Ph.D Thesis | |

Ph.D Student | Zigelman Anna |
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Subject | Coupled Grain Boundary and Exterior Surface Motion |

Department | Department of Applied Mathematics |

Supervisor | PROFESSOR EMERITUS Amy Novick-Cohen |

The coupled motion of grain boundaries and exterior surfaces, which join along "groove roots," is a common phenomenon in polycrystalline specimens and is critical in determining the stability properties of the resultant material. We focused on this coupled motion in the context of the "half loop" geometry, where a U-shaped grain extends in an otherwise single crystal. The two grains are of the same material and differ only in their relative crystalline orientation. The interface between the two grains, known as the grain boundary, contacts the exterior surface along a "groove root" where various balance laws hold. This geometry contains two types of motion; one is motion by mean curvature of the grain boundary, and the other is motion by surface diffusion of the exterior surfaces. These motions can be modeled by a system of time dependent nonlinear PDEs.

There is a dimensionless physical parameter m, defined as the ratio of the free
energy of the grain boundary to the free energy of the exterior surface, which
is influential in the boundary conditions, and m > 0 causes the appearance
of "thermal grooving" along the intersection of the grain boundary
with the exterior surface. Typically, 0 < m < 1/3, and m can be taken to
be a small positive parameter.

In my M.Sc. Thesis we considered thin polycrystalline specimens and by
appealing to asymptotic analysis, we found a linear first order correction to
the problem described above.

During my PhD research, we used asymptotic analysis to find a first order correction to the same problem for thick specimens. We proved existence for the governing equation for the shape of the grain boundary in the thick case. We also proved existence of a solution to fourth order PDE system for the perturbations in heights of the exterior surfaces which couple along the groove root via boundary conditions. We solved this system numerically and observed that there exist two regimes for evolution, a Mullins’s like grooving regime and a traveling wave like regime.

Moreover, we decoupled the leading order PDE describing the shape of the thermal groove and the rate at which the tip of the interior grain recedes. We employed PDE's theory regarding existence of Green's function and existence of solutions to Volterra equation of the first kind. Thus we obtained an expression for the rate at which the tip of the interior grain recedes, then calculated this rate numerically.