[cig-commits] r16907 - in short/3D/PyLith/trunk/doc/userguide: . boundaryconditions

Sun Jun 6 09:47:35 PDT 2010

Author: brad
Date: 2010-06-06 09:47:34 -0700 (Sun, 06 Jun 2010)
New Revision: 16907

Modified:
   short/3D/PyLith/trunk/doc/userguide/boundaryconditions/boundaryconditions.lyx
   short/3D/PyLith/trunk/doc/userguide/userguide.lyx
Log:
Fixed typos.

Modified: short/3D/PyLith/trunk/doc/userguide/boundaryconditions/boundaryconditions.lyx
===================================================================

--- short/3D/PyLith/trunk/doc/userguide/boundaryconditions/boundaryconditions.lyx	2010-06-05 21:57:55 UTC (rev 16906)
+++ short/3D/PyLith/trunk/doc/userguide/boundaryconditions/boundaryconditions.lyx	2010-06-06 16:47:34 UTC (rev 16907)
@@ -4428,7 +4428,7 @@
 \end_layout
 
 \begin_layout Standard
-Dynamic fault interfaces use the FaultCohesive Dyn object to specify a fault
+Dynamic fault interfaces use the FaultCohesiveDyn object to specify a fault
  constitutive model to govern the fault tractions (friction) and the resulting
  slip.
  When friction is large enough such that there is no sliding on the fault,
@@ -4839,6 +4839,8 @@
 \end_inset
 
 .
+ As long as the fault is locked, the initial state variables are zero, so
+ specifying the initial state variables for slip-weakening friction is rate.
 \end_layout
 
 \begin_layout Standard
@@ -5133,9 +5135,9 @@
  normal traction that depends on a state variable,
 \begin_inset Formula \begin{gather}
 T_{f}=\begin{cases}
-T_{c}-\mu_{f}T_{n} & T\leq0\\
+T_{c}-\mu_{f}T_{n} & T_{n}\leq0\\
 0 & T_{n}>0\end{cases}\\
-\mu_{f}=a\sinh^{-1}\left(\frac{1}{2}\frac{V}{V_{0}}\exp\left(\mu_{0}+\frac{b}{a}\log\left(\frac{V_{0}\theta}{L}\right)\right)\right)\\
+\mu_{f}=a\sinh^{-1}\left(\frac{1}{2}\frac{V}{V_{0}}\exp\left(\frac{1}{a}\left(\mu_{0}+b\ln\left(\frac{V_{0}\theta}{L}\right)\right)\right)\right)\\
 \frac{d\theta}{dt}=1-\frac{V\theta}{L}\end{gather}
 
 \end_inset
@@ -5162,22 +5164,41 @@
 
  is a state variable.
  We have used the regularization of the coefficient of friction proposed
- by ?? to permit zero slip rate.
- Following ??, we integrate the evolution equation for the state variable
- keeping slip rate constant to get
-\begin_inset Formula \[
-\theta(t+\Delta t)=\theta(t)\exp\left(\frac{-V\Delta t}{L}\right)+\frac{L}{V}\left(1-\exp\left(-\frac{V\theta}{L}\right)\right).\]
+ by Ben-Zion and Rice 
+\begin_inset CommandInset citation
+LatexCommand cite
+key "BenZion:Rice:1997"
 
