3108 lines
66 KiB
Plaintext
3108 lines
66 KiB
Plaintext
#LyX 1.6.2 created this file. For more info see http://www.lyx.org/
|
|
\lyxformat 345
|
|
\begin_document
|
|
\begin_header
|
|
\textclass scrartcl
|
|
\use_default_options true
|
|
\begin_modules
|
|
endnotes
|
|
foottoend
|
|
hanging
|
|
linguistics
|
|
logicalmkup
|
|
minimalistic
|
|
braille
|
|
theorems-std
|
|
\end_modules
|
|
\language english
|
|
\inputencoding auto
|
|
\font_roman default
|
|
\font_sans default
|
|
\font_typewriter default
|
|
\font_default_family default
|
|
\font_sc false
|
|
\font_osf false
|
|
\font_sf_scale 100
|
|
\font_tt_scale 100
|
|
|
|
\graphics default
|
|
\paperfontsize default
|
|
\spacing onehalf
|
|
\use_hyperref false
|
|
\papersize letterpaper
|
|
\use_geometry true
|
|
\use_amsmath 1
|
|
\use_esint 1
|
|
\cite_engine natbib_authoryear
|
|
\use_bibtopic false
|
|
\paperorientation portrait
|
|
\leftmargin 2cm
|
|
\topmargin 2cm
|
|
\rightmargin 2cm
|
|
\bottommargin 2cm
|
|
\headheight 2cm
|
|
\headsep 1cm
|
|
\footskip 1cm
|
|
\secnumdepth 3
|
|
\tocdepth 3
|
|
\paragraph_separation indent
|
|
\defskip medskip
|
|
\quotes_language english
|
|
\papercolumns 1
|
|
\papersides 1
|
|
\paperpagestyle default
|
|
\tracking_changes false
|
|
\output_changes false
|
|
\author ""
|
|
\author ""
|
|
\end_header
|
|
|
|
\begin_body
|
|
|
|
\begin_layout Title
|
|
Summary of constitutive_phenoPowerlaw
|
|
\end_layout
|
|
|
|
\begin_layout Author
|
|
YUN JO RO
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
This document contains information for constitutive_phenoPowerlaw.f90.
|
|
This constitutive subroutine is modified from the current contitutive_phenomeno
|
|
logical.f90.
|
|
We introduce slip and twin family as additional index (or input) for each
|
|
crystal structure in lattice.f90 subroutine (e.g., for HCP crystal: slip and
|
|
twin system has four faimilies, respectively).
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Section
|
|
State Variables in constitutive_phenoPowelaw.f90
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
The current State variables in constitutive_phenoPowerlaw are
|
|
\begin_inset Quotes eld
|
|
\end_inset
|
|
|
|
slip resistance
|
|
\begin_inset Formula $\left(s^{\alpha}\right)$
|
|
\end_inset
|
|
|
|
|
|
\begin_inset Quotes erd
|
|
\end_inset
|
|
|
|
,
|
|
\begin_inset Quotes erd
|
|
\end_inset
|
|
|
|
twin resistance
|
|
\begin_inset Formula $\left(s^{\beta}\right)$
|
|
\end_inset
|
|
|
|
|
|
\begin_inset Quotes erd
|
|
\end_inset
|
|
|
|
,
|
|
\begin_inset Quotes eld
|
|
\end_inset
|
|
|
|
cumulative shear strain
|
|
\begin_inset Formula $\left(\gamma^{\alpha}\right)$
|
|
\end_inset
|
|
|
|
|
|
\begin_inset Quotes erd
|
|
\end_inset
|
|
|
|
, and
|
|
\begin_inset Quotes eld
|
|
\end_inset
|
|
|
|
twin volume fraction
|
|
\begin_inset Formula $\left(f^{\beta}\right)$
|
|
\end_inset
|
|
|
|
|
|
\begin_inset Quotes erd
|
|
\end_inset
|
|
|
|
.
|
|
Superscript
|
|
\begin_inset Formula $\alpha$
|
|
\end_inset
|
|
|
|
and
|
|
\begin_inset Formula $\beta$
|
|
\end_inset
|
|
|
|
denote to slip and twin systems, respectively, in this entire document.
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Section
|
|
Considered Deformation Mechanisms
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
Table
|
|
\begin_inset CommandInset ref
|
|
LatexCommand ref
|
|
reference "Flo:DeformationSystemTable"
|
|
|
|
\end_inset
|
|
|
|
lists slip/twin systems for the
|
|
\begin_inset Quotes eld
|
|
\end_inset
|
|
|
|
hex (hcp)
|
|
\begin_inset Quotes erd
|
|
\end_inset
|
|
|
|
case.
|
|
\begin_inset VSpace medskip
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
\begin_inset Float table
|
|
placement tbph
|
|
wide false
|
|
sideways false
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
\align center
|
|
\begin_inset Tabular
|
|
<lyxtabular version="3" rows="9" columns="4">
|
|
<features>
|
|
<column alignment="center" valignment="middle" width="0">
|
|
<column alignment="center" valignment="middle" width="0">
|
|
<column alignment="center" valignment="middle" width="0">
|
|
<column alignment="center" valignment="middle" width="0">
|
|
<row>
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
\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
|
|
No.
