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Micromechanics: overall properties of heterogeneous materials

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Bibliographic Details
Main Authors:	Nemat-Nasser, Siavouche (Author), Hori, Muneo (Author)
Format:	Book
Language:	English
Published:	Amsterdam [u.a.] Elsevier 1999
Edition:	2., rev. ed.
Subjects:	Heterogenität Inhomogener Festkörper Elastizität Mikromechanik
Online Access:	Inhaltsverzeichnis
Physical Description:	XXIV, 786 S. Ill., graph. Darst.
ISBN:	0444500847

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adam_text	MICROMECHANICS: OVERALL PROPERTIES OF HETEROGENEOUS MATERIALS SIA NEMAT-NASSER DEPARTMENT OF APPLIED MECHANICS AND ENGINEERING SCIENCES UNIVERSITY OF CALIFORNIA, SAN DIEGO LA JOLLA, CA 6 92093-0416, USA MUNEO HORI EARTHQUAKE RESEARCH INSTITUTE UNIVERSITY OF TOKYO TOKYO, JAPAN SECOND REVISED EDITION 1999 ELSEVIER AMSTERDAM - LAUSANNE - NEW YORK - OXFORD - SHANNON - SINGAPORE - TOKYO TABLE OF CONTENTS PREFACE V TABLE OF CONTENTS IX PARTI OVERALL PROPERTIES OF HETEROGENEOUS MATERIALS PRECIS: PART 1 CHAPTER I AGGREGATE PROPERTIES AND AVERAGING METHODS 9 SECTION 1. AGGREGATE PROPERTIES 11 1.1. REPRESENTATIVE VOLUME ELEMENT (RVE) 11 1.2. SCOPE OF THE BOOK 16 1.3. DESCRIPTION OF RVE 19 1.4. REFERENCES 23 SECTION 2. AVERAGING METHODS 27 2.1. AVERAGE STRESS AND STRESS RATE 27 2.2. AVERAGE STRAIN AND STRAIN RATE 29 2.3. AVERAGE RATE OF STRESS-WORK 31 2.3.1. UNIFORM BOUNDARY TRACTIONS 33 2.3.2. LINEAR BOUNDARY VELOCITIES 34 2.3.3. OTHER USEFUL IDENTITIES 34 2.3.4. VIRTUAL WORK PRINCIPLE 35 2.4. INTERFACES AND DISCONTINUITIES 35 2.5. POTENTIAL FUNCTION FOR MACRO-ELEMENTS 38 2.5.1. STRESS POTENTIAL 40 2.5.2. STRAIN POTENTIAL 41 2.5.3. RELATION BETWEEN MACROPOTENTIALS 42 2.5.4. ON DEFINITION OF RVE 44 2.5.5. LINEAR VERSUS NONLINEAR RESPONSE 45 2.5.6. GENERAL RELATIONS BETWEEN MACROPOTENTIALS 45 2.5.7. BOUNDS ON MACROPOTENTIAL FUNCTIONS 50 TABLE OF CONTENTS 2.6. STATISTICAL HOMOGENEITY, AVERAGE QUANTITIES, AND OVERALL PROPERTIES 53 2.6.1. LOCAL AVERAGE FIELDS 55 2.6.2. LIMITING PROCESS AND LIMIT FIELDS 58 2.7. NONMECHANICAL PROPERTIES 59 2.7.1. AVERAGING THEOREMS 59 2.7.2. MACROPOTENTIALS 60 2.7.3. BASIC INEQUALITIES 61 2.8. COUPLED MECHANICAL AND NONMECHANICAL PROPERTIES 63 2.8.1. FIELD EQUATIONS 63 2.8.2. AVERAGING THEOREMS 65 2.8.3. STRESS/ELECTRIC-FIELD POTENTIAL 66 2.8.4. STRAIN/ELECTRIC-DISPLACEMENT POTENTIAL 67 2.8.5. BASIC INEQUALITIES 68 2.9. REFERENCES 71 CHAPTER II ELASTIC SOLIDS WITH MICROCAVITIES AND MICROCRACKS 73 SECTION 3. LINEARLY ELASTIC SOLIDS 75 3.1. HOOKE S LAW AND MATERIAL SYMMETRY 75 3.1.1. ELASTIC MODULI 75 3.1.2. ELASTIC COMPLIANCES 77 3.1.3. ELASTIC SYMMETRY 78 3.1.4. PLANE STRAIN/PLANE STRESS 82 3.2. RECIPROCAL THEOREM, SUPERPOSITION, AND GREEN S FUNCTION 86 3.2.1. RECIPROCAL THEOREM 87 3.2.2. SUPERPOSITION 87 3.2.3. GREEN S FUNCTION 88 3.3. REFERENCES 91 SECTION 4. ELASTIC SOLIDS WITH TRACTION-FREE DEFECTS 93 4.1. STATEMENT OF PROBLEM AND NOTATION 93 4.2. AVERAGE STRAIN FOR PRESCRIBED MACROSTRESS 95 4.3. OVERALL COMPLIANCE TENSOR FOR POROUS ELASTIC SOLIDS 97 4.4. AVERAGE STRESS FOR PRESCRIBED MACROSTRAIN 98 4.5. OVERALL ELASTICITY TENSOR FOR POROUS ELASTIC SOLIDS 