From c74772beba718bdd28d25ee4a8628c7caaa3083d Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Sun, 3 Mar 2019 13:08:58 -0500 Subject: initialized repo --- .gitignore | 8 ++ figs/applied_voltage.pdf | Bin 0 -> 38811 bytes figs/concrete.jpg | Bin 0 -> 5051021 bytes figs/concrete.xcf | Bin 0 -> 35984577 bytes figs/crossover-moment.pdf | Bin 0 -> 34042 bytes figs/dist-all.pdf | Bin 0 -> 15269 bytes figs/dist-large.pdf | Bin 0 -> 15278 bytes figs/dist-med.pdf | Bin 0 -> 15282 bytes figs/dist-small.pdf | Bin 0 -> 15277 bytes figs/med_square.pdf | Bin 0 -> 14664 bytes figs/multi-all.pdf | Bin 0 -> 23797 bytes figs/multi-carriers.pdf | Bin 0 -> 9035 bytes figs/multi-q1.pdf | Bin 0 -> 1616 bytes figs/multi-q2.pdf | Bin 0 -> 3218 bytes figs/multi-q3.pdf | Bin 0 -> 5797 bytes figs/multi-q4.pdf | Bin 0 -> 3626 bytes figs/multi-weighted.pdf | Bin 0 -> 12129 bytes figs/nuke-allclusters.pdf | Bin 0 -> 40764 bytes figs/nuke-avalanche.pdf | Bin 0 -> 39493 bytes figs/nuke-backbone.pdf | Bin 0 -> 40378 bytes figs/nuke-broken.pdf | Bin 0 -> 38703 bytes figs/nuke-clusters.pdf | Bin 0 -> 39041 bytes figs/nuke-spanning.pdf | Bin 0 -> 40456 bytes figs/nuke-voro.pdf | Bin 0 -> 39039 bytes figs/nuke_square.pdf | Bin 0 -> 15341 bytes figs/nuke_voronoi.pdf | Bin 0 -> 38559 bytes figs/perc-allclusters.pdf | Bin 0 -> 100639 bytes figs/perc-avalanche.pdf | Bin 0 -> 23557 bytes figs/perc-backbone.pdf | Bin 0 -> 29352 bytes figs/perc-clusters.pdf | Bin 0 -> 42284 bytes figs/perc-plain.pdf | Bin 0 -> 39024 bytes figs/perc-spanning.pdf | Bin 0 -> 82060 bytes figs/perc-voronoi.pdf | Bin 0 -> 23521 bytes figs/perc_square.pdf | Bin 0 -> 12072 bytes figs/plain_square.pdf | Bin 0 -> 15354 bytes figs/plain_voronoi.pdf | Bin 0 -> 38861 bytes figs/plot_nuke-MvL.pdf | Bin 0 -> 14992 bytes figs/plot_nuke-gvx.pdf | Bin 0 -> 53851 bytes figs/plot_nuke-hvx.pdf | Bin 0 -> 34248 bytes figs/plot_nukegvx.pdf | Bin 0 -> 37359 bytes figs/plot_perc-MvL.pdf | Bin 0 -> 15322 bytes figs/plot_perc-gvx.pdf | Bin 0 -> 35776 bytes figs/plot_perc-lvL.pdf | Bin 0 -> 18087 bytes figs/plot_perc-nvs.pdf | Bin 0 -> 37759 bytes figs/plot_percgvx.pdf | Bin 0 -> 38230 bytes figs/plot_skinnyx.pdf | Bin 0 -> 20478 bytes figs/plot_skinnyx_broken.pdf | Bin 0 -> 46786 bytes figs/prop-allclusters.pdf | Bin 0 -> 93447 bytes figs/prop-backbone.pdf | Bin 0 -> 35476 bytes figs/prop-broken.pdf | Bin 0 -> 32938 bytes figs/prop-clusters.pdf | Bin 0 -> 89439 bytes figs/prop-lastavalanche.pdf | Bin 0 -> 34301 bytes figs/prop-none.pdf | Bin 0 -> 38902 bytes figs/prop-spanning.pdf | Bin 0 -> 37020 bytes figs/shekhawat.png | Bin 0 -> 317585 bytes figs/skinny.pdf | Bin 0 -> 31639 bytes figs/skinny_long_clusters.pdf | Bin 0 -> 64197 bytes figs/skinny_short_clusters.pdf | Bin 0 -> 61658 bytes figs/uncut_square.pdf | Bin 0 -> 15306 bytes mm_2019.tex | 277 +++++++++++++++++++++++++++++++++++++++++ 60 files changed, 285 insertions(+) create mode 100644 .gitignore create mode 100644 figs/applied_voltage.pdf create mode 100644 figs/concrete.jpg create mode 100644 