  A linear bubble model of grain growth is introduced to study the conditions
under which an isolated grain can grow to a size much larger than the
surrounding matrix average (abnormal growth). We first consider the case of
bubbles of two different types such that the permeability of links joining
unlike bubbles is larger than that of like bubbles (a simple model of grain
boundary anisotropy). Stable abnormal growth is found both by mean field
analysis and direct numerical solution. We next study the role of grain
boundary pinning (e.g., due to impurities or precipitate phases) by introducing
a linear bubble model that includes lower and upper thresholds in the driving
force for bubble growth. The link permeability is assumed finite for driving
forces above the upper threshold, zero below the lower threshold, and
hysteretic in between. Abnormal growth is also observed in this case.
