# [cig-commits] r14215 - in doc/snac: . figures

echoi at geodynamics.org echoi at geodynamics.org
Tue Mar 3 14:57:10 PST 2009

Author: echoi
Date: 2009-03-03 14:57:10 -0800 (Tue, 03 Mar 2009)
New Revision: 14215

doc/snac/figures/MohrCoulomb_S1S3.pdf
doc/snac/figures/MohrCoulomb_S1S3.tex
Modified:
doc/snac/snac.lyx
Log:
* Removed an unfinished sentence.

* Expanded the section on the plastic computations.

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Property changes on: doc/snac/figures/MohrCoulomb_S1S3.pdf
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Name: svn:mime-type
+ application/octet-stream

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--- doc/snac/figures/MohrCoulomb_S1S3.tex	                        (rev 0)
+++ doc/snac/figures/MohrCoulomb_S1S3.tex	2009-03-03 22:57:10 UTC (rev 14215)
@@ -0,0 +1,46 @@
+\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}{0.5}{x 3 div 1.1547 add}
+  \psplot[linecolor=red, linestyle=dotted]{0.5}{1.732051}{x 3 div 1.1547 add}
+% tension limit
+  \psplot[linecolor=red]{0.5}{1.3213672}{1.3213672}
+  \psplot[linestyle=dashed]{1.3213672}{2.3}{1.3213672}
+  \psplot[linestyle=dashed]{1.732051}{2.3}{1.732051}
+% boundary between tension and shear
+  \psplot[linecolor=blue]{0.1}{0.5}{x -6.16227766 mul 4.402506 add}
+% Pressure axis
+  \psplot[linecolor=red, linestyle=dotted]{-3.0}{1.732051}{x}
+% Coordinate Axes
+  \psaxes[ticks=none, labels=none]{->}(0,0)(-3.0,-2.0)(3.0,3.0)
+
+% Labels
+  \uput[u](2.8,0){{\huge $\sigma_1$}}
+  \uput[l](0,2.8){{\huge $\sigma_3$}}
+
+  \uput[u]{45}(-1.0,-1.0){{\Large $\sigma_1=\sigma_3$}}
+
+  \uput[u]{18}(-3.0,0.15){{\Large $f_s=0$}}
+  \uput[u]{18}(-2.25,0.4){{\Large $(-)$}}
+  \uput[dl]{18}(-2.25,0.4){{\Large $(+)$}}
+
+  \uput[ur]{0}(1.3,1.3213672){{\Large $f_t=0$}}
+  \uput[u]{0}(0.8,1.3213672){{\Large $(+)$}}
+  \uput[d]{0}(0.8,1.3213672){{\Large $(-)$}}
+
+  \uput[d]{-80}(0.4,3.0){{\Large $h=0$}}
+  \uput[d]{0}(0.6,2.2){{\Large $(+)$}}
+  \uput[d]{0}(0.15,2.2){{\Large $(-)$}}
+
+  \uput[r]{0}(2.3,1.732051){{\huge $\frac{2c}{\tan\phi}$}}
+  \uput[r]{0}(2.3,1.3213672){{\huge $\sigma_t$}}
+
+\end{pspicture}
+
+\end{document}

Modified: doc/snac/snac.lyx
===================================================================
--- doc/snac/snac.lyx	2009-03-03 22:54:14 UTC (rev 14214)
+++ doc/snac/snac.lyx	2009-03-03 22:57:10 UTC (rev 14215)
@@ -983,13 +983,12 @@
First, as in the case of an under-damped oscillator, SNAC's solutions will
exhibit artificial oscillations, but these are only transient and do not
affect the static equilibrium.
- Since the state [TODO something here, sentence is unfinished].
Another notable artifact is the randomness in the magnitude of residual
forces.
Suppose the magnitude of assembled residual force is very small compared
to those of the contributing internal and external forces.
- It means that many significant figures in floating-point numbers representing
- the contributing forces are lost during the assemblage.
+ It means that many significant figures in floating-point numbers are lost
+ during the assemblage.
The residual force ends up with having random numbers in the floating number
corresponding to it.
This is related to a fundamental issue of representing real numbers with
@@ -1426,7 +1425,67 @@
\end_inset

