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Nonstandard Integral Equations for the Harmonic Oscillations of Thin Plates

In: Integral Methods in Science and Engineering

Author

Listed:
  • G. R. Thomson

    (A.C.C.A.)

  • C. Constanda

    (The University of Tulsa)

  • D. R. Doty

    (The University of Tulsa)

Abstract

In [ThCo97] and [ThCo09a] the problems of high frequency harmonic oscillations of thin elastic plates with Dirichlet, Neumann, and Robin boundary conditions were investigated by means of a classical indirect boundary integral equation method. This method was not entirely satisfactory since, for the exterior problems, it produced integral equations with nonunique solutions for certain values of the oscillation frequency, although the actual boundary value problems always had at most one solution. When a direct method was employed (see [ThCo99] and [ThCo10]), it was found that uniqueness could be guaranteed only if a pair of integral equations was derived for each exterior problem. The classical techniques did not seem to offer any answer to the question of whether the solutions could be obtained from single, uniquely solvable equations. Below we propose a modified indirect boundary integral equation method, based on constructing a matrix of fundamental solutions satisfying a dissipative (or Robin-type) condition on a curve interior to the scatterer, which answers the above question in the affirmative.

Suggested Citation

  • G. R. Thomson & C. Constanda & D. R. Doty, 2013. "Nonstandard Integral Equations for the Harmonic Oscillations of Thin Plates," Springer Books, in: Christian Constanda & Bardo E.J. Bodmann & Haroldo F. de Campos Velho (ed.), Integral Methods in Science and Engineering, edition 127, chapter 0, pages 311-328, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4614-7828-7_22
    DOI: 10.1007/978-1-4614-7828-7_22
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