A size-dependent nonlinear microbeam model based on the micropolar elasticity theory

A size-dependent microbeam model including von Karman geometric nonlinearity is developed based on the micropolar elasticity theory. The governing equations and the boundary conditions are obtained using Hamilton's principle. In the present model, the microrotation quantity and two new material parameters (characteristic length of material and second shear modulus) are introduced. The size effects, nonlinear static and dynamic behaviors are investigated by directly applying the formulas derived. Special attention is paid on the operating mechanism of microrotation and effects of the second shear modulus on the system. The numerical results for the static bending problem reveal that (1) geometric nonlinearity and the size effect enhance the stiffness of the beam; (2) the microrotation has an opposite motion trend to those of deflection and macroangle of the centroidal axis with changing beam thickness or second shear modulus. A multiple-scale method is employed to obtain an approximate analytical solution for the natural frequency and time response of the beam. It shows that (1) the second shear modulus has significant effects on the lower-order frequency, while the effect on the higher-order frequency is negligible; (2) for increasing the natural frequency of the beam, geometric nonlinearity plays the essential role for the beam with larger thickness, while the size effect plays the essential role for the beam with smaller thickness.