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\begin{document}

\title{Summary of constitutive\_phenoPowerlaw}


\author{YUN JO RO}

\maketitle
This document contains information for constitutive\_phenoPowerlaw.f90.
This constitutive subroutine is modified from the current contitutive\_phenomenological.f90.
We introduce slip and twin family as additional index (or input) for
each crystal structure in lattice.f90 subroutine (e.g., for HCP crystal:
slip and twin system has four faimilies, respectively). 


\section{State Variables in constitutive\_phenoPowelaw.f90}

The current State variables in constitutive\_phenoPowerlaw are {}``slip
resistance $\left(s^{\alpha}\right)$'', ''twin resistance $\left(s^{\beta}\right)$'',
{}``cumulative shear strain $\left(\gamma^{\alpha}\right)$'', and
{}``twin volume fraction $\left(f^{\beta}\right)$''. Superscript
$\alpha$ and $\beta$ denote to slip and twin systems, respectively,
in this entire document. 


\section{Considered Deformation Mechanisms}

Table \ref{Flo:DeformationSystemTable} lists slip/twin systems for
the {}``hex (hcp)'' case.\medskip{}


%
\begin{table}[tbph]
\centering{}\begin{tabular}{|c|c|c|c|}
\hline 
 &amp;  &amp;  &amp; No. of slip system\tabularnewline
\hline 
slip system &amp; basal &amp; $\left\{ 0001\right\} \left\langle 1\bar{2}10\right\rangle $ &amp; 3\tabularnewline
\cline{2-4} 
 &amp; prism &amp; $\left\{ 10\bar{1}0\right\} \left\langle 1\bar{2}10\right\rangle $ &amp; 3\tabularnewline
\cline{2-4} 
 &amp; pyr &lt;a&gt; &amp; $\left\{ 10\bar{1}1\right\} \left\langle 1\bar{2}10\right\rangle $ &amp; 6\tabularnewline
\cline{2-4} 
 &amp; pyr &lt;c+a&gt; &amp; $\left\{ 10\bar{1}1\right\} \left\langle 2\bar{1}\bar{1}3\right\rangle $ &amp; 12\tabularnewline
\hline
twin system &amp; tensile (T1) &amp; $\left\{ 10\bar{1}2\right\} \left\langle \bar{1}011\right\rangle $ &amp; 6\tabularnewline
\cline{2-4} 
 &amp; compressive (C1) &amp; $\left\{ 11\bar{2}2\right\} \left\langle 11\bar{2}\bar{3}\right\rangle $ &amp; 6\tabularnewline
\cline{2-4} 
 &amp; tensile (T2) &amp; $\left\{ 11\bar{2}1\right\} \left\langle \bar{1}\bar{1}26\right\rangle $ &amp; 6\tabularnewline
\cline{2-4} 
 &amp; compressive (C1) &amp; $\left\{ 10\bar{1}1\right\} \left\langle 10\bar{1}\bar{2}\right\rangle $ &amp; 6\tabularnewline
\hline
\end{tabular}\caption{Implemented deformation mechanims in $\alpha$-Ti }
\label{Flo:DeformationSystemTable}
\end{table}

\begin{itemize}
\item Slip/twin system for HCP are illustrated in Figures \ref{Fig:slipSystemHCP}
and \ref{Fig:twinSystemHCP}.
\end{itemize}
%
\begin{figure}
\begin{centering}
\includegraphics[clip,scale=0.25]{figures/slipSystemForHCP}
\par\end{centering}

\caption{Drawing for slip system for HCP. Burgers vectors were scaled.}
\label{Fig:slipSystemHCP}


\end{figure}
%
\begin{figure}
\begin{centering}
\includegraphics[clip,scale=0.25]{figures/twinSystemForHCP}
\par\end{centering}

\caption{Drawing for twin system for HCP ($\alpha$- Ti). Twin directions are
not scaled yet. }
\label{Fig:twinSystemHCP}


\end{figure}


\clearpage{}


\section{Kinetics}

Shear strain rate due to slip is described by following eqation \citet{Salem2005,Wu2007}:\begin{equation}
\dot{\gamma}^{\alpha}=\dot{\gamma_{o}}\left|\frac{\tau^{\alpha}}{s^{\alpha}}\right|^{n}sign\left(\tau^{\alpha}\right)\label{eq:slipStrainRate}\end{equation}


, where $\dot{\gamma}^{\alpha}$; shear strain rate, $\dot{\gamma}_{o}$;
reference shear strain rate, $\tau^{\alpha}$; resolved shear stress
on the slip system, $n$; stress exponent, and $s^{\alpha}$; slip
resistance.

