  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).
