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

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There was a factor of 2 missing in Eq. (36) in the original paper [1] and this error in Eq. (36) has propagated throughout the rest of the paper [1]. For the sake of brevity, only the affected equations are listed in the following:(36)U=2·M2l2EI=π2lEI(37)π2lEI=EBlq(38)lq=π2EIEB1l(39)lT=l+π2EIEB1l≥2l·π2EIEB1l=2πEIEB(40)lcr=4πEIEB=4πreqx=12.5664reqx(44)U1=2·M12l12EI=EIl1φ12 U2=2·M22l22EI=EIl2φ22(45)EI(φ12l1+φ22l2)=EBlq(46)lq=EIEB(φ12l1+φ22l2)lT=l1+l2+lq=l1+sinφ11-sinφ1φ1-π2φ1l1+EIEB(φ12l1+φ1(φ1-π2)sinφ11-sinφ1l1)=l1(1+sinφ11-sinφ1φ1-π2**φ1)+EIEB1l1(φ12+φ1(φ1-π2)sinφ11-sinφ1) ≥2EIEB(1+sinφ11-sinφ1φ1-π2φ1)(φ12+φ1(φ1-π2)sinφ11-sinφ1)=2φ1EIEB(sinφ11-sinφ1+φ1-π2φ1)(1-sinφ1sinφ1+φ1-π2φ1)(50)lcr=4EIEBMinπ2<φ<πh(φ)(53)lcr=13.4487reqx(58)U=4k2EI[K(k)]2l[1-1k2K(k)-E(k)K(k)](59)4k2EI[K(k)]2l[1-1k2K(k)-E(k)K(k)]=EBlq(61)cE=4k2[K(k)]2EIEB[1-1k2K(k)-E(k)K(k)](62)lcr=8kK(k)req1-1k2K(k)-E(k)K(k)x(63)lcr=10.6038reqx(70)lcrL,f=1425.7A° lcrU,f=2790.73A°(71)lcrreq=4πx=12.5664x (a circle and a straight line)(72)lcrreq=13.4487x (two circles and a straight line)(73)lcrreq=10.6038x (an elliptic function and a straight line)(74)lcrL,e=2616.79A° lcrU,e=3318.84A°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 lcr. The nondimensional parameter x in the above equations was introduced in Eq. (33) of [1].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. (70) and (74). 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. 7Normalized critical threshold length lcr/req as a function of x obtained by the force methodFig. 9Normalized critical threshold length lcr/req as a function of x obtained by the force method and the energy methodThere were typographical errors in the sentence just above Eq. (40) and in Eq. (54) in the original paper [1]. λcr in the sentence just above Eq. (40) should be changed to lcr, and Eq. (54) should be corrected as follows:(54)sinθ2=ksn[K(k)-bs]