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The propagation of a crack into a material whose mechanical and thermal properties depend on the temperature field developed around the running crack tip is studied in the present work. This is achieved by employing the Non Steady Heat Source model, for the calculation of the temperature field surrounding the crack tip region, the dynamic version of the T-criterion of failure and the experimental data concerning the variation of thermal and mechanical properties of the material with temperature. It is proved that in the immediate vicinity of the crack tip the material properties show steep gradient. Thus, if a failure criterion is to be applied for the prediction of the path of an already traveling crack, then the current values of the mechanical properties should be taken into account. Otherwise, fallacious conclusions may be drawn. The results obtained are in satisfactory agreement with the limited experimental evidence available from literature concerning both the intensity of the temperature field developed and, also, the direction of the crack propagation.