## 2 Fundamental of FDEM
### Fracture Constitutive Model for the Joint Element  
In the FDEM, the initiation and propagation of cracks are controlled by the constitutive fracture model of the joint element. The four nodes of the joint element are shared with the two adjacent triangular elements on both sides. Therefore, the initiation and propagation of cracks are only along the boundary of the triangular element. According to the relative displacement state of the nodes of the joint element and the fracture constitutive model, it can be judged whether the joint element is yielded, the I-type tension failure, II-type shear failure, or I-II mixed tension-shear failure [20].  
The relationship between the normal and tangential traction force and the normal opening and tangential slip is shown in Equations (2) and (3). These two equations mainly  
Figure 1: The connection relationship between the triangular element and joint element in FDEM [20].  
represent the softening behavior in the latter part of the tensile and shear peaks, as shown in Figure 2.  
\[\sigma=\left\{\begin{array}{cc}p_{f}o,&o<0\\ f(D)T_{s},&o\geq 0\end{array}\right. \tag{2}\]  
where \(P_{f}\) is the penalty parameter of the joint element, \(o\) is the normal opening, \(T_{s}\) is the tensile strength.  
\[\tau=\left\{\begin{array}{cc}(c-\sigma\tan\phi)f(D),&\sigma<0\\ cf(D),&\sigma\geq 0\end{array}\right. \tag{3}\]  
where \(D\) is the damage variable of the joint element, which can be specifically calculated.  
When the joint element is in an elastic state:  
\[D=0,\ \ \ o\leq o_{p}\ \text{and}\ |s|=0 \tag{4}\]  
when the joint element is in a pure tension state:  
\[D=\frac{o-o_{p}}{o_{1}-o_{p}},\ \ \ o>o_{p}\ \text{and}\ |s|=0 \tag{5}\]  
where \(o_{p}\) is the corresponding normal opening amount when the joint element is at the peak tensile stress (equal to the tensile strength), \(o_{t}\) is the critical opening amount when the joint element is tensile-fractured. When the opening amount of the joint element reaches the critical opening amount \(o_{t}\), the damage variable \(D\) is \(1\). At this moment, the tensile stress is reduced to \(0\) and a tensile crack occurs. \(o_{t}\) can be calculated according to the fracture energy of Mode I according to the following formula:  
\[G_{fI}=\int_{0}^{o_{t}}\sigma(o)do \tag{6}\]  
For pure shear state:  
\[D=\frac{|s|-s_{p}}{s_{t}-s_{p}},\ \ \ |s|>s_{p}\ \text{and}\ o\leq 0 \tag{7}\]  
where \(s_{p}\) is the corresponding tangential slip when the joint element is at the peak shear stress (equal to the shear strength), and \(s_{t}\) is the critical tangential slip when the joint element is tensile-fractured. When the tangential slip reaches the critical slip \(s_{t}\), the damage variable \(D\) is \(1\). At this moment, the tangential shear stress is reduced to \(0\) and a shear crack occurs. \(s_{t}\) can be calculated according to the fracture energy of Mode II by:  
\[G_{fII}=\int_{0}^{s_{t}}\tau(s)ds \tag{8}\]  
For the mixed tensile-shear state:  
\[D=\sqrt{\left(\frac{o-o_{p}}{o_{t}-o_{p}}\right)^{2}+\left(\frac{|s|-s_{p}}{s _{t}-s_{p}}\right)^{2}}\rho>o_{p}\ \text{or}\ |s|>s_{p} \tag{9}\]  
According to Equations (4), (5), (7), and (9), the damage degree of the joint element can be calculated. If the damage variable \(D\) is greater than \(1\), set \(D\) to1. At this time, the joint element breaks, and a crack occurs.