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Constitutive Modeling and Simulation of Shape Memory
Polymers
Defense Proposal
ADVISOR: DR I.J. RAODATE : 11/17/2008
MAHESH KHANOLKAR
Outline Introduction
What are shape memory materials. Different types of shape memory materials How shape memory polymers work
Modeling Natural Configurations Thermo-mechanical Framework Model development
Glassy SMP Model Application of Crystallizable SMP Model
Simulations and Results Conclusions
What Are Shape Memory Materials?
“Remember” the original shape even after undergoing significant deformation
Revert back to original shape by a suitable trigger Most common trigger: heating above a recovery
temperature, TR Other triggers: Magnetic fields, electromagnetic radiation
etc.
Trigger
Overview of SMP’s
Mechanism for “remembering” original shape and transient shape.
Common mechanisms: Entanglements, Crosslinks and hard-domains.
Transient shape fixed usually with crystalline phase or the glassy state.
Revert back to original shape by heating. Heating above Tm (if the crystalline phase is used to
fix the transient shape) Heating above Tg (if the glassy phase is used to fix the
transient shape)
How Shape Memory Polymers Work
Original: Crystalline Hard domains(Physical cross-links)
Temporary: Crystallites
Original: Chemical Cross-LinksTemporary: Crystallites
Original: Chemical Cross-LinksTemporary: Glassy Phase
Lendlein et al.
Shape Memory Alloys (SMA) - Extensive work has been carried out in the last 10
years. - Constitutive equations and modeling fairly well
developed.
Shape Memory Polymers (SMP) - Advantages - SM effect can be seen for large deformation - Manufacturing methods are conventional and cheap - Bio-compatible - Recovery temperature can be adjusted
- Disadvantage - Actuation force (SMP) << Actuation force (SMA)
Types of Shape Memory Materials
Shape Memory Polymers Representative Application
Biodegradable Shape Memory Polymerfor Suturing wounds. (Langer 2002)
Shape Memory Polymers Representative Application
Time series photographs that show the recovery of a shape-memory tube. (a)- (f) Start to finish of the process takes a total of 10 s at 50°C (Marc Behl et
al 2007).
Shape Memory Polymers
Applications SMP fibers for comfort wear MEMS devices, temperature
sensors Damping elements Intravenous needles and
implantable Medical device Films and fibers used in
insulation applications Rewritable digital storage
devices Morphing Aircraft Wings many more…
Shape Memory Mechanism in CSMP’s
Deform
Cool
Unload
Heat
Amorphous polymer
Cross-link
Crystallite
Legend
Melting Crystallization
T > Tr
T < Tr
State 1
State 4
State 2
State 3
Stretch
Nom
inal
Str
ess
1
2
3
4
Shape Memory Mechanism in GSMP’s
Deform
Cool
Unload
Heat
Amorphous polymer
Cross-link
Glassy polymer
Legend
Glass Transition
T > Tr
T < Tr
State 1
State 4
State 2
State 3
StretchN
omin
al S
tres
s
1
2
3
4
Modeling (Salient Features)
Constitutive Modeling – Mathematical description of how a material responds to deformations.
It is a relation between two physical quantities (often described by tensors).
Modeling of polymers– Write equations for stress tensor in terms of deformation gradient.
Change in Entropy and internal energy is macroscopic manifestation of changes in microstructure.
Non-linear response.
Modeling (Salient Features)
Above Tr the material behavior is rubber like Hard domains act as cross-links in thermoplastic SMP’s Chemical cross-links in the case of thermoset SMP’s
Cooling in deformed shape causes partial crystallization / glass transition
Crystallization – drop in stress Glass-Transition- stress remains constant or increases Semi-crystalline polymer is anisotropic
Unloading the the specimen below Tr, a small recovery strain observed.
Heating above Tr, return to original shape
Modeling Framework
Need to account for the influence of each phase
Amorphous rubbery phase above the recovery temperature.
Semi-crystalline polymers: amorphous and crystalline phases
Glassy polymers: amorphous and glassy phases (mixture region)
Each phase can have its own stress free state
Modeling - Natural Configurations
In most traditional approaches the response of the material is assumed to be known from a single configuration.
