Strongly non-local modelling of dislocation transport and pile-up

Abstract

The purpose of this work is the continuum modelling of transport and pile-up of infinite discrete dislocation walls driven by non-local interaction and external loading. To this end, the underlying model for dislocation wall interaction is based on the non-singular Peierls–Nabarro (PN) model for the dislocation stress field. For simplicity, attention is restricted to walls consisting of single-sign dislocations and to continuous wall distributions on a single glide plane. In this context, the influence of strongly non-local (SNL; long-range) interaction, and its approximation as weakly non-local (WNL; short-range) are studied in the context of interaction- and external-load-driven wall pile-up at a boundary. The pile-up boundary is modelled via a spatially dependent dislocation mobility which decreases to zero at the boundary. The pile-up behaviour predicted by the current SNL-based continuous wall distribution modelling is consistent with that predicted by discrete wall distribution modelling. Both deviate substantially from the pile-up behaviour predicted by WNL-based continuous wall distribution modelling. As such, it is clearly essential to account in continuum models for the intrinsic SNL character of the interaction between same-sign dislocations ‘close’ to the boundary. Gradient-based WNL ‘approximation’ of this interaction is not justified.

Keywords:

• Peierls-Nabarro modelling
• strongly non-local interaction
• transport
• pile-up

Acknowledgements

We thank the two reviewers of the first version of this paper for their constructive comments and criticism which have led to its improvement. J. Mianroodi and B. Svendsen gratefully acknowledge partial financial support for Project A10 of the Cooperative Research Center 761 ‘ab initio steel’ of the German National Science Foundation (DFG).

Notes

No potential conflict of interest was reported by the authors.

1 in the current work corresponds to in the work of [11, Chapter 2].

2 Limits in (4(4) ) are assumed infinite because the model (7(7) ) for has bounded support (see Figure 2).

3 An infinite isotropic linear elastic medium and the Frenkel model for lattice resistance.

4 The coefficient in Equation (12) of [11, Chapter 2] differs by a factor of two from that in (12(12) ) due to different stress scaling.