commit 49c88bd8a879487f69798fd57c98e291ea11e7f8
parent 9af4c78d1c79e35a777d652691aad0825cc34c4e
Author: Anders Damsgaard <anders@adamsgaard.dk>
Date: Wed, 9 Oct 2019 14:43:05 +0200
Work on results section
Diffstat:
1 file changed, 48 insertions(+), 33 deletions(-)
diff --git a/continuum-granular-manuscript1.tex b/continuum-granular-manuscript1.tex
@@ -289,6 +289,21 @@ The simulated velocities are for the most part far greater than any glacial sett
Parameter values and their references are listed in Table~\ref{tab:params}.
For the first experiment with variable water pressure, we apply a water-pressure forcing amplitude of 50 kPa that modulates effective stress at the top around 100 kPa (Fig.~\ref{fig:stick_slip}).
+\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 shear stress $\tau(z=L_z)$ (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]
{\scriptsize
\begin{tabular}{lllllll}
@@ -326,34 +341,6 @@ For the first experiment with variable water pressure, we apply a water-pressure
\section{Results}%
\label{sec:results}
-% Calibration of A against prior experiments
-
-% Unfortunately, there isn't a laboratory experiment in the literature where the
-% effects of normal stress are analysed for changes in strain distribution in
-% the till. So we will have to do with my discrete-element simulations.
-
-% In the DEM, low normal stresses produce shallow deformation and higher normal
-% stresses deepen deformation.
-
-% There are field observations from glaciers indicating similar trends, but the
-% physical setting is less well controlled.
-
-% By plugging in the corresponding stresses and material properties to the
-% non-local continuum model, we can almost exactly replicate the DEM result.
-
-% The simulations for this figure took about two months to compute with a
-% powerful graphics processing unit, whereas this was done in a fraction of a
-% second.
-
-% Stress dependence of sediment advection is very interesting because it is
-% relevant for non-planar ice-bed interfaces and theories of landform
-% instability (drumlins, ribbed moraines, etc)
-
-
-
-
-
-
\begin{figure*}[htbp]
\begin{center}
\includegraphics[width=15cm]{experiments/fig1.pdf}
@@ -362,7 +349,6 @@ For the first experiment with variable water pressure, we apply a water-pressure
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 continuum model.
- The friction value can be shifted up and down by adjusting $\mu_\text{s}$ in Eqs.~\ref{eq:g_local} and~\ref{eq:cooperativity}.
Here, $\mu_\text{s} = 0.5$ and $\sigma_\text{n}' = 100$ kPa.
\textbf{c:}
Mohr-Coulomb analysis of till samples in laboratory experiments \citep[after][]{Iverson2010}.
@@ -372,6 +358,14 @@ For the first experiment with variable water pressure, we apply a water-pressure
\end{center}
\end{figure*}
+We first compare the modeled mechanical behavior to various tills tested in laboratory settings (Fig.~\ref{fig:rate_dependence}), after \citet{Iverson2010}.
+Over five orders of 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}.
+Our model is effectively rate-independent over most of the range, but higher $b$ values provide larger frictional resistance at extreme shear-strain rates (Fig.~\ref{fig:rate_dependence}b).
+The modeled friction value can be shifted up and down by adjusting $\mu_\text{s}$ in Eqs.~\ref{eq:g_local} and~\ref{eq:cooperativity}.
+Tills are Mohr-Coulomb materials, where the shear stress linearly depends on effective normal stress (Fig.~\ref{fig:rate_dependence}c).
+Our modified NGF model can simulate any combination of effective friction (or friction angle $\varphi = \tan^{-1}(\mu_s)$) and cohesion (Fig.~\ref{fig:rate_dependence}d).
+
\begin{figure*}[htbp]
\begin{center}
\includegraphics[width=0.48\textwidth]{experiments/damsgaard2013-fig8.pdf}\\
@@ -382,19 +376,32 @@ For the first experiment with variable water pressure, we apply a water-pressure
\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}.
+By inserting relevant material parameters for grain size, friction, stress, and shear velocity (Table~\ref{tab:params}), we almost exactly replicate the strain distribution with the NGF model (Fig.~\ref{fig:strain_distribution}).
+Sediment advection is pressure dependent, with low effective normal stresses producing shallow deformation, and high effective normal stresses deepening the material mobilization.
+The DEM results took more than two months of computational time, whereas the continuum model is completed in a fraction of a second, albeit without detail of individual particle kinematics.
+
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=15cm]{experiments/fig3.pdf}
- \caption{\label{fig:till_simulation}%
+ \caption{\label{fig:parameter_test}%
Analysis of parameter influence on steady-state strain distribution and bulk friction during shear.
- All experiments are at a shear rate of 300 m a$^{-1}$ and a normal stress of $\sigma_\text{n}'$ = 100 kPa.
Parameter values marked with an asterisk (*) are used outside of the individual parameter sensitivity tests.
}
\end{center}
\end{figure}
-Figure~\ref{fig:stick_slip} shows the water-pressure forcing and observed shear dynamics over a simulation time of seven days.
-The shear velocities during the first cycle ($t<1$ d) is slightly different from later cycles ($t>1$ d) since the model is initialized with a hydrostatic water-pressure distribution.
+Figure~\ref{fig:parameter_test} is a systematic analysis of parameter influence under a constant shear rate.
+All experiments are at a shear rate of 300 m a$^{-1}$ and a normal stress of $\sigma_\text{n}'$ = 100 kPa.
+The grain size $d$ has a major influence on the strain distribution, where finer materials show deeper deformation.
+The material is slightly weaker with larger grain sizes.
+The shear zone is more narrow with higher material static friction coefficients ($\mu_\text{s}$), as the material is less willing to fail.
+Our implementation of cohesion does not influence strain.
+Static friction and cohesion both linearly scale the bulk friction, as expected with Mohr-Coulomb materials (see also Fig.~\ref{fig:rate_dependence}).
+The non-local amplitude $A$ slightly changes the curvature of the shear strain profile, but does not affect the overall friction.
+There is a significant strengthening when the bed thickness $L_z$ begins to constrict the shear zone thickness.
\begin{figure}[htbp]
\begin{center}
@@ -409,6 +416,10 @@ The shear velocities during the first cycle ($t<1$ d) is slightly different from
\end{center}
\end{figure}
+Next we vary the top water pressure and observe the shear dynamics over a simulation time of seven days Figure~\ref{fig:stick_slip}.
+We perform tests under both stress- and rate-controlled configurations.
+The shear velocities during the first cycle ($t<1$ d) is slightly different from later cycles ($t>1$ d) since the model is initialized with a hydrostatic water-pressure distribution.
+
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.49\textwidth]{experiments/fig5.pdf}
@@ -476,6 +487,10 @@ Practically all of the shear strain through a perturbation cycle occurs above th
\citet{Tulaczyk1999} and \citet{Tulaczyk2000} demonstrated that diffusion of pore-pressure variations into the bed can distribute strain away form the ice-bed interface.
+% Stress dependence of sediment advection is very interesting because it is
+% relevant for non-planar ice-bed interfaces and theories of landform
+% instability (drumlins, ribbed moraines, etc)
+
\section{Conclusion}%
\label{sec:conclusion}