## 2 Numerical Models
### Double Cantilecer Beam Model  
The first of the two models, whose results are presented in this study, is the double cantilever beam (DCB), which is commonly used to assess the interlaminar fracture toughness of composite materials [33]. This model, whose geometry and boundary conditions are illustrated in Figure 3a, was used to assess the feasibility of the contemporary use of xFEM and cohesive elements for modeling crack propagation within the adhesive layer bonding the two adherends of DCB. In addition, as briefly explained earlier, the case was used to tune the materials properties that are required as input by both xFEM and cohesive elements.  
The overall model's specimen dimensions are 150 mm \(\times\) 25 mm \(\times\) 9 mm, with the initial crack length of 50 mm embedded within the mid-plane of the adhesive, in one end of the specimen, (see Figure 3a). This model is a simplification of the hybrid composite used for the experimental tests, consisting of a hybrid magnesium sheet and FRP forming the upper adherend, and biaxial FRP forming the lower adherend. It should be noted that, to further simplify the analysis (without compromising the overall accuracy), each of the two 4-mm thick adherends was homogenized into an equivalent elstic material. In this way, the equivalent materials had the same flexural stiffness as the combined hybrid materials, but the analysis would concume significantly less CPU. The adopted scheme also facilitates more effective debugging. The 1-mm thick adhesive layer was modeled in three ways, by using (i) a combination of elastic and cohesive elements, as shown in Figure 3b, (ii) xFEM elements only, as shown in Figure 3c, and (iii) a combination of xFEM and cohesive elements, as shown in Figure 3d. These models are referred to as COHESIVE, XFEM, and MIXED, respectively, hereafter. The same cohesive material model was used in conjunction with both cohesive and xFEM elements; moreover, the xFEM elements were also assigned elastic model properties. In other words, the elements defined as xFEM would initially behave elastically until the stresses reach to a level at which xFEM's enrichment is activated, thereby using the assigned cohesive properties. It should be noted that one could also assign other material models (e.g., elasto-plastic) to the xFEM elements instead of the elastic model [32].  
The generation of the precrack, for the XFEM and MIXED models, was done using the *BOUNDARY_PRECRACK keyword, which enriches the elements to account for the presence of an initial crack. Note that the conventional practice in fracture mechanics, that is, having a series of  
Figure 3: (**a**) DCB specimen’s geometry, boundary conditions (not to scale), and zoom-up of the mesh around the crack-tip in the (**b**) COHESIVE model, (**c**) XFEM model, and (**d**) MIXED model.  
disconnected adjacent layers of elements to model the crack, cannot be used in conjunction with xFEM elements. This is because xFEM element formulation allows for the crack to propagate only within the element. For the COHESIVE case, however, the crack was generated as done conventionally, that is by simply deleting the appropriate number of elements corresponding to the location of the actual crack/delamination. Therefore, to maintain consistency of the results when comparing the results generated by the three models, only the elements forming a portion of the adhesive that would be cracking (i.e., at the midplane of adhesive) were modeled by the cohesive material model, while the remaining portions were modeled with the elastic model, hereafter referred to as \"elastic element\". This approach also saves the CPU time.  
Finally, the adhesive layer was discretized with seven layers of elements as shown in Figure 3 and its density was kept constant along the bond length. The mesh density was stablished upon conducting a convergence study by which a reasonable accuracy could be attained by consuming an optimal CPU time.