Nanoscale Continuum Modelling of Carbon Nanotubes by Polyhedral Finite Elements

Table 1

Comparative study on Young’s modulus.

Method	References	Year	Remarks	Wall thickness (nm)	Young’s modulus (TPa)
Experimental	Wong et al. [3]	1997	—	—	1.28
AM	Hernández et al. [14]	1998	Tight binding (MD)	0.34	1.24
	Goze et al. [15]	1999	Tight binding (MD)	0.34	1.24
	Jin and Yuan [16]	2003	Energy approach (MD)	0.34	1.238
	Jin and Yuan [16]	2003	Force approach (MD)	0.34	1.35
	Peng et al. [17]	2006	Ab initio	0.34	1.23–1.36
	Cai et al. [18]	2009	Tight binding (MD) and ab initio	—	0.95–1.33 (MD) and 1.02 (ab initio)
CM	Giannopoulos et al. [19]	2008	FEM spring elements	0.34	1.2478
NCM	Rafiee and Heidarhaei [20]	2012	FEM nonlinear spring elements	0.34	1.325
	[21–29]	2003–2015	FEM beam element	0.34	0.81–1.4
	Al-Kharusi et al. [30]	2016	FEM beam element	Other than 0.34	1.27 ± 0.02
	Current work	2016	Polyhedral FEM	0.34	0.953–1.22 (Armchair, n = 6, 7, 8. Length = 0.74 nm–6.27 nm). 1.268–1.41; (Armchair, n = 3, 4, 5. Length = 6.27 nm).; 0.965–1.338; (Zigzag, n = 6, 7, 8. Length = 1.14 nm–10.72 nm).; 1.257–1.338; (Zigzag, n = 6, 7, 8. Length = 10.72 nm).; Average steady state value: 1.2 for armchair; Average steady state value: 1.3 for zigzag
Combined (AM and NCM)	Cheng et al. [21]	2009	MD and FEM (using spring and beam elements)	0.34	1.2, 1.4

Notes: AM: atomistic modelling, CM: continuum modelling, and NCM: nanoscale continuum modelling.