Static And Dynamic Response Of Functionally Graded Material Beams And Circular Plates Subjected To Follower Forces

Functionally graded material (FGM) has been regarded as one of the advanced inhomogeneous composite materials, usually made from metal and ceramic, taking advantage of the merits of constituent materials adequately which has found wider application foreground in many fields such as aerospace, mechanical, electronics, nuclear engineering, civil engineering, and so on. It is of great important to analyze the static and dynamic mechanical behavior of non-conservative and conservative FGM structures and the problem of static deformation and dynamic response of FGM structures have been extensively investigated and become one of the most important issues in solid mechanics. Now, in the study of FGM structures, the static researches are more than the dynamic ones, and also the researchers on conservative systems are more than these on non-conservative ones. Especially, because of strong nonlinearity and continuous change in position of material properties, the investigations on the dynamics of FGM structures under non-conservative follower force are very few. In this paper, for a non-conservative system of FGM beams and circular plates subjected to distributed follower force, a more comprehensive study, especially quantitative research on static and dynamic characteristics are carried out, and then some useful results are obtained.The main contents of this dissertation are involved as the following four parts.(I) Firstly, the basic equations of static and dynamic response under distributed follower force of FGM beams, are presented. By assuming that the amplitude of the beam vibration is small and its response is harmonic, the above non-linear partial differential equations are reduced to two sets of coupled ordinary differential equations, the one is for postbuckling and the other is for the dynamic response. As a special case, these equations mentioned above are easily degraded to basic equations subjected to conservative load. On this basis, the numerical methods which can be used to deal with the static and dynamic response of FGM beams under non-conservative distributed follower force are established. The equilibrium paths as well as the post-buckling configurations of the deformed beam are presented. The effects of material gradient property and boundary conditions on the buckling deformation of the FGM beam are discussed in details. The numerical results show that features of the equilibrium paths of FGM beams subjected to a non-conservative load are different from those to a conservative one. Next, we further study the post-buckling behavior of FGM Timoshenko beams subjected to a distributed tangential follower force along the central axis. By accurately considering the axial extension and transverse shear deformation, geometrically nonlinear governing equations for FGM beams under non-conservative force are formulated. The numerical results indicate that the post-buckling behavior of FGM Timoshenko and Euler beam are significantly different.(Ⅱ) The free vibrations in the vicinity of post-buckling of FGM Euler beam subject to axially distributed follow force are investigated. By employing the numerical shooting technique to solve the ordinary differential equations obtained in chapter2, Characteristic curves of the first three natural frequencies of the FGM beam versus the dimensionless load parameter are illustrated. The effect of the parameters of material gradients and load on the responses of vibration of the beam was analyzed and discussed. The numerical results show that the three lowest frequencies of the pre-buckled beam decrease with an increment of load parameter for both hinged-fixed beam and hinged-pinned beam. The first natural frequency has a strong nonlinearity. However, when the beam is in post-buckled state, the frequency versus the load parameter is showing a different variation. The frequency-load relationship is between the pure ceramics and metal.(Ⅲ) Based on the assumption of straight normal line and by employing geometrically nonlinear theory for the extensible beams, governing equations of large static deformation of curved elastic FGM beam subjected to a distributed tangential follower force along the central axis are established. In the mathematical model, not only the effects of the axial elongation and the initial curvature of the curved FGM beam on the deformation are accurately taken into account but also the tension-bending coupling is considered. By using shooting method, the nonlinear plane bending of a semicircle curved FGM beam made of metal and ceramic subjected to both tangentially distributed follower force along the axial line and distributed follower force along the radial direction are analyzed. The equilibrium paths and configurations of the deformed FGM beams with different material gradient properties, varying with the load parameter in a large range, are presented, and compared with those of the curved beams made of pute metal and ceramic. Then governing equations of large static deformation of FGM elastic composite curved beams subjected a non-conservative distributed tangential load along axis are established. As a numerical example, the nonlinear plane bending of large static deformations of a cantilever composite beam composed of straight beam and one fourth circle part is presented. The results indicate that the basic theory and numerical method we built can be used to analyze the problem of FGM curved beams subjected to arbitrarily distributed load(conservative and non-conservative).(Ⅳ) Based on the classical plate theory, governing equations of post-buckling of a functionally graded circular plate subjected to uniformly distributed follower compressive force are established. It is assumed that the material properties of the beam vary continuously in the thickness direction. By using shooting method to solve the ordinary differential equations with clamped and simply supported boundary in the middle plane, the response of post-buckling of the FGM circular plate is obtained. Effects of material gradient parameter on the critical buckling, post-buckling behaviors of the FGM plate are discussed in details.The research results will be helpful to us to further understand the characteristics of static and dynamic responses of FGM structures, and improve the research achievement of non-conservative system.