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 dOmega ± 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