Based on previous work for the static problem, in this paper we first derive one form of dynamic finite-strain shell equations for incompressible hyperelastic materials that involve three shell constitutive relations. In order to single out the bending effect as well as to reduce the number of shell constitutive relations, a further refinement is performed, which leads to a refined dynamic finite-strain shell theory with only two shell constitutive relations (deducible from the given three-dimensional (3D) strain energy function) and some new insights are also deduced. By using the weak formulation of the shell equations and the variation of the 3D Lagrange functional, boundary conditions and the two-dimensional (2D) shell virtual work principle are derived. As a benchmark problem, we consider the extension and inflation of an arterial segment. The good agreement between the asymptotic solution based on the shell equations and that from the 3D exact one gives verification of the former. The refined shell theory is also applied to study the plane-strain vibrations of a pressurized artery, and the effects of the axial pre-stretch, pressure and fibre angle on the vibration frequencies are investigated in detail.

中文翻译：

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不可压缩超弹性材料的改进动态有限应变壳理论：方程和二维壳虚功原理

基于先前针对静态问题的工作，在本文中，我们首先推导出一种形式的不可压缩超弹性材料的动态有限应变壳方程，其中涉及三个壳本构关系。为了挑出弯曲效应并减少壳本构关系的数量，进行了进一步的细化，这导致了一个细化的动态有限应变壳理论，只有两个壳本构关系（从给定的三维（3D）应变能函数）和一些新的见解也被推导出来。通过使用壳方程的弱公式和 3D 拉格朗日泛函的变化，导出了边界条件和二维 (2D) 壳虚功原理。作为基准问题，我们考虑动脉段的延伸和膨胀。基于壳方程的渐近解与 3D 精确解之间的良好一致性验证了前者。细化壳理论还应用于研究加压动脉的平面应变振动，并详细研究了轴向预拉伸、压力和纤维角度对振动频率的影响。