Molecular dynamics investigation of mechanical properties of single-layer phagraphene

Abstract

Phagraphene is a very attractive two-dimensional (2D) full carbon allotrope with very interesting mechanical, electronic, optical, and thermal properties. The objective of this study is to investigate the mechanical properties of this new graphene like 2D material. In this work, mechanical properties of phagraphene have been studied not only in the defect-free form, but also with the critical defect of line cracks, using the classical molecular dynamics simulations. Our study shows that the pristine phagraphene in zigzag direction experience a ductile behavior under uniaxial tensile loading and the nanosheet in this direction are less sensitive to temperature changes as compared to the armchair direction. We studied different crack lengths to explore the influence of defects on the mechanical properties of phagraphene. We also investigated the temperature effect on the mechanical properties of pristine and defective phagraphene. Our classical atomistic simulation results confirm that larger cracks can reduce the strength of the phagraphene. Moreover, it was shown the temperature has a considerable weakening effect on the tensile strength of phagraphene. The results of this study may be useful for the design of nano-devices using the phagraphene.

Keywords

phaqraphene / mechanical properties / crack propaqation / molecular dynamics / thermal effects

Introduction

Two-dimensional (2D) materials are considered as the new class of materials for numerous and diverse applications ranging from nanoelectronics to aerospace structures. 2D materials have keep attracting the interest of researchers of different fields since the isolation of graphene from graphite reported in 2004 [¹–⁴]. The monolayer graphene with a honeycomb structure is the strongest carbon allotrope with heavily \sigma bands which are even stronger than bonds in diamond. The lattice of carbon atoms in graphene yields the thinnest, lightest, and strongest materials which show unique mechanical and thermal properties. Graphene can withstand a tensile strength of 130 GPa [⁵] which is to date the highest reported experimentally value. Interestingly, graphene exhibit the highest measured thermal conductivity among all the materials ever known [⁶]. The applications of graphene and other 2D materials have been the topic of numerous studies during the last decade [⁷–¹⁰]. In this regard, graphene and other 2D materials because of their uniquely high mechanical properties and thermal conductivity have been widely employed in the fabrication of advanced polymer composites with enhanced mechanical and thermal conduction properties [¹¹–²²]. Due to the excellent physical properties of 2D materials, they are potential candidates to improve numerous current technologies in various systems such as nanoelectronics, supercapacitors, rechargeable metal-ion batteries, optoelectronics, superconductors and mechanically robust components [²⁰–²⁹].

The interest toward the 2D materials are mainly due to their extremely high ratio of surface area to mass [³⁰‒³³] which makes them ideal for applications in energy conversion and storage, chemical functionalization and sensing and mechanical load and heat transfer. The fascinating properties of 2D materials can be further tailored using the foreign atoms or fabricating heterostructures. For example, graphene is a semimetal with a zero band gap which makes it difficult to be used in electronic circuits as a transistor [³⁴]. Interestingly, the zero band gap in graphene can be modified by defect engineering, chemical doping using the nitrogen or boron atoms or fabrication of 2D heterostructures in which another 2D materials, like hexagonal boron-nitride are used [³⁵–⁴¹].

Recently, phagraphene [⁴²] a new full carbon 2D material composed of 5, 6, and 7 carbon rings was theoretically realized. According to the theoretical results this new planar carbon allotrope is energetically comparable to graphene [⁴²] and such that phagraphene from the energy point of view is more favorable than other 2D carbon allotropes proposed so far (except the graphene). The high stability of phagraphene is due to its sp²-hybridization and also its dense atomic packing [⁴²]. Recent theoretical studies have confirmed that phagraphene can show very attractive electronic, magnetic and thermal conduction properties and such that it might be useful for the design of novel nano-devices [⁴³–⁴⁶]. Nevertheless, to the best of our knowledge there exist only a single research paper [⁴⁶] on the mechanical properties of phagraphene which was limited to study the mechanical response of pristine film and only at the room temperature. In this regard, effects of the mechanical defects and temperature on the mechanical properties of phagraphene have not been explored so far. We remind that there exist always defects in the fabricated nano-materials. These defects can extremely reduce the ultimate tensile strength of the material [⁴⁷–⁵¹]. Therefore, it is critical to study the mechanical properties of the materials in presence of defects such as notch and line crack. It should be noted that the fracture is dependent on the size of the crack. There are different methods to study the fracture mechanism such as fracture mechanic from the continuum point of view. In the continuum level, large crack lengths can be captured and analyzed efficiently on the basis of fracture mechanics theories [⁵²–⁵⁶]. Furthermore, the temperature effect on the mechanical properties is always worthy to investigate because it defines the temperature range that a material can be used without facing degradation. In current study, we investigate the mechanical properties of the phagraphene with the defects of line crack. We study the propagation of the crack for the different crack lengths diameters at different temperatures up to 1000 K. The information provided by this study can be very useful to use phagraphene in future nano-devices.

