The mechanics and thermodynamics of continua:
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| Main Authors: | , , |
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| Format: | Book |
| Language: | English |
| Published: |
Cambridge [u.a.]
Cambridge Univ. Press
2010
|
| Edition: | 1. publ. |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XXI, 694 S. graph. Darst. |
| ISBN: | 9780521405980 |
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| 100 | 1 | |a Gurtin, Morton E. |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a The mechanics and thermodynamics of continua |c Morton E. Gurtin ; Eliot Fried ; Lallit Anand |
| 250 | |a 1. publ. | ||
| 264 | 1 | |a Cambridge [u.a.] |b Cambridge Univ. Press |c 2010 | |
| 300 | |a XXI, 694 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 650 | 4 | |a Continuum mechanics - Mathematics | |
| 650 | 4 | |a Thermodynamics - Mathematics | |
| 650 | 4 | |a Mathematik | |
| 650 | 4 | |a Continuum mechanics |x Mathematics | |
| 650 | 4 | |a Thermodynamics |x Mathematics | |
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| adam_text | Contents
Preface
page
xix
PARTI. VECTOR AND TENSOR ALGEBRA
1
1
Vector Algebra
......................................3
1.1
Inner Product. Cross Product
3
1.2
Cartesian Coordinate Frames
6
1.3
Summation Convention. Components of a Vector and a Point
6
2
Tensor Algebra
......................................9
2.1
What Is a Tensor?
9
2.2
Zero and Identity Tensors. Tensor Product of Two
Vectors. Projection Tensor. Spherical Tensor
10
2.3
Components of a Tensor
11
2.4
Transpose of a Tensor. Symmetric and Skew Tensors
12
2.5
Product of Tensors
13
2.6
Vector Cross. Axial Vector of a Skew Tensor
15
2.7
Trace of a Tensor.
Deviatone
Tensors
16
2.8
Inner Product of Tensors. Magnitude of a Tensor
17
2.9
Invertible Tensors
19
2.10
Determinant of a Tensor
21
2.11
Cofactor of a Tensor
22
2.12
Orthogonal Tensors
25
2.13
Matrix of a Tensor
26
2.14
Eigenvalues and Eigenvectors of a Tensor. Spectral Theorem
28
2.15
Square Root of a Symmetric, Positive-Definite Tensor.
Polar Decomposition Theorem
31
2.16
Principal Invariants of a Tensor. Cayley-Hamilton Equation
35
PARTII.
VECTOR AND TENSOR ANALYSIS
39
3
Differentiation
......................................41
3.1
Differentiation of Functions of a Scalar
41
3.2
Differentiation of Fields. Gradient
43
3.3
Divergence and Curl. Vector and Tensor Identities
46
3.4
Differentiation of a Scalar Function of a Tensor
49
v¡
Contents
4
Integral
Theorems...................................52
4.1
The
Divergence
Theorem 52
4.2 Line Integrals.
Stokes
Theorem 53
PART III. KINEMATICS
59
5 Motion
of a Body....................................
61
5.1
Reference Body.
Material Points 61
5.2
Basic Quantities Associated with the Motion of a Body
61
5.3
Convection of Sets with the Body
63
6
The Deformation Gradient
..............................64
6.1
Approximation of a Deformation by a Homogeneous
Deformation
64
6.1.1
Homogeneous Deformations
64
6.1.2
General Deformations
65
6.2
Convection of Geometric Quantities
66
6.2.1
Infinitesimal Fibers
66
6.2.2
Curves
67
6.2.3
Tangent Vectors
67
6.2.4
Bases
68
7
Stretch, Strain, and Rotation
.............................69
7.1
Stretch and Rotation Tensors. Strain
69
7.2
Fibers. Properties of the Tensors
U
and
С
70
7.2.1
Infinitesimal Fibers
70
7.2.2
Finite Fibers
71
7.3
Principal Stretches and Principal Directions
73
8
Deformation of Volume and Area
.........................75
8.1
Deformation of Normals
75
8.2
Deformation of Volume
76
8.3
Deformation of Area
77
9
Material and Spatial Descriptions of Fields
...................80
9.1
Gradient, Divergence, and Curl
80
9.2
Material and Spatial Time Derivatives
81
9.3
Velocity Gradient
82
9.4
Commutator Identities
84
9.5
Particle Paths
85
9.6
Stretching of Deformed Fibers
85
10
Special Motions
.....................................86
10.1
Rigid Motions
86
10.2
Motions Whose Velocity Gradient is Symmetric and
Spatially Constant
87
11
Stretching and Spin in an Arbitrary Motion
...................89
11.1
Stretching and Spin as Tensor Fields
89
11.2
Properties of
D
90
11.3
Stretching and Spin Using the Current Configuration as
Reference
92
Contents
vii
12 Material
and Spatial Tensor Fields. Fullback and Pushforward
Operations
........................................95
12.1
Material and Spatial Tensor Fields
95
12.2
Pullback and Pushforward Operations
95
13
Modes of Evolution for Vector and Tensor Fields
...............98
13.1
Vector and Tensor Fields That Convect With the Body
98
13.1.1
Vector Fields That Convect as Tangents
98
13.1.2
Vector Fields That Convect as Normals
99
13.1.3
Tangentially Convecting Basis and Its Dual Basis.
