The effect of thickness on the mechanics of nanobeams

Nonlocal strain gradient models are being used more and more extensively in examining the size-dependent effects on the statical and dynamical behaviors of micro/nano-structures. However, for the fake of simplification, the size-dependent effects are often assumed to be neglected in the thickness direction of beams and plates. This simplification is originally aiming at reducing the complexity of the studied beam and plate problems while retaining all the most important mechanical features. Based on the nonlocal strain gradient model without the thickness effect, the size-dependent effect may reveal stiffness-softening or stiffness-hardening effect, which only depends on the values of strain-gradient and stress-gradient parameters but is independent on the geometric feature. In this paper, a nonlocal strain gradient beam model incorporating the thickness effect is developed for the size-dependent buckling analysis of nanobeams, and closed-form solutions are derived for post-buckling configuration and critical buckling force. When incorporating the size-dependent effect of thickness, it is found that the stiffness-softening and stiffness-hardening effects depend not only on the ratio of stress-gradient parameter to strain-gradient parameter, but also on the geometric feature (slenderness ratio). Interestingly, the stiffness-softening effect can be only found for the longitudinal dispersion relation of Silicon, while the stiffness-hardening effect may be observed for the buckling behaviors of nanobeams made of Silicon due to the size-dependent effect of thickness.