[cig-commits] r14626 - doc/snac
echoi at geodynamics.org
echoi at geodynamics.org
Wed Apr 8 12:07:50 PDT 2009
Author: echoi
Date: 2009-04-08 12:07:50 -0700 (Wed, 08 Apr 2009)
New Revision: 14626
Modified:
doc/snac/snac.lyx
Log:
Changed soem wording and fixed typos in the "Hardening/Softening" sectin.
Modified: doc/snac/snac.lyx
===================================================================
--- doc/snac/snac.lyx 2009-04-08 18:58:11 UTC (rev 14625)
+++ doc/snac/snac.lyx 2009-04-08 19:07:50 UTC (rev 14626)
@@ -1714,15 +1714,17 @@
\end_layout
\begin_layout Standard
-The plastic strain increment,
-\begin_inset Formula $\Delta\varepsilon^{p}=\beta\frac{\partial g}{\partial\sigma}$
+The plastic strain increment is
+\begin_inset Formula \begin{equation}
+\Delta\varepsilon^{p}=\beta\frac{\partial g}{\partial\sigma}\,,\label{eq:plastic flow rule}\end{equation}
+
\end_inset
-, where
+where
\begin_inset Formula $\beta$
\end_inset
- is a proortionality constant.
+ is a proortionality constant to be determined.
\begin_inset Formula $\beta$
\end_inset
@@ -1838,7 +1840,15 @@
\end_layout
\begin_layout Standard
-Here are some preliminary definitions:
+Here are some preliminary definitions.
+ From (
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "eq:plastic flow rule"
+
+\end_inset
+
+), the plastic strain rate is given as
\end_layout
\begin_layout Standard
@@ -1868,7 +1878,7 @@
\begin_layout Standard
\begin_inset Formula \begin{align}
-\dot{\epsilon}^{p*} & =\frac{\partial\epsilon^{p*}}{\partial\epsilon_{i}}\dot{\epsilon}_{i}^{p}\nonumber \\
+\dot{\epsilon}^{p*} & =\frac{\partial\epsilon^{p*}}{\partial\epsilon_{i}^{p}}\dot{\epsilon}_{i}^{p}\nonumber \\
& =\dot{\beta}\frac{\partial\epsilon^{p*}}{\partial\epsilon_{i}^{p}}\frac{\partial G(\mathbf{\sigma},\epsilon^{p*})}{\partial\sigma_{i}}\label{eq:internal variable rate}\\
& =-\dot{\beta}r(\mathbf{\sigma},\epsilon^{p*})\nonumber \end{align}
@@ -1879,11 +1889,15 @@
\end_inset
has been defined above.
- Finally, we require that the updated stress stay on the yield surface (discrete
- consistency condition): i.e.,
+
\end_layout
\begin_layout Standard
+We require that the updated stress stay on the yield surface (discrete consisten
+cy condition): i.e.,
+\end_layout
+
+\begin_layout Standard
\begin_inset Formula \begin{equation}
\bar{F}(\Delta\beta)\equiv F(\mathbf{\sigma}(\Delta\beta),\epsilon^{p*}(\Delta\beta))=0,\label{eq:discrete consistency condition}\end{equation}
@@ -1893,7 +1907,7 @@
\begin_inset Formula $\Delta\beta$
\end_inset
- is the increment of the consistency paramter during the interval between
+ is the increment of the consistency paramter during a time interval between
\begin_inset Formula $t_{n}$
\end_inset
@@ -1903,7 +1917,7 @@
\end_inset
.
