PSI - Issue 79

Domenico Ammendolea et al. / Procedia Structural Integrity 79 (2026) 467–474

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model actively monitors the extension of the stretching segment ( L c ) and the occurrence of crack branching. The first check is vital to ensure realistic curved crack trajectories, which the model approximates as a series of linear segments. Specifically, the current length of the stretching segment is checked against a predefined maximum threshold ( L lim ). If the critical threshold is met ( L c > L lim ), the current analysis halts immediately. A new geometry is then instantaneously generated, which is based on the configuration of the system at the moment of the pause. In this updated geometry, a new stretching segment is defined, and the analysis restarts. Crucially, it uses the displacement solution from the paused step as its initial condition (Figure 2-d). The possibility of crack branching is then determined by evaluating the branching fracture function (as defined in the given Eq. (11)). If the branching condition is met, the analysis is again paused. The system creates a new geometry and introduces two short segments at the crack front that represent the splitting These new segments have a user-defined length ( L b ) and an initial inclination angle ( α b ) relative to the original crack plane. When the analysis restarts, these branching segments function as new stretching segments (Figure 2-e). Crucially, their subsequent propagation direction is evaluated using Eq.(8). Because the Moving Mesh (MM) technique modifies the orientation of these segments consistent with the computed propagation directions, it automatically corrects the initial trial angle ( α b ). This leverages the MM capability to simplify numerical implementation by avoiding the need to define complex theoretical criteria for the branching angle explicitly. The numerical analysis then continues, following these steps until the body collapses.

Figure 2. Snapshots of some of the key stages utilized by the proposed model to replicate the crack propagation phenomenon: (a) Geometry of a pre-cracked body; (b) Generation of the stretching segment; (c) Crack propagation; (d) Geometry update following the maximum extension of the stretching segment; (e) Geometry update following the occurrence of a crack branching event.

4. Results Figure 3-a shows a rectangular plate subjected to uniform tensile traction on its upper and lower boundaries. The material is glass, characterized by standard elastic properties E =32 GPa, ν =0.2, and ρ =2450 kg/m 3 ), and a dynamic crack initiation toughness ( K Id = 0.33 MP am 1/2 ). Figure 3-b depicts the computational mesh adopted in the numerical simulation. Consistent with established experimental and numerical evidence, the velocity criterion was chosen for branching, with the critical branching velocity ( V c ) set to 0.6 c R , where c R is the Rayleigh wave speed of the glass. The time history of the crack-tip propagation velocity (as shown in Figure 4) provides key validation data. The simulation using the proposed method predicts that crack propagation initiates at 10.3 µ s. The velocity then follows a non-linear trajectory: an initial sharp increase to 1080 m/s is followed by a temporary drop, after which the