An Approximate Theory of Linear Waves in an Elastic Layer and Its Relation to Microstructured Solids

  • Manfred BraunEmail author
  • Merle Randrüüt
Part of the Mathematics of Planet Earth book series (MPE, volume 6)


For the dynamics of an infinite elastic layer or plate , an approximate model is introduced. The governing equations are obtained by Lagrange’s method with suitable restrictions imposed to the transverse deformation of the layer. It turns out that the resulting equations, in the one-dimensional case, are of the same kind as those introduced by Engelbrecht and Pastrone (Proc. Estonian Acad. Sci. Phys. Math. 52(1), 12–20 (2003)) for the dynamics of a microstructured solid in the sense of Mindlin. These equations cover a wide range of possible applications, but usually, there is no validation as to what extent they describe the actual behaviour in reality. The approximate layer model is governed by the same system of equations. It may serve as a test that allows to compare the approximation with the known results of the exact theory. To this end, the propagation of harmonic waves in the layer is considered. The dispersion curves of the approximate and exact theories are compared for some values of Poisson’s ratio . It is shown that the main, acoustical branch fits well for wavelengths above three times the plate thickness. The optical branches of the approximate model deviate from their exact counterparts but exhibit qualitatively the same behaviour.


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  1. Achenbach, J.D.: Wave Propagation in Elastic Solids. North-Holland, Amsterdam (1973)zbMATHGoogle Scholar
  2. Brillouin, L.: Wave Propagation in Periodic Structures. Electric Filters and Crystal Lattices. Second edition, with corrections and additions, Dover Publications, New York (1953)Google Scholar
  3. Engelbrecht, J., Pastrone, F.: Waves in microstructured solids with strong nonlinearities in microscale. Proc. Estonian Acad. Sci. Phys. Math. 52(1), 12–20 (2003)MathSciNetzbMATHGoogle Scholar
  4. Engelbrecht, J., Berezovski, A., Pastrone, M., Braun, M.: Waves in microstructured materials and dispersion. Phil. Mag. 85(33-35), 4127–4141 (2005). CrossRefGoogle Scholar
  5. Engelbrecht, J., Pastrone, F., Braun, M., Berezovski, A.: Hierarchies of waves in nonclassical materials. In: Delsanto, P.P. (ed.), The Universality of Nonclassical Nonlinearity: Applications to Non-destructive Evaluations and Ultrasonics, pp. 29–47. Springer, New York (2006). CrossRefGoogle Scholar
  6. Graff, K.F.: Wave Motion in Elastic Solids. Dover Publications, New York (1991)zbMATHGoogle Scholar
  7. Lamb, H.: On the flexure of an elastic plate. Proc. Lond. Math. Soc. 21(1), 70–90 (1889). MathSciNetCrossRefGoogle Scholar
  8. Lamb, H.: On waves in an elastic plate. Proc. Roy. Soc. A 93(648), 114–128 (1917)CrossRefGoogle Scholar
  9. Miklowitz, J.: The Theory of Elastic Waves and Wave Guides. North-Holland, Amsterdam (1978)zbMATHGoogle Scholar
  10. Mindlin, R.D.: An Introduction to the Mathematical Theory of Vibrations of Elastic Plates. J. Yang (ed.) World Scientific, New Jersey (2006). CrossRefGoogle Scholar
  11. Mindlin, R.D.: Waves and vibrations in isotropic elastic plates. In: Goodier, J.N., Hoff, N.J. (eds.) Structural Mechanics, pp. 199–232. Pergamon Press, New York (1960)Google Scholar
  12. Mindlin, R.D.: Microstructure in linear elasticity. Arch. Ration. Mech. Anal. 16(1), 51–78 (1964). CrossRefGoogle Scholar
  13. Mindlin, R.D., Medick, M.A.: Extensional vibrations of elastic plates. Report, Columbia University, Department of Civil Engineering and Engineering Mechanics, New York (1958).
  14. Mindlin, R.D., Medick, M.A.: Extensional vibrations of elastic plates. J. Appl. Mech. 26, 561–569 (1959)MathSciNetGoogle Scholar
  15. Onoe, M., NcNiven, H.D., Mindlin, R.D.: Dispersion of axially symmetric waves in elastic rods. J. Appl. Mech. 29(4), 729–734 (1962). CrossRefGoogle Scholar
  16. Peets, T.: Dispersion Analysis of Wave Motion in Microstructured Solids. Ph.D. Thesis, Tallinn University of Technology, Tallinn (2011)Google Scholar
  17. Peets, T., Randrüüt, M., Engelbrecht, J.: On modelling dispersion in microstructured solids. Wave Motion 45(4), 471–480 (2008). MathSciNetCrossRefGoogle Scholar
  18. Pochhammer, L.A.: Über die Fortpflanzungsgeschwindigkeiten kleiner Schwingungen in einem unbegrenzten isotropen Kreiscylinder. J. Reine Angew. Math. 81, 324–336 (1876)MathSciNetzbMATHGoogle Scholar
  19. Randrüüt, M.: Wave Propagation in Microstructured Solids: Solitary and Periodic Waves. Ph.D. Thesis, Tallinn University of Technology, Tallinn (2010)Google Scholar
  20. [Lord] Rayleigh: On waves propagated along plane surfaces of an elastic solid. Proc. Lond. Math. Soc. 17(1), 4–11 (1885). MathSciNetCrossRefGoogle Scholar
  21. [Lord] Rayleigh: On the free vibrations of an infinite plate of homogeneous isotropic elastic matter. Proc. Lond. Math. Soc. 20(1), 225–234 (1889). MathSciNetCrossRefGoogle Scholar
  22. Rose, J.L.: Ultrasonic Guided Waves in Solid Media. Cambridge University Press, Cambridge (2014).

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Lehrstuhl für Mechanik und RobotikUniversität Duisburg–EssenDuisburgGermany

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