Constitutive modeling of viscoelastic behavior of CNT/Polymer composites

K. Yazdchi1, M. Salehi2

1- Multi scale Mechanics (MSM), Faculty of Engineering Technology, University of Twente, Enschede, The Netherlands

k.yazdchi@utwente.nl2- Mechanical engineering Department, Amirkabir University of

Technology, Tehran, Iran

Presentation Outline

• Geometric Structure, Mechanical Properties

and Applications of SWCNTs

• Micromechanical analysis

• Predicting mechanical properties, using

equivalent continuum model (ECM)

• Numerical results

• Conclusions

Geometric Structure of SWCNTs

Source: http://www.photon.t.u-tokyo.ac.jp/~maruyama/agallery/agallery.html

Geometric Structure of SWCNTs

21 manaCh

22

1

2

3sin

nmnm

m

O

C

ArmchairZigzag

Wrapping (10,0) SWNT (Zigzag)

Ch = (10,0)

Wrapping (10,10) SWNT (Armchair)

Ch = (10,10)

Wrapping (10,5) SWNT (Chiral)

Ch = (10,5)

Mechanical Properties of CNTs

High Elastic Modulus (1 TPa)

Strength 100 times greater than steel (up to 50

GPa) at one sixth the Weight

High Strain to Failure (10%-30%)

High Electrical and Thermal Conductivity

High Aspect Ratio (1000)

Excellent Resilience and Toughness

Excellent Optical and Transport Properties

Low Density (1.3 g/cm^3)

CNTs Applications

Reinforcement Elements

Aerospace, Automobile, Medicine, or

Chemical Industry

Sensors and Actuators

Space Elevator

CNT Nano-Gear and Puncher

CNT Transistor

Defect and Junction Devices

Micromechanical analysis

Representative Volume Element (RVE)

Macroscale Microscale

Zoom

Inclusions Voids

Micromechanical analysis (Modeling Procedures)

Step 1

Step 2

Step 3

Homogenization scheme

Micromechanical analysis (Stress & Strain Averages)

,1w

wdVx

V

ww

dVxV

1

Inclusions

Matrix

00

0

11

1

01

11

wwV

w

V

www

dVxdVxV

dVxV

0101 wwwvv

00

0

11

1

01

11

wwV

w

V

www

dVxdVxV

dVxV

01

01 wwwvv

Micromechanical analysis (Homogenized elastic operator)

0101

:1::1: 01110111 w

el

w

el

w

el

w

el

wCvCvCvCv

Assume that each phase of this RVE obeys Hooke’s law:

1

1

1

:0 v

Avww

w

On the other hand:

w

el

w

elelel

wCACCvC :::0110

ACCvCC elelelel:0110 ?

Micromechanical analysis (Voigt Assumption)

01 www

elelel

w

elel

wCvCvCCvCv 01110111 1:1

4IAVoigt

0 0Upper Bound

Micromechanical analysis (Reuss Assumption)

01 www

0 0

01

11 1:wwww

el

wvvD

01

:1: 0111 w

el

w

el

wDvDv

elelelDvDvD 0111 1

11

01

1

11 1

elelelCvCvC

Lower Bound

Micromechanical analysis (Mori Tanaka (M-T) scheme)

• The most popular

• For composites with moderate volume fractions of inclusions (25% - 30%)

• Takes into account the interaction between inclusions

Heterogeneous RVE

Step 1

Step 2

Associated Isolated Inclusion Medium

0 //0

//0

• For composites with transversely isotropic, spheroidal inclusions, unidirectional reinforcements

Equivalent Homogeneous Medium

Micromechanical analysis (Viscoelasticity )

Dynamic Correspondence Principle (DCP):

Elastic Solution Viscoelastic Solution

ssEs^^^

E

Time Domain Frequency Domain

LCT

Inverse LCT iws

Numerical Results• Straight NTs (Effects of Waviness is ignored)

• Perfectly aligned or completely randomly oriented

nanotubes

• Matrix is linearly viscoelastic and isotropic and

effective continuum fiber is elastic and transversely

isotropic

• Perfect bounding between NT and polymer

• Assume a SWCNT, the non-bulk local polymer around

the NT, and the NT/polymer interface layer collectively as

an effective continuum fiber

• Mechanical properties of NT and polymer are

independent from temperature

Numerical Results (Analytical Formulation)