figs/concrete.xcf create mode 100644 figs/crossover-moment.pdf create mode 100644 figs/dist-all.pdf create mode 100644 figs/dist-large.pdf create mode 100644 figs/dist-med.pdf create mode 100644 figs/dist-small.pdf create mode 100644 figs/med_square.pdf create mode 100644 figs/multi-all.pdf create mode 100644 figs/multi-carriers.pdf create mode 100644 figs/multi-q1.pdf create mode 100644 figs/multi-q2.pdf create mode 100644 figs/multi-q3.pdf create mode 100644 figs/multi-q4.pdf create mode 100644 figs/multi-weighted.pdf create mode 100644 figs/nuke-allclusters.pdf create mode 100644 figs/nuke-avalanche.pdf create mode 100644 figs/nuke-backbone.pdf create mode 100644 figs/nuke-broken.pdf create mode 100644 figs/nuke-clusters.pdf create mode 100644 figs/nuke-spanning.pdf create mode 100644 figs/nuke-voro.pdf create mode 100644 figs/nuke_square.pdf create mode 100644 figs/nuke_voronoi.pdf create mode 100644 figs/perc-allclusters.pdf create mode 100644 figs/perc-avalanche.pdf create mode 100644 figs/perc-backbone.pdf create mode 100644 figs/perc-clusters.pdf create mode 100644 figs/perc-plain.pdf create mode 100644 figs/perc-spanning.pdf create mode 100644 figs/perc-voronoi.pdf create mode 100644 figs/perc_square.pdf create mode 100644 figs/plain_square.pdf create mode 100644 figs/plain_voronoi.pdf create mode 100644 figs/plot_nuke-MvL.pdf create mode 100644 figs/plot_nuke-gvx.pdf create mode 100644 figs/plot_nuke-hvx.pdf create mode 100644 figs/plot_nukegvx.pdf create mode 100644 figs/plot_perc-MvL.pdf create mode 100644 figs/plot_perc-gvx.pdf create mode 100644 figs/plot_perc-lvL.pdf create mode 100644 figs/plot_perc-nvs.pdf create mode 100644 figs/plot_percgvx.pdf create mode 100644 figs/plot_skinnyx.pdf create mode 100644 figs/plot_skinnyx_broken.pdf create mode 100644 figs/prop-allclusters.pdf create mode 100644 figs/prop-backbone.pdf create mode 100644 figs/prop-broken.pdf create mode 100644 figs/prop-clusters.pdf create mode 100644 figs/prop-lastavalanche.pdf create mode 100644 figs/prop-none.pdf create mode 100644 figs/prop-spanning.pdf create mode 100644 figs/shekhawat.png create mode 100644 figs/skinny.pdf create mode 100644 figs/skinny_long_clusters.pdf create mode 100644 figs/skinny_short_clusters.pdf create mode 100644 figs/uncut_square.pdf create mode 100644 mm_2019.tex diff --git a/.gitignore b/.gitignore new file mode 100644 index 0000000..4215af7 --- /dev/null +++ b/.gitignore @@ -0,0 +1,8 @@ +*.aux +*.log +*.nav +*.out +mm_2019.pdf +*.snm +*.synctex.gz +*.toc diff --git a/figs/applied_voltage.pdf b/figs/applied_voltage.pdf new file mode 100644 index 0000000..c70a219 Binary files /dev/null and b/figs/applied_voltage.pdf differ diff --git a/figs/concrete.jpg b/figs/concrete.jpg new file mode 100644 index 0000000..64c1f08 Binary files /dev/null and b/figs/concrete.jpg differ diff --git a/figs/concrete.xcf b/figs/concrete.xcf new file mode 100644 index 0000000..c5ba283 Binary files /dev/null and b/figs/concrete.xcf differ diff --git a/figs/crossover-moment.pdf b/figs/crossover-moment.pdf