In this paper, a four-node $$C^{0}$$ tetrahedral element for the modified couple stress theory is proposed. Since the governing equations are the fourth-order differential equations, the first-order derivative of displacement or rotation should be approximated by a continuous function. In the proposed element, nodal rotations are defined using the node-based smoothing technique. Continuous rotation fields are defined with the shape functions and nodal rotations. Both the displacement field and the rotation field are expressed solely in terms of the displacement degrees of freedom. The element stiffness matrix is calculated using the newly defined rotation field. To prevent the increase of calculation cost due to increase of the bandwidth of the stiffness matrix, the preconditioned conjugate gradient method is introduced. The performance of the proposed element is evaluated through various numerical examples.

中文翻译：

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基于节点平滑技术的四节点 C$$^{0}$$0 四面体单元，用于修正偶应力理论

本文提出了修正偶应力理论的四节点$$C^{0}$$四面体单元。由于控制方程是四阶微分方程，位移或旋转的一阶导数应近似为连续函数。在提议的元素中，节点旋转是使用基于节点的平滑技术定义的。连续旋转场由形状函数和节点旋转定义。位移场和旋转场都仅用位移自由度表示。使用新定义的旋转场计算单元刚度矩阵。为了防止刚度矩阵带宽增加导致计算成本增加，引入了预处理共轭梯度法。