2Dmechanics.com

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Table 4 Mechanical properties of various 2D crystals. The derived method in the last column (experiment or theory) refers to the modulus and strength.

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3

2D Material
(nL=No layers)

E2D
(N/m)

E
(GPa)

Strength
(GPa)

Fracture Strain (%)

Interlayer shear modulus (GPa)

Bending rigidity
(eV)

Poisson’s ratio

Method

4

Graphene

1L[1]

340

1000

130

30

-

1-2.4[2, 3]

0.13-0.20[4-7]

Exp.

5

2L[8]

698

1040

126

-

0.40[9]

3.35[9]

«

Exp.

6

3L[8]

986

980

101

-

0.49

6.92

«

Exp.

7

4L[10]

-

1020

-

0.47

12.5

«

Theory

8

5L[10]

-

1020

-

0.4

18.1

«

Theory

9

6L

-

0.36

28.29

«

Exp.

10

CVD (1L)[11]

324-328

1000

90-99

-

«

Exp.

11

MoS2[12]

1L

180

270

22

6-11

-

9.61-10.2[2, 13]

0.27[14]

Exp.

12

2L

260

200

21

6-11

19[15]

74[16]

«

Exp.

13

h-BN[17]

1-9L

-

865

70.5

17

7.7[18]

0.86-1.54[19-22]

0.21[21]

Exp.

14

WS2[23]

1L

177

272

-

13.4[2]

0.21[24]

Exp.

15

WSe2[25]

ML

596-1615**

167.3

12.4

7.3

-

11.9[2]

0.19[24]

Exp.

16

MoSe2[26]

1L

-

177.2

4.8

-

18.7[15]

6.39-10.14[27]

0.23

Exp.

17

MoTe2

1L

79.9-94.07[24]

-

21.7[15]

-

0.24[24]

Theory

18

WTe2[28]

1L

71.29/106.54**

-

19

-

0.26/0.38

Theory

19

Black Phosphorus

1L[29]

23/92.3

41.3/106.4

-

48/11

-

0.40/0.93

Theory

20

FL[30]

-

46-276

2.1-25

8-17

-

«

Exp.

21

Borophene[31]

1L(α=0)⁑

212/212

-

~22/13♂

~9/15‖

-

0.79/0.79

0.14/0.14

Theory

22

Triangular

399/163

-

20.9/12.2♂

8.7/14.3‖

-

4.76/1.39

−0.23/0

Theory

23

v1/5

196/208

-

0.52/0.54

0.11/0.12

Theory

24

v1/6

189/210

-

16.4/15.4♂

12.5/10.6‖

-

0.56/0.39

0.15/0.17

Theory

25

v1/8

216/222

-

0.74/0.59

0.17/0.18

Theory

26

v1/12

208/161

-

1.33/0.92

0.08/0.09

Theory

27

Silicene

1L

61.7/59[32-35]

-

5.85-4.78♂

18/9

-

38.63[35]

0.29/0.33[36]

Theory

28

Stanene

1L

25.2/23.5[34, 37]

-

2.6/2.2♂[36]

17/18*

-

0.36/0.42[36]

Theory

29

Germanene

1L

44/43.4[34, 36]

-

4.7/4.1♂[36]

20/20.5*

-

0.29/0.35[36]

Theory

30

Be5C2[38]

1L

32.68/130.03

32.9/130.89

-

−0.041/−0.16

Theory

31

Bi2Se3[39]

7-12L

-

17.86-25.45

-

4-8.3

-

0.27[40]

Exp.

32

33

34

♂ These values correspond to 2D stress in units Nm−1. ‖ These strains correspond to the peak stress where phase transition occurs.

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⁑ All borophenes are monolayers but with different HH concentration as discussed in the main text. * The tensile strain at the peak stress (not at fracture).

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** With the ‘-‘ we denote range of values between minimum and maximum while with ‘/’ we separate the armchair and zigzag directions.

