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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.

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