动载荷下不可压缩超弹性球形薄膜的若干定性性质

对于由横观各向同性不可压缩的Rivlin-Saunders材料组成的球形薄膜，研究了薄膜的内、外表面在周期阶梯载荷作用下的轴对称变形的非线性动力学特性．通过令球形结构的厚度趋近于1，得到了近似描述薄膜径向对称运动的二阶非线性常微分方程．详细讨论了解的定性性质．特别地，给出了球形薄膜随时间的运动产生非线性周期振动的可控性条件，证明了在某些情形下周期振动的振幅会出现“∞”型同宿轨道以及周期振动的振幅会出现不连续增长现象，并给出了相应的数值模拟．

Some Qualitative Properties of Incompressible Hyperelastic Spherical Membranes Under Dynamic Loads

The nonlinear dynamic properties of axisymm etric deformation were examined for a sphericalm embrane composed of a transversely isotropic incompressible Rivlin-Saundersmaterial, where the membrane was subjected to periodic step loads at its inner and outer surfaces. A second order nonlinear ordinary differential equation that approxmiately describes the radially symmetric motion of the membrane was obtained by setting the thickness of the spherical structure close to 1 and the qualitative properties of the solutions were discussed in detail. In particular, the conditions that control the nonlinear periodic oscillation of the spherical membrane were proposed. Under certain cases, it was proved that the oscillating form of the spherical membrane would present a homoclinic orbit of type "∞" and that the growth of the amplitude of the periodic oscillation was discontinuous, and numerical results were also provided.