Thin membranes are notoriously sensitive to instabilities under mechanical loading, and need sophisticated analysis methods. Although analytical results are available for several special cases and assumptions, numerical approaches are normally needed for general descriptions of non-linear response and stability. The paper uses the case of a thin spherical hyper-elastic membrane subjected to internal gas over-pressure to investigate how stability conclusions are affected by chosen material models and kinematic discretizations. For spherical symmetry, group representation theory leads to linearized modes on the uniformly stretched sphere, with eigenvalues obtained from the mechanics of a thin membrane. A complete three-dimensional geometric description allows non-axisymmetric shear modes of the sphere, and such instabilities are shown to exist. When the symmetry of the continuous sphere is broken by discretized models, group representation theory gives predictions on the effects on the critical states. Numerical simulations of the pressurized sphere show and verify stability conclusions for sets of meshing strategies and hyper-elastic models.

中文翻译：

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高度对称系统的计算稳定性研究：加压球形膜

众所周知，薄膜对机械载荷下的不稳定性非常敏感，需要复杂的分析方法。尽管对于一些特殊情况和假设可以获得分析结果，但通常需要数值方法来对非线性响应和稳定性进行一般描述。该论文使用了承受内部气体超压的薄球形超弹性膜的案例，来研究所选材料模型和运动学离散化如何影响稳定性结论。对于球对称性，群表示理论导致均匀拉伸球体上的线性化模式，其特征值从薄膜力学中获得。完整的三维几何描述允许球体的非轴对称剪切模式，并且显示存在这种不稳定性。当连续球体的对称性被离散模型破坏时，群表示理论对临界状态的影响给出了预测。加压球体的数值模拟显示并验证了一组网格划分策略和超弹性模型的稳定性结论。