Kaushi
k
Bhattachary
a
Divisio
n
o
f
Engineerin
g
&
Applie
d
Science
,
Californi
a
Institut
e
o
f
Technology
,
Pasadena
,
C
A
91125
Crystallographi
c
Attribute
s
o
f
a
Shape-Memor
y
Allo
y
Shape-memory
alloys
are
attractive
for
many
potential
applications.
In
an
attempt
to
provide
ideas
and
guidelines
for
the
development
of
new
shape-memory
alloys,
this
paper
reports
on
a
series
of
investigations
that
examine
the
reasons
in
the
crystallography
that
make
(i)
shape-memory
alloys
special
amongst
martensites
and
(ii)
Nickel-Titanium
special
among
shape-memory
alloys.
1
Introductio
n
Shape-memor
y
effec
t
(SME
)
i
s
th
e
abilit
y
o
f
certai
n
metalli
c
alloy
s
t
o
recover
,
o
n
heating
,
apparentl
y
plasti
c
deformatio
n
sustaine
d
belo
w
a
critica
l
temperature
.
Shape-memor
y
alloy
s
(SMAs
)
als
o
displa
y
othe
r
relate
d
phenomen
a
suc
h
a
s
super
-
o
r
pseudo-elasticity
.
Al
l
thes
e
mak
e
SMA
s
ver
y
appealin
g
fo
r
man
y
potentia
l
applications
.
Thoug
h
a
sizabl
e
numbe
r
o
f
SMA
s
ar
e
known
,
application
s
hav
e
essentiall
y
bee
n
limite
d
t
o
nickel
-
titaniu
m
(a
t
composition
s
clos
e
t
o
equiatomic
)
du
e
t
o
a
variet
y
o
f
reasons
.
Th
e
hig
h
cos
t
o
f
Ni-Ti
,
a
s
wel
l
a
s
th
e
narro
w
temper
-
atur
e
rang
e
i
n
whic
h
i
t
ca
n
b
e
used
,
limi
t
thes
e
applications
.
Therefore
,
i
t
i
s
importan
t
t
o
improv
e
know
n
material
s
an
d
de
-
velo
p
ne
w
SMAs
.
Thi
s
pape
r
summarize
s
th
e
result
s
o
f
a
lin
e
o
f
researc
h
motivate
d
b
y
thes
e
concerns
.
Th
e
goa
l
i
s
t
o
under
-
stan
d
i
f
ther
e
ar
e
reason
s
i
n
th
e
crystallograph
y
tha
t
mak
e
1
.
SMA
s
specia
l
amon
g
martensite
s
an
d
2
.
Ni-T
i
specia
l
amongs
t
SMAs
.
I
n
doin
g
so
,
i
t
hope
s
t
o
provid
e
idea
s
an
d
guideline
s
fo
r
th
e
developmen
t
o
f
ne
w
SMAs
.
Th
e
phenomenolog
y
o
f
SM
E
i
s
wel
l
understoo
d
i
n
a
quaUta
-
tiv
e
fashion
,
se
e
fo
r
exampl
e
Sabur
i
an
d
Nenn
o
(1981
)
an
d
Wayma
n
(1992)
.
Th
e
hear
t
o
f
thi
s
effec
t
lie
s
i
n
th
e
reversibl
e
o
r
"thermoelastic
"
martensiti
c
transformatio
n
tha
t
thes
e
crys
-
tallin
e
solid
s
undergo
.
A
martensiti
c
transformatio
n
i
s
a
temper
-
ature-induce
d
first-order
diffusionles
s
phas
e
transformatio
n
be
-
twee
n
th
e
hig
h
temperatur
e
austenite
phas
e
an
d
th
e
lo
w
temper
-
atur
e
martensite
phase
.
I
n
typica
l
SMAs
,
th
e
lattic
e
o
f
th
e
austenit
e
ha
s
highe
r
symmetr
y
tha
n
tha
t
o
f
th
e
martensite
.