Covariant and
Contravariant
Components of Spatial Fields
99
13.1.4
Covariant and
Contravariant
Convection of Tensor Fields
102
13.2
Corotational Vector and Tensor Fields
105
14
Motions with Constant Velocity Gradient
...................107
14.1
Motions
107
15
Material and Spatial Integration
..........................109
15.1
Line Integrals
109
15.2
Volume and Surface Integrals
109
15.2.1
Volume Integrals
110
15.2.2
Surface Integrals 111
15.3
Localization of Integrals 111
16
Reynolds Transport Relation. Isochoric Motions
..............113
17
More Kinematics
....................................115
17.1
Vorticity
115
17.2
Transport Relations for Spin and Vorticity
115
17.3
Irrotational Motions
117
17.4
Circulation
118
17.5
Vortex Lines
120
17.6
Steady Motions
121
17.7
A Class of Natural Reference Configurations for Fluids
122
17.8
The Motion Problem
122
17.8.1
Kinematical Boundary Conditions
122
17.8.2
The Motion Problem in a Fixed Container
123
17.8.3
The Motion Problem in All of Space. Solution with
Constant Velocity Gradient
123
PART IV. BASIC MECHANICAL PRINCIPLES
125
18
Balance of Mass
....................................127
18.1
Global Form of Balance of Mass
127
18.2
Local Forms of Balance of Mass
128
18.3
Simple Consequences of Mass Balance
129
19
Forces and Moments. Balance Laws for Linear and Angular
Momentum
.......................................131
19.1
Inerţial
Frames. Linear and Angular Momentum
131
19.2
Surface Tractions. Body Forces
132
19.3
Balance Laws for Linear and Angular Momentum
134
Contents
19.4 Balance
of
Forces and Moments
Based on the
Generalized Body Force
136
19.5
Cauchy s Theorem for the Existence of Stress
137
19.6
Local Forms of the Force and Moment Balances
139
19.7
Kinetic Energy. Conventional and Generalized External
Power Expenditures
141
19.7.1
Conventional Form of the External Power
142
19.7.2
Kinetic Energy and
Inerţial
Power
142
19.7.3
Generalized Power Balance
143
19.7.4
The Assumption of Negligible
Inerţial
Forces
144
20
Frames of Reference
.................................146
20.1
Changes of Frame
146
20.2
Frame-Indifferent Fields
147
20.3
Transformation Rules for Kinematic Fields
148
20.3.1
Material Time-Derivatives of Frame-Indifferent Tensor
Fields are Not Frame-Indifferent
151
20.3.2
The Corotational, Covariant, and
Contravariant
Rates of
a Tensor Field
151
20.3.3
Other Relations for the Corotational Rate
152
20.3.4
Other Relations for the Covariant Rate
153
20.3.5
Other Relations for the
Contravariant
Rate
154
20.3.6
General
Tensorial
Rate
155
21
Frame-Indifference Principle
............................157
21.1
Transformation Rules for Stress and Body Force
157
21.2
Inerţial
Body Force in a Frame That Is Not
Inerţial
159
22
Alternative Formulations of the Force and Moment Balances
......161
22.1
Force and Moment Balances as a Consequence of
Frame-Indifference of the Expended Power
161
22.2
Principle of Virtual Power
163
22.2.1
Application to Boundary-Value Problems
165
22.2.2
Fundamental Lemma of the Calculus of Variations
167
23
Mechanical Laws for a Spatial Control Volume
................168
23.1
Mass Balance for a Control Volume
169
23.2
Momentum Balances for a Control Volume
169
24
Referential Forms for the Mechanical Laws
..................173
24.1
Piola
Stress. Force and Moment Balances
173
24.2
Expended Power
175
25
Further Discussion of Stress
............................177
25.1
Power-Conjugate Pairings. Second
Piola
Stress
177
25.2
Transformation Laws for the
Piola
Stresses
178
PARTV. BASIC THERMODYNAMICAL PRINCIPLES
181
26
The First Law: Balance of Energy
.........................183
26.1
Global and Local Forms of Energy Balance
184
26.2
Terminology for Extensive Quantities
185
Contents ix
27
The Second Law:
Nonnegative
Production of Entropy
...........186
27.1
Global Form of the Entropy Imbalance
187
27.2
Temperature and the Entropy Imbalance
187
27.3
Free-Energy Imbalance. Dissipation
188
28
General Theorems
...................................190
28.1
Invariant Nature of the First Two Laws
190
28.2
Decay Inequalities for the Body Under Passive
