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.

CAVITY FORMATION AND ITS VIBRATION FOR A CLASS OF GENERALIZED INCOMPRESSIBLE HYPER-ELASTIC MATERIALS

The problem of radial symmetric motion for a solid sphere composed of a class of generalized incompressible neo-Hookean materials, subjected to a suddenly applied surface tensile dead load, is examined.The analytic solutions for this problem and the motion equation of cavity that describes cavity formation and growth with time are obtained. The e?ect of radial perturbation of the materials on cavity formation and its motion is discussed. The plane of the perturbation parameters of the materials is divided into four regions. The existential conditions and qualitative properties of solutions of the motion equation of the cavity are studied in di?erent parameters’ regions in detail. It is proved that the cavity motion with time is a nonlinear periodic vibration. The vibration center is then determined.

QUALITATIVE STUDY OF CAVITATED BIFURCATION FOR A CLASS OF INCOMPRESSIBLE GENERALIZED NEO-HOOKEAN SPHERES

The problem of spherical cavitated bifurcation was examined for a class of incompressible generalized neo-Hookean materials, in which the materials may be viewed as the homogeneous incompressible isotropic neo-Hookean material with radial perturbations. The condition of void nucleation for this problem was obtained. In contrast to the situation for a homogeneous isotropic neo-Hookean sphere, it is shown that not only there exists a secondary turning bifurcation point on the cavitated bifurcation solution which bifurcates locally to the left from trivial solution, and also the critical load is smaller than that for the material with no perturbations, as the parameters belong to some regions. It is proved that the cavitated bifurcation equation is equivalent to a class of normal forms with single-sided constraints near the critical point by using singularity theory. The stability of solutions and the actual stable equilibrium state were discussed respectively by using the minimal potential energy principle.

具有缺陷的不可压缩neo-Hookean球壳的动力学行为的定性分析

研究了一类具有缺陷的不可压缩超弹性材料球壳的径向对称运动问题,该类材料可以看作是带有径向摄动的均匀各向同性不可压缩的neo_Hookean材料.得到了描述球壳内表面运动的二阶非线性常微分方程,并给出了方程的首次积分.通过对微分方程的解的动力学行为的分析,讨论了材料的缺陷参数和球壳变形前的内外半径的比值对解的定性性质的影响,并给出了相应的数值算例.特别地,对于一些给定的参数,证明了存在一个正的临界值,当内压与外压之差小于临界值时,球壳内表面随时间的演化是非线性周期振动;当内压与外压之差大于临界值时,球壳的内表面半径随时间的演化将无限增大,即球壳最终将被破坏.

Deformation analysis of an incompressible composite cylindrical tube subjected to end axial loads and internal constraint

In this paper,the problem of axially symmetric deformation is examined for a composite cylindrical tube under equal axial loads acting on its two ends,where the tube is composed of two different incompressible neo-Hookean materials.Significantly,the implicit analytical solutions describing the deformation of the tube are proposed.Numerical simulations are given to further illustrate the qualitative properties of the solutions and some meaningful conclusions are obtained.In the tension case,with the increasing axial loads or with the decreasing ratio of shear moduli of the outer and the inner materials,it is proved that the tube will shrink more along the radial direction and will extend more along the axial direction.Under either tension or compression,the deformation along the axial direction is obvious near the two ends of the tube,while in the rest,the change is relatively small.Similarly,for a large domain of the middle part,the axial elongation is almost constant;however,the variation is very fast near the two ends.In addition,the absolute value of the axial displacement increases gradually from the central cross-section of the tube and achieves the maximum at the two endpoints.

Axisymmetric 3:1 internal resonance of thin-walled hyperelastic cylindrical shells under both axial and radial excitations

Nonlinear vibration with axisymmetric 3:1 internal resonance is investigated for an incompressible neo-Hookean hyperelastic cylindrical shell under both axial and radial harmonic excitations.A full nonlinear strain-displacement relation is derived from the large deflection theory of thin-walled shells.A set of nonlinear differential equations describing the large deflection vibration are formulated by the Lagrange equation and the assumption of small strains.Steady-state responses of the system are predicted via the harmonic balance method with the arc length continuation,and their stabilities are determined via the modified sorting method.The effects of excitations on the steady-state responses are analyzed.The results reveal a crucial role played by the phase difference in the structural response,and the phase difference can effectively control the amplitude of vibration.