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Instability of functionally graded micro-beams via micro-structure-dependent beam theory

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Abstract

This paper focuses on the buckling behaviors of a micro-scaled bi-directional functionally graded (FG) beam with a rectangular cross-section, which is now widely used in fabricating components of micro-nano-electro-mechanical systems (MEMS/NEMS) with a wide range of aspect ratios. Based on the modified couple stress theory and the principle of minimum potential energy, the governing equations and boundary conditions for a micro-structure-dependent beam theory are derived. The present beam theory incorporates different kinds of higher-order shear assumptions as well as the two familiar beam theories, namely, the Euler-Bernoulli and Timoshenko beam theories. A numerical solution procedure, based on a generalized differential quadrature method (GDQM), is used to calculate the results of the bi-directional FG beams. The effects of the two exponential FG indexes, the higher-order shear deformations, the length scale parameter, the geometric dimensions, and the different boundary conditions on the critical buckling loads are studied in detail, by assuming that Young’s modulus obeys an exponential distribution function in both length and thickness directions. To reach the desired critical buckling load, the appropriate exponential FG indexes and geometric shape of micro-beams can be designed according to the proposed theory.

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Authors and Affiliations

  1. School of Mechanical Science and Engineering, Huazhong University of Science and Technology, Wuhan, 430074, China

    Xiaobai Li, Li Li & Yujin Hu

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  1. Xiaobai Li
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  2. Li Li
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  3. Yujin Hu
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Correspondence to Yujin Hu.

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Citation: LI, X. B., LI, L., and HU, Y. J. Instability of functionally graded micro-beams via microstructure-dependent beam theory. Applied Mathematics and Mechanics (English Edition), 39(7), 923–952 (2018) https://doi.org/10.1007/s10483-018-2343-8

Project supported by the National Natural Science Foundation of China (Nos. 51375184 and 51605172) and the Fundamental Research Funds for the Central Universities (No. 2015MS014)

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Li, X., Li, L. & Hu, Y. Instability of functionally graded micro-beams via micro-structure-dependent beam theory. Appl. Math. Mech.-Engl. Ed. 39, 923–952 (2018). https://doi.org/10.1007/s10483-018-2343-8

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  • DOI: https://doi.org/10.1007/s10483-018-2343-8

Keywords

Chinese Library Classification

2010 Mathematics Subject Classification

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