Mechanical Response Analysis Of The FGM Beam And Plate On Frational Order Viscoelastic Foundation Subjected To Dynamic Loads

It is well known that the results of mechanical response of beam and plate on elastic foundation have been used widely in practical engineering. But sometimes the foundation will show the nature of viscoelasticity, so the study of mechanical response of beam and plate on viscoelastic foundation can simulate actual working conditions more accurate. Many scholars have carried out some research on this problem. However, there are a lot of mechanical models of viscoelasticity. Study on mechanical response of beam and plate on different model of viscoelastic foundation can verify the theory and play an important role in practical applications. Functionally Graded Material as a new type of composite material overcome many practical problems with its excellent performance. Therefore, the research on the mechanical response of FGM beams and plates has become the frontier of the research of mechanics. In this paper, mechanical response of FGM beams and plates on the fractional order viscoelastic foundation subjected to dynamic loads is studied.Exact solution for an infinite FGM beam on frational order viscoelastic foundation subjected to dynamic loads. Firstly, based on the fractional Zener model, the governing differential equations of motion for an infinite FGM beam resting on a frational order viscoelastic foundation subjected to dynamic loads are formulated. The Fourier and Laplace transforms are utilized to simplify the governing differential equation of the beam to an algebraic equation, so that the solution can be conveniently obtained in the frequency domain. The inverse Laplace transform, inverse Fourier transform, and convolution theorem are employed to convert the solution into the time domain and the exact solution for responses of deflection, velocity, acceleration, bending moment, and shearing force of the FGM beam are obtained. Finally, the dynamics responses of the FGM beams on elastic foundation subjected to shock loads are calculated and vertical velocity and bending moment responses curves at the position of x =0 are shown.Analytical solution for an infinite FGM plate on viscoelastic foundation subjected to impulsive loads. Based on the fractional Zener model, the dynamic equations of an infinite FGM plate resting on a viscoelastic foundation under impulsive loads were established. By using the triangular series method and Laplace-Fourier integral transformation method,the problem for the dynamic response of an infinite FGM plate resting on a viscoelastic foundation subjected to impulsive loads was developed. Final, the analytical solutions for deflection, velocity, acceleration responses of FGM plate were deduced. The velocity response and acceleration response of an infinite FGM plate resting on an elastic foundation subjected to impulsive loads were obtained by numerical integration algorithm. Numerical results for different gradient index were presented and compared with each other. In addition, the effect of stiffness coefficient on velocity was studied.