Advertisement

Wave Solutions in Linear Viscoelastic Materials

  • A. Morro
Part of the International Centre for Mechanical Sciences book series (CISM, volume 356)

Abstract

Dissipative materials are modelled as linear viscoelastic and the thermodynamic restrictions are investigated in detail. For definiteness, attention is mainly confined to solids. Some concepts about thermodynamics and dissipativity in materials with memory are re-visited. Next wave propagation is considered and emphasis is given to the consequences of thermodynamic restrictions on the amplitude evolution. Waves are described as surface discontinuities, time-harmonic waves, rays. The known properties of surface discontinuities are recalled to emphasize the exponential attenuation induced by thermodynamics. Particular attention is then addressed to the description of time-harmonic inhomogeneous waves and asymptotic rays in isotropic and anisotropic solids. Recent results are exhibited along with some work in progress.

Keywords

Plane Wave Wave Solution Thermodynamic Restriction Viscoelastic Body Inhomogeneous Wave 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Fabrizio, M. and A. Morro: Mathematical Problems in Linear Viscoelasticity, SIAM, Philadelphia, 1992.CrossRefMATHGoogle Scholar
  2. 2.
    Caviglia, G. and A. Morro: Inhomogeneous Waves in Solids and Fluids, World Scientific, Singapore, 1992.CrossRefMATHGoogle Scholar
  3. 3.
    Boltzmann, L.: Zur Theorie der elastichen Nachwirkung, Sitzber. Kaiserl. Akad. Wiss. Wien, Math.-Naturw. Kl. 70 (1874), 275–300.Google Scholar
  4. 4.
    Lamb, J. and J. Richter: Anisotropic acoustic attenuation with new measurements for quartz at room temperatures, Proc. R. Soc. London A 293 (1966), 479–492.CrossRefGoogle Scholar
  5. 5.
    Truesdell, C.: Rational Thermodynamics, Springer, Berlin, 1984.CrossRefMATHGoogle Scholar
  6. 6.
    Morro, A. and M. Vianello: Minimal and maximal free energy for materials with memory, Boll. Un. Mat. Ital. A 4 (1990), 45–55.MathSciNetMATHGoogle Scholar
  7. 7.
    Day, W. A.: An objection to using entropy as a primitive concept in continuum thermodynamics, Acta Mech. 27 (1977), 251–255.CrossRefMATHGoogle Scholar
  8. 8.
    Caviglia, G. and A. Morro: Dissipative effects on wave propagation in viscoelastic solids, Eur. J. Mech. A/Solids 12 (1993), 387–402.MATHGoogle Scholar
  9. 9.
    Leitman, M. J. and G. M. C. Fisher: The linear theory of viscoelasticity, in: Encyclopedia of Physics, vol. VIa/3 (Ed. C. Truesdell ), Springer, Berlin, 1973.Google Scholar
  10. 10.
    Chen, P. and A. Morro: On induced discontinuities in a class of linear materials with internal state parameters, Meccanica 22 (1987), 14–18.CrossRefMATHGoogle Scholar
  11. 11.
    Fisher, G. M. C. and M. E. Gurtin: Wave propagation in the linear theory of viscoelasticity, Quart. Appl. Math. 23 (1965), 257–263.MATHGoogle Scholar
  12. 12.
    Morro, A.: Temperature waves in rigid materials with memory, Meccanica 12 (1977), 73–77.Google Scholar
  13. 13.
    Fisher, G. M. C.: The decay of plane waves in the linear theory of viscoelasticity, Brown University, Report NONR 562 (40)/2, Providence, RI, 1965.Google Scholar
  14. 14.
    Lockett, F. J.: The reflection and refraction of waves at an interface between viscelastic materials, J. Mech. Phys. Solids 10 (1962), 58–64.MathSciNetGoogle Scholar
  15. 15.
    Buchen, P. W.: Plane waves in linear viscoelastic media, Geophys. J. R. Astr. Soc. 23 (1971), 531–542.CrossRefMATHGoogle Scholar
  16. 16.
    Currie, P. K., M. A. Hayes and P. M. O’Leary: Viscoelastic Rayleigh waves, Quart. Appl. Math. 35 (1977), 35–53.MATHGoogle Scholar
  17. 17.
    Currie, P. K. and P. M. O’Leary: Viscoelastic Rayleigh waves II, Quart. Appl. Math. 35 (1978), 445–454.MATHGoogle Scholar
  18. 18.
    Boulanger, Ph. and M. Hayes: Inhomogeneous plane waves in viscous fluids,/Continuum Mech. Thermodyn. 2 (1990), 1–9.MathSciNetMATHGoogle Scholar
  19. 19.
    Poirée, B.: Complex harmonic plane waves, in: Physical Acoustics (Eds. O. Leroy and M. A. Breazeale), Plenum, New York, 1991.Google Scholar
  20. 20.
    Aki, K. and P. G. Richards: Quantitative Seismology I Freeman, San Francisco, 1980; p. 170.Google Scholar
  21. 21.
    Morro, A.: Wave propagation in dissipative solids, Proc. Workshop “Advanced mathematical tools in metrology”, Turin, 1993.Google Scholar
  22. 22.
    Hayes, M.: Inhomogeneous plane waves, Arch. Rational Mech. Anal. 85 (1984), 41–79.Google Scholar
  23. 23.
    Fedorov, F. I.: Theory of Elastic Waves in Crystals, Plenum, New York, 1968.CrossRefGoogle Scholar
  24. 24.
    Musgrave, M. J. P.: Crystal Acoustics, Holden Day, San Francisco, 1970.MATHGoogle Scholar
  25. 25.
    Crampin, S.: A review of wave motion in anisotropic and cracked elastic media, Wave Motion 3 (1981), 343–391.CrossRefMATHGoogle Scholar
  26. 26.
    Kerner, C., B. Dyer and M. Worthington: Wave propagation in a vertical transversely isotropic medium: field experiment and model study, Geophys. J. 97 (1989), 295–305.CrossRefGoogle Scholar
  27. 27.
    Chadwick, P.: Wave propagation in transversely isotropic elastic media I. Homogenous plane waves, Proc. Roy. Soc. London A 422 (1989), 23–66.Google Scholar
  28. 28.
    Hayes, M. A. and R. S. Rivlin: Longitudinal waves in a linear viscoelastic material, ZAMP 23 (1972), 153–156.CrossRefMATHGoogle Scholar
  29. 29.
    Hayes, M. A. and R. S. Rivlin: Plane waves in linear viscoelastic materials, Q. Appl. Math. 32 (1974), 113–121.MATHGoogle Scholar
  30. 30.
    Mainardi, F.: Wave propagation in viscoelastic media, Pitman, Boston, 1982.MATHGoogle Scholar
  31. 31.
    Caviglia, G. and A. Morro: Inhomogeneous waves in anisotropie dissipative solids,Continuum Mech. Thermodyn., to appear.Google Scholar
  32. 32.
    Caviglia, G. and A. Morro: Asymptotic ray theory in heterogeneous viscoelastic solids, Q. Jl Mech. appl. Math. 46 (1993), 569–582.MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Caviglia, G. and A. Morro: Asymptotic rays in pre-stressed, anisotropie, dissipative solids,in preparation.Google Scholar
  34. 34.
    Bleistein, N.: Mathematical Methods for Wave Phenomena, Academic Press, London, 1984.MATHGoogle Scholar

Copyright information

© Springer-Verlag Wien 1995

Authors and Affiliations

  • A. Morro
    • 1
  1. 1.DIBEUniversity of GenoaGenoaItaly

Personalised recommendations