|
|
of slip system
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
</row>
|
|
<row>
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
slip system
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
basal
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
\begin_inset Formula $\left\{ 0001\right\} \left\langle 1\bar{2}10\right\rangle $
|
|
\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
|
|
3
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
</row>
|
|
<row>
|
|
<cell alignment="center" valignment="top" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
prism
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
\begin_inset Formula $\left\{ 10\bar{1}0\right\} \left\langle 1\bar{2}10\right\rangle $
|
|
\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
|
|
3
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
</row>
|
|
<row>
|
|
<cell alignment="center" valignment="top" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
pyr <a>
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
\begin_inset Formula $\left\{ 10\bar{1}1\right\} \left\langle 1\bar{2}10\right\rangle $
|
|
\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
|
|
6
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
</row>
|
|
<row>
|
|
<cell alignment="center" valignment="top" bottomline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
pyr <c+a>
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
\begin_inset Formula $\left\{ 10\bar{1}1\right\} \left\langle 2\bar{1}\bar{1}3\right\rangle $
|
|
\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
|
|
12
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
</row>
|
|
<row>
|
|
<cell alignment="center" valignment="top" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
twin system
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
tensile (T1)
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
\begin_inset Formula $\left\{ 10\bar{1}2\right\} \left\langle \bar{1}011\right\rangle $
|
|
\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
|
|
6
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
</row>
|
|
<row>
|
|
<cell alignment="center" valignment="top" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
compressive (C1)
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
\begin_inset Formula $\left\{ 11\bar{2}2\right\} \left\langle 11\bar{2}\bar{3}\right\rangle $
|
|
\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
|
|
6
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
</row>
|
|
<row>
|
|
<cell alignment="center" valignment="top" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
tensile (T2)
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
\begin_inset Formula $\left\{ 11\bar{2}1\right\} \left\langle \bar{1}\bar{1}26\right\rangle $
|
|
\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
|
|
6
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
</row>
|
|
<row>
|
|
<cell alignment="center" valignment="top" bottomline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
compressive (C1)
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
\begin_inset Formula $\left\{ 10\bar{1}1\right\} \left\langle 10\bar{1}\bar{2}\right\rangle $
|
|
\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
|
|
6
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
</row>
|
|
</lyxtabular>
|
|
|
|
\end_inset
|
|
|
|
|
|
\begin_inset Caption
|
|
|
|
\begin_layout Plain Layout
|
|
Implemented deformation mechanims in
|
|
\begin_inset Formula $\alpha$
|
|
\end_inset
|
|
|
|
-Ti
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\begin_inset CommandInset label
|
|
LatexCommand label
|
|
name "Flo:DeformationSystemTable"
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Itemize
|
|
Slip/twin system for HCP are illustrated in Figures
|
|
\begin_inset CommandInset ref
|
|
LatexCommand ref
|
|
reference "Fig:slipSystemHCP"
|
|
|
|
\end_inset
|
|
|
|
and
|
|
\begin_inset CommandInset ref
|
|
LatexCommand ref
|
|
reference "Fig:twinSystemHCP"
|
|
|
|
\end_inset
|
|
|
|
.
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
\begin_inset Float figure
|
|
wide false
|
|
sideways false
|
|
status collapsed
|
|
|
|
\begin_layout Plain Layout
|
|
\align center
|
|
\begin_inset Graphics
|
|
filename figures/slipSystemForHCP.jpg
|
|
lyxscale 20
|
|
scale 25
|
|
clip
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
\begin_inset Caption
|
|
|
|
\begin_layout Plain Layout
|
|
Drawing for slip system for HCP.
|
|
Burgers vectors were scaled.
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\begin_inset CommandInset label
|
|
LatexCommand label
|
|
name "Fig:slipSystemHCP"
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\begin_inset Float figure
|
|
wide false
|
|
sideways false
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
\align center
|
|
\begin_inset Graphics
|
|
filename figures/twinSystemForHCP.jpg
|
|
lyxscale 20
|
|
scale 25
|
|
clip
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
\begin_inset Caption
|
|
|
|
\begin_layout Plain Layout
|
|
Drawing for twin system for HCP (
|
|
\begin_inset Formula $\alpha$
|
|
\end_inset
|
|
|
|
- Ti).
|
|
Twin directions are not scaled because the
|
|
\begin_inset Quotes eld
|
|
\end_inset
|
|
|
|
twin Burgers vector
|
|
\begin_inset Quotes erd
|
|
\end_inset
|
|
|
|
magnitude is too small to show in the current figures.
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\begin_inset CommandInset label
|
|
LatexCommand label
|
|
name "Fig:twinSystemHCP"
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\begin_inset Newpage clearpage
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
The Table
|
|
\begin_inset CommandInset ref
|
|
LatexCommand ref
|
|
reference "Tab:TwinBurgersVector"
|
|
|
|
\end_inset
|
|
|
|
shows the twin Burgers vector and its maganitude.
|
|
The magnitude of twin Burgers vector for each system is not shown in Figure
|
|
|
|
\begin_inset CommandInset ref
|
|
LatexCommand ref
|
|
reference "Fig:twinSystemHCP"
|
|
|
|
\end_inset
|
|
|
|
since the scale of twin Burgers vector is too small to put in figures.
