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%
% poster_05.2014.tex - description
%
% Created by on Tue May 13 12:26:55 PDT 2014.
% Copyright (c) 2014 pants productions. All rights reserved.
%
\documentclass[portrait]{a0poster}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage[]{amsmath}
\usepackage{amssymb,latexsym,mathtools,multicol,lipsum,wrapfig}
\usepackage[font=normalsize,labelfont=bf]{caption}
\usepackage{tgheros}
\usepackage[helvet]{sfmath}
\usepackage[export]{adjustbox}
\renewcommand*\familydefault{\sfdefault}
\mathtoolsset{showonlyrefs=true}
\setlength\textwidth{194pc}
\begin{document}
\setlength\columnseprule{2pt}
\setlength\columnsep{5pc}
\renewenvironment{figure}
{\par\medskip\noindent\minipage{\linewidth}}
{\endminipage\par\medskip}
\renewcommand\section[1]{
\vspace{3pc}
\noindent\huge\textbf{#1}\large
\vspace{1.5pc}
}
\newcommand\unit[1]{\hat{\vec{#1}}}
\renewcommand\vec[1]{\boldsymbol{\mathbf{#1}}}
\newcommand\norm[1]{\|#1\|}
\def\rr{\rho}
\newcommand\abs[1]{|#1|}
\def\dd{\mathrm d}
\def\rec{\mathrm{rec}}
\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}
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