[1] Simulation of damage evolution in composites: A phase-field model

S B Biner and S Y Yu

In this study, a phase-field model is introduced for the simulation of damage evolution in composite systems. The damage evolution in discontinuously reinforced and laminated composites, including elastic–plastic cases, was studied parametrically in order to establish its viability. The algorithm offers significant advantages to describe the microstructure and topological changes associated with the damage evolution in comparison to conventional simulation algorithms, due to the absence of formal meshing. The known damage mechanisms for specific composite systems naturally evolve from the algorithm with a simple evolution equation.

N Ng et al

Ferroelectric nanostructures can be formed by local switching of domains using techniques such as piezo-force microscopy (PFM). Understanding the dependence of the switching behavior on the lateral size of the electrode is important to determine the minimum feature size for writing ferroelectric nanostructures. To understand these lateral size effects, we use the time-dependent Ginzburg–Landau equations in a two-dimensional square to rectangle ferroelectric transition to simulate localized switching of domains for PFM-type and parallel-plate capacitor configurations. Our investigations indicate that fringing electric fields lead to switching via intermediate 90° domains even in the absence of substrate or clamping effects for films of sufficient thicknesses, and via 180° rotations at smaller thicknesses. The voltage required to switch the domain increases by decreasing the lateral size, and at very small lateral sizes the coercive voltage becomes so large that it becomes virtually impossible to switch the domain.

[3] Phase field modeling of grain boundary migration with solute drag

J Li et al

The grain boundary (GB) motion in the presence of GB segregation is investigated by means of phase field simulations. It is found that the solute concentration at the moving GB may increase with increasing velocity and becomes larger than the equilibrium value, which is unexpected according to the solute drag theory proposed by Cahn, but has been observed in some experiments. A non-linear relation between the driving force (curvature) and the GB velocity is found in two cases: (1) the GB motion undergoes a transition from the low-velocity extreme to the high-velocity extreme; (2) the GB migrates slowly in a strongly segregating system. The first case is consistent with the solute drag theory of Cahn. As for the second case, which is unexpected according to solute drag theory, the non-linear relation between the GB velocity and curvature comes from two sources: the non-linear relation of the solute drag force with GB velocity, and the variation in GB energy with curvature. It is also found that, when the diffusivity is spatially inhomogeneous, the kinetics of GB motion is different from that with a constant diffusivity.

B-K Yoon et al

The effect of an intergranular amorphous film on grain growth behavior has been studied in a faceted model system, BaTiO3. We prepared two kinds of samples with and without intergranular amorphous films but with the same grain size and density. During annealing the samples at 1350 °C in air, abnormal grain growth occurred in samples with intergranular amorphous films while grain growth was inhibited in samples with dry boundaries, indicating the presence of a pinning force in samples with dry boundaries. To compare the mobilities of dry and wet boundaries, single crystal and polycrystal bilayer samples with or without amorphous films were prepared and annealed at 1340 °C. In contrast to the observed grain growth behavior in polycrystals, the growth of the single crystal into the polycrystal with dry boundaries was faster than that into the polycrystal with wet boundaries, demonstrating the higher mobility of a dry boundary, unlike the conventional understanding.

## A phase field review

### February 18, 2009

A review with emphasis on solidification problems:

The phase field technique for modeling multiphase materials

I Singer-Loginova and H M Singer

Abstract. This paper reviews methods and applications of the phase field technique, one of the fastest growing areas in computational materials science. The phase field method is used as a theory and computational tool for predictions of the evolution of arbitrarily shaped morphologies and complex microstructures in materials. In this method, the interface between two phases (e.g. solid and liquid) is treated as a region of finite width having a gradual variation of different physical quantities, i.e. it is a diffuse interface model. An auxiliary variable, the phase field or order parameter , is introduced, which distinguishes one phase from the other. Interfaces are identified by the variation of the phase field. We begin with presenting the physical background of the phase field method and give a detailed thermodynamical derivation of the phase field equations. We demonstrate how equilibrium and non-equilibrium physical phenomena at the phase interface are incorporated into the phase field methods. Then we address in detail dendritic and directional solidification of pure and multicomponent alloys, effects of natural convection and forced flow, grain growth, nucleation, solid–solid phase transformation and highlight other applications of the phase field methods. In particular, we review the novel phase field crystal model, which combines atomistic length scales with diffusive time scales. We also discuss aspects of quantitative phase field modeling such as thin interface asymptotic analysis and coupling to thermodynamic databases. The phase field methods result in a set of partial differential equations, whose solutions require time-consuming large-scale computations and often limit the applicability of the method. Subsequently, we review numerical approaches to solve the phase field equations and present a finite difference discretization of the anisotropic Laplacian operator.

