# [cig-commits] r14177 - in doc/snac: . figures

echoi at geodynamics.org echoi at geodynamics.org
Sun Mar 1 10:52:08 PST 2009

Author: echoi
Date: 2009-03-01 10:52:08 -0800 (Sun, 01 Mar 2009)
New Revision: 14177

doc/snac/figures/MohrCoulomb_sigmatau.pdf
doc/snac/figures/MohrCoulomb_sigmatau.tex
Modified:
doc/snac/snac.lyx
Log:
Added more explanations about the plastic constitutive model with a new figure.

===================================================================
(Binary files differ)

Property changes on: doc/snac/figures/MohrCoulomb_sigmatau.pdf
___________________________________________________________________
Name: svn:mime-type
+ application/octet-stream

===================================================================
--- doc/snac/figures/MohrCoulomb_sigmatau.tex	                        (rev 0)
+++ doc/snac/figures/MohrCoulomb_sigmatau.tex	2009-03-01 18:52:08 UTC (rev 14177)
@@ -0,0 +1,28 @@
+\documentclass[12pt]{article}
+\usepackage{pst-all}
+\usepackage{pst-pdf}
+
+\begin{document}
+
+\psset{xunit=1in,yunit=1in, runit=1in, linewidth=1.5pt}
+\begin{pspicture}(-3.5,-2)(3,3)
+% Yield function
+  \psplot[linecolor=red]{-3.0}{1.666667}{x -0.6 mul 1.0 add}
+% Mohr circle
+  \pscircle(-1.28,0){1.516}
+% Arc for the friction angle
+  \psarc(1.666667,0){0.5}{150}{180}
+% Randial line to the intersection
+  \psline[linestyle=dashed](-1.28,0)(-0.5,1.3)
+% Coordinate Axes
+  \psaxes[ticks=none, labels=none]{->}(0,0)(-3.0,-2.0)(3.0,3.0)
+
+% Labels
+  \uput[u](2.8,0){{\huge $\sigma$}}
+  \uput[l](0,2.8){{\huge $\tau$}}
+  \uput[ul](1.1,0){{\huge $\phi$}}
+  \uput[dr](0.236,0){{\huge $\sigma_3$}}
+  \uput[dl](-2.796,0){{\huge $\sigma_1$}}
+\end{pspicture}
+
+\end{document}

Modified: doc/snac/snac.lyx
===================================================================
--- doc/snac/snac.lyx	2009-03-01 16:00:58 UTC (rev 14176)
+++ doc/snac/snac.lyx	2009-03-01 18:52:08 UTC (rev 14177)
@@ -1366,7 +1366,7 @@
\end_inset

is inside or on the yield surface, i.e.,
-\begin_inset Formula $f\left(\sigma^{n+1}\right)\leq0,$
+\begin_inset Formula $f\left(\sigma^{n+1}\right)\geq0,$
\end_inset

where
@@ -1400,7 +1400,7 @@

\begin_layout Standard
\begin_inset Formula $$-f\left(\sigma^{n+1}\right)=\tau+q_{\phi}\sigma_{p}-C\label{eq:frictional materials yield}$$
+f\left(\sigma^{n+1}\right)=q_{\phi}\sigma_{p}+C-\tau\label{eq:frictional materials yield}

\end_inset

@@ -1425,62 +1425,132 @@
\end_inset

is cohesive strength of the material.
- In the case of Drucker-Prager material,
-\begin_inset Formula $\tau$
+ In the case of Mohr-Coulomb material,
+\begin_inset Formula $q_{\phi}$
\end_inset

- is defined as the second invariant of the deviatoric tensor
+ is given as
+\begin_inset Formula $\tan\phi$
+\end_inset
+
+,
\end_layout

\begin_layout Standard
\begin_inset Formula $$-\tau=\sqrt{\frac{1}{2}S_{ij}S_{ij}},\label{eq:Drucker-Prager material}$$
+\tau=\frac{1}{2}\left(\sigma_{3}-\sigma_{1}\right)\cos\phi\text{ and }\sigma_{p}=\frac{1}{2}\left(\sigma_{3}+\sigma_{1}\right)-\frac{1}{2}\left(\sigma_{3}-\sigma_{1}\right)\sin\phi,\label{eq:Mohr-Coulomb material}

