## 2 The Microstructure Model Considering Nonaffine Deformations
### Local and Global Constitutive Relations
\begin{bmatrix}a&0&0\\ 0&-\frac{\updelta}{2}&0\\ 0&0&-\frac{\updelta}{2}\end{bmatrix} , \tag{19}\]  
in which \(a\) is the parameter reflecting the degree of anisotropy. The material is isotropic under the case of \(a=0\). The macroscale stress measures mentioned in Section 2.2 can then be obtained in terms of macroscale deformations, i.e., global constitutive relations, by the combination of Equations (16) and (17):  
\[\mathbf{\sigma}=\frac{\partial W}{\partial\upvarepsilon}=\frac{N_{p}\iint\limits \partial w^{\varepsilon}\xi(\gamma,\ \varphi)\mathrm{d}\gamma\mathrm{d}\varphi}{V \partial\upvarepsilon}=\left(N_{p}l^{2}\iint\limits_{s}\mathbf{K}^{\updelta} \otimes\mathbf{n}\otimes\mathbf{n}\zeta\sin\gamma\mathrm{d}\gamma\mathrm{d}\varphi \right):\upvarepsilon=\mathbb{C}^{4}:\upvarepsilon, \tag{20}\]  
\[\mathbf{\tau}=\frac{\partial W}{\partial\mathbf{R}}=\frac{N_{p}\iint\limits_{V} \partial w^{\varepsilon}\xi(\gamma,\ \varphi)\mathrm{d}\gamma\mathrm{d}\varphi}{ \partial\mathbf{R}}=\left(N_{p}l^{2}\iint\limits_{s}\mathbf{K}^{\updelta} \otimes\mathbf{n}\otimes\mathbf{n}\zeta\sin\gamma\mathrm{d}\gamma\mathrm{d}\varphi \right):\mathbf{R}=\mathbb{E}^{4}:\mathbf{R}, \tag{21}\]  
\[\mathbb{T}^{3}=\frac{\partial W}{\partial(\
abla\mathbf{A})}=\frac{N_{p}\iint \limits_{V}\partial w^{\varepsilon}\xi(\gamma,\ \varphi)\mathrm{d}\gamma\mathrm{d}\varphi}{ \partial(\
abla\mathbf{A})}=\left(\frac{N_{p}l^{4}}{4}\iint\limits_{s} \mathbf{K}^{\updelta}\otimes\mathbf{n}\otimes\mathbf{n}\otimes\mathbf{n}\otimes\mathbf{n} \otimes\mathbf{n}\zeta\sin\gamma\mathrm{d}\gamma\mathrm{d}\varphi\right): \mathrm{d}(\
abla\mathbf{A})=\mathbb{H}^{6}:\!\mathrm{d}(\
abla\mathbf{A}), \tag{22}\]  
\[\mathbf{\mu}=\frac{\partial W}{\partial\mathbf{\kappa}}=\frac{N_{p}\iint\limits \partial w^{\varepsilon}\xi(\gamma,\ \varphi)\mathrm{d}\gamma\mathrm{d}\varphi}{V \partial\mathbf{\kappa}}=\left(N_{p}l^{2}\iint\limits_{s}\mathbf{G}^{\updelta }\otimes\mathbf{n}\otimes\mathbf{n}\zeta\sin\gamma\mathrm{d}\gamma\mathrm{d}\varphi \right):\mathbf{\kappa}=\mathbb{J}^{4}:\mathbf{\kappa}, \tag{23}\]  
\[\mathbf{\upsilon}=\frac{\partial W}{\partial\mathbf{B}}=\frac{N_{p}\iint\limits \partial w^{\varepsilon}\xi(\gamma,\ \varphi)\mathrm{d}\gamma\mathrm{d}\varphi}{V \partial\mathbf{D}}=\left(N_{p}l^{2}\iint\limits_{s}\mathbf{G}^{\upvarepsilon }\otimes\mathbf{n}\otimes\mathbf{n}\zeta\sin\gamma\mathrm{d}\gamma\mathrm{d}\varphi \right):\mathbf{D}=\mathbb{M}^{4}:\mathbf{D}, \tag{24}\]  
\[\mathbb{V}^{3}=\frac{\partial W}{\partial(\
abla\mathbf{B})}=\frac{N_{p}\iint \limits_{V}\partial w^{\varepsilon}\xi(\gamma,\ \varphi)\mathrm{d}\gamma\mathrm{d}\varphi}{ \partial(\
abla\mathbf{B})}=\left(\frac{N_{p}l^{4}}{4}\iint\limits_{s} \mathbf{G}^{\updelta}\otimes\mathbf{n}\otimes\mathbf{n}\otimes\mathbf{n}\otimes\mathbf{n} \zeta\sin\gamma\mathrm{d}\gamma\mathrm{d}\varphi\right):\mathrm{d}(\
abla \mathbf{B})=\mathbb{N}^{6}:\!\mathrm{d}(\
abla\mathbf{B}), \tag{25}\]  
where \(N_{p}\) denotes the number of contacts per volume and \(l\) is the length of the branch vector. It can be seen that the present microstructure model consists of six sets of constitutive equations. Equations (20)-(22) are associated with particle displacement, and Equations (23)-(25) are associated with particle rotation. The expressions of constitutive tensors \(\mathbb{C}^{4}\), \(\mathbb{E}^{4}\), \(\mathbb{H}^{6}\), \(\mathbb{J}^{4}\), \(\mathbb{M}^{4}\), and \(\mathbb{N}^{6}\) in macroscale constitutive relations can be found in Appendix B.