commit d11cc47f2f2165b487410373ee5cc7876cafddbb
parent e656b2632a4c7d73766c6aa1608c80453cc88363
Author: Anders Damsgaard <anders@adamsgaard.dk>
Date: Mon, 8 Jul 2019 12:14:39 +0200
Add cohesion to granular flow equations
Diffstat:
2 files changed, 5 insertions(+), 4 deletions(-)
diff --git a/continuum-granular-manuscript1.tex b/continuum-granular-manuscript1.tex
@@ -66,16 +66,17 @@ We assume that the elasticity is negligible and set the total shear rate $\dot{\
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.
+We expand the fluidity term in \citet{Henann2013} to account for material cohesion.
The local fluidity is defined as:
\begin{equation}
g_\text{local}(\mu, \sigma_\text{n}') =
\begin{cases}
- \sqrt{d^2 \sigma_\text{n}' / \rho_\text{s}} (\mu - \mu_\text{s})/(b\mu) &\text{if } \mu > \mu_\text{s} \text{, and}\\
- 0 &\text{if } \mu \leq \mu_\text{s}.
+ \sqrt{d^2 \sigma_\text{n}' / \rho_\text{s}} ((\mu - C/\sigma_\text{n}' - \mu_\text{s})/(b\mu) &\text{if } \mu - C/\sigma_\text{n}') > \mu_\text{s} \text{, and}\\
+ 0 &\text{if } \mu - C/\sigma_\text{n}' \leq \mu_\text{s}.
\end{cases}
\label{eq:g_local}
\end{equation}
-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.
+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.
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}),
@@ -83,7 +84,7 @@ For steady flow the non-locality is determined by a Poisson-type equation where
\end{equation}
where
\begin{equation}
- \xi(\mu) = \frac{Ad}{\sqrt{|\mu - \mu_\text{s}|}}.
+ \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 $A$ [-].
diff --git a/experiments/fig3.pdf b/experiments/fig3.pdf
Binary files differ.
