Stress intensity factor analysis of epoxy/SWCNTs based on global-local multiscale method

1 Introduction

Over the past two decades, carbon nanotubes (CNTs) have attracted much attention among researchers of different fields due to their novelty and exceptionally high stiffness, strength, and resilience [1]. As a matrix material, epoxy has been widely applied in composites structures, especially in the automobile and aerospace industries. In recent years, many researchers have found through large numbers of experiments that adding nanoparticles to epoxy has some positive effects on the fracture toughness and the energy release rate [2–6].

The supreme mechanical properties of CNTs rendered them as a potential candidate for reinforcing agents for a new generation of polymeric composites [7–10]. There is some evidence in the literature [11–13] mentioning the significant enhancement in mechanical properties of polymeric resins by utilizing small amounts of CNTs. Consequently, prediction of mechanical properties of CNT-based composites plays an important role in understanding their behavior and can pave the road toward their industrial application. CNT is a reinforcing agent at nanoscale, whereas mechanical properties of CNT-reinforced polymer composites are characterized at microscale.

Whereas a significant number of the existing studies have been focused on the stiffness and strength of CNT-reinforced composites, relatively few research studies have dealt with the fracture behavior of these nanocomposites in the presence of a preexisting crack. Recent investigations have shown that CNTs when aligned perpendicular to cracks are able to slow the crack growth by bridging up the crack faces [14–16]. The effect of CNT dimensions on the fracture properties of nanocomposites is another important issue that requires careful investigation. A few researchers have investigated the effect of CNT dimensions on different properties of nanocomposites [17, 18].

In this research, a multiscale analysis and finite element (FE) method are proposed to study the interaction of nanotubes and matrix at the nanoscale near a crack tip. The effects of the chirality, length, and radius of single-walled CNTs (SWCNTs) in a polymer matrix in the presence of van der Waals (vdW) interaction as interphase region on the fracture behavior were studied.

2 Modeling

Multiscale simulation, from the nanoscale to the macroscale or in the reverse order, has the capability to help better understand the toughness mechanism. Multiscale modeling is accomplished with either of two opposite approaches. One way is starting from the representative volume element (RVE) or unit cell and moving up to higher scales [19, 20], i.e., the bottom-up or upscaling or micromechanics analysis method. The other one starts from the structural scale, moving down to lower scales [21], i.e., the top-down or downscaling or global-local method. The advantage here is that the model can accurately reflect the load and boundary conditions according to the experimental setup. In this research, top-down method is utilized for stress intensity factor analysis of epoxy/SWCNTs.

The stress intensity factor of the epoxy/SWCNT nanocomposites was measured using the compact tension (CT) model according to ASTM D5045. First, the half CT model is analyzed on the basis of an effective modulus of elasticity. Reaction forces that are calculated on the cut boundary of the appropriately loaded global model are specified as boundary conditions for the local model. Due to the large difference in dimensional scales from the CT sample to the nanoscale RVE model, several local models with smaller sizes are built. Through data transfer among several local models, the size of the last submodel approaches the nanoscale.

The three-dimensional (3D) model is developed using the ANSYS commercial FE code. Isotropic properties are considered for the first step of modeling (half CT sample) [22]. Load and boundary conditions are applied to the CT model as shown in Figure 1. Reaction forces calculated from crack tip elements in the first step are implemented to the second step. Although a two- or three-level model can yield sufficiently proper results, by increasing the steps the accuracy of analysis will be improved. Then, a six-level FE model is simulated. In the smallest local model, RVE models are considered. The opening mode for bridging condition (SWCNT is orthogonal to crack and crack is in middle of RVE) in epoxy/SWCNT 0.1 wt% is considered. The crack propagation in a toughened material is slowed by the crack bridging mechanism. In this mechanism, CNT bridges on the path of the crack and resists the growth of crack.

Figure 1

Fracture simulation of epoxy/SWCNT by global-local method.

