General Transmission Problems in the Theory of Elastic Oscillations of Anisotropic Bodies (Mixed Interface Problems)

Jentsch, Lothar; Natroshvili, David; Wendland, Wolfgang L.

doi:10.1006/jmaa.1999.6360

Cited by 14 publications

(19 citation statements)

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“…This is certainly the case for such problems as transmission problems, the elastic echo problem, the asymptotic repartition of energy and inverse problems. It would be also interesting to extend our work to the case of anisotropic media, for which we can beneÿt of the fundamental contributions made by Jentsch-Natroshvili [17][18][19] and the georgian school.…”

Section: Resultsmentioning

confidence: 99%

The scattering theory of C. Wilcox in elasticity

Mabrouk¹,

Helali

2002

Math Methods in App Sciences

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SUMMARYWe extend the abstract time-dependent scattering theory of C.H. Wilcox to the case of elastic waves. Most of the results are proved with the minimal assumption that the obstacle satisÿes the energy local compactness condition (ELC Notations: Let us ÿrst explain the notations which are used throughout the text. The space R 3 is endowed with the canonical basis {e 1 ; e 2 ; e 3 }, the origin O and the coordinate system (x 1 ; x 2 ; x 3 ). The canonical scalar product of R 3 is denoted by a dot. By a '·' we mean the contracted product of tensors. Vectors and matrices are typed in bold style.I is the identity map or matrix or tensor. When we want more precision, we use the notation id X for the identity map in the space X . For a matrix or a map A; t A means the transpose. For any open set ⊆ R 3 , we introduce the classical spaces of scalar complex functions: 3 . * = − curlcurl + ( + ) graddiv is the LamÃ e operator, ; are the LamÃ e constants, c 1 = + 2 ; c 2 = √ are the speeds of the longitudinal and shear waves and we set c jl = c j =c l ; k j (!) = c j =! are wave numbers. T(u; n) = 2 (9u=9n) + n div u + n ∧ curl u is the traction vector at the boundary point where the normal is n.

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Section: Resultsmentioning

confidence: 99%

The scattering theory of C. Wilcox in elasticity

Mabrouk¹,

Helali

2002

Math Methods in App Sciences

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“…Comparing (18), (19) shows that the left-hand side of (19) is real-valued and real-analytic in R 3 \0. Similar to the fundamental solution, the multipolar series (19) has the following properties: Proposition 4.3.…”

Section: Multipolar Expansions For Singular Operatorsmentioning

confidence: 99%

“…Similar to the fundamental solution, the multipolar series (19) has the following properties: Proposition 4.3. At any fixed regular point y ∈ ∂Ω, the series on the right-hand side of (19) converges to the distribution defined by its kernel V: a) in the weak topology in…”

Section: Multipolar Expansions For Singular Operatorsmentioning

confidence: 99%

“…Routine procedure, similar to one exploited for obtaining (17), (19), produces the kernel W in the form of multipolar series:…”

Section: Multipolar Expansions For Hyper-singular Operatorsmentioning

confidence: 99%

“…where in contrast to (17), (19), the series on the right-hand side of (24) contains harmonics of the even order only, because of positive homogeneity of the amplitude with respect to the ξ-variable. Matrix coefficients W nk are determined by integration of the amplitude (23) on the unit sphere in R 3 similarly to obtaining V nk .…”

Section: Multipolar Expansions For Hyper-singular Operatorsmentioning

confidence: 99%

See 2 more Smart Citations

Fundamental and singular solutions of Lamé equations for media with arbitrary elastic anisotropy

Kuznetsov

2005

Quart. Appl. Math.

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Non-local approach in mathematical problems of fluid-structure interaction

Jentsch

Natroshvili

1999

Math. Meth. Appl. Sci.

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Three‐dimensional mathematical problems of interaction between elastic and scalar oscillation fields are investigated. An elastic field is to be defined in a bounded inhomogeneous anisotropic body occupying the domain Ω¯1⊂ℝ3 while a physical (acoustic) scalar field is to be defined in the exterior domain Ω¯2=ℝ3\Ω1 which is filled up also by an anisotropic (fluid) medium. These two fields satisfy the governing equations of steady‐state oscillations in the corresponding domains together with special kinematic and dynamic transmission conditions on the interface ∂Ω1. The problems are studied by the so‐called non‐local approach, which is the coupling of the boundary integral equation method (in the unbounded domain) and the functional‐variational method (in the bounded domain). The uniqueness and existence theorems are proved and the regularity of solutions are established with the help of the corresponding Steklov–Poincaré type operators and on the basis of the Gårding inequality and the Lax–Milgram theorem. In particular, it is shown that the physical fluid–solid acoustic interaction problem is solvable for arbitrary values of the frequency parameter. Copyright © 1999 John Wiley & Sons, Ltd.

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General Transmission Problems in the Theory of Elastic Oscillations of Anisotropic Bodies (Mixed Interface Problems)

Cited by 14 publications

References 3 publications

The scattering theory of C. Wilcox in elasticity

The scattering theory of C. Wilcox in elasticity

Fundamental and singular solutions of Lamé equations for media with arbitrary elastic anisotropy

Non-local approach in mathematical problems of fluid-structure interaction

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