Nemat-Nasser / Hori / Achenbach Micromechanics

Overall Properties of Heterogeneous Materials
1. Auflage 2013
ISBN: 978-1-4832-9151-2
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band Volume 37, 708 Seiten, Web PDF

Reihe: North-Holland Series in Applied Mathematics and Mechanics

A comprehensive overview is given in this book towards a fundamental understanding of the micromechanics of the overall response and failure modes of advanced materials, such as ceramics and ceramic and other composites. These advanced materials have become the focus of systematic and extensive research in recent times. The book consists of two parts. The first part reviews solids with microdefects such as cavities, cracks, and inclusions, as well as elastic composites. To render the book self-contained, the second part focuses on the fundamentals of continuum mechanics, particularly linear elasticity which forms the basis for the development of small deformation micromechanics.In Part 1, a fundamental and general framework for quantitative, rigorous analysis of the overall response and failure modes of microstructurally heterogeneous solids is systematically developed. These expressions apply to broad classes of materials with inhomogeneities and defects. While for the most part, the general framework is set within linear elasticity, the results directly translate to heterogeneous solids with rate-dependent or rate-independent inelastic constituents. This application is specifically referred to in various chapters. The general exact correlations obtained between the overall properties and the microstructure are then used together with simple models, to develop techniques for direct quantitative evaluation of the overall response which is generally described in terms of instantaneous overall moduli or compliance. The correlations among the corresponding results for a variety of problems are examined in great detail. The bounds as well as the specific results, include new observations and original developments, as well as an in-depth account of the state of the art.Part 2 focuses on Elasticity. The section on variational methods includes some new elements which should prove useful for application to advanced modeling, as well as solutions of composites and related heterogeneous bodies. A brief modern version of elements in vector and tensor algebra is provided which is particularly tailored to provide a background for the rest of this book.The data contained in this volume as Part 1 includes new results on many basic issues in micromechanics, which will be helpful to graduate students and researchers involved with rigorous physically-based modeling of overall properties of heterogeneous solids.

Autoren/Hrsg.

Nemat-Nasser, S.

Hori, M.

Achenbach, J. D.

