Free vibration of a sandwich beam with functionally graded materials face sheets and three types of auxetic cores using fifth-order shear deformation theory for general strain

In this research, based on fifth-order shear deformation theory for general strain, free vibration of a sandwich beam with (FGM) face sheets and three types of auxetic cores, including concave hexagonal (Type 1), conventional re-entrant hexagonal (Type 2), and hexa-chiral (Type 3), is investigated. The governing equations of motion are derived using the variational method, thus, the mass and stiffness matrices are derived for a sandwich beam. The effects of various parameters, such as aspect ratio, thickness ratio, power index of FGM, angle between vertical and inclined members, three types of auxetic cores, and material properties, are examined. The Types 3 and 2 exhibit the highest and lowest natural frequency, respectively, that the auxetic effect is highly dependent on the core geometric parameters. It is concluded that the natural frequency of the system for the conventional re-entrant hexagonal auxetic core with L/h=8 and (P=10) in comparison to the concave hexagonal core increases by 1.6%. for L/h=2, the natural frequency corresponding to Type 3 increases by 8.9% and 11.7% compared with Types 1 and 2, respectively. With an increase in L/h=20, the natural frequency of Type 3 further increases relative to Types 1 and 2 by 9.1% and 15.2%, respectively.