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

@@ -1455,12 +1514,16 @@
\end_inset

).
- The actual form of the yield function used in SNAC is
+ The actual form of the yield function for
+\emph on
+shear failure
+\emph default
+ used in SNAC is
\end_layout

\begin_layout Standard
\begin_inset Formula $$-f(\sigma_{1},\sigma_{3})=\sigma_{1}-N_{\phi}\sigma_{3}+2C\sqrt{N_{\phi}},\label{eq:Mohr-Coulomb yield function}$$
+f_{s}(\sigma_{1},\sigma_{3})=\sigma_{1}-N_{\phi}\sigma_{3}+2C\sqrt{N_{\phi}},\label{eq:Mohr-Coulomb yield function}

\end_inset

@@ -1472,10 +1535,88 @@
\begin_inset Formula $\sqrt{N_{\phi}}=\frac{\cos\phi}{1-\sin\phi}$
\end_inset

+ (Figure
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "fig:MC yield envelop in S1-S3"
+
+\end_inset
+
+).
+ The tensile yield function is defined as
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $$+f_{t}(\sigma_{3})=\sigma_{3}-\sigma_{t},\label{eq:Mohr-Coulomb tensile yield function}$$
+
+\end_inset
+
+where
+\begin_inset Formula $\sigma_{t}$
+\end_inset
+
+ is the tension cut-off.
+ If the tension cut-off is given as a parameter, a smaller value between
+ the theoretical limit,
+\begin_inset Formula $2C/\tan\phi$
+\end_inset
+
+, and the given value is assigned to
+\begin_inset Formula $\sigma_{t}$
+\end_inset
+
.
+ To make sure a unique decision on the mode of yielding, shear vs.
+ tensile, we define additional function,
+\begin_inset Formula $h(\sigma_{1},\sigma_{3})$
+\end_inset
+
+, which bisects the obtuse angle made by two yield functions (Figure
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "fig:MC yield envelop in S1-S3"
+
+\end_inset
+
+) as
\end_layout

\begin_layout Standard
+\begin_inset Formula $$+f_{h}(\sigma_{1},\sigma_{3})=\sigma_{3}-\sigma_{t}+(\sqrt{N_{\phi}^{2}+1}+N_{\phi})(\sigma_{1}-N_{\phi}\sigma_{t}+2c\sqrt{N_{\phi}}),\label{eq:Mohr-Coulomb h function}$$
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Once yielding is declared (i.e.,
+\begin_inset Formula $f_{s}<0$
+\end_inset
+
+ or
+\begin_inset Formula $f_{t}>0$
+\end_inset
+
+), then shear or tensile failure is finally decided based on the value of
+
+\begin_inset Formula $h$
+\end_inset
+
+:
+\emph on
+shear if
+\begin_inset Formula $h<0$
+\end_inset
+
+ and tensile otherwise
+\emph default
+.
+\end_layout
+
+\begin_layout Standard
\begin_inset Float figure
wide false
sideways false
@@ -1483,8 +1624,7 @@

\begin_layout Plain Layout
\begin_inset Graphics
-	filename figures/MohrCoulomb_sigmatau.pdf
-	width 7cm
+	filename figures/MohrCoulomb_S1S3.pdf

\end_inset

@@ -1497,24 +1637,11 @@
\begin_layout Plain Layout
\begin_inset CommandInset label
LatexCommand label
-name "fig:MC yield envelop in sigma-tau"
+name "fig:MC yield envelop in S1-S3"

\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
-
-.
+Yield functions that are used to declare yielding in shear and tensile mode.
\end_layout

\end_inset
@@ -1534,12 +1661,17 @@
\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 for the Mohr-Coulomb model can be given as
+ As in the definitions of yield functions, the plastic flow potential for
+ the
+\emph on
+shear
+\emph default
+ failure in the Mohr-Coulomb model can be given as
\end_layout

\begin_layout Standard
\begin_inset Formula $$-g\left(\sigma_{n+1}\right)=\sigma_{1}-N_{\psi}\sigma_{3}\label{eq:plastic flow potential for plastic flow}$$
+g_{s}\left(\sigma_{1},\sigma_{3}\right)=\sigma_{1}-N_{\psi}\sigma_{3}\label{eq:plastic flow potential for shear failure}

