Thermomechanics of Metallic Solids Undergoing Martensitic Phase Transformations

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Oct 29, 2013 (3 years and 10 months ago)

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KMM European Doctoral Programme
KMM EDP
Thermomechanics of Metallic Solids
Undergoing Martensitic Phase
Transformations
KMM EDP accredited courses in the Spring Semester 2008
Warsaw, 2008
Bogdan Raniecki
- Selected Aspects-



European Doctoral Programme
in Knowledge-based Multifunctional Materials

(KMM EDP)

European Doctoral Programme in Knowledge-based Multifunctional Materials
(KMM EDP) has been created within the framework of the KMM Network
of Excellence and, then, will continue in European Virtual Institute on Knowledge-
based Multifuctional Materials AISBL (KMM-VIN) once the KMM-NoE project has
been finished.

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Universities/Institutes conducting doctoral studies and entitled to confer PhD degree
in Materials Science and Engineering, Mechanics of Materials or equivalent domains
following national classifications.

A direct objective is to offer KMM PhD students a framework to attend specialized
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degree. Some of them may also make a challenge for the KMM Label of Excellence.

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ƒ a label of the European Institute of Innovation and Technology for the PhD
degrees issued within the European Doctoral Programme KMM may be sought
when rules for EIT labelling become operating;
ƒ a label for the European Doctorate in Knowledge-based Multifunctional Materials
will be sought, together with other European projects in advanced materials, when
an institutional framework for European Doctorate is defined.
The tools for implementation of KMM European Doctoral Programme are Intensive
Sessions organized by the KMM Integrated Post-Graduate School, accredited
compact-style KMM courses, as well as programmes of PhD students exchange and
co-supervising of the theses. In the spring semester three sessions of accredited
doctoral courses were offered by the KMM EDP Partners (Politecnico di Milano,
Technische Universität Wien and IPPT). This session is the last one in this semester.

CONTENTS


PART I

Selected Aspects of Thermodynamics of Interfaces in Coherent Phases


A. Preliminaries .................................................................................……..2
A.1 Elements of Kinematics in Continuum Mechanics ....................................………2
A.2 Basic Theorems .........................................................................................……….3
1. Elements of Interfacial Deformation ............................................……….4

1.1. Compatibility Relations – Material Description.............................................…....4
1.2. Compatibility Relations – Semi-spacial Description......................................……6
1.3. Interfacial deformation produced by moving interface

…...........................……..7
2. Fundamental Balance Laws in Continuum Thermodynamics.
Closed Systems

......................................................................................13
2.1. Symbolic Global Law ......................................................................................…

13
2.2. Identification for Closed System

..................................................................……14
2.3. Global Laws...................................................................................................…...14
2.4. Local Laws.................................................................................................……...15
3. Thermodynamic Driving Force. Qualitative Analysis of Interface
Local Equilibrium........................................................................……..16
3.1. Definition of Thermodynamic Driving Force .............................................…….17
3.2. Reversible Processes - Phase Equilibriums

................................................…….18
3.3. Remark on Caloric Relation

...................................................................……......18
3.4. Other Definition of Driving Force............................................................……...19
3.5. Solid With Instantaneous Elasticity ...........................................................…….19
4. Thermodynamics of Coherent Phases Under Small Deformations.
Selected Aspects

............................................................................……24
4.1. Free energy Function. Double-well Energy........................................……......…24
4.2. Thermodynamic Driving Force Function

...............................................………..25
4.3. Thermodynamically admissible zone vector. Basic Relations

.................………26
4.4. Static and Dynamic Phase Equilibriums.............................................……..........28
4.5. On thermodynamic Stability of Inclusions Under Uniform Temperature

………30
4.6. One-dimensional Problem Relevant to Laminates

...................................……...32
4.7. Thin-plate Theory. Phase Transition Criteria

..........................................……….34
4.8. On Equilibrium Instability of Plate-like Inclusions

................................………..37
4.9. Thermal Instability of Plane Martensitic Interface

..................................……….38
4.10. Thermodynamic Driving Force in Inviscid Fluids..............................…………39

PART II

Selected Concepts of Micromechanics

5. Concept of Macroscopic Small Deformation Measures

.....................….41
5.1. Geometrical Characteristics

.........................................................................…….41
5.2. Macroscopic Small Deformations

................................................................……42
6. Concept of Macroscopic Stress.....................................................……...43
6.1. Static Global Equilibrium

............................................................................…….43
6.2. Concept of Macroscopic Stress

..............................................................……......44
7. On the Concept of Overall (Macroscopic) Moduli in Linear Elasticity...45
7.1. Hill’s Postulate of work compatibility.........................................................…….45
7.2. Notion of Overall (Macroscopic) Moduli

....................................................……47
7.3. Notion of the Representative Volume Element (RVE)

................................……50
8. Concept of Macroscopic Elastic and Phase Strains

.........................……52
8.1. Partitioning of Actual Fields......................................................................……...52
8.2. Concept of Macroscopic Elastic Strain.......................................................……..54
8.3. Concept of Macroscopic Phase- Strain (Inelastic Strain)

..........................……...55
8.4. Macroscopic Elastic Strain Energy............................................................……...56

PART III

Phenomenological Thermodynamics of One-way Shape Memory Effect

9. Macroscopic Specific Free Energy ................................................…….58
9.1. Local Fields of Free Energy

........................................................................…….58
9.2. Macroscopic Free Energy of Phase Mixture

...............................................…….58
10. RVE as a Three- Phase Mixture

...............................................………...59
10.1. Limit Phase Strain

….........................................................................................59
10.2. Macroscopic Gibbs Potential ...................................................................…….60
10.3. Concept of Phase Strain Potential

.............................................................……60
10.4. Gibbs Function of partially unconstrained equilibrium states

.................……..61
11. Thermostatic of RL Model...................................................……………63
11.1. Gibbs Function. Fundamental Differential Equations of State........…………..63
11.2. Phase Strain Potential. Relations between Coherency Coefficients

…………..64
11.3. First and Second Laws of Thermodynamics in Quasistatic
Homogeneous Processes...............................................................…………...65
11.4. Caloric Equation

…………................................................................................65
11.5. Examples of Partial Equilibrium Properties

……………..................................66
11.6. Exclusion Criteria. Regions of 3-Phase Metastability

......................………….71
12. Formal Equations of Transformation Kinetics........................………….74
12.1. Kinetics of p.t. of oriented martensite

......................................……………….74
12.2. Kinetics of p.t. of self-accommodating martensite........................……………75
13. Analytical Simulation of 1-D Fundamental Tests.................…………...76
13.1. Isostress 1D tensile tests

.....................................................…………………...76
13.2. Isothermal 1D tension tests...............................................…………………….76
Notations
1
,( ),identity tensors
2
vector product,permutation symbol
,,det( ) determinant of
either or,either or
tr
ij im in in jm
ijk j k ijk
i j kk
i i ij ij ik kj ijmn mnkl
T
a b
a b tr A
AB A B A B A B
  
 
   
   
  
  

