In 1938 Zener showed that for an isotropic homogeneous solid, the elastothermodynamic damping may be predicted using either a mechanical description or an entropic description. The objective of this paper is to extend Zener's result to a composite material with anisotropic subregions and thermally imperfect interfaces. Specifically, we will show that for time-harmonic vibrations, the external work lost per cycle throughout the body is equal to the entropy produced per cycle throughout the body times the equilibrium temperature. However, contrary to intuition we will show that the work lost per unit volume per cycle at a point is not equal to the entropy produced per unit volume per cycle at that point times the equilibrium temperature.
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