100 4.6. REFERENCES 102 TABLE OF CONTENTS XI SECTION 5. ELASTIC SOLIDS WITH MICROCAVITIES 103 5.1. EFFECTIVE MODULI OF AN ELASTIC PLATE CONTAINING CIRCULAR HOLES 103 5.1.1. ESTIMATES OF THREE-DIMENSIONAL MODULI FROM TWO-DIMENSIONAL RESULTS 104 5.1.2. EFFECTIVE MODULI: DILUTE DISTRIBUTION OF CAVITIES 106 5.1.3. EFFECTIVE MODULI: SELF-CONSISTENT ESTIMATES 111 5.1.4. EFFECTIVE MODULI IN X 3 -DIRECTION 113 5.2. EFFECTIVE BULK MODULUS OF AN ELASTIC BODY CONTAINING SPHERICAL CAVITIES 115 5.3. ENERGY CONSIDERATION AND SYMMETRY PROPERTIES OF TENSOR H 117 5.4. CAVITY STRAIN 118 5.5. REFERENCES 119 SECTION 6. ELASTIC SOLIDS WITH MICROCRACKS 121 6.1. OVERALL STRAIN DUE TO MICROCRACKS 121 6.2. OVERALL COMPLIANCE AND MODULUS TENSORS OF HOMO- GENEOUS LINEARLY ELASTIC SOLIDS WITH MICROCRACKS 123 6.3. EFFECTIVE MODULI OF AN ELASTIC SOLID CONTAINING ALIGNED SLIT MICROCRACKS 124 6.3.1. CRACK OPENING DISPLACEMENTS 125 6.3.2. EFFECTIVE MODULI: DILUTE DISTRIBUTION OF ALIGNED MICROCRACKS 125 6.3.3. EFFECTIVE MODULI: DILUTE DISTRIBUTION OF ALIGNED FRICTIONAL MICROCRACKS 129 6.4. EFFECTIVE MODULI OF AN ELASTIC SOLID CONTAINING RANDOMLY DISTRIBUTED SLIT MICROCRACKS 131 6.4.1. EFFECTIVE MODULI: RANDOM DILUTE DISTRIBUTION OF OPEN MICROCRACKS 131 6.4.2. EFFECTIVE MODULI: SELF-CONSISTENT ESTIMATE 135 6.4.3. EFFECTIVE MODULI IN ANTIPLANE SHEAR: RANDOM DILUTE DISTRIBUTION OF FRICTIONLESS MICROCRACKS 137 6.4.4. PLANE STRESS, PLANE STRAIN, AND THREE-DIMENSIONAL OVERALL MODULI 140 6.4.5. EFFECT OF FRICTION AND LOAD-INDUCED ANISOTROPY 141 6.5. EFFECTIVE MODULI OF AN ELASTIC BODY CONTAINING ALIGNED PENNY-SHAPED MICROCRACKS 147 6.5.1. CRACK-OPENING-DISPLACEMENTS 147 6.5.2. EFFECTIVE MODULI: DILUTE DISTRIBUTION OF ALIGNED MICROCRACKS 147 6.6. EFFECTIVE MODULI OF AN ELASTIC BODY CONTAINING RANDOMLY DISTRIBUTED PENNY-SHAPED MICROCRACKS 151 6.6.1. DILUTE OPEN MICROCRACKS WITH PRESCRIBED DISTRIBUTION 151 6.6.2. EFFECTIVE MODULI: RANDOM DILUTE DISTRIBUTION OF MICROCRACKS 154 6.6.3. EFFECTIVE MODULI: SELF-CONSISTENT ESTIMATES 158 XLL TABLE OF CONTENTS 6.7. EFFECTIVE MODULI OF AN ELASTIC BODY CONTAINING PENNY-SHAPED MICROCRACKS PARALLEL TO AN AXIS 162 6.8. INTERACTION EFFECTS 167 6.8.1. CRACK-OPENING-DISPLACEMENTS AND ASSOCIATED STRAINS 168 6.8.2. DILUTE DISTRIBUTION OF PARALLEL CRACK ARRAYS 170 6.8.3. RANDOMLY ORIENTED OPEN SLIT CRACK ARRAYS PARALLEL TO AN AXIS 172 6.9. BRITTLE FAILURE IN COMPRESSION 174 6.9.1. INTRODUCTORY COMMENTS 174 6.9.2. BRIDGMAN PARADOXES 176 6.9.3. A NEW LOOK AT MICROCRACKING IN COMPRESSION 180 6.9.4. MODEL CALCULATIONS: AXIAL SPLITTING 184 6.9.5. MODEL CALCULATIONS: FAULTING 187 6.9.6. MODEL CALCULATIONS: BRITTLE-DUCTILE TRANSITION 188 6.10. DYNAMIC BRITTLE FAILURE IN COMPRESSION 193 6.10.1. STRAIN-RATE EFFECT ON BRITTLE FAILURE IN COMPRESSION 195 6.10.2. ILLUSTRATIVE EXAMPLES OF DYNAMIC BRITTLE FAILURE IN COMPRESSION 197 6.11. REFERENCES 200 CHAPTER III ELASTIC SOLIDS WITH MICRO-INCLUSIONS 207 SECTION 7. OVERALL ELASTIC MODULUS AND COMPLIANCE TENSORS 209 7.1. MACROSTRESS PRESCRIBED 209 7.2. MACROSTRAIN PRESCRIBED 212 7.3. EIGENSTRAIN AND EIGENSTRESS TENSORS 213 7.3.1. EIGENSTRAIN 215 7.3.2. EIGENSTRESS 216 7.3.3. UNIFORM EIGENSTRAIN AND EIGENSTRESS 216 7.3.4. CONSISTENCY CONDITIONS 