The transition from subsonic to supersonic cracks
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We present the full analytical solution for steady-state in-plane crack motion in a brittle triangular lattice. This allows quick numerical evaluation of solutions for very large systems, facilitating comparisons with continuum fracture theory. Cracks that propagate faster than the Rayleigh wave speed have been thought to be forbidden in the continuum theory, but clearly exist in lattice systems. Using our analytical methods, we examine in detail the motion of atoms around a crack tip as crack speed changes from subsonic to supersonic. Subsonic cracks feature displacement fields consistent with a stress intensity factor. For supersonic cracks, the stress intensity factor disappears. Subsonic cracks are characterized by small-amplitude, high-frequency oscillations in the vertical displacement of an atom along the crack line, while supersonic cracks have large-amplitude, low-frequency oscillations. Thus, while supersonic cracks are no less physical than subsonic cracks, the connection between microscopic and macroscopic behavior must be made in a different way. This is one reason supersonic cracks in tension had been thought not to exist. In continuum fracture theory, the energy flowing into the crack tip becomes negative or imaginary for crack speeds faster than the Rayleigh wave speed. This would suggest that supersonic cracks are not physically allowed. In response to this, we study the energy flow in our supersonic solutions in the lattice. First, we construct an energy flux vector in the lattice analogous to the Poynting vector in electromagnetism. This allows us to calculate the energy flow at each atom in the lattice. We find that there is positive energy flux into the crack tip for both subsonic and supersonic solutions in the lattice.