PSI - Issue 41

Fabrizio Greco et al. / Procedia Structural Integrity 41 (2022) 576–588 Author name / Structural Integrity Procedia 00 (2019) 000–000

583

8

Fig. 3. A schematic of the most relevant steps of the propagation procedure. (a) Input geometry. (b) Creating stretching segment. (c) Crack tip motion. (d) Geometry updating.

The DSIFs are mandatory for evaluating propagation direction and assessing crack initiation conditions through the fracture function I F f (Eq. (8)). When I F f =0, dynamic crack propagation occurs, and the process is governed by the function P F f (Eq. (9)). Then, the computational node of the crack tip advances at a velocity a  , along the direction identified by the instantaneous value of propagation angle c  (Fig. 3-c). In particular, a  is computed by solving the optimization problem defined through Eq. (11). Noting that the movement of the mesh node of the crack tip distorts the adjacent finite elements, thus potentially compromising the accuracy of the numerical solution. To avoid such a drawback, the code re-meshes the computational domain when the distortion of finite elements is excessive. In the proposed method, the distortion entity is evaluated through the (nonnegative) first invariant of the isochoric Green–Lagrange strain tensor associated with the configured computational mesh. Besides assessing the regularity of the computational mesh during the crack growth process, the code checks the angle variation of the stretching segment regarding the initial pre-crack. This control needs for configuring proper reproduction of curved crack paths. Specifically, a tolerance value for the angle variation is imposed (Toll.  c ) so that when  c =Toll.  c , the code arrests the analysis. Subsequently, the mesh configuration associated with the last step of the analysis is assumed as novel geometry for the model (Fig. 3-d). The new geometry represents a novel starting point for the analysis and the code creates a new stretching segment. The novel geometry is re-meshed, and the analysis continues until either a crack arrest condition or a collapse event undergoes. In the former case, the analysis continues under the conditions dictated by the fracture function I F f , whereas in the latter case, the analysis stops. 4 Results This section reports numerical results to assess the accuracy and efficiency of the proposed modeling approach. The numerical investigation is performed regarding the pre-cracked rectangular plate depicted in Fig. 4-a. The dimensions of the plate are the following: length L=10 m, height 2H=4 m, and thickness B=1 m. Externally, the plate presents line constraints along the vertical boundaries that permit vertical sliding only. Besides, a uniform and distributed traction σ 0 = 500 MPa acts on the upper edge of the plate. The Young’s Modulus ( E ), Poisson’s ratio (ν), mass density (ρ), dilatational wave speed (c d ), and Rayleigh wave speed (c r ) are equal to E =210 GPa, ν=0.3, ρ=8000 kg/m 3 , c d =5944.5 m/s, and c r =2942.8 m/s. The fracture behavior of the plate is investigated under the assumption that the crack front advances horizontally at a constant velocity a  =1500 m/s. Fig. 4-b reports the mesh configurations adopted for the analysis together with a zoomed view of the crack tip region (the blue line identifies the stretching segment). The first mesh ( i.e. , mesh M1) comprises 432 triangular elements, finely arranged around the crack tip and somewhat coarser elsewhere.