PSI - Issue 39

M.R.M. Aliha et al. / Procedia Structural Integrity 39 (2022) 393–402 Author name / Structural Integrity Procedia 00 (2021) 000–000

396 4

= −1 � 3 2 + � 4 + 8 2 2 2 +9 2 �

(3) where the crack propagation angle is measured with respect to the crack plane. The crack propagation in the straight ahead direction is indicated by = 0 . If > 0 , the result is < 0 , but if < 0 , the result is > 0 . 3.2. Hybrid XFEM-CZM approaches Belytschko and Black [14] outlined the essential characteristics of the recently developed extended finite element method (XFEM), which models crack development and separation between crack faces using local enrichment functions for nodal displacements. When the crack path/tip changes location and direction owing to loading circumstances, the XFE methode creates the appropriate enrichment functions for the nodal points of the finite elements surrounding the crack path/tip [15]. Without background experience of the crack path, XFEM can be used to initiate and propagate cohesive cracks in an adhesive layer [16]. In recent years, the cohesive zone model (CZM) has widely been utilized to simulate the fracture initiation, propagation, and failure. The traction-separation constitutive law is used to forecast damage and failure in cohesive elements (surfaces) [8,17]. In this study, a bi-linear constitutive law is employed to predict the damage and failure of the cohesive surfaces at the interfaces. This law is shown in Figure 2 in the form of a traction-separation response. The quadratic nominal stress criterion was used for monitoring damage initiation. Based on this criterion, damage initiates when the summation of the normalized stresses reaches unity, according to Equation (4). In the quadratic nominal stress criterion, the tension and shear traction-separation response are considered to be dependent on each other that make this criterion a mode dependent model. The bi-linear cohesive zone model is completely defined by damage evolution. In this study power law criteria were used for analyzing damage evolution according to Equation (5) [8,18]. � ⟨ ⟩ � 2 + � � 2 + � � 2 = 1 (4) � � + � � + � � = 1 (5) is the normal contact stress in the pure mode I. , and are the shear contact stress along the first and second shear directions, respectively. , , and are the peak values of the contact stress. “< >” is a Macaulay bracket, demonstrating that under pure compression, no damage occurs. , , and are the tensile and shear strain energy release rate, respectively. , , and are the critical values of strain energy release rate. is the power law exponent for mixed mode fracture, which in this study is considered equal to one. Damage initiation and evolution behavior of the XFEM enrichment zone, similar to cohesive surfaces, are based on the quadratic nominal stress criterion and the power law criteria respectively, and follow the linear damage curve as shown in Figure 2. A combined XFEM and surface-based CZM model can forecast crack growth in adhesive areas near to the interface, which is difficult to predict with a basic XFEM model, as well as mesh independent crack growth in the adhesive layer and interface, which is unachievable with only cohesive elements [16,17].