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References: activated links redirecting to the journal website

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[1] Lee C, Wei X, Kysar J W and Hone J 2008 Measurement of the elastic properties and intrinsic strength of monolayer graphene science 321 385-8

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[2] Zhao J, Deng Q, Ly T H, Han G H, Sandeep G and Rümmeli M H 2015 Two-dimensional membrane as elastic shell with proof on the folds revealed by three-dimensional atomic mapping Nat Commun 6 8935

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[3] Lu Q, Arroyo M and Huang R 2009 Elastic bending modulus of monolayer graphene Journal of Physics D: Applied Physics 42 102002

42

[4] Liu F, Ming P and Li J 2007 Ab initio calculation of ideal strength and phonon instability of graphene under tension Phys Rev B 76 064120

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[5] Blakslee O, Proctor D, Seldin E, Spence G and Weng T 1970 Elastic constants of compression‐annealed pyrolytic graphite Journal of Applied Physics 41 3373-82

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[6] Bosak A, Krisch M, Mohr M, Maultzsch J and Thomsen C 2007 Elasticity of single-crystalline graphite: Inelastic x-ray scattering study Phys Rev B 75 153408

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[7] Kalosakas G, Lathiotakis N, Galiotis C and Papagelis K 2013 In-plane force fields and elastic properties of graphene Journal of Applied Physics 113 134307

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[8] Lee C, Wei X, Li Q, Carpick R, Kysar J W and Hone J 2009 Elastic and frictional properties of graphene physica status solidi (b) 246 2562-7

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[9] Chen X, Yi C and Ke C 2015 Bending stiffness and interlayer shear modulus of few-layer graphene Applied Physics Letters 106 101907

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[10] Wei X D, Meng Z X, Ruiz L, Xia W J, Lee C, Kysar J W, Hone J C, Keten S and Espinosa H D 2016 Recoverable Slippage Mechanism in Multilayer Graphene Leads to Repeatable Energy Dissipation Acs Nano 10 1820-8

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[11] Lee G-H, Cooper R C, An S J, Lee S, van der Zande A, Petrone N, Hammerberg A G, Lee C, Crawford B and Oliver W 2013 High-strength chemical-vapor–deposited graphene and grain boundaries Science 340 1073-6

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[12] Bertolazzi S, Brivio J and Kis A 2011 Stretching and breaking of ultrathin MoS2 ACS nano 5 9703-9

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[13] Jiang J-W, Qi Z, Park H S and Rabczuk T 2013 Elastic bending modulus of single-layer molybdenum disulfide (MoS2): finite thickness effect Nanotechnology 24 435705

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[14] Feldman J 1976 Elastic constants of 2H-MoS2 and 2H-NbSe2 extracted from measured dispersion curves and linear compressibilities Journal of Physics and Chemistry of Solids 37 1141-4

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[15] Kuzuba T and Ishii M 1989 Elastic Constant c44 of Uniaxial Layered Crystals physica status solidi (b) 155

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[16] Elder R M, Neupane M R and Chantawansri T L 2015 Stacking order dependent mechanical properties of graphene/MoS2 bilayer and trilayer heterostructures Applied Physics Letters 107 073101

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[17] Falin A, Cai Q, Santos E J, Scullion D, Qian D, Zhang R, Yang Z, Huang S, Watanabe K and Taniguchi T 2017 Mechanical properties of atomically thin boron nitride and the role of interlayer interactions Nat Commun 8 15815

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[18] Bosak A, Serrano J, Krisch M, Watanabe K, Taniguchi T and Kanda H 2006 Elasticity of hexagonal boron nitride: Inelastic x-ray scattering measurements Phys Rev B 73 041402

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[19] Singh S K, Neek-Amal M, Costamagna S and Peeters F 2013 Thermomechanical properties of a single hexagonal boron nitride sheet Phys Rev B 87 184106

