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Although the small-scale effect and the material nonlinearity signi cantly impact the mechanical properties of nanobeams, their combined effects have not attracted researchers' attention. In the present paper, we propose two new nonlinear nonlocal Euler-Bernoulli theories to model nanobeam's mechanical properties corresponding to extensible or inextensible locus. Two new theories consider the material nonlinearity and the small-scale effect induced by the nonlocal effect. The new models are used to analyze the static bending and the forced vibration for single-walled carbon nanotubes (SWCNTs). The results indicate that the material nonlinearity and the nonlocal effect signi cantly impact SWCNT's mechanical properties. Therefore, neglecting the two factors may cause qualitative mistakes.

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Based on Hamilton's principle and the modified couple stress theory, a Bernoulli-Euler microbeam model is developed with the finite rotation of the cross-section. The present model includes three couple-stress-induced nonlinear terms, and these nonlinear terms have a significant influence on the mechanic response of the beam.