## 4 Numerical Modeling with Anisotropic Slate  
### CZM and Its Finite Element Formulation  
The hybrid finite and cohesive element method (FCCM) by inserting cohesive elements into finite elements is widely used for applying to the fracture simulation of kinds of materials that including composites and various rocks [41; 42; 43; 44; 45; 46]. When a cohesive zone formulation is employed to model cracking, it implies that the notion of a cohesive force ahead of the crack that prevents propagation is introduced. The micro-mechanisms of material deterioration and fracture are thus embedded into the constitutive law that relates the cohesive traction \(\sigma\) with the local separation. Damage is restricted to evolve along the intrinsic cohesive interfaces, where cohesive elements could be randomly distributed in the mesh generated. In Figure 10, \(\Delta_{m}^{f}\) and \(\Delta_{m}\) are the critical displacement of failure and the effective opening displacement of the crack, respectively; \(f_{t}\) represents the tensile strength (cohesive) on the crack surface. In the fracture model of the continuous-discontinuous method, the mode I peak streng th is defined by a constant tensile strength, \(f_{t}\), while the mode II peak strength, \(f_{s}\), is computed according to the Mohr-Coulomb criterion. The complete breakage of the crack element and consequently the nucleation of a discrete crack are accomplished after dissipating the material fracture energy release rate, \(G_{c}\). After breakage of the cohesive surface, the  
Figure 10: Illustration of the CZM and its form of separation law. Figure 9: Empirical fitting relationship between tensile strength and \(K_{IC}\) of the carbonaceous slate. crack element is removed from the simulation and model transition from a continuum to discontinuum is thus locally completed. Damage initiation is referred to the beginning of deterioration of the response of a material point, which indicates the degradation of interface elements in the slate in the present work. The process of degradation begins when the stresses and/or strains satisfy certain damage initiation principles that we use. In the present work, the damage is assumed to initiate when the maximum nominal stress component ratio reaches a dimensionless value of one, which can be read as  
\[\text{MAX}\bigg{\{}\frac{\langle T_{n}\rangle}{T_{nmax}},\frac{T_{s}}{T_{smax}},\frac{T_{t}}{T_{tmax}}\bigg{\}}\geq 1 \tag{5}\]  
where \(T_{n}\), \(T_{s}\), and \(T_{t}\) represent the normal stress and the two tangential stresses of the cohesive element, respectively; \(T_{tmax}\), \(T_{smax}\), and \(T_{tmax}\) is the normal strength and the two shear strengths of the cohesive element. In addition, it is assumed that the cohesive element does not become damaged under pure compressions so that we introduce MacAulay parentheses \(<\) \(>\), in which \(\langle T_{n}\rangle\) can be read as follows;  
\[\langle T_{n}\rangle=\left\{ \begin{array}{ll}T_{n},&T_{n}\geq 0\\ 0,&T_{n}<0\end{array} \right. \tag{6}\]  
The cohesive constitutive law is completed by an appropriate equation governing the evolution of damage \(D\). It is of note that \(D\) remains constant if the traction on the interface is less than its current strength, if the interface is unloaded, or if \(D\) reaches 1. Otherwise, \(D\) has to evolve so that the strength of the interface decreases progressively from its initial value to zero as the effective displacement \(\Delta_{\text{m}}=\left(\left\langle\Delta_{\text{n}}\right\rangle^{2}+\Delta_ {\text{s}}^{2}+\Delta_{\text{t}}^{2}\right)^{1/2}\) increases from \(\Delta_{m}=\Delta_{m}^{0}\) (the effective displacement at damage initiation) to \(\Delta_{m}=\Delta_{m}^{f}\) (the effective displacement at complete failure) by an exponential form.