Harmonic vibration of cusped plates in the N-th approximation of Vekua’s hierarchical models

• Autorzy: Chinchaladze N.
• Konferencja: Solid Mechanics Conference (38 ; 27-31.08.2012 ; Warsaw, Poland)
• Języki publikacji: EN

Abstrakty

EN

In this paper elastic cusped symmetric prismatic shells (i.e., plates of variable thickness with cusped edges) in the N-th approximation of Vekua’s hierarchical models are considered. The well-posedness of the boundary value problems (BVPs) under the reasonable boundary conditions at the cusped edge and given displacements at the non-cusped edge is studied in the case of harmonic vibration. The classical and weak setting of the BVPs in the case of the N-th approximation of hierarchical models is considered. Appropriate weighted functional spaces are introduced. Uniqueness and existence results for the variational problem are proved. The structure of the constructed weighted space is described and its connection with weighted Sobolev spaces is established.

Słowa kluczowe

EN

cusped plates; cusped prismatic shells; degenerate elliptic systems; weighted spaces; Hardy's inequality; Korn’s weighted inequality

• Wydawca: Instytut Podstawowych Problemów Techniki PAN
• Czasopismo: Archives of Mechanics
• Rocznik: 2013
• Tom: Vol. 65, nr 5
• Strony: 345--365
• Opis fizyczny: Bibliogr. 21 poz.

Twórcy

autor

Chinchaladze N.

• Iv. Javakhishvili Tbilisi State University,Faculty of Exact and Natural Sciences and I. Vekua Institute of Applied Mathematics 2 University st., 0186 Tbilisi, Georgia

Bibliografia

• 1. G.V. Jaiani, Solution of some Problems for a Degenerate Elliptic Equation of Higher Order and their Applications to Prismatic Shells, Tbilisi University Press, 1982 [in Russian].
• 2. G. Jaiani, Cusped shell-like structures, SpringerBriefs in Applied Science and Technology, Springer, Heidelberg, 2011.
• 3. E. Makhover, Bending of a plate of variable thickness with a cusped edge, Scientific Notes of Leningrad State Ped. Institute. 17, 2, 28–39, 1957 [in Russian].
• 4. S.G. Mikhlin, Variational methods in mathematical physics, Nauka, Moscow, 1970 [in Russian].
• 5. A.R. Khvoles, The general representation for solutions of equilibrium equations of prismatic shell with variable thickness, Seminar of the Institute of Applied Mathematics of Tbilisi State University, Annot. of Reports, 5, 19–21, 1971 [in Russian].
• 6. I.N. Vekua, On a method of computing prismatic shells , Akad. Nauk Gruzin. SSR. Trudy Tbiliss. Mat. Inst. Razmadze, 21, 191–259, 1955 [in Russian].
• 7. I.N. Vekua, Theory of thin shallow shells of variable thickness , Akad. Nauk Gruzin. SSR Trudy Tbiliss. Mat. Inst. Razmadze, 30, 3–103, 1965 [in Russian].
• 8. N. Chinchaladze, R.P. Gilbert, Cylindrical vibration of an elastic cusped plate under action of an incompressible fluid in case of N= 0 approximation of I.Vekua’s hierarchical models, Complex Variables, 50, No. 7-11, 479–496, 2005.
• 9. N. Chinchaladze, G. Jaiani, R. Gilbert, S. Kharibegashvili, D. Natroshvili, Existence and uniqueness theorems for cusped prismatic shells in the N-th hierarchical model, Mathematical Methods in Applied Sciences.31, 11, 1345–1367, 2008.
• 10. G. Devdariani, The first boundary value problem for a degenerate elliptic system, Bulletin TICMI, 5, 23–24, 2001; for electronic version see: http://www.emis.de/journals/TICMI
• 11. G.V. Jaiani, S.S. Kharibegashvili, D.G. Natroshvili, W.L. Wendland, Two-dimensional Hierarchical Models for Prismatic Shells with Thickness Vanishing at the Boundary, Journal of Elasticity, 77, 2, 95–122, 2004.
• 12. G.V. Jaiani, B.W. Schulze, Some degenerate elliptic systems and applications to cusped plates, Mathematische Nachrichten, 280, 4, 388–407, 2007.
• 13. N. Chinchaladze, R.P. Gilbert, Harmonic vibration of prismatic shells in zero approximation of Vekua’s hierarchical models, Applicable Analysis: An International Journal, Doi:10.1080/00036811.2012.731502, i-first 2012.
• 14. N. Chinchaladze, Harmonic vibration of a cusped plates in the first approximation of Vekua’s hierarchical models, Proceedings of I. Vekua Institute of Applied Mathematics, 2012, accepted for publication.
• 15. D. Natroshvili, S. Kharibegashvili, Investigation of hyperbolic systems with order degeneration arising in I.Vekua’s hierarchical models, Applicable Analysis: An International Journal, Doi:10.1080/00036811.2012.746961, accepted for publication.
• 16. I.N. Vekua, Shell Theory: General Methods of Construction, Monographs, Advanced Texts and Surveys in Pure and Applied Mathematics, 25. Pitman (Advanced Publishing Program), Boston, MA. 1985 Harmonic vibration of cusped plates in the N-th approximation... 365
• 17. G.V. Jaiani, On a mathematical model of bars with variable rectangular cross-sections, ZAMM-Zeitschrift für Angewandte Mathematik und Mechanik, 81, 3, 147–173, 2001.
• 18. G.V. Jaiani, S.S. Kharibegashvili, D.G. Natroshvili, W.L. Wendland, Hierarchical models for elastic cusped plates and beams, Lecture Notes of TICMI, 4, 2003, for electronic version see: http://www.emis.de/journals/TICMI.
• 19. G. Devdariani, G.V. Jaiani, S.S. Kharibegashvili, D. Natroshvili, The first boundary value problem for the system of cusped prismatic shells in the first approximation, Appl. Math. Inform., 5, 2, 26–46, 2000.
• 20. M.I. Vishik, Boundary value problems for elliptic equations degenerating of the boundary of domain, Math. Sb., 35, (77), No. 3, 513–568, 1954 [in Russian].
• 21. H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, Berlin, DVW, 1978.