new file mode 100644 index 0000000..a9661f3 Binary files /dev/null and b/figs/crossover-moment.pdf differ diff --git a/figs/dist-all.pdf b/figs/dist-all.pdf new file mode 100644 index 0000000..e112af3 Binary files /dev/null and b/figs/dist-all.pdf differ diff --git a/figs/dist-large.pdf b/figs/dist-large.pdf new file mode 100644 index 0000000..a3e5413 Binary files /dev/null and b/figs/dist-large.pdf differ diff --git a/figs/dist-med.pdf b/figs/dist-med.pdf new file mode 100644 index 0000000..f950bc2 Binary files /dev/null and b/figs/dist-med.pdf differ diff --git a/figs/dist-small.pdf b/figs/dist-small.pdf new file mode 100644 index 0000000..f14bcab Binary files /dev/null and b/figs/dist-small.pdf differ diff --git a/figs/med_square.pdf b/figs/med_square.pdf new file mode 100644 index 0000000..878a8f6 Binary files /dev/null and b/figs/med_square.pdf differ diff --git a/figs/multi-all.pdf b/figs/multi-all.pdf new file mode 100644 index 0000000..08491bb Binary files /dev/null and b/figs/multi-all.pdf differ diff --git a/figs/multi-carriers.pdf b/figs/multi-carriers.pdf new file mode 100644 index 0000000..1f73d59 Binary files /dev/null and b/figs/multi-carriers.pdf differ diff --git a/figs/multi-q1.pdf b/figs/multi-q1.pdf new file mode 100644 index 0000000..675c600 Binary files /dev/null and b/figs/multi-q1.pdf differ diff --git a/figs/multi-q2.pdf b/figs/multi-q2.pdf new file mode 100644 index 0000000..43cebdf Binary files /dev/null and b/figs/multi-q2.pdf differ diff --git a/figs/multi-q3.pdf b/figs/multi-q3.pdf new file mode 100644 index 0000000..635c20e Binary files /dev/null and b/figs/multi-q3.pdf differ diff --git a/figs/multi-q4.pdf b/figs/multi-q4.pdf new file mode 100644 index 0000000..0c1e07d Binary files /dev/null and b/figs/multi-q4.pdf differ diff --git a/figs/multi-weighted.pdf b/figs/multi-weighted.pdf new file mode 100644 index 0000000..3d275ed Binary files /dev/null and b/figs/multi-weighted.pdf differ diff --git a/figs/nuke-allclusters.pdf b/figs/nuke-allclusters.pdf new file mode 100644 index 0000000..8b8a2ee Binary files /dev/null and b/figs/nuke-allclusters.pdf differ diff --git a/figs/nuke-avalanche.pdf b/figs/nuke-avalanche.pdf new file mode 100644 index 0000000..438db1b Binary files /dev/null and b/figs/nuke-avalanche.pdf differ diff --git a/figs/nuke-backbone.pdf b/figs/nuke-backbone.pdf new file mode 100644 index 0000000..538a71d Binary files /dev/null and b/figs/nuke-backbone.pdf differ diff --git a/figs/nuke-broken.pdf b/figs/nuke-broken.pdf new file mode 100644 index 0000000..cda0c04 Binary files /dev/null and b/figs/nuke-broken.pdf differ diff --git a/figs/nuke-clusters.pdf b/figs/nuke-clusters.pdf new file mode 100644 index 0000000..7bfdccb Binary files /dev/null and b/figs/nuke-clusters.pdf differ diff --git a/figs/nuke-spanning.pdf b/figs/nuke-spanning.pdf new file mode 100644 index 0000000..97df079 Binary files /dev/null and b/figs/nuke-spanning.pdf differ diff --git a/figs/nuke-voro.pdf b/figs/nuke-voro.pdf new file mode 100644 index 