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
TABLE OF CONTENTS
v
ix
PARTI
OVERALL PROPERTIES OF
HETEROGENEOUS MATERIALS
PRECIS: PART 1
CHAPTER I AGGREGATE PROPERTIES
AND AVERAGING METHODS
9
SECTION 1. AGGREGATE PROPERTIES
1.1. REPRESENTATIVE VOLUME ELEMENT (RVE)
1.2. SCOPE OF THE BOOK
1.3. DESCRIPTION OF RVE
1.4. REFERENCES
11
11
16
19
23
SECTION 2. AVERAGING METHODS
2.1. AVERAGE STRESS AND STRESS RATE
2.2. AVERAGE STRAIN AND STRAIN RATE
2.3. AVERAGE RATE OF STRESS-WORK
2.3.1. Uniform Boundary Tractions
2.3.2. Linear Boundary Velocities
2.3.3. Other Useful Identities
2.3.4. Virtual Work Principle
2.4. INTERFACES AND DISCONTINUITIES
2.5. POTENTIAL FUNCTION FOR MACRO-ELEMENTS
2.5.1. Stress Potential
2.5.2. Strain Potential
2.5.3. Relation between Macropotentials
2.5.4. On Definition of RVE
2.5.5. Linear Versus Nonlinear Response
2.5.6. General Relations Between Macropotentials
2.5.7. Bounds on Macropotential Functions
27
27
29
31
33
34
34
35
35
38
40
41
42
44
45
45
50
TABLE OF CONTENTS
2.6. STATISTICAL HOMOGENEITY, AVERAGE QUANTITIES, AND
OVERALL PROPERTIES
2.6.1. Local Average Fields
2.6.2. Limiting Process and Limit Fields
2.7. NONMECHANICAL PROPERTIES
2.7.1. Averaging Theorems
2.7.2. Macropotentials
2.7.3. Basic Inequalities
2.8. COUPLED MECHANICAL AND NONMECHANICAL PROPERTIES
2.8.1. Field Equations
2.8.2. Averaging Theorems
2.8.3. Stress/Electric-field Potential
2.8.4. Strain/Electric-displacement Potential
2.8.5. Basic Inequalities
2.9. REFERENCES
53
55
58
59
59
60
61
63
63
65
66
67
68
71
CHAPTER II ELASTIC SOLIDS WITH
MICROCAVITIES AND MICROCRACKS
73
SECTION 3. LINEARLY ELASTIC SOLIDS
3.1. HOOKE' S LAW AND MATERIAL SYMMETRY
3.1.1. Elastic Moduli
3.1.2. Elastic Compliances
3.1.3. Elastic Symmetry
3.1.4. Plane Strain/Plane Stress
3.2. RECIPROCAL THEOREM, SUPERPOSITION, AND GREEN'S
FUNCTION
3.2.1. Reciprocal Theorem
3.2.2. Superposition
3.2.3. Green's Function
3.3. REFERENCES
75
75
75
77
78
82
SECTION 4. ELASTIC SOLIDS WITH TRACTION-FREE DEFECTS
4.1. STATEMENT OF PROBLEM AND NOTATION
4.2. AVERAGE STRAIN FOR PRESCRIBED MACROSTRESS
4.3. OVERALL COMPLIANCE TENSOR FOR POROUS ELASTIC
SOLIDS
4.4. AVERAGE STRESS FOR PRESCRIBED MACROSTRAIN
4.5. OVERALL ELASTICITY TENSOR FOR POROUS ELASTIC
SOLIDS
4.6. REFERENCES
93
93
95
86
87
87
88
91
97
98
100
102
TABLE OF CONTENTS
SECTION 5. ELASTIC SOLIDS WITH MICROCAVITIES
XI
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 x3-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
6.1. OVERALL STRAIN DUE TO MICROCRACKS
6.2. OVERALL COMPLIANCE AND MODULUS TENSORS OF HOMOGENEOUS LINEARLY ELASTIC SOLIDS WITH MICROCRACKS
6.3. EFFECTIVE MODULI OF AN ELASTIC SOLID CONTAINING
ALIGNED SLIT MICROCRACKS
6.3.1. Crack Opening Displacements
6.3.2. Effective Moduli: Dilute Distribution of Aligned Microcracks
6.3.3. Effective Moduli: Dilute Distribution of Aligned Frictional
Microcracks
6.4. EFFECTIVE MODULI OF AN ELASTIC SOLID CONTAINING
RANDOMLY DISTRIBUTED SLIT MICROCRACKS
6.4.1. Effective Moduli: Random Dilute Distribution of Open
Microcracks
6.4.2. Effective Moduli: Self-Consistent Estimate
