Stress distribution in carbon nanotubes with bending fracture
Introduction
Carbon nanotubes are known for their potential in various applications owning to its exceptional properties. It was reported that the Young’s modulus can be as high as 1 TPa and the failure strain up to 5% [1], [2], [3]. Conventional reinforcing fibers, such as carbon fibers, are typically brittle and cannot survive a large extension or bending without breakage. In comparison, CNTs are composed of graphene sheets which are exceptionally resilient, and thus the elastic range of CNTs is remarkably large [4], [5]. While CNTs are resilient along the axial direction, they are susceptible to inter-layer stresses inside the CNT.
For a CNT to be stressed, the load must be applied on surface and then transferred into inner layers through the interlayer shear stress. The mechanism of load transfer between the graphite layers associated with non-fractured as well as fractured has been an important research topic studied both theoretically and experimentally [6], [7], [8], [9], [10], [11], [12]. Once a CNT is loaded, it could undergo fracture that renders the material unusable. To exploit its potential in many applications, the knowledge of CNT deformation and fracture is essential [13], [14]. Nikiforov utilized the molecular dynamics to study the bending behavior of CNTs [15]. They predicted and characterized a Fourier-type rippling mode that dominates the incipient nonlinear elastic response of MWCNTs with closed cores. Wang used the molecular mechanics simulations to study the local buckling of a SWNT under bending [16]. The critical deformation at the force location was observed and the indentations or kinks at the onset of the local buckling were described. Belytschko observed a measured drop of the effective bending stiffness of MWNTs with larger diameters [17]. In addition, ripples were also predicted for MWNTs subjected to torsion. Falvo studied the resilience of MWNTs by applying a repeated bending motion. They observed reversible, periodic buckling of nanotubes [18]. Xu Guo investigated the bending stiffness of SWCNTs and some related issues by the combined use of the molecular-mechanics model and the deformation mapping technique [19]. An analytical expression for the bending stiffness of SWCNTs based on the molecular mechanics model under infinitesimal deformation was presented. It shows that the bending stiffness of SWCNTs is approximately proportional to the cube of the tube radius. Still, the progress in the analysis of the bending behavior has been slow, especially for MWCNTs.
Along with the efforts on studying stress transfer in MWCNTs, the cracking mechanisms of MWCNTs are also investigated experimentally. Yu first conducted tensile tests and observed pullout of the inner CNT [20], and this was termed as sword-in-sheath. Recently, Kuo adopted a special approach to incur bending fracture in CNTs [21], [22]. They observed two major modes of bending fracture, including the cone-shaped and the shear-cut. Because the fracture modes in CNTs are unusual and unprecedented in micron-scale filaments, such as carbon fibers, it is necessary that these modes be well characterized analytically.
In this work, new formulations based on the experimental observations have been developed to study the deformation and stresses in a CNT under bending. The previous microscopic observations showed that many CNTs were partially fractured. Some outer graphene layers of a CNT were fractured, while the inner layers remain intact. This indicated that the crack starts from the outermost layer and follows a stepwise, stop-and-go manner to the inner layers. The crack grows when the stress and the stored strain energy are sufficiently high. Once the energy is dissipated, it stops and can proceed only more energy is supplied from external loads. This stop-and-go manner allows us to analyze the detailed stress distribution within the layers for any stage of crack growth. This information is important as it dominates the nucleation and development of a crack. Both non-fractured and partially fractured CNTs are examined, referring to the cases of crack initiation and propagation, respectively. The shear-lag model and the finite element method are employed to study the stress distribution in partially fractured CNTs. Without loss of generality, the influence of layer number is analyzed up to ten. The finite element method is used to calculate the crack growth angle.
Section snippets
Experimental observation
According to the previous experimental observation, CNTs showed unique fracture modes when bent to fracture [21]. Similar to the sword-and-sheath type of tensile fracture, the bending fracture is separated into two regions: the outer-tube and the inner-tube. The outer-tube is failed in tensile rupture, similar to the sheath. The inner-tube is fractured and pulled out. Unlike the sword, the pullout has an inclined side surface. The inclined crack propagation was due to the stepwise fracture in
Model transformation
Despite the gap in the layered structure, there exist some mechanisms for load transfer between graphene layers [23], [24]. If the structure of a CNT is commensurate (with parallel chiral vectors), the efficiency of load transfer through shear is optimum as explained by Zalamea et al. [9] and Kolmogorov and Crespi [25]. To study of the load transfer among the layers, the discrete and layered CNT is converted into the equivalent continuum configuration [9], [10], [11], [12], as shown in Fig. 2.
Non-fractured CNTs
The continuum mechanics and the shear-lag model are adopted to relate the stress and deformation [9], [10], [11], [12]. In this work, the bending moment is applied on the outermost graphene layer [15], [26]. This moment induces the normal stress σ01 at the edge y = L/2. As shown in Fig. 3, a four-layer model is first studied. The CNT geometry structure was symmetric with respect to both x and y axes. In the figure, Ro.1 and Ri.7 represent the outer radius of outermost layer and inner radius of
Non-fractured CNTs
To compare the results from the shear-lag model, a finite element software (ANSYS Workbench) was employed. In the simulation, each graphite layer was treated as a continuum thin layer [9], [33]. The original 3-D structure of the CNTs is used for simulation. To simulate the extent of interlayer interaction, a thin layer representing for the effective interphase was introduced between the adjacent layers. In this paper, the interphase thickness is selected as p = 0.01 nm [10].
In the FEA simulation,
Bending behavior of non-fractured CNTs
The distribution of the normalized normal stress in tensile side of each layer in a ten-layered CNT is shown in Fig. 7. A moment is applied at the end of layer 1. The results of the saturation stress from both methods are very close. The stresses vary more drastically near the end where the load is applied. At the end region, the stress in layer 2 can grow even higher than the saturation stress. In general, both are in good agreement. Fig. 8 shows the stress along the hoop direction. The hoop
Conclusion
Based on the experimental observation of CNT bending fracture, a model for crack growth has been proposed. Stress distribution in each graphene layer is first analyzed by both the shear-lag and the finite element method. The same approaches are then applied to the CNTs with some outer layers fractured. The influences of the interlayer cohesion, expressed in terms of the interlayer shear modulus and the Young’s modulus, has been examined. When the aspect ratio is small, the effect of interlayer
Acknowledgement
The authors would like to thank the National Science Council of Taiwan, R.O.C. for the support of this research (NSC 100-2221-E-035-034-MY3).
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