Thi
s
give
s
ris
e
t
o
mor
e
tha
n
on
e
varian
t
o
f
martensite
;
variant
s
ar
e
identica
l
crysta
l
lattice
s
whic
h
ar
e
oriente
d
differentl
y
wit
h
re
-
spec
t
t
o
th
e
austenite
.
Conside
r
a
specime
n
o
f
a
give
n
shap
e
i
n
Fig
.
1(a)
.
I
t
i
s
i
n
th
e
austenit
e
phase
.
Suppos
e
momentaril
y
tha
t
ou
r
specime
n
i
s
a
singl
e
crystal
.
O
n
cooling
,
th
e
austenit
e
transform
s
t
o
martens
-
ite
.
I
n
particular
,
i
t
transform
s
t
o
a
coheren
t
fine-scale
micro
-
structur
e
involvin
g
th
e
differen
t
variant
s
i
n
suc
h
a
manne
r
tha
t
ther
e
i
s
n
o
macroscopi
c
chang
e
i
n
shap
e
(Fig
.
l{b)).
Thi
s
i
s
know
n
a
s
self-accommodation.
Whe
n
load
s
ar
e
applie
d
t
o
th
e
martensite
,
i
t
deform
s
b
y
convertin
g
on
e
varian
t
t
o
anothe
r
an
d
formin
g
a
ne
w
coheren
t
fine-scale
microstructur
e
(Fig
.
1(c))
.
O
n
heating
,
eac
h
varian
t
transform
s
bac
k
t
o
austenite
.
Sinc
e
ther
e
i
s
onl
y
on
e
varian
t
o
f
austenite
,
al
l
th
e
strai
n
i
s
recovere
d
an
d
th
e
specime
n
return
s
t
o
it
s
origina
l
shap
e
(Fig
.
1(a))
.
Notic
e
tha
t
th
e
strain
s
ar
e
recoverabl
e
becaus
e
th
e
deformatio
n
belo
w
th
e
transformatio
n
temperatur
e
i
s
no
t
du
e
t
o
slip
,
bu
t
rathe
r
du
e
t
o
th
e
rearrangemen
t
o
f
martensiti
c
variants
.
Notic
e
als
o
tha
t
onl
y
certai
n
strain
s
ar
e
recovered
:
thos
e
tha
t
ca
n
b
e
Contribute
d
b
y
th
e
Material
s
Divisio
n
fo
r
publicatio
n
i
n
th
e
JOURNA
L
O
F
ENGI
-
NEERIN
G
MATERIAL
S
AN
D
TECHNOLOGY
.
Manuscrip
t
receive
d
b
y
th
e
Material
s
Divisio
n
Februar
y
11
,
1998
;
revise
d
manuscrip
t
receive
d
Ma
y
20
,
1998
.
Gues
t
Editors
;
H
.
Sehitogl
u
an
d
Y
.
Chumlyakov
.
achieve
d
b
y
th
e
rearrangemen
t
o
f
martensiti
c
variants
.
Large
r
strain
s
introduc
e
stress
,
givin
g
rise
t
o
lattic
e
defect
s
an
d
nonre
-
coverability
.
I
f
ou
r
specime
n
i
s
a
polycrystal
,
th
e
situatio
n
i
s
mor
e
compli
-
cated
.
A
t
th
e
hig
h
temperatur
e
i
t
consist
s
o
f
a
numbe
r
o
f
grain
s
o
f
austenite
.
A
s
i
t
i
s
cooled
,
eac
h
grai
n
transform
s
t
o
a
self
-
accommodate
d
microstructur
e
o
f
variants
.
A
s
th
e
polycrysta
l
i
s
deformed
,
eac
h
grai
n
trie
s
t
o
accommodat
e
th
e
strai
n
b
y
ad
-
justin
g
it
s
microstructur
e
o
f
martensiti
c
variants
.
However
,
eac
h
grai
n
i
s
capabl
e
o
f
sustainin
g
a
differen
t
clas
s
o
f
microstructure
s
du
e
t
o
th
e
varyin
g
orientation
.