I1
a b
a b A= A A
A B AB
A
1 1 1
1 1
anspose of either or
1
inverse of either or ( )
2
either or
T T T
ij ji ijmn mnij
ij jk ik mnij ijkl mk nl ml nk
A A A A
A A A A    
  
 
 
     
I
A,
A A
A A= 1 A A=
KMM European Doctoral Programme
PART I
Selected Aspects of Thermodynamics
of Interfaces in Coherent Phases
KMM EDP accredited courses in the Spring Semester 2008
A. PRELIMINARIES
2
(,)
-1
(,) (,),(,)
(,) (,),det( ) 0
(,),( ) (,),( )/
(,) ( )
,( )
0.5[ ( ) ( )],
i i K
iK i K
ij i k j
t
i
i
T
i
i
x x X t t t
F t x t X J
t t t L x x
t
A t A t A
A t
t t x
t t tr div
x
= ⇔
= ∂ ∂ ≡ >

= = = ∂ ∂

∂ ∂ ∂
= = + =
∂ ∂ ∂

+ ≡

 
X x
x = x X X = x
X X/F
x
X x,X
X x,
L x,FF
D= L x,L x,D=
v v v
v
v
v
v
F(X,t+dt)
F(X,t)
X
x(X,t)
A.1 Elements of Kinematics in Continuum Mechanics 1
F
- gradient of deformation
- particle velocity
L
- spatial gradient of velocity
- material derivative
- strain rate (stretching)
v
x(X,t+dt)
1+ L(X,t)dt
i
x
A

D
(A.1)
Fig. A1
3
Suggested reading:[1] [3]
F(t)
F(t+dt)
1+L(t)dt
time t
0
V
0
time t
V(t)
t+dt
V(t+dt)= V(t)+
n(t)
S(t)
m(t)
S
0
n
0
m
0
m(t+dt)= m(t)+
n(t+dt)= n(t)+
S(t+dt)=S(t)+
dt

n
Sdt

Vdt

(t)dtm
V
0
, V(t), V(t+dt) -
material volume
--------------------------
S
0
, S(t), S(t+dt) –
material surface area
--------------------------
m
0
, m(t), m(t+dt) –
unit vector along
a material fiber.
-------------------------
n
0
, n(t), n(t+dt) –
unit vector normal
to a material surface.
-----------------------------
F(t), F(t+dt), –
deformation tensors.
--------------------------
deformation-rate tensor
--------------------------------
- strain
rate tensor (stretching)
---------------------------
-1
( ) ( ) ( )t t t=

L F F
0
V( )/V det[ ( )] V( )/V( )/( ) ( )t J t t t J J tr t tr t= = ⇒ = = =
 
F L D
0
S( ) S S( ) [ ( ) ( ) ( ) ( )]S( )t Jv t tr t t t t t= ⇒ = − ⋅n n

D D
1
0
( ) ( ) ( ) ( ) ( ) [ ( ) ( ) ( )] ( )t t t t t t t t tυ

= ⇒ = − ⋅m m m m m m m

F L D
1
0
( ) ( ) ( ) [ ( ) ( ) ( )] ( ) ( ) ( )
T T
t v t t t t t t t t
− −
= ⇒ = ⋅ −n n n n n n n

F D L
2 -1
0 0
,v = ⋅n nC
T
C = F F
2
0 0
υ = ⋅m mC
Homogeneous Deformation (motion)
2
T
D= L+ L
(A.5)
(A.2)
(A.3)
Fig. A2
(A.4)
A.1 Elements of Kinematics in Continuum
Mechanics 2
4
V
n
S (t)
S (t)
S (t)
δh
V (t)
Equation of moveable surface of jump discontinuity of
particle velocity . - unit normal to , - normal
speed of displacement of in the laboratory.
∂R
i
- material surface (i=1.2)
S (t) – non-material surface
S (t)
w
v
1/2
δ
δ
/,lim
( )
t
f
f f h
w w
t t
δ
∂ ∂ ∂
= − ⋅ =
∂ ∂ ∂x x
v v
V
S (t)
1/2
/
( )
f f f∂ ∂ ∂
= ⋅
∂ ∂ ∂x x x
v
1 2
(,) 0 [ (,) 0 for R, (,) 0 for R ]f t f t f t= > ∈ < ∈x x x x x
S (t)
1 2 1 2
1 2
( )
(,)
,
i
i
R R R R R R t
A t
dV An d A d A A A
x
= + ∂ =∂ +∂

= − ≡ −

∫ ∫ ∫
   
x
S S
S
i
v
+
-
( ) ( )
1 2
,
A
A A A
+ −
= =
Green-Gauss theorem
( ) ( ) ( ) ( )
( ) ( ) ( )
1 1 2 2 1
(,)
(,) ( )
,
R t R t R t t
R t R t t
d A
A t dV dV A d A w d
dt t
d
A t dV A Adiv dV Ac d
dt
c w c w c


= + ⋅ − ⇒

= + −
= − ⋅ = − ⋅ +
∫ ∫ ∫ ∫
∫ ∫ ∫
 

 
 
x
x
S S
S
S
S
v
v v
v n
v
v v vv v =
S (t+δt)
S (t)
v
S (t)
Hadamar theorem
c
i

speeds of propagation of
measured by observer placed on
material particle in front of
(i=1) and behind (i=2) of
- distance

δh(t)
A.2 Basic Theorems
Fig.A3
Fig.A4
(A.6)
(A.7)
(A.8)
5
1.ELEMENTS OF INTERFACIAL
DEFORMATION
(Deformation produced by moving interface)
6
x=
χ
1
(X,t)
x=
χ
2
(X,t)
R
0
2
(t)
R
0
1
(t)
R
1
(t)
R
2
(t)
n
0
S (t)
V (t)
V
0
(t)
n(t)
S
0
(t)
x
j
, X
K
X
x
R
0
R(t)
Actual configuration (time t)
Reference configuration, e.g.,
parent phase state at t < t
n

R(t)

R
0
Fig.1.1
1.1 COMPATIBILITY RELATIONS – Material Description 1
2
1
1
2
+
S (t) - actual position of
interface (closed non-
material surface
- unit normal to S (t)
R
1
(t) – actual region occuppied
by parent phase (phase
Nr 1)
R
2
(t) – actual region occuppied
by product phase (phase
Nr 1)
∂R(t)–actual external material
surface of two-phase
body R(t)=R
1
(t)+ R
2
(t)
- unit normal to ∂R(t)
S
0
(t), R
0
1
(t), R
0
2
(t) – images of S (t), R
1
(t) and R
2
(t) in
reference configuration
V (t)
n(t)
1 2
(,) 0 [ 0 for R, 0 for R ]f t f f= > ∈ < ∈x x x
∂R
0,
R
0
[= R
0
1
(t)+ R
0
1
(t)] time independent images of ∂R(t) and R(t) in ref. conf.
V
0
(t) – unit normal to S
0
(t),
n
0
-
time independent unit normal to ∂R
0
.
Description of motion and
deformation of two-phase body:
Symbolic equation of S (t):
Equation of S
0
(t):
0
1 1
0
2 2
(,) for R
(,)
(,) for R
t
t
t
⎧ ∈

