climb equation corrected

minor phrase polishing

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

@@ -1,177 +1,184 @@
-\documentclass{beamer}
-
-\usepackage{amsmath,amssymb,amsfonts}
-\usepackage{bm}
-\usepackage{array}
-%\include{Shortcuts}
-\newcommand{\ie}{\textit{i.e.}}
-\newcommand{\eg}{\textit{e.g.}}
-\newcommand{\vect}[1]{\ensuremath{\mathbf{#1}}}
-\newcommand{\tensII}[1]{\ensuremath{\mathbf{#1}}}
-\newcommand{\tensIV}[1]{\ensuremath{\mathbb{#1}}}
-\newcommand{\slip}[1]{\ensuremath{#1^{\alpha}}}
-\newcommand{\slipslip}[1]{\ensuremath{#1^{\alpha\alpha}}}
-\newcommand{\slipt}[1]{\ensuremath{#1^{\tilde\alpha}}}
-\newcommand{\slipslipt}[1]{\ensuremath{#1^{\alpha\tilde\alpha}}}
-\newcommand{\twin}[1]{\ensuremath{#1^{\beta}}}
-\newcommand{\twint}[1]{\ensuremath{#1^{\tilde\beta}}}
-\newcommand{\twintwint}[1]{\ensuremath{#1^{\beta\tilde\beta}}}
-\newcommand{\sliptwin}[1]{\ensuremath{#1^{\alpha\beta}}}
-\newcommand{\twinslip}[1]{\ensuremath{#1^{\beta\alpha}}}
-
-\usetheme{mpie}
-\setbeamertemplate{blocks}[rounded][shadow=true]
-
-\title{Dislocation structure and kinetics in slip-twin model}
-\date{MSU Twin Meeting, Duesseldorf -- October 6$^{\textsf{th}}$, 2009}
-
-\begin{document}
-
-\frame{\titlepage}
-
-\frame {
- \frametitle{Dislocation structure parametrization}
-
- \begin{block}{Internal variables:}
-  \begin{itemize}
-  \item<1-> $\slip N$ edge dislocation densities $\slip\varrho_{\text{edge}}$
-  \item<1-> $\slip N$ dipole densities $\slip\varrho_{\text{dipole}}$
-  \end{itemize}
- \end{block}
-
- \begin{block}{Derived measures:}
-  \begin{itemize}
-  \item<1-> $\slip\tau_{\mathrm{c}}$ threshold shear stress
-  \item<1-> $\slip\lambda$ mean distance between 2 obstacles seen by a dislocation
-  \end{itemize}
- \end{block}
-}
-
-\frame {
- \frametitle{Dislocation structure parametrization}
-
- \begin{block}{Threshold stress $\slip\tau$:}
-  \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
-  \end{equation}
- \end{block}
-
- \begin{block}{with:}
-  \begin{itemize}
-  \item<1-> $G_{\text{iso}}$ Isotropic shear modulus
-  \item<1-> $\slip b$ Burgers vector of slip system $\alpha$
-  \item<1-> $\slipslipt\xi$ interaction strength (Kubin et al. 2008)
-  \end{itemize}
- \end{block}
-}
-
-\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
-  \end{equation}
- \end{block}
-
- \begin{block}{Velocity $\slip v_{\text{glide}}$:}
-  \begin{equation}
-  \slip v_{\text{glide}} = v_0\,
-  \exp{\left[-\dfrac{Q}{k_{\text{B}}\,T}\,\left(1-\left(\dfrac{|\slip\tau|}{\slip\tau_{\text{c}}}\right)^p\right)^q\right]} \operatorname{sign}(\slip\tau) \nonumber
-  \end{equation}
- \end{block}
-
- \begin{block}{with:}
-  \begin{itemize}
-  \item<1-> $v_0$ Velocity pre-factor
-  \item<1-> $Q$ Activation energy for dislocation glide
-  \item<1-> $k_{\text{B}}\,T$ Boltzmann energy
-  \end{itemize}
- \end{block}
-}
-
-\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}
-}
-
-\frame {
- \frametitle{Dislocation dipole formation}
-
- \begin{block}{Dipole formation:}
