Skip to main content

Wave fronts in second-order elasticity determined by perturbation method applied to the eikonal equation

Abstract

In Marasco and Romano (Math Comput Model 49(7–8)1504–1518, 2009), Marasco (Math Comput Model 49(7–8):1644–1652, 2009; Int J Eng Sci 47(4):499–511, 2009), we have proposed a perturbation method to determine the speed and the amplitude of the acceleration waves in a second-order elastic body. In this paper, using the above results, we apply a perturbation procedure to analyze the evolution of the wave front of an acceleration wave in the same class of elastic materials. In particular, a second-order approximate solution of the eikonal equation is determined introducing a suitable system of coordinates. The general results are applied to an infinitesimal deformation, and the analytical solution of the eikonal equation is compared with the exact numerical one.

This is a preview of subscription content, access via your institution.

References

  1. Marasco A., Romano A.: On the acceleration waves in second-order elastic, isotropic, compressible, and homogeneous materials. Math. Comput. Model. 49(7–8), 1504–1518 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  2. Marasco A.: On the first-order speeds in any directions of acceleration waves in prestressed second-order isotropic, compressible, and homogeneous materials. Math. Comput. Model. 49(7–8), 1644–1652 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  3. Marasco A.: Second-order effects on the wave propagation in elastic, isotropic, incompressible, and homogeneous media. Int. J. Eng. Sci. 47(4), 499–511 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  4. Romano, A., Marasco, A.: Continuum Mechanics: Advanced Topics and Research Trends. Birkhauser, Boston-Basel-Berlin. ISBN 978-0-8176-4869-5 (2010)

  5. Truesdell C., Noll W.: The nonlinear field theories of mechanics. Handbuch der Physik, Band III/3. Springer, Berlin (1965)

    Google Scholar 

  6. Wang C., Truesdell C.: Introduction to Rational Elasticity. Noordhoff, Groningen (1973)

    MATH  Google Scholar 

  7. Ericksen J.L.: On the propagation of waves in isotropic incompressible perfectly elastic materials. J. Ration. Mech. Anal. 2, 329–337 (1953)

    MathSciNet  MATH  Google Scholar 

  8. Hill R.: Acceleration waves in solids. J. Mech. Phys. Solids. 10, 1–16 (1962)

    MathSciNet  ADS  MATH  Article  Google Scholar 

  9. Scott N.H.: Acceleration waves in constrained elastic materials. Arch. Ration. Mech. Anal. 58(1), 57–75 (1975)

    MATH  Article  Google Scholar 

  10. Scott N.H.: Acceleration waves in incompressible solids viscoelastic waves. Quart. J. Mech. Appl. Math. 29, 295–310 (1976)

    MATH  Article  Google Scholar 

  11. Cohen H., Wang C.C.: Principal waves in monotropic laminated bodies. Arch. Ratio. Mech. Anal. 137(1), 27–47 (1997)

    MathSciNet  MATH  Article  Google Scholar 

  12. Gültop T.: Existence of shear bands in hyperelastic solids. Mech. Res. Commun. 29/5, 431–436 (2002)

    Article  Google Scholar 

  13. Gültop T.: On the propagation of acceleration waves in incompressible hyperelastic solids. J. Sound Vib. 264(2), 377–389 (2003)

    ADS  Article  Google Scholar 

  14. Major M.: Velocity of acceleration wave propagating in hyperelastic Zahorski and Mooney–Rivlin materials. J. Theor. Appl. Mech. 43(4), 777–787 (2005)

    Google Scholar 

  15. Ciarletta M., Straughan B., Zampoli V.: Thermo-poroacoustic acceleration waves in elastic materials with voids without energy dissipation. Int. J. Eng. Sci. 45(9), 736–743 (2007)

    MathSciNet  MATH  Article  Google Scholar 

  16. Nowack R.L.: Wavefronts and solutions of the eikonal equation. Geophys. J. Int. 110(1), 55–62 (1992)

    ADS  Article  Google Scholar 

  17. Fu Z., Jeong W.-K., Pan Y., Kirby R.M., Whitaker R.T.: A fast iterative method for solving the eikonal equation on triangulated surfaces. SIAM J. Sci. Comput. 33, 2468–2488 (2011)

    MathSciNet  MATH  Article  Google Scholar 

  18. Kim S.: Wavefronts of linear elastic waves: local convexity and modeling. Wave Motion. 32(3), 203–216 (2000)

    MathSciNet  MATH  Article  Google Scholar 

  19. dell’Isola F., Ruta G., Batra R.: Second-order solution of Saint-Venant’s problem for an elastic pretwisted bar using Signorini’s perturbation method. J. Elast. 49, 113–127 (1997)

    MathSciNet  Article  Google Scholar 

  20. Batra R., dell’Isola F., Vidoli S.: A second-order solution of Saint-Venant’s problem for a piezoelectric circular bar using Signorini’s perturbation method. J. Elast. 52, 75–90 (1998)

    MathSciNet  MATH  Article  Google Scholar 

  21. Iaccarino G.L., Marasco A., Romano A.: Signorini’s method for live loads and 2-nd order effects. Int. J. Eng. Sci. 44(5–6), 312–324 (2006)

    MathSciNet  MATH  Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to A. Marasco.

Additional information

Communicated by Francesco dell'Isola and Samuel Forest.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Marasco, A., Romano, A. Wave fronts in second-order elasticity determined by perturbation method applied to the eikonal equation. Continuum Mech. Thermodyn. 25, 229–242 (2013). https://doi.org/10.1007/s00161-012-0243-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00161-012-0243-z

Keywords

  • Wave fronts
  • Eikonal equation
  • Nonlinear elasticity
  • Perturbation method
  • Second-order effects