Thermal vibration characteristics of pre/post-buckled bi-directional functionally graded tapered microbeams based on mod
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ORIGINAL ARTICLE
Thermal vibration characteristics of pre/post‑buckled bi‑directional functionally graded tapered microbeams based on modified couple stress Reddy beam theory Mohamed A. Attia1 · Salwa. A. Mohamed2 Received: 28 July 2020 / Accepted: 28 September 2020 © Springer-Verlag London Ltd., part of Springer Nature 2020
Abstract This paper investigates the nonlinear vibration characteristics of pre- and post-buckled nonuniform bi-directional functionally graded (BDFG) microbeams subjected to nonlinear thermal loading. The formulations derived herein are based on thermoelastic constitutive relations of the higher-order Reddy beam theory in conjunction with the modified couple stress theory, von Karman nonlinear strains, and physical neutral plane concept, for the first time. Power-law model is employed to describe the continuous variation of temperature-dependent thermomechanical material properties of BDFG in both the thickness and length directions. Variable microstructure length scale parameter and Poisson’s ratio in both directions are considered. The variable coefficient nonlinear governing equations and associated boundary conditions of nonuniform BDFG microbeam in the thermal environment are derived utilizing the Hamilton principle. The obtained equations are discretized using the differential quadrature method accounting for various boundary conditions, and iterative Newton’s method is adopted to solve the resulting nonlinear algebraic equations. The developed model and solution procedure are validated by comparing the predicted results with those in the available literature. To this end, extensive parametric studies are carried out to explore the influence of linear and nonlinear temperature profiles, microstructure length scale parameter ratio, microstructure length scale parameter-to-thickness ratio, gradient indices in thickness and length directions, material-temperature dependency, and taperness parameters on the nonlinear vibration response and higher-order mode frequencies of tapered BDFG microbeams accounting different boundary conditions. Keywords Bi-directional functionally graded (BDFG) microbeams · Modified couple stress theory · Nonlinear vibration · Stability analysis · Nonlinear thermal environment · Differential quadrature method (DQM)
1 Introduction Compared to conventional materials, the functionally graded materials (FGMs) have a heterogeneous microstructure as their thermomechanical properties vary continuously and gradually along a certain direction(s). The spatial variation of FGMs can be tailored to attain the required properties, i.e. mechanical, thermal, electrical, magnetic, etc., * Mohamed A. Attia [email protected]; [email protected] 1
Department of Mechanical Design and Production Engineering, Faculty of Engineering, Zagazig University, Zagazig 44519, Egypt
Engineering Mathematics Department, Faculty of Engineering, Zagazig University, Zagazig 44519, Egypt
2
for a particular engineering application [1–11]. FGMs structures may fail under thermal loading by r
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