Molecular dynamics modelling

The classical molecular dynamics simulations were used to study the mechanical properties of phagraphene nanosheets including the defect of crack with different lengths. The molecular dynamics simulations were performed through the open-source software of LAMMPS (Large-scale Atomic/Molecular Massively Parallel Simulator) [⁵⁷]. Tersoff potential with optimized Tersoff parameters for graphene were used in the simulations which models the atomic interactions between carbon atoms. This modified Tersoff potential has been used in many studies of the properties of graphene. The accuracy of the optimized Tersoff potential for the phagraphene has been verified in order to ensure the stability of the phagraphene crystal structure. The results of our study were depicted through the open-source visualization software called OVITO [⁵⁸]. The calculated stresses, strains, and the atomistic positions of the atoms were recorded via OVITO. The structure of the phagraphene is shown in Fig. 1 that is similar to graphene includes two major orientations of armchair and zigzag. The pentagonal, hexagonal, and heptagonal carbon rings are also depicted in the figure. Carbon rings are connected together with strong polar covalent bonds. The structure of the phagraphene and the boundary conditions are periodic.

The simulations performed in both directions of armchair and zigzag for the pristine phagraphene and the phagraphene nanosheet with the defect of line crack. The uniaxial tensile stresses were applied in order to explore the ultimate tensile strength of the phagraphene nanosheet. The effect of temperature on the models also investigated in a range of 200 K to 1000 K. The mechanical properties of the nanosheets were explored under uniaxial tension loading. A monolayer phagraphene nanosheet contains 21,600 atoms were considered for the molecular dynamic simulations. The boundary conditions are periodic in the in-plane direction. First of all, the Nosé-Hoover barostat and thermostat (NPT) method were performed for relaxation of simulation box in order to ensure the stress free condition in the first step. An engineering strain rate of 10⁸ s^‒1 was applied in every small time steps of 0.25 fs. As a consequence of the applied strain rate, the nanosheet elongates in every time step of the simulation. The positions of the atoms were rescaled at every time step in order to avoid the void formation or fast unexpected stretch of atomic bond. Virial stresses were calculated over every time stem based on Virial theorem [⁵⁹]. The stresses should be averaged over every time steps of the simulation to obtain the stress values.

Results and discussion

To validate of our models, we compared our simulations with the work of Pereira et al. [⁴⁶]. It was shown at room temperature for a pristine phagraphene, the tensile strength is 82 GPa in zigzag direction and 86 GPa in armchair direction. The models were stretched with the same strain rate of the previous study [⁴⁶]. The obtained results were in a very good agreement with their work. In our study we considered a nanosheet with the dimensions of 300×180 Å made from the monolayer phagraphene with a thickness of 3.2 Å. The models were simulated at different temperatures form 200 K to 1000 K. The strain rates applied in the molecular dynamic simulations are usually high as compared with experimental setups. Nevertheless we used the strain rate of 10⁸ s^‒1, in accordance with the work by Pereira et al. [⁴⁶] which very accurately reproduces the mechanical properties of pristine graphene reported experimentally by Lee et al. [⁶⁰].

At first, the pristine phagraphene were stretched in two main directions at room temperature. Figure 2 shows the predicted stress-strain curves of phagraphene which show that the ultimate tensile stresses occur at two different points for the armchair and zigzag loading directions. This indicates that the mechanical properties of phagraphene are anisotropic along the two main orientations of its structure. As well, phagraphene elongates more in the zigzag direction and shows a unique behavior after the linear elastic zone up to the rupture of the nanosheet. This behavior can be more explained with the process of the nanosheet rupture under the uniaxial tension loading. In this figure phagraphene shows a more ductile behavior in the zigzag direction. Point A is a turning-point in which the slope of the stress-strain curve decreases and in point B the slope of the stress-strain curve increases up to the end of the rupture process. The reason of such a behavior should be investigated in the structure of phagraphene with 5, 6, and 7 carbon rings.