,2

,2

,2

,

,

,

31

^

44

^

31

^

23

^

23

^

22

^

23

^

66

^

23

^

12

^

44

^

12

^

33

^

22

^

22

^

23

^

11

^

12

^

33

^

33

^

23

^

22

^

22

^

11

^

12

^

22

^

33

^

12

^

22

^

12

^

11

^

11

^

11

^

L

LLL

L

LLL

LLL

LLL

Transversely isotropic

^^^^^^

2,2,,,2 pmnlkL

,,,

,,

12

^^

23

^^

11

^^

12

^^

23

^^

GpGmLn

LlKk

Numerical Results (Modeling the interphase region)

Bulk polymer

Interphase

CNT

R

05.0

fr

t

1

fr

t

Carbon fibers

Carbon Nanotubes!!

Numerical Results (Modeling the interphase region)

Multiscale modeling

Numerical Results (Completely randomly oriented nanotubes)

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

0 5 10

t (hour)

1/E

(1/

GP

a)

Aspect ratio = 5

Aspect ratio = 50

Aspect ratio = 5000

0.2

0.4

0.6

0.8

1

1.2

1.4

0 5 10

t (hour)

1/G

(1/

GP

a)

CNT volumefraction = 1%

CNT volumefraction = 5%

CNT volumefraction = 10%

Isotropic composites

^^

,GK

44

^

11

^

,LLOR

Numerical Results (Perfectly aligned)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 2 4 6 8 10

t (hour)

Cre

ep C

omp

lian

ce (

1/G

Pa)

M22, CNT Volumefraction = 1%

M22, CNT Volumefraction = 5%

M22, CNT Volumefraction = 10%

k23, CNT Volumefraction = 1%

k23, CNT Volumefraction = 5%

k23, CNT Volumefraction = 10%

M44, CNT Volumefraction = 1%

M44, CNT Volumefraction = 5%

M44, CNT Volumefraction = 10%

Transversely isotropic composites

Numerical Results (Perfectly aligned)

Transversely isotropic composites

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0 2 4 6 8 10

t (hour)

Cre

ep C

ompl

ianc

e (1

/GP

a)

M11, CNT Volumefraction = 1%

M11, CNT Volumefraction = 5%

M11, CNT Volumefraction = 10%

Numerical Results (Perfectly aligned)

Transversely isotropic composites

0.10.150.2

0.250.3

0.350.4

0.450.5

0.55

0 2 4 6 8 10

t (hour)

Cre

ep C

ompl

ianc

e (1

/GP

a) M22, Aspect ratio = 5

M22, Aspect ratio = 50

M22, Aspect ratio = 500

k23, Aspect ratio = 5

k23, Aspect ratio = 50

k23, Aspect ratio = 500

Numerical Results (Perfectly aligned)

Transversely isotropic composites

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 2 4 6 8 10

t (hour)

Cre

ep C

ompl

ianc

e (1

/GP

a) M11, Aspect ratio = 5

M11, Aspect ratio = 50

M11, Aspect ratio = 500

M44, Aspect ratio = 5

M44, Aspect ratio = 50

M44, Aspect ratio = 500

Conclusions

The parameters which affect the mechanical properties of NTRPCs are NT aspect ratio, volume fraction and orientation.

For composites having unidirectionally aligned nanotubes (transversely isotropic), numerical results indicate that the increase of the nanotube aspect ratio and volume fraction significantly enhances their axial creep resistance but has insignificant influences on their transverse, shear and plane strain bulk creep compliances.

The effect of the nanotube orientation on the shear compliances is negligibly small.

Conclusions

For composites with aligned or randomly oriented nanotubes, all the compliances are found to decrease monotonically with the increase of the nanotube volume fraction.

For composites having randomly oriented NTs (isotropic) with increasing the aspect ratio or NT volume fraction, the axial and shear creep compliances will decreases also the effect of aspect ratio in comparison with volume fraction is negligible.

The model proposed in the foregoing is simple and very economical to employ, particularly in viscoelastic behaviour of nanocomposites, compared with other methods.

Suggestions

The effects of NT waviness and agglomeration and also temperature on the viscoelastic behavior of NTRPCs.

Find new methods in modeling the interphase region (such as MD, etc).

The effect of anisotropic properties of CNTs, 3D modelling, end caps and any possible relative motion between individual shells or tubes in a MWNT and an NT bundle Voids and Defects will be studied in the future.

Use other Micromechanical models and compare the results with experimental data.

Thank you for your attention

Any Questions?