Carbon nanotubes (CNTs) have been regarded as an ideal reinforcements of high-performance composites with enormous applications. In this paper, the effects of pinhole defect are investigated for carbon nanotube based nanocomposites using a 3D representative volume element (RVE) with long CNTs. The CNT is modeled as a continuum hollow cylindrical shape elastic material with pinholes in it. These defects are considered on the single wall (CNTs). The mechanical properties such as Young’s modulus of elasticity are evaluated for various pinhole locations and number of defects. The influence of the pinhole defects on the nanocomposite is studied under an axial load condition. Numerical equations are used to extract the effective material properties for the different geometries of RVEs with nondefective CNTs. The field-emission microscopy (FEM) results obtained for nondefective CNTs are consistent with the analytical results for cylindrical RVEs, which validate the proposed model. It is observed that the presence of pinhole defects significantly reduces the effective reinforcement when compared with nondefective nanotubes, and this reinforcement decreases with the increase in the number of pinhole defects. It is also found from the simulation results that the geometry of RVE does not have much significance on the stiffness of nanocomposites.
Introduction
Owing to their remarkable properties, carbon nanotubes have been employed in many diverse areas of applications. Carbon nanotubes, discovered first by Iijima in 1991 (1), 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. The use of CNTs for biomedical applications is acquiring more and more substantiating evidence for potential development. Carbon nanotubes are strong, capable of being shaped into 3D architectures, and promising in the construction of engineered products for biological applications. Carbon is an inert material and thus is generally biocompatible. The studies on cell uptake, applications in vaccine delivery, interaction with nucleic acids, and impact on health show remarkable areas for innovative therapies achievable with CNTs. Carbon nanotubes may play an integral role as a unique biomaterial for creating and monitoring engineered tissue (2). Carbon nanotubes may be an important tissue engineering material for an improved tracking of cells, sensing of microenvironments, and scaffolding for incorporating with the host’s body. The optical properties of nanotubes impart promising advantages to their use in imaging applications within live cells and tissues. Although the toxicological studies on pristine CNTs (3) are contradictory, showing a certain degree of risk, it is becoming evident that functionalized CNTs have reduced toxic effects. The mechanical properties of nanotubes (4,5,6,7,8) have been extensively studied. Its very small diameter, large aspect ratio, and extremely high strength and stiffness naturally make the CNT become a kind of most attractive reinforcement for nanocomposites. 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% and 42% and the tensile strength by 25% (9). The mechanical-load carrying capacities of CNTs in nanocomposites have also been demonstrated in experiments (9,10) and preliminary simulations (11,12). The measured specific tensile strength of a single layer of a multiwalled carbon nanotube can be as high as 100 times that of steel, and the graphene sheet (in-plane) is as stiff as diamond at low strain. These mechanical properties motivate further study of possible applications for lightweight and high strength materials (13). However, there are many parameters that may influence the mechanical properties of nanocomposites, such as the dispersion, alignment, waviness, and defects of CNTs. Similar to any of the many man-made materials used today, CNTs are also susceptible to various kinds of defects. Experimental observations have revealed that topological defects are commonly present in CNTs (14). Defects degrade the mechanical performance of CNTs since they alter not only their inelastic properties but also their elastic properties, such as the Young’s modulus and the Poisson’s ratio. The longitudinal and transverse stiffness as well as the flexural rigidity in tension, torsion, and bending are consequently being altered. Therefore, it is necessary to investigate the effects of these factors on the macroscopic properties of composites. Hirai et al. (15) showed that in the case of the defective single wall nanotubes (SWNTs), the yield tensile strength decreases to about 80% that of the nondefective SWNT by a single type-1 pinhole defect. The Halpin–Tsai concept has been applied for simulating the mechanical responses of various composites under different boundary as well as loading conditions (16). This concept can also be applied successfully for carbon nanotube based nanocomposites. CNTs are treated as solids in cylindrical shapes with a representative volume element, which is employed to study the interactions of the nanotube with the matrix and to investigate the effective material properties of the nanocomposites (17) and validated by Halpin–Tsai equations. However, work has been found on the effect of defects on buckling of CNTs (18), where the influence of two types of pinhole defects was studied on the buckling strength of SWCNTs and significant reduction in buckling properties was observed due to the presence of large pinhole defects. The effect of single atomic vacancy defects on the buckling of SWCNTs has been studied using molecular dynamics simulation (MDS) and continuum beam models (19).
The concept of unit cells or RVEs has been applied in this paper to study the CNT based composites at the nanoscale. In this unit cell or RVE approach, a single nanotube with surrounding matrix material can be modeled with properly applied boundary and interface conditions to account for the effects of the surrounding materials. An evaluation of effective elasticity property has been done for nondefective long as well as short CNTs with hexagonal RVE under an axial stretch (20). In this work, the coupled effects of multiple defects and location of defects made in CNTs are investigated for the nanocomposites stiffness. Analytical equations given by Halpin and Tsai are used to extract the effective material properties for the cylindrical and hexagonal RVEs with CNTs under axial loading condition.
