[cig-commits] r16876 - short/3D/PyLith/trunk/doc/userguide/governingeqns

Wed Jun 2 17:53:16 PDT 2010

Author: brad
Date: 2010-06-02 17:53:16 -0700 (Wed, 02 Jun 2010)
New Revision: 16876

Modified:
   short/3D/PyLith/trunk/doc/userguide/governingeqns/governingeqns.lyx
Log:
Added small strain formulation.

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

--- short/3D/PyLith/trunk/doc/userguide/governingeqns/governingeqns.lyx	2010-06-02 23:20:55 UTC (rev 16875)
+++ short/3D/PyLith/trunk/doc/userguide/governingeqns/governingeqns.lyx	2010-06-03 00:53:16 UTC (rev 16876)
@@ -1,4 +1,4 @@
-#LyX 1.6.5 created this file. For more info see http://www.lyx.org/
+#LyX 1.6.4 created this file. For more info see http://www.lyx.org/
 \lyxformat 345
 \begin_document
 \begin_header
@@ -1397,7 +1397,7 @@
 
 \begin_layout Standard
 We find the system Jacobian matrix by making use of the temporal discretization
- and isolating the term for the increment in the displacment field at time
+ and isolating the term for the increment in the displacement field at time
  
 \begin_inset Formula $t$
 \end_inset
@@ -1508,18 +1508,17 @@
 \end_layout
 
 \begin_layout Section
-Formulation for Rigid Body Motion and Small Strains
+Small Strain Formulation
 \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.
+ the assumptions associated with infinitesimal strains no longer hold.
  This is often the case for problems when one wants to include the effects
- of deformation on the overburden pressure.
+ of gravitational body forces and 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 
+ strain tensor and the Second Piola-Kirchhoff stress tensor 
 \begin_inset CommandInset citation
 LatexCommand cite
 key "Bathe:1995"
@@ -1541,13 +1540,17 @@
 \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,
+ The Second Piola-Kirchhoff stress tensor, 
+\begin_inset Formula $S_{ij}$
+\end_inset
+
+, is related to the Green-Lagrange strain tensor through the elasticity
+ constants,
 \end_layout
 
 \begin_layout Standard
-\begin_inset Formula \[
-S_{ij}=C_{ijkl}\varepsilon_{kl},\]
+\begin_inset Formula \begin{equation}
+S_{ij}=C_{ijkl}\varepsilon_{kl},\end{equation}
 
 \end_inset
 
@@ -1575,28 +1578,114 @@
 
 \end_inset
 
-.
+, where 
+\begin_inset Formula $\delta\varepsilon_{ij}$
+\end_inset
+
+ is the 
+\begin_inset Quotes eld
+\end_inset
+
+virtual
+\begin_inset Quotes erd
+\end_inset
+
+ strain.
  Using the definition of the Green-Lagrangian strain, we have
-\begin_inset Formula \[
-\int_{V}S_{ij}\delta\varepsilon_{ij}\: dV=\int_{V}\frac{1}{2}S_{ij}(\delta u_{i,j}+\delta u_{j,i}+u_{k,i}\delta u_{k,j}+\]
+\begin_inset Formula \begin{equation}
+\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}+u_{k,j}\delta u_{k,i})\: dV.\end{equation}
 
 \end_inset
 
- 
+Writing the displacements in terms of the basis functions and forcing the
+ terms associated with the arbitrary weighting function (
+\begin_inset Quotes eld
+\end_inset
+
+virtual
+\begin_inset Quotes erd
+\end_inset
+
+ strain) to zero yields the elastic term in the residual,
+\begin_inset Formula \begin{equation}
+r_{i}^{n}=\int_{V}S_{ij}(N_{,i}^{n}+(\sum_{m}a_{k}^{m}N_{,j}^{m})N_{,i}^{n})\: dV.\end{equation}
+
+\end_inset
+
+Thus, we have one additional term (the second term) compared with the residual
+ for infinitesimal strains.
+ Just as in the infinitesimal formulation, we evaluate the integral over
+ the volume using numerical quadrature with sums over the quadrature points
+ of each cell.
 \end_layout
 
 \begin_layout Subsection
 Quasi-static Problems
 \end_layout
 
+\begin_layout Standard
+The system Jacobian for quasi-static problems includes terms associated
+ with elasticity.
+ For the small strain formulation, we write the elasticity term at time
+ 
+\begin_inset Formula $t+\Delta t$
+\end_inset
+
+ and consider the first terms of the Taylor series expansion,
+\begin_inset Formula \begin{equation}
+\int_{v}S_{ij}(t+\Delta t)\delta\varepsilon_{ij}(t+\Delta t)\: dV=\int_{V}(S_{ij}(t)\delta\varepsilon_{ij}(t)+dS_{ij}(t)\delta\varepsilon_{ij}(t)+S_{ij}(t)d\delta\varepsilon_{ij}(t))\: dV.\end{equation}
+
+\end_inset
+
+We approximate the increment in the stress tensor using the elastic constants,
+\begin_inset Formula \begin{equation}
+dS_{ij}=C_{ijkl}d\varepsilon_{kl},\end{equation}
+
+\end_inset
+
+and the increment in the 
+\begin_inset Quotes eld
+\end_inset
+
+virtual
+\begin_inset Quotes erd
+\end_inset
+
+ strain via
+\begin_inset Formula \begin{equation}
+d\delta\varepsilon_{ij}=\frac{1}{2}(du_{k,i}\delta u_{k,j}+du_{k,j}\delta u_{k,i}).\end{equation}
+
+\end_inset
+
+We associate the system Jacobian with the terms involving the increment
+ in displacements.
+ After substituting in the expressions for the increment in the stresses
+ and the increment in the 
+\begin_inset Quotes eld
+\end_inset
+
+virtual
+\begin_inset Quotes erd
+\end_inset
+
+ strains, we have
+\begin_inset Formula \begin{equation}
+A_{ij}^{nm}=\int_{V}\frac{1}{4}C_{ijkl}(N_{,k}^{m}+(\sum_{r}a_{p}^{r}N_{,l}^{r})N_{,k}^{m})(N_{,i}^{n}+(\sum_{r}a_{p}^{r}N_{,j}^{r})N_{,i}^{n})+\frac{1}{2}S_{kl}N_{,l}^{m}N_{,l}^{n}\delta_{ij}\: dV.\end{equation}
+
+\end_inset
+
+The small strain formulation produces additional terms associated with the
+ elastic constants and new a new term associated with the stress tensor.
+\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.
+The system Jacobian matrix in dynamic problems does not include any terms
+ associated with elasticity, so the system Jacobian matrix in the small
+ strain formulation matches the one used in the infinitesimal strain formulation.
 \end_layout
 
 \end_body

