On Nonlocal Vertical and Horizontal Bending of a Micro-Beam

The vertical and horizontal bending of micro-beams subjected to axial compressive and transverse concentrated loadings is a common lateral deformation in micro-/nano-engineering that plays a significant role in the design and optimization of micro-/nano-devices. The present study aims to investigate it using the nonlocal theory. For this purpose, the simplified mathematical model is developed, and the nonlocal differential constitutive equation is applied. Since the mechanical properties of micro-beams are different from those of macro-beams, the non-classical nonlocal bending moment is introduced to improve the classical bending formulation in order to adapt to the vertical and horizontal bending of micro-beams. The effects of the external load, external size, structural stiffness, and internal characteristic scale on the vertical and horizontal bending deformation including the midpoint deflection and critical compression are presented. The present analytical model and results are validated by the finite element method. It is shown that the critical compression decreases with increasing the internal characteristic scale. Moreover, the midpoint deflection varies remarkably with respect to the axial and transverse loadings, structural stiffness, internal characteristic scale, and external size. An obvious nonlocal scale effect is found, in which the internal characteristic scale cannot be neglected compared with the external size. Besides, a threshold value of the structural stiffness is determined, and the connotation of the nonlocal interaction requires that the structural stiffness shall not be lower than that threshold. A mutual restriction between structural stiffness and external loadings is observed in the vertical and horizontal bending. In particular, it is further proved that the classical continuum mechanics can not be used in micro-/nano-scaled mechanics through a strange phenomenon that is contrary to mechanical common sense in the calculation example. The study is expected to be beneficial to the design and application of micro-beams subjected to the vertical and horizontal bending.