6.4.3. Effective Moduli in Antiplane Shear: Random Dilute
Distribution of Frictionless Microcracks
6.4.4. Plane Stress, Plane Strain, and Three-Dimensional Overall
Moduli
6.4.5. Effect of Friction and Load-Induced Anisotropy
6.5. EFFECTIVE MODULI OF AN ELASTIC BODY CONTAINING
ALIGNED PENNY-SHAPED MICROCRACKS
6.5.1. Crack-Opening-Displacements
6.5.2. Effective Moduli: Dilute Distribution of Aligned Microcracks
6.6. EFFECTIVE MODULI OF AN ELASTIC BODY CONTAINING
RANDOMLY DISTRIBUTED PENNY-SHAPED MICROCRACKS
6.6.1. Dilute Open Microcracks with Prescribed Distribution
6.6.2. Effective Moduli: Random Dilute Distribution of Microcracks
6.6.3. Effective Moduli: Self-Consistent Estimates
121
121
123
124
125
125
129
131
131
135
137
140
141
147
147
147
151
151
154
158
Xll
TABLE OF CONTENTS
6.7. EFFECTIVE MODULI OF AN ELASTIC BODY CONTAINING
PENNY-SHAPED MICROCRACKS PARALLEL TO AN AXIS
6.8. INTERACTION EFFECTS
6.8.1. Crack-Opening-Displacements and Associated Strains
6.8.2. Dilute Distribution of Parallel Crack Arrays
6.8.3. Randomly Oriented Open Slit Crack Arrays Parallel to an Axis
6.9. BRITTLE FAILURE IN COMPRESSION
6.9.1. Introductory Comments
6.9.2. Bridgman Paradoxes
6.9.3. A New Look at Microcracking in Compression
6.9.4. Model Calculations: Axial Splitting
6.9.5. Model Calculations: Faulting
6.9.6. Model Calculations: Brittle-Ductile Transition
6.10. DYNAMIC BRITTLE FAILURE IN COMPRESSION
6.10.1. Strain-rate Effect on Brittle Failure in
Compression
6.10.2. Illustrative Examples of Dynamic Brittle
Failure in Compression
6.11. REFERENCES
197
200
CHAPTER III ELASTIC SOLIDS WITH
MICRO-INCLUSIONS
207
SECTION 7. OVERALL ELASTIC MODULUS AND COMPLIANCE
TENSORS
7.1. MACROSTRESS PRESCRIBED
7.2. MACROSTRAIN PRESCRIBED
7.3. EIGENSTRAIN AND EIGENSTRESS TENSORS
7.3.1. Eigenstrain
7.3.2. Eigenstress
7.3.3. Uniform Eigenstrain and Eigenstress
7.3.4. Consistency Conditions
7.3.5. H- and J-Tensors
7.3.6. Eshelby's Tensor for Special Cases
7.3.7. Transformation Strain
7.4. ESTIMATES OF OVERALL MODULUS AND COMPLIANCE
TENSORS: DILUTE DISTRIBUTION
7.4.1. Macrostress Prescribed
7.4.2. Macrostrain Prescribed
7.4.3. Equivalence between Overall Compliance and Elasticity
Tensors
7.5. ESTIMATES OF OVERALL MODULUS AND COMPLIANCE
162
167
168
170
172
174
174
176
180
184
187
188
193
195
209
209
212
213
215
216
216
218
220
221
223
225
226
227
228
TABLE OF CONTENTS
TENSORS: SELF-CONSISTENT METHOD
7.5.1. Macrostress Prescribed
7.5.2. Macrostrain Prescribed
7.5.3. Equivalence of Overall Compliance and Elasticity Tensors
Obtained by Self-Consistent Method
7.5.4. Overall Elasticity and Compliance Tensors for Polycrystals
7.6. ENERGY CONSIDERATION AND SYMMETRY OF OVERALL
ELASTICITY AND COMPLIANCE TENSORS
7.6.1. Macrostrain Prescribed
7.6.2. Macrostress Prescribed
7.6.3. Equivalence of Overall Compliance and Elasticity Tensors
Obtained on the Basis of Elastic Energy
7.6.4. Certain Exact Identities Involving Overall Elastic Energy
7.7. REFERENCES
SECTION 8. EXAMPLES OF ELASTIC SOLIDS WITH ELASTIC
MICRO-INCLUSIONS
8.1. RANDOM DISTRIBUTION OF SPHERICAL MICRO-INCLUSIONS
