The purpose of this study is to establish an Inside Temperature (IT) method for estimating temperatures in steady wave fronts in a thermoviscous material. A fundamental assumption that the material in the wave front, was approximately in an equilibrium state was used in this method. A further assumption that heat transport was neglected was used in the ITdQ=0 method, while in the ITIM method, the work done by the thermal stress was offset by heat transport. Two irreversible thermodynamic equations for the temperature in the wave front derived were connected with the Hugoniot function and the Mie-Grüneisen equation, respectively. To verify the efficacy of the IT method, three temperature distributions were estimated qualitatively using an equation for entropy including no assumption on heat transport, that including the assumption used in the ITdQ=0 method, and that in the ITIM method. These three distributions suggested that the temperatures were overestimated by the ITdQ=0 method, while the ITIM method was effective for shock compressions where the effect of viscosity was distinguished.

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