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slight polishing of documentation for phenopowerlaw

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documentation/ConstitutiveLaw/phenoPowerlaw/ConstitutivePhenoPowerLaw.pdf
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documentation/ConstitutiveLaw/phenoPowerlaw/ConstitutivePhenoPowerLaw.tex
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%% LyX 1.6.2 created this file. For more info, see http://www.lyx.org/. %% 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. %% Do not edit unless you really know what you are doing.
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bookmarksnumbered=true,% generate bookmarks with numbers
pagebackref=true,% generate backref in biblio
colorlinks=true,%
linkcolor=DarkBlue,citecolor=DarkBlue,urlcolor=DarkBlue%
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\usepackage{babel}
\begin{document} \begin{document}
\title{Summary of constitutive\_phenoPowerlaw} \title{Summary of constitutive\_phenoPowerlaw}
\author{YunJo Ro \and Philip Eisenlohr}
\author{YUN JO RO}
\maketitle \maketitle
\begin{abstract}
This document contains information for constitutive\_phenoPowerlaw.f90. This document contains information for constitutive\_phenoPowerlaw.f90.
This constitutive subroutine is modified from the current contitutive\_phenomenological.f90. This constitutive subroutine is modified from the current contitutive\_phenomenological.f90.
We introduce slip and twin family as additional index (or input) for We introduce slip and twin family as additional index (or input) for
each crystal structure in lattice.f90 subroutine (e.g., for HCP crystal: each crystal structure in lattice.f90 subroutine (e.g., for HCP crystal:
slip and twin system has four faimilies, respectively). slip and twin system has four families, respectively).
\end{abstract}
\section{State Variables in constitutive\_phenoPowerlaw.f90}
\section{State Variables in constitutive\_phenoPowelaw.f90}
The current State variables in constitutive\_phenoPowerlaw are {}``slip The current State variables in constitutive\_phenoPowerlaw are {}``slip
resistance $\left(s^{\alpha}\right)$'', ''twin resistance $\left(s^{\beta}\right)$'', resistance $\left(s^{\alpha}\right)$'', ''twin resistance $\left(s^{\beta}\right)$'',
@ -64,65 +66,86 @@ the {}``hex (hcp)'' case.\medskip{}
% %
\begin{table}[tbph] \begin{table}[tbph]
\centering{}\begin{tabular}{|c|c|c|c|} \centering
\hline \begin{tabular}{cccc}
& & & No. of slip system\tabularnewline \toprule
\hline \textbf{type} & \textbf{system} & \textbf{plane / direction} & \textbf{multiplicity}\\
slip system & basal & $\left\{ 0001\right\} \left\langle 1\bar{2}10\right\rangle $ & 3\tabularnewline \midrule
\cline{2-4} slip & basal & $\left\{ 0001\right\} \left\langle 1\bar{2}10\right\rangle $ & 3\\
& prism & $\left\{ 10\bar{1}0\right\} \left\langle 1\bar{2}10\right\rangle $ & 3\tabularnewline & prism & $\left\{ 10\bar{1}0\right\} \left\langle 1\bar{2}10\right\rangle $ & 3\\
\cline{2-4} & pyr \hkl<a> & $\left\{ 10\bar{1}1\right\} \left\langle 1\bar{2}10\right\rangle $ & 6\\
& pyr <a> & $\left\{ 10\bar{1}1\right\} \left\langle 1\bar{2}10\right\rangle $ & 6\tabularnewline & pyr \hkl<c+a> & $\left\{ 10\bar{1}1\right\} \left\langle 2\bar{1}\bar{1}3\right\rangle $ & 12\\
\cline{2-4} \midrule
& pyr <c+a> & $\left\{ 10\bar{1}1\right\} \left\langle 2\bar{1}\bar{1}3\right\rangle $ & 12\tabularnewline twin & T1 & $\left\{ 10\bar{1}2\right\} \left\langle \bar{1}011\right\rangle $ & 6\\
\hline & C1 & $\left\{ 11\bar{2}2\right\} \left\langle 11\bar{2}\bar{3}\right\rangle $ & 6\\
twin system & tensile (T1) & $\left\{ 10\bar{1}2\right\} \left\langle \bar{1}011\right\rangle $ & 6\tabularnewline & T2 & $\left\{ 11\bar{2}1\right\} \left\langle \bar{1}\bar{1}26\right\rangle $ & 6\\
\cline{2-4} & C2 & $\left\{ 10\bar{1}1\right\} \left\langle 10\bar{1}\bar{2}\right\rangle $ & 6\\
& compressive (C1) & $\left\{ 11\bar{2}2\right\} \left\langle 11\bar{2}\bar{3}\right\rangle $ & 6\tabularnewline \bottomrule
\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 } \end{tabular}\caption{Implemented deformation mechanims in $\alpha$-Ti }
\label{Flo:DeformationSystemTable} \label{Flo:DeformationSystemTable}
\end{table} \end{table}
\begin{itemize} Slip/twin system for HCP are illustrated in Figures \ref{fig: dislocation slip systems}
\item Slip/twin system for HCP are illustrated in Figures \ref{Fig:slipSystemHCP} and \ref{fig: twinning systems}.
and \ref{Fig:twinSystemHCP}.
