## 2 Numerical Models
### Material Model for Cohesive and xFEM Elements  
In LS-DYNA, only one material model is currently available for use in conjunction with the xFEM formulation, which is: *MAT_COHESIVE_TH. This is a cohesive material law proposed by [PERSON] and [PERSON] [29] with tri-linear traction-separation behavior (see Figure 1a), where the maximum traction stress and normal or tangential ultimate displacements are the governing and required parameters [30]. The model accepts only one value of the maximum stress; therefore, it could be either the maximum normal stress or maximum shear stress, accordingly. In this study, because mode I fracture is the dominant failure mode, the maximum tensile stress is chosen as the maximum stress governing the failure of the material. Note that the lack of differentiation between the maximum normal and shear stresses is an important limitation of this model. This cohesive model is based on the non-dimensional parameters \(\lambda_{1}\), \(\lambda_{2}\), and \(\lambda_{\text{fail}}\), defining the traction-separation law behavior, as shown in Figure 1. These parameters correspond to various segments of the traction-separation curve (i.e., the peak traction, the beginning of softening segment, and the final failure, respectively). In other words, these parameters are used to represent a measure of the global dimensionless separation, \(\lambda\), mathematically represented for the two-dimensional case as follows:  
\[\lambda=\sqrt{\left(\frac{\langle\delta_{N}\rangle}{\delta_{N}^{fail}}\right) ^{2}+\left(\frac{\delta_{T}}{\delta_{T}^{fail}}\right)^{2}}, \tag{1}\]  
where \(\delta_{N}\) and \(\delta_{T}\) are the normal and tangent separation displacements, \(\delta_{N}^{fail}\) and \(\delta_{T}^{fail}\) are the respective separation values at failure, and the operator \(\langle\cdot\rangle\) refers to the Mc-Cauley brackets, used to differentiate the behavior under tension and compression. The stress state is computed, using the trilinear traction-separation law parameters (see Figure 1), as follows:  
\[\sigma(\lambda)=\left\{ \begin{array}{l}\sigma_{max}\frac{\lambda_{1}}{ \lambda_{final}},\lambda<\frac{\lambda_{1}}{\lambda_{fail}};\\ \sigma_{max},\frac{\lambda_{1}}{\lambda_{fail}}<\lambda<\frac{\lambda_{2}}{ \lambda_{fail}}\\ \sigma_{max}\frac{1-\lambda_{2}}{1-\frac{\lambda_{2}}{\lambda_{fail}}},\frac{ \lambda_{1}}{\lambda_{fail}}<\lambda<1;\end{array} \right. \tag{2}\]  
where \(\sigma_{max}\) refers to the maximum tensile or shear stress, as mentioned previously. Using a potential function, \(\varphi\), defined as:  
\[\varphi(\delta_{N},\delta_{T})=\delta_{N}^{fail}\int\limits_{0}^{\lambda} \sigma(\hat{\lambda})d\hat{\lambda}, \tag{3}\]  
and the normal surface traction, \(\sigma_{N}\), and the tangential surface traction, \(\sigma_{T}\), are expressed by the following derivatives:  
\[\sigma_{N}=\frac{\partial\varphi}{\partial\delta_{N}},\sigma_{T}=\frac{ \partial\varphi}{\partial\delta_{T}}, \tag{4}\]  
Finally, the development of the derivatives leads to the traction vector, expressed as:  
\[\left\{ \begin{array}{l}\sigma_{N}\\ \sigma_{T}\end{array} \right\}=\frac{\sigma(\lambda)}{\lambda}\left[ \begin{array} []{cc}\frac{1}{\lambda_{N}^{fail}}&0\\ 0&\frac{1}{\lambda_{T}^{fail}}\end{array} \right]\left\{ \begin{array}{l} \langle\delta_{N}\rangle\\ \delta_{T}\end{array} \right\} \tag{5}\]  
This model is totally reversible, in other words, the loading and unloading follow the same path. In addition, the difference in behavior between tension and compression is accounted for, with the following equation describing the behavior for \(\delta_{N}<0\):  
\[\sigma_{N}=\kappa\frac{\sigma_{max}}{\delta_{N}^{fail}\frac{\lambda_{1}}{ \lambda_{fail}}}\delta_{N} \tag{6}\]  
where \(\kappa\) is the penetration stiffness multiplier, defined by the user. As mentioned previously, the tri-linear behavior is controlled by the three non-dimensional parameters, which affect term A in the following equation.