Evaluation of the Mechanical Properties of Carbon

Unnati A Joshi et. al. / International Journal of Engineering Science and Technology Vol. 2(5), 2010, 1098-1107 

Evaluation of the Mechanical Properties of Carbon Nanotube Based Composites by Finite Element Analysis Unnati A Joshi*, Preeti Joshi, S. P. Harsha, Satish C Sharma Vibration & Noise Control Laboratory Mechanical & Industrial Engineering Department Indian Institute of Technology Roorkee Roorkee – 247667, Uttarakhand, India Abstract Carbon nanotubes (CNTs) possess extremely high stiffness, strength and resilience, and may provide ultimate reinforcing materials for the development of nano composites. In this paper, the effective material properties of CNT-based composites are evaluated using a square representative volume element (RVE) based on the continuum mechanics and with the finite element method (FEM). Formulas to extract the effective material constants from solutions for the square RVEs under the axial stretch load is derived based on the elasticity theory. An extended rule of mixtures, based on the strength of materials theory for estimating the effective Young’s modulus in the axial direction of the RVE, is applied for comparisons of FEM results. It has been observed that the addition of the CNTs in a matrix at volume fractions of only about 3.6%, the stiffness of the composite is increased by 33% for long CNT 10, whereas there is no much improvement in stiffness has been noticed in case of short CNTS at at E ⁄E 10. Effectiveness of composites is evaluated in terms of various dimensions like thickness, diameter and E ⁄E length of CNT. These results suggest that short CNTs in a matrix may not be as effective as long CNTs in reinforcing a composite. Keywords: Carbon nanotubes, Effective material properties, nano composites, Finite element method. I. INTRODUCTION Carbon nanotubes, discovered first by Iijima in 1991 [6], possess exceptionally high stiffness, strength and resilience, as well as superior electrical and thermal properties, which may become the ultimate reinforcing materials for the development of an entirely new class of composites. It has been demonstrated that with just 1% (by weight) of CNTs added in a matrix, the stiffness of the resulting composite can increase between 36-42% and the tensile strength by 25% [10]. The mechanical-load carrying capacities of CNTs in nanocomposites have also been demonstrated in experiments [10,1] and preliminary simulations [3,7]. All these studies show the great potentials of CNT based composites as well as the enormous challenges in the development of such nanocomposites. However, much work still need to be done before the potentials of the CNT-based composites can be fully realized in real engineering applications. Evaluating the effective material properties of such nanoscale materials is one of the challenging tasks for the development of nanocomposites. Computational approaches, based on the molecular dynamics (MD) approach (for smaller scales) and continuum mechanics approach (for larger scales), can play significant roles in the areas of characterizing CNT-based composites [2]. The discrete models, such as molecular dynamics models, are applied widely in the nanoscale research. In these MD approaches, the atoms are considered as individual particles and the forces among them are calculated using potential theories. However, these nanoscale simulations are currently limited to very small length and time scales and cannot deal with the larger length scales needed in characterizations of nano composites, due to the limitations of current computing power (e.g., a cube with an edge length of only 1 μm could contain up to 1012atoms). The continuum mechanics approach has also been applied successfully for simulating the mechanical responses of individual carbon nanotubes which are treated as beams, thin shells or solids in cylindrical shapes [14, 4, 12, and 11]. The current results using the continuum approaches have indicated that continuum mechanics can be applied to

