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Advanced Texts in Physics
This program of advanced texts covers a broad spectrum of topics which are of current and emerging interest in physics. Each book provides a comprehensive and yet accessible introduction to a field at the forefront of modern research. As such, these texts are intended for senior undergraduate and graduate students at the MS and PhD level; however, research scientists seeking an introduction to particular areas of physics will also benefit from the titles in this collection.
Peter Haupt
Continuum Mechanics and Theory of Materials
Translated from German by Joan A. Kurth
With 73 Figures
Springer
Professor Dr. Peter Haupt Institute of Mechanics University of Kassel Monchebergstrasse 7 34109 Kassel Germany
Translator Joan A. Kurth Silberbornweg 5 34346 Hannoversch Munden Germany
ISSN 1439-2674
ISBN 978-3-662-04111-6 ISBN 978-3-662-04109-3 (eBook) DOI 10.1007/978-3-662-04109-3
Library of Congress Cataloging-in-Publication Data applied for. Die Deutsche Bibliothek - CIP-Einheitsaufnahme Haupt, Peter: Continuum mechanics and theory of materials I Peter Haupt. Trans!. from German by Joan A. Kurth. - Berlin; Heidelberg; New York; Barcelona; Hong Kong; London; Milan; Paris; Singapore; Tokyo: Springer, 2000 (Advanced texts in physics)
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© Springer-Verlag Berlin Heidelberg 2000
Originally published by Springer-Verlag Berlin Heidelberg New York in 2000.
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Preface
This exposition of the theory of materials has its origins in the lectures I gave at the universities of Darmstadt and Kassel from 1978 onwards. Research projects carried out during the same period have been the source of extensive refinements to the subject-matter. The reason for adding yet another book to the existing wealth of volumes dealing with continuum mechanics was my desire to describe the phenomenological theory of material properties from my own point of view. As a result, it is without doubt a subjectively inspired and incomplete work. This particularly applies to the selection of quotations from the literature.
The text has been influenced and enhanced by the numerous discussions I had the privilege of holding with students and experts alike. I should like to thank them all sincerely for their contributions and encouragement.
My special thanks go to my academic teachers Rudolf Trostel 1 and Hubertus 1. Weinitschke,2 whose stimulating lectures convinced me at the time that continuum mechanics is a field of science worth pursuing.
I greatly appreciate the long and amicable collaboration with Babis Tsakmakis and Manfred Korzen, during which a number of indispensable fundamental aspects emerged.
Valuable inspiration regarding the development of the thermomechanical theory of materials was given by Roman Bonn, Markus Horz, Marc Kamlah and Alexander Lion. It was Lion's skill that provided the link between the theoretical modelling and experimental investigation of material behaviour. The fact that he received active support from the Institute of Mechanics'
1 TROSTEL [1966, 1990, 1993]. 2 WEINITSCHKE [1968].
VIII Preface
own laboratory is mainly due to Lothar Schreiber, who supplied efficient measurement techniques and the optimisation procedures required for the experimental identification of material parameters. Stefan Hartmann and Georg Liihrs investigated the properties of constitutive models and related field equations in terms of their numerical solutions.
I am grateful to Kolumban Hutter for his critical perusal of an earlier version of the manuscript. This led to many improvements and encouraged me to submit this work for publication.
My young colleagues, who checked the final version, proposed a lot of beneficial rectifications. I duly thank Stefan Hartmann, Dirk Helm, Thomas Kersten, Alexander Lion, Anton Matzenmiller, Lothar Schreiber and Konstantin Sedlan.
In particular, I would like to thank Mrs Joan Kurth for undertaking the laborious task of translating the work into the "language of modern science", which she did with considerable sensitivity and forbearance.
Finally, I wish to express my thanks to Springer-Verlag for agreeing to publish the book and for their kind cooperation.