Figure 3 shows the progress of the rupture of the pristine phagraphene nanosheet under normal tension in the zigzag direction at room temperature. Figure 3(a) shows the initial steps of the simulation where the stresses are not yet increased. In Fig. 3(b), the atomic bond between two pentagonal carbon ring breaks and form a bigger carbon ring with ten carbon atoms. Figure 3(b) has been shown partially in an inset in order to illustrate the carbon atom rings. The formation of these carbon rings is related to the point A in Fig. 2. The big carbon rings increases with the loading and spread in the whole nanosheet (Fig. 3(c)). From the point B up to the complete rupture of the nanosheet in point C, the stresses increase and the stress distribution in the nanosheet has a uniform change. The process of the pristine phagraphene under tensile stress provides a more ductile failure as compared to the graphene and also phagraphene in the armchair direction in which the breakage of the bonds and the rupture occur very close together.

In the next step of our study, the pristine models were stretched at the temperatures of 200, 300, 500, 800, and 1000 K in order to study the effect of temperature on mechanical properties of the phagraphene nanosheet. Figure 4 shows the stress-strain curves of the simulations in the directions of zigzag and armchair. As can be seen, the tensile strength of the nanosheet decreases when the temperature rises. It can be described according to the strength of the atomic bonds. At higher temperatures, atomic bonds are weaker, therefore, the material can withstand lower tensions. Furthermore, in the zigzag direction, the effect of the ductile fracture which was discussed in the previous section decreases as the temperature increases. Ductile fracture obviously occurs at 200 K in which there are less molecular motions and there are stronger atomic bonds between carbon atoms. It is also worth to note that the strain energy in zigzag directions is more than the armchair directions. It can be shown from the area under the stress-strain curves in the zigzag direction that is more from the case of armchair direction. In other words, in the zigzag direction, the nanosheet absorbs more energy until it reaches to the point of fracture.

From the stress-strain curves, we can draw the curve of the maximum stresses and the related maximum strains. The values of the maximum stresses and strain for the defect-free phagraphene at different temperatures are illustrated in the Fig. 5. It shows in Fig. 5(a) that in both armchair and zigzag direction, the tensile strength decreases when the temperature increases. The tensile strength of the pristine at the zigzag direction decreases 47% while the decrement is 65% in the armchair direction. It means the pristine phagraphene is more sensitive to the temperature changes in armchair direction, however it can tolerate more tensile strength at lower temperatures. Furthermore, the nanosheet in zigzag direction can withstand more tensile stress at temperatures higher than 500 K that shows it is stiffer at higher temperatures. Figure 5(b) shows the strain of the pristine graphene at the maximum tensile stress. It shows that at different temperatures in both armchair and zigzag directions, the strain decreases with the temperature rise.

In the next step of our study, we focus on the influence of the line cracks on the thermo-mechanical properties of the phagraphene nanosheet. The cracks in the nanosheet were modelled through the districted atomic bonds in the crack zone. The atomic bonds were disconnected in crack faces in order to simulate a line crack in the nanosheet. Different crack lengths were selected, i.e., L/9, L/6, L/4, and L/3 where L is the width of the nanosheet. The cracks were modelled in the center of the nanosheet. Figure 6 shows the tensile strength of phagraphene nanosheets with different crack lengths at different temperatures. All models were stretched in two main orientations of the phagraphene. It shows that the tensile strength of the nanosheet decreases when the temperature increases. As well, the tensile strength decreases with the increment of crack lengths. Furthermore, similarly to the case of pristine phagraphene, the defective nanosheets in zigzag direction have more tensile strength at high temperatures. Results shown in Fig. 6 confirm that the thermo-mechanical properties of the defective phagraphene nanosheet are dependant to the loading direction along the armchair and zigzag. In another words, phagraphene shows anisotropic mechanical properties.

