PSI - Issue 47

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Domenico Ammendolea et al. / Procedia Structural Integrity 47 (2023) 488–502 Author name / Structural Integrity Procedia 00 (2019) 000–000

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Figure 2. A pre-cracked solid domain subjected to quasi-static external loadings.

The governing equations describing the kinematics of the moving mesh problem are usually referred to as smoothing or rezoning methods. Such methods adjust the position of the nodes inside the computational mesh with great regularity, preventing finite elements from excessive distortions. Among the several smoothing methods available in the literature, the proposed model employs the Laplacian rezoning approach, which consists of solving the following differential equations inside the computational domain: M M    x 0 (5) where, 1 2 [ ] x x  x is the vector identifying the spatial position occupied by the i -th node of the computational mesh at the current configuration of the solid domain. Eq. (5) is to be solved with the following boundary conditions: on ; at C F C      x 0 x n  (6) In particular, the first condition states that the nodal mesh displacement function   x x X (being x the spatial coordinate and X the material coordinate of the mesh nodes) is zero for the mesh nodes along the external boundary of the computational domain. On the other hand, the second condition imposes that the mesh node of the crack tip must advance along the propagation direction (identified by the versor C n ) of a displacement increment F   . In the proposed method the propagation direction C n and, more generally, the conditions of crack nucleation are defined by means of the group of equations associated with the Fracture Mechanics problem. In particular, such conditions are expressed in an incremental form through the following Karush ‐ Kuhn ‐ Tucker (KKT) relationships: 0; 0; 0 F F F F f f        (7) In Eq. (7), ( , , , ( )) F F I II C C f f K K K a   is a proper fracture function defining the crack onset conditions. Nothing that this fracture function depends on the mixed-mode SIFs at the crack front ( i.e. , , I II K K ), the kinking angle ( C  ) evaluated with respect to the crack tip coordinate system, and the material fracture toughness ( ( ) C K a ), whose value is dependent on the length of the internal crack ( a ), according to the adopted R -curve approach. In the proposed model, the maximum hoop stress criterion developed by Erdogan and Sin (Erdogan and Sih, 1963) is used for defining the fracture function F f . The SIFs computing and the definition of the material fracture toughness will be described in detail in the next subsections. Returning to Eqs. (5) and (6), by using Eq. (2) and the Lagrangian Multipliers Method (LMM) to account for the boundary conditions, the weak form of the moving mesh problem can be expressed as follows:                1 1 0 C R R R R R R C C F C C J d J d                                                 Ψ Ψ X X x J x J λ x λ x λ n x n λ n x n  (8)