Twin volume fraction rate is described by following eqation \citet{Salem2005,Wu2007}:

\begin{equation}
\dot{f}^{\beta}=\frac{\dot{\gamma_{o}}}{\gamma^{\beta}}\left|\frac{\tau^{\beta}}{s^{\beta}}\right|^{n}\mathbb{\mathcal{H}}\left(\tau^{\beta}\right)\label{eq:twinVolrate}\end{equation}


, where $\dot{f}^{\beta}$; twin volume fraction rate, $\dot{\gamma}_{o}$;
reference shear strain rate, $\gamma^{\beta}$;shear strain due to
mechanical twinning, $\tau^{\beta}$; resolved shear stress on the
twin system, and $s^{\beta}$; twin resistance. $\mathcal{H}$ is
Heaviside function. 


\section{Structure Evolution}

In this present section, we attempt to show how we establish the relationship
between the evolutoin of slip/twin resistance and the evolution of
shear strain/twin volume fraction. 


\subsection{Interaction matrix. }

Conceptual relationship between the evolution of state and kinetic
variables is shown in Equation \ref{eq:InteractionMatrix}.

\begin{equation}
\left[\begin{array}{c}
\dot{s}^{\alpha}\\
\dot{s}^{\beta}\end{array}\right]=\left[\begin{array}{cc}
M_{\mathrm{slip-slip}} &amp; M_{\mathrm{slip-twin}}\\
M_{\mathrm{twin-slip}} &amp; M_{\mathrm{twin-twin}}\end{array}\right]\left[\begin{array}{c}
\dot{\gamma}^{\alpha}\\
\gamma^{\beta}\cdot\dot{f}^{\beta}\end{array}\right]\label{eq:InteractionMatrix}\end{equation}


Four interaction martices are followings; i) slip-slip interaction
matrix $\left(M_{\mathrm{{\scriptstyle slip-slip}}}\right)$, ii)
slip-twin interaction matrix $\left(M_{\mathrm{slip-twin}}\right)$,
iii) twin-slip interaction matrix $\left(M_{\mathrm{twin-slip}}\right)$,
and iv) twin-twin interaction matrix $\left(M_{\mathrm{twin-twin}}\right)$. 

Detailed interaction type matrices in Equation \ref{eq:InteractionMatrix}
will be further discussed in the following Section.


\subsection{Interaction type matrix}

Following sections are sparated into four based on each interaction
type matrix alluded. Numbers in Tables \ref{Flo:SlipSlipIntTypeTable},
\ref{Flo:SlipTwinIntTypeTable}, \ref{Flo:TwinSlipIntTypeTable},
and \ref{Flo:TwinTwinIntTypeTable} denote the type of interaction
between deformation systems (The first column vs. The first row).


\subsubsection{Slip-Slip interaction type matrix}
\begin{itemize}
\item There are 20 types of slip-slip interaction as shown in Table \ref{Flo:SlipSlipIntTypeTable}. 
\item In Table \ref{Flo:SlipSlipIntTypeTable}, types of latent hardening
among slip systems are listed. 
\item Actual slip-slip interaction type matrix, $M_{\mathrm{slip-slip}}^{'}$,
is listed in Equation \ref{eq:SlipSlipIntMatrix}.
\end{itemize}
%
\begin{table}[H]
\begin{centering}
\begin{tabular}{|&gt;{\centering}m{0.8in}|&gt;{\centering}m{0.7in}|&gt;{\centering}m{0.6in}|&gt;{\centering}m{0.6in}|&gt;{\centering}m{0.7in}|}
\hline 
 &amp; basal &amp; prism &amp; pyr &lt;a&gt; &amp; pyr&lt;c+a&gt;\tabularnewline
\hline 
basal &amp; 1, 5 &amp; 9 &amp; 12 &amp; 14\tabularnewline
\hline 
prism &amp; 15 &amp; 2, 6 &amp; 10 &amp; 13\tabularnewline
\hline 
pyr &lt;a&gt; &amp; 18 &amp; 16 &amp; 3, 7 &amp; 11\tabularnewline
\hline 
pyr &lt;c+a&gt; &amp; 20 &amp; 19 &amp; 17 &amp; 4, 8\tabularnewline
\hline
\end{tabular}
\par\end{centering}

