commit 142b1a670939e25cf295e867b026092516f3e939
parent 74c363cc6a9ce131e137a595138379090a66bc78
Author: Anders Damsgaard <anders@adamsgaard.dk>
Date: Tue, 8 Oct 2019 15:05:13 +0200
Finish first draft of intro
Diffstat:
1 file changed, 66 insertions(+), 14 deletions(-)
diff --git a/continuum-granular-manuscript1.tex b/continuum-granular-manuscript1.tex
@@ -87,32 +87,58 @@ In these forms the shear stress is still limited to the Mohr-Coulomb value at hi
However, the Coulomb-frictional parameterizations do not describe the actual sediment deformation, but describe the basal friction felt by the flowing ice.
Shear deformation is known to deepen under increasing effective normal stress \citep{Fischer1997, Iverson1999, Boulton2001, Damsgaard2013}, and this may be a primary ingredient for growth of subglacial topography \citep[e.g.][]{Schoof2007}.
In order to model soft-bed sliding and till continuity, a model is necessary that accurately describes subglacial shear strain while being in accordance to Mohr-Coulomb friction and sediment near-plastcitiy.
-The discrete element model for sediment deformation by \citet{Damsgaard2013} includes Coulomb-frictional physics and sediment strain distribution, but is far too costly for coupled ice-till computations.
+The discrete element model for sediment deformation by \citet{Damsgaard2013} includes Coulomb-frictional physics and sediment strain distribution, but is far too detailed and costly for coupled ice-till computations.
+\subsection{Insights from granular physics}%
+Soils, tills, and other sediments are granular materials, consisting of discrete grains interacting with frictional losses.
+A key characteristic of granular materials is the ability to change mechanical phase \citep[jammed, flowing, in suspension, e.g.][]{Jaeger1996, deGennes1999, Forterre2008}.
+The holy grail within the field of granular physics is to find a unifying mathematical framework describing behavior across all phases.
+The fundamental understanding of the strength of granular materials goes all the way back to the 18th century.
+The Mohr-Coulomb constitutive relation postulates a linear relationship betweeen effective normal stress on a shear zone and the maximum shear stress it can support.
+For most sedimentary materials cohesion C is close to zero.
+Importantly, the Mohr-Coulomb relationship only describes the yield point of inertialess deformation, and there is no length scale to shear zones.
-% mu(I)
-% GDR-MiDi2004
-% Jop2005
-% Pouliquen2006
-% Forterre2008
-
-% non-local mu(I)
-% Henann2013, Kamrin2015, Henann2016
-
-
-However, with $\mu(I)$ being a \emph{local} rheology, the local stress state and material properties alone define the local strain rate $\mu(I)$.
+\citet{Bagnold1954} realized that granular flows show a complex rate dependence.
+It was later shown that a dimensionless inertia number $I$ summarizes the mechanical behavior of cohesionless, critical-state, granular flows \citep{GDR-MiDi2004}:
+\begin{equation}
+ I = \frac{\dot{\gamma} d}{\sqrt{\sigma_\text{n}/\rho_\text{s}}},
+ \label{eq:inertia_number}
+\end{equation}
+where $d$ is the representative grain size, $\sigma_\text{n}$ is the normal stress, and $\rho_\text{s}$ is the density of the grain material.
+$I$ describes the relative importance of the microscopic and macroscopic time scales.
+For example, if a granular material is sheared quickly, $I$ goes up as inertia increases and grains spend less time in a locked arrangement.
+On the contrary, if normal stress increases $I$ goes down as grain mobility in the shear zone is restricted.
+For a given material, the critical-state ratio between shear stress and normal stress $\mu = \tau/\sigma_\text{n}$ depends non-linearly on $I$ \citep{daCruz2005, Jop2006}.
+The dependence allows for empirical relationships where $\mu(I)$ takes form of a highly non-linear Bingham rheology.
+Tuned to experiments on simple granular materials, the material is more rate dependent at large inertia numbers (i.e., ``landslide regime'').
+With smaller inertia numbers, the behavior smoothly transitions to a ``pseudo-static regime'' of near rate independence.
+Similarly, shear-zone porosity was found to linearly depend on $I$ \citep{Pouliquen2006}.
+The relationships for strength and porosity act as constitutive relations, making continuum modeling of granular flows possible.
+\citet{Pailha2008} and \citet{Pailha2009} demonstrated simple coupling of the continuum granular model to pore-pressure dynamics.
+However, these models are \emph{local}, meaning that the spatially local stress state determines the local strain-rate response alone.
+Compared to glacier models, this formulation corresponds to the shallow-ice approximation where local ice-surface slope is the sole factor for local shear-strain rate.
+But, like many glacial settings, granular deformation is often \emph{non-local}.
+For example, granular shear zones have a minimum width, dependent on grain characteristics \citep[e.g.][]{Nedderman1980, Kamrin2018}.
Furthermore, $\mu(I)$ rheology does not work for slow flows as the thickness of shear bands depends on the shear velocity and vanishes in the quasi-static limit \citep{Forterre2008}.
+\citet{Henann2013} presented the non-local granular fluidity (NGF) model where a \emph{fluidity} field accounts for the non-local effects on deformation.
+The NGF model builds on the previous $\mu(I)$ rheology, but accurately describes strain distribution in a variety of experimental settings.
+However, the NGF model assumes all the material to be in the critical state and a uniform porosity throughout the domain.
+The fluidity acts as a state variable, describing the phase transition between jammed and flowing parts.
+The resultant granular rheology is based on observations, non-dimensionalized on the base of material properties.
+The \citet{Henann2013} is fundamentally different than viscous rheologies proposed for glacial tills, but is more akin to statistical mechanics.
+The NGF model allows upscaling of the discrete element method \citep[e.g.][]{Cundall1979, Damsgaard2013}, while remaining true to the physics.
+However, the NGF model is dry, and in the context of subglacial mechanics, dry models are generally not useful.
+In this paper we expand the steady-state NGF continuum model for granular flow by \citet{Henann2013} with cohesion and a coupling to pore-pressure diffusion, and analyze how fluid-pressure perturbations affect strain distribution and material stability.
\section{Methods}%
\label{sec:methods}
\subsection{Granular flow}%
\label{sub:granular_flow}
-We expand a steady-state continuum model for granular flow by \citet{Henann2013} with material cohesion and a coupling to pore-pressure diffusion.
-The sediment deforms as a highly nonlinear Bingham material with yield beyond the Mohr-Coulomb failure limit.
+In the NGF model, the sediment deforms as a highly nonlinear Bingham material with yield beyond the Mohr-Coulomb failure limit \citep[e.g.][]{Henann2013, Henann2016}.
We assume that the elasticity is negligible and set the total shear rate $\dot{\gamma}$ to consist of a plastic contribution $\dot{\gamma}^\text{p}$:
\begin{equation}
\dot{\gamma} \approx \dot{\gamma}^\text{p} = g(\mu_\text{c}, \sigma_\text{n}') \mu,
@@ -301,6 +327,32 @@ 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]
