Objective:

The aim of this study is three-fold:

• To present a general higher-order shell theory to analyze the large deformation of thin or thick shell structures made of general compressible or incompressible hyperelastic materials.

• To utilize the orthonormal (Cartan's) moving frame in the formulation of shell theory in contrast to the classical covariant coordinate frame which makes the numerical model of the shell theory computationally very efficient. 

• To present the nonlinear weak-form finite element model for the given shell theory. The displacement field of the line normal to the shell reference surface is approximated by the general Taylor series/Legendre polynomials.

Motivation:

The key challenge in any shell theory is to obtain various kinematic relations in terms of the approximated displacement field in the assumed coordinate system efficiently. Since the governing equation is given in the tangent space, which is Euclidean, the best way to describe the kinematics is via orthonormal or Cartan’s moving frame. Kinematic quantities such as the gradient of the displacement field or the determinant of deformation gradient have a large number of terms in the natural covariant frame than the orthonormal frame due to non-identity covariant metric tensor, which ultimately results in an inefficient numerical model for analysis. Efficient representation of such kinematic terms is of crucial importance, especially for nonlinear hyperelastic material models. Hence, I have used the method of exterior calculus in conjugation with Cartan’s moving frame for the formulation of shell theory as well as its nonlinear finite element model. Also, the methodology developed herein is very much algorithmic, and hence it can also be applied for any arbitrary interpolated surfaces with equal ease.

Overview:

The governing equation of the shell has been derived in the general surface coordinates for the general hyperelastic material model. The weak-form finite element model for the presented shell theory is also developed. The higher-order nature of the approximation of the displacement field makes the theory suitable for analyzing thin as well as thick shell structures. Moreover, the formulation presented in this study can be specialized for various nonlinear hyperelastic constitutive models, for examples, neo-Hookean material, Mooney-Rivlin material, Generalized power-law neo-Hookean material, and so on. This shell theory of soft material can be used in many bio-mechanical applications (such as analysis of arteries or veins, heart valve, aneurysm,  etc.). Although, the study has been done for isotropic homogeneous material, it can also be extended for the layered shell structures, functionally graded shell, anisotropic material, or other smart materials, which have many real life applications.

Hyperboloidal shell under point loads (compressible neo-Hookeam material model):

Equilibrium path for Circular arc-shaped shell stripe (Saint Venant–Kirchhoff nonlinear material model): 

1. A. Arbind, J. N. Reddy, and A. R. Srinivasa. A general higher-order shell theory for isotropic hyperelastic materials using orthonormal moving frame . International Journal for Numerical Methods in Engineering,  accepted, (2020). [link]

2. A. Arbind, J. N. Reddy, and A. R. Srinivasa. A general higher-order shell theory for isotropic hyperelastic material using Cartan's moving frame : incompressible material." to be submitted, (2020).