## 3 Results and Discussion
### Stress Distribution of the Adhesive  
The stress distribution analysis for single lap joints using four different modeling methods revealed distinct behaviors across different adherend thicknesses and overlap lengths. Initially, all methods produced similar peel and shear stresses in the elastic region for a 3 mm thick substrate with a 25 mm overlap. However, significant differences emerged in the maximum force region. The SRCE method exhibited a notably high peel stress of 50 MPa, in contrast to the cohesive element methods, which produced peel stresses around 22 MPa as shown in Figure 13c. This discrepancy highlights the continuum solid element method's inability to accurately capture failure or post-failure effects, underscoring the limitations of this approach in simulating adhesive joint behavior under high load conditions. At the point of maximum force, the three cohesive element methods produced shear stresses at 28 MPa, which aligns with the adhesive's shear stress threshold, indicating imminent failure. This consistency among cohesive methods demonstrates their efficacy in capturing the initiation and progression of failure within the adhesive layer. All these findings were further validated through an analytical formulation [29; 30; 31] showing the same shear stress distribution and validating the predictive capability of these cohesive models. In contrast, the continuum method's peel stress at this stage remained significantly lower at the middle of the lap joints, reflecting its inadequacy in representing the true  
Figure 13: Adhesive stress distributions at the mid-thickness of the adhesive layer at the elastic region and maximum force for a 3 mm thick and 25 mm overlap lap joint for: (**a**) peel stress at elastic region; (**b**) shear stress at elastic region; (**c**) peel stress at the maximum force; (**d**) shear stress at maximum force. stress state of the joint. This finding underscores the importance of cohesive elements in accurately modeling the complex failure mechanisms of adhesive joints. In Figure 13a,c, the observed trend where peel stresses started at approximately 30 MPa in the elastic region and decreased to 22 MPa at maximum force indicates that peel stresses are initially highest and reduce towards failure. This behavior is consistent across different overlap lengths and thicknesses, though the pattern varies slightly (Figure 14). This absence of a wavy pattern highlights the influence of overlap length on the initial stress distribution, with shorter overlaps uniformly distributing the stresses across the different regions of the overlap area. A closer examination of Figure 13a reveals that the continuum approach consistently displays the same amplitude fluctuations with the other cohesive zone (CZ) models. Since SRCE is the most streamlined cohesive representation by assigning an equivalent traction-separation law to the entire adhesive layer, it naturally smooths out localized effects. Moreover, all the CZ-based methods show slightly higher peak stresses than the analytical model, which is common because the finite penalty stiffness of cohesive elements may introduce additional compliance, altering the stress distribution. Nonetheless, these differences remain small, indicating that the chosen cohesive stiffness is sufficiently high to mimic near-elastic behavior before failure initiation, a key objective in cohesive zone modeling, where the elastic branch of the traction-separation law is not meant to replicate the entire adhesive material's elasticity in detail, but rather to enable subsequent failure prediction. Figure 13c demonstrates a similar trend: the two middle and interfacial CZ approaches with comparable through-thickness representations exhibit closely matched amplitudes, while the SRCE model still yields a slightly smoother distribution. In this mid-load regime, the differences among the CZ approaches remain modest, reflecting the fact that none of the joints have fully transitioned into damage-dominated behavior.