Modeling of CNT based composites: Numerical Issues N. Chandra and C. Shet

1. Modeling of CNT based composites: Numerical Issues N. Chandra and C. Shet FAMU-FSU College of Engineering, Florida State University, Tallahassee, FL 32310 AMML

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 AMML

3. Shear Lag Model * (a) (b) 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. AMML

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. AMML

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. 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 AMML

7. 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

8. Variable Original Modified Ec • 104.4 • 1540 522 (GPa) Failure Strength (MPa) 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. 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 Experimental [1] Young’s modulus is 105 GPa and failure strength is around 515 MPa [1]Dunn, M.L. and Ledbetter, H., Elastic-plastic behavior of textured short-fiber composites, Acta mater. 1997; 45(8):3327-3340

9. 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 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

10. Longitudinal Stress in fiber at different strain level Interface strength = 5000 MPa Interface strength = 50 MPa AMML

11. Shear Stress in fiber at different strain level Interface strength = 5000 MPa Interface strength = 50 MPa AMML

12. Critical Bond Length l/2 Table 1. Critical bond lengths for short fibers of length 200andfor different interface strengths and interface displacement parameter dmax1 value 0.15.

13. 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

14. 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 AMML

15. 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

16. 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.

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 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.

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. AMML