\caption{Slip-slip interaction type}
\label{Flo:SlipSlipIntTypeTable}
\end{table}


\begin{equation}
M_{\mathrm{slip-slip}}^{'}=\left[\begin{array}{ccc|ccc|cccccc|cccccccccccc}
1 &amp; 5 &amp; 5 &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot\\
 &amp; 1 &amp; 5 &amp; \cdot &amp; 9 &amp; \cdot &amp; \cdot &amp; \cdot &amp; 12 &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; 14 &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot\\
 &amp;  &amp; 1 &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot\\
\hline \cdot &amp; \cdot &amp; \cdot &amp; 2 &amp; 6 &amp; 6 &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot\\
\cdot &amp; 15 &amp; \cdot &amp;  &amp; 2 &amp; 6 &amp; \cdot &amp; \cdot &amp; 10 &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; 13 &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot\\
\cdot &amp; \cdot &amp; \cdot &amp;  &amp;  &amp; 2 &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot\\
\hline \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; 3 &amp; 7 &amp; 7 &amp; 7 &amp; 7 &amp; 7 &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot\\
\cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp;  &amp; 3 &amp; 7 &amp; 7 &amp; 7 &amp; 7 &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot\\
\cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp;  &amp;  &amp; 3 &amp; 7 &amp; 7 &amp; 7 &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; 11 &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot\\
\cdot &amp; 18 &amp; \cdot &amp; \cdot &amp; 16 &amp; \cdot &amp;  &amp;  &amp;  &amp; 3 &amp; 7 &amp; 7 &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot\\
\cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp;  &amp;  &amp;  &amp;  &amp; 3 &amp; 7 &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot\\
\cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp;  &amp;  &amp;  &amp;  &amp;  &amp; 3 &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot\\
\hline \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; 4 &amp; 8 &amp; 8 &amp; 8 &amp; 8 &amp; 8 &amp; 8 &amp; 8 &amp; 8 &amp; 8 &amp; 8 &amp; 8\\
\cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp;  &amp; 4 &amp; 8 &amp; 8 &amp; 8 &amp; 8 &amp; 8 &amp; 8 &amp; 8 &amp; 8 &amp; 8 &amp; 8\\
\cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp;  &amp;  &amp; 4 &amp; 8 &amp; 8 &amp; 8 &amp; 8 &amp; 8 &amp; 8 &amp; 8 &amp; 8 &amp; 8\\
\cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp;  &amp;  &amp;  &amp; 4 &amp; 8 &amp; 8 &amp; 8 &amp; 8 &amp; 8 &amp; 8 &amp; 8 &amp; 8\\
\cdot &amp; 20 &amp; \cdot &amp; \cdot &amp; 19 &amp; \cdot &amp; \cdot &amp; \cdot &amp; 17 &amp; \cdot &amp; \cdot &amp; \cdot &amp;  &amp;  &amp;  &amp;  &amp; 4 &amp; 8 &amp; 8 &amp; 8 &amp; 8 &amp; 8 &amp; 8 &amp; 8\\
\cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp;  &amp;  &amp;  &amp;  &amp;  &amp; 4 &amp; 8 &amp; 8 &amp; 8 &amp; 8 &amp; 8 &amp; 8\\
\cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp;  &amp;  &amp;  &amp;  &amp;  &amp;  &amp; 4 &amp; 8 &amp; 8 &amp; 8 &amp; 8 &amp; 8\\
\cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp;  &amp;  &amp;  &amp;  &amp;  &amp;  &amp;  &amp; 4 &amp; 8 &amp; 8 &amp; 8 &amp; 8\\
\cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp;  &amp;  &amp;  &amp;  &amp;  &amp;  &amp;  &amp;  &amp; 4 &amp; 8 &amp; 8 &amp; 8\\
\cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp;  &amp;  &amp;  &amp;  &amp;  &amp;  &amp;  &amp;  &amp;  &amp; 4 &amp; 8 &amp; 8\\
\cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp;  &amp;  &amp;  &amp;  &amp;  &amp;  &amp;  &amp;  &amp;  &amp;  &amp; 4 &amp; 8\\
\cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp;  &amp;  &amp;  &amp;  &amp;  &amp;  &amp;  &amp;  &amp;  &amp;  &amp;  &amp; 4\end{array}\right]\label{eq:SlipSlipIntMatrix}\end{equation}


