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Modeling of CNT based composites

N. Chandra and C. Shet

FAMU-FSU College of Engineering, Florida State University, Tallahassee, FL 32310

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Answer: Currently NO!!!Researcher Matrix Vol% CNT

Exptl CalculationSeriesParallel

Schaddler ‘98 Epoxy 2.85 (tension) 1.13 9.60 1.03

Epoxy 2.85 (comp) 1.4 9.60 1.03

Andrews ‘99 Petroleumpitch

0.33 1.20 9.09 1.003

1.62 2.29 12.46 1.016

Gong ‘00 Epoxy 0.57 1.12 4.98 1.0057

0.57 1.25 4.98 1.0057

Qian ‘00

(With surf actant)

Polystyrene 0.49 1.24 4.9151 1.0049

Ma’00 PET 3.6 1.4 4.564 1.037

Andrews’02 Polystyrene 2.5 1.22 14.86 1.035.0 1.28 28.73 1.0510.0 1.67 56.46 1.1115.0 2.06 84.18 1.1825.0 2.50 139.64 1.33

PPA 0.50 1.17 5.16 1.011.50 1.33 13.49 1.022.50 1.50 21.81 1.035.00 2.50 42.62 1.05

EC EM

EC EM

C f f m mE V E V E

Parallel modelUpper Bound

1 f m

C f m

V VE E E

Series modelLower Bound

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Inter

face

Properties affected

Fatigue/Fracture Thermal/electronic/magnetic

Factors affecting interfacial properties

Trans. & long.Stiffness/strength

Interfacial chemistry Mechanical effects

Origin: Chemical reaction during thermal-mechanical Processing and service conditions, e.g. Aging, Coatings, Exposures at high temp..

Issues: Chemistry and architecture effects on mechanical properties.

Approach: Analyze the effect of size of reaction zone and chemical bond strength (e.g. SCS-6/Ti matrix and SCS-6/Ti matrix )

Residual stress

Origin: CTE mismatch between fiber and matrix.

Issues: Significantly affects the state of stress at interface and hence fracture process

Approach: Isolate the effects of residual stress state by plastic straining of specimen; and validate with numerical models.

Asperities

Origin: Surface irregularities inherent in the interfaceIssues: Affects interface fracture process through mechanical loading and frictionApproach: Incorporate roughness effects in the interface model; Study effect of generating surface roughness using: Sinusoidal functions and fractal approach; Use push-back test data and measured roughness profile of push-out fibers for the model.

Metal/ceramic/polymer

CNTs

H. Li and N. Chandra, International Journal of Plasticity, 19, 849-882, (2003).

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Functionalized Nanotubes• Change in hybridization (SP2 to SP3)• Experimental reports of different

chemical attachments• Application in composites, medicine,

sensors • Functionalized CNT are possibly

fibers in composites

108o120o

Graphite Diamond

How do fiber properties differ with chemical modification of surface?

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Stress(Gpa)

0 0.01 0.02 0.03 0.04

5

10

15

20

25

30

35

(10,10) CNT 0.84 T Pa(10,10) CNT with vinyl 0.92 T Pa(10,10) CNT with butyl 1.03 T Pa

Functionalized nanotubes• Increase in stiffness observed by functionalizing

Stiffness increase is more for higher number of chemical attachments

Stiffness increase higher for longer chemical attachments

Volume for Stress Calculation

Vinyl and ButylHydrocarbonsT=77K and 3000KLutsko stress

N. Chandra, S. Namilae, Physical Review B, 69 (9), 09141, (2004)

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Atom Number

Radius

100 200 300 400

6.6

6.7

6.8

6.9

7

7.1

7.2

7.3

with vinyl attachmentswithout attachments

• Increased radius of curvature at the attachment because of change in hybridization

• Radius of curvature lowered in adjoining area

Sp3 Hybridization here

Higher stress atthe location ofattachment

Stress (GPa) Stress (GPa)

Stress (GPa) Stress (GPa) Stress (GPa)

(a) (b) (c)

(d) (e) (f)

Radius variation

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Evolution of defects in functionalized CNT

• Defects Evolve at much lower strain of 6.5 % in CNT with chemical attachments

Onset of plastic deformation at lower strain. Reduced fracture strain

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Different Fracture Mechanisms

