The size effect from the nanoscale to the Hall-Petch effect 60 years of faulty fitting of data – A Bayesian approach

CCMMP Seminars
Prof David Dunstan
Dr A. J. Misquitta
October 2nd, 2014 at 14:00
GO Jones Room 610

Very often, a good theoretical fit to data is published with the claim that the data supports the theory. This can be dangerous – it is necessary always to ask what else fits. In this seminar, we look at an example of 60 years of such faulty reasoning and we consider criteria for the choice between theories. 
Classical metallurgy achieves control of the strength of metals firstly by alloying, and secondly by a variety of processes which control dislocation motion. These include strain-hardening, precipitation hardening, and grain-refining.
For more than half a century the strength of metals has been known to vary with the inverse square-root of the grain size – the Hall-Petch effect – yet a definitive explanation is still lacking. More recently, structure size effects have become apparent in a wide range of micromechanical testing configurations, from nano-indentation to micropillar compression. We have argued that these are all manifestations of a single underlying size effect, fundamentally due to the relationship between dislocation curvature and stress, resulting in a higher stress required to drive dislocation extension or multiplication events in small volumes.1 
We critically review the classic experimental evidence for the Hall-Petch effect, expressed as σHP = σ0 + kHP d–½  where σ is the yield or flow stress. Evidence for the inverse-square-root dependence is in fact weak or non-existent.2 The data fit just as well the expression σCT = σ0 + kCT d–1log d.  Semi-Bayesian and Bayesian criteria of choice between models are discussed. We conclude that the Hall-Petch effect is not an inverse-square root dependence on grain size and is the same size effect as seen in nanostructures and described by critical thickness theory.

1  D.J. Dunstan and A.J. Bushby, 2013, The scaling exponent in the size effect of small scale plastic deformation, Int. J. Plasticity 40, 152-162.
2  D.J. Dunstan and A.J. Bushby, 2014, Grain size dependence of the strength of metals: The Hall-Petch effect does not scale as the inverse square root of grain size, Int. J. Plasticity 53, 56-65.