Dynamical response of spherical thin shell composed of incompressible hyperelastic material of Rivlin type

DOI：10.7511/dllgxb201805001

 作者 单位 赵振涛,袁学刚,张洪武,赵巍,张文正

基于非线性弹性动力学理论，研究了由不可压缩超弹性材料组成的球形薄壳在突加常值荷载作用下的动力响应．材料的本构关系采用了一类多项式形式的Rivlin类模型，并建立相应问题的数学模型；求得了描述球形薄壳径向对称运动的二阶非线性常微分方程．通过对微分方程的定性分析，讨论了材料参数及应变能函数的高阶项对方程平衡点个数的影响．对于给定结构参数和材料参数，证明了存在临界荷载，方程的相图会出现非对称的“∝”型或“∞”型同宿轨道，薄壳结构的周期和振幅出现跳跃以及结构被破坏等现象；定性指出了当应变能函数中两个不变量的阶数超过特定值时，薄壳结构不再出现新的动力响应．

Based on the theory of nonlinear elastodynamics, the dynamical response is investigated for a spherical thin shell composed of an incompressible hyperelastic material under a suddenly applied constant load. The constitutive relation, i. e., a class of the Rivlin models with polynomial forms, is used to establish the mathematical model of the corresponding problem. A second-order nonlinear ordinary differential equation describing the radially symmetric motion of the spherical thin shell is obtained. Significantly, 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 structure 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 when the degrees of the two invariants in the strain energy function exceed certain values, the new dynamic responses of thin shell will no longer appear.