Everted Deformation,Stability And Vibration Of Two Hyperelastic Axisymmetric Structures

Hyperelastic materials,such as rubber,rubber-like and several biological soft tissues,are widely used in aerospace,engineering design,industrial building,medical and health and biomechanics due to their high elasticity and resilience properties.However,cylindrical and spherical structures composed of hyperelastic materials are commonly used in our actual life and social production,for example,rubber rings and rods for architectural design and aerospace aircraft,rubber tubes for transiting liquid and air,the balloon catheters for clinical intervention,blood vessel for biological soft tissues,and so on.Because of the unique properties and the widely applied backgrounds,the problems of deformation,stability and motion for materials and structures based on the hyperelastic constitutive relations are always the focuses of experts and scholars at home and abroad.In this dissertation,based on the theory of finite deformation,stability and dynamics in nonlinear elasticity,the problems of the finite deformation of everted cylindrical shells and spherical shells,the stability of cylindrical shells of the prestressed state by eversion and the nonlinear radial oscillation of cylindrical and spherical membranes,are examined for hyperelastic materials.Some improtant conclusions are obtained,as follows.1)The finite deformations of several everted cylindrical shells composed of hyperelastic materials are studied.The mathematical model is reduced to a class of boundary value problems of an nonlinear differential equation.For incompressible materials,by using the semi-inverse method,implicit analytical solutions are derived by the incompressible constraint condition.For compressible materials,firstly,the exact solutions are derived for a class of special harmonic materials by the classical analysis method;secondly,because the classical methods are not suitable,so the numerical solutions are derived for a general compressible material by the improved shooting method.Compared with the analytic results,it is proved that the improved shooting method is effective.It can be seen that the initial thickness and material parameters are the main factors for the axially stretch of hyperelastic cylindrical shells,specially eversion produces stresses.2)In order to reveal the mechanics mechanism of eversion deformation for axisymmetric structures fully,several classes of hyperelastic spherical shells are studied.Due to the different structures,the established mathematical model is slightly different.By using the methods for cylindrical shells,the analytical and numerical solutions of the corresponding problems for spherical shells are obtained for complicated strain energy fuctions.Moreover,the improved shooting method for the eversion of axisymmetric structures composed of compressible hyperelastic materials is applicable.By contrast,it is found that for isotropic materials,the thinner the cylindrical or spherical shell is,the larger the inner radius is,the stress distribution of everted cylindrical and spherical shells are basically identical.However,the influences of the material parameters on the axial stretch of cylindrical shells are significant,the influence of anisotropic parameter on the inner radius of spherical shell is obvious.3)The stability of an incompressible hyperelastic everted cylindrical shell under an axial load is studied.In terms of the theory of small deformations superposed on large elastic deformations,the nonlinear partial differential equation and the boundary conditions with respect to the incremental displacement are formulated.Then the approximate analytical solution and the control equations of instability by Bessel functions are obtained.By using numerical simulations,the influences of the initial thickness and the ratio of outer radius and length on the critical values of the governing deformed parameter are discussed,and the finite deformation of the cylindrical shell at the state of instability is also shown.It is shown that whether the shell is everted or not,the thinner or wider the cylindrical shell is,the more instable the structure is,and the shell suffers instability more easily in high mode numbers.Especially,the everted cylindrical shell is more easily instable than the undeformed shell.4)The nonlinear dynamic behaviors of cylindrical and spherical membranes composed of incompressible hyperelastic materials subjected to three kinds of loads on their inner surfaces are studied.A second-order nonlinear ordinary differential equation that approximately describes the radial vibration of membranes with the corresponding initial and boundary conditions are obtained by asymptotic expansion.Through qualitative analyses and numerical simulations,the influence factors of vibration for the cylindrical and spherical membranes are discussed.It is shown that under the constant load,the motion of the membrane presents a nonlinear periodic vibration.The phase diagram is a smooth closed curve,and the ?-type homoclinic orbit may appear when the load reaches a certain value in certain cases.Under the periodic step load or the periodic load,the membrane presents a nonlinear nonperiodic vibration,the phase diagram is no longer a smooth closed curve.