PSI - Issue 2_B

Chyanbin Hwu et al. / Procedia Structural Integrity 2 (2016) 1327–1334 H wu and Yeh / Structural Integrity Procedia 00 (2016) 000–000

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model was proposed to estimate the stiffness of nanomaterials (Hwu and Yeh, 2014). In that model the potential energy describing the interactions of atoms is not restricted to the harmonic potential function, and hence its deriving stress-strain relation is not restricted to be linear. By taking proper potential energy function such as modified Morse potential function, and applying proper strain field for uniform tension, a nonlinear stress-strain diagram can be plotted for the carbon nanotubes. Through this diagram, the tensile strength can be predicted from the zero derivative of stress with respect to strain (Yeh and Hwu, 2016). To predict fracture toughness of carbon nanotubes, a new parameter called the strain intensity factor was introduced to be the counterpart of the stress intensity factor. The near tip solution of linear elastic fracture mechanics is then rewritten in terms of strain intensity factor to locate the atoms in the deformed state of the cracked specimen. After setting the proper deformation field, the changes of bond distance and bond angle between atoms can be obtained. With this information, the potential energy within the region of representative volume can be calculated. By treating this potential energy as the strain energy in the deformed cracked specimen, and using the well-known relation between strain energy release rate and stress intensity factor, a nonlinear generalized stress strain diagram which showing the relation between stress intensity factor and strain intensity factor, can be plotted for the carbon nanotubes. The estimated fracture toughness can then be obtained from the maximum point of this diagram. To know whether our prediction is stable with respect to the crack increment, crack length and tube radius, prediction based upon different parameters were presented in our recent study (Yeh and Hwu, 2016). The numerical results show that our prediction falls in the reasonable range set by the other methods. Following the success of our recent study, in this paper attention is focused on the proof of correctness with proper representative volume and the comparison of various crack simulation in carbon nanotubes. 2. Modified molecular-continuum model A modified molecular-continuum model was proposed to estimate the fracture toughness of nanomaterials (Yeh and Hwu, 2016). To be specifically employed to the cases of carbon nanotubes discussed in this paper, the procedure of this model is stated as follows. (1) Select an appropriate representative volume element (RVE), which is set to be a circular region with center at the crack tip and radius 0 r a η = (see Fig. 1), where a is the half-length of crack and

10 2 , mode I 5 3 6 2 , mode II 9 ν ν ν ν  −  −  +  +

(1)

, for plane stress condition.

= 

η

ν is the Poisson’s ratio of carbon nanotubes, which can be estimated by using the molecular-continuum model (Hwu and Yeh, 2014). The results estimated by this model is shown in Table 1, from which we see that 1 / 1.008 z z θ θ ν ν < < . Due to the small difference between z θ ν and z θ ν , the carbon nanotubes can be treated as an isotropic material and the Poisson’s ratio ν in eq. (1) was taken to be the value of z θ ν shown in Table 1.

2 a

a η

Fig. 1. RVE for the estimation of fracture toughness.