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\begin{document}
\title{Phase field model for rapid solidification of Ni-superalloy}
\author[1]{Toni Ivas}%
\affil[1]{EPFL}%
\vspace{-1em}
\date{\today}
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\section*{Introduction}
{\label{231647}}\par\null
Nickel based superalloys are high temperature materials used for turbine
parts, airplane parts etc. CM247LC is alloy used for polycrystalline
turbine blades and it is strengthen by~ and gamma-prime precipitation.
Selective laser melting (SLM) is used to selectively melt the powder in
layer by layer fashion, and thus create 3D parts. Advantage of SLM to
more traditional manufacturing methods is freedom of the geometry and
possibility of very complex cooling channels, which is of course
interesting for high temperature applications. Heat treatment if often
used to reduce the residual stresses,~
\par\null
Phase field models are used to model microstructure evolution during the
solidification process. The application of the phase field model to
additive manufacturing was recently done by~\cite{Keller_2017} for
Inconel 625 Ni-superalloy.
\par\null
\subsection*{Numerical methods}
{\label{100663}}\par\null
We used the Abaqus for modeling the heat transfer in Additive
Manufacturing process. The heat transfer in SLM process is modelled
using the heat equation with Fouriers law and laser heat source:
\par\null
\begin{equation}\label{eq:heat}
\frac{\partial (\rho c_p T)}{\partial t} = \nabla (\kappa \nabla T) + S
\end{equation}
in which $\rho$ is density, $c_p$ is specific heat capacity, $\kappa$ is thermal conductivity and $T$ is the temperature of system. All material parameters are temperature dependent and were obtained from literature data (refs). The source term $S$ in Eq (\ref{eq:heat}) is modeled using the Goldak source term \cite{Goldak_1986} defined by:
\begin{equation}\label{eq:goldak}
S=\frac{6\sqrt{3}P\eta f}{abc \pi \sqrt{\pi}}exp[-(3(x+u*t)^2/a^2 +3(y+v*t)^2/b^2+3(z+w*t)^2/c^2)]
\end{equation}
Equation (\ref{eq:goldak}) represents the volumetric source were
$P$ is power of the laser, $\eta$ is process efficiency, and $x,y,z$ are coordinates of the double ellipsoid model; $a$,$b$,and $c$ are the ellipsoid axes
which represent width, depth and tail of the heat source. In most cases is $a$ axis set to deposition half width and $b$ to the melt pool depth. The velocities of the laser beam are denoted by $u,v,w$.
The initial condition of 298 K were assumed at time $t = 0$. The bottom and sides of the system shown in Fig.(\ref{}) were isolated and top surface we implemented the boundary conditions as:
\begin{equation}\label{eq:top}
(-\kappa \nabla T)*\bar{n} = h(T-T_e)+\epsilon\sigma(T^4-T^4_e)
\end{equation}
where $h$ represents the heat convection coefficient, $\epsilon$ is thermal radiation coefficient, and $\sigma$ is Stefan-Boltzmann constant.
The terms on the right side of Eq.(\ref{eq:top}) are heat convection loss due to flowing of the gas, and radiation loss due to Stefan-Boltzman law.
The interaction of Nd-YAG laser with wavelength ($\lambda$=1064 nm) used in our system (Concept M2) with the CM247LC is modeled using Eq.(\ref{eq:goldak}). The power of the laser in M2 system is $P$ = 200 W and laser beam diameter is 100 $\mu m$.
The build-up process of during the additive manufacturing process is modeled using so called "birth" element method as described in \cite{Michaleris_2014}. In this model the elements are inactive at start of the AM process and are subsequently activated as material is deposited to
the substrate. We use Abaqus user subroutine OUTVOLACTIVATED to activate the elements. The activation in SLM process is done after deposition of the powder layer.
\section*{~Calphad modeling}
{\label{580022}}\par\null
\section*{Finite dissipation model of rapid
solidification}
{\label{882699}}
The Phase-field model used in our study is based on the finite
dissipation model developed by Steinbach et al.~\cite{Zhang_2015,Steinbach_2012}. In
this model, the rapid phase transformations are modeled using two
concentrations fields which are linked by kinetic equation. The rate
exchange between two phase is controlled using so called interface
pemeability P.
\par\null
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