Instability in dynamic fracture and the failure of the classical theory of cracks

TL;DR

This paper develops a new model for high-speed crack propagation in brittle materials, revealing an oscillatory instability driven by elastic nonlinearity that challenges classical fracture theories.

Contribution

The authors introduce a two-dimensional dynamic fracture model accounting for elastic nonlinearity, predicting crack paths and instabilities beyond classical assumptions.

Findings

Cracks undergo an oscillatory instability controlled by elastic nonlinearity.

The instability occurs above a critical velocity with a wavelength proportional to fracture energy over elastic modulus.

Results align quantitatively with experimental observations.

Abstract

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.