[cig-commits] r11335 - in short/3D/PyLith/trunk/doc/userguide: . analyticalsolns analyticalsolns/figs
willic3 at geodynamics.org
willic3 at geodynamics.org
Wed Mar 5 15:16:00 PST 2008
Author: willic3
Date: 2008-03-05 15:16:00 -0800 (Wed, 05 Mar 2008)
New Revision: 11335
Added:
short/3D/PyLith/trunk/doc/userguide/analyticalsolns/
short/3D/PyLith/trunk/doc/userguide/analyticalsolns/analyticalsolns.lyx
short/3D/PyLith/trunk/doc/userguide/analyticalsolns/figs/
short/3D/PyLith/trunk/doc/userguide/analyticalsolns/figs/consttract.eps
short/3D/PyLith/trunk/doc/userguide/analyticalsolns/figs/consttract.fig
Modified:
short/3D/PyLith/trunk/doc/userguide/userguide.lyx
Log:
Added analytical solution for rectangular region subject to constant
traction BC.
Added: short/3D/PyLith/trunk/doc/userguide/analyticalsolns/analyticalsolns.lyx
===================================================================
--- short/3D/PyLith/trunk/doc/userguide/analyticalsolns/analyticalsolns.lyx (rev 0)
+++ short/3D/PyLith/trunk/doc/userguide/analyticalsolns/analyticalsolns.lyx 2008-03-05 23:16:00 UTC (rev 11335)
@@ -0,0 +1,322 @@
+#LyX 1.5.3 created this file. For more info see http://www.lyx.org/
+\lyxformat 276
+\begin_document
+\begin_header
+\textclass book
+\begin_preamble
+
+\end_preamble
+\language english
+\inputencoding latin1
+\font_roman default
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+\paperorientation portrait
+\leftmargin 1in
+\topmargin 1in
+\rightmargin 1in
+\bottommargin 2in
+\secnumdepth 3
+\tocdepth 3
+\paragraph_separation indent
+\defskip medskip
+\quotes_language english
+\papercolumns 1
+\papersides 2
+\paperpagestyle default
+\tracking_changes false
+\output_changes false
+\author ""
+\author ""
+\end_header
+
+\begin_body
+
+\begin_layout Chapter
+\begin_inset LatexCommand label
+name "cha:Analytical-Solns"
+
+\end_inset
+
+Analytical Solutions
+\end_layout
+
+\begin_layout Section
+\begin_inset LatexCommand label
+name "sec:TractionProblems"
+
+\end_inset
+
+Traction Problems
+\end_layout
+
+\begin_layout Standard
+Computation of analytical solutions for elastostatic problems over regular
+ domains is a relatively straightforward procedure.
+ These problems are typically formulated in terms of a combination of displaceme
+nt and traction boundary conditions, and such problems provide a good test
+ of the code accuracy, as well as specifically testing the implementation
+ of traction boundary conditions.
+ We present here two simple problems for this purpose.
+\end_layout
+
+\begin_layout Subsection
+Solutions Using Polynomial Stress Functions
+\end_layout
+
+\begin_layout Standard
+Our derivation follows the procedures outlined in
+\begin_inset LatexCommand cite
+key "Timoshenko:Goodier:1987"
+
+\end_inset
+
+, and we restrict ourselves to two-dimensional problems.
+ Any problem in elastostatics must satisfy the equilibrium equations
+\begin_inset Formula \begin{gather}
+\frac{\partial\sigma_{xx}}{\partial x}+\frac{\partial\sigma_{xy}}{\partial y}+X=0\\
+\frac{\partial\sigma_{yy}}{\partial y}+\frac{\partial\sigma_{xy}}{\partial x}+Y=0,\nonumber \end{gather}
+
+\end_inset
+
+where
+\shape italic
+X
+\shape default
+ and
+\shape italic
+Y
+\shape default
+ are the body force components in the
+\shape italic
+x
+\shape default
+ and
+\shape italic
+y
+\shape default
+-directions, respectively, and the stress components are given by
+\begin_inset Formula $\sigma$
+\end_inset
+
+.