Inhaltsverzeichnis

1;Front Cover;1
2;Micromechanics: Overall Properties of Heterogeneous Materials;4
3;Copyright Page;5
4;Table of Contents;8
5;PREFACE;6
6;PART: OVERALL PROPERTIES OF HETEROGENEOUS SOLIDS;22
7;PREFACE TO PART 1;24
7.1;CHAPTER I. AGGREGATE PROPERTIES AND AVERAGING METHODS;30
7.1.1;SECTION 1. AGGREGATE PROPERTIES;32
7.1.1.1;1.1. REPRESENTATIVE VOLUME ELEMENT ;32
7.1.1.2;1.2. SCOPE OF THE BOOK;37
7.1.1.3;1.3. DESCRIPTION OF RVE;40
7.1.1.4;1.4. REFERENCES;44
7.2;SECTION 2. AVERAGING METHODS;48
7.2.1;2.1. AVERAGE STRESS AND STRESS RATE;48
7.2.2;2.2. AVERAGE STRAIN AND STRAIN RATE;50
7.2.3;2.3. AVERAGE RATE OF STRESS-WORK;52
7.2.4;2.4. INTERFACES AND DISCONTINUITIES;56
7.2.5;2.5. POTENTIAL FUNCTION FOR MACRO-ELEMENTS;59
7.2.6;2.7. NONMECHANICAL PROPERTIES;79
7.2.7;2.8. REFERENCES;84
7.3;CHAPTER II. ELASTIC SOLIDS WITH MICROCAVITIES AND MICROCRACKS;86
7.3.1;SECTION 3. LINEARLY ELASTIC SOLIDS;88
7.3.1.1;3.1. HOOKE'S LAW AND MATERIAL SYMMETRY;88
7.3.1.2;3.2. RECIPROCAL THEOREM, SUPERPOSITION, AND GREEN'SFUNCTION;99
7.3.1.3;3.3. REFERENCES;104
7.3.2;SECTION 4. ELASTIC SOLIDS WITH TRACTION-FREE DEFECTS;106
7.3.2.1;4.1. STATEMENT OF PROBLEM AND NOTATION;106
7.3.2.2;4.2. AVERAGE STRAIN FOR PRESCRIBED MACROSTRESS;108
7.3.2.3;4.3. OVERALL COMPLIANCE TENSOR FOR POROUS ELASTIC SOLIDS;110
7.3.2.4;4.4. AVERAGE STRESS FOR PRESCRIBED MACROSTRAIN;111
7.3.2.5;4.5. OVERALL ELASTICITY TENSOR FOR POROUS ELASTIC SOLIDS;113
7.3.2.6;4.6. REFERENCES;115
7.3.3;SECTION 5. ELASTIC SOLIDS WITH MICROCAVITIES;116
7.3.3.1;5.1. EFFECTIVE MODULI OF AN ELASTIC PLATE CONTAINING CIRCULAR HOLES;116
7.3.3.2;5.2. EFFECTIVE BULK MODULUS OF AN ELASTIC BODY CONTAINING SPHERICAL CAVITIES;128
7.3.3.3;5.3. ENERGY CONSIDERATION AND SYMMETRY PROPERTIES OF TENSORH;130
7.3.3.4;5.4. CAVITY STRAIN;131
7.3.3.5;5.5. REFERENCES;132
7.3.4;SECTION 6. ELASTIC SOLIDS WITH MICROCRACKS;134
7.3.4.1;6.1. OVERALL STRAIN DUE TO MICROCRACKS;134
7.3.4.2;6.2. OVERALL COMPLIANCE AND MODULUS TENSORS OF HOMOGENEOUS LINEARLY ELASTIC SOLIDS WITH MICROCRACKS;136
7.3.4.3;6.3. EFFECTIVE MODULI OF AN ELASTIC SOLID CONTAINING ALIGNED SLIT MICROCRACKS;137
7.3.4.4;6.4. EFFECTIVE MODULI OF AN ELASTIC SOLID CONTAINING RANDOMLY DISTRIBUTED SLIT MICROCRACKS;144
7.3.4.5;6.5. EFFECTIVE MODULI OF AN ELASTIC BODY CONTAINING ALIGNED PENNY-SHAPED MICROCRACKS;160
7.3.4.6;6.6. EFFECTIVE MODULI OF AN ELASTIC BODY CONTAINING RANDOMLY DISTRIBUTED PENNY-SHAPED MICROCRACKS;164
7.3.4.7;6.7. EFFECTIVE MODULI OF AN ELASTIC BODY CONTAINING PENNY-SHAPED MICROCRACKS PARALLEL TO AN AXIS;175
7.3.4.8;6.8. INTERACTION EFFECTS;180
7.3.4.9;6.9. BRITTLE FAILURE IN COMPRESSION;187
7.3.4.10;6.10. REFERENCES;206
7.4;CHAPTER III. ELASTIC SOLIDS WITH MICRO-INCLUSIONS;212
7.4.1;SECTION 7. OVERALL ELASTIC MODULUS AND COMPLIANCE TENSORS;214
7.4.1.1;7.1. MACROSTRESS PRESCRIBED;214
7.4.1.2;7.2. MACROSTRAIN PRESCRIBED;217
7.4.1.3;7.3. EIGENSTRAIN AND EIGENSTRESS TENSORS;218
