The thermodynamic behaviour of two-dimensional single-component elastic crystalline solids is developed: the surface Euler's equation, the surface Gibbs equation, the surface Gibbs-Duhem equation, and the conditions to be expected at equilibrium, including the stress-deformation behaviour of the crystal. The analysis recognizes that the surface Helmholtz free energy is an explicit function of the lattice vectors defining the crystalline structure. As an application, we obtain the stress-deformation behaviour of single-wall carbon nanotubes which are composed of a regular two-dimensional array of hexagonal lattices of carbon atoms. Using two potentials, Tersoff -Brenner  and Brenner et al.  to describe interatomic potentials and hence the specific surface Helmholtz free energy, we compute the surface elastic properties for the single-wall carbon nanotubes. These are compared with the available experimental values. 2005 Taylor & Francis Group Ltd.