Phase-field study on effect of mechanical stresson dendrite fragmentation
H. Kashima, T. Takaki*, T. Fukui, K. Morinishi
Kyoto Institute of Technology, Matsugasaki, Sakyoku, Kyoto,606-8585 Japan*takaki@kit.ac.jp, http://www.cis.kit.ac.jp/~takaki/
NUMERICAL CONDITIONSINTROUCTION
Fundamental understanding of the relationship between solidification variables and the resulting
microstructure is essential for the development of optimum methods for quality casting. The
microstructures of casting product consist of columnar grains formed around the wall and equiaxed
grains formed in the middle of the mold. Generally, the equiaxed grains have more dominant effect on
the mechanical properties of the product than the columnar grains. Therefore, it is crucial for industrial
applications to understand the formation mechanism of equiaxed grain structure. Although themechanical fragmentation of columnar dendrite due to melt flow and gravity is thought to be a possible
nucleation mechanism of equiaxed grains, it has not been clarified yet.
In this study, we investigate stress concentration, which is the indicator of mechanicalfragmentation, occurred at dendrite neck due to gravity applying to the vertical direction of dendrite
growth direction. The coupling simulations by phase-field method and finite element method areperformed.

The left figure shows computational
domain and initial conditions for phasefield simulations of dendrite growth.
Initially, all regions are filled with theundercooled liquid and dimensionlessundercooling is set to be ΔT=0.3 andΔT=0.5 . One circular seed with radius
r=10Δx is putted on the origin (0,0) or theleft bottom corner. By considering
symmetrical shape of dendrite, a half
region in y-direction is simulated.
The information of phase-field istransferredtothefiniteelement
simulations to evaluate the stress
distributions due to gravity. The left figure
shows an example of finite element modeland boundary condition corresponding to
the dendritic morphology shown in leftupper figure.

Phase-field simulation

ΔT=0.3Finite element simulation

NUMERICAL MODELSThe dendrite growth from undercooled pure Ni is simulated by a quantitative phase-field model. Then,the calculated dendritic morphologies are transformed to the finite element simulation and themechanical stress due to gravity are simulated.

<parameter>< Phase-field method >Interface energy γ = 0.37 J/m2
Young’s modulus E ＊Poisson’s ratio ν = 0.3
Strength of anisotropy ζ = 0.1

Evolution equations of phase-field variable φ and temperature T⎡∂φ∂ ⎛ ∂a ∂φ ⎞ ∂ ⎛ ∂a ∂φ ⎞ 115L(T − Tm ) ⎫⎤2 ⎧⎟⎟ + ⎜ a= M φ ⎢a 2∇ 2φ − ⎜⎜ a(1 − φ 2 )⎟ + W (1 − φ )⎨φ −⎬⎥∂t∂∂∂∂∂∂θ
θ
xyyx
44WTm⎝⎠⎝⎠⎩⎭⎦⎣∂T
1 L ∂p= D∇ 2T +∂t
2 c p ∂t

φ : phase-field variable(φ = 1 in solid, φ = -1 in liquid)
Here,Surface anisotropy

a=1 3δγ,2 b

D :thermal diffusivity
cp: specific heat

L : latent heat

L :latent heat

CORRECT COEFFICIENT TO
EVALLUATE MAXIMUM STRESS

Cantilever
Diffuse interface FE simulation
φ=0γ : strength of anisotropy
k : mode numberθ : angle between surface normal and x-axis

a (θ ) = a{1 + ς cos(kθ )}Phase-field parametersΤ : temperaturea : gradient coefficientΤm : melting temperature＊ E (φ ) = 1 ⎛ tanh φ + 1⎞ E⎜⎟ s2⎝2τ
⎠Normal FE simulationa, W and M φcan be related to the material parameters by following equations.

W=6γb

δ

Mesh size
ΔT=0.5 Δx = 3nm
ΔT=0.3 Δx = 13nm
Time increment
Δt = Δx2/6D, Mφ =4 2b 2Tmκ
3a1a2δ 2 L2

δ : interface thickness
γ : interface energyκ : thermal conductivityb : constant related to δ

In the present finite element simulations withdiffuse interface, the maximum stress isunderestimated comparing to the normal
simulations where only solid region is meshed.
Therefore, the finite element results need to becorrected. So, as shown in upper figure, thecorrection coefficients are calculated by thesimple cantilever simulations beforehand.

< Finite element method >Finite element simulations are performed as liner elastic plane stress problem.

SIMULATION RESULTS

hunderestimate

Δx : Lattice size

DENDRITE MORPHOLOGIESRELATIONSHIPS BETWEEN DENDRITE
ARM LENGTH AND MAXIMUM STRESSAND STRESS DISTRIBUTIONS<cantilever model>

cantilever
FEMcantilever
FEMThe maximum stress in thecantilever model is calculated byσ max =close-up view

6 N∑ ( xi − x0 ) fi
wh 2 i =1

w: thickness of dendrite (w=1)
N : number of nodefi : nodal force

hxi

6μm
These figures show the dendrite morphologies and stress distributions at an identical length for ΔT=0.5 and 0.3. The radius of curvature and the minimum cross section are smaller for ΔT=0.5 than ΔT=0.3.
Therefore, the maximum equivalent stress in case of ΔT=0.5 is higher than that of ΔT=0.3.

x0CONCLUTIONS
These figures show the relationship between dendrite arm length and maximum stress at dendrite
neck and dendrite morphologies for each time step. From the dendritic morphologies, although theradius of curvature at the dendrite neck becomes larger with the dendrite growth, the ratio of it to therepresentative size of dendrite becomes small. Therefore, it is concluded that the dendritic morphology
approaches to a shape, which is easy to induce the stress concentration, with growth. As a result, thedifferences between the cantilever model and the present FE simulation increase with dendrite growth.

The stress concentration occurred at the neck of dendrite that grows to the vertical direction for gravity
was investigated by coupling phase-field method and finite element method. As a result, it was clearlyshown that the maximum stress at dendrite neck increase with dendrite growth and the increasing ratio ishigher for high undercooling condition than lower one. Furthermore, to evaluate the maximum stress inthe dendrite with high accuracy, the simple cantilever model was concluded to be not enough and finiteelement method which can directly treat the dendrite morphology is much more powerful.
The developed coupling model of phase-field method and finite element method is anticipated to be
applied to the study of mechanical dendrite fragmentation due to the gravity and melt flow.

FUTURE WORK
We will try to develop coupling model of phase-field method,
finite element method and Navier-Stokes equation, and to estimate
stress distribution in dendrite due to melt flow and gravity force.
The 16th International Conference
on Crystal Growth(ICCG-16)