\end{itemize}
% %..............FIG...............
% === SEM ===
\begin{figure} \begin{figure}
\begin{centering} \centering
\includegraphics[clip,scale=0.25]{figures/slipSystemForHCP} \subfloat[Basal \hkl<a> slip]{%
\par\end{centering} \label{fig: dislocation slip basal}%
\includegraphics{slipSystem_basal}}
\caption{Drawing for slip system for HCP. Burgers vectors were scaled.} \quad
\label{Fig:slipSystemHCP} \subfloat[Prismatic \hkl<a> slip]{%
\label{fig: dislocation slip prism}%
\includegraphics{slipSystem_prismA}}
\quad
\subfloat[Pyramidal \hkl<a> slip]{%
\label{fig: dislocation slip pyramidal a}%
\includegraphics{slipSystem_pyrA}}
\quad
\subfloat[Pyramidal \hkl<c+a> slip]{%
\label{fig: dislocation slip pyramidal ca}%
\includegraphics{slipSystem_pyrCA}}
\quad
\caption{
Dislocation slip systems considered for hexagonal lattice structure.}
\label{fig: dislocation slip systems}
\end{figure} \end{figure}
% %...................................
%..............FIG...............
% === SEM ===
\begin{figure} \begin{figure}
\begin{centering} \centering
\includegraphics[clip,scale=0.25]{figures/twinSystemForHCP} \subfloat[Extension (T1)]{%
\par\end{centering} \label{fig: twin T1}%
\includegraphics{twinSystem_T1}}
\caption{Drawing for twin system for HCP ($\alpha$- Ti). Twin directions are \quad
not scaled yet. } \subfloat[Contraction (C1)]{%
\label{Fig:twinSystemHCP} \label{fig: twin C1}%
\includegraphics{twinSystem_C1}}
\quad
\subfloat[Extension (T2)]{%
\label{fig: twin T2}%
\includegraphics{twinSystem_T2}}
\quad
\subfloat[Contraction (C2)]{%
\label{fig: twin C2}%
\includegraphics{twinSystem_C2}}
\quad
\caption{
Mechanical twinning systems considered for hexagonal lattice structure. Burgers vectors are not drawn to scale.}
\label{fig: twinning systems}
\end{figure} \end{figure}
%...................................
\clearpage{}
\section{Kinetics} \section{Kinetics}
Shear strain rate due to slip is described by following eqation \citet{Salem2005,Wu2007}:\begin{equation} Shear strain rate due to slip is described by following equation \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} \dot{\gamma}^{\alpha}=\dot{\gamma_{o}}\left|\frac{\tau^{\alpha}}{s^{\alpha}}\right|^{n}sign\left(\tau^{\alpha}\right)\label{eq:slipStrainRate}\end{equation}
@ -131,7 +154,7 @@ reference shear strain rate, $\tau^{\alpha}$; resolved shear stress
on the slip system, $n$; stress exponent, and $s^{\alpha}$; slip on the slip system, $n$; stress exponent, and $s^{\alpha}$; slip
resistance. resistance.
Twin volume fraction rate is described by following eqation \citet{Salem2005,Wu2007}: Twin volume fraction rate is described by following equation \citet{Salem2005,Wu2007}:
\begin{equation} \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} \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}
@ -196,22 +219,19 @@ is listed in Equation \ref{eq:SlipSlipIntMatrix}.
% %
\begin{table}[H] \begin{table}[H]
\begin{centering} \begin{centering}
\begin{tabular}{|>{\centering}m{0.8in}|>{\centering}m{0.7in}|>{\centering}m{0.6in}|>{\centering}m{0.6in}|>{\centering}m{0.7in}|} \begin{tabular}{ccccc}
\hline \toprule
& basal & prism & pyr <a> & pyr<c+a>\tabularnewline & basal & prism & pyr \hkl<a> & pyr\hkl<c+a>\\
\hline \midrule
basal & 1, 5 & 9 & 12 & 14\tabularnewline basal & 1, 5 & 9 & 12 & 14\\
\hline prism & 15 & 2, 6 & 10 & 13\\
prism & 15 & 2, 6 & 10 & 13\tabularnewline pyr \hkl<a> & 18 & 16 & 3, 7 & 11\\
\hline pyr \hkl<c+a> & 20 & 19 & 17 & 4, 8\\
pyr <a> & 18 & 16 & 3, 7 & 11\tabularnewline \bottomrule
\hline
pyr <c+a> & 20 & 19 & 17 & 4, 8\tabularnewline
\hline
\end{tabular} \end{tabular}
\par\end{centering} \par\end{centering}
\caption{Slip-slip interaction type} \caption{Slip--slip interaction type}
\label{Flo:SlipSlipIntTypeTable} \label{Flo:SlipSlipIntTypeTable}
\end{table} \end{table}
@ -259,18 +279,15 @@ is listed in Equation \ref{eq:SlipTwinIntMatrix}.
