An Activated Elastic Bar: Effective Properties

  • Konstantin A. Lurie
Part of the Advances in Mechanics and Mathematics book series (AMMA, volume 15)


This chapter mostly concerns the wave propagation through a special type of DM—the first rank laminates in 1D + time. Such laminates are classified into regular and irregular, depending on the characteristic pattern produced by material and structural parameters. For regular laminates, the characteristics of the same family do not collide, for irregular laminates, they do. Homogenization applies to regular activated laminates, and their effective properties are discussed in detail. Special attention is paid to the analysis of energy balance through homogenization.


  1. 1.
    Bakhvalov, N.S., Panasenko, G.P.: Homogenization: Averaging Processes in Periodic Media - Mathematical Problem in the Mechanics of Composite Materials, p. 408. Kluwer, Dordrecht (1989)Google Scholar
  2. 2.
    Casaldo-Diaz, J., Couce-Calvo, J., Maestre, F., Martin-Gomez, J.D.: Homogenization and corrector for the wave equation with discontinuous coefficients in time. J. Math. Anal. Appl. 379 (2), 664–681 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Dong, Q.L., Cao, L.Q.: Multiscale asymptotic expansions and numerical algorithms for the wave equations of second order with rapidly oscillating coefficients. Appl. Numer. Math. 59(12), 3008–3032 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Gel’fand, I.M.: Some problems of the theory of quasilinear equations. Office of Technical Services, U. S. Department of Commerce, Washington (1960). Translation of an article in Uspekhi matematicheskikh nauk, vol. XIV, no. 2, 87–158 (1959)Google Scholar
  5. 5.
    Loitsianskii, L.G., Lurie, A.I.: A Course of Theoretical Mechanics (in Russian), vol. 2, 7th edn., p.719. Drofa, Moscow (2005)Google Scholar
  6. 6.
    Lurie, K.A.: On homogenization of activated laminaes in 1D-space and time. Z. f\(\ddot{u}\) r Angew. Math. Mecf. 89(4), 333–340 (2009)Google Scholar
  7. 7.
    Shui, L.-Q., Yuz, Z.-F., Liu, Y.-S., Liu, Q.-C., Guo, J.-J.: One-dimensional linear elastic waves at moving property interface. Wave Motion 51, 1179–1192 (2014)CrossRefGoogle Scholar
  8. 8.
    Shui, L.-Q., Yuz, Z.-F., Liu, Y.-S., Liu, Q.-C., Guo, J.-J., He, X.-D.: Novel composites with asymmetrical elastic wave properties. Compos. Sci. Technol. 113, 19–30 (2015)CrossRefGoogle Scholar
  9. 9.
    To, H.T.: Homogenization of dynamic laminates. J. Math. Anal. Appl. 354(2), 518–538 (2009)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Konstantin A. Lurie
    • 1
  1. 1.Department of Mathematical SciencesWorcester Polytechnic InstituteWorcesterUSA

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