Oscillatory and tip-splitting instabilities in 2D dynamic fracture: The roles of intrinsic material length and time scales

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Abstract

Recent theoretical and computational progress has led to unprecedented understanding of symmetry-breaking instabilities in 2D dynamic fracture. At the heart of this progress resides the identification of two intrinsic, near crack tip length scales — a nonlinear elastic length scale and a dissipation length scale ξ — that do not exist in Linear Elastic Fracture Mechanics (LEFM), the classical theory of cracks. In particular, it has been shown that at a propagation velocity v of about 90% of the shear wave-speed, cracks in 2D brittle materials undergo an oscillatory instability whose wavelength varies linearly with , and at larger loading levels (corresponding to yet higher propagation velocities), a tip-splitting instability emerges, both in agreements with experiments. In this paper, using phase-field models of brittle fracture, we demonstrate the following properties of the oscillatory instability: (i) It exists also in the absence of near-tip elastic nonlinearity, i.e. in the limit 0, with a wavelength determined by the dissipation length scale ξ. This result shows that the instability crucially depends on the existence of an intrinsic length scale associated with the breakdown of linear elasticity near crack tips, independently of whether the latter is related to nonlinear elasticity or to dissipation. (ii) It is a supercritical Hopf bifurcation, featuring a vanishing oscillations amplitude at onset. (iii) It is largely independent of the phenomenological forms of the degradation functions assumed in the phase-field framework to describe the cohesive zone, and of the velocity-dependence of the fracture energy Γ(v) that is controlled by the dissipation time scale in the Ginzburg–Landau-type evolution equation for the phase-field. These results substantiate the universal nature of the oscillatory instability in 2D. In addition, we provide evidence indicating that the tip-splitting instability is controlled by the limiting rate of elastic energy transport inside the crack tip region. The latter is sensitive to the wave-speed inside the dissipation zone, which can be systematically varied within the phase-field approach. Finally, we describe in detail the numerical implementation scheme of the employed phase-field fracture approach, allowing its application in a broad range of materials failure problems.

Section snippets

Background and motivation

Materials failure, which is mainly mediated by crack propagation, is an intrinsically complex phenomenon that couples dynamic processes at length and time scales that are separated by many orders of magnitude, giving rise to a wealth of emergent behaviors. Crack initiation and dynamics are of prime fundamental and practical importance, and have been intensively studied in the last few decades (Freund, 1998, Broberg, 1999). Despite some significant progress, our understanding of many basic

A nonlinear phase-field approach to dynamic fracture: resolving physically-relevant, intrinsic material length scales

The nonlinear phase-field approach to dynamic fracture, to be employed in this paper, has been introduced in quite some detail in Chen et al. (2017) and studied in Chen et al. (2017) and Lubomirsky et al. (2018). Its presentation is repeated here for completeness, and in order to further highlight its physical content and potential utility. This phase-field approach is a Lagrangian field theory that is designed to incorporate the intrinsic material length scales and ξ, and to allow for high

The oscillatory instability: The 0 limit, supercritical Hopf bifurcation and independence of Γ(v)

One of the major achievements of the phase-field approach presented in the previous section is related to the high-velocity 2D oscillatory instability, shown in  Fig. 1a–b and briefly discussed earlier in Section 1. To apply the phase-field framework to a given physical problem, one needs to specify the relevant elastic strain energy density functional estrain, the degradation functions g(ϕ), f(ϕ) and w(ϕ), the system’s geometry and the applied boundary conditions. As the 2D oscillatory

The ultra-high velocity tip-splitting instability: Relations to the wave-speed inside the dissipation zone

As discussed above in relation to Figs. 1c,f and 2c, upon increasing the driving force WΓ0 for fracture, cracks are predicted to accelerate faster and to yet higher velocities, and feature a tip-splitting instability, either after the onset of oscillations or even prior to it. This behavior is supported by experiments, cf.  Fig. 1d. The observation of tip-split crack states, together with the previously discussed oscillatory crack states, allow one to construct a comprehensive phase diagram

Discussion and concluding remarks

In this paper, we used phase-field simulations to investigate the role of intrinsic material length and time scales on the emergence of oscillatory and tip-splitting instabilities in 2D dynamic fracture. The two basic length scales, which are absent in LEFM, include the scale ξ of the dissipation zone where elastic energy is transformed irreversibly into new fracture surfaces and a nonlinear length that is a measure of the distance from the crack tip at which elastic nonlinearity becomes

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

This research was supported by a grant from the United States-Israel Binational Science Foundation (BSF, Grant No. 2018603), Jerusalem, Israel, and the United States National Science Foundation (NSF, Grant No. 1827343). E.B. also acknowledges support from the Ben May Center for Chemical Theory and Computation and the Harold Perlman Family .

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