Free Vibration And Post-buckling Of Functionally Graded Material Beam And Bar

Functionally graded materials(FGMs)are new type of non-uniform composites,with a high temperature and corrosion resistance because of its gradient of material composition and physical parameters.Therefore,it is widely used in aerospace,civil engineering,machinery and other fields.In recent years,many mechanics have embarked on a study of FGM structures.This paper will analyze the buckling and dynamic response of FGM beam and bar using the numerical method and analytic method.The main works are as follows:1.Analysis of free vibration of functionally graded Timoshenko beam.Based on the first-order shear deformation theory,the free vibration eigenvalues of the functionally gradient Timoshenko beam are studied by analytic method.Differential equation of motion of the functional gradient Timoshenko beam is deduced using the Hamilton principle.The free vibration equation of the FGM beam is solved and the degradation of the FGM beam is made into the homogeneous material beam.Compared with the numerical solution of the existing literature,the feasibility of the method and the correctness of the natural frequency are verified.Finally,the effects of gradient index and slenderness on the free vibration and natural frequency of FGM beams are discussed and modal analysis is carried out.2.The post-buckling analysis of functionally graded material bars with variable section compressionBased on the linear mixing rate and Mori-Tanaka two micromechanical models,the post-buckling response of the functionally graded variable cross-section bar was analyzed.The post-buckling differential equation of FGM variable cross-section compression bar is numerically solved by using the geometric nonlinear theory of axis extensibility.The influence of cross-section variation coefficient and gradient index on the critical load of the compression bar is studied under two models,and the post-buckling equilibrium path of the compression bar under different turning angles is plotted when the cross-section variation coefficient and the gradient index are fixed.The critical load of the functionally graded material press bar decreases with the increase of the gradient index when the cross-sectional variation coefficient is constant.When gradient index p> 2,the critical load tends to be stable.The critical load increases with the increase of the cross-sectional variation coefficient when the gradient index is constant.And the critical load of the Mori-Tanaka model is slightly larger than the critical load of the linear mixing model.The numerical results of this paper can provide the theoretical basis for the stability of the design bar.