  Cracks, the major vehicle for material failure, tend to accelerate to high
velocities in brittle materials. In three-dimensions, cracks generically
undergo a micro-branching instability at about 40% of their sonic limiting
velocity. Recent experiments showed that in sufficiently thin systems cracks
unprecedentedly accelerate to nearly their limiting velocity without
micro-branching, until they undergo an oscillatory instability. Despite their
fundamental importance and apparent similarities to other instabilities in
condensed-matter physics and materials science, these dynamic fracture
instabilities remain poorly understood. They are not described by the classical
theory of cracks, which assumes that linear elasticity is valid inside a
stressed material and uses an extraneous local symmetry criterion to predict
crack paths. Here we develop a model of two-dimensional dynamic brittle
fracture capable of predicting arbitrary paths of ultra-high-speed cracks in
the presence of elastic nonlinearity without extraneous criteria. We show, by
extensive computations, that cracks undergo a dynamic oscillatory instability
controlled by small-scale elastic nonlinearity near the crack tip. This
instability occurs above a ultra-high critical velocity and features an
intrinsic wavelength that increases proportionally to the ratio of the fracture
energy to an elastic modulus, in quantitative agreement with experiments. This
ratio emerges as a fundamental scaling length assumed to play no role in the
classical theory of cracks, but shown here to strongly influence crack
dynamics. Those results pave the way for resolving other long-standing puzzles
in the failure of materials.
