Finite difference method for sixth-order derivatives of differential equations in buckling of nanoplates due to coupled surface energy and non-local elasticity theories

Document Type: Reasearch Paper


Department of Mechanical Engineering, Isfahan University of Technology, Isfahan 84156-83111, Iran.



In this article, finite difference method (FDM) is used to solve sixth-order derivatives of differential equations in buckling analysis of nanoplates due to coupled surface energy and non-local elasticity theories. The uniform temperature change is used to study thermal effect. The small scale and surface energy effects are added into the governing equations using Eringen’s non-local elasticity and Gurtin-Murdoch’s theories, respectively. Two different boundary conditions including simply-supported and clamped boundary conditions are investigated. The numerical results are presented to demonstrate the difference between buckling obtained by considering the surface energy effects and that obtained without the consideration of surface properties. The results show that the finite difference method can be used as a powerful method to determine the mechanical behavior of nanoplates. In addition, this method can be used to solve higher-order derivatives of differential equations with different types of boundary condition with little computational effort. Moreover, it is observed that the effects of surface properties tend to increase in thinner and larger nanoplates; and vice versa.


Main Subjects

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