Homogenization of Heterogeneous Thin and Thick Plates

Karam Sab is a Professor and head of Laboratoire Navier, which is a joint research unit between Ecole des Ponts ParisTech, IFSTTAR and CNRS. The research concerns the mechanics and physics of materials and structures, in geotechnics, and their applications to in particular civil engineering and petroleum geophysics. His main research topics concern the homogenization of heterogeneous materials and plates: Random materials, Representative Volume Element (RVE), Simulation, Higher order models (Cosserat), Multi-layered plates, Periodic plates, Random plates, Shear effects. The applications concern solid foams, aggregate composites, bituminous materials, cementitious materials, masonry walls, reinforced structures, functionally graded materials, sandwich plates, and space frames. Arthur Lebee is Researcher at Laboratoire Navier, which is a joint research unit between Ecole des Ponts ParisTech, IFSTTAR and CNRS. The research concerns the mechanics and physics of materials and structures, in geotechnics, and their applications to in particular civil engineering and petroleum geophysics.

Introduction 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