[cig-commits] r15374 - short/3D/PyLith/trunk/doc/userguide/materials
willic3 at geodynamics.org
willic3 at geodynamics.org
Mon Jun 22 21:51:06 PDT 2009
Author: willic3
Date: 2009-06-22 21:51:06 -0700 (Mon, 22 Jun 2009)
New Revision: 15374
Modified:
short/3D/PyLith/trunk/doc/userguide/materials/materials.lyx
Log:
Put in new equations for computing elasticity constitutive matrix for
viscoelastic case.
Modified: short/3D/PyLith/trunk/doc/userguide/materials/materials.lyx
===================================================================
--- short/3D/PyLith/trunk/doc/userguide/materials/materials.lyx 2009-06-23 04:09:46 UTC (rev 15373)
+++ short/3D/PyLith/trunk/doc/userguide/materials/materials.lyx 2009-06-23 04:51:06 UTC (rev 15374)
@@ -1,4 +1,4 @@
-#LyX 1.6.2 created this file. For more info see http://www.lyx.org/
+#LyX 1.6.0 created this file. For more info see http://www.lyx.org/
\lyxformat 345
\begin_document
\begin_header
@@ -504,14 +504,6 @@
\end_inset
-\end_layout
-
-\begin_layout Plain Layout
-\align center
-\begin_inset VSpace defskip
-\end_inset
-
-
\begin_inset Tabular
<lyxtabular version="3" rows="5" columns="3">
<features>
@@ -664,18 +656,12 @@
\begin_layout Plain Layout
\family typewriter
-total_strain, stress, viscous_strain_1,
+total_strain, stress, viscous_strain_1,viscous_strain_2,
\end_layout
\begin_layout Plain Layout
\family typewriter
-viscous_strain_2,
-\end_layout
-
-\begin_layout Plain Layout
-
-\family typewriter
viscous_strain_3
\end_layout
@@ -760,10 +746,6 @@
\end_inset
-\begin_inset VSpace defskip
-\end_inset
-
-
\begin_inset Tabular
<lyxtabular version="3" rows="3" columns="4">
<features>
@@ -1084,10 +1066,6 @@
\end_inset
-\begin_inset VSpace defskip
-\end_inset
-
-
\begin_inset Tabular
<lyxtabular version="3" rows="4" columns="2">
<features>
@@ -1471,10 +1449,6 @@
\end_inset
-\begin_inset VSpace defskip
-\end_inset
-
-
\begin_inset Tabular
<lyxtabular version="3" rows="4" columns="2">
<features>
@@ -1768,10 +1742,6 @@
\end_inset
-\begin_inset VSpace defskip
-\end_inset
-
-
\begin_inset Tabular
<lyxtabular version="3" rows="3" columns="3">
<features>
@@ -2749,7 +2719,7 @@
as
\begin_inset Formula \begin{gather}
-^{t+\Delta t}\underline{S}=\frac{1}{\left(a_{E}+\alpha\Delta t^{\tau}\gamma\right)}\left[^{t+\Delta t}\underline{e}^{\prime\prime}-\Delta t^{\tau}\gamma\left(1-\alpha\right)^{t}\underline{S}+a_{E}\underline{S}^{I}\right]\,,\label{eq:70}\end{gather}
+F=^{t+\Delta t}S_{i}\left(a_{E}+\alpha\Delta t^{\tau}\gamma\right)-^{t+\Delta t}e_{i}^{\prime\prime}+\Delta t^{\tau}\gamma\left(1-\alpha\right)^{t}S_{i}-a_{E}S_{i}^{I}=0\label{eq:70}\end{gather}
\end_inset