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
\begin_inset Float table
|
|
placement tbph
|
|
wide false
|
|
sideways false
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
\align center
|
|
\begin_inset Tabular
|
|
<lyxtabular version="3" rows="5" columns="5">
|
|
<features>
|
|
<column alignment="center" valignment="middle" width="0">
|
|
<column alignment="center" valignment="middle" width="0">
|
|
<column alignment="center" valignment="middle" width="0">
|
|
<column alignment="center" valignment="middle" width="0">
|
|
<column alignment="center" valignment="middle" width="0">
|
|
<row>
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
Twin Burgers vector
|
|
\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
|
|
Magnitude
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
</row>
|
|
<row>
|
|
<cell alignment="center" valignment="top" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
twin system
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
tensile (T1)
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
\begin_inset Formula $\left\{ 10\bar{1}2\right\} \left\langle \bar{1}011\right\rangle $
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
\begin_inset Formula $b_{T}=\frac{3-\left(\frac{c}{a}\right)^{2}}{3+\left(\frac{c}{a}\right)^{2}}\left\langle \bar{1}011\right\rangle $
|
|
\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
|
|
\begin_inset Formula $|b_{T}|=\frac{3-\left(\frac{c}{a}\right)^{2}}{\sqrt{3+\left(\frac{c}{a}\right)^{2}}}\cdot a$
|
|
\end_inset
|
|
|
|
, 0.20
|
|
\begin_inset Formula $\cdot a$
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
</row>
|
|
<row>
|
|
<cell alignment="center" valignment="top" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
compressive (C1)
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
\begin_inset Formula $\left\{ 11\bar{2}2\right\} \left\langle 11\bar{2}\bar{3}\right\rangle $
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
\begin_inset Formula $b_{T}=\frac{\left(\frac{c}{a}\right)^{2}-2}{3\cdot\left(\left(\frac{c}{a}\right)^{2}+1\right)}\left\langle 11\bar{2}\bar{3}\right\rangle $
|
|
\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
|
|
\begin_inset Formula $|b_{T}|=\frac{\left(\frac{c}{a}\right)^{2}-2}{\sqrt{\left(\frac{c}{a}\right)^{2}+1}}\cdot a$
|
|
\end_inset
|
|
|
|
, 0.30
|
|
\begin_inset Formula $\cdot a$
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
</row>
|
|
<row>
|
|
<cell alignment="center" valignment="top" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
tensile (T2)
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
\begin_inset Formula $\left\{ 11\bar{2}1\right\} \left\langle \bar{1}\bar{1}26\right\rangle $
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
\begin_inset Formula $b_{T}=\frac{1}{3\cdot\left(4\cdot\left(\frac{c}{a}\right)^{2}+1\right)}\left\langle \bar{1}\bar{1}26\right\rangle $
|
|
\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
|
|
\begin_inset Formula $|b_{T}|=\frac{1}{\sqrt{1+4\cdot\left(\frac{c}{a}\right)^{2}}}\cdot a$
|
|
\end_inset
|
|
|
|
, 0.24
|
|
\begin_inset Formula $\cdot a$
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
</row>
|
|
<row>
|
|
<cell alignment="center" valignment="top" bottomline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
compressive (C1)
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
\begin_inset Formula $\left\{ 10\bar{1}1\right\} \left\langle 10\bar{1}\bar{2}\right\rangle $
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
\begin_inset Formula $b_{T}=\frac{4\cdot\left(\frac{c}{a}\right)^{2}-9}{4\cdot\left(\frac{c}{a}\right)^{2}+3}\left\langle 10\bar{1}\bar{2}\right\rangle $
|
|
\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
|
|
\begin_inset Formula $|b_{T}|=\frac{\sqrt{2}\cdot\left(4\cdot\left(\frac{c}{a}\right)^{2}-9\right)}{\sqrt{3\cdot\left(4\cdot\left(\frac{c}{a}\right)^{2}+3\right)}}\cdot a$
|
|
\end_inset
|
|
|
|
, 0.28
|
|
\begin_inset Formula $\cdot a$
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
</row>
|
|
</lyxtabular>
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
\begin_inset Caption
|
|
|
|
\begin_layout Plain Layout
|
|
Twin Burgers vector,
|
|
\begin_inset Formula $\frac{c}{a}=1.587$
|
|
\end_inset
|
|
|
|
.
|
|
Equations in Table are adopted from reference
|
|
\begin_inset CommandInset citation
|
|
LatexCommand citet
|
|
key "Christian1995"
|
|
|
|
\end_inset
|
|
|
|
.
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\begin_inset CommandInset label
|
|
LatexCommand label
|
|
name "Tab:TwinBurgersVector"
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
\begin_inset Newpage clearpage
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Section
|
|
Kinetics
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
Shear strain rate due to slip is described by following eqation
|
|
\begin_inset CommandInset citation
|
|
LatexCommand citet
|
|
key "Salem2005,Wu2007"
|
|
|
|
\end_inset
|
|
|
|
:
|
|
\begin_inset Formula \begin{equation}
|
|
\dot{\gamma}^{\alpha}=\dot{\gamma_{o}}\left|\frac{\tau^{\alpha}}{s^{\alpha}}\right|^{n}sign\left(\tau^{\alpha}\right)\label{eq:slipStrainRate}\end{equation}
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
, where
|
|
\begin_inset Formula $\dot{\gamma}^{\alpha}$
|
|
\end_inset
|
|
|
|
; shear strain rate,
|
|
\begin_inset Formula $\dot{\gamma}_{o}$
|
|
\end_inset
|
|
|
|
; reference shear strain rate,
|
|
\begin_inset Formula $\tau^{\alpha}$
|
|
\end_inset
|
|
|
|
; resolved shear stress on the slip system,
|
|
\begin_inset Formula $n$
|
|
\end_inset
|
|
|
|
; stress exponent, and
|
|
\begin_inset Formula $s^{\alpha}$
|
|
\end_inset
|
|
|
|
; slip resistance.
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
Twin volume fraction rate is described by following eqation
|
|
\begin_inset CommandInset citation
|
|
LatexCommand citet
|
|
key "Salem2005,Wu2007"
|
|
|
|
\end_inset
|
|
|
|
:
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
\begin_inset Formula \begin{equation}
|
|
\dot{f}^{\beta}=\frac{\dot{\gamma_{o}}}{\gamma^{\beta}}\left|\frac{\tau^{\beta}}{s^{\beta}}\right|^{n}\mathbb{\mathcal{H}}\left(\tau^{\beta}\right)\label{eq:twinVolrate}\end{equation}
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
, where
|
|
\begin_inset Formula $\dot{f}^{\beta}$
|
|
\end_inset
|
|
|
|
; twin volume fraction rate,
|
|
\begin_inset Formula $\dot{\gamma}_{o}$
|
|
\end_inset
|
|
|
|
; reference shear strain rate,
|
|
\begin_inset Formula $\gamma^{\beta}$
|
|
\end_inset
|
|
|
|
;shear strain due to mechanical twinning,
|
|
\begin_inset Formula $\tau^{\beta}$
|
|
\end_inset
|
|
|
|
; resolved shear stress on the twin system, and
|
|
\begin_inset Formula $s^{\beta}$
|
|
\end_inset
|
|
|
|
; twin resistance.
|
|
|
|
\begin_inset Formula $\mathcal{H}$
|
|
\end_inset
|
|
|
|
is Heaviside function.
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Section
|
|
Structure Evolution
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
In this present section, we attempt to show how we establish the relationship
|
|
between the evolutoin of slip/twin resistance and the evolution of shear
|
|
strain/twin volume fraction.
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Subsection
|
|
Interaction matrix.
