manus_continuum_granular1

manuscript files for first continuum-till paper
git clone git://src.adamsgaard.dk/manus_continuum_granular1
Log | Files | Refs

commit da1d5f353273020082a98f2264312a6c7a46f83c
parent 1c9826a8e16240f07e80fdef3387db4dcca36c52
Author: Anders Damsgaard <anders@adamsgaard.dk>
Date:   Thu, 27 Jun 2019 10:54:40 +0200

Add Iverson figure and improve methods section

Diffstat:
AIverson2010-fig2a.png | 0
Mcontinuum-granular-manuscript1.tex | 23++++++++++++++++-------
2 files changed, 16 insertions(+), 7 deletions(-)

diff --git a/Iverson2010-fig2a.png b/Iverson2010-fig2a.png Binary files differ. diff --git a/continuum-granular-manuscript1.tex b/continuum-granular-manuscript1.tex @@ -53,18 +53,19 @@ We show that past pulses in water pressure can transfer shear away from the ice- \section{Methods}% \label{sec:methods} -We expand a steady-state continuum model for granular flow by \citet{Henann2013} with a coupling to pore-pressure diffusion. \subsection{Granular flow}% \label{sub:granular_flow} -In our model, the sediment deforms as a highly nonlinear Bingham material with yield beyond the Mohr-Coulomb failure limit for a cohesionless material. +We expand a steady-state continuum model for granular flow by \citet{Henann2013} with a coupling to pore-pressure diffusion. +The sediment deforms as a highly nonlinear Bingham material with yield beyond the Mohr-Coulomb failure limit for a cohesionless material. 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, \label{eq:shear_strain_rate} \end{equation} where $\mu = \tau/\sigma_\text{n}'$ is the dimensionless ratio between shear stress ($\tau$ [Pa]) and effective normal stress ($\sigma_\text{n}' = \sigma_\text{n} - p_f$ [Pa]). -Water pressure is $p_\text{f}$ [Pa] and $g$ [s$^{-1}$] is the granular fluidity. The fluidity consists of local and non-local components. +Water pressure is $p_\text{f}$ [Pa] and $g$ [s$^{-1}$] is the granular fluidity. +The fluidity consists of local and non-local components. The local fluidity is defined as: \begin{equation} g_\text{local}(\mu, \sigma_\text{n}') = @@ -74,7 +75,8 @@ The local fluidity is defined as: \end{cases} \label{eq:g_local} \end{equation} -For steady flow the non-locality is determined by the cooperativity length $\xi$: +where $d$ [m] is the representative grain diameter, $\mu_\text{s}$ [-] is the static Coulomb yield coefficient, and $b$ [-] is the non-linear rate dependence beyond yield. +For steady flow the non-locality is determined by a Poisson-type equation where strain is spread in space, as scaled by the cooperativity length $\xi$: \begin{equation} \nabla^2 g = \frac{1}{\xi^2(\mu)} (g - g_\text{local}), \label{eq:g} @@ -84,6 +86,8 @@ where \xi(\mu) = \frac{Ad}{\sqrt{|\mu - \mu_\text{s}|}}. \label{eq:cooperativity} \end{equation} +The non-locality scales with nonlocal amplitude $A$ [-]. +It is worth noting that the above formulation distributes strain in space based on material properties and stress, as observed in simple granular materials \citep[e.g.][]{Damsgaard2013}. \subsection{Fluid-pressure evolution}% \label{sub:fluid_pressure_evolution} @@ -98,8 +102,6 @@ The fluid pressure is used to determine the effective normal stress used in the \subsection{Numerical solution procedure}% \label{sub:numerical_solution_procedure} -The material parameters consist of nonlocal amplitude $A$ [-], representative grain diameter $d$ [m], static yield coefficient $\mu_\text{s}$ [-], and the rate dependence beyond yield $b$ [-]. -Hydraulic properties are given by the permeability $k$ [m$^2$], the fluid viscosity $\mu_\text{f}$ [Pa s], and the adiabatic fluid compressibility $\beta_\text{f}$ [Pa$^{-1}$]. These parameters do not change over the course of a simulation, and are kept constant everywhere in the domain where the material is of identical origin. The above formulation is applicable to any spatial dimensionality, for the purposes of this study we apply it in a 1D spatial reference system. @@ -159,11 +161,16 @@ For the first experiment with variable water pressure, we apply a water-pressure \begin{figure}[htbp] \begin{center} + \includegraphics[width=6.5cm]{Iverson2010-fig2a.png}\\ \includegraphics[width=7.5cm]{experiments/fig1.pdf} \caption{\label{fig:rate_dependence}% + \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. - Plot limits equal to \citet{Iverson2010}. The effective normal stress is held constant at $\sigma_\text{n}' = 100$ kPa. + \textbf{TODO:} Redraw the Iverson 2010 figure. + \textbf{TODO:} Add "a" and "b" labels. } \end{center} \end{figure} @@ -223,6 +230,8 @@ As long as fluid and diffusion properties are constant, The above relation implies that the amplitude in water-pressure forcing does not influence the maximum depth of slip. Figure~\ref{fig:skin_depth} shows the skin depth for water under a range of permeabilities and forcing frequencies. The stick-slip experiments (Fig.~\ref{fig:stick_slip} to~\ref{fig:stick_slip_depth_normalized}) correspond to a skin depth of 2.2 meter. +Practically all of the shear strain through a perturbation cycle occurs within the upper half to the skin depth ($\sim$1 m in Fig.~\ref{fig:stick_slip_depth}). + \section{Conclusion}% \label{sec:conclusion}