Dynamical Dislocation Theory of Crystal Plasticity. I. The Yield Stress

Abstract

The derivation is outlined of a general relation connecting the macroscopic plastic strain rate with the configuration and velocity distribution of dislocations in a monocrystal with a single active glide system. This can be simplified for particular cases, the most simple being the case of parallel straight dislocation lines. A new form of the dynamical theory of plasticity is applied to detailed calculations and compared with previous work. The major modification is the use of the equation v=v* exp[−D/τ] to express the quasi-viscous behavior of dislocation velocities. Here, v=dislocation velocity, v*= terminal velocity, D=characteristic drag stress, and τ=resolved shear stress. Also, a physical relation between dislocation density and plastic strain is derived. The nonlinear differential equation that results from combining the strain-rate equation with an equation that describes the loading machine is integrated. Approximate analytic solutions, and numerical calculations, are given for the upper yield stress and the initial portion of the stress-strain curve. The effects of initial dislocation density, characteristic drag stress, crosshead speed, machine stiffness, and dislocation multiplication rate on the upper yield stress are calculated. It is shown that the upper yield stress exceeds the cohesive stress at a critical crosshead speed. Delay times are also calculated for rapid load application followed by constancy of the load and compared with experiments for LiF. The effects of strain hardening are considered in a companion paper.