Static bending model and size effect of plane orthotropic functionally graded beam based on a new modified couple stress theory

Based on the new couple stress theory, a new model of plane orthotropic functionally graded beam model was proposed. The two material scale parameters included in the model enables present model to describe different bending stiffness enhancements in two orthogonal directions caused by scale effect. The equilibrium equations and related boundary conditions were derived using the principle of minimum potential energy. Analytical solution of bending problem was developed for a bending cantilever beam with concentrate force applied on the free end. The governing equations and solutions of present model have similar form with classical beam model, except one additional term related to scale effect in rigidity. Numerical results indicate that deflection of the beam predicted by present model is smaller than that of the classical one, which means the scale effect can be captured. The scale effect will increase with the decrease of the geometrical size of the beam, but diminish little by little when the geometrical size is much larger than the material scale parameter.