Continuum scale modeling of phase transformation

Taking a continuum-mechanical perspective, the isothermal model in Hallberg et al. (2007) introduces the volume fraction of martensite as an internal variable. Along with a transformation condition, dependent on the state of deformation and on temperature, this allows the evolution of the martensitic phase to be traced. The presence of a transformation condition allows establishment of a transformation potential surface, much like the yield condition and yield surface found in plasticity theory. The transformation surface is illustrated in deviatoric stress space and in the meridian plane below.

Transformation surface in the deviatoric and in the meridian plane, respectively.
Transformation surface in the deviatoric and in the meridian plane, respectively.

Depending on which one is active, the yield and transformation conditions determine the response of the material. The relative influence of austenite and martensite on mechanical material properties is considered through a homogenization procedure, based on the phase fractions.

The above isothermal model is further elaborated in Hallberg et. al (2010b), where full thermo-mechanical coupling is considered. These models are suitable for large-scale simulations of metal forming processes involving materials exposed to martensitic phase transformation. The application to sheet metal forming is illustrated below by images from simulations of a deep-drawing process.

Volume fraction of martensite in a stainless steel sheet during deep-drawing at different temperatures. Note that three drawing stages are shown at each temperature. a) T=213K, b) T=233K, c) T=293K and d) T=313K.
Volume fraction of martensite in a stainless steel sheet during deep-drawing at different temperatures. Note that three drawing stages are shown at each temperature. a) T=213K, b) T=233K, c) T=293K and d) T=313K.