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The paper presents a multi-scale mathematical model dedicated to a comprehensive simulation of resistance heating combined with the melting and controlled cooling of steel samples. Experiments in order to verify the formulated numerical model were performed using a Gleeble 3800 thermo-mechanical simulator. The model for the macro scale was based upon the solution of Fourier-Kirchhoff equation as regards predicting the distribution of temperature fields within the volume of the sample. The macro scale solution is complemented by a functional model generating voluminal heat sources, resulting from the electric current flowing through the sample. The model for the micro-scale, concerning the grain growth simulation, is based upon the probabilistic Monte Carlo algorithm, and on the minimization of the system energy. The model takes into account the forming mushy zone, where grains degrade at the melting stage – it is a unique feature of the micro-solution. The solution domains are coupled by the interpolation of node temperatures of the finite element mesh (the macro model) onto the Monte Carlo cells (micro model). The paper is complemented with examples of resistance heating results and macro- and micro-structural tests, along with test computations concerning the estimation of the range of zones with diverse dynamics of grain growth.
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