A thermodynamically consistent, novel multiphase phase field approach for stress-and temperature-induced martensitic phase transformations at finite strains and with interfacial stresses has been developed. The model considers a single order parameter to describe the austenite↔martensitic transformations, and another N order parameters describing N variants and constrained to a plane in an N-dimensional order parameter space. In the free energy model coexistence of three or more phases at a single material point (multiphase junction), and deviation of each variant-variant transformation path from a straight line have been penalized. Some shortcomings of the existing models are resolved. Three different kinematic models (KMs) for the transformation deformation gradient tensors are assumed: (i) In KM-I the transformation deformation gradient tensor is a is a linear function of the Bain tensors for the variants. (ii) In KM-II the natural logarithms of the transformation deformation gradient is taken as a linear combination of the natural logarithm of the Bain tensors multiplied with the interpolation functions. (iii) In KM-III it is derived using the twinning equation from the crystallographic theory. The instability criteria for all the phase transformations have been derived for all the kinematic models, and their comparative study is presented. A large strain finite element procedure has been developed and used for studying the evolution of some complex microstructures in nanoscale samples under various loading conditions. Also, the stresses within variant-variant boundaries, the sample size effect, effect of penalizing the triple junctions, and twinned microstructures have been studied. The present approach can be extended for studying grain growth, solidifications, para↔ferro electric transformations, and diffusive phase transformations.

The coupled motion of a closed non-circular grain boundary (GB) in a bicrystal, with both isotropic and anisotropic GB energies, is studied using the level set method. The kinetic relations, obtained within the framework of linear irreversible thermodynamics, govern the overall dynamics including normal motion (migration) of the GB, viscous sliding along the GB, and tangential motion of the grains which is geometrically coupled with the migration. The shape accommodation necessary to maintain coherency of relatively rotating and non-deforming grains is accomplished by allowing for diffusion along the GB. We solve the governing equations for the coupled motion to determine the shape and the misorientation evolution of an isolated GB under various constitutive assumptions. First, assuming both GB energy and kinetic coefficients to be isotropic, we study the interplay between kinetic coefficients for initially circular, near-circular, and non-circular GBs, as well as the role of stress and initial conditions on the GB dynamics. Next, we study the influence of anisotropy in GB energy, mobility, and geometric coupling for various combinations of parameters and initial conditions. Allowing for geometric coupling can in fact lead to distinctly different shapes than what are usually predicted on the basis of migration alone. Our numerical scheme provides a general framework to study these and other related problems of GB motion.

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a b s t r a c tGrain rotation and grain boundary (GB) sliding are two important mechanisms for grain coarsening and plastic deformation in nanocrystalline materials. They are in general coupled with GB migration and the resulting dynamics, driven by capillary and external stress, is significantly affected by the presence of junctions. Our aim is to develop and apply a novel continuum theory of incoherent interfaces with junctions to derive the kinetic relations for the coupled motion in a tricrystalline arrangement. The considered tricrystal consists of a columnar grain embedded at the center of a non-planar GB of a much larger bicrystal made of two rectangular grains. We examine the shape evolution of the embedded grain numerically using a finite difference scheme while emphasizing the role of coupled motion as well as junction mobility and external stress. The shape accommodation at the GB, necessary to maintain coherency, is achieved by allowing for GB diffusion along the boundary.