Kassel, August 1999 Peter Haupt
Contents
Introduction
1 Kinematics 1. 1 Material Bodies 1. 2 Material and Spatial Representation 1. 3 Deformation Gradient 1. 4 Strain Tensors 1. 5 Convective Coordinates 1. 6 Velocity Gradient 1. 7 Strain Rate Tensors 1. 8 Strain Rates in Convective Coordinates 1. 9 Geometric Linearisation 1. 10 Incompatible Configurations
1. 10. 1 Euclidean Space 1. 10. 2 Non-Euclidean Spaces 1. 10. 3 Conditions of Compatibility
2 Balance Relations of Mechanics 2. 1 Preliminary Remarks 2.2 Mass
2. 2. 1 Balance of Mass: Global Form 2. 2. 2 Balance of Mass: Local Form
2. 3 Linear Momentum and Rotational Momentum 2. 3. 1 Balance of Linear Momentum
and Rotational Momentum: Global Formulation 2. 3. 2 Stress Tensors 2. 3. 3 Stress Tensors in Convective Coordinates 2. 3. 4 Local Formulation of the Balance
of Linear Momentum and Rotational Momentum 2. 3. 5 Initial and Boundary Conditions
1
7 7
19 23 32 38 42 46 50 53 57 58 63 64
75 75 78 78 79 84
84 90 95
95 101
x
2. 4 Conclusions from the Balance Equations of Mechanics 2. 4. 1 Balance of Mechanical Energy 2.4. 2 The Principle of d'Alembert 2. 4. 3 Principle of Virtual Work 2. 4. 4 Incremental Form of the Principle of d'Alembert
Contents
104 105 109 114 115
3 Balance Relations of Thermodynamics 3.1 Preliminary Remarks
119 119 120 125 130 132 132
3.2 Energy 3. 3 Temperature and Entropy 3. 4 Initial and Boundary Conditions 3. 5 Balance Relations for Open Systems
3. 5. 1 Transport Theorem 3. 5. 2 Balance of Linear Momentum
for Systems with Time-Dependent Mass 3. 5. 3 Balance Relations: Conservation Laws 3. 5. 4 Discontinuity Surfaces and Jump Conditions 3.5. 5 Multi-Component Systems (Mixtures)
3. 6 Summary: Basic Relations of Thermomechanics
4 Objectivity 4. 1 Frames of Reference 4. 2. Affine Spaces 4. 3 Change of Frame: Passive Interpretation 4. 4 Change of Frame: Active Interpretation 4. 5 Objective Quantities 4. 6 Observer-Invariant Relations
5 Classical Theories of Continuum Mechanics 5. 1 Introduction 5. 2 Elastic Fluid 5. 3 Linear-Viscous Fluid 5. 4 Linear-Elastic Solid 5. 5 Linear-Viscoelastic Solid 5. 6 Perfectly Plastic Solid 5. 7 Plasticity with Hardening 5. 8 Visco plasticity with Elastic Range 5. 9 Remarks on the Classical Theories
135 137 140 144 153
155 155 156 159 162 164 171
177 177 178 182 185 188 207 211 224 229
6 Experimental Observation and Mathematical Modelling 231 6. 1 General Aspects 231 6. 2 Information from Experiments 235
6. 2. 1 Material Properties of Steel XCrNi 18.9 235 6. 2. 2 Material Properties of Carbon-Black-Filled Elastomers 243
6. 3 Four Categories of Material Behaviour 249 6. 4 Four Theories of Material Behaviour 251 6. 5 Contribution of the Classical Theories 253
Contents XI
7 General Theory of Mechanical Material Behaviour 255 7. 1 General Principles 255 7. 2 Constitutive Equations 259
7. 2. 1 Simple Materials 259 7. 2. 2 Reduced Forms of the General Constitutive Equation 263 7. 2. 3 Simple Examples of Material Objectivity 268 7. 2. 4 Frame-Indifference and Observer-In variance 269
7. 3 Properties of Material Symmetry 273 7. 3. 1 The Concept of the Symmetry Group 273 7. 3. 2 Classification of Simple Materials into Fluids and Solids 278
7. 4 Kinematic Conditions of Internal Constraint 285 7. 4. 1 General Theory 285 7. 4. 2 Special Conditions of Internal Constraint 288
