commit 5fa37d2267bb472efeb45862d99e55a2e3430540
parent 949b94d32770a4b2dbbfe8828f43082448b05ce3
Author: Anders Damsgaard <anders@adamsgaard.dk>
Date: Tue, 19 Nov 2019 14:16:33 +0100
Update parameter table
Diffstat:
2 files changed, 16 insertions(+), 17 deletions(-)
diff --git a/continuum-granular-manuscript1.tex b/continuum-granular-manuscript1.tex
@@ -308,18 +308,19 @@ For the first experiment with variable water pressure, we apply a water-pressure
\begin{table*}[htbp]
{\scriptsize
- \begin{tabular}{lllllll}
+ \begin{tabular}{llllllll}
\toprule
- Parameter & Symbol & Units & DEM particles & Storglaci\"aren & WIS & Two Rivers \\
+ Parameter oo2ae
+ Symbol & Units & DEM particles & Caesar Till & Storglaci\"aren & Two Rivers & WIS \\
\midrule
- Friction coefficient & $\mu_\text{s}$ & -- & $0.404_\text{a}$ & $0.494_\text{b}$ & 0.443$_\text{c}$ & $0.321_\text{b}$ \\
- Cohesion & $C$ & kPa & $0_\text{a}$ & $5.0_\text{b}$ & 3.0$_\text{c}$ & $14_\text{b}$ \\
- Representative grain size & $d$ & m & $0.04$ & $1.0\times10^{-3}$ & $1.0\times10^{-3}$ & $1.0\times10^{-3}$ \\
- Hydraulic permeability & $k$ & m$^2$ & $2.0\times10^{-17}_\text{d}$ & $1.3\times10^{-14}_\text{e}$ & $4.9\times10^{-17}_\text{f}$ & $1.3\times10^{-17}_\text{?}$ \\
- Nonlocal amplitude & $A$ & -- & $0.40_\text{a}$ & $0.40_\text{a}$ & $0.40_\text{a}$ & $0.40_\text{a}$ \\
- Nonlinear rate dependence beyond yield & $b$ & -- & $0.9377_\text{g}$ & $0.9377_\text{g}$ & $0.9377_\text{g}$ & $0.9377_\text{g}$ \\
- Grain material density & $\rho_\text{s}$ & kg m$^{-3}$ & $2.6\times10^3_\text{a}$ & $2.6\times10^3_\text{g}$ & $2.6\times10^3_\text{g}$ & $2.6\times10^3_\text{g}$ \\
- Porosity & $\phi$ & -- & $0.25_\text{a}$ & $0.22_\text{h}$ & $0.35_\text{?}$ & $0.37_\text{h}$ \\
+ Friction coefficient & $\mu_\text{s}$ & -- & $0.404_\text{a}$ & $0.532_\text{}$ & $0.494_\text{b}$ & $0.321_\text{b}$ & 0.443$_\text{c}$ \\
+ Cohesion & $C$ & kPa & $0_\text{a}$ & $-6.5_\text{}$ & $5.0_\text{b}$ & $14_\text{b}$ & 3.0$_\text{c}$ \\
+ Representative grain size & $d$ & m & $0.04$ & -- & $1.0\times10^{-3}$ & $1.0\times10^{-3}$ & $1.0\times10^{-3}$ \\
+ Hydraulic permeability & $k$ & m$^2$ & $2.0\times10^{-17}_\text{d}$ & -- & $1.3\times10^{-14}_\text{e}$ & $1.3\times10^{-17}_\text{?}$ & $4.9\times10^{-17}_\text{f}$ \\
+ Nonlocal amplitude & $A$ & -- & $0.40_\text{a}$ & -- & $0.40_\text{a}$ & $0.40_\text{a}$ & $0.40_\text{a}$ \\
+ Nonlinear rate dependence & $b$ & -- & $0.9377_\text{g}$ & -- & $0.9377_\text{g}$ & $0.9377_\text{g}$ & $0.9377_\text{g}$ \\
+ Grain material density & $\rho_\text{s}$ & kg m$^{-3}$ & $2.6\times10^3_\text{a}$ & -- & $2.6\times10^3_\text{g}$ & $2.6\times10^3_\text{g}$ & $2.6\times10^3_\text{g}$ \\
+ Porosity & $\phi$ & -- & $0.25_\text{a}$ & -- & $0.22_\text{h}$ & $0.37_\text{h}$ & $0.35_\text{?}$ \\
\bottomrule
\end{tabular}
}
@@ -340,7 +341,6 @@ For the first experiment with variable water pressure, we apply a water-pressure
}
\end{table*}
-
\section{Results}%
\label{sec:results}
@@ -367,12 +367,12 @@ 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}).
+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}.
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), making the model under these conditions rate strengthening.
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:rate_dependence}d), which is useful as these parameters are often constrained from Mohr-Coulomb tests on till samples.
+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}
@@ -383,11 +383,10 @@ Our model can simulate any combination of effective friction (or friction angle
\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 $A$.
-By inserting relevant material parameters for grain size, friction, stress, and shear velocity (DEM, Table~\ref{tab:params}), we almost exactly replicate the strain distribution with the NGF model (Fig.~\ref{fig:strain_distribution}).
+By inserting relevant material parameters for grain size, friction, stress, and shear velocity (DEM, Table~\ref{tab:params}), the NGF model model approximates the strain distribution well (Fig.~\ref{fig:strain_distribution}).
Both models show that 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 and adjustment towards the critical state.
@@ -407,8 +406,8 @@ Several observations emerge from this parameter sensitivity analysis.
The representative 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 after yielding.
-Static friction and cohesion both linearly scale the bulk friction, as expected with Mohr-Coulomb materials (see also Fig.~\ref{fig:rate_dependence}).
+Our implementation of cohesion does not influence strain after yield.
+Static friction and cohesion both linearly scale the bulk friction, as expected with Mohr-Coulomb materials (see also Fig.~\ref{fig:mohr_coulomb}).
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.
diff --git a/experiments/fig-mohr_coulomb.pdf b/experiments/fig-mohr_coulomb.pdf
Binary files differ.
