The reformulated micro-beam models by incorporating the general strain gradient elasticity theory (GSGET)

Abstract Two independent material parameters (Lame constants) are incorporated in classical isotropic elasticity theory, while how many independent material parameters are incorporated in the higher-order isotropic elasticity theory? It is answered by Zhou's General Strain Gradient Elasticity Theory (GSGET), which firstly proved that only three material parameters are independent. In this paper, the static and dynamic analyses of micro-beam models are proposed based on the GSGET, which can capture the size effects of micro-structures. Then the variational formulations of the micro-beams including Bernoulli-Euler and Timoshenko beams are deduced by taking advantage of Hamilton's principle. Governing equations, initial conditions, and all possible boundary conditions are derived subsequently. What's more, the static bending and free vibration problems for a simply supported beam have been analytically solved, and the effects of normalized maximum deflection, stiffness, and natural frequency are studied by comparing the present results with the classical and couple stress results respectively. The results indicate that the GSGET should be used to characterize the size effects of mechanical properties for micro-structures.