Fracture Behavior Different

• Fracture happens by formation of defects, coalescence of defects and final separation of damaged region in defect free CNT

• In Functionalized CNT it happens in a brittle manner by breaking of bonds

S. Namilae, N. Chandra, Chemical Physics Letters, 387, 4-6, 247-252, (2004)

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Interfacial shear

Displacement (A)

Reaction(eV/A)

5 10 15-1

0

1

2

3

4

5

6

7

8

Typical interface shear force pattern. Note zero force afterFailure (separation of chemical attachment)

After Failure

Max load

250,000 steps

Interfacial shear measured as reaction force of fixed atoms

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Matrix

Debonding and Rebonding

Energy for debonding of chemical attachment 3eV Strain energy in force-displacement plot 20 ± 4 eV

Energy increase due to debonding-rebonding

Matrix

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T

T

Interfaces are modeled as cohesive zones using a potential function

( , ) ( , , , )n t n t n tf ,n t are work of normal and

tangential separation

are normal and tangential displacement jump ,n t

The interfacial tractions aregiven by

,n tn t

n t

T T

Interfacial traction-displacement relationship are obtained using molecular dynamics simulation based on EAM functions

1.X.P. Xu and A Needleman, Modelling Simul. Mater. Sci. Eng.I (1993) 111-1322.N. Chandra and P.Dang, J of Mater. Sci., 34 (1999) 655-666

Grain boundaryinterface

Mechanics of Interfaces in CompositesAtomic Simulations

Reference

Formulations

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Debonding and Rebonding of Interfaces

displacement (A)

Force(eV/A)

0 5 10 15-2

-1

0

1

2

3

4

5

6

7

8 RebondingDebonding

Failure

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Prelude 2 Cohesive Zone Model

CZM is represented by traction-displacement jump curves to model the separating surfaces

AdvantagesCZM can create new surfaces. Maintains continuity conditions mathematically, despite the physical separation. CZM represents physics of the fracture process at the atomic scale.Eliminates singularity of stress and limits it to the cohesive strength of the the material.It is an ideal framework to model strength, stiffness and failure in an integrated manner.

T or T f , , (or )n t max max n tT Tt nStiffness of cohesive zone k = or

t n

N. Chandra et.al, Int. J. Solids Structures, 37, 461-484, (2002).

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Finite element simulation: Composite stiffness

Volume % CNT

ElasticModulus(GPa)

0 5 10 15 20

5

10

15

20

25

30

35

Interface Strength = 5 MPa

Interface Strength = 5 GPa

Interface Strength = 50 MPa

Interface Strength = 500 MPa

Perfect Interface

Matrix Elastic Modulus (GPa)

Com

positeElasticModulus(GPa)

0 5 100

5

10

15

20

25

30

35

40

Interface strength= 50 MPa

Interface strength= 500 MPa

Interface strength= 5 GPa

Interface strength= 5 MPa

Perfect Interface

Pure Matrix

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Fig. Shear lag model for aligned short fiber composites. (a) representative short fiber (b) unit cell for analysis

e

e Fiber

Matrix

l

dD

r

z

ee

(a)

(b)

Shear Lag Model *Prelude 1

sfs

Td 4 4 4 k u udz d d h d

The governing DE

Whose solution is given by

Where

Disadvantages• The interface stiffness is dependent on Young’s modulus of matrix and fiber, hence it may not represent exact interface property.•k remains invariant with deformation• Cannot model imperfect interfaces

f f 1 2E C cosh( z)+C sinh( z) e

f

4kdE

m2GInterface property k =d ln(D / d)

*Original model developed by Cox [1] and Kelly [2]

[1]         Cox, H.L., J. Appl. Phys. 1952; Vol. 3: p. 72 [2]         Kelly, A., Strong Soilids, 2nd Ed., Oxford University Press, 1973, Chap. 5.