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
Conceptual relationship between the evolution of state and kinetic variables
|
|
is shown in Equation
|
|
\begin_inset CommandInset ref
|
|
LatexCommand ref
|
|
reference "eq:InteractionMatrix"
|
|
|
|
\end_inset
|
|
|
|
.
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
\begin_inset Formula \begin{equation}
|
|
\left[\begin{array}{c}
|
|
\dot{s}^{\alpha}\\
|
|
\dot{s}^{\beta}\end{array}\right]=\left[\begin{array}{cc}
|
|
M_{\mathrm{slip-slip}} & M_{\mathrm{slip-twin}}\\
|
|
M_{\mathrm{twin-slip}} & M_{\mathrm{twin-twin}}\end{array}\right]\left[\begin{array}{c}
|
|
\dot{\gamma}^{\alpha}\\
|
|
\gamma^{\beta}\cdot\dot{f}^{\beta}\end{array}\right]\label{eq:InteractionMatrix}\end{equation}
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
Four interaction martices are followings; i) slip-slip interaction matrix
|
|
|
|
\begin_inset Formula $\left(M_{\mathrm{{\scriptstyle slip-slip}}}\right)$
|
|
\end_inset
|
|
|
|
, ii) slip-twin interaction matrix
|
|
\begin_inset Formula $\left(M_{\mathrm{slip-twin}}\right)$
|
|
\end_inset
|
|
|
|
, iii) twin-slip interaction matrix
|
|
\begin_inset Formula $\left(M_{\mathrm{twin-slip}}\right)$
|
|
\end_inset
|
|
|
|
, and iv) twin-twin interaction matrix
|
|
\begin_inset Formula $\left(M_{\mathrm{twin-twin}}\right)$
|
|
\end_inset
|
|
|
|
.
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
Detailed interaction type matrices in Equation
|
|
\begin_inset CommandInset ref
|
|
LatexCommand ref
|
|
reference "eq:InteractionMatrix"
|
|
|
|
\end_inset
|
|
|
|
will be further discussed in the following Section.
|
|
\end_layout
|
|
|
|
\begin_layout Subsection
|
|
Interaction type matrix
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
Following sections are sparated into four based on each interaction type
|
|
matrix alluded.
|
|
Numbers in Tables
|
|
\begin_inset CommandInset ref
|
|
LatexCommand ref
|
|
reference "Flo:SlipSlipIntTypeTable"
|
|
|
|
\end_inset
|
|
|
|
,
|
|
\begin_inset CommandInset ref
|
|
LatexCommand ref
|
|
reference "Flo:SlipTwinIntTypeTable"
|
|
|
|
\end_inset
|
|
|
|
,
|
|
\begin_inset CommandInset ref
|
|
LatexCommand ref
|
|
reference "Flo:TwinSlipIntTypeTable"
|
|
|
|
\end_inset
|
|
|
|
, and
|
|
\begin_inset CommandInset ref
|
|
LatexCommand ref
|
|
reference "Flo:TwinTwinIntTypeTable"
|
|
|
|
\end_inset
|
|
|
|
denote the type of interaction between deformation systems (The first column
|
|
vs.
|
|
The first row).
|
|
\end_layout
|
|
|
|
\begin_layout Subsubsection
|
|
Slip-Slip interaction type matrix
|
|
\end_layout
|
|
|
|
\begin_layout Itemize
|
|
There are 20 types of slip-slip interaction as shown in Table
|
|
\begin_inset CommandInset ref
|
|
LatexCommand ref
|
|
reference "Flo:SlipSlipIntTypeTable"
|
|
|
|
\end_inset
|
|
|
|
.
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Itemize
|
|
In Table
|
|
\begin_inset CommandInset ref
|
|
LatexCommand ref
|
|
reference "Flo:SlipSlipIntTypeTable"
|
|
|
|
\end_inset
|
|
|
|
, types of latent hardening among slip systems are listed.
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Itemize
|
|
Actual slip-slip interaction type matrix,
|
|
\begin_inset Formula $M_{\mathrm{slip-slip}}^{'}$
|
|
\end_inset
|
|
|
|
, is listed in Equation
|
|
\begin_inset CommandInset ref
|
|
LatexCommand ref
|
|
reference "eq:SlipSlipIntMatrix"
|
|
|
|
\end_inset
|
|
|
|
.
