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authorJaron Kent-Dobias <jaron@kent-dobias.com>2019-03-03 13:08:58 -0500
committerJaron Kent-Dobias <jaron@kent-dobias.com>2019-03-03 13:08:58 -0500
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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}
+