%% LyX 1.6.2 created this file. For more info, see http://www.lyx.org/. %% Do not edit unless you really know what you are doing. \documentclass[english]{scrartcl} \usepackage[T1]{fontenc} \usepackage[latin9]{inputenc} \usepackage[letterpaper]{geometry} \geometry{verbose,tmargin=2cm,bmargin=2cm,lmargin=2cm,rmargin=2cm,headheight=2cm,headsep=1cm,footskip=1cm} \usepackage{array} \usepackage{float} \usepackage{endnotes} \usepackage{graphicx} \usepackage{setspace} \usepackage{amssymb} \usepackage[authoryear]{natbib} \onehalfspacing %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% LyX specific LaTeX commands. %% Because html converters don't know tabularnewline \providecommand{\tabularnewline}{\\} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Textclass specific LaTeX commands. \let\footnote=\endnote \usepackage{braille} \newcommand{\braillenormal}[1] {\setlength{\brailleunit}{2.4mm}\braille{#1}} % With \brailleunit == 0.75ex, the braille letters will % approximately match the other letters in size. \newcommand{\brailletext}[1] {\setlength{\brailleunit}{0.75ex}\braille{#1}} \usepackage{babel} \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 & & & No. of slip system\tabularnewline \hline slip system & basal & $\left\{ 0001\right\} \left\langle 1\bar{2}10\right\rangle $ & 3\tabularnewline \cline{2-4} & prism & $\left\{ 10\bar{1}0\right\} \left\langle 1\bar{2}10\right\rangle $ & 3\tabularnewline \cline{2-4} & pyr & $\left\{ 10\bar{1}1\right\} \left\langle 1\bar{2}10\right\rangle $ & 6\tabularnewline \cline{2-4} & pyr & $\left\{ 10\bar{1}1\right\} \left\langle 2\bar{1}\bar{1}3\right\rangle $ & 12\tabularnewline \hline twin system & tensile (T1) & $\left\{ 10\bar{1}2\right\} \left\langle \bar{1}011\right\rangle $ & 6\tabularnewline \cline{2-4} & compressive (C1) & $\left\{ 11\bar{2}2\right\} \left\langle 11\bar{2}\bar{3}\right\rangle $ & 6\tabularnewline \cline{2-4} & tensile (T2) & $\left\{ 11\bar{2}1\right\} \left\langle \bar{1}\bar{1}26\right\rangle $ & 6\tabularnewline \cline{2-4} & compressive (C1) & $\left\{ 10\bar{1}1\right\} \left\langle 10\bar{1}\bar{2}\right\rangle $ & 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}} & M_{\mathrm{slip-twin}}\\ M_{\mathrm{twin-slip}} & 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}{|>{\centering}m{0.8in}|>{\centering}m{0.7in}|>{\centering}m{0.6in}|>{\centering}m{0.6in}|>{\centering}m{0.7in}|} \hline & basal & prism & pyr & pyr\tabularnewline \hline basal & 1, 5 & 9 & 12 & 14\tabularnewline \hline prism & 15 & 2, 6 & 10 & 13\tabularnewline \hline pyr & 18 & 16 & 3, 7 & 11\tabularnewline \hline pyr & 20 & 19 & 17 & 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 & 5 & 5 & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot\\ & 1 & 5 & \cdot & 9 & \cdot & \cdot & \cdot & 12 & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & 14 & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot\\ & & 1 & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot\\ \hline \cdot & \cdot & \cdot & 2 & 6 & 6 & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot\\ \cdot & 15 & \cdot & & 2 & 6 & \cdot & \cdot & 10 & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & 13 & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot\\ \cdot & \cdot & \cdot & & & 2 & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot\\ \hline \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & 3 & 7 & 7 & 7 & 7 & 7 & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot\\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & & 3 & 7 & 7 & 7 & 7 & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot\\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & & & 3 & 7 & 7 & 7 & \cdot & \cdot & \cdot & \cdot & \cdot & 11 & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot\\ \cdot & 18 & \cdot & \cdot & 16 & \cdot & & & & 3 & 7 & 7 & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot\\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & & & & & 3 & 7 & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot\\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & & & & & & 3 & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot\\ \hline \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & 4 & 8 & 8 & 8 & 8 & 8 & 8 & 8 & 8 & 8 & 8 & 8\\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & & 4 & 8 & 8 & 8 & 8 & 8 & 8 & 8 & 8 & 8 & 8\\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & & & 4 & 8 & 8 & 8 & 8 & 8 & 8 & 8 & 8 & 8\\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & & & & 4 & 8 & 8 & 8 & 8 & 8 & 8 & 8 & 8\\ \cdot & 20 & \cdot & \cdot & 19 & \cdot & \cdot & \cdot & 17 & \cdot & \cdot & \cdot & & & & & 4 & 8 & 8 & 8 & 8 & 8 & 8 & 8\\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & & & & & & 