A discrete dislocation plasticity study of the micro-cantilever size effect

Micro-cantilevers are increasingly used to extract elastic and plastic material properties through controlled bending using a nanoindenter. Focused Ion Beam milling can be used to produce small scale single crystal cantilevers with cross-sectional dimensions on the order of microns, and electron backscatter diffraction (EBSD) allows cantilevers to be milled from a grain with a desired crystal orientation. Micro-cantilever bending experiments suggest that sufficiently smaller cantilevers are stronger, which is generally believed to be related to the effect of the neutral axis on the evolution of the dislocation structure. A planar model of discrete dislocation plasticity was used to simulate end-loaded cantilevers to interpret the behaviour observed in experiments. The model allowed correlation of the initial dislocation source density and resulting slip band spacing to the experimental load displacement curve. There are similarities between the predictions of this model and those of earlier discrete dislocation plasticity models of pure bending. However, there are notable differences, including a strong source density dependence of the size effect that cannot be explained by geometrically necessary dislocation (GND) arguments, and the effect of the cantilever stress distribution on the locations of soft pile-ups. The planar model was used to identify zero resolved shear stress isolines, rather than the neutral axis, as controlling the soft pile-up location, and source spacing as limiting the slip band spacing in the observed size effect; strengthening was much greater in the source-limited regime. The effect of sample dimensions and dislocation source density were investigated and compared to small scale mechanical tests conducted on titanium and zirconium. The calculations predict a scaling exponent n≈1 for the dependence of stress on size if size is normalised by the average source spacing and a term representing the size-independent flow stress is included, whereas the simple power-law form ordinarily used to fit experimental data significantly underestimates n.