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

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 σG<b/3L (L=linelength, b=Burgersvector, G=shearmodulus, σ=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 2Wₖ (Wₖ=kinkenergy). A linear range, which is confined to stresses σG<(10^-1b/L)(sinφ+5kTGb³) (${φ}=angle{{0ex}{0ex}}against{{0ex}{0ex}}close{-}packed{{0ex}{0ex}}direction$, $T=temperature$), is followed by a region with ${{{∂}}²{ε}}{{∂}{{σ}}²}<0$ (${ε}=strain$). This region corresponds to the restricted motion of geometrical kinks. After passing through an inflection point, which is roughly determined by ${{σ}}{G}={α}(b/L)({2{W}ₖ}{G{b}³})$ (${α}=numericalfactorbetween1and2$), a region with ${{{∂}}²{ε}}{{∂}{{σ}}²}>0$ 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^∘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.