## 2 Numerical Models
### Material Model for Cohesive and xFEM Elements
These parameters are related to the material's fracture toughness, \(\mathrm{G_{IC}}\) or \(\mathrm{G_{IIC}}\) in the following manner:  
\[\mathrm{G_{iC}}=A\delta_{i}, \tag{7}\]  
where the subscript \"i\" relates to the normal or tangential directions and \(A\) is the area under the normalized traction-separation curve (see Figure 1a,b). The cohesive parameters used in this investigation, as reported in the Appendix A, were obtained by calibrating the trial values in such a way that the numerical simulation-produced results would closely match the results obtained through the actual testing of the double cantilever beam (DCB) specimen, using the load-opening curve as the criterion, as shown in Figure 1c. It should be noted that the experimental test data of DCB was obtained under a static loading, while the simulation of the 3D-FML specimen of our interest, as will be presented later, was carried out when the specimen was subjected to an impact loading state; therefore, there would be some discrepancies between the evaluated values and those exhibited by the actual specimen dynamically. The establishment of the CZM parameters as explained is based on matching the overall behavior, which would not include while the fluctuations that could potentially develop locally. However, in this paper, the authors' intent is to demonstrate the feasibility of the described method, not its accuracy. The selected calibration method is meant to simply establish the values of the cohesive zone's parameters used to facilitate the simulation. Moreover, the xFEM formulation is currently only available under the dynamic, explicit solution scheme of LS-DYNA. Therefore, to ensure a reasonable solution time, all the simulations were run in dynamic mode, with the simulated event being in the order of a millisecond. It is also worth mentioning that, during the calibration, no significant difference was found when reducing the tri-linear law to a bi-linear one (see Figure 1b), which facilitates a more CPU-efficient numerical solution. Consequently, \(\lambda_{1}\) and \(\lambda_{2}\) were both set to 0.5, thereby reducing the tri-linear model to a bi-linear model. Note that some researchers [31; 32] have recommended the use of an initially-rigid cohesive law (i.e., \(\lambda_{1}=\lambda_{2}\cong 0\)) for obtaining a more reliable estimation of the state of stress prior to the onset of a crack. However, numerical instabilities were encountered when the approach was adopted in this study. Moreover, this is not to say that adaptation of the tri-linear law and/or the initially rigid cohesive response would lead to similar issues when simulating other cases. It should be noted that the main objective of the study presented here is to demonstrate the potential of the xFEM method in simulating the response of a complex material system under a relatively complex loading state, as opposed to targeting the degree of accuracy that could be attained when using the technique. Consequently, no further calibration effort was expended towards this issue. Finally, as mentioned earlier, this cohesive model is the only one available for use within xFEM in LS-DYNA. Therefore, for the sake of consistency, this model was also used with the cohesive elements, even though other cohesive models are available that could potentially produce more accurate predictions in the case of mixed mode fracture. Figure 1: (**a**) Representation of the traction-separation law of the “COHESIVE_TH material model, and (**b**) the bi-linear traction-separation cohesive model used in this investigation. Note that A refers to the total area under the curve. (**c**) the load-opening curves of the double cantilever beam (DCB) experimental test used for establishing the cohesive zone method (CZM) parameters.