A non-linear analysis of the bending modulus of carbon nanotubes with rippling deformations

Recently, some experimental techniques involving bending of cantilevered nanotubes have been used to deduce their elastic bending modulus. Using these methods it was found that the value of elastic bending modulus decreases dramatically from about 1 TPa for D less than 10 nm to 100 GPa for D larger than 20 nm. This discovery defines usual belief that the bending modulus is a material’s intrinsic property, independent of the size of the nanotubes. This paper concerned with the non-linear moment–curvature relationship during bending of nanotubes, and the effect of ripple formation on the bending modulus. By using an advanced finite element analysis package, ABAQUS, a non-linear bending moment–curvature relationship of carbon nanotubes, which is the appearance of a rippling mode on the inner arc of the bent nanotubes, is simulated from the three-dimensional orthotropic theory of finite elasticity deformation. Utilizing the non-linear bending moment–curvature relationship, and a non-linear vibration analysis method, we capture that the rippling deformation can indeed result in effective bending modulus of carbon nanotubes decrease substantially with increasing diameter. The result carried out may be used to explain some experiment phenomena, and also guides further experiment investigating the bending behaviors of carbon nanotubes.