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<p>This study investigates the size-dependent eigen-buckling behavior of Kirchhoff microplates within the framework of a modified strain gradient theory (MSGT), which introduces three intrinsic material length scale parameters and leads to a sixth-order governing differential equation. Using the variational principle, both classical and non-classical boundary conditions are systematically derived in a concise form. The direct separation of variables method is extended to solve the eigenvalue problem arising from the MSGT model. In particular, for configurations where two adjacent edges are clamped and the remaining edges are either clamped or simply supported, closed-form analytical solutions which have previously been considered unattainable are achieved for the first time. A comprehensive parametric study is performed to examine the influence of various boundary conditions and material length scale parameters on the critical buckling loads, thereby elucidating the inherent size-dependent stability behavior at micro- and nano-scales. These results provide critical insights into the eigen-buckling characteristics of Kirchhoff microplates and further contribute to the theoretical foundation for the design and analysis of micro-/nano-scale systems.</p>

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A novel analytical solution method for eigen-buckling of microplates based on a modified strain gradient theory

  • 期刊论文

  • 发表日期：

    2026年1月5日

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  • Yifeng Tian,
  • Wei Xiang

摘要

This study investigates the size-dependent eigen-buckling behavior of Kirchhoff microplates within the framework of a modified strain gradient theory (MSGT), which introduces three intrinsic material length scale parameters and leads to a sixth-order governing differential equation. Using the variational principle, both classical and non-classical boundary conditions are systematically derived in a concise form. The direct separation of variables method is extended to solve the eigenvalue problem arising from the MSGT model. In particular, for configurations where two adjacent edges are clamped and the remaining edges are either clamped or simply supported, closed-form analytical solutions which have previously been considered unattainable are achieved for the first time. A comprehensive parametric study is performed to examine the influence of various boundary conditions and material length scale parameters on the critical buckling loads, thereby elucidating the inherent size-dependent stability behavior at micro- and nano-scales. These results provide critical insights into the eigen-buckling characteristics of Kirchhoff microplates and further contribute to the theoretical foundation for the design and analysis of micro-/nano-scale systems.