commit 49d40537f70fb8b80551c219e0cbc399759db60c
parent 3167f948531dcb7e8b54cf5e212323572fc4249f
Author: Anders Damsgaard <anders@adamsgaard.dk>
Date: Mon, 25 Nov 2019 12:55:18 +0100
Update figure and write caption for new figure 1
Diffstat:
3 files changed, 9 insertions(+), 51 deletions(-)
diff --git a/continuum-granular-manuscript1.tex b/continuum-granular-manuscript1.tex
@@ -304,25 +304,16 @@ A rate-controlled setup is the opposite end member, where changes in bed frictio
\includegraphics[width=0.49\textwidth]{experiments/fig-mohr_coulomb.pdf}
\includegraphics[width=0.49\textwidth]{experiments/fig-strain_distribution.pdf}
\caption{\label{fig:validation}%
- Experimental setup and model validation.
- a)
+ Comparison between shear experiments on subglacial till and the non-local granular fluidigy (NGF) model applied in this study.
+ a) Experimental setup for the NGF model in one-dimensional shear.
+ The upper boundary is constant friction $\mu$ (stress controlled), or constant velocity $v_x$ (velocity controlled).
+ b) Rate dependence of critical-state till friction in laboratory experiments \citep[after][]{Iverson2010}, and influence of rate-dependence factor $b$ in Eq.~\ref{eq:g_local} on friction in the NGF continuum model with $\mu_\text{s} = 0.5$ and $\sigma_\text{n}' = 100$ kPa.
+ c) Mohr-Coulomb analysis of till samples in laboratory experiments and the NGF model.
+ d) Modeled strain distribution under varying effective normal stress ($\sigma_\text{n}'$) with the discrete-element method \citep[DEM,][]{Damsgaard2013} and the NGF model.
}
\end{center}
\end{figure*}
-\begin{figure*}[htbp]
- \begin{center}
- \includegraphics{experimental-setup.pdf}
- \caption{\label{fig:experimental-setup}%
- Experimental setup for 1d shear experiments.
- The upper boundary conditions for the granular solver are fixed applied friction $\mu(z=L_z)$ (stress controlled), or fixed shear velocity $v_x(z=L_z)$ (rate controlled).
- The upper normal stress ($\sigma_\text{n}$) is constant, but effective normal stress ($\sigma'_\text{n}$) varies if $p_\text{f}$ changes.
- The granular phase lower boundary condition is no slip.
- For the fluid solver, the top is fixed fluid pressure $p_\text{f}(z=L_z)$, which can be constant or vary in time.
- The lower fluid boundary condition is a constant hydrostatic gradient (von Neumann, $dp_\text{f}/dz(z=0) = \rho_\text{f}G$).
- }
- \end{center}
-\end{figure*}
\begin{table*}[htbp]
@@ -354,30 +345,6 @@ A rate-controlled setup is the opposite end member, where changes in bed frictio
\section{Results}%
\label{sec:results}
-\begin{figure*}[htbp]
- \begin{center}
- \includegraphics[width=7.5cm]{experiments/fig-rate_dependence.pdf}
- \caption{\label{fig:rate_dependence}%
- \nocite{Tika1996}
- \textbf{a:}
- Rate dependence in till friction from laboratory experiments \citep[after][]{Iverson2010}.
- \textbf{b:}
- Influence of rate-dependence factor $b$ in Eq.~\ref{eq:g_local} on post-failure friction in the current continuum model.
- Here, $\mu_\text{s} = 0.5$ and $\sigma_\text{n}' = 100$ kPa.
- }
- \end{center}
-\end{figure*}
-
-\begin{figure*}[htbp]
- \begin{center}
- \includegraphics[width=7.5cm]{experiments/fig-mohr_coulomb.pdf}
- \caption{\label{fig:mohr_coulomb}%
- Mohr-Coulomb analysis of till samples in laboratory experiments \citep[after][]{Iverson2010},
- and the continuum model presented here, with simulation parameters listed in Table~\ref{tab:params}.
- }
- \end{center}
-\end{figure*}
-
We first compare the modeled mechanical behavior to various tills tested in laboratory settings (Fig.~\ref{fig:rate_dependence} and~\ref{fig:mohr_coulomb}, after \citet{Iverson2010}).
Over five orders of strain-rate magnitude, some tills show slight rate weakening and others are slightly rate strengthening (Fig.~\ref{fig:rate_dependence}a).
Shear-strain rates up to 5.000 a$^{-1}$ are realistic for natural glacier systems \citep{Cuffey2010}.
@@ -385,15 +352,6 @@ Our model is effectively rate-independent over most of the range, but higher $b$
The modeled friction value can be linearly scaled by adjusting $\mu_\text{s}$ in Eqs.~\ref{eq:g_local} and~\ref{eq:cooperativity}.
Our model can simulate any combination of effective friction (or friction angle $\varphi = \tan^{-1}(\mu_s)$) and cohesion (Fig.~\ref{fig:mohr_coulomb}), which is useful as these parameters are often constrained from Mohr-Coulomb tests on till samples.
-\begin{figure*}[htbp]
- \begin{center}
- \includegraphics[width=0.48\textwidth]{experiments/fig-strain_distribution.pdf}
- \caption{\label{fig:strain_distribution}%
- Modeled strain distribution under varying effective normal stress ($\sigma_\text{n}'$) with the discrete-element method (DEM, marked with symobo \citep[DEM, marked with symbols,][]{Damsgaard2013}, and the continuum model presented here (continuous lines).
- }
- \end{center}
-\end{figure*}
-
The NGF model contains parameter $A$ for adjusting the degree of material non-locality (Eq.~\ref{eq:cooperativity}).
Unfortunately, no laboratory experiment exists in the literature where the effects of normal stress are analysed for changes in strain distribution in the till.
Instead, we compare the modeled strain distribution with discrete-element results from \citet{Damsgaard2013} which allow us to calibrate $A$.
diff --git a/experimental-setup.pdf b/experimental-setup.pdf
Binary files differ.
diff --git a/experimental-setup.svg b/experimental-setup.svg
@@ -439,7 +439,7 @@
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@@ -581,12 +581,12 @@
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