## 2 Two-Dimensional Ordinary State-Based Peridynamic Model for DCB Modeling  
### Governing Equation  
The two-dimensional OSB PD composite model used in the present study for DCB modeling is obtained by reformulating the previous OSB PD lamina model ([PERSON] and [PERSON] [27]) in x-z direction. The detailed derivation process is given in Appendix A. The governing equation of the present two-dimensional OSB PD composite model can be written as  
\[\rho_{(k)}\ddot{\mathbf{u}}_{(k)}=\sum_{j=1}^{\infty}\big{[}t_{(k)(j)}- \mathbf{t}_{(j)(k)}\big{]}V_{(j)}+\mathbf{b}_{(k)}, \tag{1}\]  
where \(\rho_{(k)}\) is the density of material point \(\mathbf{x}_{(k)}\), \(\ddot{\mathbf{u}}_{(k)}\) is instantaneous acceleration of \(\mathbf{x}_{(k)}\), as shown in Figure 1, \(\mathbf{b}_{(k)}\) is the external load density, and \(\mathbf{t}_{(k)(j)}\) and \(\mathbf{t}_{(j)(k)}\) are PD force density between \(\mathbf{x}_{(k)}\) and \(\mathbf{x}_{(j)}\). The PD force density can be expressed as \[\mathbf{t}_{(k)(j)}=A_{(k)(j)}\frac{\mathbf{y}_{(j)}-\mathbf{y}_{(k)}}{\left| \mathbf{y}_{(j)}-\mathbf{y}_{(k)}\right|}, \tag{2}\]  
\[\mathbf{t}_{(j)(k)}=B_{(j)(k)}\frac{\mathbf{y}_{(k)}-\mathbf{y}_{(j)}}{\left| \mathbf{y}_{(k)}-\mathbf{y}_{(j)}\right|}, \tag{3}\]  
with  
\[A_{(k)(j)}=2 ad\frac{\delta}{\left|\mathbf{x}_{(j)}-\mathbf{x}_{(k)}\right|} \Lambda_{(k)(j)}\theta_{(k)}+2\delta bs_{(k)(j)}+2\delta(\mu_{F}b_{F}+\mu_{Z}b_ {Z})s_{(k)(j)}, \tag{4}\]  
\[B_{(j)(k)}=2 ad\frac{\delta}{\left|\mathbf{x}_{(k)}-\mathbf{x}_{(j)}\right|} \Lambda_{(j)(k)}\theta_{(j)}+2\delta bs_{(j)(k)}+2\delta(\mu_{F}b_{F}+\mu_{Z} b_{Z})s_{(j)(k)}, \tag{5}\]  
and  
\[s_{(k)(j)}=\frac{\left|\mathbf{y}_{(j)}-\mathbf{y}_{(k)}\right|-\left| \mathbf{x}_{(j)}-\mathbf{x}_{(k)}\right|}{\left|\mathbf{x}_{(j)}-\mathbf{x}_ {(k)}\right|}, \tag{6}\]  
and  
\[\mu_{F}=\left\{\begin{array}{ll}1&\quad(\mathbf{x}_{(j)}-\mathbf{x}_{(k)}) \parallel\text{fiber direction}\\ 0&\quad\text{otherwise}\end{array}\right., \tag{7}\]  
\[\mu_{Z}=\left\{\begin{array}{ll}1&\quad(\mathbf{x}_{(j)}-\mathbf{x}_{(k)}) \bot\text{fiber direction}\\ 0&\quad\text{otherwise}\end{array}\right., \tag{8}\]  
where \(s_{(k)(j)}\) is the stretch of bonds, and \(\delta\) is the radius of the horizon zone. The direction cosines of the relative position vectors between the material points \(\mathbf{x}_{(k)}\) and \(\mathbf{x}_{(j)}\) in the undeformed and deformed states are defined as  
\[\Lambda_{(k)(j)}=\frac{\mathbf{y}_{(j)}-\mathbf{y}_{(k)}}{\left|\mathbf{y}_{ (j)}-\mathbf{y}_{(k)}\right|}\cdot\frac{\mathbf{x}_{(j)}-\mathbf{x}_{(k)}}{ \left|\mathbf{x}_{(j)}-\mathbf{x}_{(k)}\right|}. \tag{9}\]  
The PD dilatation \(\theta_{(k)}\) can be expressed as  
\[\theta_{(k)}=d\sum_{j=1}^{\infty}\delta s_{(k)(j)}\Lambda_{(k)(j)}V_{(j)}. \tag{10}\]  
Figure 1: Peridynamic model notations.  
The PD material parameters \(a,d\) characterize the effect of dilatation, and \(b,b_{F},b_{Z}\) are associated with deformation of material points in arbitrary directions, fiber direction, and thickness direction, respectively. These parameters are related to material properties of CFRP composite and horizon radius. The detailed derivation procedures to get these PD material parameters are illustrated in Appendix A.2. These PD material parameters are related to composite material parameters as  
\[a=\frac{1}{2}(Q_{13}-Q_{55}), \tag{11}\]  
\[d=\frac{2}{\pi\delta^{3}}, \tag{12}\]  
\[b=\frac{6Q_{55}}{\pi\delta^{4}}, \tag{13}\]  
\[b_{F}=\frac{Q_{11}-Q_{13}-2Q_{55}}{2\delta\sum\limits_{j=1}^{J}\Big{|}x_{(j)}-x _{(k)}\Big{|}V_{(j)}}, \tag{14}\]  
\[b_{Z}=\frac{Q_{33}-Q_{13}-2Q_{55}}{2\delta\sum\limits_{j=1}^{J}\Big{|}x_{(j)}-x _{(k)}\Big{|}V_{(j)}}, \tag{15}\]  
where \(Q_{11}\), \(Q_{33}\), \(Q_{13}\), and \(Q_{55}\) are coefficients of stiffness matrix \([Q]\) presented in Appendix A.1.