Nonlinear dynamics of two-directional functionally graded microbeam with geometrical imperfection using unified shear deformable beam theory - Liu, Hu (2022)
In this paper, the nonlinear dynamic response of two-directional functionally graded (2D-FG) microbeam incorporating geometrically imperfect effect is investigated via a unified shear deformable beam theory, which can degenerate into several previous theories including Euler-Bernoulli, Timoshenko, and Reddy shear deformable beam theories. The material properties of 2D-FG microbeam are assumed to be varied continually along both the axial and thickness directions following the power-law distribution, and two patterns of material distribution along the thickness direction are considered. Both the global and localized imperfection modes are taken into account by the product of trigonometric and hyperbolic functions. Employing modified couple stress theory and the Hamilton's principle, the governing equations of the imperfect 2D-FG microbeam are derived and solved with the aid of the differential quadrature method. The influences of material distribution pattern, geometrical imperfection shape and amplitude, axial and thickness power-law indices, as well as length scale parameter and boundary condition on the nonlinear vibration performance of the 2D-FG microbeam are examined in detail. It is expected that the presented numerical results can be used to guide the optimal design of multi-functional micro-structures.