Operation of a 3D Frank–Read source in a stress gradient and implications for size-dependent plasticity

The operation of a single Frank–Read (FR) source in the presence of a spatial stress gradient is studied using 3D discrete-dislocation dynamics (DDD) simulations and analytic models. Under a sufficiently large stress gradient, the FR source shows a new stable configuration controlled by the low-stress region of the graded stress field. Successive emissions from the source generate a growing dislocation pile-up that exerts an increasing back stress on the source, leading to hardening. The operation of the FR source in the gradient field can be well-approximated by a single critical stress at a unique critical location, allowing for the development of an analytic model of the source operation. These results are applied to rationalize the size-scaling of strength measured in simulations of single-crystal beam bending using DDD and in experiments on bending of single-crystal cantilever beams. This work demonstrates that size effects in plasticity emerge naturally from the mechanics of dislocation sources operating within a stress gradient, and in this case the relevant material length scale is the length of the FR source.