Table Of ContentLecture Notes in Applied
and Computational Mechanics
Volume 54
Series Editors
Prof. Dr.-Ing. Friedrich Pfeiffer
Prof. Dr.-Ing. Peter Wriggers
Lecture Notes in Applied and Computational Mechanics
Edited by F. Pfeiffer and P. Wriggers
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Vol. 54: Sanchez-Palencia, E., Vol. 42: Hashiguchi, K.
Millet, O., Béchet, F. Elastoplasticity Theory
Singular Problems in Shell Theory 432 p. 2009 [978-3-642-00272-4
265 p. 2010 [978-3-642-13814-0]
Vol. 41: Browand, F., Ross, J., McCallen, R. (Eds.)
Vol. 53: Litewka, P. Aerodynamics of Heavy Vehicles II: Trucks, Buses,
Finite Element Analysis of Beam-to-Beam Contact and Trains
159 p. 2010 [978-3-642-12939-1] 486 p. 2009 [978-3-540-85069-4]
Vol. 52: Pilipchuk, V. N. Vol. 40: Pfeiffer, F.
Nonlinear Dynamics:Between Linear and Impact Limits Mechanical System Dynamics
364 p. 2010 [978-3-642-12798-4] 578 p. 2008 [978-3-540-79435-6]
Vol. 51: Besdo, D., Heimann, B., Klüppel, M., Vol. 39: Lucchesi, M., Padovani, C., Pasquinelli, G., Zani, N.
Kröger, M., Wriggers, P., Nackenhorst, U. Masonry Constructions: Mechanical
Elastomere Friction Models and Numerical Applications
249 p. 2010 [978-3-642-10656-9] 176 p. 2008 [978-3-540-79110-2]
Vol. 50: Ganghoffer, J.-F., Pastrone, F. (Eds.) Vol. 38: Marynowski, K.
Mechanics of Microstructured Solids 2 Dynamics of the Axially Moving Orthotropic Web
102 p. 2010 [978-3-642-05170-8] 140 p. 2008 [978-3-540-78988-8]
Vol. 49: Hazra, S.B. Vol. 37: Chaudhary, H., Saha, S.K.
Large-Scale PDE-Constrained Optimization Dynamics and Balancing of Multibody Systems
in Applications 200 p. 2008 [978-3-540-78178-3]
224 p. 2010 [978-3-642-01501-4]
Vol. 36: Leine, R.I.; van de Wouw, N.
Vol. 48: Su, Z.; Ye, L. Stability and Convergence of Mechanical Systems
Identification of Damage Using Lamb Waves with Unilateral Constraints
346 p. 2009 [978-1-84882-783-7] 250 p. 2008 [978-3-540-76974-3]
Vol. 47: Studer, C. Vol. 35: Acary, V.; Brogliato, B.
Numerics of Unilateral Contacts and Friction Numerical Methods for Nonsmooth Dynamical Systems:
191 p. 2009 [978-3-642-01099-6] Applications in Mechanics and Electronics
545 p. 2008 [978-3-540-75391-9]
Vol. 46: Ganghoffer, J.-F., Pastrone, F. (Eds.)
Vol. 34: Flores, P.; Ambrósio, J.; Pimenta Claro, J.C.;
Mechanics of Microstructured Solids
Lankarani Hamid M.
136 p. 2009 [978-3-642-00910-5]
Kinematics and Dynamics of Multibody Systems
with Imperfect Joints: Models and Case Studies
Vol. 45: Shevchuk, I.V. 186 p. 2008 [978-3-540-74359-0
Convective Heat and Mass Transfer in Rotating Disk
Systems Vol. 33: Niesony, A.; Macha, E.
300 p. 2009 [978-3-642-00717-0] Spectral Method in Multiaxial Random Fatigue
146 p. 2007 [978-3-540-73822-0]
Vol. 44: Ibrahim R.A., Babitsky, V.I., Okuma, M. (Eds.)
Vibro-Impact Dynamics of Ocean Systems and Related Vol. 32: Bardzokas, D.I.; Filshtinsky, M.L.;
Problems Filshtinsky, L.A. (Eds.)
