Bifurcation And Chaos Of Functionally Graded Materials Circular Plates

Functionally graded material(FGM) is a new type of composite consisting of two or more phases, which can be designed its composition varies in some spatial direction by special manufacturing process. Usually, functionally graded materials are made from a mixture metal and ceramics. It is known that ceramic constituent of the material has excellent characteristics in heat-resistance while the metal part keeps a certain extent of strength and toughness. Compared with the homogeneous materials, FGM can take a more complex nonlinear dynamics due to the affect of the mechanical and thermal loads. Research on the nonlinear dynamics of FGMs plates has an important theoretical and engineering application. The aim of the thesis focuses on the bifurcation and chaos behavior of functionally graded circular plate by theoretical analysis and numerical calculation.Based on the theory of plates and shells, the nonlinear forced axisymmetric vibration of a thin circular functionally graded plate in thermal environment is formulated firstly. For a circular plate with clamped immovable edge, the nondimensional Duffing vibration equation are derived by using Galerkin's approach.Secondly, considering two cases of primary resonance and combination resonance, a multi-scale method is utilized to obtain the bifurcation equation, respectively. Using singularity theory, the transition sets in parameter and the bifurcation diagrams are plotted under some conditions for unfolding parameters, and then discuss the corresponding bifurcation behaviors of singularity. Numerical simulations including global bifurcation diagrams, wave forms, phase portraits, Poincare map and amplitude-frequency curves are plotted. The results show that the vibration of functionally graded circular plate should be influenced by the material volume fraction index, temperature and excitation. It is observed that periodic, multi-periodic and chaotic motions exist for the FGM circular plate under certain conditions.Finally, the Duffing system with different external forcing terms is investigated. The criterion of existence of chaos under the periodic perturbation is given by using Melnikov method. Numerical simulations including homoclinic bifurcation curves, homoclinic bifurcation surfaces, global bifurcation diagrams and maximum Lyapunov exponents are given to illustrate the theoretical analysis, and to expose the influence of material volume fraction index and temperature etc.