Errata to Approximate Solutions for a Self-Folding Problem of Carbon Nanotubes

Mikata, Yozo

doi:10.1115/1.4007336

There was a factor of 2 missing in Eq. in the original paper [1] and this error in Eq. has propagated throughout the rest of the paper [1]. For the sake of brevity, only the affected equations are listed in the following:

U = 2 \cdot \frac{M^{2} l}{2 EI} = \frac{π^{2}}{l} EI

(36)

\frac{π^{2}}{l} EI = E_{B} l_{q}

(37)

l_{q} = \frac{π^{2} EI}{E_{B}} \frac{1}{l}

(38)

l_{T} = l + \frac{π^{2} EI}{E_{B}} \frac{1}{l} \geq 2 \sqrt{l \cdot \frac{π^{2} EI}{E_{B}} \frac{1}{l}} = 2 π \sqrt{\frac{EI}{E_{B}}}

(39)

l_{cr} = 4 π \sqrt{\frac{EI}{E_{B}}} = 4 π r_{eq} \sqrt{x} = 12.5664 r_{eq} \sqrt{x}

(40)

U_{1} = 2 \cdot \frac{M_{1}^{2} l_{1}}{2 EI} = \frac{EI}{l_{1}} φ_{1}^{2} U_{2} = 2 \cdot \frac{M_{2}^{2} l_{2}}{2 EI} = \frac{EI}{l_{2}} φ_{2}^{2}

(44)

EI (\frac{φ_{1}^{2}}{l_{1}} + \frac{φ_{2}^{2}}{l_{2}}) = E_{B} l_{q}

(45)

l_{q} = \frac{EI}{E_{B}} (\frac{φ_{1}^{2}}{l_{1}} + \frac{φ_{2}^{2}}{l_{2}})

(46)

\begin{matrix} l_{T} = l_{1} + l_{2} + l_{q} \\ = l_{1} + \frac{\sin φ_{1}}{1 - \sin φ_{1}} \frac{φ_{1} - \frac{π}{2}}{φ_{1}} l_{1} + \frac{EI}{E_{B}} (\frac{φ_{1}^{2}}{l_{1}} + \frac{φ_{1} (φ_{1} - \frac{π}{2})}{\frac{\sin φ_{1}}{1 - \sin φ_{1}} l_{1}}) \\ = l_{1} (1 + \frac{\sin φ_{1}}{1 - \sin φ_{1}} \frac{φ_{1} - \frac{π}{2**}}{φ_{1}}) + \frac{EI}{E_{B}} \frac{1}{l_{1}} (φ_{1}^{2} + \frac{φ_{1} (φ_{1} - \frac{π}{2})}{\frac{\sin φ_{1}}{1 - \sin φ_{1}}}) \geq 2 \sqrt{\frac{EI}{E_{B}} (1 + \frac{\sin φ_{1}}{1 - \sin φ_{1}} \frac{φ_{1} - \frac{π}{2}}{φ_{1}}) (φ_{1}^{2} + \frac{φ_{1} (φ_{1} - \frac{π}{2})}{\frac{\sin φ_{1}}{1 - \sin φ_{1}}})} = 2 φ_{1} \sqrt{\frac{EI}{E_{B}} (\frac{\sin φ_{1}}{1 - \sin φ_{1}} + \frac{φ_{1} - \frac{π}{2}}{φ_{1}}) (\frac{1 - \sin φ_{1}}{\sin φ_{1}} + \frac{φ_{1} - \frac{π}{2}}{φ_{1}})} \end{matrix}

l_{cr} = 4 \sqrt{\frac{EI}{E_{B}}} \underset{\frac{π}{2} < φ < π}{Min} h (φ)

(50)

l_{cr} = 13.4487 r_{eq} \sqrt{x}

(53)

U = 4 k^{2} EI \frac{[K (k)]^{2}}{l} [1 - \frac{1}{k^{2}} \frac{K (k) - E (k)}{K (k)}]

(58)

4 k^{2} EI \frac{[K (k)]^{2}}{l} [1 - \frac{1}{k^{2}} \frac{K (k) - E (k)}{K (k)}] = E_{B} l_{q}

(59)

c_{E} = 4 k^{2} [K (k)]^{2} \frac{EI}{E_{B}} [1 - \frac{1}{k^{2}} \frac{K (k) - E (k)}{K (k)}]

