WIAS Preprint No. 1283, (2007)

A mathematical model for case hardening of steel


  • Fasano, Antonio
  • Hömberg, Dietmar
    ORCID: 0000-0001-9460-5729
  • Panizzi, Lucia

2010 Mathematics Subject Classification

  • 35K60 35R05 82B26


  • Heat treatment, phase transitions, coupled PDE, case hardening




A mathematical model for the case hardening of steel is presented. Carbon is dissolved in the surface layer of a low-carbon steel part at a temperature sufficient to render the steel austenitic, followed by quenching to form a martensitic microstructure. The model consists of a nonlinear evolution equation for the temperature, coupled with a nonlinear evolution equation for the carbon concentration, both coupled with two ordinary differential equations to describe the evolution of phase fractions. We investigate questions of existence and uniqueness of a solution and finally present some numerical simulations.

Appeared in

  • Math. Models Methods Appl. Sci., 19 (2009) pp. 2101--2126.

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