ABSTRACT
This chapter considers the mathematical modeling of the free vibration problem of GPLR nanocomposite beams resting on an elastic substrate given the influences of thermal gradient and magnetic field on the vibrational response of the system. The beam-type element is of length L and thickness h that is assumed to rest on a Winkler-Pasternak elastic foundation. The GPLs are assumed to be dispersed in the polymeric matrix via UD pattern and the equivalent thermo-mechanical material properties of the constituent materials are developed using the Halpin-Tsai micromechanical method, as explained in Section 2.3.2. The motion equations of the problem will be derived based on expanding Hamilton's principle for refined shear deformable beams. Because of the effects of magnetic field on the dynamic responses of the nanocomposite structure, the magnetic induction relations of Maxwell must be written for the case of a beam-type element to reach the final version of the governing equations of the problem. It is worth mentioning that in the present chapter, only UTR-type thermal loading will be discussed. The governing equations of the problem can be found by referring to Eqs. (2.163)–(2.165). In addition to the above explanations, the magnetic field is assumed to be uniform and it will be applied in the longitudinal direction, parallel with the beam's axis. The governing equations will be solved using Galerkin's method, a powerful solution that can cover the effects of various types of BCs on the mechanical response of the continuous systems. It will be shown below that increasing the intensity of the magnetic field can help to improve the frequency range that can be supported by the nanocomposite beam. Also, thermal losses can play a crucial role in reducing the thermo-magnetically affected dynamic responses of the system.
