## 1 Introduction  
Cross-laminated timber (CLT) has emerged as a viable structural material in modern construction due to its favorable mechanical properties, ease of prefabrication, and suitability for large-scale applications. Its layered configuration, provides enhanced stiffness and dimensional stability, making it particularly attractive for multi-story buildings and modular construction systems. CLT is composed of an odd number of orthogonally arranged timber layers bonded with adhesives, providing high load-bearing capacity and structural stability. Its modular design allows for prefabrication and rapid on-site assembly, enhancing construction efficiency while reducing material waste and energy consumption. To ensure the safe and efficient use of CLT in structural applications, it is essential to understand its failure mechanisms. Therefore, computational simulations have become indispensable tools in this context, enabling virtual testing under various loading and degradation scenarios, while extensive research exists for composite materials like fiber-reinforced plastics, the study of fracture resistance in CLT, especially under nonlinear phenomena such as buckling, remains limited. Nevertheless, methodologies such as multisurface failure criteria [1; 2], lattice models [3], cohesive damage models [4; 5; 6; 7], finite fracture mechanics concepts [8; 9], and the extended finite element method (XFEM) [10; 11] offer valuable frameworks that can be adapted to wooden materials. Rolling shear failure, a critical phenomenon in CLT panels, has been previously investigated with good correlation between experimental and numerical results [5; 12; 13]. Additional studies have addressed the influence of shear deformation, number of layers, and geometric features on buckling behavior [14; 15]. However, the coupled effect of buckling and crack propagation, particularly delamination, has received little attention in the literature. One of the main challenges in modeling this interaction is the disparity in scale: buckling manifests at the structural level, while delamination occurs at the material level. Accurately capturing these multi-scale phenomena requires fine-meshed, high-resolution models and significant computational resources. This complexity calls for advanced strategies such as domain decomposition and global-local modeling. Cohesive zone modeling (CZM) has advanced significantly in recent years, offering reliable tools for simulating fracture in heterogeneous materials. Research has focused on providing a foundation for consistent energy dissipation in fracture simulations [16], mixed-mode fracture [17], Mode I delamination coupled with fiber bridging [18], arbitrary or complex crack propagation paths [19; 20], rate-dependent cohesive laws [21], adaptive meshing [22], parameter identification [23; 24], integration with multiscale and non-intrusive methods [25; 26; 27; 28], combining cohesive elements with XFEM [29], the prediction of fatigue life of adhesive joints for automotive applications [30], and temperature-dependent material properties [31], among others. These developments have improved the capacity to simulate complex failure mechanisms efficiently and accurately. This study aims to evaluate the applicability of a primal-dual global-local strategy [32; 33] to simulate crack growth in a double-cantilever beam (DCB) test. To mitigate the numerical complexities associated with the anisotropic behavior of CLT, the analysis is initially implemented using an isotropic material, allowing for a more robust evaluation of the method's performance and convergence behavior. Is important to mention that in engineering practice, the modeling of CLT or any other wall type elements are made using 2D shell elements, with rotational and translational degrees of freedom per node. Therefore, to capture the correct behavior of the adhesive analysis using CZM or another interface model technique, a coupled analysis between 2D shell elements and 3D complex nonlinear behavior must be considered.