Frequency analysis of sandwich beam with FG carbon nanotubes face sheets and flexible core using high-order element

In the present paper, the vibrational behavior of sandwich beam with a flexible core and anti-symmetric functionally graded carbon nanotubes face sheet is investigated. Carbon nanotubes are considered as functional graded materials in the thickness of the faces and their properties change along the thickness of the face sheets. For the modeling of sandwich beam behavior, the Euler-Bernoulli theory is used for face sheets and the semi-3D elasticity is used for the core, which allowed us to investigate the flexibility of the core. Differential equations of motions are derived using the virtual displacement principle. In this research, a high-order element is presented for solving equations of motion, and then by using this element, the finite element formulation has been extracted and solved. Numerical results are obtained for various boundary conditions, which include simply support, clamped, free-clamped, and simply support-clamped. Also, different volumes of carbon nanotubes have been investigated for different distributions. The results showed that the distribution of the FG-X pattern carbon nanotube leads to the highest natural frequency of the beam. The main conclusion of this research is that, in most cases the FG-O pattern has the lowest natural frequencies and in some cases the FG-? pattern has the lowest natural frequencies. In other words, generally, it can be say that the lowest natural frequencies of the sandwich beam with functionally graded carbon nanotubes faces depend on the boundary conditions, thickness ratio, and also the volume fraction of carbon nanotubes. In this paper also the effect of geometric angles of the beam, such as the thickness of the core and face sheet thickness on natural frequency of the system is also investigated.