Advertisement

Waves in Inhomogeneous Solids

  • Arkadi BerezovskiEmail author
  • Mihhail Berezovski
  • Jüri Engelbrecht
Chapter

Abstract

The paper aims at presenting a numerical technique used in simulating the propagation of waves in inhomogeneous elastic solids. The basic governing equations are solved by means of a finite-volume scheme that is faithful, accurate, and conservative. Furthermore, this scheme is compatible with thermodynamics through the identification of the notions of numerical fluxes (a notion from numerics) and of excess quantities (a notion from irreversible thermodynamics). A selection of one-dimensional wave propagation problems is presented, the simulation of which exploits the designed numerical scheme. This selection of exemplary problems includes (i) waves in periodic media for weakly nonlinear waves with a typical formation of a wave train, (ii) linear waves in laminates with the competition of different length scales, (iii) nonlinear waves in laminates under an impact loading with a comparison with available experimental data, and (iv) waves in functionally graded materials.

Keywords

Pulse Shape Riemann Problem Computational Cell Jump Relation Thermoelastic 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.
    Achenbach, J.D.: Wave Propagation in Elastic Solids. North-Holland, Amsterdam (1973) zbMATHGoogle Scholar
  2. 2.
    Bale, D.S., LeVeque, R.J., Mitran, S., Rossmanith, J.A.: A wave propagation method for conservation laws and balance laws with spatially varying flux functions. SIAM J. Sci. Comp. 24, 955–978 (2003) CrossRefMathSciNetGoogle Scholar
  3. 3.
    Bedford, A., Drumheller, D.S.: Introduction to Elastic Wave Propagation. Wiley, New York (1994) Google Scholar
  4. 4.
    Berezovski, A., Berezovski, M., Engelbrecht, J.: Numerical simulation of nonlinear elastic wave propagation in piecewise homogeneous media. Mater. Sci. Eng. A418, 364–369 (2006) Google Scholar
  5. 5.
    Berezovski A, Berezovski, M., Engelbrecht, J., Maugin, G.A.: Numerical simulation of waves and fronts in inhomogeneous solids. In: Nowacki, W.K., Zhao, H. (eds.) Multi-Phase and Multi-Component Materials under Dynamic Loading, pp. 71-80. Inst. Fundam. Technol. Research, Warsaw (2007) Google Scholar
  6. 6.
    Berezovski, A., Maugin, G.A.: Simulation of thermoelastic wave propagation by means of a composite wave-propagation algorithm. J. Comp. Physics 168, 249–264 (2001) zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Berezovski, A., Maugin, G.A.: Thermoelastic wave and front propagation. J. Thermal Stresses 25, 719–743 (2002) CrossRefMathSciNetGoogle Scholar
  8. 8.
    Berezovski, A., Maugin, G.A.: Stress-induced phase-transition front propagation in thermoelastic solids. Eur. J. Mech. A/Solids 24, 1–21 (2005) zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Billingham, J., King, A.C.: Wave Motion. Cambridge University Press (2000) Google Scholar
  10. 10.
    Chakraborty, A., Gopalakrishnan, S.: Various numerical techniques for analysis of longitudinal wave propagation in inhomogeneous one-dimensional waveguides. Acta Mech. 162, 1–27 (2003) zbMATHCrossRefGoogle Scholar
  11. 11.
    Chakraborty, A., Gopalakrishnan, S.: Wave propagation in inhomogeneous layered media: solution of forward and inverse problems. Acta Mech. 169, 153–185 (2004) zbMATHCrossRefGoogle Scholar
  12. 12.
    Chen, X., Chandra, N.: The effect of heterogeneity on plane wave propagation through layered composites. Comp. Sci. Technol. 64, 1477–1493 (2004) zbMATHCrossRefGoogle Scholar
  13. 13.
    Chen, X., Chandra, N., Rajendran, A.M.: Analytical solution to the plate impact problem of layered heterogeneous material systems. Int. J. Solids Struct. 41, 4635–4659 (2004) zbMATHCrossRefGoogle Scholar
  14. 14.
    Chiu, T.-C., Erdogan, F.: One-dimensional wave propagation in a functionally graded elastic medium. J. Sound Vibr. 222, 453–487 (1999) CrossRefGoogle Scholar
  15. 15.
    Engelbrecht, J., Berezovski, A., Pastrone, F., Braun, M.: Waves in microstructured materials and dispersion. Phil. Mag. 85, 4127–4141 (2005) CrossRefGoogle Scholar
  16. 16.
    Fogarthy, T., LeVeque, R.J.: High-resolution finite-volume methods for acoustics in periodic and random media. J. Acoust. Soc. Am. 106, 261–297 (1999) Google Scholar
