Added summary of the constitutive model for mechanical twinning as implemented in DisloTwin.f90 (both pdf and corresponding tex source)

documentation/ConstitutiveLaw/DisloTwinLaw/Tex Files/[06_10_2009]_MSU_TwinMeeting.tex

@@ -22,16 +22,24 @@
 \usetheme{mpie}
 \setbeamertemplate{blocks}[rounded][shadow=true]
 
-\title{Dislocation structure and kinetics in slip-twin model}
+\title{Dislocation glide and deformation twinning as implemented in DisloTwin.f90}
 \date{MSU Twin Meeting, D\"usseldorf -- October 6\textsuperscript{th}, 2009}
 
 \begin{document}
 
 \frame{\titlepage}
 
-\frame {
+\section[Outline]{}
- \frametitle{Dislocation structure parametrization}
+\frame{\tableofcontents}
 
+\section{Microstructure Parametrization}
+\frame {
+ \frametitle{}
+ \begin{block}{\begin{center}PART I\end{center}} \begin{center}Microstructure Parametrization\end{center} \end{block}
+}
+\subsection{Dislocation structure}
+\frame {
+ \frametitle{Dislocation structure}
  \begin{block}{Internal variables:}
   \begin{itemize}
   \item<1-> $\slip N$ edge dislocation densities $\slip\varrho_{\text{edge}}$
@@ -41,24 +49,46 @@
 
  \begin{block}{Derived measures:}
   \begin{itemize}
-  \item<1-> $\slip\tau_{\mathrm{c}}$ threshold shear stress
+  \item<1-> $\slip\tau_{\text{c}}$ threshold shear stress fordislocation glide
   \item<1-> $\slip\lambda$ mean distance between 2 obstacles seen by a dislocation
   \end{itemize}
  \end{block}
 }
 
+\subsection{Mechanical twins}
 \frame {
- \frametitle{Dislocation structure parametrization}
+ \frametitle{Morphology and topology of mechanical twins}
+ \begin{block}{Internal variables:}
+  \begin{itemize}
+  \item<1-> $\twin N$ twin volume fractions $\twin f$
+  \item<1-> ($\twin N$ twin mean thicknesses $\twin s$)
+  \end{itemize}
+ \end{block}
 
- \begin{block}{Threshold stress $\slip\tau$:}
+ \begin{block}{Derived measures:}
+  \begin{itemize}
+  \item<1-> $f$ total twin volume fraction
+  \item<1-> $\twin l=\frac{\twin s\,(1-f)}{\twin f}$ mean distance between neighboring twins $\beta$
+  \item<1-> $\twin\tau_{\text{c}}$ threshold shear stress for twinning
+  \item<1-> $\twin\lambda$ mean distance between 2 obstacles seen by a growing twin
+  \end{itemize}
+ \end{block}
+}
+
+\subsection{Threshold stresses}
+\frame {
+ \frametitle{Threshold stress for glide activity}
+ \begin{block}{Threshold stress $\slip\tau_{\text{c}}$:}
   \begin{equation}
-  \slip\tau_{\text{c}} = G_{\text{iso}}\,\slip b\,\sqrt{\sum_{\tilde\alpha\,=\,1}^{\slip N}\,\slipslipt\xi\,(\slipt\varrho_{\text{edge}} + \slipt\varrho_{\text{dipole}})} \nonumber
+  \slip\tau_{\text{c}} = k_{\text{friction}}\,G_{\text{iso}}\,\sqrt{c}\,+\,G_{\text{iso}}\,\slip b\,\sqrt{\sum_{\tilde\alpha\,=\,1}^{\slip N}\,\slipslipt\xi\,(\slipt\varrho_{\text{edge}} + \slipt\varrho_{\text{dipole}})} \nonumber
   \end{equation}
  \end{block}
 
  \begin{block}{with:}
   \begin{itemize}
-  \item<1-> $G_{\text{iso}}$ Isotropic shear modulus
+  \item<1-> $G_{\text{iso}}$ isotropic shear modulus
+  \item<1-> $c$ carbon concentration (at.\%) 
+  \item<1-> $k_{\text{friction}}$ adjusting parameter for solute atome friction stress
   \item<1-> $\slip b$ Burgers vector of slip system $\alpha$
   \item<1-> $\slipslipt\xi$ interaction strength (Kubin et al. 2008)
   \end{itemize}
@@ -66,8 +96,69 @@
 }
 
