Presentation

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


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).


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


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


-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

)


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

)


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


Applied load to the Element


Step 1 Large Deformation on the single element


Step 2 Constraining the element to retain its temporary shape


Step 3 Removing load – Small amount of strain recovery


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.

THANK YOU

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