Phase-field approach to surface-induced phase transformations and dislocations

Martensitic phase transformations (PTs) play a very important part in material science, being responsible for formation of unique microstructure, mechanical properties, and material phenomena in steels, shape memory alloys and ceramics. In particular, surface-induced PTs and pretransformations, surface energy, surface tension, interface, and scale effect at the nanoscale play essential roles in thermodynamics, kinetics, and nanostructures. In addition, various material phenomena are related to the interaction of martensitic PTs and plastic deformation due to twinning and dislocations and are of fundamental and technological importance. Examples are: heat and thermomechanical treatment of material to obtain desired structure and properties; pseudoelasticity, pseudoplasticity, shape memory effect; transformation-induced plasticity (TRIP); and synthesis of materials under high pressure with large plastic deformations, e.g., during ball milling; and PTs during friction, surface treatment, and projectile penetration. The two main approaches to study martensitic PTs are the sharp interface approach (SIA) and the phase field approach (FPA). In the SIA, the interface between two phases is considered as a single surface across which there is a jump in all thermomechanical properties. In contrast, in the FPA, the interface has a finite width across which properties smoothly vary from one phase to another. Each phase is shown by an order parameter which varies from 0, corresponding to austenite (A), to 1, corresponding to martensite (M). The PFA is broadly used to study the martensitic PTs, however the current FP models cannot describe a lot of basic physics. In our work, we advanced the PFA to martensitic PTs in three important directions: the potential is developed that introduces the surface tension at interfaces; a mixed term in gradient energy is introduced to control the martensite-martensite interface energy independent of that for austenite-martensite; and a noncontradictory expression for variable surface energy is suggested. The problems of surface-induced pretransformation, barrierless multivariant nucleation, and the growth of an embryo in a nanosize sample are solved to elucidate the effect of the above contributions. Also, an in-detail study of M-M interface width, energy, and surface tension, as well as the effect of finite element discretization on the width and the energy, and the formation of martensitic nanostructures in the trasnforming grain is presented. In addition, the external surface layer, as a new key parameter in surface-induced PTs, is introduced in PFA, and the effect of the width of this layer and internal stresses on surface-induced pretransformation and PTs is revealed. In addition to study martensitic PTs using FPA, we used FPA to study dislocations evolution. The current PFA to dislocations, which is based on a formalism similar to the PFA for martensitic PTs, suffers from several main drawbacks. In our work, the PF theory to dislocations is conceptually advanced in the following directions: (a) Large strain formulation is developed. (b) A new local potential is developed to eliminate stress-dependence of the Burgers vector and to reproduce desired local stress-strain curve, as well as the desired, mesh-independent, dislocation height for any dislocation orientation. (c) A new gradient energy is defined to exclude localization of dislocation within height smaller than the prescribed height but does not produce artificial interface energy and dislocation widening. After developing the most advanced PFA to PTs and dislocations, we developed a new PF theory to coupled evolution of PTs and dislocations and the following problems of the interaction of PTs and dislocations are studied: hysteretic behavior and propagation of A-M interface with incoherency dislocations for temperature-induced PT; evolution of phase and dislocation structures for stress-induced PT, and the growth and arrest of martensitic plate for temperature-induced PT

Phase field approach to interaction of phase transformations and plasticity at large strains