@@ -2759,13 +2729,26 @@
\end_inset
-The derivative is then
+The derivative of this function with respect to
+\begin_inset Formula $^{t+\Delta t}e_{k}^{\prime\prime}$
+\end_inset
+
+ is
\begin_inset Formula \begin{gather}
-\frac{\partial{}^{t+\Delta t}S_{i}}{\partial{}^{t+\Delta t}e_{k}^{\prime\prime}}=\frac{1}{a_{E}+\alpha\Delta t^{\tau}\gamma}\nonumber \\
-\left\langle \delta_{ik}-\frac{\partial^{\tau}\gamma}{\partial{}^{t+\Delta t}e_{k}^{\prime\prime}}\left\{ \Delta t\left(1-\alpha\right)^{t}S_{i}+\frac{\alpha\Delta t}{a_{E}+\alpha\Delta t^{\tau}\gamma}\left[^{t+\Delta t}e_{i}^{\prime\prime}-\Delta t^{\tau}\gamma\left(1-\alpha\right)^{t}S_{i}+a_{E}S_{i}^{I}\right]\right\} \right\rangle \,.\label{eq:72}\end{gather}
+\frac{\partial F}{\partial^{t+\Delta t}e_{k}^{\prime\prime}}=-\delta_{ik}\:,\label{eq:72}\end{gather}
\end_inset
+and the derivative with respect to
+\begin_inset Formula $^{t+\Delta t}S_{i}$
+\end_inset
+
+ is
+\begin_inset Formula \begin{gather}
+\frac{\partial F}{\partial^{t+\Delta t}S_{i}}=a_{E}+\alpha\Delta t^{\tau}\gamma+\frac{\partial^{\tau}\gamma}{\partial^{t+\Delta t}S_{i}}\Delta t\left[\alpha^{t+\Delta t}S_{i}+\left(1-\alpha\right)^{t}S_{i}\right]\:.\label{eq:73}\end{gather}
+
+\end_inset
+
From Equation
\begin_inset CommandInset ref
LatexCommand ref
@@ -2773,47 +2756,57 @@
\end_inset
+ and Equation
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "eq:49"
+
+\end_inset
+
,
\begin_inset Formula \begin{gather}
-\frac{\partial^{\tau}\gamma}{\partial{}^{t+\Delta t}e_{k}^{\prime\prime}}=\frac{\left(n-1\right)\sqrt{^{\tau}J_{2}^{\prime}}^{n-2}}{2\eta^{n}}\frac{\partial\sqrt{^{\tau}J_{2}^{\prime}}}{\partial{}^{t+\Delta t}e_{k}^{\prime\prime}}=K_{1}\frac{\partial\sqrt{^{\tau}J_{2}^{\prime}}}{\partial{}^{t+\Delta t}e_{k}^{\prime\prime}}\,.\label{eq:73}\end{gather}
+^{\tau}\gamma=\frac{1}{2\eta^{n}}\left[\alpha\sqrt{^{t+\Delta t}J_{2}^{\prime}}+\left(1-\alpha\right)\sqrt{^{t}J_{2}^{\prime}}\right]^{n-1}\:.\label{eq:74}\end{gather}
\end_inset
-We first note that
+Then
\begin_inset Formula \begin{gather}
-\sqrt{^{t+\Delta t}J_{2}^{\prime}}=\frac{1}{\alpha}\left[\sqrt{^{\tau}J_{2}^{\prime}}-\left(1-\alpha\right)\sqrt{^{t}J_{2}^{\prime}}\right]\,,\label{eq:74}\end{gather}
+\frac{\partial^{\tau}\gamma}{\partial{}^{t+\Delta t}S_{i}}=\frac{\partial^{\tau}\gamma}{\partial\sqrt{^{t+\Delta t}J_{2}^{\prime}}}\frac{\partial\sqrt{^{t+\Delta t}J_{2}^{\prime}}}{\partial^{t+\Delta t}S_{l}}\label{eq:75}\\
+=\frac{\alpha\left(n-1\right)\sqrt{^{\tau}J_{2}^{\prime}}^{n-2}{}^{t+\Delta t}T_{i}}{4\eta^{n}}\,,\nonumber \end{gather}
\end_inset
- which allows us to rewrite Equation