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
\begin_inset Float table
|
|
placement H
|
|
wide false
|
|
sideways false
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
\align center
|
|
\begin_inset Tabular
|
|
<lyxtabular version="3" rows="5" columns="5">
|
|
<features>
|
|
<column alignment="center" valignment="middle" width="0.8in">
|
|
<column alignment="center" valignment="middle" width="0.7in">
|
|
<column alignment="center" valignment="middle" width="0.6in">
|
|
<column alignment="center" valignment="middle" width="0.6in">
|
|
<column alignment="center" valignment="middle" width="0.7in">
|
|
<row>
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
basal
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
prism
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
pyr <a>
|
|
\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
|
|
pyr<c+a>
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
</row>
|
|
<row>
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
basal
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
1, 5
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
9
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
12
|
|
\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
|
|
14
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
</row>
|
|
<row>
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
prism
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
15
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
2, 6
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
10
|
|
\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
|
|
13
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
</row>
|
|
<row>
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
pyr <a>
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
18
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
16
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
3, 7
|
|
\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
|
|
11
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
</row>
|
|
<row>
|
|
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
pyr <c+a>
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
20
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
19
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
17
|
|
\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
|
|
4, 8
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
</row>
|
|
</lyxtabular>
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
\begin_inset Caption
|
|
|
|
\begin_layout Plain Layout
|
|
Slip-slip interaction type
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\begin_inset CommandInset label
|
|
LatexCommand label
|
|
name "Flo:SlipSlipIntTypeTable"
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
\begin_inset Formula \begin{equation}
|
|
M_{\mathrm{slip-slip}}^{'}=\left[\begin{array}{ccc|ccc|cccccc|cccccccccccc}
|
|
1 & 5 & 5 & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot\\
|
|
& 1 & 5 & \cdot & 9 & \cdot & \cdot & \cdot & 12 & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & 14 & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot\\
|
|
& & 1 & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot\\
|
|
\hline \cdot & \cdot & \cdot & 2 & 6 & 6 & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot\\
|
|
\cdot & 15 & \cdot & & 2 & 6 & \cdot & \cdot & 10 & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & 13 & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot\\
|
|
\cdot & \cdot & \cdot & & & 2 & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot\\
|
|
\hline \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & 3 & 7 & 7 & 7 & 7 & 7 & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot\\
|
|
\cdot & \cdot & \cdot & \cdot & \cdot & \cdot & & 3 & 7 & 7 & 7 & 7 & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot\\
|
|
\cdot & \cdot & \cdot & \cdot & \cdot & \cdot & & & 3 & 7 & 7 & 7 & \cdot & \cdot & \cdot & \cdot & \cdot & 11 & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot\\
|
|
\cdot & 18 & \cdot & \cdot & 16 & \cdot & & & & 3 & 7 & 7 & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot\\
|
|
\cdot & \cdot & \cdot & \cdot & \cdot & \cdot & & & & & 3 & 7 & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot\\
|
|
\cdot & \cdot & \cdot & \cdot & \cdot & \cdot & & & & & & 3 & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot\\
|
|
\hline \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & 4 & 8 & 8 & 8 & 8 & 8 & 8 & 8 & 8 & 8 & 8 & 8\\
|
|
\cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & & 4 & 8 & 8 & 8 & 8 & 8 & 8 & 8 & 8 & 8 & 8\\
|
|
\cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & & & 4 & 8 & 8 & 8 & 8 & 8 & 8 & 8 & 8 & 8\\
|
|
\cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & & & & 4 & 8 & 8 & 8 & 8 & 8 & 8 & 8 & 8\\
|
|
\cdot & 20 & \cdot & \cdot & 19 & \cdot & \cdot & \cdot & 17 & \cdot & \cdot & \cdot & & & & & 4 & 8 & 8 & 8 & 8 & 8 & 8 & 8\\
|
|
\cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & & & & & & 4 & 8 & 8 & 8 & 8 & 8 & 8\\
|
|
\cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & & & & & & & 4 & 8 & 8 & 8 & 8 & 8\\
|
|
\cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & & & & & & & & 4 & 8 & 8 & 8 & 8\\
|
|
\cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & & & & & & & & & 4 & 8 & 8 & 8\\
|
|
\cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & & & & & & & & & & 4 & 8 & 8\\
|
|
\cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & & & & & & & & & & & 4 & 8\\
|
|
\cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & & & & & & & & & & & & 4\end{array}\right]\label{eq:SlipSlipIntMatrix}\end{equation}
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
\begin_inset VSpace vfill
|
|
\end_inset
|
|
|
|
|
|
\begin_inset VSpace vfill
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Subsubsection
|
|
Slip-Twin interaction type matrix
|
|
\end_layout
|
|
|
|
\begin_layout Itemize
|
|
There are 16 types of slip-twin interaction in Table
|
|
\begin_inset CommandInset ref
|
|
LatexCommand ref
|
|
reference "Flo:SlipTwinIntTypeTable"
|
|
|
|
\end_inset
|
|
|
|
.
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Itemize
|
|
Meaning of T1, C1, T2, C2 is listed in Table
|
|
\begin_inset CommandInset ref
|
|
LatexCommand ref
|
|
reference "Flo:DeformationSystemTable"
|
|
|
|
\end_inset
|
|
|
|
.
|
|
\end_layout
|
|
|
|
\begin_layout Itemize
|
|
Actual slip-twin interaction type matrix,
|
|
\begin_inset Formula $M_{\mathrm{slip-twin}}^{'}$
|
|
\end_inset
|
|
|
|
, is listed in Equation
|
|
\begin_inset CommandInset ref
|
|
LatexCommand ref
|
|
reference "eq:SlipTwinIntMatrix"
|
|
|
|
\end_inset
|
|
|
|
.
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
\begin_inset Float table
|
|
placement H
|
|
wide false
|
|
sideways false
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
\align center
|
|
\begin_inset Tabular
|
|
<lyxtabular version="3" rows="5" columns="5">
|
|
<features>
|
|
<column alignment="center" valignment="middle" width="0.8in">
|
|
<column alignment="center" valignment="middle" width="0.7in">
|
|
<column alignment="center" valignment="middle" width="0.6in">
|
|
<column alignment="center" valignment="middle" width="0.6in">
|
|
<column alignment="center" valignment="middle" width="0.7in">
|
|
<row>
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
T1
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
C1
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
T2
|
|
\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
|
|
C1
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
</row>
|
|
<row>
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
basal
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
1
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
2
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
3
|
|
\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
|
|
4
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
</row>
|
|
<row>
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
prism
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
5
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
6
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
7
|
|
\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
|
|
8
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
</row>
|
|
<row>
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
pyr <a>
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
9
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
10
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
11
|
|
\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
|
|
12
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
</row>
|
|
<row>
|
|
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
pyr <c+a>
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
13
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
14
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
15
|
|
\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
|
|
16
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
</row>
|
|
</lyxtabular>
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
\begin_inset Caption
|
|
|
|
\begin_layout Plain Layout
|
|
Slip-twin interaction type
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\begin_inset CommandInset label
|
|
LatexCommand label
|
|
name "Flo:SlipTwinIntTypeTable"
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
\begin_inset Formula \begin{equation}
|
|
M_{\mathrm{slip-twin}}^{'}=\left[\begin{array}{c|c|c|c}
|
|
1 & 2 & 3 & 4\\
|
|
\hline 5 & 6 & 7 & 8\\
|
|
\hline 9 & 10 & 11 & 12\\
|
|
\hline 13 & 14 & 15 & 16\end{array}\right]\label{eq:SlipTwinIntMatrix}\end{equation}
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Subsubsection
|
|
Twin-Slip interaction type matrix
|
|
\end_layout
|
|
|
|
\begin_layout Itemize
|
|
There 16 types of twin-slip interaction in Table
|
|
\begin_inset CommandInset ref
|
|
LatexCommand ref
|
|
reference "Flo:TwinSlipIntTypeTable"
|
|
|
|
\end_inset
|
|
|
|
.
