Brief mM Manual
Introduction
The elementary objects treated by the code are discrete dislocation lines embedded
into an elastic continuum. To obtain a better efficiency without loss of
accuracy, time and space are discretized, as well as the orientations (i.e.,
the characters) of the dislocation lines. The continuous shapes of these lines
are decomposed into a set of straight segments lying on a lattice homothetic to
that of the considered crystalline material. The parameter of this underlying lattice is
adjusted to the treatment of the smallest length scale of interest for a given
problem. The dislocations move by discrete jumps on the underlying lattice,
which allows including the slip geometry for a range of different
crystallographic structures (fcc, bcc, dc, hcp) and cutting-off very small
length scales below which elasticity theory breaks down. The line orientations
are discretized in such a way as to include at least the screw and edge
directions in each potential slip plane, the character of the junctions formed
when two attractive, non-coplanar segments cross each other and the glide
direction of all the discrete dislocation characters in all potential slip
planes.
The elastic properties of dislocations (line tension, dislocation-dislocation
interactions, etc ...) and the computation of the effective (net) force on each
dislocation segment directly follow from the classical theory of dislocations.
The equation of motion of each segment is computed at its mid-point, using
classical methods similar to those employed in MD simulations. Algorithmic
parameters like the time step, the minimum length of the segments and their
maximum travel distance during a step are optimized in such a way as to
reproduce known solutions for typical elastic problems like the critical stress
for a Frank-Read source.
Dislocation core properties, which are relevant of an atomic-scale treatment, cannot be
accounted for in a purely elastic framework. The most two important ones are
the stress vs. mobility relation and the energetics of the cross-slip
mechanism. They are incorporated through the use of "local rules"
that apply to each segment at each step of the simulation. These rules are
determined from the available information on core mechanisms, as obtained from
continuum modeling, experiment and, increasingly in the past few years,
atomistic modeling.
Periodic boundary conditions are implemented in order to ensure that the mean-free path of dislocations in the
simulation is larger than their "physical" mean-free path. The
dimensions of the elementary simulated cell depends on the problem investigated
and can go up to typically 10-20 cubic microns for simulations of single crystals. A simple but efficient
method, deriving from order-N algorithms for the many-body problem, is used for
the computation of long range elastic interactions of dislocations. All the
critical parts of the simulation are numerically optimized and the simulation
can be run on clusters thanks to MPI libs.
The output of the simulation yields
information about the microstructure, local quantities of interest (internal
stresses, dislocation densities, slip systems activity) and on the global
mechanical response. This code is perfectly adapted for performing mass
simulations on single crystals.
When coupled to a finite element code, thus replacing the usual
constitutive relations, it can be used to investigate more complex materials
and loading conditions (this coupling is not presently included in the standard
distribution of ‘mM’).
Download a paper presenting the mean features of the simulation code mM!
Last update: 3/11/11