[cig-commits] r15749 - doc/geodynamics.org/benchmarks/trunk/long
luis at geodynamics.org
luis at geodynamics.org
Mon Oct 5 15:39:31 PDT 2009
Author: luis
Date: 2009-10-05 15:39:30 -0700 (Mon, 05 Oct 2009)
New Revision: 15749
Modified:
doc/geodynamics.org/benchmarks/trunk/long/drucker-prager.html
doc/geodynamics.org/benchmarks/trunk/long/drucker-prager.rst
Log:
Updated long/drucker-prager.rst
Modified: doc/geodynamics.org/benchmarks/trunk/long/drucker-prager.html
===================================================================
--- doc/geodynamics.org/benchmarks/trunk/long/drucker-prager.html 2009-10-05 22:39:24 UTC (rev 15748)
+++ doc/geodynamics.org/benchmarks/trunk/long/drucker-prager.html 2009-10-05 22:39:30 UTC (rev 15749)
@@ -290,16 +290,21 @@
<div class="section" id="analytic-treatment">
<h1>Analytic Treatment</h1>
<p>For the Drucker-Prager rehology in 2D, we can write the yielding relation
-as $$sigma_{ns}=sigma_{nn}tanvarphi+C,$$ where $sigma_{ns}$ is the
-shear stress perpendicular to the fault plane, $sigma_{nn}$ is the shear
-stress parallel to the fault plane, $varphi$ is the internal angle of
-friction, and $C$ is the cohesion. Decomposing this into principal
-stresses $sigma_{I}$, $sigma_{II}$, and $sigma_{III}$ gives
-$$sin(2Theta)(sigma_{I}-sigma_{III})/2=tanvarphileft((sigma_{I}+sigma_{III})/2+cos(2Theta)(sigma_{I}-sigma_{III})/2right)+C,$$
-where $Theta$ is the angle that the fault makes relative to the maximum
+as</p>
+<blockquote>
+[;sigma_{ns}=sigma_{nn}tanvarphi+C,;]</blockquote>
+<p>where [;sigma_{ns};] is the shear stress perpendicular to the fault plane,
+[;sigma_{nn};] is the shear stress parallel to the fault plane, [;varphi;]
+is the internal angle of friction, and [;C;] is the cohesion. Decomposing this
+into principal stresses [;sigma_{I};], [;sigma_{II};], and [;sigma_{III};]
+gives</p>
+<blockquote>
+[;sin(2Theta)(sigma_{I}-sigma_{III})/2=tanvarphileft((sigma_{I}+sigma_{III})/2+cos(2Theta)(sigma_{I}-sigma_{III})/2right)+C,;]</blockquote>
+<p>where [;Theta;] is the angle that the fault makes relative to the maximum
shear stress. Assuming that the fault forms where the shear stress
-$sigma_{I}-sigma_{III}$ is a minimum, a little algebra gives us
-$$Theta=pmleft(frac{pi}{4} + frac{varphi}{2}right).$$</p>
+[;sigma_{I}-sigma_{III};] is a minimum, a little algebra gives us</p>
+<blockquote>
+[;Theta=pmleft(frac{pi}{4} + frac{varphi}{2}right).;]</blockquote>
<p>Using this, we can construct a simple plasticity experiment and make sure
that Gale gives the correct faulting angle.</p>
</div>
@@ -310,24 +315,40 @@
the strain rate invariant to find incipient faults. We do not take any
time steps, removing any confounding effects they may cause.</p>
<div class="figure">
-<img alt="images/Mohr_Coulomb_setup.epsFigure[fig:Mohr-Coulomb-setup]Thesetupfortheshorteningexperiment.Theboxis1unitonaside,andthelowviscosityregionhasaradiusof0.01(itssizeisexaggerated)." src="images/Mohr_Coulomb_setup.epsFigure[fig:Mohr-Coulomb-setup]Thesetupfortheshorteningexperiment.Theboxis1unitonaside,andthelowviscosityregionhasaradiusof0.01(itssizeisexaggerated)." />
+<img alt="images/Mohr_Coulomb_setup.eps" src="images/Mohr_Coulomb_setup.eps" />
