XXII ICTAM, 25–29 August 2008, Adelaide, Australia

DEFORMATION SIMULATION UNDER CONSTANT VOLUME CONDITIONUSING PHASE FIELD CRYSTAL METHOD
Tomoyuki Hirouchi∗ , Tomohiro Takaki∗∗ & Yoshihiro Tomita∗∗ Graduate School of Engineering, Kobe University, Rokkodai, Nada, Kobe, 657-8501, Japan∗∗ Graduate School of Science and Technology, Kyoto Institute of Technology, Kyoto, 606-8585, Japan

Summary We proposed a new technique for the deformation simulation using the phase field crystal (PFC) method, which keeps
constant volume during deformation, and investigated the deformation behaviors of nano single crystal and bicrystal materials. We
confirmed that the PFC model could reproduce the deformation simulation under the constant volume condition. Furthermore, weobserved that grain boundary could be the source of dislocation emission and absorption.

INTRODUCTION
It is very important to predict macroscopic properties of industrial metallic materials depending on microstructures so as
to improve the properties and/or create novel functions of such materials. Recently, the phase field crystal (PFC) method[1, 2] is developed as a new numerical approach in atomic level. In this method, a free energy is minimized by a periodic
order parameter, which represents a local-averaged atomic density field. Therefore, this formulation can describe physicalsystems in diffusive time and interatomic length scales. In this sense, the PFC model is anticipated as a model that bridges
between a molecular description and a continuum field theory, or a multi-scale model.
In general, it is well known that metallic materials actually keep the volume constant during plastic deformation. However,in the previous PFC simulations of tensile deformation [2, 3], the volume of the polycrystalline structure increased during
the deformation simulation because it was difficult to set the boundary condition properly. Therefore, it is very important
to develop an appropriate numerical condition for PFC deformation simulation.
In this study, we propose a new technique applying deformation by controlling computational grid sizes. By applying
this technique, we study the deformation behaviors of nano single crystal and bicrystal structures to demonstrate basic
properties of the proposed technique.
PHASE FILED CRYSTAL METHOD
Unlike conventional phase field models, a free energy in the PFC model is minimized by a periodic order parameter. Thisorder parameter, or phase field ϕ, is defined as a local-time-averaged atomic density field. Therefore, the phase field is aconserved field. Figure 1(a) schematically shows the phase field profile for the PFC model. In the steady solid phase, ϕhas the periodic profile with a wavelength a to express atoms fixed in a periodic arrangement. Here, the local maxima of
ϕ correspond to atomic positions. On the other hand, atomic positions in the liquid phase depend on time. Therefore, ϕ in
the liquid phase becomes constant, ϕ = ϕ0 , as a local-time-averaged value.
Here, the dimensionless free energy functional F minimized by the phase field profile that generates a triangular lattice
structure is given by [1, 2],[]∫ϕϕ4
F = dV{α + (1 + ∇2 )2 }ϕ +(1)24where α is the driving force related to temperature. This free energy gives rise to a phase diagram of solid (triangular),liquid (constant), striped phases and the coexistence of each phase as shown in Fig. 1(b). Since ϕ is a conserved value,
the time evolution equation is derived from the functional derivative by the Chan-Hilliard equation,∂ϕ= ∇2 [{α + (1 + ∇2 )2 }ϕ + ϕ3 ]∂t(2)To perform deformation simulations using the PFC method, we propose a new technique enabling constant volume deformation by controlling computational grid sizes. Here, in the case of two-dimensional tensile deformation simulation,
the grid size in x-direction, ∆x, is increased and that in y-direction, ∆y, is decreased by following a volume constantcondition. Therefore, this technique enables constant volume deformation under periodic boundary condition.
RESULTS AND DISCCUSSIONFirst, to confirm the propriety of the proposed technique, we perform the uniaxial tensile deformation simulation of √thesingle crystal at constant strain rate. Computation is performed on a domain of 880 × 20 lattices with ∆x=∆y=4π/ 3(a/10). Model parameters are set to (α, ϕ0 ) = (-0.25, -0.285). Figure 2 shows atomic positions at the strained values.
The data clearly shows that the response of the system satisfies linear elasticity. Thus, we could confirm the proprietyof our technique. Second, to present the role of the grain boundary in plastic deformation, we perform the deformation
simulation of the bicrystal. Figure 3(a) shows the free energy contribution at the initial condition. Crystal orientations of20 and 40 degrees are selected to construct the high angle grain boundaries. Model parameters are set to (α, ϕ0 ) = (-0.25,-0.285). The boundary conditions are periodic conditions in all direction. In Fig. 3(a), we can confirm two dislocations
and grain boundaries as high energy. Figures. 3(b)-(d) show the deformation behavior of the bicrystal. As strain increases,the convex part of the grain boundary X in square region of Fig. 3(a) becomes sharp and the dislocation A is emitted fromthere in Fig. 3(b). We understand that this occur because of the stress concentration on there. Next, the dislocation A
is absorbed into the grain boundary Y as shown in Figs. 3(c)-(d). On the other hand, the grain boundary Y starts togenerate serration and to nucleate dislocations in Fig. 3(c). Thus, we observed that grain boundary could create andabsorb dislocations. These behaviors of grain boundary are proved in molecular dynamics simulations [4, 5]. Therefore,
we consider that the proposed technique can be used as the analysis tool simulating plastic deformation.
20Atomic position

Constant phase

1614

α-0.2

used parameter

Stripedphasearulng eia as
Tr phTria
ph nguas lareφ = φ0

Displacement

aInterface

12

Strain rate
5.17 10 -610864-0.4

Solid phase

Strain
0.03
0.02
0.01

1802

Liquid phase-0.4-0.2

00.2

0.4

0

Averaged density, φ0243648

607284Position

(b)

(a)

12Figure 1. (a) Phase field profile for PFC model and(b) phase diagram in two dimensional problem.

Figure 2. Atomic displacement distribution
under tension.

Crystal orientation

138.6π 600 lattices)

20deg

A

Grain boundary X
40deg

Grain boundary YA
20deg

138.6π (600 lattices)(a) ε = 0.000(b) ε = 0.075(c) ε = 0.085(d) ε = 0.100

0.021 0.025 0.029 0.033 0.037 0.041
free energy

Figure 3. (a) Initial condition of the bicrystal, and deformation state with (b) ε = 0.075, (c) ε = 0.085 and (d) ε = 0.100.

CONCLUSIONSIn this work, we proposed a new technique enabling constant volume deformation simulation using the PFC model andstudied the deformation behaviors of nanocrystalline materials to present the propriety of the proposed technique. In thecase of the single crystal, we confirmed the propriety from the elastic response of the system. Moreover, in the case of thebicrystal, we observed the emission and absorption of the dislocations from grain boundary. Therefore, we conclude that
this method enables us to simulate plastic deformation.
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[4] L. Capolungo, D. E. Spearot, M. Cherkaoui, D. L. McDowell, J. Qu and K. I. Jacob.: Dislocation nucleation from bicrystal interfaces and grainboundary ledges: Relationship to nanocrystalline deformation, J. Mech. Phys. Solids 55, 2300-2327, 2007
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