From c74772beba718bdd28d25ee4a8628c7caaa3083d Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Sun, 3 Mar 2019 13:08:58 -0500
Subject: initialized repo

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+
+\documentclass[fleqn,aspectratio=169]{beamer}
+
+
+\setbeamerfont{frametitle}{family=\bf}
+\setbeamerfont{title}{family=\bf}
+\setbeamerfont{author}{family=\bf}
+\setbeamerfont{normal text}{family=\rm}
+\setbeamertemplate{navigation symbols}{}
+
+\usepackage{textcomp,rotating}
+
+\title{Scaling and spatial correlations\\ in the quasibrittle process zone}
+%\subtitle{``Broke again\ldots''}
+\author{Jaron Kent-Dobias \and James P Sethna}
+\institute{Cornell University}
+\date{}
+
+\begin{document}
+
+\begin{frame}
+  \maketitle
+\end{frame}
+
+\begin{frame}
+  \frametitle{Quasibrittle materials \& fracture}
+  \begin{columns}
+    \begin{column}{0.35\textwidth}
+      Brittle with quenched disorder
+      
+      \bigskip
+
+      Process zone of correlated microfracture, large as meters
+
+      \bigskip
+
+      Size and boundary effects dominate statistics of fracture
+
+      \bigskip
+
+      Depending on substance and scale, fracture can look clean or crumbly
+
+    \end{column}
+    \begin{column}{0.6\textwidth}
+      \includegraphics[width=\textwidth]{figs/concrete}
+    \end{column}
+  \end{columns}
+\end{frame}
+
+\begin{frame}
+  \frametitle{Crossover theory}
+  \begin{columns}
+    \begin{column}{0.5\textwidth}
+      \hfill\tiny Shekhawat \textit{et al}, Phys Rev Lett \textbf{110} 185505\hspace{2em}\\
+      \includegraphics[width=\textwidth]{figs/shekhawat}
+    \end{column}
+    \begin{column}{0.5\textwidth}
+      Previous work suggests a \emph{scaling crossover} between fracture regimes
+
+      \bigskip
+
+      Crumbly regime controlled by percolation fixed point, clean by nucleation
+
+      \bigskip
+
+      Avalanches dominate intermediate disorder, vanish when first fractures system
+
+      \bigskip
+
+      Analogous idea for crack structure: crumbly crack surface coarse-grains to clean one through microcracked crossover
+    \end{column}
+  \end{columns}
+\end{frame}
+
+\begin{frame}
+  \frametitle{Our simulations: Fuse networks}
+  \begin{columns}
+    \begin{column}{0.5\textwidth}
+      \begin{overprint}
+        \onslide<1,3>\includegraphics[width=\textwidth]{figs/plain_square}
+        \onslide<2>\includegraphics[width=\textwidth]{figs/uncut_square}
+        \onslide<4>\includegraphics[width=\textwidth]{figs/perc_square}
+        \onslide<5>\includegraphics[width=\textwidth]{figs/med_square}
+        \onslide<6>\includegraphics[width=\textwidth]{figs/nuke_square}
+        \onslide<7>\includegraphics[width=\textwidth]{figs/nuke_voronoi}
+      \end{overprint}
+    \end{column}
+    \begin{column}{0.5\textwidth}
+      \vspace{0.5em}
+
+      Resistive fuses as scalar elastic analogue
+
+      \bigskip
+
+      Quenched disorder via breaking thresholds $0<x\leq1$ distributed by
+        $p(x)=\beta x^{\beta -1}$
+
+      \bigskip
+
+      Fractured adiabatically: fuse with largest threshold to current ratio broken
+
+      \bigskip
+
+      \alert<4>{Small $\beta$} is disordered, \alert<6->{large $\beta$} ordered
