This work provides the mathematical foundation for and the mechanical analysis of a fully nonlinear, geometrically exact shell theory. This formulation of the shell equations has the advantage that, although the exact nonlinear geometric structure of the idealized two-dimensional body is maintained, complex differential geometry concepts such as covariant derivatives, Riemannian connections or Christoffel symbols are avoided. Furthermore, a weak or variational statement of the local balance equations is developed that is ideally suited for numerical solution techniques.;The analysis begins with a derivation of the integrated or resultant momentum balance equations governing the deformation of a three-dimensional body. These resultant equations comprise an infinite set of balance equations and correspond to the resultant balance of linear momentum plus the successive moments of linear momentum about the resultant shell mid-surface. In addition, an infinite set of integrated balance of angular momentum equations result.;Next, the single director shell kinematic assumption is introduced. The single director shell assumption is the defining feature of the subsequent analysis. With this assumption, the complete kinematic description of the shell and the complete (finite) set of shell momentum balance equations are determined.;For elastic material response, the shell resultant stored-energy function is derived and then defined in terms of the three-dimensional stored-energy function. The shell stress and stress couple resultants are given by hyper-elastic constitutive equations. Theoretically, this three-dimensional definition can be used to obtain shell resultant constitutive equations for any three-dimensional material model. The particular case of a compressible Mooney-Rivlin type material model is considered. From this model, the linear isotropic, neo-Hookean and incompressible Mooney-Rivlin models are obtained as particular cases.;A variational formulation of the shell equations is then developed. The weak form of the momentum balance equations is particularly important for the numerical implementation of the shell equations. Furthermore, the consistent tangent operator for the momentum balance equations is derived. The tangent operator is a vital component of the nonlinear iterative solution procedure and is required for such topics as bifurcation analysis.;The last two chapters investigate linear shell theory and extended kinematical models for finite elasticity and nonlinear shells. A development of the linear theory allows for quantitative comparison between this work and classical shell theories. The extended kinematical models introduce point-wise rotations as independent variables. For shells, a drill rotation formulation of the geometrically exact shell model results.
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