Boundary Conditions
191
28.2.1
Isolated Body
191
28.2.2
Boundary Essentially at Constant Pressure and
Temperature
192
29
A Free-Energy Imbalance for Mechanical Theories
.............194
29.1
Free-Energy Imbalance. Dissipation
194
29.2
Digression: Role of the Free-Energy Imbalance within
the General Thermodynamic Framework
195
29.3
Decay Inequalities
196
30
The First Two Laws for a Spatial Control Volume
..............197
31
The First Two Laws Expressed Referentially
.................199
31.1
Global Forms of the First Two Laws
200
31.2
Local Forms of the First Two Laws
201
31.3
Decay Inequalities for the Body Under Passive
Boundary Conditions
202
31.4
Mechanical Theory: Free-Energy Imbalance
204
PART VI. MECHANICAL AND THERMODYNAMICAL
LAWS AT A SHOCK WAVE
207
32
Shock Wave Kinematics
...............................209
32.1
Notation. Terminology
209
32.2
Hadamard s Compatibility Conditions
210
32.3
Relation Between the Scalar Normal Velocities VR and V
212
32.4
Transport Relations in the Presence of a Shock Wave
212
32.5
The Divergence Theorem in the Presence of a Shock Wave
215
33
Basic Laws at a Shock Wave: Jump Conditions
................216
33.1
Balance of Mass and Momentum
216
33.2
Balance of Energy and the Entropy Imbalance
218
PART
VII.
INTERLUDE: BASIC HYPOTHESES FOR
DEVELOPING PHYSICALLY MEANINGFUL
CONSTITUTIVE THEORIES
221
34
General Considerations
...............................223
35
Constitutive Response Functions
.........................224
36
Frame-Indifference and Compatibility with Thermodynamics
......225
Contents
PART
VIII. RIGID
HEAT CONDUCTORS
227
37
Basic
Laws
........................................229
38
General
Constitutive Equations
..........................230
39
Thermodynamics and Constitutive Restrictions: The
Coleman-Noll Procedure
..............................232
40
Consequences of the State Restrictions
.....................234
41
Consequences of the Heat-Conduction Inequality
..............236
42
Fourier s Law
......................................237
PART IX. THE MECHANICAL THEORY OF COMPRESSIBLE
AND INCOMPRESSIBLE FLUIDS
239
43
Brief Review
......................................241
43.1
Basic Kinematical Relations
241
43.2
Basic Laws
241
43.3
Transformation Rules and Objective Rates
242
44
Elastic Fluids
......................................244
44.1
Constitutive Theory
244
44.2
Consequences of Frame-Indifference
244
44.3
Consequences of Thermodynamics
245
44.4
Evolution Equations
246
45
Compressible, Viscous Fluids
...........................250
45.1
General Constitutive Equations
250
45.2
Consequences of Frame-Indifference
251
45.3
Consequences of Thermodynamics
253
45.4
Compressible, Linearly Viscous Fluids
255
45.5
Compressible Navier-Stokes Equations
256
45.6
Vorticity Transport Equation
256
46
Incompressible Fluids
................................259
46.1
Free-Energy Imbalance for an Incompressible Body
259
46.2
Incompressible, Viscous Fluids
260
46.3
Incompressible, Linearly Viscous Fluids
261
46.4
Incompressible Navier-Stokes Equations
262
46.5
Circulation. Vorticity-Transport Equation
263
46.6
Pressure
Poisson
Equation
265
46.7
Transport Equations for the Velocity Gradient,
Stretching, and Spin in a Linearly Viscous,
Incompressible Fluid
265
46.8
Impetus-Gauge Formulation of the Navier-Stokes
Equations
267
46.9
Perfect Fluids
268
Contents xi
PART X. MECHANICAL THEORY OF ELASTIC SOLIDS
271
47
Brief Review
...................................... 273
47.1
Kinematical Relations
273
47.2
Basic Laws
273
47.3
Transformation Laws Under a Change in Frame
274
48
Constitutive Theory
.................................. 276
48.1
Consequences of Frame-Indifference
276
48.2
Thermodynamic Restrictions
278
48.2.1
The Stress Relation
278
48.2.2
Consequences of the Stress Relation
280
48.2.3
Natural Reference Configuration
280
49
Summary of Basic Equations. Initial/Boundary- Value Problems
..... 282
49.1
Basic Field Equations
282
49.2
A Typical Initial/Boundary-Value Problem
283
50
Material Symmetry
.................................. 284
50.1
The Notion of a Group.
