[cig-commits] r18623 - short/3D/PyLith/trunk/doc/userguide/materials
brad at geodynamics.org
brad at geodynamics.org
Tue Jun 14 15:36:19 PDT 2011
Author: brad
Date: 2011-06-14 15:36:19 -0700 (Tue, 14 Jun 2011)
New Revision: 18623
Modified:
short/3D/PyLith/trunk/doc/userguide/materials/materials.lyx
Log:
Cleanup of cross references in materials. Moved tables to correct sections.
Modified: short/3D/PyLith/trunk/doc/userguide/materials/materials.lyx
===================================================================
--- short/3D/PyLith/trunk/doc/userguide/materials/materials.lyx 2011-06-14 21:39:11 UTC (rev 18622)
+++ short/3D/PyLith/trunk/doc/userguide/materials/materials.lyx 2011-06-14 22:36:19 UTC (rev 18623)
@@ -1592,7 +1592,7 @@
).
Future code versions may include alternative formulations for the various
- material models (
+ material models (Appendix
\begin_inset CommandInset ref
LatexCommand ref
reference "cha:Alternative-Formulations"
@@ -1611,7 +1611,8 @@
\begin_inset Formula $\sigma_{33}$
\end_inset
-), which is generally nonzero.
+), which is generally nonzero, using the same method as the other tensor
+ components.
To allow for the specification of this initial stress component, an additional
state variable corresponding to
\begin_inset Formula $\sigma_{33}^{I}$
@@ -2121,7 +2122,7 @@
Linear viscoelastic models are obtained by various combinations of a linear
elastic spring and a linear viscous dashpot in series or parallel.
The simplest example is probably the linear Maxwell model, which consists
- of a spring in series with a dashpot, as shown in
+ of a spring in series with a dashpot, as shown in Figure
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:Material-models"
@@ -2194,7 +2195,7 @@
\begin_layout Standard
The most general form of linear viscoelastic model is the generalized Maxwell
model, which consists of a spring in parallel with a number of Maxwell
- models (last model in
+ models (see Figure
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:Material-models"
@@ -2220,12 +2221,7 @@
\begin_inset Formula $\mu_{1}=\infty$
\end_inset
-.
- Due to its flexibility, the generalized Maxwell model is the model used
- in PyLith, although at present the number of Maxwell models in parallel
- is limited to three.
- We have also included the Maxwell model, since this is more computationally
- efficient if the additional models are not needed.
+ (or a very large number).
\end_layout
\begin_layout Subsection
@@ -2250,12 +2246,12 @@
\end_inset
.
- PyLith versions greater than 1.1 include the specific case of a standard
- Maxwell model as well as a model consisting of a spring in parallel with
- three Maxwell models.
- A number of common material models may be obtained from this model by setting
- the shear moduli of various springs to zero, such as the Maxwell model,
- the Kelvin model, and the standard linear solid.
+ PyLith includes the specific case of a spring in parallel with three Maxwell
+ models.
+ As described in the previous paragraph, a number of common material models
+ may be obtained from this model by setting the shear moduli of various
+ springs to zero, such as the Maxwell model, the Kelvin model, and the standard
+ linear solid.
We follow formulations similar to those used by Zienkiewicz and Taylor
\begin_inset CommandInset citation
@@ -2641,6 +2637,471 @@
\end_layout
+\begin_layout Standard
+\noindent
+\align center
+\begin_inset Float table
+placement H
+wide false
+sideways false
+status open
+
+\begin_layout Plain Layout
+\noindent
+\align center
+\begin_inset Caption
+
+\begin_layout Plain Layout
+\begin_inset CommandInset label
+LatexCommand label
+name "tab:linearMaxwell"
+
+\end_inset
+
+Values in spatial database used as parameters in the linear Maxwell viscoelastic
+ material constitutive model.
+\end_layout
+
+\end_inset
+
+
+\begin_inset Tabular
+<lyxtabular version="3" rows="5" columns="2">
+<features tabularvalignment="middle">
+<column alignment="center" valignment="middle" width="0.85in">
+<column alignment="center" valignment="middle" width="2.47in">
+<row>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\series bold
+Parameter
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\align center
+
+\series bold
+Name in Spatial Database
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $v_{p}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+Vp
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $v_{s}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+Vs
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $\rho$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+density
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $\eta$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+viscosity
+\end_layout
+
+\end_inset
+</cell>
+</row>
+</lyxtabular>
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\noindent
+\align center
+\begin_inset Float table
+placement H
+wide false
+sideways false
+status open
+
+\begin_layout Plain Layout
+\noindent
+\align center
+\begin_inset Caption
+
+\begin_layout Plain Layout
+\begin_inset CommandInset label
+LatexCommand label
+name "tab:genMaxwell"
+
+\end_inset
+
+Values in spatial database used as parameters in the generalized linear
+ Maxwell viscoelastic material constitutive model.