The crack propagation also depend on the direction of the crack with respect to the direction of applied load [⁶¹]. We have studied the fracture mode I in which the line crack is normal to the uniaxial tensile loading. Stress intensity factor (SIF) is an important factor in crack propagation. The SIF at the ultimate tensile strength (critical SIF) can be calculated as the following [⁶²]:

(1){{\displaystyle K}}_{IC}={{\displaystyle \sigma }}_f\sqrt{\mathrm{2}h\tan \left({\displaystyle \frac{\mathrm{\pi }a}{\mathrm{2}h}}\right)},

where {{\displaystyle \sigma }}_f is the maximum tensile stress, and \mathrm{2}h is the width of the nanosheet, and \mathrm{2}a is the initial crack length. Figure 7 shows the calculated critical SIF for the defective phagraphene nanosheet with the crack length of L/6 in two main orientations of the phagraphene structure at different temperatures. The simulations were done several times in order to obtain the range of the error of the results. The deviations of the results are illustrated with the error bars on the figure. The trend of the critical SIF shows that it decreases with the increasing of temperature. The critical SIF depends on the geometry of the crack and the ultimate strength as well. At higher temperatures, the atomic bonds are weaker which leads to the lower tensile strength. The simulations in the zigzag direction show that the critical SIF at higher temperatures is more than critical SIF for the armchair directions. The range of the difference in critical SIF in armchair direction is larger. The smaller range of the critical SIF in the zigzag direction shows that the nanosheet in this direction is less sensitive to the temperature rises.

The crack propagation of a nanosheet with the crack length of L/9 in armchair direction at 300 K is illustrated in Fig. 8. The fracture mode I has been considered to illustrate the crack propagation of the nanosheet under uniaxial tensile loading. Figure 8(a) depicts the initial steps of the loading. These steps can be found in the linear zone of the stress-strain diagram. The high stresses increase in the crack tips while there are smaller stresses in the crack faces, see Figs. 8(b) and 8(c). The rupture of the whole nanosheet along the direction of the line crack is shown in Fig. 8(d). The corresponding strain values to each step are given on the figure. In order to have a better visualization in the crack zone, insets of the Figs. 8(a) and 8(b) are depicted on the figure.

Conclusions

Our study was focused on the mechanical properties of the monolayer phagraphene nanosheets. We considered the pristine nanosheets and defective nanosheets with the line cracks. The models were stretched in two directions of armchair and zigzag. They were tested at a temperature range of 200 K to 1000 K. Our results for the pristine models showed that the phagraphene nanosheets in the zigzag direction have a ductile fracture behavior under the uniaxial tension loading. It was shown the start point of the fracture and the complete rupture are not very close together. The reasons of this phenomenon were extensively discussed according to the formation of the rings of carbon rings with ten carbon atoms. It was shown for the pristine models that the increase of temperature decrease the ultimate tensile strength and the engineering strain at the maximum tensile stress. As well, it was shown that the models in zigzag direction have higher tensile strength at higher temperature as compared to the models in armchair direction. The crack models were simulated with different line crack lengths at different temperatures. It was shown that the temperature rise has a strong weakening effect on tensile strength of the nanosheet. It was also shown that when the crack length increases the ultimate tensile strength decreases. The critical stress intensity factor was calculated for a crack model in two main orientations of phagraphene nanosheet. It was shown that the critical intensity factor decreases with the temperature increment. It was shown that at higher temperatures there exist higher values of critical stress intensity factor for the nanosheets in zigzag direction. The crack propagation of a model in armchair direction was discussed. It was shown that in the crack tips the stress values are higher than other points while crack faces experience smaller amounts of stress. Our results can be a useful guide for the thermo-mechanical design of nano-devices made from phagraphene nanosheets.

References

[1] Novoselov K S, Geim A K, Morozov S V, Jiang D, Zhang Y, Dubonos S V, Grigorieva I V, Firsov A A. Electric field effect in atomically thin carbon films. Science, 2004, 306(5696): 666–669

[2] Geim A K, Novoselov K S. The rise of graphene. Nature Materials, 2007, 6(3): 183–191

[3] Stankovich S, Dikin D A, Piner R D, Kohlhaas K A, Kleinhammes A, Jia Y, Wu Y, Nguyen S B T, Ruoff R S. Synthesis of graphene-based nanosheets via chemical reduction of exfoliated graphite oxide. Carbon, 2007, 45(7): 1558–1565

[4] Stankovich S, Piner R D, Chen X, Wu N, Nguyen S T, Ruoff R S. Stable aqueous dispersions of graphitic nanoplatelets via the reduction of exfoliated graphite oxide in the presence of poly (sodium 4-styrenesulfonate). Journal of Materials Chemistry, 2006, 16(2): 155–158

