[cig-commits] r7305 - in short/3D/PyLith/trunk/doc/userguide: .
materials runpylith
brad at geodynamics.org
brad at geodynamics.org
Tue Jun 19 12:40:49 PDT 2007
Author: brad
Date: 2007-06-19 12:40:49 -0700 (Tue, 19 Jun 2007)
New Revision: 7305
Modified:
short/3D/PyLith/trunk/doc/userguide/TODO
short/3D/PyLith/trunk/doc/userguide/materials/materials.lyx
short/3D/PyLith/trunk/doc/userguide/runpylith/runpylith.lyx
short/3D/PyLith/trunk/doc/userguide/userguide.lyx
Log:
Added discussion of 1-D elastic materials. Reorganized materials chapter (separate by elastic/viscoelastic instead of dimension).
Modified: short/3D/PyLith/trunk/doc/userguide/TODO
===================================================================
--- short/3D/PyLith/trunk/doc/userguide/TODO 2007-06-19 19:09:21 UTC (rev 7304)
+++ short/3D/PyLith/trunk/doc/userguide/TODO 2007-06-19 19:40:49 UTC (rev 7305)
@@ -7,16 +7,12 @@
5. Preface (should be able to mostly copy from PyLith 0.8) [Charles]
DONE
-6. Intro [MOSTLY DONE]
+6. Intro [DONE (for now)]
- Needs more work on PyLith design section.
-
Figures
- Workflow [Brad]
+ PyLith Diagram [future: Brad]
- PyLith Diagram [Brad]
-
Pyre
PyLith
Modified: short/3D/PyLith/trunk/doc/userguide/materials/materials.lyx
===================================================================
--- short/3D/PyLith/trunk/doc/userguide/materials/materials.lyx 2007-06-19 19:09:21 UTC (rev 7304)
+++ short/3D/PyLith/trunk/doc/userguide/materials/materials.lyx 2007-06-19 19:40:49 UTC (rev 7305)
@@ -1,4 +1,4 @@
-#LyX 1.4.3 created this file. For more info see http://www.lyx.org/
+#LyX 1.4.4 created this file. For more info see http://www.lyx.org/
\lyxformat 245
\begin_document
\begin_header
@@ -57,53 +57,9 @@
\end_layout
\begin_layout Section
-1-D Material Models
+Elastic Material Models
\end_layout
-\begin_layout Subsection
-Elastic
-\begin_inset Quotes eld
-\end_inset
-
-Linear
-\begin_inset Quotes erd
-\end_inset
-
- Strain
-\end_layout
-
-\begin_layout Subsection
-Elastic
-\begin_inset Quotes eld
-\end_inset
-
-Linear
-\begin_inset Quotes erd
-\end_inset
-
- Stress
-\end_layout
-
-\begin_layout Section
-2-D Material Models
-\end_layout
-
-\begin_layout Subsection
-Elastic Plane Strain
-\end_layout
-
-\begin_layout Subsection
-Elastic Plane Stress
-\end_layout
-
-\begin_layout Section
-3-D Material Models
-\end_layout
-
-\begin_layout Subsection
-Isotropic Elastic Material
-\end_layout
-
\begin_layout Standard
The generalized form of Hooke's law relating stress and strain for linear
elastic materials is
@@ -117,16 +73,32 @@
Due to symmetry considerations, however, the 81 components of the elasticity
matrix are reduced to 21 independent components for the most general case
- of anisotropic elasticity; for the case of isotropic elasticity, the number
- of independent components reduces to two.
+ of anisotropic elasticity.
Representing the stress and strain in terms of vectors, the constitutive
relation may be written
\begin_inset Formula \begin{gather}
-\overrightarrow{\sigma}=\underline{C}\overrightarrow{\epsilon\,}.\label{eq:7}\end{gather}
+\overrightarrow{\sigma}=\underline{C}\overrightarrow{\epsilon\,},\label{eq:7}\end{gather}
\end_inset
-This model may be characterized by two parameters, Lame's constants
+where
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \begin{gather}
+\underline{C}=\left[\begin{array}{cccccc}
+C_{1111} & C_{1122} & C_{1133} & C_{1112} & C_{1123} & C_{1113}\\
+C_{1122} & C_{2222} & C_{2233} & C_{2212} & C_{2223} & C_{2213}\\
+C_{1133} & C_{2233} & C_{3333} & C_{3312} & C_{3323} & C_{3313}\\
+C_{1112} & C_{2212} & C_{3312} & C_{1212} & C_{1223} & C_{1213}\\
+C_{1123} & C_{2223} & C_{3323} & C_{1223} & C_{2323} & C_{2313}\\
+C_{1113} & C_{2213} & C_{3313} & C_{1213} & C_{1223} & C_{1313}\end{array}\right]\:.\label{eq:8}\end{gather}
+
+\end_inset
+
+For the case of isotropic elasticity, the number of independent components
+ reduces to two, and the model can be characterized by two parameters, Lame's
+ constants
\begin_inset Formula $\mu$
\end_inset
@@ -135,11 +107,11 @@
\end_inset
.
