Interaction between phase transformations and dislocations at the nanoscale. Part 2: Phase field simulation examples

Abstract

The complete system of phase field equations for coupled martensitic phase transformations (PTs), dislocation evolution, and mechanics at large strains is presented. Finite element method (FEM) is utilized to solve this system for two important problems. The first one is related to the simulation of shear strain-induced PT at the evolving dislocation pile-ups in a nanosized bicrystal. Plasticity plays a dual part in the interaction with PT. Dislocation pile-ups produce strong stress tensor concentrators that lead to barrierless martensite (M) nucleation. On the other hand, plasticity in the transforming grain relaxes these stress concentrators suppressing PT. The final stationary M morphology is governed by the local thermodynamic equilibrium, either at the interfaces or in terms of stresses averaged over the martensitic region or the entire grain. This is very surprising because of strong heterogeneity of stress fields and is in contrast to previous statements that phase equilibrium conditions do not enter the description of strain-induced PTs. The second problem is devoted to martensitic plate propagation through a bicrystal during temperature-induced PT. For elastic growth (without dislocations) and a large thermal driving force, a complex transformation path with plate branching and direct and reverse PTs is observed, which still ends with the same stationary nanostructure as for a smaller driving force and a traditional transformation path. Sharp grain boundary arrests plate growth at a relatively small driving force, exhibiting an athermal friction. For elastoplastic growth, the generation of dislocations produces athermal friction and arrests the plate below some critical driving force, leading to a morphological transition from plate to lath M. The width of the martensitic plate increases in comparison with elastic growth * Corresponding author. Email address: vlevitas@iastate.edu, tel. (515)-294-9691 (V. I. Levitas) 2 due to internal stress relaxation. Plate growth is accompanied by the nucleation of dislocations within M and remaining in M, the nucleation of dislocations at the tip of a plate and spreading them in austenite (A), and passing some dislocations through M, then through a M-A interface, and then through A. Due to the existence of a stationary equilibrium M nanostructure and concentration for each temperature, for a large enough observation time one observes athermal, rate-and time-independent kinetics, even while local kinetics is rate dependent. In the final structure, most dislocations are in M despite its having a yield strength three times larger than for A, which is consistent with experiments.

Phase field simulations of plastic strain-induced phase transformations under high pressure and large shear

“…In fact, the uniaxial compression reduces the PT pressure in comparison with hydrostatic conditions without shear, see, e.g., [91,94,96,99]. Alternative boundary conditions-free lateral surfaces-, have been applied for an exploratory study of the shearinduced PT at zero pressure in [8]. Still, results obtained in [64], again due to their generic character, were applied in [103] for the interpretation of experiments on high pressure torsion, in [112] on PTs in nanomaterials under mechanical loading, and in [111] on shock-induced amorphization in Si.…”

“…We will utilize a theory for interaction between PTs and dislocations developed in [62] and corresponding numerical approach presented in [8]. This theory synergistically combines fully geometrically nonlinear theory for martensitic PTs [32] and dislocations [60] and corresponding numerical approaches presented in [45,61].…”

“…This theory synergistically combines fully geometrically nonlinear theory for martensitic PTs [32] and dislocations [60] and corresponding numerical approaches presented in [45,61]. The complete system of equations from [8,62] is presented in Box 1. Also, for those who need to solve similar problems for infinitesimal strains, a simplified small strain version is presented as well.…”

“…Dislocational plasticity significantly changes PT thermodynamics, nucleation and growth kinetics, microstructure, and even type of PT. For example, from one side dislocations relax elastic stresses that appear due to heterogeneous distribution of the transformation strain, which promotes PT [2,4,6,8]. From the other side, dislocations produce an athermal threshold for interface propagation and can arrest growth of martensite and lead to morphological transition from plate to lath martensite [1,2,9].…”

“…An additional interaction between dislocations and PT occurs through stress fields generated by their eigen strains and is determined by a solution of coupled PFA and mechanical problems. This theory was applied in [8,59,63,64] to solutions of some material problems, including revealing an athermal hysteresis which depends on the ratio of the phase interface width and the magnitude of the Burgers vector of a dislocation, finding a mechanism of semicoherent interface motion, dislocation inheritance by propagating phase interface, and the temperature-induced nucleation, growth, and arrest of M plate in an A bicrystal. Some earlier PFAs on interaction of PTs and discrete dislocations include analytical [65] and numerical [66,67] solutions for M nucleation on dislocations, which were introduced through their stationary stress fields, or which belong to the moving phase interface only [68] and consequently do not involve phase field equations for dislocations and their inheritance during PT.…”