# Impact Fracture and Fragmentation of Glass via the 3D Combined Finite-Discrete Element Method +
## 3 Generalized Traction-Separation Model for Fracture and Fragmentation
\begin{array}{c}\sigma=\frac{\delta_{n}}{\delta_{1}^{2}}f(\delta)\\ \tau_{1}=\frac{\beta_{1}^{2}}{\alpha_{1}^{2}}\frac{\delta_{t_{1}}}{\delta}f( \delta)\\ \tau_{2}=\frac{\beta_{2}^{2}}{\alpha_{2}^{2}}\frac{\delta_{t_{2}}}{\delta}f( \delta)\end{array}  \tag{9}\]  
The energy release calculated from Equation (5) is  
\[G=\int\mathbf{p}\cdot\mathrm{d}\mathbf{\delta}=\int\
abla\varphi(\delta)\cdot \mathrm{d}\mathbf{\delta}=\int f(\delta)\mathrm{d}\delta \tag{10}\]  
which indicates that the total energy release for a growing crack does not depend on the opening path, i.e., the fracture energy release rates calculated from Equation (5) for mode I and mode II are the same. In order to introduce different fracture energy release rates for different modes, following the approach of [PERSON] and [PERSON] [43], the tractions are modified to  
\[ \begin{array}{c}\sigma=\frac{\delta_{n}}{\delta_{2}}f(\delta)\\ \tau_{1}=\frac{\beta_{1}^{2}}{\alpha_{1}}\frac{\delta_{t_{1}}}{\delta}f(\delta )\\ \tau_{2}=\frac{\beta_{2}^{2}}{\alpha_{2}^{2}}\frac{\delta_{t_{2}}}{\delta}f( \delta)\end{array}  \tag{11}\]which needs four material parameters (\(\alpha_{1}\), \(\beta_{1}\), \(\alpha_{2}\), and \(\beta_{2}\)) and one function \(f(\delta)\) to define the material's traction-separation law. It is noted that the traction-separation law introduced in [43] can be retrieved from the proposed model when \(\alpha_{1}=\alpha_{2}\) and \(\beta_{1}=\beta_{2}\). For pure tension, one has  
\[ \begin{array}{c}\delta=\delta_{n}\\ \sigma=f(\delta)\end{array}  \tag{12}\]  
which implies that the traction-separation curve can be determined through pure tension. Equation (12) yields  
\[ \begin{array}{c}\delta_{nc}=\delta_{c}\\ \sigma_{c}=f(0)\\ G_{\rm I}=\int\sigma\mathrm{d}\delta_{n}=\int f(\delta)\mathrm{d}\delta\end{array}  \tag{13}\]  
where \(\delta_{c}\) is a critical value at which point the function \(f(\delta_{c})=0\), while \(\delta_{nc}=\delta_{c}\) is the maximum tensile separation at which point the tensile stress is \(\sigma=0\). \(\sigma_{c}\) is the tensile strength and \(G_{\rm I}\) is the fracture energy release rate for mode I fracture. For pure shear in direction \(\mathbf{t}_{1}\), one has  
\[ \begin{array}{c}\delta=\frac{\beta_{1}}{\alpha_{1}}\delta_{t_{1}}\\ \tau_{1}=\beta_{1}f(\delta)\end{array}  \tag{14}\]  
which yields  
\[ \begin{array}{c}\delta_{t_{1}c}=\frac{\alpha_{1}}{\beta_{1}}\delta_{c}= \frac{\alpha_{1}}{\beta_{1}}\delta_{nc}\\ \tau_{1c}=\beta_{1}f(0)=\beta_{1}\sigma_{c}\\ G_{\rm II_{1}}=\int\tau_{1}\mathrm{d}\delta_{t_{1}}=\alpha_{1}\int f(\delta) \mathrm{d}\delta=\alpha_{1}G_{\rm I}\end{array}  \tag{15}\]  
where \(\delta_{t_{1}c}\) and \(\tau_{1c}\) are the maximum shear separation and the shear strength in direction \(\mathbf{t}_{1}\), respectively, while \(G_{\rm II_{1}}\) is the fracture energy release rate with respect to \(\mathbf{t}_{1}\). Equation (15) implies that the material parameter \(\alpha_{1}\) is the ratio of \(G_{\rm II_{1}}\) and \(G_{\rm I}\), while the parameter \(\beta_{1}\) is the ratio of shear strength \(\tau_{1c}\) and tensile strength \(\sigma_{c}\). Similarly, for pure shear in direction \(\mathbf{t}_{2}\), one has  
\[ \begin{array}{c}\delta=\frac{\beta_{2}}{\alpha_{2}}\delta_{t_{2}}\\ \tau_{2}=\beta_{2}f(\delta)\end{array}  \tag{16}\]  
which yields  
\[ \begin{array}{c}\delta_{t_{2}c}=\frac{\alpha_{2}}{\beta_{2}}\delta_{c}= \frac{\alpha_{2}}{\beta_{2}}\delta_{nc}\\ \tau_{2c}=\beta_{2}f(0)=\beta_{2}\sigma_{c}\\ G_{\rm II_{2}}=\int\tau_{2}\mathrm{d}\delta_{t_{2}}=\alpha_{2}\int f(\delta) \mathrm{d}\delta=\alpha_{2}G_{\rm I}\end{array}  \tag{17}\]  
where \(\delta_{t_{2}c}\) and \(\tau_{2c}\) are the maximum shear separation and the shear strength in direction \(\mathbf{t}_{2}\), respectively, while \(G_{\rm II_{2}}\) is the fracture energy release rate with respect to \(\mathbf{t}_{2}\).