(61)

l_{cr} = 8 kK (k) r_{eq} \sqrt{1 - \frac{1}{k^{2}} \frac{K (k) - E (k)}{K (k)}} \sqrt{x}

(62)

l_{cr} = 10.6038 r_{eq} \sqrt{x}

(63)

l_{cr}^{L, f} = 1425.7 \overset{°}{A} l_{cr}^{U, f} = 2790.73 \overset{°}{A}

(70)

\frac{l_{cr}}{r_{eq}} = 4 π \sqrt{x} = 12.5664 \sqrt{x} (a circle and a straight line)

(71)

\frac{l_{cr}}{r_{eq}} = 13.4487 \sqrt{x} (two circles and a straight line)

(72)

\frac{l_{cr}}{r_{eq}} = 10.6038 \sqrt{x} (an elliptic function and a straight line)

(73)

l_{cr}^{L, e} = 2616.79 \overset{°}{A} l_{cr}^{U, e} = 3318.84 \overset{°}{A}

(74)

Because the values of the estimates by the energy method have changed due to the corrections of the formulas (71), (72), and (73), Table 3 and Eq. (74) in Ref. [1] are changed for correction. Corresponding to the changes in Table 3, we have added two more cases for Tables 1 and 2 to produce higher estimates for the critical threshold length $l_{cr}$ ⁠. The nondimensional parameter x in the above equations was introduced in Eq. (33) of [1].

Table 1

Coefficient c in the critical threshold length formula

	Case 1	Case 2	Case 3	Case 4	Case 5	Case 6	Case 7	Case 8
α	0.5	0.7	0.8	0.9	0.95	0.99	0.99	0.99
β	1.5	2	2	2	2	2	3	4
c	6.87147	8.96696	10.2646	12.9326	16.294	27.8624	31.8945	35.1045

Table 2

Estimates of the critical threshold length by the force method

	Case 1	Case 2	Case 3	Case 4	Case 5	Case 6	Case 7	Case 8
$l_{cr} / r_{eq}$	54.6587	71.3271	81.6492	102.871	129.61	221.63	253.70	279.24
$l_{cr}$ (⁠ $\overset{°}{A}$ ⁠)	601.245	784.598	898.14	1131.58	1425.7	2437.93	2790.73	3071.6

Table 3

Estimates of the critical threshold length by the energy method

	Circle	Two circles	Elliptic function
$l_{cr} / r_{eq}$	281.92	301.713	237.89
$l_{cr}$ (⁠ $\overset{°}{A}$ ⁠)	3101.11	3318.84	2616.79

It is interesting to note that Eq. (71) is exactly the same analytical expression for the critical threshold length developed by Zhou et al. [2], even though their result was obtained by a different method.

Because of the changes brought about due to the corrections of the formulas, one sentence in the paragraph between Eq. (65) and Eq. (66) in Sec. 6 of Ref. [1] should be changed as follows: “Even though there is no rational way to single out a ‘true’ solution from the above cases, let us use Cases 5 and 7 as our lower and upper estimates, respectively.” The changes in the lower and upper estimates for the critical threshold length of the (5,5) armchair carbon nanotube are reflected in the updated values in Eqs. and . Similarly, updated normalized critical threshold lengths obtained by the force method and the energy method are shown in the revised Figs. 7 and 9.

Fig. 7

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Normalized critical threshold length $l_{cr} / r_{eq}$ as a function of x obtained by the force method

Fig. 9

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Normalized critical threshold length $l_{cr} / r_{eq}$ as a function of x obtained by the force method and the energy method

There were typographical errors in the sentence just above Eq. and in Eq. in the original paper [1]. λ_cr in the sentence just above Eq. should be changed to

l_{cr}

⁠, and Eq. should be corrected as follows:

\sin \frac{θ}{2} = k sn [K (k) - bs]

(54)

References

1.

Mikata

,

Y.

,

2010

, “

Approximate Solutions for a Self-Folding Problem of Carbon Nanotubes

,”

ASME J. Eng. Mater. Technol.

,

132

, p.

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2.

Zhou

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W.

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Huang

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Y.

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Liu

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B.

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Hwang

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K. C.

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Zuo

,

J. M.

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Buehler

,

M. J.

, and

Gao

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H.

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2007

, “

Self-Folding of Single- and Multiwall Carbon Nanotubes

,”

Appl. Phys. Lett.

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Crossref

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