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The purpose of this study was to establish a method for estimating temperature distributions in steady-plane-wave fronts in a thermoviscous material using a Hugoniot function. To this end, under the fundamental assumption that the material in the wave front is approximately in an equilibrium state, two irreversible thermodynamic equations for temperature in the wave front were derived. In the first equation, heat transport was neglected, and in the second equation, the work done by thermal stress was offset by heat transport. The temperature distributions were evaluated qualitatively under the assumption of heat transport. This evaluation indicated that the second equation was effective if the effects of viscosity were large. These two equations were applied to the shock compressions up to 140 GPa of yttria-doped tetragonal zirconia. The second equation sufficiently predicted temperature behind the shock and also fairly accurately predicted temperatures in the shock front. The influence of heat transport on both temperatures was also examined. © 2001 American Institute of Physics.