Dynamic Analysis of Sigmoid Bidirectional FG Microbeams under Moving Load and Thermal Load: Analytical Laplace Solution

This paper presents for the first time a closed-form solution of the dynamic response of sigmoid bidirectional functionally graded (SBDFG) microbeams under moving harmonic load and thermal environmental conditions. The formulation is established in the context of the modified couple stress theory to integrate the effects of microstructure. On the basis of the elasticity theory, nonclassical governing equations are derived by using Hamilton’s principle in combination with the parabolic higher-order shear deformation theory considering the physical neutral plane concept. Sigmoid distribution functions are used to describe the temperature-dependent thermomechanical material of bulk continuums of the beam in both the axial and thickness directions, and the gradation of the material length scale parameter is also considered. Linear and nonlinear temperature profiles are considered to present the environmental thermal loads. The Laplace transform is exploited for the first time to evaluate the closed-form solution of the proposed model for a simply supported (SS) boundary condition. The solution is verified by comparing the predicted fundamental frequency and dynamic response with the previously published results. A parametric study is conducted to explore the impacts of gradient indices in both directions, graded material length scale parameters, thermal loads, and moving speed of the acted load on the dynamic response of microbeams. The results can serve as a principle for evaluating the multi-functional and optimal design of microbeams acted upon by a moving load.