[cig-commits] r16255 - short/3D/PyLith/trunk/doc/userguide/materials
willic3 at geodynamics.org
willic3 at geodynamics.org
Thu Feb 11 19:34:45 PST 2010
Author: willic3
Date: 2010-02-11 19:34:45 -0800 (Thu, 11 Feb 2010)
New Revision: 16255
Modified:
short/3D/PyLith/trunk/doc/userguide/materials/materials.lyx
Log:
Finished putting in derivatives for tangent constitutive matrix.
I need to implement them and double-check, because they are pretty ugly.
Modified: short/3D/PyLith/trunk/doc/userguide/materials/materials.lyx
===================================================================
--- short/3D/PyLith/trunk/doc/userguide/materials/materials.lyx 2010-02-10 00:45:22 UTC (rev 16254)
+++ short/3D/PyLith/trunk/doc/userguide/materials/materials.lyx 2010-02-12 03:34:45 UTC (rev 16255)
@@ -4419,8 +4419,7 @@
\end_inset
-The terms involving only strains are easily computed.
- For the dilatational portion,
+First considering the dilatational portion, we have
\begin_inset Formula \begin{equation}
\frac{\partial\phantom{}^{t+\Delta t}e_{k}^{\prime}}{\partial\phantom{}^{t+\Delta t}e_{l}}=\delta_{kl}\:,\label{eq:119}\end{equation}
@@ -4436,19 +4435,6 @@
\end_inset
-For the volumetric portion,
-\begin_inset Formula \begin{equation}
-\frac{\partial\phantom{}^{t+\Delta t}\theta^{\prime}}{\partial\phantom{}^{t+\Delta t}\epsilon_{j}}=\frac{1}{3}\left[\begin{array}{ccc}
-1 & 1 & 1\\
-1 & 1 & 1\\
-1 & 1 & 1\end{array}\right]\:.\label{eq:121}\end{equation}
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
From equation
\begin_inset CommandInset ref
LatexCommand ref
@@ -4458,7 +4444,7 @@
, we have
\begin_inset Formula \begin{equation}
-\frac{\partial\phantom{}^{t+\Delta t}S_{i}}{\partial\phantom{}^{t+\Delta t}e_{j}^{\prime}}=\frac{1}{a_{E}}\left(\delta_{ij}-\frac{\partial\Delta e_{i}^{P}}{\partial\phantom{}^{t+\Delta t}e_{j}^{\prime}}\right)\:,\label{eq:122}\end{equation}
+\frac{\partial\phantom{}^{t+\Delta t}S_{i}}{\partial\phantom{}^{t+\Delta t}e_{j}^{\prime}}=\frac{1}{a_{E}}\left(\delta_{ij}-\frac{\partial\Delta e_{i}^{P}}{\partial\phantom{}^{t+\Delta t}e_{j}^{\prime}}\right)\:,\label{eq:121}\end{equation}
\end_inset
@@ -4471,7 +4457,7 @@
we have
\begin_inset Formula \begin{equation}
-\frac{\partial\Delta e_{i}^{P}}{\partial\phantom{}^{t+\Delta t}e_{j}^{\prime}}=\frac{\partial\lambda}{\partial\phantom{}^{t+\Delta t}e_{j}^{\prime}}\frac{1}{\sqrt{2}\,\phantom{}^{t+\Delta t}d}\left(^{t+\Delta t}e_{i}^{\prime}+a_{E}S_{i}^{I}\right)+\frac{\lambda}{\sqrt{2}}\left[\frac{-1}{^{t+\Delta t}d^{2}}\frac{\partial\phantom{}^{t+\Delta t}d}{\partial\phantom{}^{t+\Delta t}e_{j}^{\prime}}\left(^{t+\Delta t}e_{i}^{\prime}+a_{E}S_{i}^{I}\right)+\frac{\delta_{ij}}{^{t+\Delta t}d}\right]\:.\label{eq:123}\end{equation}
