PSI - Issue 52

Roman Vodička et al. / Procedia Structural Integrity 52 (2024) 242 – 251

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R. Vodicˇka / Structural Integrity Procedia 00 (2023) 000–000

Fig. 6. Crack propagation velocity in terms of the Rayleigh wave speed v R .

(a)

(c)

(b)

Fig. 7. Distributions of the phase-field variable α (a), the stress trace [MPa] (b), and norm of deviatoric stress [MPa] (c) in the block at the instant t = 40 µ s for the inviscid case. Displacements are magnified 100 times in the case of α . (a) (b) (c)

Fig. 8. Distributions of the phase-field variable α (a), the stress trace [MPa] (b), and norm of deviatoric stress [MPa] (c) in the block at the instant t = 40 µ s for the case with τ r1 = 1 µ s. Displacements are magnified 100 times in the case of α . (a) (b) (c)

Fig. 9. Distributions of the phase-field variable α a), the stress trace [MPa] (b) and norm of deviatoric stress [MPa] (c) in the block at the instant t = 40 µ s for the case with τ r1 = 1 µ s and τ r2 = 1 µ s. Displacements are magnified 100 times in the case of α .

and t = 24 . 2 µ s or t = 24 . 5 µ s for the other options. Due to used rheologies, the cases with viscosity show slower wave propagation and also the positions of the crack bifurcation points are di ff erent: in inviscid case the bifurcation occurs at the instant t = 58 µ s at the position of x 1 = 77 . 5 mm, while changing just τ r1 = 1 µ s provides the data for bifurcation as: t = 60 µ s and x 1 = 70 . 5 mm. Adding also another damping parameters even (slightly) modifies the previous to get t = 61 µ s and x 1 = 69 . 5mm. Two instants were chosen to demonstrate the actual distribution of the phase-field variable α and pertinent stress state expressed in terms of stress trace tr σ and the norm of the deviatoric stress dev σ . Deformations are magnified in drawings of α to see the opening character of the crack which is distinguished as a white zone pertinent to α = 0. In the first instant shown in Figs. 7, 8 and 9 for various inner material parametrs, the crack still propagates in a straight direction. There is a stress concentration close to the crack tip whose critical value corresponds to the value derived in Vodicˇka (2023) for the case of no viscosity: Considering the present material characteristics the critical stresses obey the inequality tr + σ 100MPa 2 + dev σ 60MPa 2 ≤ 1. From the observation that the influence of shear is not too big, the maximal stress trace is about 10 MPa. The same variables are observed at an instant after the crack bifurcation corresponding to t = 80 µ s in Figs. 10, 11 and 12. The slower crack propagation of the viscid cases is clearly visible. The bifurcated crack makes changes in the stress distribution and may provide a zone with high shear stress where a crack might have been initiated. Therefore, it