Finite Rotation Shells
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From the reviews:
"This outstanding, very important and useful book presets a modern continuum mechanics and mathematical framework to study the physical behaviours of shells, and to formulate and evaluate finite element procedures. . The monograph is written in an accessible and self-contained manner. It will be of interest to mathematicians and engineers working in solid mechanics. Moreover, it would be good as a textbook for graduate courses in mechanics of elastic shells (specializing in shells, finite elements and applied numerical methods)." (Ján Lovísek, Mathematical Reviews, Issue 2012 g)
"The objective is to provide a comprehensive introduction to finite rotation shells and to nonlinear shell finite elements. The book is intended for both teaching and self-study, and emphasizes fundamental aspects and techniques of the subject. This book also contains several topics related to nonlinear shells such as the parametrization of finite rotations, the methods of inclusion of the drilling rotation, various methods of treating the normal strain, and the mixed finite elements." (V. Leontiev, Zentralblatt MATH, Vol. 1201, 2011)
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Content
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
