The aim of this study is three-fold: (i) to present a general higher-order shell theory to analyze large deformations of thin or thick shell structures made of general compressible hyperelastic materials; (ii) to utilize the orthonormal or Cartans moving frame in the formulation of shell theory in contrast to the classical tensorial covariant coordinate system; and (iii) to present the nonlinear weak-form Galerkin finite element model for the given shell theory. The displacement field of a point on the line normal to the shell reference surface is approximated by the Taylor series or Legendre polynomials. The kinematics of motion in the assumed coordinate system is derived using the tools of exterior calculus. The use of an orthonormal moving frame makes it possible to represent kinematic quantities, e.g., determinant of the deformation gradient, in a far more efficient manner than the classical tensorial representation of the same with covariant bases. The manipulation of the various tensor used in the kinematics and dynamics of the structures can be carried out with ease and a more computationally efficient manner. The governing equation of the shell has been derived in the general surface coordinates. The methodology developed herein is very much algorithmic, and hence it can also be applied for any arbitrary interpolated surfaces with equal ease. The higher-order nature of the approximation of the displacement field makes the theory suitable for analyzing thick and thin shell structures. The compressible hyperelastic material model used as the constitutive relation of the material. The formulation presented herein can be specialized for various nonlinear hyperelastic constitutive models suitable for use, for example, in bio-mechanics and other soft-material problems (e.g., neo-Hookean material, Mooney-Rivlin material, Generalized power-law neo-Hookean material, and so on).

中文翻译：

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使用正交移动坐标系的可压缩各向同性超弹性材料的一般高阶壳理论

本研究的目的有三个：(i) 提出一般的高阶壳理论来分析由一般可压缩超弹性材料制成的薄或厚壳结构的大变形；(ii) 与经典的张量协变坐标系相比，在壳理论的表述中利用正交或 Cartans 移动坐标系；(iii) 给出给定壳理论的非线性弱形式伽辽金有限元模型。与壳参考表面垂直的线上点的位移场由泰勒级数或勒让德多项式近似。假定坐标系中的运动学是使用外部微积分工具推导出来的。使用正交运动坐标系可以表示运动量，例如变形梯度的行列式，以比具有协变基的经典张量表示更有效的方式。在结构的运动学和动力学中使用的各种张量的操作可以轻松且计算效率更高的方式进行。壳的控制方程已经在一般表面坐标中导出。这里开发的方法是非常算法化的，因此它也可以同样轻松地应用于任何任意插值表面。位移场近似的高阶性质使得该理论适用于分析厚壳和薄壳结构。用作材料本构关系的可压缩超弹性材料模型。