Continuum treatments of lattice defects such as dislocations and fracture cracks do not predict the resistance to the defect mobility which is due to the Peierls energy in the case of the dislocation. The discrete character of the lattice in the case of the fracture crack leads to a stress stability range for the crack above and below the Griffith stress over which the crack is stable or ``lattice trapped''. We develop here two essentially qualitative theoretical treatments of the lattice structure of cracks. In the first, we carry out a lattice sum of the bond energies of atoms facing each other across the crack plane under certain assumptions regarding the crack shape and atomic force laws. In the second, we introduce a one-dimensional lattice model of a crack which can be solved exactly. In both of these treatments the range of stress over which the crack is lattice trapped appears to be of the order of magnitude of the Griffith stress itself. As in the case of dislocations the lattice trapping is a strong function of the ``width'' of the elastic singularity at the tip, and our prediction is that one should expect materials to exist in which lattice trapping is not only observable, but is an important effect. We also find that because of lattice trapping, the true surface energy is a lower bound to the mechanical surface energy as expressed in the Griffith-stress relation.

A quasianalytic solution for the atomic displacements of a discrete two-dimensional lattice containing a crack is obtained, We assume that the force laws are linear up to a critical displacement when the bond snaps, which is the basic assumption of the lattice static approximation. When compared to the classic Griffith continuum description, new results are: (i) a predicted and observable lattice trapping of the crack, (ii) difficulties with the interpretations of the crystal surface energy in a cleavage experiment, and (iii) a predicted characteristic crack creep phenomenon under external constant stress. The present theory shows how two separate "surface energies" are inferred from the stress to open and to close a crack, and on our model these energies differ from one another by a large factor of 5.7. The thermodynamic "surface energy" is not related to either of these quantities. Experimental verification of the lattice trapping of cracks is thought to be most readily and directly obtained by observations of the creep of a crack under high vacuum conditions.

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