218 7.3.5. H- AND J-TENSORS 220 7.3.6. ESHELBY S TENSOR FOR SPECIAL CASES 221 7.3.7. TRANSFORMATION STRAIN 223 7.4. ESTIMATES OF OVERALL MODULUS AND COMPLIANCE TENSORS: DILUTE DISTRIBUTION 225 7.4.1. MACROSTRESS PRESCRIBED 226 7.4.2. MACROSTRAIN PRESCRIBED 227 7.4.3. EQUIVALENCE BETWEEN OVERALL COMPLIANCE AND ELASTICITY TENSORS 228 7.5. ESTIMATES OF OVERALL MODULUS AND COMPLIANCE TABLE OF CONTENTS XLLL TENSORS: SELF-CONSISTENT METHOD 229 7.5.1. MACROSTRESS PRESCRIBED 230 7.5.2. MACROSTRAIN PRESCRIBED 231 7.5.3. EQUIVALENCE OF OVERALL COMPLIANCE AND ELASTICITY TENSORS OBTAINED BY SELF-CONSISTENT METHOD 231 7.5.4. OVERALL ELASTICITY AND COMPLIANCE TENSORS FOR POLYCRYSTALS 233 7.6. ENERGY CONSIDERATION AND SYMMETRY OF OVERALL ELASTICITY AND COMPLIANCE TENSORS 235 7.6.1. MACROSTRAIN PRESCRIBED 236 7.6.2. MACROSTRESS PRESCRIBED 237 7.6.3. EQUIVALENCE OF OVERALL COMPLIANCE AND ELASTICITY TENSORS OBTAINED ON THE BASIS OF ELASTIC ENERGY 238 7.6.4. CERTAIN EXACT IDENTITIES INVOLVING OVERALL ELASTIC ENERGY 240 7.7. REFERENCES 242 SECTION 8. EXAMPLES OF ELASTIC SOLIDS WITH ELASTIC MICRO-INCLUSIONS 245 8.1. RANDOM DISTRIBUTION OF SPHERICAL MICRO-INCLUSIONS 245 8.1.1. EFFECTIVE MODULI: DILUTE DISTRIBUTION OF SPHERICAL INCLUSIONS 246 8.1.2. EFFECTIVE MODULI: SELF-CONSISTENT ESTIMATES 248 8.2. EFFECTIVE MODULI OF AN ELASTIC PLATE CONTAINING ALIGNED REINFORCING-FIBERS 250 8.2.1. EFFECTIVE MODULI: DILUTE DISTRIBUTION OF FIBERS 254 8.2.2. EFFECTIVE MODULI: SELF-CONSISTENT ESTIMATES 255 8.2.3. EFFECTIVE MODULI IN ANTIPLANE SHEAR: DILUTE-DISTRIBUTION AND SELF-CONSISTENT ESTIMATES 256 8.3. THREE-DIMENSIONAL ANALYSIS OF PLANE STRAIN AND PLANE STRESS STATES 259 8.3.1. REDUCTION OF THREE-DIMENSIONAL MODULI TO TWO-DIMENSIONAL MODULI 259 8.3.2. TWO-DIMENSIONAL NOMINAL ESHELBY TENSOR 260 8.3.3. COMPUTATION OF NOMINAL ESHELBY TENSOR FOR PLANE STRESS 261 8.4. REFERENCES 262 SECTION 9. UPPER AND LOWER BOUNDS FOR OVERALL ELASTIC MODULI 265 9.1. HASHIN-SHTRIKMAN VARIATIONAL PRINCIPLE 267 9.1.1. MACROSTRESS PRESCRIBED 267 9.1.2. MACROSTRAIN PRESCRIBED 271 9.2. UPPER AND LOWER BOUNDS FOR ENERGY FUNCTIONALS 275 9.2.1. STIFF MICRO-INCLUSIONS 276 9.2.2. COMPLIANT MICRO-INCLUSIONS 278 9.2.3. BOUNDS FOR ELASTIC STRAIN AND COMPLEMENTARY ELASTIC ENERGIES 278 9.3. GENERALIZED BOUNDS ON OVERALL ENERGIES 280 XIV TABLE OF CONTENTS 9.3.1. CORRELATION TENSORS 281 9.3.2. UPPER AND LOWER BOUNDS ON OVERALL ENERGIES 283 9.3.3. SUBREGION APPROXIMATION METHOD 286 9.4. DIRECT ESTIMATES OF OVERALL MODULI 287 9.4.1. BOUNDARY-VALUE PROBLEMS FOR EQUIVALENT HOMOGENEOUS SOLID 288 9.4.2. SIMPLIFIED INTEGRAL OPERATORS 290 9.4.3. APPROXIMATE CORRELATION TENSORS 291 9.4.4. OPTIMAL EIGENSTRAINS AND EIGENSTRESSES 294 9.5. GENERALIZED VARIATIONAL PRINCIPLES; EXACT BOUNDS 296 9.5.1. GENERALIZATION OF ENERGY FUNCTIONALS AND BOUNDS 296 9.5.2. INEQUALITIES AMONG GENERALIZED ENERGY FUNCTIONALS 302 9.5.3. FUNCTIONALS WITH SIMPLIFIED INTEGRAL OPERATORS 303 9.5.4. EXACT BOUNDS BASED ON SIMPLIFIED FUNCTIONALS 310 9.5.5. CALCULATION OF BOUNDS 314 9.5.6. ALTERNATIVE FORMULATION OF EXACT INEQUALITIES: DIRECT EVALUATION OF EXACT B OUNDS 316 9.6. UNIVERSAL BOUNDS FOR OVERALL MODULI 320 9.6.1. EQUIVALENCE OF TWO APPROXIMATE FUNCTIONALS 321 