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%  poster_05.2014.tex - description
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%  Created by  on Tue May 13 12:26:55 PDT 2014.
%  Copyright (c) 2014 pants productions. All rights reserved.
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\documentclass[portrait]{a0poster}

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\usepackage[]{amsmath}
\usepackage{amssymb,latexsym,mathtools,multicol,lipsum,wrapfig}
\usepackage[font=normalsize,labelfont=bf]{caption}
\usepackage{tgheros}
\usepackage[helvet]{sfmath}
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\setlength\textwidth{194pc}
\begin{document}

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  {\par\medskip\noindent\minipage{\linewidth}}
    {\endminipage\par\medskip}

    \renewcommand\section[1]{
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    \noindent\huge\textbf{#1}\large
    \vspace{1.5pc}
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\def\CC{{C\kern-.05em\lower-.4ex\hbox{\cpp +\kern-0.05em+}} }
\font\cpp=cmr24
\def\max{\mathrm{max}}

% Our mysterious boy
\def\urusi{URu$_{\text2}$Si$_{\text2}$}

\def\e{{\text{\textsc{elastic}}}} % "elastic"
\def\i{{\text{\textsc{int}}}} % "interaction"

\def\Dfh{D$_{\text{4h}}$}

% Irreducible representations (use in math mode)
\def\Aog{{\text A_{\text{1g}}}}
\def\Atg{{\text A_{\text{2g}}}}
\def\Bog{{\text B_{\text{1g}}}}
\def\Btg{{\text B_{\text{2g}}}}
\def\Eg {{\text E_{\text  g}}}
\def\Aou{{\text A_{\text{1u}}}}
\def\Atu{{\text A_{\text{2u}}}}
\def\Bou{{\text B_{\text{1u}}}}
\def\Btu{{\text B_{\text{2u}}}}
\def\Eu {{\text E_{\text  u}}}

% Variables to represent some representation
\def\X{\text X}
\def\Y{\text Y}

% Units
\def\J{\text J}
\def\m{\text m}
\def\K{\text K}
\def\GPa{\text{GPa}}
\def\A{\text{\r A}}

% Other
\def\op{\textsc{op}} % order parameter
\def\ho{\textsc{ho}} % hidden order
\def\rus{\textsc{rus}} % resonant ultrasound spectroscopy 
\def\Rus{\textsc{Rus}} % Resonant ultrasound spectroscopy 
\def\afm{\textsc{afm}} % antiferromagnetism 
\def\recip{{\{-1\}}} % functional reciprocal

  \noindent\hspace{177pc}\includegraphics[width=18pc]{CULogo-red120px.eps}
  \vspace{-24.5pc}\\
  \Huge \textbf{Elastic properties of hidden order in URu$_{\text2}$Si$_{\text2}$ are reproduced by\\ staggered nematic order}
  \bigskip\\
  \huge \textbf{Jaron~Kent-Dobias, Michael Matty \& Brad J Ramshaw}
  \vspace{1pc}


  \begin{multicols}{2}

    \section{Resonant ultrasound spectroscopy}
    \Large

    \begin{wrapfigure}{r}{.25\textwidth}
      \centering
      \includegraphics[width=0.25\textwidth]{rus_resonances.jpg}
      \caption{Resonances }
    \end{wrapfigure}

    Strain measures the displacement of material from its equilibrium configuration. 

    \section{Fuse Networks}

    \begin{wrapfigure}{l}{.25\textwidth}
      \centering
      \includegraphics[width=0.55\columnwidth]{urusi_modes.png}
      \captionof{figure}{
        The crystal structure of \urusi\ and the influence of irreducible strains on it.
      }
      \label{nets}
    \end{wrapfigure}

    \Large

    We model quasibrittle cracking and fracture using simulations of fuse
    networks, which are electrical systems of resistive fuses whose random
    current thresholds $t$ are cumulatively distributed by $t^\beta$.  $\beta$
    parameterizes the amount of disorder in the system: large
    $\beta$ corresponds to vanishing disorder, small $\beta$ to very large
    disorder.  Fracture is performed adiabatically: the fuse whose ratio of
    current to threshold is largest breaks, and the current across the
    networks is recomputed.  In order to reduce lattice effects, which become large for
    small disorder (see Figure \ref{nets}), we use voronoi meshes for our fuse networks.


