## 3 Results and Discussion
### Stress Distribution of the Adhesive
During the initial stages of loading, shear stresses are highest at the edges of the overlap, forming a characteristic \"U-shaped\" distribution across the width of the joint. As the load increases, these edge regions exceed the cohesive zone's traction threshold, initiating local damage and triggering the softening branch of the traction-separation law. As a result, the degraded edges can no longer carry as much load, causing the shear stress to redistribute inward toward the middle of the overlap. Consequently, at higher load levels (Figure 13d), the shear stress profile appears inverted with the peak shifting to the central region. This shift in stress concentration reflects a common cohesive damage progression: once the outer edges fail, the surviving material in the middle is forced to carry an increasing share of the load, thereby raising its stress level. Papers on cohesive zone modeling of single lap joints similarly report this pattern, attributing it to a combination of localized debonding at the edges and the adhesive's progressive softening response [22]. Understanding this transition underscores why joint designs or reinforcements targeting edge-strength improvements can delay edge damage and maintain a more uniform stress distribution, potentially enhancing overall joint strength and durability. The numerical results for different overlap lengths (Figure 16) for a 3 mm thick adherend reveal that shorter overlaps exhibit higher stress concentrations, primarily at the edges. When the overlap is small, a large bending effect develops because the eccentric load path has less adhesive area over which to distribute the forces, causing the peel stress at the edges to spike. Meanwhile, longer overlaps may display a \"W-shaped\" distribution early in the loading process, with pronounced tensile peaks near the ends of the overlap and a slightly compressed region at mid-span. As damage initiates at the overstressed edges, the load shifts toward the center of the joint, progressively flattening the outer peaks and producing a more \"U-shaped\" profile at higher loads. This transition mirrors the cohesive zone model's softening behavior, wherein edge regions degrade first and compel the still-intact central region to carry a greater share of the load. Furthermore, [PERSON] and [PERSON]'s analysis [29] indicates that, in the absence of significant external peel forces, the net integral of peel stresses across the bond line must approximately balance to zero. Hence, an elevated tensile stress at the overlap edges is typically offset by a correspondingly compressive region, which is often observed in the central part of the overlap. Such behavior appears in our simulations, where shorter overlaps show large positive shear stresses and negative stresses in the middle, consistent with the need for equilibrium in bending moments. Although the analytical approach sometimes predicts slightly higher peak stresses than the numerical model (e.g., a 10% difference for the 10 mm overlap scenario in the shear stress contour), the overall distribution trends remain remarkably similar, underscoring the continued relevance of [PERSON]'s classical solution. This stress distribution analysis, validated by the analytical formulation, underscores the critical impact of adherend thickness, overlap length, and modeling method on the mechanical performance and failure behavior of single lap joints. Cohesive element approaches consistently demonstrated superior accuracy in capturing adhesive failure mechanisms, while the single-row continuum method fell short in representing post-failure behavior. The analytical results not only confirmed the trends observed in the numerical simulations but also provided a theoretical foundation for understanding the mechanics of the joints, emphasizing the robustness of the cohesive models used in this study. This section highlights the importance of considering adherend thickness and overlap length in joint design to optimize stress distribution and enhance joint reliability.