Effect of Axial Stresses of the Core on the Free Vibration Response of a Sandwich Beam with FG Carbon Nanotube Faces and Stiff and Flexible Cores

In this article, a vibrational behavior of sandwich beams with stiff and flexible cores and face sheets reinforced with carbon nanotubes is investigated. Carbon nanotubes are used as materials with properties varying along the thickness. In order to model the behavior of faces, the Timoshenko beam’s theory is employed and also for modeling the behavior of the core, three-dimensional elasticity is used. The axial stresses of the core are considered in this model and therefore it is suitable for modelling two types of stiff and flexible cores. The equations of motion are derived using the variations of energy, and the Navier method is used to solve the equations of motion. Results are presented for different volumes of carbon nanotubes with different distributions along the thickness of the faces. In the case of stiff core, results show that the FG-V distribution has the highest natural frequency and the FG-Λ distribution has the lowest natural frequency in all cases. For flexible core, the FG-X distribution leads to the highest natural frequency and also the FG-O distribution has the  lowest natural frequency. Furthermore, results indicate that an extended high-order sandwich panel theory is a suitable model for analysis of stiff and flexible core sandwich panels. It must be mentioned for the cores made of stiff materials, the normal stress along the length of the core must be considered. It is due to the fact that the obtained results show that ignoring the normal stress along the length of the core leads to the large difference in the natural frequency of the system. In this article, due to the high order displacement field of the core, the flexibility of the core can be seen in the modeling. Additionally, since the term σc xx of the core is considered in the strain energy, a stiff core can be modeled. In many works the axial stresses of the core is removed from equations, therefore according to the results of sandwich beam with stiff core, lots of errors will be observed. Therefore, a proposed theory in this research can easily model a sandwich beam with two types of stiff and flexible cores. Since the Timoshenko beam theory is also implemented for modeling faces, different pattern of CNTs can be investigated accurately.