## 2 The Microstructure Model Considering Nonaffine Deformations  
### Macroscale and Microscale Deformations  
A material point composed of a large number of particles with the volume \(V\) and bounded by surface \(S\) can be generally viewed as a continuum (see Figure 1). In classical continuum theory or other enhanced continuum models, the macroscale deformations are generally considered to be homogeneous within the material point, which can be generally denoted by strain \(\mathbf{\varepsilon}\) for linear deformation and curvature \(\mathbf{\kappa}\) for bending (or twist) deformation, defined as:  
\[\mathbf{\varepsilon}=\frac{1}{2}\Big{(}\
abla\mathbf{u}+(\
abla\mathbf{u})^{\mathrm{T}} \Big{)},\mathbf{\kappa}=\frac{1}{2}\Big{(}\
abla\mathbf{\bar{\omega}}+(\
abla\mathbf{ \bar{\omega}})^{\mathrm{T}}\Big{)} \tag{1}\]where \(u\) denotes the macroscale displacement vector, and \(\varpi\) denotes the macroscale rotation vector. As we know, however, numerous experimental and numerical results have shown that particles inside the granular media undergo significant nonaffine motions under external loading because of its discrete nature and microstructure. When considering the microstructure effect and this nonaffine deformation of microstructure, we introduce a second rectangular coordinate system \(y\), a so-called microscale coordinate system parallel to the macroscale coordinate system \(x\), the origin of which is attached to the barycenter of the microstructure unit (or unit cell) \(P\), as shown in Figure 1. Hence, the kinematics of particles within a unit cell are the function of not only the macroscale coordinate system \(x\) but also the microscale coordinate system \(y\). At the microscale level, the translation \(v\) and the rotation \(\omega\) of particle \(n\), according to [18,19], can be approximated from the deformation of its neighboring particle \(m\) using a polynomial expansion:  
\[v^{n}(x,y^{n})=\
u^{m}(x,y^{m})+\mathbf{A}(x)\cdot(y^{n}-y^{m})+\frac{1}{2} \mathbb{A}^{3}(x)\cdot(y^{n}-y^{m})\cdot(y^{n}-y^{m}), \tag{2}\]  
\[\omega^{n}(x,y^{m})=\omega^{m}(x,y^{m})+\mathbf{B}(x)\cdot(y^{n}-y^{m})+\frac {1}{2}\mathbb{B}^{3}(x)\cdot(y^{n}-y^{m})\cdot(y^{n}-y^{m}), \tag{3}\]  
where \(\mathbf{A}\) and \(\mathbf{B}\) are microscale strain and curvature, respectively; \(\mathbb{A}^{3}\) and \(\mathbb{B}^{3}\) are the gradient of microscale strain and curvature, respectively. For simplicity, in this paper, these deformations are assumed to be only dependent on the macroscale coordinate system and hence they are macroscale measures. Due to the difference between macroscale and microscale deformations, e.g., the difference of \(\mathbf{A}\) and \(\varepsilon\), here we introduce relative deformation tensors to account for these differences, given by  
\[\mathbf{R}=\varepsilon-\mathbf{A},\;\mathbb{R}^{3}=\
abla\mathbf{A}-\mathbb{A }^{3};\;\mathbf{D}=\kappa-\mathbf{B},\;\mathbb{D}^{3}=\
abla\mathbf{B}- \mathbb{B}^{3}, \tag{4}\]  
To simplify the problem, the higher-order relative deformation tensors \(\mathbb{D}^{3}\) and \(\mathbb{R}^{3}\) are assumed to be zero. In other words, the gradient of strain and curvature are homogeneous in the material point. By substituting Equation (4) into Equations (2) and (3), we can obtain the microscale relative displacement \(\Delta^{c}\) and rotation \(\Theta^{c}\) between two particles in terms of macroscale deformations:  
\[ \begin{array}{l}\Delta^{c}=v^{n}-v^{m}=\mathbf{A}\cdot(y^{n}-y^{m})+\frac{1 }{2}\mathbb{A}^{3}\cdot(y^{n}-y^{m})\cdot(y^{n}-y^{m})\\ =\varepsilon\cdot\mathcal{F}-\mathbf{R}\cdot\mathcal{F}+\frac{1}{2}\
abla \mathbf{A}:\mathbf{L}^{c}=\Delta^{\alpha\alpha}-\Delta^{\alpha\tau}+\Delta^{ \alpha\xi}\end{array} , \tag{5}\]  
Figure 1: Macroscale material point and its granular microstructure.