[5] Wang X, Zhi L, Müllen K. Transparent, conductive graphene electrodes for dye-sensitized solar cells. Nano Letters, 2008, 8(1): 323–327

[6] Ghosh S, Calizo I, Teweldebrhan D, Pokatilov E P, Nika D L, Balandin A A, Bao W, Miao F, Lau C N. Extremely high thermal conductivity of graphene: prospects for thermal management applications in nanoelectronic circuits. Applied Physics Letters, 2008, 92(15): 151911

[7] Geim A K. Graphene: status and prospects. Science, 2009, 324(5934): 1530–1534

[8] Novoselov K S. Graphene: Materials in the Flatland. Nobel Lecture, 2010, 106–131

[9] Geim A K. Nobel Lecture: random walk to graphene. Reviews of Modern Physics, 2011, 83(3): 851–862

[10] Mortazavi B, Pereira L F C, Jiang J W, Rabczuk T. Modelling heat conduction in polycrystalline hexagonal boron-nitride films. Scientific Reports, 2015, 5(1): 13228

[11] Eigler S. Graphene. An Introduction to the Fundamentals and Industrial Applications. Edited by Madhuri Sharon and Maheshwar Sharon. Angewandte Chemie International Editon. Wiley-Blackwell, 2016, doi: 10.1002/anie.201602067

[12] Sainsbury T, Gnaniah S, Spencer S J, Mignuzzi S, Belsey N A, Paton K R, Satti A. Extreme mechanical reinforcement in graphene oxide based thin-film nanocomposites via covalently tailored nanofiller matrix compatibilization. Carbon, 2017, 114: 367–376

[13] Kim Y, Lee J, Yeom M S, Shin J W, Kim H, Cui Y, Kysar J W, Hone J, Jung Y, Jeon S, Han S M. Strengthening effect of single-atomic-layer graphene in metal–graphene nanolayered composites. Nature Communications, 2013, 4, doi: 10.1038/ncomms3114

[14] Mortazavi B, Hassouna F, Laachachi A, Rajabpour A, Ahzi S, Chapron D, Toniazzo V, Ruch D. Experimental and multiscale modeling of thermal conductivity and elastic properties of PLA/expanded graphite polymer nanocomposites. Thermochimica Acta, 2013, 552: 106–113

[15] Mortazavi B, Rabczuk T. Multiscale modeling of heat conduction in graphene laminates. Carbon, 2015, 85: 1–7

[16] Malekpour H, Chang K H, Chen J C, Lu C Y, Nika D L, Novoselov K S, Balandin A A. Thermal conductivity of graphene laminate. Nano Letters, 2014, 14(9): 5155–5161

[17] Mortazavi B, Yang H, Mohebbi F, Cuniberti G, Rabczuk T. Graphene or h-BN paraffin composite structures for the thermal management of Li-ion batteries: a multiscale investigation. Applied Energy, 2017, 202: 323–334

[18] Msekh M A, Silani M, Jamshidian M, Areias P, Zhuang X, Zi G, He P, Rabczuk T. Predictions of J integral and tensile strength of clay/epoxy nanocomposites material using phase field model. Composites. Part B, Engineering, 2016, 93: 97–114

[19] Almasi A, Silani M, Talebi H, Rabczuk T. Stochastic analysis of the interphase effects on the mechanical properties of clay/epoxy nanocomposites. Composite Structures, 2015, 133: 1302–1312

[20] Vu-Bac N, Silani M, Lahmer T, Zhuang X, Rabczuk T. A unified framework for stochastic predictions of mechanical properties of polymeric nanocomposites. Computational Materials Science, 2015, 96: 520–535

[21] Silani M, Talebi H, Ziaei-Rad S, Kerfriden P, Bordas S P A, Rabczuk T. Stochastic modelling of clay/epoxy nanocomposites. Composite Structures, 2014, 118: 241–249

[22] Hamdia K M, Msekh M A, Silani M, Vu-Bac N, Zhuang X, Nguyen-Thoi T, Rabczuk T. Uncertainty quantification of the fracture properties of polymeric nanocomposites based on phase field modeling. Composite Structures, 2015, 133: 1177–1190

[23] Mortazavi B, Dianat A, Cuniberti G, Rabczuk, T. Application of silicene, germanene and stanene for Na or Li ion storage: a theoretical investigation. Electrochimica Acta, 2016, 213: 865–870

