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

Adali, S.

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

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Adali S.

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Warianty tytułu

Zasady wariacyjne i naturalne warunki brzegowe dla wielowarstwowych ortotropowych paneli grafenowych poddanych drganiom, sformułowane w ramach nielokalnej teorii sprężystości

Języki publikacji

Abstrakty

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.

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.

Słowa kluczowe

variational formulation multilayered graphene sheets non-local theory vibrations semi-inverse method

Wydawca

Polskie Towarzystwo Mechaniki Teoretycznej i Stosowanej

Czasopismo

Journal of Theoretical and Applied Mechanics

Rocznik

2011

Tom

Vol. 49 nr 3

Strony

621--639

Opis fizyczny

Bibliogr. 51 poz., rys.

Twórcy

autor

Adali S.

School of Mechanical Engineering, University of KwaZulu-Natal, Durban, South Africa, adali@ukzn.ac.za

Bibliografia

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