Nonlinear Spectral Finite Element Model for Analysis of Wave Propagation in Solid with Internal Friction and Dissipation

  • D. Roy Mahapatra
  • S. Gopalakrishnan
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2668)


A geometrically non-linear Spectral Finite Flement Model (SFEM) including hysteresis, internal friction and viscous dissipation in the material is developed and is used to study non-linear dissipative wave propagation in elementary rod under high amplitude pulse loading. The solution to non-linear dispersive dissipative equation constitutes one of the most difficult problems in contemporary mathematical physics. Although intensive research towards analytical developments are on, a general purpose cumputational discretization technique for complex applications, such as finite element, but with all the features of travelling wave (TW) solutions is not available. The present effort is aimed towards development of such computational framework. Fast Fourier Transform (FFT) is used for transformation between temporal and frequency domain. SFEM for the associated linear system is used as initial state for vector iteration. General purpose procedure involving matrix computation and frequency domain convolution operators are used and implemented in a finite element code. Convergnence of the spectral residual force vector ensures the solution accuracy. Important conclusions are drawn from the numerical simulations. Future course of developments are highlighted.


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  1. 1.
    Samsonov, A.M.: Strain Solitons in Solids, Monographs and Surveys in Pure and Applied Mathematics. 117 (2001) Chapman & Hall/CRCGoogle Scholar
  2. 2.
    Clarkson, P.A. and Kruskal, M.D.: New similarity reduction of Boussinesq equation. J. Mathematical Physics 30(10) (1989) 2201–2213MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Clarkson, P.A. and Winternitz, P.: Physica D 49 (1991) 257MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Pupkins, D.S. and Atluri, S.N.: Non-linear analysis of wave propagation using transform methods. Computational Mechanics 11 (1993) 207–227CrossRefGoogle Scholar
  5. 5.
    Roy Mahapatra, D. and Gopalakrishnan, S.: A spectral finite element model for analysis of axial-flexural-shear coupled wave propagation in laminated composite beams. Composite Structures 59(1) (2003) 67–88CrossRefGoogle Scholar
  6. 6.
    Balachandran, B. and Khan, K.A.: Spectral analysis of nonlinear interactions. Mechanical Systems and Signal Processing 10(6), (1996) 711–727CrossRefGoogle Scholar
  7. 7.
    Rushchitsky, J.J.: Interaction of waves in solid mixtures. Appl. Mech. Rev. 52(2) (1999) 35–74CrossRefGoogle Scholar
  8. 8.
    Vollmann, J. and Dual, J.: High-resolution analysis of the complex wave spectrum in a cylindrical shell containing and viscoelastic medium. Part I. Theory and experimental results. J. Acoust. Soc. America, 102(2) (1997) 896–920CrossRefGoogle Scholar
  9. 9.
    McDanel, J.G., Dupont, P. and Salvino, L.: A wave approach to estimating frequency-dependent damping under transient loading. J. Sound and Vibration, 231(2) (2000) 433–449CrossRefGoogle Scholar
  10. 10.
    Zakharov, V.E. and Shabat, A.B.: Exact theory of two-dimensional focusing and one-dimensional self-modulation in non-linear media. Soviet Physics, JETP, 34 (1972) 62–69MathSciNetGoogle Scholar
  11. 11.
    Ostrovsky, L.A. and Potapov, A.I.: Modulated Waves. Johns Hopkins University Press, Washington, 1999MATHGoogle Scholar
  12. 12.
    Doyle, J.F.: Wave Propagation in Structures. Springer-Verlag, 1997Google Scholar
  13. 13.
    Roy Mahapatra, D. and Gopalakrishnan, S.: A spectral finite element for analysis of wave propagation in uniform composite tubes, J. of Sound and Vibration (in press) 2003Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • D. Roy Mahapatra
    • 1
  • S. Gopalakrishnan
    • 1
  1. 1.Department of Aerospace EngineeringIndian Institute of ScienceBangaloreINDIA

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