Dynamic simulation of solution hardening

The flow stress for the motion of a dislocation through a random array of weak obstacles of finite interaction range, and in the presence of viscous forces, has been calculated by integrating numerically the equation of motion in a digital computer. Through a normalization of the coordinates and time it is shown that the normalized critical stress S is a function of only two parameters: a normalized obstacle depth η0 and a normalized viscous damping γ. Numerical values of S were obtained for stepwise changes in η0 and γ and for a set of boundary conditions compatible with real experiments. For γ≳3, S becomes independent of γ and of the initial conditions for the dislocation motion. The results reproduce the analytical dependences of the theories that have been developed for extreme values of η0, providing furthermore the proportionality constants and the extent of η0 for which these theories are applicable. For γ<3, S is a function of η0 and γ and of the initial conditions for the dislocation motion: For a dislocation starting from rest there is an upper critical stress to initiate the motion, lower in value than S (γ≳3), while for a dislocation already in motion there is an even lower critical stress at which the dislocation stops moving. The latter corresponds to that calculated by the previous inertial theories, while the former had not been accounted for previously.