 \frame {
- \frametitle{Orowan's kinetics}
+ \frametitle{Threshold stress for twinning:}
+ \begin{block}{Threshold stress $\twin\tau_{\text{c}}$:}
+  \begin{equation}
+  \twin\tau_{\text{c}} = \frac{\gamma_{\text{sfe}}}{3\,\twin b}\,+\,\frac{G_{\text{iso}}\,\twin b}{L_0} \nonumber
+  \end{equation}
+ \end{block}
 
+ \begin{block}{with:}
+  \begin{itemize}
+  \item<1-> $\gamma_{\text{sfe}}$ temperature-dependant stacking fault energy
+  \item<1-> $\twin b$ Burgers vector of twin system $\beta$
+  \item<1-> $L_0$ twin source length
+  \end{itemize}
+ \end{block}
+}
+
+\subsection{Mean free distances}
+\frame {
+ \frametitle{Dislocation mean free distance between two obstacles}
+ \begin{block}{Harmonic averaging:}
+  \begin{eqnarray}
+  \frac{1}{\slip\lambda} & = & 
+  \frac{1}{d_{\text{grain}}}
+  \,+\,\frac{\sqrt{\slip\varrho_{\text{edge}}\,+\,\slip\varrho_{\text{dipole}}}}{k_\lambda}
+  \,+\,\frac{1}{\sliptwin d} \nonumber \\
+  & = &
+  \frac{1}{d_{\text{grain}}}
+  \,+\,\frac{\sqrt{\slip\varrho_{\text{edge}}\,+\,\slip\varrho_{\text{dipole}}}}{k_\lambda} \,+\,\sum_{\beta\,=\,1}^{\slip{N}}\,\sliptwin{I}\,\frac{1}{\twin l} \nonumber
+  \end{eqnarray}
+ \end{block}
+
+ \begin{block}{with:}
+  \begin{itemize}
+  \item<1-> $d_{\text{grain}}$ grain size
+  \item<1-> $\sliptwin{I}$ slip--twin interactions (0 if $\alpha$,$\beta$ coplanars or cross-slip; 1 otherwise)
+  \end{itemize}
+ \end{block}
+}
+
+\frame {
+ \frametitle{Twin mean free distance between two obstacles}
+ \begin{block}{Harmonic averaging:}
+  \begin{eqnarray}
+  \frac{1}{\twin\lambda} & = & \frac{1}{d_{\text{grain}}} + \dfrac{1}{\twin d} \nonumber \\
+  & = & \dfrac{1}{d_{\mathrm{grain}}} + \sum_{\tilde\beta\,=\,1}^{\twin{N}}\,\twintwint{I}\,\dfrac{1}{\twint l} \nonumber
+  \end{eqnarray}
+ \end{block}
+
+ \begin{block}{with:}
+  \begin{itemize}
+  \item<1-> $\twintwint{I}$ twin--twin interactions (0 if $\beta$,$\tilde\beta$ coplanars; 1 otherwise)
+  \end{itemize}
+ \end{block}
+}
+
+\section{Kinetics}
+\frame {
+ \frametitle{}
+ \begin{block}{\begin{center}PART II\end{center}} \begin{center}Kinetics\end{center} \end{block}
+}
+\subsection{Thermally-activated dislocation motion}
+\frame {
+ \frametitle{Orowan's kinetics}
  \begin{block}{Shear rate $\slip{\dot\gamma}$:}
   \begin{equation}
   \slip{\dot\gamma} = \slip\varrho_{\text{edge}}\,\slip b\,\slip v_{\text{glide}} \nonumber
@@ -83,33 +174,63 @@
 
  \begin{block}{with:}
   \begin{itemize}
-  \item<1-> $v_0$ Velocity pre-factor
+  \item<1-> $v_0$ velocity pre-factor
-  \item<1-> $Q$ Activation energy for dislocation glide
+  \item<1-> $Q$ activation energy for dislocation glide
   \item<1-> $k_{\text{B}}\,T$ Boltzmann energy
   \end{itemize}
  \end{block}
 }
 
