Qualitative Analyses Of Micro-Void Motion Centered At Incompressible Hyperelastic Spheres

Rubber and rubber-like materials are typical hyperelastic materials.They are widely used in the fields of engineering and national defense because of their superior properties,such as high elasticity and corrosion resistance,and so on.Particularly,the stability of hyperelastic materials has been paid great attention due to the dual nonlinear effects of materials and geometries.In this thesis,mathematical models of micro-void centered at incompressible hyperelastic spheres under periodic perturbation loads and structural damping are formulated,and the dynamic problems of hyperelastic structures are transformed into the problems of initial and boundary conditions of nonlinear differential equations,and the dynamic characteristics of micro-void are analyzed in detail.The main research contents are as follows:1.For the neo-Hookean material model,a nonlinear ordinary differential equation describing the radially symmetric motion of the micro-void is formulated in terms of the equilibrium differential equation and the initial-boundary conditions.Through qualitatively analyzing the solutions of the differential equation,some interesting qualitative behaviors of micro-void are given.(1)For constant loads,the effects of material parameters and structural parameters on equilibrium points of the system are discussed,especially,the secondary turning bifurcation of micro-void and the periodic and amplitude jumps of the system are analyzed.(2)For periodic perturbation loads,the quasi-periodic and chaotic motions of the micro-void are discussed in terms of the secondary turning bifurcation by using the numerical calculation,the existence conditions of chaos are given,and the effects of periodic perturbation loads on the chaotic motions of the micro-void are further analyzed.2.For the Gent-Thomas material model,the mathematical model describing the radially symmetric motion of micro-void is established by using the energy variational principle.And some meaningful conclusions have been drawn.(1)Under constant loading without damping,the bifurcation behavior of the micro-void motion is discussed,and the influences of physical parameters on the number of equilibrium points in the system are given.Under constant loading with damping,the effects of structural damping on the domains of attraction of the system are analyzed in detail.(2)Under periodic loading without damping,the quasiperiodic and chaotic motions of the micro-void are investigated,and the effects of perturbation parameters on the chaotic motions of the micro-void are further analyzed.Under periodic loading with damping,the periodic and quasi-periodic motions of micro-void near the center are discussed.The critical value for chaos is obtained by using the Melnikov method and numerical calculation,and the bifurcation characteristics of micro-void motion near the saddle point are analyzed by using the bifurcation diagrams.In particular,it is found that the motions of micro-void will present a process of alternating periods to chaos to periods,and that it will enter chaotic motion in the form of period doubling bifurcation and returns to periodic motion in the form of inverse period doubling bifurcation.