|
|
\end_layout
|
|
|
|
\begin_layout Itemize
|
|
Meaning of T1, C1, T2, C2 is listed in Table
|
|
\begin_inset CommandInset ref
|
|
LatexCommand ref
|
|
reference "Flo:DeformationSystemTable"
|
|
|
|
\end_inset
|
|
|
|
.
|
|
\end_layout
|
|
|
|
\begin_layout Itemize
|
|
Actual twin-slip interaction type matrix,
|
|
\begin_inset Formula $M_{\mathrm{twin-slip}}^{'}$
|
|
\end_inset
|
|
|
|
, is listed in Equation
|
|
\begin_inset CommandInset ref
|
|
LatexCommand ref
|
|
reference "eq:TwinSlipIntMatrix"
|
|
|
|
\end_inset
|
|
|
|
.
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
\begin_inset Float table
|
|
placement H
|
|
wide false
|
|
sideways false
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
\align center
|
|
\begin_inset Tabular
|
|
<lyxtabular version="3" rows="5" columns="5">
|
|
<features>
|
|
<column alignment="center" valignment="middle" width="0.8in">
|
|
<column alignment="center" valignment="middle" width="0.7in">
|
|
<column alignment="center" valignment="middle" width="0.6in">
|
|
<column alignment="center" valignment="middle" width="0.6in">
|
|
<column alignment="center" valignment="middle" width="0.7in">
|
|
<row>
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
basal
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
prism
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
pyr <a>
|
|
\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
|
|
pyr <c+a>
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
</row>
|
|
<row>
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
T1
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
1
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
5
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
9
|
|
\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
|
|
13
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
</row>
|
|
<row>
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
C1
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
2
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
6
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
10
|
|
\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
|
|
14
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
</row>
|
|
<row>
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
T2
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
3
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
7
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
11
|
|
\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
|
|
15
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
</row>
|
|
<row>
|
|
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
C2
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
4
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
8
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
12
|
|
\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
|
|
16
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
</row>
|
|
</lyxtabular>
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
\begin_inset Caption
|
|
|
|
\begin_layout Plain Layout
|
|
Twin-slip interaction type
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\begin_inset CommandInset label
|
|
LatexCommand label
|
|
name "Flo:TwinSlipIntTypeTable"
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
\begin_inset Formula \begin{equation}
|
|
M_{\mathrm{twin-slip}}^{'}=\left[\begin{array}{c|c|c|c}
|
|
1 & 5 & 9 & 13\\
|
|
\hline 2 & 6 & 10 & 14\\
|
|
\hline 3 & 7 & 11 & 15\\
|
|
\hline 4 & 8 & 12 & 16\end{array}\right]\label{eq:TwinSlipIntMatrix}\end{equation}
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Subsubsection
|
|
Twin-twin interaction type matrix
|
|
\end_layout
|
|
|
|
\begin_layout Itemize
|
|
There are 20 types of twin-twin interaction as shown in Table
|
|
\begin_inset CommandInset ref
|
|
LatexCommand ref
|
|
reference "Flo:TwinTwinIntTypeTable"
|
|
|
|
\end_inset
|
|
|
|
.
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Itemize
|
|
In Table
|
|
\begin_inset CommandInset ref
|
|
LatexCommand ref
|
|
reference "Flo:TwinTwinIntTypeTable"
|
|
|
|
\end_inset
|
|
|
|
, types of latent hardening among twin systems are listed.
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Itemize
|
|
Actual twin-twin interaction type marix,
|
|
\begin_inset Formula $M_{\mathrm{twin-twin}}^{'}$
|
|
\end_inset
|
|
|
|
, is listed in Equation
|
|
\begin_inset CommandInset ref
|
|
LatexCommand ref
|
|
reference "eq:TwinTwinIntMatrix"
|
|
|
|
\end_inset
|
|
|
|
.
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
\begin_inset Float table
|
|
placement H
|
|
wide false
|
|
sideways false
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
\align center
|
|
\begin_inset Tabular
|
|
<lyxtabular version="3" rows="5" columns="5">
|
|
<features>
|
|
<column alignment="center" valignment="middle" width="0.8in">
|
|
<column alignment="center" valignment="middle" width="0.7in">
|
|
<column alignment="center" valignment="middle" width="0.6in">
|
|
<column alignment="center" valignment="middle" width="0.6in">
|
|
<column alignment="center" valignment="middle" width="0.7in">
|
|
<row>
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
T1
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
C1
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
T2
|
|
\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
|
|
C2
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
</row>
|
|
<row>
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
T1
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
1, 5
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
9
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
12
|
|
\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
|
|
14
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
</row>
|
|
<row>
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
C1
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
15
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
2, 6
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
10
|
|
\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
|
|
13
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
</row>
|
|
<row>
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
T2
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
18
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
16
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
3, 7
|
|
\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
|
|
11
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
</row>
|
|
<row>
|
|
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
C2
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
20
