Table 6 Critical buckling strain results of \(\left(n,n/2\right)\) 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, 1)	(4, 2)	(6, 3)	(8, 4)	(10, 5)	(14, 7)	(18, 9)	(22, 11)	(26, 13)	(30, 15)
	Diameter	0.2072	0.4144	0.6215	0.8287	1.0359	1.4502	1.8646	2.2789	2.6933	3.1076
	Variation percentage	0	100	200	300	400.01	600.01	800.01	1000	1200	1400
Infinite length	Critical strain	0.3156	0.1251	0.0744	0.051	0.0371	0.0214	0.013	0.0097	0.0105	0.0091
	Variation percentage	0	− 152.35	− 323.91	− 518.9	− 750.57	− 1373	− 2333	− 3147	− 2905	− 3368
	Ratio of strain changes to diameter changes		− 1.5234	− 1.6195	− 1.73	− 1.8764	− 2.287	− 2.916	− 3.146	− 2.421	− 2.405
2a	Critical strain	0.5386	0.1611	0.0961	0.068	0.0518	0.0342	0.0245	0.0183	0.0141	0.0111
	Length to diameter ratio (\(L/D\))	35.63229	17.81614	11.87934	8.909147	7.127145	5.091029	3.959568	3.239725	2.741251	2.375792
	Variation percentage	0	− 234.42	− 460.77	− 692	− 939	− 1474	− 2097	− 2837	− 3734	− 4770
	Ratio of strain changes to diameter changes		− 2.3441	− 2.3038	− 2.307	− 2.3475	− 2.456	− 2.621	− 2.837	− 3.111	− 3.407
3a	Critical strain	0.4603	0.1522	0.0922	0.0653	0.0496	0.0323	0.0228	0.0168	0.0129	0.0098
	Length to diameter ratio (\(L/D\))	59.38702	29.69351	19.79886	14.84855	11.87855	8.48503	6.599265	5.39953	4.568741	3.959644
	Variation percentage	0	− 202.44	− 399.23	− 605.3	− 828.38	− 1325	− 1918	− 2638	− 3479	− 4621
	Ratio of strain changes to diameter changes		− 2.0243	− 1.9961	− 2.018	− 2.0709	− 2.208	− 2.397	− 2.638	− 2.899	− 3.301
5a	Critical strain	0.377	0.1382	0.0835	0.0586	0.044	0.0276	0.0187	0.0133	0.0092	0.0093
	Length to diameter ratio (\(L/D\))	106.8969	53.44843	35.63802	26.72744	21.38144	15.27309	11.8787	9.719176	8.223752	7.127375
	Variation percentage	0	− 172.83	− 351.32	− 543.5	− 756.8	− 1268	− 1917	− 2737	− 4011	− 3954
	Ratio of strain changes to diameter changes		− 1.7282	− 1.7566	− 1.812	− 1.8919	− 2.114	− 2.396	− 2.737	− 3.343	− 2.824
8a	Critical strain	0.3677	0.1382	0.084	0.0585	0.0438	0.0275	0.0184	0.0131	0.0099	0.0092
	Length to diameter ratio (\(L/D\))	178.1611	89.08055	59.39659	44.54565	35.63566	25.4551	19.7978	16.1986	13.70623	11.87894
	Variation percentage	0	− 165.96	− 337.64	− 528.5	− 738.83	− 1239	− 1894	− 2713	− 3625	− 3896
	Ratio of strain changes to diameter changes		− 1.6595	− 1.6882	− 1.762	− 1.847	− 2.065	− 2.367	− 2.713	− 3.021	− 2.783
2z	Critical strain	0.3944	0.1239	0.0737	0.051	0.0381	0.0235	0.0156	0.0108	0.0091	0.0093
	Length to diameter ratio (\(L/D\))	34.28721	17.14361	11.43091	8.572837	6.858104	4.898848	3.810099	3.117429	2.637772	2.286108
	Variation percentage	0	− 218.36	− 435.2	− 673.3	− 934.6	− 1578	− 2423	− 3548	− 4253	− 4141
	Ratio of strain changes to diameter changes		− 2.1835	− 2.176	− 2.244	− 2.3364	− 2.63	− 3.029	− 3.548	− 3.544	− 2.958
3z	Critical strain	0.3285	0.1155	0.0686	0.0469	0.0343	0.0202	0.0125	0.0091	0.0093	0.0092
	Length to diameter ratio (\(L/D\))	54.85951	27.42975	18.28944	13.71653	10.97296	7.838153	6.096155	4.987885	4.220432	3.657771
	Variation percentage	0	− 184.38	− 378.88	− 600.6	− 856.92	− 1530	− 2520	− 3506	− 3436	− 3467
	Ratio of strain changes to diameter changes		− 1.8437	− 1.8943	− 2.002	− 2.1422	− 2.55	− 3.15	− 3.506	− 2.863	− 2.476
5z	Critical strain	0.322	0.1193	0.071	0.0484	0.0354	0.0207	0.0127	0.0091	0.009	0.009
	Length to diameter ratio (\(L/D\))	96.00381	48.00191	32.00642	24.00385	19.20262	13.71672	10.66823	8.728768	7.385731	6.401078
	Variation percentage	0	− 169.88	− 353.49	− 564.9	− 810.95	− 1456	− 2444	− 3450	− 3466	− 3474
	Ratio of strain changes to diameter changes		− 1.6987	− 1.7674	− 1.883	− 2.0273	− 2.427	− 3.054	− 3.45	− 2.888	− 2.481
9z	Critical strain	0.3256	0.1236	0.074	0.0505	0.037	0.022	0.0136	0.0098	0.0097	0.0091
	Length to diameter ratio (\(L/D\))	178.2932	89.14662	59.44064	44.57869	35.66209	25.47398	19.81249	16.21061	13.71639	11.88775
	Variation percentage	0	− 163.45	− 340.11	− 544.5	− 779.86	− 1381	− 2296	− 3232	− 3260	− 3462
	Ratio of strain changes to diameter changes		− 1.6345	− 1.7005	− 1.815	− 1.9496	− 2.302	− 2.869	− 3.232	− 2.716	− 2.473

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Table 6 Critical buckling strain results of \(\left(n,n/2\right)\) nanotubes with finite and infinite lengths obtained by wrapping armchair and zigzag nanosheets.

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