Semi-analytical solutions for static and dynamic responses of bi-directional functionally graded nonuniform nanobeams wi
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ORIGINAL ARTICLE
Semi‑analytical solutions for static and dynamic responses of bi‑directional functionally graded nonuniform nanobeams with surface energy effect Rabab A. Shanab1 · Mohamed A. Attia2 Received: 1 July 2020 / Accepted: 21 October 2020 © Springer-Verlag London Ltd., part of Springer Nature 2020
Abstract In this paper, the bending, buckling, and vibration behaviors of bi-directional functionally graded (BDFG) nonuniform micro/nanobeams are investigated. A new Euler–Bernoulli beam model is developed for BDFG tapered micro/nanobeams using Gurtin–Murdoch surface elasticity theory and modified couple stress theory to capture the effects of surface energy and microstructure stiffening, respectively. The present formulation accounts for the physical neutral surface. The material properties of the bulk and surface continuums of the nanobeam are assumed to vary along the thickness and length directions according to power law. Also, the cross section is assumed to vary linearly along the length direction. Hamilton principle is employed to derive the nonclassical equations of motions and boundary conditions. The generalized differential quadrature method (GDQM) is employed to accurately evaluate the variable coefficients of the obtained governing equations. Then after, the Navier’s method is employed for the simply supported BDFG nanobeam for its static bending deflection, critical buckling load, and fundamental frequency. The proposed model is validated by comparing the obtained results with available literature. Effects of different geometrical and material parameters on static and dynamic behaviors of small-scale BDFG nanobeams with the simultaneous effects of microstructure and surface elasticity are comprehensively studied and discussed. Results disclose that the nonuniformity parameters, aspect ratio, dimensionless material length-scale parameter, surface stress, surface elasticity, and gradient indices have a significant effect on the bending, buckling, and free vibration responses of BDFG tapered micro/ nanobeams. Keywords Bi-directional functionally graded material · Nonuniform nanobeams · Surface elasticity theory · Modified couple stress theory · Semi-analytical solution
1 Introduction Functionally graded materials (FGMs) are a subclass of composite materials, which are designed to achieve the optimal distribution of constituent materials suitable for certain applications. The superior properties of FGMs such * Rabab A. Shanab [email protected]
Mohamed A. Attia [email protected]; [email protected]
1
Engineering Mathematics Department, Faculty of Engineering, Zagazig University, Zagazig 44519, Egypt
Department of Mechanical Design and Production Engineering, Faculty of Engineering, Zagazig University, Zagazig 44519, Egypt
2
as designability, lower weight, higher fracture toughness, enhanced thermal properties, lower stress intensity factor, reduced residual thermal stress, reduced interface problems, smaller stress concentration, enhanced corrosion resistance damage resistance,
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