Stress Analysis During Phase Transformation

During phase transformation of a material local changes of temperature induce phase transformation which cause changes in local properties such as specific volume, modulus, etc. The newly transformed material introduces a strain under stress free condition, a situation similar to the thermal strain induced in a material due to temperature change. The transformation strain, being constrained by the surrounding material, generates a stress field, the analysis of which is the subject matter of this paper. When the material under stress is elastic and under static condition, the problem is commonly addressed as the Eshelby problem. However, the more general types of problem such as occurring in material processing are elastic-plastic and dynamic, they need a fresh look. It is noted that the transformation strains in general are both dilatational and shear in nature. The shear strain has been treated in metallurgical literatures from the crystal lattice point of view. Its effect on the bulk behavior is still a subject of study at the present time. In this paper only the dilatational strain will be discussed. The formulation of the overall problem of quenching of a hollow cylinder in terms of an initial-boundary value problem of a coupled field of temperature and stress will be used to illustrate the ideas involved in this subject matter. It would be a problem of thermal stress alone if phase transformation were not involved. As the temperature of the cylinder drops below the critical temperature in the outer surface regions of the cylinder and gradually spread to the central region, the energy and the momentum balance equations are solved in time sequence with the simultaneous introduction of the latent heat and the dilatational strain due to phase transformation.