Invariance
Under a Group
284
50.2
The Symmetry Group
Q
285
50.2.1
Proof That
Q
Is a Group
288
50.3
Isotropy
288
50.3.1
Free Energy Expressed in Terms of Invariants
290
50.3.2
Free Energy Expressed in Terms of Principal stretches
292
51
Simple Shear of a Homogeneous,
Isotropie
Elastic Body
.......... 294
52
The Linear Theory of Elasticity
.......................... 297
52.1
Small Deformations
297
52.2
The Stress-Strain Law for Small Deformations
298
52.2.1
The Elasticity Tensor
298
52.2.2
The Compliance Tensor
300
52.2.3
Estimates for the Stress and Free Energy
300
52.3
Basic Equations of the Linear Theory of Elasticity
302
52.4
Special Forms for the Elasticity Tensor
302
52.4.1 Isotropie
Material
303
52.4.2
Cubic Crystal
304
52.5
Basic Equations of the Linear theory of Elasticity for an
Isotropie
Material
306
52.5.1
Statical Equations
307
52.6
Some Simple Statical Solutions
309
52.7
Boundary-Value Problems
310
52.7.1
Elastostatics
310
52.7.2
Elastodynamics
313
52.8
Sinusoidal Progressive Waves
313
53
Digression: Incompressibility
............................ 316
53.1
Kinematics of Incompressibility
316
53.2
Indeterminacy of the Pressure. Free-Energy Imbalance
317
53.3
Changes in Frame
Contents
54
Incompressible
Elastic
Materials
.........................319
54.1
Constitutive
Theory
319
54.1.1
Consequences of Frame-Indifference
319
54.1.2
Domain of Definition of the Response Functions
320
54.1.3
Thermodynamic Restrictions
321
54.2
Incompressible
Isotropie
Elastic Bodies
323
54.3
Simple Shear of a Homogeneous,
Isotropie,
Incompressible Elastic Body
324
55
Approximately Incompressible Elastic Materials
...............326
PART XI. THERMOELASTICITY
331
56
Brief Review
......................................333
56.1
Kinematical Relations
333
56.2
Basic Laws
333
57
Constitutive Theory
..................................335
57.1
Consequences of Frame-Indifference
335
57.2
Thermodynamic Restrictions
336
57.3
Consequences of the Thermodynamic Restrictions
338
57.3.1
Consequences of the State Relations
338
57.3.2
Consequences of the Heat-Conduction Inequality
339
57.4
Elasticity Tensor. Stress-Temperature Modulus. Heat Capacity
341
57.5
The Basic Thermoelastic Field Equations
342
57.6
Entropy as Independent Variable. Nonconductors
343
57.7
Nonconductors
346
57.8
Material Symmetry
346
58
Natural Reference Configuration for a Given Temperature
........348
58.1
Asymptotic Stability and its Consequences. The Gibbs Function
348
58.2
Local Relations at a Reference Configuration that is
Natural for a Temperature
ůo
349
59
Linear Thermoelasticity
...............................354
59.1
Approximate Constitutive Equations for the Stress and Entropy
354
59.2
Basic Field Equations of Linear Thermoelasticity
356
59.3 Isotropie
Linear Thermoelasticity
356
PART
XII.
