Nonlinear Problems And Bifurcations In Solid Mechanics

By comparing theory modeling with experiment research, some nonlinear problems in solid mechanics are studied in this paper. At first basic concepts and history of nonlinear differential dynamic system, movement stability, structure stability and bifurcation are introduced, and then the conditions giving rise to static bifurcations such as saddle-node bifurcation, transcritical bifurcation and pitchfork bifurcation of one dimensional differential dynamic system are deduced and non-hyperbolic point bifurcation property investigation method and Hopf bifurcation are studied.Then theoretical Analyses on the nonlinear response of a clamped-sliding buckled beam to a harmonic axial excitation are presented. We deduce the governing movement equation of the buckled beam and calculate the local bifurcation values of the system by multiscale method, then use numerical method to analyse the global bifurcation phenomena and present the list of the displacement extremum of the buckled beam's crosswise perturbation, describe the mode to chaos from period-doubling bifurcation of a nonlinear dynamic system under period excitation which indicates that period-doubling bifurcation is an approach to chaos. The configurations of its period attractors, chaos attractors and period-doubling attractors are simulated and the transient chaos phenomena are observed. It's proved that there exists period-doubling, chaos movement and other complicated dynamic behaviors such as the period windows between transient chaos and chaos.After the theoretical analyses, we do some corresponding experiments to check up the reliability of the theoretical analyses. The experiments present the movement course of the fundamental parametric resonance and principle parametric resonance, exhibit the time histories and frequency spectra diagrams and then post the characteristics of nonlinear vibrations of the parametric system under this type of boundary conditions. The experiment data accord with the theoretical and simulation results. Moreover problems of nonlinear vibration of some structures such as beam, arch and shell are summed up to dynamical models possessed material, geometric and boundary conditions nonlinearities. The model's mathematical form is nonlinear partial differential equations with space-time variable derivatives. Using physical and geometry relations of the models, we change them to nonlinear ordinary differential dynamic equations, then use Melnikov method to study the dynamical behavior of this type of equations, seek the critical value when Smale horseshoes chaos takes place, employ numerical method to simulate the dynamic phenomena at the values near the critical value. The results match those analysed by Melnikov method.