[24] Shirazi A H N, Abadi, R, Izadifar M, Alajlan N, Rabczuk T.Mechanical responses of pristine and defective C3N nanosheets studied by molecular dynamics simulations. Computational Materials Science, 2018, 147, 316–321

[25] Mahmood J, Lee E K, Jung M, Shin D, Choi H J, Seo J M, Jung S M, Kim D, Li F, Lah M S, Park N, Shin H J, Oh J H, Baek J B. Two-dimensional polyaniline (C3N) from carbonized organic single crystals in solid state. Proceedings of the National Academy of Sciences of the United States of America, 2016, 113(27): 7414–7419

[26] Mortazavi B. Ultra high stiffness and thermal conductivity of graphene like C3N. Carbon, 2017, 118: 25–34

[27] Mortazavi B, Rahaman O, Rabczuk T, Pereira L F C. Thermal conductivity and mechanical properties of nitrogenated holey graphene. Carbon, 2016, 106: 1–8

[28] Mortazavi B, Rémond Y, Ahzi S, Toniazzo V. Thickness and chirality effects on tensile behavior of few-layer graphene by molecular dynamics simulations. Computational Materials Science, 2012, 53(1): 298–302

[29] Mortazavi B, Rahaman O, Dianat A, Rabczuk T. Mechanical responses of borophene sheets: a first-principles study. Physical Chemistry Chemical Physics, 2016, 18(39): 27405–27413

[30] Mortazavi B, Dianat A, Rahaman O, Cuniberti G, Rabczuk T. Borophene as an anode material for Ca, Mg, Na or Li ion storage: a first-principle study. Journal of Power Sources, 2016, 329: 456–461

[31] Liu Y, Peng X. Recent advances of supercapacitors based on two-dimensional materials. Applied Material Today, 2017, 8: 104–115

[32] Mortazavi B, Shahrokhi M, Rabczuk T, Pereira L F C. Electronic, optical and thermal properties of highly stretchable 2D carbon Ene-yne graphyne. Carbon, 2017, 123: 344–353

[33] Liu W R. Graphene-based energy devices. In: Rashid bin Mohd Yusoff, ed. Graphene-Based Energy Devices. Wiley-VCH, 2015, 85–122

[34] Xia F, Farmer D B, Lin Y M, Avouris P. Graphene field-effect transistors with high on/off current ratio and large transport band gap at room temperature. Nano Letters, 2010, 10(2): 715–718

[35] Panchakarla L S, Subrahmanyam K S, Saha S K, Govindaraj A, Krishnamurthy H R, Waghmare U V, Rao C N R. Synthesis, structure, and properties of boron- and nitrogen-doped graphene. Advanced Materials, 2009, 21: 4726–4730

[36] Lherbier A, Blase X, Niquet Y M, Triozon F, Roche S. Charge transport in chemically doped 2D graphene. Physical Review Letters, 2008, 101(3): 036808

[37] Liu Z, Ma L, Shi G, Zhou W, Gong Y, Lei S, Yang X, Zhang J, Yu J, Hackenberg K P, Babakhani A, Idrobo J C, Vajtai R, Lou J, Ajayan P M. In-plane heterostructures of graphene and hexagonal boron nitride with controlled domain sizes. Nature Nanotechnology, 2013, 8(2): 119–124

[38] Van Tuan D, Kotakoski J, Louvet T, Ortmann F, Meyer J C, Roche S. Scaling properties of charge transport in polycrystalline graphene. Nano Letters, 2013, 13(4): 1730–1735

[39] Mortazavi B, Rahaman O, Makaremi M, Dianat A, Cuniberti G, Rabczuk T. First-principles investigation of mechanical properties of silicene, germanene and stanene. Physica E: Low-Dimensional Systems and Nanostructures, 2017, 87: 228–232.

[40] Cresti A, Nemec N, Biel B, Niebler G, Triozon F, Cuniberti G, Roche S. Charge transport in disordered graphene-based low dimensional materials. Nano Research, 2008, 1(5): 361–394

[41] Mortazavi B, Lherbier A, Fan Z, Harju A, Rabczuk T, Charlier J C. Thermal and electronic transport characteristics of highly stretchable graphene kirigami. Nanoscale, 2017, 9(42): 16329–16341

[42] Wang Z, Zhou X F, Zhang X, Zhu Q, Dong H, Zhao M, Oganov A R. Phagraphene: a low-energy graphene allotrope composed of 5-6-7 carbon rings with distorted dirac cones. Nano Letters, 2015, 15(9): 6182–6186

