Autonomy and Singularity in Dynamic Fracture

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.

4 pages, 2 figures, related papers: arXiv:0902.2121 and arXiv:0807.4868

References in corpus (5)

• The Breakdown of Linear Elastic Fracture Mechanics near the Tip of a Rapid Crack
• Weakly Nonlinear Theory of Dynamic Fracture
• The 1/r singularity in weakly nonlinear fracture mechanics
• Dynamic Crack Tip Equation of Motion: High-speed Oscillatory Instability
• Elastic nonlinearities in a one-dimensional model of fracture

Cited by in corpus (5)

• The Dynamics of Rapid Fracture: Instabilities, Nonlinearities and Length Scales
• Oscillatory and tip-splitting instabilities in 2D dynamic fracture: The roles of intrinsic material length and time scales
• Dynamic stability of crack fronts: Out-of-plane corrugations
• A nonlinear symmetry breaking effect in shear cracks
• Nonlinear extension of Kolosov-Muskhelishvili stress function formalism