WIAS Preprint No. 766, (2002)

A mathematical model for induction hardening including mechanical effects


  • Hömberg, Dietmar

2010 Mathematics Subject Classification

  • 74F05 74F10 74D10 77N25

2008 Physics and Astronomy Classification Scheme

  • 64.70.Kg 81.40.Gh


  • joule heating, thermoviscoelasticity, phase transitions


In most structural components in mechanical engineering, there are surface parts, which are particularly stressed. The aim of surface hardening is to increase the hardness of the corresponding boundary layers by rapid heating and subsequent quenching. This heat treatment leads to a change in the microstructure, which produces the desired hardening effect. The mathematical model accounts for electromagnetic effects that lead to the heating of the workpiece as well as thermomechanical effects that cause the hardening of the workpiece. The new contribution of this paper is that we put a special emphasis on the thermomechanical effects caused by the phase transitions. We formulate a consistent model which takes care of effects like transformation strain and transformation plasticity induced by the phase transitions and allows for physical parameters depending on the respective phase volume fractions. The coupling between the electromagnetic and the thermomechanical part of the model is given through the temperature-dependent electric conductivity on the one hand and through the Joule heating term on the other hand, which appears in the energy balance and leads to the rise in temperature. Owing to the quadratic Joule heat term and a quadratic mechanical dissipation term in the energy balance, we obtain a parabolic equation with L1 data. We prove existence of a weak solution to the complete system using a truncation argument.

Appeared in

  • Nonlinear Anal. Real World Appl., 5 (2004), pp. 55-90

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