Generalizations of the theory to finite convex specimens.

The goal of this paper is to explore how the time cone theory can be made exact when the specimen is finite and convex and when and are functions of time alone. We let the time cone for the point continue to be the set of all points in time and space , including points outside the specimen, that, had nucleation occurred there and then, would have resulted in transformation at the point if there were no interference from other grains. Extending the cone beyond the specimen boundary is tantamount to assuming that the grains that might have nucleated outside the specimen can grow into the specimen, which causes no problem if the nucleation rate is set to outside the specimen and at times before the experiment began. By assuming that the specimen is convex in shape (and time), we insure that there is no interference from the specimen boundary from the nucleation event at to the point at time . With this assumption, time cones for finite specimens remain exactly as for an infinite specimen, equation (2.3); if V is constant, a cone in space-time with sides sloping by , and if V is time dependent a conical surface of revolution with sides sloping by 1/V(t). We let the nucleation rate be whatever it is in the specimen and identically outside the specimen. We then calculate the expected number of nuclei in the time cone by integrating the nucleation rate as before in equation (2.6). If the nucleation rate is constant in the specimen, the expected number of nuclei remains simply proportional to the ``volume'' of the intersection of the nucleation time period in the space of the specimen with the time cone. will be a function of the position of the apex, and the time period of the nucleation and the geometry of the specimen. The expected fraction untransformed will then be given by equation 2.10.

• Specific Geometries.