The application of variational and Lagrangian thermodynamics is extended to problems of heat conduction with melting boundaries. The physical properties of the conducting material are considered to be temperature dependent. In particular, the material Alumina (Al2/O3) representing a class of ceramic materials for which the effective conductivity keff = kphonon + kphoton is subjected to above treatment and the temperature distribution and the melting rate are found. The results are compared with the constant-conductivity case.

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