commit 9af4c78d1c79e35a777d652691aad0825cc34c4e
parent a468d014e67e2d3f8f8253cf50a0ffcf5fafd4aa
Author: Anders Damsgaard <anders@adamsgaard.dk>
Date: Wed, 9 Oct 2019 10:24:51 +0200
Add figure with experimental setup
Diffstat:
3 files changed, 721 insertions(+), 14 deletions(-)
diff --git a/continuum-granular-manuscript1.tex b/continuum-granular-manuscript1.tex
@@ -112,8 +112,8 @@ On the contrary, if normal stress increases $I$ goes down as grain mobility in t
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}.
+With smaller inertia numbers, the behavior smoothly transitions to a ``pseudo-static regime'' of near rate independence ($I < 10^{-3}$).
+Similarly, shear-zone porosity was in monodisperse materials 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.
@@ -125,7 +125,7 @@ Furthermore, $\mu(I)$ rheology does not work for slow flows as the thickness of
\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 fluidity acts as a state variable, describing the phase transition between jammed and flowing parts \citep[e.g.][]{Zhang2017}.
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.
@@ -139,7 +139,7 @@ In this paper we expand the steady-state NGF continuum model for granular flow b
\subsection{Granular flow}%
\label{sub:granular_flow}
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 $\dot{\gamma}^\text{p}$:
+We assume that 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,
\label{eq:shear_strain_rate}
@@ -159,7 +159,7 @@ The local contribution to fluidity is defined as:
where $d$ [m] is the representative grain diameter, $\mu_\text{s}$ [-] is the static Coulomb yield coefficient, $C$ [Pa] is the material cohesion, and $b$ [-] is the non-linear rate dependence beyond yield.
The failure point is determined by the Mohr-Coulomb constituent relation.
Beyond failure, the flow is governed by a Poisson-type equation that distributes strain in space according to material properties and stress state.
-The flow non-locality is summarized by the cooperativity length $\xi$:
+The degree of non-locality is scaled by the cooperativity length $\xi$:
\begin{equation}
\nabla^2 g = \frac{1}{\xi^2(\mu)} (g - g_\text{local}),
\label{eq:g}
@@ -169,8 +169,8 @@ where
\xi(\mu) = \frac{Ad}{\sqrt{|(\mu - C/\sigma_\text{n}') - \mu_\text{s}|}}.
\label{eq:cooperativity}
\end{equation}
-The non-locality scales with nonlocal amplitude $A$ [-].
-In the above mathematical framework, the material slightly strengthens when the shear zone size is restricted by thickness of the granular bed.
+The non-locality scales with nonlocal amplitude $A$ [-] and grain size $d$.
+In the above framework, the material strengthens when the shear zone size is restricted by thickness of the granular bed.
\subsection{Fluid-pressure evolution}%
\label{sub:fluid_pressure_evolution}
@@ -187,21 +187,20 @@ The fluid pressure is used to determine the effective normal stress used in the
\label{sub:limitations_of_the_continuum_model}
The presented model considers the material to be in the critical (steady) state throughout the domain.
Consequently, porosity is prescribed as a constant and material-specific parameter.
-For that reason the model is not able to simulate uniaxial compaction or shear-induced dilation \citep[e.g.][]{Iverson2000, Iverson2010-2, Damsgaard2015} or compaction \citep[e.g.][]{Dewhurst1996}.
+For that reason the model is not able to simulate uniaxial compaction or shear-induced volume changes \citep[e.g.][]{Iverson2000, Iverson2010-2, Damsgaard2015} or compaction \citep[e.g.][]{Dewhurst1996}.
A transient granular model with state-dependent porosity is currently under development.
-The strain distribution is in the presented model dependent on the representative grain size $d$.
-However, it is uncomfortable to describe the grain size distribution with a single value for diamictons such as most subglacial tills \citep[e.g.][]{Hooke1995}.
-Future research will investigate how wide grain-size distributions affect strain distribution.
+The representative grain size $d$ scales the non-locality and strain distribution.
+However, it is awkward to describe grain size distributions of diamictons with a single value \citep[e.g.][]{Hooke1995}.
We expect that a volumetrically dominant grain size dominates the strain distribution, outside of effects of ploughing by large clasts \citep[e.g.][]{Tulaczyk1999}.
+%Future research will investigate how wide grain-size distributions affect strain distribution.
\subsection{Numerical solution procedure}%
\label{sub:numerical_solution_procedure}
-We apply the model in a one-dimensional setup where simple shear occurs along the horizontal axis $x$.
-The vertical axis is denoted $z$.
+We apply the model in a one-dimensional setup where simple shear occurs along a horizontal axis $x$, orthogonal to a vertical axis $z$.
The spatial domain is $L_z = 8$ m long and is discretized into cells with equal size to the representative grain size $d$.
The upper boundary, i.e.\ the ``ice-bed interface'', exerts effective normal stress and shear stress on the granular assemblage.
-The miniscule contribution to material strength through fluid viscosity is neglected.
+We neglect the miniscule contribution to material shear strength by water viscosity.
The effective normal stress is found by adding the lithostatic contribution that increases with depth to the normal stress applied from the top:
\begin{equation}
\sigma_\text{n}(z)
diff --git a/experimental-setup.pdf b/experimental-setup.pdf
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diff --git a/experimental-setup.svg b/experimental-setup.svg
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