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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. Answer: Currently NO!!!. Parallel model Upper Bound. Series model Lower Bound. Factors affecting interfacial properties. Asperities .

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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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Amml 1316641

Modeling of CNT based composites

N. Chandra and C. Shet

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


Answer currently no

Answer: Currently NO!!!

Parallel model

Upper Bound

Series model

Lower Bound


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Factors affecting interfacial properties


Interfacial chemistry

Mechanical effects

Origin:Surface irregularities inherent in the interface

Issues: Affects interface fracture process through mechanical loading and friction

Approach: 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.

Residual stress

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 )

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.






Properties affected

Trans. & long.





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

Functionalized nanotubes

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

  • How do fiber properties differ with chemical modification of surface?


Functionalized nanotubes1

Functionalized nanotubes

Vinyl and Butyl


T=77K and 3000K

Lutsko stress

  • Increase in stiffness observed by functionalizing

Stiffness increase is more for higher number of chemical attachments

Stiffness increase higher for longer chemical attachments


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

Radius variation

Sp3 Hybridization here

Radius variation

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

  • Radius of curvature lowered in adjoining area


Evolution of defects in functionalized cnt

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


Different fracture mechanisms

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)

Interfacial shear

Interfacial shear

Interfacial shear measured as reaction force of fixed atoms

Max load

Typical interface shear force pattern. Note zero force after

Failure (separation of chemical attachment)

After Failure

250,000 steps


Debonding and rebonding

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


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Mechanics of Interfaces in Composites


Atomic Simulations

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

Interfaces are modeled as cohesive zones using a potential function

are work of normal and tangential separation

are normal and tangential displacement jump

The interfacial tractions are

given by

Grain boundary




1.X.P. Xu and A Needleman, Modelling Simul. Mater. Sci. Eng.I (1993) 111-132

2.N. Chandra and P.Dang, J of Mater. Sci., 34 (1999) 655-666

Debonding and rebonding of interfaces




Debonding and Rebonding of Interfaces


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Prelude 2

Cohesive Zone Model

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

  • Advantages

  • CZM 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.

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


Finite element simulation composite stiffness

Finite element simulation: Composite stiffness


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Shear Lag Model *



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

Prelude 1

  • 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

*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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Modified Shear lag Model

The governing DE

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

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

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

Comparison between Original and Modified Shear Lag Model

  • The parameter b 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


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

Comparison with Experimental Result

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

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

With N=5, and k0 = 1, k1 = 10, k2 = -36, k3 = 72, k4 = -59, k5 = 12.

Taking smax = 1.8 sy, where sy is yield stress of matrix and dmax =0.06 dc

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Variable Original Modified Experiment

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.



  • 104.4 105

  • 1540 522 515




Comparison (contd.)

The constitutive behavior of 6061-T6 Al matrix [21] can be represented by

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

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

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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

Comparison with Numerical Results


  • 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

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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

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

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

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

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

  • 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.


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Critical Bond Length


Table 1. Critical bond lengths for short fibers of length 200andfor different interface strengths and interface displacement parameter dmax1 value 0.15.

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interface strength is 5000MPa

Variation of Critical Bond Length

with interface property

  • Critical bond length varies with interface property (Cohesive zone parameters (smax, dmax1)

  • 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 dmax1 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 exceedssemi-fiber length for some values dmax1 tending the fiber ineffective in transferring the load

interface strength is 50MPa

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Fiber volume fraction = 0.02

Fiber volume fraction = 0.05

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

Effect of interface strength on strength of Composites

  • 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

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Effect of interface displacement parameter dmax1

on strength and stiffness

Fig. Variation of yield strength of the composite material with

interface displacement parameter dmax1 for different interface strengths.

Fig. Variation of stiffness of composite material with interface displacement parameter dmax1 for different interface strengths.

  • As the slope of T-d curve decreases (with increase in dmax1), the overall interface property is weakened and hence the stiffness and strength reduces with increasing values of dmax1.

  • 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.

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Effect of length of the fiber on strength and stiffness

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

Fig. Variation of yield strength of the composite material with different fiber lengths and 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 interfacethe 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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  • 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.


To 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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