Communications in Mathematical Sciences

Volume 16 (2018)

Number 2

A continuum model for distributions of dislocations incorporating short-range interactions

Pages: 491 – 522



Xiaohua Niu (Department of Mathematics, Hong Kong University of Science and Technology, Kowloon, Hong Kong)

Yichao Zhu (State Key Laboratory of Structural Analysis for Industrial Equipment, Dept. of Engineering Mechanics and International Research Center for Computational Mechanics, Dalian University of Technology, Dalian, Liaoning, China)

Shuyang Dai (School of Mathematics and Statistics, Wuhan University, Wuhan, Hubei, China)

Yang Xiang (Department of Mathematics, Hong Kong University of Science and Technology, Kowloon, Hong Kong)


Dislocations are the main carriers of the permanent deformation of crystals. For simulations of engineering applications, continuum models where material microstructures are represented by continuous density distributions of dislocations are preferred. It is challenging to capture in the continuum model the short-range dislocation interactions, which vanish after the standard averaging procedure from discrete dislocation models. In this study, we consider systems of parallel straight dislocation walls and develop continuum descriptions for the short-range interactions of dislocations by using asymptotic analysis. The obtained continuum short-range interaction formulas are incorporated in the continuum model for dislocation dynamics based on a pair of dislocation density potential functions that represent continuous distributions of dislocations. This derived continuum model is able to describe the anisotropic dislocation interaction and motion. Mathematically, these short-range interaction terms ensure strong stability property of the continuum model that is possessed by the discrete dislocation dynamics model. The derived continuum model is validated by comparisons with the discrete dislocation dynamical simulation results.


discrete dislocation model, continuum theory, short-range interaction, asymptotic analysis, level set method

2010 Mathematics Subject Classification

35Q74, 41A60, 74C99

This work was partially supported by the Hong Kong Research Grants Council General Research Fund 606313 and HKUST Postdoctoral Fellowship Matching Fund. The work of Y.C.Z was partially supported by Natural Science Foundation of China (NSFC) under the contract no. 11772076. The work of S.Y.D was supported by Natural Science Foundation of China (NSFC) no. 11701433.

Received 6 July 2017

Accepted 14 January 2018

Published 14 May 2018