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# Variational equations of asymptotic theory for multilayer thin plates

 Authors: Dimitrienko Yu.I., Gubareva E.A., Yurin Yu.V. Published: 04.09.2015 Published in issue: #4(61)/2015 DOI: 10.18698/1812-3368-2015-4-67-87 Category: Mechanics | Chapter: Mechanics of Deformable Solid Body Keywords: multilayer thin plates, asymptotic plate theory, asymptotic averaging method, asymptotic expansions, Lagrange variational principle, Hellinger-Reissner principle, Hermann principle

The article presents a derivation of the Lagrange type variational equation for thin multilayer plates based on the variational Lagrange principle for the threedimensional equations of the elasticity theory with the help of the theory of the asymptotic small-parameter expansions. The parameter was the ratio of a thickness and a typical length of the plate without introducing any hypotheses about the nature of the distribution of both stresses and displacements in thickness. It is shown that the variational equation is equivalent to the differential equation system of the Kirchhoff -Love plate theory. The developed asymptotic plate theory provides a mathematically rigorous (in the asymptotic sense) justification of the classical Kirchhoff - Love plate theory, but unlike the Kirchhoff-Love plate model, the developed asymptotic theory allows finding the distribution of all six components of the stress tensor. Variational principles of Hellinger-Reissner and Hermann types were derived for the asymptotic theory of thin plates.

## References

[1] Grigolyuk E.I., Kulikov G.M. The Generalized Model of the Mechanics of Thin-Walled Structures Made of Composite Materials. Composite Mechanics and Design, 1988, no. 4, pp. 698-704 (in Russ.).

[2] Zveryaev E.M., Makarov G.I. The General Method of Constructing Theories Like Tymoshenko. Prikl. Mat. Mekh. [J. Appl. Math. Mech.], 2008, vol. 72, iss. 2, pp. 308321 (in Russ.).

[3] Zveryaev E.M. Analysis of the Hypotheses Used in the Theory of Beams and Slabs. Prikl. Mat. Mekh. [J. Appl. Math. Mech.], 2003, vol. 67, iss. 3, pp. 472-483 (in Russ.).

[4] Sheshenin S.V. Asymptotic analysis for periodic in plane plates. Izv. RAN. MTT [Proc. of the Russ. Acad. Sci. Mech. Rigid Body], 2006, no. 6, pp. 71-79 (in Russ.).

[5] Sheshenin S.V., Khodos O.A. Effective Stiffness of Corrugated Plates. Vychislitel’naya mekhanika sploshnoi sredy [Computational Continuum Mechanics], 2011, vol. 4, no. 2, pp. 128-139 (in Russ.).

[6] Kohn R.V., Vogelyus M. A new model of thin plates with rapidly varying thickness. Int. J. Solids and Struct., 1984, vol. 20, no. 4, pp. 333-350.

[7] Panasenko G.P., Reztsov M.V. Averaging the Three-Dimensional Problem of Elasticity Theory in an Inhomogeneous Plate. Dokl. AN SSSR [Reports of Acad. Sci. USSR], 1987, vol. 294, no. 5, pp. 1061-1065 (in Russ.).

[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 structures with initial stresses. Springer-Verlag, Berlin, Heidelberg, 2004, 228 p.

[10] Dimitrienko Yu.I. Asymptotic Theory of Multilayer Thin Plates. Vestn. Mosk. Gos. Tekh. Univ. im. N.E. Baumana, Estestv. Nauki [Herald of the Bauman Moscow State Tech. Univ., Nat. Sci.], 2012, no.3, pp. 86-100 (in Russ.).

[11] Dimitrienko Yu.I., Yakovlev D.O. Asymptotic theory of thermoelasticity of multilayer composite plates. Mekhanika kompozitsionnykh materialov i konstruktsiy [Journal on Composite Mechanics and Design], 2014, vol. 20, no. 2, pp. 259-282 (in Russ.).

[12] Dimitrienko Yu.I., Gubareva E.A., Sborschikov S.V. Asymptotic theory of constructive-orthotropic plates with two-periodic structure. Mat. modelirovanie i chislennye metody [Math. Modeling and Computational Methods], 2014, no. 1, pp. 36-57 (in Russ.).