7.4.1.4;7.4. ESTIMATES OF OVERALL MODULUS AND COMPLIANCE TENSORS: DILUTE DISTRIBUTION;230
7.4.1.5;7.5. ESTIMATES OF OVERALL MODULUS AND COMPLIANCE TENSORS: SELF-CONSISTENT METHOD;234
7.4.1.6;7.6. ENERGY CONSIDERATION AND SYMMETRY OF OVERALL ELASTICITY AND COMPLIANCE TENSORS;240
7.4.1.7;7.7. REFERENCES;247
7.4.2;SECTION 8. EXAMPLES OF ELASTIC SOLIDS WITH ELASTIC MICRO-INCLUSIONS;250
7.4.2.1;8.1. RANDOM DISTRIBUTION OF SPHERICAL MICRO-INCLUSIONS;250
7.4.2.2;8.2. EFFECTIVE MODULI OF AN ELASTIC PLATE CONTAINING ALIGNED REINFORCING-FIBERS;255
7.4.2.3;8.3. THREE-DIMENSIONAL ANALYSIS OF PLANE STRAIN AND PLANE STRESS STATES;264
7.4.2.4;8.4. REFERENCES;267
7.4.3;SECTION 9. UPPER AND LOWER BOUNDS FOR OVERALL ELASTIC MODULI;270
7.4.3.1;9.1. HASHIN-SHTRIKMAN VARIATIONAL PRINCIPLE;272
7.4.3.2;9.2. UPPER AND LOWER BOUNDS FOR ENERGY FUNCTIONALS;280
7.4.3.3;9.3. GENERALIZED BOUNDS ON OVERALL ENERGIES;285
7.4.3.4;9.4. DIRECT ESTIMATES OF OVERALL MODULI;292
7.4.3.5;9.5. GENERALIZED VARIATIONAL PRINCIPLES; EXACT BOUNDS;301
7.4.3.6;9.6. UNIVERSAL BOUNDS FOR OVERALL MODULI;325
7.4.3.7;9.7 BOUNDS FOR OVERALL NONMECHANICAL MODULI;334
7.4.3.8;9.8. REFERENCES;343
7.4.4;SECTION 10. SELF-CONSISTENT, DIIYERENTIAL, AND RELATED AVERAGING METHODS;346
7.4.4.1;10.1. SUMMARY OF EXACT RELATIONS BETWEEN AVERAGE QUANTITIES;346
7.4.4.2;10.2. SELF-CONSISTENT METHOD;350
7.4.4.3;10.3. DIFFERENTIAL SCHEME;354
7.4.4.4;10.4. TWO-PHASE MODEL AND DOUBLE-INCLUSION METHOD;361
7.4.4.5;10.5. EQUIVALENCE AMONG ESTIMATES BY DILUTE DISTRIBUTION, SELF-CONSISTENT, DIFFERENTIAL, AND DOUBLEINCLUSION METHODS;378
7.4.4.6;10.6. OTHER AVERAGING SCHEMES;380
7.4.5;SECTION 11. ESHELBY'S TENSOR AND RELATED TOPICS;388
7.4.5.1;11.1. EIGENSTRAIN AND EIGENSTRESS PROBLEMS;388
7.4.5.2;11.2. ESHELBY'S TENSOR;393
7.4.5.3;11.3. SOME BASIC PROPERTIES OF ESHELBY'S TENSOR;398
7.4.5.4;11.4. RELATIONS AMONG AVERAGE QUANTITIES;403
7.4.5.5;11.5. REFERENCES;408
7.5;CHAPTER IV. SOLIDS WITH PERIODIC MICROSTRUCTURE;410
7.5.1;SECTION 12. GENERAL PROPERTIES AND FIELD EQUATIONS;412
7.5.1.1;12.1. PERIODIC MICROSTRUCTURE AND RVE;412
7.5.1.2;12.2. PERIODICITY AND UNIT CELL;413
7.5.1.3;12.3. FOURIER SERIES;415
7.5.1.4;12.4. HOMOGENIZATION;419
7.5.1.5;12.5. TWO-PHASE PERIODIC MICROSTRUCTURE;430
7.5.1.6;12.6. ELASTIC INCLUSIONS AND CAVITIES;435
7.5.1.7;12.7. PERIODICALLY DISTRIBUTED MICROCRACKS;441
7.5.1.8;12.8. APPLICATION TO NONLINEAR COMPOSITES;452
7.5.1.9;12.9. REFERENCES;455
7.5.2;SECTION 13. OVERALL PROPERTIES OF SOLIDS WITH PERIODIC MICROSTRUCTURE;458
7.5.2.1;13.1. GENERAL EQUIVALENT HOMOGENEOUS SOLID;459
7.5.2.2;13.2. HASHIN-SHTRIKMAN VARIATIONAL PRINCIPLE APPLIED TO PERIODIC STRUCTURES;467
7.5.2.3;13.3. APPLICATION OF FOURIER SERIES EXPANSION TO ENERGY FUNCTIONALS;476
7.5.2.4;13.4. EXAMPLE: ONE-DIMENSIONAL PERIODIC MICROSTRUCTURE;482
7.5.2.5;13.5. PIECEWISE CONSTANT APPROXIMATION AND UNIVERSAL BOUNDS;488
7.5.2.6;13.6. EXAMPLES;496
7.5.2.7;13.7. REFERENCES;501
7.5.3;SECTION 14. MIRROR-IMAGE DECOMPOSITION OF PERIODIC FIELDS;502
