In this work, we propose a non-classical Kirchhoff plate model to investigate surface energy and gradient elasticity effects on the static bending and free vibration behavior of micro-plates. Gurtin-Murdoch surface elasticity theory and a single-parameter gradient elasticity theory are combined to capture three types of size effects. The equations of motion and related boundary conditions of the model are obtained by the energy variational principle. A C²-type differential quadrature finite element is constructed to solve the resulting sixth-order boundary value problem of micro-plates. Theoretical model validation and solution method verification are made through comparison with the available literature. Finally, the new plate model is applied to analyze the mechanical behavior of micro-plates and to carry out detailed parametric studies. Our results demonstrate that the combined effects of three physical reasons can result in not only the stiffening or softening behavior of static deflections and vibration frequencies but also the significant change in the vibration mode shapes of micro-plates. Further, the introduction of strain gradient effect can introduce a boundary layer at the simply supported edge.

在这项工作中，我们提出了一个非经典的Kirchhoff板模型，以研究表面能和梯度弹性对微板的静态弯曲和自由振动行为的影响。Gurtin-Murdoch表面弹性理论和单参数梯度弹性理论相结合以捕获三种类型的尺寸效应。通过能量变分原理获得了模型的运动方程和相关的边界条件。A C ²为了解决微板的六阶边值问题，构造了一种类型的微分正交有限元。通过与现有文献进行比较来进行理论模型验证和求解方法验证。最后，将新的板模型应用于分析微板的力学行为并进行详细的参数研究。我们的结果表明，三种物理原因的综合作用不仅会导致静态挠度和振动频率的变硬或变软，而且还会导致微板的振动模式形状发生重大变化。此外，应变梯度效应的引入可以在简单支撑的边缘处引入边界层。