+\subsection{Twin kinetics}
+\frame {
+ \frametitle{Twin nucleation law}
+ \begin{block}{Shear rate $\twin{\dot\gamma}$:}
+  \begin{equation}
+  \twin{\dot\gamma} = \twin\gamma_{\text{c}}\,\twin{\dot f} = \twin\gamma_{\text{c}}\,(1-f)\,\twin V\,\twin{\dot N} \nonumber
+  \end{equation}
+ \end{block}
+
+ \begin{block}{Nucleation rate $\twin{\dot N}$:}
+  \begin{equation}
+  \twin{\dot N} = \dot{N}_0\,\exp\left[-\left(\frac{\twin\tau_{\text{c}}}{\twin\tau}\right)^r\right] \nonumber
+  \end{equation}
+ \end{block}
+
+ \begin{block}{with:}
+  \begin{itemize}
+  \item<1-> $\twin\gamma_{\text{c}}$ characteristical twin shear
+  \item<1-> $\twin V$ volume of grown-up twins
+  \item<1-> $\dot{N}_0$ constant twin nucleation rate per time and volume
+  \end{itemize}
+ \end{block}
+}
+
+\frame {
+ \frametitle{Spontaneous twin growth}
+ \begin{block}{Volume of grown-up twins $\twin V$:}
+  \begin{equation}
+  \twin V = \frac{\pi}{6}\,\twin s\,{\twin\lambda}^2 \nonumber
+  \end{equation}
+ \end{block}
+}
+
+\section{Evolution laws for microstructure}
+\frame {
+ \frametitle{}
+ \begin{block}{\begin{center}PART III\end{center}} \begin{center}Evolution laws for microstructure\end{center} \end{block}
+}
+\subsection{Multiplication and annihilation mechanisms}
 \frame {
  \frametitle{Dislocation multiplication}
-
  \begin{block}{Multiplication:}
   \begin{equation}
   \slip{\dot\varrho_{\text{multiplication}}} = \dfrac{|\slip{\dot\gamma}|}{\slip b\,\slip\lambda} \nonumber
   \end{equation}
  \end{block}
-
- \begin{block}{Multiplication constant:}
-  \begin{equation}
-  \slip\lambda = k_{\lambda} \left(\slip\varrho\right)^{-1/2}
-\nonumber
-  \end{equation}
- \end{block}
 }
 
 \frame {
- \frametitle{Dislocation dipole formation}
+ \frametitle{Dipole formation}
-
  \begin{block}{Dipole formation:}
   \begin{equation}
   \slip{\dot\varrho_{\text{formation}}} = 2\,\dfrac{2\,\operatorname{max}(\slip{\hat d},\slip{\check d})}{\slip b}\,\dfrac{\slip\varrho_{\text{edge}}}{2}\,|\slip{\dot\gamma}| \nonumber
@@ -125,7 +246,6 @@
 
 \frame {
  \frametitle{Spontaneous annihilation of 2 single dislocations}
-
  \begin{block}{Single--single annihilation:}
   \begin{equation}
   \slip{\dot\varrho_{\text{single--single}}} = 2\,\dfrac{2\,\slip{\check d}}{\slip b}\,\dfrac{\slip\varrho_{\text{edge}}}{2}\,|\slip{\dot\gamma}| \nonumber
@@ -141,7 +261,6 @@
 
 \frame {
  \frametitle{Spontaneous annihilation of one single dislocation with a dipole constituent}
-
  \begin{block}{Single--dipole constituent annihilation:}
   \begin{equation}
   \slip{\dot\varrho_{\text{single--dipole}}} = 2\,\dfrac{2\,\slip{\check d}}{\slip b}\,\dfrac{\slip\varrho_{\text{dipole}}}{2}\,|\slip{\dot\gamma}| \nonumber
@@ -150,8 +269,7 @@
 }
 
 \frame {
- \frametitle{Dislocation dipole climb}
+ \frametitle{Dipole climb}
-
  \begin{block}{Dipole climb:}
   \begin{equation}
   \slip{\dot\varrho_{\text{climb}}} = \slip\varrho_{\text{dipole}}\,\dfrac{2\,v_{\text{climb}}}{(\slip{\hat d}-\slip{\check d})/2} \nonumber
@@ -165,9 +283,9 @@
  \end{block}
 }
 
+\subsection{Evolution of dislocation densities}
 \frame {
  \frametitle{Evolution of dislocation densities}
-
  \begin{block}{Edge dislocation density rate:}
   \begin{equation}
   \slip{\dot\varrho_{\text{edge}}} = \slip{\dot\varrho_{\text{multiplication}}} - \slip{\dot\varrho_{\text{formation}}} - \slip{\dot\varrho_{\text{single--single}}} \nonumber