% %
\begin{table}[H] \begin{table}[H]
\begin{centering} \begin{centering}
\begin{tabular}{|>{\centering}m{0.8in}|>{\centering}m{0.7in}|>{\centering}m{0.6in}|>{\centering}m{0.6in}|>{\centering}m{0.7in}|} \begin{tabular}{ccccc}
\hline \toprule
& T1 & C1 & T2 & C1\tabularnewline & T1 & C1 & T2 & C1\\
\hline \midrule
basal & 1 & 2 & 3 & 4\tabularnewline basal & 1 & 2 & 3 & 4\\
\hline prism & 5 & 6 & 7 & 8\\
prism & 5 & 6 & 7 & 8\tabularnewline pyr \hkl<a> & 9 & 10 & 11 & 12\\
\hline pyr \hkl<c+a> & 13 & 14 & 15 & 16\\
pyr <a> & 9 & 10 & 11 & 12\tabularnewline \bottomrule
\hline
pyr <c+a> & 13 & 14 & 15 & 16\tabularnewline
\hline
\end{tabular} \end{tabular}
\par\end{centering} \par\end{centering}
@ -298,18 +315,15 @@ is listed in Equation \ref{eq:TwinSlipIntMatrix}.
% %
\begin{table}[H] \begin{table}[H]
\begin{centering} \begin{centering}
\begin{tabular}{|>{\centering}m{0.8in}|>{\centering}m{0.7in}|>{\centering}m{0.6in}|>{\centering}m{0.6in}|>{\centering}m{0.7in}|} \begin{tabular}{ccccc}
\hline \toprule
& basal & prism & pyr <a> & pyr <c+a>\tabularnewline & basal & prism & pyr \hkl<a> & pyr \hkl<c+a>\\
\hline \midrule
T1 & 1 & 5 & 9 & 13\tabularnewline T1 & 1 & 5 & 9 & 13\\
\hline C1 & 2 & 6 & 10 & 14\\
C1 & 2 & 6 & 10 & 14\tabularnewline T2 & 3 & 7 & 11 & 15\\
\hline C2 & 4 & 8 & 12 & 16\\
T2 & 3 & 7 & 11 & 15\tabularnewline \bottomrule
\hline
C2 & 4 & 8 & 12 & 16\tabularnewline
\hline
\end{tabular} \end{tabular}
\par\end{centering} \par\end{centering}
@ -338,18 +352,15 @@ is listed in Equation \ref{eq:TwinTwinIntMatrix}.
% %
\begin{table}[H] \begin{table}[H]
\begin{centering} \begin{centering}
\begin{tabular}{|>{\centering}m{0.8in}|>{\centering}m{0.7in}|>{\centering}m{0.6in}|>{\centering}m{0.6in}|>{\centering}m{0.7in}|} \begin{tabular}{ccccc}
\hline \toprule
& T1 & C1 & T2 & C2\tabularnewline & T1 & C1 & T2 & C2\\
\hline \midrule
T1 & 1, 5 & 9 & 12 & 14\tabularnewline T1 & 1, 5 & 9 & 12 & 14\\
\hline C1 & 15 & 2, 6 & 10 & 13\\
C1 & 15 & 2, 6 & 10 & 13\tabularnewline T2 & 18 & 16 & 3, 7 & 11\\
\hline C2 & 20 & 19 & 17 & 4, 8\\
T2 & 18 & 16 & 3, 7 & 11\tabularnewline \bottomrule
\hline
C2 & 20 & 19 & 17 & 4, 8\tabularnewline
\hline
\end{tabular} \end{tabular}
\par\end{centering} \par\end{centering}
@ -454,7 +465,7 @@ contribution, and $\Gamma=\sum_{\alpha}\gamma^{\alpha}$.
% %
\begin{figure}[tbph] \begin{figure}[tbph]
\begin{centering} \begin{centering}
\includegraphics[clip,scale=0.8]{figures/ExpectedMaterialConfigFile}\caption{Expected of phenomenological modelling parameters.} \includegraphics[clip,scale=0.6]{figures/ExpectedMaterialConfigFile}\caption{Expected of phenomenological modelling parameters.}
\label{Fig:ModelParameters} \label{Fig:ModelParameters}
\par\end{centering} \par\end{centering}
@ -467,9 +478,9 @@ is the sequence of numbering in Tables \ref{Flo:SlipSlipIntTypeTable},
\ref{Flo:SlipTwinIntTypeTable}, \ref{Flo:TwinSlipIntTypeTable}, \ref{Flo:SlipTwinIntTypeTable}, \ref{Flo:TwinSlipIntTypeTable},
and \ref{Flo:TwinTwinIntTypeTable} above. and \ref{Flo:TwinTwinIntTypeTable} above.
\end{itemize} \end{itemize}
\clearpage{}
\bibliographystyle{plain} \clearpage{}
\addcontentsline{toc}{section}{\refname}\bibliography{MPIEyjr} \bibliographystyle{plainnat}
\bibliography{MPIEyjr}
\end{document} \end{document}

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