PSI - Issue 41

J.E.S.M. Silva et al. / Procedia Structural Integrity 41 (2022) 36–47 Silva et al. / Structural Integrity Procedia 00 (2019) 000 – 000

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Fig. 4 – Boundary conditions applied to the model and close-up of the bond-line partitions.

3.2. CZM description Adhesive joints are commonly loaded in mixed mode, i.e., a combination of normal and shear stresses. Cohesive formulations can represent the damage under such conditions. The cohesive law can have different shapes, such as bi-linear or triangular, linear-parabolic, trapezoidal, and exponential (Alfano, 2006). The triangular law provides a good representation of the actual behaviour of adhesives in bonded joints (Campilho et al., 2013), and requires fewer parameters than other cohesive laws, easing the experimental work necessary to determine those parameters, and consequently its application within computational software (Campilho et al., 2013). Under pure mode (i.e., mode-I or traction), the cohesive traction ( t n ) increases proportionally to the displacement (  n ) up to reaching a maximum or peak value ( t n 0 at  n 0 ), being the proportionality constant the initial stiffness ( K 0 ). Once the peak value is reached, t n decreases as  n increases until t n =0 at  n f . T he shape of this decrement gives the name to the cohesive law’s shape, being linear for the triangular one. In addition, the area beneath the t n -  n curve corresponds to the tensile fracture energy ( G IC ) dissipated in the process (Alfano, 2006). The process is equivalent for mode-II or shear, being the corresponding terms t s 0 ,  s 0   s f , and G IIC . In pure more for the adhesive layer, K 0 takes the value of E and 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 the stiffness in traction ( E ) and shear ( G ) moduli with the strains in the respective directions. This formulation is described in more detail in the literature (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, and damage propagation is defined by a linear energetic criterion (Campilho et al., 2013). 4. Results 4.1. Validation with experimental data The Results Section begins with the validation of the static CZM approach for bonded tubular joint design, to enable the numerical TSJ study that follows purely numerically. Fig. 5 shows the comparison between the average experimental P m (with the respective standard deviation) and numerical predictions as a function of L O . The numerical CZM results are close to the experiments for the most part. For L O =20 mm, the joint sustains a P m =27.2 kN, while the numerical offset is 6.1%. For L O =40 mm, the joint performance improved due to the adhesive ’ s moderate ductility and inherent plasticization. This allows a good stress distribution after plasticization onset reaching a P m =39.1 kN (an improvement of 43.4% over L O =20 mm). The numerical prediction for L O =40 mm showed a deviation of 2.9%. As a result, it is confirmed that the triangular law is suitable to model moderately ductile adhesives like the Araldite ® 2015 (Campilho et al., 2011c). The employed CZM model with triangular softening results in the reduction of transmitted stresses right after the damage initiation criterion is triggered. Although this does not exactly correspond to the behaviour of this adhesive, fracture growth in CZM modelling consists of an energetic approach which enables the method to work well for this type of adhesive (Fernandes and Campilho, 2017, 2019).