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The behaviour and failure of carbon nanotubes (CNTs) under cyclic loading conditions are of high interest because of the potential use of CNTs as nanoscale building blocks of synthetic biomaterials, electromagnetic devices or polymer composites. Very limited work has been reported on this topic, although substantial research has been published on the mechanical strength, plastic deformation and failure mechanisms of CNTs. The authors show theoretically for the first time the stress-strain behaviour of CNTs under cyclic tensile and compressive loads by numerical simulation. Nonlinear elasticity, preconditioning (stress softening) and hysteresis characteristics have been reported most recently on multiwalled CNT blocks. It is found that elastic instability (local and global buckling) has a large influence on the stress-strain behaviour of CNTs under cyclic compression, whereas the cyclic strength remains unchanged under cyclic loading. The residual defect-free morphological deformation is considered as the primary mechanism responsible for the cyclic failure of CNTs.