Influence of the Peierls Potentials on the Reversible Stress-Strain Relation for Dislocations

Abstract

The mechanical equation of state of kinked dislocations is considered. Contrary to dislocation strings, which follow under the assumption of constant line tension a linear stress-strain relation for stresses $\frac{\ensuremath{\sigma}}{G}<\frac{b}{3L}$ ($L=\mathrm{line}\mathrm{length}$, $b=\mathrm{Burgers}\mathrm{vector}$, $G=\mathrm{shear}\mathrm{modulus}$, $\ensuremath{\sigma}=\mathrm{stress}$), one finds significant nonlinearities in the reversible stress-strain relation of kinked dislocations. The physical reason for the nonlinearities can be ascribed to the fact that, owing to the Peierls potentials, the energy of a dislocation increases in multiples of the double-kink energy $2{W}_{k}$ (${W}_{k}=\mathrm{kink}\mathrm{energy}$). A linear range, which is confined to stresses $\frac{\ensuremath{\sigma}}{G}<({10}^{\ensuremath{-}1}\frac{b}{L})(sin\ensuremath{\phi}+\frac{5kT}{G{b}^{3}})$ ($\ensuremath{\phi}=\mathrm{a}\mathrm{n}\mathrm{g}\mathrm{l}\mathrm{e}\phantom{\rule{0ex}{0ex}}\mathrm{a}\mathrm{g}\mathrm{a}\mathrm{i}\mathrm{n}\mathrm{s}\mathrm{t}\phantom{\rule{0ex}{0ex}}\mathrm{c}\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{e}\ensuremath{-}\mathrm{p}\mathrm{a}\mathrm{c}\mathrm{k}\mathrm{e}\mathrm{d}\phantom{\rule{0ex}{0ex}}\mathrm{d}\mathrm{i}\mathrm{r}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}$, $T=\mathrm{temperature}$), is followed by a region with $\frac{{\ensuremath{\partial}}^{2}\ensuremath{\epsilon}}{\ensuremath{\partial}{\ensuremath{\sigma}}^{2}}<0$ ($\ensuremath{\epsilon}=\mathrm{strain}$). This region corresponds to the restricted motion of geometrical kinks. After passing through an inflection point, which is roughly determined by $\frac{\ensuremath{\sigma}}{G}=\ensuremath{\alpha}(\frac{b}{L})(\frac{2{W}_{k}}{G{b}^{3}})$ ($\ensuremath{\alpha}=\mathrm{numerical}\mathrm{factor}\mathrm{between}1\mathrm{and}2$), a region with $\frac{{\ensuremath{\partial}}^{2}\ensuremath{\epsilon}}{\ensuremath{\partial}{\ensuremath{\sigma}}^{2}}>0$ follows. It is caused by double-kink generation. If the measuring time is too short for thermally activated double kink generation, the inflection point is determined by the stress which is required for stress-assisted thermally activated double kink generation. At $T=0\ifmmode^\circ\else\textdegree\fi{}$K, the stress of the inflection point provides a measure for the Peierls stress. It is suggested that evidence for the Peierls potentials can be established through a verification of the nonlinear-stress-strain relation by the following experiments: (a) The restricted motion of geometric kinks should be detectable beyond the stress for activating Frank-Read sources as a decrease of the modulus defect with increasing stress amplitude. (b) The double-kink-generation peaks should, in undeformed material, rise out of the background in high-amplitude measurements. (c) The double-kink-generation peaks should be found in undeformed pure material by applying a static-bias stress. (d) In deformed material, high-amplitude oscillations should cause an increase of the peak height before the peak starts to shift to lower temperatures.