PSI - Issue 72

A.F.L. Macedo et al. / Procedia Structural Integrity 72 (2025) 61–68

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2.3. G I and G II estimation

2.3.1. DCB and ENF G IC and G IIC of the pure mode tests were obtained from the DCB and ENF tests, respectively, enabling creating the fracture envelopes. Only the CBBM was applied. This method does not need to measure a , inferring an equivalent crack ( a eq ) from the current compliance ( C=  /P ), and considers the fracture process zone (FPZ), which is particularly large for ductile adhesives. The fact that a was calculated, rather than estimated, eliminates measurement errors.

2.3.2. SLB Six methods were then identified, among them the CBBM applied to SLB specimens:

 Model 1 (Oliveira et al. (2009)) is based on the Timoschenko beam theory, including the C established by the Castigliano theorem. The total fracture energy ( G T ) in an SLB specimen is determined by the Irwin-Kies equation (Irwin and Kies 1954), and the mode partitioning is carried out by the Szekrényes and Uj model [17];  Model 2 (Szekrényes and Uj (2004) ) is supported by Timoschenko’s beam theory and Winkler's foundation analysis. The model considers transverse shear and the elastic effects of the foundation (it also includes a shear correction factor), separating expressions for G I and G II ;  Model 3 (Zhu (2009)) is a continuation of the study by Szekrényes and Uj (2007). C and G I / G II for the SLB specimen are determined from the Euler-Bernoulli and Timoshenko beam theories. The formulation is based on a Winkler-Pasternak Foundation analysis, including Saint-Venant effects at the crack tip;  Model 4 (Kim et al. (2011)) applies the Irwin-Kies equation (Irwin and Kies 1954) to an unbalanced SLB specimen (with different h ). A relationship is established between C and a experimentally, including bending and shear at the crack tip. The expressions of G I and G II are derived from the load decomposition into tensile and shear;  Model 5 (Szekrényes and Uj (2007)) evolves from model 2, based on beam theory, taking into account the Saint Venant effect and shear deformation at the crack tip. The final expressions are divided into G I and G II ;  Model 6/CBBM (Fernández et al. (2013)) applies the beam theory of Szekrényes and Uj (2004). However, it applies a eq and not a , which is calculated from the P -  curves. The benefits of this model were previously discussed. 2.4. Abaqus ® pre-processing The SLB tests were numerically replicated in software Abaqus ® , employing CZM to estimate the cohesive laws and validate the crack propagation criteria observed experimentally. The simulations accounted for geometric non linearity, and the mesh was refined particularly around the crack growth area and the points of contact with the loading cylinders (as illustrated in Fig. 3, along with the boundary conditions).

Fig. 3. Mesh details at the crack tip and boundary conditions (specimen with t A =0.1 mm).

The two-dimensional models utilized plane-strain four-node solid elements (CPE4 in Abaqus ® ) for the adherends and a single layer of four-node cohesive elements with a triangular law (COH2D4 in Abaqus ® ) to model the adhesive layer. Mesh refinement was biased, with six elements across the thickness of the adherends, showing greater detail at the free faces. The boundary conditions included clamping the edge cylinders in the plane, constraining the horizontal