Description
Book SynopsisThis book gives new insight on plate models in the linear elasticity framework tacking into account heterogeneities and thickness effects. It is targeted to graduate students how want to discover plate models but deals also with latest developments on higher order models. Plates models are both an ancient matter and a still active field of research. First attempts date back to the beginning of the 19th century with Sophie Germain. Very efficient models have been suggested for homogeneous and isotropic plates by Love (1888) for thin plates and Reissner (1945) for thick plates. However, the extension of such models to more general situations --such as laminated plates with highly anisotropic layers-- and periodic plates --such as honeycomb sandwich panels-- raised a number of difficulties. An extremely wide literature is accessible on these questions, from very simplistic approaches, which are very limited, to extremely elaborated mathematical theories, which might refrain the beginner. Starting from continuum mechanics concepts, this book introduces plate models of progressive complexity and tackles rigorously the influence of the thickness of the plate and of the heterogeneity. It provides also latest research results. The major part of the book deals with a new theory which is the extension to general situations of the well established Reissner-Mindlin theory. These results are completely new and give a new insight to some aspects of plate theories which were controversial till recently.
Table of ContentsIntroduction xi
Chapter 1. Linear Elasticity 1
1.1. Notations 1
1.2. Stress 3
1.3. Linearized strains 6
1.4. Small perturbations 8
1.5. Linear elasticity 8
1.6. Boundary value problem in linear elasticity 10
1.7. Variational formulations. 11
1.7.1. Compatible strains and stresses 11
1.7.2. Principle of minimum of potential energy 13
1.7.3. Principle of minimum of complementary energy 14
1.7.4. Two-energy principle 15
1.8. Anisotropy 15
1.8.1. Voigt notations 15
1.8.2. Material symmetries 17
1.8.3. Orthotropy 20
1.8.4. Transverse isotropy 22
1.8.5. Isotropy 23
Part 1. Thin Laminated Plates 27
Chapter 2. A Static Approach for Deriving the Kirchhoff–Love Model for Thin Homogeneous Plates 29
2.1. The 3D problem 29
2.2. Thin plate subjected to in-plane loading 32
2.2.1. The plane-stress 2D elasticity problem 33
2.2.2. Application of the two-energy principle 34
2.2.3. In-plane surfacic forces on ∂Ω ± 336
2.2.4. Dirichlet conditions on the lateral boundary of the plate 38
2.3. Thin plate subjected to out-of-plane loading 40
2.3.1. The Kirchhoff–Love plate model 41
2.3.2. Application of the two-energy principle 47
Chapter 3. The Kirchhoff–Love Model for Thin Laminated Plates 53
3.1. The 3D problem 53
3.2. Deriving the Kirchhoff–Love plate model 55
3.2.1. The generalized plate stresses 55
3.2.2. Static variational formulation of the Kirchhoff–Love plate model 56
3.2.3. Direct formulation of the Kirchhoff–Love plate model 58
3.3. Application of the two-energy principle 59
Part 2. Thick Laminated Plates 65
Chapter 4. Thick Homogeneous Plate Subjected to Out-of-Plane Loading 67
4.1. The 3D problem 67
4.2. The Reissner–Mindlin plate model. 69
4.2.1. The 3D stress distribution in the Kirchhoff–Love plate model 69
4.2.2. Formulation of the Reissner–Mindlin plate model 71
4.2.3. Characterization of the Reissner–Mindlin stress solution 72
4.2.4. The Reissner–Mindlin kinematics 73
4.2.5. Derivation of the direct formulation of the Reissner–Mindlin plate model 74
4.2.6. The relations between generalized plate displacements and 3D displacements 76
Chapter 5. Thick Symmetric Laminated Plate Subjected to Out-of-Plane Loading 81
5.1. Notations 81
5.2. The 3D problem 82
5.3. The generalized Reissner plate model 85
5.3.1. The 3D stress distribution in the Kirchhoff–Love plate model 85
5.3.2. Formulation of the generalized Reissner plate model 90
5.3.3. The subspaces of generalized stresses 91
5.3.4. The generalized Reissner equilibrium equations 95
5.3.5. Characterization of the generalized Reissner stress solution 97
5.3.6. The generalized Reissner kinematics 98
5.3.7. Derivation of the direct formulation of the generalized Reissner plate model 100
5.3.8. The relationships between generalized plate displacements and 3D displacements 102
5.4. Derivation of the Bending-Gradient plate model 106
5.5. The case of isotropic homogeneous plates 109
5.6. Bending-Gradient or Reissner–Mindlin plate model? 111