Modified Shear lag Model

sfs

Td 4 4 4 k u udz d d h d

f f 1 2E C cosh( z)+C sinh( z) e

e

2 2f fe i

4d4k kThen ( solid fibers) (hollow fiber)dE Ed d

The governing DE

If the interface between fiber and matrix is represented by cohesive zone, then

s f m

max max

T k u v ,

where interface stiffness k k(T , )

Evaluating constants by using boundary conditions, stresses in fiber is given by

o

f off f f

f

1 cosh( z)E E d E 1 , 1 1. - cosh( z)

l l Ecosh lcosh2 2

e e e

e

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Comparison between Original and Modified Shear Lag Model

StrainSt

ress

0 0.001 0.002 0.0030

50

100

150

200

250

300

350

Original shear lag model

CZM based shear lag model

200 k'

16.7 k'

5 k'

1.11 k'

max

ct

k' =

Variation of stress-strain response in the elastic limit with respect to parameter

• The parameter defined by defines the interface strength in two models through variable k.• In original model

• In modified model interface stiffness is given by slope of traction-displacement curve given by

• In original model k is invariant with loading and it cannot be varied•In modified model k can be varied to represent a range of values from perfect to zero bonding

f

4kdE

m2Gk =d ln(D / d)

T Tt nk = or t n

Comparison with Experimental Result

o

ff f m m

l2 tanh 2E E 1 , E ,

l2

e

e e

The average stress in fiber and matrix far a applied strain e is given by

Then by rule of mixture the stress in composites can be obtained as

c f m f f(1 V ) V

max

max c

T

n

Fig. A typical traction-displacement curve used for interface between SiC fiber and 6061-Al matrix

For SiC-6061-T6-Al composite interface is modeled by CZM model given by

maxmax , ( )n, ( ) n maxn maxmaxmax

T , ki i 1n N Nmaxk ,( max) i k ,( max)max i n max i n

c c ci 0 i 1

where , andn max

area undet T- curve as 2.224 max c

With N=5, and k0 = 1, k1 = 10, k2 = -36, k3 = 72, k4 = -59, k5 = 12. Takingmax = 1.8y, where y is yield stress of matrix and max =0.06 c

Fig.. Comparison of experimental [1] stress-strain curve for Sic/6061-T6-Al composite with stress-strain curves predicted from original shear lag model and CZM based Shear lag model.

Strain

Stre

ss(M

Pa)

0 0.005 0.01 0.015 0.02 0.025 0.030

200

400

600

800

1000

1200

1400

1600

1800 SiC/6061-T6 Al (Experiment)SiC/6061-T6 Al (Predicted-CZM based shear lag model)SiC/6061-T6 Al (Predicted-original shear lag model)SiC6061-T6 Al

Fiber

Original shear lag model

Matrix

New model (CZM-Shear lag)

[1]         Dunn, M.L. and Ledbetter, H., Elastic-plastic behavior of textured short-fiber composites, Acta mater. 1997; 45(8):3327-3340

The constitutive behavior of 6061-T6 Al matrix [21] can be represented by Comparison (contd.)

ny ph e

yield stress =250 MPa, and hardening parameters h = 173 MPa, n = 0.46. Young’s modulus of matrix is 76.4 GPa. Young’s modulus of SiC fiber is Ef of 423 GPa

Result comparison

Ec 115 104.4 105

1540 522 515

(GPa)

FailureStrength(MPa)

Variable Original Modified Experiment

FEAModel

•The CNT is modeled as a hollow tube with a length of 200 , outer radius of 6.98 and thickness of 0.4 . • CNT modeled using 1596 axi-symmetric elements.• Matrix modeled using 11379 axi-symmetric elements.•Interface modeled using 399 4 node axisymmetric CZ elements with zero thickness

Comparison with Numerical Results

Fig. (a) Finite element mesh of a quarter portion of unit model (b) a enlarged portion of the mesh near the curved cap of CNT

tt=1max1 m ax2

m ax

T t

nn=1max

max

T n(a) (b)

A

B C

DA1

B1

C1

max , ( )t max1max1

T ( max 2)t max max11max , ( )t max 21 max 2

max n, ( )maxmaxTn 1max

, ( max)n1 max, n t

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Position along the length of fiber (m)

Long

itudi

nals

tress

inth

efib

er(M

Pa)

2E-09 4E-09 6E-09 8E-09

300

600

900

1200

1500

1800

2100

2400

2700

3000

3300

3600

FEM SimulationAnalytical Solution

e

e

e

rz

z=0 z=1E-08 m

Position along length of the fiber (m)

Long

itudi

nals

tress

inth

efib

er(M

Pa)

