Based upon the principle of minimum total potential energy for consistent couple stress theory, a Ritz variational approach is developed using tensor product B-splines for both two- and three-dimensional elastostatic analysis. The underlying theory is size-dependent due to incorporation of the energy conjugate skew-symmetric couple-stress and mean curvature tensors, in addition to the force-stress and strain conjugate pair of classical theory. The use of B-splines as the basis functions can assure the required $C^1$ continuity of the displacement field, while also permitting higher order representations. Both displacements and rotations are essential boundary conditions in this theory. Displacements can be enforced in the usual way, but rotations require special treatment to maintain symmetry and positive definiteness of the stiffness matrix for well-posed elastostatic problems. Several computational examples are considered to validate the formulation, illustrate convergence characteristics, and investigate mechanical behavior under consistent couple stress theory. All previous numerical analysis of consistent couple stress theory has been limited to plane strain problems. Thus, the extension here to three-dimensions is of great importance, especially because the behavior in several cases shows significant deviation from the plane strain solutions. Consequently, the variational formulations and computational methodology presented in this paper can play a critical role in assessing the predictive capability of consistent couple stress theory and in understanding size-dependent elastic response.

基于一致偶应力理论的最小总势能原理，使用张量积B样条开发了一种Ritz变分方法，用于二维和三维弹性静力分析。除了经典理论中的力-应力和应变共轭对之外，由于结合了能量共轭偏对称对称应力和平均曲率张量，因此基础理论与尺寸有关。使用B样条作为基础函数可以确保所需的$C^{\mathrm{1个}}$位移场的连续性，同时还允许更高阶的表示。位移和旋转都是该理论中必不可少的边界条件。位移可以按通常的方式进行，但是旋转需要特殊处理，以保持刚度矩阵的对称性和正定性，以解决弹性弹力问题。。在一致耦合应力理论下，考虑了几个计算示例来验证公式，说明收敛特性并研究力学行为。先前所有的一致偶应力理论的数值分析都局限于平面应变问题。因此，这里扩展到三维非常重要，尤其是因为在某些情况下，其行为显示出与平面应变解的显着偏差。因此，本文介绍的变分公式和计算方法可以在评估一致偶应力理论的预测能力以及理解尺寸依赖性弹性响应中发挥关键作用。