## 2 The Microstructure Model Considering Nonaffine Deformations
### Governing Equations and Boundary Conditions  
Generally, the governing equations and boundary conditions for a continuum are derived based on [PERSON]'s principle. Therefore, according to the above analysis of macroscale deformation and microscale kinematics and motivated by [PERSON] (2018), we give the expressions of the variations of kinetic energy \(\delta T\), deformation energy \(\delta W\), and the work done by external forces (stresses) \(\delta Q\) as follows:  
\[\delta T=-\int\limits_{V}\left(\rho\ddot{\mathbf{u}}\cdot\delta\mathbf{u}+\rho\ddot{ \mathbf{A}}\cdot\mathbf{\alpha}:\delta\mathbf{A}+\rho\beta\ddot{\mathbf{\omega}}\cdot \delta\mathbf{\omega}+\rho\gamma\ddot{\mathbf{B}}:\delta\mathbf{B}\right)\mathrm{d}V, \tag{7}\]  
\[\delta W=\int\limits_{V}\left[\mathbf{\sigma}:\delta\mathbf{\varepsilon}+\mathbf{\tau}: \delta\mathbf{R}+\mathbb{T}^{3}\vdots\delta(\
abla\mathbf{A})+\mathbf{\mu}:\delta \mathbf{\kappa}+\mathbf{\upsilon}:\delta\mathbf{D}+\mathbb{V}^{3}\vdots\delta( \
abla\mathbf{B})\right]\mathrm{d}V, \tag{8}\]  
\[\delta Q=\int\limits_{V}\left(\mathbf{f}\cdot\delta\mathbf{u}+\mathbf{\Phi}:\delta \mathbf{A}+\mathbf{k}\cdot\delta\mathbf{\omega}+\mathbf{\Pi}:\delta\mathbf{B}\right) \mathrm{d}V+\int\limits_{S}\left(\mathbf{t}\cdot\delta\mathbf{u}+\mathbf{\Upsilon}:\delta \mathbf{A}+\mathbf{g}\cdot\delta\mathbf{\omega}+\mathbf{\Gamma}:\delta\mathbf{B}\right) \mathrm{d}S, \tag{9}\]  
where \(\rho=\frac{1}{V^{\prime}}\int\limits_{V^{\prime}}\rho_{\mathrm{m}}\mathrm{d}V^ {\prime}\) is the macroscale density, \(\rho_{\mathrm{m}}\) is the microscale (particle) density, and \(V^{\prime}\) is the volume of unit cell (microstructure). \(\mathbf{\alpha}=\frac{1}{V^{\prime}}\int\limits_{V^{\prime}}\mathbf{y}\otimes\mathbf{y} \mathrm{d}V^{\prime}\) is the average second order moment of inertia. \(\beta=\frac{1}{V^{\prime}}\int\limits_{V^{\prime}}\mathbf{y}\cdot\mathbf{y}\mathrm{d} V^{\prime}\) and \(\gamma=\frac{1}{V^{\prime}}\int\limits_{V^{\prime}}(\mathbf{y}\cdot\mathbf{y})^{2} \mathrm{d}V^{\prime}\) are the average second and fourth order polar moments of inertia, respectively. Considering one particle as a unit cell for monodisperse granular assemblies, we have \(\beta=\frac{2}{5}R^{2}\) and \(\gamma=\frac{12}{35}R^{4}\), where \(R\) is the radius of particles. The \(\mathbf{\sigma}\) and \(\mathbf{\mu}\) denote the Cauchy stress tensor and couple stress tensor conjugated to the strain \(\mathbf{\varepsilon}\) and curvature \(\mathbf{\kappa}\), respectively; \(\mathbf{\tau}\) and \(\mathbf{\upsilon}\) are relative stress and couple stress tensors, respectively; \(\mathbf{\mathbb{T}}^{3}\) and \(\mathbb{V}^{3}\) are higher-order stress measures conjugated to the macroscale deformation measures \(\
abla\mathbf{A}\) and \(\
abla\mathbf{B}\), respectively. \(\mathbf{f}\) and \(\mathbf{t}\) are the volume force and surface traction, respectively; \(\mathbf{\Phi}\) and \(\mathbf{\Psi}\) are the double body force and surface traction, respectively; \(\mathbf{k}\) and \(\mathbf{g}\) denote the body moment and surface moment, respectively; \(\mathbf{\Pi}\) and \(\mathbf{\Gamma}\) denote the double body moment and surface moment, respectively.  
Using the [PERSON]'s principle \(\int_{0}^{t}(\delta Q-\delta W+\delta T)\mathrm{d}t=0\) and Gauss divergence theorem, we can easily obtain the following governing equations and boundary conditions for the present microstructure model:  
\[\
abla\cdot(\mathbf{\sigma}+\mathbf{\tau})+\mathbf{f}=\rho\ddot{\mathbf{u}},\] (10a) \[\mathbf{\tau}+\
abla\cdot\mathbb{T}^{3}+\mathbf{\Phi}=\rho\mathbf{\alpha} \ddot{\mathbf{A}},\] (10b) \[\
abla\cdot(\mathbf{\mu}+\mathbf{\upsilon})+\mathbf{k}=\rho\mathbf{\beta}\mathbf{ \varpi},\] (10c) \[\mathbf{\upsilon}+\
abla\cdot\mathbb{V}^{3}+\mathbf{\Pi}=\rho\gamma\ddot{ \mathbf{B}},\] (10d) \[(\mathbf{\sigma}+\mathbf{\tau})\cdot\widetilde{\mathbf{n}}=\mathbf{t},\] (11a) \[\mathbb{T}^{3}\cdot\widetilde{\mathbf{n}}=\mathbf{\Psi},\] (11b) \[(\mathbf{\mu}+\mathbf{\upsilon})\cdot\widetilde{\mathbf{n}}=\mathbf{g},\] (11c) \[\mathbb{V