Th
e
recoverabl
e
strain
s
i
n
a
polycrysta
l
ar
e
th
e
macroscopi
c
average
s
o
f
thos
e
inhomoge
-
neou
s
strai
n
fields
tha
t
ma
y
b
e
accommodate
d
i
n
eac
h
grai
n
b
y
th
e
rearrangemen
t
o
f
martensit
e
variants
.
I
n
summary
,
SM
E
ca
n
onl
y
b
e
observe
d
i
n
martensiti
c
materi
-
al
s
tha
t
ar
e
self-accommodating
,
tha
t
ca
n
for
m
a
larg
e
clas
s
o
f
coheren
t
microstructure
s
an
d
tha
t
ca
n
defor
m
b
y
changin
g
microstructur
e
a
t
relativel
y
smal
l
stresses
.
Ou
r
tas
k
i
s
t
o
find
quantitatively
an
y
restriction
s
tha
t
thes
e
requirement
s
impos
e
o
n
th
e
crystallograph
y
o
f
th
e
material
.
2
Theor
y
o
f
Martensit
e
Microstructur
e
Erickse
n
(1980
,
1984
,
1986)
,
Jame
s
(1984)
,
Bal
l
an
d
Jame
s
(1987
,
1992
)
an
d
other
s
hav
e
develope
d
a
theor
y
i
n
th
e
frame
-
wor
k
o
f
finite
thermoelasticit
y
t
o
describ
e
th
e
behavio
r
o
f
mar
-
tensiti
c
materials
.
Se
e
Bal
l
an
d
Jame
s
(1998
)
o
r
Bhattachary
a
(1998
)
fo
r
detaile
d
explanation
.
W
e
assum
e
tha
t
th
e
behavio
r
o
f
th
e
materia
l
i
s
describe
d
b
y
a
(Helmholtz
)
fre
e
energ
y
den
-
sit
y
whic
h
depend
s
o
n
temperatur
e
an
d
deformatio
n
gradient
.
Abov
e
th
e
transformatio
n
temperature
,
th
e
fre
e
energ
y
densit
y
ha
s
a
n
absolut
e
minimu
m
a
t
th
e
austenit
e
state
;
belo
w
th
e
trans
-
formatio
n
temperature
,
i
t
ha
s
a
n
absolut
e
minimu
m
a
t
th
e
mar
-
tensit
e
state
.
Th
e
energ
y
densit
y
satisfie
s
al
l
requirement
s
o
f
materia
l
frame-indifferenc
e
an
d
materia
l
symmetry
.
Conside
r
a
singl
e
crysta
l
o
f
austenit
e
a
t
th
e
transformatio
n
temperature
.
Choos
e
thi
s
a
s
th
e
referenc
e
configuratio
n
an
d
de
-
scrib
e
al
l
othe
r
configuration
s
o
f
th
e
crysta
l
a
s
deformation
s
o
f
thi
s
referenc
e
configuration
.
Fo
r
example
,
th
e
transformatio
n
ma
y
b
e
describe
d
b
y
th
e
homogeneou
s
deformatio
n
y
=
t/*"x
.
[/"
'
i
s
know
n
a
s
th
e
transformatio
n
o
r
th
e
Bai
n
matrix
.
How
-
ever
,
du
e
t
o
th
e
chang
e
i
n
symmetry
,
ther
e
ar
e
k
variant
s
o
f
martensit
e
wit
h
transformatio
n
matrice
s
[/'"
,
t/*^'
,
.
.
.
,
f/**'
.
W
e
not
e
tha
t
k
an
d
th
e
matrice
s
t/*"
,
[/'^'
,
.
.
. ,
[/*'
*
ar
e
know
n
fo
r
an
y
give
n
material
:
the
y
ma
y
b
e
calculate
d
fro
m
th
e
chang
e
i
n
symmetr
y
an
d
th
e
chang
e
i
n
lattic
e
parameter
s
durin
g
trans
-
formation
.
Se
e
Tabl
e
1
fo
r
a
fe
w
importan
t
specia
l
case
s
(onl
y
[/<
"
i
s
shown
;
t/"'
,
t/"
\
...