ISSN: 0975-5462

1098

Unnati A Joshi et. al. / International Journal of Engineering Science and Technology Vol. 2(5), 2010, 1098-1107 models with dimensions of a few hundred nanometers and larger, where averaging of material properties can be done properly for CNTs. Simulation results obtained using the continuum mechanics approach should also be interpreted correctly. Attentions should be paid to overall deformations or load transfer mechanisms, rather than to local stresses, such as those at the interface between the CNTs and matrix. The modelling considerations in characterizing CNT-based composites using the continuum approach. It is proposed that the 3-D elasticity models, instead of beam or shell models, should be employed for modeling the CNTs embedded in a matrix, in order to ensure the accuracy and compatibility between the models for the CNTs and matrix. A method based on the elasticity theory for evaluating effective material properties of CNT-based composites using the representative volume elements (RVEs) (Fig. 1) is established. Formulas to extract the effective material properties from numerical solutions for the square RVEs under three loading cases are derived. Analytical results (extended rule of mixtures) based on the strength of materials theory to estimate the effective Young’s modulus in the axial direction, which can help validate the numerical solutions, are also derived for both long and short CNT cases in [7]. Numerical results using the finite element method (FEM) for the square RVEs show significant increases of the stiffness in the CNT direction of the nano composites under various combinations of the CNT and matrix material properties [7].

(a) Cylindrical RVE

(b) Square RVE

(c) Hexagonal RVE

Fig. 1: Three representative volume elements (RVE) for the analysis of CNT-based nanocomposites.

In this paper, the work initiated in [07] is extended to square RVEs (Fig. 1(b)) for the evaluations of effective material properties of the CNT-based composites. Formulas based on the elasticity theory for extracting the effective material properties from solutions of the square RVEs are derived and numerical studies using the FEM are conducted. The material properties Young’s modulus and poison’s ratio are also evaluated by changing the thickness and diameter of the CNT for long and short CNTs (with hemispherical end caps and without using hemispherical end caps) in the axial direction. II. FORMULAS FOR EXTRACTING THE EFFECTIVE MATERIAL CONSTANTS To derive the formulas for extracting the equivalent material constants, a homogenized elasticity model of the square RVE (Fig. 2) is considered. The elasticity model is filled with a single, transversely isotropic material that has five independent material constants. The four effective material constants (Young’s moduli Ex, Ez and Poisson‘s ratios νxy , νzx relating the normal stress and strain components) will be determined (see Fig. 2 for the orientation of the coordinates).

Fig. 2: A square RVE containing a short CNT shown in a cut through view

ISSN: 0975-5462

1099

Unnati A Joshi et. al. / International Journal of Engineering Science and Technology Vol. 2(5), 2010, 1098-1107 The general 3-D strain–stress relation relating the normal stresses (σ , σ , σ ) and strains ( transversely isotropic material can be written as 1 1

,

,

) for a

1 1

Upon expanding above matrix, the following set of equations is: (i) -

(ii) (iii)

To determine the four unknown material constants (E , E , υ , υ ) four equations will be needed. Only the loading case for square RVE under an axial stretch ΔL has been considered to evaluate two material constants (E , υ in axial direction. A. Square RVE under an axial stretch ∆L:

Fig. 3: The square RVE used to evaluate the effective material properties of the CNT-based composites: Under axial stretch ΔL

In this load case (Fig. 3), the stress and strain components on the lateral surface are : ∆L ∆a σ σ 0, , along x a L a ∆a along y a and a Where Δa is the change of dimension of the cross section under the stretch ΔL in the z-direction (Δa < 0, if ΔL > 0). 0, on Integrating and averaging the equation (iii) obtained from eq. (1), after substituting the values of the plane Z=L/2 one has immediately: σ L σ 2 E ∆L Where, the averaged value of stress is given by 1 σ σ x, y, L 2 dx dy A A with ‘A’ being the area of the end surface. σ can be evaluated for the RVE using the FEM results. Using the equation (i) or (ii) from Eq. (1) and the result (2), one has along x a. ∆L υ ∆a σ υ L a Thus, obtains an expression for the Poisson’s ratio

ISSN: 0975-5462

1100

Unnati A Joshi et. al. / International Journal of Engineering Science and Technology Vol. 2(5), 2010, 1098-1107

 zx

a  a

L L

(3)

Eqs. (2) and (3) can be applied to estimate the effective Young’s modulus Ez and Poison’s ratio  zy   zx , once the contraction a and the stress

 ave

in load case is obtained.

