Mathematical Methods For Dynamic Responses Of A Class Of Hyperelastic Thin-walled Shells

Due to the high elasticity and excellent corrosion resistance,hyperelastic thin-walled shells are widely applied in many fields,including aerospace,marine engineering and auto-mobile industry.In addition,the dynamic characteristics of shells,especially the nonlinear responses,are related to its security and reliability.Thus,it is necessary to investigate the dynamic characteristics of hyperelastic thin-walled shells.In this paper,new methods are developed and applied to study the problems,some interesting phenomena are found.By formulating the mathematical model and considering the nonlinearity of structure and ma-terial,the dynamic problems of hyperelastic thin-walled shells are described as nonlinear differential equations,and the dynamic characteristics of shells are studied.The main works of this thesis are given as follows:(1)Based on Donnell's nonlinear shallow shell theory,hyperelastic constitutive rela-tion and Lagrange equations,the mathematical model is established.Combining with the two kinds of displacement functions which satisfy the geometric boundary conditions,the nonlinear differential equations describing the motion of shells are obtained,respectively.In addition,the equations are simplified by using the degree-of-freedom condensation method,and new differential equations are obtained by nondimensionalizing the simplified equations.(2)For the dynamic response equation of thin-walled shells given by displacemen-t functions of the first kind,by introducing new parameter transformations and the mod-ified Lindstedt-Poincare method,the perturbation analysis of nonlinear differential equa-tions describing the free vibration and forced vibration of walled-thin shells is carried out,the corresponding amplitude-frequency and phase-frequency response curves are presented.Numerical results demonstrate that the geometrically nonlinearity characteristic caused by the large deflection vibration leads to a hardening behavior,while the nonlinearity of hyper-elastic material weakens the behavior.(3)For the dynamic response equation of thin-walled shells given by displacement func-tions of the second kind,the fourth-order Runge-Kutta method is used to solve the equations numerically.Based on the bifurcation diagrams and the Poincare sections,the chaotic recog-nition describing the radial vibration of shells are illustrated.The influences of the amplitude and frequency of the external excitation,structural parameters and material parameters on the radial motion of the shell are examined.Numerical results show that:(i)There exists a crit-ical value for the excitation amplitude,when the value is larger than the critical value,the motions of shells alternate between chaotic motions and periodic motions;(ii)In terms of the results of multimodal expansion,it is found that the response of shells to radial motion is more regular than that without considering the coupling between modes,while there are more interesting phenomena for the uncoupled case.