modeling of cnt based composites: numerical issues n. chandra and c. shet

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Modeling of CNT based composites: Numerical Issues

N. Chandra and C. Shet

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

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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.

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

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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 4k u u

dz 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

4k

dE

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.

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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

Modified Shear lag Model

sfs

Td 4 4 4k u u

dz 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

StrainS

tres

s0 0.001 0.002 0.003

0

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

4k

dE

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

Str

ess

(MP

a)

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)

SiC

6061-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 comparisonExperimental [1] Young’s modulus is 105 GPa

and failure strength is around 515 MPa

Ec 115 104.4

1540 522

(GPa)

FailureStrength(MPa)

Variable Original Modified

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)

Lon

gitu

dina

lstr

ess

inth

efi

ber

(MP

a)

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

300

600

900

1200

1500

1800

2100

2400

2700

3000

3300

3600

FEM Simulation

Analytical Solution

e

e

e

rz

z=0 z=1E-08 m

Position along length of the fiber (m)

Lon

gitu

dina

lstr

ess

inth

efi

ber

(MP

a)

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

250

500

750

1000

1250

1500

1750

2000

e

e

e

FEM Simulation

Analytical 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


She

arst

ress

inth

efi

ber

(MP

a)

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

10

20

30

40

50

60

70

80

90

100

FEM Simulation

Analytical Solution

e

e

e

rz

z=0 z=1E-08 m


She

arst

ress

inth

efi

ber

(MP

a)

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

2

4

6

8

10

12

14

16

FEM Simulation

Analytical Solution

e

e

e

rz

z=0 z=1E-08 m

Interface strength = 5000 MPa Interface strength = 50 MPa

Critical Bond Length

2 2e i f o

ce 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 interface

d ,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

Cri

tical

Bon

dL

engt

h(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

200

600

1000

5000

200

600

1000

5000

}}

Lengths of

Tubular

Fibers in A

Lengths of Solid

Cylindrical

fibers 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

Cri

tical

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 of

Tubular

Fibers in A

Lengths of Solid

Cylindrical

Fibers in A

max1

Bond length limitfor fibers of length 5000 A

o


o


o

o

o

Variation of Critical Bond Length with interface property

interface strength is 50MPa

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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 ofthematrixEm (in GPa)

0.02 0.03 0.05

Ec(elastic)/Em Ec(elastic)/EmEc(elastic)/Em

1.18 1.28 1.46

Interfacestrength

Tmax (in MPa)

2.46 3.17 4.61

4.98 6.99 10.96

1.05 1.07 1.13

1.5 1.74 2.24

2.38 3.07 4.45

0.99 0.986 0.98

1.05 1.08 1.13

1.18 1.27 1.45

Volumefraction

3.5

10

70

50

500

5000

50

500

5000

50

500

5000

50

500

5000

200

0.984 0.977 0.96

1.005 1.009 1.015

1.053 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

Effect of interface strength on strength of Composites

0.02 0.03 0.05

Volumefraction

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

Str

ain

0 0.025 0.05 0.075 0.10

400

800

1200

1600

2000

2400

2800

3200

3600

Interface strength = 5000 MPa


Interface strength= 50 MPa

Ec/Em = 10.96

Ec/Em = 4.61

Ec/Em = 1.46

Fiber volume fraction = 0.02

Strain

Str

ess

0 0.02 0.04 0.06 0.08 0.10

200

400

600

800

1000

1200

1400

1600



Interface strength= 50 MPa

Ec/Em = 4.97

Ec/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.

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 100000

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

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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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.

modeling of cnt based composites: numerical issues n. chandra and c. shet

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