Simulation Model
The influences of defect type and the number of defects are investigated for SWNTs. In this study, small SWNTs with 1.5 nm diameter and 15.0 nm length are investigated. Based on the above discussed models, the effective moduli of elasticity are calculated for various conditions.
Numerical Example
To evaluate the effective material constants of the CNT based nanocomposites, the deformations and stresses are computed first for the loading case (Fig. 2), as described in Sec. 2. The FEM results are processed, and Eqs. 1,2 are applied to extract the effective Young’s moduli for the CNT based composite. Cylindrical and hexagonal RVEs are studied with long CNTs under axial load condition. In all the cases, quadratic solid (brick) elements are employed for the 3D models, which offer higher accuracy in FEM stress analysis.
Cylindrical RVE
The dimensions of the cylindrical RVE and the pinholed CNT through the RVE length are as follows:
For the matrix: length and outer .
For the CNT: length , outer radius , and inner radius .
The Young’s moduli and Poison’s ratio used for the CNT and the matrix are as follows:
CNT: .
Diameter of the type-1 pinhole, 0.4 nm.
Diameter of the type-2 pinhole 0.8 nm.
Hexagonal RVE
A hexagonal RVE with a pinholed CNT is considered through the RVE length. The dimensions used are as follows:
For the matrix: length and side of hexagonal base .
For the CNT: length , outer radius , and inner radius .
The Young’s moduli and Poison’s ratio used for the CNT and the matrix are as follows:
CNT: .
Diameter of the type-1 pinhole, 0.4 nm.
Diameter of the type-2 pinhole, 0.8 nm.
Results and Discussion
The stiffness is evaluated under an axial stretch. These characteristics are important for utilizing the CNT as mechanical probes. In the simulation, both ends of the SWNT are kept free, which defines the boundary condition, and are stretched along the axial direction. Figure 7 shows the comparison between the FEM results and the results obtained by the Halpin–Tsai model for nondefective SWNT. The FEM results are found to be in close approximation with Halpin and Tsai’s results, which validate the proposed FEM model.
Table 1 presents the variation in effective reinforcing stiffness values of nanocomposites for the different values of , where shows the modulus of elasticity of the CNT and shows the modulus of elasticity for the matrix material. Here, the effective elasticity values are also compared between type-1 and type-2 pinholes. From the results, it is very clear that the increase in the size of the pinhole decreases the effective reinforcement. The reason for this decrement is that more material is removed in type-2 pinhole than in type-1 pinhole, which leads to the reduction in the effective stiffness of the CNT based nanocomposite.
FEM results for cylindrical RVE | |||
for type-1 pinhole | for type-2 pinhole | ||
1 | 0.976551 | 0.974209 | |
5 | 0.934057 | 0.925518 | |
10 | 0.914394 | 0.901941 | |
50 | 0.887133 | 0.867043 | |
100 | 0.88239 | 0.860479 | |
150 | 0.880713 | 0.858099 | |
200 | 0.879858 | 0.856869 |
FEM results for cylindrical RVE | |||
for type-1 pinhole | for type-2 pinhole | ||
1 | 0.976551 | 0.974209 | |
5 | 0.934057 | 0.925518 | |
10 | 0.914394 | 0.901941 | |
50 | 0.887133 | 0.867043 | |
100 | 0.88239 | 0.860479 | |
150 | 0.880713 | 0.858099 | |
200 | 0.879858 | 0.856869 |
For the hexagonal RVE, the FEM results are given in Table 2. Here, the effective elasticity modules of the nondefective CNT based nanocomposites are compared with the defective ones. Both types of pinhole defects are considered. It is found from the results that the nondefective CNT based nanocomposite gives the highest effective stiffness of nanocomposite compared with the defective ones.
FEM results for Hexagonal RVE | ||||
without pinhole | for type-1 pinhole | for type-2 pinhole | ||
1 | 1.201808 | 1.201317 | 1.198533 | |
5 | 1.985933 | 1.984308 | 1.966055 | |
10 | 2.959954 | 2.957552 | 2.916439 | |
50 | 10.72715 | 10.72129 | 10.46999 | |
100 | 20.43016 | 20.42081 | 19.89462 | |
150 | 30.13252 | 30.11976 | 29.31645 | |
200 | 39.83468 | 39.81852 | 38.73749 |
FEM results for Hexagonal RVE | ||||
without pinhole | for type-1 pinhole | for type-2 pinhole | ||
1 | 1.201808 | 1.201317 | 1.198533 | |
5 | 1.985933 | 1.984308 | 1.966055 | |
10 | 2.959954 | 2.957552 | 2.916439 | |
50 | 10.72715 | 10.72129 | 10.46999 | |
100 | 20.43016 | 20.42081 | 19.89462 | |
150 | 30.13252 | 30.11976 | 29.31645 | |
200 | 39.83468 | 39.81852 | 38.73749 |
Figure 4 shows the stress distribution pattern for the cylindrical RVE with 11 type-2 pinhole defects. In Fig. 4, the highest of the stresses are shown at both ends, as both are pulled axially. Stress concentration can be visualized around all the pinholes. Deformed areas of composite with hexagonal RVE having five type-1 pinhole defects are shown in Fig. 5. Also, the ends of the composite exhibit an expanded material under the effect of axial stretch.
