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Higher order solution to the Euler buckling threshold for compressible hyperelastic bilayers

可压缩双层结构屈曲临界载荷的高阶渐近解

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Abstract

This study is concerned with the Euler buckling of a bilayer structure composed of two compressible hyperelastic materials subjected to an axial compression. Within the framework of finite elasticity, an exact bifurcation condition is derived based on the linearized incremental equation by employing the neo-Hookean and the Blatz-Ko models. Focusing on thin bilayers, an asymptotic analysis is carried out to deduce two-term (for the neo-Hookean material) and three-term (for the Blatz-Ko material) solutions for the axial and lateral stretches where global buckling takes place. These explicit formulas illustrate a clear picture of how the material and geometric nonlinearities are combined together to affect the critical buckling load and further unravel that the material nonlinearity and the Poisson’s ratio all appear in the higher order terms. A detailed parametric study is performed to elucidate the effect of the material and geometrical parameters on the onset of buckling in virtue of the exact solution, from which the accuracy of the asymptotic solution is examined as well. It is found that the derived asymptotic solution offers extremely precise prediction even for moderately thick bilayers. Compared with the leading-order approximation, which can be obtained by use of the Euler-Bernoulli beam equations, we furnish some higher-order corrections. It is expected that these concise formulas will pave a convenient way to regulate buckling instability in bilayer structures, and further supply a benchmark for estimating the accuracy of certain reduced models.

摘要

本文研究了由可压缩超弹性材料组成的双层结构在轴向压缩下的屈曲失稳问题. 使用neo-Hookean本构模型和Blatz-Ko本构模型, 在有限变形理论框架下推导了欧拉屈曲的临界失稳条件. 对于细长结构, 采用摄动法得到了屈曲临界压缩比的二阶(neo-Hookean材料)和三阶(Blatz-Ko材料)渐近解析解. 基于精确分岔条件分析了结构几何参数和材料参数对失稳临界载荷的影响, 同时校核了渐近解的准确性和适用范围. 相较于根据欧拉伯努利梁理论得到的一阶近似解, 本文给出了一些高阶修正项, 提高了渐近解的精度, 进一步发现材料非线性和泊松比仅对渐近解的高阶项有影响. 研究结果可指导双层结构屈曲失稳的调控, 还可以作为基准问题为新型简化理论和数值方法的精度校核提供参考.

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant Nos. 12072227 and 12021002).

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Authors and Affiliations

  1. Department of Mechanics, School of Mechanical Engineering, Tianjin University, Tianjin, 300354, China

    Yang Liu  (刘洋)

  2. Tianjin Key Laboratory of Modern Engineering Mechanics, Tianjin, 300354, China

    Yang Liu  (刘洋)

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  1. Yang Liu  (刘洋)
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Correspondence to Yang Liu  (刘洋).

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Liu, Y. Higher order solution to the Euler buckling threshold for compressible hyperelastic bilayers. Acta Mech. Sin. 39, 422379 (2023). https://doi.org/10.1007/s10409-023-22379-x

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