PSI - Issue 66

Umberto De Maio et al. / Procedia Structural Integrity 66 (2024) 495–501 Author name / Structural Integrity Procedia 00 (2025) 000–000

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2. Adaptive cohesive fracture modeling The main concept of the proposed numerical procedure for simulating crack propagation in quasi-brittle materials consists of two main stages. First, using the moving mesh technique (ALE formulation), a boundary element of the mesh representing a segment of the crack is detected and aligned with the direction of fracture propagation. This is done in accordance with the inter-element crack approach, once a predefined stress threshold for crack initiation is reached. In order to describe the nonlinear crack propagation, a zero-thickness cohesive interface element is dynamically inserted along this chosen boundary in the second stage. It is controlled by a traction-separation law. The cohesive interface model used in this study adopts a variational approach, written considering a cracked body, formulated in the referential configuration (also called mesh frame). To restate the standard weak form of the boundary value problem (BVP), usually written in the material configuration, the transformation rule of the spatial derivates from the material to he referential system are employed together with the Jacobian matrix (Donea et al., 2004; Amini and Shahani, 2013; Ponthot and Belytschko, 1998). It follows that the solving equation of the quasi static equilibrium problem can be expressed as:             R R R R R R R 1 R R 1 R R R R R R : d d d d c N c R R J J J J V                                   C u J v J t v f v p v v      (1) where � � � / � and � � � / � are the Jacobian related to the volume and area (representing the ratio of differential length), respectively, while Ω � , Γ � � , Γ �� and are the volume, the cohesive and Neumann boundary in the referential configuration. Moreover, the vectors � and � are unknown displacement field and arbitrary virtual displacement field, respectively, which belong to the set of kinematically admissible displacements U and to set of kinematically admissible variations of the approximated displacement field V , respectively. It is worth noting that the second term of Eq. (1) represents the virtual work of the cohesive tractions during the crack propagation process. Cohesive behavior is described through a traction-separation law with exponential softening (Campilho et al., 2013), incorporating both normal and shear displacements. It is expressed as follows:

0

K

p n 

n  n 



    

s 



 max

;

t

t

w







n 

(2)





0

w

n

s





w

max

w

max

max

were the exponential softening law � ��� � , depicted in Fig. 1, depends on the tensile strength � and fracture energy � of the considered material.

Fig. 1. CZM: (a) Exponential softening law and (b) the displacement and force components acting along the cracked boundary.