In this study, we integrate the advantages of differential quadrature method (DQM) and finite element method (FEM) to construct a C¹‐type four‐node quadrilateral element with 48 degrees of freedom (DOF) for strain gradient Mindlin micro‐plates. This element is free of shape functions and shear locking. The C¹‐continuity requirements of deflection and rotation functions are accomplished by a fourth‐order differential quadrature (DQ)‐based geometric mapping scheme, which facilitates the conversion of the displacement parameters at Gauss‐Lobatto quadrature (GLQ) points into those at element nodes. The appropriate application of DQ rule to non‐rectangular domains is proceeded by the natural‐to‐Cartesian geometric mapping technique. Using GLQ and DQ rules, we discretize the total potential energy functional of a generic micro‐plate element into a function of nodal displacement parameters. Then, we adopt the principle of minimum potential energy to determine element stiffness matrix, mass matrix, and load vector. The efficacy of the present element is validated through several examples associated with the static bending and free vibration problems of rectangular, annular sectorial, and elliptical micro‐plates. Finally, the developed element is applied to study the behavior of freely vibrating moderately thick micro‐plates with irregular shapes. It is shown that our element has better convergence and adaptability than that of Bogner‐Fox‐Schmit (BFS) one, and strain gradient effects can cause a significant increase in vibration frequencies and a certain change in vibration mode shapes.

中文翻译：

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中等厚度微板的应变梯度微分正交有限元

在这项研究中，我们结合了差分正交方法（DQM）和有限元方法（FEM）的优势，为应变梯度Mindlin微板构建了一个具有48个自由度（DOF）的C ¹型四节点四边形元素。该元件没有形状功能和剪切锁定。的Ç ¹偏转和旋转函数的连续性要求是通过基于四阶微分正交（DQ）的几何映射方案实现的，该方案简化了高斯-洛巴托正交（GLQ）点处的位移参数到单元节点处的位移参数的转换。DQ规则在非矩形域中的适当应用是通过自然到笛卡尔的几何映射技术进行的。使用GLQ和DQ规则，我们将通用微板元件的总势能函数离散为节点位移参数的函数。然后，我们采用最小势能原理确定单元刚度矩阵，质量矩阵和载荷矢量。通过与矩形，环形扇形和椭圆形微板的静态弯曲和自由振动问题相关的几个示例验证了本元件的有效性。最后，将开发的元素用于研究自由振动的不规则形状的中等厚度的微板的行为。结果表明，我们的单元具有比Bogner-Fox-Schmit（BFS）单元更好的收敛性和适应性，并且应变梯度效应会导致振动频率显着增加，并且振动模式形状也会发生某些变化。