For the modeling of the CNT bonds, the 3D elastic BEAM4 element is used. The specific element is a uniaxial element with tension, compression, torsion, and bending capabilities. It has six degrees of freedom at each node: translations in the nodal x, y, and z directions and rotations about the nodal x, y, and z axes. The properties of these elements are obtained by linking the potential energy of bonds and the strain energy of mechanical elements. A circular beam of length l, diameter d, Young’s modulus E, and shear modulus G representing the covalent bond between carbon atoms is considered [1]. For matrix modeling, SOLID45 element was utilized [16]. The Young’s modulus of epoxy is given as 2.599 MPa [23]. The vdW interactions between carbon atoms of CNT and the nodes of the inner surface of the surrounding resin are modeled by using 3D nonlinear spring (COMBIN39) based on Lennard-Jones potential [16, 24, 25].

3 Results and discussion

3.1 The effect of chirality on stress intensity factor

The 3D FE model is proposed to study the effect of chirality of nanotubes on the stress intensity factor of RVE (Figure 2). CNTs possessing similar radius with (10, 10) CNT but with different chirality are represented in Table 1. These nanotubes have the same length that is equal to 100 Å. Stress intensity factors of RVEs containing SWCNT with different chirality subjected to similar loading are given in Table 1.

Table 1

Stress intensity factor of RVEs having CNTs possessing similar radius and length.

KI (MPam)	Length (Å)	Radius (Å)	Chirality
0.420	99.68	6.785	(10,10)
0.433	100.293	6.796	(11,9)
0.391	100.245	6.830	(12,8)
0.395	100.279	6.886	(13,7)
0.387	100.211	6.682	(14,5)
0.428	100.293	6.796	(15,4)
0.394	100.288	6.694	(16,2)
0.425	100.181	6.659	(17,0)

Figure 2

(A) FE meshes of RVE (iso view), (B) displacement, and (C) stress contours of RVE.

Figure 3 shows that RVEs with chiral nanotubes (14, 5) and (11, 9) have minimum and maximum stress intensity factors, respectively. These results originated from the fact that nanotube (14, 5) has the highest Young’s modulus compared to others. In addition, a comparison of armchair (10, 10) and zigzag (17, 0) nanotubes shows that armchair CNT has lower stress intensity factor.

Figure 3

Stress intensity factor of RVEs having CNTs possessing similar radius and length.

3.2 The effect of CNT length on stress intensity factor

The effect of length on the stress intensity factor of RVE is investigated by modeling (10, 10), (17, 0) and (14, 5) nanotubes with different lengths (25, 50, 75, 100 Å) as given in Table 2. The stress intensity factors are represented in Figure 4 showing that with increasing the length, the stress intensity factor decreases. Furthermore, for different lengths of nanotube, the chiral nanotube (14, 5) has the lowest stress intensity factor.

Figure 4

Stress intensity factor of RVEs with (10, 10), (17, 0), (14, 5) nanotubes for length of (25, 50, 75, 100 Å).

Table 2

Stress intensity factor of RVEs with (10, 10), (17, 0), (14, 5) nanotubes for length of (25, 50, 75, 100 Å) K_I (MPam)

(14, 5)	(17, 0)	(10, 10)	Chirality
			Length (Å)
0.5010	0.5310	0.5140	25
0.4820	0.5140	0.4990	50
0.4420	0.4760	0.4710	75
0.3870	0.4270	0.4200	100

4 Conclusion

The global-local multiscale method is proposed for analysis of the stress intensity factor of epoxy/SWCNT. Results show that CNTs have evident influences on the crack resistance of nanocomposites. Also, chirality has a significant effect on the stress intensity factor of RVE. Results show that in a constant load, chiral nanotubes (14, 5) and (11, 9) have minimum and maximum stress intensity factors, respectively. Moreover, by increasing the length of RVE in a constant chirality, the stress intensity factor decreases, and consequently, the crack resistance improves. In addition, in a certain length, when radiuses of armchair and zigzag nanotubes increase, the stress intensity factor decreases. Multiscale simulations from macro to nano or the reverse improved the specification of toughness mechanisms.