Dynamic Behaviors Of Incompressible Nonlinearly Elastic Spherical Structures Under Dynamic Loads

Hyperelastic materials are also called the Green elastic materials,whose constitutive relations can be completely described by their strain energy functions.Their spherical structures(such as hollow sphere,spherical shell and spherical membrane)have excellent mechanical properties,large volume,light weight and low cost,which are widely used in modern architecture,aerospace,medical and health fields.Particularly,these structures will inevitably be subjected to various dynamic loads(such as constant load,periodic step load and harmonically periodic load,etc.)in the process of application,therefore the deformation,motion,instability and failure problems must be taken into consideration in the design of hyperelastic spherical structures.The modeling and analyzing of incompressible nonlinearly elastic spherical structures under dynamic loads are carried out in this dissertation.The bifurcation and chaos phenomena of hyperelastic and visco-hyperelastic structures are mainly focused.The details are given as follows:1)For the spherical membrane composed of isotropic incompressible hyperelastic materials,the nonlinear dynamic behaviors of the structure under periodic loads are discussed.The following two aspects are mainly concerned,(?)for the Mooney-Rivlin material model,the governing equation describing the radially symmetric motion of the structure is derived in terms of the incompressible constraints and stress boundary conditions,and the qualitative and quantitative analyses of dynamic behaviors of the system are presented for different parameters.Significantly,for the given constant load,the system is integrable.In certain cases,the system has asymmetric homoclinic orbits of "?" type.For the given periodically perturbed load superposed on the constant one,the system is approximately integrable.The level curves near the center point break into islands,the spherical membrane performs a quasi-periodic motion,the envelope of the time history curve exhibits periodicity;the Poincare section and the maximal Lyapunov characteristic exponent are used to show the existence of chaos near the saddle point,and the evolution processes of the motion from periodic to quasi-periodic and chaotic are presented.(?)for Rivlin-Saunders material model of the power-law type,based on the variational principle,the governing equation describing the problem is obtained with the spherically symmetric deformation assumption.Then,the dynamic characteristics of the system are qualitatively analyzed in detail in terms of different values of material parameters.Particularly,for the given constant load,the parameter spaces describing the bifurcation behaviors of equilibrium curves are established;for the periodically perturbed load,the quasiperiodic and chaotic behaviors are discussed for the systems with two and three equilibrium points,respectively.2)For the spherical shell composed of isotropic incompressible hyperelastic materials,the nonlinear dynamic behaviors of the structure under dynamic loads are discussed.The following two aspects are mainly concerned,(i)for the Rivlin models with polynomial forms,a second-order nonlinear ordinary differential equation describing the radially symmetric motion of the spherical shell is obtained.The effects of material parameters and higher order terms in the strain energy function on the number of equilibrium points are discussed by qualitatively analyzing the differential equation.For the given structural and material parameters,interestingly,it is shown that there exist critical loads,the phase diagrams may be the asymmetric homoclinic orbits of the "?" type or the "?" type,the jumping phenomena of the period and the amplitude may occur,in certain cases,the structure may be destroyed.It should be qualitatively pointed out that the dynamic responses of the structure can be qualitatively shown by the lower order terms of the strain energy function,while adding the higher order term can more accurately describe it.(ii)For the Yeoh material model,based on the variational principle,the second order nonlinear ordinary differential equation describing the problem considering the structural damping is obtained with the spherically symmetric deformation assumption.Then,the dynamic behaviors,such as quasiperiodic and chaotic motions,are discussed under different loading types.For constant loads,the first integral of the integrable Hamiltonian system without damping is given and it is numerically proved that there exists an asymmetric "?" homoclinic.Moreover,with large pretensions,the structural response will appear local hardening,and the basins of attraction are given with the structural damping.For periodic loads,there exist quasiperiodic oscillation in the approximately integrable Hamiltonian system and limit cycles with the damping.Based on the homoclinic,the criterion for chaos is discussed by the Melnikov method combined with the numerical calculation and the chaos is further analyzed with the Poincare section and the phase plane.3)For spherical shells composed of a class of incompressible visco-hyperelastic materials,the dynamic behaviors of the structure under periodic loads uniformly distributed at the inner and outer surfaces are studied.The coupled integro-differential equations describing the radial motion of structures are obtained by using the energy variational principle,and the complex dynamic behaviors of structures under constant and periodic loads are discussed,respectively.In particular,to consider the influence of thickness,a more general model for studying the dynamic problems of viscoelastic hyperelastic structures is proposed and valided in terms of the finite element method;Due to both the geometrical and physical nonlinearities,there exists an asymmetric homoclinic orbits for the hyperelastic structure.Under constant loads,the system converges to a stable equilibrium point,and the convergence position and speed are closely related to both the initial condition and the viscosity because of the existence of different basins.Under periodic loads,by numerical analyses,parametric studies are carried out to illustrate the effects of viscosity,load amplitude,external frequency and initial condition.