Key role of elastic vortices in the initiation of intersonic shear cracks

Sergey G. Psakhie, Evgeny V. Shilko, Mikhail V. Popov, and Valentin L. Popov

Phys. Rev. E 91, 063302 – Published 4 June 2015

Abstract

Using the particle-based method of movable cellular automata, we analyze the initiation and propagation of intersonic mode II cracks along a weak interface. We show that the stress concentration in front of the crack tip, which is believed to be the mechanism of acceleration of the crack beyond the speed of shear waves, is due to the formation of an elastic vortex. The vortex develops in front of the crack during the short initial period of crack propagation. It expands and moves away from the crack tip and finally detaches from it. Maximum stress concentration in the vortex is achieved at the moment of detachment of the vortex. The crack can accelerate towards the longitudinal wave speed if the magnitude of shear stresses in the elastic vortex reaches the material shear strength before vortex detachment. We have found that for given material parameters, the condition for the unstable accelerated crack propagation depends only on the ratio of the initial crack length to its width (e.g., due to surface roughness).

Received 23 August 2014
Revised 10 December 2014

DOI:https://doi.org/10.1103/PhysRevE.91.063302

Authors & Affiliations

Sergey G. Psakhie^1,2,3,4,*, Evgeny V. Shilko^1,3, Mikhail V. Popov^3,5, and Valentin L. Popov^3,4,5

¹Institute of Strength Physics and Materials Science, Russian Academy of Sciences, Tomsk, Russia
²Skolkovo Institute of Science and Technology, 143025 Moscow, Russia
³Tomsk State University, 634050 Tomsk, Russia
⁴Tomsk Polytechnic University, Tomsk, 634050 Tomsk, Russia
⁵Berlin University of Technology, 10623 Berlin, Germany

^*Corresponding author: sp@ispms.tsc.ru

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Issue

Vol. 91, Iss. 6 — June 2015

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Images

Figure 1
Schematic representation of the two-dimensional model and the loading conditions. Horizontal solid bold lines delineate the upper and lower external boundaries. Vertical dashed bold lines delineate the vertical faces to which periodic boundary conditions in the horizontal direction are applied. Longitudinal shear loading is introduced by horizontal displacement of the upper and lower external boundaries in opposite directions at a constant velocity $V_{load}$ . The vertical positions of the upper and lower boundaries are fixed.
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Figure 2
Velocity field near the tip of the propagating shear crack (a) $0.75 μ s$ , (b) $3.75 μ s$ , and (c) 7.5 μs after the beginning of propagation. Velocity vectors are shown for every third particle in the ensemble. The horizontal arrows mark the position of the plane of the crack and of its right tip. The dashed lines mark intact part of the interface (ahead of the crack tip). In this example the initial crack length was 0.6 mm, the height of the slab was 15 mm, the size of a movable cellular automaton was 0.1 mm, the shear strain rate was $10^{- 2} c^{- 1}$ .
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Figure 3
Snapshots of the distribution of equivalent stress near the right tip of a growing shear crack (a) 3.75 μs and (b) 7.5 μs after growth start. The system parameters are the same as in Fig. 2. The stress distributions correspond to the time moments shown in Figs. 2 and 2. The images demonstrate stress patterns before (a) and after (b) detachment of the elastic vortex from the crack. The white lines depict particle velocities.
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Figure 4
(a) Snapshots of the velocity field and (b) the distribution of equivalent stress near the tip of a growing shear crack 15 μs after growth start. The system and initial crack parameters are the same as in Fig. 2. In Fig. 4 velocity vectors of automata belonging to the top and bottom halves of the slab are marked by purple and blue colors respectively. In Fig. 4 black arrows indicate the first (1) and the second (2) elastic vortices. The images demonstrate velocity and stress patterns just after detachment of the second elastic vortex from the crack.
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Figure 5
Velocity field near the tip of the propagating shear crack (a) 0.1 μs and (b) 0.75 μs after the moment of nucleation of the secondary crack at a small distance ahead of the main crack. In this example the initial crack length was 0.5 mm. Other parameters of the model and the value of shear strain rate are the same as in the example shown in Figs. 2, 3, 4.
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Figure 6
Dependence of shear strength of the interface with initial crack $τ_{0}$ on dimensionless geometrical crack parameter $P$ . Shear strength $τ_{0}$ is normalized to the value of strength of the intact interface $τ_{i s}$ . Points show numerically determined values of shear strength for the interface with different kinds of cracks (section and two notches with doubly different thicknesses). Solid line shows empirically determined approximation (1).
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