Autonomy and singularity in dynamic fracture.
The recently developed weakly nonlinear theory of dynamic fracture predicts 1/r corrections to the standard asymptotic linear elastic 1/(square root of 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/(square root of 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 solution, are in favorable agreement with recent near-tip measurements.
Bouchbinder
Eran
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Department of Chemical Physics, Weizmann Institute of Science, Rehovot, Israel.