9.6.2. SUMMARY OF EXACT INEQUALITIES 322 9.6.3. UNIVERSAL BOUNDS FOR OVERALL MODULI OF ELLIPSOIDAL RVE (1) 323 9.6.4. UNIVERSAL BOUNDS FOR OVERALL MODULI OF ELLIPSOIDAL RVE (2) 327 9.6.5. RELATION BETWEEN UNIVERSAL BOUNDS AND ESTIMATED BOUNDS 328 9.7. BOUNDS FOR OVERALL NONMECHANICAL MODULI 330 9.7.1. GENERALIZED HASHIN-SHTRIKMAN VARIATIONAL PRINCIPLE 331 9.7.2. CONSEQUENCE OF UNIVERSAL THEOREMS 333 9.7.3. UNIVERSAL BOUNDS FOR OVERALL CONDUCTIVITY 335 9.8. BOUNDS FOR OVERALL MODULI OF PIEZOELECTRIC RVE S 339 9.8.1. GENERALIZED HASHIN-SHTRIKMAN VARIATIONAL PRINCIPLE 339 9.8.2. CONSEQUENCE OF UNIVERSAL THEOREMS 343 9.8.3. COMMENTS ON COMPUTING BOUNDS FOR OVERALL MODULI 346 9.9. REFERENCES 349 SECTION 10. SELF-CONSISTENT, DIFFERENTIAL, AND RELATED AVERAGING METHODS 353 10.1. SUMMARY OF EXACT RELATIONS BETWEEN AVERAGE QUANTITIES 353 10.1.1. ASSUMPTIONS IN DILUTE-DISTRIBUTION MODEL 354 10.1.2. DILUTE DISTRIBUTION: MODELING APPROXIMATION 356 10.2. SELF-CONSISTENT METHOD 357 10.3. DIFFERENTIAL SCHEME 361 10.3.1. TWO-PHASE RVE 362 10.3.2. MULTI-PHASE RVE 364 10.3.3. EQUIVALENCE BETWEEN OVERALL ELASTICITY AND COMPLIANCE TENSORS 367 TABLE OF CONTENTS XV 10.4. TWO-PHASE MODEL AND DOUBLE-INCLUSION METHOD 368 10.4.1. BASIC FORMULATION: TWO-PHASE MODEL 369 10.4.2. COMMENTS ON TWO-PHASE MODEL 373 10.4.3. RELATION WITH HASHIN-SHTRIKMAN BOUNDS 374 10.4.4. GENERALIZATION OF ESHELBY S RESULTS 375 10.4.5. DOUBLE-INCLUSION METHOD 378 10.4.6. MULTI-INCLUSION METHOD 381 10.4.7. MULTI-PHASE COMPOSITE MODEL 382 10.4.8. BOUNDS ON OVERALL MODULI BY DOUBLE-INCLUSION METHOD 384 10.5. EQUIVALENCE AMONG ESTIMATES BY DILUTE DISTRI- BUTION, SELF-CONSISTENT, DIFFERENTIAL, AND DOUBLE- INCLUSION METHODS 386 10.6. OTHER AVERAGING SCHEMES 388 10.6.1. COMPOSITE-SPHERES MODEL 389 10.6.2. THREE-PHASE MODEL 390 10.7. REFERENCES 394 SECTION 11. ESHELBY S TENSOR AND RELATED TOPICS 397 11.1. EIGENSTRAIN AND EIGENSTRESS PROBLEMS 397 11.1.1. GREEN S FUNCTION FOR INFINITE DOMAIN 398 11.1.2. THE BODY-FORCE PROBLEM 399 11.1.3. THE EIGENSTRAIN- OR EIGENSTRESS-PROBLEM 400 11.2. ESHELBY S TENSOR 402 11.2.1. UNIFORM EIGENSTRAINS IN AN ELLIPSOIDAL DOMAIN 402 11.2.2. ESHELBY S TENSOR FOR AN ISOTROPIC SOLID 403 11.2.3. ESHELBY S TENSOR FOR ANISOTROPIC MEDIA 406 11.3. SOME BASIC PROPERTIES OF ESHELBY S TENSOR 407 11.3.1. SYMMETRY OF THE ESHELBY TENSOR 407 11.3.2. CONJUGATE ESHELBY TENSOR 408 11.3.3. EVALUATION OF AVERAGE QUANTITIES 409 11.4. RELATIONS AMONG AVERAGE QUANTITIES 412 11.4.1. GENERAL RELATIONS 412 11.4.2. SUPERPOSITION OF UNIFORM STRAIN AND STRESS FIELDS 414 11.4.3. PRESCRIBED BOUNDARY CONDITIONS 415 11.5. REFERENCES 417 CHAPTER IV SOLIDS WITH PERIODIC MICROSTRUCTURE 419 SECTION 12. GENERAL PROPERTIES AND FIELD EQUATIONS 421 12.1. PERIODIC MICROSTRUCTURE AND RVE 421 12.2. PERIODICITY AND UNIT CELL 422 12.3. FOURIER SERIES 424 XVI TABLE OF CONTENTS 12.3.1. DISPLACEMENT AND STRAIN FIELDS 425 12.3.2. STRESS FIELD 427 12.4. HOMOGENIZATION 428 12.4.1. PERIODIC EIGENSTRAIN AND EIGENSTRESS FIELDS 428 12.4.2. GOVERNING EQUATIONS 429 12.4.3. PERIODIC INTEGRAL OPERATORS 430 12.4.4. ISOTROPIC MATRIX 432 12.4.5. CONSISTENCY CONDITIONS 433 12.4.6. ALTERNATIVE FORMULATION 435 12.5. TWO-PHASE PERIODIC MICROSTRUCTURE 439 