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the phenomenological and curve fi tting models. The Young's modulus of the low
temperature martensite is lower than that of austenite. In all models it is assumed
to be linearly decreasing with increasing martensite fraction (
x
):
(9 )
Thermodynamic models are based on potential energy functions. In these models,
the possible states are mathematically represented by 'wells', being local minima
of potential energy with respect to the shear length. The transformation dynam-
ics are described by the probability of a crystal being in one well to overcome
the energy barrier to jump to the next using Boltzmann statistics. One of the ear-
lier models for SMA behavior by Achenbach [88] is such a model. Others are by
Seelecke [87] and Massad
et al
. [ 89 ].
Phenomenological models, like those by Tanaka [90], Liang and Rogers [91]
and Brinson [92-94] are also based on thermodynamic potential formulations, but
in these models often Gibs and Helmholtz free energy functions are employed
because they do not rely on entropy as an internal parameter [19]. So-called hard-
ening functions are assumed to describe the transformation dynamics. With these
models, the martensite fraction of the material is determined by using the
s
,
T
-state
of the material. The models differ in the way that transition areas are modeled.
Tanaka derives the following constitutive relation from the Helmholtz free energy:
E
() (1
x
=−
x
)
E
+
x
E
A
M
d
s
=
Ed
e
Θ+Ω
d
T
d
x
( 10 )
In this equation
E
refers to the modulus of elasticity,
Θ
is related to the CTE and
Ω
is called the 'transformation tensor'. Equation (10) can be written in integral
form, with constant material properties:
ss
−= − +Θ− +Ω−
E
(
ee
)
(
T
T
)
(
xx
)
( 11)
0
0
0
0
Tanaka only distinguishes between austenite and martensite and models the stress-
temperature dependency of the martensite fraction
x
with an exponential function.
For the AM transition:
a
aMTb
(
−+
)
s
M
x
=−
1
exp
for
s
≥
(
TM
−
)
(12 )
Ms
M
s
b
M
and for the MA transition:
a
aATb
(
−+
)
s
A
x
=
exp
for
s
≤
(
TA
−
)
( 13 )
As
A
s
b
A
The coeffi cients
a
M
,
b
M
,
a
A
and
b
A
are dependent on the transitions' start and fi n-
ish temperatures and the stress dependency of these temperatures, the Clausius
Clapeyron constants
C
A
and
C
M
. Assuming that the transition is complete with
99% conversion:
2ln10
,
a
A
a
=
b
=
(14 )
A
A
AA
−
C
f
s
A
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