=




X X
x = X
X X
χ
χ
χ
0
(,) [ (,),] 0f t f t t= =X X
χ
-
+
-
-
parent phase state
t
n

nucleation
, t > t
n
grow
n
0 t<t≤
(1.1)
(1.2)
(1.3)
7
1.1 COMPATIBILITY RELATIONS – Material Description 2
Coherency Conditions:Suggested reading:[1]
is continuous function of both arguments
-------------------------------------------------------
First derivatives:deformation gradient and particle velocity may suffer
jump discontinuities at
(,)tx = X
χ
1 2
(,) (,) 0 for ( )t t t≡ − = ∈ 
0
SX X Xχ χ χ
/

∂ ∂
F
X
χ
/t
=
∂ ∂ ≡

v
χ
χ
( )t∈
0
SX
1 1 2 2 1 2
1 1 2 2 1 2
/for ( ) 0/ for ( ) 0, 0
/for ( ) 0,/for ( ) 0, 0
t t
t t t t
≡ ∂ ∂ ∈ + ≡ ∂ ∂ ∈ − = − ≠
≡ ∂ ∂ ∈ + ≡ ∂ ∂ ∈ − = − ≠
 
 
F X X , F X X F F F
X X
χ
χ
χ χ
0 0
0 0
S S
S Sv v v v v
Mapping , ,
have continuous first derivatives
and its inverse is one-to one⇒X x
0
0 det,R R(t)J
<
≡ ∞ ∈ ∈F < X,x
Here, particles of R
0
1
(parent phase) approaching (+) side of ,
particles of R
0
2
(product phase) approaching (- )side of
( ) 0t∈ +
0
SX
( )t
0
S
( ) 0t∈ −
0
SX
( )t
0
S
(,) ( 1,2)
i
t i =Xχ
(1.4)
(1.5)
8
1.1 COMPATIBILITY RELATIONS – Material Description 3
1/2
0 0 0 0 0
/,( ),( )
K
K K K
f f f f f
t
X X X
∂ ∂ ∂ ∂ ∂
= = ∈
∂ ∂ ∂ ∂ ∂
X
X X
0
S
0
V
0 0
0
/,( )
f f
c t
t
∂ ∂
= − ∈
∂ ∂
0
SX
X
- equation of Lagrangean
interface (in the selected reference conf.),
-unit normal to Lagrangean interface
c
0
- normal speed of Lagrangean interface,
0
(,) 0f t
=
X
0
V
0
0 0
/0,( )
f
df d c dt t

= ⋅ − = ∈

X X
X
0
S
0
V
1 2
(,) for ( ) 0,(,) for ( ) 0A t A t A t A t≡ ∈ + ≡ ∈ +X X X X
0 0
S S
1 2
//0dA dA dA A d A t dt= − = ∂ ∂ ⋅ ∂ ∂ =     X X +
A- any property of a material continuous
on interface A
1
=A
2
and having
discontinuous first derivatives there.
- Lagrangea multipliers
- jump of def. tensor
- jump of particle
velocity at the interface
provided that (2.8) holds,
0
0
[ ] 0d c dtλ ⋅ − =X
0
V
- geomerical compatibility relation
- kinematical compatibility relation
Implication of the coherency condition with reference to deformation tensor and particle velocity [ ]:
0
/A λ∂ ∂ = − X
0
V
0
0
/
A
t c
λ
∂ ∂ = 
0
0
//,/A t c A Aλ

∂ = − ∂ ∂ ⋅ = − ∂ ∂ ⋅     X X
0 0
V V
0 0

λ
0 0 0
0 0
, , c c= − ⊗ = ⇒ = − = −         F F Fλ λ λ
0 0 0
V V Vv v
1 2
=
 
F
F - F
1 2
=
− v v v
(1.6)
(1.7)
(1.8)
(1.9)
(1.10)
(1.11)
0
(,) 0f t
=
⇒X
9
X
x=X
1.2 COMPATIBILITY RELATIONS – Semi- Spatial Description 1
Updated Lagrangean description of parent phase
deformation only:
At generic instant
t
of a process the current
configuration of the parent phase Nr 1 coincides with the
actual configuration
,
Hence, at time
t
the Lagrangean interface coincides with the
actual one ,
- equation of interface, and
R
1
(t)=R
0
1
(t)
R
1
(t)=R
0
1
(t)
x=X
R
2
(t)=R
0
2
(t)
x≠X
x
x=
X
R
1
(t)=R
0
1
(t)
R
1
(t)=R
0
1
(t)
V (t) = V
0
(t)
( )tn
x
= χ
2
(X,t)
X
1
(,)t
=
x = X X
χ
R
1
(t)=R
0
1
(t)
2
(,)t

x = X X
χ
V (
x
,t) = V
0
(
X
,t)-unit normal
1 2
0
=
− ≠ v v v
1 2

F
= 1, F 1
0
(,) (,) 0f t f t
=
=
X x
S
0
(t)=S (t)
V (t) = V
0
(t)
COMPATIBILITY RELATIONS [ ]
2 1
,c

= F 1 =− λ λV v
1 1
c w
=
− ⋅
v
vv
- normal speed of displacement of in the laboratory.
1/2
δ
δ
/,li m
( )
t
f
f f h
w w
t t
δ

∂ ∂
= − ⋅ =
∂ ∂ ∂x x
v v
1/2
/
( )
f
f f

∂ ∂
= ⋅

∂ ∂x x x
v
S
(t+
δ
t)
S
(t)
S
(t)
δ
h
V
(t)
S
(t)
w
v
c
1

speed of propagation of
interface measured by an
observer placed on material
particle in front of interface
(1.12)
(1.13)
Fig.1.2
Fig.1.3
Fig.1.4
10
S
(t-0)
S
(t+0)
V
(t+0)
1
2
S
(t-0)
2
, , >0,| | 1g g
λ
λ
⊗ = =F 1 =− λ λV V V
2
F
2
2
1
det( ) 1
V
V
Δ
= = + ⋅ ⇒
Δ
F λ V
2
1
1
V
V
δ
Δ

