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Finite Rotation Shells: Basic Equations and Finite Elements for Reissner Kinematics

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This book treats the formulation and finite elements for shells, and its subject is defined by the following topics:

  1. Computational mechanics of non-linear shells,

  2. Shell equations: Reissner kinematics, finite rotations, finite strains,

  3. Shell finite elements: 4-node, enhanced or mixed or mixed/enhanced,

  4. Drilling rotation: drilling Rotation Constraint or Allman shape functions,

  5. Normal strain: recovered or parameterized,

  6. Constitutive equations: incremental, plane stress or 3D.

 A wide range of applications of shell elements implies that they should be versatile, i.e. account for finite rotations and strains, admit the incorporation of various constitutive laws and enable convenient linking with other elements. This is a serious challenge, which requires various aspects of the element's formulation to be very advanced.

The basic information on linear shell elements may be found in some textbooks on FEs, but this book contains several advanced topics related to non-linear shells, such as e.g. the parametrization of finite rotations, the methods of including the drilling rotation, various methods of treating the normal strain, and the enhanced as well as mixed and mixed/enhanced finite elements.

Some of these topics have been a subject of my research for years, and all the described methods have been implemented in my own elements, and tested. Therefore, I believe that my understanding of this complicated subject is correct and mature enough to be canned, and presented to others.

Author(s): K. Wisniewski
Series: Lecture Notes on Numerical Methods in Engineering and Scienc
Edition: 1st Edition.
Publisher: Springer
Year: 2010

Language: English
Pages: 494
Tags: Математика;Вычислительная математика;Метод конечных элементов;

