diff --git a/trunk/documentation/ConstitutiveLaw/powerLaw/ConstitutivePhenoPowerLaw.lyx b/trunk/documentation/ConstitutiveLaw/powerLaw/ConstitutivePhenoPowerLaw.lyx index 08d6f52b1..99d44e275 100644 --- a/trunk/documentation/ConstitutiveLaw/powerLaw/ConstitutivePhenoPowerLaw.lyx +++ b/trunk/documentation/ConstitutiveLaw/powerLaw/ConstitutivePhenoPowerLaw.lyx @@ -74,7 +74,15 @@ logical.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). - The current State variables in constitutive_phenoPowerlaw are + +\end_layout + +\begin_layout Section +State Variables in constitutive_phenoPowelaw.f90 +\end_layout + +\begin_layout Standard +The current State variables in constitutive_phenoPowerlaw are \begin_inset Quotes eld \end_inset @@ -132,7 +140,15 @@ twin volume fraction \end_inset denote to slip and twin systems, respectively, in this entire document. - Table + +\end_layout + +\begin_layout Section +Considered Deformation Mechanisms +\end_layout + +\begin_layout Standard +Table \begin_inset CommandInset ref LatexCommand ref reference "Flo:DeformationSystemTable" @@ -823,7 +839,7 @@ reference "eq:InteractionMatrix" 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}\\ -\dot{f}^{\beta}\end{array}\right]\label{eq:InteractionMatrix}\end{equation} +\gamma^{\beta}\cdot\dot{f}^{\beta}\end{array}\right]\label{eq:InteractionMatrix}\end{equation} \end_inset diff --git a/trunk/documentation/ConstitutiveLaw/powerLaw/ConstitutivePhenoPowerLaw.pdf b/trunk/documentation/ConstitutiveLaw/powerLaw/ConstitutivePhenoPowerLaw.pdf index adf6c4a92..7fbb36896 100644 Binary files a/trunk/documentation/ConstitutiveLaw/powerLaw/ConstitutivePhenoPowerLaw.pdf and b/trunk/documentation/ConstitutiveLaw/powerLaw/ConstitutivePhenoPowerLaw.pdf differ diff --git a/trunk/documentation/ConstitutiveLaw/powerLaw/ConstitutivePhenoPowerLaw.tex b/trunk/documentation/ConstitutiveLaw/powerLaw/ConstitutivePhenoPowerLaw.tex new file mode 100644 index 000000000..9b33ae827 --- /dev/null +++ b/trunk/documentation/ConstitutiveLaw/powerLaw/ConstitutivePhenoPowerLaw.tex @@ -0,0 +1,475 @@ +%% 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}