\end_inset

where
-\begin_inset Formula $S$
+\begin_inset Formula $\sigma_{1}\leq\sigma_{2}\leq\sigma_{3}$
\end_inset

- is the deviatoric part of stress tensor.
- Similarly in the case of Mohr-Coulomb material,
+ are the principal stresses of stress tensor (Figure
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "fig:MC yield envelop in sigma-tau"
+
+\end_inset
+
+).
+ The actual form of the yield function used in SNAC is
\end_layout

\begin_layout Standard
\begin_inset Formula $$-\tau=\frac{1}{2}\left(\sigma_{3}-\sigma_{1}\right)\text{ and }\sigma_{p}=\frac{1}{2}\left(\sigma_{3}+\sigma_{1}\right),\label{eq:Mohr-Coulomb material}$$
+f(\sigma_{1},\sigma_{3})=\sigma_{1}-N_{\phi}\sigma_{3}+2C\sqrt{N_{\phi}},\label{eq:Mohr-Coulomb yield function}

\end_inset

where
-\begin_inset Formula $\sigma_{1}\leq\sigma_{2}\leq\sigma_{3}$
+\begin_inset Formula $N_{\phi}=\frac{1+\sin\phi}{1-\sin\phi}$
\end_inset

- are the principal stresses of stress tensor.
-
+ and
+\begin_inset Formula $\sqrt{N_{\phi}}=\frac{\cos\phi}{1-\sin\phi}$
+\end_inset
+
+.
\end_layout

\begin_layout Standard
+\begin_inset Float figure
+wide false
+sideways false
+status open
+
+\begin_layout Plain Layout
+\begin_inset Graphics
+	filename figures/MohrCoulomb_sigmatau.pdf
+	width 7cm
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Plain Layout
+\begin_inset Caption
+
+\begin_layout Plain Layout
+\begin_inset CommandInset label
+LatexCommand label
+name "fig:MC yield envelop in sigma-tau"
+
+\end_inset
+
+A diagram showing a Mohr-Coulomb yield envelope with
+\begin_inset Formula $\tan\phi=0.6$
+\end_inset
+
+ and a non-zero cohesion as well as a Mohr circle corresponding to the principal
+ stresses,
+\begin_inset Formula $\sigma_{1}$
+\end_inset
+
+ and
+\begin_inset Formula $\sigma_{3}$
+\end_inset
+
+.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
In general, flow rule for frictional materials is non-associative, i.e., flow
direction differs from the normal of the yield surface normal.
- The plastic flow potential can be given as
+ The plastic flow potential for the Mohr-Coulomb model can be given as
\end_layout

\begin_layout Standard
\begin_inset Formula $$-g\left(\sigma_{n+1}\right)=\tau+q_{\psi}\sigma_{p}\label{eq:plastic flow potential for plastic flow}$$
+g\left(\sigma_{n+1}\right)=\sigma_{1}-N_{\psi}\sigma_{3}\label{eq:plastic flow potential for plastic flow}

\end_inset

where
-\begin_inset Formula $q_{\psi}$
+\begin_inset Formula $N_{\psi}=\frac{1+\sin\psi}{1-\sin\psi}$
\end_inset

- is a function of the dilation angle
+ and
\begin_inset Formula $\psi$
\end_inset

-.
+ is the dilation angle.

\end_layout

@@ -1512,16 +1582,16 @@

\begin_layout Standard
The plastic strain increment,
-\begin_inset Formula $\Delta\varepsilon^{p}=\beta\frac{\partial g}{\partial\sigma}$
+\begin_inset Formula $\Delta\varepsilon^{p}=\lambda\frac{\partial g}{\partial\sigma}$
\end_inset

, where
-\begin_inset Formula $\beta$
+\begin_inset Formula $\lambda$
\end_inset

is the magnitude of the plastic flow.

-\begin_inset Formula $\beta$
+\begin_inset Formula $\lambda$
\end_inset

is determined such that updated stress state is on the yield surface, i.e.,