PSI - Issue 33

A.F.M.V. Silva et al. / Procedia Structural Integrity 33 (2021) 138–148 Silva et al. / Structural Integrity Procedia 00 (2019) 000–000

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t SE =2 mm and L O =10 mm. On the one hand, the geometry evaluation consisted of separately varying L O between 10 and 20 mm, and also t SE between 1 and 4 mm. Although this joint geometry is similar to the SLJ used in the validation process, thus making this validation possible, there are few specificities of the tubular joint geometry and modelling assumptions that make it worth studying: asymmetric stress distribution arising from the different cross-sectional area between tubes, and use of two-dimensional (2D) axisymmetric models to equate the computational burden to that of SLJ 2D modelling, instead of using full 3D models.

Fig. 2. Tubular joint geometry and boundary conditions.

2.3. Abaqus ® modelling The validation and numerical CZM study were performed in Abaqus ® explicit (due to the dynamic load application), including a triangular CZM to model crack growth in the adhesive layer. The code is geometrically non linear and it automatically defined the optimal time step for all joints to assure that the critical step time was not exceeded. Actually, the step interval is calculated with the global estimation method, based on the maximum dilatational wave speed of the model. This value is updated for all increments depending on the finite elements’ distortion (Abaqus® 2013). Due to the revolved shape of the tubular joints, axisymmetric models were created, based on a 2D shape that is subsequently revolved over a central axis. The adhesive layer was modelled with COHAX4 cohesive elements and the adherends with CAX4 solid elements, both with explicit formulation (Barbosa et al. 2018). The triangular CZM formulation relies on tensile and shear pure-mode laws, which are input by the user, and selected mixed-mode criteria to infer damage initiation and growth under mixed-mode conditions. This model is described in detail in the next Section. The adhesive layer and respective ability to undergo damage and crack propagation was emulated with a single layer of CZM elements with a height equal to t A , which allows simulating the adhesive layer’s behavior in a macro-scale, thus not considering stress variations along the thickness. Two type of models were used: stress analysis and strength/energy prediction. The meshes for the strength/energy models were populated with a minimum element size of 0.2 mm (equal to t A ) at the overlap edges. On the other hand, the bias function reduced the mesh refinement at the neighboring regions. This procedure resulted in a refined mesh where stresses are known to vary (e.g., the overlap edges of the adhesive layer and the vicinity of the adhesive in the adherends), but less refinement where the stress fields show less variations due to the absence of geometric discontinuities. The models for stress analysis were typically 10 times more refined for accurate stress estimation. Fig. 3 shows an example mesh for a strength/energy prediction model with L O =10 mm. Fig. 2 depicts the boundary conditions, which consist of clamping one joint edge (in this case the leftmost one) and radially restraining the other edge (rightmost one). The impact loading (with U =40 J) was simulated by adding an artificial mass at the right edge of the tubular joints (Fig. 2). By the knowledge that, at impact, U = U c =½ mv 2 , and predefining a velocity field to the mass of 1.75 m/s, the mass volume and density are varied to achieve U =40 J. Thus, in this work, m and v were tuned to achieve this specific U .

Fig. 3. Mesh details at the overlap for the tubular joint with L O =10 mm.