Finite Rotation Shells

This book covers theoretical and computational aspects of non-linear shells. Several advanced topics of shell equations and finite elements, not included in standard textbooks on finite elements, are addressed.

Key features include: several sets of 3D equations with the rotations introduced by either the polar decomposition equation or the rotation constraint equation; shell equations based on Reissner kinematics for finite rotations and strains, formulated in terms of different strains and stresses; a comprehensive account of finite rotations, including their properties and parameterization, as well as the algorithmic issues pertaining to rotation parameters; a comprehensive description and evaluation of several enhanced, mixed, and mixed/enhanced 4-node elements; a selection of useful remedies for such problems as: poor accuracy of in-plane shear strain, transverse shear locking, over-stiffening of warped elements, locking in sinusoidal bending, and deterioration of accuracy for extremely thin elements; a large set of numerical benchmarks for finite rotation shells; an extensive bibliography and comprehensive index.

Shells have been a subject of the author's research for years, and all the methods described in the book have been implemented and tested in the field.

The book can be useful for graduate students, professional engineers, and researchers specializing in shells, Finite Elements and applied numerical methods.

Résumé
The objective of this book is to provide a comprehensive introduction to ?nite rotation shells and to non-linear shell ?nite elements. It is divided into 5 parts: I. Preliminaries (20 pages), II. Shell equations (104 pages), III. Finite rotations for shells (103 pages), IV. Four-node shell elements (189 pages), and V. Numerical examples (41 pages). Additional numerical examples are presented in Parts III and IV. The bibliography includes 270 entries. The book is intended for both teaching and self-study, and emphasizes fundamental aspects and techniques of the subject. Some familiarity with non-linear mechanics and the ?nite element method is assumed. Shell elements are a subject of active research which results in many publications every year and several conferences and sessions are held r- ularly, among them, two large international conferences: \Computation of Shell and Spatial Structures" and \Shell Structures. Theory and - plications" (SSTA). The literature is voluminous, not easy to follow and evaluate, and the subject is dicult to comprehend. I hope that this will be facilitated by the book. I would like to express my gratitude to several persons who helped me in my professional life, in this way contributing to the book. I thank Prof. R.L. Taylor from the University of California at Berkeley, Prof. B. Schre er from the University of Padua, and Prof. J.T. Santos from the Instituto Superior Tecnico at Lisbon, for hosting and supporting me when I was a post-doctoral researcher.

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Proviosional Table of contents (October 2009)

I PRELIMINARIES; 1 Introduction; 1.1 Subject of this book; 1.2 Notation; 2 Operations on tensors and their representations; 2.1 Cartesian bases; 2.2 Normal bases; 2.3 Gradients and derivatives; II SHELL EQUATIONS; 3 Rotations for 3D Cauchy continuum; 3.1 Polar decomposition of deformation gradient; 3.2 Rotation Constraint equation; 3.3 Interpretation of rotation Q; 3.4 Rate form of RC equation ; 3.5 Rotations calculated from the RC equation; 4 3D formulations with rotations; 4.1 Governing equations; 4.2 4-F formulation for nominal stress; 4.3 3-F formulation for nominal stress; 4.4 3-F and 2-F formulations for Biot stress; 4.5 3-F and 2-F formulations for 2nd Piola-Kirchhoff stress; 4.6 2-F formulation with unconstrained rotations; 5 Basic geometric definitions for shells; 5.1 Coordinates and position vector; 5.2 Basic geometric definitions; 5.3 Example: Geometrical description of cylinder; 6 Shells with Reissner kinematics and drilling rotation; 6.1 Kinematics; 6.2 Rotation Constraint for shells; 6.3 Shell strains; 6.4 Virtual work equation for shell; 6.5 Local shell equations; 6.6 Enhanced shell kinematics; 7 Shell-type constitutive equations; 7.1 Constitutive equations for 3D shells; 7.2 Reduced shell constitutive equations; 7.3 Shear correction factor; III FINITE ROTATIONS FOR SHELLS; 8 Parametrization of finite rotations; 8.1 Basic properties of rotations; 8.2 Parametrization of rotations; 8.3 Composition of rotations; 9 Algorithmic schemes for finite rotations; 9.1 Increments of rotation vectors in two tangent planes; 9.2 Variation of rotation tensor; 9.3 Algorithmic schemes for finite rotations; 9.4 Angular velocity and acceleration; IV FOUR-NODE SHELL ELEMENTS; 10 Basic relations for 4-node shell elements; 10.1 Bilinear isoparametric approximations; 10.2 Geometry and bases of shell element ; 10.3 Jacobian matrices; 10.4 Deformation gradient, FTF and QTF products; 10.5 Numerical integration of shell elements; 10.6 Newton method and tangent operator; 11 Plane 4-node elements (without drilling rotation); 11.1 Basic equations; 11.2 Displacement element Q4; 11.3 Solution of FE equations for problems with additional variables; 11.4 Enhanced strain elements based on potential energy; 11.5 Mixed Hellinger-Reissner and Hu-Washizu elements; 11.6 Modification of FTF product; 12 Plane 4-node elements with drilling rotation; 12.1 Basic relations for drill RC equation; 12.2 Difficulties in approximation of drill RC; 12.3 Implementation of drill RC in finite elements; 12.4 EADG method for formulations with rotations; 12.5 Mixed HW and HR functionals with rotations; 12.6 2D+drill elements for bi-linear shape functions; 12.7 2D+drill elements for Allman shape functions; 12.8 Numerical tests; 13 Modification of transverse shear stiffness of shell element; 13.1 Treatment of transverse shear stiffness of beams ; 13.2 Treatment of transverse shear stiffness of shell; 14 Warped 4-node shell element; 14.1 Definition of warpage ; 14.2 Warped element with modifications; 14.3 Substitute flat element and warpage correction; 14.4 Membrane locking of curved shell elements ; 14.5 Remarks on approximation of curved surfaces by 4-node elements ; V NUMERICAL EXAMPLES; 15 Numerical tests; 15.1 Characteristics of tested shell elements; 15.2 Elementary and linear tests; 15.3 Nonlinear tests; References; Author index; Subject Index