Three-fold symmetry restrictions on two-dimensional micropolar materials

Analysis of the mechanical properties of engineering materials with microstructure generally requires modification of the concept of a simple material. One approach is the theory of micropolar materials which introduces an independent rotation of a material element and the resulting stress and strain tensors are generally non-symmetric. In two-dimensional material models these microstructures are often represented by geometries which exhibit three-fold symmetry in the plane. In this work we investigate the form of the constitutive relations which this three-fold symmetry imposes. We show that three-fold symmetry requires both the stress and couple stress tensors to be isotropic in the plane. We obtain specific constitutive relations for an equilateral triangle structure and for a hexagonal or honeycomb structure and compare these results with the results of previous investigations of these two-dimensional material models.