## 3 Finite Element Expressions
\begin{bmatrix} \boldsymbol{L}_{1}^{T}\hat{\boldsymbol{D}}_{1}\boldsymbol{L}_{1}&\boldsymbol{ \theta}_{8\times 2}\\ \boldsymbol{\theta}_{8\times 2}&\boldsymbol{L}_{2}^{T}\hat{\boldsymbol{D}}_{2} \boldsymbol{L}_{2}\end{bmatrix}  \tag{13}\]  
with  
\[\boldsymbol{L}= \begin{bmatrix}\boldsymbol{L}_{1}&\boldsymbol{\theta}_{8 \times 2}\\ \boldsymbol{\theta}_{2\times 8}&\boldsymbol{L}_{2}\end{bmatrix} ,\boldsymbol{L}_{1}= \begin{bmatrix}\boldsymbol{\lambda}\\ \boldsymbol{\lambda}\\ \boldsymbol{\lambda}\\ \boldsymbol{\lambda}\\ \boldsymbol{\lambda}\end{bmatrix} ,\text{and}\ \boldsymbol{L}_{2}= \boldsymbol{\lambda}= \begin{bmatrix}\cos\theta&\sin\theta\\ -\sin\theta&\cos\theta\end{bmatrix}  \tag{14}\]  
where \(\theta\) is the included angle between the global coordinate system and the local coordinate system. Using Equations (11) and (12), the material stiffness matrix \(\boldsymbol{K}_{ij}\) is expressed in the following form:  
\[\boldsymbol{K}_{ij}=\boldsymbol{B}_{i}^{T}\boldsymbol{D}\boldsymbol{B}_{j}= \begin{pmatrix}\boldsymbol{B}_{i1}^{T}\boldsymbol{D}_{1}\boldsymbol{B}_{j1}& \boldsymbol{B}_{i1}^{T}\boldsymbol{D}_{1}\boldsymbol{B}_{j2}\\ \boldsymbol{B}_{i2}^{T}\boldsymbol{D}_{1}\boldsymbol{B}_{j1}&\boldsymbol{B}_{i 2}^{T}\boldsymbol{D}_{2}\boldsymbol{B}_{j2}+\boldsymbol{B}_{i3}^{T}\boldsymbol{ D}_{2}\boldsymbol{B}_{j3}\end{pmatrix}  \tag{15}\]  
In this case, the Cosserat strain tensor will be divided into two parts, symmetry and asymmetry [26]:  
\[\left\{ \begin{array}{l}e_{ij}^{sym}=(u_{j,i}+u_{i,j})/2,\kappa_{i3}=\omega_{i}^ {c}\\ e_{ij}^{attn}=(u_{j,i}-u_{i,j})/2+e_{ij}\omega^{c}\end{array} \right. \tag{20}\]  
where \(e_{ij}\) is the alternating tensor, and \"sym\" and \"ant\" of the superscript denote symmetry and skew symmetry, respectively. Similarly, the Cosserat stress tensor will be divided into two parts, i.e., symmetry and asymmetry:  
\[\left\{ \begin{array}{l}\sigma_{ij}^{sym}=D_{ijkl}^{sym}e_{kl}^{sym}\\ \sigma_{ij}^{ant}=(D_{ijkl}-D_{ijkl}^{sym})e_{kl}^{att}\\ m_{i3}=B_{i}\kappa_{i3}\end{array} \right. \tag{21}\]  
where \(D_{ijkl}\) and \(B_{i}\) are material stiffness tensors. By using symmetrical and skew-symmetric stress tensors, three invariant expressions of the stress deviator tensors \(I_{1},J_{2}\) and \(J_{3}\) are modified in the classical continuum theory. From Equation (21), we obtain  
\[\left\{ \begin{array}{l}I_{1}=\sigma_{11}+\sigma_{22}+\sigma_{33}\\ J_{2}=J_{2}^{sym}+J_{2}^{att}+J_{2}^{m}\\ J_{3}=s_{11}^{sym}s_{22}^{sym}s_{33}^{sym}-s_{12}^{sym}s_{21}^{sym}s_{33}^{ sym}\end{array} \right. \tag{22}\]  
where  
\[\left\{ \begin{array}{l}J_{2}^{sym}=[{(s_{11}^{sym})}^{2}+{(s_{22}^{sym})}^{ 2}+{(s_{33}^{sym})}^{2}+{(s_{12}^{sym})}^{2}+\\ {(s_{21}^{sym})}^{2}+{(s_{23}^{sym})}^{2}+{(s_{32}^{sym})}^{2}+{(s_{13}^{sym}) }^{2}]/2\\ J_{2}^{attn}=[{(s_{21}^{att})}^{2}+{(s_{21}^{att})}^{2}+{(s_{23}^{att})}^{2}+{( s_{23}^{att})}^{2}]/2\\ J_{2}^{m}=m_{12}m_{21}\\ s_{ij}^{sym}=\sigma_{ij}^{sym}-\delta_{ij}I_{1}/3,\;s_{ij}^{att}=\sigma_{ij}^{ ant}\\ \sigma_{ij}^{sym}=\left\{\begin{array}{l}\sigma_{ij}\\ {(\sigma_{ij}+\sigma_{ji})}/2\,,i\
eq j\end{array}\right.\end{array} \right. \tag{23}\]  
In the Cosserat continuum medium, the M-C and D-P yield criteria are assumed to both be expressed as invariants of the stress tensor, and their cross-sections in the principal stress space and \(\pi\)-plane can be found in Figure 3. Similar to the classical continuum, the M-C and D-P yield criteria were established in the Cosserat medium by using three invariants of the stress tensor, which can be written as  
\[\left\{ \begin{array}{l}f=\frac{1}{3}I_{1}\sin\varphi+\sqrt{J_{2}}(\cos \theta-\frac{1}{\sqrt{3}}\sin\theta\sin\varphi)-c\cos\varphi=0\\ f=\alpha I_{1}+\sqrt{J_{2}}-k=0\end{array} \right.