III. RULES OF MIXTURES BASED ON THE STRENGTH OF MATERIALS THEORY Simple rules of mixtures can be established based on the strength of materials theory. These rules of mixtures can be applied to verify the numerical results for the effective Young’s moduli in the CNT axial direction. More general theories and extended results, in the context of fiber-reinforced composites, can be found in Refs. [5, 8, 9]. A. CNT through the length of the RVE

Fig. 3(a): CNT through the length of the RVE

Simplified strength of materials models based on the square RVEs for estimating the effective Young’s modulus in the CNT direction. (a) CNT through the length of the RVE. This is the case when the CNT is relatively long (with large aspect ratio) and therefore a segment can be modeled using an RVE. For the square RVE, the volume fraction of the CNT (a tube, Fig.3 (a)) is defined by: 2

Vt 

2

 ( r0  ri ) 2 4a 2  ri

(4)

The effective Young’s modulus Ez in the axial direction is found to be (5) E z  E tV t  E m (1  V t ) where, E t is the Young’s modulus of the CNT and E m is the Young’s modulus of the matrix. B. CNT inside the RVE

Fig. 3(b): CNT inside the RVE

In this case (Fig. 3(b)), the RVE can be divided into two segments: one segment accounting for the two ends with total length Le and Young’s modulus Em; and another segment accounting for the centre part with length L and an effective Young’s modulus E c . Note that the two hemispherical end caps of the CNT have been ignored in this derivation. Since the centre part is a special case of Fig. 3(a), its effective Young’s modulus is found to be

ISSN: 0975-5462

1101


E c  E tV t  E m (1  V t )

(6)

using Eq. (5) in which the volume fraction of the CNT given by Eq. (4) is computed based on the center part of the RVE (with length Lc) only.

1 1  L  1  L  A   m  e   c  e   E z E  L  E  L  Ac 

(7)

Where, A= 4a2, Ac = 4a2 – πri2, Eqs. (5) and (7) will be applied to compare the FEM estimates of the effective Young’s moduli in the axial direction. IV. NUMERICAL EXAMPLES Several square RVE models for single-walled CNTs in a matrix material are studied using the FEM in this section, in order to evaluate the effective material constants of the CNT-based nanocomposite. The deformations and stresses are computed first for the first loading cases (Fig. 3) as described in Section 2. The FEM results are then processed and Eqs. (2), (3), (5) and (6) are applied to extract the effective Young’s moduli and Poisson’s ratios for the CNTbased composite. Two numerical examples are studied, one on RVEs with long CNTs and the other on an RVE with a short CNT. In all the cases, quadratic solid (brick) elements are employed for the 3-D models, which offer higher accuracy in FEM stress analysis. A. Long CNT through the length of a square RVE Consider a RVE for a long CNT all the way through the RVE length. The dimensions are: For matrix: Length L = 100nm , outer radius a=10nm. For CNT: Length L = 100nm, outer radius ro = 5nm inner radius ri = 4.6nm The Young’s moduli and Poison’s ratio used for the CNT and matrix are: CNT: Et = 1000nN/nm2 , νt =0.3; Matrix: Em = 5,20,100 and 200nN /nm2, νm =0.3. The results for the effective material constants of the CNT-based composite from the 3-D FEM model are given in Table 1. The effective Young’s modulus Ez estimated by the strength of materials solutions is also listed in Table 1 for comparison. A Finite element mesh of a 3-D FEM model for the square RVE with a long CNT (CNT thickness=0.4 nm) is shown in fig. 4(a). TABLE1 COMPUTED EFFECTIVE MATERIAL CONSTANTS FOR CNT THROUGH THE LENGTH OF THE SQUARE

5 10 50 200

ISSN: 0975-5462

FEM results for square RVE

ERM results

1.1423 1.3520 2.696 8.210

1.1446 1.3255 2.7723 8.190

0.408 0.486 0.845 1.644

1102


Fig.4 (a): A Finite element mesh of a 3-D FEM model for the square RVE with a long CNT (b)-(g) Plot of stress distributions for the square RVE under an axial stretch ∆L