8.1.1. Effective Moduli: Dilute Distribution of Spherical Inclusions
8.1.2. Effective Moduli: Self-Consistent Estimates
8.2. EFFECTIVE MODULI OF AN ELASTIC PLATE CONTAINING
ALIGNED REINFORCING-FIBERS
8.2.1. Effective Moduli: Dilute Distribution of Fibers
8.2.2. Effective Moduli: Self-Consistent Estimates
8.2.3. Effective Moduli in Antiplane Shear: Dilute-Distribution and
Self-Consistent Estimates
8.3. THREE-DIMENSIONAL ANALYSIS OF PLANE STRAIN AND
PLANE STRESS STATES
8.3.1. Reduction of Three-Dimensional Moduli to Two-Dimensional
Moduli
8.3.2. Two-Dimensional Nominal Eshelby Tensor
8.3.3. Computation of Nominal Eshelby Tensor for Plane Stress
8.4. REFERENCES
Xlll
229
230
231
231
233
235
236
237
238
240
242
245
245
246
248
250
254
255
256
259
259
260
261
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
10.1. SUMMARY OF EXACT RELATIONS BETWEEN AVERAGE
QUANTITIES
10.1.1. Assumptions in Dilute-Distribution Model
10.1.2. Dilute Distribution: Modeling Approximation
10.2. SELF-CONSISTENT METHOD
10.3. DIFFERENTIAL SCHEME
10.3.1. Two-Phase RVE
10.3.2. Multi-Phase RVE
10.3.3. Equivalence between Overall Elasticity and Compliance
Tensors
353
353
354
356
357
361
362
364
367
TABLE OF CONTENTS
XV
10.4. TWO-PHASE MODEL AND DOUBLE-INCLUSION METHOD
10.4.1. Basic Formulation: Two-Phase Model
10.4.2. Comments on Two-Phase Model
10.4.3. Relation with Hashin-Shtrikman Bounds
10.4.4. Generalization of Eshelby's Results
10.4.5. Double-Inclusion Method
10.4.6. Multi-Inclusion Method
10.4.7. Multi-Phase Composite Model
10.4.8. Bounds on Overall Moduli by Double-Inclusion Method
10.5. EQUIVALENCE AMONG ESTIMATES BY DILUTE DISTRIBUTION, SELF-CONSISTENT, DIFFERENTIAL, AND DOUBLEINCLUSION METHODS
10.6. OTHER AVERAGING SCHEMES
10.6.1. Composite-Spheres Model
10.6.2. Three-Phase Model
10.7. REFERENCES
368
369
373
374
375
378
381
382
384
SECTION 11. ESHELBY'S TENSOR AND RELATED TOPICS
11.1. EIGENSTRAIN AND EIGENSTRESS PROBLEMS
11.1.1. Green's Function for Infinite Domain
11.1.2. The Body-Force Problem
11.1.3. The Eigenstrain- or Eigenstress-Problem
11.2. ESHELBY'S TENSOR
11.2.1. Uniform Eigenstrains in an Ellipsoidal Domain
11.2.2. Eshelby' s Tensor for an Isotropic Solid
11.2.3. Eshelby's Tensor for Anisotropic Media
11.3. SOME BASIC PROPERTIES OF ESHELBY'S TENSOR
11.3.1. Symmetry of the Eshelby Tensor
11.3.2. Conjugate Eshelby Tensor
11.3.3. Evaluation of Average Quantities
11.4. RELATIONS AMONG AVERAGE QUANTITIES
11.4.1. General Relations
11.4.2. Superposition of Uniform Strain and Stress Fields
11.4.3. Prescribed Boundary Conditions
11.5. REFERENCES
397
397
398
399
400
402
402
403
406
407
407
408
409
412
412
414
415
417
CHAPTER IV SOLIDS WITH PERIODIC
MICROSTRUCTURE
419
SECTION 12. GENERAL PROPERTIES AND FIELD EQUATIONS
12.1. PERIODIC MICROSTRUCTURE AND RVE
12.2. PERIODICITY AND UNIT CELL
12.3. FOURIER SERIES
421
421
422
424
386
388
389
390
394
XVI
TABLE OF CONTENTS
12.3.1. Displacement and Strain Fields
12.3.2. Stress Field
12.4. HOMOGENIZATION
12.4.1. Periodic Eigenstrain and Eigenstress Fields
12.4.2. Governing Equations
