Summary. We outline a procedure for obtaining solutions of certain boundary value problems of a recently proposed theory of gradient elasticity in terms of solutions of classical elasticity. The method is applied to illustrate, among other things, how the gradient theory can remove the strain singularity from some typical examples of the classical theory.

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An elastic model is presented to study infinitesimal buckling of a double-walled carbon nanotube under axial compression. A simple formula is derived for the critical axial strain, which clearly indicates the role of the van der Waals forces between the inner and outer tubes characterized by two parameters. In particular, the analysis shows that inserting an inner tube into a single-walled nanotube does not increase the critical axial strain as compared to the single-walled nanotube under otherwise identical conditions, despite the fact that the total critical axial force of the double-walled nanotube could be increased due to an increase in the cross-sectional area.

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