The Effect of Carbon Nanotube Wall Thickness on 

Elastic Modulus of Nanocomposite 

N. Kordani1*, R. Adibipour2, A. Sadough Vanini2 
1Department of Mechanical Engineering, University of Mazandaran, Mazandaran, Iran 

2Department of Mechanical Engineering, Amirkabir University of Technology, Tehran, Iran 

Abstract- This paper focuses on effect of carbon nanotubes physical parameter on elastic 

modulus of nanocomposites. The remarkable properties of at least some of nano particles 

have led to high research in the field of nanocomposites, especially carbon nanotubes. In 

this paper, polymer matrixes and carbon nanotubes are interest.  At the nano scale, the 

structure of the carbon nanotube strongly influences the overall properties of the 

composite. Some well-known theories such as Halpin-Tsai equation, shear lag model and 

modified mixture of low were employed to consider the efficient of carbon nanotubes 

physical parameter on elastic modulus of nanocomposites. According to the results, 

addition of volume fraction of carbon nanotubes caused a reduction of elastic modulus. 

The nanocomposite elastic properties are particularly sensitive to the nanotube diameter, 

with increasing on diameter and wall thickness of carbon nanotube the elastic modulus 

decreases and when length of carbon nanotube is increasing, the elastic modulus 

increases. 

Keywords: Mechanical Properties, Nanocomposites, Polymer Matrix Composites, Aspect 

Ratio, Carbon Nanotube 

1. Introduction

The development of nano-particle reinforced polymer composites is newly one of the

most favorable approaches in the field of future engineering applications. The remarkable 

properties of at least some of these nano particles have led to high research in the field of 

nanocomposites, especially carbon nanotubes (Sreejarani & Ray, 2011; Baharvandi & et. 

al., 2017; Kordani & Sadough, 2014).  

Of the various nano-particles, carbon nanotubes (CNTs) have attracted great interest 

newly as structural reinforcements because of their unique properties. CNTs with their 

notable mechanical properties such as low density, high aspect ratio, high strength and 

stiffness, excellent electrical and chemical resistance are a potential candidate as 

reinforcement for polymeric materials. The addition of only small quantity of nano 

particle (specially CNTs) leads to improved mechanical properties of matrix (Baharvandi 

& et. al., 2016; Guozhong, 2004; Harris, 1999). The most important of properties of 

Single-Walled Carbon Nanotubes (SWCNT) and Multi-Walled Carbon Nanotubes 

(MWCNT) are collected on Table 1.  

Table 1 Typical properties of SWCNT and MWCNT. 

Property SWCNT MWCNT Ref. 

Diameter (nm) 0.4-5 5–50 (Micah & et.al, 2009) 

Aspect ratio 100–10,000 100–10,000 (Saleh & Sundararaj, 2011) 

Density (𝑔 𝑐𝑚3⁄ ) ~1.3 ~1.75 (Saleh & Sundararaj, 2011) 
Tensile strength (GPa) 50–500 10–60 (Saleh & Sundararaj, 2011) 

Elastic modulus (TPa) ~1 (from 1 to 5) 0.2-0.95 (Meo & Rossi, 2006; Yu & 
et.al, 2000) 

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Corresponding Author:naser.kordani@umz.ac.ir 

mailto:naser.kordani@umz.ac.ir


Failure strain (%) ~5 − 10 10.5, up to 12 (Liu & et.al, 2005; Chowdhury 
& et.al, 2012) 

CNTs have many structures, differing in length, diameter, thickness, spiral types 

and number of layers that are used to modify the other materials. MWNTs and 

SWNTs are the most popular type of CNTs. In 1991, MWNTs consist of many 

coaxial graphite cylindrical tubes and in 1993, SWNTs with one graphite cylindrical 

tube were discovered by Iijima. MWCNTs and SWCNTs were discovered in the soot 

of the arc-discharge method and using of metal catalysts in the arc-discharge method 

(Natsuki & Tantrakan, 2004; Khare & Bose, 2005). TEM micrograph of a MWNT is 

shown in figure 1. 

Figure 1 TEM micrograph of a MWNT with measurements of outside diameter, inside 

diameter and wall thickness (Thostenson & et. al, 2001). 

2. Influence of Physical parameters of CNTs on Elastic modulus of nanocomposite

At the nano scale, the structure of the carbon nanotube strongly influences the

overall properties of the composite. For design purposes, we need to have simple and 

rapid calculative procedures for estimating the effective properties. Some well-

known theories such as Halpin-Tsai equation, shear lag model and modified mixture 

of low were used. Their equations depend on a parameter which is considered, table. 

2. 

