#### 539.3 Asymptotic theory of constructive-orthotropic plates with two-periodic structures

##### Dimitrienko Y. I. (Bauman Moscow State Technical University), Gubareva E. A. (Bauman Moscow State Technical University), Sborschikov S. V. (Bauman Moscow State Technical University)

###### MULTILAYER PLATES, TWO-PERIODIC STRUCTURE, HONEYCOMB SANDWICH PANELS CONSTRUCTIVE-ORTHOTROPIC PLATES, TWO-PERIODIC STRUCTURE, ASYMPTOTIC EXPANSIONS, LOCAL PROB-LEMS.

doi: 10.18698/2309-3684-2014-1-3656

The theory of thin constructive-orthotropic plates with a two-periodic structure was suggested. Examples of such structures are honeycomb sandwich panels and backed plates. The theory is based on equations of a three-dimensional elasticity theory with the help of asymptotic expansions in terms of a small parameter being the ratio of a plate thickness and a characteristic length without introducing any hypotheses on a distribution character for displacements and stresses through the thickness. Local problems were formulated for finding stresses in all structural elements of a plate. It was shown that the global (averaged by the certain rules) equations of the plate theory are similar to equations of the

[1] Grigolyuk E.I., Kulikov G.M. Mekhanika kompozitsionnykh materialov – Composite Mechanics and Design, 1988, 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] Zveryaev E.M., Makarov G.I. PMM — J. Appl. Math. Mech., 2008, vol. 72, iss. 2, pp. 308–321.
[5] Zveriaev E. M. PMM — J. Appl. Math. Mech., 2003, vol. 67, iss. 3, pp. 472–483.
[6] Kohn R.V., Vogelyus M. Int. J. Solids and Struct., 1984, vol. 20, no. 4, pp. 333–350.
[7] Panasenko G.P., Reztsov M.V. Dokl. AN SSSR — Reports of Acad. Sci. USSR, 1987, vol. 294, no. 5, pp. 1061–1065.
[8] Levinski T., Telega J.J. Plates, laminates and shells. Asymptotic analysis and homogenization. Singapore, London, World Sci. Publ., 2000, 739 p.
[9] Kolpakov A. G. Homogenized models for thin-walled nonhomogeneous struc-tures with initial stresses. Springer Verlag, Berlin, Heidelberg, 2004, 228 p.
[10] Pobedrya B.E. Mekhanika kompozitsionnykh materialov [Mechanics of com-posite materials]. Moscow, Lomonosov MSU Publ., 1984, 336 p.
[11] Bakhvalov N.S., Panasenko G.P. Osrednenie protsessov v periodicheskikh sredakh [Averaging of processes in periodic media]. Moscow, Nauka Publ., 1984.
[12] Sanches-Palensiya E. Neodnorodnye sredy i teoriya kolebaniy [Non-uniform media and theory of oscillations]. Moscow, Mir Publ., 1984.
[13] Dimitrienko Yu.I., Kashkarov A.V. Vestnik MGTU im. N.E. Baumana. Seriya Estestvennye Nauki — Herald of Bauman Moscow State Technical University, Natural Science Series, 2002, no. 2.
[14] Dimitrienko Yu.I., Sokolov A.P. Vestnik MGTU im. N.E. Baumana. Seriya Estestvennye nauki — Herald of Bauman Moscow State Technical University, Natural Science Series, 2008, no. 2, pp. 57–67.
[15] Dimitrienko Yu.I., Sokolov A.P. Izvestiya Rossiyskoi akademii nauk. Seriya fizicheskaya – Proc. Russ. Acad. Sci., 2011, vol. 75, no. 11, pp. 1549–1554.
[16] Dimitrienko Yu.I., Sokolov A.P. Matematicheskoe modelirovanie — Mathe-matical Modeling, 2009, vol .21, no. 4, pp. 96–110.
[17] Dimitrienko Yu.I., Sokolov A.P. Vestnik MGTU im. N.E. Baumana. Seriya Estestvennye nauki — Herald of Bauman Moscow State Technical University, Natural Science Series, 2008, no. 2, pp. 57–67.
[18] Dimitrienko Yu.I., Sborshchikov S.V., Sokolov A.P. Mekhanika kompo-zitsionnykh materialov i konstruktsiy — Composite Mechanics and design, 2013, vol. 19, no. 3, pp. 365–383.
[19] Dimitrienko Yu.I., Sborshchikov S.V., Sokolov A.P., Shpakova Yu.V. Vychislitel'naya mekhanika sploshnoi sredy — Computational continuum me-chanics, 2013, vol. 6, no. 4, pp. 389–402. doi: 10.7242/1999–6691/2013.6.4.43
[20] Dimitrienko Yu.I. Vestnik MGTU im. N.E. Baumana. Seriya Estestvennye nauki — Herald of Bauman Moscow State Technical University, Natural Sci-ence Series, 2012, no. 3, pp. 86–100.
[21] 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
[22] Dimitrienko Yu.I. Mekhanika sploshnoi sredy [Continuum mechanics]. Vol. 1. Tenzornyi analiz [Tensor analysis]. Moscow, Bauman MSTU Publ., 367 p.
[23] Dimitrienko Yu.I. Mekhanika sploshnoi sredy [Continuum mechanics]. Vol. 4. Osnovy mekhaniki tverdogo tela [Fundamentals of solid mechanics]. Moscow, Bauman MSTU Publ., 2013, 624 p.

Dimitrienko Y., Gubareva E., Sborschikov S. Asymptotic theory of constructive-orthotropic plates with two-periodic structures. Маthematical Modeling and Coтputational Methods, 2014, №1 (1), pp. 36-56