PSI - Issue 39

Arturo Pascuzzo et al. / Procedia Structural Integrity 39 (2022) 649–662 Author name / Structural Integrity Procedia 00 (2019) 000–000

656 8

E M ′

E M ′

, act aux I −

, act aux II −

(

)

(

)

act

aux I −

act

aux II −

0 and

0

(18)

K

K

K

K

=

=

=

=

I

II

II

I

aux I −

aux II −

2

2

K

K

I

II

Figure 3. J -integral and a schematic of the arbitrary function q(x 1 , x 2 )

3. Numerical implementation The proposed approach is implemented in commercially available software, that is, Comsol Multi-physics (COMSOL (2018)). This software offers a flexible and easy-to-use FE environment to analyze different structural problems. Besides, it provides practical apps that permit the linkage with other specialist software, such as those devoted to design, mathematics, and programming. In the present case, the LiveLink for MatLab application is used to link Comsol with MatLab, thus extending the basic Comsol functionalities with MatLab programming tools (Lonetti and Pascuzzo (2014), Bruno et al. (2016), Lonetti and Pascuzzo (2016), Lonetti et al. (2016), Lonetti et al. (2019), Lonetti and Pascuzzo (2020)). In particular, using the Livelink platform, a MatLab script to manage the propagation process has been configured (Greco et al. (2013), Greco et al. (2018a)). In the following, a description of the propagation process configured by the proposed modeling approach is reported. Figure 4 depicts some sketches to support the discussion. Initially, the script code executes preliminary operations on the geometry of the problem worth being analyzed (Figure 4-a and b). Specifically, the initial pre-crack, represented through a polyline, is split into two pieces by adding an extra node close to the crack tip. The result is a short segment that stretches during the propagation process, whereas the remaining portion of the polyline keeps fix. Once that the preliminary geometric operations are concluded, the code meshes the geometry, assigns the boundary conditions, and starts the analysis. The external loads are increased progressively and, for each incremental step, the fracture variables at the crack front are extracted using the M -integral. Hence, a preliminary value for the kinking angle is computed ( θ ) and the conditions of crack nucleation are assessed. The load increases until crack nucleation conditions are satisfied. When such conditions are met, then c θ θ = , and the propagation phase takes place. Hence, the mesh node of the crack tip is moved along the direction identified by the kinking angle (Figure 4-c). It is worth noting that the motion of the mesh causes distortions for the finite elements around the crack tip region. Relevant distortions affect negatively the accuracy of the proposed modelling approach. To avoid this circumstance, the code executes a remeshing of the computational mesh when the element distortions, evaluated using the (nonnegative) first invariant of the isochoric Green–Lagrange strain tensor relative to the mesh frame, reaches a pre-fixed threshold. The propagation phase continues until that the angle variation of the stretching segment regarding the initial pre crack reaches a tolerance value (Toll. θ ). When θ =Toll. θ , the propagation is stopped, and the code updates the geometry of the model by adopting the configuration gained in the last step of the analysis. The new geometry represents a novel starting point for the analysis. Therefore, starting from this configuration, a new stretching segment