\vfill{}
\vfill{}



\subsubsection{Slip-Twin interaction type matrix}
\begin{itemize}
\item There are 16 types of slip-twin interaction in Table \ref{Flo:SlipTwinIntTypeTable}. 
\item Meaning of T1, C1, T2, C2 is listed in Table \ref{Flo:DeformationSystemTable}.
\item Actual slip-twin interaction type matrix, $M_{\mathrm{slip-twin}}^{'}$,
is listed in Equation \ref{eq:SlipTwinIntMatrix}.
\end{itemize}
%
\begin{table}[H]
\begin{centering}
\begin{tabular}{|&gt;{\centering}m{0.8in}|&gt;{\centering}m{0.7in}|&gt;{\centering}m{0.6in}|&gt;{\centering}m{0.6in}|&gt;{\centering}m{0.7in}|}
\hline 
 &amp; T1 &amp; C1 &amp; T2 &amp; C1\tabularnewline
\hline 
basal &amp; 1 &amp; 2 &amp; 3 &amp; 4\tabularnewline
\hline 
prism &amp; 5 &amp; 6 &amp; 7 &amp; 8\tabularnewline
\hline 
pyr &lt;a&gt; &amp; 9 &amp; 10 &amp; 11 &amp; 12\tabularnewline
\hline 
pyr &lt;c+a&gt; &amp; 13 &amp; 14 &amp; 15 &amp; 16\tabularnewline
\hline
\end{tabular}
\par\end{centering}

\caption{Slip-twin interaction type}
\label{Flo:SlipTwinIntTypeTable}
\end{table}


\begin{equation}
M_{\mathrm{slip-twin}}^{'}=\left[\begin{array}{c|c|c|c}
1 &amp; 2 &amp; 3 &amp; 4\\
\hline 5 &amp; 6 &amp; 7 &amp; 8\\
\hline 9 &amp; 10 &amp; 11 &amp; 12\\
\hline 13 &amp; 14 &amp; 15 &amp; 16\end{array}\right]\label{eq:SlipTwinIntMatrix}\end{equation}



\subsubsection{Twin-Slip interaction type matrix}
\begin{itemize}
\item There 16 types of twin-slip interaction in Table \ref{Flo:TwinSlipIntTypeTable}.
\item Meaning of T1, C1, T2, C2 is listed in Table \ref{Flo:DeformationSystemTable}.
\item Actual twin-slip interaction type matrix, $M_{\mathrm{twin-slip}}^{'}$,
is listed in Equation \ref{eq:TwinSlipIntMatrix}.
\end{itemize}
%
\begin{table}[H]
\begin{centering}
\begin{tabular}{|&gt;{\centering}m{0.8in}|&gt;{\centering}m{0.7in}|&gt;{\centering}m{0.6in}|&gt;{\centering}m{0.6in}|&gt;{\centering}m{0.7in}|}
\hline 
 &amp; basal &amp; prism &amp; pyr &lt;a&gt; &amp; pyr &lt;c+a&gt;\tabularnewline
\hline 
T1 &amp; 1 &amp; 5 &amp; 9 &amp; 13\tabularnewline
\hline 
C1 &amp; 2 &amp; 6 &amp; 10 &amp; 14\tabularnewline
\hline 
T2 &amp; 3 &amp; 7 &amp; 11 &amp; 15\tabularnewline
\hline 
C2 &amp; 4 &amp; 8 &amp; 12 &amp; 16\tabularnewline
\hline
\end{tabular}
\par\end{centering}

\caption{Twin-slip interaction type}
\label{Flo:TwinSlipIntTypeTable}
\end{table}


\begin{equation}
M_{\mathrm{twin-slip}}^{'}=\left[\begin{array}{c|c|c|c}
1 &amp; 5 &amp; 9 &amp; 13\\
\hline 2 &amp; 6 &amp; 10 &amp; 14\\
\hline 3 &amp; 7 &amp; 11 &amp; 15\\
\hline 4 &amp; 8 &amp; 12 &amp; 16\end{array}\right]\label{eq:TwinSlipIntMatrix}\end{equation}