⋅ −
Δ
λ V =
Phase dilatation coefficient
- δ
δ
>0 – rarefaction,
δ
<0 – compation,
δ
=0 – incompressibility and simple shear
( ) under actual state (stress, temperature etc.) of both phases.
If then – simple extension
[simple rarefaction (
δ
>0) or simple compaction (
δ
<0 )] in the direction of V .
ΔV
1
- volume of A
1
ΔV
2
- volume of A
2
=sign( ) , | |,( )g
λ
δ
δ δ= =λV V V
0
λ
λ
⋅ = ⇒ ⊥V V V V
GEOMETRICAL COMPATIBILITY RELATION –
Further aspects
1.2 COMPATIBILITY RELATIONS – Semi- Spatial Description 2
Since det(F
2
)>0 we have
δ
>
-
1
. Likewise, since
we have the following conditions for admissible values of
parameters g and
δ
1/1g
λ
δ− ≤ ⋅ = ≤V V
| | and >-1g
δ
δ≤
δ
g0
1
-1
1
simple
extension
simple
shear
(1.14)
(1.15)
(1.16)
(1.17)
Fig.1.5
Fig.1.6
11
S (t-0)
S (t+0)
V (t+0)
1
2
S (t-0)
1 1
, >0,| | 1c c g g
λ
λ
= = =  V Vv λ
2
F
1
2
1
ρ
δ
ρ
= −
1 1
c cδ = ⋅ > −  Vv
i)In the course of simple shear (
δ
=0) the component of
particle velocity normal to the interface does not suffer
discontinuity ( )
ii) In the course of simple extension
the particle velocity component tangent to the interface
is continuous.
Interfacial mass balance in absence of diffusion
1
2 2 1 1 1 1 1 2 2 1
2
, ,( 1) ( )c W c W c c c c
ρ
δ
ρ
≡ − ⋅ ≡ − ⋅ = − = ⋅ − ⋅ − ⇒ 
v v
v v v v vV V V = V =
1 1 2 2
other form of local mass balance c c
ρ
ρ=
1.2 COMPATIBILITY RELATIONS – Semi- Spatial Description 3
KINEMATICALL COMPATIBILITY REALATION –
Further aspects
1 2
⋅ ⋅V = Vv v
=sign( )
λ
δV V
(1.18)
(1.19)
(1.20)
(1.21)
Fig.1.5
12
1.3 Interfacial deformation produced by moving interface 1
V
V
λ
λ
S
(t-0)
V
II
V
III
S-
shear plane
H -
plane
2 2
2 0 0
,,| |,| |, | | 1,| | 1l l l l= = = = ⋅ = = ⋅ =x = F X X = N,x m X x N N N m m m
-1
2 2
,,/(1 )g f f g
λ
λ
δ
= + ⊗ = − ⊗ = +F 1 F 1V V V V
Invariant plane strain:
- zone axis. Denote by
k
0
normal to any plane
parallel to the zone axis, , since
all such planes are invariant
-
0
0
λ
⋅ =k V
2 0 0 0 0
( )
T
f
λ

∝ − ⋅ =k F k = k k kV V
λ
V
If X is a material fiber of habit plane H (X N ) then x=F
2
X = X and l=l
0
. All such
fibers are invariant (remain undeformed and do not change their orientation in a laboratory)
0

⋅V V ==
PROPERTY 1:
PROPERTY 2:
2 2
2 2
,,,
sh ex
III III
g
δ
γ γ δ γ δ= ≡ + ⊗ ≡ + ⊗ = +F 1 F 1V + V V V V Vλ
PROPERTY 3:
The local (coherent phase) deformation may be regarded as simple shear followed by simple
extension :
2 2 2
ex sh
=
F
F F
(1.22)
(1.23)
(1.24)
(1.25)
(1.26)
Fig.1.7
-
13
1.3 Interfacial deformation produced by moving interface 2
V
λ
χ
0
V
II
S-
shear plane
H -
plane
2
2 2 2
( )
T
g g
λ
λ
≡ + ⊗ + ⊗ + ⊗C F F = 1 V V V V V V
2 0 0
2 2
,,| |,| |
| | 1,| | 1
l l l l= = =
= ⋅ = = ⋅ =
x = F X X = N,x m X x
N N N m m m
Stretch of a material fiber :
2
2
0 0
( ) 1 ( )
( )/ stretch of material fiber l l l
Λ − = ⋅
Λ = −
N N C - 1 N
N X = N
( )Λ N
0
lX = N
V
λ
V
III
S
(t)
X
0
α

0
α
0 0
0 0 0 0
( ) ( ) ( ) cos sin
cos,cos(90 ) sin,cos
III
III
g
λ
λ
γ δ γ α δ χ
α
χ χ α
⋅ = ⋅ = ⋅ + ⋅ = +

= ⋅ ⋅ − = ⋅
N N N N
N N = N =
V V V
V V V
λ
2
( ) 1 ( )( )C g

Λ − = ⋅ ⋅N N NV V
(2 )/
g
C
λ

= +V V V
2 2 2
| 2 | 4 4 ( ) 4 4,| | 1C g g g g
λ
λ
δ

≡ + = + ⋅ + = + + =V V V V V
2 2 2
2 2 2 2 2
0 0 0 0 0 0
( ) 1 2 ( )( ) ( )
2 sin cos (2 ) sin 2 sin cos sin
g g
g g
λ
γ
χ α δ δ γ χ χ α χ
Λ − = ⋅ ⋅ + ⋅ =
′= + + + = +
N N N N
V V V
cf. (1.24)
(1.27)
(1.28)
cf.(1.25)
(1.29)
Fig.1.8
(1.30)
(1.31)
-
initial angle between N and , - the angle between zone axis and the direction N
-
the initial angle between N and invariant H-plane (Fig.1.8)
0
α
0
α

λ
V
0
χ
I
II
V
14
1.3 Interfacial deformation produced by moving interface 3
2 2 2 1/2
0 0 0
( ) [1 2 sin cos (2 )sin ] -Christian (2002) [2]γ χ α δ δ γ χ
Λ
= + + + +N
2 2 1/2
0 0 0
( ) [1 2 sin cos sin ]
Schmid and Boazs (1936) [8]
γ χ α γ χΛ = + +

N
This is the correction of slightly inaccurate relation that can
be found in Otsuka et al. (1976) [ 9 ] and Otsuka and Wayman (1977) [10]
]
(1.32)
(1.33)
(1.34)
(1.35)
2 2 1/2
0 0 0
( ) [1 2 sin cos sin ]g gχ α χ

Λ = + +N
Simple coherent shear or twinning (
δ
= 0)
- the initial angle
between N and invariant H-plane,
- initial angle
between N and
- the angle between N
and shear plane
-the angle between
and the perpedicular projecton of N
on shear plane
S
- the angle between plane
containing and N and shear
plane
S
I
II
V
β
0
ς
0 0
cos cos cos,sin cos sin
α
ς β χ ς β= =
0 0 0
sin sin cos
χ
α ς=
2 2 2 2 2
2 2 2 2
0 0 0 0
( ) 1 cos [ sin 2 (2 )sin ]
( ) 1 cos [ sin 2 (2 )sin cos ]
ςγ β δ δ γ β
ς
γ α δ δ γ α ς
Λ − = + + +
Λ − = + + +
N
N
0
( ) | cos,sin,cos cos
( sin ) cos
II II II
III II III
ς
ς ς β
ς α
− ⋅ = ⋅ = =
= ⋅ − = ⋅
| N N N
N N =
V V V
V V V
0
[ ( ) ] | ( ) | cos(90 )
cos sin sin
II II II II
β
ς β χ
⋅ ⋅ − ⋅ = − ⋅ −
= =
N = N N N NV V V V V V
0 0
0 0 0
| ( ) | sin,sin
| ( ) | cos sin cos
III III
III III
α
χ
ς α ς
− ⋅ = ⋅ = =
− ⋅ =
N N N
N N
V V V
V V
Fig.1.9
I
II
V
I
II
V
0
(,)
I
II
α
=
 N V
0
(,) 90
χ
=
− N V
0
(,)
H
χ
=
 N
(,)
ς
=

