Gavin R. Thomson

A matrix of fundamental solutions in the theory of plate oscillations

Abstract A matrix of fundamental solutions is constructed for the operator governing the high-frequency oscillations of a plate with transverse shear deformation. Important properties of the corresponding wave functions and layer potentials are discussed, as an essential step in the development of numerical boundary element methods.

Uniqueness of solution for the Robin problem in high-frequency vibrations of elastic plates

The interior and exterior Robin boundary value problems are considered for the high-frequency harmonic oscillations of a plate with transverse shear deformation, and the uniqueness of their solutions is investigated in terms of eigenfrequencies and far-field radiation conditions.

The Transmission Problem

We now examine the transmission problem (also referred to as the inclusion or boundary–contact problem) in which the plate consists of two elastic materials in separate subdomains with conditions specified on the interface.

The direct method for harmonic oscillations of elastic plates with Robin boundary conditions

The interior and exterior Robin boundary value problems for the model of flexural vibrations of plates with transverse shear deformation are solved by means of layer potentials in conjunction with the integral representation formulas for the solutions.

Scattering of High-Frequency Flexural Oscillations in Thin Plates

The direct boundary equation method is used to find representations for the solutions of the exterior Dirichlet and Neumann problems in the case of high-frequency flexural oscillations in plates with transverse shear deformation. The method leads to integral equations with unique solutions for all frequencies.

Nonstandard Integral Equations for the Harmonic Oscillations of Thin Plates

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.

The Eigenfrequencies of the Dirichlet and Neumann Problems for an Oscillating Finite Plate

The Green’s tensors are constructed and the existence of discrete real spectra is proved for the displacement and traction boundary value problems in the case of a finite elastic plate with transverse shear deformation undergoing high-frequency vibrations.

Uniqueness of analytic solutions for stationary plate oscillations in an annulus

Abstract The equations governing the harmonic oscillations of a plate with transverse shear deformation are considered in an annular domain. It is shown that under nonstandard boundary conditions where both the displacements and tractions are zero on the internal boundary curve, the corresponding analytic solution is zero in the entire domain. This property is then used to prove that a boundary value problem with Dirichlet or Neumann conditions on the external boundary and Robin conditions on the internal boundary has at most one analytic solution.

Integral equations of the first kind in the theory of oscillating plates

A modified matrix of fundamental solutions is used to derive and solve first-kind integral equations for the problem of high-frequency harmonic oscillations of an infinite elastic plate with a hole when Dirichlet or Neumann conditions are prescribed on the boundary curve.

Smoothness properties of Newtonian potentials in the study of elastic plates

The Hölder continuity and differentiability are investigated of Newtonian potentials arising in the theory of bending of elastic plates with transverse shear deformation. These properties play an essential role in the study by means of boundary integral equation techniques of boundary value problems for the equilibrium and harmonic oscillation states, and in the construction of associated boundary element methods.