QUALITATIVE ANALYSIS OF DYNAMICAL BEHAVIOR FOR AN IMPERFECT INCOMPRESSIBLE NEO-HOOKEAN SPHERICAL SHELL

The radial symmetric motion problem was examined for a spherical shell composed of a class of imperfect incompressible hyper-elastic materials, in which the materials may be viewed as the homogeneous incompressible isotropic neo-Hookean material with radial perturbations. A second-order nonlinear ordinary differential equation that describes the radial motion of the inner surface of the shell was obtained. And the first integral of the equation was then carded out. Via analyzing the dynamical properties of the solution of the differential equation, the effects of the prescribed imperfection parameter of the material and the ratio of the inner and the outer radii of the underformed shell on the motion of the inner surface of the shell were discussed, and the corresponding numerical examples were carded out simultaneously. In particular, for some given parameters, it was proved that, there exists a positive critical value, and the motion of the inner surface with respect to time will present a nonlinear periodic oscillation as the difference between the inner and the outer presses does not exceed the critical value. However, as the difference exceeds the critical value, the motion of the inner surface with respect to time will increase infinitely. That is to say, the shell will be destroyed ultimately.

Some qualitative properties of incompressible hyperelastic spherical membranes under dynamic loads

Some nonlinear dynamic properties of axisymmetric deformation are ex- amined for a spherical membrane composed of a transversely isotropic incompressible Rivlin-Saunders material. The membrane is subjected to periodic step loads at its inner and outer surfaces. A second-order nonlinear ordinary differential equation approximately describing radially symmetric motion of the membrane is obtained by setting the thick- ness of the spherical structure close to one. The qualitative properties of the solutions are discussed in detail. In particular, the conditions that control the nonlinear periodic oscillation of the spherical membrane are proposed. In certain cases, it is proved that the oscillating form of the spherical membrane would present a homoclinic orbit of type ＂∞＂, and the amplitude growth of the periodic oscillation is discontinuous. Numerical results are provided.

DYNAMICAL FORMATION OF CAVITY IN A COMPOSED HYPER-ELASTIC SPHERE

The dynamical formation of cavity in a hyper_elastic sphere composed of two materials with the incompressible strain energy function, subjected to a suddenly applied uniform radial tensile boundary dead_load, was studied following the theory of finite deformation dynamics. Besides a trivial solution corresponding to the homogeneous static state, a cavity forms at the center of the sphere when the tensile load is larger than its critical value. An exact differential relation between the cavity radius and the tensile land was obtained. It is proved that the evolution of cavity radius with time displays nonlinear periodic oscillations. The phase diagram for oscillation, the maximum amplitude, the approximate period and the critical load were all discussed.

Dynamical cavitation and oscillation of an anisotropic incompressible hyper-elastic sphere

Dynamical cavitation and oscillation of an anisotropic two-family fiber-reinforced incompressible hyper-elastic sphere subjected to a suddenly applied constant boundary dead load are examined within the framework of finite elasto-dynamics.An exact differential equation between the radius of the cavity and the applied load is obtained.The curves for the variation of the maximum radius of the cavity with the load and the phase diagrams are obtained by vibration theories and numerical computation.It is shown that there exists a critical value for the applied load.When the applied load is larger than the critical value,a spherical cavity will suddenly form at the center of the sphere.It is proved that the evolution of the cavity radius with time follows that of nonlinear periodic oscillation,and oscillation of the anisotropic sphere is not the same as that of the isotropic sphere.

DYNAMIC CHARACTERISTICS IN INCOMPRESSIBLE HYPERELASTIC CYLINDRICAL MEMBRANES

In this paper, the dynamic characteristics are examined for a cylindrical membrane composed of a transversely isotropic incompressible hyperelastic material under an applied uniform radial constant pressure at its inner surface. A second-order nonlinear ordinary differential equation that approximately describes the radial oscillation of the inner surface of the membrane with respect to time is obtained. Some interesting conclusions are proposed for different materials, such as the neo-Hookean material, the Mooney-Rivlin material and the Rivlin-Saunders material. Firstly, the bifurcation conditions depending on the material parameters and the pressure loads are determined. Secondly, the conditions of periodic motion are presented in detail for membranes composed of different materials. Meanwhile, numerical simulations are also provided.