N
S
N
15
1.3 Interfacial deformation produced by moving interface 4
0
χ
( )t
χ
X=l
0
N
V
λ
1 1 1
2 0 2 0 2
1
0 2
( ) ( ) ( ) ( )
sin,( ) sin ( ),
l l t t l l t t
t t f
λ
χ χ
− − −

= = ⇒ ⋅ = ⋅
⋅ ⋅ = − ⋅ ⇒
X F x,N F m N F m
N m F 1
V V
V = V = V V
0
0
1 sin ( )
( ) (1 ) sin
l t
l t
χ
δ
χ
=
+
1
0 2 0
( ) ( ),( ) cos ( ),cos
III III III
l l t t t tα α

⋅ = ⋅ ⋅ = ⋅ = ⇒N F m m NV V V V
V
I
I
V
x=l (t) m
S-plane
H-plane
α
0
( )t
α
0
0 0
cos ( )
( ) cos sin
l t
l t
α
α
γ χ
=
+
PROPERTY 4:
In the course of straining material fiber rotates relative to the habit plane ( its crystalographic
orientation changes).
Fig.1.10
(1.36)
(1.37)
Return to (1.30) :

2
( ) 1 ( )( )C g

Λ − = ⋅ ⋅
N N NV V
(2 )/
g
C
λ

= +V V V
I
II
V
16
1.3 Interfacial deformation produced by moving interface 5
PROPERTY 6:
2 2
0 0 0 0
( )( ) 0 cos/sin (2 )/2 or cos/sin/2
g
α χ δ δ γ γ α χ′⋅ ⋅ < ⇒ < − + + < −N NV V
Any material fiber will decsease in length ( ) after
transformation provided that (1.32) holds.This requires the projection of X onto shear
plane
S
to lie within the wedge region shown in Fig. 1.11.
0
l=
X
N
(1.38)
( ) 1
Λ
<N
S-
shear plane
V
V
V
III
Fig.1.11
0
( ) 1Λ <N
( ) 1
Λ
<N
PROPERTY 5:
Any material fiber laying in the plane H
c
defined by unit normal does not change
its lenght during transformation (since for all such fibers) .
0
l=
X
N
V
0⋅ N =V
intersection
with H
c
- plane
intersection
with H- plane
17
Let unit vectors and represent two arbitrary fibers of the plane H
c
(cf. (1.30)),
Since the fibers preserve their lengths the cosine of their angles after transformation are, cf.
(1.27)
( ) ( ) ( ) ( ) ( ) ( )
2 0,2 0
I I I II II II
g g
λ
λ
⋅ ⋅ + ⋅ ⋅ ⋅ + ⋅N N N N N NV = V V = V = V V =
( )
I
N
( )
I
I
N
The second term on the right hand side of the last equality vavishes , on account of (2.39). Hence
( ) ( ) ( ) ( ) ( ) ( )
cos(,) cos(,)
I
II I II I II
m m N N= = ⋅N N
PROPERTY 7:
All fibers belonging to the plane H
c
retain not only their lengthes but also mutual orientations.
The plane H
c
may then be regarded as conjugate habit (invariant with respect to strainning) plane
(1.39)
(1.40)
(1.41)
1.3 Interfacial deformation produced by moving interface 6
( ) ( )
0 0 2 2
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
0 0 2 2
,,
,,,
I I I II II II I I II II
I
I I I I II II II II II I I II II
l l
l l l l
= = = =
= = = =
X N X N,x F X x F X
x m = m x m = m m F N m F N
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
2 2 2
( ) ( ) ( ) ( ) 2 ( ) ( )
cos(,) ( ) ( )
{ ( )( ) ( )( ) ( )( )}
I II I II I II I II I II
I II II I II I
m m
g g g
λ λ
=
⋅ ⋅ = ⋅ = ⋅ +
+ ⋅ ⋅ + ⋅ ⋅ + ⋅ ⋅
m m = F N F N N C N N N
N N N N N NV V V V V V
18
1.3 Interfacial deformation produced by moving interface 7
Conjugate invariant plane strain deformation-Algebraic aspects:
It has been just shown that the deformation leaves the fibers of the conjugate habit
plane
H
c
unstrained, although the plane is admitted to undergo rigid body rotation in a laboratory. It is
therefore expedient to find the conjugate plane strain deformation,say (cf. (1.30)
that leaves the fibers of both planes
H
c
and
H
undistorted and retains
H
c
-
plane orientation in the
laboratory ( may cause rigid body rotation of H-plane). This will be possible provided that there exist
rigid body rotation, say ( ) such that
Supposing that g, and occurring in (1.22 ) are known, from (1.42)-(1.43) and (1.22) we find
as follows ( is defined in (1.30) ) :
g
λ

+ ⊗
2
F 1 V V
2

+ ⊗F 1 Vλ
2
F
R
1T

=R R
2 2

R
F = F
2 2 2 2 2 2
T T
≡ ≡C F F = C F F
V
λ
V
λ
(1.42)
V
2 2
2 2 2 2
det det,2 | | 2 | |tr tr= ⇒ ⋅ ⋅ = ⇒ ⋅ + ⋅ + ⇒V = V V = VF F C Cλ λ λ λ λ λ
,cos(90 )/
g
g
λ λ λ
ψ δ
= ⋅ = ⋅ − =
λ V V V V V =
2 2
g g
λ λ λ λ
=
⇒ ⊗ + ⊗ + ⋅ = ⊗ + ⊗ + ⋅ ⇒C C V V V V V V V V V V V V
(1.44)
[2(1 ) ]/
g
C
λ λ
δ

= + −V V V
1
2 2
2 [2(1 ) 2 ( )]/
g
g C
λ λ λ λ
δ
− ∗
+ + ⊗ − ⊗ − ⊗ + ⊗F F = 1R = V V V V V V V V
(2 )/cos 0C
λ λ
δ ϕ


⋅ + = >V V = V V =
(1.45)
(1.46)
C
*
is
defined in
(1.31)
(1.43)
19
H
c
-
plane
1.3 Interfacial deformation produced by moving interface 8
V
Q
c
λ
V
λ
V
III
90-ψ
S
-
shear plane
H -
plane
Conjugate invariant plane strain
deformation – Trigonometry aspects:
- conjugate zone axis.
V
ψ
ψ
ϕ
ϕ
Q
λ
V
P
c
9
0-ψ
Fig.1.12
0
λ
λ
V
P
( ) 1
Λ
<N
2
sin/,
sin( ) (2 )/( )
g
g
g C
λ
λ
ψ δ
ϕ ψ δ

⋅ = ⋅ = =
⋅ + = +
V V V V
V V =
cos (2 )/C
λ λ
ϕ δ

⋅ ⋅ = = +V V = V V
Example 2
:Given g>0 and
ψ
(g sin
ψ
>-1) :find
δ
and
ϕ
.
2
sin,4 4 sin cos (2 sin )/
g
C g g g Cδ ψ ψ ϕ ψ