+ In the problems considered here, we neglect body forces, so
+\shape italic
+X
+\shape default
+ and
+\shape italic
+Y
+\shape default
+ disappear from the equilibrium equations.
+ The solution must also satisfy the boundary conditions, given as surface
+ tractions over the surface of the body.
+ Finally, the solution must satisfy the conditions of compatibility, which
+ may be expressed as:
+\begin_inset Formula \begin{equation}
+\left(\frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{2}}{\partial y^{2}}\right)\left(\sigma_{xx}+\sigma_{yy}\right)=0.\end{equation}
+
+\end_inset
+
+To compute the displacement field, it is also necessary to specify displacement
+ boundary conditions.
+\end_layout
+
+\begin_layout Standard
+Equations (1) may be satisfied by taking any function
+\begin_inset Formula $\phi$
+\end_inset
+
+ of
+\shape italic
+x
+\shape default
+ and
+\shape italic
+y
+\shape default
+, and letting the stress components be given by the following expressions:
+\begin_inset Formula \begin{equation}
+\sigma_{xx}=\frac{\partial^{2}\phi}{\partial y^{2}},\;\sigma_{yy}=\frac{\partial^{2}\phi}{\partial x^{2}},\;\sigma_{xy}=-\frac{\partial^{2}\phi}{\partial x\partial y}.\end{equation}
+
+\end_inset
+
+The solution must also satisfy the compatibility equations.
+ Substituting equations (3) into equation (2), we find that the stress function
+
+\begin_inset Formula $\phi$
+\end_inset
+
+ must satisfy
+\begin_inset Formula \begin{equation}
+\frac{\partial^{4}\phi}{\partial x^{4}}+2\frac{\partial^{4}\phi}{\partial x^{2}\partial y^{2}}+\frac{\partial^{4}\phi}{\partial y^{4}}=0.\end{equation}
+
+\end_inset
+
+A relatively easy way to solve a number of problems is to select expressions
+ for
+\begin_inset Formula $\phi$
+\end_inset
+
+ consisting of polynomial functions of
+\shape italic
+x
+\shape default
+ and
+\shape italic
+y
+\shape default
+ of second degree or higher.
+ Any polynomial of second or third degree will satisfy equation (4).
+ For polynomials of higher degree, they must be substituted into equation
+ (4) to determine what restrictions must be placed on the solution.
+\end_layout
+
+\begin_layout Subsection
+Constant Traction Applied to a Rectangular Region
+\end_layout
+
+\begin_layout Standard
+For this problem, a constant normal traction,
+\begin_inset Formula $\overrightarrow{N}$
+\end_inset
+
+, is applied along the positive
+\shape italic
+x
+\shape default
+-edge of the rectangular region (
+\begin_inset Formula $x=x_{0}$
+\end_inset
+
+), and roller boundary conditions are applied along the left and bottom
+ boundaries (
+\begin_inset LatexCommand ref
+reference "fig:Const-tractions"
+
+\end_inset
+
+).
+ Since the tractions are constant, we assume a second degree polynomial
+ for the stress function:
+\begin_inset Formula \begin{equation}
+\phi_{2}=\frac{a_{2}}{2}x^{2}+b_{2}xy+\frac{c_{2}}{2}y^{2}.\end{equation}
+
+\end_inset
+
+This yields the following expressions for the stresses:
+\begin_inset Formula \begin{equation}
+\sigma_{xx}=\frac{\partial^{2}\phi_{2}}{\partial y^{2}}=c_{2}=N,\;\sigma_{yy}=a_{2}=0,\;\sigma_{xy}=-b_{2}=0.\end{equation}
+
+\end_inset
+
+From Hooke's law, we have:
+\begin_inset Formula \begin{equation}
+\epsilon_{xx}=\frac{\partial u}{\partial x}=\frac{\left(1-\nu^{2}\right)N}{E},\;\epsilon_{yy}=\frac{\partial v}{\partial y}=\frac{-\nu\left(1+\nu\right)N}{E},\;\epsilon_{xy}=\frac{1}{2}\left(\frac{\partial v}{\partial x}+\frac{\partial u}{\partial y}\right)=0.\end{equation}
+
+\end_inset
+
+The strain components are thus easily obtained from the stress components.