280 p. 2009 [978-3-642-00628-9] Mathematical Methods in Electro-Magneto-Elasticity
530 p. 2007 [978-3-540-71030-1]
Vol.43: Ibrahim, R.A. Vol. 31: Lehmann, L. (Ed.)
Vibro-Impact Dynamics Wave Propagation in Infinite Domains
312 p. 2009 [978-3-642-00274-8]] 186 p. 2007 [978-3-540-71108-7]
Singular Problems in
Shell Theory
Computing and Asymptotics
Evariste Sanchez-Palencia, Olivier Millet,
Fabien Béchet
123
Prof. Evariste Sanchez-Palencia Dr. Fabien Bechet
Institut Jean Le Rond d’Alembert Metz University
4 place Jussieu LPMM
75252 Paris Cedex 05 Ile du Saulcy
France 57045 Metz Cedex 01
E-mail: [email protected] France
E-mail: [email protected]
Prof. Olivier Millet
La Rochelle University
LEPTIAB
Avenue Michel Crépeau
17000 La Rochelle
France
E-mail: [email protected]
ISBN: 978-3-642-13814-0 e-ISBN: 978-3-642-13815-7
DOI 10.1007/ 978-3-642-13815-7
Lecture Notes in Applied and Computational Mechanics ISSN 1613-7736
e-ISSN 1860-0816
Library of Congress Control Number: 2010928271
© Springer-Verlag Berlin Heidelberg 2010
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Contents
Introduction................................................. 1
1 Geometric Formalism of Shell Theory ................... 13
1.1 Introduction .......................................... 13
1.2 Recall on Surface Theory............................... 13
1.2.1 Mapping - CovariantBasis ....................... 13
1.2.2 First Fundamental Form of the Surface
S - ContravariantBasis .......................... 14
1.2.3 Second Fundamental Form ....................... 15
1.3 Classification of Surfaces ............................... 16
1.4 Differentiation on the Surface S ......................... 18
1.5 Surface Rigidity ....................................... 21
1.5.1 Deformation of a Surface ......................... 21
1.5.2 The Rigidity System and Its Characteristic
Curves......................................... 22
1.5.3 Handling Systems of Equations with Various
Orders: Indices of Equations and Unknowns ........ 25
1.6 The Koiter Shell Model ................................ 26
1.7 The Limit Membrane Model ............................ 29
1.7.1 The Membrane Model ........................... 29
1.7.2 The System of Membrane Tension ................. 30
1.7.3 Back to the Membrane System.................... 31
2 Singularities and Boundary Layers in Thin Elastic Shell
Theory .................................................. 33
2.1 Introduction .......................................... 33
2.2 Geometrically Rigid Surfaces ........................... 34
2.2.1 Inextensional Displacements ...................... 34
2.2.2 Examples of Geometrically Rigid Surface ........... 35
VI Contents
2.3 Limit Behavior of Koiter Model ........................ 37
2.3.1 The Limit Membrane Problem .................... 37
2.3.2 Boundary Layers and Singularities................. 38
2.3.3 Convergence to the Membrane Model in the
Inhibited Case .................................. 38
2.3.4 A More General Result of Convergence............. 40
2.3.5 Convergence to the Pure Bending Model in the
Non-inhibited Case .............................. 43
2.4 Complements on Nagdhi Model and its Limits ............ 45
2.5 Reduction of the Membrane System to one PDE for Each
Component of the Displacement......................... 47
2.5.1 Case of the Normal Displacement u ............... 48
3
2.5.2 Tangential Displacements u and u ............... 49
1 2
2.6 Structure of the Displacement Singularities when the
Loading is Singular along a Curve ....................... 50
2.6.1 Singularity along a Non-characteristic Line ......... 53
2.6.2 Singularity along a Characteristic Line ............. 56
2.6.3 Summary of the Results.......................... 61
2.7 Pseudo-reflections for Hyperbolic Shells .................. 63
2.8 Thickness of the Layers ................................ 63
2.8.1 Case of a Layer along a Non-characteristic Line ..... 64
2.8.2 Case of a Layer along a Characteristic Line ......... 65
2.9 Conclusion ........................................... 67
3 Anisotropic Error Estimates in the Layers ............... 69
3.1 Introduction .......................................... 69
3.2 Estimate for Galerkin Approximation in Singular
Perturbation and Penalty Problems...................... 70
3.2.1 Degradation of the Estimate in a Singular
Perturbation Problem............................ 72
