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A probabilistic explanation for the size-effect in crystal plasticity

In this work, the well known power-law relation between strength and sample size, $d^{-n}$, is derived from the knowledge that a dislocation network exhibits scale-free behaviour and the extreme value statistical properties of an arbitrary distribution of critical stresses. This approach yields $n=(τ+1)/(α+1)$, where $α$ reflects the leading order algebraic exponent of the low stress regime of the critical stress distribution and $τ$ is the scaling exponent for intermittent plastic strain activity. This quite general derivation supports the experimental observation that the size effect paradigm is applicable to a wide range of materials, differing in crystal structure, internal microstructure and external sample geometry.