12.5.1. AVERAGE EIGENSTRAIN FORMULATION 439 12.5.2. MODIFICATION FOR MULTI-PHASE PERIODIC MICROSTRUCTURE 442 12.5.3. PROPERTIES OF THE G-INTEGRAL 442 12.6. ELASTIC INCLUSIONS AND CAVITIES 444 12.6.1. ELASTIC SPHERICAL INCLUSIONS 445 12.6.2. ELASTIC ELLIPSOIDAL INCLUSIONS 447 12.6.3. CYLINDRICAL VOIDS 448 12.7. PERIODICALLY DISTRIBUTED MICROCRACKS 450 12.7.1. LIMIT OF ESHELBY S SOLUTION 451 12.7.2. THE G-INTEGRAL FOR A CRACK 453 12.7.3. PIECEWISE CONSTANT DISTRIBUTION OF EIGENSTRAIN 454 12.7.4. STRESS INTENSITY FACTOR OF PERIODIC CRACKS 457 12.7.5. ILLUSTRATIVE EXAMPLES 459 12.8. APPLICATION TO NONLINEAR COMPOSITES 461 12.9. REFERENCES 464 SECTION 13. OVERALL PROPERTIES OF SOLIDS WITH PERIODIC MICROSTRUCTURE 467 13.1. GENERAL EQUIVALENT HOMOGENEOUS SOLID 468 13.1.1. NOTATION AND INTRODUCTORY COMMENTS 468 13.1.2. MACROFIELD VARIABLES AND HOMOGENEOUS SOLUTIONS 469 13.1.3. PERIODIC MICROSTRUCTURE VERSUS RVE 471 13.1.4. UNIT CELL AS A BOUNDED BODY 472 13.1.5. EQUIVALENT HOMOGENEOUS SOLID FOR PERIODIC MICROSTRUCTURE 473 13.2. HASHIN-SHTRIKMAN VARIATIONAL PRINCIPLE APPLIED TO PERIODIC STRUCTURES 476 13.2.1. SELF-ADJOINTNESS 476 13.2.2. HASHIN-SHTRIKMAN VARIATIONAL PRINCIPLE AND BOUNDS ON OVERALL MODULI 478 13.2.3. EQUIVALENCE OF TWO ENERGY FUNCTIONALS 479 13.2.4. ALTERNATIVE FORMULATION OF EXACT BOUNDS 482 13.3. APPLICATION OF FOURIER SERIES EXPANSION TO ENERGY FUNCTIONALS 485 13.3.1. FOURIER SERIES REPRESENTATION OF EIGENSTRESS 485 TABLE OF CONTENTS XV11 13.3.2. TRUNCATED FOURIER SERIES OF EIGENSTRESS FIELD 487 13.3.3. MATRIX REPRESENTATION OF EULER EQUATIONS 488 13.4. EXAMPLE: ONE-DIMENSIONAL PERIODIC MICROSTRUCTURE 491 13.4.1. EXACT SOLUTION 491 13.4.2. EQUIVALENT HOMOGENEOUS SOLID WITH PERIODIC EIGENSTRESS FIELD 493 13.4.3. HASHIN-SHTRIKMAN VARIATIONAL PRINCIPLE 494 13.5. PIECEWISE CONSTANT APPROXIMATION AND UNIVERSAL BOUNDS 497 13.5.1. PIECEWISE CONSTANT APPROXIMATION OF EIGENSTRESS FIELD 497 13.5.2. COMPUTATION OF ENERGY FUNCTIONS AND UNIVERSAL BOUNDS 499 13.5.3. GENERAL PIECEWISE CONSTANT APPROXIMATION OF EIGENSTRESS FIELD 502 13.6. EXAMPLES 505 13.6.1. EXAMPLE (1): ONE-DIMENSIONAL PERIODIC STRUCTURE 505 13.6.2. EXAMPLE (2): THREE-DIMENSIONAL PERIODIC STRUCTURE 506 13.7. REFERENCES 510 SECTION 14. MIRROR-IMAGE DECOMPOSITION OF PERIODIC FIELDS 511 14.1. MIRROR IMAGES OF POSITION VECTORS AND VECTORS 511 14.2. MIRROR-IMAGE SYMMETRY/ANTISYMMETRY OF TENSOR FIELDS 516 14.2.1. MIRROR-IMAGE (MI) SYM/ANT OF TENSOR FIELDS 516 14.2.2. MI SYM/ANT DECOMPOSITION OF TENSOR FIELDS 517 14.2.3. COMPONENTS OF MI SYM/ANT PARTS 519 14.2.4. OPERATIONS ON MI SYM/ANT PARTS OF TENSOR FIELDS 520 14.3. MIRROR-IMAGE SYMMETRY AND ANTISYMMETRY OF FOURIER SERIES 521 14.3.1. MI SYM/ANT OF COMPLEX KERNEL 521 14.3.2. MI SYM/ANT OF FOURIER SERIES 522 14.4. BOUNDARY CONDITIONS FOR A UNIT CELL 526 14.4.1. SYMMETRY OF UNIT CELL 526 14.4.2. MI SYM/ANT FIELDS FOR A SYMMETRIC UNIT CELL 527 14.4.3. SURFACE DATA FOR MI SYM/ANT SET OF PERIODIC FIELDS IN A SYMMETRIC UNIT CELL 529 14.4.4. HOMOGENEOUS FIELDS 531 14.5. FOURIER SERIES EXPANSION OF MI SYM/ANT SET OF PERIODIC FIELDS 532 14.5.1. MI SYM/ANT DECOMPOSITION OF GOVERNING FIELD EQUATIONS 532 14.5.2. ISOTROPIC EQUIVALENT HOMOGENEOUS SOLID 535 14.6. APPLICATION