Communications in Mathematical Sciences

Volume 10 (2012)

Number 4

A continuum model for the dynamics of dislocation arrays

Pages: 1081 – 1103



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

Xiaohong Zhu (Department of Mathematics, Jinan University, Guangzhou, China)


We derive a continuum model for the dynamics of a dislocation array that consists of dislocations in different slip planes. In the continuum model, the dislocation array is represented by a continuous surface, of which there are many dislocations in a unit area at the scale of the continuum model. The continuum model is derived rigorously from the discrete model of the dynamics of the constituent dislocations in the array using asymptotic analysis. The obtained continuum model contains an integral over the dislocation array surface representing the long-range interaction of dislocations, and a local term that comes from the line tension effect of dislocations. The sizedependent effect due to dislocation line tension is accurately incorporated in the continuum model. Well-posedness of the continuum model is examined. A generalization to dislocation arrays in an elastically anisotropic medium is discussed.


dislocation dynamics, Peach-Koehler force, continuum model, elasticity, plasticity

2010 Mathematics Subject Classification

35Q74, 74A10, 74A50

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