4 & 8 & 8 & 8 & 8 & 8 & 8\\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & & & & & & & 4 & 8 & 8 & 8 & 8 & 8\\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & & & & & & & & 4 & 8 & 8 & 8 & 8\\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & & & & & & & & & 4 & 8 & 8 & 8\\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & & & & & & & & & & 4 & 8 & 8\\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & & & & & & & & & & & 4 & 8\\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & & & & & & & & & & & & 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}{|>{\centering}m{0.8in}|>{\centering}m{0.7in}|>{\centering}m{0.6in}|>{\centering}m{0.6in}|>{\centering}m{0.7in}|} \hline & T1 & C1 & T2 & C1\tabularnewline \hline basal & 1 & 2 & 3 & 4\tabularnewline \hline prism & 5 & 6 & 7 & 8\tabularnewline \hline pyr & 9 & 10 & 11 & 12\tabularnewline \hline pyr & 13 & 14 & 15 & 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 & 2 & 3 & 4\\ \hline 5 & 6 & 7 & 8\\ \hline 9 & 10 & 11 & 12\\ \hline 13 & 14 & 15 & 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}{|>{\centering}m{0.8in}|>{\centering}m{0.7in}|>{\centering}m{0.6in}|>{\centering}m{0.6in}|>{\centering}m{0.7in}|} \hline & basal & prism & pyr & pyr \tabularnewline \hline T1 & 1 & 5 & 9 & 13\tabularnewline \hline C1 & 2 & 6 & 10 & 14\tabularnewline \hline T2 & 3 & 7 & 11 & 15\tabularnewline \hline C2 & 4 & 8 & 12 & 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 & 5 & 9 & 13\\ \hline 2 & 6 & 10 & 14\\ \hline 3 & 7 & 11 & 15\\ \hline 4 & 8 & 12 & 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}{|>{\centering}m{0.8in}|>{\centering}m{0.7in}|>{\centering}m{0.6in}|>{\centering}m{0.6in}|>{\centering}m{0.7in}|} \hline & T1 & C1 & T2 & C2\tabularnewline \hline T1 & 1, 5 & 9 & 12 & 14\tabularnewline \hline C1 & 15 & 2, 6 & 10 & 13\tabularnewline \hline T2 & 18 & 16 & 3, 7 & 11\tabularnewline \hline C2 & 20 & 19 & 17 & 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 & 5 & 5 & 5 & 5 & 5 & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot\\ & 1 & 5 & 5 & 5 & 5 & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot\\ & & 1 & 5 & 5 & 5 & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot\\ & & & 1 & 5 & 5 & \cdot & \cdot & \cdot & 9 & \cdot & \cdot & \cdot & \cdot & \cdot & 12 & \cdot & \cdot & \cdot & \cdot & \cdot & 14 & \cdot & \cdot\\ & & & & 1 & 5 & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot\\ & & & & & 1 & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot\\ \hline \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & 2 & 6 & 6 & 6 & 6 & 6 & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot\\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & & 2 & 6 & 6 & 6 & 6 & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot\\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & & & 2 & 6 & 6 & 6 & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot\\ \cdot & \cdot & \cdot & 15 & \cdot & \cdot & & & & 2 & 6 & 6 & \cdot & \cdot & \cdot & 10 & \cdot & \cdot & \cdot & \cdot & \cdot & 13 & \cdot & \cdot\\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & & & & & 2 & 6 & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot\\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & & & & & & 2 & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot\\ \hline \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & 3 & 7 & 7 & 7 & 7 & 7 & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot\\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & & 3 & 7 & 7 & 7 & 7 & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot\\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & & & 3 & 7 & 7 & 7 & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot\\ \cdot & \cdot & \cdot & 18 & \cdot & \cdot & \cdot & \cdot & \cdot & 16 & \cdot & \cdot & & & & 3 & 7 & 7 & \cdot & \cdot & \cdot & 11 & \cdot & \cdot\\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & & & & & 3 & 7 & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot\\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & & & & & & 3 & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot\\ \hline \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & 4 & 8 & 8 & 8 & 8 & 8\\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & & 4 & 8 & 8 & 8 & 8\\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & & & 4 & 8 & 8 & 8\\ \cdot & \cdot & \cdot & 20 & \cdot & \cdot & \cdot & \cdot & \cdot & 19 & \cdot & \cdot & \cdot & \cdot & \cdot & 17 & \cdot & \cdot & & & & 4 & 8 & 8\\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & & & & & 4 & 8\\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & & & & & & 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}