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1

Variational principles and natural boundary conditions for multilayered orthotropic graphene sheets undergoing vibrations and based on nonlocal elastic theory

Adali S.

Journal of Theoretical and Applied Mechanics

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2011

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Vol. 49 nr 3

621-639

EN

Variational principles are derived for multilayered orthotropic graphene sheets undergoing transverse vibrations based on the nonlocal elastic theory of orthotropic plates which provide a continuum model for graphene sheets. The variational formulation allows the derivation of natural boundary conditions which are expressed in the form of a set of coupled equations for multilayered sheets as opposed to uncoupled boundary conditions applicable to simply supported and clamped boundaries and also in the case of a formulation based on the local (classical) elasticity theory. For the free vibrations case, the Rayleigh quotient is derived. The methods for the variational formulation use techniques of calculus of variations and the semi-inverse method for deriving variational integrals. Variational formulations provide the basis for a number of approximate and numerical methods of solutions and improve the understanding of the physical phenomena.

PL

W pracy zajęto się problemem drgań poprzecznych ortotropowych paneli grafenowych, dla których sformułowano zasady wariacyjne na podstawie nielokalnej teorii sprężystości, co pozwoliło na budowę ciągłego modelu takich struktur. Formuła wariacyjna umożliwiła konstrukcję naturalnych warunków brzegowych wyrażonych zbiorem sprzężonych równań opisujących grafenowe panele wielowarstwowe w odróżnieniu od rozprzężonych warunków brzegowych stosowanych jedynie do zamocowań typu swobodne podparcie lub zamurowanie, jednocześnie przy zastosowaniu lokalnej (klasycznej) teorii sprężystości. Dla przypadku drgań swobodnych wyznaczono iloraz Rayleigha układu z grafenu. W prezentowanym sformułowaniu użyto odpowiednich technik obliczania funkcjonałów i półodwrotnej metody wyznaczania całek. Wykazano, że postać wariacyjna stanowi podstawę dla numerycznych metod poszukiwania przybliżonych rozwiązań i pogłębia zrozumienie zachodzących zjawisk fizycznych w takich układach.

2

Variational principles of bending problems of thin plates

He Ji-Huan

Archives of Mechanics

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2001

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Vol. 53, nr 6

631-642

EN

In this paper, via the semi-inverse method proposed previously by the present author, a family of variational principles of bending problems of plates is derived directly from their governing equations and boundary conditions, without using the Lagrange multiplier method. In this method, an energy-like trial functional is constructed with a certain unknown function, which can be identified step by step. A new generalized variational principle is obtained.

3

Semi-inverse solutions in nonlinear theory of elastic shells

Zubov L. M.

Archives of Mechanics

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2001

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Vol. 53, nr 4-5

599-610

EN

The semi-inverse method is applied to the solution of static problems in the nonlinear theory of elastic shells. This method consists in construction of particular solutions in such a way that the initial system of equations is reduced to a system of a smaller number of independent variables. Two nonlinear models, one of the Love type and another of the Cosserat type, are considered. For these models, several two-parameter families of finite deformations are found; then the system of partial differential equations of equilibrium reduces to a system of nonlinear ordinary differential equations. The semi-inverse solutions found are valid for prismatic and toroidal shells, as well as for shells of revolution. These solutions are of practical significance. They describe torsion of a prismatic shell under large angles of twist, strong flexure of a thin-walled cylinder of arbitrary cross-section, spatial bending of a shell having the shape of a sector of a surface of revolution, straightening of a toroidal shell to a cylindrical surface, and some other types of large deformations.