- In terms of these two parameters, the stress-strain matrix,
+ For this case the stress-strain matrix,
\begin_inset Formula $\underline{C}$
\end_inset
-, may be written:
+, becomes
\end_layout
\begin_layout Standard
@@ -154,10 +126,6 @@
\end_inset
-
-\end_layout
-
-\begin_layout Standard
Lame's constants are related to the density (
\begin_inset Formula $\rho$
\end_inset
@@ -174,21 +142,100 @@
\end_layout
\begin_layout Standard
-\begin_inset Formula $ $
+\begin_inset Formula \begin{equation}
+\begin{aligned}\mu= & \rho v_{s}^{2}\\
+\lambda= & \rho v_{p}^{2}-2\mu\end{aligned}
+\end{equation}
+
\end_inset
-\begin_inset Formula \[
-\begin{aligned}\mu= & \rho v_{s}^{2}\\
-\lambda= & \rho v_{p}^{2}-2\mu\end{aligned}
-\]
+\end_layout
+\begin_layout Subsection
+1-D Elastic Material Models
+\end_layout
+
+\begin_layout Subsubsection
+Elastic
+\begin_inset Quotes eld
\end_inset
+Linear
+\begin_inset Quotes erd
+\end_inset
+ Strain
\end_layout
+\begin_layout Standard
+For purely 1-D axial deformation, Hooke's law simplifies to
+\begin_inset Formula \begin{equation}
+\sigma_{11}=(\lambda+2\mu)\epsilon_{11},\end{equation}
+
+\end_inset
+
+ with
+\begin_inset Formula \begin{gather}
+\sigma_{22}=\sigma_{33}=\lambda\epsilon_{11},\\
+\sigma_{12}=\sigma_{23}=\sigma_{13}=0.\end{gather}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Subsubsection
+Elastic
+\begin_inset Quotes eld
+\end_inset
+
+Linear
+\begin_inset Quotes erd
+\end_inset
+
+ Stress
+\end_layout
+
+\begin_layout Standard
+For deformation where the tractions are confined to the axial direction,
+ Hooke's law simplifies to
+\begin_inset Formula \begin{equation}
+\sigma_{11}=\frac{\mu(3\lambda+2\mu)}{\lambda+\mu}\epsilon_{11},\end{equation}
+
+\end_inset
+
+with
+\begin_inset Formula \begin{gather}
+\epsilon_{22}=\epsilon_{33}=-\frac{\lambda}{2(\lambda+\mu)}\epsilon_{11},\\
+\sigma_{22}=\sigma_{33}=\sigma_{12}=\sigma_{23}=\sigma_{13}=0.\end{gather}
+
+\end_inset
+
+
+\end_layout
+
\begin_layout Subsection
+2-D Elastic Material Models
+\end_layout
+
+\begin_layout Subsubsection
+Elastic Plane Strain
+\end_layout
+
+\begin_layout Subsubsection
+Elastic Plane Stress
+\end_layout
+
+\begin_layout Subsection
+3-D Elastic Material Models
+\end_layout
+
+\begin_layout Subsubsection
+Isotropic
+\end_layout
+
+\begin_layout Section
Viscoelastic Materials
\end_layout
@@ -503,7 +550,7 @@
\end_layout
-\begin_layout Subsubsection
+\begin_layout Subsection
Definitions
\end_layout
@@ -753,7 +800,7 @@
\end_layout
-\begin_layout Subsubsection
+\begin_layout Subsection
Effective Stress Formulations for Viscoelastic Materials
\end_layout
@@ -855,7 +902,7 @@
\end_layout
-\begin_layout Subsubsection
+\begin_layout Subsection
Linear Maxwell Viscoelastic Material
\end_layout
Modified: short/3D/PyLith/trunk/doc/userguide/runpylith/runpylith.lyx
===================================================================
--- short/3D/PyLith/trunk/doc/userguide/runpylith/runpylith.lyx 2007-06-19 19:09:21 UTC (rev 7304)
+++ short/3D/PyLith/trunk/doc/userguide/runpylith/runpylith.lyx 2007-06-19 19:40:49 UTC (rev 7305)
@@ -1,4 +1,4 @@
-#LyX 1.4.3 created this file. For more info see http://www.lyx.org/
+#LyX 1.4.4 created this file. For more info see http://www.lyx.org/
\lyxformat 245
\begin_document
\begin_header
@@ -803,13 +803,16 @@
\begin_layout Standard
\align center
+
+\hfill
+
\begin_inset Graphics
filename figs/tri3.eps
\end_inset
- \InsetSpace ~
-\InsetSpace ~
+
+\hfill
\begin_inset Graphics
filename figs/quad4.eps
@@ -817,6 +820,8 @@
\end_inset
+\hfill
+
\end_layout
\begin_layout Caption
@@ -848,20 +853,25 @@
\begin_layout Standard
\align center
+
+\hfill
+
\begin_inset Graphics
filename figs/tet4.eps
\end_inset
-\InsetSpace ~
-\InsetSpace ~
+\hfill
+
\begin_inset Graphics
filename figs/hex8.eps
\end_inset
+\hfill
+
\end_layout
\begin_layout Caption
Modified: short/3D/PyLith/trunk/doc/userguide/userguide.lyx
===================================================================
--- short/3D/PyLith/trunk/doc/userguide/userguide.lyx 2007-06-19 19:09:21 UTC (rev 7304)
+++ short/3D/PyLith/trunk/doc/userguide/userguide.lyx 2007-06-19 19:40:49 UTC (rev 7305)
@@ -1,4 +1,4 @@
-#LyX 1.4.3 created this file. For more info see http://www.lyx.org/
+#LyX 1.4.4 created this file. For more info see http://www.lyx.org/
\lyxformat 245
\begin_document
\begin_header
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