\subsubsection{Twin-twin interaction type matrix}
\begin{itemize}
\item There are 20 types of twin-twin interaction as shown in Table \ref{Flo:TwinTwinIntTypeTable}. 
\item In Table \ref{Flo:TwinTwinIntTypeTable}, types of latent hardening
among twin systems are listed. 
\item Actual twin-twin interaction type marix, $M_{\mathrm{twin-twin}}^{'}$,
is listed in Equation \ref{eq:TwinTwinIntMatrix}.
\end{itemize}
%
\begin{table}[H]
\begin{centering}
\begin{tabular}{|&gt;{\centering}m{0.8in}|&gt;{\centering}m{0.7in}|&gt;{\centering}m{0.6in}|&gt;{\centering}m{0.6in}|&gt;{\centering}m{0.7in}|}
\hline 
 &amp; T1 &amp; C1 &amp; T2 &amp; C2\tabularnewline
\hline 
T1 &amp; 1, 5 &amp; 9 &amp; 12 &amp; 14\tabularnewline
\hline 
C1 &amp; 15 &amp; 2, 6 &amp; 10 &amp; 13\tabularnewline
\hline 
T2 &amp; 18 &amp; 16 &amp; 3, 7 &amp; 11\tabularnewline
\hline 
C2 &amp; 20 &amp; 19 &amp; 17 &amp; 4, 8\tabularnewline
\hline
\end{tabular}
\par\end{centering}

\caption{Twin-twin interaction type}
\label{Flo:TwinTwinIntTypeTable}
\end{table}


\begin{equation}
M_{\mathrm{twin-twin}}^{'}=\left[\begin{array}{cccccc|cccccc|cccccc|cccccc}
1 &amp; 5 &amp; 5 &amp; 5 &amp; 5 &amp; 5 &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot\\
 &amp; 1 &amp; 5 &amp; 5 &amp; 5 &amp; 5 &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot\\
 &amp;  &amp; 1 &amp; 5 &amp; 5 &amp; 5 &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot\\
 &amp;  &amp;  &amp; 1 &amp; 5 &amp; 5 &amp; \cdot &amp; \cdot &amp; \cdot &amp; 9 &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; 12 &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; 14 &amp; \cdot &amp; \cdot\\
 &amp;  &amp;  &amp;  &amp; 1 &amp; 5 &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot\\
 &amp;  &amp;  &amp;  &amp;  &amp; 1 &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot\\
\hline \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; 2 &amp; 6 &amp; 6 &amp; 6 &amp; 6 &amp; 6 &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot\\
\cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp;  &amp; 2 &amp; 6 &amp; 6 &amp; 6 &amp; 6 &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot\\
\cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp;  &amp;  &amp; 2 &amp; 6 &amp; 6 &amp; 6 &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot\\
\cdot &amp; \cdot &amp; \cdot &amp; 15 &amp; \cdot &amp; \cdot &amp;  &amp;  &amp;  &amp; 2 &amp; 6 &amp; 6 &amp; \cdot &amp; \cdot &amp; \cdot &amp; 10 &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; 13 &amp; \cdot &amp; \cdot\\
\cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp;  &amp;  &amp;  &amp;  &amp; 2 &amp; 6 &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot\\
\cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp;  &amp;  &amp;  &amp;  &amp;  &amp; 2 &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot\\
\hline \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; 3 &amp; 7 &amp; 7 &amp; 7 &amp; 7 &amp; 7 &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot\\
\cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp;  &amp; 3 &amp; 7 &amp; 7 &amp; 7 &amp; 7 &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot\\
\cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp;  &amp;  &amp; 3 &amp; 7 &amp; 7 &amp; 7 &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot\\
\cdot &amp; \cdot &amp; \cdot &amp; 18 &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; 16 &amp; \cdot &amp; \cdot &amp;  &amp;  &amp;  &amp; 3 &amp; 7 &amp; 7 &amp; \cdot &amp; \cdot &amp; \cdot &amp; 11 &amp; \cdot &amp; \cdot\\
\cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp;  &amp;  &amp;  &amp;  &amp; 3 &amp; 7 &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot\\
\cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp;  &amp;  &amp;  &amp;  &amp;  &amp; 3 &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot\\
\hline \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; 4 &amp; 8 &amp; 8 &amp; 8 &amp; 8 &amp; 8\\
\cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp;  &amp; 4 &amp; 8 &amp; 8 &amp; 8 &amp; 8\\
\cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp;  &amp;  &amp; 4 &amp; 8 &amp; 8 &amp; 8\\
\cdot &amp; \cdot &amp; \cdot &amp; 20 &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; 19 &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; 17 &amp; \cdot &amp; \cdot &amp;  &amp;  &amp;  &amp; 4 &amp; 8 &amp; 8\\
\cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp;  &amp;  &amp;  &amp;  &amp; 4 &amp; 8\\
\cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp; \cdot &amp;  &amp;  &amp;  &amp;  &amp;  &amp; 4\end{array}\right]\label{eq:TwinTwinIntMatrix}\end{equation}



