<i>Theory of Elasticity of an Anisotropic Elastic Body</i>

Lekhnitskii, S.G.; Fern, pN M Ferreira J M; Brandstatter, J.J; Dill, Ellis H.

doi:10.1063/1.3051394

Cited by 1,026 publications

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“…Figures 5 and 6). Viewing the domain Ω as a solid, we consider the spectral problem of elasticity theory in a matrix form (see [5,6]):…”

Section: Setting Of the Problem Let π Be A Domain In Euclidean Spacementioning

confidence: 99%

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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“…Figures 5 and 6). Viewing the domain Ω as a solid, we consider the spectral problem of elasticity theory in a matrix form (see [5,6]):…”

Section: Setting Of the Problem Let π Be A Domain In Euclidean Spacementioning

confidence: 99%

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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“…Inserting the results (5) into the original ansatz (3) yields the asymptotic representation for the vector displacement field (6) where again f>" = f>"(i) is any unit vector fixed with respect to the local ( er, e0, ez) basis. Note that the phase velocity Cv(f>") is not spatially varying, although the p" · -; term is.…”

Section: Ray Theorymentioning

confidence: 99%

“…The constants a;j and Gap appearing in the above field equations come from the constitutive law as follows, see for example [5] or [6]. The a;j give the normal polar stresses in terms of the normal polar strains, and thus are the components of a matrix whose inverse a;:/ = S;j have components Sn = 1/ En sl2 = -VrfJ/ Eo = -Vorl Er = s21' etc.…”

Section: Field Equationsmentioning

confidence: 99%

Pulse Propagation in Cylindrically Wound Thick Composite Specimens

Budreck

1990

Review of Progress in Quantitative Nondestructive Evaluation

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“…To make this sure, however, we must check other necessary conditions still not considered. So we review here the full set of conditions to characterize the elastic transverse isot ropy with respect to z axis (8,9).…”

Section: Transverse Isotropymentioning

confidence: 99%

Studies on the Handling of Urethane Foam. (Part 9). Elastic Anisotropy in Flexible Polyurethane Foam.

Omura

1993

Sen-i Gakkaishi

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The anisotropic elasticity of some flexible foamed materials of polyurethane was investigated by experiment. The data of compression and shearing in various directions suggest that the tested materials are transversely isotropic around the principal direction, i.e., the direction in which the constituent foams of elongated shapes are oriented on the average. Based on this and another empirical symmetry, a formula was derived, by which the compression modulus in any direction can be estimated from the compression moduli in two axes, the principal one and any normal to it.

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Theory of Elasticity of an Anisotropic Elastic Body

Cited by 1,026 publications

References 0 publications

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

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

Pulse Propagation in Cylindrically Wound Thick Composite Specimens

Studies on the Handling of Urethane Foam. (Part 9). Elastic Anisotropy in Flexible Polyurethane Foam.

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