A simple stochastic model for yielding in specimens with limited number of dislocations

A simple statistical model is developed based on a random distribution and orientation of dislocations in order to explain recent experimental observations of the strength of small specimens containing a limited number of dislocations. Two different types of randomness are introduced, namely, randomness in the spatial location of the dislocations and randomness in the stress needed to activate them. For convenience, the randomness in the activation stress is modeled by assigning a random Schmid factor to the dislocations. In contrast to previous stochastic models, the current model predicts the yield strength not only in the presence of dislocations but also in their absence. Furthermore, the model predicts the scatter in the yield strength in addition to the mean. The model is found to quantitatively explain the yield strength and scatter in micro-compression/tension tests of Mo-alloy fibers using dislocation densities and arrangements measured by transmission electron microscopy. The results of Brenner's classic tensile tests on metallic whiskers are qualitatively reconciled. The model adds credence to the notion that “smaller is stronger” from a purely statistical point of view.