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

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

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. AMML

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

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

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 AMML

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

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 1540 522 (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):3327-3340

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 6.98 and thickness of 0.4 . CNT modeled using 1596 axi-symmetric elements. Matrix modeled using 11379 Interface modeled using 399 4 node axisymmetric CZ elements with zero thickness

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

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

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 0.15.

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

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% AMML

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

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.

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.

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