,
[/<'
*
ma
y
b
e
obtaine
d
fro
m
i
t
b
y
symmetry)
.
Therefore
,
th
e
identit
y
matri
x
/
describe
s
th
e
austenit
e
stat
e
whil
e
th
e
matrice
s
[/*"
,
t/'^',.
.
.
,
t/'*
'
describ
e
th
e
martensit
e
state
.
Further
,
frame-indifferenc
e
say
s
tha
t
a
rigi
d
rotatio
n
doe
s
no
t
chang
e
th
e
energ
y
o
f
a
crysta
l
(o
r
i
n
othe
r
words
,
a
rigi
d
rotatio
n
doe
s
no
t
chang
e
th
e
stat
e
o
f
th
e
crystal)
.
Therefore
,
w
e
conclud
e
tha
t
th
e
deformatio
n
gradient
s
corre
-
spondin
g
t
o
th
e
Journa
l
o
f
Engineerin
g
Material
s
an
d
Technolog
y
Copyrigh
t
©
199
9
b
y
ASM
E
JANUAR
Y
1999
,
Vol
.
12
1
/
9
3
Downloaded From: http://materialstechnology.asmedigitalcollection.asme.org/ on 10/08/2013 Terms of Use: http://asme.org/terms
Fig
.
1
Th
e
shape-memor
y
effec
t
Fig
.
2
Schemati
c
vie
w
o
f
a
wedge-lil<
e
microstructur
e
1
.
Austenit
e
state
s
ar
e
R
fo
r
an
y
rotatio
n
matri
x
R;
2
.
Martensit
e
state
s
ar
e
/?f/<"
,
RU^^\
...
,
o
r
«!/<*
'
fo
r
an
y
rotatio
n
matri
x
R.
(1
)
Bal
l
an
d
Jame
s
(1987
)
a
s
wel
l
a
s
Chipo
t
an
d
Kinderlehre
r
(1988
)
showe
d
tha
t
energ
y
minimizatio
n
wit
h
suc
h
a
n
energ
y
densit
y
lead
s
t
o
fine-scale
microstructure
.
Roughly
,
th
e
ide
a
i
s
th
e
following
.
Whe
n
subjecte
d
t
o
certai
n
boundar
y
conditions
,
th
e
materia
l
trie
s
t
o
minimiz
e
it
s
energ
y
b
y
makin
g
mixture
s
o
f
th
e
differen
t
variant
s
o
f
martensit
e
whil
e
tryin
g
t
o
satisf
y
al
l
th
e
coherenc
e
requirements
.
Thi
s
lead
s
t
o
"minimizin
g
se
-
quences
"
whic
h
ar
e
interprete
d
a
s
fine-scale
microstructure
.
Bal
l
an
d
Jame
s
(1987
)
als
o
showe
d
tha
t
th
e
well-know
n
phe
-
nomenologica
l
o
r
crystallographi
c
theor
y
o
f
martensit
e
follow
s
a
s
a
consequenc
e
o
f
thi
s
theory
.
Sinc
e
then
,
ther
e
ha
s
bee
n
muc
h
progres
s
i
n
understandin
g
martensiti
c
microstructure
,
an
d
analytica
l
tool
s
Uk
e
th
e
"Youn
g
measure
"
an
d
averag
e
compat
-
ibilit
y
condition
s
lik
e
th
e
"minor
s
relations
"
hav
e
bee
n
devel
-
oped
.
3
Th
e
Wedge-Lik
e
Microstructur
e
I
t
i
s
ver
y
commo
n
t
o
observ
e
a
wedge-lik
e
o
r
spear-lik
e
microstructur
e
i
n
SMAs
.
Whe
n
th
e
allo
y
i
s
coole
d
fro
m
abov
e
th
e
transformatio
n
temperature
,
wedge-shape
d
region
s
o
f
mar
-
tensit
e
gro
w
int
o
th
e
austenite
.
A
s
show
n
i
n
Fig
.