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[20] Thomas S, Ajith K, Chandra S and Valsakumar M 2015 Temperature dependent structural properties and bending rigidity of pristine and defective hexagonal boron nitride Journal of Physics: Condensed Matter 27 315302

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[21] Kudin K N, Scuseria G E and Yakobson B I 2001 C 2 F, BN, and C nanoshell elasticity from ab initio computations Phys Rev B 64 235406

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[22] Wu J, Wang B, Wei Y, Yang R and Dresselhaus M 2013 Mechanics and mechanically tunable band gap in single-layer hexagonal boron-nitride Materials Research Letters 1 200-6

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[23] Liu K, Yan Q, Chen M, Fan W, Sun Y, Suh J, Fu D, Lee S, Zhou J and Tongay S 2014 Elastic properties of chemical-vapor-deposited monolayer MoS2, WS2, and their bilayer heterostructures Nano Lett 14 5097-103

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[24] Fan Z, Wei-Bing Z and Bi-Yu T 2015 Electronic structures and elastic properties of monolayer and bilayer transition metal dichalcogenides MX2 (M= Mo, W; X= O, S, Se, Te): a comparative first-principles study Chinese Physics B 24 097103

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[25] Zhang R, Koutsos V and Cheung R 2016 Elastic properties of suspended multilayer WSe2 Applied Physics Letters 108 042104

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[26] Yang Y C, Li X, Wen M R, Hacopian E, Chen W B, Gong Y J, Zhang J, Li B, Zhou W, Ajayan P M, Chen Q, Zhu T and Lou J 2017 Brittle Fracture of 2D MoSe2 Adv Mater 29

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[27] Lai K, Zhang W-B, Zhou F, Zeng F and Tang B-Y 2016 Bending rigidity of transition metal dichalcogenide monolayers from first-principles Journal of Physics D: Applied Physics 49 185301

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[28] Torun E, Sahin H, Cahangirov S, Rubio A and Peeters F 2016 Anisotropic electronic, mechanical, and optical properties of monolayer WTe2 Journal of Applied Physics 119 074307

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[29] Jiang J-W and Park H S 2014 Mechanical properties of single-layer black phosphorus Journal of Physics D: Applied Physics 47 385304

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[30] Wang J-Y, Li Y, Zhan Z-Y, Li T, Zhen L and Xu C-Y 2016 Elastic properties of suspended black phosphorus nanosheets Applied Physics Letters 108 013104

69

[31] Zhang Z H, Yang Y, Penev E S and Yakobson B I 2017 Elasticity, Flexibility, and Ideal Strength of Borophenes Advanced Functional Materials 27

70

[32] Zhao H J 2012 Strain and chirality effects on the mechanical and electronic properties of silicene and silicane under uniaxial tension Phys Lett A 376 3546-50

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[33] Qin R, Wang C H, Zhu W J and Zhang Y L 2012 First-principles calculations of mechanical and electronic properties of silicene under strain Aip Adv 2

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[34] John R and Merlin B 2016 Theoretical Investigation of Structural, Electronic, and Mechanical Properties of Two Dimensional C, Si, Ge, Sn Crystal Structure Theory and Applications 5 43

73

[35] Roman R E and Cranford S W 2014 Mechanical properties of silicene Comp Mater Sci 82 50-5

74

[36] Mortazavi B, Rahaman O, Makaremi M, Dianat A, Cuniberti G and Rabczuk T 2017 First-principles investigation of mechanical properties of silicene, germanene and stanene Physica E 87 228-32

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[37] Tao L, Yang C, Wu L, Han L, Song Y, Wang S and Lu P 2016 Tension-induced mechanical properties of stanene Modern Physics Letters B 30 1650146

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[38] Wang Y, Li F, Li Y and Chen Z 2016 Semi-metallic Be5C2 monolayer global minimum with quasi-planar pentacoordinate carbons and negative Poisson's ratio Nat Commun 7

77

[39] Yan H, Vajner C, Kuhlman M, Guo L, Li L, Araujo P T and Wang H-T 2016 Elastic behavior of Bi2Se3 2D nanosheets grown by van der Waals epitaxy Applied Physics Letters 109 032103