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "eq:68"
+Where
+\begin_inset Formula \begin{gather}
+^{t+\Delta t}T_{i}=^{t+\Delta t}S_{i}\:;\:\:1\leq i\leq3\label{eq:76}\\
+^{t+\Delta t}T_{i}=2^{t+\Delta t}S_{i}\:;\:\:\textrm{otherwise.}\nonumber \end{gather}
\end_inset
- as
-\begin_inset Formula \begin{gather}
-\frac{a^{2}}{\alpha^{2}}\left[\sqrt{^{\tau}J_{2}^{\prime}}-\left(1-\alpha\right)\sqrt{^{t}J_{2}^{\prime}}\right]^{2}-b+c\,^{\tau}\gamma-d^{2}\,^{\tau}\gamma^{2}=\frac{a^{2}}{\alpha^{2}}K_{2}^{2}-b+c\,^{\tau}\gamma-d^{2}\,^{\tau}\gamma^{2}=F=0\,.\label{eq:75}\end{gather}
+Then using Equations
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "eq:72"
\end_inset
-The derivatives of this function are
-\begin_inset Formula \begin{gather}
-\frac{\partial F}{\partial\sqrt{^{\tau}J_{2}^{\prime}}}=\frac{2aK_{2}}{\alpha^{2}}\left(\alpha\Delta tK_{1}K_{2}+a\right)+K_{1}\left(c-2d^{2}\,^{\tau}\gamma\right)\nonumber \\
-\frac{\partial F}{\partial{}^{t+\Delta t}e_{k}^{\prime\prime}}=-\frac{^{t+\Delta t}e_{k}^{\prime\prime}}{2}+a_{E}S_{k}^{I}-\Delta t\left(1-\alpha\right)^{t}S_{k}\,^{\tau}\gamma\,.\label{eq:76}\end{gather}
+,
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "eq:73"
\end_inset
-Then using the quotient rule for derivatives,
-\begin_inset Formula \begin{gather}
-\frac{\partial\sqrt{^{\tau}J_{2}^{\prime}}}{\partial{}^{t+\Delta t}e_{k}^{\prime\prime}}=\frac{\left[\frac{^{t+\Delta t}e_{k}^{\prime\prime}}{2}+a_{E}S_{k}^{I}-\Delta t\left(1-\alpha\right)^{t}S_{k}\,^{\tau}\gamma\right]}{\frac{2aK_{2}}{\alpha^{2}}\left(\alpha\Delta tK_{1}K_{2}+a\right)+K_{1}\left(c-2d^{2}\,^{\tau}\gamma\right)}\,.\label{eq:77}\end{gather}
+,
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "eq:75"
\end_inset
-This yields
+, and the quotient rule for derivatives of an implicit function,
\begin_inset Formula \begin{gather}
-\frac{\partial\sqrt{^{\tau}J_{2}^{\prime}}}{\partial{}^{t+\Delta t}e_{k}^{\prime\prime}}=\frac{K_{1}\left[\frac{^{t+\Delta t}e_{k}^{\prime\prime}}{2}+a_{E}S_{k}^{I}-\Delta t\left(1-\alpha\right)^{t}S_{k}\,^{\tau}\gamma\right]}{\frac{2aK_{2}}{\alpha^{2}}\left(\alpha\Delta tK_{1}K_{2}+a\right)+K_{1}\left(c-2d^{2}\,^{\tau}\gamma\right)}\,.\label{eq:78}\end{gather}
+\frac{\partial^{t+\Delta t}S_{i}}{\partial{}^{t+\Delta t}e_{k}^{\prime\prime}}=\frac{\delta_{ik}}{a_{E}+\alpha\Delta t\left[^{\tau}\gamma+\frac{^{\tau}S_{i}\left(n-1\right){}^{t+\Delta t}T_{i}\sqrt{^{\tau}J_{2}^{\prime}}^{n-2}}{4\sqrt{^{t+\Delta t}J_{2}^{\prime}}\eta^{n}}\right]}\,.\label{eq:77}\end{gather}
\end_inset
@@ -2821,11 +2814,10 @@
\begin_inset Formula $\left(n=1\right)$
\end_inset
-, this derivative is zero.
