[cig-commits] r14632 - doc/snac

[sue at geodynamics.org]

Author: sue
Date: 2009-04-08 14:19:01 -0700 (Wed, 08 Apr 2009)
New Revision: 14632

Modified:
   doc/snac/snac.lyx
Log:
fixed typos and standardized eq. references

Modified: doc/snac/snac.lyx
===================================================================
--- doc/snac/snac.lyx	2009-04-08 21:00:35 UTC (rev 14631)
+++ doc/snac/snac.lyx	2009-04-08 21:19:01 UTC (rev 14632)
@@ -1724,7 +1724,7 @@
 \begin_inset Formula $\beta$
 \end_inset
 
- is a proortionality constant to be determined.
+ is a proportionality constant to be determined.
  
 \begin_inset Formula $\beta$
 \end_inset
@@ -1822,7 +1822,7 @@
 
 \begin_layout Standard
 If strain hardening (including softening as a negative hardening) is considered,
- a more general return maping is required.
+ a more general return mapping is required.
  For completeness, we review the 
 \emph on
 cutting-plane
@@ -1907,7 +1907,7 @@
 \begin_inset Formula $\Delta\beta$
 \end_inset
 
- is the increment of the consistency paramter during a time interval between
+ is the increment of the consistency parameter during a time interval between
  
 \begin_inset Formula $t_{n}$
 \end_inset
@@ -1955,14 +1955,15 @@
 
 \end_inset
 
-From eq.(
+From Eq.
+ 
 \begin_inset CommandInset ref
 LatexCommand ref
 reference "eq:internal variable rate"
 
 \end_inset
 
-),
+,
 \end_layout
 
 \begin_layout Standard
@@ -1971,28 +1972,30 @@
 
 \end_inset
 
-By substituting (
+By substituting Eqs.
+ 
 \begin_inset CommandInset ref
 LatexCommand ref
 reference "eq:stress derivative w.r.t. consistency parameter"
 
 \end_inset
 
-) and (
+ and 
 \begin_inset CommandInset ref
 LatexCommand ref
 reference "eq:internal var increment derivative w.r.t. consistency parameter increment"
 
 \end_inset
 
-) into (
+ into Eq.
+ 
 \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}
 
@@ -2002,8 +2005,8 @@
 \end_layout
 
 \begin_layout Standard
-Now we present the cutting-plane algorithm for updating stress, internal
- variable and consistency parameter iteratively in case of a general non-linear
+Now we present the cutting-plane algorithm for updating the stress, internal
+ variable and consistency parameters iteratively in case of a general non-linear
  hardening.
 \end_layout
 
@@ -2054,33 +2057,33 @@
 
 \begin_layout Standard
 This algorithm is just a standard Newton method applied to the discrete
- consistency equation (
+ consistency equation 
 \begin_inset CommandInset ref
 LatexCommand ref
 reference "eq:discrete consistency condition"
 
 \end_inset
 
-).
+.
  Thus, it should be obvious that the plastic correction is accomplished
  by the single step update if the yield function is a linear function of
  the consistency variable.
- In that case, eq.
- (
+ In that case, Eqs.
+ 
 \begin_inset CommandInset ref
 LatexCommand ref
 reference "eq:flow parameter for shear failure"
 
 \end_inset
 
-) and (
+ 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
@@ -2094,7 +2097,7 @@
 \end_inset
 
 .
- From eq.
+ From Eq.
  
 \begin_inset CommandInset ref
 LatexCommand ref
@@ -2152,14 +2155,13 @@
 \end_inset
 
 1.
- 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, 
+ 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 approximately linear w.rt.
- the consistency paramter.
- The current consistency parameter computaiton implemented in SNAC is thus
+ is approximately linear with regard to the consistency parameter.
+ The current consistency parameter computation implemented in SNAC is thus
  justified.
  However, if any non-linearity is introduced such as friction angle varying
  with the internal variable and non-linearly changing cohesion, the interation