ABSTRACT
This chapter presents an analytical solution for the damped dynamic bending behaviors of GPLR nanocomposite rectangular plates. The damping effects are generated due to the internal damping of the polymeric nanocomposite and these effects are covered in this chapter by implementing a viscoelastic model for the linear elastic nanocomposite materials. The effective material properties of the nanocomposite in the elastic domain can be easily generated using the fundamentals of the Halpin-Tsai micromechanical method for the case of a single-layered GPLR nanocomposite (for more information, refer to Section 2.3.2). The refined-type HSDT of the plates will be utilized to derive the governing equations of the plate, as expressed in Eqs. (2.222)–(2.225). However, note that the aforementioned set of governing equations belongs to the dynamic problem of an elastic GPLR nanocomposite and these equations must be changed to reach the governing equations for the damped bending analysis of a viscoelastic GPLR nanocomposite plate. This transfer can be simply accomplished using the instructions presented in Section 2.3.6 for viscoelastic nanocomposite materials. Based on the referenced section, the cross-sectional rigidities of the plate must be multiplied by the term https://www.w3.org/1998/Math/MathML"> ( 1 + g ∂ ∂ t ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429316791/7c807740-5d49-4cfe-8851-729db7fee972/content/math21_1.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> to derive the governing equations of a viscoelastic nanocomposite plate. In addition, the influence of the applied transverse bending force must be considered as mentioned at the end of Section 2.2.7 for the transient analysis of a refined plate. Following the previous steps, the final governing equations will be obtained and they will be solved analytically by combining the Navier-type solution with the concept of the Laplace transformation to produce the time-dependent deflection of the viscoelastic nanocomposite plate. In the chapter, a group of numerical examples will clarify the effects of the material's internal damping on the dynamic responses of the nanocomposite plate.
