PSI - Issue 33

Domenico Ammendolea et al. / Procedia Structural Integrity 33 (2021) 858–870

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Domenico Ammendolea et al./ Structural Integrity Procedia 00 (2019) 000 – 000

3 Numerical implementation The proposed scheme is implemented in Comsol (COMSOL (2018)), a commercially available software that provides a powerful FE environment for analyzing multi-physics problems easily. In the present case, the structural and heat transfer equations, pre-available in the software, are integrated into a unique scheme together with a moving mesh strategy consistent with the ALE formulation. Besides, Comsol makes available advanced linking apps for integrating the main functionalities of other specialist software, such as those devoted to design, mathematics, and programming. Among the various options, the proposed scheme uses the Livelink for MatLab platform that links Comsol to MatLab, thus offering the possibility of handling Comsol through traditional MatLab commands ( i.e. , scripts or functions) (Bruno et al. (2016), Lonetti and Pascuzzo (2014), Lonetti and Pascuzzo (2020), Lonetti et al. (2019), Lonetti et al. (2016)). Specifically, this capability has been used to set out a user-made script that manages the various steps involved in the propagation process automatically, thus creating an integrated framework where the ALE and the M -integral method work coordinately. The following describes the main steps of the propagation process configured by the proposed modeling approach. In particular, the prominent actions are depicted in Fig. 3. The first step regards the definition of the geometry of the problem under investigation. The geometry has a pre crack (represented by means of a polyline) that departs from the external boundary and culminates to an initial crack tip. Once crack onset conditions occur, the crack tip moves according to the fracture condition defined by one of the classic fracture criteria. Because the movement of the crack tip must develop while preserving the initial shape of the pre-crack, an extra node is added to the polyline near the crack tip (see Fig. 3-b). Such an extra node splits the polyline into two pieces. The longest one, colored in gray in Fig. 3-b, represents the pre-crack that stands fixed during mesh movement. The remaining one is a short segment (highlighted in blue in Fig. 3-b) that stretches during the propagation process, thus reproducing the crack advance. Once that the geometry is built, the code meshes the domain (see Fig. 3-c) and assigns boundary conditions. Next, the analysis starts, and the external actions increase progressively. For each incremental step, the M -integral extracts the SIFs and compute the kinking angle. Meanwhile, it assesses crack onset conditions. When crack conditions are satisfied ( i.e. , f F = 0), the propagation process starts, and the crack tip moves along with the direction identified by the kinking angle (  C ) (Fig. 3-c). The movement of mesh points occurs consistently with the ALE formulation. It is worth noting that the elements around the crack tip distort because of the motion of the mesh points. When distorts are relevant, the procedure re-meshes the computational domain to ensure regularity. In the proposed scheme, the rate of finite element distortion is evaluated using the (nonnegative) first invariant of the isochoric Green – Lagrange strain tensor relative to the mesh frame. This invariant corresponds to the Jacobian of the mapping function introduce in Eq. (5). A user-defined threshold for element distortion is fixed, so thus each time that the fixed threshold is reached, a remeshing action takes place.

Fig. 3. Description of the propagation procedure. (a) Input the geometry. (b) Creating stretching segment. (c) Crack tip motion