12.4.3. Periodic Integral Operators
12.4.4. Isotropic Matrix
12.4.5. Consistency Conditions
12.4.6. Alternative Formulation
12.5. TWO-PHASE PERIODIC MICROSTRUCTURE
12.5.1. Average Eigenstrain Formulation
12.5.2. Modification for Multi-Phase Periodic Microstructure
12.5.3. Properties of the g-Integral
12.6. ELASTIC INCLUSIONS AND CAVITIES
12.6.1. Elastic Spherical Inclusions
12.6.2. Elastic Ellipsoidal Inclusions
12.6.3. Cylindrical Voids
12.7. PERIODICALLY DISTRIBUTED MICROCRACKS
12.7.1. Limit of Eshelby's Solution
12.7.2. The g-Integral for a Crack
12.7.3. Piecewise Constant Distribution of Eigenstrain
12.7.4. Stress Intensity Factor of Periodic Cracks
12.7.5. Illustrative Examples
12.8. APPLICATION TO NONLINEAR COMPOSITES
12.9. REFERENCES
425
427
428
428
429
430
432
433
435
439
439
442
442
444
445
447
448
450
451
453
454
457
459
461
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
13.3.2. Truncated Fourier Series of Eigenstress Field
13.3.3. Matrix Representation of Euler Equations
13.4. EXAMPLE: ONE-DIMENSIONAL PERIODIC
MICROSTRUCTURE
13.4.1. Exact Solution
13.4.2. Equivalent Homogeneous Solid with Periodic Eigenstress
Field
13.4.3. Hashin-Shtrikman Variational Principle
13.5. PIECEWISE CONSTANT APPROXIMATION AND UNIVERSAL
BOUNDS
13.5.1. Piecewise Constant Approximation of Eigenstress Field
13.5.2. Computation of Energy Functions and Universal Bounds
13.5.3. General Piecewise Constant Approximation of Eigenstress
Field
13.6. EXAMPLES
13.6.1. Example (1): One-Dimensional Periodic Structure
13.6.2. Example (2): Three-Dimensional Periodic Structure
13.7. REFERENCES
SECTION 14. MIRROR-IMAGE DECOMPOSITION OF PERIODIC
FIELDS
14.1. MIRROR IMAGES OF POSITION VECTORS AND VECTORS
14.2. MIRROR-IMAGE SYMMETRY/ANTISYMMETRY OF TENSOR
FIELDS
14.2.1. Mirror-Image (MI) Sym/Ant of Tensor Fields
14.2.2. MI Sym/Ant Decomposition of Tensor Fields
14.2.3. Components of MI Sym/Ant Parts
14.2.4. Operations on MI Sym/Ant Parts of Tensor Fields
14.3. MIRROR-IMAGE SYMMETRY AND ANTISYMMETRY OF
FOURIER SERIES
14.3.1. MI Sym/Ant of Complex Kernel
14.3.2. MI Sym/Ant of Fourier Series
14.4. BOUNDARY CONDITIONS FOR A UNIT CELL
14.4.1. Symmetry of Unit Cell
14.4.2. MI Sym/Ant Fields for a Symmetric Unit Cell
14.4.3. Surface Data for MI Sym/Ant Set of Periodic Fields in a
Symmetric Unit Cell
'
14.4.4. Homogeneous Fields
14.5. FOURIER SERIES EXPANSION OF MI SYM/ANT SET OF
PERIODIC FIELDS
14.5.1. MI Sym/Ant Decomposition of Governing Field Equations
14.5.2. Isotropic Equivalent Homogeneous Solid
14.6. APPLICATION OF HASHIN-SHTRIKMAN VARIATIONAL
PRINCIPLE
XV11
487
488
491
491
493
494
497
497
499
502
505
505
506
510
511
511
516
516
517
519
520
521
521
522
526
526
527
529
531
532
532
535
537
XV111
TABLE OF CONTENTS
14.6.1. Inner Product of Stress and Strain
14.6.2. Application of MI Sym/Ant Decomposition to Energy
Functional
14.6.3. Application of MI Sym/Ant Decomposition to Quadratic
Forms
14.6.4. Two-Phase Periodic Structure
14.7. REFERENCES
APPENDIX A APPLICATION TO INELASTIC HETEROGENEOUS SOLIDS
A. 1. SOURCES OF INELASTICITY