+\end_layout
+
+\begin_layout Standard
+To obtain the displacements, we must integrate equation (7) and make use
+ of the displacement boundary conditions.
+ Integrating the first two of these, we obtain
+\begin_inset Formula \begin{equation}
+u=\frac{\left(1-\nu^{2}\right)Nx}{E}+f\left(y\right),\; v=\frac{-\nu\left(1+\nu\right)Ny}{E}+f\left(x\right).\end{equation}
+
+\end_inset
+
+Substituting these into the third expression, we obtain
+\begin_inset Formula \begin{equation}
+\frac{\partial f\left(x\right)}{\partial x}=-\frac{\partial f\left(y\right)}{\partial y},\end{equation}
+
+\end_inset
+
+which means that both
+\begin_inset Formula $f\left(x\right)$
+\end_inset
+
+ and
+\begin_inset Formula $f\left(y\right)$
+\end_inset
+
+ must be constant.
+ We solve for these constants using the displacement boundary conditions
+ along
+\begin_inset Formula $x=x_{0}$
+\end_inset
+
+ and
+\begin_inset Formula $y=y_{0}$
+\end_inset
+
+.
+ Doing this, we obtain the following expressions for the displacement components
+:
+\begin_inset Formula \begin{equation}
+u=\frac{\left(1-\nu^{2}\right)N}{E}\left(x-x_{0}\right),\; v=\frac{\nu\left(1+\nu\right)N}{E}\left(y_{0}-y\right).\end{equation}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Float figure
+wide false
+sideways false
+status open
+
+\begin_layout Standard
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Caption
+
+\begin_layout Standard
+\begin_inset LatexCommand label
+name "fig:Const-tractions"
+
+\end_inset
+
+Problem with constant traction boundary conditions applied along right edge.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Graphics
+ filename figs/consttract.eps
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_body
+\end_document
Added: short/3D/PyLith/trunk/doc/userguide/analyticalsolns/figs/consttract.eps
===================================================================
--- short/3D/PyLith/trunk/doc/userguide/analyticalsolns/figs/consttract.eps (rev 0)
+++ short/3D/PyLith/trunk/doc/userguide/analyticalsolns/figs/consttract.eps 2008-03-05 23:16:00 UTC (rev 11335)
@@ -0,0 +1,258 @@
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Added: short/3D/PyLith/trunk/doc/userguide/analyticalsolns/figs/consttract.fig
===================================================================
--- short/3D/PyLith/trunk/doc/userguide/analyticalsolns/figs/consttract.fig (rev 0)
+++ short/3D/PyLith/trunk/doc/userguide/analyticalsolns/figs/consttract.fig 2008-03-05 23:16:00 UTC (rev 11335)
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Modified: short/3D/PyLith/trunk/doc/userguide/userguide.lyx
===================================================================
--- short/3D/PyLith/trunk/doc/userguide/userguide.lyx 2008-03-05 23:13:36 UTC (rev 11334)
+++ short/3D/PyLith/trunk/doc/userguide/userguide.lyx 2008-03-05 23:16:00 UTC (rev 11335)
@@ -269,7 +269,13 @@
\end_inset
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+
+
\begin_inset Include \input{license.lyx}
preview false
@@ -440,5 +446,21 @@
, 154 pp.
\end_layout
+\begin_layout Bibliography
+\begin_inset LatexCommand bibitem
+label "9"
+key "Timoshenko:Goodier:1987"
+
+\end_inset
+
+Timoshenko, S.P.
+ and J.N.
+ Goodier (1987),
+\shape italic
+Theory of Elasticity, Third Edition
+\shape default
+, McGraw-Hill, New York, 567 pp.
+\end_layout
+
\end_body
\end_document
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