  17. 17.
    Godlewski, E., Raviart, P.-A.: Numerical Approximation of Hyperbolic Systems of Conservation Laws. New York, Springer (1996) zbMATHGoogle Scholar
  18. 18.
    Grady, D.: Scattering as a mechanism for structured shock waves in metals. J. Mech. Phys. Solids 46, 2017–2032 (1998) zbMATHCrossRefGoogle Scholar
  19. 19.
    Graff, K.F.: Wave Motion in Elastic Solids. Oxford University Press (1975) Google Scholar
  20. 20.
    Guinot, V.: Godunov-type Schemes: An Introduction for Engineers. Elsevier, Amsterdam (2003) Google Scholar
  21. 21.
    Hirai, T.: Functionally graded materials. In: Processing of Ceramics. Vol. 17B, Part 2, pp. 292-341. VCH Verlagsgesellschaft, Weinheim (1996) Google Scholar
  22. 22.
    Hoffmann, K.H., Burzler, J.M., Schubert, S.: Endoreversible thermodynamics. J. Non-Equil. Thermodyn. 22, 311–355 (1997) CrossRefGoogle Scholar
  23. 23.
    Langseth, J.O., LeVeque, R.J.: A wave propagation method for three-dimensional hyperbolic conservation laws. J. Comp. Physics 165, 126–166 (2000) zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    LeVeque, R.J.: Wave propagation algorithms for multidimensional hyperbolic systems. J. Comp. Physics 131, 327–353 (1997) zbMATHCrossRefGoogle Scholar
  25. 25.
    LeVeque, R.J.: Balancing source terms and flux gradients in high-resolution Godunov methods: the quasi-steady wave-propagation algorithm. J. Comp. Physics 148, 346–365 (1998) CrossRefMathSciNetGoogle Scholar
  26. 26.
    LeVeque, R.J.: Finite volume methods for nonlinear elasticity in heterogeneous media. Int. J. Numer. Methods in Fluids 40, 93–104 (2002) zbMATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    LeVeque, R.J.: Finite Volume Methods for Hyperbolic Problems. Cambridge University Press (2002) Google Scholar
  28. 28.
    LeVeque, R.J., Yong, D.H.: Solitary waves in layered nonlinear media. SIAM J. Appl. Math. 63, 1539–1560 (2003) zbMATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Liska, R., Wendroff, B.: Composite schemes for conservation laws. SIAM J. Numer. Anal. 35, 2250–2271 (1998) zbMATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Markworth, A.J., Ramesh, K.S., Parks, W.P.: Modelling studies applied to functionally graded materials. J. Mater. Sci. 30, 2183–2193 (1995) CrossRefGoogle Scholar
  31. 31.
    Meurer, T., Qu, J., Jacobs, L.J.: Wave propagation in nonlinear and hysteretic media – a numerical study. Int. J. Solids Struct. 39, 5585–5614 (2002) zbMATHCrossRefGoogle Scholar
  32. 32.
    Muschik, W., Berezovski, A.: Thermodynamic interaction between two discrete systems in non-equilibrium. J. Non-Equilib. Thermodyn. 29, 237–255 (2004) zbMATHCrossRefGoogle Scholar
  33. 33.
    Rokhlin, S.I., Wang, L.: Ultrasonic waves in layered anisotropic media: characterization of multidirectional composites. Int. J. Solids Struct. 39, 5529–5545 (2002) zbMATHCrossRefGoogle Scholar
  34. 34.
    Santosa, F., Symes, W.W.: A dispersive effective medium for wave propagation in periodic composites. SIAM J. Appl. Math. 51, 984–1005 (1991) zbMATHCrossRefMathSciNetGoogle Scholar
  35. 35.
    Suresh, S., Mortensen, A.: Fundamentals of Functionally Graded Materials. The Institute of Materials, IOM Communications, London (1998) Google Scholar
  36. 36.
    Toro, E.F.: Riemann Solvers and Numerical Methods for Fluid Dynamics. Springer, Berlin (1997) zbMATHGoogle Scholar
  37. 37.
    Toro, E.F. (ed.): Godunov Methods: Theory and Applications. Kluwer, New York (2001) zbMATHGoogle Scholar
  38. 38.
    Wang, L., Rokhlin, S.I.: Recursive geometric integrators for wave propagation in a functionally graded multilayered elastic medium. J. Mech. Phys. Solids 52, 2473–2506 (2004) zbMATHCrossRefMathSciNetGoogle Scholar
  39. 39.
    Zhuang, S., Ravichandran, G., Grady, D.: An experimental investigation of shock wave propagation in periodically layered composites. J. Mech. Phys. Solids 51, 245–265 (2003) CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Arkadi Berezovski
    • 1
    Email author
  • Mihhail Berezovski
    • 1
  • Jüri Engelbrecht
    • 1
  1. 1.Centre for Nonlinear StudiesInstitute of Cybernetics at Tallinn University of TechnologyTallinnEstonia

Personalised recommendations