0000000..7267262 Binary files /dev/null and b/figs/nuke-voro.pdf differ diff --git a/figs/nuke_square.pdf b/figs/nuke_square.pdf new file mode 100644 index 0000000..9732c68 Binary files /dev/null and b/figs/nuke_square.pdf differ diff --git a/figs/nuke_voronoi.pdf b/figs/nuke_voronoi.pdf new file mode 100644 index 0000000..db57e53 Binary files /dev/null and b/figs/nuke_voronoi.pdf differ diff --git a/figs/perc-allclusters.pdf b/figs/perc-allclusters.pdf new file mode 100644 index 0000000..9c6792f Binary files /dev/null and b/figs/perc-allclusters.pdf differ diff --git a/figs/perc-avalanche.pdf b/figs/perc-avalanche.pdf new file mode 100644 index 0000000..de77b20 Binary files /dev/null and b/figs/perc-avalanche.pdf differ diff --git a/figs/perc-backbone.pdf b/figs/perc-backbone.pdf new file mode 100644 index 0000000..cbe400e Binary files /dev/null and b/figs/perc-backbone.pdf differ diff --git a/figs/perc-clusters.pdf b/figs/perc-clusters.pdf new file mode 100644 index 0000000..6e9f641 Binary files /dev/null and b/figs/perc-clusters.pdf differ diff --git a/figs/perc-plain.pdf b/figs/perc-plain.pdf new file mode 100644 index 0000000..9f46d7a Binary files /dev/null and b/figs/perc-plain.pdf differ diff --git a/figs/perc-spanning.pdf b/figs/perc-spanning.pdf new file mode 100644 index 0000000..a20231e Binary files /dev/null and b/figs/perc-spanning.pdf differ diff --git a/figs/perc-voronoi.pdf b/figs/perc-voronoi.pdf new file mode 100644 index 0000000..a2302f0 Binary files /dev/null and b/figs/perc-voronoi.pdf differ diff --git a/figs/perc_square.pdf b/figs/perc_square.pdf new file mode 100644 index 0000000..bf534a1 Binary files /dev/null and b/figs/perc_square.pdf differ diff --git a/figs/plain_square.pdf b/figs/plain_square.pdf new file mode 100644 index 0000000..802c7a3 Binary files /dev/null and b/figs/plain_square.pdf differ diff --git a/figs/plain_voronoi.pdf b/figs/plain_voronoi.pdf new file mode 100644 index 0000000..6e2990b Binary files /dev/null and b/figs/plain_voronoi.pdf differ diff --git a/figs/plot_nuke-MvL.pdf b/figs/plot_nuke-MvL.pdf new file mode 100644 index 0000000..301f3b5 Binary files /dev/null and b/figs/plot_nuke-MvL.pdf differ diff --git a/figs/plot_nuke-gvx.pdf b/figs/plot_nuke-gvx.pdf new file mode 100644 index 0000000..388f79f Binary files /dev/null and b/figs/plot_nuke-gvx.pdf differ diff --git a/figs/plot_nuke-hvx.pdf b/figs/plot_nuke-hvx.pdf new file mode 100644 index 0000000..0b673ff Binary files /dev/null and b/figs/plot_nuke-hvx.pdf differ diff --git a/figs/plot_nukegvx.pdf b/figs/plot_nukegvx.pdf new file mode 100644 index 0000000..223a7dd Binary files /dev/null and b/figs/plot_nukegvx.pdf differ diff --git a/figs/plot_perc-MvL.pdf b/figs/plot_perc-MvL.pdf new file mode 100644 index 0000000..f1ff4e5 Binary files /dev/null and b/figs/plot_perc-MvL.pdf differ diff --git a/figs/plot_perc-gvx.pdf b/figs/plot_perc-gvx.pdf new file mode 100644 index 0000000..0ec8404 Binary files /dev/null and b/figs/plot_perc-gvx.pdf differ diff --git a/figs/plot_perc-lvL.pdf