## 3 Modelling Framework
### Traction-Separation Law
#### 3.3.2 Damage Evolution (Separation Propagation)  
Once the initiation criterion is met, the damage evolution law describes the rate at which the cohesive stiffness degrades. The overall damage at the contact point is represented by \(D\), a scalar damage variable. It starts with a value of 0 and develops from 0 to 1 during the loading process. The fracture energy, which is the area under the curve in a graph of traction versus separation, or the maximal displacement must be specified. The normal and shear stresses components are affected by the damage according to [17]. In this paper, the fracture energy is defined for damage evolution and it is assumed to be a linear softening for the damage evolution zone. To describe the evolution of damage under  
Figure 4: Bilinear traction-separation law.  
a combination of normal and shear separations across the interface, an effective separation is defined according to [24] as:  
\[\delta_{m}=\sqrt{\langle\delta_{n}\rangle^{2}+\delta_{s}{}^{2}+\delta_{t}{}^{2}} \tag{5}\]  
\[D=\frac{\delta_{m}^{*}}{\delta_{m}^{u}}\left(\frac{\delta_{m}^{u}-\delta_{m}^{0} }{\delta_{m}^{*}-\delta_{m}^{0}}\right) \tag{6}\]  
\[t_{n}=\left\{\begin{array}{cc}(1-D)\overline{t}_{n}&if\ t_{n}\geq 0\\ \overline{t}_{n}&if\ t_{n}<0\ (compression)\end{array}\right. \tag{7}\]  
\[t_{s}=(1-D)\overline{t}_{s} \tag{8}\]  
\[t_{t}=(1-D)\overline{t}_{t} \tag{9}\]  
\(\delta_{m}\) represents the effective separation, \(\delta_{m}^{*}\) is the maximum effective separation during the loading, \(\delta_{n}\), \(\delta_{s}\), and \(\delta_{t}\) are the normal, shear, and tangential components of the interfacial separation directions, respectively. \(\delta_{m}^{u}\) is the effective separation just before unloading, and \(\delta_{m}^{0}\) is the effective separation at the damage initiation stage. Where \(\overline{t}_{n}\), \(\overline{t}_{s}\), and \(\overline{t}_{t}\) are the contact stress components predicted by the elastic traction-separation behavior for the current separations without damage. A Macaulay bracket \(\langle\rangle\) is used to prevent compressive damage.  
The hatched area below the curve in Figure 4 provides the energy dissipated to produce a new pair of fully disconnected surfaces, so-called, interface fracture toughness. A mixed-mode propagation response of the cohesive interface can be generated due to the positioning of the interface for capsules with respect to the applied load. The propagation mixed mode involves different energies associated with the debonding capability in normal (\(n\)) and parallel directions to the interface (\(s\), \(t\)). The maximum fracture separation is then calculated by [24]:  
\[\delta_{n}^{*}=\frac{2G_{n}^{*}}{t_{n}^{0}} \tag{10}\]  
\[\delta_{s}^{*}=\frac{2G_{s}^{*}}{t_{s}^{0}} \tag{11}\]  
\[\delta_{t}^{*}=\frac{2G_{t}^{*}}{t_{t}^{0}} \tag{12}\]  
This paper assumes that the interaction between the energies of each individual mode (i.e., \(n\), \(s\), and \(t\)) follows the power-law fracture criterion [24] given by:  
\[\left\{\frac{G_{n}}{G_{n}^{*}}\right\}^{\alpha}+\left\{\frac{G_{s}}{G_{s}^{*} }\right\}^{\alpha}+\left\{\frac{G_{t}}{G_{t}^{*}}\right\}^{\alpha}=1 \tag{13}\]  
where \(G_{n}\), \(G_{s}\), and \(G_{t}\) are the energy release rates calculated from the traction and normal, shear, and tangential displacements during interface opening, respectively, and the power \(\alpha\) is a cohesive property parameter that describes the interaction between modes. The critical interface toughness for each direction is represented by the properties \(G_{n}^{*}\), \(G_{s}^{*}\), and \(G_{t}^{*}\). In this paper, it is assumed that the critical fracture toughness is the same in all directions. In addition, taking into account the effect of this parameter on the response, a value \(\alpha=1\) has been used [24].