-  \begin{equation}
-  \slip{\dot\varrho_{\text{formation}}} = 2\,\dfrac{2\,\slip{\hat d}}{\slip b}\,\dfrac{\slip\varrho_{\text{edge}}}{2}\,|\slip{\dot\gamma}| \nonumber
-  \end{equation}
- \end{block}
-
- \begin{block}{Length $\slip{\hat d}$:}
-  \begin{equation}
-  \slip{\hat d} = \dfrac{1}{8\,\pi}\,\dfrac{G_{\text{iso}}\,\slip b}{1-\nu}\,\dfrac{1}{|\slip\tau|} \nonumber
-  \end{equation}
- \end{block}
-}
-
-\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
-  \end{equation}
- \end{block}
-
- \begin{block}{Length $\slip{\check d}$:}
-  \begin{equation}
-  \slip{\check d} \propto \slip b \nonumber
-  \end{equation}
- \end{block}
-}
-
-\frame {
- \frametitle{Spontaneous annihilation of one single dislocation and one 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
-  \end{equation}
- \end{block}
-}
-
-\frame {
- \frametitle{Dislocation dipole climb}
-
- \begin{block}{Dipole climb:}
-  \begin{equation}
-  \slip{\dot\varrho_{\text{climb}}} = \slip\varrho_{\text{dipole}}\,\dfrac{4\,v_{\text{climb}}}{\slip{\hat d}+\slip{\check d}} \nonumber
-  \end{equation}
- \end{block}
-
-  \begin{block}{Climb velocity $\slip v_{\text{climb}}$:}
-  \begin{equation}
-  \slip v_{\text{climb}} = \dfrac{D\,\slip\Omega}{\slip b\,k_{\text{B}}\,T}\,\dfrac{G_{\text{iso}}\,\slip b}{2\,\pi\,(1-\nu)}\,\dfrac{2}{\slip{\hat d}+\slip{\check d}} \nonumber
-  \end{equation}
- \end{block}
-}
-
-\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
-  \end{equation}
- \end{block}
-
-\begin{block}{Dislocation dipole density rate:}
-  \begin{equation}
-  \slip{\dot\varrho_{\text{dipole}}} = \slip{\dot\varrho_{\text{formation}}} - \slip{\dot\varrho_{\text{single-dipole}}} - \slip{\dot\varrho_{\text{climb}}} \nonumber
-  \end{equation}
- \end{block}
-}
-
+\documentclass{beamer}
+
+\usepackage{amsmath,amssymb,amsfonts}
+\usepackage{bm}
+\usepackage{array}
+%\include{Shortcuts}
+\newcommand{\ie}{\textit{i.e.}}
+\newcommand{\eg}{\textit{e.g.}}
+\newcommand{\vect}[1]{\ensuremath{\mathbf{#1}}}
+\newcommand{\tensII}[1]{\ensuremath{\mathbf{#1}}}
+\newcommand{\tensIV}[1]{\ensuremath{\mathbb{#1}}}
+\newcommand{\slip}[1]{\ensuremath{#1^{\alpha}}}
+\newcommand{\slipslip}[1]{\ensuremath{#1^{\alpha\alpha}}}
+\newcommand{\slipt}[1]{\ensuremath{#1^{\tilde\alpha}}}
+\newcommand{\slipslipt}[1]{\ensuremath{#1^{\alpha\tilde\alpha}}}
+\newcommand{\twin}[1]{\ensuremath{#1^{\beta}}}
+\newcommand{\twint}[1]{\ensuremath{#1^{\tilde\beta}}}
+\newcommand{\twintwint}[1]{\ensuremath{#1^{\beta\tilde\beta}}}
+\newcommand{\sliptwin}[1]{\ensuremath{#1^{\alpha\beta}}}
+\newcommand{\twinslip}[1]{\ensuremath{#1^{\beta\alpha}}}
+
+\usetheme{mpie}
+\setbeamertemplate{blocks}[rounded][shadow=true]
+
+\title{Dislocation structure and kinetics in slip-twin model}
+\date{MSU Twin Meeting, D\"usseldorf -- October 6\textsuperscript{th}, 2009}
+
+\begin{document}
+
+\frame{\titlepage}
+
+\frame {
+ \frametitle{Dislocation structure parametrization}
+
+ \begin{block}{Internal variables:}
+  \begin{itemize}