Figure 6 presents the maximum principle stress pattern for the hexagonal RVE with seven type-2 pinhole defects. Figure 7presents the graph of effective elasticity moduli of the nanocomposite with a cylindrical RVE, which validates the FEM model with Halpin–Tsai model.
Figure 8 shows the comparison between the nondefective and defective CNTs for the cylindrical RVE against different ratios of . It also clearly describes that composites with nondefective CNT have higher tensile strength compared with the composites containing defective CNT. A decreasing trend is observed for type-2 pinholes when compared with type-1 pinholes. Due to the change in the location of pinhole defects, much significant effect is not observed on Young’s modulus of nanocomposites, which is very clear from Fig. 9. A comparison between the results of the current FEM model and the model of Hirai et al. (15) is presented in Fig. 10. Both the curves represent the tensile stiffness values for different numbers of defectives, where the FEM results are clearly found to be in good agreement with the results of the model of Hirai et al. A decreasing trend of effective elasticity modulus for the cylindrical RVE with type-1 pinhole defects over type-2 pinhole defects is presented in Fig. 11. For both types of defects, the diagram also shows the decrement in elasticity moduli by increasing the number of pinhole defects.
Comparison for effective reinforcement stiffness between the nondefective and defective CNTs with hexagonal RVEs against different ratios of is shown in Fig. 12, while Fig. 13 shows the effect of the number of pinhole defects on effective modulus of elasticity for the hexagonal RVE. It is very clear from the plot that by increasing the number of defects in CNT, the effective elastic strength of the composite decreases.
Figure 14 shows the column chart presentation for modulus of elasticity of the pinholed nanocomposite with the hexagonal RVE. In this chart, the different colored contours show the different intensities of effective elasticity moduli for type-2 pinhole defects. All three parameters, , , and number of pinhole defects are taken care of simultaneously. Figure 15 presents the comparison between the elasticity modulus values of the cylindrical as well as the hexagonal RVEs. Very little difference is observed in stiffness values between the cylindrical and hexagonal RVEs; i.e., the geometry of RVE does not have much effect on the properties of the defective CNT based nanocomposites.
Table 3 indicates the comparison between the results for the cylindrical and hexagonal RVEs. The effective Young’s modulus is considered as a parameter. The results of both types of pinhole defects are tabulated for axial loading condition. shows the elasticity of the CNT with reference to the elasticity of the matrix material, where seven different values are considered, as given in Table 3. By increasing the value of , a decreasing trend is observed for all defective, pinholed CNT cases.
FEM results for hexagonal RVE | FEM results for cylindrical RVE | ||||
Pinhole type-1 | Pinhole type-2 | Pinhole type-1 | Pinhole type-2 | ||
1 | 1.201317 | 1.198533 | 1.193529 | 1.190667 | |
5 | 1.984308 | 1.966055 | 1.978262 | 1.960176 | |
10 | 2.957552 | 2.916439 | 2.953054 | 2.91284 | |
50 | 10.72129 | 10.46999 | 10.72679 | 10.48388 | |
100 | 20.42081 | 19.89462 | 20.43804 | 19.93052 | |
150 | 30.11976 | 29.31645 | 30.14861 | 29.37448 | |
200 | 39.81852 | 38.73749 | 39.85906 | 38.81762 |
FEM results for hexagonal RVE | FEM results for cylindrical RVE | ||||
Pinhole type-1 | Pinhole type-2 | Pinhole type-1 | Pinhole type-2 | ||
1 | 1.201317 | 1.198533 | 1.193529 | 1.190667 | |
5 | 1.984308 | 1.966055 | 1.978262 | 1.960176 | |
10 | 2.957552 | 2.916439 | 2.953054 | 2.91284 | |
50 | 10.72129 | 10.46999 | 10.72679 | 10.48388 | |
100 | 20.42081 | 19.89462 | 20.43804 | 19.93052 | |
150 | 30.11976 | 29.31645 | 30.14861 | 29.37448 | |
200 | 39.81852 | 38.73749 | 39.85906 | 38.81762 |
Conclusions
In this paper, micromechanics simulation is applied on SWNTs with pinhole defects using a continuum approach based on the Halpin–Tsai equations to analyze the degradation of the mechanical property such as effective elasticity modulus due to pinhole defects in CNT. Two different types of pinhole defects having different dimensions are considered. These defects play a critical role in determining the reinforcing efficiency of CNT reinforced nanocomposites. Different defect configurations are studied to check the degree of degradation of effective elasticity properties for nanocomposites. It is found that even a single pinhole defect adversely affects the effective reinforcement in composites. As the dimension of the type-2 pinhole is greater than the type-1 pinhole, more degradation in elasticity property is observed. From the simulation results, it is observed that by increasing the number of defects, the effective elastic moduli decrease. Much significance is not found for change in geometry of RVEs. The proposed current FEM model is validated by analytical equations of Halpin and Tsai, where values are found to be in good agreement with each other. Also, the FEM model of the pinholed CNT is validated with that of model given by Hirai et al.