Well known that a body can be stress-free in more than one configuration
Solid which can exist in two different phases (e.g. Austenite and Martensite) with different symmetries.
Polymers, which can exist in the amorphous and crystalline phase
Deform Unload
Modeling - Natural Configurations
Rk
c(t)k
p(t)kG
RkF
p(t)F
Natural configurations associate with a viscoelastic melt
Modeling - Glassy SMP (Amorphous Rubbery Phase)
Model as an incompressible hyperelastic material Stress is given by:
Based on Rubber elasticity: entropic in origin.
2a a
a
Tap
T I F F
C
aF c
a
Modeling – Glassy SMP
100 % conversion into glass during vitrification
Glassy phase is viscoelastic
Glassy phase is formed in stressed state
Little Change in length on cooling, iso-stress, Mather(2006)
Modeling – Glassy SMP(Mixture of rubbery and glassy phase)
Stress in nascent glass = stress in rubbery phase Stress is given by:
1 2 2a a g g
a g
gT Tap
T I F F F FC C
aF
c
g
gF
Current configuration of glassy phaseg
Current configuration of amorphous phasecNatural configuration of amorphous phasea
Modeling – Glassy SMP(Mixture of rubbery and glassy phase)
Constrained Cooling below the glass transition temperature.
Increase in thermal stress.
Natural configurations evolve as the material is cooled/deformed.
Natural configuration associated with the previously formed material shifts to a new position.
Increase in mechanical deformation gradient, decrease in thermal deformation gradient, so that the total deformation gradient remains constant (constrained cooling).
Modeling – Glassy SMP(Mixture of rubbery and glassy phase)
Natural Configurations associated with the glassy-rubbery phase solid phase mixture
RK
1c (t)K2c (t)K
p (t)2kFp (t)1
kF
1p (t)K
1p (t)K
2p (t)K
RKF
Modeling – Glassy SMP Cycle - Equations
Loading: where T is the stress in the rubbery part of the polymer and µa is the modulus
Cooling:
2 1a a
a
T
22
2 20
( )1 1(1 ) (t)
(t) (t) (t) ( )
tg g
aa a g
dT d
B B d
where 1 ( ( ) ( ))B t 1
m( ) m( ) ( )The Deformation Gradient is given as : ( ) ( ) ( )t t F F F
Modeling – Glassy SMP Cycle - Equations
Unloading
Melting
2 22
0
1 1 1(1 ) (t) ( )
(t) (t) ( )
tg
a ga a g
dT d
B d
n( ) m( )Total Deformation Gradient : ( ) (t) (t)unloadt F F F
22
2 20
( )1 1(1 ) (t) 0
(t) (t) (t) ( )
tg g
aa a g
dd
B B d
Modeling – Glassy SMP Cycle
Stress–strain–temperature diagram illustrating the thermo mechanical behavior of a shape memory polymer under different
strain/stress constraint conditions
Simulation and Results(Uniaxial Deformation Cycle GSMP)
0 1 2 3 4 5 6 7 8 9 100
0.5
1
1.5
2
2.5
3
3.5
4
4.5
Strain (%)
Str
ess
(MP
a)
Equation 1Equation 2Equation 3Equation 4
Stress vs Strain for the complete SMP Cycle
a
TL (K) 273
Tg (K) 343
TH (K) 358
(Mpa) 8.8 MPa
(Mpa) 750 MPag
Simulation and Results(Uniaxial Deformation Cycle GSMP)
0.8 0.85 0.9 0.95 1 1.050
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Temperature (T/Tg)
Str
ess
(MP
a)
Mathematical ModelExperimental Data
Stress vs Temperature
Simulation and Results(Uniaxial Deformation Cycle GSMP)
-10 -8 -6 -4 -2 0 2 4 6 8 10-1
0
1
2
3
4
5
6
7
8
Strain (%)
Str
ess
(MP
a)
Tension (358 K)Experimental DataTension (273 K)Experimental DataCompression (358 K)Experimental Data
Stress vs Strain plot (Yiping Liu et al, 2005)
Nanoparticle Reinforced Glassy SMP
Reinforcing Glassy SMP with nanoparticles increases its stiffness.