+<p class="caption">Figure [fig:Mohr-Coulomb-setup]</p>
+<div class="legend">
+The setup for the shortening experiment. The box is 1 unit on a side,
+and the low viscosity region has a radius of 0.01 (its size is
+exaggerated).</div>
</div>
</div>
<div class="section" id="numerical-results">
<h1>Numerical Results</h1>
<p>Figure [fig:Mohr-Coulomb-sri] shows the results for three different
-resolutions for $varphi=45^{deg}$. There is not much difference between
-the medium ($256 times 256$) and high ($512 times 512$) results,
+resolutions for [;varphi=45^{deg};]. There is not much difference between
+the medium ([;256 times 256;]) and high ([;512 times 512;]) results,
suggesting that we have sufficient resolution. Figure
[fig:Mohr-Coulomb-comparison] shows a plot of the numerical vs. analytic
results for several different angles for medium resolution. This gives us
confidence that, at least in compression(sp?) in 2D, our Drucker-Prager
implementation gives the correct results.</p>
<div class="figure">
-<img alt="images/Mohr_coulomb_resolutions.pngFigure[fig:Mohr-Coulomb-sri]Strainrateinvariantfortheshorteningexperimentwhere$\varphi=45^{\deg}$withthreedifferentresolutions:$128\times128$,$256\times256$,$512\times512$.Anydifferencesbetweenthemediumandhighresolutionrunsareswampedbyuncertaintiesindeterminingtheoverallangleoffaulting." src="images/Mohr_coulomb_resolutions.pngFigure[fig:Mohr-Coulomb-sri]Strainrateinvariantfortheshorteningexperimentwhere$\varphi=45^{\deg}$withthreedifferentresolutions:$128\times128$,$256\times256$,$512\times512$.Anydifferencesbetweenthemediumandhighresolutionrunsareswampedbyuncertaintiesindeterminingtheoverallangleoffaulting." />
+<img alt="images/Mohr_coulomb_resolutions.png" src="images/Mohr_coulomb_resolutions.png" />
+<p class="caption">Figure [fig:Mohr-Coulomb-sri]</p>
+<div class="legend">
+Strain rate invariant for the shortening experiment where
+$varphi=45^{deg}$ with three different resolutions:
+$128 times 128$, $256 times 256$, $512 times 512$.
+Any differences between the medium and high resolution runs are swamped
+by uncertainties in determining the overall angle of faulting.</div>
</div>
<div class="figure">
-<img alt="images/mohr_coulomb_angles.epsFigure[fig:Mohr-Coulomb-comparison]Numericalvs.analyticresultsforfaultanglesasafunctionofinternalangleoffriction." src="images/mohr_coulomb_angles.epsFigure[fig:Mohr-Coulomb-comparison]Numericalvs.analyticresultsforfaultanglesasafunctionofinternalangleoffriction." />
+<img alt="images/mohr_coulomb_angles.eps" src="images/mohr_coulomb_angles.eps" />
+<p class="caption">Figure [fig:Mohr-Coulomb-comparison]</p>
+<div class="legend">
+Numerical vs. analytic results for fault angles as a function of
+internal angle of friction.</div>
</div>
</div>
</div>
Modified: doc/geodynamics.org/benchmarks/trunk/long/drucker-prager.rst
===================================================================
--- doc/geodynamics.org/benchmarks/trunk/long/drucker-prager.rst 2009-10-05 22:39:24 UTC (rev 15748)
+++ doc/geodynamics.org/benchmarks/trunk/long/drucker-prager.rst 2009-10-05 22:39:30 UTC (rev 15749)
@@ -6,17 +6,24 @@
------------------
