A dislocation-dipole in one-dimensional lattice model

ABSTRACT

A family of equilibria corresponding to a dislocation–dipole, with variable separation between the two dislocations of opposite sign, is constructed in a one-dimensional lattice model. A continuous path is constructed, in the space of lattice configurations, connecting certain members of this family; eventually leading to an energy landscape, using the concept of order-parameter. The energy landscape is found to exhibit the familiar Peierls relief for the variation of energy associated with certain sequential transitions between these equilibria. The results allow an interpretation in terms of quasi-statically separating pair of dislocations of opposite sign from the viewpoint of the original Frenkel–Kontorova model albeit a piecewise-quadratic onsite potential is employed in the paper. The closed form expressions are provided after an application of detailed analytical means wherein an analysis of the effect of an intermediate spinodal region is included.

KEYWORDS:

• Dislocation mechanics
• defects in solids
• lattice defects
• nanoscale
• Peierls barrier
• Frenkel–Kontorova model

Acknowledgments

This work has been available free of peer review on the arXiv (2009.05591) since 15/09/2019. The author thanks the anonymous reviewers for their constructive comments, elaborate criticism, and suggestions to improve the manuscript.

Disclosure statement

No potential conflict of interest was reported by the author.

Notes

1 Truskinovsky, L., Private communication (2006)

2 Throughout the paper, we use the notation $\{.\}$ to denote sets (containing finite or countably infinite number of elements).

3 In this paper, the value 0.5 of structural constant $\kappa$ is often used.

4 $\mathrm{\Theta }\left(0\right)$ does not affect the explicit results presented in this paper so it can be left undefined as well or assumed to belong to the interval $[0,1]$; for simplicity and symmetry related reasons, however, it has been assumed to be 1/2 in (7(7) $\mathrm{\Theta }\left(x\right):=+1\mathrm{f}\mathrm{o}\mathrm{r}x>0,\mathrm{\Theta }\left(x\right):=0,x<0,\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{\Theta }\left(0\right):=1/2.$(7) ). The reader should also note that the analysis of the effect of this choice has not been carried in this paper. Implications of the choice are not so simple usually, the analysis of dynamics necessitates a careful assessment of the same issue; for example, see [42] for an instance of its non-trivial implications for the traveling wave solutions of moving kink in the Frenkel–Kontorova model representing dislocation motion under external stress.

5 Here ${\sigma }_M$ is synonymous to the Maxwell stress in the terminology of phase transitions [43,44].

6 The word ‘state’ and ‘configuration’ are used as synonyms in this paper.

Additional information

Funding

The partial support of Science and Engineering Research Board (SERB) MATRICS [grant number MTR/2017/000013] is gratefully acknowledged.