Stress Resultant Geometrically Exact Shell Theory for Finite Rotations and Its Finite Element Implementation

Abstract

This article reviews the significant progress on shell problem theoretical foundation and numerical implementation attained over a period of the last several years. First, a careful consideration of the three-dimensional finite rotations is given including the choice of optimal parameters, their admissible variations and the much revealing relationship between different parameters. A non-conventional derivation of the stress resultant shell theory is presented, which makes use of the virtual work principle and local Cartesian frames. The presented derivation introduces no simplifying hypotheses regarding the shell balance equations, hence the resulting shell theory is referred to as being geometrically exact. The strain measures energy-conjugate to the chosen stress resultants are identified and the nature of the stress resultants with respect to the three-dimensional stress tensor is explained along with the resulting constitutive restrictions. Comments are made regarding a rather useful extension of the shell theory which accounts for the rotational degree of freedom about the director, the so-called drilling rotation. A linear shell theory is obtained as a very useful by-product of the present work, by linearizing the present nonlinear shell theory about the reference configuration. It is shown that this non-conventional approach not only clarifies an often confusing derivation of the linear shell theory, but also leads to a novel linear shell theory capable of delivering significantly improved results and essentially exact solutions to the standard linear benchmark problems. Another important aspect of the nonlinear shell problem solution, the finite element approximation of the shell theory, is also discussed. The model problem of assumed shear strain interpolation is used to illustrate that numerical implementation which preserves the salient features of the theoretical formulation often brings an improved final result. For the selected rotation parameterization and the finite element interpolation, the issues of the consistent linearization of the nonlinear shell problem are addressed. In a number of numerical simulations, the latter is proved to play a crucial role not only in ensuring the robust performance of the Newton solution procedure, but also in linear and nonlinear buckling problems of shells. Several directions for future research are pointed out and some contemporary works of special interest are listed. There are 127 references at the end of the article.