Fracture strength of Graphene at high temperatures: data driven investigations supported by MD and analytical approaches

Abstract

The extraordinary opto-electronic and mechanical properties of Graphene makes it a popular material for several applications. However, defects like: cracks, and voids are unavoidable during its production, which can lead to poor properties. Furthermore, the fracture properties degrades at higher temperatures. In this study, the fracture strength of Graphene is investigated as a function of temperature, considering the influence of lattice orientation, initial crack size and its orientation. As a first step, an analytical model is developed to estimate the fracture strength of Graphene with respect to temperature, considering the above parameters. Later on, molecular dynamics simulations are performed with an included initial edge crack in ten different sizes and four orientations, at three particular lattice orientations, and operating at thirteen different temperatures. Finally, a deep machine learning model is developed to estimate the fracture strength of defective Graphene. Results from molecular dynamics simulations are used to train the developed deep machine learning model. Furthermore, the training is enhanced using transfer learning, where the weights and biases for the data set considering $0^\circ$ lattice orientation are adopted in training the networks for $13.9^\circ$ and $30^\circ$ lattice orientations. Results from the developed deep machine learning model are validated by comparing them with the results from the analytical and molecular dynamics models and a good agreement is observed. Thus, a deep machine learning model has been proposed here to estimate the fracture strength of defective Graphene. The developed model serves as a tool for quick estimation fracture strength of defective Graphene.

Appendix A: Correlation to Griffith’s theory

According to Griffith’s theory (Griffith 1921), brittle fracture occurs when the released strain energy exceeds the surface energy required for an infinitesimal extension of the crack. Consider a domain having an initial edge crack of size a. Based on the Griffith criterion the critical stress at the onset of fracture, $\sigma _f$, can be expressed as a function of a, as (Peng et al. 2014; Javvaji et al. 2016):

$$\begin{aligned} \sigma _f = \sqrt{\frac{2\gamma \text {E}}{\pi a}}, \end{aligned}$$

(A.1)

where E is Young’s modulus and $\gamma$ is the surface energy, which is the edge energy for a 2D material like Graphene. The left-hand side of Eq. (A.1) contains only the computed fracture strength and crack size, whereas the right-hand side depends on material parameters and the term $1/\sqrt{a}$. Therefore, based on Eq. (A.1), the Griffith theory of brittle fracture is applicable to Graphene, if the estimated pairs of $\sigma _f$ and a results in a constant product of $\sigma _f\sqrt{a}$.

However, in the present study, the fracture strength of Graphene as a function of temperature and strain rate is given by Eq. (18). The fracture strength in Eq. (18) is a function of fracture strength of pristine Graphene, crack tip radius, fracture quantum, crack orientation and initial crack size. The fracture strength of pristine Graphene is a function of the temperature. For a given domain with initial crack orientation, the parameters w, $\rho$ and $\theta$ in Eq. (18) will be constant. As a result, the temperature dependent fracture strength $\sigma ^{\text {CL},{\dot{\varepsilon }},T}_f$ in Eq. (18) can be expressed as:

$$\begin{aligned} \sigma ^{\text {CL},{\dot{\varepsilon }},T}_f = \frac{\text {constant}}{\sqrt{a+a'_\text {fq}/2}}. \end{aligned}$$

(A.2)

As $a'_\text {fq} \ll a$, the term $\sqrt{a+a'_\text {fq}/2}$ can be replaced with a. Therefore, the temperature dependent fracture strength in Eq. (A.2) will be inversely proposal to $\sqrt{a}$.

Figure 21 shows the variation of natural logarithm of fracture strength ($\ln ({\sigma _f}$)) with respect to the natural logarithm of initial crack length ($\ln ({a}$)), when the lattice is oriented along the 0$^\circ$, 13.9$^\circ$ and 30$^\circ$ directions and crack orientations equal to 0$^\circ$ and 30$^\circ$.

Fig. 21

Variation of natural logarithm of fracture strength ($\ln ({\sigma _f}$)) with respect to the natural logarithm of initial crack length ($\ln ({a}$)), considering a $\alpha$ = 0$^\circ$ and $\theta$ = 0$^\circ$, b $\alpha$ = 13.9$^\circ$ and $\theta$ = 30$^\circ$ and c $\alpha$ = 30$^\circ$ and $\theta$ = 30$^\circ$

The slope of each curve in Fig. 21 is observed to be negative with increase in crack size, indicating the inverse relationship between fracture strength and crack size. The slope of an ideal brittle material satisfying Griffith’s criteria will be equal to − 0.5. The results in Fig. 21a–c are observed to be closely following the Griffith’s theory with an average slope of − 0.45. Also, the trend is in agreement with published results in Javvaji et al. (2016). Therefore, the Griffith’s criteria is observed to be satisfied at higher temperatures.