Peierls Stress and Creep of a Linear Chain

The Frenkel-Kontorova dislocation model is modified by replacing the sinusoidal substrate force by one which is piecewise linear. Exact solutions are found for the static configuration and for the Peierls stress, ${\sigma }_P$. Good agreement is found between these values of ${\sigma }_P$ and those obtained previously for a two-dimensional Rosenstock-Newell model. The atoms of the linear chain are then considered in random motion corresponding to thermal equilibrium and under an applied stress $\sigma <{\sigma }_P$. The time required for motion of the dislocation from one position of stable equilibrium to an adjacent one is computed by means of a rate-theory formulation adapted to the present type of problem in which the positions of all the atoms in the chain are required to vary in passing over the potential barrier. The theoretical transition times for an infinite chain are compared with analog computer results for a six-atom chain and reasonably good agreement is found.