Static, dynamic and buckling behavior of functionally graded beams with tunable inclusions

The introduction of microstructurally tunable materials endorses functionally graded (FG) beams with distinctly inspiring functionalities and mechanical properties. In order to facilitate the multiscale design of FG beams and evaluations of corresponding mechanical behavior, analytical solutions are developed in the present research on the static, dynamic and buckling responses of FG beams reinforced with graphene platelets, porosity or origami inclusions, etc. To facilitate the analytical derivation as well as numerical extension, this paper establishes a theoretical framework, firstly formulating the principle of virtual work and reciprocal theorem of work for the FG beams, and then deriving and proving the principles of minimum potential and complementary energies, the former of which is used for the establishment of the variational principles for the static bending, free vibration and buckling phenomenon in the present work. Following the proposed variational principles and considering Timoshenko beam theory, the strong-form partial differential governing equations as well as the corresponding boundary conditions can be obtained for static bending, free vibration and buckling responses of FG beams. The Ritz method is finally employed to obtain the explicit expressions of transverse bending and rotation as well as the frequencies and critical loads for FG beams with different boundary conditions. What's more, it should be noted that the FG of the beams is achieved through tailoring micromechanical parameters, such as volume fraction and distribution of reinforcement, which are usually unevenly distributed. The mechanical responses of FG beams are also predicted through micromechanics approaches, such as Halpin-Tsai expression. Conventional GPL distributions as well as graphene origami (GOri) distributions with negative Poisson's ratios (NPR) are also presented to validate the analytical derivation and solutions. Finally, the proposed solutions are employed to investigate the effect of the microstructural parameters on the effective behavior of FG beams.