3.2.2 Degradation of the Estimate in a Penalty Problem... 72
3.3 Interpolation Error for Isotropic Meshes in Layers ......... 73
3.3.1 The Basic F. E. Interpolation Error Estimate ....... 73
3.3.2 Case of a Layer: Interpolation Error for Isotropic
Meshes......................................... 74
3.4 Interpolation Error for Anisotropic Meshes in Layers....... 76
3.5 Galerkin Error Estimates in a Layer ..................... 78
3.6 First Remarks on Approximations in Layers .............. 80
3.7 Estimates for Significant Entities in the Layer: Local
Locking in Layers ..................................... 82
3.8 Conclusion ........................................... 85
4 Numerical Simulation with Anisotropic Adaptive
Mesh .................................................... 87
4.1 Introduction .......................................... 87
Contents VII
4.2 Review on the Numerical Locking ....................... 88
4.2.1 Introduction .................................... 88
4.2.2 Locking in the Non-inhibited Case (Classical
Locking Associated with a Limit Constraint) ....... 88
4.2.3 Locking in the Inhibited Case (Singular
Perturbations) .................................. 93
4.3 Shell Element and Associated Discrete Problem ........... 94
4.3.1 The Shell Element D.K.T......................... 95
4.3.2 Discretization of Naghdi Model ................... 96
4.3.3 Adaptive Mesh Strategy: BAMG .................. 98
4.3.4 Coupling BAMG-MODULEF for Shell
Computations................................... 100
4.4 Membrane and Bending Energies Computation with
MODULEF........................................... 101
4.4.1 Implementation Procedure in MODULEF .......... 101
4.4.2 Validation on Simple Examples ................... 102
4.5 Conclusion ........................................... 105
5 Singularities of Parabolic Inhibited Shells................ 107
5.1 Introduction .......................................... 107
5.2 Study of the Singularities and of their Propagation ........ 108
5.2.1 Singularity along a Characteristic Line ............. 109
5.2.2 Singularity along a Non-characteristic Line ......... 111
5.3 Example of a Half-Cylinder ............................. 114
5.3.1 Geometric Description of the Cylinder ............. 114
5.3.2 Constitutive Law................................ 116
5.3.3 Loading and Boundary Conditions................. 116
5.3.4 Singularities of the Displacements ................. 119
5.4 Numerical Simulations with Anisotropic Adaptive Mesh .... 124
5.4.1 Remark for the Interpretation of the Numerical
Results in Terms of Singularities .................. 125
5.4.2 Convergence of the Adaptive Mesh Procedure....... 126
5.4.3 Computing the Displacements..................... 127
5.4.4 Influence of the Relative Thickness ε............... 129
5.4.5 Localization of Membrane and Bending Energies .... 131
5.5 Comparison between Uniform and Adapted Meshes ........ 133
5.6 Numerical Study of Singularities on Non-characteristic
Lines ................................................ 135
5.7 Singularity along a Boundary ........................... 136
5.7.1 Theoretical Considerations ....................... 137
5.7.2 Numerical Simulations ........................... 137
5.8 Singularities due to the Shape of the Domain ............. 142
5.8.1 Conclusion ..................................... 144
VIII Contents
6 Singularities of Hyperbolic Inhibited Shells .............. 147
6.1 Introduction .......................................... 147
6.2 The Limit Problem for a Hyperbolic Inhibited Shell........ 147
6.2.1 Example of a Hyperbolic Paraboloid............... 148
6.2.2 Singularities of the Displacements due to a Loading
Singular on the Line y1 =0....................... 149
6.2.3 Three Cases of Loading .......................... 151
6.2.4 The Singularities of the Resulting Displacements .... 154
6.3 Numerical Computations Using Adaptive Meshes.......... 154
6.3.1 Numerical Results for Loading A .................. 154
6.3.2 Results for the Loading B ........................ 158
6.3.3 Results for the Loading C ........................ 161
6.4 Some Examples Including Pseudo-reflections .............. 163
6.4.1 Reflection of a Characteristic Layer................ 163
6.4.2 Reflection of a Non-characteristic Layer ............ 165
6.4.3 Reflection of a Characteristic Layer when the