0 2.5E-09 5E-09 7.5E-090

250

500

750

1000

1250

1500

1750

2000

e

e

e

FEM SimulationAnalytical Solution

rz

z=0 z=1E-08 m

Longitudinal Stress in fiber at different strain level

Interface strength = 5000 MPa Interface strength = 50 MPa

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Shear Stress in fiber at different strain level

Position along the length of fiber (m)

Shea

rstre

ssin

the

fiber

(MPa

)

2E-09 4E-09 6E-09 8E-09

10

20

30

40

50

60

70

80

90

100

FEM SimulationAnalytical Solution

e

e

e

rz

z=0 z=1E-08 m

Position along the length of fiber (m)

Shea

rstre

ssin

the

fiber

(MPa

)

2E-09 4E-09 6E-09 8E-09

2

4

6

8

10

12

14

16

FEM SimulationAnalytical Solution

e

e

e

rz

z=0 z=1E-08 m

Interface strength = 5000 MPa Interface strength = 50 MPa

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Table : Variation of Young’s modulus of the composite with matrix young’s modulus, volume fraction and interface strength

Young’sModulus ofthe matrixEm (in GPa)

0.02 0.03 0.05

Ec(elastic)/Em Ec(elastic)/EmEc(elastic)/Em1.18 1.28 1.46

Interfacestrength

Tmax (in MPa)

2.46 3.17 4.614.98 6.99 10.961.05 1.07 1.131.5 1.74 2.242.38 3.07 4.450.99 0.986 0.981.05 1.08 1.131.18 1.27 1.45

Volumefraction

3.5

10

70

50500

500050500500050

5005000

505005000

2000.984 0.977 0.961.005 1.009 1.0151.053 1.075 1.13

Effect of interface strength on stiffness of Composites

Young’s Modulus (stiffness) of the composite not only increases with matrix stiffness and fiber volume fraction, but also with interface strength

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Conclusion

1. The critical bond length or ineffective fiber length is affected by interface strength. Lower the interface strength higher is the ineffective length.

2. In addition to volume fraction and matrix stiffness, interface property, length and diameter of the fiber also affects elastic modulus of composites.

3. Stiffness and yield strength of the composite increases with increase in interface strength.

4. In order to exploit the superior properties of the fiber in developing super strong composites, interfaces need to be engineered to have higher interface strength.

Critical Bond Length 2 2

e i f oc

e f

f oc

f

d dl (hollow fiber)

d 2 (max)

(max)dl (solid fiber)2 (max)

o

ff f y

z 0

f

e i

1 cosh( z)E

(max) E 1lcosh 2

(max) Shear strength of the interfaced ,d are external and internal diameters respectively

e

e

o

f

f

lc

r

Matrix

Fiber

Interface Shear Traction Variation

Longitudinal fiber stress Variation

Bond Length

z

l/2

Table 1. Critical bond lengths for short fibers of length 200 and for different interface strengths and interface displacement parameter max1 value 0.15.

Interface strengthTmax in MPa

Critical bond length lc in Ao

5000

500

50

3.23

26.4

74.7

Hollow cylindrical fiber Solid cylindrical fiber

24.4

73.08

91.4

Criti

calB

ond

Leng

th(A

)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

5

10

15

20

25

30

max1

200600

1000500020060010005000

}}

Lengths ofTubularFibers in A

Lengths of SolidCylindricalfibers in A

oo

o

• Critical bond length varies with interface property (Cohesive zone parameters (max , max1)•When the external diameter of a solid fiber is the same as that of a hollow fiber, then, for any given length the load carried by solid fiber is more than that of hollow fiber. Thus, it requires a longer critical bond length to transfer the load •At higher max1 the longitudinal fiber stress when the matrix begins to yield is lower, hence critical bond length reduces•For solid cylindrical fibers, at low interface strength of 50 MPa, when the fiber length is 600

and above, the critical bond length on each end of the fiber exceeds semi-fiber length for some values max1 tending the fiber ineffective in transferring the load

interface strength is 5000MPa

Crit

ical

bond

leng

th(A

)

