[cig-commits] r6459 - short/3D/PyLith/trunk/doc/userguide

willic3 at geodynamics.org willic3 at geodynamics.org
Wed Mar 28 22:17:57 PDT 2007 

Author: willic3
Date: 2007-03-28 22:17:57 -0700 (Wed, 28 Mar 2007)
New Revision: 6459

Modified:
   short/3D/PyLith/trunk/doc/userguide/qstatic-smalldef.lyx
Log:
Partially finished quasi-static equations.  Still need to finish
time integration section and possibly change earlier equations to
explicitly show nonlinear constitutive relations.



Modified: short/3D/PyLith/trunk/doc/userguide/qstatic-smalldef.lyx
===================================================================
--- short/3D/PyLith/trunk/doc/userguide/qstatic-smalldef.lyx	2007-03-29 04:38:09 UTC (rev 6458)
+++ short/3D/PyLith/trunk/doc/userguide/qstatic-smalldef.lyx	2007-03-29 05:17:57 UTC (rev 6459)
@@ -407,7 +407,7 @@
 
 \begin_layout Standard
 \begin_inset Formula \begin{equation}
-\sigma_{ij}=h_{ij}\left(\varepsilon_{ij},q_{k}\right).\end{equation}
+\sigma_{ij}=h_{ij}\left(\varepsilon_{ij},q_{k}\right)+\sigma_{ij}^{0}.\end{equation}
 
 \end_inset
 
@@ -434,6 +434,17 @@
 \end_inset
 
 .
+ The 
+\begin_inset Formula $\sigma_{ij}^{0}$
+\end_inset
+
+are the initial stresses in the material.
+ To simplify our derivations, we consider the initial stresses to be included
+ in the additional variables, 
+\begin_inset Formula $q_{k}$
+\end_inset
+
+.
  The strain tensor is given by
 \end_layout
 
@@ -446,8 +457,9 @@
 The stress can be either a linear or nonlinear function of strain and the
  additional variables, and in some cases the additional variables are not
  needed.
- In linear elastic behavior, for example, the stress is linearly dependent
- on the strain and there are no additional variables.
+ In linear elastic behavior in the absence of initial stresses, for example,
+ the stress is linearly dependent on the strain and there are no additional
+ variables.
 \end_layout
 
 \begin_layout Subsubsection
@@ -464,7 +476,7 @@
 
 \begin_layout Standard
 \begin_inset Formula \begin{equation}
-\boldsymbol{\underline{\sigma}}=\boldsymbol{\underline{h}}\left(\boldsymbol{\underline{\varepsilon}},\boldsymbol{q}\right).\end{equation}
+\boldsymbol{\underline{\sigma}}=\boldsymbol{\underline{h}}\left(\boldsymbol{\underline{\varepsilon}},\boldsymbol{q}\right)+\boldsymbol{\sigma}_{0}.\end{equation}
 
 \end_inset
 
@@ -476,7 +488,7 @@
 \begin_inset Formula $\boldsymbol{q}$
 \end_inset
 
-represent additional variables upon which the stress depends.
+ represent additional variables upon which the stress depends.
  These additional variables follow the evolution equations
 \end_layout
 
@@ -491,6 +503,17 @@
 \end_inset
 
 .
+ The 
+\begin_inset Formula $\underline{\boldsymbol{\sigma}}_{0}$
+\end_inset
+
+are the initial stresses in the material.
+ To simplify our derivations, we consider the initial stresses to be included
+ in the additional variables, 
+\begin_inset Formula $\boldsymbol{q}$
+\end_inset
+
+.
  The strain tensor is given by
 \end_layout
 
@@ -503,14 +526,19 @@
 The stress can be either a linear or nonlinear function of strain and the
  additional variables, and in some cases the additional variables are not
  needed.
- In linear elastic behavior, for example, the stress is linearly dependent
- on the strain and there are no additional variables.
+ In linear elastic behavior in the absence of initial stresses, for example,
+ the stress is linearly dependent on the strain and there are no additional
+ variables.
 \end_layout
 
-\begin_layout Subsection*
+\begin_layout Subsection
 Derivation of Equilibrium Equations
 \end_layout
 
+\begin_layout Subsubsection
+Index Notation
+\end_layout
+
 \begin_layout Standard
 Consider volume 
 \begin_inset Formula $V$
@@ -718,7 +746,7 @@
 
 \end_inset
 
- Consider the divergence theorem applied to the dot product of the stress
+Consider the divergence theorem applied to the dot product of the stress
  tensor and the trial function, 
 \begin_inset Formula $\sigma_{ij}\phi_{i}$
 \end_inset
@@ -863,7 +891,7 @@
 
