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\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}

