Archive of Applied Mechanics

, Volume 74, Issue 5–6, pp 359–374 | Cite as

Lower-dimensional long wave dynamic models for idealised fibre-reinforced elastic structures

  • L.Yu. Kossovitch
  • R.R. Moukhomodiarov
  • G.A. RogersonEmail author


The dispersion of harmonic waves in an idealised fibre-reinforced elastic layer is investigated. Guided by a numerical and asymptotic long-wave investigation of the dispersion relation, appropriate scales are introduced to help elucidate features of long wave high- and low-frequency motion. In the former case, the stress–strain-state is determined in terms of the long-wave amplitude, appropriate leading-order and refined second-order governing equations being obtained from the second- and third-order problems, respectively. At each order the dispersion relation associated with the governing equation agrees with the appropriate expansion of the “exact” dispersion relation. With respect to low-frequency motion, the long wave limit of anti-symmetric motion is non-zero. This contrasts with the classical case and also indicates that inextensible fibres preclude classical bending. The asymptotic long-wave low frequency stress–strain-state is determined in terms of the governing extensions and mid-surface deflection in the symmetric and anti-symmetric cases, respectively. Appropriate leading and second-order governing equations are also found for these functions. The second-order equations act both to refine the stress–strain-state and also provide the leading-order governing equation in the vicinity of the appropriate quasi wave front. This phenomenon is illustrated by considering a problem concerning shock edge loading of a semi-infinite layer.


Dispersion Relation Harmonic Wave Elastic Layer Wave Dynamic Elastic Structure 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Kossovich, L.Yu.; Moukhomodiarov, R.R.; Rogerson, G.A.: Analysis of the dispersion relation for an incompressible transversely isotropic elastic plate. Acta Mech 153 (2002) 89–111Google Scholar
  2. 2.
    Kossovich, L.Yu.; Moukhomodiarov, R.R.; Rogerson, G.A.: Long wave asymptotic integration in incompressible transversely isotropic elastic structures. Acta Mech 159 (2002) 53–64Google Scholar
  3. 3.
    Spencer, A.J.M.: Deformations of fibre-reinforced materials. Clarendon Press. Oxford 1972Google Scholar
  4. 4.
    Green, A.E.: Boundary layer equations in the linear theory of thin elastic shells. Proc R Soc Lond A 269 (1963) 481–491Google Scholar
  5. 5.
    Goldenveiser, A.L.: An application of asymptotic integration of the equations of elasticity to derive an approximate theory for plate bending. Prik Mat Mekhan 21 (1962) 668–686Google Scholar
  6. 6.
    Friedrichs, K.O.; Dressler, R.F.: A boundary layer theory for elastic plates. Commun Pure Appl Math 14 (1966) 1–33Google Scholar
  7. 7.
    Kaplunov, J.D.; Kossovich, L.Yu.; Nolde, E.V.: Dynamics of thin-walled elastic bodies. Academic New York, 1998Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • L.Yu. Kossovitch
    • 1
  • R.R. Moukhomodiarov
    • 1
  • G.A. Rogerson
    • 2
    Email author
  1. 1.Department of Mathematical Theory of Elasticity and BiomechanicsSaratov State UniversitySaratovRussia
  2. 2.Centre of Applied Mathematics, School of Computing, Science and EngineeringUniversity of SalfordUK

Personalised recommendations