Free flexural vibration of geometrically imperfect functionally graded microbeams

Free flexural vibration characteristics of functionally graded (FG) microbeams with geometric imperfection are explored numerically, taking into account the size effect phenomenon based on modified couple stress theory. This theory employs only one material length scale parameter to interpret the size-dependent mechanical behavior of microstructures. The mechanical and physical properties of FG microbeam are assumed to vary smoothly and continuously through the thickness direction according to a power-law distribution. Hamilton's principle in conjunction with Euler-Bernoulli beam theory is used to establish the coupled longitudinal-transverse equations of motion and associated boundary conditions. A weighted-residual method is utilized to evaluate the size-dependent free flexural vibration behavior of FG microbeams with clamped-clamped, clamped-pinned, and pinned-pinned boundary conditions. The influences of different dimensionless parameters i.e., maximum imperfection amplitude-to-length ratio, length-to-thickness ratio, flexural rigidity ratio, and power-law index on the flexural frequencies and mode shapes of FG microbeams are investigated. The mode veering phenomenon is also explored. Finally, the role of longitudinal displacement in the free flexural vibration of the geometrically imperfect FG microbeams is examined.