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+1,277 @@ + +\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} + |