The results reveal that the increase of the stiffness of the composite can be significant in the CNT axial direction. With a volume fraction of the CNT being only about 3.6%, the stiffness of the composite in the axial direction Ez can increase by about 33% compared with that of the matrix, when Et / Em = 10. For comparison, the effective material constants obtained using the cylindrical RVE which is of the same size (same length L and the diameter of the cylindrical RVE = 2a of the square RVE), is listed in Table 1. It is seen that the cylindrical RVE overestimates the Young’s moduli. This is explained that the cylindrical RVE overestimates the volume fraction of the CNT due to the negligence of the small amount of matrix material (at the four corners of the square RVE) in the cylindrical RVE. Plot of the first principal stresses for the 3-D long CNT through the length of the RVE under the axial stretch ΔL at Et/Em = 10 is shown in fig. 4(b). B. A short CNT inside the square RVE The dimensions for the RVE are the same as in the previous example, except for the total length of the CNT, which is 50 nm (including the two end hemispherical caps).

ISSN: 0975-5462

1103

Unnati A Joshi et. al. / International Journal of Engineering Science and Technology Vol. 2(5), 2010, 1098-1107 TABLE 2 COMPUTED EFFECTIVE MATERIAL CONSTANTS FOR CNT INSIDE THE SQUARE

FEM results for square RVE

ERM results

5 10

0.980 1.046

0.976 1.049

50

1.197

0.300 0.321 0.431

200

1.389

0.522

1.396

1.169

With the addition of the CNT in a matrix at the volume fractions of 3.6%, the stiffness of the composite increased by 33% as compared with that of the matrix in the case of long CNT at E ⁄E 10, whereas in case of short CNT 10 due to the small volume fraction of the CNT addition, the stiffness in the axial direction is moderate at E ⁄E (about 1.6%). These results suggest that short CNTs in a matrix may not be as effective as long CNTs in reinforcing a composite. Computed effective material constants for CNT inside the square and cylindrical RVE is listed in table 2.The strength of materials solution for the stiffness in the axial direction (Ez), using the extended rule of mixtures (Fig. 3(b) and Eq. (7)), is quite close to the FEM solution which is based on 3-D elasticity, with a difference of only about 1%. Therefore, the extended rule of mixtures (Eq. (7) may serve as a quick tool to estimate the stiffness of the CNT-based composites in the axial direction when the CNTs are relatively short, while the conventional rule of mixtures (Eq. (5)) can continue to serve in cases when the CNTs are relatively long. Comparison of FEM results for long and short CNT using square RVE is shown in fig. 5(a). Plot of the first principal stresses for the 3-D short CNT 10 is shown in fig. 4(c). inside the RVE under the axial stretch ∆L at E ⁄E C. Effect of CNT dimensions on the mechanical properties of carbon nanotube composites Taking into consideration of the effect of CNT dimensions (thickness and diameter) the material properties Young’s modulus and Poisson’s ratio are evaluated. The material properties are also evaluated for short CNT with and without using hemispherical end caps. The effect of CNT thickness: Changing CNT thickness for both long and short CNT is considered and the material properties are evaluated for 10. square RVE at E ⁄E The dimensions are: For matrix: Length L For CNT: Length L

100 nm, outer radius a 100 nm, outer radius r

10 nm. 5 nm, inner radius r

4.6 nm (thickness = 0.4,0.6,0.8,1,1.2).

The Young’s moduli and Poison’s ratio used for the CNT and matrix are: CNT: E Matrix: E

1000 nN/nm ( Gpa , υ =0.3; 100 nN/nm Gpa , υ =0.3.

