Understanding Size Effects in Small Scale Deformation: A Statistical Perspective

Recent experimental observations of micro-compression / tension tests indicate that as the size of test specimen decreases the yield strength increases. This raises a fundamental question: Why is smaller stronger? Is there a fundamental relationship between the size of a specimen and its intrinsic strength? This simple question pushes the limit of the current understanding of the physical mechanisms underlying material deformation, especially at small scales. In order to explain the experimental observations of the strength of small specimens containing a limited number of dislocations, a simple statistical model is developed. Two different types of randomness are introduced, viz., randomness in the spatial location of 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 the previous stochastic models, the current model not only predicts the yield strength in the presence of dislocations but also in their absence. Furthermore, the model has the capability to predict the scatter in the yield strength in addition to the mean. Monte Carlo simulations are also performed for comparison. Interestingly, the model adds credence to the notion that “smaller is stronger” from a purely statistical point of view. 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 TEM. Furthermore, the model is extended to spherical indentation pop-in which is an analogous size dependent problem in small scale mechanics. In this case, the model predicts the load and maximum shear stress at pop-in as a function of indenter radius and is found to closely match the experimental results on single crystal molybdenum using a dislocation density estimated by micro-focus x-ray techniques. In summary, the current work provides possible explanations for the strength and scatter in strength of small specimens from a purely statistical perspective.