Mixed interface problems for anisotropic elastic bodies

Natroshvili, David

doi:10.1007/bf02262859

Cited by 11 publications

(9 citation statements)

References 5 publications

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“…We note that by the same methods a wide class of three-dimensional solid-solid interaction problems has been investigated in [36,47,28,32,33] (see also the references therein).…”

Section: Communicated By E Meistermentioning

confidence: 99%

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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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“…We note that by the same methods a wide class of three-dimensional solid-solid interaction problems has been investigated in [36,47,28,32,33] (see also the references therein).…”

Section: Communicated By E Meistermentioning

confidence: 99%

“…We can apply results and arguments of [10,9,32,47,12,13,33] to formulate the above lemmata in the Sobolev ¼Q N , Bessel-potential HQ N and Bessov BQ NO spaces. Namely, the following propositions hold.…”

Section: Integral Representation Formulae and Potentialsmentioning

confidence: 99%

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

Jentsch

Natroshvili

1999

Math. Meth. Appl. Sci.

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“…By the equality (2.12) the vector [¹u]\ 1 is defined correctly (cf. [5,26]). For regular vectors f, g3 C(S)…”

Section: 4mentioning

confidence: 99%

“…Due to Lemma 3.2 (see (3.20)) the oscillation potentials (5.2) and (5.3) have the same regularity properties as the corresponding potentials of statics constructed by the fundamental matrix (x) (see [25,26]). For the readers' convenience we formulate them in the form of the following theorems.…”

Section: Uniqueness Theoremsmentioning

confidence: 99%

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Boundary Integral Equation Method in the Steady State Oscillation Problems for Anisotropic Bodies

Natroshvili¹

1997

Math. Meth. Appl. Sci.

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The three‐dimensional steady state oscillation problems of the elasticity theory for homogeneous anisotropic bodies are studied. By means of the limiting absortion principle the fundamental matrices maximally decaying at infinity are constructed and the generalized Sommerfeld–Kupradze type radiation conditions are formulated. Special functional spaces are introduced in which the basic and mixed exterior boundary value problems of the steady state oscillation theory have unique solutions for arbitrary values of the oscillation parameter. Existence theorems are proved by reduction of the original boundary value problems to equivalent boundary integral (pseudodifferential) equations. © 1997 by B.G. Teubner Stuttgart‐John Wiley & Sons, Ltd.

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Non ‐ Classical Mixed Interface Problems for Anisotropic Bodies

Jentsch

Natroshvili

1996

Mathematische Nachrichten

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The paper deals with the three-dimensional mathematical problems of the elasticity theory of anisotropic piece-wise homogeneous bodies. Non -classical mixed type boundary value problems are studied when on one part (Sl) of the interface the rigid contact conditions (jumps of displacement and stress vectors) are given, while conditions of the non-classical interface Problem ' H or Problem Q are imposed on the remaining part (Sz) of the interface, i. e., in both cases jumps of the normal components of displacement and stress vectors are known on SZ and, in addition, in the first one (Problem 'H) the tangent components of the displacement vector and in the second one (Problem Q) the tangent components of the stress vector are given from the both sides on SZ.The investigation is carried out by the potential method and the theory of pseudodifferential equations on manifolds with boundary.

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Mixed interface problems for anisotropic elastic bodies

Cited by 11 publications

References 5 publications

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

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

Boundary Integral Equation Method in the Steady State Oscillation Problems for Anisotropic Bodies

Non ‐ Classical Mixed Interface Problems for Anisotropic Bodies

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