    \section{Homogeneous Scaling}

    \begin{wrapfigure}{}{0.6\columnwidth}
      \centering
      \includegraphics[width=0.6\columnwidth]{paper/phase_diagram_experiments}

      \vspace{1em}

      \includegraphics[width=0.3\columnwidth]{paper/phases_scalar}\hspace{-0.75em}
      \includegraphics[width=0.3\columnwidth]{paper/phases_vector}
      \captionof{figure}{
        Phase diagrams for (a) \urusi\ from Phys Rev B \textbf{77} 115117 (2008) (b) mean
        field theory of a one-component ($\Bog$ or $\Btg$) Lifshitz point (c) mean
        field theory of a two-component ($\Eg$) Lifshitz point. Solid lines denote
        continuous transitions, while dashed lines denote first order transitions.
      }
      \label{phase_diagram}
    \end{wrapfigure}

    \Large

    The problem of fracture in fuse networks was unresolved until recently.
    For low disorder, fracture is nucleation-like, similar to that of ordinary
    brittle systems.  At large disorder, fracture occurs after a very large
    amount of uncorrelated damage, and appears percolation-like.
    Sethna and Shekhawat developed a theory which unifies these behaviors with
    an {\sc rg} crossover at intermediate disorder characterized by mean-field
    avalanches.  The percolation-like behavior at high disorder was shown to
    be unstable under course-graining, and therefore any nonzero $\beta$
    will cause nucleated fracture at a sufficiently large system size (see
    Figure \ref{ashivni}).

    In unpublished work, Shekhawat and Sethna found a scaling form for the
    distribution of network strengths $\sigma_\max$, the largest current
    applied to the network before it has broken.  It is given by
    \begin{equation}
      P(\sigma_\max\mid\beta, L,u)=\sigma_\max^{-\tau_\sigma}\mathcal P(\beta
      L^{1/\nu_f},\sigma_\max L^\delta,uL^{-\Delta/\nu_f})\label{form}
    \end{equation}
    where $\tau_\sigma$, $\nu_f$, $\delta$ and $\Delta$ are universal
    exponents, $L$ is the system size, and $u$ is an irrelevant scaling
    variable.  

    \begin{figure}
      \vspace{1pc}
      \centering
      \includegraphics[width=\columnwidth]{paper/fig-stiffnesses.pdf}
      \captionof{figure}{Fractured fuse networks at various $\beta$.  Each colored region shows a
        contiguous cracked cluster.  The black region shows the surface of
        the spanning crack.}
      \label{homo}
      \vspace{1pc}
    \end{figure}


    \section{Scaling in the Process Zone}


    \Large
    We have made progress for developing a scaling theory of damage and stress
    in the process zone of quasibrittle cracks.  We have essentially taken the
    scaling behavior of \eqref{form} as an ansatz for how the corresponding
    qualities of a critically semi-cracked network should scale.  We have
    begun demonstrating the validity of this theory.  Figure \ref{notches}
    shows critically cracked networks at constant $\beta L^{1/\nu_f}$, an
    invariant scaling combination under this theory.  As can be seen in Figure
    \ref{collapse}, the disorder-averaged stress profiles caused by each
    collapses nicely.


    \section{Next Steps}


    \Large

    The voronoi networks we are able to generate allow us great flexibility
    for future multiscale computational modelling.  Once we have hashed out
    our scaling theory more thoroughly, we plan on using it to probe much
    larger systems than previously feasible using networks whose fuse density
    becomes smaller in regions of less importance.  This should allow us to
    see cleaner stress and damage scaling and cleanly stitch our discrete system to a
    continuum approximation.

  \end{multicols}
\end{document}