Loading “Touches” The Non-characteristic
Boundary ...................................... 168
6.5 Conclusion ........................................... 170
7 Singularities of Elliptic Well-Inhibited Shells............. 171
7.1 Introduction .......................................... 171
7.2 Existence of Logarithmic Point Singularities at the
Corners of the Loading Domain ......................... 171
7.2.1 Model Problem of Second Order .................. 173
7.2.2 The Membrane Problem Δ2u =C f3(θ) ......... 175
3 4
7.2.3 Particular Case when the Logarithmic Point
Singularity Vanishes ............................. 177
7.2.4 Existence Condition of a Logarithmic Singularity.... 177
7.3 Example of an Elliptic Paraboloid ....................... 181
7.3.1 Geometric Properties ............................ 182
7.3.2 Numerical Results ............................... 183
7.3.3 Mesh Adaptation................................ 184
7.3.4 Thickness of the Internal Layer along y1 =0.5 ...... 187
7.3.5 The Logarithmic Singularity at the Corner ......... 189
7.3.6 Membrane and Bending Energies.................. 192
7.4 Conclusion ........................................... 193
8 Generalities on Boundary Conditions for Equations
and Systems: Introduction to Sensitive Problems ........ 195
8.1 Introduction .......................................... 195
8.2 The Cauchy Problem for Equations and Systems .......... 196
8.2.1 Generalities..................................... 196
8.2.2 Role of the Characteristics........................ 197
Contents IX
8.2.3 Normal Form of a Hyperbolic System: Riemann
Invariants ...................................... 199
8.2.4 Elliptic Equations or Systems ..................... 201
8.3 Boundary Value Problems for Elliptic Equations and
Systems .............................................. 204
8.3.1 Regularity of the Solution ........................ 204
8.3.2 The Shapiro–LopatinskiiCondition ................ 206
8.4 The Shapiro–Lopatinskii Condition and the Membrane
Problem ............................................. 207
8.5 Sensitive Problems .................................... 210
8.5.1 Elliptic Shell Clamped by a Part Γ of the
0
Boundary and Free by the Rest Γ ................ 210
1
8.5.2 Qualitative Description of the Solution of Sensitive
Problems....................................... 212
8.5.3 Heuristic Treatment of the Problem................ 214
8.6 Conclusion ........................................... 216
9 Numerical Simulations for Sensitive Shells............... 219
9.1 Introduction .......................................... 219
9.2 First Examples of Numerical Computations for Sensitive
Problems (Ill-Inhibited Shells) .......................... 220
9.3 Asymptotic Process when ε Tends to Zero ................ 222
9.4 Influence of the Free Edge Length ....................... 225
9.5 Energy Repartition in Sensitive Problems................. 228
9.6 Influence of the Loading Domain ........................ 229
9.7 Conclusion ........................................... 232
10 Examples of Non-inhibited Shell Problems
(Non-geometrically Rigid Problems)..................... 235
10.1 Examples of Partially Non-inhibited Shells................ 236
10.1.1 First Case: α=0 and β =0.25.................... 236
10.1.2 Second Case: α=0.25 and β =0.25 ............... 238
10.2 Propagationof Singularities in the Partially Non-inhibited
Regions .............................................. 240
10.2.1 Loading Applied in the Inhibited Area ............. 240
10.2.2 Loading Domain Tangent to the Non-inhibited
Area........................................... 243
10.2.3 Loading Partially Applied in the Non-inhibited
Area........................................... 244
10.3 Conclusion ........................................... 245
References................................................... 247
APPENDICES
A Characteristics of the Membrane System ................ 253
X Contents
B Reduced Membrane and Koiter Equations............... 255
B.1 Membrane Problem.................................... 255
B.1.1 Case of the Normal Displacement u ............... 256
3
B.1.2 ReducedEquationfortheTangentialDisplacements
u and u ...................................... 259
1 2
B.2 Koiter Problem ....................................... 260
Index........................................................ 263
