Boundary-value problems for the system of elasticity theory in unbounded domains. Korn's inequalities

Кондратьев, В А; Oleinik, O A

doi:10.1070/rm1988v043n05abeh001945

Cited by 220 publications

(185 citation statements)

References 6 publications

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“…Moreover, the set ϑ j is star-shaped with respect to a ball of radius j −2 with a common for all j constant . Thus, a result in [36] (see also [15, Ch. 2, §1]) assures the Korn inequality…”

Section: Proposition 2 the Anisotropic Weighted Inequalitymentioning

confidence: 99%

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Waves and Radiation Conditions in a Cuspidal Sharpening of Elastic Bodies

Kozlov

Назаров

2017

J Elast

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Elastic bodies with cuspidal singularities at the surface are known to support wave processes in a finite volume which lead to absorption of elastic and acoustic oscillations (this effect is recognized in the engineering literature as Vibration Black Holes). With a simple argument we will give the complete description of the phenomenon of wave propagation towards the tip of a three-dimensional cusp and provide a formulation of the Mandelstam (energy) radiation conditions based on the calculation of the Umov-Poynting vector of energy transfer. Outside thresholds, in particular, above the lower bound of the continuous spectrum, these conditions coincide with ones due to the traditional Sommerfeld principle but also work at the threshold frequencies where the latter principle cannot indicate the direction of wave propagation. The energy radiation conditions supply the problem with a Fredholm operator of index zero so that a solution is determined up to a trapped mode with a finite elastic energy and exists provided the external loading is orthogonal to these modes. An example of a symmetric cuspidal body is presented which supports an infinite sequence of eigenfrequencies embedded into the continuous spectrum and the corresponding trapped modes with the exponential decay at the cusp tip. We also determine (unitary and symmetric) scattering matrices of two types and derive a criterion for the existence of a trapped mode with the power-low behavior near the tip.

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Section: Proposition 2 the Anisotropic Weighted Inequalitymentioning

confidence: 99%

“…The variational formulation of the problem (1.10), (1.11) runs as follows: to find non-trivial vector function u ∈ H 1 (Ω) 3 If the domain Ω is Lipschitz, the Korn inequality, see, e.g., [36],…”

Section: Energy Space and Korn's Inequalitymentioning

confidence: 99%

Waves and Radiation Conditions in a Cuspidal Sharpening of Elastic Bodies

Kozlov

Назаров

2017

J Elast

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“…Largely, the dependence of the Korn constant c Ξ on the domain Ξ is not known; in [37] it was proved only that if a body Ξ is star-shaped relative to the ball B R , then the constant c in the inequality…”

Section: ])mentioning

confidence: 99%

“…(see [36,37,6], and also relation (3.1)). In the second term on the left, we keep only the derivative ∂ z U and perform differentiation and integration with respect to z ∈ (0, H).…”

Section: Spectrum Of the Pencil On The Periodicity Cellmentioning

confidence: 99%

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Gap opening in the essential spectrum of the elasticity theory problem in a periodic half-layer

Назаров

2010

St. Petersburg Math. J.

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Abstract. Rayleigh waves are studied in an elastic half-layer with a periodic end and rigidly clamped faces. It is established that the essential spectrum of the corresponding problem of elasticity theory has a band structure, and an example of a waveguide is presented in which a gap opens in the essential spectrum; i.e., an interval arises that contains points of an at most discrete spectrum. §1. Introduction 1. Preamble. In the case of a homogeneous isotropic elastic half-space, surface waves were discovered by Lord Rayleigh [1], and since then many investigations devoted to similar effects have appeared (see a survey of modern literature in [2], and also the paper [3], which is absent in [2]). A Rayleigh wave is a plane wave of the forma vector-valued factor U (z) decaying exponentially as z = x 3 → −∞. The arising of such waves explains specific wave processes in elastic bodies.The wave number k ∈ R + = [0, +∞) determines a frequency cutoff ω † (k) above which, i.e., for ω ≥ ω † (k), the wave (1.1) exists necessarily. In the present paper we deal with a problem related to a similar phenomenon. Namely, we study an elastic, but not necessarily homogeneous and isotropic cushion Ω 0 having the form of a half-layer with a periodic end and rigidly clamped side faces (see Figure 1, where the clamped surface is shadowed). Some Rayleigh waves decaying exponentially as z → −∞ can propagate along the end of the cushion, and if Ω 0 is a cylinder ∆ × R, then we have a single cutoff ω † > 0; i.e., the corresponding operator of the elasticity theory system acquires a continuous spectrum [ω † , +∞). Our main goal in this paper is to show that, in the periodic case, a gap can open in the essential spectrum; i.e., an interval can exist the ends of which belong to the continuous spectrum, but inside which only points of the discrete spectrum may occur. Some of the results were announced earlier in [4].

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Korn's Inequalities for Junctions of Spatial Bodies and Thin Rods

Назаров¹

1997

Math. Meth. Appl. Sci.

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Boundary-value problems for the system of elasticity theory in unbounded domains. Korn's inequalities

Cited by 220 publications

References 6 publications

Waves and Radiation Conditions in a Cuspidal Sharpening of Elastic Bodies

Waves and Radiation Conditions in a Cuspidal Sharpening of Elastic Bodies

Gap opening in the essential spectrum of the elasticity theory problem in a periodic half-layer

Korn's Inequalities for Junctions of Spatial Bodies and Thin Rods

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