PSI - Issue 41

T.F.C. Pereira et al. / Procedia Structural Integrity 41 (2022) 14–23 Pereira et al. / Structural Integrity Procedia 00 (2019) 000 – 000

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shapes are the triangular (also known as bilinear) (Alfano and Crisfield 2001), parabolic (Allix and Corigliano 1996), polynomial (Chen 2002), trapezoidal (Kafkalidis and Thouless 2002), and exponential (Chandra et al. 2002). The triangular law requires fewer parameters, easing its application, but it has the drawback of underpredicting the behavior of ductile materials (Liljedahl et al. 2006). In this work, three cohesive laws were evaluated: triangular, exponential, and trapezoidal, as shown in Fig. 3.

Fig. 3 – Triangular, exponential, and trapezoidal cohesive laws (adapted from (Alfano 2006)).

Under pure mode (i.e., mode-I or traction), the tensile cohesive stress ( t n ) increases proportionally to the tensile displacement (  n ) until it reaches a maximum or peak value ( t n 0 at  n 0 ), being the proportionality constant the initial stiffness ( K 0 ). For the triangular and exponential laws, once the peak value is reached, t n decreases as  n increases until t n =0 at the tensile failure displacement (  n f ), as shown in Fig. 3 (a). In the trapezoidal law (Fig. 3 b), at the end of the elastic stage, t n remains constant until reaching the stress softening onset displacement (  n s ), and then it decreases linearly until  n f . Disregarding the CZM law shape, the area beneath the t n -  n curve corresponds to G IC (Alfano 2006). The process is equivalent for mode-II or shear, being the corresponding terms t s , t s 0 ,  s 0 ,  s s   s f , and G IIC . Under tensile and shear pure mores for the adhesive layer, K 0 takes the value of E and shear modulus ( G ), respectively. Under mixed mode, both mode-I and mode-II contribute to material degradation until fracture. In this case, the stiffness ( K COH ) is a matrix relating E and G moduli with the strains in the respective directions. This formulation is described with more detail in the literature (Campilho et al. 2013, de Sousa et al. 2017). Finally, damage initiation is defined by a quadratic criterion combining the proportions of traction and shear stresses within an elliptic envelope, noting that compression does not contribute to damage initiation, while damage growth is defined by a linear energetic function (Campilho et al. 2013). 3. Results 3.1. CZM validation with experiments To validate the CZM model used in the numerical analysis, validation with SLJ experimental data was done first, using DIN 55 Si7 steel adherends. Additional details are given in Valente et al. (2019). For CZM validation, P m and the displacement at P m (  P m ) were compared with experiments. Fig. 4 shows the load-displacement ( P -  ) curve (a) and P m and  P m (b) comparison between the experimental tests and numerical predictions.

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Fig. 4 – Comparison between the test and CZM data: P -  curves (a) and P m and  P m (b).