PSI - Issue 52

Made Wiragunarsa et al. / Procedia Structural Integrity 52 (2024) 583–593

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Wiragunarsa et al. / Structural Integrity Procedia 00 (2023) 000–000

1 2

U n + 1 i

( U n

i + U ∗∗ i )

(16)

=

The time step ∆ t = t n + 1 - t n is calculated based on the Courant-Friedrichs-Lewy (CFL) condition as follow

∆ t = α CFL

h C p , max

(17)

where h is the smoothing length and C p , max is the maximum p -wave speed. The value of α CFL is varied from 0 to 1.

3.3. Crack modelling

Two crack modelling techniques are compared in this research. The first technique is particle deletion, as presented in Figure 3(a). When the crack propagates, or the fracture criterion is reached, the particle with the highest value of the maximum principal stress will be deleted. Particles with red rings are deleted in the previous step, indicating the crack path. The second technique is the interaction deletion, as shown in Figure 3(b). The crack is located between the particles. All of the interaction pairs that intersect the crack line will be removed. When the crack propagates, the interaction that has the highest value of the average maximum principal stress 1 2 ( σ i 1 + σ j 1 ) will be deactivated. Therefore, the two particles in the interaction pair will not a ff ect each other. The details of the procedure are illustrated in Figure 4. Particles with red rings are identified as the crack surface. In the particle deletion technique, as illustrated in Figure 4(a), the nearest particle across the crack line has a distance 2 h . Therefore, the interaction strength is already zero, and the crack can be modelled straightforwardly. On the other hand, the interaction deletion technique, as illustrated in Figure Figure 4(b), has a more complex algorithm. The nearest particle where the interaction line intersects the crack line has a distance h . It means the interaction strength is not zero. In order to model the crack, the particle at the crack surface must be identified. Then, an additional algorithm must be introduced to determine which interaction is active. In the figure, the interaction can be summarised as follow

• L 1 is deactivated • L 2 is active • L 3 is deactivated

• L 4 is active • L 5 is active

The algorithm to identify the active interaction is more complex, especially when the crack direction is deflected. In the fatigue crack growth problems, crack propagation is governed by Paris’s equation for particle deletion and interaction deletion frameworks. When the stress intensity factor exceeds fracture toughness, crack propagation becomes rapid.

4. Numerical examples

4.1. Crack growth direction analysis

Qualitative analysis is performed using a rectangular plate with a centre crack, as illustrated in Figure 5. The test aims to analyse the crack growth direction using the particle deletion and the interaction deletion frameworks. The plate is made of aluminium material with Young’s modulus E = 71 . 7 GPa , Poisson’s ratio ν = 0 . 33, density ρ = 2810 kg / m 3 , ultimate strength σ u = 572 MPa , and fracture toughness K IC = 25 MPa √ m . The plate has dimensions L = 4 m , W = 1 m , and various crack orientations. Then, the plate is pulled by nominal stress σ until the stress intensity factor K I = K IC . Once the fracture criterion is reached, the crack will propagate abruptly, and the crack growth direction follows the particle with the maximum principal stress greater than or equal to the ultimate strength. The results are presented in Figure 6. The contour indicates the crack growth direction only. In the particle deletion