## 4 Conclusions  
In this study, a finite element analysis of single lap joints was conducted to investigate how variations in adheren thickness and overlap length influence joint performance and failure modes. Four different adhesive modeling approaches, single-row cohesive element (SRCE), single-row continuum element, interfacial element, and middle cohesive element, were compared to highlight their relative strengths and limitations. The single-row  
Figure 18: Strain contour comparison of 10 mm overlap adheren surface at **(a)**\(\delta\) = 0.12 mm for 0.5 mm thick adheren; **(b)**\(\delta\) = 0.12 mm for 3 mm thick adheren; **(c)**\(\delta\) = 0.3 mm for 0.5 mm thick adheren; **(d)**\(\delta\) = 0.3 mm for 3 mm thick adheren. The dashed black outline indicates the overlap region. continuum element method served as a baseline to verify the elastic response and assess the compliance of other cohesive element strategies. Although it cannot predict failure, it aligns closely with analytical models in the pre-failure region. Comparing the four adhesive modeling approaches revealed some differences. The SRCE and middle cohesive element methods proved particularly effective at simulating the initiation and propagation of adhesive failure, offering a detailed portrayal of how failure evolved. Notably, although the middle cohesive element approach allows for a more refined through-thickness representation, this study showed that SRCE still captured the global response and failure mechanisms accurately for relatively thin adhesive layers. Meanwhile, the interfacial element method showed a tendency for larger displacements at failure due to the double cohesive layering, highlighting the need to carefully calibrate such models when interface-specific failures are the primary concern. In addition, adherend thickness and overlap length play critical roles in determining the failure patterns. For instance, thicker adherends (2 mm and 3 mm) often exhibited cohesive failure within the adhesive layer when subjected to longer overlaps, underscoring the stronger load-bearing capacity of thicker materials. Conversely, thinner adherends (1 mm and especially 0.5 mm) tended to yield first, thereby preventing the adhesive from experiencing full cohesive damage in many cases. This difference highlights the importance of selecting optimal adherend dimensions in applications where joint reliability is paramount: thicker aluminum adherends tended toward cohesive failure within the adhesive layer, whereas thinner aluminum adherends often yield first, preventing full adhesive damage. Moreover, shorter overlaps produced elevated peel and shear stresses near the overlap edges, which aligns closely with classical theories on bending-induced stress concentrations. The numerical results matched analytical predictions within approximately 10%, underscoring the variations in the modeling strategies. From a practical standpoint, these findings underscore the fact that smaller overlap lengths can lead to stress concentrations that may trigger premature failures unless designers incorporate mitigating features or select higher-strength adhesive systems. Overall, these contributions enhance the existing body of knowledge on SLJ mechanics. While prior literature has addressed certain aspects of single lap joint behavior, this study provides a head-to-head comparison of multiple adhesive modeling techniques, thereby offering practitioners a clearer roadmap for choosing the most suitable numerical strategy. Looking ahead, there is potential to expand this framework by examining thicker bond lines, multi-material adherends, or varying thermal conditions to assess how each modeling approach scales under more complex scenarios. Ultimately, the work presented here reaffirms the importance of both geometric parameters and cohesive zone modeling in designing reliable adhesive joints and lays the groundwork for future explorations of bonded structures in advanced engineering applications. To build on these insights, several avenues for future work are recommended.