Table 5 Critical buckling strain results of armchair nanotubes with finite and infinite lengths obtained by wrapping armchair and zigzag nanosheets.

From: Analyzing fine scaling quantum effects on the buckling of axially-loaded carbon nanotubes based on the density functional theory and molecular mechanics method

 

Chirality

(2, 2)

(3, 3)

(4, 4)

(5, 5)

(6, 6)

(8, 8)

(11, 11)

(15, 15)

(18, 18)

(23, 23)

Diameter

0.2713

0.4069

0.5425

0.6781

0.8138

1.085

1.4919

2.0344

2.4413

3.1194

Variation percentage

0

50.002

100

150

200

300.01

450.01

650.01

800.01

1050

Infinite length

Critical strain

0.2378

0.1501

0.1089

0.0827

0.0669

0.0458

0.0312

0.0194

0.0116

0.0169

Variation percentage

0

 − 58.46

 − 118.33

 − 187.4

 − 255.46

 − 418.9

 − 661.9

 − 1129

 − 1955

 − 1304.6

Ratio of strain changes to diameter changes

 

 − 1.169

 − 1.1832

 − 1.249

 − 1.2773

 − 1.396

 − 1.471

 − 1.737

 − 2.444

 − 1.2425

2a

Critical strain

0.3456

0.181

0.1279

0.1011

0.0827

0.0603

0.0415

0.0303

0.0224

0.0156

Length to diameter ratio (\(L/D\))

27.21345

18.14453

13.60924

10.88779

9.072266

6.804618

4.94873

3.629085

3.024213

2.366805

Variation percentage

0

 − 90.98

 − 170.29

 − 241.8

 − 317.73

 − 473.4

 − 731.9

 − 1040

 − 1446

 − 2113.9

Ratio of strain changes to diameter changes

 

 − 1.819

 − 1.7028

 − 1.612

 − 1.5886

 − 1.578

 − 1.627

 − 1.6

 − 1.808

 − 2.0132

3a

Critical strain

0.3078

0.1723

0.1258

0.0992

0.0805

0.0582

0.04

0.0268

0.021

0.0142

Length to diameter ratio (\(L/D\))

45.35566

30.24082

22.68201

18.14628

15.12041

11.341

8.247865

6.048461

5.040343

3.944666

Variation percentage

0

 − 78.62

 − 144.63

 − 210.2

 − 282.16

 − 428.6

 − 669.7

 − 1048

 − 1366

 − 2064.5

Ratio of strain changes to diameter changes

 

 − 1.572

 − 1.4462

 − 1.402

 − 1.4108

 − 1.429

 − 1.488

 − 1.612

 − 1.708

 − 1.9661

5a

Critical strain

0.2661

0.1602

0.1177

0.0911

0.074

0.0526

0.0354

0.0229

0.0169

0.0108

Length to diameter ratio (\(L/D\))

81.64036

54.4336

40.82771

32.66337

27.2168

20.41385

14.84619

10.88725

9.072638

7.100414

Variation percentage

0

 − 66.13

 − 126.02

 − 192.2

 − 259.57

 − 406.3

 − 652

 − 1062

 − 1473

 − 2368.6

Ratio of strain changes to diameter changes

 

 − 1.323

 − 1.2602

 − 1.282

 − 1.2978

 − 1.354

 − 1.449

 − 1.633

 − 1.841

 − 2.2558

8a

Critical strain

0.2642

0.162

0.1194

0.0916

0.0747

0.053

0.0393

0.0227

0.0167

0.0104

Length to diameter ratio (\(L/D\))

136.067

90.72249

68.04605

54.43884

45.36124

34.02302

24.7436

18.14539

15.12103

11.834

Variation percentage

0

 − 63.07

 − 121.17

 − 188.3

 − 253.77

 − 398.4

 − 571.7

 − 1065

 − 1480

 − 2447.3

Ratio of strain changes to diameter changes

 

 − 1.261

 − 1.2116

 − 1.255

 − 1.2688

 − 1.328

 − 1.27

 − 1.638

 − 1.85

 − 2.3308

2z

Critical strain

0.2591

0.1401

0.1004

0.0789

0.0636

0.0452

0.0319

0.0193

0.0141

0.0106

Length to diameter ratio (\(L/D\))

26.18618

17.4596

13.0955

10.47679

8.729798

6.547751

4.761921

3.492091

2.910052

2.27746

Variation percentage

0

 − 84.99

 − 158.1

 − 228.4

 − 307.23

 − 473.6

 − 711.7

 − 1242

 − 1737

 − 2337.3

Ratio of strain changes to diameter changes

 

 − 1.7

 − 1.5809

 − 1.523

 − 1.5361

 − 1.579

 − 1.581

 − 1.91

 − 2.172

 − 2.2259

3z

Critical strain

0.2279

0.1337

0.0973

0.0749

0.0603

0.0418

0.0268

0.016

0.011

0.0091

Length to diameter ratio (\(L/D\))

41.89786

27.93534

20.95279

16.76285

13.96767

10.4764

7.61907

5.587343

4.656081

3.643935

Variation percentage

0

 − 70.42

 − 134.2

 − 204.2

 − 278.01

 − 445.9

 − 749.1

 − 1323

 − 1964

 − 2418.2

Ratio of strain changes to diameter changes

 

 − 1.408

 − 1.342

 − 1.361

 − 1.39

 − 1.486

 − 1.665

 − 2.036

 − 2.455

 − 2.303

5z

Critical strain

0.2302

0.1397

0.1022

0.0779

0.0628

0.0435

0.028

0.0166

0.0112

0.0092

Length to diameter ratio (\(L/D\))

73.32101

48.88668

36.66726

29.33489

24.44334

18.33363

13.33333

9.777817

8.148114

6.376864

Variation percentage

0

 − 64.8

 − 125.23

 − 195.5

 − 266.53

 − 428.8

 − 722.4

 − 1283

 − 1952

 − 2402

Ratio of strain changes to diameter changes

 

 − 1.296

 − 1.2522

 − 1.303

 − 1.3326

 − 1.429

 − 1.605

 − 1.974

 − 2.439

 − 2.2875

9z

Critical strain

0.2365

0.1458

0.1068

0.0814

0.0659

0.0458

0.0334

0.0175

0.0121

0.009

Length to diameter ratio (\(L/D\))

136.1679

90.78978

68.09652

54.47922

45.39489

34.04826

24.76195

18.15885

15.13225

11.84278

Variation percentage

0

 − 62.19

 − 121.4

 − 190.7

 − 258.91

 − 416.5

 − 608.8

 − 1248

 − 1851

 − 2515.6

Ratio of strain changes to diameter changes

 

 − 1.244

 − 1.2139

 − 1.271

 − 1.2945

 − 1.388

 − 1.353

 − 1.92

 − 2.314

 − 2.3958

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