## 2 Numerical and Analytical Methods Description
### Implementation of the CZ Model in Finite Element Analysis  
The CZ model represents the initial elastic response of an interface layer until the local stress reaches a critical value, at which point damage initiates and cracks can propagate. In this framework, zero-thickness cohesive elements are inserted along potential fracture paths, often at the interface between the adherends [39]. These elements are governed by traction-separation laws, which correlate cohesive traction (stresses) with relative displacements across the interface. During the early stages of loading, the stiffness of these elements is high to approximate a near-elastic response. As the stress approaches the critical value, the element stiffness gradually degrades, simulating the progression of damage until ultimate failure. This sequence--from initial elastic behavior through damage initiation and propagation--captures the essential fracture mechanics of bonded interfaces in single lap joints and other adhesive structures. The traction-separation law (Figure 4) is defined by three key parameters: the maximum traction (T\({}_{\text{max}}\)), the corresponding separation at maximum traction (\(\delta_{\text{max}}\)), and the critical separation (\(\delta_{\text{c}}\)) at which complete decohesion takes place. Although several traction-separation curve shapes exist such as bilinear, trapezoidal, or exponential, this study specifically adopts a bilinear law. During the analysis, as the applied load grows, the cohesive parts undergo a progressive increase in tractions and separations until they reach the maximum traction, which signifies the beginning of damage. After reaching the critical separation, the cohesive elements deteriorate in accordance with the stated softening law, resulting in full separation. This methodology enables the precise replication of both the commencement and expansion of fractures, offering in-depth understanding of the causes of failure in adhesive connections. A quadratic stress criterion is optimal for simulating a single lap joint because it effectively captures the mixed-mode nature of the stresses experienced in such joints [22]. SLJ often encounter a combination of tensile (mode I) and shear (mode II) stresses due to the eccentric load path and the resulting peel and shear forces. The quadratic stress criterion integrates the contributions of both modes by employing a combined stress approach, where the failure criterion is defined as the square root of the sum of the squares of the normalized mode I and mode II stresses. This allows for a more accurate representation of the interaction between different stress components and their cumulative effect on the joint's failure. Moreover, the quadratic stress criterion ensures a more realistic simulation of the failure envelope for adhesive joints, accounting for the non-linear interaction between tensile and shear stresses [22]. By considering the normal and shear stress components simultaneously through a squared, normalized summation that reaches a critical threshold, the criterion captures how multiple stress modes collectively contribute to damage onset. Once damage initiates, the model uses a power-law formulation for the evolution of damage, relating the energy dissipated in each stress mode to its corresponding critical fracture energy. In this study, the key parameters for the cohesive zone model (CZM) including critical fracture energies and maximum stresses were selected based on established literature for aluminum adherend systems [28; 42]. During loading, partial fracture energy is consumed in the normal and shear directions. When the sum of these mode-based energy terms, weighted by an exponent (\(\alpha\)) that dictates their mixed-mode interaction, meets or exceeds unity, the adhesive fully debonds [43]. This approach aligns well with experimental observations of progressive failure, in which local cracking initiates at high-stress regions and then spreads as more fracture energy becomes available.