SPECIES DIFFUSION COUPLED TO ELASTICITY
361
60
Balance Laws for Forces, Moments, and the Conventional
External Power
.....................................363
61
Mass Balance for a Single Diffusing Species
..................364
62
Free-Energy Imbalance Revisited. Chemical Potential
...........366
63
Multiple Species
....................................369
63.1
Species Mass Balances
369
63.2
Free-Energy Imbalance
370
Contents xiii
64
Digression:
The Thermodynamic Laws in the Presence of Species
Transport
........................................371
65
Referential Laws
....................................374
65.1
Single Species
374
65.2
Multiple Species
376
66
Constitutive Theory for a Single Species
....................377
66.1
Consequences of Frame-Indifference
377
66.2
Thermodynamic Restrictions
378
66.3
Consequences of the Thermodynamic Restrictions
380
66.4
Fick s Law
382
67
Material Symmetry
..................................385
68
Natural Reference Configuration
.........................388
69
Summary of Basic Equations for a Single Species
..............390
70
Constitutive Theory for Multiple Species
....................391
70.1
Consequences of Frame-Indifference and Thermodynamics
391
70.2
Fick s Law
393
70.3
Natural Reference Configuration
393
71
Summary of Basic Equations for
N
Independent Species
.........396
72
Substitutional Alloys
.................................398
72.1
Lattice Constraint
398
72.2
Substitutional Flux Constraint
399
72.3
Relative Chemical Potentials. Free-Energy Imbalance
399
72.4
Elimination of the Lattice Constraint.
Larché-Cahn
Differentiation
400
72.5
General Constitutive Equations
403
72.6
Thermodynamic Restrictions
404
72.7
Verification of (t)
406
72.8
Normalization Based on the Elimination of the Lattice
Constraint
406
73
Linearization
......................................408
73.1
Approximate Constitutive Equations for the Stress,
Chemical Potentials, and Fluxes
408
73.2
Basic Equations of the Linear Theory
410
73.3 Isotropie
Linear Theory
41
1
PART
XIII.
THEORY OF ISOTROPIC PLASTIC SOLIDS
UNDERGOING SMALL DEFORMATIONS
415
74
Some Phenomenological Aspects of the Elastic-Plastic
Stress-Strain Response of Polycrystalline Metals
...............417
74.1 Isotropie
and Kinematic Strain-Hardening
419
Contents
75
Formulation
of the Conventional Theory. Preliminaries
.......... 422
75.1
Basic Equations
422
75.2
Kinematical Assumptions that Define Plasticity Theory
423
75.3
Separability Hypothesis
424
75.4
Constitutive Characterization of Elastic Response
424
76
Formulation of the
Mises
Theory of Plastic Flow
............... 426
76.1
General Constitutive Equations for Plastic Flow
427
76.2
Rate-Independence
428
76.3
Strict Dissipativity
430
76.4
Formulation of the
Mises
Flow Equations
431
76.5
Initializing the
Mises
Flow Equations
434
76.5.1
Flow Equations With Y(S
)
not Identically Equal to
S
434
76.5.2
Theory with Flow Resistance as Hardening Variable
435
76.6
Solving the Hardening Equation. Accumulated Plastic
Strain is the Most General Hardening Variable
435
76.7
Flow Resistance as Hardening Variable, Revisited
439
76.8
Yield Surface. Yield Function. Consistency Condition
439
76.9
Hardening and Softening
443
77
Inversion of the
Mises
Flow Rule:
Ép
in Terms of
È
and
Τ
........ 445
78
Rate-Dependent Plastic Materials
........................ 449
78.1
Background
449
78.2
Materials with Simple Rate-Dependence
449
78.3
Power-Law Rate-Dependence
452
79
Maximum Dissipation
................................ 454
79.1
Basic Definitions
454
79.2
Warm-up: Derivation of the
Mises
Flow Equations Based
on Maximum Dissipation
456
79.3
More General Flow Rules. Drucker s Theorem
458
79.3.1
Yield-Set Hypotheses
458
79.3.2
Digression: Some Definitions and Results Concerning
Convex Surfaces
460
79.3.3
Drucker s Theorem
461
79.4
The Conventional Theory of Perfectly Plastic Materials
Fits within the Framework Presented Here
462
80
Hardening Characterized by a Defect Energy
.................465
80.1
Free-Energy Imbalance Revisited
465
80.2
Constitutive Equations. Flow Rule
466
81
The Thermodynamics of Mises-Hill Plasticity
.................469
81.1
Background
469
81.2
Thermodynamics
470
81.3
Constitutive Equations
470
81.4
Nature of the Defect Energy
472
81.5
The Flow Rule and the Boundedness Inequality
473
81.6
Balance of Energy Revisited
473
81.7
Thermally Simple Materials
475
Contents xv
81.8 Determination
of the Defect Energy by the Rosakis
Brothers,
Hodowany,
and Ravichandran
476
81.9
Summary of the Basic Equations
477
82
Formulation of Initial/Boundary- Value Problems for the
Mises
Flow Equations as Variational Inequalities
...................479
82.1
Reformulation of the
Mises
Flow Equations in Terms of
Dissipation
479
82.2
The Global Variational Inequality
482
82.3
Alternative Formulation of the Global Variational
Inequality When Hardening is Described by a Defect Energy
483
PART
XIV.
