# On the yield strength of periodic dislocation structures

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## Abstract

In his seminal paper of 1934, Taylor not only introduced the notion of dislocation, but also explained work hardening by dislocation interactions. To determine the critical shear stress needed to put dislocations in motion he considered a periodic set of edge dislocations with one positive and one negative dislocation in a cell. He found the critical stress to be of the form \(\alpha _{0}\mu b/\epsilon \), where \(\mu \) is shear modulus, *b* the interatomic distance, \(\epsilon \) the cell size, \(\alpha _{0}\) some dimensionless constant. Since the dislocation density \(\rho \) in this problem is \(2/\epsilon ^{2}\), the critical stress can be written in the form of a relation of yield stress \(\sigma _{Y}\) and the dislocation density \(\rho , \sigma _{Y}=\alpha \mu \sqrt{\rho }b, \alpha =\alpha _{0}/\sqrt{2}\). This formula is proved to be widely applicable and often referred to as Taylor’s relation. We discuss in this paper whether the formula for critical stress, \(\sigma _{Y}=\alpha _{0}\mu b/\epsilon \), yields Taylor’s relation for a large number of edge dislocations in a periodic cell. This would be the case if the constant \(\alpha _{0}\) grows with the number of dislocations in a cell *m* as \(\sqrt{m}\). Then, since \(\rho =m/\epsilon ^{2}\), Taylor’s relation follows from the formula for the critical stress of periodic dislocation structures indeed. We give here an analysis of previously reported numerical simulations which indicates that \(\alpha _{0}\) appears to be finite as *m* increases. In other words, for 2D periodic dislocation ensembles, the strengthening coefficient \(\alpha \) seems to be decaying as \(1/\sqrt{m}\) and the proper relation for the yield strength of 2D periodic structures is \(\sigma _{Y}=\alpha _{0}\mu b/\epsilon \). Thus, the yield strength depends on the cell size, and 2D periodic dislocation structures follow the similitude principle of cell structures rather than Taylor’s glide resistance relation.

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## Notes

### Acknowledgements

It is gratefully acknowledged that computations were performed at Wayne State University High Performance Computing Grid.

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