Some exact results for the velocity of cracks propagating in non-linear elastic models

TL;DR

This paper provides analytical and numerical insights into the velocity of cracks in a non-linear elastic model, revealing that crack velocity remains below sound speed and approaches it at uniform breakdown strain.

Contribution

It presents the first analytical calculation of crack velocity near critical strains in a piece-wise linear elastic model, supported by numerical simulations.

Findings

Crack velocity is lower than sound speed in the model.

Velocity approaches sound speed at uniform breakdown strain.

Analytical results agree well with numerical simulations.

Abstract

We analyze a piece-wise linear elastic model for the propagation of a crack in a stripe geometry under mode III conditions, in the absence of dissipation. The model is continuous in the propagation direction and discrete in the perpendicular direction. The velocity of the crack is a function of the value of the applied strain. We find analytically the value of the propagation velocity close to the Griffith threshold, and close to the strain of uniform breakdown. Contrary to the case of perfectly harmonic behavior up to the fracture point, in the piece-wise linear elastic model the crack velocity is lower than the sound velocity, reaching this limiting value at the strain of uniform breakdown. We complement the analytical results with numerical simulations and find excellent agreement.