A quantitative multi-phase field model of polycrystalline alloy solidification

N Ofori-Opoku and N Provatas

A multi-phase field model for quantitative simulations of polycrystalline solidification of binary alloys is introduced. During the free-growth stage of solidification, the model exploits the thin-interface analysis developed by Karma [3] in order to realistically capture bulk phase diffusion and the sharp interface corrections predicted by traditional models of solidification. During grain boundary coalescence, the model is constructed to reproduce the properties of repulsive grain boundaries described by Rappaz et al. [29]. The model provides a very simple mechanism for decoupling of solute and concentration fields at steady state, an important feature for calculating grain boundary energies.

Use of the Frank–Bilby equation for calculating misfit dislocation arrays in interfaces

J B Yang et al

The Frank–Bilby equation has been utilized to develop a general approach in which a simple criterion is proposed to classify interfaces with discrete misfit dislocation arrays into four types so that these misfit dislocation arrays can be conveniently characterized with uniform formulae. The relation connecting misfit dislocation configurations with the matrices in the Frank–Bilby equation, the special interfaces consistent with the Δg-parallelism and the continuity of misfit dislocations at intersection edges of interfaces are discussed.

The mechanical size effect as a mean-field breakdown phenomenon: Example of microscale single crystal beam bending

E Demir et al

Single crystalline copper beams with thicknesses between 0.7 and 5 μm are manufactured with a focused ion beam technique and bent in a nanoindenter. The yield strengths of the beams show a mechanical size effect (smaller-is-stronger). The geometrically necessary dislocation (GND) densities estimated from misorientation maps do not explain the observed size effect. Also, accumulation of GNDs principally requires pre-straining. We hence introduce a mean-field breakdown theory and generalize it to small-scale mechanical tests other than bending. The mean-field breakdown limit is defined in terms of a microstructural correlation measure (characteristic dislocation bow-out length) below which the local availability of dislocation sources and not the density of GNDs dominates the mechanical size effect. This explains why a size dependence can occur for samples that are not pre-strained (by using a very small critical strain to define the yield strength). After pre-straining, when GNDs build up, they can contribute to the flow stress. The mean-field breakdown theory can also explain the large scatter typically observed in small-scale mechanical tests as the availability of sufficiently soft sources at scales around or below the correlation length does not follow statistical laws but highly depends on the position where the probe is taken.

Kinetics of isothermal phase transformations above and below the peritectic temperature: Phase-field simulations

G Boussinot et al

We present phase-field simulations of isothermal phase transformations in the peritectic system below and above the peritectic temperature. The physical processes involved are of different natures, involving either a triple junction or a liquid-film-migration (LFM) mechanism. Below the peritectic temperature, one of the solid phases steadily grows along the other. Above the peritectic temperature the phase transformation proceeds via the LFM mechanism. To the best of our knowledge, this mechanism has not been discussed in the literature as a generic process of phase transitions in peritectic systems. In addition to the LFM process, we also simulate melting along the solid–solid interface. Finally, we make a simplified linear stability analysis of the liquid film, supporting our simulation results.

Phase field modeling of defects and deformation

Y Wang and J Li

New perspectives on the phase field approach in modeling deformation and fracture at the fundamental defect level are reviewed. When applied at sub-angstrom length scales the phase field crystal (PFC) model is able to describe thermally averaged atomic configurations of defects and defect processes on diffusional timescales. When applied at individual defect levels the microscopic phase field (MPF) model is a superset of the Cahn–Hilliard description of chemical inhomogeneities and the Peierls (cohesive zone) description of displacive inhomogeneities. A unique feature associated with the MPF model is its ability to predict fundamental properties of individual defects such as size, formation energy, saddle point configuration and activation energy of defect nuclei, and the micromechanisms of their mutual interactions, directly using ab initio calculations as model inputs. When applied at the mesoscopic level the coarse grained phase field (CGPF) models have the ability to predict the evolution of microstructures consisting of a large assembly of both chemically and mechanically interacting defects through coupled displacive and diffusional mechanisms. It is noted that the purpose of the MPF model is fundamentally different from that of the CGPF models. The latter have been used primarily to study microstructural evolution with user-supplied linear response rate laws, defect energies and mobilities. Combined phase field simulations hold great promise in modeling deformation and fracture with complex microstructural and chemical interactions.

Morphological instabilities and alignment of lamellar eutectics during directional solidification under a strong magnetic field

X Li et al

The effects of a strong magnetic field on Al–Al2Cu and Pb–Sn lamellar eutectics during directional solidification have been investigated experimentally. The results show that the application of a strong magnetic field caused “tilting and oscillatory” morphological instabilities and deformation of the eutectic lamellae. Moreover, it was found that the Al–Al2Cu eutectic grain became aligned under a strong magnetic field and that with an increase in the magnetic field intensity this alignment was gradually enhanced. Further, the stresses caused by the magnetization force and the thermoelectric magnetic force during directional solidification under a strong magnetic field were analyzed and it was found that they are likely responsible for the “tilting and oscillatory” morphological instabilities and deformation. This is experimental evidence that the stresses imposed on a solid are capable of inducing the morphological instabilities of lamellar eutectics. The magnetic crystalline anisotropy of the Al2Cu phases and the growth relationship between the primary Al2Cu phase and the eutectic phases was investigated and it was found that the Al2Cu phase had a remarkable magnetic crystalline anisotropy which determined the growth of the Al–Al2Cu eutectic grain. Thus, alignment of the Al–Al2Cu eutectic grain under a strong magnetic field may be attributed to the magnetic crystalline anisotropy of the Al2Cu phase. Based on the growth behaviour of Al–Al2Cu and Pb–Sn lamellar eutectics under a strong magnetic field, an alignment model of lamellar eutectics during directional solidification under a strong magnetic field is proposed.

Overview of constitutive laws, kinematics, homogenization and multiscale methods in crystal plasticity finite-element modeling: Theory, experiments, applications

F Roters et al

This article reviews continuum-based variational formulations for describing the elastic–plastic deformation of anisotropic heterogeneous crystalline matter. These approaches, commonly referred to as crystal plasticity finite-element models, are important both for basic microstructure-based mechanical predictions as well as for engineering design and performance simulations involving anisotropic media. Besides the discussion of the constitutive laws, kinematics, homogenization schemes and multiscale approaches behind these methods, we also present some examples, including, in particular, comparisons of the predictions with experiments. The applications stem from such diverse fields as orientation stability, microbeam bending, single-crystal and bicrystal deformation, nanoindentation, recrystallization, multiphase steel (TRIP) deformation, and damage prediction for the microscopic and mesoscopic scales and multiscale predictions of rolling textures, cup drawing, Lankfort (r) values and stamping simulations for the macroscopic scale.