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Asymptotics of solutions to the spectral elasticity problem for a spatial body with a thin coupler. (English. Russian original) Zbl 1414.74005

Sib. Math. J. 53, No. 2, 274-290 (2012); translation from Sib. Mat. Zh. 53, No. 2, 345-364 (2012).
Summary: We construct asymptotics for the eigenvalues and vector eigenfunctions of the elasticity problem for an anisotropic body with a thin coupler (of diameter \(h\)) attached to its surface. In the spectrum we select two series of eigenvalues with stable asymptotics. The first series is formed by eigenvalues \(O(h^2)\) corresponding to the transverse oscillations of the rod with rigidly fixed ends, while the second is generated by the longitudinal oscillations and twisting of the rod, as well as eigenoscillations of the body without the coupler. We check the convergence theorem for the first series and derive the error estimates for both series.

MSC:

74B05 Classical linear elasticity
74K30 Junctions
74E10 Anisotropy in solid mechanics
74G10 Analytic approximation of solutions (perturbation methods, asymptotic methods, series, etc.) of equilibrium problems in solid mechanics
35Q74 PDEs in connection with mechanics of deformable solids
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References:

[1] Nazarov S. A., Asymptotic Theory of Thin Plates and Rods. Reduction of Dimension and Integral Estimates [in Russian], Nauchnaya Kniga, Novosibirsk (2002).
[2] Lekhnitskiĭ S. G., Elasticity of an Anisotropic Body [in Russian], Nauka, Moscow (1977).
[3] Ladyzhenskaya O. A., Boundary Value Problems of Mathematical Physics, Springer-Verlag, New York etc. (1985). · Zbl 0588.35003
[4] Nečas J., Les Méthodes in Théorie des Équations Elliptiques, Masson-Academia, Paris and Prague (1967).
[5] Kondrat’ev V. A. and Oleĭnik O. A., ”Boundary value problems for a system of elasticity in unbounded domains. Korn inequalities,” Russian Math. Surveys, 43, No. 5, 65–119 (1988). · Zbl 0669.73005 · doi:10.1070/RM1988v043n05ABEH001945
[6] Beale J. T., ”Scattering frequencies of resonators,” Comm. Pure Appl. Math., 26, No. 4, 549–563 (1973). · Zbl 0254.35094 · doi:10.1002/cpa.3160260408
[7] Arsen’ev A. A., ”The existence of resonance poles and scattering resonances in the case of boundary conditions of the second and third kind,” USSR Comput. Math. Math. Phys., 16, No. 3, 171–177 (1976). · Zbl 0352.35039 · doi:10.1016/0041-5553(76)90212-3
[8] Gadyl’shin R. R., ”On the eigenvalues of a’ dumb-bell with a thin handle’,” Izv. Math., 69, No. 2, 265–329 (2005). · Zbl 1075.35023 · doi:10.1070/IM2005v069n02ABEH000530
[9] Nazarov S. A. and Slutskiĭ A. S., ”One-dimensional equations of deformation of thin slightly curved rods. Asymptotic analysis and justification,” Izv.: Math., 64, No. 3, 531–562 (2000). · Zbl 0996.74051 · doi:10.1070/IM2000v064n03ABEH000290
[10] Birman M. Sh. and Solomyak M. Z., Spectral Theory of Selfadjoint Operators in Hilbert Space, D. Reidel Publ. Co., Dordrecht (1987).
[11] Nazarov S. A., ”The Korn inequalities which are asymptotically sharp for thin domains,” Vestnik St. Petersburg Univ. Math., 25, No. 2, 18–22 (1992). · Zbl 0783.73012
[12] Cioranescu D., Oleinik O. A., and Tronel G., ”Korn’s inequalities for frame type structures and junctions with sharp estimates for the constants,” Asymptotic Anal., 8, 1–14 (1994). · Zbl 0791.73010
[13] Nazarov S. A., ”Justification of the asymptotic theory of thin rods. Integral and pointwise estimates,” J. Math. Sci., 97, No. 4, 4245–4279 (1999). · doi:10.1007/BF02365044
[14] Nazarov S. A. and Slutskiĭ A. S., ”Korn’s inequality for an arbitrary system of distorted thin rods,” Siberian Math. J., 43, No. 6, 1069–1079 (2002). · doi:10.1023/A:1021121402082
[15] Nazarov S. A., ”Korn inequalities for elastic junctions of massive bodies, thin plates, and rods,” Russian Math. Surveys, 63, No. 1, 35–107 (2008). · Zbl 1155.74027 · doi:10.1070/RM2008v063n01ABEH004501
[16] Kondrat’ev V. A., ”Boundary value problems for elliptic equations in domains with conical or angular points,” Trudy Moskov. Mat. Obshch., 16, 209–292 (1967).
[17] Maz’ya V. G. and Plamenevskiĭ B. A., ”On coefficients in asymptotic expansions of solutions to elliptic boundary value problems in domains with conical points,” Math. Nachr., Bd 76, 29–60 (1977). · Zbl 0359.35024 · doi:10.1002/mana.19770760103
[18] Nazarov S. A. and Plamenevsky B. A., Elliptic Problems in Domains with Piecewise Smooth Boundaries, Walter de Gruyter, Berlin and New York (1994). · Zbl 0806.35001
[19] Vishik M. I. and Lyusternik L. A., ”Regular degeneration and a boundary layer for linear differential equations with a small parameter,” Uspekhi Mat. Nauk, 12, No. 5, 3–122 (1957). · Zbl 0087.29602
[20] Nazarov S. A., ”Uniform estimates of remainders in asymptotic expansions of solutions to the problem on eigen-oscillations of a piezoelectric plate,” J. Math. Sci., 114, No. 5, 1657–1725 (2003). · doi:10.1023/A:1022364812273
[21] Nazarov S. A., ”Korn’s inequalities for junctions of spatial bodies and thin rods,” Math. Methods Appl. Sci., 20, No. 3, 219–243 (1997). · Zbl 0880.35040 · doi:10.1002/(SICI)1099-1476(199702)20:3<219::AID-MMA854>3.0.CO;2-C
[22] Kozlov V. A., Maz’ya V. G., and Movchan A. B., ”Asymptotic representation of elastic fields in a multi-structure,” Asymptotic Anal., 11, 343–415 (1995). · Zbl 0846.73009
[23] Kozlov V. A., Maz’ya V. G., and Movchan A. B., ”Fields in non-degenerate 1D–3D elastic multi-structures,” Quart. J. Mech. Appl. Math., 54, 177–212 (2001). · Zbl 0988.74014 · doi:10.1093/qjmam/54.2.177
[24] Kozlov V. A., Maz’ya V. G., and Movchan A. B., Asymptotic Analysis of Fields in Multi-Structures, Clarendon Press, Oxford (1999).
[25] Nazarov S. A., ”Asymptotic analysis and modeling of the jointing of a massive body with thin rods,” J. Math. Sci., 127, No. 5, 2172–2263 (2003).
[26] Nazarov S. A., ”Junctions of singularly degenerating domains with different limit dimensions. 1,” J. Math. Sci., 80, No. 5, 1989–2034 (1996). · Zbl 0862.35027 · doi:10.1007/BF02362511
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