Autonomy and Singularity in Dynamic Fracture

TL;DR

This paper investigates the physical and mathematical properties of a new $1/r$ singularity in dynamic fracture theory, demonstrating its relation to autonomy, Newton's equations, and experimental observations.

Contribution

It clarifies the conditions under which the $1/r$ singularity maintains autonomy and satisfies Newton's equations in dynamic fracture theory.

Findings

The $1/r$ singularity does not automatically satisfy autonomy.

Enforcing Newton's equation ensures the singularity's autonomy.

The linear momentum carried by the $1/r$ fields vanishes.

Abstract

The recently developed weakly nonlinear theory of dynamic fracture predicts $1/r$ corrections to the standard asymptotic linear elastic $1/\sqrt{r}$ displacement-gradients, where $r$ is measured from the tip of a tensile crack. We show that the $1/r$ singularity does not automatically conform with the notion of autonomy (autonomy means that any crack tip nonlinear solution is uniquely determined by the surrounding linear elastic $1/\sqrt{r}$ fields) and that it does not automatically satisfy the resultant Newton's equation in the crack parallel direction. We show that these two properties are interrelated and that by requiring that the resultant Newton's equation is satisfied, autonomy of the $1/r$ singular solution is retained. We further show that the resultant linear momentum carried by the $1/r$ singular fields vanishes identically. Our results, which reveal the physical and mathematical nature of the new solution, are in favorable agreement with recent near tip measurements.