+
+      \medskip
+
+    \begin{overprint}
+      \onslide<1-3>\includegraphics[width=\textwidth]{figs/dist-all}
+      \onslide<4>\includegraphics[width=\textwidth]{figs/dist-small}
+      \onslide<5>\includegraphics[width=\textwidth]{figs/dist-med}
+      \onslide<6->\includegraphics[width=\textwidth]{figs/dist-large}
+    \end{overprint}
+    \end{column}
+  \end{columns}
+\end{frame}
+
+\begin{frame}
+  \frametitle{Properties of interest}
+  \begin{columns}
+    \begin{column}{0.5\textwidth}
+      Many spatial properties to study:
+      \begin{itemize}
+        \item\alert<3>{backbone}
+        \item\alert<4>{spanning cluster}
+        \item\alert<5,8->{all clusters}
+        \item\alert<6>{non-spanning clusters}
+        \item\alert<7>{final avalanche}
+      \end{itemize}
+
+      \medskip
+
+      Focus on $g(\Delta x,\Delta y)$: probability that site displaced by $(\Delta x, \Delta y)$ is in same cluster
+    \end{column}
+    \begin{column}{0.5\textwidth}
+      \begin{overprint}
+        \onslide<1>\includegraphics[width=\textwidth]{figs/prop-none}
+        \onslide<2>\includegraphics[width=\textwidth]{figs/prop-broken}
+        \onslide<3>\includegraphics[width=\textwidth]{figs/prop-backbone}
+        \onslide<4>\includegraphics[width=\textwidth]{figs/prop-spanning}
+        \onslide<5>\includegraphics[width=\textwidth]{figs/prop-allclusters}
+        \onslide<6>\includegraphics[width=\textwidth]{figs/prop-clusters}
+        \onslide<7>\includegraphics[width=\textwidth]{figs/prop-lastavalanche}
+        \onslide<8->\includegraphics[width=\textwidth]{figs/prop-allclusters}
+      \end{overprint}
+    \end{column}
+  \end{columns}
+\end{frame}
+
+\begin{frame}
+  \frametitle{Fixed points}
+  \framesubtitle{$\pmb\beta\pmb=\pmb0$ -- percolation}
+  \begin{columns}
+    \begin{column}{0.5\textwidth}
+      \begin{overprint}
+        \onslide<1>\includegraphics[width=\textwidth]{figs/perc-plain}
+        \onslide<2>\includegraphics[width=\textwidth]{figs/perc-voronoi}
+        \onslide<3>\includegraphics[width=\textwidth]{figs/perc-backbone}
+        \onslide<4>\includegraphics[width=\textwidth]{figs/perc-spanning}
+        \onslide<5,8->\includegraphics[width=\textwidth]{figs/perc-allclusters}
+        \onslide<6>\includegraphics[width=\textwidth]{figs/perc-clusters}
+        \onslide<7>\includegraphics[width=\textwidth]{figs/perc-avalanche}
+      \end{overprint}
+    \end{column}
+    \begin{column}{0.5\textwidth}
+      As $\beta\to0$, $L\to\infty$, reduces (almost) exactly to percolation
+      \begin{itemize}
+        \item\alert<3>{backbone --- $\ell(L)\sim L^{d_{\text{min}}}$}
+        \item\alert<4>{spanning cluster --- $M(L)\sim L^{d_f}$}
+        \item\alert<5,8->{clusters --- $g(r)\sim|r|^{-2(d-d_f)}$}
+        \item\alert<6>{non-spanning clusters --- $n^c_s\sim s^{-\tau}$}
+        \item\alert<7>{final avalanche --- $n^a_s=\delta_{1s}$}
+      \end{itemize}
+
+      \ \\
+
+      \includegraphics[width=\textwidth]{figs/plot_percgvx}
+    \end{column}
+  \end{columns}
+\end{frame}
+
+\begin{frame}
+  \frametitle{Fixed points}
+  \framesubtitle{$\pmb\beta=\infty$ -- nucleation}
