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Develops shell equations with drilling rotation from various sets of 3D equations with rotations,
Expounds the role of the Rotation Constraint equation as a means to introduce the drilling rotation,
Describes and evaluates several enhanced, mixed, and mixed/enhanced 4-node elements with drilling rotations,
Presents a fairly complete set of numerical test for 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.
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