Table 2  Theories on mechanical properties of nanocomposite with their parameter 

Model Equation Ref. 

Halpin-Sai 
𝐸𝑐 = 𝐸𝑚 (

1+𝜁𝜂𝑣𝑓

1−𝜂𝑣𝑓
), 𝜂 =

𝐸𝑓

(𝐸𝑚)
−1

𝐸𝑓

(𝐸𝑚)
+𝜁

, 𝜁 = 2𝑙 𝑑⁄
(Halpin & Tsai, 1967) 

Shear-Lag 

𝐸𝑐 = 𝜂𝐸𝑓 𝑣𝑓 + 𝐸𝑚 . (1 − 𝑣𝑓 ),

𝜂 = 1 −
tanh (𝑘

𝑙

𝑑
)

𝑘
𝑙

𝑑

, 𝑘 = √
2𝐸𝑚

𝐸𝑓(1+𝜈)ln (
1

𝑉𝑓⁄
)

(Kashyap & et.al, 2011) 

Modified Mixture 

Low 

𝐸𝑐 = 𝑋1𝜂𝐸𝑓 𝑣𝑓 + 𝐸𝑚 𝑣𝑚,

 𝜂 = 1 −
𝑡𝑎𝑛ℎ (𝑘

𝑙

𝑑
)

𝑘
𝑙

𝑑

, 𝑋1(2𝐷, 3𝐷) = 3/8,1/5

(Crutis & et.al, 1978) 

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Where 𝐸𝑐 , 𝐸𝑓 , 𝐸𝑚 are the modulus of nanocomposite, CNTs and matrix. ζ is called 

shape factor that is depend on the particular elastic property. L is average length and d is 

average diameter of nanotube and 𝜈 is poisson ratio. 𝑣𝑓  is volume fraction. 

Base on experimental results, Thostenson and Chou ploted a linear line through the 

data that shows relationship between the nanotube diameter and wall thickness, figure 2. 

According their study at smaller nanotube diameters this relationship between the 

nanotube diameter and wall thickness begins to deviate from the linear curve fit 

(Thostenson & et. al, 2001). 

Figure 2 linear relationship between wall thickness and nanotube diameter by 

(Thostenson & et. al, 2001). 

𝑇 = −3.0793 + 0.4796𝑑, 𝑅 = 0.98687 (1)

 In which, T is wall thickness and d is Nanotube diameter. R is the error of this curve 

fitting. Eq. (1), Substituted into shape factor of Halpin-Tsai equation to show the effective 

of wall thikness on elastic modulus of nanocomposite. 

𝜁 =
0.9592𝑙

𝑇 + 3.0793

(2)

We also used Eq. (2) for other theories to consider the effect of wall thickness of 

CNTs. Halpin-Tsai equation and experimental results by Montazeri et al. (2010), were 

used to consider the influence of nanotube diameter, length and volume fraction on the 

elastic modulus of nanocomposite. Results are shown in figures 3 and 4. As shown in 

figures 3 and 4, increasing on diameter and wall thickness make the elastic modulus of 

nanocomposite decreases and increasing on weight percent and wall length of CNTs make 

the elastic modulus of nanocomposite increases. 

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Figure 3 The effect of nanotube diameter, d, length, L and weight percent, wt% on the 

elastic modulus of nanocomposite. 

Figure 4 The effect of nanotube wall thickness, T, length, L and weight percent, wt% on 

the elastic modulus of nanocomposite. 

By using the theories on table 2 and experimental results by (Montazeri & et al., 

2010), effect of diameter on the composite elastic modulus of nanocomposite are 

predicted, and results are shown in figure 5. Modified mixture of low in 2D and 3D are 

inefficient to consider the wall thickness of CNTs on elastic modulus of nanocomposite.  

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Figure 5  Effect of diameter on the elastic modulus of nanocomposite by using Halpin-Tsai equation 
(H-T), shear-Lag theory (SH-L), and modified mixture of low in 2 dimension(MML,2D) and 3 

dimension(MML,3D). 

3. Conclusion

In this paper by using the experimental data, Halpin-Tsai Equation, shear lag model

and modified mixture of low, effects of reinforcement, length and diameter of carbon 

nanotube on the mechanical properties were investigated. 

    According to the results, additional CNTs weight percent caused an increase on elastic 

modulus. The elastic properties of nanocomposite are particularly sensitive to the 

nanotube diameter. By increasing the diameter and wall thickness of carbon nanotube, 

the elastic modulus decreases and when length of carbon nanotube is increasing, the 

elastic modulus increases. Effect of geometry parameter on elastic properties of 

nanocomposite is better shown by Halpin-Tsai equation and shear-Lag theory. 

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