Environmental Engineering Reference
In-Depth Information
−
Δ=
1
x
( 30)
t0
cos(
aTM
(
−
)
+
1
T
x
M
f
2
Else:
Δ=
0
( 31)
T x
Subscript 't' denotes temperature induced martensite. If the temperature is below
M
f
the detwinning is only stress-dependent. But if the temperature is between
M
f
and
M
s
, the model takes into account the formation of detwinned martensite due
to cooling through the AM transition zone. This is captured in the
Tx
parameter.
If the stress is below
c
s
s
only twinned martensite is formed. For the formation of
detwinned martensite above
M
s
Brinson derives the following.
For
T
>
M
s
and
Δ
cr
cr
s
+
CTM
(
−
)
<
s
<
s
s CTM
+
(
−
):
s
M
s
f
M
s
⎡
⎤
1
−
x
1
+
x
p
s0
cr
s0
x
=
cos
(
s
−
s
−
CTM
(
−
))
+
(32)
⎢
⎥
s
f
M
s
cr
cr
2
2
ss
−
⎣
⎦
s
f
The function for the formation of austenite above
A
s
is the same as with Liang and
Rogers, but a function for the split in stress and temperature induced martensite
is added:
x
[
]
0
x
=
cos(
aT A
(
−
)
−
s
C
)
+
1
( 33)
A
s
A
2
x
s0
xx
=−
(
xx
−
)
( 34)
s
s0
0
x
0
x
t0
xx
=−
(
xx
−
)
( 35)
t
t0
0
x
0
Like with the model of Tanaka, the different phase regions can be plotted on the
T
,
s
-plane (see Fig. 14). In a later publication Bekker and Brinson [95] introduce
so-called switching points. At these switching points, the phase transition is either
complete or the
s , T
-path reverses. In the model, the start fractions
x
s0
and
x
t0
are
then reset. This way, uncompleted transitions and embedded loops can be modeled.
These models are very insightful in understanding the underlying mechanisms of
the SME and superelasticity because they map the martensite fraction based on the
actual parameters on which it is actually depending: stress and temperature. And
more importantly: they seem to predict the SMA behavior well [96]. However, the
models provide the strain as a function of temperature, stress and load history. Invert-
ing the model is not possible and a solution must be found iteratively. Leo presents a
similar model in [97].
With the curve fi tting models, like those by Spies [98] and van der Wijst [99], the
force-displacement behavior is derived directly from the stress-strain path. The
temperature dependency of this path is taken into account by linearizing the
e
,
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