PSI - Issue 61

G.J.C. Pinheiro et al. / Procedia Structural Integrity 61 (2024) 71–78 Pinheiro et al. / Structural Integrity Procedia 00 (2019) 000 – 000

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2.3. Numerical modelling The numerical analysis was carried out in Abaqus ® . This software was chosen because of the possibility of using an integrated XFEM module, thus making it possible to simulate the damage and predict the strength of the joints. Bidimensional geometries of the joints were created using 3-node plane-strain CPE3 elements to model the adherent in the adherent-adhesive interfaces, and 4-node plane-strain CPE4 elements with full integration to model the remaining parts of the joint. A more refined mesh was used near the interface and, in the adherent s’ length direction, an increasing mesh refinement was considered, using a bias ratio, from the ends of the joint to the adhesive layer, as shown in Fig. 2.

Fig. 2. Details of the created mesh at the bonding region (  =45º).

Both the adhesive layer and the adherents were simulated with solid homogeneous sections. A triangular traction separation law was used to simulate the behavior of the latter. Besides, to predict the failure initiation of the adhesive, three stress failure criteria (QUADS, MAXS and MAXPS) and three strain failure criteria (QUADE, MAXE and MAXPE) were employed. Besides, to simulate damage propagation, an energetic criterion was used with different power laws (0.5, 1 and 2). Aiming to replicate the experimental conditions in the tests, the following boundary conditions were considered: the joints were restrained at one end (left) and restrained in the transverse direction of the joint at the opposite end. A normal displacement was also applied at the latter end. 2.4. XFEM description The XFEM incorporates enrichment functions into the finite element framework (Pike and Oskay 2015). These enrichment functions enable modelling displacement jumps that occur between crack faces during crack propagation. Abaqus ® offers six crack initiation criteria, described next, which follow a linear elastic behavior of the elements. The Maximum Principal Stress (MAXPS) and Maximum Principal Strain (MAXPE) criteria rely on the following functions, where  max and  0 max are the current and maximum principal stress, and  max and  0 max represent the current and maximum principal strain Purely compressive stress states do not induce damage, as indicated by the Macaulay brackets. For the MAXPS and MAXPE criteria, crack growth is defined as orthogonal to the direction of maximum principal stress or strain. Consequently, in cases involving mixed-mode loading, cracks tend to propagate rapidly toward the adherents. For these two criteria, the maximum load ( P m ) is considered to occur at first crack formation within the adhesive layer. The next expressions relate to the Maximum Nominal Stress (MAXS) and Maximum Nominal Strain (MAXE) criteria. In these expressions, t n and t s represent the existing normal and shear stresses, while t n 0 and t s 0 are the normal and shear strengths. The strains in the 2 nd expression have identical meaning max max 0 0 max max or f f                         = = . (1)

t t

   

    

    

n     s 0 0 ,

    

t

(2)

.

n max , 

or

max

f

f

s

=

=

0 0 n s t

n

s

Lastly, the Quadratic Nominal Stress (QUADS) and Quadratic Nominal Strain (QUADE) criteria are given by