A.2. RATE-INDEPENDENT PHENOMENOLOGICAL PLASTICITY
A.2.1. Constitutive Relations: Smooth Yield Surface
A.2.2. Flow Potential and Associative Flow Rule
A.2.3. The J2-FI0W Theory with Isotropic Hardening
A.2.4. The J2-FI0W Theory with Kinematic Hardening
A.2.5. The J2-FI0W Theory with Dilatancy and
Pressure Sensitivity
A.2.6. Constitutive Relations: Yield Vertex
A.2.7. Crystal Plasticity
A.2.8. Aggregate Properties
A.3. RATE-DEPENDENT THEORIES
A.3.1. Rate Dependent J2-Plasticity
A.3.2. Empirical Models
A.3.3. Physically-based Models
A.3.4. Drag-controlled Plastic Flow
A.3.5. Viscoplastic J2-FI0W Theory
A.3.6. Nonlinear Viscoplastic Model
A. 3.7. Rate-Dependent Crystal Plasticity
A.4. REFERENCES
537
538
540
543
546
547
547
548
549
550
551
552
553
554
556
557
558
559
559
560
563
566
566
5 67
568
APPENDIX B HOMOGENIZATION THEORY
B.I. SUMMARY OF AVERAGE FIELD THEORY
B.2. SUMMARY OF HOMOGENIZATION THEORY
B.3. EXTENSION OF HOMOGENIZATION THEORY
B.4. EFFECT OF STRAIN GRADIENT
B.5. REFERENCES
573
573
575
578
580
584
APPENDIX C UNIFORM FIELD THEORY
C.I. APPLICATION OF UNIFORM FIELD THEORY TO
THERMOELASTICITY OF HETEROGENEOUS SOLIDS
C.2. VERIFICATION OF AVERAGE FIELD THEORY
C.3. APPLICATION OF UNIFORM FIELD THEORY TO
COMPOSITES WITH ALIGNED FIBERS
C.4. REFERENCES
587
587
589
592
594
TABLE OF CONTENTS
APPENDIX D IMPROVABLE BOUNDS ON OVERALL
PROPERTIES OF HETEROGENEOUS FINITE SOLIDS
D. 1. BOUNDS ON POTENTIALS FOR GENERAL BOUNDARY DATA
D. 1.1. Weak Kinematical or Statistical Admissibility
D. 1.2. Bounds on Potentials
D. 1.3. Calculation of Bounds on Overall Potentials
D.I.4. Bounds by Discretization
D.2. LINEAR COMPOSITES
D.2.1. Examples of Closed-form Bounds
D.3. REFERENCES
XIX
595
595
595
597
599
602
602
604
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
15.1. VECTOR SPACE
15.2. ELEMENTARY CONCEPTS IN THREE-DIMENSIONAL
SPACE
15.2.1. Rectangular Cartesian Coordinates
15.2.2. Transformation of Coordinates
15.3. TENSORS IN THREE-DIMENSIONAL VECTOR SPACE
15.3.1. Vector as First-Order Tensor
15.3.2. Second-Order Tensor
15.3.3. Higher-Order Tensors
15.3.4. Remarks on Second-Order Tensors
15.4. DEL OPERATOR AND THE GAUSS THEOREM
15.5. SPECIAL TOPICS IN TENSOR ALGEBRA
15.5.1. Second-Order Base Tensors
15.5.2. Matrix Operations for Second- and Fourth-Order Tensors
15.5.3. Second-Order Symmetric Base Tensors
15.5.4. Matrix Operations for Second- and Fourth-Order Symmetric
Tensors
15.6. SPECTRAL REPRESENTATION OF FOURTH-ORDER
SYMMETRIC TENSORS
15.7. CYLINDRICAL AND SPHERICAL COORDINATES
15.8. REFERENCES
623
623
642
645
649
SECTION 16. KINEMATIC FOUNDATIONS
16.1. DEFORMATION AND STRAIN MEASURES
16.2. INFINITESIMAL STRAIN MEASURE
16.2.1. Extension, Shear Strain, and Rotation
16.2.2. Pure Deformation
16.2.3. Compatibility Conditions
16.2.4. Two-Dimensional Case
16.3. REFERENCES
651
651
654
655
656
660
663
664
SECTION 17. DYNAMIC FOUNDATIONS
17.1. EULER'S LAWS
667
667
624
624
627
627
627
628
630
630
632
635
635
636
637
638
XX11
TABLE OF CONTENTS
17.2. TRACTION VECTORS AND STRESS TENSOR
17.2.1. Traction Vectors
17.2.2. Stress Tensor
17.2.3. Cauchy's Laws