The computed material constants for long and short CNT using square RVE is listed in table 3. It is observed that as the thickness of the CNT increases the stiffness of the composite is also increased. The FEM results and ERM results for long and short CNT are quite close to each other. Comparison of FEM results for long and short CNT using square RVE with change in thickness is shown in fig.5 (b). Plot of the first principal stresses for the 3-D long 10 is shown in fig. 4(d) and and short CNT at (t=1.2 nm) using square RVE under the axial stretch ∆L at E ⁄E 4(e). The effect of CNT diameter: Changing CNT diameter for both long and short CNT is considered and the material properties are evaluated for square RVE at E ⁄E 10.

ISSN: 0975-5462

1104

Unnati A Joshi et. al. / International Journal of Engineering Science and Technology Vol. 2(5), 2010, 1098-1107 The dimensions are: For matrix: Length L For CNT: Length L

100 nm, outer radius a 100 nm, outer radius r

10 nm. 5,6,7,8,9 nm, inner radius r

4.6 nm (thickness = 0.4).

The Young’s moduli and Poison’s ratio used for the CNT and matrix are: CNT: E

1000 nN/nm (Gpa , υ =0.3;

Matrix: E

100 nN/nm Gpa , υ =0.3.

The computed material constants for long and short CNT using square RVE is listed in table 4.It is observed that in the case of long CNT with change in diameter of the CNT the stiffness of the composite is increased whereas in the case of short CNT with increase in diameter of the CNT the stiffness of the composite is decreasing due to change in the cross sectional area of CNT in a matrix. Comparison of FEM results for long and short CNT using square RVE with change in diameter is shown in fig. 5(c). Plot of the first principal stresses for the 3-D long and short CNT at 10 is shown in fig. 4(f) and 4(g). (d=18 nm) using square RVE under the axial stretch ∆L at E ⁄E D. CNT inside the Square RVE with and without hemispherical end caps This is the case where the dimensions for the RVE are the same as in the previous example, except for the total length of the CNT, which is 50 nm (without using hemispherical end caps). Comparison of FEM results for long and short CNT using square RVE with and without using end caps is shown in fig. 5(d). The computed effective material constants for short CNT using square RVE with and without end caps is listed in table 5.The material properties evaluated with and without end caps gives some difference in FEM results due to small change in effective length.

a

b

c

ISSN: 0975-5462

d

1105


e

Fig. 5: (a) ,(b),(c),(d),(e) Comparison of FEM results for long and short CNT using square RVE

Et/Em

10

Et/Em

10

Et/Em

5 10 50 200

TABLE 3 COMPUTED MATERIAL CONSTANTS FOR LONG AND SHORT CNT USING SQUARE RVE

CNT thickness 0.4 0.6 0.8 1 1.2

FEM results for square RVE ERM (long CNT) results Ez/Em Ez/Em 1.3520 0.486 1.3255 1.4625 0.530 1.4697 1.6177 0.566 1.6038 1.7201 0.59 1.7272 1.8237 0.665 1.8414

FEM results for square RVE (short CNT) Ez/Em Ez/Em 1.0460 0.321 1.0967 0.341 1.1498 0.365 1.2013 0.378 1.2352 0.390

ERM results 1.049 1.1096 1.1599 1.2032 1.2403

TABLE 4 COMPUTED MATERIAL CONSTANTS FOR LONG AND SHORT CNT USING SQUARE RVE

CNT diameter 10 12 14 16 18

FEM results for square RVE (long CNT) Ez/Em 1.3520 0.340 1.4398 0.356 1.5898 0.365 1.8263 0.379 2.1935 0.388

ERM results Ez/Em 1.3255 1.4367 1.5844 1.8073 2.1873

FEM results for square RVE (short CNT) Ez/Em Ez/Em 1.0460 0.321 1.0330 0.336 1.0100 0.357 0.9839 0.397 0.9437 0.421

