An advanced phase-field approach to martensitic phase transformations (PTs) at finite strains are developed at nanoscale that satisfies crystal lattice instability conditions obtained by atomistic simulations. Theory includes a fully geometrically nonlinear formulation for the general case of finite elastic and transformational strains as well as anisotropic and different elastic properties of phases. Material parameters are calibrated, in particular, based on the crystal lattice instability conditions from atomistic simulations for martensitic PTs between cubic Si I and tetragonal Si II phases under complex triaxial compression-tension loading. Finite element algorithms and numerical procedures are developed and implemented in the code deal.II. Various 3D problems on lattice instabilities and following nanostructure evolution in single-crystal silicon are solved for compression in one direction under lateral stresses and analyzed.

Stress-induced martensitic phase transformations (PTs) at a stationary 60◦ dislocation in single- crystalline Si are modeled. Finite element method (FEM) simulations elucidate two different mechanisms of nucleation and nanostructure evolution for two different stress-hysteresis cases. For a traditional finite-stress-hysteresis region, the PT starts with the barrierless nucleation of a thermodynamically-equilibrium-incomplete embryo, which loses its stability and grows forming a propagating martensitic band with distinct interfaces. However, in the unique zero-stress-hysteresis region, where PT for defect-free crystal occurs homogeneously through intermediate phases without nucleation, interfaces, and growth, the PT starts at a dislocation but spreads quasi-homogeneously, without interfaces, similar to the defect-free case; Despite large normal stresses produced by dis- location in the range of ±(6 − 12) GPa, a relatively small reduction in macroscopic PT stress by 1.6 GPa is quantitatively explained by mutually compensating contributions of stresses into lattice instability criterion.

Two different definition of crystal lattice instability conditions, namely phase transformation instability and elastic instability are compared. It is numerically showed that while the phase transformation instability is independent of the prescribed stress measure and occurs at a specific strain, the elastic instability depends on the stress measure and occurs at different strains when different stress measures are prescribed. Besides, it is revealed that although reaching the critical load predicted by the phase transformation instability criteria leads to the initiation of the phase transformation (PT), it is not sufficient for the completion of the PT. For a homogeneous crystal lattice, the PT is not completed unless the elastic instability critical load corresponding to the prescribed stress measure is exceeded. Nevertheless, in the case of heterogeneous crystal when a specific stress measure can not be prescribed within the bulk, no matter which stress measure is prescribed, the elastic instability corresponding to the stress measure that occurs at lower strain is the chief elastic instability point. That which instability point occurs first depends on the loading condition. For instance, during compressive loading, elastic instability predicted by Cauchy stress occurs first, however, in tensile loading SPK stress predicts the elastic instability truly.

An scale-free phase-field model to martensitic phase transformations (PTs) at finite strains is developed as an essential generalization of models in Levitas et al. (2004); Idesman et al. (2005). An advanced expression for athermal threshold in terms of stress tensor components as well as the concentration of phases enables the model to reproduce lattice instability conditions obtained by experiments or atomistic simulations. In order for the model to be scale-independent, the gradient term in the free energy is excluded, so that the model is applicable for any scale greater than 100 nm. It is shown that although finer mesh can produce more detailed microstructure, the solution becomes mesh independent after a certain mesh size. Strain rate dependence of the solutions is analyzed in detail.