## 2 Numerical and Analytical Methods Description
The adhesive, on the other hand, was modeled in four different ways to compare the similarities and differences in capturing various failure modes, in accordance with ASTM D5573-99 [38]. The considered adhesive representations were modeled as single-row cohesive elements (SRCE), continuum single-row solid elements (Continuum), interface cohesive elements between substrate and adhesive (Inter-CZ), and cohesive elements in \begin{table}
\begin{tabular}{c c c} \hline \hline  & **Aluminum Alloy (MPa)** & **FM 73M OST** \\ \hline Elastic modulus & 70,000 & 2000 \\ Poisson’s ratio & 0.3 & 0.38 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Aluminum alloy and adhesive elastic properties. Figure 1: Geometric characteristics of the single lap joint. the middle of the adhesive solid elements layer (Middle-CZ). The models were constructed using eight-node brick elements (C3D8) for the adherends. For the initial elastic stress analysis, a three-dimensional finite element mesh with standard continuum elements was employed to capture the baseline stress distribution in both the adherends and the adhesive layer. Subsequently, for the damage and strength analysis, 8-node cohesive interface elements were introduced at a prescribed distance from the adhesive/adherent interface to simulate the onset and propagation of failure. These cohesive elements, commonly used in fracture mechanics and detailed in the works of [PERSON] et al. (2019, 2020) and others, incorporate traction-separation laws that govern damage initiation and progression. To ensure reliable stress distribution and failure predictions in the single lap joint models, a mesh sensitivity study was conducted. Initially, a coarse mesh at the adhesive region (0.5 mm \(\times\) 0.5 mm \(\times\) 0.2 mm elements) was used to establish baseline computational cost and to pinpoint regions of highest stress gradients, primarily at the overlap edges. Subsequently, the mesh was incrementally refined in these critical zones and monitored changes in peak stress, strain distribution, and predicted failure loads. Once the refinement produced less than a 3-5% variation in the failure load using 0.2 mm \(\times\) 0.2 mm \(\times\) 0.2 mm elements, it was deemed sufficiently converged. Simultaneously, the computational run times were tracked, noting that excessive refinement in low-gradient regions (0.1 \(\times\) 0.1 \(\times\) 0.2 mm elements) unnecessarily inflated computational costs without improving accuracy. As a result, the final mesh struck a balance between accuracy and efficiency by using finer elements only in the overlap region and coarser elements elsewhere. This approach follows established guidelines for performing convergence studies and optimizing mesh design in finite element models. The finite element models were constructed using eight-node brick elements for the adherends and were also considered for the adhesive in places where the adhesive layer is treated as a continuum. Fully integrated elements give better results and capture bending but at a high computational cost. SLJ generally has high stress concentrations at the edges due to the sharp gradients, and 0.2 \(\times\) 0.2 \(\times\) 0.2 mm elements were used for the adhesive layer to account for this. In addition, the same mesh size was used across the different overlaps and the different thicknesses of the adherends, resulting in meshes containing on the order of 200,000 elements. For the adherends, a biased seeding was used towards the overlap region to capture stress concentrations and variations due to the bending because of the applied displacement (Figure 2). The load eccentricity makes the single lap joint bend at the edges of the overlap, which can be attributed to the asymmetry of the joint.