[13] Dimitrienko Yu.I., Fedonyuk N.N., Gubareva E.A., Sborschikov S.V., Prozorovskiy A.A., Erasov V.S., Yakovlev N.O. Modeling and development of three-layer sandwich composite materials with honecomb core. Vestn. Mosk. Gos. Tekh. Univ. im. N.E. Baumana, Estestv. Nauki [Herald of the Bauman Moscow State Tech. Univ., Nat. Sci.], 2014, no. 5, pp. 66-82 (in Russ.).

[14] Dimitrienko Yu.I., Gubareva E.A., Fedonyuk N.N., Yakovlev D.O. A method of calculation of energy dissipation in hybrid composite structures. Izv. Vyssh. Uchebn. Zaved., Mashinostr. [Proc. Univ., Mech. Eng.], 2014, no. 11, pp. 23-34 (in Russ.).

[15] Dimitrienko Yu.I., Gubareva E.A., Fedonyuk N.N., Sborschikov S.V. Modeling of elasic-dissipative properties of laminated fibrous composites. Jelektr. nauchno-tehn. Izd. "Inzhenernyy zhurnal: nauka i innovacii" [El. Sc.-Techn. Publ. "Eng. J.: Science and Innovation"], 2014, no. 4 (28). URL: http://engjournal.ru/catalog/mathmodel/material/1234.html

[16] Dimitrienko Yu.I., Gubareva E.A., Sborschikov S.V., Fedonyuk N.N. Simulation of Viscoelastic Properties of Fibrous Laminated Polymer Composite Materials. Jelektr. Nauchno-Tehn. Izd. "Nauka i obrazovanie" [El. Sc.-Tech. Publ. "Science and Education"], 2014, no. 11. URL: http://technomag.bmstu.ru/doc/734246.html (accessed 19.01.2015). DOI: 10.7463/1114.0734246

[17] Dimitrienko Yu.I., Gubareva E.A., Yakovlev D.O. The Asymptotic Theory of Viscoelasticity of Multilayer Thin Composite Plates. Jelektr. Nauchno-Tehn. Izd. "Nauka i obrazovanie" [El. Sc.-Tech. Publ. "Science and Education"], 2014, no. 10, pp. 359-382. URL: http://technomag.bmstu.ru/doc/730105.html (accessed 19.01.2015). DOI: 10.7463/1014.0730105

[18] Dimitrienko Yu.I., Gubareva E.A., Yurin Yu.V. Asymptotic theory of thermocreep for multilayer thin plates. Mat. modelirovanie i chislennye metody [Mathematical Modeling and Computational Methods], 2014, no. 4, pp. 36-57 (in Russ.).

[19] Dimitrienko Yu.I., Yakovlev D.O. Comparison analysis of asymptotic theory of multilayer composite plates and three-dimensional theory of elastisity. Jelektr. nauchno-tehn. Izd. "Inzhenernyy zhurnal: nauka i innovacii" [El. Sc.-Techn. Publ. "Eng. J.: Science and Innovation", 2013, iss. 12. URL: http://engjournal.ru/catalog/mathmodel/technic/899.html

[20] Dimitrienko Yu.I. Mekhanika sploshnoi sredy [Continuum mechanics]. Vol. 4. Osnovy mekhaniki tverdogo tela [Fundamentals of solid mechanics]. Moscow, MGTU im. N.E. Baumana Publ., 2013, 624p.

[21] Dimitrienko Yu.I. Mekhanika sploshnoi sredy. V 4 t. [Continuum mechanics. In 4 vol.]. Vol. 1. Tenzornyi analiz [Tensor analysis]. Moscow, MGTU im. N.E. Baumana Publ., 367 p.

[22] Dimitrienko Yu.I. Tenzornoye ischislenie [Tensor calculus]. Moscow, Vish. Shk. Publ., 2001. 576 p.

[23] Alfutov N.A., Zinoviev P.A., Popov B.G. Raschet mnogosloynih plastin i obolochek iz kompzitscionnih materialov [Calculation of multilayer composite plates and shells]. Moscow, Mashiostroenie Publ., 1980. 324 p.

[24] Belkin A.E., Gavrushin S.S. Raschet plastin metodom konechnih elementov [Calculation of plates by finite element method]. Moscow, MGTU im. N.E. Baumana Publ., 2008. 232 p.

[25] Popov B.G. Raschet mnogosloynih konstrukchiy variachionno-raznostnimi metodami [Calculation of multilayer structures by variation-matrix methods]. Moscow, MGTU im. N.E. Baumana Publ., 1993. 294p.