Skew-symmetric couple stress theory may permit the exploration of solids and fluids at the finest scales for which continuum mechanics applies. This theory is founded upon a true continuum representation. As a result, for solids, the rotation equals one half the curl of the displacement field. In addition, the couple-stress tensor reduces to skew-symmetric form and can be written as a polar couple-stress vector, which is energy conjugate to the mean-curvature polar vector field. The resulting theory is fully self-consistent and parsimonious, requiring only a single extra material parameter, a length scale, for the linear elastic isotropic and cubic single crystal cases. Previous finite element formulations for this theory have required either Lagrange multipliers or penalty parameters to enforce rotation-displacement compatibility, while maintaining C⁰ continuity. Here, we introduce a novel mixed C⁰ variational principle written in terms of displacement and couple-stress polar vectors that avoids any extraneous contributions to the energy functional. This stationary principle, in turn, provides the basis for a robust finite element method, which is developed in this paper for planar quasistatic size-dependent response of linear elastic isotropic media, and then generalized to consider a cubic single crystal example. The finite element implementation uses standard linear three-node triangles for the displacements and tangential edge elements on the triangles to represent the divergence-free couple-stresses. The formulation, however, is quite comprehensive in nature and thus can be extended to examine a broad range of problems, such as those associated with the dynamic response of non-centrosymmetric anisotropic bodies in three-dimensions.

斜对称偶应力理论可能允许在连续介质力学适用的最精细尺度上探索固体和流体。该理论建立在真正的连续统表示之上。因此，对于固体，旋转等于位移场旋度的二分之一。此外，耦合应力张量简化为偏对称形式，可以写成极耦合应力向量，它与平均曲率极向量场的能量共轭。由此产生的理论是完全自洽和简约的，对于线弹性各向同性和立方单晶情况，只需要一个额外的材料参数，一个长度尺度。该理论的先前有限元公式需要拉格朗日乘子或惩罚参数来强制旋转位移兼容性，C ⁰连续性。在这里，我们介绍一种新颖的混合C ⁰根据位移和耦合应力极向量编写的变分原理，避免了对能量泛函的任何无关紧要的贡献。反过来，这种平稳原理为稳健的有限元方法提供了基础，该方法在本文中开发用于线性弹性各向同性介质的平面准静态尺寸相关响应，然后推广到考虑立方单晶示例。有限元实现使用标准线性三节点三角形作为位移和三角形上的切向边缘元素来表示无发散耦合应力。然而，该公式在本质上非常全面，因此可以扩展到检查范围广泛的问题，例如与三维非中心对称各向异性体的动态响应相关的问题。