MSE1: Discrete and continuum dislocation modelling

Background

Although the word crystalline is more likely to make one think of salt than steel, metals too have a crystal structure. Almost all metals and alloys – including copper, zinc, brass and steel – are composed of a huge number of crystalline grains, each one a regular array of atoms extending for much less than a millimetre. This crystal structure is essential to the way that metals deform under stress. Stress-induced changes to the crystalline grains can lead to fatigue, permanent deformation and ultimately failure of metallic devices. 

When scientists first observed that metals are crystalline in the 1920s, they faced a major theoretical problem. It is possible to calculate the theoretical shear strength of a perfect crystal, but experimental measurements showed that real metals are thousands of times weaker than these perfect crystals. It was observed in the 1930s that the crystal structure of a metal contains mobile line defects called dislocations. As each dislocation moves, it rearranges the crystal structure, leading to permanent, plastic changes in the metal. These dislocations move and interact in response to shear stress, and this accounts for why metals are so much weaker under shear stress than predicted from models of a perfect crystal. Understanding how these dislocations behave is important to studying the behaviour of metals.

Techniques and Challenges

While it’s now more than 75 years since dislocations were first proposed as the carriers of metal plasticity, there is still plenty of room for improving the understanding of their behaviour.

Mathematically, there are two general approaches to modelling dislocation behaviour: discrete models and continuum models. A discrete model accounts for and tracks each individual dislocation. Since dislocations in metals are incredibly common, and the interactions between these dislocations are complex, tracking individual dislocations in a reasonably-sized sample is not practical. On large-scales, it becomes more practical to use a continuum description of dislocation interactions. A continuum model considers the dislocation density over a sample of metal and is mathematically more elegant than discrete models. However, continuum descriptions also have their own set of problems.

Many dislocations are statistically stored in structures like dipolar lattices (see Figures 1 and 2), meaning the dislocation density is effectively zero. While these types of dislocations generally have little impact on stress and deformation, they become important when further stress is applied and large numbers of dislocations are released. A continuum model breaks down in such a situation.

Another difficulty with conventional continuum dislocation models is that sometimes it is important to consider individual dislocations. For example, around a locked dislocation, a pile-up of dislocations may occur. The continuum dislocation density model works throughout most of the pile-up region, but breaks down at the head and tail of the pile-up where the density of dislocations approaches infinity or zero. 

In this project, researchers at the Oxford Centre for Collaborative Applied Mathematics (OCCAM) bridged the gap between the discrete and continuum models by using the mathematical technique of asymptotic analysis to derive continuum equations for dislocation densities directly from the laws of discrete dislocation interactions. 

Results

Unlike existing continuum methods, this approach to dislocation structures can describe structures like dipolar mats that have zero effective dislocation density. Moreover, the new method clearly shows the regions where it is necessary to treat dislocations as discrete entities and, using matched asymptotics, employs simplified equations in these regions.

The new analysis showed that the density of dislocations in a dipolar mat is governed by a differential equation. Solutions to this differential equation, and the discrete equations obtained in boundary layer regions, performed well when compared to the full discrete solution.

The Future

Importantly, the results suggest that the method of discrete-to-continuum asymptotics is applicable to a wide range of problems involving interacting particles, not just problems involving dislocation interactions. The methods are already being applied to analyse the mechanical behaviour of chains of neodymium magnets.

The discrete-to-continuum methods pioneered in this project will continue to reveal rich and surprising mathematical structures in discrete problems, and they also promise to be useful in a wide variety of situations.