+  \item<1-> $\slip N$ edge dislocation densities $\slip\varrho_{\text{edge}}$
+  \item<1-> $\slip N$ dipole densities $\slip\varrho_{\text{dipole}}$
+  \end{itemize}
+ \end{block}
+
+ \begin{block}{Derived measures:}
+  \begin{itemize}
+  \item<1-> $\slip\tau_{\mathrm{c}}$ threshold shear stress
+  \item<1-> $\slip\lambda$ mean distance between 2 obstacles seen by a dislocation
+  \end{itemize}
+ \end{block}
+}
+
+\frame {
+ \frametitle{Dislocation structure parametrization}
+
+ \begin{block}{Threshold stress $\slip\tau$:}
+  \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
+  \end{equation}
+ \end{block}
+
+ \begin{block}{with:}
+  \begin{itemize}
+  \item<1-> $G_{\text{iso}}$ Isotropic shear modulus
+  \item<1-> $\slip b$ Burgers vector of slip system $\alpha$
+  \item<1-> $\slipslipt\xi$ interaction strength (Kubin et al. 2008)
+  \end{itemize}
+ \end{block}
+}
+
+\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
+  \end{equation}
+ \end{block}
+
+ \begin{block}{Velocity $\slip v_{\text{glide}}$:}
+  \begin{equation}
+  \slip v_{\text{glide}} = v_0\,
+  \exp{\left[-\dfrac{Q}{k_{\text{B}}\,T}\,\left(1-\left(\dfrac{|\slip\tau|}{\slip\tau_{\text{c}}}\right)^p\right)^q\right]} \operatorname{sign}(\slip\tau) \nonumber
+  \end{equation}
+ \end{block}
+
+ \begin{block}{with:}
+  \begin{itemize}
+  \item<1-> $v_0$ Velocity pre-factor
+  \item<1-> $Q$ Activation energy for dislocation glide
+  \item<1-> $k_{\text{B}}\,T$ Boltzmann energy
+  \end{itemize}
+ \end{block}
+}
+
+\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}
+
+ \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
+  \end{equation}
+ \end{block}
+
+ \begin{block}{Upper stability limit for dipoles $\slip{\hat d}$:}
+  \begin{equation}
+  \slip{\hat d} = \dfrac{1}{8\,\pi}\,\dfrac{G_{\text{iso}}\,\slip b}{1-\nu}\,\dfrac{1}{|\slip\tau|} \nonumber
+  \end{equation}
+ \end{block}
+}
+
+\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
+  \end{equation}
+ \end{block}
+
+ \begin{block}{Lower stability limit of dipoles $\slip{\check d}$:}
+  \begin{equation}
+  \slip{\check d} \propto \slip b \nonumber
+  \end{equation}
+ \end{block}
+}
+
+\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
+  \end{equation}
+ \end{block}
+}
+
+\frame {
+ \frametitle{Dislocation 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
+  \end{equation}
+ \end{block}
+
+  \begin{block}{Climb velocity $\slip v_{\text{climb}}$:}
+  \begin{equation}
+  \slip v_{\text{climb}} = \dfrac{D\,\slip\Omega}{\slip b\,k_{\text{B}}\,T}\,\dfrac{G_{\text{iso}}\,\slip b}{2\,\pi\,(1-\nu)}\,\dfrac{1}{(\slip{\hat d}+\slip{\check d})/2} \nonumber
+  \end{equation}
+ \end{block}
+}
+
+\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
+  \end{equation}
+ \end{block}
+
+\begin{block}{Dislocation dipole density rate:}
+  \begin{equation}
+  \slip{\dot\varrho_{\text{dipole}}} = \slip{\dot\varrho_{\text{formation}}} - \slip{\dot\varrho_{\text{single--dipole}}} - \slip{\dot\varrho_{\text{climb}}} \nonumber
+  \end{equation}
+ \end{block}
+}
+
 \end{document}