Rubbery Phase:
Glassy phase:
where
K is the concentration of nanoparticles
2 1(1 )a a
a
T k
22
2
( )(t)
(t)g
m B
22 212
0
1 1(1 ) (t) (t)
(t) (t)
tga
a ma a a m
kk dd
d
T
Simulation and Results(Uniaxial Deformation Cycle GSMP) Effect of Nanoreinforcemnts
Elastic moduli of the SMP and SMP composite at 26 and 118°C(Yiping Liu et al 2003) .
Simulation and Results(Uniaxial Deformation Cycle GSMP) Effect of Nanoreinforcemnts
0 0.5 1 1.5 2 2.5 30
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Strain (%)
Str
ess
(M
Pa
)
Model(0 wt% Nanoparticle)Experimental Data(0 wt% Nanoparticle)Model(20 wt% Nanoparticle)Experimental Data(20 wt% Nanoparticle)
Stress vs Strain Above the glass transition
Torsion of a Cylinder
,
Z,
.
r r R
k
z Z
ak
1 0 0
0 1 ,
0 0 1
rk
F
a
T 2 2k
1 0 0
= 0 1 ,
0 1
r k rk
rk
B FF F
Undeformed Cylinder Deformation after applying Torsion
Motion:
Deformation gradient:
c( ) c( )
2 2k k
1 0 0 1 0 0
0 1 , 0 1
0 0 1 0 1
r k r k r k
r k
F B
M
(in sec-2) (MPa) (MPa)
0.33 120 1200 0.00007 0.256 50
0 t s tms Ga c
Simulation and Results:Torsion of a Cylinder
Moment vs Time (Torsion of a cylinder)
0 100 200 300 400 500 600 700 800 9000
10
20
30
40
50
60
70
80
90
100
Time (seconds)
Mom
ent
(N-m
)
Simulation and Results:Torsion of a Cylinder
Moment vs Shear (Torsion of a cylinder)
0 50 100 150 200 2500
10
20
30
40
50
60
70
80
90
100
Shear
Mom
ent
(N-m
)
Simulation and Results:Torsion of a Cylinder
Shear vs Time (Torsion of a cylinder)
0 200 400 600 800 1000 1200 1400 16000
50
100
150
200
250
Time (seconds)
Sh
ear
Simulation and Results:Large Deformation on a single cubic element using UMAT (ABAQUS)
Shape memory cycle on a single element using an UMAT (User Defined Material)
A strain of 100% (large deformation) has been applied to the element and the resulting deformation is seen
Steps involved in the shape memory cycle.o Large Deformation on the single element o Constraining the element to retain its temporary
shape o Removing load – Small amount of strain recovery o Return to Original Shape
Simulation and Results:Large Deformation on a single cubic element using UMAT (ABAQUS)
Applied load to the Element
Simulation and Results:Large Deformation on a single cubic element using UMAT (ABAQUS)
Step 1 Large Deformation on the single element
Simulation and Results:Large Deformation on a single cubic element using UMAT (ABAQUS)
Step 2 Constraining the element to retain its temporary shape
Simulation and Results:Large Deformation on a single cubic element using UMAT (ABAQUS)
Step 3 Removing load – Small amount of strain recovery
Simulation and Results:Large Deformation on a single cubic element using UMAT (ABAQUS)
Step 4 Back to Original Shape
Conclusion and Future Work Developed a model for SMP’s undergoing glass transition
using the notion of natural configurations.
Developed model takes in to account the thermal expansion of polymers.
Reinforcements have a impact (increased modulus) on SMP’s.
Illustrated application of a CSMP Model for a non-homogenous deformation (Torsion of a cylinder).
Illustrated application of CSMP Model using ABAQUS.
Further develop GSMP model within a full thermodynamic framework.
Solve inhomogeneous boundary value problems of importance.
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