Asymptotics of the Deflection of a Cruciform Junction of Two Narrow Kirchhoff Plates

 
PIIS004446690000452-8-1
DOI10.31857/S004446690000452-8
Publication type Article
Status Published
Authors
Affiliation:
Petersburg State University
Institute of Problems of Mechanical Engineering, Russian Academy of Sciences
Journal nameZhurnal vychislitelnoi matematiki i matematicheskoi fiziki
EditionVolume 58 Issue 7
Pages1197-1218
Abstract

Two two-dimensional plates with bending described by Sophie Germain’s equation with the biharmonic operator are joined in the form of a cross with clamped ends, but with free lateral sides outside the cross core. Asymptotics of the deflection of the junction with respect to the relative width of the plates regarded as a small parameter is constructed and justified. The variational-asymptotic model includes a system of two ordinary differential equations of the fourth and second orders with Dirichlet conditions at the endpoints of the one-dimensional cross and the Kirchhoff transmission conditions at its center. They are derived by analyzing the boundary layer near the crossing of the plates and mean that the deflection and the angles of rotation at the central point are continuous and that the total bending force and the principal torques vanish. Possible generalizations of the asymptotic analysis are discussed.

Keywordscruciform junction of narrow plates, asymptotics, one-dimensional model, boundary layer, Kirchhoff transmission conditions
Received15.08.2018
Publication date11.10.2018
Cite   Download pdf To download PDF you should sign in
Размещенный ниже текст является ознакомительной версией и может не соответствовать печатной

views: 385

Readers community rating: votes 0

1. Mikhlin S.G. Variatsionnye metody v matematicheskoj fizike. M.: Nauka, 1970.

2. Ladyzhenskaya O.A. Kraevye zadachi matematicheskoj fiziki. M.: Nauka, 1973.

3. Lions Zh.-L., Madzhenes Eh. Neodnorodnye granichnye zadachi i ikh prilozheniya. M.: Mir. 1971.

4. Williams M.L. Stress singularities resulting from various boundary conditions in angular corners of plate in extension // J. Appl. Mech. 1952. V.19, 4. P. 526–528.

5. Parton V.Z., Perlin P.I. Metody matematicheskoj teorii uprugosti. M.: Nauka, 1981.

6. Nazarov S.A., Plamenevsky B.A. Elliptic problems in domains with piecewise smooth boundaries. Berlin, New York: Walter de Gruyter. 1994.

7. Nazarov S.A. Asimptoticheskaya teoriya tonkikh plastin i sterzhnej. Ponizhenie razmernosti i integral'nye otsenki. Novosibirsk: Nauchnaya kniga, 2002.

8. Gaudiello A., Panasenko G.P., Piatnitski A. Asymptotic analysis and domain decomposition for a biharmonic problem in a thin multi-structure // Communications in Contemporary Mathematics. 2015. V. 18. P. 1–27.

9. Nazarov S.A. Polinomial'noe svojstvo samosopryazhennykh ehllipticheskikh kraevykh zadach i algebraicheskoe opisanie ikh atributov // Uspekhi matem. nauk. 1999. T. 54, 5. S. 77–142.

10. Nazarov S.A. Samosopryazhennye ehllipticheskie kraevye zadachi. Polinomial'noe svojstvo i formal'no polozhitel'nye operatory // Problemy matem. analiza. Vyp. 16. SPb: izd-vo SPbGU. 1997. S. 167–192.

11. Ciarlet P.G. Plates and junctions in elastic multi-structures: An asymptotic analysis. Paris: Masson. 1988.

12. Sanchez-Hubert J., Sanchez-Palencia E. Coques élastiques minces: Propriétés asymptotiques. Paris: Masson, 1997.

13. Nazarov S.A. Obschaya skhema osredneniya samosopryazhennykh ehllipticheskikh sistem v mnogomernykh oblastyakh, v tom chisle tonkikh // Algebra i analiz. 1995. T. 7. 5. S. 1–92.

14. Nazarov S.A. Struktura reshenij ehllipticheskikh kraevykh zadach v tonkikh oblastyakh // Vestnik LGU. Seriya 1. 1982. Vyp. 2 ( 7). S. 65–68.

15. Leora S.N., Nazarov S.A., Proskura A.V. Vyvod predel'nykh uravnenij dlya ehllipticheskikh kraevykh zadach v tonkikh oblastyakh pri pomoschi EhVM // Zhurnal vychisl. matem. i matem. fiz. 1986. T. 26. № 7. S. 1032–1048.

16. Mazja W.G., Nazarov S.A., Plamenewski B.A. Asymptotische Theorie elliptischer Randwertaufgaben in singulär gestörten Gebieten. 1, 2. Berlin: Akademie-Verlag. 1991. (Anglijskij perevod: Maz’ya V., Nazarov S., Plamenevskij B. Asymptotic theory of elliptic boundary value problems in singularly perturbed domains. Vol. 1, 2. Basel: Birkhäuser Verlag, 2000.)

17. Nazarov S.A. Nesamosopryazhennye ehllipticheskie zadachi s polinomial'nym svojstvom v oblastyakh, imeyuschikh tsilindricheskie vykhody na beskonechnost' // Zapiski nauchn. seminarov peterburg. otdeleniya matem. instituta RAN. 1997. T. 249. S. 212–230.

18. Nazarov S.A. Properties of spectra of boundary value problems in cylindrical and quasicylindrical domains // Sobolev Spaces in Mathematics. V. II (Maz’ya V., Ed.) International Mathematical Series , Vol. 9. New York: Springer, 2008. P. 261–309.

19. Kondrat'ev V.A. Kraevye zadachi dlya ehllipticheskikh uravnenij v oblastyakh s konicheskimi ili uglovymi tochkami // Trudy Moskovsk. matem. obschestva. 1963. T. 16. S. 219–292.

20. Maz'ya V.G., Plamenevskij B.A. O koehffitsientakh v asimptotike reshenij ehllipticheskikh kraevykh zadach v oblasti s konicheskimi tochkami // Math. Nachr. 1977. Bd. 76. S. 29–60.

21. Van Dajk M.D. Metody vozmuschenij v mekhanike zhidkostej. M.: Mir, 1967.

22. Il'in A.M. Soglasovanie asimptoticheskikh razlozhenij reshenij kraevykh zadach. M.: Nauka, 1989.

23. Vishik M.I., Lyusternik L.A. Regulyarnoe vyrozhdenie i pogranichnyj sloj dlya linejnykh differentsial'nykh uravnenij s malym parametrom // Uspekhi matem. nauk. 1957. T. 12. № 5. S. 3–122.

24. Maz'ya V.G., Nazarov S.A. O paradokse Sapondzhyana–Babushki v zadachakh teorii tonkikh plastin // Doklady AN ArmSSR. 1984. T. 78, 3. S. 127–130.

25. Nazarov S.A., Sweers G. A hinged plate equation and iterated Dirichlet Laplace operator on domains with concave corners // J. of Differential Equations. 2007. V. 233. № 1. P. 151–180.

26. Nazarov S.A. Asimptotika sobstvennykh znachenij zadachi Dirikhle na skoshennom T-obraznom volnovode // Zhurnal vychisl. matem. i matem. fiz. 2014. T. 54. № 5. C. 793–814.

Система Orphus

Loading...
Up