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
Factors affecting interfacial properties
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.
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.
Trans. & long.
H. Li and N. Chandra, International Journal of Plasticity, 19, 849-882, (2003).
Vinyl and Butyl
T=77K and 3000K
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)
Sp3 Hybridization here
Onset of plastic deformation at lower strain. Reduced fracture strain
Fracture Behavior Different
S. Namilae, N. Chandra, Chemical Physics Letters, 387, 4-6, 247-252, (2004)
Interfacial shear measured as reaction force of fixed atoms
Typical interface shear force pattern. Note zero force after
Failure (separation of chemical attachment)
Mechanics of Interfaces in Composites
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
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
Cohesive Zone Model
N. Chandra et.al, Int. J. Solids Structures, 37, 461-484, (2002).
Shear Lag Model *
Fig. Shear lag model for aligned short fiber composites. (a) representative short fiber (b) unit cell for analysis
*Original model developed by
Cox  and Kelly 
Cox, H.L., J. Appl. Phys. 1952; Vol. 3: p. 72
Kelly, A., Strong Soilids, 2nd Ed., Oxford University Press, 1973, Chap. 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
Variation of stress-strain response in the elastic limit with respect to parameter b
Comparison between Original and Modified Shear Lag Model
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
Variable Original Modified Experiment
Fig.. Comparison of experimental  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.
The constitutive behavior of 6061-T6 Al matrix  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
Dunn, M.L. and Ledbetter, H., Elastic-plastic behavior of textured short-fiber composites, Acta mater. 1997; 45(8):3327-3340
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
Longitudinal Stress in fiber at different strain level
Interface strength = 5000 MPa Interface strength = 50 MPa
Shear Stress in fiber at different strain level
Interface strength = 5000 MPa Interface strength = 50 MPa
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
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.
interface strength is 5000MPa
Variation of Critical Bond Length
with interface property
interface strength is 50MPa
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
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.
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
To model the interface as cohesive zones, which facilitates to introduce a range of interface properties varying from zero binding to perfect binding