SMALL DEFORMATION, ISOTROPIC PLASTICITY
BASED ON THE PRINCIPLE OF VIRTUAL POWER
485
83
Introduction
.......................................487
84
Conventional Theory Based on the Principle of Virtual Power
......489
84.1
General Principle of Virtual Power
489
84.2
Principle of Virtual Power Based on the Codirectionality
Constraint
493
84.2.1
Genera] Principle Based on Codirectionality
493
84.2.2
Streamlined Principle Based on Codirectionality
495
84.3
Virtual External Forces Associated with Dislocation Flow
496
84.4
Free-Energy Imbalance
497
84.5
Discussion of the Virtual-Power Formulation
498
85
Basic Constitutive Theory
..............................499
86
Material Stability and Its Relation to Maximum Dissipation
.......501
PART XV. STRAIN GRADIENT PLASTICITY BASED ON THE
PRINCIPLE OF VIRTUAL POWER
505
87
Introduction
.......................................507
88
Kinematics
........................................509
88.1
Characterization of the Burgers Vector
509
88.2
Irrotational Plastic Flow
511
89
The Gradient Theory of Aifantis
.........................512
89.1
The Virtual-Power Principle of Fleck and Hutchinson
512
89.2
Free-Energy Imbalance
515
89.3
Constitutive Equations
516
89.4
Flow Rules
517
89.5
Microscopically Simple Boundary Conditions
518
89.6
Variational Formulation of the Flow Rule
519
89.7
Plastic Free-Energy Balance
520
89.8
Spatial Oscillations. Shear Bands
521
89.8.1
Oscillations
521
89.8.2
Single Shear Bands and Periodic Arrays of Shear Bands
521
xv¡
Contents
90
The Gradient Theory of Gurtin and Anand
..................524
90.1
Third-Order Tensors
524
90.2
Virtual-Power Formulation: Macroscopic and
Microscopic Force Balances
525
90.3
Free-Energy Imbalance
528
90.4
Energetic Constitutive Equations
528
90.5
Dissipative Constitutive Equations
530
90.6
Flow Rule
532
90.7
Microscopically Simple Boundary Conditions
533
90.8
Variational Formulation of the Flow Rule
534
90.9
Plastic Free-Energy Balance. Flow-Induced Strengthening
535
90.10
Rate-Independent Theory
536
PART
XVI.