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
19
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
17
|
|
\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
|
|
4, 8
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
</row>
|
|
</lyxtabular>
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
\begin_inset Caption
|
|
|
|
\begin_layout Plain Layout
|
|
Twin-twin interaction type
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\begin_inset CommandInset label
|
|
LatexCommand label
|
|
name "Flo:TwinTwinIntTypeTable"
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
\begin_inset Formula \begin{equation}
|
|
M_{\mathrm{twin-twin}}^{'}=\left[\begin{array}{cccccc|cccccc|cccccc|cccccc}
|
|
1 & 5 & 5 & 5 & 5 & 5 & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot\\
|
|
& 1 & 5 & 5 & 5 & 5 & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot\\
|
|
& & 1 & 5 & 5 & 5 & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot\\
|
|
& & & 1 & 5 & 5 & \cdot & \cdot & \cdot & 9 & \cdot & \cdot & \cdot & \cdot & \cdot & 12 & \cdot & \cdot & \cdot & \cdot & \cdot & 14 & \cdot & \cdot\\
|
|
& & & & 1 & 5 & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot\\
|
|
& & & & & 1 & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot\\
|
|
\hline \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & 2 & 6 & 6 & 6 & 6 & 6 & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot\\
|
|
\cdot & \cdot & \cdot & \cdot & \cdot & \cdot & & 2 & 6 & 6 & 6 & 6 & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot\\
|
|
\cdot & \cdot & \cdot & \cdot & \cdot & \cdot & & & 2 & 6 & 6 & 6 & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot\\
|
|
\cdot & \cdot & \cdot & 15 & \cdot & \cdot & & & & 2 & 6 & 6 & \cdot & \cdot & \cdot & 10 & \cdot & \cdot & \cdot & \cdot & \cdot & 13 & \cdot & \cdot\\
|
|
\cdot & \cdot & \cdot & \cdot & \cdot & \cdot & & & & & 2 & 6 & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot\\
|
|
\cdot & \cdot & \cdot & \cdot & \cdot & \cdot & & & & & & 2 & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot\\
|
|
\hline \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & 3 & 7 & 7 & 7 & 7 & 7 & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot\\
|
|
\cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & & 3 & 7 & 7 & 7 & 7 & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot\\
|
|
\cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & & & 3 & 7 & 7 & 7 & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot\\
|
|
\cdot & \cdot & \cdot & 18 & \cdot & \cdot & \cdot & \cdot & \cdot & 16 & \cdot & \cdot & & & & 3 & 7 & 7 & \cdot & \cdot & \cdot & 11 & \cdot & \cdot\\
|
|
\cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & & & & & 3 & 7 & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot\\
|
|
\cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & & & & & & 3 & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot\\
|
|
\hline \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & 4 & 8 & 8 & 8 & 8 & 8\\
|
|
\cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & & 4 & 8 & 8 & 8 & 8\\
|
|
\cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & & & 4 & 8 & 8 & 8\\
|
|
\cdot & \cdot & \cdot & 20 & \cdot & \cdot & \cdot & \cdot & \cdot & 19 & \cdot & \cdot & \cdot & \cdot & \cdot & 17 & \cdot & \cdot & & & & 4 & 8 & 8\\
|
|
\cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & & & & & 4 & 8\\
|
|
\cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & & & & & & 4\end{array}\right]\label{eq:TwinTwinIntMatrix}\end{equation}
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Subsection
|
|
Prefactor (nonlinear factor)
|
|
\end_layout
|
|
|
|
\begin_layout Subsubsection
|
|
Prefactors for slip resistance
|
|
\begin_inset Formula $\left(s^{\alpha}\right)$
|
|
\end_inset
|
|
|
|
;
|
|
\begin_inset Formula $M_{\mathrm{slip-slip}}$
|
|
\end_inset
|
|
|
|
and
|
|
\begin_inset Formula $M_{\mathrm{slip-twin}}$
|
|
\end_inset
|
|
|
|
|
|
\begin_inset CommandInset citation
|
|
LatexCommand citet
|
|
key "Wu2007"
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
\begin_inset Formula $M_{\mathrm{slip-slip}}$
|
|
\end_inset
|
|
|
|
and
|
|
\begin_inset Formula $M_{\mathrm{slip-twin}}$
|
|
\end_inset
|
|
|
|
use for slip resistance evolution
|
|
\begin_inset Formula $\left(\dot{s}^{\alpha}\right)$
|
|
\end_inset
|
|
|
|
.
|
|
Equation
|
|
\begin_inset CommandInset ref
|
|
LatexCommand ref
|
|
reference "eq:SlipResisEvolutionEq"
|
|
|
|
\end_inset
|
|
|
|
is for a slip resistance rate evolution.
|
|
This currently shows the prefactor for
|
|
\begin_inset Quotes eld
|
|
\end_inset
|
|
|
|
slip-slip interaction matrix,
|
|
\begin_inset Formula $M_{\mathrm{slip-slip}}$
|
|
\end_inset
|
|
|
|
|
|
\begin_inset Quotes erd
|
|
\end_inset
|
|
|
|
.
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
\begin_inset VSpace medskip
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
|
|
\family roman
|
|
\series medium
|
|
\shape up
|
|
\size normal
|
|
\emph off
|
|
\bar no
|
|
\noun off
|
|
\color none
|
|
\begin_inset Formula \begin{equation}
|
|
M_{\mathrm{slip-slip}}=h_{\mathrm{slip}}\left(1+C\cdot F^{b}\right)\left(1-\frac{s^{\alpha}}{s_{so}^{\alpha}+s_{\mathrm{pr}}\cdot\sqrt{F}}\right)\cdot M_{\mathrm{slip-slip}}^{'}\label{eq:SlipResisEvolutionEq}\end{equation}
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
\begin_inset VSpace medskip
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
, where
|
|
\begin_inset Formula $h_{\mathrm{slip}}$
|
|
\end_inset
|
|
|
|
represent a hardening rate, and
|
|
\begin_inset Formula $S_{\mathrm{so}}^{\alpha}$
|
|
\end_inset
|
|
|
|
saturation slip resistance for slip system without mechanical twinning
|
|
|
|
\begin_inset Formula $\left(\sum_{\beta}f^{\beta}=0\right)$
|
|
\end_inset
|
|
|
|
, respectively.
|
|
And,
|
|
\begin_inset Formula $F$
|
|
\end_inset
|
|
|
|
is
|
|
\begin_inset Formula $\sum_{\beta}f^{\beta}$
|
|
\end_inset
|
|
|
|
, and
|
|
\begin_inset Formula $N^{S}$
|
|
\end_inset
|
|
|
|
is the total number of slip system.
|
|
\begin_inset Formula $C$
|
|
\end_inset
|
|
|
|
,
|
|
\begin_inset Formula $s_{\mathrm{pr}}$
|
|
\end_inset
|
|
|
|
, and
|
|
\begin_inset Formula $b$
|
|
\end_inset
|
|
|
|
are coefficients to introduce the effect of interaction between slip and
|
|
mechanical twin in Equation
|
|
\begin_inset CommandInset ref
|
|
LatexCommand ref
|
|
reference "eq:SlipResisEvolutionEq"
|
|
|
|
\end_inset
|
|
|
|
.
|
|
\end_layout
|
|
|
|
\begin_layout Itemize
|
|
Slip-twin interaction matrix,
|
|
\begin_inset Formula $M_{\mathrm{slip-twin}}$
|
|
\end_inset
|
|
|
|
, has not been implemented with any prefactor in the present version.
