# Impact Fracture and Fragmentation of Glass via the 3D Combined Finite-Discrete Element Method +
## 3 Generalized Traction-Separation Model for Fracture and Fragmentation  
In the cohesive zone models, the bonding stresses along the damage zone or discrete crack are defined as functions of the local separation (displacement). As shown in Figure 1a, the local separation at any point on the surfaces of a crack can be divided into three components  
\[\delta=\delta_{n}\mathbf{n}+\delta_{t_{1}}\mathbf{t}_{1}+\delta_{t_{2}} \mathbf{t}_{2} \tag{2}\]  
where \(\mathbf{n}\), \(\mathbf{t}_{1}\), and \(\mathbf{t}_{2}\) are the normal and tangential material axes which define a local coordinate systems moving and rotating with the material, while \(\delta_{n}\), \(\delta_{t_{1}}\), and \(\delta_{t_{2}}\) are the normal and tangential separations at any point in the damage zone, respectively. Accordingly, the traction vector \(\mathbf{p}\) is also divided into three components with respect to the material axis \(\mathbf{n}\), \(\mathbf{t}_{1}\), and \(\mathbf{t}_{2}\) (Figure 1b),  
\[\mathbf{p}=\sigma\mathbf{n}+\tau_{1}\mathbf{t}_{1}+\tau_{2}\mathbf{t}_{2} \tag{3}\]  
where \(\sigma\), \(\tau_{1}\), and \(\tau_{2}\) are the normal and tangential stresses in the direction of \(\mathbf{n}\), \(\mathbf{t}_{1}\), and \(\mathbf{t}_{2}\). One approach that can be used to construct a relationship between traction and separation is to establish a potential-based solution that defines the traction vector as a function of the interfacial potential gradient with respect to the components of the separation vec  
Figure 1: Separation and traction at a point on the surfaces of a crack; (**a**) separation vector divided into three components, (**b**) traction vector divided into three components. tor [35; 36]. The interfacial potential is defined as a function of the components of the separation vector as  
\[\mathbf{p}=\
abla\varphi(\delta_{n_{t}}\delta_{t_{1}},\delta_{t_{2}})=\frac{ \partial\varphi}{\partial\delta_{n}}\mathbf{n}+\frac{\partial\varphi}{ \partial\delta_{t_{1}}}\mathbf{t_{1}}+\frac{\partial\varphi}{\partial\delta_{ t_{2}}}\mathbf{t_{2}} \tag{4}\]  
One can further assume that the interfacial potential is a function of an intermediate single variable \(\delta\) which is a function of the separation vector, thus  
\[\mathbf{p}=\
abla\varphi(\delta)=\bigg{(}\frac{\partial\delta}{\partial\delta _{n}}\mathbf{n}+\frac{\partial\delta}{\partial\delta_{t_{1}}}\mathbf{t_{1}}+ \frac{\partial\delta}{\partial\delta_{t_{2}}}\mathbf{t_{2}}\bigg{)}f(\delta) \tag{5}\]  
where  
\[\delta=\sqrt{\delta_{n}^{2}+\frac{\beta_{1}^{2}}{\alpha_{1}^{2}}\delta_{t_{1}} ^{2}+\frac{\beta_{2}^{2}}{\alpha_{2}^{2}}\delta_{t_{2}}^{2}} \tag{6}\]  
is defined as the weighted combination of the components of the separation vector, where the \(\alpha_{1}\), \(\alpha_{2}\), \(\beta_{1}\), and \(\beta_{2}\) are the material parameters which will be discussed later in this section. The derivatives of \(\delta\) with respect to \(\delta_{n}\), \(\delta_{t_{1}}\), and \(\delta_{t_{2}}\) are  
\[ \begin{array}{c}\frac{\partial\delta}{\partial\delta_{n}}=\frac{\delta_{n}} {\delta}\\ \frac{\partial\delta}{\partial\delta_{t_{1}}}=\frac{\beta_{1}^{2}}{\alpha_{1}^ {2}}\frac{\delta_{t_{1}}}{\alpha_{1}^{2}}\\ \frac{\partial\delta}{\partial\delta_{t_{2}}}=\frac{\delta_{2}^{2}}{\alpha_{2}^ {2}}\frac{\delta_{t_{2}}}{\delta}\end{array}  \tag{7}\]  
While  
\[f(\delta)=\frac{\mathrm{d}\varphi(\delta)}{\mathrm{d}\delta} \tag{8}\]  
defines the shape of the traction-separation curve and should be determined according to experimental data. Theoretically, \(f(\delta)\) could be in any form, such as a linear function, bi-linear function, or a general nonlinear function, however different types of materials should use different traction-separation curves [41; 42]. The discussion of the influence of the shape of the traction-separation curves on the numerical results is out of the scope of this work. The interested reader can find more information in [41; 42]. The components of the traction vector are  
\[