M
• The SMA material model can be used with rod, 2-D solid, 3-D olid and shell elements. It is available only for impli
(Solution 601).
Shape memory alloy materials can undergo solid-to-solid phase
) with a monoclinic crystal structure in several variants.
hase are shown, as well as stress- and temperature-dependent temperatures or high stresses. Upon heating from low temperature
the t
tem ransformation is 100% complete at
tem erature A . If the material is then cooled again, the austenite sta
tra ur
tem ng
the slo
respectively. A typical variation of volume fraction of martensite in the
P
Output variables: The following gaskets output variables are available: Gasket pressure, Gasket closure strain, Gasket yield stress, Gasket plastic closure strain, Gasket status.
• Note that all these output variables are scalar quantities.
m 01 only)
• The Shape Memory Alloy (SMA) material model is intended to model the superelastic effect (SE) and the shape memory effect (S E) of shape-memory alloys. It is defined using the MATSMA material entry.
s cit analysis
•
transformations induced by stress or temperature. The high temperature phase is called austenite (A) with a body-centered cubic structure and the low-temperature phase is called martensite (M
Fig.3.9-1 shows a schematic SMA stress-temperature diagram.
The martensite phase in two generalized variants and the austenite p
transformation conditions. The martensite phase is favored at low material begins transforming from martensite to austenite a
perature As. The t
p f
rts transforming back to martensite at temperature Ms. This nsformation is 100% complete at temperature Mf. These fo
peratures are also stress dependent with high stresses favori martensite phase. This stress dependence is assumed linear with pe CM and CA for the martensite and austenite temperatures, SMA material with temperature is shown in Fig. 3.9-2.
Detwinned martensite
Figure 3.9-1: SMA stress-temperature phase diagram , T seff
CM CM CA CA
Twinned
Austenite
martensite
Mf Ms As Af Temperature
Figure 3.9-2: Volume fraction of martensite vs. temperature Temperature, T
emVolumfractionofmartensite,
Mf As
Ms Mss Af Afs Mfs Ass
0.0 1.0
Transformation atseff= 0 Transformation atseff> 0
al stress-strain curve is shown in Fig.
0.4 0.6 0.8
0.2
• A typical uniaxial isotherm
.9-3.
oys e
3
T > Af s
T < As s
(a)
Figure 3.9-3: Schematic of stress-strain curves for shape-memory all (a) superelasticity; and (b) shape memory effect
(b) e
from M
ffect is evident when the material is deformed at temperature T<As
ed in Fig.3.9-3(b). A residual transformation strain remains after unloading; however heating the material to
mperature above Af leads to thermally induced M→A ation and the recovery of transformation strain.
• Both shape memory effects due to transformation from martensite to austenite and due to re-orientation of the martensite are captured by modeling the twinned and detwinned martensites as different phases.
• The SMA material model is based on the following equations:
The total strain,
• The superelastic effect is evident when the material is deformed at temperature T>Af and is displayed in Fig.3.9-3(a). The stress cycle application induces transformations from A→M and then
→A to exhibit the hysteresis loop. The shape memory e
and is display te
transform
e t θ
ε ε = + + ε ε
where
ε
e=
elastic strainε
θ=
thermal strainε
t=
transformation strain; to be evaluated The one-dimensional macro-scale model,s t
ξ ξ ξ
= + ; 0 ≤ ξ ≤1 ξ + ξΑ= 1εt = εtmax ξs
((1 )EA EM)( θ tmax s)
σ
= −ξ
+ξ ε
−ε
−ε ξ
ξt = twinned martensite volume fraction ξs = detwinned martensite volume fraction ξA = austenite volume fractionεtmax = maximum recoverable residual strain; a material
) property usually obtained from a simple tension test when the material is fully detwinned martensite (ξs = 1 The flow rule of three-dimensional constitutive model,
max
t t t
ij s nij
ε ξ ε
∆ = ∆ 3 ij t
ij
n s
2
σ
⎛ ⎞
= ⎜ ⎟; for the martensitic transformation
⎝ ⎠ 3 2
t t ij
ij t
n
ε
ε
⎛ ⎞
= ⎜⎜ ⎟⎟; for the reverse transformation
⎝ ⎠ where
s
ij = deviatoric stresses3 2s sij ij
σ
= is the effective von Mises stress3 2
t t t
ij ij
ε
=ε ε
is the effective transformation strainThis results in the following equation for deviatoric stress calculation:
( ) ( ) ( )
1
t t
t t t t t
ij t t ij ij
s E
ξ
ε ε
ν ξ
+∆ +∆ +∆
+∆ ′′
= − ∆
+
where
t t t t t t
ij ij ij
ε ε ε
+∆ ′′= +∆ ′ −
Four phase transformation conditions,
1. Starting condition for the martensitic transformation
3 2 (
S )
M M S
f = J −C
θ
−M2. Ending condition for the martensitic transformation
3 2 (
f )
M M f
f = J −C
θ
−M3. Starting condition for the reverse transformation
3 2 (
AS A S)
f = J −C
θ
−A4. Ending condition for the reverse transformation
3 2 ( )
Af A f
f = J −C
θ
−Aacco, 1999),
The phase transformation rate using linear kinetic rule (Auricchio and E. S
f f
R f f
ξ
ξ∆ = ∆
2 2
3 (
2 tJ t J t t t )
f c
t t t
θ θ
− +∆
∆ =
o martensite transformation,
σ
+∆ − −
+∆
where, for the austenite t
- ,
f Mf M
f = f c= C and Rξ = 1 - ξ
and for the reverse martensite to austenite transformation,
f,
f A A
f = f c= C and R = ξ ξ
Evolution of single-variant detwinned martensite:
Martensite re-orientation is based on the following condition
3 2
R R R
f = J −C
θ σ
− whereσ
R =material yield property atθ
= 0CR = slope of yield function temperature variation Austenite to martensite transformation leads to
ξ
s =ξ
Martensite to austenite transformation leads to proportional transformation of the twinned and detwinned phases:
s
• Computational steps for the stress-integration of the SM m del are as follows (Kojic and Bathe, 2005):
1. Cal ia
A o
culate the tr l deviatoric stresses, assuming no additional phase transformation or re-orientation,
( ) ( )
2. Check for martensitic re-orientation, 0
fR > and t
ξ
s < tξ
and tξ
<1for nite to martensite transformation, Check auste
0 f f
f s
M M < and t
ξ
<1 and ∆ >f 0Check for martensite to austenite transformation,
f s 0
A A
f f < and t
ξ
>0 and ∆ <f 03. In case of martensitic re-orientation solve the following overning equation:
g
The martensite reorientation calculation step is optional; it is activated when σR > 0 is input.
4. In case of austenite to martensite transformation, solve the following governing equation:
2 2
. In case of martensite to austenite transformation, solve the overning equation (3.9-2) with
5 g
Rξ =
ξ
,f Af
f = f and = A
e the consistent tangent constitutive matrix.
perature-Dependent or Shape-Memory Alloys: Constitutive
Modelling, Finite-Element Implementation and
(1999)
c C
6. Update history-dependent variables for this time step/iteration step.
7. Calculat
ref. M. Kojic and K.J. Bathe, Inelastic Analysis of Solids and Structures, Springer, 2005
ref. F. Auricchio and E. Sacco, “A Tem Beam f
Numerical Simulation”, Computer Methods in Applied Mechanics and Engineering, Vol. 174, pp. 171-190