 , we have
 \begin_inset Formula \begin{multline}
-\sum_{elements}(\int_{V^{e}}\frac{1}{2}\sigma_{ij}(N_{,j}^{p}+N_{,i}^{p})\, dV+\int_{V^{e}}\rho N_{}^{p}\sum_{q}N_{}^{q}\ddot{u}_{i}^{q}\: dV-\int_{V^{e}}N_{}^{p}f_{i}\: dV-\int_{S_{T}^{e}}N_{}^{p}T_{i}\, dS)=0.\end{multline}
+\sum_{elements}(\int_{V^{e}}\frac{1}{2}\sigma_{ij}(N_{,j}^{p}+N_{,i}^{p})\, dV-\int_{V^{e}}N_{}^{p}f_{i}\: dV-\int_{S_{T}^{e}}N_{}^{p}T_{i}\, dS)=0.\end{multline}
 
 \end_inset
 
@@ -880,7 +908,7 @@
 
 \begin_layout Standard
 \begin_inset Formula \begin{gather}
-\nabla\cdot\underline{\sigma}+\overrightarrow{f}=\rho\frac{\partial^{2}\overrightarrow{u}}{\partial t^{2}}\text{ in }V,\\
+\nabla\cdot\underline{\sigma}+\overrightarrow{f}=0\text{ in }V,\\
 \underline{\sigma}\cdot\overrightarrow{n}=\overrightarrow{T}\text{ on }S_{T},\\
 \overrightarrow{u}=\overrightarrow{u}^{o}\text{ on }S_{u},\\
 \underline{\sigma}=\underline{\sigma}^{T}\text{ (symmetric).}\end{gather}
@@ -903,8 +931,8 @@
 
  Hence our weak form is
 \begin_inset Formula \begin{gather}
-\int_{V}\left(\nabla\cdot\underline{\sigma}+\overrightarrow{f}-\rho\frac{\partial^{2}\overrightarrow{u}}{\partial t^{2}}\right)\cdot\overrightarrow{\phi}\, dV=0\text{, or }\\
-\int_{V}(\nabla\cdot\underline{\sigma})\cdot\overrightarrow{\phi}\: dV+\int_{V}\overrightarrow{f}\cdot\overrightarrow{\phi}\: dV-\int_{V}\rho\frac{\partial^{2}\overrightarrow{u}}{\partial t^{2}}\cdot\overrightarrow{\phi}\: dV=0.\end{gather}
+\int_{V}\left(\nabla\cdot\underline{\sigma}+\overrightarrow{f}\right)\cdot\overrightarrow{\phi}\, dV=0\text{, or }\\
+\int_{V}(\nabla\cdot\underline{\sigma})\cdot\overrightarrow{\phi}\: dV+\int_{V}\overrightarrow{f}\cdot\overrightarrow{\phi}\: dV=0.\end{gather}
 
 \end_inset
 
@@ -933,7 +961,7 @@
 
 Substituting into the weak form gives
 \begin_inset Formula \begin{equation}
--\int_{V}\underline{\sigma}:\nabla\overrightarrow{\phi}\, dV+\int_{S}\underline{\sigma}\cdot\overrightarrow{n}\cdot\overrightarrow{\phi}\, dS+\int_{V}\overrightarrow{f}\overrightarrow{\phi}\, dV-\int_{V}\rho\frac{\partial^{2}\overrightarrow{u}}{\partial t^{2}}\cdot\overrightarrow{\phi}\, dV=0.\end{equation}
+-\int_{V}\underline{\sigma}:\nabla\overrightarrow{\phi}\, dV+\int_{S}\underline{\sigma}\cdot\overrightarrow{n}\cdot\overrightarrow{\phi}\, dS+\int_{V}\overrightarrow{f}\overrightarrow{\phi}\, dV=0.\end{equation}
 
 \end_inset
 
@@ -971,7 +999,7 @@
 
 Substituting into the first term gives
 \begin_inset Formula \begin{equation}
--\int_{V}\frac{1}{2}\underline{\sigma}:(\nabla+\nabla^{T})\overrightarrow{\phi}\, dV+\int_{S}\underline{\sigma}\cdot\overrightarrow{n}\cdot\overrightarrow{\phi}\, dS+\int_{V}\overrightarrow{f}\cdot\overrightarrow{\phi}\, dV-\int_{V}\rho\frac{\partial^{2}\overrightarrow{u}}{\partial t^{2}}\cdot\overrightarrow{\phi}\, dV=0.\end{equation}
+-\int_{V}\frac{1}{2}\underline{\sigma}:(\nabla+\nabla^{T})\overrightarrow{\phi}\, dV+\int_{S}\underline{\sigma}\cdot\overrightarrow{n}\cdot\overrightarrow{\phi}\, dS+\int_{V}\overrightarrow{f}\cdot\overrightarrow{\phi}\, dV=0.\end{equation}
 
