PSI - Issue 52

Roman Vodička et al. / Procedia Structural Integrity 52 (2024) 242 – 251 R. Vodicˇka / Structural Integrity Procedia 00 (2023) 000–000

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(a)

(c)

(b)

Fig. 3. Strain energy ω P distribution at selected instants of the simple stress-pulse configuration for various viscosity parameters: both zero (a), τ r1 = 1 µ s (b), τ r2 = 1 µ s (c).

(a)

(c)

(b)

Fig. 4. Phase-field parameter distribution at the central part of the domain at selected instants of the simple stress-pulse configuration for various viscosity parameters: both zero (a), τ r1 = 1 µ s (b), τ r2 = 1 µ s (c).

(a)

(b)

50

50

f ( t )

Γ c

20 20

f ( t )

Fig. 5. A block under tension (a) exposed to a force load (b).

wave in Fig. 3 (b)) the energy of the wave was able to degrade only partially. The values of the phase-field parameter α close to the central line can be read in Fig. 4. This crack propagation occurred at the instant of the waves bump which can be checked by the key showing the time instants pertinent to the plotted curves close to the aforementioned instant of t = 0 . 276ms. Dynamic crack propagation is studied in a typical problem of a block loaded by a force load. A pre-crack is considered to span across a middle of the domain width, shown as Γ c in Fig. 5. Actually applied loading is also shown in the same graphics. The particular geometry is motivated by similar analysis in Borden et al. (2012); Li et al. (2023) where the crack bifurcation point was observed under conditions of dynamic crack propagation. The initial elastic properties are: K p = 22 . 22GPa, µ = 13 . 33 GPa, the mass density is ρ = 2450 kgm − 3 . These parameters cause the P-wave to propagate at the velocity of 3 . 81kms − 1 , while the Rayleigh wave speed is 2 . 13kms − 1 . The additional parameters of the rheological model are γ = 1, τ r1 and τ r2 are either zero or 1 µ s giving various material options. The mesh is regular made of square elements of the size h = 0 . 5mm. The fracture energy in the domain is G I c = 3Jm − 2 , and it is slightly modified for the hear mode: G II c = 2 G I c . The phase-field length parameter is set to ε = 2 mm. The PFM degradation function Φ is chosen simply the same as in the previous example. The loading f ( t ) according to the graph in Fig. 1 is applied with a refined time step of 0 . 1 µ s, after an initiation period it is kept constant. The processes related to the crack propagation are studied in terms of the phase-field variable, which provides actual position of the crack at each instant, and also its changing values determine the velocity at which the crack propagates. The graphs in Fig. 6 demonstrate the velocity of the crack propagation expressed in terms of the Rayleigh wave speed. First, there is observed an instant when the crack starts to propagate: t = 21 . 9 µ s in the inviscid case