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\makeatletter


\begin{document}

\title{Supplemental Material: Effects of inertia on the steady-shear rheology of
disordered solids}


\author{Alexandre Nicolas$^{\S}$, Jean-Louis Barrat}


\affiliation{LIPhy, Universit\'e Grenoble-Alpes \& CNRS, F-38000 Grenoble,
France}


\author{J\"org Rottler$^{\S}$ ($^{\S}$: equal contribution)}


\affiliation{Department of Physics and Astronomy, The University of British
Columbia,
6224 Agricultural Road, Vancouver, British Columbia V6T 1Z4, Canada}

\maketitle 


\section{I. Velocity autocorrelation functions}

Figure~\ref{fig:vacf_quiescent} presents the transverse velocity
autocorrelation functions for different damping strengths, in the quiescent
system.
\begin{figure}[H]
\begin{centering}
%\subfloat[$\dot{\gamma}=0$ and $T_{0}=0.16$.]
{\begin{centering}
\includegraphics[width=8cm]{vacf1}
\par\end{centering}

}
\par\end{centering}

\begin{centering}
{\begin{centering}
\includegraphics[width=8cm]{vacf2}
\par\end{centering}

}
\par\end{centering}
\begin{centering}

\caption{ \label{fig:vacf_quiescent}Transverse velocity
autocorrelation functions for particle mass $m=1$ and different damping
magnitudes $\zeta$ in (top) quiescent systems, \emph{i.e.}, $\mathrm{Ei}=0$, at
$T_{0}=0.16$ and
(bottom) sheared systems, with  Ei and $T_{0}$ as
indicated in the legend. Note that the values of $T_{0}$ used for the flows at
$\zeta=1$ were chosen to match the kinetic temperatures of the strongly
underdamped flows ($\zeta=10^{-3}$) at the same rescaled shear rate Ei. The
dashed black line is an envelope fit with the function
$\mathrm{exp}(\frac{-t}{0.1\tau_\mathrm{vib}})$}
\end{centering}
\end{figure}


\section{II. Kinetic energy distributions}

Figure~\ref{fig:ke} presents the distribution of the kinetic energies among the
particles in the system.

\begin{figure}[H]
\begin{centering}
\includegraphics[width=9cm]{distke}
\par\end{centering}

\caption{\label{fig:ke}Distribution of the kinetic energies $e_K=
\frac{1}{2} m v_i^2$ of the
individual particles in the athermal and thermostatted systems (as indicated
in the legend). Solid black lines are fits to Boltzmann distributions $\propto
e^{-e_K/T_K}$}
\end{figure}

\section{III. Robustness of our findings: test of a different thermostat}

To test the robustness of our results, we subsitute a Langevin thermostat for
the Dissipative Particle Dynamics scheme used in the main text. 
Figure~\ref{fig:Langevin} proves that, for low enough damping strength, the
athermal Langevin flow curve also becomes nonmonotonic, with a rate-weakening
region. More importantly, the
thermostatting procedure described in the main text is also operational here,
as evidenced in the figure. This confirms the connection between the
underdamped rheology and the extent of kinetic heating.


\begin{figure}[H]
\begin{centering}
\includegraphics[width=8cm]{flow_langevin}
\par\end{centering}
\caption{\label{fig:Langevin} Flow curves obtained with a Langevin thermostat
(\emph{black})
in an underdamped athermal system and (\emph{red}) in the associated (less
underdamped)
thermostatted
system.}
\end{figure}

\section{IV. Qualitative explanation for the nonmonotonic flow curves in the
highly underdamped regime}

As can be seen in Fig.~2 of the paper, the flow curves at large enough
$\mathrm{Q}$ are nonmonotonic and thus
display a minimum $\left(\mathrm{Ei}_{m},\Sigma_{m}\right)$, where $\mathrm{Ei}$
is the rescaled shear rate. At this point, using the chain rule,
\begin{eqnarray*}
0 & = & \frac{d\Sigma}{d\mathrm{Ei}}\left(\mathrm{Ei}_{m}\right)\\
 & = & \frac{\partial\Sigma}{\partial
T_{K}}\Big|_{\mathrm{Ei}_{m}}\cdot\frac{dT_{K}}{d\mathrm{Ei}}+\frac{
\partial\Sigma}{\partial\mathrm{Ei}}\Big|_{T_{K}=T_{K}\left(\mathrm{Ei}_{m}
\right)},
\end{eqnarray*}
(As a technical detail, note that, in practice, the second partial
derivative - at fixed kinetic temperature - can only be calculated
in a system with stronger damping thermostatted to
$T_{0}=T_{K}\left(\mathrm{Ei}_{m}\right)$).
Thus, the presence of a minimum results from a competition between
two (antagonistic) effects:

(i) an increase of the sample temperature with the shear-rate, which
bends the flow curve downwards

(ii) an intrinsic rate effect at large shear rates, whereby the driving
gets so strong that particles no longer remain at the bottom of the
energy wells (collisions, etc.).

When $\mathrm{Q}$ increases, \emph{i.e.}, when the damping weakens,
the temperature effect (i) gets stronger, so that the minimum is expected
to be deeper, which is indeed confirmed by Fig. 2 of the main text.


\section{V. Deviations from the predicted behaviors observed at high kinetic
temperatures}

At high kinetic temperatures $T_K$(\emph{i.e.}, in particular at large shear
rates and/or large values of Q), Figs.~2 and 3 of the main text show that
deviations appear from the Chattoraj-Lemaitre formula for the
temperature-dependence of the stress (Eq.~3), and, to a much lower extent, from
the scaling prediction represented in Fig.~3.

This is not surprising: Chattoraj and Lemaitre explicitly derived their for
formula in the low-temperature limit, and our scaling prediction was based on
the hypothesis that regions yield (undergo a plastic event) at a local stress
$\Sigma_0$, whereas, owing to thermal activation, the local stress at yield may
be significantly lower at high $T_K$. 



\end{document}
