## 2 Two-Dimensional Ordinary State-Based Peridynamic Model for DCB Modeling
### Revised Energy-Based Failure Criteria  
Following the approach for deriving the relationship between critical bond breakage work and energy release rate by [PERSON] et al. [15], revised two-dimensional energy-based failure criteria for mode-I delamination growth of CFRP composites are proposed. The influence of the revised energy-based failure criteria will be investigated in Section 4.3.  
This approach assumes that the energy consumed by a growing delamination front equals the work required, per unit delamination front area, to separate two halves of a body across a plane (Figure 2). Suppose a plane \(A\) separates two halves of a two-dimensional body \(B\) into \(B_{+}\) and \(B_{-}\). The delamination front area \(a\) is on the plane. Consider a mode-I delamination motion with velocity field on Figure 2. The total energy \(E\) absorbed by \(P\) in this motion is  
\[E=\int_{0}^{t}W_{abs}(P)dt^{\prime}=\int_{0}^{t_{0}}\int_{P}\int_{B}t_{z}(v_{z }{}^{\prime}-v_{z})dV^{\prime}dVdt. \tag{16}\]  
Figure 2: Computation of total energy absorbed by \(P\) for delamination front area \(a\).  
The assumed critical bond breakage work \(w_{\text{IC}}\) in this motion is  
\[\int_{0}^{t_{0}}t_{z}({v_{z}}^{\prime}-v_{z})dt=w_{\text{IC}}. \tag{17}\]  
Therefore,  
\[\begin{array}{ll}E&=w_{\text{IC}}\int_{P}\int_{B}dV^{\prime}dV=w_{\text{IC}}( \int_{P_{-}}\int_{B_{+}\backslash P_{+}}dV^{\prime}dV+\int_{P_{+}}\int_{B_{-} \backslash P_{-}}dV^{\prime}dV+\int_{P_{-}}\int_{P_{+}}dV^{\prime}dV)\\ &=w_{\text{IC}}(2\int_{P_{-}}\int_{B_{+}}dV^{\prime}dV-\int_{P_{-}}\int_{P_{+} }dV^{\prime}dV)\\ &=w_{\text{IC}}(2a\int_{0}^{\delta}\int_{z}^{\delta}\int_{-\cos^{-1}(z/\xi)} ^{\cos^{-1}(z/\xi)}\xi d\theta d\xi dz-a\int_{0}^{\delta}a(\delta-z)dz)\\ &=w_{\text{IC}}(\frac{4}{3}a\delta^{3}-\frac{1}{2}a^{2}\delta^{2})\end{array}, \tag{18}\]  
For Equation (18), as stated by [PERSON] et al. [15], bonds that do not cross plane \(A\) have no contributions to the integrand. However, in the present study, bonds that cross plane \(A\) are separated into three parts: bonds connecting \(P_{-}\) and \(B_{+}\backslash P_{+}\), bonds connecting \(P_{+}\) and \(B_{-}\backslash P_{-}\), and bonds connecting \(P_{-}\) and \(P_{+}\), as shown in Figure 2. The bond breakage work between \(P_{-}\) and \(P_{+}\) is calculated only once compared with the original procedure in [15]. Adding these three parts of the bond breakage work together gives the revised total absorbed energy \(E\) shown in Equation (18).  
Assuming this total absorbed energy equals the critical energy release rate times the area of delamination front,  
\[G_{\text{IC}}=E/a=w_{\text{IC}}(\frac{4}{3}\delta^{3}-\frac{1}{2}a\delta^{2}). \tag{19}\]  
Thus, the critical bond breakage work for mode-I delamination is related to critical energy release rate,  
\[w_{\text{IC}}=\frac{G_{\text{IC}}}{\frac{4}{3}\delta^{3}-\frac{1}{2}a\delta^{ 2}}. \tag{20}\]  
For two-dimensional numerical modeling, the delamination front area \(a\) can be set as the discretized grid spacing \(dx\),  
\[w_{\text{IC}}=\frac{G_{\text{IC}}}{\frac{4}{3}\delta^{3}-\frac{1}{2}dx\delta^{ 2}}. \tag{21}\]  
From the above derivation, revised energy-based failure criteria for mode-I delamination growth are proposed as  
\[\left|\frac{w_{1}}{w_{\text{IC}}}\right|\geq 1,\ w_{\text{IC}}=\frac{G_{\text{IC} }}{\frac{4}{3}\delta^{3}-\frac{1}{2}dx\delta^{2}},\ w_{1}=\int_{0}^{t}t_{z}({v_{z}} ^{\prime}-v_{z})dt=t_{z}(w^{\prime}-w). \tag{22}\]  
Local damage at a material point is defined as the weighted ratio of the number of eliminated interactions to the total number of initial interactions of the material point with its family members [28].  
The status variable for delamination damage, \(\mu\), is defined as  
\[\mu=\left\{\begin{array}{ll}1,&\left|\frac{w_{1}}{w_{\text{IC}}}\right|<1 \quad\text{no damage}\\ 0,&\left|\frac{w_{1}}{w_{\text{IC}}}\right|\geq 1\quad\text{ damage}\end{array}\right.. \tag{23}\]