OF HASHIN-SHTRIKMAN VARIATIONAL PRINCIPLE 537 XV111 TABLE OF CONTENTS 14.6.1. INNER PRODUCT OF STRESS AND STRAIN 537 14.6.2. APPLICATION OF MI SYM/ANT DECOMPOSITION TO ENERGY FUNCTIONAL 538 14.6.3. APPLICATION OF MI SYM/ANT DECOMPOSITION TO QUADRATIC FORMS 540 14.6.4. TWO-PHASE PERIODIC STRUCTURE 543 14.7. REFERENCES 546 APPENDIX A APPLICATION TO INELASTIC HETERO- GENEOUS SOLIDS 547 A. 1. SOURCES OF INELASTICITY 547 A.2. RATE-INDEPENDENT PHENOMENOLOGICAL PLASTICITY 548 A.2.1. CONSTITUTIVE RELATIONS: SMOOTH YIELD SURFACE 549 A.2.2. FLOW POTENTIAL AND ASSOCIATIVE FLOW RULE 550 A.2.3. THE J2-FI0W THEORY WITH ISOTROPIC HARDENING 551 A.2.4. THE J2-FI0W THEORY WITH KINEMATIC HARDENING 552 A.2.5. THE J2-FI0W THEORY WITH DILATANCY AND PRESSURE SENSITIVITY 553 A.2.6. CONSTITUTIVE RELATIONS: YIELD VERTEX 554 A.2.7. CRYSTAL PLASTICITY 556 A.2.8. AGGREGATE PROPERTIES 557 A.3. RATE-DEPENDENT THEORIES 558 A.3.1. RATE DEPENDENT J 2 -PLASTICITY 559 A.3.2. EMPIRICAL MODELS 559 A.3.3. PHYSICALLY-BASED MODELS 560 A.3.4. DRAG-CONTROLLED PLASTIC FLOW 563 A.3.5. VISCOPLASTIC J2-FI0W THEORY 566 A.3.6. NONLINEAR VISCOPLASTIC MODEL 566 A. 3.7. RATE-DEPENDENT CRYSTAL PLASTICITY 5 67 A.4. REFERENCES 568 APPENDIX B HOMOGENIZATION THEORY 573 B.I. SUMMARY OF AVERAGE FIELD THEORY 573 B.2. SUMMARY OF HOMOGENIZATION THEORY 575 B.3. EXTENSION OF HOMOGENIZATION THEORY 578 B.4. EFFECT OF STRAIN GRADIENT 580 B.5. REFERENCES 584 APPENDIX C UNIFORM FIELD THEORY 587 C.I. APPLICATION OF UNIFORM FIELD THEORY TO THERMOELASTICITY OF HETEROGENEOUS SOLIDS 587 C.2. VERIFICATION OF AVERAGE FIELD THEORY 589 C.3. APPLICATION OF UNIFORM FIELD THEORY TO COMPOSITES WITH ALIGNED FIBERS 592 C.4. REFERENCES 594 TABLE OF CONTENTS XIX APPENDIX D IMPROVABLE BOUNDS ON OVERALL PROPERTIES OF HETEROGENEOUS FINITE SOLIDS 595 D. 1. BOUNDS ON POTENTIALS FOR GENERAL BOUNDARY DATA 595 D. 1.1. WEAK KINEMATICAL OR STATISTICAL ADMISSIBILITY 595 D. 1.2. BOUNDS ON POTENTIALS 597 D. 1.3. CALCULATION OF BOUNDS ON OVERALL POTENTIALS 599 D.I.4. BOUNDS BY DISCRETIZATION 602 D.2. LINEAR COMPOSITES 602 D.2.1. EXAMPLES OF CLOSED-FORM BOUNDS 604 D.3. REFERENCES 611 TABLE OF CONTENTS XXI PART 2 INTRODUCTION TO BASIC ELEMENTS OF ELASTICITY THEORY PRECIS: PART 2 617 CHAPTER V FOUNDATIONS 621 SECTION 15. GEOMETRIC FOUNDATIONS 623 15.1. VECTOR SPACE 623 15.2. ELEMENTARY CONCEPTS IN THREE-DIMENSIONAL SPACE 624 15.2.1. RECTANGULAR CARTESIAN COORDINATES 624 15.2.2. TRANSFORMATION OF COORDINATES 627 15.3. TENSORS IN THREE-DIMENSIONAL VECTOR SPACE 627 15.3.1. VECTOR AS FIRST-ORDER TENSOR 627 15.3.2. SECOND-ORDER TENSOR 628 15.3.3. HIGHER-ORDER TENSORS 630 15.3.4. REMARKS ON SECOND-ORDER TENSORS 630 15.4. DEL OPERATOR AND THE GAUSS THEOREM 632 15.5. SPECIAL TOPICS IN TENSOR ALGEBRA 635 15.5.1. SECOND-ORDER BASE TENSORS 635 15.5.2. MATRIX OPERATIONS FOR SECOND- AND FOURTH-ORDER TENSORS 636 15.5.3. SECOND-ORDER SYMMETRIC BASE TENSORS 637 15.5.4. MATRIX OPERATIONS FOR SECOND- AND FOURTH-ORDER SYMMETRIC TENSORS 638 15.6. SPECTRAL REPRESENTATION OF FOURTH-ORDER SYMMETRIC TENSORS 642 15.7. CYLINDRICAL AND SPHERICAL COORDINATES 645 15.8. REFERENCES 649 SECTION 16. KINEMATIC FOUNDATIONS 651 16.1. DEFORMATION AND STRAIN MEASURES 651 16.2. INFINITESIMAL