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The rate of thermal expansion of a thin metallic slab of arbitrary shape

In a previous paper the rate of thermal expansion of a long, slender insulating bar has been worked out. Our present aim is to extend that work to the thermal expansion rate of not only a long metallic bar, but to further generalize it to a thin metallic slab of arbitrary shape. Assuming that the thickness of the slab is small compared to the linear dimension of its area we again take advantage of the two distinct, disparate timescales to turn the familiar problem of thermal expansion into a time-dependent problem of the rate of the expansion. Based on the previously established finite physical momentum of an acoustic phonon when translational invariance is broken, we show that the combined pressure of the phonons and the free electrons due to their outward momenta would suffer a Doppler reduction as the specimen expands upon heating. This Doppler reduction gives rise to damping of the expanding motion, thus yielding as a first result the time of thermal expansion of a long slender metal bar. The generalization to the important case of a thin metallic slab of any shape is then worked out in detail before a concluding section containing a long physical discussion and summary.