## 2 Generation of Mesostructures and Numerical Models
### Numerical Model
#### 2.4.2 Constitutive Response of the CIEs  
Because the cohesive crack model can simulate the softening and plastic behavior of the material in the crack tip, it can accurately simulate the damage and fracture process of the material. The constitutive response of CIE is defined in terms of stress (traction) versus displacement (separation) of adjacent virtual surfaces. Normal traction, \(\mathsf{t}_{\mathsf{n}}\), and tangential traction, \(\mathsf{t}_{\mathsf{r}}\), act on adjacent virtual surfaces, and their corresponding separations are crack opening displacement, \(\delta_{\mathsf{n}}\), and crack sliding displacement, \(\delta_{\mathsf{t}}\). In this paper, the control equations of the bilinear traction separation law, which are widely used in finite element software, are defined as:  
\[\mathsf{t}_{\mathsf{n}}=\left\{ \begin{array}{cc}\frac{\mathrm{t}_{\mathsf{n }}^{0}}{\delta_{\mathsf{n}}^{0}}\delta_{\mathsf{n}}&(\delta_{\mathsf{n}}\leq \delta_{\mathsf{n}}^{0})\\ \mathrm{t}_{\mathsf{n}}^{0}\frac{\delta_{\mathsf{n}}^{0}-\delta_{\mathsf{n}}} {\delta_{\mathsf{n}}^{0}-\delta_{\mathsf{n}}^{0}}&(\delta_{\mathsf{n}}>\delta _{\mathsf{n}}^{0})\end{array} \right.\qquad\mathsf{t}_{\mathsf{r}}=\left\{ \begin{array}{cc}\frac{\mathrm{t}_{\mathsf{n}}^{0}}{\delta_{\mathsf{n}}} \delta_{\mathsf{t}}&(\delta_{\mathsf{t}}\leq\delta_{\mathsf{t}}^{0})\\ \mathrm{t}_{\mathsf{t}}^{0}\frac{\delta_{\mathsf{t}}^{0}-\delta_{\mathsf{t}}} {\delta_{\mathsf{t}}^{0}-\delta_{\mathsf{t}}^{0}}&(\delta_{\mathsf{t}}>\delta _{\mathsf{t}}^{0})\end{array} \right. \tag{3}\]  
where \(\mathrm{t}_{\mathsf{n}}^{0}\) and \(\mathrm{t}_{\mathsf{t}}^{0}\) are the maximum normal traction and the maximum tangential traction, respectively, \(\delta_{\mathsf{n}}^{0}\) and \(\delta_{\mathsf{t}}^{0}\) are normal separation and tangential separation when the normal stress and tangential stress are at their maximum, respectively, and \(\delta_{\mathsf{n}}^{\mathrm{f}}\) and \(\delta_{\mathsf{t}}^{\mathrm{f}}\) are the normal and shear separations when debonding is completed, respectively. Under stretching, shearing, or their combined action, the CIE will be damaged. The quadratic nominal stress law is adopted as the basis for judging the damage initiation in this paper, as follows:  
\[\left\{\frac{\left\langle\mathsf{t}_{\mathsf{n}}\right\rangle}{\mathrm{t}_{ \mathsf{n}}^{0}}\right\}^{2}+\left\{\frac{\mathsf{t}_{\mathsf{t}}}{\mathrm{t}_ {\mathsf{t}}^{0}}\right\}^{2}=1 \tag{4}\]  
where \(\left\langle\right\rangle\) is the Macaulay bracket and  
\[\left\langle\mathsf{t}_{\mathsf{n}}\right\rangle=\left\{ \begin{array}{cc} \mathsf{t}_{\mathsf{n}},&\mathsf{t}_{\mathsf{n}}\geq 0(\mathsf{tension})\\ 0,&\mathsf{t}_{\mathsf{n}}<0(\mathsf{compression})\end{array} \right. \tag{5}\]  
When the CIE develops into the softening state, the damage evolution is characterized by the scalar damage index, \(\mathsf{D}\), which is a function of the effective relative separation, \(\delta_{\mathsf{m}}\), combining the effects of \(\delta_{\mathsf{n}}\) and \(\delta_{\mathsf{t}}\):  
\[\delta_{\mathsf{m}}=\sqrt{\left\langle\delta_{\mathsf{n}}\right\rangle^{2}+ \delta_{\mathsf{t}}^{2}} \tag{6}\]  
The damage variable, \(\mathsf{D}\), of the linear softening law, which monotonically increases from \(0\) to \(1\) under further loading after the initiation of damage, is calculated as:  
\[\mathsf{D}=\left\{ \begin{array}{cc}\delta_{\mathsf{m}}^{\mathrm{f}}(\delta_ {\mathsf{m}}^{\mathrm{max}}-\delta_{\mathsf{m}}^{0}),&\delta_{\mathsf{m}}^{ \mathrm{max}}\geq\delta_{\mathsf{m}}^{0}\\ 0,&\delta_{\mathsf{m}}^{\mathrm{max}}\leq\delta_{\mathsf{m}}^{0}\end{array} \right. \tag{7}\]  
where \(\delta_{\mathsf{m}}^{0}\) and \(\delta_{\mathsf{m}}^{f}\) are the effective separations at the damage initiation and complete failure, respectively, and \(\delta_{\mathsf{m}}^{max}\) is the maximum effective relative separation during the entire loading process. Once the damage variable, \(\mathsf{D}\), is determined, the tractions can be calculated as:  
\[\mathsf{t}_{\mathsf{n}}=\left\{