= = + + = +,
Simple shear
δ
= 0:
2
cos 2/4,0 org
ϕ
ψ ψ π= + = =
Simple extension :
ϕ
=0 and
[ or
]
| |,1g
δ
δ= > −
/2 (simple rarefaction, )
λ
λ
ψ
π= V = V = V = V
3/2 (simple compation, )
λ
λ
ψ
π= − −V = V = V = V
Example 1
:Show the identities:
2 2 2 2 2
4
sin cos ( ),cos ( ) 4cos 4[1 (/) ]
g
C gϕ ψ ϕ ψ ϕ ψ δ

= + + = = −
(1.47)
(1.48)
(1.49)
20
Direction
of interface
migration
Fig. 1.13
- Simple coherent shear
δ
=
ψ
= 0, g = 0.5
a) parent phase reference
configuration
b) actual configuration – simple
shear
c) actual configuration –
conjugate simple shear
( cf. Bilby & Christian (1956)
[ 6-7 ] )
a)
2
F
R
V
1.3 Interfacial deformation produced by moving interface 9
2
D
2
A
A
2
C
B
C
2
A
2
C
2
D
B
2
2
B
λ
V
D
V
Direction
of interface
migration –
conjugate
simple
coherent
shear
2
F
b)
c)
Example 3
:
Simple coherent shear
λ
V
Surface
relief
H
c
-
planes
H-
planes
Distortion of straight
reference mark K-K’.
Tilds along AB and
CD representig surface
relief.
S
-
plane
K
K’
T’
T
R
A
B
C
D
N
M
P
H-p
.
H-plane
Specimen
surface
Fig. 1.14. The shape deformation produced by
martensite plate ABCDMNPR
(after Bilby
&Christian (1956) [ 6-7 ]
21
where (cf. (1.31))
1.3 Interfacial deformation produced by moving interface 10
Example 4
:
Calculate the extremes of [cf. (1.29)-(1.30)]
under constrain and find principal stretches in shear plane
S
assuming that they are different. Determine corresponding principal axes of strain in
S
represented by unit normals . Show that
2 2 2
( ) 1 2 ( )( ) ( ) ( )( )g g C g
λ

Λ − = ⋅ ⋅ + ⋅ = ⋅ ⋅N N N N N NV V V V V
1⋅N N =
1 3
Λ
> Λ
(1) (3)
N, N
1 3
(1) (3)
( )/2 1,( )/2 1
( ),( )
C g C g
∗ ∗
Λ = + > Λ = − <
∝ + ∝N NV V V - V
2 2 2
| 2 | 4 4 ( ) 4 4,| | 1C g g g g
λ
λ
δ

≡ + = + ⋅ + = + + =V V V V V
1 3 1 3
,1g
δ
Λ
−Λ = Λ Λ = +
Thus, principal directions intersect the angle between habit plane Hand its conjugate H
c
–plane,
what is marked by dotted lines in Figs. 1.11 and 1.12.
In the limit we have simple compaction, and simple rarefaction occur when .
1
1Λ →
3
1Λ →
(1.50)
22
Example 5 –
Mathematical Aspect – Inverse problem(see e.g. Bhattacharya (2003) [4]):
Given representing invariant plane strain (one principal value equals 1, smallest
principal values less than 1, largest principal values greater than 1). Find
and having the form (1.22)
1
and (1.42), respectively.
Hints:
a) Determine the principal directions and (cf. (1.50)
1
) as well as principal values of C ,
and principal stretches , . Determine stretch tensor .
b) Find
δ
, g , and using (1.50) (1.44) and (1.31).
c) Once and are known use (1.46) to determine and polar the rotation .
1.3 Interfacial deformation produced by moving interface 11
2 2
= ≡C C C
2
( )
U U = C
2
F
2
F
(1)
N
(3)
N
1
Λ
3
Λ
V
λ
V
R
2
F
2
F
1
2 2

=R F U
23
2. FUNDAMENTAL BALANCE LAWS IN
CONTINUUM THERMODYNAMICS
*CLOSED SYSTEMS
*
24
2.1 SYMBOLIC GLOBAL LAW
[1]
( )
( ) ( )
( ) (,) (,)
(,) (,) (,)
g
R t
g
R t R t
t t t dV
d
t t dS f t dV
dt
ψ ψ
ψ ρ ψ
ψ ρ

=
= ⋅ + ⇒

∫ ∫
x x
j x x xn
( ) ( )
{ ( ) } 0
R t t
div div f dV c dS
ψ ψ ψ
ρψ ψρ ρ ρ ρψ+ + − − − + ⋅ =
∫ ∫
   j jv
S
v
( ) 0 ( ) ( 1,2)
0 ( )
i
div div f for R t i
c for t
ψ ψ
ψ
ρψ ψρ ρ ρ
ρψ
+ + − − = ∈ =
+ ⋅ = ∈
 
 
j x
j x
v
Sv
2.2 SYMBOLIC LOCAL LAW:
- total amount of
symbolic quatity at time t
ψ- specific (per unit of mass)
value of the same symbolic
quantity
ρ
- mass density,
- symbolic vector flux
ofψ through the outer
surface ∂R
i
- the supply ofψ
within the region R,
- particle velocity,
- mobile surface of
discontinuity of fields: ,
ψ,
ρ,
- unit normal to ,
c -speed of propagation of
g
ψ
ψ
j
f
ψ
v
( )tS
v
v
( )tS
ψ
j
( )tS


for Clausius –Duhem Inq. (2d law of thermodynamic)


for Clausius –Duhem Inq.
for Clausius –Duhem Inq.


S (t)
V (t)
n (t)
Use was made of
(A7)- (A8)

(2.1)
(2.2)
(2.3)
Fig. A3
25
(After [ ])
2.2. IDENTIFICATION FOR CLOSED SYSTEMS
g
ψ
ψ
ψ
j
s
S
–total entropy
2d law of
thermod.
E
k
+U
kinetic+
intern. energy
Energy
- moment of
momentum
Moment of
momentum
-
total
momentum
Momentum
001
M –
total mass
Mass
Balance of
( )
(,),
R t
M
t dVρ=

x
( )
(,),
R t
t dVρ=

P x v
P
( )
( )
p
R t
dVρ= ×

x vL
p
L
/2u
+
⋅v v
f
ψ
v
σ
b
×
x v
×
x
σ
×
x b

qv
σ
r

+b v
/T

q
/r T
( )
(/2),
k
R t
E
dVρ= ⋅

v v
( )
,
R t
U udVρ=

( )R t
S sdVρ=

ρ
- mass density
-Cauchy’s stress
-velocity
- body forces
u
- specific internal
energy
- heat flux
s
- specific
entropy
r
- heat sources
- vector prod.
σ
v
b
q
(2.4)
(2.5)
×
a b
26
2.3 GLOBAL LAWS
0
dM
dt
=
( ) ( )
( )
(,) (,)
R t R t
d t
t dS t dV
dt
ρ