78

[40] Gao X, Zhou M, Cheng Y and Ji G 2016 First-principles study of structural, elastic, electronic and thermodynamic properties of topological insulator Bi2Se3 under pressure Philosophical Magazine 96 208-22


1		2D Material	Toughness	Stress intensity factor
2			(energy release rate)	KIC
3			(Jm^−2)	(MPa√m)
4		Graphene (1L)	15.9	4
5		Graphene (2L)	-	8.5
6		Graphene (3L)	-	9.17-16.8
7		MoS2	33.48-39.73	1.3-1.8
8		h-BN	0.038-0.072	5.56/5.35
9		Black Phosphorus	5.66 to 16.66	-
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13	References: activated links redirecting to the journal website
14	[1] Zhang P, Ma L L, Fan F F, Zeng Z, Peng C, Loya P E, Liu Z, Gong Y J, Zhang J N, Zhang X X, Ajayan P M, Zhu T and Lou J 2014 Fracture toughness of graphene Nat Commun 5
15	[2] Jang B, Kim B, Kim J-H, Lee H-J, Sumigawa T and Kitamura T 2017 Asynchronous cracking with dissimilar paths in multilayer graphene Nanoscale
16	[3] Wang S, Qin Z, Jung G S, Martin-Martinez F J, Zhang K, Buehler M J and Warner J H 2016 Atomically Sharp Crack Tips in Monolayer MoS2 and Their Enhanced Toughness by Vacancy Defects ACS nano 10 9831-9
17	[4] Wang X, Tabarraei A and Spearot D E 2015 Fracture mechanics of monolayer molybdenum disulfide Nanotechnology 26 175703
18	[5] Becton M and Wang X 2015 Grain-size dependence of mechanical properties in polycrystalline boron-nitride: a computational study Phys Chem Chem Phys 17 21894-901
19	[6] Tabarraei A and Wang X 2015 A molecular dynamics study of nanofracture in monolayer boron nitride Materials Science and Engineering: A 641 225-30
20	[7] Liu N, Hong J, Pidaparti R and Wang X 2016 Fracture patterns and the energy release rate of phosphorene Nanoscale 8 5728-36


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2	Review: Tailoring the mechanical properties of 2D materials and heterostructures
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5		Charalampos Androulidakis(1), Kaihao Zhang(1), Matthew Robertson(1), and Sameh Tawfick(1,2)
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8		1. Department of Mechanical Science and Engineering, University of Illinois Urbana-Champaign, 1206 W. Green St., Urbana, IL 61801
9		2. The Beckman Institute for Advanced Science and Technology, University of Illinois Urbana-Champaign, 1206 W. Green St., Urbana, IL 61801.
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1	Review Papers Related to Mechanical Propeties
2		Mechanics	Wrinkling	Friction
3	2D Materials	[1] [3] [4] [9] [10]	[6] [10]	[8]
4	2D Heterostructures	[1] [10]	[10]
5	Graphene	[2] [10]	[5] [10]	[7]
6	MoS2	[10]	[10]
7	BN	[10]
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15	1. Review: Tailoring the mechanical properties of 2D materials and heterostructures
16	2. Graphene mechanics: current status and perspectives
17	3. A review on mechanics and mechanical properties of 2D materials—Graphene and beyond
18	4.Mechanical Properties of 2D Materials Studied by In Situ Microscopy Techniques
19	5. Wrinkled, rippled and crumpled graphene: an overview of formation mechanism, electronic properties, and applications
20	6. Mechanical instability driven self-assembly and architecturing of 2D materials
21	7. Graphene: a new emerging lubricant
22	8. 2D-nanomaterials for controlling friction and wear at interfaces
23	9. Mechanical properties of graphene and graphene-based nanocomposites
24	10. Strain Engineering of 2D Materials: Issues and Opportunities at the Interface