Cover
......Page 1
Finite Rotation Shells......Page 2
Contents......Page 4
Preface......Page 10
Part I
PRELIMINARIES......Page 12
1.1 Subject of the book......Page 13
1.2 Notation......Page 15
2.1 Cartesian bases......Page 17
2.2 Normal bases......Page 21
2.3 Gradients and derivatives......Page 30
Part II
SHELL EQUATIONS......Page 32
3.1 Polar decomposition of deformation gradient......Page 33
3.2 Rotation Constraint equation......Page 35
3.3 Interpretation of rotation Q......Page 37
3.4 Rate form of RC equation......Page 39
3.5 Rotations calculated from the RC equation......Page 40
4:
3D formulations with rotations......Page 41
4.1 Governing equations......Page 42
4.2 4-F formulation for nominal stress......Page 48
4.3 3-F formulation for nominal stress......Page 51
4.4 3-F and 2-F formulations for Biot stress......Page 53
4.5 3-F and 2-F formulations for second Piola–Kirchhoff stress......Page 56
4.6 2-F formulation with unconstrained rotations......Page 59
5.1 Coordinates and position vector......Page 60
5.2 Basic geometric definitions......Page 63
5.3 Example: Geometrical description of cylinder......Page 68
6.1 Kinematics......Page 71
6.2 Rotation Constraint for shells......Page 76
6.3.1 Non-symmetric relaxed right stretch strain......Page 77
6.3.2 Symmetric relaxed right stretch strain......Page 79
6.3.3 Green strain......Page 80
6.3.4 Transverse shear strains satisfying RC. Kirchhoff kinematics......Page 84
6.3.5 Rotation as an intermediate variable symmetrizing strain......Page 86
6.3.6 In-plane deformation with drilling rotation......Page 87
6.3.7 Forward-rotated shell strains variations......Page 89
6.4.1 Virtual work of Biot stress......Page 91
6.4.2 Virtual work of second Piola–Kirchhoff stress......Page 93
6.4.3 Variation of RC term......Page 94
6.4.4 Virtual work of body forces and external forces......Page 95
6.4.5 Virtual work equation for shell......Page 96
6.5 Local shell equations......Page 97
6.6.1 Two normal stretches......Page 102
6.6.2 In-plane twist rotation......Page 103
6.6.3 Warping parameters for cross-section......Page 105
6.6.4 Shift of the reference surface......Page 106
7.1 Constitutive equations for 3D shells......Page 108
7.1.1 Incremental 3D constitutive equations......Page 109
7.1.2 Incremental constitutive equations for shell resultants......Page 110
7.1.3 General form of constitutive equations for shell resultants......Page 112
7.2 Reduced shell constitutive equations......Page 118
7.2.1 Reduced constitutive equations for ZNS condition......Page 119
7.2.2 Reduced constitutive equations for incompressibility condition......Page 126
7.3 Shear correction factor......Page 131
Part III
FINITE ROTATIONS FOR SHELLS......Page 136
8.1 Basic properties of rotations......Page 137
8.1.1 Rotation tensor......Page 138
8.1.2 Rotation of vector about axis......Page 139
8.1.3 Rotation of a triad of vectors......Page 145
8.2 Parametrization of rotations......Page 148
8.2.1 Six parameters......Page 150
8.2.2 Five parameters......Page 151
8.2.3 Four parameters: Euler parameters (quaternions)......Page 155
8.2.4 Three parameters: rotation pseudo-vectors......Page 157
8.2.5 Three parameters: Euler angles......Page 171
8.2.6 Two parameters: constrained rotations of shell director......Page 175
8.3.1 Composition of rotation tensors......Page 181
8.3.2 Composition of Euler parameters (quaternions)......Page 183
8.3.3 Composition of rotation pseudo-vectors......Page 184
8.3.4 Composition of Euler angles......Page 188
9: Algorithmic schemes for finite rotations
......Page 189
9.1.1 Operator T......Page 190
9.1.2 Differential ÂT......Page 199
9.2.1 Variation of rotation tensor for additive composition......Page 200
9.2.2 Variation of rotation tensor for multiplicative composition......Page 201
9.2.3 Relations between variations for various composition rules......Page 203
9.2.4 Second variation of rotation tensor......Page 209
9.3 Algorithmic schemes for finite rotations......Page 213
9.3.1 Scheme 1: formulation in TISO(3)......Page 215
9.3.2 Scheme 2: formulation in TRrefSO(3)......Page 216
9.3.3 Scheme 3: formulation in TRSO(3)......Page 219
9.3.4 Symmetry of tangent operator for structures with rotational dofs......Page 220
9.3.5 Example: twisted ring by 3D beam element......Page 221
9.4.1 Basic definitions......Page 223
9.4.2 Angular velocity and acceleration for parametrizations......Page 225
9.4.3 Examples of updates for rigid body motion......Page 229
Part IV
FOUR-NODE SHELL ELEMENTS......Page 240
10.1 Bilinear isoparametric approximations......Page 241
10.2 Geometry and bases of shell element......Page 243
10.3 Jacobian matrices......Page 251
10.4 Deformation gradient, FTF and QTF products......Page 260
10.5 Numerical integration of shell elements......Page 266
10.6 Newton method and tangent operator......Page 272
11.1 Basic equations......Page 279
11.2 Displacement element Q4......Page 283
11.3 Solution of FE equations for problems with additionalvariables......Page 285
11.4 Enhanced strain elements based on potential energy......Page 288
11.4.1 ID4 element......Page 289
11.4.2 EAS4 element......Page 294
11.4.3 EADG4 element......Page 299
11.5 Mixed Hellinger–Reissner and Hu–Washizu elements......Page 301
11.5.1 Assumed stress HR elements: PS and HR5-S......Page 312
11.5.2 Assumed stress and strain HW14-SS element......Page 318
11.6 Modification of FTF product......Page 321
12: Plane four-node elements with drilling
rotation......Page 323
12.1 Basic relations for drill RC equation......Page 324
12.2 Difficulties in approximation of drill RC......Page 328
12.3 Implementation of drill RC in finite elements......Page 332
12.3.1 Selected methods to include the drill RC......Page 334
12.3.2 Selection of regularization parameter for drill RC......Page 343
12.4 EADG method for formulations with rotations......Page 348
12.5 Mixed HW and HR functionals with rotations......Page 350
12.6.1 EADG4 elements based on potential energy......Page 352
12.6.2 Assumed stress HR5-S elements......Page 353
12.6.3 Assumed stress/enhanced strain HR7-S elements......Page 354
12.6.4 Assumed stress and strain HW14-SS elements......Page 355
12.6.5 Assumed stress and strain/enhanced strain HW18-SS elements......Page 356
12.7.1 Allman-type shape functions......Page 358
12.7.2 EADG2x enhancement of Allman quadrilateral......Page 367
12.7.3 Special techniques for Allman quadrilateral......Page 368
12.7.4 Allman+EADG2x elements......Page 370
12.8.1 Comparison of various elements......Page 371
12.8.2 Selection of the value of regularization parameter......Page 372
13.1 Treatment of transverse shear stiffness of beams......Page 376
13.1.1 Reduced integration of transverse shear energy......Page 377
13.1.2 Residual Bending Flexibility (RBF) correction......Page 381
13.1.3 Scaling down of transverse shear stiffness......Page 385
13.1.4 Numerical tests for beams......Page 387
13.1.5 Curvature correction......Page 390
13.2 Treatment of transverse shear stiffness of shells......Page 394
13.2.2 Assumed Natural Strain method......Page 395
13.2.3 RBF correction for shells......Page 399
13.2.4 Miscellaneous topics......Page 404
14.1 Definition of warpage......Page 407
14.2 Warped element with modifications......Page 411
14.3 Substitute flat element and warpage correction......Page 414
14.4 Membrane locking of curved shell elements......Page 419
14.5 Remarks on approximation of curved surfaces byfour-node elements......Page 426
Part V
NUMERICAL EXAMPLES......Page 430
15.1 Characteristics of tested shell elements......Page 431
15.2.1 Eigenvalues of a single element......Page 433
15.2.3 Constant strain patch tests......Page 435
15.2.4 Distortion test......Page 438
15.2.5 Warped single element......Page 439
15.2.6 Straight cantilever beam......Page 441
15.2.7 Cook’s membrane......Page 443
15.2.8 Curved beam......Page 444
15.2.9 Pinched cylinder with end diaphragms......Page 445
15.2.10 Raasch’s hook......Page 446
15.3.1 Slender cantilever under in-plane shear......Page 447
15.3.3 Torsion of a plate strip......Page 449
15.3.4 L-shaped plate......Page 451
15.3.5 Twisted beam......Page 452
15.3.6 Hinged cylindrical panel......Page 454
15.3.7 Slit open annular plate......Page 455
15.3.8 Pinched hemispherical shell with hole......Page 456
15.3.9 Pinched clamped cylinder......Page 458
15.3.10 Stretched cylinder with free ends......Page 459
15.3.11 Pinched spherical shell......Page 461
15.3.12 Short channel section beam......Page 462
15.3.13 Long channel section beam......Page 463
15.3.14 Hyperboloidal shell......Page 464
15.3.15 Twisted ring......Page 465
References......Page 468
Author index......Page 485
Subject index......Page 489