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Subsubsection
|
|
Prefactors for twin resistance
|
|
\begin_inset Formula $\left(s^{\beta}\right)$
|
|
\end_inset
|
|
|
|
;
|
|
\begin_inset Formula $M_{\mathrm{twin-slip}}$
|
|
\end_inset
|
|
|
|
and
|
|
\begin_inset Formula $M_{\mathrm{twin-twin}}$
|
|
\end_inset
|
|
|
|
|
|
\begin_inset CommandInset citation
|
|
LatexCommand citet
|
|
key "Salem2005"
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
\begin_inset Formula $M_{\mathrm{twin-sli}p}$
|
|
\end_inset
|
|
|
|
and
|
|
\begin_inset Formula $M_{\mathrm{twin-twin}}$
|
|
\end_inset
|
|
|
|
use for twin resistance evolution
|
|
\begin_inset Formula $\left(\dot{s}^{\beta}\right)$
|
|
\end_inset
|
|
|
|
.
|
|
Twin-twin and twin-slip interaction matrices are described in Equations
|
|
|
|
\begin_inset CommandInset ref
|
|
LatexCommand ref
|
|
reference "eq:TwinTwinContributionToTwinResis"
|
|
|
|
\end_inset
|
|
|
|
and
|
|
\begin_inset CommandInset ref
|
|
LatexCommand ref
|
|
reference "eq:TwinSlipContributionToTwinResis"
|
|
|
|
\end_inset
|
|
|
|
.
|
|
|
|
\begin_inset VSpace medskip
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
\begin_inset Formula \begin{equation}
|
|
M_{\mathrm{twin-twin}}=h_{\mathrm{tw}}\cdot F^{d}\cdot M_{\mathrm{twin-twin}}^{'}\label{eq:TwinTwinContributionToTwinResis}\end{equation}
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
,where
|
|
\begin_inset Formula $h_{\mathrm{tw}}$
|
|
\end_inset
|
|
|
|
and
|
|
\begin_inset Formula $d$
|
|
\end_inset
|
|
|
|
are coefficients for twin-twin contribution.
|
|
|
|
\begin_inset Formula $F$
|
|
\end_inset
|
|
|
|
is
|
|
\begin_inset Formula $\sum_{\beta}f^{\beta}$
|
|
\end_inset
|
|
|
|
.
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
\begin_inset VSpace medskip
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
\begin_inset Formula \begin{equation}
|
|
M_{\mathrm{twin-slip}}=h_{\mathrm{tw-sl}}\cdot\Gamma^{e}\cdot M_{\mathrm{twin-slip}}^{'}\label{eq:TwinSlipContributionToTwinResis}\end{equation}
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
,where
|
|
\begin_inset Formula $h_{\mathrm{tw-sl}}$
|
|
\end_inset
|
|
|
|
and
|
|
\begin_inset Formula $e$
|
|
\end_inset
|
|
|
|
are coefficients for twin-slip contribution
|
|
\family roman
|
|
\series medium
|
|
\shape up
|
|
\size normal
|
|
\emph off
|
|
\bar no
|
|
\noun off
|
|
\color none
|
|
, and
|
|
\begin_inset Formula $\Gamma=\sum_{\alpha}\gamma^{\alpha}$
|
|
\end_inset
|
|
|
|
.
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
\begin_inset Newpage clearpage
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Section
|
|
Material Parameters (Material Configuration file)
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
\begin_inset Float figure
|
|
placement tbph
|
|
wide false
|
|
sideways false
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
\align center
|
|
\begin_inset Graphics
|
|
filename figures/ExpectedMaterialConfigFile.jpg
|
|
lyxscale 30
|
|
scale 80
|
|
clip
|
|
|
|
\end_inset
|
|
|
|
|
|
\begin_inset Caption
|
|
|
|
\begin_layout Plain Layout
|
|
Expected of phenomenological modelling parameters.
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\begin_inset CommandInset label
|
|
LatexCommand label
|
|
name "Fig:ModelParameters"
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Itemize
|
|
The sequence for hardening coefficients in Figure
|
|
\begin_inset CommandInset ref
|
|
LatexCommand ref
|
|
reference "Fig:ModelParameters"
|
|
|
|
\end_inset
|
|
|
|
is the sequence of numbering in Tables
|
|
\begin_inset CommandInset ref
|
|
LatexCommand ref
|
|
reference "Flo:SlipSlipIntTypeTable"
|
|
|
|
\end_inset
|
|
|
|
,
|
|
\begin_inset CommandInset ref
|
|
LatexCommand ref
|
|
reference "Flo:SlipTwinIntTypeTable"
|
|
|
|
\end_inset
|
|
|
|
,
|
|
\begin_inset CommandInset ref
|
|
LatexCommand ref
|
|
reference "Flo:TwinSlipIntTypeTable"
|
|
|
|
\end_inset
|
|
|
|
, and
|
|
\begin_inset CommandInset ref
|
|
LatexCommand ref
|
|
reference "Flo:TwinTwinIntTypeTable"
|
|
|
|
\end_inset
|
|
|
|
above.
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
\begin_inset Newpage clearpage
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
\begin_inset CommandInset bibtex
|
|
LatexCommand bibtex
|
|
bibfiles "MPIEyjr"
|
|
options "bibtotoc,plain"
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\end_body
|
|
\end_document
|