Trapped modes in an elastic plate with a hole

Abstract

Abstract. For an infinite linear elastic plate with stress-free boundary, the trapped modes arising around holes in the plate are investigated. These are L 2 -eigenvalues of the elastostatic operator in the punched plate subject to Neumann type stress-free boundary conditions at the surface of the hole. It is proved that the perturbation gives rise to infinitely many eigenvalues embedded into the essential spectrum. The eigenvalues accumulate to a positive threshold. An estimate of the accumulation rate is given. §1. Introduction We consider a homogeneous and isotropic linear elastic medium in the domainThen the quadratic forminduces the elastostatic operator A for materials with zero Poisson ratio; A corresponds to the differential expressionwith stress-free (Neumann-type) boundary conditions. 1 The operator A has purely absolutely continuous spectrum filling the nonnegative half-line. Now, consider a punched plate Ω c = G \ Ω with a hole Ω = Ω 0 × J, where Ω 0 ⊂ R 2 is a bounded Lipschitz domain. Let A Ω c be the elastostatic operator corresponding to the differential expression (1.2) on the outer domain Ω c subject to stress-free (Neumanntype) boundary conditions. This geometric perturbation does not change the location of the essential spectrum, but it gives rise to a somewhat unexpected trapping effect. As the main result of this paper we prove the existence of infinitely many eigenvalues {ν k } k∈N of A Ω c , embedded into the essential spectrum, which accumulate at a certain threshold Λ > 0, and we compute the accumulation rate of these trapped modes. We establish the formula2010 Mathematics Subject Classification. Primary 74B05. Key words and phrases. Elasticity operator, trapped modes. 1 Here we have chosen a suitable set of units such that Young's modulus E fulfills E = 2. Moreover, we point out that the special choice of zero Poisson ratio is essential for the results of this paper.

On Well-Posedness of Scattering Problems in a Kirchhoff--Love Infinite Plate

“…As our uniqueness result excludes a possible countable set of frequencies, it is natural to wonder if such exceptional frequencies actually exist, that is if trapped modes can really occur for the free plate. Let us mention some contributions which prove existence of trapped modes in situations which are close to ours, for example the case of a simply supported strip in the presence of a free obstacle [16] or the case of an infinite thick plate in the presence of a free obstacle and ν = 0 [9]. It is interesting to note that for a circular hole, the trapped modes disappear when the thickness of such thick plate tends to 0, in view of Theorem 9.…”

“…Hence, by defining u pr and u ev by ( 8), we see that the equation ∆ 2 u − k 4 u = 0 is equivalent to the decomposition u = u pr + u ev where u pr and u ev satisfy (9). In order to obtain the series expansions (7) of u pr and u ev , we consider their respective Fourier series with respect to θ.…”

Embedded Eigenvalues for an Elastic Strip with Cracks

“…chosen similar to (3.4), see [3] or [5] for further details. This introduces more freedom for the choice of the generalised eigenfunction φ.…”

Eigenvalue asymptotics for an elastic strip and an elastic plate with a crack

“…The symmetry decomposition of the elasticity operator for vanishing Poisson's ratio goes back to [2], where the existence of embedded eigenvalues was shown for the elastic semistrip. These results have been generalised to the case of an elastic strip or plate with zero Poisson's ratio where material properties are perturbed [3,4], where a hole is cut out [5], or strips and plates having a crack [1]. In [3,4] an asymptotic formula for the convergence of the eigenvalues to some spectral threshold was proved.…”

“…(4) Let ω = 0, ξ = 0: In this case all solutions have to be linear in both components. (5) Let ω = 0, ξ = 0: Then we have β = γ = iξ and we get…”