STRAIN MEASURE 654 16.2.1. EXTENSION, SHEAR STRAIN, AND ROTATION 655 16.2.2. PURE DEFORMATION 656 16.2.3. COMPATIBILITY CONDITIONS 660 16.2.4. TWO-DIMENSIONAL CASE 663 16.3. REFERENCES 664 SECTION 17. DYNAMIC FOUNDATIONS 667 17.1. EULER S LAWS 667 XX11 TABLE OF CONTENTS 17.2. TRACTION VECTORS AND STRESS TENSOR 669 17.2.1. TRACTION VECTORS 669 17.2.2. STRESS TENSOR 671 17.2.3. CAUCHY S LAWS 672 17.2.4. PRINCIPAL STRESSES 673 17.3. GEOMETRICAL REPRESENTATION OF STRESS TENSOR 674 17.3.1. MOHR S CIRCLE 675 17.3.2. QUADRATIC FORM 676 17.4. REFERENCES 677 SECTION 18. CONSTITUTIVE RELATIONS 679 18.1. STRAIN ENERGY DENSITY 679 18.1.1. CONSERVATION LAWS 679 18.1.2. STRAIN ENERGY DENSITY FUNCTION W 681 18.2. LINEAR ELASTICITY 682 18.2.1. ELASTICITY 682 18.2.2. LINEAR ELASTICITY 683 18.3. ELASTICITY AND COMPLIANCE TENSORS 684 18.3.1. POSITIVE-DEFINITENESS 684 18.3.2. STRONG ELLIPTICITY 685 18.4. REFERENCES 686 CHAPTER VI ELASTOSTATIC PROBLEMS OF LINEAR ELASTICITY 687 SECTION 19. BOUNDARY-VALUE PROBLEMS AND EXTREMUM PRINCIPLES 689 19.1. BOUNDARY-VALUE PROBLEMS 689 19.2. KINEMATICALLY AND STATICALLY ADMISSIBLE FIELDS 691 19.2.1. KINEMATICALLY ADMISSIBLE DISPLACEMENT FIELD 691 19.2.2. STATICALLY ADMISSIBLE STRESS FIELD 692 19.3. POTENTIAL ENERGY 693 19.3.1. VIRTUAL WORK PRINCIPLE 693 19.3.2. VARIATIONAL PRINCIPLE FOR KINEMATICALLY ADMISSIBLE DISPLACEMENT FIELDS 694 19.3.3. MINIMUM POTENTIAL ENERGY 694 19.4. COMPLEMENTARY ENERGY ,- 696 19.4.1. VIRTUAL WORK PRINCIPLE FOR VIRTUAL STRESS 696 19.4.2. VARIATIONAL PRINCIPLE FOR STATICALLY ADMISSIBLE STRESS FIELDS 697 19.4.3. MINIMUM COMPLEMENTARY ENERGY 697 19.5. GENERAL VARIATIONAL PRINCIPLES 699 19.5.1. GENERAL POTENTIAL ENERGY 699 19.5.2. JUMP CONDITIONS AT DISCONTINUITY SURFACES 701 TABLE OF CONTENTS XX111 19.6. REFERENCES 704 SECTION 20. THREE-DIMENSIONAL PROBLEMS 705 20.1. HELMHOLTZ S DECOMPOSITION THEOREM 705 20.2. WAVE EQUATIONS 706 20.3. PAPKOVICH-NEUBER REPRESENTATION 709 20.3.1. PAPKOVICH-NEUBER REPRESENTATION 709 20.3.2. GALERKIN VECTOR 711 20.4. CONCENTRATED FORCE IN INFINITE AND SEMI-INFINITE SOLIDS 712 20.4.1. GREEN S SECOND IDENTITY 712 20.4.2. INFINITELY EXTENDED SOLID 712 20.4.3. SEMI-INFINITE BODY WITH NORMAL CONCENTRATED FORCES 714 20.4.4. SEMI-INFINITE BODY WITH TANGENTIAL CONCENTRATED FORCES 717 20.5. REFERENCES 720 SECTION 21. SOLUTIONS OF SINGULAR PROBLEMS 723 21.1. AIRY S STRESS FUNCTION 723 21.1.1. SOLUTION TO EQUILIBRIUM EQUATIONS 723 21.1.2. GOVERNING EQUATION FOR AIRY S STRESS FUNCTION 724 21.1.3. ANALYTIC FUNCTIONS 725 21.1.4. BI-HARMONIC FUNCTIONS 726 21.2. GREEN S FUNCTION AND DISLOCATION 728 21.2.1. GREEN S FUNCTION 728 21.2.2. DISLOCATION 731 21.2.3. CENTER OF DILATATION AND DISCLINATION 732 21.3. THE HILBERT PROBLEM 734 21.3.1. HOLOMORPHIC FUNCTIONS 734 21.3.2. THE CAUCHY INTEGRAL 735 21.3.3. THE HILBERT PROBLEM 736 21.3.4. EXAMPLES 738 21.4. TWO-DIMENSIONAL CRACK PROBLEMS 739 21 A.I. CRACK AND DISLOCATIONS 740 21.4.2. INTEGRAL EQUATION FOR DISLOCATION DENSITY 740 21.4.3. EXAMPLE 741 21.4.4. ALTERNATIVE INTEGRAL EQUATION FOR CRACK PROBLEM 742 21.4.5. FINITE-PART INTEGRAL 744 21.5. ANISOTROPIC CASE 745 21.5.1. AIRY S STRESS FUNCTION AND MUSKHELISHVILI S COMPLEX POTENTIALS FOR ANISOTROPIC MATERIALS 746 21.5.2. DISLOCATION IN ANISOTROPIC MEDIUM 748 21.5.3. CRACK IN ANISOTROPIC MEDIUM 751 21.5.4. FULL OR PARTIAL CRACK BRIDGING 753 21.6. DUALITY PRINCIPLES IN ANISOTROPIC ELASTICITY 754 XXIV TABLE OF CONTENTS 21.6.1. A GENERAL DUALITY PRINCIPLE 21.6.2. AN EXAMPLE 21.6.3. DUAL BOUNDARY CONDITIONS 21.6.4. FUNDAMENTAL ELASTICITY MATRIX WITH REPEATED EIGENVALUES 21.6.5. EXAMPLES OF DUALITY 21.7. REFERENCES AUTHOR INDEX SUBJECT INDEX 759 760 763 765 767 768 771 779