 \end_inset
 
+ to permit zero slip rate.
+ Following Kaneko 
+\shape italic
+et al.
+
+\shape default
+ 
+\begin_inset CommandInset citation
+LatexCommand cite
+key "Kaneko:etal:2008"
+
+\end_inset
+
+, we integrate the evolution equation for the state variable keeping slip
+ rate constant to get
+\begin_inset Formula \begin{equation}
+\theta(t+\Delta t)=\theta(t)\exp\left(\frac{-V(t)\Delta t}{L}\right)+\frac{L}{V(t)}\left(1-\exp\left(-\frac{V(t)\Delta t}{L}\right)\right).\end{equation}
+
+\end_inset
+
 As the slip rate approaches zero, the first exponential term approaches
  1.
  Using the first three terms of the Taylor series expansion of the second
  exponential yields
-\begin_inset Formula \[
+\begin_inset Formula \begin{equation}
 \theta(t+\Delta t)=\begin{cases}
-\theta(t)\exp\left(-\frac{V\Delta t}{L}\right)+\Delta t-\frac{1}{2}\frac{V\Delta t^{2}}{L} & \frac{V\Delta t}{L}<0.00001\\
-\theta(t)\exp\left(-\frac{V\Delta t}{L}\right)+\frac{L}{V}\left(1-\exp\left(-\frac{V\theta}{L}\right)\right) & \frac{V\Delta t}{L}\ge0.00001\end{cases}.\]
+\theta(t)\exp\left(-\frac{V(t)\Delta t}{L}\right)+\Delta t-\frac{1}{2}\frac{V(t)\Delta t^{2}}{L} & \frac{V(t)\Delta t}{L}<0.00001\\
+\theta(t)\exp\left(-\frac{V(t)\Delta t}{L}\right)+\frac{L}{V(t)}\left(1-\exp\left(-\frac{V(t)\Delta t}{L}\right)\right) & \frac{V(t)\Delta t}{L}\ge0.00001\end{cases}.\end{equation}
 
 \end_inset
 
@@ -5395,7 +5416,11 @@
 \begin_inset Text
 
 \begin_layout Plain Layout
-Cofficient for the ?? term, 
+Coefficient for the 
+\begin_inset Formula $\ln$
+\end_inset
+
+ slip rate term, 
 \begin_inset Formula $a$
 \end_inset
 
@@ -5428,7 +5453,11 @@
 \begin_inset Text
 
 \begin_layout Plain Layout
-Coefficient for the ?? term, 
+Coefficient for the 
+\begin_inset Formula $\ln$
+\end_inset
+
+ state variable term, 
 \begin_inset Formula $b$
 \end_inset
 

Modified: short/3D/PyLith/trunk/doc/userguide/userguide.lyx
===================================================================
--- short/3D/PyLith/trunk/doc/userguide/userguide.lyx	2010-06-05 21:57:55 UTC (rev 16906)
+++ short/3D/PyLith/trunk/doc/userguide/userguide.lyx	2010-06-06 16:47:34 UTC (rev 16907)
@@ -1,4 +1,4 @@
-#LyX 1.6.4 created this file. For more info see http://www.lyx.org/
+#LyX 1.6.5 created this file. For more info see http://www.lyx.org/
 \lyxformat 345
 \begin_document
 \begin_header
@@ -707,9 +707,9 @@
 
 Liu, P., R.J.
  Archuleta, S.H.
- Hartzell (2006).
- Prediction of broadband ground-motion time histories: Hybrid low/high-frequency
- method with correlated random source parameters, 
+ Hartzell (2006), Prediction of broadband ground-motion time histories:
+ Hybrid low/high-frequency method with correlated random source parameters,
+ 
 \shape italic
 Bull.
  Seismol.
@@ -719,5 +719,40 @@
 , 96, 2118-2130.
 \end_layout
 
+\begin_layout Bibliography
+\begin_inset CommandInset bibitem
+LatexCommand bibitem
+label "20"
+key "BenZion:Rice:1997"
+
+\end_inset
+
+Ben-Zion, Y.
+ and J.R.
+ Rice (1997), Dynamic simulations of slip on a smooth fault in an elastic
+ solid, J.
+ Geophys.
+ Res., 102, 17,771–17,784.
+\end_layout
+
+\begin_layout Bibliography
+\begin_inset CommandInset bibitem
+LatexCommand bibitem
+label "21"
+key "Kaneko:etal:2008"
+
+\end_inset
+
+> Kaneko, Y., N.
+ Lapusta, and J.-P.
+ Ampuero (2008), Spectral element modeling of spontaneous earthquake rupture
+ on rate and state faults: Effect of velocity-strengthening friction at
+ shallow depths, 
+\shape italic
+Journal of Geophysical Research
+\shape default
+, 113, B09317, doi:10.1029/2007JB005553.
+\end_layout
+
 \end_body
 \end_document

