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