\end_inset

@@ -1552,10 +1684,19 @@
\end_inset

is the dilation angle.
-
+ Likewise, the tensile flow potential is given as
\end_layout

\begin_layout Standard
+\begin_inset Formula $$+g_{t}\left(\sigma_{3}\right)=\sigma_{3}-\sigma_{t}\label{eq:plastic flow potential for tensile failure}$$
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
In the presence of plasticity, the total strain,
\begin_inset Formula $\Delta\varepsilon$
\end_inset
@@ -1565,34 +1706,34 @@

\begin_layout Standard
\begin_inset Formula $$-\Delta\varepsilon=\Delta\varepsilon^{ev}+\Delta\varepsilon^{p},\label{eq:total strain in presence of plasticity}$$
+\Delta\varepsilon=\Delta\varepsilon^{e}+\Delta\varepsilon^{p},\label{eq:total strain in presence of plasticity}

\end_inset

where
-\begin_inset Formula $\Delta\varepsilon^{ev}$
+\begin_inset Formula $\Delta\varepsilon^{e}$
\end_inset

- is the viscoelastic, and
+ is the elastic, and
\begin_inset Formula $\Delta\varepsilon^{p}$
\end_inset

is the plastic strain increment.
-
+ We assume that if plastic yielding implies negligible viscous flow.
\end_layout

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

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

is the magnitude of the plastic flow.

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

is determined such that updated stress state is on the yield surface, i.e.,
@@ -1600,26 +1741,62 @@
\begin_inset Formula $f\left(\sigma^{n}+\Delta\sigma^{n}\right)=0$
\end_inset

+, where
+\begin_inset Formula $\Delta\sigma^{n}=\mathbf{C}:(\Delta\varepsilon^{n}-\Delta\varepsilon^{p})$
+\end_inset
+
.
- Linearizing the yield condition we obtain
+ In the principal component representation,
+\begin_inset Formula $\sigma_{A}=a_{AB}^{e}e_{B}$
+\end_inset
+
+ where
+\begin_inset Formula $\sigma_{A}$
+\end_inset
+
+ and
+\begin_inset Formula $\epsilon_{A}$
+\end_inset
+
+ are principal stress and strain, respectively, and
+\begin_inset Formula $\mathbf{a}^{e}$
+\end_inset
+
+ is a corresponding elastic stiffness matrix of which components are given
+ in terms of Lame's constants:
+\begin_inset Formula $a_{AB}^{e}=\lambda+2\mu\delta_{AB}$
+\end_inset
+
+.
+ By expanding each yield function using
+\begin_inset Formula $\sigma^{TR}=\sigma^{n}+\mathbf{C}:\Delta\varepsilon^{n}$
+\end_inset
+
+, we get the following formulae for
+\begin_inset Formula $\beta$
+\end_inset
+
+:
\end_layout

-\begin_layout Standard
+\begin_layout Itemize
+In case of shear failure
\begin_inset Formula $$-\beta=\frac{f}{G+\kappa q_{\phi}q_{\psi}}\label{eq:linearizing yield condition}$$
+\beta=\frac{\sigma_{1}^{TR}-N_{\phi}\sigma_{3}^{TR}+2c\sqrt{N_{\phi}}}{a_{1B}^{e}\frac{\partial g_{s}}{\partial\sigma_{B}}-N_{\phi}a_{3B}^{e}\frac{\partial g_{s}}{\partial\sigma_{B}}}\label{eq:flow parameter for shear failure}

\end_inset

-which provides stress correction.
- More details of calculations for the Mohr-Coulomb model are available in
- Chapter
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "cha:Benchmark-Problems"

+\end_layout
+
+\begin_layout Itemize
+In case of tensile failure
+\begin_inset Formula $$+\beta=\frac{\sigma_{3}^{TR}-\sigma_{t}}{\frac{\partial g^{t}}{\partial\sigma_{B}}}\label{eq:flow parameter for tensile failure}$$
+
\end_inset

-.
+
\end_layout

\begin_layout Subsection