BEYOND THE ATOMIC SCALE

As larger systems are being considered, atomic scale simulations quickly become impossible: doubling the characteristic length scale typically means increasing the volume, and thus the number of atoms, by a factor eight. At present, simulations with up to 100 million atoms are possible using parallel supercomputers (Abraham 1997; Abraham et al. 1997; Zhou et al. 1997), corresponding to a system of less than 100nm cubed. If a larger system is to be studied, simulations which consider individual atoms cannot in practise be carried out. In this section we give a short summary over a few methods that go beyond an atomic description to describe the material at a coarser length scale. See also the review by Carlsson and Thomson (1998), where simulation techniques at different length scales are discussed in the context of how to calculate fracture toughness.

Simulations based on individual dislocations.

The idea behind ``dislocation dynamics'' is to use individual dislocations as ``fundamental particles'' in the simulations, i.e. to describe the material as consisting of a collections of dislocations that interact with each other through their elastic fields. This is a sensible idea when studying properties dominated by the dislocation dynamics, but cannot be expected to work without modifications in cases where other defect types (e.g. cracks and grain boundaries) play a significant role.

In principle the methods developed for atomistic simulations can be used, provided one replaces the interatomic interactions with the formula for interactions between dislocations. There are, however, a number of complications:

1.

Dislocations are lines, not points, i.e. the position of a dislocation cannot be described by a single set of coordinates.

2.

Both the force acting between dislocations, and the equation of motion for a single dislocation are highly anisotropic.

3.

Interactions between dislocations are long ranged, the force decays as 1/r, whereas the forces between atoms in a metal decay exponentially due to screening effects. It is thus not possible to introduce a cutoff distance beyond which dislocations do not interact. Attempts to introduce such cutoffs have been shown to cause the emergence of spurious structures with a characteristic length scale equal to the chosen cutoff (Gulluoglu, Srolovitz, LeSar and Lomdahl 1989; Holt 1970).

4.

Although the long-range elastic forces between dislocations are well known, the parameters describing the short-range interactions (dislocation crossing, cross slip, annihilation etc.) are presently not well known, and will have to be extracted from experiments, come from theoretical arguments or come from atomic-scale simulations.

Barts and Carlsson (1995) have developed a method for simulating the interactions of a large number of dislocations in two-dimensional systems. In two dimensions the dislocations are points, eliminating the first problem mentioned above, and simplifying the interactions significantly. The drawback is of course that the simulations are limited to problems that are essentially two-dimensional, i.e. where the dislocations are parallel and straight. One such problem, which has been successfully modelled, is polygonisation in single glide (Barts and Carlsson 1997). They solve the problem of the long range interactions between dislocations by using a Fourier transform of the long range part of the interactions, resulting in a method where the required computer time scales linearly with the number of dislocations.

Similar methods can be developed in three dimensions, as demonstrated by Kubin and coworkers (Kubin, Canova, Condat, Devincre, Pontikis and Bréchet 1992; Devincre and Kubin 1997). They solve the problem of representing the position of the dislocation strings by modelling the dislocations as consisting of a chain of straight segments of pure screw or edge dislocations, where each segment can be oriented along a finite number of directions. The interactions between the dislocations are then described by the interactions of the individual segments. The method has been used to study the influence of cross slip on work hardening (Devincre and Kubin 1994) and on pattern formation during cyclic deformation (Kratochvíl, Saxlovà, Devincre and Kubin 1997).

Simulations based on dislocation densities.

When the structures under consideration become larger, even dislocation based simulations become unwieldy. It is tempting to attempt to make a model based on a continuum description of the dislocation density, i.e. by replacing the positions of individual dislocations by a continuous dislocation density. A simple scalar density is clearly insufficient to describe the dislocations; as a dislocation is described by two vectors (the sense and the Burgers vector), the density becomes a tensor (Nabarro 1987).

As shown by Barts and Carlsson (1997) a continuum description based on the dislocation density tensor alone is insufficient to describe a relatively simple patterning process such as polygonisation. This is because the stress field from a dislocation wall is caused by the discrete dislocation positions, and vanish in a continuum model. They propose that a continuum model should contain additional variables describing the local environment of the dislocations. It is at present unclear if such a description can be made.

For an overview of the work that has been done on continuum modelling of dislocations, see the review by Selitser and Morris (1994).

Simulations based on grains.

For some types of simulations, an even coarser length scale can be used: the scale set by the size of the gains. In typical metals this will be of the order of 100$\mu$m. At this length scale one may consider the grains as the fundamental unit of the simulation. This approach has successfully been used e.g. to study recrystallization and the evolution of texture during growth (Juul Jensen 1997a,b and references therein).

Finite-element methods.

For engineering purposes, the relevant length scale will be set by the physical dimensions of the sample. Typically simulations ignore the grain structure of the material, which is treated as homogeneous (but possibly anisotropic). A continuum description based on the stress and strain fields is used, and the field equations are typically solved using finite element methods.