Several Mechanic Problems Of Carbon Nanotubes Based On Continuum Theory

The reported research on carbon nanotubes(CNTs) is introduced, and then statics, buckling and dynamics of CNTs are studied on the basis of micropolar and nonlocal elasticity appropriate to microscopic mechanics,respectively.The difference of micropolar elasticity from classical continuum is that the former incorporates micro-rotation in its deformation formulation and couple stresses in its stresses. Based on micropolar elastic theory, the constitutive equation, which incorporates length of C-C bond and thickness of wall, is derived. A carbon nanotube is divided into elements, and the principle of the virtual work is applied to obtain the equilibrium equation of an element.The stiffness matrix of the element can be obtained from the variation of energy with respect to displacement, and then the total stiffness matrix can be obtained by assembling stiffness matrices of all the elements. Displacements, stresses and strains are obtained when the carbon nanotube is pressured, bent and twisted by applying different forces on the single-walled carbon nanotube and double-walled carbon nanotube, respectively. The obtained results are discussed and compared with the literatures. They are in good agreements.The difference of nonlocal elasticity from classical continuum is that long range force is incorporated in the former, namely, stress at a point depends on strain in a region near that point. With the help of this idea, in space any field can be expressed by a weighted average with respect to its volume. A right hexagonal element is considered as a typical element of the carbon nanotube, the average stress, which is regarded as the nonlocal stress, can be obtained by expanding the stress field of the point in the element into Taylor series and then taking the average of stresses with respect to the typical element volume. After the relationship of the nonlocal stress and the classical stress having been obtained, the constitutive equation can be derived. In the nonlocal elasticity the constitutive equation of the carbon nanotube incorporates the internal characteristic size. Based on the above constitutive equation, the effect of the small scalar size on the radial and axial buckling stresses of the single-walled, double-walled and triad-walled carbon nanotubes are investigated. Results denote that effect of the buckling modes and the small scalar size on the buckling stress is interrelation. When diameter of a carbon nanotube is fixed, the effect of the small scalar size will decreases with the increase of length of a carbon nanotube, the effect of the small scalar size will disappear when length of the carbon nanotube exceeds a critical value. In this case, the half wavenumber for axial buckling is independent of the small scalar size, and the effect of small scalar size will increase as the wavenumber in perimeter direction increases. When length of a carbon nanotube is given, the effect of the small scalar size will decreases with the increase of diameter of a carbon nanotube, the effect of the small scalar size will disappear when diameter of the carbon nanotube exceeds a critical value. In this case, the buckling wavenumber in perimeter direction of the carbon nanotube is independent of the effect of small scalar size, and the effect of the small scalar size will increase as the halfwavenumber of the axial buckling increases. When length and diameter of the carbon nanotube are both within the range of their critical values, the effect of small scalar size will increase with the increase of the buckling modes.Based on the nonlocal elasticity, dynamical properties of the carbon nanotube are investigated. Results obtained show that the size and vibration modes of the carbon...