ИСТИНА

A model of a two-dimensional defectless medium is formulated as a special case of the general theory of a three-dimensional medium with fields of conserved dislocations having adhesive properties with a surface that limits the medium. The potential energy in the general theory is the sum of the volume and the superficial integral from the corresponding energy densities. In a limiting case, when the thickness of a shell is equal to zero, the volume portion of potential energy becomes zero. As a result, the potential energy of such an object is defined only by the surface potential energy. A single wall nanotube (SWNT) is examined as an example of such a two-dimensional medium. The problems concerning an SWNT axially deforming as well as the torsional case are examined. The general statement of an axisymmetric problem within the ideal theory of adhesion is formulated. Special cases involving a SWNT deforming axially and in torsion are studied: a case with ideal adhesion, the quasiclassical case, and a case with a large SWNT radius. It is shown that the case with ideal adhesion corresponds to the membrane theory of a cylindrical shell. It is shown that the particular case of the quasiclassical theory of a cylindrical shell is not a logical next step from the general theory when the moduli of ideal adhesion are partially considered and, to a lesser extent, the moduli of purely gradient adhesion. The characteristic feature of all statements is the fact that the mechanical properties of a SWNT are not defined by the “volumetric” moduli but by the adhesive moduli, which have different physical dimensions that coincide with the dimensions of the corresponding stiffness in the classical and nonclassical shells.