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Size-dependent postbuckling for microbeams: analytical solutions using a reformulated strain gradient elasticity theory

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Abstract

This paper presents closed-form solutions for exploring size-dependent postbuckling of Euler–Bernoulli and Timoshenko microbeams using a novel strain gradient elasticity theory. To capture the size effects of microbeams, a reformulated strain gradient elasticity theory incorporating strain gradient and couple stress effects simultaneously with only one material parameter for each effect is employed. The governing equations and boundary conditions of both Euler–Bernoulli and Timoshenko microbeams are derived using the principle of minimum potential energy and von Kármán geometric nonlinearity theory. These models can be degenerated into modified couple stress theory if the strain gradient material length-scale parameter is taken to be zero. The analytical solutions of the critical buckling load are derived for both Euler–Bernoulli and Timoshenko microbeams. The numerical results are depicted to illustrate the effects of material length-scale parameter, length-to-thickness ratio and beam thickness on the postbuckling response.

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant No. 12002086, 52075465), the Hunan Provincial Natural Science Foundation of China (2021JJ30649), the Hunan Science Foundation for Distinguished Young Scholars (Grant No.2019JJ20015), the science and technology innovation Program of Hunan Province (Grant No. 2020RC4038) and the Education Department of Hunan Province (Grant No. 20B565). The financial supports are gratefully acknowledged.

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Authors and Affiliations

  1. School of Mechanical Engineering and Mechanics, Xiangtan University, Hunan, 411105, People’s Republic of China

    Shuohui Yin, Zhibing Xiao, Xuefei Wang & Jingang Liu

  2. Foshan Green Intelligent Manufacturing Research Institute of Xiangtan University, Guangdong, 528311, People’s Republic of China

    Shuohui Yin

  3. Jiangsu Key Laboratory of Engineering Mechanics, School of Civil Engineering, Southeast University, Nanjing, 210096, Jiangsu, People’s Republic of China

    Gongye Zhang

  4. Duy Tan Research Institute for Computational Engineering (DTRICE), Duy Tan University, 6 Tran Nhat Duat, 1 dist., Ho Chi Minh City, 700000, Vietnam

    Tinh Quoc Bui

  5. Faculty of Civil Engineering, Duy Tan University, Da Nang, 550000, Vietnam

    Tinh Quoc Bui

  6. Hunan National Applied Mathematics Center, Hunan, 411105, People’s Republic of China

    Jingang Liu

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  1. Shuohui Yin
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Yin, S., Xiao, Z., Zhang, G. et al. Size-dependent postbuckling for microbeams: analytical solutions using a reformulated strain gradient elasticity theory. Acta Mech 233, 5045–5060 (2022). https://doi.org/10.1007/s00707-022-03360-x

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  • DOI: https://doi.org/10.1007/s00707-022-03360-x

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