## 2 Numerical and Analytical Methods Description
### Description of the Modeling Approaches  
The complexity of adhesive behavior under different loading conditions necessitates the use of advanced modeling techniques. This section explores four major adhesive model representations combining both local and continuum approaches as shown in Figure 3. The first approach represents the adhesive layer using a finite-thickness single-row cohesive element configuration, wherein a single layer of cohesive elements (SRCE) is placed between the adherends ([PERSON], 2017, 2018). These elements are assigned a small finite thickness to approximate the real bond line. A bilinear traction-separation law governs the onset and progression of damage, allowing this method to capture post-failure fracture when the local stresses exceed critical values ([PERSON], 2017). However, while computationally efficient and straightforward to implement, this model may not fully capture complex through-thickness effects in a thicker adhesive layer.  
In the SRCE approach, Abaqus allows the user to specify a finite thickness, enabling the element itself to represent both the bulk adhesive layer and its overall failure mode  
Figure 3: Different adhesive model representations for (**a**) single-row cohesive element (SRCE); (**b**) continuum solid elements; (**c**) interface cohesive elements (Inter-CZ); (**d**) middle cohesive elements (Middle-CZ).  
Figure 2: Example of finite element mesh for (**a**) 25 mm overlap; (**b**) 3D mesh representation.  
through a single feature in the model. This finite element configuration incorporates the adhesive's elastic modulus to reflect the mechanical behavior across the thickness in an average sense. Thus, the effective elastic stiffness within the cohesive zone model is defined directly from the adhesive modulus divided by the actual thickness dimension. By contrast, the continuum single-row element approach models the adhesive as a single layer of 3D elements, to which elastic properties are assigned to reflect the stiffness of the adhesive layer. Without a means to represent plasticity or damage, the continuum single-row element approach serves simply as a reference for the response of the SLJ at low applied loads; this approach is unsuitable for tracking post-failure crack propagation. Where a more refined depiction of fracture is needed, the middle cohesive element strategy (Middle-CZ) inserts a zero-thickness cohesive interface within an otherwise continuum adhesive mesh, facilitating progressive fracture through the adhesive's interior [39]. This hybrid approach is particularly useful for capturing damage that initiates away from the adherend-adhesive interface or when the adhesive thickness is substantial enough to warrant more detailed modeling.  
Finally, the interfacial cohesive element technique (Inter-CZ) locates zero-thickness cohesive layers at the boundaries between the adherends and the adhesive itself, an approach well suited for conditions in which debonding consistently begins at or near the interface [40; 41]. While interfacial cohesive methods precisely track fracture at the interface, they can neglect any potential bulk adhesive damage unless cohesive elements are also introduced elsewhere in the bond line. Although each modeling method presents distinct advantages and limitations, their collective application provides a view of adhesive joint performance under various loading conditions, enabling tailored modeling choices that align with specific design objectives and experimental observations.