LARGE-DEFORMATION THEORY OF ISOTROPIC
PLASTIC SOLIDS
539
91
Kinematics
........................................541
91.1
The
Kröner
Decomposition
541
91.2
Digression: Single Crystals
543
91.3
Elastic and Plastic Stretching and Spin. Plastic Incompressibility
543
91.4
Elastic and Plastic Polar Decompositions
544
91.5
Change in Frame Revisited in View of the
Kröner
Decomposition
546
92
Virtual-Power Formulation of the Standard and Microscopic
Force Balances
.....................................548
92.1
Internal and External Expenditures of Power
548
92.2
Principle of Virtual Power
549
92.2.1
Consequences of Frame-Indifference
550
92.2.2
Macroscopic Force Balance
551
92.2.3
Microscopic Force Balance
551
93
Free-Energy Imbalance
...............................553
93.1
Free-Energy Imbalance Expressed in Terms of the
Cauchy Stress
553
94
Two New Stresses
...................................555
94.1
The Second
Piola
Elastic-Stress Te
555
94.2
The
Mandel
Stress Me
556
95
Constitutive Theory
..................................557
95.1
General Separable Constitutive Theory
557
95.2
Structural Frame-Indifference and the Characterization
of Polycrystalline Materials Without Texture
559
95.3
Interaction of Elasticity and Plastic Flow
562
95.4
Consequences of Rate-Independence
563
95.5
Derivation of the
Mises
Flow Equations Based on
Maximum-Dissipation
564
96
Summary of the Basic Equations. Remarks
..................566
97
Plastic Irrotationality: The Condition
W
= 0.................567
Contents
98
Yield Surface. Yield Function. Consistency Condition
...........569
99
ID^I in Terms of
É
and Me
............................. 571
99.1
Some Important Identities
571
99.2
Conditions that Describe Loading and Unloading
571
99.3
The Inverted Flow Rule
574
99.4
Equivalent Formulation of the Constitutive Equations
and Plastic
Mises
Flow Equations Based on the Inverted
Flow Rule
574
100
Evolution Equation for the Second
Piola
Stress
............... 576
101
Rate-Dependent Plastic Materials
........................ 579
101.1
Rate-Dependent Flow Rule
579
101.2
Inversion of the Rate-Dependent Flow Rule
579
101.3
Summary of the Complete Constitutive Theory
580
PART
XVII.
THEORY OF SINGLE CRYSTALS UNDERGOING
SMALL DEFORMATIONS
583
102.1
Introduction
583
102
Basic Single-Crystal Kinematics
.......................... 586
103
The Burgers Vector and the Flow of Screw and Edge Dislocations
... 588
103.1
Decomposition of the Burgers Tensor
G
into
Distributions of Edge and Screw Dislocations
588
103.2
Dislocation Balances
590
103.3
The Tangential Gradient Va on the Slip Plane
ГГ
590
104
Conventional Theory of Single-Crystals
..................... 593
104.1
Virtual-Power Formulation of the Standard and
Microscopic Force Balances
593
104.2
Free-Energy Imbalance
596
104.3
General Separable Constitutive Theory
596
104.4
Linear Elastic Stress-Strain Law
597
104.5
Constitutive Equations for Flow with Simple Rate-Dependence
597
104.6
Power-Law Rate Dependence
601
104.7
Self-Hardening, Latent-Hardening
601
104.8
Summary of the Constitutive Theory
602
105
Single-Crystal Plasticity at Small Length-Scales: A
Small-Deformation Gradient Theory
...................... 604
105.1
Virtual-Power Formulation of the Standard and
Microscopic Force Balances of the Gradient Theory
604
105.2
Free-Energy Imbalance
607
105.3
Energetic Constitutive Equations. Peach-Koehler Forces
607
105.4
Constitutive Equations that Account for Dissipation
609
105.5
Viscoplastic Flow Rule
612
105.6
Microscopically Simple Boundary Conditions
615
105.7
Variational Formulation of the Flow Rule
616
105.8
Plastic Free-Energy Balance
617
105.9
Some Remarks
618
xv¡¡¡
Contents
PART
XVIII.
