Edge resonance in an elastic semi-strip

Roitberg, Inna Ya.

doi:10.1093/qjmam/51.1.1

Cited by 56 publications

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“…The associated angular frequency ω e would be the corresponding eigenvalue. It is worth reiterating that the existence of such mode has been proven by Roitberg et al (1998) for the case of zero Poisson's ratio.…”

Section: Article Submitted To Royal Societymentioning

confidence: 80%

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Eigenvalue of a semi-infinite elastic strip

2006

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A semi-infinite elastic strip, subjected to traction free boundary conditions, is studied in the context of in-plane stationary vibrations. By using normal (RayleighLamb) mode expansion the problem of existence of the strip eigenmode is reformulated in terms of the linear dependence within infinite system of normal modes. The concept of Gram's determinant is used to introduce a generalized criterion of linear dependence, which is valid for infinite systems of modes and complex frequencies. Using this criterion, it is demonstrated numerically that in addition to the edge resonance for the Poisson ratio ν = 0, there exists another value of ν ≈ 0.22475 associated with an undamped resonance. This resonance is best explained physically by the orthogonality between the edge mode and the first Lamé mode. A semi-analytical proof for the existence of the edge resonance is then presented for both described cases with the help of the augmented scattering matrix formalism.

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Section: Article Submitted To Royal Societymentioning

confidence: 80%

“…The proof for the existence of pure eigenmode by Roitberg et al (1998) has been enabled by the special symmetry of the governing elasticity operator for zero Poisson's ratio. The crucial simplification cames from the fact that when ν = 0 the fundamental mode of an elastic strip becomes non-dispersive, with α 1 = ±K/ √ 2.…”

Section: (B) Orthogonalitymentioning

confidence: 99%

Eigenvalue of a semi-infinite elastic strip

2006

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“…This paper is related to an extensive series of works on trapped modes in perturbed quantum [1]- [5], acoustic [6]- [10], and elastic waveguides [11]- [15].…”

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confidence: 99%

Trapped modes in an elastic plate with a hole

Förster

Weidl

2012

St. Petersburg Math. J.

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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.

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“…The basic idea of the method of [7] consists precisely in creating a positive threshold, and the subsequent study of the discrete spectrum of the corresponding boundary-value problem formed artificially (see also [9,13] in connection with problems of elasticity theory).…”

Section: Branching Many-dimensional Waveguide; the Dirichlet Problemmentioning

confidence: 99%

Discrete spectrum of cranked, branching, and periodic waveguides

Назаров

2012

St. Petersburg Math. J.

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Abstract. The variational method is applied to the study of the spectrum of the Laplace operator with mixed boundary conditions and with Dirichlet conditions in planar or multidimensional domains (waveguides) with cylindrical or periodic exits to infinity. The planar waveguides of constant width are discussed completely, such as cranked, broken, smoothly bent, or branching waveguides. For them, the existence of eigenvalues below the continuous spectrum threshold is established. A similar result is obtained for the multidimensional cranked and branching waveguides, and also for some periodic ones. Several open questions are stated; in particular, they concern problems with Neumann boundary conditions, full multiplicity of the discrete spectrum, and planar waveguides with piecewise constant boundary.

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Edge resonance in an elastic semi-strip

Cited by 56 publications

References 0 publications

Eigenvalue of a semi-infinite elastic strip

Eigenvalue of a semi-infinite elastic strip

Trapped modes in an elastic plate with a hole

Discrete spectrum of cranked, branching, and periodic waveguides

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