Abstract
The symplectic approach was utilized to derive solutions to the orthotropic micropolar plane stress problem. The Hamiltonian canonical equation was first obtained by applying Legendre’s transformation and the Hamiltonian mixed energy variational principle. Then, by using the method of separation of variables, the eigenproblem of the corresponding homogeneous Hamiltonian canonical equation was derived. Subsequently, the corresponding eigensolutions for three kinds of homogeneous boundary conditions were derived. According to the adjoint symplectic orthogonality of the eigensolutions and expansion theorems, the solutions to this plane stress problem were expressed as a series expansion of these eigensolutions. The numerical results for the orthotropic micropolar plane stress problem under various boundary conditions were presented and validated using the finite element method, which confirmed the convergence and accuracy of the proposed approach. We also investigated the relationship between the size-dependent behaviour and material parameters using the proposed approach. Furthermore, this approach was applied to analyze lattice structures under an equivalent micropolar continuum approximation.
摘要
本文采用辛弹性力学方法得到正交各向异性微极平面应力问题的解析解. 首先, 应用勒让德变换和哈密顿混合能量变分原理得 到哈密顿正则方程. 然后, 利用分离变量法, 导出了相应的齐次哈密顿正则方程的本征问题. 最后, 推导了三种齐次边界条件下问题的 相应本征解. 根据本征解的共轭辛正交性和展开定理, 正交各向异性微极平面应力问题的解可以表示为这些本征解的级数展开. 本文 给出了各种边界条件下平面应力问题的数值结果, 并用有限元法证明了该方法的收敛性和准确性. 同时使用该方法研究了尺寸效应和 材料尺度参数之间的关系. 并且, 在等效微极连续介质近似下, 使用辛方法分析格子结构的力学行为.
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Acknowledgements
This work was supported by the National Key R&D Program of China (Grant No. 2022YFB4201200), Technology Major Project (Grant No. J2019-IV-0019-0087), and National Science and Technology Major Project (Grant No. J2019-IV-0019-0087).
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Author contributions Long Chen: Methodology, Software, Visualization, Writing — original draft, Writing — review & editing. Zhaofei Tang: Methodology, Writing — review & editing. Qiong Wu: Methodology, Software. Qiang Gao: Conceptualization, Methodology, Writing — review & editing.
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Chen, L., Tang, Z., Wu, Q. et al. Symplectic solutions for orthotropic micropolar plane stress problem. Acta Mech. Sin. 41, 423548 (2025). https://doi.org/10.1007/s10409-024-23548-x
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DOI: https://doi.org/10.1007/s10409-024-23548-x