[cig-commits] r16874 - in short/3D/PyLith/trunk/doc/userguide: . governingeqns

[brad at geodynamics.org]

Author: brad
Date: 2010-06-02 16:06:50 -0700 (Wed, 02 Jun 2010)
New Revision: 16874

Modified:
   short/3D/PyLith/trunk/doc/userguide/governingeqns/governingeqns.lyx
   short/3D/PyLith/trunk/doc/userguide/userguide.lyx
Log:
Updated.

Modified: short/3D/PyLith/trunk/doc/userguide/governingeqns/governingeqns.lyx
===================================================================
--- short/3D/PyLith/trunk/doc/userguide/governingeqns/governingeqns.lyx	2010-06-02 22:14:58 UTC (rev 16873)
+++ short/3D/PyLith/trunk/doc/userguide/governingeqns/governingeqns.lyx	2010-06-02 23:06:50 UTC (rev 16874)
@@ -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
@@ -1507,5 +1507,97 @@
 .
 \end_layout
 
+\begin_layout Section
+Formulation for Rigid Body Motion and Small Strains
+\end_layout
+
+\begin_layout Standard
+In some crustal deformation problems sufficient deformation may occur that
+ the assumptions associated with infinitesimal strains assumed in the preceding
+ sections no longer hold.
+ This is often the case for problems when one wants to include the effects
+ of deformation on the overburden pressure.
+ In such cases we want to account for both rigid body motion and small strains.
+ The elasticity formulation in PyLith for small strains uses the Green-Lagrange
+ strain tensor and the Second Piola-Kirchoff stress tensor 
+\begin_inset CommandInset citation
+LatexCommand cite
+key "Bathe:1995"
+
+\end_inset
+
+.
+ The Green-Lagrange strain provides a measure of the strain relative to
+ the original, undeformed configuration.
+\begin_inset Formula \begin{gather}
+\varepsilon_{ij}=\frac{1}{2}(u_{i,j}+u_{j,i}+u_{k,i}u_{k,i}),\text{ or}\\
+\varepsilon_{ij}=X_{ji}X_{ij}-\delta_{ij},\text{ where}\\
+X_{ij}=x_{i,j}(t)=\frac{\partial}{\partial x_{j}}(x_{i}(0)+u_{i}(t)),\end{gather}
+
+\end_inset
+
+and 
+\begin_inset Formula $X_{ij}$
+\end_inset
+
+ is the deformation tensor.
+ The Second Piola-Kirchoff stress tensor, S_{ij}, is related to the Green-Lagran
+ge strain tensor through the elasticity constants,
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \[
+S_{ij}=C_{ijkl}\varepsilon_{kl},\]
+
+\end_inset
+
+in the same manner as in the infinitesimal strain formulation.
+\end_layout
+
+\begin_layout Standard
+The elasticity integral in the finite-element formulation includes additional
+ terms when we account for small strains.
+ Recognizing the similarity between the weighting function and an increment
+ in strain in the infinitesimal formulation (many finite-element texts derive
+ the finite-element formulation for elasticity using the Principle of Virtual
+ Work), we replace 
+\begin_inset Formula $\int_{V}\sigma_{ij}\phi_{i,j}\: dV$
+\end_inset
+
+ with 
+\begin_inset Formula $\int_{V}S_{ij}\delta\varepsilon_{ij}\: dV$
+\end_inset
+
+ in equation 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "eq:elasticity:integral"
+
+\end_inset
+
+.
+ Using the definition of the Green-Lagrangian strain, we have
+\begin_inset Formula \[
+\int_{V}S_{ij}\delta\varepsilon_{ij}\: dV=\int_{V}\frac{1}{2}S_{ij}(\delta u_{i,j}+\delta u_{j,i}+u_{k,i}\delta u_{k,j}+\]
+
+\end_inset
+
+ 
+\end_layout
+
+\begin_layout Subsection
+Quasi-static Problems
+\end_layout
+
+\begin_layout Subsection
+Dynamic Problems
+\end_layout
+
+\begin_layout Standard
+The system Jacobian matrix in dynamic problems does not include the elasticity
+ term, so the system Jacobian matrix in the small strain formulation matches
+ the one used in the infinitesimal strain formulation.
+\end_layout
+
 \end_body
 \end_document

Modified: short/3D/PyLith/trunk/doc/userguide/userguide.lyx
===================================================================
--- short/3D/PyLith/trunk/doc/userguide/userguide.lyx	2010-06-02 22:14:58 UTC (rev 16873)
+++ short/3D/PyLith/trunk/doc/userguide/userguide.lyx	2010-06-02 23:06:50 UTC (rev 16874)
@@ -671,5 +671,21 @@
 , 157–165.
 \end_layout
 
+\begin_layout Bibliography
+\begin_inset CommandInset bibitem
+LatexCommand bibitem
+label "18"
+key "Bathe:1995"
+
+\end_inset
+
+Bathe, K.-J.
+ (1995), 
+\shape italic
+Finite-Element Procedures
+\shape default
+, Prentice Hall, Upper Saddle River, New Jersey, 1037 pp.
+\end_layout
+
 \end_body
 \end_document