 \end_inset
 
@@ -990,8 +1018,8 @@
 
 ,
 \begin_inset Formula \begin{multline}
--\int_{V}\frac{1}{2}\underline{\sigma}:(\nabla+\nabla^{T})\overrightarrow{\phi}\, dV+\int_{S_{T}}\underline{\sigma}\cdot\overrightarrow{n}\cdot\overrightarrow{\phi}\, dS+\int_{S_{u}}\underline{\sigma}\cdot\overrightarrow{n}\cdot\overrightarrow{\phi}\, dS+\int_{V}\overrightarrow{f}\cdot\overrightarrow{\phi}\, dV\\
--\int_{V}\rho\frac{\partial^{2}\overrightarrow{u}}{\partial t^{2}}\cdot\overrightarrow{\phi}\, dV=0,\end{multline}
+-\int_{V}\frac{1}{2}\underline{\sigma}:(\nabla+\nabla^{T})\overrightarrow{\phi}\, dV+\int_{S_{T}}\underline{\sigma}\cdot\overrightarrow{n}\cdot\overrightarrow{\phi}\, dS+\int_{S_{u}}\underline{\sigma}\cdot\overrightarrow{n}\cdot\overrightarrow{\phi}\, dS+\int_{V}\overrightarrow{f}\cdot\overrightarrow{\phi}\, dV=0\\
+\\\end{multline}
 
 \end_inset
 
@@ -1004,7 +1032,7 @@
 
 so that the equation reduces to
 \begin_inset Formula \begin{equation}
--\int_{V}\frac{1}{2}\underline{\sigma}:(\nabla+\nabla^{T})\overrightarrow{\phi}\: dV+\int_{S_{T}}\overrightarrow{T}\cdot\overrightarrow{\phi}\, dS+\int_{V}\overrightarrow{f}\cdot\overrightarrow{\phi}\, dV-\int_{V}\rho\frac{\partial^{2}\overrightarrow{u}}{\partial t^{2}}\cdot\overrightarrow{\phi}\, dV=0.\end{equation}
+-\int_{V}\frac{1}{2}\underline{\sigma}:(\nabla+\nabla^{T})\overrightarrow{\phi}\: dV+\int_{S_{T}}\overrightarrow{T}\cdot\overrightarrow{\phi}\, dS+\int_{V}\overrightarrow{f}\cdot\overrightarrow{\phi}\, dV=0.\end{equation}
 
 \end_inset
 
@@ -1012,7 +1040,7 @@
  Discretizing into finite-elements separates the integral over the domain
  and boundaries into a sum of integrals over elements and element boundaries,
 \begin_inset Formula \begin{equation}
-\sum_{elements}(\int_{V^{e}}\frac{1}{2}\underline{\sigma}:(\nabla+\nabla^{T})\overrightarrow{\phi}\, dV+\int_{V^{e}}\rho\frac{\partial^{2}\overrightarrow{u}}{\partial t^{2}}\cdot\overrightarrow{\phi}\, dV-\int_{V^{e}}\overrightarrow{f}\cdot\overrightarrow{\phi}\, dV-\int_{S_{t}^{e}}\overrightarrow{T}\cdot\overrightarrow{\phi}\, dS)=0.\end{equation}
+\sum_{elements}(\int_{V^{e}}\frac{1}{2}\underline{\sigma}:(\nabla+\nabla^{T})\overrightarrow{\phi}\, dV-\int_{V^{e}}\overrightarrow{f}\cdot\overrightarrow{\phi}\, dV-\int_{S_{t}^{e}}\overrightarrow{T}\cdot\overrightarrow{\phi}\, dS)=0.\end{equation}
 
 \end_inset
 
@@ -1067,7 +1095,7 @@
 
 Substituting into the integral equation yields
 \begin_inset Formula \begin{multline}
-\sum_{elements}(\int_{V^{e}}\frac{1}{2}\underline{\sigma}:(\nabla+\nabla^{T})\underline{N}\, dV+\int_{V^{e}}\rho\underline{N}\cdot\underline{N}\cdot\frac{\partial^{2}\overrightarrow{u^{e}}}{\partial t^{2}}\: dV-\int_{V^{e}}\underline{N}\cdot\overrightarrow{f^{e}}^{}\, dV-\int_{S_{T}}\underline{N}\cdot\overrightarrow{T^{e}}_{}\, dS)=0\end{multline}
+\sum_{elements}(\int_{V^{e}}\frac{1}{2}\underline{\sigma}:(\nabla+\nabla^{T})\underline{N}\, dV-\int_{V^{e}}\underline{N}\cdot\overrightarrow{f^{e}}^{}\, dV-\int_{S_{T}}\underline{N}\cdot\overrightarrow{T^{e}}_{}\, dS)=0\end{multline}
 
 \end_inset

More information about the cig-commits mailing list