2
,
th
e
wedg
e
consist
s
o
f
tw
o
set
s
o
f
fine
martensiti
c
twin
s
(fin
e
alternatin
g
band
s
o
f
tw
o
martensit
e
variants
)
separate
d
b
y
a
midrib
.
Thi
s
microstructur
e
provide
s
a
n
eas
y
wa
y
fo
r
th
e
initiatio
n
o
f
trans
-
formatio
n
an
d
i
s
thu
s
importan
t
fo
r
thermoelasticit
y
an
d
revers
-
ibilit
y
o
f
transformatio
n
(Otsuk
a
an
d
Shimizu
,
1969)
.
T
o
chec
k
whethe
r
a
materia
l
ca
n
for
m
a
wedge
,
i
t
i
s
necessar
y
t
o
enforc
e
tw
o
conditions
:
(/
)
th
e
deformatio
n
gradient
s
withi
n
th
e
wedg
e
Tabl
e
1
Som
e
transformation
s
an
d
transformatio
n
matrice
s
Transformatio
n
Transformatio
n
matri
x
Cubi
c
t
o
tetragona
l
Eg
:
Ni-A
l
Cubi
c
t
o
orthorhombi
c
Eg
:
y!
Cu-Al-N
i
Cubi
c
t
o
monoclinic-
I
Eg
:
Ni-T
i
Cubi
c
t
o
monoclinic-I
I
Eg
:
Cu-Zn-A
l
1
2
1
2
a
0
0
'a
S
^
0
'a
S
\e
'a
S
fi
0
a
0
6
a
0
6
a
e
S
7
0
0
0
13
0
0
P
e
P
0
0
/
9
shoul
d
correspon
d
t
o
martensit
e
variant
s
whil
e
th
e
deformatio
n
gradient
s
outsid
e
th
e
wedg
e
shoul
d
correspon
d
t
o
th
e
austenit
e
an
d
(ii
)
compatibilit
y
condition
s
hav
e
t
o
b
e
satisfie
d
a
t
th
e
five
interfaces—tw
o
twi
n
interfaces
,
tw
o
austenite-martensit
e
interface
s
an
d
on
e
midrib
.
Bhattachary
a
(1991
)
showe
d
tha
t
th
e
five
compatibilit
y
con
-
dition
s
ar
e
too
restrictive
fo
r
an
y
arbitrar
y
materia
l
t
o
satisfy
.
Fo
r
example
,
a
materia
l
tha
t
undergoe
s
a
cubi
c
t
o
tetragona
l
transformatio
n
ca
n
for
m
a
wedg
e
i
f
an
d
onl
y
i
f
th
e
materia
l
parameter
s
a
an
d
P
(cf
.
Tabl
e
1
)
satisf
y
th
e
conditio
n
(
1
-
py
+
4/3^(
1
-t
-
p^)
(
1
-
P^Y
+
8/3
"
(2
)
Thi
s
describe
s
a
curv
e
i
n
a-P
spac
e
(Fig
.
3)
,
an
d
onl
y
thos
e
material
s
whos
e
measure
d
parameter
s
li
e
o
n
th
e
curv
e
ca
n
for
m
a
wedge
.
Similarly
,
i
n
material
s
tha
t
underg
o
a
cubi
c
t
o
ortho
-
rhombi
c
transformation
,
th
e
materia
l
ca
n
for
m
a
wedg
e
i
f
an
d
onl
y
i
f
th
e
measure
d
materia
l
parameter
s
a,
P
an
d
6
li
e
o
n
a
certai
n
famil
y
o
f
surface
s
i
n
a-pS
space
.
Furthermore
,
th
e
the
-
or
y
predict
s
variou
s
geometrica
l
feature
s
o
f
th
e
wedge
.
Material
s
whic
h
ar
e
know
n
t
o
for
m
a
wedg
e
satisf
y
thes
e
condition
s
ver
y
closely
.
W
e
highligh
t
Ni-A
l
an
d
Cu-Al-N
i
i
n
Tabl
e
2
;
als
o
se
e
Fig
.
3
.
