet  Google Scholar 

  14. A. M. Samsonov, Strain soltons in solids and how to construct them

    Google Scholar 

  15. A. M. Samsonov, G. V. Dreiden, A. V. Porubov, and I. V. Semenova, Longitudinal-strain soliton focusing in a narrowing nonlinearly elastic rod, Phys. Rev. B, 57 (1998), 5778-5787.

    Article  Google Scholar 

  16. A. V. Porubov, A. M. Samsonov, M. G. Velarde, and A.V. Bukhan-ovsky, Strain solitary waves in an elastic rod embedded in another elasti external medium with sliding, Phys. Rev. E, 58 (1998), 3854-3864.

    Article  MathSciNet  Google Scholar 

  17. C. I. Christov and M. G. Velarde, Dissipative solitons, Physica D, 86(1995), 323-347.

    Article  MathSciNet  MATH  Google Scholar 

  18. A. D. Kerr, Elastic and viscoelastic foundation models, J. Appl. Mech., 31 (1964), 491-498.

    Article  MATH  Google Scholar 

  19. G.A. Maugin and W. Muschik, Thermodynamics with internal variables. Part I: General Concepts. J. Non-Equil. Thermodyn., 19 (1994), 217-249

    MATH  Google Scholar 

  20. G.A. Maugin and W. Muschik, Thermodynamics with internal variables.Part II: Applications. J. Non-Equil. Thermodyn., 19 (1994), 250-289.

    Google Scholar 

  21. J. Engelbreght and M. Braun, Nonlinear waves in nonlocal media, Appl. Mech. Rev., 51:8 (1998), 475-488.

    Article  Google Scholar 

  22. P. Cermelli and F. Pastrone, Growth and decay of waves in microstruc-tured solids, Proc. Estonian Acad. Sci. Phys. Math., 46 (1997), 32-40.

    MathSciNet  MATH  Google Scholar 

  23. A. C. Eringen, Theory of micropolar elasticity, in: H. Liebowitz, ed. Fracture. An advanced treatise. V.2. Mathematical fundamentals, Academic Press, New York, 1968, pp. 622-729.

    Google Scholar 

  24. R. D. Mindlin, Microstructure in linear elasticity, Archive of Rational Mechanics and Analysis, 1 (1964), 51-78.

    MathSciNet  Google Scholar 

  25. W. Nowacki, Theory of Elasticity, Mir, Moscow, 1975, (in Russian).

    Google Scholar 

  26. G. N. Savin, A. A. Lukashev and E. M. Lysko, Elastic wave propagation in a solid with microstructure, Soviet Appl. Mech., 15 (1973), 725-.

    Google Scholar 

  27. G. N. Savin, A. A. Lukashev, E. M. Lysko, S. V. Veremeenko, and G.G. Agasiev, Elastic wave propagation in the Cosserat continuum with constrained particle rotation, Soviet Appl. Mech., 15, (1973).

    Google Scholar 

  28. V. I. Erofeev and A. I. Potapov, Longitudinal strain waves in non-linearly elastic media with couple stresses, Int. J. Nonl. Mech., 28:4 (1993), 483-488.

    Article  MATH  Google Scholar 

  29. V. I. Erofeev, Wave processes in solids with microstructure, Moscow University Publ., 1999, (in Russian).

    Google Scholar 

  30. A. V. Porubov, Strain solitary waves in an elastic rod with microstructure, Rendiconti del Seminario Matematico dell'Universita' e Politecnico di Torino, 2001, (in press).

    Google Scholar 

  31. F. D. Murnaghan, Finite Deformations of an Elastic Solid, J. Wiley, New York, 1951.

    MATH  Google Scholar 

  32. E. J. Parkes and B. R. Duffy, An automated tanh-function method for finding solitary wave solutions to nonlinear evolution equations, Computer Phys. Comm., 98, (1996), 288-300.

    Article  MATH  Google Scholar 

  33. K. W. Chow, A class of exact, periodic solutions of nonlinear envelope equations, J. Math. Phys., 36 (1995), 4125-4137.

    Article  MathSciNet  MATH  Google Scholar 

  34. A. Nakamura, A direct method of calculating periodic wave solutions to nonlinear evolution equations,J. Phys. Soc. Japan., 47 (1979), 1701-1705.

    Article  MathSciNet  Google Scholar 

  35. A. Nakamura, A direct method of calculating periodic wave solutions to nonlinear evolution equations, I. Exact two-periodic wave solution, J. Phys. Soc. Japan., 47 (1979), 1701-1705.

    Article  MathSciNet  Google Scholar 

  36. N. A. Kostov, I. M. Uzunov, New kinds of periodical waves in birefrin-gent optical fibers, Opt. Comm., 89 (1992), 389-392.

    Article  Google Scholar 

  37. D. F. Parker and E. N. Tsoy, Explicit Solitary and Periodic Solutions for Optical Cascading, J. Eng. Math., 36 (1999), 149-162.