7.5.3.1;14.1. MIRROR IMAGES OF POSITION VECTORS AND VECTORS;502
7.5.3.2;14.2. MIRROR-IMAGE SYMMETRY/ANTISYMMETRY OF TENSOR FIELDS;507
7.5.3.3;14.3. MIRROR-IMAGE SYMMETRY AND ANTISYMMETRY OF FOURIER SERIES;512
7.5.3.4;14.4. BOUNDARY CONDITIONS FOR A UNIT CELL;517
7.5.3.5;14.5. FOURIER SERIES EXPANSION OF MI SYM/ANT SET OF PERIODIC FIELDS;523
7.5.3.6;14.6. APPLICATION OF HASHIN-SHTRIKMAN VARIATIONAL PRINCIPLE;528
7.6;APPENDIX A: APPLICATION TO INELASTIC HETEROGENEOUS SOLIDS;538
7.6.1;A.1. SOURCES OF INELASTICITY;538
7.6.2;A.2. RATE-INDEPENDENT PHENOMENOLOGICAL PLASTICITY;539
7.6.3;A.3. RATE-DEPENDENT THEORIES;549
7.6.4;A.4. REFERENCES;552
8;PART 2: INTRODUCTION TO BASIC ELEMENTS OF ELASTICITY THEORY;558
8.1;PREFACE TO PART 2;560
8.1.1;CHAPTER V. FOUNDATIONS;562
8.1.1.1;SECTION 15. GEOMETRIC FOUNDATIONS;564
8.1.1.1.1;15.1. VECTOR SPACE;564
8.1.1.1.2;15.2. ELEMENTARY CONCEPTS IN THREE-DIMENSIONAL SPACE;565
8.1.1.1.3;15.3. TENSORS IN THREE-DIMENSIONAL VECTOR SPACE;568
8.1.1.1.4;15.4. DEL OPERATOR AND THE GAUSS THEOREM;573
8.1.1.1.5;15.5. SPECIAL TOPICS IN TENSOR ALGEBRA;576
8.1.1.1.6;15.6. SPECTRAL REPRESENTATION OF FOURTH-ORDER SYMMETRIC TENSORS;583
8.1.1.1.7;15.7. CYLINDRICAL AND SPHERICAL COORDINATES;586
8.1.1.1.8;15.8. REFERENCES;590
8.1.1.2;SECTION 16. KINEMATIC FOUNDATIONS;592
8.1.1.2.1;16.1. DEFORMATION AND STRAIN MEASURES;592
8.1.1.2.2;16.2. INFINITESIMAL STRAIN MEASURE;595
8.1.1.2.3;16.3. REFERENCES;605
8.1.1.3;SECTION 17. DYNAMIC FOUNDATIONS;608
8.1.1.3.1;17.1. EULER'S LAWS;608
8.1.1.3.2;17.2. TRACTION VECTORS AND STRESS TENSOR;610
8.1.1.3.3;17.3. GEOMETRICAL REPRESENTATION OF STRESS TENSOR;615
8.1.1.3.4;17.4. REFERENCES;618
8.1.1.4;SECTION 18. CONSTITUTIVE RELATIONS;620
8.1.1.4.1;18.1. STRAIN ENERGY DENSITY;620
8.1.1.4.2;18.2. LINEAR ELASTICITY;623
8.1.1.4.3;18.3. ELASTICITY AND COMPLIANCE TENSORS;625
8.1.1.4.4;18.4. REFERENCES;627
8.1.2;CHAPTER VI. ELASTOSTATIC PROBLEMS OF LINEAR ELASTICITY;628
8.1.2.1;SECTION 19. BOUNDARY-VALUE PROBLEMS AND EXTREMUM PRINCIPLES;630
8.1.2.1.1;19.1. BOUNDARY-VALUE PROBLEMS;630
8.1.2.1.2;19.2. KINEMATICALLY AND STATICALLY ADMISSIBLE FIELDS;632
8.1.2.1.3;19.3. POTENTIAL ENERGY;634
8.1.2.1.4;19.4. COMPLEMENTARY ENERGY;637
8.1.2.1.5;19.5. GENERAL VARIATIONAL PRINCIPLES;640
8.1.2.1.6;19.6. REFERENCES;645
8.1.2.2;SECTION 20. THREE-DIMENSIONAL PROBLEMS;646
8.1.2.2.1;20.1. HELMHOLTZ' S DECOMPOSITION THEOREM;646
8.1.2.2.2;20.2. WAVE EQUATIONS;647
8.1.2.2.3;20.3. PAPKOVICH-NEUBER REPRESENTATION;650
8.1.2.2.4;20.4. CONCENTRATED FORCE IN INFINITE AND SEMI-INFINITE SOLIDS;653
8.1.2.2.5;20.5. REFERENCES;661
8.1.2.3;SECTION 21. SOLUTION OF SINGULAR PROBLEMS;664
8.1.2.3.1;21.1. AIRY'S STRESS FUNCTION;664
8.1.2.3.2;21.2. GREEN'S FUNCTION AND DISLOCATION;669
8.1.2.3.3;21.3. THE HILBERT PROBLEM;675
8.1.2.3.4;21.4. TWO-DIMENSIONAL CRACK PROBLEMS;680
8.1.2.3.5;21.5. ANISOTROPIC CASE;686
8.1.2.3.6;21.6. REFERENCES;695
9;AUTHOR INDEX;698
10;SUBJECT INDEX;704