Modeling of discrete dislocation dynamics by means of the mathematical theory of evolving curves

Abstract: Dislocation dynamics (DD) has become a standard tool for analyzing deformation microstructures. Fundamentals of dislocation theory have been established in 1930s and later verified by first TEM experiments in 1950s. The ultimate objective of DD is to fill the gap between fully continuous models in the macroscopic scale based on crystal plasticity, and atomistic models in nanoscale usually treated by the molecular dynamics methods.

In crystalline solids, dislocations represent irregularities or defects in the crystal structure. Such defects are usually of the line character and the characteristic unit of length of such defects is microns. These line defects are the subject of both external forces applied on the crystal as well as various internal mechanisms, and they evolve in the so-called slip planes given by particular crystallographic orientation.

Our research interest is in the detailed and precise modeling of fundamental mechanisms involving single or very few dislocations. Such a topic is typically referred to as a discrete dislocation dynamics (DDD).

In our approach to DDD, a single dislocation carrying the plastic flow in its respective slip plane is represented as an evolving planar curve. The experimentally observed mechanism how dislocations evolve is of a non-Newtonian type and can be schematically described as

normal velocity = curvature + external force,

where the curvature term approximates the self-stress of the dislocation generated by a line tension. We treat this kind of evolution equation by the direct (Lagrangian) approach resulting in the system of two degenerate parabolic equations for parametrization of the single dislocation curve. We solve this problem numerically by means of the flowing finite-volume method, and we improve the stability of the numerical scheme by the suitable choice of the external tangential velocity which helps to redistribute the discretization points along the dislocation in an appropriate way.

We demonstrate our approach in several computational examples covering the fundamental dislocation mechanisms and then we focus on the dislocation cross-slip, which is considered as one of the key mechanisms of microplasticity and still an open problem in general. We show how we modify our method by the geodesic description of curves to treat the cross-slip within our framework.

MathematicsPhysics

Audience: researchers in the topic

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Nečas Seminar on Continuum Mechanics

Series comments: This seminar was founded on December 14, 1966.

Faculty of Mathematics and Physics, Charles University, Sokolovská 83, Prague 8. If not written otherwise, we will meet on Mondays at 15:40 in lecture hall K3 (2nd floor)

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