Homogenization of Reticulated Structures

1 Homogenization in Perforated Media.- 1. The General Method of Homogenization.- 1.1 The One-Dimensional Periodic Case.- 1.2 A Model Example: the Thermal Problem.- 1.3 Perforated Domains.- 2. The Homogeneous Neumann Problem.- 2.1 Perforated Domains and Variational Formulation.- 2.2 Multiple-Scale Method.- 2.3 Extension Operators.- 2.4 Convergence Theorems.- 2.5 Domains with Nonisolated Holes.- 2.6 Error Estimates.- 3. Other Boundary Value Problems.- 3.1 The Dirichlet Problem.- 3.2 Fourier Conditions.- 3.3 Eigenvalue Problem.- 2 Lattice-Type Structures.- 1. The Two-Dimensional Case.- 1.1 Statement of the Problem and the Main Theorem.- 1.2 Proof of the Main Theorem: Technique of Dilatations.- 1.3 Superposition Method.- 1.4 Error Estimates.- 2. The Three-Dimensional Case.- 2.1 Honeycomb Structures.- 2.2 Reinforced Structures.- 3. Complex Structures and Loss of Ellipticity.- 3.1 A General Method for Diagonal Bars.- 3.2 Linearized Elasticity and Loss of Ellipticity.- 3.3 Examples.- 4. Other Boundary Conditions.- 4.1 The Dirichlet Problem.- 4.2 Fourier Conditions.- 4.3 Eigenvalue Problem.- 3 Thermal Problems for Gridworks.- 1. Statement of the Problem.- 2. Case e = k?.- 2.1 Change of Scale.- 2.2 Limit for ? ? 0.- 2.3 Limit for ? ? 0.- 3. Case ? ? e.- 3.1 The Multiple-Scale Method.- 3.2 The Variational Method.- 3.3 Limit for e ? 0.- 3.4 Limit for ? ? 0.- 4. Case e ? ?.- 4.1 Limit for e ? 0.- 4.2 Limit for ? ? 0.- 4.3 Limit for ? ? 0.- 4.4 Comparison of the Different Limits.- 4 Elasticity Problems for Gridworks.- 1. Statement of the Problem.- 2. Limit Plate Behavior.- 2.1 Main Result.- 2.2 A Priori Estimates and Limits of Displacements.- 2.3 Limits of Stresses and Moments and Limit Equations.- 3. Homogenization Result.- 4. Final Explicit Result and Loss of Ellipticity.- 5. Case ? ? e.- 5.1 Limit for ? ? 0.- 5.2 Limit for e ? 0.- 5.3 Limit for ? ? 0.- 5.4 Loss of Ellipticity.- 6. Plates Without Loss of Ellipticity.- 7. Time-Dependent Plates Models: An Experimental Result.- 5 Thermal Problems for Thin Tall Structures.- 1. Statement of the Problem.- 2. Case e = ??.- 2.1 Limit for ? ? 0.- 2.2 Limit for ? ? 0.- 3. Case ? ? e.- 3.1 Limit for ? ? 0.- 3.2 Limit for e ? 0, Then for ? ? 0.- 3.3 Limit for ? ? 0.- 3.4 Limit for e ? 0.- 4. Case e ? ?.- 5. Comparison of Limit Systems and Solutions.- 6. Numerical Results for a Two-Dimensional Case.- 6.1 Limit for ? ? 0.- 6.2 Limit for e ? 0.- 6.3 Numerical Computation of the Homogenized Solution.- 6.4 Limit for ? ? 0.- 6.5 Numerical Computation of the Solution V?.- 7. Generalization for a Three-Dimensional Structure.- 7.1 Case e = ??.- 7.2 Case ? ? e.- 7.3 Case e ? ?.- 6 Elasticity Problems for Thin Tall Structures.- 1. Statement of the Problem.- 1.1 Geometric Assumptions.- 1.2 Variational Formulation.- 2. Limit for e ? 0.- 2.1 Change of Scale.- 2.2 Assumptions on the Data.- 2.3 Limit for e ? 0: Beam Behaviour.- 3. Limit for e ? 0: Homogenization.- 4. Limit for ? ? 0.- 5. Applications to Other Structures.- 5.1 Towers.- 5.2 Tall Structures with Oblique Bars.- Final Comments.- References.