Further
,
th
e
geometri
c
detail
s
o
f
th
e
observe
d
wedge
s
agre
e
ver
y
wel
l
wit
h
th
e
predictions
;
fo
r
exam
-
pl
e
th
e
theor
y
predict
s
tha
t
wedge
s
wit
h
Typ
e
I
twin
s
i
n
Cu
-
Al-N
i
resembl
e
th
e
bottom-righ
t
o
f
Fig
.
2
whil
e
thos
e
wit
h
Typ
e
I
I
twin
s
th
e
top-righ
t
i
n
agreemen
t
wit
h
observations
.
Thi
s
calculatio
n
show
s
tha
t
microstructur
e
ofte
n
depend
s
crit
-
icall
y
o
n
th
e
lattic
e
parameters
:
a
smal
l
chang
e
i
n
lattic
e
parame
-
ter
s
ca
n
resul
t
i
n
a
significantly
differen
t
clas
s
o
f
microstructure
s
an
d
consequentl
y
significantl
y
affec
t
macroscopi
c
properties
.
Therefore
,
i
t
suggest
s
tha
t
th
e
abilit
y
o
f
a
materia
l
t
o
displa
y
0.
8
0.
9
1.
1
1.
2
d
7
2
P
a
—
b-
-
c
—
d
-
e
-
f
-
g-
-
h-
-
i
-
Wedg
e
possibl
e
N
o
volum
e
chang
e
(Self-accommodating
)
NiM
n
NiZnC
u
NiA
l
NiZnS
i
InT
l
FeAl
C
FeP
t
FeCr
C
FeNi
C
Th
e
axi
s
o
f
monoclini
c
symmetr
y
i
s
(110)cubi
c
((100>cubic
)
i
n
monoclinic
-
I
(monoclinic-II)
.
Fig
.
3
Th
e
specia
l
relation
s
o
n
th
e
transformatio
n
strai
n
fo
r
wedg
e
an
d
self-accommodatio
n
i
n
a
cubi
c
t
o
tetragona
l
transformation
.
Th
e
mea
-
sure
d
lattic
e
parameter
s
o
f
som
e
alloy
s
ar
e
als
o
shown
.
Wedge
s
hav
e
bee
n
observe
d
i
n
alloy
s
b
,
d
,
f
an
d
i
whil
e
alloy
s
a-
e
an
d
g
ar
e
self
-
accommodatin
g
(se
e
Bhattachary
a
(1991
,
1992
)
fo
r
details
an
d
refer
-
ences)
.
9
4
/
Vol
.
121
,
JANUAR
Y
199
9
Transaction
s
o
f
th
e
ASM
E
Downloaded From: http://materialstechnology.asmedigitalcollection.asme.org/ on 10/08/2013 Terms of Use: http://asme.org/terms
Tabl
e 2 Wedg
e an
d self-accommodation
: Compariso
n o
f
theor
y an
d experiment
. Wedge
s ar
e see
n in Ni-A
l an
d Cu
-
Al-Ni
; al
l thre
e alloy
s ar
e self-accommodatin
g (se
e Bhatta
-
chary
a (1991,1992
) fo
r detail
s an
d references
)
Materia
l
Observe
d
parameter
s
Parameter
s
for a wedg
e
Ni-Al
Cu-Al-N
i
Ni-Ti
a =
0.939
2
0
=
1.130
2
a =
1.042
5
P =
0.917
8
6
=
0.019
4
a =
0.944
5
0
=
1.130
2
a
=
1.042
5
0
=
0.919
3
6
=
0.019
4
Observe
d
volum
e chang
e
0.305
%
0.297
%
0.023
%
the SM
E ma
y depen
d criticall
y o
n th
e lattic
e parameter
s an
d
consequentl
y on composition
.
Tabl
e 3 Summar
y of experimenta
l observation
s of tensil
e
recoverabl
e strain
s (se
e Bhattachary
a an
d Koh
n (1996
) fo
r
detail
s an
d references
)
Materia
l
Singl
e
crysta
l
Polycrysta
l
Ni-A
l
CuAlNi/Cu-Zn-A
l
Ni-T
i
0-13
%
Almos
t non
e
2-9
%
Typicall
y -2%
; up to i
ribbon
s
3-10
% 5-8
% in wires/sheet
s
% in specia
l
to solv
e al
l th
e compatibilit
y conditions—canno
t be use
d here
.