- For later uses, we further derive the derivative of
+ For later uses, we also compute the derivative of
\begin_inset Formula $\bar{F}$
\end_inset
@@ -1921,7 +1935,7 @@
\end_inset
Since
-\begin_inset Formula $\sigma_{n+1}=\mathbf{a}^{e}\cdot(\epsilon_{n+1}-\epsilon_{n+1}^{p})=\mathbf{a}^{e}\cdot(\epsilon_{n+1}-(\epsilon_{n}^{p}+\Delta\beta\partial_{\sigma}G(\sigma_{n+1},\epsilon_{n+1}^{p*}))$
+\begin_inset Formula $\sigma_{n+1}=\mathbf{a}^{e}\cdot(\epsilon_{n+1}-\epsilon_{n+1}^{p})=\mathbf{a}^{e}\cdot[\epsilon_{n+1}-(\epsilon_{n}^{p}+\Delta\beta\partial_{\sigma}G(\sigma_{n+1},\epsilon_{n+1}^{p*}))]$
\end_inset
and
@@ -1932,7 +1946,7 @@
\begin_inset Formula $\epsilon_{n}^{p}$
\end_inset
- is constant during an time interval,
+ is constant during a time interval,
\end_layout
\begin_layout Standard
@@ -1941,14 +1955,14 @@
\end_inset
-From
+From eq.(
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:internal variable rate"
\end_inset
-,
+),
\end_layout
\begin_layout Standard
@@ -1957,28 +1971,28 @@
\end_inset
-By substituting
+By substituting (
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:stress derivative w.r.t. consistency parameter"
\end_inset
-
+) and (
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:internal var increment derivative w.r.t. consistency parameter increment"
\end_inset
- into
+) into (
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:yield function derivative w.r.t. internal variable increment"
\end_inset
-, we get
+), we get
\begin_inset Formula \begin{equation}
\frac{d\bar{F}}{d\Delta\beta}=\frac{\partial F}{\partial\mathbf{\sigma}}\cdot\left(-\mathbf{a}^{e}\cdot\frac{\partial G}{\partial\mathbf{\sigma}}\right)+\frac{\partial F}{\partial\epsilon^{P*}}(-r)\end{equation}
@@ -1988,13 +2002,13 @@
\end_layout
\begin_layout Standard
-Now we present the algorithm of updating stress, internal variable and consisten
-cy parameter iteratively in case of a general non-linear hardening.
-
+Now we present the cutting-plane algorithm for updating stress, internal
+ variable and consistency parameter iteratively in case of a general non-linear
+ hardening.
\end_layout
\begin_layout Enumerate
-Initialization
+Initialization:
\begin_inset Formula \begin{align*}
\epsilon_{n+1}^{p(0)} & =\epsilon_{n}^{p}\\
\epsilon_{n+1}^{p*(0)} & =\epsilon_{n}^{p*}\\
@@ -2025,11 +2039,7 @@
\end_layout
\begin_layout Enumerate
-Incremental upate for
-\begin_inset Formula $k+1$
-\end_inset
-
-:
+Incremental upate:
\begin_inset Formula \begin{align*}
\Delta^{2}\beta & =\frac{F_{n+1}^{(k)}}{-\frac{\partial F}{\partial\Delta\beta}}=\frac{F_{n+1}^{(k)}}{\frac{\partial F}{\partial\sigma}\cdot\mathbf{a}\cdot\frac{\partial G}{\partial\sigma}+\frac{\partial F}{\partial\epsilon^{p*}}r}\\
\epsilon_{n+1}^{p(k+1)} & =\epsilon_{n+1}^{p(k)}+\Delta^{2}\beta\frac{\partial G}{\partial\sigma}(\sigma_{n+1}^{(k)},\epsilon_{n+1}^{p*(k)})\\
@@ -2053,27 +2063,24 @@
).
Thus, it should be obvious that the plastic correction is accomplished
- by the single step update if the yield function,
-\begin_inset Formula $F$
-\end_inset
-
-, is a linear function of the consistency variable.
- In this case, eq.
- [
+ by the single step update if the yield function is a linear function of
+ the consistency variable.
+ In that case, eq.
+ (
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:flow parameter for shear failure"
\end_inset
-,
+) and (
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:flow parameter for tensile failure"
\end_inset
-] are immediately retrieved.
+) are immediately retrieved.
\end_layout
\begin_layout Standard
@@ -2145,17 +2152,18 @@
\end_inset
1.
- This analysis and the strain weakening only through piecewise linear variation
- in cohesion (i.e.,
+ Since the dilation angle is often set to be zero or a small value and the
+ cohesion is always defined as a piecewise linear function,
\begin_inset Formula $F$
\end_inset
- is linear w.rt.
- the consistency paramter) justify the consistency parameter computaiton
- currenctly implemented in SNAC.
+ is approximately linear w.rt.
+ the consistency paramter.
+ The current consistency parameter computaiton implemented in SNAC is thus
+ justified.
However, if any non-linearity is introduced such as friction angle varying
- with the internal variable and cohesion as a non-linear function of the
- internal variable, the interation presented above must be performed.
+ with the internal variable and non-linearly changing cohesion, the interation
+ presented above should be performed.
\end_layout
\begin_layout Subsection
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