= ≡ +
∫ ∫
P
F x b xnσ
( ) ( )
0
( ) ( )
( )
(,),
p
n n
R t R t
d t
dS t dV
dt
ρ

= ≡ × + × ≡
∫ ∫
L
M t tx x b x n
σ
( )
( )
( ) ( )
( ) ( )
( )
( ) (,) (,)
( ) (,) (,)
m
k
m
R t R t
R t R t
d E U
W Q
dt
W t t dS t dV
Q t t dS r t dV
ρ
ρ


+
= +
= ⋅ +
= − ⋅ +
∫ ∫
∫ ∫
σv n
n
x b x
q x x
Moment of Momentum↓
Mass↓
Momentum↓
Energy↓
( ) ( )
(,)
R t R t
dS r
dS dV
dt T t T
ρ


≥ − +


q
x
n
Clausius-Duhem Inequality↓
- resultant force
- resultant moment
- body surface
tractions
- total mechanical
power
Q
- rate of total heat
added
- heat added
- heat emitted
T
- temperature
F
0
M
( )n
t
( )m
W
(2.6)
(2.7)
(2.8)
(1.9)
(2.9)
0

>q n
0

<q n
27
2.4 LOCAL LAWS
0div tr
ρ
ρ ρ ρ+ = + 
D
=v
C-D Ineq
.
Energy
M. of mom.
Momentum
Mass
0div
ρ
ρ− =

bv
σ

ij ji
σ
σ=
0u div r
ρ
ρ− ⋅ − = D+ q
σ
(/)/0
s
s div T r Tρ ρ
( )
σ
≡ + − ≥

q
1 2
R
or R∈ ∈x x
C-D Ineq
.
Energy
M. of mom.
Momentum
Mass
Interface -
( ) (unit normal )t∈

S vx
1 1 2 2
1/c c c c
ρ
ρ ρ ρ ρ≡ = ⇒ ⋅ = −   vv
( ) ( )
0,cρ + = =   t t vv
v v
σ
ij ji
σ
σ=
( )
( 2) 0c uρ + ⋅ − ⋅ − ⋅ =   v tqv v/v
v
( )
/0
s
T c sρΔ ≡ ⋅ − ≥   vq
ρ
- mass density
- particle velocity
- strain rate
- Cauchy’s stress
- body forces
u
- specific internal energy
- heat flux
r
- heat sources
- density of entropy
production (per current
volume)
c
i
-
speeds of propagation of
interface measured by
observer placed on
material particle in front
of (i=1) and behind
(i=2) of interface
s
- specific entropy
- interfacial entropy
production
( )tS
( )
s
Δ
( )
s
σ
q
σ
b
D
v
(2.10)
(2.11)
28
3. THERMODYNAMIC DRIVING
FORCE.
Qualitative Analysis of Interface Local
Equilibrium.
29
C-D Ineq
.
Energy
M. of mom.
Momentum
Mass
Interface -
( ) (unit normal )t∈

Sx v
( ) ( )
0,cρ + = =   t t vv
v v
σ
ij ji
σ
σ=
( )
( 2) 0c uρ + ⋅ − ⋅ − ⋅ =   v tqv v/v
v
( )
/0
s
T c sρΔ ≡ ⋅ − ≥   vq
(2.11)
SHORT RECAPITULATION
, >0,| | 1g g
λ
λ
=
=V Vλ
2 1
,c⊗ = F 1 =− λ λV v
1 1
c w= − ⋅
v
vv
(1.12)-(1.14)
S (t-0)
S (t+0)
V (t+0)
S (t-0)
1
2
2
F
Fig.2.5
V (t)
n (t)
Fig. A3
ρ
- mass density, - particle
velocity, - Cauchy’s stress
u - specific internal energy
- heat flux, r - heat sources
-interfacial entropy
production
c
i
- speeds of propagation of
interface measured by observer
placed on material particle in
front of (i=1) nad behind (i=2)
of interface
s
- specific entropy
heat emitted from
new shell
1 1 2 2
1/c c c c
ρ ρ ρ ρ ρ
≡ = ⇒ ⋅ = −   vv
S (t)
σ
q
( )
s
Δ
v
1 2
A
A A

− 
=
〈〈 〉〉 + 〈〈 〉〉     AB A B A B
1 2
( )/2〈〈 〉〉 ≡ +A A A
(3.1)
1 2
( )/2〈〈 〉〉 ≡ +A A A
0

> q v
30
3.1 Definition of Thermodynamic Driving Force (Coherent Phases) 1
i) We shall express balance of energy in equivalent forms. First multiply balance of momentum(2.11)
2
by
and use the identity , following from (3.1) when A=B=v , to get
2

= 〈〈 〉〉 ⋅   v v v v
( ) ( ) ( )
0.5 0.5 0c cρ ρ⋅ +〈〈 〉〉 ⋅ = ⋅ + ⋅ − ⋅ 〈〈 〉〉 =         t t t
V V V
v v v v v v v
Here
, ,
and we have applied identity (3.1) to the product
Introduce the specific free energy ,eliminate between (2.11)
3
and (3.2),and use
compatibility relation (1.12) to get equivalent forms of energy balance
( ) ( ) ( )
1 2 1 2
0.5( ) 0.5( )〈〈 〉〉 = + = +t t t σ σ
V V V
V V
( )
⋅ t
V
v
u Ts
φ
= −

 v v
(3.2)
( ) ( ) ( ) ( ) ( )
1 1 1 1 1
( ) ( ) ( ) ( )
1
0,
0
c u q c u c q q
c c Ts q c c Ts c q
ρ ρ
ρ φ ρ ρ φ ρ
− + ⋅ 〈〈 〉〉 = + ⋅ 〈〈 〉〉 − = ≡ ⋅
+ − + ⋅ 〈〈 〉〉 = + + ⋅ 〈〈 〉〉 =−
         
             
λ
λ
t t q
t t
V V V V V
V V V V
Vv
v
(3.3)
Since we regard existence of the interface as a consequence of athermal martensitic transformation,the
temperature T on both sides of interface is assumed to be the same,
1 2
0 ( ) (unit normal )T T T T t= ⇔ = = ∈  x S
v
(3.4)
ii) Eliminate from C-D inequality (2.11)
5
using (3.3)
2
and write T in the form
( )
q 
V
( )
s
Δ
( )
1 1
0
s f
T cρΔ = Σ ≥
2 2
( | | )g λ= = ⋅
λ
λ
(3.5)
To get the last equality use was made of momentum balance which may be rewritten as
on account of (1.12) (Raniecki-Tanaka,1994, [22]).is called thermodynamical driving force (t.d.f.) of
interface migration [20] .
f
Σ
( ) ( ) 2
2 1 1 1
cρ= +t t
V V
λ
(3.6)
( )
1
1
f
φ
ρ
Σ = + ⋅ 〈〈 〉〉  tλ
V

( ) 2
1 1
1
1 1
2
f

ρ
Σ
= + ⋅ + ⋅ 
λ
λλt
V
1 2
( )/2〈〈 〉〉= +v v v
31
T.d. f. can be written in a few other equivalent forms. For example, denote by and by
specific first Piola- Kirchhoff stress and specific energy-momentum tensor (Eshelby (1975)[17 ], Chadwick
(1975)[24]), respectively. In Lagrangea description (Abeyaratne, Knowles 1988-1990 [20])we have expression
/
T
ρ