+\end_layout
+
+\end_inset
+
+
+\begin_inset Tabular
+<lyxtabular version="3" rows="10" columns="2">
+<features tabularvalignment="middle">
+<column alignment="center" valignment="middle" width="0.85in">
+<column alignment="center" valignment="middle" width="2.47in">
+<row>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\series bold
+Parameter
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\align center
+
+\series bold
+Name in Spatial Database
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $v_{p}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+Vp
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $v_{s}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+Vs
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $\rho$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+density
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $\mu_{1}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+shear-ratio-1
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $\mu_{2}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+shear-ratio-2
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $\mu_{3}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+shear-ratio-3
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $\eta_{1}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+viscosity-1
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $\eta_{2}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+viscosity-2
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $\eta_{3}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+viscosity-3
+\end_layout
+
+\end_inset
+</cell>
+</row>
+</lyxtabular>
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
\begin_layout Subsection
\begin_inset CommandInset label
LatexCommand label
@@ -2663,14 +3124,14 @@
may be employed for both a linear Maxwell model and a power-law Maxwell
model.
Note that this formulation is not presently employed for linear viscoelastic
- models (
+ models (see Appendix
\begin_inset CommandInset ref
LatexCommand ref
reference "cha:Alternative-Formulations"
\end_inset
-), but it may be employed in future versions.
+), but it is used for power-law viscoelastic materials.
For the viscoelastic materials considered here, the viscous volumetric
strains are zero (incompressible flow), and it is convenient to separate
the general stress-strain relationship at time
@@ -2867,7 +3328,7 @@
\end_inset
-The first term of
+The first term of Equation
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:58"
@@ -2875,7 +3336,7 @@
\end_inset
depends on the particular constitutive relationship, and the complete tangent
- matrix may then be obtained from
+ matrix may then be obtained from Equation
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:57"
@@ -2985,7 +3446,7 @@
\end_inset
-Note that equations
+Note that Equations
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:65"
@@ -2999,14 +3460,14 @@
\end_inset
- are consistent, since equation
+ are consistent, since Equation
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:65"
\end_inset
- may be obtained from equation
+ may be obtained from Equation
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:66"
@@ -3085,7 +3546,7 @@
\end_inset
-Substituting equations
+Substituting Equations
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:70"
@@ -3099,7 +3560,7 @@
\end_inset
- into equation
+ into Equation
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:64"
@@ -3225,7 +3686,7 @@
\begin_inset Formula $A_{T}$
\end_inset
- using
+ using Equation
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:75"
@@ -3288,7 +3749,7 @@
.
The user then specifies either a reference stress or a reference strain
rate, and a database suitable for PyLith is generated.
- This utility is described more fully in
+ This utility is described more fully in Section
\begin_inset CommandInset ref
LatexCommand ref
reference "sub:Tutorial-Step08-Power-law"
@@ -3422,7 +3883,7 @@
\end_inset
, and the total stresses may be found by combining the deviatoric and volumetric
- components from
+ components from Equation
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:42"
@@ -3434,7 +3895,7 @@
\begin_layout Standard
To compute the tangent stress-strain relation, we need to compute the first
- term in
+ term in Equation
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:58"
@@ -3554,7 +4015,7 @@
\begin_inset Formula $\left(n=1\right)$
\end_inset
-, this equation is identical to the linear formulation in
+, this equation is identical to the linear formulation in Section
\begin_inset CommandInset ref
LatexCommand ref
reference "sub:Effective-Stress-Formulation-Maxwell"
@@ -3626,7 +4087,7 @@
\begin_layout Standard
To compute the zero of the effective stress function using Newton's method,
- we require the derivative of equation
+ we require the derivative of Equation
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:84"
@@ -3645,11 +4106,6 @@
\end_layout
\begin_layout Standard
-The effective stress formulation is the method used in PyLith 1.4 and above
- for power-law viscoelasticity.
-\end_layout
-
-\begin_layout Standard
\noindent
\align center
\begin_inset Float table
@@ -3666,471 +4122,6 @@
\begin_layout Plain Layout
\begin_inset CommandInset label
LatexCommand label
-name "tab:linearMaxwell"
-
-\end_inset
-
-Values in spatial database used as parameters in the linear Maxwell viscoelastic
- material constitutive model.
-\end_layout
-
-\end_inset
-
-
-\begin_inset Tabular
-<lyxtabular version="3" rows="5" columns="2">
-<features tabularvalignment="middle">
-<column alignment="center" valignment="middle" width="0.85in">
-<column alignment="center" valignment="middle" width="2.47in">
-<row>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\series bold
-Parameter
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-\align center
-
-\series bold
-Name in Spatial Database
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-\begin_inset Formula $v_{p}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-Vp
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-\begin_inset Formula $v_{s}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-Vs
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-\begin_inset Formula $\rho$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-density
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-\begin_inset Formula $\eta$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-viscosity
-\end_layout
-
-\end_inset
-</cell>
-</row>
-</lyxtabular>
-
-\end_inset
-
-
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-\noindent
-\align center
-\begin_inset Float table
-placement H
-wide false
-sideways false
-status open
-
-\begin_layout Plain Layout
-\noindent
-\align center
-\begin_inset Caption
-
-\begin_layout Plain Layout
-\begin_inset CommandInset label
-LatexCommand label
-name "tab:genMaxwell"
-
-\end_inset
-
-Values in spatial database used as parameters in the generalized linear
- Maxwell viscoelastic material constitutive model.