+\frac{\partial\Delta e_{i}^{P}}{\partial\phantom{}^{t+\Delta t}e_{j}^{\prime}}=\frac{\partial\lambda}{\partial\phantom{}^{t+\Delta t}e_{j}^{\prime}}\frac{1}{\sqrt{2}\,\phantom{}^{t+\Delta t}d}\left(^{t+\Delta t}e_{i}^{\prime}+a_{E}S_{i}^{I}\right)+\frac{\lambda}{\sqrt{2}}\left[\frac{-1}{^{t+\Delta t}d^{2}}\frac{\partial\phantom{}^{t+\Delta t}d}{\partial\phantom{}^{t+\Delta t}e_{j}^{\prime}}\left(^{t+\Delta t}e_{i}^{\prime}+a_{E}S_{i}^{I}\right)+\frac{\delta_{ij}}{^{t+\Delta t}d}\right]\:.\label{eq:122}\end{equation}
\end_inset
@@ -4481,14 +4467,14 @@
is
\begin_inset Formula \begin{equation}
-\frac{\partial\phantom{}^{t+\Delta t}d}{\partial\phantom{}^{t+\Delta t}e_{j}^{\prime}}=\frac{2a_{E}T_{j}^{I}+\phantom{}^{t+\Delta t}E_{j}}{2\phantom{}^{t+\Delta t}d}\:,\label{eq:124}\end{equation}
+\frac{\partial\phantom{}^{t+\Delta t}d}{\partial\phantom{}^{t+\Delta t}e_{j}^{\prime}}=\frac{a_{E}T_{j}^{I}+\phantom{}^{t+\Delta t}E_{j}}{\phantom{}^{t+\Delta t}d}\:,\label{eq:123}\end{equation}
\end_inset
where
\begin_inset Formula \begin{align}
T_{j}^{I} & =S_{j}^{I}\;\mathrm{and}\;\phantom{}^{t+\Delta t}E_{j}=\phantom{}^{t+\Delta t}e_{j}^{\prime}\:;\; j=1,2,3\nonumber \\
-T_{j}^{I} & =2S_{j}^{I}\;\mathrm{and}\;\phantom{}^{t+\Delta t}E_{j}=2\phantom{}^{t+\Delta t}e_{j}^{\prime}\:;\; j=4,5,6\:.\label{eq:125}\end{align}
+T_{j}^{I} & =2S_{j}^{I}\;\mathrm{and}\;\phantom{}^{t+\Delta t}E_{j}=2\phantom{}^{t+\Delta t}e_{j}^{\prime}\:;\; j=4,5,6\:.\label{eq:124}\end{align}
\end_inset
@@ -4498,25 +4484,24 @@
:
\begin_inset Formula \begin{equation}
-\frac{\partial\lambda}{\partial\phantom{}^{t+\Delta t}e_{j}^{\prime}}=\frac{2a_{E}a_{m}}{6\alpha_{f}\alpha_{g}a_{E}+a_{m}}\left(\frac{3\alpha_{f}}{a_{m}}\frac{\partial\phantom{}^{t+\Delta t}\theta^{\prime}}{\partial\phantom{}^{t+\Delta t}\epsilon_{k}}\frac{\partial\phantom{}^{t+\Delta t}\epsilon_{k}}{\partial\phantom{}^{t+\Delta t}e_{l}}\frac{\partial\phantom{}^{t+\Delta t}e_{l}}{\partial\phantom{}^{t+\Delta t}e_{j}^{\prime}}+\frac{1}{\sqrt{2}a_{E}}\frac{\partial\phantom{}^{t+\Delta t}d}{\partial\phantom{}^{t+\Delta t}e_{j}^{\prime}}\right)\:.\label{eq:126}\end{equation}
+\frac{\partial\lambda}{\partial\phantom{}^{t+\Delta t}e_{j}^{\prime}}=\frac{2a_{E}a_{m}}{6\alpha_{f}\alpha_{g}a_{E}+a_{m}}\left(\frac{3\alpha_{f}}{a_{m}}\frac{\partial\phantom{}^{t+\Delta t}\theta^{\prime}}{\partial\phantom{}^{t+\Delta t}\epsilon_{k}}\frac{\partial\phantom{}^{t+\Delta t}\epsilon_{k}}{\partial\phantom{}^{t+\Delta t}e_{l}}\frac{\partial\phantom{}^{t+\Delta t}e_{l}}{\partial\phantom{}^{t+\Delta t}e_{j}^{\prime}}+\frac{1}{\sqrt{2}a_{E}}\frac{\partial\phantom{}^{t+\Delta t}d}{\partial\phantom{}^{t+\Delta t}e_{j}^{\prime}}\right)\:.\label{eq:125}\end{equation}