No matte
r ho
w man
y microstructure
s w
e check
, ther
e ar
e alway
s
others
. Therefore
, w
e nee
d method
s to analyz
e genera
l classe
s
of microstructures
. It is her
e tha
t tool
s lik
e th
e "Youn
g mea
-
sure
"
an
d averag
e compatibilit
y condition
s lik
e th
e "minor
s
relations
" ar
e ver
y useful
.
4 Self-Accommodatio
n
Self-accommodatio
n is th
e abilit
y o
f a SM
A to transfor
m
fro
m th
e austenit
e to th
e martensit
e wit
h no macroscopi
c chang
e
in shape
. Apar
t fro
m bein
g an inheren
t par
t of SME
, Wayma
n
(1992
) an
d other
s hav
e emphasize
d th
e rol
e of self-accommoda
-
tio
n as a necessar
y conditio
n fo
r SME
.
No
t ever
y materia
l tha
t undergoe
s a martensiti
c transforma
-
tio
n is self-accommodating
. Consider
, fo
r example
, a materia
l
wher
e th
e volum
e of th
e martensit
e is smalle
r tha
n tha
t of th
e
austenite
. Clearly
, th
e transformatio
n lead
s to a chang
e in vol
-
um
e (unles
s th
e materia
l is subjecte
d to larg
e stresses
) an
d th
e
materia
l is no
t self-accommodating
. So
, wha
t ar
e th
e condition
s
tha
t guarante
e tha
t a materia
l is self-accommodating
?
Fro
m th
e argument
s above
, it is clea
r tha
t volum
e preservin
g
transformatio
n is a necessar
y conditio
n fo
r self-accommodation
.
Bhattachary
a (1992
) showe
d tha
t if th
e symmetr
y of th
e austen
-
ite is cubic
, the
n thi
s conditio
n is als
o
sufficient
fo
r self-accom
-
modation
. If on th
e othe
r hand
, th
e symmetr
y of th
e austenit
e
is not
cubic
, th
e lattic
e parameter
s of th
e materia
l hav
e to satisf
y
additiona
l restriction
s whic
h ar
e extremel
y stringen
t an
d "non
-
generic.
" Therefore
, material
s wit
h cubi
c austenit
e hav
e to sat
-
isfy a
rathe
r mil
d constraint
, whil
e material
s wit
h non-cubi
c
austenit
e hav
e to satisf
y ver
y restrictiv
e non-generi
c condition
s
in orde
r to be self-accommodating
. It is highl
y unlikel
y tha
t
realisti
c material
s wil
l satisf
y suc
h nongeneri
c conditions
. Con
-
sequently
, onl
y material
s wit
h cubi
c austenit
e whic
h underg
o
almos
t volum
e preservin
g transformatio
n ar
e self-accommodat
-
ing an
d henc
e shape-memor
y materials
. Thi
s conclusio
n is in
agreemen
t wit
h experimenta
l observation
s as highlighte
d in Ta
-
ble 2 an
d Fig
. 3; als
o se
e Bhattachary
a (1992
) fo
r extensiv
e
comparison
.
Ther
e is als
o a ver
y interestin
g an
d probabl
y importan
t coin
-
cidence
. Conside
r a cubi
c to tetragona
l transformation
. Volum
e
preservin
g transformatio
n correspond
s
to
a^p
=
1
whic
h de
-
scribe
s a curv
e in
a-P
(transformatio
n strain
) space
. A
s show
n
in Fig
. 3, thi
s curv
e is ver
y clos
e to th
e curv
e on whic
h material
s
can for
m a wedg
e (se
e Eq
. (2)
) whe
n
a
an
d /? ar
e clos
e to
1,
th
e rang
e of experimenta
l interest
. Similarly
, in a cubi
c to
orthorhombi
c transformation
, th
e volum
e preservin
g surfac
e is
clos
e to th
e wedg
e formin
g surface
s in th
e
a-pS
space
. Thus
,
man
y material
s wit
h cubi
c austenit
e undergoin
g volum
e pre
-
servin
g transformation
s ma
y als
o displa
y a wedge
!