+  \begin{columns}
+    \begin{column}{0.5\textwidth}
+      As $L\to\infty$, $\beta\to\infty$, reduces to nucleated crack propagation
+      \begin{itemize}
+        \item\alert<3>{backbone}
+        \item\alert<4>{spanning cluster}
+        \item\alert<5,8->{clusters}
+        \item\alert<6>{non-spanning clusters}
+        \item\alert<7>{final avalanche}
+      \end{itemize}
+
+      \ \\
+
+      \includegraphics[width=\textwidth]{figs/plot_nukegvx}
+    \end{column}
+    \begin{column}{0.5\textwidth}
+      \begin{overprint}
+        \onslide<1>\includegraphics[width=\textwidth]{figs/nuke-voro}
+        \onslide<2>\includegraphics[width=\textwidth]{figs/nuke-broken}
+        \onslide<3>\includegraphics[width=\textwidth]{figs/nuke-backbone}
+        \onslide<4>\includegraphics[width=\textwidth]{figs/nuke-spanning}
+        \onslide<5,8->\includegraphics[width=\textwidth]{figs/nuke-allclusters}
+        \onslide<6>\includegraphics[width=\textwidth]{figs/nuke-clusters}
+        \onslide<7>\includegraphics[width=\textwidth]{figs/nuke-avalanche}
+      \end{overprint}
+    \end{column}
+  \end{columns}
+\end{frame}
+
+\begin{frame}
+  \frametitle{Self-similarity vs.\ self-affinity}
+
+  Voltage (strain) applied along one direction---we should expect anisotropy!
+  
+  \bigskip
+
+  Clean cracks are \emph{self-affine}, self-similar under different rescalings of $x$ and $y$
+
+
+  \begin{overprint}
+    \onslide<1>\centering\includegraphics[height=\textwidth, angle=90]{figs/skinny}
+    \onslide<2>\centering\includegraphics[height=\textwidth, angle=90]{figs/skinny_long_clusters}
+    \onslide<3>\centering\includegraphics[height=\textwidth, angle=90]{figs/skinny_short_clusters}
+  \end{overprint}
+
+\end{frame}
+
+\begin{frame}
+  \frametitle{Scaling theory}
+
+  Old: quantities like moments of $g$ depend on $L\beta^\nu$
+
+  \bigskip
+
+  New: quantities like moments of $g$ depend on $L_x\beta^{\nu_x}$ and $L_y\beta^{\nu_y}$
+
+  \bigskip
+
+  \centering
+
+  \huge FIGURE: EXAMPLE SCALING OF $x$ AND $y$ MOMENTS OF $g$ SHOWING DIFFERENT $\nu$
+
+
+\end{frame}
+
+\begin{frame}
+  \frametitle{Continuing work}
+  \begin{columns}
+    \begin{column}{0.5\textwidth}
+      Need to make consistent cross-property measurement of exponents; working on expected form of scaling functions through different regiemes
+
+      \bigskip
+
+      How does the established multifractal distribution of bond currents (stresses) affect this analysis, if at all? 
+  \end{column}
+  \begin{column}{0.5\textwidth}
+    \begin{overprint}
+      \onslide<1>\includegraphics[width=\textwidth]{figs/multi-all}
+      \onslide<2>\includegraphics[width=\textwidth]{figs/multi-carriers}
+      \onslide<3>\includegraphics[width=\textwidth]{figs/multi-weighted}
+      \onslide<4>\includegraphics[width=\textwidth]{figs/multi-q1}
+      \onslide<5>\includegraphics[width=\textwidth]{figs/multi-q2}
+      \onslide<6>\includegraphics[width=\textwidth]{figs/multi-q3}
+      \onslide<7>\includegraphics[width=\textwidth]{figs/multi-q4}
+    \end{overprint}
+  \end{column}
+
+  \end{columns}
+
+\end{frame}
+
+\end{document}
+
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