Dislocations, as one of main structural defects, have an important role to determine materials proprties and behaviors. Indeed, dislocations are the sliding of crystalline planes over each other which result in plastic defermation. Due to their inelastic local strain, they are cucial in structure stress relaxation. Since dislocations are defined at the nanoscale and their evolution occurs in the picosecond scale, their nucleation and evolution are considered in nano structures which have attracted a great attention in researchers. In this research, the nucleation and evolution of edge dislocations are studied in single andi-crystalat the nanoscale using the phase field method. Mechanical loadings consist of inelastic misfit strains and boundary loadings. The dislocation nucleation and evolution are determined by defining proper terms in the Helmholtz energy and minimizing it through the Ginzburg-Landau equation. The present study is indeed the coupled mechanics-phase field problem which is implemented by the FEM method using the COMSOL Multiphysics code. The results of periodic crystalline energy, equivalent geometry and periodic burgers vector models have been analyzed and the appropriate physical model has been obtained for the simulation of parallel slip planes. Various functions are presented for the gradient energy normal to the slip planes ans the results have been compared. Various types of functions for the plastic strain are compared and the appropriate physical model is obtained. The effect of type and order of the FEM discretization is studied. The mathematical model and analytical process are verified by the comparison of the dislocation height and the distance between dislocation cores between the present study and previous models. Various and important problems such as dislocation nucleation and evolution near the martensite lath with different shear and volumetric strains, along A-M sharp and diffuse interface with various thicknesess, in single andi-crystalunder the prescribed displacement and also along inclined slip systems have been studied. Surface and size effect on the dislocation evolution have been studied. The obtained results present comprehensive information for the mathematical modeling and numerical simulation of the dislocation evolution and its interaction with other structural changes. Keywords: dislocations, phase field method, elastic strain, inelastic strain, single crystal, bi-crystals