Thermodynamically consistent phase field approach (PFA) for multivariant martensitic phase transformations (PTs) and twinning for large strains is developed [1, 2]. Thermodynamic potential in hyperspherical order parameters is introduced, which allowed us to describe each martensite‑martensite (i.e., twin) interface with a single order parameter [3]. These theories are utilized for finite element simulation of various important problems [1‑4]. Phase field approach to dislocation evolution was developed during the last decade and it is widely used for the simulation of plasticity at the nanoscale. Despite significant success, there are still a number of points for essential improvement. In our study [5], a new PFA to dislocation evolution is developed. It leads to a well-posed formulation and mesh-independent solutions and is based on fully large-strain formulation. Our local potential is designed to eliminate stress-dependence of the Burgers vector and to reproduce desired local stress–strain curve, as well as to obtain the mesh-independent dislocation height H for any dislocation orientation. The gradient energy contains an additional term, which excludes localization of dislocation within height smaller than H but disappears at the boundary of dislocation and the rest of the crystal; thus, it does not produce interface energy and does not lead to a dislocation widening. Problems for nucleation and evolution of multiple dislocations along the multiple slip systems are studied. The interaction between PT and dislocations is the most basic problem in the study of martensite nucleation and growth. Here, a PFA is developed to a coupled evolution of martensitic PTs and dislocations [6], including inheritance of dislocation during direct and reverse PTs. A complete system of equations, including Ginzburg–Landau equations is presented. It is applied to studying the hysteretic behavior and propagation of an austenite‑martensite interface with incoherency dislocations, the growth and arrest of martensitic plate for temperature-induced PTs, the evolution of phase and dislocation structures for stress-induced PTs, and the evolution of dislocations and high pressure phase in a nanograined material under pressure and shear [6, 7]. REFERENCES [1] Levitas, V.I., Levin, V.A., Zingerman, K.M., Freiman, E.I. Phys. Rev. Lett. 2009, 103, 025702. [2] Levitas, V.I. Int. J. Plasticity. 2013, 49, 85‑118. [3] Levitas, V.I., Roy, A.M., Preston, D.L. Phys. Rev. B. 2013, 88, 054113. [4] Levin, V.A., Levitas, V.I., Zingerman, K.M., Freiman, E.I. Int. J. Solids & Struct. 2013, 50, 2914‑2928. [5] Levitas, V.I., Javanbakht, M. Phys. Rev. B., Rapid Commun. 2012, 86, 140101. [6] Levitas, V.I., Javanbakht, M. Appl. Phys. Lett. 2013, 102, 251904. [7] Levitas, V.I., Javanbakht, M. Nanoscale. 2014, 6, 162‑166

Phase field approach to interaction of phase transformation and dislocation evolution

Phase field approach to coupled evolution of martensitic phase transformations (PTs) anddislocation is developed. A fully geometrically nonlinear formulation is utilized. The finite element method procedure is developed and applied to study the hysteretic behavior and propagation of an austenite (A)–martensite (M) interface with incoherency dislocations, the growth and arrest of martensitic plate for temperature-induced PT, and the evolution of phase and dislocation structures for stress-induced PT. A similar approach can be developed for the interaction of dislocations with twins and diffusive PTs described by Cahn-Hilliard theory

Coupled phase field and nonlocal integral elasticity analysis of stress-induced martensitic transformations at the nanoscale: boundary effects, limitations and contradictions

In this paper, the coupled phase field and local/nonlocal integral elasticity theories are used for stress-induced martensitic phase transformations (MPTs) at the nanoscale to investigate the limitations and contradictions of the nonlocal integral elasticity, which are due to the fact that the support of the nonlocal kernel exceeds the integration domain, i.e., the boundary effect. Different functions for the nonlocal kernel are compared. In order to compensate the boundary effect, a new nonlocal kernel, i.e., the compensated two-phase kernel, is introduced, in which a local part is added to the nonlocal part of the two-phase kernel to account for the boundary effect. In contrast to the previously introduced modified kernel, the compensated two-phase kernel does not lead to a purely nonlocal behavior in the core region, and hence no singular behavior, and consequently, no computational convergence issue is observed. The nonlinear finite element approach and the COMSOL code are used to solve the coupled system of Ginzburg–Landau and local/nonlocal integral elasticity equations. The numerical implementation of the phase field-local elasticity equations and the 2D nonlocal integral elasticity are verified. Boundary effect is investigated for MPT with both homogeneous and nonhomogeneous stress distributions. For the former, in contrast to the local elasticity, a nonhomogeneous phase transformation (PT) occurs in the nonlocal case with the two-phase kernel. Using the compensated two-phase kernel results in a homogeneous PT similar to the local elasticity. For the latter, the sample transforms to martensite except the adjacent region to the boundary for the local elasticity, while for the two-phase kernel, the entire sample transforms to martensite. The solution of the compensated two-phase kernel, however, is very similar to that of the local elasticity. The applicability of boundary symmetry in phase field problems is also investigated, which shows that it leads to incorrect results within the nonlocal integral elasticity. This is because when the symmetric portions of a sample are removed, the corresponding nonlocal effects on the remaining portion are neglected and the symmetric boundaries violate the normalization condition. An example is presented in which the results of a complete model with the two-phase kernel are different from those of its one-fourth model. In contrast, the compensated two-phase kernel can generate similar solutions for both the complete and one-fourth models. However, in general, none of the nonlocal kernels can overcome this issue. Therefore, the symmetrical models are not recommended for nonlocal integral elasticity based phase field simulations of MPTs. The current study helps for a better study of nonlocal elasticity based phase field problems for various phenomena such as various PTs