Modeling size-dependent behaviors of axially functionally graded Bernoulli-Euler micro-beam

This work focus on the mechanical behaviors, which are related to the size effect, functionally graded (FG) effect and Poisson effect, of an axially functionally graded (AFG) micro-beam whose elastic modulus varies according to sinusoidal law along its axial direction. The displacement field of the AFG micro-beam is set according to the Bernoulli-Euler beam theory. Employing the modified couple stress theory (MCST), the components of strain, curvature, stress and couple stress are expressed by the second derivative of the deflection of the AFG micro-beam. A size-dependent model related to FG effect and Poisson effect, which includes the formulations of bending stiffness, deflection, normal stress and couple stress, is developed to predict the mechanical behaviors of the AFG micro-beam by employing the principle of minimum potential energy. The mechanical behaviors of a simply supported AFG micro-beam are numerically investigated using the developed model for demonstrating the size effects, FG effects and Poisson effects of the AFG micro-beam. Results show that the mechanical behaviors of AFG micro-beams are distinctly size-dependent only when the ratio of micro-beam height to material length-scale parameter is small enough. The FG parameter is an important factor that determines and regulates the size-dependent behaviors of AFG micro-beams. The influences of Poisson’s ratio on the mechanical behaviors of AFG micro-beams are not negligible, and should be also considered in the design and analysis of an AFG micro-beam. This work supplies a theoretical basis and a technical reference for the design and analysis of AFG micro-beams in the related regions.