For the Drucker-Prager rehology in 2D, we can write the yielding relation
-as $$\sigma_{ns}=\sigma_{nn}\tan\varphi+C,$$ where $\sigma_{ns}$ is the
-shear stress perpendicular to the fault plane, $\sigma_{nn}$ is the shear
-stress parallel to the fault plane, $\varphi$ is the internal angle of
-friction, and $C$ is the cohesion. Decomposing this into principal
-stresses $\sigma_{I}$, $\sigma_{II}$, and $\sigma_{III}$ gives
-$$\sin(2\Theta)(\sigma_{I}-\sigma_{III})/2=\tan\varphi\left((\sigma_{I}+\sigma_{III})/2+\cos(2\Theta)(\sigma_{I}-\sigma_{III})/2\right)+C,$$
-where $\Theta$ is the angle that the fault makes relative to the maximum
+as
+
+ [;\sigma_{ns}=\sigma_{nn}\tan\varphi+C,;]
+
+where [;\sigma_{ns};] is the shear stress perpendicular to the fault plane,
+[;\sigma_{nn};] is the shear stress parallel to the fault plane, [;\varphi;]
+is the internal angle of friction, and [;C;] is the cohesion. Decomposing this
+into principal stresses [;\sigma_{I};], [;\sigma_{II};], and [;\sigma_{III};]
+gives
+
+ [;\sin(2\Theta)(\sigma_{I}-\sigma_{III})/2=\tan\varphi\left((\sigma_{I}+\sigma_{III})/2+\cos(2\Theta)(\sigma_{I}-\sigma_{III})/2\right)+C,;]
+
+where [;\Theta;] is the angle that the fault makes relative to the maximum
shear stress. Assuming that the fault forms where the shear stress
-$\sigma_{I}-\sigma_{III}$ is a minimum, a little algebra gives us
-$$\Theta=\pm\left(\frac{\pi}{4} + \frac{\varphi}{2}\right).$$
+[;\sigma_{I}-\sigma_{III};] is a minimum, a little algebra gives us
+ [;\Theta=\pm\left(\frac{\pi}{4} + \frac{\varphi}{2}\right).;]
+
Using this, we can construct a simple plasticity experiment and make sure
that Gale gives the correct faulting angle.
@@ -30,7 +37,9 @@
time steps, removing any confounding effects they may cause.
.. figure:: images/Mohr_Coulomb_setup.eps
+
Figure [fig:Mohr-Coulomb-setup]
+
The setup for the shortening experiment. The box is 1 unit on a side,
and the low viscosity region has a radius of 0.01 (its size is
exaggerated).
@@ -39,8 +48,8 @@
-----------------
Figure [fig:Mohr-Coulomb-sri] shows the results for three different
-resolutions for $\varphi=45^{\deg}$. There is not much difference between
-the medium ($256 \times 256$) and high ($512 \times 512$) results,
+resolutions for [;\varphi=45^{\deg};]. There is not much difference between
+the medium ([;256 \times 256;]) and high ([;512 \times 512;]) results,
suggesting that we have sufficient resolution. Figure
[fig:Mohr-Coulomb-comparison] shows a plot of the numerical vs. analytic
results for several different angles for medium resolution. This gives us
@@ -48,7 +57,9 @@
implementation gives the correct results.
.. figure:: images/Mohr_coulomb_resolutions.png
+
Figure [fig:Mohr-Coulomb-sri]
+
Strain rate invariant for the shortening experiment where
$\varphi=45^{\deg}$ with three different resolutions:
$128 \times 128$, $256 \times 256$, $512 \times 512$.
@@ -56,7 +67,9 @@
by uncertainties in determining the overall angle of faulting.
.. figure:: images/mohr_coulomb_angles.eps
+
Figure [fig:Mohr-Coulomb-comparison]
+
Numerical vs. analytic results for fault angles as a function of
internal angle of friction.
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