Pull-in and freestanding instability of actuated functionally graded nanobeams including surface and stiffening effects
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ORIGINAL ARTICLE
Pull‑in and freestanding instability of actuated functionally graded nanobeams including surface and stiffening effects Rasha M. Abo‑Bakr1 · Mohamed A. Eltaher2,3 · Mohamed A. Attia3 Received: 29 May 2020 / Accepted: 11 August 2020 © Springer-Verlag London Ltd., part of Springer Nature 2020
Abstract Because of fasttechnological development, electrostatic nanoactuator devices like nanosensors, nanoswitches, and nanoresonators are highly considered by scientific community. Thus, this article presents a new solution technique in solving highly nonlinear integro-differential equation governing electrically actuated nanobeams made of functionally graded material. The modified couple stress theory and Gurtin–Murdoch surface elasticity theory are coupled together to capture the size effects of the nanoscale thin beam in the context of Euler–Bernoulli beam theory. For accurate modelling, all the material properties of the bulk and surface continuums of the FG nanoactuator are varied continuously in thickness direction according to power law. The nonlinearity arising from the electrostatic actuation, fringing field, mid-plane stretching effect, axial residual stress, Casimir dispersion, and van der Waals forces are considered in mathematical formulation. The nonlinear nonclassical equilibrium equation of FG nanobeam-based actuators and associated boundary conditions are exactly derived using Hamilton principle. The new solution methodology is combined from three phases. The first phase applies Galerkin method to get an integro-algebraic equation. The second one employs particle swarm optimization method to approximate the integral terms (i.e. electrostatic force, fringing field, and intermolecular forces) to non-integral cubic algebraic equation. Then, solved the system easily in last phase. The resulting algebraic model provides means for obtaining critical deflection, pull-in voltage, detachment length, minimum gap, and freestanding effects. A reasonable agreement is found between the results obtained from the present method and those in the available literature. A parametric study is performed to investigate the effects of the gradient index, material length scale parameter, surface energy, intermolecular forces, initial gap, and beam length on the pull-in response and freestanding phenomena of fully clamped and cantilever FG nanoactuators. Keywords Pull-in instability · Freestanding phenomenon · FG nanobeams · Modified couple stress theory · Surface energy · Galerkin method · Particle swarm optimization method
1 Introduction Recent advances in the application of nanotechnology have resulted in the manufacture of nanoelectromechanical devices. The attractiveness of them is due to their excellent * Mohamed A. Eltaher [email protected]; [email protected] Mohamed A. Attia [email protected] 1
Mathematics Department, Faculty of Science, Zagazig University, Zagazig 44519, Egypt
2
Mechanical Department, Faculty of Engineering, King Abdulaziz University, P.O. Box 80204, Jeddah, Saudi Arabia
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