Skip to main content
SpringerLink
Log in

Dispersive Elastic Waves

  • Chapter
  • First Online:
Internal Variables in Thermoelasticity

Part of the book series: Solid Mechanics and Its Applications ((SMIA,volume 243))

  • 690 Accesses

Abstract

Several well-known dispersive wave propagation models together with generalization of the Mindlin-type models are derived by using internal variables. The adopted phenomenological approach is based on the material formulation of continuum mechanics and provides the full thermodynamic consistency due to the dual internal variables concept.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Askes H, Aifantis EC (2006) Gradient elasticity theories in statics and dynamics-a unification of approaches. Int J Fract 139(2):297–304

    Article  MATH  Google Scholar 

  2. Askes H, Metrikine AV (2005) Higher-order continua derived from discrete media: continualisation aspects and boundary conditions. Int J Solids Struct 42(1):187–202

    Article  MATH  Google Scholar 

  3. Askes H, Metrikine AV, Pichugin AV, Bennett T (2008) Four simplified gradient elasticity models for the simulation of dispersive wave propagation. Philos Mag 88(28–29):3415–3443

    Article  Google Scholar 

  4. Berezovski A, Engelbrecht J, Maugin GA (2008) Numerical simulation of waves and fronts in inhomogeneous solids. World Scientific, Singapore

    Book  MATH  Google Scholar 

  5. Berezovski A, Engelbrecht J, Maugin GA (2009) Internal variables and generalized continuum theories. In: IUTAM symposium on progress in the theory and numerics of configurational mechanics. Springer, pp 149–158

    Google Scholar 

  6. Berezovski A, Engelbrecht J, Peets T (2010) Multiscale modeling of microstructured solids. Mech Res Commun 37(6):531–534

    Article  MATH  Google Scholar 

  7. Coleman BD, Gurtin ME (1967) Thermodynamics with internal state variables. J Chem Phys 47(2):597–613

    Article  Google Scholar 

  8. Engelbrecht J (2015) Questions about elastic waves. Springer, Berlin

    Book  MATH  Google Scholar 

  9. Engelbrecht J, Pastrone F (2003) Waves in microstructured solids with nonlinearities in microscale. Proc Estonian Acad Sci Phys Math 52(1):12–20

    MathSciNet  MATH  Google Scholar 

  10. Engelbrecht J, Cermelli P, Pastrone F (1999) Wave hierarchy in microstructured solids. Geometry, continua and microstructure. Hermann Publication, Paris, pp 99–111

    Google Scholar 

  11. Engelbrecht JJ, Berezovski A, Pastrone F, Braun M (2005) Waves in microstructured materials and dispersion. Philos Mag 85(33–35):4127–4141

    Google Scholar 

  12. Maugin GA (1990) Internal variables and dissipative structures. J Non-Equilib Thermodyn 15(2):173–192

    Article  Google Scholar 

  13. Maugin GA (1993) Material inhomogeneities in elasticity. CRC Press, Boca Raton

    Book  MATH  Google Scholar 

  14. Maugin GA (1999) Nonlinear waves in elastic crystals. Oxford University Press, Oxford on Demand

    MATH  Google Scholar 

  15. Maugin GA (2006) On the thermomechanics of continuous media with diffusion and/or weak nonlocality. Arch Appl Mech 75(10–12):723–738

    Article  MATH  Google Scholar 

  16. Maugin GA, Muschik W (1994) Thermodynamics with internal variables. Part I. General concepts. J Non Equilib Thermodyn 19:217–249

    MATH  Google Scholar 

  17. Mindlin RD (1964) Micro-structure in linear elasticity. Arch Rationol Mech Anal 16(1):51–78

    MathSciNet  MATH  Google Scholar 

  18. Muschik W (1990) Internal variables in non-equilibrium thermodynamics. J Non-Equilib Thermodyn 15(2):127–137

    Article  Google Scholar 

  19. Papargyri-Beskou S, Polyzos D, Beskos D (2009) Wave dispersion in gradient elastic solids and structures: a unified treatment. Int J Solids Struct 46(21):3751–3759

    Google Scholar 

  20. Pastrone F, Cermelli P, Porubov A (2004) Nonlinear waves in 1-D solids with microstructure. Mater Phys Mech 7:9–16

    Google Scholar 

  21. Rice JR (1971) Inelastic constitutive relations for solids: an internal-variable theory and its application to metal plasticity. J Mech Phys Solids 19(6):433–455

    Article  MATH  Google Scholar 

  22. Ván P, Berezovski A, Engelbrecht J (2008) Internal variables and dynamic degrees of freedom. J Non-Equilib Thermodyn 33(3):235–254

    Article  MATH  Google Scholar 

Download references

Acknowledgements

This chapter is derived in part from the article published in Acta Mech. (2011) 220: 349–363. Copyright\(\copyright \) Springer-Verlag, available online: https://link.springer.com/article/10.1007/s00707-011-0468-0

Author information

Authors and Affiliations

  1. Department of Cybernetics, School of Science, Tallinn University of Technology, Tallinn, Estonia

    Arkadi Berezovski

  2. Institute of Particle and Nuclear Physics, MTA WIGNER Research Centre for Physics, Budapest, Hungary

    Peter Ván

  3. Department of Energy Engineering, Montavid Thermodynamic Research Group, Budapest University of Technology and Economics, Budapest, Hungary

    Peter Ván

Authors
  1. Arkadi Berezovski
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Peter Ván
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Arkadi Berezovski .

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this chapter

Cite this chapter

Berezovski, A., Ván, P. (2017). Dispersive Elastic Waves. In: Internal Variables in Thermoelasticity. Solid Mechanics and Its Applications, vol 243. Springer, Cham. https://doi.org/10.1007/978-3-319-56934-5_6

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-56934-5_6

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-56933-8

  • Online ISBN: 978-3-319-56934-5

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation