Martensitic phase transformation is a first order, displacive and diffusionless transformation which occurs between austenite and martensite phases. Austenite is a cubic phase which is stable at high temperatures while martensite is tetragonal and is stable at low temperatures. The formation of martensite requires a rapid cooling. The main aim of this study is to investigate martensitic phase transformations in polar coordinate at nanoscale which is the base for the analysis of different transformations from martensitic ones to surface melting and so on in nanostructures with a circular cross section such as nanotubes and nanoparticles. First, the phase field equations, i.e., the Ginzburg-Landau kinetics equations, and the energy functions are described in detail. Next, due to the nonlinearity of kinetics equations, the finite element form of these equations are derived using the nonlinear finite elemente method. Then, elasticity equations consisting of the equilibrium (neglecting the mass), kinematics and consititutive equations are presented in the polar coordinatefor plane stress problems and their finite element form is derived. Finally, the elasticity and the phase field equations which are coupled through the phase dependence of strain and the presence of the elasticity energy in the phase field energy are solved using the nonlinear finite elemente method and the Matlab code. Different explicit and implicit numerical methods are used for the time dependent system of equations. Several examples of phase transformations such as pure thermal hase transformation (without mechanics) and also stress- and thermal induced phase transformations for a circular sample with one and two martensitic variants are presented The developed code provides a proper tool to study different transformation phenomena in samples with complex geometries and more variants and under different mechanical loadings. Keywords: martensitic transformation, polar coordinate, small strain, phase field, finite elemente.
