Dislocation Dynamics and Cooperative Behaviour of Dislocations

Abstract

We present a theoretical model of repeated yielding which reproduces many experimentally observed features apart from demonstrating how the temporal behavior emerges as a natural consequence of the cooperative behavior of defects. The starting point for building such a model is our earlier work on a statistical description of dislocation dynamics which explains creep in simple materials. The model leads to an alternate but equivalent description which allows us to quantify the mobile dislocation density in terms of a coupled set of equations for the mobile and the inmobile components. We then include another type of dislocation and some transformation between them. This leads to a coupled set of nonlinear differential equations for the three dislocation densities. We show that for a range of values of the rate constants, limit cycle solutions are exhibited leading to jumps on creep curve. Approximate closed form solutions are also obtained. The model is extended to the constant Strain rate case by coupling the above equations to the machine equation. The temporal ordering of repeated yielding naturally follows. Several such features as bounds on the strain rate, bounds on the concentration of solute atoms, the negative strain rate dependence of the flow stress, the dependence of the amplitude on the strain rate, strain etc., emerge from the model. The model also exhibits period doubling bifurcation with an exponent value same as that for the quadratic map. Finally we report the effect fluctuations during a single yield dron.