Static bending of perforated nanobeams including surface energy and microstructure effects
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ORIGINAL ARTICLE
Static bending of perforated nanobeams including surface energy and microstructure effects Alaa A. Abdelrahman1 · Norhan A. Mohamed2 · Mohamed A. Eltaher3,4 Received: 23 June 2020 / Accepted: 14 August 2020 © Springer-Verlag London Ltd., part of Springer Nature 2020
Abstract This article aims to present comprehensive model and analytical solution to investigate the static bending behavior of regularly squared cutout perforated thin/thick nanobeams incorporating the coupled effect of the microstructure and surface energy for the first time. The perforation influence is considered to be deriving equivalent geometrical and material characteristics. The modified couple stress theory is adopted to incorporate the microstructure effect while the modified Gurtin–Murdoch surface elasticity model is employed to incorporate the surface stress effect in perforated nanobeams. A variational formulation based on minimization of the total potential energy principle is employed to derive the equilibrium equations of perforated nanobeams based on both Euler–Bernoulli and Timoshenko beams theories are developed to investigate the associated effect of the shear deformation due to perforation process. Additionally, Poisson’s effect is also incorporated. Analytical closed-form for the non-classical bending profiles as well as the rotational displacement are developed for both beam theories considering the simultaneous effect of both couple stress and surface stress for both uniformly distributed and concentrated loading patterns. The verification of the developed model is verified and compared with previous works, and an excellent agreement is obtained. The applicability of the developed model is demonstrated and applied to study and analyze the nonclassical bending behavior of regularly squared perforated simply supported beams under different loading conditions. Additionally, effects of the perforation configuration parameters, beam size as well as beam aspect ratio on the bending behavior of perforated beams in the presence of microstructure and surface stress effects are also investigated and analyzed. The obtained results reveal that both couple stress and surface stress significantly affect the bending behavior of regularly squared cutout perforated beam structures. Results obtained are supportive for the design, analysis and manufacturing of perforated NEMS applications. Keywords Modified couple stress · Gurtin–Murdoch model · Perforated beam · Filling ratio · Shear deformation · Bending profile · Closed-form
1 Introduction * Mohamed A. Eltaher [email protected]; [email protected] 1
Mechanical Design & Production Department, Faculty of Engineering, Zagazig University, P. O. Box 44519, Zagazig, Egypt
2
Department of Engineering Mathematics, Faculty of Engineering, Zagazig University, P. O. Box 44519, Zagazig, Egypt
3
Mechanical Engineering Department, Faculty of Engineering, King Abdulaziz University, P. O. Box 80204, Jeddah, Saudi Arabia
4
Design & Production Dept., Faculty of Engineering,
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