5.6.1. When does the Bending-Gradient model degenerate into the Reissner–Mindlin’s model? 112
5.6.2. The shear compliance projection of the Bending-Gradient model onto the Reissner–Mindlin model 113
5.6.3. The shear stiffness projection of the Bending-Gradient model onto the Reissner–Mindlin model 115
5.6.4. The cylindrical bending projection of the Bending-Gradient model onto the Reissner–Mindlin model 116
Chapter 6. The Bending-Gradient Theory 117
6.1. The 3D problem 117
6.2. The Bending-Gradient problem 119
6.2.1. Generalized stresses 119
6.2.2. Equilibrium equations 121
6.2.3. Generalized displacements 122
6.2.4. Constitutive equations 122
6.2.5. Summary of the Bending-Gradient plate model 123
6.2.6. Field localization 123
6.3. Variational formulations 125
6.3.1. Minimum of the potential energy 126
6.3.2. Minimum of the complementary energy 127
6.4. Boundary conditions 128
6.4.1. Free boundary condition 129
6.4.2. Simple support boundary condition 130
6.4.3. Clamped boundary condition 131
6.5. Voigt notations 131
6.5.1. In-plane variables and constitutive equations 131
6.5.2. Generalized shear variables and constitutive equations 132
6.5.3. Field localization 135
6.6. Symmetries 136
6.6.1. Transformation formulas 136
6.6.2. Orthotropy 139
6.6.3. π/2 invariance 140
6.6.4. Square symmetry 140
6.6.5. Isotropy 140
6.6.6. The remarkable case of functionally graded materials 142
Chapter 7. Application to Laminates 145
7.1. Laminated plate configuration 145
7.2. Localization fields 146
7.2.1. In-plane stress unit distributions (bending stress) 147
7.2.2. Transverse shear unit distributions (generalized shear stress) 148
7.3. Distance between the Reissner–Mindlin and the Bending-Gradient model 149
7.4. Cylindrical bending 150
7.4.1. Closed-form solution for the Bending-Gradient model 152
7.4.2. Comparison of field distributions 155
7.4.3. Empirical error estimates and convergence rate 160
7.4.4. Influence of the bending direction 161
7.5. Conclusion 163
Part 3 Periodic Plates 167
Chapter 8. Thin Periodic Plates 169
8.1. The 3D problem 169
8.2. The homogenized plate problem 173
8.3. Determination of the homogenized plate elastic stiffness tensors 174
8.4. A first justification: the asymptotic effective elastic properties of periodic plates 181
8.5. Effect of symmetries 184
8.5.1. Symmetric periodic plate 185
8.5.2. Material symmetry of the homogenized plate 186
8.5.3. Important special cases 187
8.5.4. Rectangular parallelepipedic unit cell 189
8.6. Second justification: the asymptotic expansion method 194
Chapter 9. Thick Periodic Plates 205
9.1. The 3D problem 206
9.2. The asymptotic solution 208
9.3. The Bending-Gradient homogenization scheme 209
9.3.1. Motivation and descrition of the approach 210
9.3.2. Introduction of corrective terms to the asymptotic solution 210
9.3.3. Identification of the localization tensors 212
9.3.4. Identification of the Bending-Gradient compliance tensor 214
Chapter 10. Application to Cellular Sandwich Panels 219
10.1. Introduction 219
10.2. Questions raised by sandwich panel shear force stiffness 220
10.2.1. The case of homogeneous cores 221
10.2.2. The case of cellular cores 223
10.3. The membrane and bending behavior of sandwich panels 225
10.3.1. The case of homogeneous cores 225
10.3.2. The case of cellular cores 226
10.4. The transverse shear behavior of sandwich panels 229
10.4.1. The case of homogeneous cores 229
10.4.2. A direct homogenization scheme for cellular sandwich panel shear force stiffness 230
10.4.3. Discussion 232
10.5. Application to a sandwich panel including Miura-ori 235
10.5.1. Folded cores 236
10.5.2. Description of the sandwich panel including the folded core 237
10.5.3. Symmetries of Miura-ori 238
10.5.4. Implementation 239
10.5.5. Results 241
10.5.6. Discussion on shear force stiffness 250
10.5.7. Consequence of skins distortion 255
10.6. Conclusion 257
Chapter 11. Application to Space Frames 259
11.1. Introduction 259
11.2. Homogenization of a periodic space frame as a thick plate 261
11.2.1. Homogenization scheme 261
11.3. Homogenization of a square lattice as a Bending-Gradient plate 268
11.3.1. The unit-cell 268
11.3.2. Kirchhoff–Love auxiliary problem 269
11.3.3. Bending-Gradient and Reissner–Mindlin auxiliary problems 270
11.3.4. Difference between Reissner–Mindlin and Bending-Gradient constitutive equation 273
11.4. Cylindrical bending of a square beam lattice 274
11.4.1. Lattice at 0° 274
11.4.2. Lattice at 45° 276
11.5. Discussion 282
11.6. Conclusion 283
Bibliography 285
Index 293
