1. 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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2. 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
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3. Shear Lag Model * Prelude 1 Disadvantages *Original model developed by
Fig. Shear lag model for aligned short fiber composites. (a) representative short fiber (b) unit cell for analysis
(a)
(b)
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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4. 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.
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5. 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

6. 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
Variation of stress-strain response in the elastic limit with respect to parameter b
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7. Comparison with Experimental Result
The average stress in fiber and matrix far a applied strain e is given by
Fig. A typical traction-displacement curve used for interface between SiC fiber and 6061-Al matrix
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

8. Result comparison Comparison (contd.)
The constitutive behavior of 6061-T6 Al matrix [21] can be represented by
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.
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
Experimental [1] Young’s modulus is 105 GPa
and failure strength is around 515 MPa Ec
104.4
(GPa)
Failure
Strength
(MPa)
Variable Original Modified
[1]         Dunn, M.L. and Ledbetter, H., Elastic-plastic behavior of textured short-fiber composites, Acta mater. 1997; 45(8):

9. Comparison with Numerical Results
FEAModel
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
The CNT is modeled as a hollow
tube with a length of 200 , outer
radius of and thickness of 0.4 .
CNT modeled using 1596
axi-symmetric elements.
Matrix modeled using 11379
Interface modeled using node
axisymmetric CZ elements with
zero thickness

10. Longitudinal Stress in fiber at different strain level
Interface strength = 5000 MPa Interface strength = 50 MPa
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11. Shear Stress in fiber at different strain level
Interface strength = 5000 MPa Interface strength = 50 MPa
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12. Critical Bond Length l/2
Table 1. Critical bond lengths for short fibers of length 200 and for different interface strengths and interface displacement parameter dmax1 value

13. Variation of Critical Bond Length with interface property
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 exceeds semi-fiber length for some values dmax1 tending the fiber ineffective in transferring the load
interface strength is 50MPa

14. Effect of interface strength on stiffness of Composites
Table : Variation of Young’s modulus of the composite with matrix young’s modulus, volume fraction and interface strength
Young’s Modulus (stiffness) of the composite not only increases with matrix stiffness and fiber volume fraction, but also with interface strength
Note 1. For a given volume fraction of fiber increase in young,s modulus, yield strength with increasing interface strength
2. For pure matrix material drop in modulus with on set of plasticity is 98%
3. For CNT based composite with weak interface drop in stiffness is 80% for 2% VF, 68% for 3% VF. 47 % for 5% VF
4. For CNT based composite with strong interface drop in stiffness is 8 % for 38%
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15. Effect of interface strength on strength of Composites
Fiber volume fraction = 0.02
Fiber volume fraction = 0.05
Table Yield strength (in MPa) of composites for different volume fraction and interface strength
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

16. Effect of interface displacement parameter dmax1
on strength and stiffness
Fig. Variation of stiffness of composite material with interface displacement parameter dmax1 for different interface strengths.
Fig. Variation of yield strength of the 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.

17. 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 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.

18. Conclusion
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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