    Article  MathSciNet  MATH  Google Scholar 

  38. A. M. Samsonov, Travelling wave solutions for nonlinear waves with dissipation, Appl. Anal., 57 (1995), 85-100.

    Article  MathSciNet  MATH  Google Scholar 

  39. A. V. Porubov, Exact travelling wave solutions of nonlinear evolution equation of surface waves in a convecting fluid, J. Phys. A: Math. Gen., 26 (1993), L797-L800.

    Article  MathSciNet  Google Scholar 

  40. A. V. Porubov, Periodical solution to the nonlinear dissipative equation for surface waves in a convective liquid layer, Phys. Lett. A, 221 (1996), 391-394.

    Article  MathSciNet  MATH  Google Scholar 

  41. A. V. Porubov and D. F. Parker, Some General Periodic Solutions to Coupled Nonlinear Schrödinger Equations, Wave Motion, 29(1999), 97-109.

    Article  MathSciNet  MATH  Google Scholar 

  42. A. V. Porubov and M. G. Velarde. Exact Periodic Solutions of the Complex Ginzburg-Landau Equation, J. Math. Phys., 40 (1999), 884-896.

    Article  MathSciNet  MATH  Google Scholar 

  43. A. V. Porubov and D. F. Parker, Some General Periodic Solutions for Optical Cascading, Submitted to Proc. Roy. Soc. A, (2001).

    Google Scholar 

  44. E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, Cambridge University Press, Cambridge, U.K., 1927.

    MATH  Google Scholar 

  45. J. Weiss, M. Tabor and G. Carnevale, The Painlevé property for partial differential equations, J. Math. Phys., 24 (1983), 522-526.

    Article  MathSciNet  MATH  Google Scholar 

  46. E. V. Sokurinskaya, “Study of nonlinear travelling strain waves in one-dimensional elastic wave guide”, Ph.D. Dissertation, Technical University, St. Petersburg (1991), (in Russian).

    Google Scholar 

  47. M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform, SIAM, Philadelphia, 1981.

    Google Scholar 

  48. P. L. Sachdev, Nonlinear diffusive waves, Cambridge University Press, Cambridge, U.K., 1987.

    MATH  Google Scholar 

  49. G. B. Whitham, Linear and Nonlinear Waves Wiley, New York, 1974.

    MATH  Google Scholar 

  50. A. V. Porubov and M. G. Velarde, Dispersive-dissipative solitons in nonlinear solids, Wave Motion, 31:3 (2000), 197-207.

    Article  MathSciNet  MATH  Google Scholar 

  51. A. V. Porubov and M. G. Velarde, Strain kinks in an elastic rod embedded in a viscoelastic medium, Wave Motion, (2001), (in press).

    Google Scholar 

  52. I. L. Kliakhandler, A. V. Porubov and M. G. Velarde, Localized finite-amplitude disturbances and selection of solitary waves, Phys. Rev. E, 62 (2000), 4959-4962.

    Article  MathSciNet  Google Scholar 

  53. A. V. Porubov, P. Cermelli, and F. Pastrone, Influence of macro- and microdissipation on nonlinear strain solitary wave evolution, (submitted for publication).

    Google Scholar 

  54. J. Engelbrecht, P. Cermelli and F. Pastrone, Wave hierarchy in microstructured solids, in: G. Maugin, ed., Geometry, Continua and Microstructure, Herman Publ., Paris, 1999, pp. 99-111.

    Google Scholar 

  55. M. Vlieg-Hultsman and W. Halford, The Korteweg-de Vries-Burgers equation: a reconstruction of exact solutions, Wave Motion, 14 (1991) 267-271.

    Article  MathSciNet  MATH  Google Scholar 

  56. A.V. Porubov and A. M. Samsonov, Long nonlinear strain waves in layered elastic half-space, Intern. J. Nonlinear Mechanics, 30 (1995), 861-877.

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Ioffe Technical Institute of the Russian Academy, St. Petersburg, 194021, Russia

    A. V. Porubov

  2. Institute for High-Performance Computing and Data Bases, P.O. box 71, St. Petersburg, 194291, Russia

    A. V. Porubov

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  1. A. V. Porubov
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Editors and Affiliations

  1. Department of Mathematics, University of Louisiana at Lafayette, Lafayette, LA, 70504-1010, USA

    C. I. Christov

  2. Institute of Structronics, 275 Slater Street, Ottawa, K1P-5H9, Canada

    A. Guran

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Porubov, A.V. (2002). Dissipative Nonlinear Strain Waves in Solids. In: Christov, C.I., Guran, A. (eds) Selected Topics in Nonlinear Wave Mechanics. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-0095-6_7

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  • DOI: https://doi.org/10.1007/978-1-4612-0095-6_7

  • Publisher Name: Birkhäuser, Boston, MA

  • Print ISBN: 978-1-4612-6609-9

  • Online ISBN: 978-1-4612-0095-6

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