ERM results 1.049 1.0392 1.0207 0.99367 0.9567

TABLE 5 COMPUTED EFFECTIVE MATERIAL CONSTANTS FOR SHORT CNT USING SQUARE RVE WITH AND WITHOUT HEMISPHERICAL END CAPS

FEM results for square RVE(long ERM CNT) results Ez/Em 0.980 1.046 1.197 1.389

Ez/Em 0.300 0.321 0.431 0.522

Ez/Em 0.976 1.049 1.169 1.396

FEM results for square RVE(short ERM CNT) results Ez/Em 0.930 0.971 1.179 1.321

0.300 0.311 0.36 0.45

0.976 1.049 1.169 1.396

V. DISCUSSIONS The effective mechanical properties of CNT based composites are evaluated using square RVEs based on 3-D elasticity theory and solved by the FEM. Formulas to extract the effective material constants from solutions for the square RVEs are established based on elasticity theory. Numerical examples using the FEM are presented, which demonstrate that the load-carrying capabilities of the CNTs in a matrix are significant. With the addition of only about 3.6% volume fraction of the CNTs in a matrix, the stiffness of the composite in the CNT axial direction can increase as much as 33% for the case of long CNT fibers. It is also found that cylindrical RVEs tend to overestimate the effective

ISSN: 0975-5462

1106

Unnati A Joshi et. al. / International Journal of Engineering Science and Technology Vol. 2(5), 2010, 1098-1107 Young’s moduli due to the fact that they overestimate the volume fractions of the CNTs in a matrix. Finally, the rules of mixtures, for both long and short CNT cases, are found to be quite accurate in estimating the effective Young’s moduli in the CNT axial direction. In the case with change in diameter of short CNT the stiffness of the matrix is decreased due to the change in the cross section area of the CNT in a matrix. In future research, the MD and continuum approach should be integrated in a multiscale modeling and simulation environment for analyzing the CNT-based composites. More efficient models of the CNTs in a matrix also need to be developed, so that a large number of CNTs, in different shapes and forms (curved or twisted), or randomly distributed in a matrix, can be modeled. REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]

Bower, C. and Rosen, R.et al. “Deformation of carbon nanotubes in nanotube-polymer composites,” Applied Physics Letters, 74, 3317- 3319 (1999). Chen, X.L. and Liu, Y.J. Continuum models of carbon nanotube-based composites by the BEM, Electronic Journal of Boundary Elements, in press (2003). Chen, X.L. and Liu, Y.J. “Square representative volume elements for evaluating the effective material properties of carbon nanotube-based composites,” Computational Materials Science, in press (2003). Govindjee, S. and Sackman, J.L.“On the use of continuum mechanics to estimate the properties of nanotubes,” Solid State Communications, 110, 227-230 (1999). Hyer, M.W. Stress Analysis of Fiber-Reinforced Composite Materials, first ed., McGraw-Hill, Boston, 1998. Iijima, S. “Helical microtubules of graphitic carbon,” Nature, 354, 56–58 (1991). Liu, Y.J. and Chen, X.L. “Evaluations of the effective materials properties of carbon nanotube-based composites using a nanoscale representative volume element,” Mechanics of Materials, 35, 69-81 (2003). Nemat-Nasser, S. and Hori, M. Micromechanics: Overall Properties of Heterogeneous Materials, second ed., Elsevier, Amsterdam, 1999. Qian, D., Dickey, E.C., Andrews, R. and Rantell, T. “Load transfer and deformation mechanisms in carbon nanotube-polystyrene composites,” Applied Physics Letters,76, 2868-2870 (2000). Qian, D., Liu, W. K. and Ruoff, R. S. “Mechanics of C60 in nanotubes,” J. Phys.Chem. B, 105, 10753-10758 (2001). Ruoff, R.S. and Lorents, D.C. 1995. Mechanical and thermal properties of carbon nanotubes. Carbon 33 (7), 925–930. Sohlberg, K. and Sumpter, B. G. et al. “Continuum methods of mechanics as a simplified approach to structural engineering of nanostructures,” Nanotechnology, 9, 30-36 (1998). Timoshenko, S.P. and Goodier, J.N. Theory of elasticity.3rd ed.McGraw-Hill International Editions. Y u, M.F., Lourie, O. and Ruoff, R.S. Science 287, 637 (2000).

ISSN: 0975-5462

1107