Multi-scale Modelling for Structures and Composites

Multi-scale Modelling for Structures and Composites

English

Description

Rod structures are widely used in modern engineering. These are bars, beams, frames and trusses of structures, gridwork, network, framework and other constructions. Numerous applications of rod structures in civil engineering, aircraft and spacecraft confirm the importance of the topic. On the other hand the majority of books on structural mechanics use some simplifying hypotheses; these hypotheses do not allow to consider some important effects, for instance the boundary layer effects near the points of junction of rods. So the question concerning the limits of applicability of structural mechanics hypotheses and the possibilities of their refinement arise. In this connection the asymptotic analysis of equations of mathematical physics, the equations of elasticity in rod structures (without these hypotheses and simplifying assumptions being imposed) is undertaken in the present book. Moreover, a lot of modern structures are made of composite materials and therefore the material of the rods is not homogeneous. This inhomogeneity of the material can generate some unexpected effects. These effects are analysed in the present book. The methods of multi-scale modelling are presented in the book by the homogenization, multi-level asymptotic analysis and the domain decomposition. These methods give an access to a new class of hybrid models combining macroscopic description with "microscopic zooms".

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Published 01 January 2005
Reads 13
EAN13 1402029829
License: All rights reserved
Language English

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Contents
Preface Notations Chapter 1. Notions and MethodIntroduction: Basic s 1.1. What is an inhomogeneous rod? 1.2.What are effective coefficients? 1.3. A scheme for calculating effective coefficients 1.4. Microscopic structure of a field 1.5. What is the homogenization method? 1.6.What is anite rod structure? 1.7. What is a lattice structure? 1.8.Advantages and disadvantages of the asymptotic approach 1.9.Appendices 1.A1.Appendix 1: What is the Poincar´eFriedrichsKorn inequalities? Chapter 2. Heterogeneous Rod 2.1. Homogenization(N.Bakhvalov’s ansatz and the boundary layer technique) 2ansatz.1.1. Bakhvalov’s 2.1.2. An example of formal asymptotic solution 2.1.3. Theboundaryconditions corrector 2to the boundar.1.4. Introduction ylayer technique s 2.1.5.Homogenization inIR 2layer correctors to homogenization.1.6. Boundary s inIR 2.2.Steadystate conductivityofa rod 2.2.1. Statement of the problem 2.2.2.Inner expansion 2layer corrector.2.3. Boundary 2.2.4. The justication ofthe asymptotic expansion 2.3. Steadystate elasticityequation in a rod 2.3.1. Formulation of the problem 2.3.2.Inner expansion 2layer corrector.3.3. Boundary 2.3.4. Theboundarylayer corrector when the left end of the bar is free 2.3.5. The boundarylayer corrector for the two bar contact problem 2.3.6. Homogenized problem ofzero order 2.3.7. Thejustification of the asymptotic expansion 2.4. Non steadystate conductivityof a rod 2.4.1. Statement of the problem 2.4.2. Inner expansion
ix xii 1 1 3 5 9 9 10 13 16 17
17 21
22 22 23 26 27 29
32 36 36 38 43 48 56 57 59 65
69
73 79 82 98 98 98
VI
2.4.3 Boundary layer corrector 2.4.4. Justication 2.5. Non steadystate elasticityof a rod 2.5.1. Statement of the problem 2.5.2.Inner expansion 2.5.3. Boundarylayer corrector 2.5.4. Justication 2.6. Contrasting coefficients(Multicomponent homogenization) 2.7. EFMODUL: a codefor cell problems 2.8. Bibliographical Remark CHeterogeneous Plathapter 3. e 3.1. Conductivityof a plate 3.1.1.Statement ofthe problem 3ex.1.2. Inner pansion 3.1.3. Boundarylayer corrector 3.1.4. Algorithm for calculatingthe eective conductivity ofa plate 3.1.5. Justication ofthe asymptotic expansion 3.2. Elasticity ofa plate 3.2.1.Statement oftheproblem 3.2.2.Inner expansion 3layer corrector.2.3. Boundary 3.2.4. ProofofTheorem3.2.1. 3.2.5. Algorithm for calculating the eective stiness ofaplate 3.3. Equivalenthomogeneous plate problem 3.4. Time dependent elasticity problem for a plate 3.5. Bibliographical Remark Chapter 4. Finite RodStructures 4.1. Definitions. Lconvergence 4.1.1. Finiterod structure 4method.1.2. Lconvergence for anite rod structure 4.2.Shape optimization ofanite rod structure 4.2.1. Stored energy as the cost 4of the set o.2.2. Simplification f Initial connite rod structures. guration 4.2.3. An iterative algorithm for the optimal design problem 4.2.4. Some results of numerical experiment 4.3.Casymptotic expansiononductivity: an 4.3.1.Construction ofasymptotic expansion 4leadin.3.2. The gterm of the asymptotic expansion 4.4. Elasticity: an asymptotic expansion 4.4.1.Construction ofasymptotic expansion 4.4.2. The leading term ofasymptotic expansion 4.5. Flows in tube structures
100 100 103 103 104 106 106 110 120 126 129 130 130 131 134
136 138 146 146 146 149 151
154 156 158 160 161 161 162 165 713 174
716
177 180 813 183 192 194 194 212 215
4.5.1. Definitions.One bundle structure 4.5.2. Tube structure with m bundles oftubes 4.6. BibliographicalRemark 4.7. Appendices 4.A1.Aestimates for traces in theppendix 1: prenodal domain 4.A2.Appendix 2: the Poincaand the Friedrichs inequalitiesfora finiterod structure 4.A3.Athe Korn inequalitppendix 3: yfor the niterod structures ChapLattice Structureter 5. s 5.1. Definition of lattice structure 5.2. Lconvergence homogenization of lattices 5.2.1. Lconvergence for the simplest lattice 5.2.2.Some auxiliary inequalities 5.2.3. FLconvergence. Relation to the Lconvergence 5of Theorem 5.2..2.4. Proof 1 5.3. Nonstationaryproblems 5.3.1. Rectangular lattice 5.3.2. Lattices: generalcase 5.3.3. Proof of Theorem 5.3.2 5.4. L and FLConvergence in elasticity 5.5.Conductivity ofa net 5.6. Elasticity ofa net 5.7. Conductivityan expansionof a lattice: 5.8. High order homogenization of elastic lattices 5.9.Random coecients on a lattice 5.9.1. TheSimpThe main resullest Lattice. t 5.9.2. Proof of theorem 5.9.1 5.10. BibliographicalRemark 5.11.Appendices 5.A1.Appendtix 1: he Poincaand the Friedrichs inequalitiesfor lattices 5.A2.Athe Korn inequalitppendix 2: yfor lattices ChapThe MultiScale Domain Decomter 6. position 6versio.1. Differential n 6.1.1. General description of the differential version 6.1.2. Model example 6.1.3. Poisson equation in a rodstructure 6.2. Variationalversion 6.2.1. General description of the variational version 6.2.2. Model example 6.2.3. Elasticity equations
VII
215 227 230 230
230
233
242 247 247 249 249 521 254 255 258 258 259 262 269 270 273 278 291 307 307 309 314 318
318
321 337 341 341 343 349 354 354 357 358
VIII
6.3. Decomposition ofaow in a tube structure 6.4. The partial homogenization 6.5. BibliographicalRemark
Bibliography Subject index
364 373 382
385 397