17.2.4. Principal Stresses
17.3. GEOMETRICAL REPRESENTATION OF STRESS TENSOR
17.3.1. Mohr's Circle
17.3.2. Quadratic Form
17.4. REFERENCES
669
669
671
672
673
674
675
676
677
SECTION 18. CONSTITUTIVE RELATIONS
18.1. STRAIN ENERGY DENSITY
18.1.1. Conservation Laws
18.1.2. Strain Energy Density Function w
18.2. LINEAR ELASTICITY
18.2.1. Elasticity
18.2.2. Linear Elasticity
18.3. ELASTICITY AND COMPLIANCE TENSORS
18.3.1. Positive-Definiteness
18.3.2. Strong Ellipticity
18.4. REFERENCES
679
679
679
681
682
682
683
684
684
685
686
CHAPTER VI ELASTOSTATIC PROBLEMS OF
LINEAR ELASTICITY
687
SECTION 19. BOUNDARY-VALUE PROBLEMS AND EXTREMUM
PRINCIPLES
19.1. BOUNDARY-VALUE PROBLEMS
19.2. KINEMATICALLY AND STATICALLY ADMISSIBLE FIELDS
19.2.1. Kinematically Admissible Displacement Field
19.2.2. Statically Admissible Stress Field
19.3. POTENTIAL ENERGY
19.3.1. Virtual Work Principle
19.3.2. Variational Principle for Kinematically Admissible
Displacement Fields
19.3.3. Minimum Potential Energy
19.4. COMPLEMENTARY ENERGY
,19.4.1. Virtual Work Principle for Virtual Stress
19.4.2. Variational Principle for Statically Admissible Stress Fields
19.4.3. Minimum Complementary Energy
19.5. GENERAL VARIATIONAL PRINCIPLES
19.5.1. General Potential Energy
19.5.2. Jump Conditions at Discontinuity Surfaces
689
689
691
691
692
693
693
694
694
696
696
697
697
699
699
701
TABLE OF CONTENTS
XX111
19.6. REFERENCES
704
SECTION 20. THREE-DIMENSIONAL PROBLEMS
20.1. HELMHOLTZ' S DECOMPOSITION THEOREM
20.2. WAVE EQUATIONS
20.3. PAPKOVICH-NEUBER REPRESENTATION
20.3.1. Papkovich-Neuber Representation
20.3.2. Galerkin Vector
20.4. CONCENTRATED FORCE IN INFINITE AND SEMI-INFINITE
SOLIDS
20.4.1. Green's Second Identity
20.4.2. Infinitely Extended Solid
20.4.3. Semi-Infinite Body with Normal Concentrated Forces
20.4.4. Semi-Infinite Body with Tangential Concentrated Forces
20.5. REFERENCES
705
705
706
709
709
711
SECTION 21. SOLUTIONS OF SINGULAR PROBLEMS
21.1. AIRY'S STRESS FUNCTION
21.1.1. Solution to Equilibrium Equations
21.1.2. Governing Equation for Airy's Stress Function
21.1.3. Analytic Functions
21.1.4. Bi-Harmonic Functions
21.2. GREEN'S FUNCTION AND DISLOCATION
21.2.1. Green's Function
21.2.2. Dislocation
21.2.3. Center of Dilatation and Disclination
21.3. THE HILBERT PROBLEM
21.3.1. Holomorphic Functions
21.3.2. The Cauchy Integral
21.3.3. The Hilbert Problem
21.3.4. Examples
21.4. TWO-DIMENSIONAL CRACK PROBLEMS
21 A.I. Crack and Dislocations
21.4.2. Integral Equation for Dislocation Density
21.4.3. Example
21.4.4. Alternative Integral Equation for Crack Problem
21.4.5. Finite-Part Integral
21.5. ANISOTROPIC CASE
21.5.1. Airy's Stress Function and Muskhelishvili's Complex
Potentials for Anisotropic Materials
21.5.2. Dislocation in Anisotropic Medium
21.5.3. Crack in Anisotropic Medium
21.5.4. Full or Partial Crack Bridging
21.6. DUALITY PRINCIPLES IN ANISOTROPIC ELASTICITY
723
723
723
724
725
726
728
728
731
732
734
734
735
736
738
739
740
740
741
742
744
745
712
712
712
714
717
720
746
748
751
753
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