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author	Nemat-Nasser, Siavouche Hori, Muneo
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discipline	Physik Werkstoffwissenschaften Chemie-Ingenieurwesen Maschinenbau
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spelling	Nemat-Nasser, Siavouche Verfasser aut Micromechanics overall properties of heterogeneous materials by Sia Nemat-Nasser ; Muneo Hori 2., rev. ed. Amsterdam [u.a.] Elsevier 1999 XXIV, 786 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Heterogenität (DE-588)4201275-2 gnd rswk-swf Inhomogener Festkörper (DE-588)4225741-4 gnd rswk-swf Elastizität (DE-588)4014159-7 gnd rswk-swf Mikromechanik (DE-588)4205811-9 gnd rswk-swf Inhomogener Festkörper (DE-588)4225741-4 s Mikromechanik (DE-588)4205811-9 s DE-604 Elastizität (DE-588)4014159-7 s 1\p DE-604 Heterogenität (DE-588)4201275-2 s 2\p DE-604 Hori, Muneo Verfasser aut HEBIS Datenaustausch Darmstadt application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008529545&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 2\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk
spellingShingle	Nemat-Nasser, Siavouche Hori, Muneo Micromechanics overall properties of heterogeneous materials Heterogenität (DE-588)4201275-2 gnd Inhomogener Festkörper (DE-588)4225741-4 gnd Elastizität (DE-588)4014159-7 gnd Mikromechanik (DE-588)4205811-9 gnd
subject_GND	(DE-588)4201275-2 (DE-588)4225741-4 (DE-588)4014159-7 (DE-588)4205811-9
title	Micromechanics overall properties of heterogeneous materials
title_auth	Micromechanics overall properties of heterogeneous materials
title_exact_search	Micromechanics overall properties of heterogeneous materials
title_full	Micromechanics overall properties of heterogeneous materials by Sia Nemat-Nasser ; Muneo Hori
title_fullStr	Micromechanics overall properties of heterogeneous materials by Sia Nemat-Nasser ; Muneo Hori
title_full_unstemmed	Micromechanics overall properties of heterogeneous materials by Sia Nemat-Nasser ; Muneo Hori
title_short	Micromechanics
title_sort	micromechanics overall properties of heterogeneous materials
title_sub	overall properties of heterogeneous materials
topic	Heterogenität (DE-588)4201275-2 gnd Inhomogener Festkörper (DE-588)4225741-4 gnd Elastizität (DE-588)4014159-7 gnd Mikromechanik (DE-588)4205811-9 gnd
topic_facet	Heterogenität Inhomogener Festkörper Elastizität Mikromechanik
url	http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008529545&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA
work_keys_str_mv	AT nematnassersiavouche micromechanicsoverallpropertiesofheterogeneousmaterials AT horimuneo micromechanicsoverallpropertiesofheterogeneousmaterials

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