[43] Liu Y, Chen Z, Hu S, Yu G, Peng Y. The influence of silicon atom doping phagraphene nanoribbons on the electronic and magnetic properties. Materials Science and Engineering B, 2017, 220: 30–36

[44] Luo A Y, Hu R, Fan Z Q, Zhang H L, Yuan J H, Yang C H, Zhang Z H. Electronic structure, carrier mobility and device properties for mixed-edge phagraphene nanoribbon by hetero-atom doping. Organic Electronics, 2017, 51: 277–286

[45] Yuan P F, Fan Z Q, Zhang Z H. Magneto-electronic properties and carrier mobility in phagraphene nanoribbons: a theoretical prediction. Carbon, 2017, 124: 228–237

[46] Pereira L F C, Mortazavi B, Makaremi M, Rabczuk T. Anisotropic thermal conductivity and mechanical properties of phagraphene: a molecular dynamics study. RSC Advances, 2016, 6(63): 57773–57779

[47] Abadi R, Uma R P, Izadifar M, Rabczuk T. The effect of temperature and topological defects on fracture strength of grain boundaries in single-layer polycrystalline boron-nitride nanosheet. Computational Materials Science, 2016, 123: 277–286

[48] Mortazavi B, Cuniberti G. Mechanical properties of polycrystalline boron-nitride nanosheets. RSC Advances, 2014, 4(37): 19137–19143

[49] Abadi R, Uma R P, Izadifar M, Rabczuk T. Investigation of crack propagation and existing notch on the mechanical response of polycrystalline hexagonal boron-nitride nanosheets. Computational Materials Science, 2017, 131: 86–99

[50] Talebi H, Silani M, Bordas S P, Kerfriden P, Rabczuk T. Molecular dynamics/XFEM coupling by a three-dimensional extended bridging domain with applications to dynamic brittle fracture. International Journal for Multiscale Computational Engineering, 2013, 11(6): 527–541

[51] Talebi H, Silani M, Rabczuk T. Concurrent multiscale modeling of three dimensional crack and dislocation propagation. Advances in Engineering Software, 2015, 80: 82–92

[52] Silani M, Talebi H, Hamouda A M, Rabczuk T. Nonlocal damage modelling in clay/epoxy nanocomposites using a multiscale approach. Journal of Computational Science, 2016, 15: 18–23

[53] Talebi H, Silani M, Bordas S P A, Kerfriden P, Rabczuk T. A computational library for multiscale modeling of material failure. Computational Mechanics, 2014, 53(5): 1047–1071

[54] Budarapu P R, Gracie R, Bordas S P A, Rabczuk T. An adaptive multiscale method for quasi-static crack growth. Computational Mechanics, 2014, 53(6): 1129–1148

[55] Mortazavi B, Cuniberti G, Rabczuk T. Mechanical properties and thermal conductivity of graphitic carbon nitride: a molecular dynamics study. Computational Materials Science, 2015, 99: 285–289

[56] Hamdia K M, Silani M, Zhuang X, He P, Rabczuk T. Stochastic analysis of the fracture toughness of polymeric nanoparticle composites using polynomial chaos expansions. International Journal of Fracture, 2017, 206(2): 215–227

[57] Plimpton S. Fast parallel algorithms for short-range molecular dynamics. Journal of Computational Physics, 1995, 117(1): 1–19

[58] Stukowski A. Visualization and analysis of atomistic simulation data with OVITO–the Open Visualization Tool. Modelling and Simulation in Materials Science and Engineering, 2010, 18(1): 015012

[59] Tsai D H. The virial theorem and stress calculation in molecular dynamics. Journal of Chemical Physics, 1979, 70(3): 1375–1382

[60] Lee C, Wei X, Kysar J W, Hone J. Measurement of the elastic properties and intrinsic strength of monolayer grapheme. Science, 2008, 321(5887): 385–388

[61] Irwin G R. Fracture . In: Flügge S, ed. Elast. Plast. / Elastizität Und Plast. Springer Berlin Heidelberg, 1958, 551–590

[62] Gross D, Seelig T. Fracture Mechanics: With An Introduction to Micromechanics (2nd ed). Heidelberg: Springer, 2011

RIGHTS & PERMISSIONS

Higher Education Press and Springer-Verlag GmbH Germany, part of Springer Nature