A. The Generalized Particle (GP) Method
3. Surface corrections
It is not always feasible to use periodic boundary conditions and not always physical. In many cases free surfaces, or at least non-periodicity, is desired. Since a surface is a discontinuity between a solid/liquid and a vacuum/gas it requires atomic resolution to be accurate. However in many cases this accuracy may not be needed. In this subsection will be discussed the cause of surface effects in higher scales and a way to minimize or circumvent them.
The same models as used in section II.A.1 are used to demonstrate the effects of having a free surface. The periodicity in the X direction was removed and the models were equilibrated at 300K and 1atm then loaded in tension after 15 ps. Figure 6 shows the configuration energy for the atomistic S1 model and the scale-2 model. It is seen that the S2 model has significantly higher energy than the S1 model.
Figure 6. Configuration energy for the pure S1 and S2 models with free surfaces in the X direction.
The reason why S2 has larger energy is due to two reasons, the surface energy and the fact that it is S2. Since it is S2, the cutoff radius is k times as large as in S1, meaning that the surface effects will be k times as deep, however the model volume is the same as in S1, this causes the surface to volume ratio to be larger in S2 than S1. Essentially, higher scales have larger surface effects than smaller scales because their surface effects go deeper into the material. Imagine an atom at a distance of r from a free surface, if r is less than the atomic cutoff radius, rc, then this atom is influenced by the free surface.
Now an S2 particle that is kr from a free surface will have about the same configuration as that atom, and will be influenced in the same way due to inverse mapping, yet the particle is twice as deep in the bulk.
When high scale general particles are exposed to a vacuum, the material discontinuity causes a greater surface effect than an atomic surface. This surface effect can affect the surrounding deformation and cause trouble in various parts of the model.
This very noticeable change in configuration energy caused by surfaces could be large enough to nucleate dislocations from the surface when the atomistic model would not.
This difference could cause incompatible failure phenomena, thus some technique should be used to minimize this effect.
An alternative to using periodic boundary conditions in an effort to reduce or eliminate the surface effect of high scales is to use a way that would reduce the force imbalance at the surface. One way to do this is to place imaginary particles at the surface so that the real particles near the surface just see other particles, making them feel as though they are part of a bulk material. The trick comes from the question of how to determine the positions of these imaginary surface particles? They could be rigidly linked to real particles on the inside. Another approach would be to restrain the displacement of the particles at the surface from moving perpendicular to the surface. This would roughly maintain their perpendicular strain to that of the inner material; for a uniform strain, there is no wide spread effect from the surface force imbalance.
In this instance the former definition will be used around the edges of the model.
These edges are exposed to a vacuum and not in periodic conditions so they will be affected by the surface effects. The surface image layer, to be most effective, should have
a depth equal to the inter-particle cutoff radius, for the same reasons as for scale interfaces. For instance if the interatomic cutoff radius is 6.5 Å and the scale ratio k=2, for S2 it would be 13.0 Å. Surface images are created before the simulation is run. The user must specify domains within-which real particles will be converted to imaginary particles and be rigidly linked to a group of the closest real particles of the same scale, see Fig. 7. So in this case, domains should be specified all along the edges of the S2 domain having a depth of 13.0 Å.
Figure 7. Example of surface images, i, and j, linking to real particles within a cutoff and angle θ. Their fixed distance from the average, re.
Surface images are still required even if the model will be connected to an FEA mesh, because the GP model must equilibrate alone first, before connecting with FEA. The Surface images will later become the WG domain during FEA connection, which will be discussed later.
When the models have surface images and are equilibrated again their configuration energy is the same as if they were equilibrated with 3D periodic boundaries. Their energy difference is shown in Fig. 8 to be zero. This is a significant improvement to the free surface case. This shows that the use of surface images effectively eliminates the free surface effects. This is useful for models with geometries that are unsuitable for periodic conditions, or that are of a nature inherently unperiodic,
such as a thin plate with an edge crack; where the use of periodic conditions would cause the model to behave as if it had a center crack that recurred at every model width.
Figure 8. The difference of configuration energy from the 3D PBC condition for the free surface models and those with surface images, showing that the surface images recover the same configuration energy as the 3D PBC condition.