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

500

1000

1500

2000

2500

o

200

600

1000

5000

200

600

1000

5000

}}

Lengths ofTubularFibers in A

Lengths of SolidCylindricalFibers in A

max1

Bond length limitfor fibers of length 5000 A

o

Bond length limitfor fibers of length 1000 A

o

Bond length limitfor fibers of length 600 A

o

o

o

Variation of Critical Bond Length with interface property

interface strength is 50MPa

ct

max1 =0.1 0.2 0.4 0.6 0.8

t

Tt

max2 = 0.9

Effect of interface strength on strength of Composites

0.02 0.03 0.05Volume

fraction

50

500

5000

InterfacesstrengthTmax (in MPa)

107

340

809

87 94

180

367

234

515

Table Yield strength (in MPa) of composites for different volume fraction and interface strength

Strain

Stra

in

0 0.025 0.05 0.075 0.10

400

800

1200

1600

2000

2400

2800

3200

3600

Interface strength = 5000 MPa

Interface strength = 500 MPa

Interface strength= 50 MPa

Ec/Em = 10.96

Ec/Em = 4.61

Ec/Em = 1.46

Fiber volume fraction = 0.02

Strain

Stre

ss

0 0.02 0.04 0.06 0.08 0.10

200

400

600

800

1000

1200

1400

1600

Interface strength = 5000 MPa

Interface strength = 500 MPa

Interface strength= 50 MPa

Ec/Em = 4.97Ec/Em = 2.46

Ec/Em = 1.18

Fiber volume fraction = 0.05

•Yield strength (when matrix yields) of the composite increases with fiber volume fraction (and matrix stiffness) but also with interface strength•With higher interface strength hardening modulus and post yield strength increases considerably

Effect of interface displacement parameter max1 on strength and stiffness

E/E

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

1

2

3

4

5

6

7

8

9

10

11

cm

max1

E = Ec m

T = 5000MPa

T = 500 MPa

T = 50 MPa

max

max

max

length = 200 E-10 mDiameter = 6.98E-10mVolume fraction = 0.05

Fig. Variation of stiffness of composite material with interface displacement parameter max1 for different interface strengths.

(com

posi

te)/

(mat

rix)

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

1

2

3

4

5

6

7

8

9

10

11

yy

max1

y y

T = 5000MPa

T = 500 MPa

T = 50 MPa

max

max

max

length = 200 E-10 mDiameter = 6.98E-10mVolume fraction = 0.05

(composite) (matrix)

Fig. Variation of yield strength of the composite material with interface displacement parameter max1 for different interface strengths.

• As the slope of T- curve decreases (with increase in max1), the overall interface property is weakened and hence the stiffness and strength reduces with increasing values of max1. •When the interface strength is 50 MPa and fiber length is small the young’s modulus and yield strength of the composite material reaches a limiting value of that of matrix material.

ct

max1 =0.1 0.2 0.4 0.6 0.8

t

Tt

max2 = 0.9

Effect of length of the fiber on strength and stiffness

Length (X 1.0 E-10 m)

(com

posi

te)/

(mat

rix)

0 2500 5000 7500 100000123456789

10111213

141516

yy

max1T = 5000MPa

T = 500 MPa

T = 50 MPa

max

max

max

Diameter = 6.98E-10mVolume fraction = 0.05

Fig. Variation of yield strength of the composite material with different fiber lengths and different interface strengths

Length ( X 1.0 E-10 m)

E/E

0 2500 5000 7500 100000

2

4

6

8

10

12

14

16

cm

T = 5000MPa

T = 500 MPa

T = 50 MPa

max

max

max

Diameter = 6.98E-10mVolume fraction = 0.05

max1

Fig. Variation of Young’s modulus of the composite material with different fiber lengths and for different interface strengths

• For a given volume fraction the composite material can attain optimum values for mechanical properties irrespective of interface strength.• For composites with stronger interface the optimum possible values can be obtained with smaller fiber length• With low interface strength longer fiber lengths are required to obtain higher composite properties. During processing it is difficult to maintain longer CNT fiber straigth.

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Objective•To develop an analytical model that can predict the mechanical properties of short-fiber composites with imperfect interfaces.•To study the effect of interface bond strength on critical bond length lc •To study the effect of bond strength on mechanical properties of composites.Approach

To model the interface as cohesive zones, which facilitates to introduce a range of interface properties varying from zero binding to perfect binding