A nanotorus is theoretically described as carbon nanotube bent into a torus (doughnut shape). Nanotori are predicted to have many unique properties, such as magnetic moments 1000 times larger than previously expected for certain specific radii. Its properties vary widely depending on radius of the torus and radius of the tube. Here, we developed a continuum model of nanotorus and obtained a closed form solutions for nanotorus deformation, vibration, and buckling and embedded in an elastic medium. The nanotorus is considered as a continuum model.

Since its discovery, nanotube has attracted great attention because of its unusual properties [

Carbon nanotubes are the strongest and stiffest materials yet discovered in terms of tensile strength and elastic modulus, respectively. This strength results from the covalent sp² bonds formed between the individual carbon atoms. In 2000, a multiwalled carbon nanotube was tested to have a tensile strength of 63 gigapascals (GPa). This, for illustration, translates into the ability to endure tension of a weight equivalent to 6422 kg on a cable with cross-section of 1 mm^{2}. Since carbon nanotubes have a low density for a solid of 1.3 to 1.4 g^{−3}, their specific strength of up to 48,000 kN^{−1} is the best of known materials, compared to high-carbon steel’s 154 kN^{−1}. Comparing its strength stress, unfortunately, CNTs are not nearly as strong under compression. Because of their hollow structure and high aspect ratio, they tend to undergo buckling when placed under compressive, torsion, or bending stress.

A nanotorus is theoretically described as carbon nanotube bent into a torus (doughnut shape). Nanotori are predicted to have many unique properties, such as magnetic moments 1000 times larger than previously expected for certain specific radii, or may be used as a black body whose emissivity or absorbance is almost of 1.0. Its properties vary widely depending on radius of the torus and radius of the tube. It is expected that the nanotorus has also have unique mechanical properties.

Comparing the comprehensive studies on the nanotube with cylindrical nanostructures, the nanotorus has not been well investigated [

Modelling the behaviour of the nanotorus is crucial for device design and interpretation of experimental results. Because experiments at the nanoscale are extremely difficult and atomistic modelling remains prohibitively expensive for large-sized atomic system, it has been accepted that continuum models will continue to play an essential role in the study of nanotorus as a toroidal shells. The validity of using continuum model for nanotube and graphene has been supported by both experimental results and molecular-dynamics simulation; all previous investigation on nanotube has indicated that the laws of continuum mechanics are still valid to some extent even in nanoscale. The successful of continuum model to nanotube gives us a confidence to predict that continuum modelling is going to be also valid for nanotori mechanics analysis [

Due to the development of nanopolymer or nanocomposites, considerable attention has turned to mechanical behaviour of single-walled nanotube embedded in a polymer or metal matrix. The Winkler elastic model will be adopted to study the nanotorus embedded in a elastic medium.

Although we wish to formulate the nanotorus using toroidal shell model, unfortunately, mathematically, the governing equations of toroidal shells are very complicated due to their strong variable confidents and singularity at turning point when

To simplify the toroidal shell modeling, we have noticed a factor that nanotubes have been constructed with length-to-diameter ratio of up to 132,000,000 : 1, which is significantly larger than any other material. It means that nanotorus’s

Following our previous work in [

The nature of toroidal shells with circular cross-section is that one principal radius is constant and another is variable. From Figures

Complete nanotorus and its continuum model.

Geometrical description of nanotorus.

Displacement of nanotorus.

Rotation of normal of nanotorus.

Internal resultant forces

The displacement type governing equations for the toroidal shells as follows

Owing to the complexity of (

As nanotube has been constructed with length-to-diameter ratio of up to 132,000,000 : 1, which is significantly larger than any other material. It means that nanotorus’s

Applying

The internal resultant forces and moments are as follows:

Then we have following closed form solution:

We have displacements as follows:

In the case of concentrated loading condition, the above solution can be written in a simpler form. Since now

As an example, we can apply the above solution (

The nanotorus under concentrated loading. The separation displacement can be found as

When the nanotube or nanotorus embedded in composite materials, the nanotorus-matric interaction can be viewed as a nanotorus buried in a elastic foundation, which can be described as a Winkler model. There are some investigations on nanotube buried in the elastic medium [

In Winkler model, the loading distribution in normal direction of the nanotorus will be in the form of

After introducing the Winkler model, the governing equation of the nanotorus embedded in elastic medium take the following form:

The advantage of using the displacement type equations is that we are able to study the deformation as well as the vibration of the shells. For literal vibration, the vibration in the direction of

With the solution of the free vibration problem, any forced vibration can be derived by using the superposition principal of vibration modes. This is a standard operation which has no any difficult.

General speaking, the buckling of toroidal shells is a very difficult problem, and no analytical solution has been obtained. By taking advantage of the displacement form of the governing equations for toroidal shells, we can find its analytical solution of the buckling problem.

The buckling equation of the slender toroidal shell is as follows:

It is worth noting that the strain

In the light of continuum model and the factor of very large length-to-diameter ratio of nanotorus, we successfully formulated the deformation, vibration, and buckling of the nanotorus embedded in an elastic medium by using toroidal shell model. The mathematical complicated equations of the general nanotorus have been reduced to simple differential equations with constant coefficients, which leads to a successful finding of closed-form solution of nanotorus. An example has been demonstrated for the case of concentrated loading condition. All solutions obtained here will be very useful for the future designing nanotori devices.

This paper was supported by South Africa National Research Foundation. This support is gratefully acknowledged. The author would like to express his thankfulness to the constructive comments and insightful suggestions given by three anonymous referees.