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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}
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\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{Introduction}
\Large
\begin{wrapfigure}{r}{.25\textwidth}
\centering
%\includegraphics[width=0.25\textwidth]{imgs/crack.jpg}
\caption{Cracking in concrete.}
\end{wrapfigure}
Understanding material cracking and fracture is necessary for
understanding the aging and failure of those materials in our buildings
and infrastructure. In ordinary brittle materials like glass, stress at
the tip of a crack causes it to quickly and cleanly propagate through the
material. In ductile materials like metals, this stress is reduced by
plastic deformation around the crack tip, forming the crack's
\textbf{process zone}. In quasi- (or disordered) brittle materials like
concrete, this stress is reduced by opening a complicated network of
microcracks in the process zone. This makes the structure of the
quasibrittle process zone and crack propagation
difficult to study by ordinary means.
\section{Fuse Networks}
\begin{wrapfigure}{l}{.25\textwidth}
\centering
%\includegraphics[width=0.12\textwidth]{imgs/square_network.pdf}
%\includegraphics[width=0.12\textwidth]{imgs/square_high_beta.pdf}\\
%\includegraphics[width=0.12\textwidth]{imgs/voronoi_network.pdf}
%\includegraphics[width=0.12\textwidth]{imgs/voronoi_high_beta.pdf}
\captionof{figure}{Contrasting the square (top) and voronoi (bottom) networks. {\bf Left:} Unbroken fuse networks. {\bf Right:} A
fracture surface in each at low disorder ($\beta=10$).}
\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}{r}{0.25\textwidth}
\centering
%\includegraphics[width=0.25\textwidth]{imgs/ashivni.png}
\captionof{figure}{The `phase diagram' for fracture in homogeneous
systems.}
\label{ashivni}
\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=0.325\textwidth]{imgs/voronoi_homo_big.pdf}
%\includegraphics[width=0.325\textwidth]{imgs/voronoi_homo_med.pdf}
%\includegraphics[width=0.325\textwidth]{imgs/voronoi_homo_lit.pdf}\\
\begin{minipage}[c]{0.325\textwidth}\centering$\pmb{\beta=3}$\end{minipage}
\begin{minipage}[c]{0.325\textwidth}\centering$\pmb{\beta=0.5}$\end{minipage}
\begin{minipage}[c]{0.325\textwidth}\centering$\pmb{\beta=0.03}$\end{minipage}\\
\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}
\begin{figure}
\centering
%\includegraphics[width=0.245\textwidth]{imgs/sample_16.pdf}
%\includegraphics[width=0.245\textwidth]{imgs/sample_32.pdf}
%\includegraphics[width=0.245\textwidth]{imgs/sample_64.pdf}
%\includegraphics[width=0.245\textwidth]{imgs/sample_128.pdf}\\
\begin{minipage}[c]{0.245\textwidth}\centering$\pmb{L=16,\;\beta=1.9}$\end{minipage}
\begin{minipage}[c]{0.245\textwidth}\centering$\pmb{L=32,\;\beta=1.2}$\end{minipage}
\begin{minipage}[c]{0.245\textwidth}\centering$\pmb{L=64,\;\beta=0.78}$\end{minipage}
\begin{minipage}[c]{0.245\textwidth}\centering$\pmb{L=128,\;\beta=0.5}$\end{minipage}\\
\captionof{figure}{Notched fuse networks at critical stress. The
disorder for each system is tuned so that $\beta L^{1/\nu_f}$ is
constant, and the statistics of each should scale trivially.}
\vspace{1pc}
\label{notches}
\end{figure}
\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.
\begin{figure}
\centering
%\includegraphics[width=0.5\textwidth]{imgs/legends.pdf}\\
\vspace{-.5em}
%\includegraphics[width=0.505\textwidth,valign=t]{imgs/sample_collapse_1.pdf}\hfill
%\includegraphics[width=0.48\textwidth,valign=t]{imgs/sample_collapse_2.pdf}\\
\vspace{-.5em}
\captionof{figure}{Disorder-averaged stress $\sigma$ as a function of
distance $x$ from the tip of a critical crack. $\beta L^{1/\nu_f}$ is
constant for each curve. \textbf{Left:} The unmodified stress.
\textbf{Right:} The stress collapsed.}
\label{collapse}
\end{figure}
\section{Next Steps}
\begin{wrapfigure}{r}{0.25\textwidth}
\centering
\vspace{-2em}
%\includegraphics[width=.25\textwidth]{imgs/voronoi_heir.pdf}
\captionof{figure}{A hierarchical voronoi lattice.}
\end{wrapfigure}
\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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