b/figs/plot_perc-lvL.pdf new file mode 100644 index 0000000..5215745 Binary files /dev/null and b/figs/plot_perc-lvL.pdf differ diff --git a/figs/plot_perc-nvs.pdf b/figs/plot_perc-nvs.pdf new file mode 100644 index 0000000..ad17e82 Binary files /dev/null and b/figs/plot_perc-nvs.pdf differ diff --git a/figs/plot_percgvx.pdf b/figs/plot_percgvx.pdf new file mode 100644 index 0000000..354c123 Binary files /dev/null and b/figs/plot_percgvx.pdf differ diff --git a/figs/plot_skinnyx.pdf b/figs/plot_skinnyx.pdf new file mode 100644 index 0000000..ed2bc71 Binary files /dev/null and b/figs/plot_skinnyx.pdf differ diff --git a/figs/plot_skinnyx_broken.pdf b/figs/plot_skinnyx_broken.pdf new file mode 100644 index 0000000..d0c49ee Binary files /dev/null and b/figs/plot_skinnyx_broken.pdf differ diff --git a/figs/prop-allclusters.pdf b/figs/prop-allclusters.pdf new file mode 100644 index 0000000..76a7e43 Binary files /dev/null and b/figs/prop-allclusters.pdf differ diff --git a/figs/prop-backbone.pdf b/figs/prop-backbone.pdf new file mode 100644 index 0000000..27a483f Binary files /dev/null and b/figs/prop-backbone.pdf differ diff --git a/figs/prop-broken.pdf b/figs/prop-broken.pdf new file mode 100644 index 0000000..6873d36 Binary files /dev/null and b/figs/prop-broken.pdf differ diff --git a/figs/prop-clusters.pdf b/figs/prop-clusters.pdf new file mode 100644 index 0000000..3ce8dfd Binary files /dev/null and b/figs/prop-clusters.pdf differ diff --git a/figs/prop-lastavalanche.pdf b/figs/prop-lastavalanche.pdf new file mode 100644 index 0000000..a6e8314 Binary files /dev/null and b/figs/prop-lastavalanche.pdf differ diff --git a/figs/prop-none.pdf b/figs/prop-none.pdf new file mode 100644 index 0000000..6e09b93 Binary files /dev/null and b/figs/prop-none.pdf differ diff --git a/figs/prop-spanning.pdf b/figs/prop-spanning.pdf new file mode 100644 index 0000000..7105c12 Binary files /dev/null and b/figs/prop-spanning.pdf differ diff --git a/figs/shekhawat.png b/figs/shekhawat.png new file mode 100644 index 0000000..d7cc4e9 Binary files /dev/null and b/figs/shekhawat.png differ diff --git a/figs/skinny.pdf b/figs/skinny.pdf new file mode 100644 index 0000000..4655c5e Binary files /dev/null and b/figs/skinny.pdf differ diff --git a/figs/skinny_long_clusters.pdf b/figs/skinny_long_clusters.pdf new file mode 100644 index 0000000..6793e8e Binary files /dev/null and b/figs/skinny_long_clusters.pdf differ diff --git a/figs/skinny_short_clusters.pdf b/figs/skinny_short_clusters.pdf new file mode 100644 index 0000000..5694b82 Binary files /dev/null and b/figs/skinny_short_clusters.pdf differ diff --git a/figs/uncut_square.pdf b/figs/uncut_square.pdf new file mode 100644 index 0000000..9181dc5 Binary files /dev/null and b/figs/uncut_square.pdf differ diff --git a/mm_2019.tex b/mm_2019.tex new file mode 100644 index 0000000..fedeace --- /dev/null +++ b/mm_2019.tex @@ -0,0 +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{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} + -- cgit v1.2.3-54-g00ecf