[01] Belov Petr A., Department of Non Classical Models of Composites & Structures, Institute of Applied Mechanics RAS, Moscow, Russia. [02] Lurie Sergey A., Department of Non Classical Models of Composites & Structures, Institute of Applied Mechanics RAS, Moscow, Russia; Department of Mathematical Physics, Computer Center of RAS, Moscow, Russia.

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 of 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 axial deforming as well as the torsional case are examined. The general statement of an axisymmetric problem within the theory of ideal 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.

Gradient Theories of Elasticity, Ideal Adhesion, Gradient Adhesion, Mechanical Properties of SWNT, Nonclassical Moduli

[01] Belov Petr A. «The theory of continuum with conserved dislocations: generalisation of Mindlin theory», «Composites and nanostructures», 2011, Volume 3, №1, pp. 24-38. [02] Mindlin R. D. «Micro-structure in linear elasticity», «Archive of Rational Mechanics and Analysis», 1964, №1, p. 51-78. [03] Toupin R. A. «Elastic materials with couple-stresses», «Archive of Rational Mechanics and Analysis», 1964, №2, p. 85-112. [04] Geim A. K., Novoselov K. S., Jiang D., Schedin F., Booth T. J., Khotkevich V. V., Morozov S. V. «Two-dimensional atomic crystals», PNAS 102, 10451 (2005) DOI:10.1073/pnas.0502848102 [05] Belov Petr A., Lurie S. A. «The theory of ideal adhesion interactions», "The mechanics of composite materials and designs", 2007, Volume 14, №3, pp. 519-536. [06] Belov Petr A. «Mechanical properties of graphene within the framework of gradient theory of adhesion», "Journal of Civil Engineering and Architecture", 2014, V. 8, №6, page 693-698. [07] Belov P. A., Lurie S. A., H. Altenbach Classification of gradient adhesion theories across length scale, Springer International Publishing Switzerland 2016, H. Altenbach and S. Forest (eds.), In book: Generalized Continua as Models for Classical and Advanced Materials, Advanced Structured Materials 42, ISBN 978-3-319-31721-2. [08] Belov PA, Lurie SA Mathematical theory of damaged media. Gradient theory of elasticity. Formulations. Hierarchy. Comparative analysis. 2014, Palmarium Academic Publishing, Germany, 336р, ISBN 978-3-639-73761-5 [09] Belov P. A., Nelyub V. A., The Timoshenko theory of plates with adhesive properties of face surfaces, Polymer Science, Series D, 2015, V8, №4, p. 307-309. [10] Sergey A. Lurie, P. A. Belov, K. D. Kharchenko, The theory of media with defect fields and models of deformation of functional layers in isotropic materials, Nanomechanics Science and Technology: An International Journal, 2015, V.6, №1, p. 1–16. [11] Sergey A. Lurie, P. A. Belov, Dmitriy B. Volkov-Bogorodsky, E. D. Lykosova, Do nanosized rods have abnormal mechanical properties? Nanomechanics Science and Technology: An International Journal, 2016, V.7, №4, p.1–35.