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

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 $\ell$ and a dissipation
length scale $\xi$ -- that do not exist in the classical theory of cracks. In
particular, it has been shown that at a high propagation velocity $v$, cracks
in 2D brittle materials undergo an oscillatory instability whose wavelength
varies linearly with $\ell$, and at yet higher propagation velocities and
larger loading levels, 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
$\ell\!\to\!0$, with a wavelength determined by the dissipation length scale
$\xi$. 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 fracture energy $\Gamma(v)$ that is controlled by a
dissipation time scale. These results substantiate the universal nature of the
oscillatory instability of ultra-high speed cracks in 2D. In addition, we
provide evidence indicating that the ultra-high velocity tip-splitting
instability is controlled by the limiting rate of elastic energy transport
inside the crack tip region. Finally, we describe in detail the numerical
implementation scheme of the employed phase-field fracture approach.