The Peierls stress for a wide dislocation

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The Peierls stress σ P of a wide dislocation with a core w atoms wide is small. However, estimates of the asymptotic dependence of σ P on w in a crystal of shear modulus μ range from σ P ≈ μ/ w to σ P ≈ μ/ exp(2 πw). Existing calculations are summarized, and some new calculations are presented, but a clear picture does not emerge. It seems that the two-dimensional Peierls model and the one-dimensional Frenkel-Kontorova (F-K) model behave rather similarly, even though the relations between w and the ratio of μ to the critical shear stress τ crit of the perfect crystal are difference in the two cases. An exponential decrease in σ P with increasing w is found in calculations in which the energy associated with a smooth potential is sampled at a set of points which are carried through the lattice by a rigid displacement function, and also in Hobart's computation for the F-K model. A power law dependence is found in models using a piecewise linear force law for which the second derivative of the potential is discontinuous. A treatment in the spirit of second-order perturbation theory, in which the displacement function is modulated as the dislocation moves through the lattice, supports Lifshits' proposal that σ P ≈ μ/ w 5 for any smooth potential. Attention is drawn to old arguments showing that the presence of kinks can lead to anelastic effects and to microstrain, but not to extensive plastic strain. The work of Dietze and of Kuhlmann-Wilsdorf shows that thermal vibrations will reduce σ P by a factor exp(− πwT/ mt m), where T m is the melting temperature and m is respectively 1.5 and less than 10 in the two calculations.