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Universal Power-Law Strengthening in Metals?

The strength of most metals used in daily life scales with either an internal or external length scale. Empirically, this is characterized by power-laws persisting to six orders of magnitude in both strength and length scale. Attempts at understanding this scaling have generally been based on a specific mechanism. However the wide applicability of material type and microstructure to this phenomenon suggests a single mechanism is unlikely to capture the observed trend. Here we develop a model which gives quantitative insight into the scaling exponent using the known universal properties of a dislocation network and the leading order stress dependence of an underlying critical stress distribution. This approach justifies a value for the scaling exponent for virtually any experimental data set within the frameworks of both Hall-Petch strengthening and the "small is stronger" paradigm of small scale plasticity.