3D DISCRETE DISLOCATION DYNAMICS APPLIED TO ... - NUMODIS

2.4 Boundary conditions 31
a mathematical form.
ic(L + i) = ic(i)
ic(−i) = ic(L − i) for i = 0, . . . (2.19)
The array ipc can be used to redirect a segment coordinates, which has left an initial volume lattice,
to an equivalent position in the initial lattice. Then it is possible to apply PBC with a simple
array reference. The orthorhombic simulation volume is readily realizable by changing the range of
periodicity according to the maximum length of the simulation volume along each axis.
Because of the subnetwork in the simulation volume (see Sec. 2.1.3), the periodicity should be a
multiple of 4xl along each axis.
2.4.2 Internal interfaces
Motivations and review of the literature
The collective behavior of dislocations in a single crystal can be simulated with the stress com-
putation and motion treatments as explained in the previous sections. More rigorous boundary
conditions need to be implemented on the method in order to treat more general cases, such as
a crystal with free surfaces, a crystal containing particles of a second-phase or a polycrystal with
grain boundaries.
A dislocation experiences forces near an interface because the dislocation energy is different in the
two mediums involved. The dislocation is attracted towards a free surface, for example, and repelled
by a rigid surface layer. These image stresses can be treated by using the superposition principle.
The effects of free surfaces were treated by Fivel et al. ([Fivel & Canova 99]). The forces ex-
erted on a free surface by dislocations are computed assuming that the dislocations are embedded
in an infinite medium. These forces are then reversed and changed into the appropriate point
forces to enforce the traction free surface condition. Applications of this method can be found in
[Fivel et al. 98].
The image stresses by a free surface is, in fact, a special case of the more general situation in which
an interface separates two materials of differing elastic constants, e.g., oxide layers and particles.
The image stresses on a dislocation in the presence of a second phase particle can be computed also
by the superposition principle. The formulation follows that of Van der Giessen and Needleman
([Giessen & Needleman 95]). Previous applications of this method to 2D cases can be found in
Cleveringa et al. ([Cleveringa et al. 97]).