**5. Conclusions**

44 Continuum Mechanics – Progress in Fundamentals and Engineering Applications

To close this mathematical formulation of the above problem, we then complement the inextensible condition of the elastic rod (Bishopp and Drucker, 1945), which is written as

> 

From Eq. (40), one can solve the corresponding values of *k*=0.82 for the given value of *d*0.

<sup>1</sup> 42 , , <sup>2</sup> *Lec <sup>a</sup> k Fk Fk R R* 

For a CNT with an initial radius *R*=3 nm, the analytical result of Eq. (43) is presented as 0.317854, which is nearly equal to the approximated solution given by Tang *et al*. (2005b) is

After the flat contact length *a* has been solved by Eq. (43), the deflection of the rod can be

 <sup>2</sup> 

 *s Fk Fk Ek Ek* , , 2, , 

0 3 6 9 12

*x*/*Lec*

Fig. 5. Cross-section shapes of CNTs with initial radii of 2, 3 and 4 nm, respectively.

2

<sup>0</sup> cos d *s*

The above displacements normalized by the elasto-cohesive length read

 

*x s* 

*Fk Fk* , ,

d 2 sin *C*

 

2 0

Then the substitution of Eq. (34) into (42) yields

*aR d L R* π <sup>0</sup> 3.035 *ec* π +0.5=0.30485.

*y* 



0

*y*/*Lec*

3

6

determined by

<sup>0</sup> sin d *s*

  2

<sup>2</sup>

<sup>0</sup> .

0

0 0 . (45)

> *R*=2 nm  *R*=3 nm  *R*=4 nm

, (44)

. (43)

 

*ak k* 4 2 2 cos

(42)

*R a <sup>R</sup> a ds*

> In this study, we demonstrated that a lot of problems dealing with the moving boundaries can be grouped into a unified frame, such as the adhesion of micro-beams and collapse of SWCNT. We first constructed the energy functional of the general system, then derived the governing equation and the transversality condition. We put this analysis method to solve the critical length and deflections of two micro-beams. Moreover, we derived the governing equation, i.e., the elastica model of the collapsed morphology for the SWCNT. Under the inextensible condition of the rod, the closed-form solutions for the flat contact segment, critical radii, and collapsed configuration were obtained in terms of elliptical integrals. It is clearly shown that our analytical solutions are in good agreement with the results of the references.

> This analysis method paves a new way to examine nano-scaled mechanics by means of continuum mechanics. The presented results are also beneficial to design and fabricate new devices, micro-sensors and advanced materials in micro/nano scale, which casts a light on enhancing their mechanical, chemical, optical and electronic properties. Furthermore, this model can be generalized to investigate both a macroscopic sheet wrapped by a liquid film and a CNT self-folded by van der Waals forces, and can be adopted to analyze the crack or contact problems.
