PSI - Issue 41

A.E.S. Pinheiro et al. / Procedia Structural Integrity 41 (2022) 60–71 Pinheiro et al. / Structural Integrity Procedia 00 (2019) 000 – 000

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P m . The second approach used plane elements in the adhesive layer, and it was used to estimate the elastic stress distributions. In all cases, the adherends were modelled with plane elements (axisymmetric elements) and were meshed with four-node quadrilateral axisymmetric elements (CAX4). Moreover, in the models for the second approach, the adhesive layer was modelled with 10 rows of elements, as shown in Fig. 4. The mesh along the overlap had a double bias with a minimum size of 0.02 mm and a maximum size of 0.1 mm. Single biases were used on the adherends’ meshes in the through -thickness direction and from the ends towards the overlap. The element sizes used were equal to those used for the overlap region. The biases allowed assigning small element sizes in the vicinities of the bond-line and its ends, and in the adhesive-adherend interfaces. This approach allowed for a suitable mesh for both geometrical cases studied ( L O =20 mm and 40 mm).

Fig. 4. Detail of the mesh at the adhesive thickness and boundary conditions applied.

Material properties were assigned accordingly to each region. The specific properties were described in Section 2.2. Regarding the boundary conditions and loads, one end of the other tube was constrained in all directions ( U R = U  = U Z =0), whilst a traction displacement,  , was applied at the end of the inner tube ( U R =  , U  = U Z =0), as shown in Fig. 4. The boundary conditions were represented on a polar coordinate system because the joints were modelled as axisymmetric. The models were simulated considering a static general approach using the standard solver, and large deformations were considered. 3. Results 3.1. Peel and shear stresses To analyze the tubular adhesive joints under tensile stresses, two analytical models were evaluated: Nayeb-Hashemi et al. (N-H) and Pugno and Carpinteri (P&C), which were compared with an elastic analysis by the FEM, regarded as the comparison standard due to the reliability of the method for these purposes (Goglio et al. 2008). The analytical models were processed in Microsoft Excel ® , using the described formulations in Section 2.4, the data provided on the geometry of the tubular joint in Section 2.1, and the material properties in section 2.2. For a critical analysis and comparison of the stresses between the analytical methods and the FEM, the axial loading with P =1000 N is simulated. This value was assigned to prevent plasticization of both the adhesive and adherends. From the presented models, the absolute  y and  xy stresses were obtained. To simplify the analysis, only the 2015 plots will be presented, although the peak stresses will be described for all adhesives. Fig. 5 compares the absolute  y stresses in the 2015 by the models of Pugno and Carpinteri, and the FEM, for L O =20 mm (a) and L O =40 mm (b).  y stresses traditionally peak at the overlap edges, which was found for the evaluated joints, but a large divergence was found, especially at the ends of the joint. Moreover, the theoretical model predicts compressive stresses at x / L O =1, which does not agree with the FEM predictions. The peak value comparisons are given in Table 3 for the three adhesives. Considering the AV138, the percentile deviation of the theoretical predictions over the FEM (   y ) was 541.5% for both L O , by excess. The differences for the 2015 were 270.7% ( L O =20 mm) and 280.4% ( L O =40 mm). The smaller difference can be justified by the smaller stiffness of this adhesive. The 7752 provides the smallest difference between the three adhesives: 101.3% for L O =20 mm and 355.9% for L O =40 mm, because of the smallest stress gradients in the overlap.