S = Fσ
φ
= −P 1 FS
2
1
0.5
f
cΣ = ⋅ + P CV V
T
C = F F
In semi-spatial description (cf. Sec. 1.2) we have (to get the second beneath equation mass balance is used)
(3.7)
( ) ( )
1 1 1 2 2 2 2 1
/,( )//
f
λ
ρ
ρ ρ= − ⊗ =S t S t
V V
V V = V V V
σ
Using (3.8), momentum balance, and the fact that the Eq. (3.7) can readily be
reduced to (3.5).In quasistatic situation the inertia forces in the momentumbalance (2.11)
2
are
negligible and the balance reduces to the equilibrium equations
Then ((Heidug &Lehner (1985)[18])
2
(2 )
g
δ⋅ + CV V = -
(3.8)
( ) ( ) ( )
1 2
0,= ⇔ = t t t
V V V
( ) ( )
1 1 1
1
1
0,
s f f
st st
T cρ φ
ρ
Δ = Σ ≥ Σ = + ⋅ = ⋅   t P
V
V Vλ
T.d.f.represents the difference between the passive work done by surface traction and the
free energy variation in the jump transition from state 1 to state 2 (1→2). Presuming that the direction
of interface migration is always towards the parent phase(c
1
>0),from (3.5) it follows that cannot be
negative.
f
Σ
( )
1
/
ρ

〈〈 〉〉tλ
V
2 1
φ
φ−
f
Σ
If state 2 is accessible from state 1 through irreversible process ( ) of isothermal phase
transformation then the reverse transformation from the same state 2 to 1 is impossible.
Fundamental feature of irreversibility (usually displayed in the form of hysteresis)
:
1
0,0
f
c
Σ
≠ ≠
(3.10)
(3.9)
3.1 Definition of Thermodynamic Driving Force (Coherent phases) 2
32
3.2 Reversible processes – Phase equilibriums
The infinitesimal motion of the interface in coherent phases and the process of jump phase transition are
reversible dynamic processes when t.d.f vanishes and
( ) ( ) 2
1 1
1 1
1 1 1
0
2
f
cφ φ
ρ ρ
Σ = + ⋅ 〈〈 〉〉 = + ⋅ + ⋅ ⋅=   λ λ λλt t
V V
Likewise, the motion of interface and the jump phase transition are reversible quasistatic processes when
and
Example 1
As an example assume that in an equilibrium state the normal to the habit plane coincides with one
of the principal directions of the Cauchy’s stress. In that case
(α=1 or 2 or 3)
where are principal stresses. Bearing in mind that (cf.(1.15), (1.20))
the Maxwell condition (3.12) of phase equilibrium reduces to classical Gibbs condition of phase
equilibrium in heterogeneous substances
( )
1
1
1
0
f
st
φ
ρ
Σ = + ⋅ =  t
V
λ
(3.12)
(3.11)
V
1 2 1
(1 ) (1 )
ρ ρ ρ

−V/=//
λ
( ) ( ) ( )
1 2
α
σ= = −t t
V V
V
( )
α
σ
( ) ( )
1 1 2 2
α α
φ
σ ρ φ σ ρ+ = +//
(α=1 or 2 or 3)
(3.13)
The relation of the type (3.11)-(3.12) together with is also referred to as Maxwell condition of
phase equilibrium.
1
0c >
1
0c
=
1
0c >
The dynamic or quasistatic processes are irreversible processes when and (or )
1
0c >
0
f
Σ
>
0
f
st
Σ >
33
3.3 Remark on caloric relation
Bearing in mind that the temperature is continuous at the interface,the jump of the normal component of
heat flux at can be found directly from (2.11)
5
and (3.5)
Consider reversible quasistatic motion of interface ( ) . If the parent phase 1 and the product
phase 2 are identified as austenite and martensite, respectively, then the term
may be identified as positive reversible latent heat of phase
transition ( p.t is exothermic). During reverse transformation parent phase 1 is identified as
martensite whereas the product phase 2 is the austenite such that
(the left hand side of equation ( 3.14 ) does not change since ) i.e., the same amount of heat which
is emmitted from the new shell during forward p.t is added to it (absorbed by shell) during
reverse p.t. The peculiarity of this reversible process is that the temperature T is not constant
during the interface migration. It will generally be the function of variable thermodynamic states on both
sides of interface following from the constrain
( )tS
( )
( )
f
q c T sρ= Σ +   
V
0
f f
st
Σ
= Σ =
A
M
A
M
0 0
1 2
( ),( ) 0
A M
T s T s s T s s s s= − = Δ Δ ≡ − > 

A
M
0
1 2
( ) 0T s T s s T s
=
− = − Δ < 

A
M
→−V V
(3.14)
→M A
0.
f
st
Σ
=
In the course of irreversible quasistatic processes we have .The term occurring in (3.14)
represents the dissipated energy. It brings the heat outside of the specimen both during forward and
reverse p.t.
0
f
st
Σ
>
f
Σ
y
V
T
A
M
( )
1
q
V
( )
2
q
V
→A M
V
T
AM
( )
2
q
V
( )
1
q
V
y
→M A
Fig.3.1
Reversible endothermic
Reversible exothermic
34
3.4 Other Definition of Driving Force
The free energyφ may be expressed as as a sum of two terms
where represents energy of a phase in some selected stress free reference state, such that it is a
function of temperature alone, and represents elastic strain energy (cf. Sec.4).
In material science literature the elastic strain energy is frequently not accounted for explicitly. The
driving force is expressed as
The term represent the mechnical work done by applied macroscopic stress [use was
made of (1.25)
1
( )].According to Christian [ 2 ] they are equivalent to Schmid’s law
modified to allow for a deformation mechanism which gives an invariant plane strain instead of simple
shear in S-plane. When in addition the difference in specific heats of both phases are negilgible the second
terms is approximately linear function of the temperature,and it is frequently regarded as the chemical free
energy change.The driving force may then be assumed to have a critical value (representing stored elastic
energy) at martensite start temperature (cooling at constant ). Patel nad Cohen (1953) [26] found
quantitative agreement between such theory nad experiment in iron-nickel alloys (cf. Christian [ 2]).
( )
Tch el
T
φ
φ φ
=
+
( )
Tch

el
φ
1 1 1
( ) ( ), ,
f Tch Tch
s
t III
T Tρ ρφ δσ τγ ρφ σ τΣ = ⋅ = = ⋅ = ⋅
      
   λσ σ σ+V + + V V V V
V V
el
φ

σ
I
II
δ γ
= V + V
λ
( )M
σ
s

σ
Example 2
For simple tension along N – axis ( ) we have.
Using (3.15) show that -cf. Fig.1.9 (Christian [ ])
Show that the net mechanical work is equal to the product of displacement
along N – axis and
δ
σ τγ⋅ =  +V
λ
σ
V
A
σ
=

 
N Nσ
(3.15)
2
0 0 0