# Impact Fracture and Fragmentation of Glass via the 3D Combined Finite-Discrete Element Method +
## 3 Generalized Traction-Separation Model for Fracture and Fragmentation
Equation (17) implies that the material parameter \(\alpha_{2}\) is the ratio of \(G_{\rm II_{2}}\) and \(G_{\rm I}\), while the parameter \(\beta_{2}\) is the ratio of shear strength \(\tau_{2c}\) and tensile strength \(\sigma_{c}\). One can further introduce the damage as  
\[d=\frac{\delta}{\delta_{c}}=\sqrt{\frac{\delta_{n}^{2}}{\delta_{c}^{2}}+\frac{ \beta_{1}^{2}\delta_{t_{1}}^{2}}{\alpha_{1}^{2}\delta_{c}^{2}}+\frac{\beta_{2} ^{2}\delta_{t_{2}}^{2}}{\alpha_{2}^{2}\delta_{c}^{2}}} \tag{18}\]  
It is noted that the damage \(d\) should be less than 1.0 since \(d\geq 1.0\) means the material is completely damaged. To stabilize the damage representation equation, \(d\) is therefore set to 1.0 as long as \(d\geq 1.0\). Substituting Equations (13), (15), and (17) into Equation (18), one gets  
\[d=\sqrt{d_{n}^{2}+d_{t_{1}}^{2}+d_{t_{2}}^{2}}=\sqrt{\left(\frac{\delta_{n}}{ \delta_{nc}}\right)^{2}+\left(\frac{\delta_{t_{1}}}{\delta_{t_{1}c}}\right)^{ 2}+\left(\frac{\delta_{t_{2}}}{\delta_{t_{2}c}}\right)^{2}} \tag{19}\]where  
\[d_{n}=\frac{\delta_{n}}{\delta_{n}c};\;d_{t_{1}}=\frac{\delta_{t_{1}}}{\delta_{t_{ 1}c}};\;d_{t_{2}}=\frac{\delta_{t_{2}}}{\delta_{t_{2}c}} \tag{20}\]  
are the damage components in \(\mathbf{n}\), \(\mathbf{t}_{1}\), and \(\mathbf{t}_{2}\), respectively. The tractions can also be reformed as functions of damage  
\[ \begin{array}{c}\sigma=\frac{\delta_{n}}{\delta_{n}}f(\delta)=\frac{d_{n}}{d }\sigma_{c}z(d)\\ \tau_{1}=\frac{\beta_{1}^{2}}{\alpha_{1}}\frac{\delta_{t_{1}}}{\delta}f(\delta )=\frac{d_{1}}{d}\tau_{1c}z(d)\\ \tau_{2}=\frac{\beta_{2}^{2}}{\alpha_{2}}\frac{\delta_{t_{2}}}{\delta}f(\delta )=\frac{d_{t_{2}}}{d}\tau_{2c}z(d)\end{array}  \tag{21}\]  
where \(z(d)\) is the normalized form of \(f(\delta)\)  
\[z(d)=\frac{1}{\sigma_{c}}f(\delta) \tag{22}\]  
In this paper, the heuristic softening function \(z(d)\) introduced in [37] was utilized  
\[z(d)=\left[1-\frac{a+b-1}{a+b}e^{d(a+cb/((a+b)(1-a-b)))}\right]\times\left[a(1 -d)+b(1-d)^{c}\right] \tag{23}\]  
where \(a\), \(b\), \(c\) are the parameters chosen to fit a particular experimental curve. To generalize the traction-separation model proposed in this work, Equation (21) is redefined as  
\[ \begin{array}{c}\sigma=\sigma_{c}g_{n}(d_{n},d)z(d)\\ \tau_{1}=\tau_{1c}g_{1}(d_{t_{1}},d)z(d)\\ \tau_{2}=\tau_{2c}g_{t_{2}}(d_{t_{2}},d)z(d)\end{array}  \tag{24}\]  
where \(g_{n}\) is a function of \(d_{n}\) and \(d\); \(g_{t_{1}}\) is a function of \(d_{t_{1}}\) and \(d\); \(g_{t_{2}}\) is a function of \(d_{t_{2}}\) and \(d\). In a simplified form, one can further assume that \(g_{n}=g_{t_{1}}=g_{t_{2}}=1\), which yields  
\[ \begin{array}{c}\sigma=\sigma_{c}z(d)\\ \tau_{1}=\tau_{1c}z(d)\\ \tau_{2}=\tau_{2c}z(d)\end{array}  \tag{25}\]  
It is worth noting that the traction-separation law implemented in the 3D combined single smeared crack model [14; 39] can be retrieved from Equation (25) when \(\delta_{t_{1}c}=\delta_{t_{2}c}\) and \(\tau_{1c}=\tau_{2c}\). Since the users have more freedom to choose different forms for the functions \(g_{n}\), \(g_{t_{1}}\), \(g_{t_{2}}\), and \(z\), the traction-separation model defined through the Equations (19), (20), and (24) is referred to as a \"generalized traction-separation law\". It is noted that, beside functions \(g_{n}\), \(g_{t_{1}}\), \(g_{t_{2}}\), and \(z\), the generalized traction-separation law needs six parameters, i.e., three critical separations \(\delta_{nc}\), \(\delta_{t_{1}c}\), \(\delta_{t_{2}c}\), and three strengths \(\sigma_{c}\), \(\tau_{1c}\), \(\tau_{2c}\). In addition, according to the derivation, one can conclude that the traction-separation law implemented in the combined single smeared crack model [37; 38; 39] and the traction-separation law defined through Equations (6) and (11) are two special cases of the proposed generalized traction-separation model.