- This relation may be used in Equation
+, this equation is identical to Equation
\begin_inset CommandInset ref
LatexCommand ref
-reference "eq:72"
+reference "eq:61"
\end_inset
@@ -2858,32 +2850,30 @@
-1 & -1 & 2 & 0 & 0 & 0\\
0 & 0 & 0 & 3 & 0 & 0\\
0 & 0 & 0 & 0 & 3 & 0\\
-0 & 0 & 0 & 0 & 0 & 3\end{array}\right]\,.\label{eq:79}\end{gather}
+0 & 0 & 0 & 0 & 0 & 3\end{array}\right]\,.\label{eq:78}\end{gather}
\end_inset
-As for the linear case, we use an altered version of the matrix appropriate
- for engineering strain measures.
- At the beginning of a time step, the strains have not yet been computed,
- and we use the following approximation:
-\begin_inset Formula \begin{gather}
-\frac{\partial{}^{t+\Delta t}S_{i}}{\partial{}^{t+\Delta t}e_{k}^{\prime\prime}}\approx\frac{\delta_{ik}}{a_{E}+\Delta t^{\tau}\gamma}\,,\label{eq:80}\end{gather}
-
+Note that if there are no deviatoric stresses at the beginning and end of
+ a time step (or if
+\begin_inset Formula $\eta$
\end_inset
-where we have neglected the changes in
-\begin_inset Formula $^{\tau}\gamma$
-\end_inset
+ goes to infinity), Equations
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "eq:77"
- due to changes in
-\begin_inset Formula $^{t+\Delta t}e_{k}^{\prime\prime}$
\end_inset
-, and we have used a value of
-\begin_inset Formula $\alpha=1$
+ and
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "eq:78"
+
\end_inset
-.
+ reduce to the elastic constitutive matrix, as expected.
\end_layout
\begin_layout Standard
@@ -2897,7 +2887,7 @@
, which may be written:
\begin_inset Formula \begin{gather}
-\frac{\partial F}{\partial\sqrt{^{t+\Delta t}J_{2}^{\prime}}}=2a^{2}\sqrt{^{t+\Delta t}J_{2}^{\prime}}+\frac{\alpha\left(n-1\right)\sqrt{^{\tau}J_{2}^{\prime}}^{n-2}}{2\eta^{n}}\left(2a\alpha\Delta t{}^{t+\Delta t}J_{2}^{\prime}+c-2d^{2}\,^{\tau}\gamma\right)\,.\label{eq:81}\end{gather}
+\frac{\partial F}{\partial\sqrt{^{t+\Delta t}J_{2}^{\prime}}}=2a^{2}\sqrt{^{t+\Delta t}J_{2}^{\prime}}+\frac{\alpha\left(n-1\right)\sqrt{^{\tau}J_{2}^{\prime}}^{n-2}}{2\eta^{n}}\left(2a\alpha\Delta t{}^{t+\Delta t}J_{2}^{\prime}+c-2d^{2}\,^{\tau}\gamma\right)\,.\label{eq:79}\end{gather}
\end_inset
@@ -2938,10 +2928,6 @@
\end_inset
-\begin_inset VSpace defskip
-\end_inset
-
-
\begin_inset Tabular
<lyxtabular version="3" rows="5" columns="2">
<features>
@@ -3112,10 +3098,6 @@
\end_inset
-\begin_inset VSpace defskip
-\end_inset
-
-
\begin_inset Tabular
<lyxtabular version="3" rows="10" columns="2">
<features>
@@ -3411,10 +3393,6 @@
\end_inset
-\begin_inset VSpace defskip
-\end_inset
-
-
\begin_inset Tabular
<lyxtabular version="3" rows="6" columns="2">
<features>
@@ -3667,10 +3645,6 @@
\end_inset
-\begin_inset VSpace defskip
-\end_inset
-
-
\begin_inset Tabular
<lyxtabular version="3" rows="3" columns="2">
<features>
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