\end_inset
The first derivative in
\begin_inset CommandInset ref
LatexCommand ref
-reference "eq:126"
+reference "eq:125"
\end_inset
- is already given in equation
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "eq:121"
+ is simply
+\begin_inset Formula \begin{equation}
+\frac{\partial\phantom{}^{t+\Delta t}\theta^{\prime}}{\partial\phantom{}^{t+\Delta t}\epsilon_{k}}=\frac{R_{k}}{3}\:.\label{eq:126}\end{equation}
\end_inset
-, the third yields the same result as equation
+The third derivative yields the same result as equation
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:119"
@@ -4526,7 +4511,7 @@
, and the fourth is given by equation
\begin_inset CommandInset ref
LatexCommand ref
-reference "eq:124"
+reference "eq:123"
\end_inset
@@ -4551,6 +4536,55 @@
\end_layout
+\begin_layout Standard
+For the volumetric portion we use equation
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "eq:113"
+
+\end_inset
+
+ to obtain
+\begin_inset Formula \begin{equation}
+\frac{\partial\phantom{}^{t+\Delta t}P}{\partial\phantom{}^{t+\Delta t}\epsilon_{j}}=\frac{1}{a_{m}}\left(\frac{R_{j}}{3}-\alpha_{g}\frac{\partial\lambda}{\partial\phantom{}^{t+\Delta t}\epsilon_{j}}\right)\:.\label{eq:128}\end{equation}
+
+\end_inset
+
+From equation
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "eq:114"
+
+\end_inset
+
+,
+\begin_inset Formula \begin{equation}
+\frac{\partial\lambda}{\partial\phantom{}^{t+\Delta t}\epsilon_{j}}=\frac{2a_{E}a_{m}}{6\alpha_{f}\alpha_{g}a_{E}+a_{m}}\left(\frac{R_{j}\alpha_{f}}{a_{m}}+\frac{1}{\sqrt{2}a_{E}}\frac{\partial\phantom{}^{t+\Delta t}d}{\partial\phantom{}^{t+\Delta t}\epsilon_{j}}\right)\:.\label{eq:129}\end{equation}
+
+\end_inset
+
+From equation
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "eq:111"
+
+\end_inset
+
+ we have
+\begin_inset Formula \begin{equation}
+\frac{\partial\phantom{}^{t+\Delta t}d}{\partial\phantom{}^{t+\Delta t}\epsilon_{j}}=\frac{a_{e}T_{j}^{I}+\phantom{}^{t+\Delta t}E_{j}}{\phantom{}^{t+\Delta t}d}\:,\label{eq:130}\end{equation}
+
+\end_inset
+
+so that
+\begin_inset Formula \begin{equation}
+\frac{\partial\phantom{}^{t+\Delta t}P}{\partial\phantom{}^{t+\Delta t}\epsilon_{j}}=\frac{1}{a_{m}}\left[\frac{R_{j}}{3}-\frac{2a_{E}a_{m}\alpha_{g}}{6\alpha_{f}\alpha_{g}a_{E}+a_{m}}\left(\frac{R_{j}\alpha_{f}}{a_{m}}+\frac{a_{E}T_{j}^{I}+\phantom{}^{t+\Delta t}E_{j}}{\sqrt{2}a_{E}\phantom{}^{t+\Delta t}d}\right)\right]\:.\label{eq:131}\end{equation}
+
+\end_inset
+
+
+\end_layout
+
\begin_layout Section
Initial State Variables
\end_layout
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