## Gauge theory of dislocations

### February 17, 2009

The gauge theory of dislocations: Static solutions of screw and edge dislocations

M Lazar and C Anastassiadis

We investigate the T(3)-gauge theory of static dislocations in continuous solids. We use the most general linear constitutive relations in terms of the elastic distortion tensor and dislocation density tensor for the force and pseudomoment stresses of an isotropic solid. The constitutive relations contain six material parameters. In this theory, both the force and pseudomoment stresses are asymmetric. The theory possesses four characteristic lengths ℓ1, ℓ2, ℓ3 and ℓ4, which are given explicitly. We first derive the three-dimensional Green tensor of the master equation for the force stresses in the translational gauge theory of dislocations. We then investigate the situation of generalized plane strain (anti-plane strain and plane strain). Using the stress function method, we find modified stress functions for screw and edge dislocations. The solution of the screw dislocation is given in terms of one independent length ℓ1 = ℓ4. For the problem of an edge dislocation, only two characteristic lengths ℓ2 and ℓ3 arise with one of them being the same ℓ2 = ℓ1 as for the screw dislocation. Thus, this theory possesses only two independent lengths for generalized plane strain. If the two lengths ℓ2 and ℓ3 of an edge dislocation are equal, we obtain an edge dislocation, which is the gauge theoretical version of a modified Volterra edge dislocation. In the case of symmetric stresses, we recover well-known results obtained earlier.

## Improved phase field microelasticity theory

### February 12, 2009

Y Shen et al

The three-dimensional phase field microelasticity theory for elastically and structurally inhomogeneous solids is improved with a simple and efficient damped iterative method. This method can be used to obtain the effective stress-free strain distribution that fully determines the stress and strain fields in the elastically and structurally inhomogeneous solids, or directly obtain the strain field from the equilibrium equation.

Got to implement this some time!

## Measurement of interfacial energies

### February 12, 2009

D Watanabe, C Watanabe and R Monzen

The coarsening theory of a spherical particle in a ternary alloy developed by Kuehmann and Voorhees (KV) has been generalized to any centro-symmetric particle. A classical thermodynamic analysis reveals that the generalized KV theory enables us to estimate the interface energy of a particle with a fixed shape, even if the shape of the particle is not controlled by minimization of the interface energy. Data on the coarsening of spherical, {0 0 1}-faceted cuboidal and {1 1 1}-faceted octahedral precipitates in a Cu–Co alloy, a Cu–Fe alloy, and three Cu–Co–Fe alloys with different Co and Fe contents during aging at 873–973 K have been collected by transmission electron microscopy and electrical resistivity. By applying the generalized KV theory to the experimental data, the energies of sphere, {0 0 1} and {1 1 1} interfaces have been determined. Their energies increase with increasing the Fe composition in the alloys.

## Observation of dislocation nucleation and escape

### February 10, 2009

A paper in the recent Nature Materials (via iMechanica):

In situ observation of dislocation nucleation and escape in a submicrometre aluminium single crystal

S H Oh et al‘Smaller is stronger’ does not hold true only for nanocrystalline materials but also for single crystals. It is argued that this effect is caused by geometrical constraints on the nucleation and motion of dislocations in submicrometre-sized crystals. Here, we report the first in situ transmission electron microscopy tensile tests of a submicrometre aluminium single crystal that are capable of providing direct insight into source-controlled dislocation plasticity in a submicrometre crystal. Single-ended sources emit dislocations that escape the crystal before being able to multiply. As dislocation nucleation and loss rates are counterbalanced at about 0.2 events per second, the dislocation density remains statistically constant throughout the deformation at strain rates of about 10

^{-4}s^{-1}. However, a sudden increase in strain rate to 10^{-3}s^{-1}causes a noticeable surge in dislocation density as the nucleation rate outweighs the loss rate. This observation indicates that the deformation of submicrometre crystals is strain-rate sensitive.