The classical theory of elasticity is an idealized model of a continuum, which works well for many engineering applications. However, with careful experiments one finds that it may fail in describing behavior in fatigue, at small scales and in structures having high stress concentration factors. Many size-dependent theories have been developed to capture these effects, one of which is the consistent couple stress theory. In this theory, couple stress ${\mu }_{ij}$ is present in addition to force stress ${\sigma }_{ij}$ and its tensor form is shown to have skew symmetry. The mean curvature ${\kappa }_{ij}$, which is defined as the skew-symmetric part of the gradient of rotations, is the correct energy conjugate of the couple stress. This mean curvature ${\kappa }_{ij}$ and strain $e_{ij}$ together contribute to the elastic energy. The scope of this paper is to extend the work to study anisotropic materials and present a corresponding finite element method. A fully displacement based finite element method for couple stress elasticity requires $C^1$ continuity. To avoid this, a mixed formulation is presented with primary variables of displacements $u_i$ and couple stress ${\mu }_i$ vectors, both of which require only $C^0$ continuity. Centrosymmetric classes of materials are considered here for which force stress and strain are decoupled from couple stress and mean curvature in the constitutive relations. Details regarding the numerical implementation are discussed and the effect of couple stress elasticity on anisotropic materials is examined through several computational examples.

弹性的经典理论是连续体的理想模型，在许多工程应用中都很好用。然而，通过仔细的实验​​，人们发现它可能无法描述疲劳，小规模和具有高应力集中系数的结构中的行为。已经开发出许多大小相关的理论来捕获这些影响，其中之一就是一致的耦合应力理论。在这个理论中，情侣压力${\mu }_{\mathrm{一世}Ĵ}$ 除了压力外还存在 ${\sigma }_{\mathrm{一世}Ĵ}$并且其张量形式显示为具有偏斜对称性。平均曲率${\kappa }_{\mathrm{一世}Ĵ}$定义为旋转梯度的斜对称部分，是耦合应力的正确能量共轭。这个平均曲率${\kappa }_{\mathrm{一世}Ĵ}$ 和应变 ${Ë}_{\mathrm{一世}Ĵ}$一起有助于弹性能。本文的范围是扩展对各向异性材料的研究工作，并提出一种相应的有限元方法。基于全位移的耦合应力弹性有限元方法需要$C^{\mathrm{1个}}$连续性。为避免这种情况，提出了一个混合公式，其中包含位移的主要变量${ü}_{\mathrm{一世}}$ 和情侣压力 ${\mu }_{\mathrm{一世}}$ 向量，两者都只需要 $C^0$连续性。此处考虑了中心对称类的材料，对于这些材料，本构关系中的力应力和应变与耦合应力和平均曲率解耦。讨论了有关数值实现的细节，并通过几个计算示例检验了耦合应力弹性对各向异性材料的影响。