-\end_layout
-
-\end_inset
-
-
-\begin_inset Tabular
-<lyxtabular version="3" rows="10" columns="2">
-<features tabularvalignment="middle">
-<column alignment="center" valignment="middle" width="0.85in">
-<column alignment="center" valignment="middle" width="2.47in">
-<row>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\series bold
-Parameter
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-\align center
-
-\series bold
-Name in Spatial Database
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-\begin_inset Formula $v_{p}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-Vp
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-\begin_inset Formula $v_{s}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-Vs
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-\begin_inset Formula $\rho$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-density
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-\begin_inset Formula $\mu_{1}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-shear-ratio-1
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-\begin_inset Formula $\mu_{2}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-shear-ratio-2
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-\begin_inset Formula $\mu_{3}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-shear-ratio-3
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-\begin_inset Formula $\eta_{1}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-viscosity-1
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-\begin_inset Formula $\eta_{2}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-viscosity-2
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-\begin_inset Formula $\eta_{3}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Plain Layout
-
-\family typewriter
-viscosity-3
-\end_layout
-
-\end_inset
-</cell>
-</row>
-</lyxtabular>
-
-\end_inset
-
-
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-\noindent
-\align center
-\begin_inset Float table
-placement H
-wide false
-sideways false
-status open
-
-\begin_layout Plain Layout
-\noindent
-\align center
-\begin_inset Caption
-
-\begin_layout Plain Layout
-\begin_inset CommandInset label
-LatexCommand label
name "tab:powerLaw"
\end_inset
@@ -4425,7 +4416,7 @@
\end_layout
\begin_layout Standard
-To perform the solution, the yield condition (
+To perform the solution, the yield condition (Equation
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:97"
@@ -4502,7 +4493,7 @@
\end_inset
.
- The yield surface defined by equations
+ The yield surface defined by Equations
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:100"
@@ -4572,7 +4563,7 @@
Since the plasticity is pressure-dependent, there are volumetric plastic
strains, unlike the viscous strains in the previous section.
- From equation
+ From Equation
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:99"
@@ -4631,7 +4622,7 @@
\family typewriter
dilatation-angle
\family default
- in
+ in Table
\begin_inset CommandInset ref
LatexCommand ref
reference "tab:druckerPrager"
@@ -4650,7 +4641,7 @@
\begin_inset Formula $\psi$
\end_inset
- in equations
+ in Equations
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:101"
@@ -4684,7 +4675,7 @@
\begin_inset Formula $\alpha_{g}$
\end_inset
- (alpha-flow), as shown in
+ (alpha-flow), as shown in Table
\begin_inset CommandInset ref
LatexCommand ref
reference "tab:material-model-output"
@@ -4700,14 +4691,14 @@
\begin_inset Formula $\lambda$
\end_inset
- by substituting
+ by substituting Equation
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:109"
\end_inset
- into equation
+ into Equation
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:105"
@@ -4738,7 +4729,7 @@
\end_inset
-and the pressure is computed from
+and the pressure is computed from Equations
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:105"
@@ -4802,7 +4793,7 @@
\end_inset
-Substituting this into
+Substituting this into Equation
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:105"
@@ -4817,14 +4808,14 @@
\end_inset
-We then use
+We then use Equation
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:108"
\end_inset
- and the second equation of
+ and the second line of Equation
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:105"
@@ -4838,7 +4829,7 @@
\end_inset
- and the first equation of
+ and the first line of Equation
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:105"
@@ -4849,14 +4840,15 @@
\end_layout
\begin_layout Standard
-To compute the elastoplastic tangent matrix we begin by writing
+To compute the elastoplastic tangent matrix we begin by writing Equation
+
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:105"
\end_inset
- as a single equation in terms of stress and strain vectors:
+ as a single expression in terms of stress and strain vectors:
\begin_inset Formula
\begin{equation}
^{t+\Delta t}\sigma_{i}=\frac{1}{a_{E}}\left(^{t+\Delta t}e_{i}^{\prime}-\Delta e_{i}^{P}\right)+S_{i}^{I}+\frac{R_{i}}{a_{m}}\left(^{t+\Delta t}\theta^{\prime}-\Delta\theta^{P}\right)+P^{I}\label{eq:117}
@@ -4885,7 +4877,7 @@
\end_layout
\begin_layout Standard
-From
+From Equations
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:16"
@@ -4914,7 +4906,7 @@
\end_inset
-and from
+and from Equations
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:15"
@@ -4936,7 +4928,7 @@
\end_inset
-From equation
+From Equation
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:116"
@@ -4985,7 +4977,7 @@
\end_inset
-Finally, from
+Finally, from Equation
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:108"
@@ -5299,7 +5291,7 @@
Values in spatial database for initial state variables for 3D problems.
2D problems use only the relevant values.
Note that initial stress and strain are available for all material models.
- Some models have additional state variables (
+ Some models have additional state variables (Table
\begin_inset CommandInset ref
LatexCommand ref
reference "tab:material-model-output"
More information about the CIG-COMMITS
mailing list