\subsection{Prefactor (nonlinear factor)}


\subsubsection{Prefactors for slip resistance $\left(s^{\alpha}\right)$; $M_{\mathrm{slip-slip}}$
and $M_{\mathrm{slip-twin}}$\citet{Wu2007}}

$M_{\mathrm{slip-slip}}$ and $M_{\mathrm{slip-twin}}$ use for slip
resistance evolution $\left(\dot{s}^{\alpha}\right)$. Equation \ref{eq:SlipResisEvolutionEq}
is for a slip resistance rate evolution. This currently shows the
prefactor for {}``slip-slip interaction matrix, $M_{\mathrm{slip-slip}}$''.

\medskip{}


\begin{equation}
M_{\mathrm{slip-slip}}=h_{\mathrm{slip}}\left(1+C\cdot F^{b}\right)\left(1-\frac{s^{\alpha}}{s_{so}^{\alpha}+s_{\mathrm{pr}}\cdot\sqrt{F}}\right)\cdot M_{\mathrm{slip-slip}}^{'}\label{eq:SlipResisEvolutionEq}\end{equation}


\medskip{}


, where $h_{\mathrm{slip}}$represent a hardening rate, and $S_{\mathrm{so}}^{\alpha}$
saturation slip resistance for slip system without mechanical twinning
$\left(\sum_{\beta}f^{\beta}=0\right)$, respectively. And, $F$ is
$\sum_{\beta}f^{\beta}$, and $N^{S}$is the total number of slip
system.$C$, $s_{\mathrm{pr}}$, and $b$ are coefficients to introduce
the effect of interaction between slip and mechanical twin in Equation
\ref{eq:SlipResisEvolutionEq}.
\begin{itemize}
\item Slip-twin interaction matrix, $M_{\mathrm{slip-twin}}$, has not been
implemented with any prefactor in the present version. 
\end{itemize}

\subsubsection{Prefactors for twin resistance $\left(s^{\beta}\right)$; $M_{\mathrm{twin-slip}}$
and $M_{\mathrm{twin-twin}}$\citet{Salem2005}}

$M_{\mathrm{twin-sli}p}$ and $M_{\mathrm{twin-twin}}$ use for twin
resistance evolution $\left(\dot{s}^{\beta}\right)$. Twin-twin and
twin-slip interaction matrices are described in Equations \ref{eq:TwinTwinContributionToTwinResis}
and \ref{eq:TwinSlipContributionToTwinResis}. \medskip{}


\begin{equation}
M_{\mathrm{twin-twin}}=h_{\mathrm{tw}}\cdot F^{d}\cdot M_{\mathrm{twin-twin}}^{'}\label{eq:TwinTwinContributionToTwinResis}\end{equation}


,where $h_{\mathrm{tw}}$ and $d$ are coefficients for twin-twin
contribution. $F$ is $\sum_{\beta}f^{\beta}$.

\medskip{}


\begin{equation}
M_{\mathrm{twin-slip}}=h_{\mathrm{tw-sl}}\cdot\Gamma^{e}\cdot M_{\mathrm{twin-slip}}^{'}\label{eq:TwinSlipContributionToTwinResis}\end{equation}


,where $h_{\mathrm{tw-sl}}$ and $e$ are coefficients for twin-slip
contribution, and $\Gamma=\sum_{\alpha}\gamma^{\alpha}$.

\clearpage{}


\section{Material Parameters (Material Configuration file)}

%
\begin{figure}[tbph]
\begin{centering}
\includegraphics[clip,scale=0.8]{figures/ExpectedMaterialConfigFile}\caption{Expected of phenomenological modelling parameters.}
\label{Fig:ModelParameters}
\par\end{centering}


\end{figure}

\begin{itemize}
\item The sequence for hardening coefficients in Figure \ref{Fig:ModelParameters}
is the sequence of numbering in Tables \ref{Flo:SlipSlipIntTypeTable},
\ref{Flo:SlipTwinIntTypeTable}, \ref{Flo:TwinSlipIntTypeTable},
and \ref{Flo:TwinTwinIntTypeTable} above.
\end{itemize}
\clearpage{}

\bibliographystyle{plain}
\addcontentsline{toc}{section}{\refname}\bibliography{MPIEyjr}

\end{document}
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