Thin sells theory based on the asymptotic analysis of three-dimensional equations of the elasticity theory

Article

Published: 14.10.2015

Authors: Dimitrienko Yu.I., Gubareva E.A., Shalygin I.S.

DOI: 10.18698/2308-6033-2015-5-1406

Category: Mechanics | Chapter: Mechanics of Deformable Solid Body

The article presents main relations for a new theory of thin multilayer anisotropic shells. The main equations of the shell theory suggested were deduced from general threedimensional theory of elasticity by means of asymptotic expansions over small parameter without any hypothesis concerning displacement and stress distribution over thickness. It is shown that the averaged problem of the shell theory developed proves to be similar to the Kirchhoff-Love shell theory, but there are some differences in constitutive relations, that contain derivatives for membrane strains. The method suggested allows one to calculate all six stress tenor components including transverse normal stresses and stresses of interlayer shear of thin elastic shells.

References
[1] Grigolyuk E.I., Kulikov G.M. Mekhanika kompozitnykh materialov - Mechanics of Composite Materials, 1988, vol. 24, no. 4, pp. 698-704.
[2] Sheshenin S.V. Izv. RAN. MTT - Proc. of the Russ. Acad. Sci. Mech. Rigid Body, 2006, no. 6, pp. 71-79.
[3] Sheshenin S.V., Khodos O.A. Vychislitel’naya mekhanika sploshnoi sredy - Computational Continuum Mechanics, 2011, vol. 4, no. 2, pp. 128-139.
[4] Nazarov S.A, Sweers G.H., Slutskij A.S. Matematicheskiy sbornik - Sbornik: Mathematics., 2011, vol. 202, no. 8, pp. 41-80.
[5] Akimova E.A., Nazarov S.A., Chechkin G.A. Asymptotics of the solution of the problem of deformation of an arbitrary locally periodic thin plate. Trans. Mosc. Math. Soc, 2004, pp. 1-29.
[6] Zveryaev E.M., Makarov G.I. PMM - J. Appl. Math. Mech., 2008, vol. 72, iss. 2, pp. 308-321.
[7] Zveryaev E.M. PMM-J. Appl. Math. Mech, 2003, vol. 67, iss. 3, pp. 472-483.
[8] Kohn R.V., Vogelyus M. Int. J. Solids and Struct, 1984, vol. 20, no. 4, pp. 333-350.
[9] Panasenko G.P., Reztsov M.V. Dokl. AN SSSR - Reports of Acad. Sci. USSR, 1987, vol. 294, no. 5, pp. 1061-1065.
[10] Levinski T., Telega J.J. Plates, laminates and shells. Asymptotic analysis and homogenization. Singapore, London, World Sci. Publ., 2000, 739 p.
[11] Kolpakov A.G. Homogenized models for thin-walled nonhomogeneous structures with initial stresses. Springer Verlag, Berlin, Heidelberg, 2004, 228 p.
[12] Pobedrya B.E. Mekhanika kompozitsionnykh materialov [Mechanics of composite materials]. Moscow, Lomonosov MST Publ., 1984, 336 p.
[13] Bakhvalov N.S., Panasenko G.P. Osrednenie protsessov v periodicheskikh sredakh [Averaging of processes in periodic media]. Moscow, Nauka Publ., 1984, 356 p.
[14] Sanchez-Palencia E. Non-Homogeneous Media and Vibration Theory. Moscow, Mir Publ., 1984, 471 p. (in Russ.).
[15] Dimitrienko Yu.I., Kashkarov A.I. Vestnik MGTU im. N.E. Baumana. Seriya Estestvennye nauki - Herald of the Bauman MSTU. Series: Natural sciences, 2002, no. 2, pp. 95-108.
[16] Dimitrienko Yu.I., Kashkarov A.I., Makashov A.A. Vestnik MGTU im. N.E. Baumana. Seriya Estestvennye nauki - Herald of the Bauman MSTU. Series: Natural sciences, 2007, no. 1, pp. 102-116.
[17] Dimitrienko Yu.I., Sokolov A.P. Matematicheskoe Modelirovanie - Mathematical Models and Computer Simulations, 2012, vol. 24, no. 5, pp. 3-20.
[18] Dimitrienko Yu.I. Thermomechanics of Composites under High Temperatures. Dordrecht; Boston; London, Kluwer Academic Publishers, 1999, 347 p.
[19] Dimitrienko Yu.I. Vestnik MGTU im. N.E. Baumana. Seriya Estestvennye nauki - Herald of the Bauman MSTU. Series: Natural sciences, 2012, no. 3, pp. 86100.
[20] Dimitrienko Yu.I., Yakovlev D.O. Mekhanika kompositsionnykh materialov i konstruktsiy - Composite Mechanics and Design, 2014, vol. 20, no. 2, pp. 260-282.
[21] Dimitrienko Yu.I., Gubareva E.A., Sborschikov S.V. Matematicheskoe modelirovanie i chislennye metody - Mathematical Modeling and Computational Methods, 2014, no. 1, pp. 36-57.
[22] Dimitrienko Yu.I., Gubareva E.A., Yakovlev D.O. Nauka i obrazovanie. El-ektronnoe nauchno-tekhnicheskoe izdanie - Science and Education. Electronic Scientific and Technical Journal, 2014, no. 10. doi: 10.7463/1014.0730105. pp. 359-382.
[23] Dimitrienko Yu.I., Gubareva E.A., Yurin Yu.V. Matematicheskoe modelirovanie i chislennye metody - Mathematical Modeling and Computational Methods, 2014, no. 4, pp.18-36.
[24] Dimitrienko Yu.I., Yakovlev D.O. Inzhenernyi zhurnal: nauka i innovatsii - Engineering Journal: Science and Innovation, 2013, iss. 12. Available at: http://engjournal.ru/catalog/mathmodel/technic/899.html
[25] Dimitrienko Yu.I. Mekhanika sploshnoy sredy. Tom 4. Osnovy mekhaniki tverdogo tela [Continuum Mechanics. Vol. 4. Fundamentals of solid mechanics]. Moscow, BMSTU Publ., 2013, 624 p.