A Simplified Multivariant SMA Model Based on Invariant Plane ...

Table 1. Interaction energy and force comparison between different models
chem surf int
Helmholtz’s free energy: Ψ = ∆G
+ W + W + �
Author Interaction energy
W int Interaction
=
1 II p
−
2V z σ ij ε ijdv
v
force
int
int W
Fi
=−
f i
∂
Magnitude
Features
∂
(unit volume)
int
Fi (Mpa)
Patoor et 1
al. (Patoor ∑ Hnm fnfm 2 nm ,
et al., 1994;
Patoor et
al., 1996)
−∑ Hnm fm
m
H 1 µ
= ≈40
1000
H 2 µ
= ≈270
150
Lu and
Weng
(Lu and
Weng,
1997)
Vivet and
Lexcellent
(Vivet and
Lexcellent,
1998)
Huang and
Brinson
(Huang et
al., 2000;
Huang and
Brinson,
1998)
1
c1( 1−
c1) L
2
p p
( I − S)
ε ε
(only 1 variant
considered)
N
∑
s=
1
−
N
f ( 1−
f ) W
s s s
N
∑ ∑
f f W
s t st
s=
1 s= 1, t= 1,
t≠s 1 p p
Ws= ε L( I −S)
ε
2
G
1
p
− σ ε f =
1
∑
2 g=
1
G
∑
2 g=
1
G
∑
m=
1
g g
p
p
ε [ LI ( − S)
ε
g
p
− f L( I −S)
ε ] f
g
m m
g
g
1
− −
2
−
1 2 ( c1) L
p p
( I S)
ε ε
−( 1−2f ) W
s s
(simplified form
when only one
variant is
considered; same
as Lu and Weng
above)
− ∂W
∂
f ( gn , )
Gao and No explicit form. Small constant,
Brinson
(this paper)
lumped into Fc
int
< 1
W s ≈ 0.468
(Interaction
between
austenite and
martensite)
–40 to 60
when µ=40
-12 to 18
when µ=12
~1
note Fc=4
typically
27
A multivariant model
Interaction matrix H as approximation:
H 1 for self-accommodating variants
H 2 for non-self-accommodating ones
Resistance force of F int increases with
increased volume fraction, causing a
continuous strain hardening effect
Only values of H an order of magnitude
lower than listed will produce a more
appropriate non-strain-hardening response.
Good qualitative agreement to uniaxial
experimental results.
A single variant model
Analytical solution just for one variant
Resistance force of F int decreases with
increased volume fraction, becoming
actually a driving force for f >0.5
Aspect ratio of thin oblate was taken to be 0,
reducing the magnitude of F int
Non-linear dependence on f of dissipation
Good agreement to uniaxial experimental
results for plates of aspect ratio 0
A single variant model
Analytical solution just for one variant
Resistance force of F int decreases with
increased volume fraction, becoming
actually a driving force for f >0.5
Very thin oblate (a3/a1≈0.001) → lower Ws
Very good agreement to uniaxial
experimental results
A multivariant model
Analytical solution without additional
approximation
Self-accommodating groups of variants
without interaction among themselves
Plate aspect ratio: a3/a1=0.01
Long time of calculation
Good qualitative agreement to uni- or multiaxial
experimental results
A multivariant model
<strong>Based</strong> on invariant plane nature of
martensitic transformation
Accounts for anisotropy in single crystal case
Very fast calculation
Excellent agreement to uni- or multi- axial
experimental results