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Interface Migration of Solid 3-D
Three-Grain Interface
by Erik Ingemann Waldorff
Introduction

Many projects have dealt with the interface migration of a surface simulated as a two dimensional case. However the effects shown by these projects might be of a different character if the interface grains are fully simulated. This project takes the similar idea of interface migration to the level of three dimensions, in order to examine the effects of the interface migration at the interface of three grains.

Setup

The three-grain interface was set up as a circular symmetric surface, which contained the three two-surface interfaces and the one common triple point interface. This was done using the commercial software finite element package I-DEAS, where the surface was modeled with triangular elements. The perimeter of the surface was set to be fully clamped in all 3 directions, which would lead to boundary effects (see conclusion), which was expected.

For the theoretical part of the programming the triangular elements utilized was selected to be constant strain triangles. Using this type of elements the in-plane forces and displacements could be transferred into the global coordinates and vice versa. The driving force of the two-surface interfaces and the triple point were set to   and   respectively.  Furthermore for the triangular elements only the energy term that is due to the elongation of the element was considered. The terms that are due to the rotation and the area swept by the motion of the element were considered small comparatively, and were thus ignored. Hence the in-plane driving force was set to be only  . The mobility of the nodes was set to m.

The parameters used for the simulation were:





The time step was selected to be 3 seconds, and the simulation was carried out for up to 10,000 iterations where it was found that further iterations would not develop the interface any further.

Simulation results

In the below streaming movie the results for a total simulation time of 0min, 5min, 10min, 20min, 30min, 40min, 50min, 100min, 200min, 300min, 400min and 500min are presented. The resolution timewise is currently being redone to make a smoother simulation (i.e. more pictures will be added per second, so check back soon).
 

Center Grain displacement

As the simulations were carried out the center grain displacement was tracked as to determine when the effect of grooving would cease, as the driving forces of the grain boundaries would equal the in-plane surface forces. The plot of the grain position vs. time is as follows:

Conclusion

From the simulations it can be seen that we can conclude the following:

1) The interface migrates as expected from a two-dimensional point of view for the first 30 min of simulation. After that the effects of the clamped boundary makes the migration shape stretch out to a linear plane. If time had permitted it, the identical simulation would have been carried out with either a) altered boundary conditions, or b) a larger disk, in order to isolate the center grain from the boundary effects.

2) As it is the case for the similar two-dimensional problem, the surface migration velocity decreases as the simulation time increases. This is most noticeable in the decrease in vertical movement rate of the center grain. This result goes hand in hand with the theory of the grooving effect reaching equilibrium with respect to energy, as interface migration is carried out over time.
 

Future work

If time had permitted it I would like to have explored the following scenarios, which could also be used for another ME599 term project:

1) Include the energy terms that are due to the rotation and the area swept by the motion of the triangular elements.
2) Try similar simulations for interface migration and diffusion.

Code

The simulation was carried out using Matlab 6.1. Click on the individual links to see the codes utilized. Two files are present: 'project5.m' is the main program, and 'loadnodes2.m' is the program that contains all nodal data.