SINGLE CRYSTALS UNDERGOING LARGE
DEFORMATIONS
621
106
Basic Single-Crystal Kinematics
.......................... 623
107
The Burgers Vector and the Flow of Screw and Edge Dislocations
... 626
107.1
Transformation of Vector Area Measures Between the
Reference, Observed, and Lattice Spaces
626
107.2
Characterization of the Burgers Vector
627
107.3
The Plastically Convected Rate of
G
629
107.4
Densities of Screw and Edge Dislocations
631
107.5
Comparison of Small- and Large-Deformation Results
Concerning Dislocation Densities
633
108
Virtual-Power Formulation of the Standard and Microscopic
Force Balances
..................................... 634
108.1
Internal and External Expenditures of Power
634
108.2
Consequences of Frame-Indifference
636
108.3
Macroscopic and Microscopic Force Balances
637
109
Free-Energy Imbalance
............................... 639
110
Conventional Theory
................................ 641
110.1
Constitutive Relations
641
110.2
Simplified Constitutive Theory
643
110.3
Summary of Basic Equations
644
111 Taylor s Model of Polycrystal
........................... 646
111.1
Kinematics of a Taylor Polycrystal
646
111
.2
Principle of Virtual Power
648
111.3
Free-Energy Imbalance
651
111.4
Constitutive Relations
651
112
Single-Crystal Plasticity at Small Length Scales: A
Large-Deformation Gradient Theory
...................... 653
112.1
Energetic Constitutive Equations. Peach-Koehler Forces
653
112.2
Dissipative Constitutive Equations that Account for
Slip-Rate Gradients
654
112.3
Viscoplastic Flow Rule
656
112.4
Microscopically Simple Boundary Conditions
657
112.5
Variational Formulation
658
112.6
Plastic Free-Energy Balance
659
112.7
Some Remarks
660
113 Isotropie
Functions
.................................. 665
113.1 Isotropie
Scalar Functions
666
113.2 Isotropie
Tensor Functions
666
113.3 Isotropie
Linear Tensor Functions
668
114
The Exponential of a Tensor
References
Index
669
683
|
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| author | Gurtin, Morton E. Fried, Eliot Anand, Lallit |
| author_facet | Gurtin, Morton E. Fried, Eliot Anand, Lallit |
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| dewey-sort | 3531 |
| dewey-tens | 530 - Physics |
| discipline | Physik |
| edition | 1. publ. |
| format | Book |
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| id | DE-604.BV035773560 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T22:04:13Z |
| institution | BVB |
| isbn | 9780521405980 |
| language | English |
| lccn | 2009010668 |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-018633204 |
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| owner_facet | DE-703 DE-20 DE-355 DE-BY-UBR DE-29T |
| physical | XXI, 694 S. graph. Darst. |
| publishDate | 2010 |
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| publisher | Cambridge Univ. Press |
| record_format | marc |
| spelling | Gurtin, Morton E. Verfasser aut The mechanics and thermodynamics of continua Morton E. Gurtin ; Eliot Fried ; Lallit Anand 1. publ. Cambridge [u.a.] Cambridge Univ. Press 2010 XXI, 694 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Continuum mechanics - Mathematics Thermodynamics - Mathematics Mathematik Continuum mechanics Mathematics Thermodynamics Mathematics Kontinuumsmechanik (DE-588)4032296-8 gnd rswk-swf Thermodynamisches System (DE-588)4185141-9 gnd rswk-swf Thermodynamik (DE-588)4059827-5 gnd rswk-swf Kontinuumsmechanik (DE-588)4032296-8 s DE-604 Thermodynamisches System (DE-588)4185141-9 s Thermodynamik (DE-588)4059827-5 s Fried, Eliot Verfasser aut Anand, Lallit Verfasser aut Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018633204&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Gurtin, Morton E. Fried, Eliot Anand, Lallit The mechanics and thermodynamics of continua Continuum mechanics - Mathematics Thermodynamics - Mathematics Mathematik Continuum mechanics Mathematics Thermodynamics Mathematics Kontinuumsmechanik (DE-588)4032296-8 gnd Thermodynamisches System (DE-588)4185141-9 gnd Thermodynamik (DE-588)4059827-5 gnd |
| subject_GND | (DE-588)4032296-8 (DE-588)4185141-9 (DE-588)4059827-5 |
| title | The mechanics and thermodynamics of continua |
| title_auth | The mechanics and thermodynamics of continua |
| title_exact_search | The mechanics and thermodynamics of continua |
| title_full | The mechanics and thermodynamics of continua Morton E. Gurtin ; Eliot Fried ; Lallit Anand |
| title_fullStr | The mechanics and thermodynamics of continua Morton E. Gurtin ; Eliot Fried ; Lallit Anand |
| title_full_unstemmed | The mechanics and thermodynamics of continua Morton E. Gurtin ; Eliot Fried ; Lallit Anand |
| title_short | The mechanics and thermodynamics of continua |
| title_sort | the mechanics and thermodynamics of continua |
| topic | Continuum mechanics - Mathematics Thermodynamics - Mathematics Mathematik Continuum mechanics Mathematics Thermodynamics Mathematics Kontinuumsmechanik (DE-588)4032296-8 gnd Thermodynamisches System (DE-588)4185141-9 gnd Thermodynamik (DE-588)4059827-5 gnd |
| topic_facet | Continuum mechanics - Mathematics Thermodynamics - Mathematics Mathematik Continuum mechanics Mathematics Thermodynamics Mathematics Kontinuumsmechanik Thermodynamisches System Thermodynamik |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018633204&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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