## 3 Results and Discussion
### Strength Prediction  
A finite element analysis was conducted to investigate the force-displacement behavior of single lap joints with various adherend thicknesses and overlap lengths. The analysis incorporated the four different modeling approaches for the adhesive layer, and the objective was to evaluate the joints' performance and failure mechanisms under various configurations. The results presented here highlight how changes in adherend thickness and overlap affect the load capacity, stiffness, and predominant failure mode in SLJ. For instance, in the case of the 3 mm thick adherend and the four different overlaps, cohesive failure was noticed in the adhesive layer, and no hardening was observed within the adherends (see Figure 10a). Because the adherend thickness is relatively large compared to the adhesive layer thickness, the adherends remain rigid throughout the loading process, causing the primary failure to be driven by the adhesive's cohesive properties rather than by plastic deformation or hardening in the adherends. Figure 9: Different representations at the adhesive mid-thickness for the (**a**) peel stress at elastic region; (**b**) peel stress at maximum force; (**c**) shear stress at elastic region; (**d**) shear stress at maximum force. The SRCE produced results nearly identical to the middle cohesive elements, as shown in Figure 10, demonstrating its reliability in capturing adhesive (cohesive) failure. The continuum solid elements adhesive model representation (continuum) only indicated the yielding of the adherend and the adhesive (the traction serving as the yield property of the adhesive); because it lacks a cohesive zone (CZ) framework, it cannot simulate failure initiation and propagation, and instead merely indicates yielding without progressing to a failure. Despite this limitation, the continuum model is included as a critical reference for verifying the pre-failure (elastic and early plastic) behavior of the other adhesive representations. By comparing the initial force-displacement response and the local stress/strain fields of the continuum model with those of the CZ-based methods, it is noticed that the cohesive element penalty stiffness is sufficiently high to approximate a nearly rigid interface during early loading. Consequently, the single-row cohesive, interfacial CZ, and middle cohesive models align closely with the continuum approach in the elastic region, typically within 5% discrepancy. This validation step addresses an important philosophical point noted in both recent and foundational works on cohesive zone modeling [46]. Because a zero-thickness interface with a finite penalty stiffness can introduce artificial compliance if the stiffness is not chosen carefully, ensuring that the CZ layer's initial slope parallels that of the full-continuum model bolsters confidence in the accuracy of the predicted joint stiffness prior to damage. Only once localized stress exceeds the adhesive's cohesive strength do the CZ-based approaches diverge by simulating the failure initiation, something the continuum model cannot replicate. As a result, the continuum approach demonstrates its usefulness for gauging elastic performance, while the CZ methods become indispensable for analyzing post-failure mechanisms. By combining both perspectives, this study ensures simula  
Figure 10: Force–displacement curves of the different adhesive model representations for (**a**) 25 mm overlap, 3 mm thick adhered; (**b**) 10 mm overlap, 3 mm thick adherend; (**c**) 25 mm overlap, 2 mm thick adherend; (**d**) 10 mm overlap, 2 mm thick adherend. tion of both the early elastic phases and the discrete failure processes inherent to single lap joint configurations.