The pseudospherical reduction of an uniaxial deformation of the Carbon nanotubes Petre P. Teodorescu, Veturia Chiroiu and Ligia Munteanu Abstract This paper is focused on the single-walled carbon nanotubes with different radius and chirality (armchair (n, n), chiral (2n, n), and zigzag (n, 0)), and perfect circular cross section, subjected to tensile loading. The aim is to determine a parametrical representation for a class of constitutive laws for which the motion equations attached to a material system can be associated to a pseudospherical surface (with negative Gaussian curvature K). A genetic algorithm is performed to study some inverse problems associated to some experimental results. Mathematics Subject Classification: 37N15, 14Rxx. Key words: Carbon nanotubes, pseudospherical surface, Guassian curvature, con- stitutive law, uniaxial deformation. 1 Introduction The carbon nanotubes as a quasi-one-dimensional structures were discovered by Iijima [1], Iijima and Ichihashi [2]. The carbon nanotube is one of the most promising building blocks for future development of functional nanostructures (Srivastava, Menon and Cho [3], Gao, Cagin and Goddard [4]). The single-walled carbon nanotubes can be regarded as a rolled-up graphite sheet in cylindrical form. Thess and co-workers [5] produced crystalline ropes of metallic carbon nanotubes with 100–500 single-walled carbon nanotubes bundled into a two-dimensional triangular lattice. These ropes are expected to have remarkable mechanical, electronic and magnetic properties. The na- notube is a cylindrical molecule composed of carbon atoms, with open or closed ends. A typical section of a single-walled carbon nanotube is illustrated in Fig. 1.1, each node being a carbon atom and lines the chemical bonds (Ruoff et al. [6]). * The Fifth Conference of Balkan Society of Geometers , August 29 - September 2, 2005, Mangalia- Romania, pp. 157-165. c  Balkan Society of Geometers, Geometry Balkan Press 2006.