On lattice trapping of cracks
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A simple analytic theory to quantify the extent of “trapping” of cracks by a crystal lattice and its dependence on the range of the interatomic force law is presented. The theory requires a priorz knowledge of only one or two lattice-dependent, but force law independent, parameters and can then reproduce nearly all previous (numerically demanding) theoretical results. Moreover, the extent of lattice trapping does not decrease monotonically with increasing range of the force law. For realistic interatomic potentials, however, lattice trapping is quite small. The analysis is then extended to the case of chemical corrosion by direct chemical attack of the crack tip bond. It is shown that similar slow crack growth thresholds can be predicted by the thermodynamic Griffith approach and a fracture criterion based on a local bond instability.
1. INTRODUCTION The traditional Griffith fracture criterion for failure in brittle materials, i.e., those materials not emitting dislocations prior to failure, equates the energy by extending required to create new crack surface, ESurr, the crack by an amount ac to the elastic energy decrease, E,I,,, in the entire material upon extension of the crack by ac at fixed strain c.’ The first of these ener= ydc, where y is the surface energy gies is simply Esurf (per unit length) of the material. The second energy is2 r
1
where c is the crack length, E is the material Young’s modulus, and uapp = E E is the applied tensile stress at infinity. Equating these energies yields the critical applied stress necessary to advance the crack,
The Griffith criterion is derived entirely from equiLibrium energy considerations: hence the opening proand if the applied stress is cess is reversible at a&,p reduced, the crack should reheal continuously. In some early lattice models of fracture, the phenomenon of lattice trapping was found.3 Lattice trapping arises if the stresses required to open and close a crack are different and occurs if only local equilibrium is considered, so that each bond/atom is in a local energy minimum but not a more global minimum. An easy way to see this effect is to consider the potential energy as a function of the crack tip displacement for various stresses in a manner analogous to the Landau picture of 1st order phase transitions with the mapping of free energy -+ potential energy, order parameter + crack tip displacement and temperature + stress, as shown in Fig. 1. In the sequence of increasing stress J. Mater. Res.. Vol. 5,No.7; Jul 1990
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from to u7shown in Fig. 1, we refer to the intact crack tip as state 1 and the fractured tip (advanced crack) as state 2. Starting from stress mlr state 1 is locally stable up to ( T g , at which point there is no barrier to falling into state 2. On decreasing stress from u7, however, one must return all the way to uz before state 2 becomes locally unstable and the crack closes by going to state 1. The stress u4 here corresponds to the Griffith critical stress, where the energies o
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