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\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.
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\end_inset
\end_layout
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No.
of slip system
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|
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slip system
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basal
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pyr
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\begin_inset Formula $\left\{ 10\bar{1}1\right\} \left\langle 2\bar{1}\bar{1}3\right\rangle $
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12
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twin system
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tensile (T1)
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\begin_inset Formula $\left\{ 10\bar{1}2\right\} \left\langle \bar{1}011\right\rangle $
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compressive (C1)
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\begin_inset Formula $\left\{ 11\bar{2}2\right\} \left\langle 11\bar{2}\bar{3}\right\rangle $
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tensile (T2)
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\begin_inset Formula $\left\{ 11\bar{2}1\right\} \left\langle \bar{1}\bar{1}26\right\rangle $
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compressive (C1)
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\begin_inset Caption
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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
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reference "Fig:twinSystemHCP"
\end_inset
.
\end_layout
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filename figures/slipSystemForHCP.jpg
lyxscale 20
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\end_layout
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\begin_inset Caption
\begin_layout Plain Layout
Drawing for slip system for HCP.
Burgers vectors were scaled.
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\begin_inset CommandInset label
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name "Fig:slipSystemHCP"
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\begin_inset Caption
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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
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LatexCommand label
name "Fig:twinSystemHCP"
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\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
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Twin Burgers vector
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Magnitude
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twin system
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tensile (T1)
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\end_inset
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\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
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\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
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compressive (C1)
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\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 $
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\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
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tensile (T2)
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\begin_inset Formula $\left\{ 11\bar{2}1\right\} \left\langle \bar{1}\bar{1}26\right\rangle $
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\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
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\begin_inset Formula $|b_{T}|=\frac{1}{\sqrt{1+4\cdot\left(\frac{c}{a}\right)^{2}}}\cdot a$
\end_inset
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compressive (C1)
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\begin_inset Formula $\left\{ 10\bar{1}1\right\} \left\langle 10\bar{1}\bar{2}\right\rangle $
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\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 $
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\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
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\end_layout
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\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
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\begin_inset CommandInset label
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name "Tab:TwinBurgersVector"
\end_inset
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\begin_layout Standard
\begin_inset Newpage clearpage
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\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
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\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
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9
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13
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\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
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placement H
wide false
sideways false
status open
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1
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13
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C1
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16
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\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
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C1
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11
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C2
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20
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17
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4, 8
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\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