Materials science is an interdisciplinary field that has emerged as one of the central pillars of the modern physical sciences. A central tenet of it is that many of the properties of a material are dominated by microstructures of the material. Therefore it is extremely important to study the formation and evolution of microstructure, so that we can get correct microstructure and gain desired properties of the material. As an attempt to simulate, at the meso-scale, the evolution of microstructure, we employ phase-field approach that is still young but has now become an important and powerful tool, to propose a new model for the microstructure evolution due to martensitic phase transitions driven by configurational forces in elastic solids. The model is an elliptic-parabolic coupled system of partial differential equations, which differs from the Allen-Cahn model by a gradient term, and renders hybrid features of a Hamiltonian equation and a parabolic one. The existence of weak solutions, traveling waves, etc. to this model in 1-D case is proved, and numerical simulations are carried out for some slightly simplified cases. Many problems on this model remain open, some are listed.