Anothe
r interestin
g an
d importan
t consequenc
e o
f self-ac
-
commodatio
n is tha
t it is
not
possible
to induc
e th
e two-wa
y
SM
E by makin
g texture
d pol
y crystals
.
Finally
, a wor
d abou
t th
e theoretica
l methods
. In th
e cas
e of
the wedge
, w
e wer
e analyzin
g a give
n microstructure
. In con
-
trast
, her
e w
e ar
e askin
g a broade
r question
: is ther
e an
y micro
-
structur
e tha
t is self-accommodating
? Therefore
, th
e metho
d
use
d to stud
y th
e earlie
r problems—writin
g dow
n an
d tryin
g
5 Geometricall
y Linea
r Theor
y of Martensit
e Micro
-
structur
e
We no
w tur
n to calculatin
g th
e recoverabl
e strain
s tha
t ar
e
inheren
t in th
e fundamenta
l crystallograph
y of a give
n material
.
Thi
s questio
n turn
s out
to b
e extremel
y difficul
t an
d at thi
s
tim
e w
e ar
e unabl
e to carr
y throug
h suc
h a calculatio
n in th
e
geometricall
y nonlinea
r theor
y tha
t w
e hav
e bee
n usin
g so far
.
However
, it is possibl
e to writ
e a "geometricall
y linear
"
versio
n of thi
s theor
y by assumin
g infinitesima
l displacements
.
Thi
s is simila
r to th
e theor
y use
d by Roitbur
d (1978
) an
d Kha
-
chaturya
n (1983
) amon
g other
s (se
e Bhattachary
a (1993
) fo
r
a detaile
d comparison)
. I believ
e tha
t in th
e problem
s discusse
d
below
, thi
s approximat
e theor
y give
s reasonabl
e result
s thoug
h
the exac
t quantitativ
e detail
s ma
y b
e different
. I
n thi
s geo
-
metricall
y linea
r theory
, microstructure
s correspon
d to con
-
tinuou
s displacement
s
u
suc
h tha
t th
e infinitesima
l strain
s e =
5(V
M
-I-
VM'^
)
"essentially
" tak
e value
s
1.
0 (Austenit
e states
) or
Here
, e**
* = [/'*
* - / ar
e th
e transformatio
n strains
.
2.
e<"
, e"'
, . . . , e'*
> (Martensit
e states)
. (3
)
6 Recoverabl
e Strain
s
Man
y alloy
s ar
e know
n to be goo
d SMA
s as singl
e crystals
,
bu
t th
e exten
t to whic
h the
y retai
n thei
r SM
E in polycrystallin
e
for
m is widel
y varie
d a
s highlighte
d b
y thre
e alloy
s in Ta
-
ble 3.
6.1 Singl
e Crystal
.
W
e defin
e th
e se
t o
f recoverabl
e
strain
s in a singl
e crystal
,
S,
as th
e se
t of al
l possibl
e strain
s tha
t
can be achieve
d by mixin
g th
e differen
t variant
s of martensit
e in
S (grai
n 1)
Fig
.
4 Th
e se
t of recoverabl
e strain
s fo
r a materia
l undergoin
g larg
e
chang
e in symmetr
y (left
) an
d smal
l chang
e in symmetr
y (right)
. Th
e
recoverabl
e strain
s (S
) in a singl
e crysta
l is determine
d by th
e transfor
-
matio
n strain
s of th
e variant
s whil
e in a polycrysta
l it Is determine
d by
the interactio
n betwee
n th
e grain
s an
d Is estimate
d by th
e se
t
T.
Journa
l of Engineerin
g Material
s an
d Technolog
y
JANUAR
Y 1999
, Vol
. 12
1 / 9
5
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