7. 5 Formulation of Material Models 291 7. 5. 1 General Aspects 291 7. 5. 2 Representation by Means of Functionals 292 7. 5. 3 Representation by Means of Internal Variables 293 7. 5. 4 Comparison 295
8 Dual Variables 297 8. 1 Tensor-Valued Evolution Equations 297
8. 1. 1 Introduction 297 8. 1. 2 Objective Time Derivatives of Objective Tensors 8. 1. 3 Example: Maxwell Fluid 8. 1. 4 Example: Rigid-Plastic Solid with Hardening
8. 2 The Concept of Dual Variables 8. 2. 1 Motivation 8. 2. 2 Strain and Stress Tensors (Summary) 8. 2. 3 Dual Variables and Derivatives
299 302 305 309 309 311 314
9 Elasticity 325 9. 1 Elasticity and Hyperelasticity 325 9. 2 Isotropic Elastic Bodies 332
9. 2. 1 General Constitutive Equation for Elastic Fluids and Solids 332
9. 2. 2 Isotropic Hyperelastic Bodies 338 9. 2. 3 Incompressible Isotropic Elastic Materials 343 9. 2. 4 Constitutive Equations of Isotropic Elasticity (Examples) 345
9. 3 Anisotropic Hyperelastic Solids 356 9. 3. 1 Approximation of the General Constitutive Equation 356 9. 3. 2 General Representation of the Strain Energy Function 359 9. 3. 2 Physical Linearisation 368
10 Viscoelasticity 375 10. 1 Representation by Means of Functionals 375
10. 1. 1 Rate-Dependent Functionals with Fading Memory Properties 376
10. 1. 2 Continuity Properties and Approximations 388
XII
10.2 Representation by Means of Internal Variables 10. 2. 1 General Concept 10. 2. 2 Internal Variables of the Strain Type 10. 2. 3 A General Model of Finite Viscoelasticity
11 Plasticity 11. 1 Rate-Independent Functionals 11.2 Representation by Means of Internal Variables 11. 3 Elastoplasticity
11. 3. 1 Preliminary Remarks 11. 3. 2 Stress-Free Intermediate Configuration 11. 3. 3 Isotropic Elasticity 11. 3. 4 Yield Function and Evolution Equations 11. 3. 5 Consistency Condition
12 Viscoplasticity 12. 1 Preliminary Remarks 12. 2 Visco plasticity with Elastic Domain
12. 2. 1 A General Constitutive Model 12. 2. 2 Application of the Intermediate Configuration
12. 3 Plasticity as a Limit Case of Viscoplasticity 12. 3. 1 The Differential Equation of the Yield Function 12. 3. 2 Relaxation Property 12. 3. 3 Slow Deformation Processes 12. 3. 4 Elastoplasticity and Arclength Representation
12. 4 A Concept for General Visco plasticity 12. 4. 1 Motivation 12. 4. 2 Equilibrium Stress and Overstress 12. 4. 3 An Example of General Viscoplasticity 12. 4. 4 Conclusions Regarding the Modelling
of Mechanical Material Behaviour
13 Constitutive Models in Thermomechanics 13. 1 Thermomechanical Consistency 13. 2 Thermoelasticity
13. 2. 1 General Theory 13. 2. 2 Thermoelastic Fluid 13. 2. 3 Linear-Thermoelastic Solids
13. 3 Thermoviscoelasticity 13. 3. 1 General Concept 13. 3. 2 Thermoelasticity as a Limit Case
of Thermoviscoelasticity 13. 3. 3 Internal Variables of Strain Type 13. 3. 4 Incorporation of Anisotropic Elasticity Properties
13. 4 Thermoviscoplasticity with Elastic Domain 13. 4. 1 General Concept 13.4. 2 Application of the Intermediate Configuration 13. 4. 3 Thermoplasticity as a Limit Case
of Thermoviscoplasticity
Contents
397 397 404 411
413 413 422 428 428 432 437 438 441
453 453 455 455 458 462 462 467 469 475 477 477 478 479
485
487 487 492 492 498 505 508 508
515 519 523 523 523 528
532
Contents
13. 5 General Thermoviscoplasticity 13. 5. 1 Small Deformations 13. 5. 2 Finite Deformations 13. 5. 3 Conclusion
13. 6 Anisotropic Material Properties 13. 6. 1 Motivation 13. 6. 2 Axes of Elastic Anisotropy 13. 6. 3 Application in Thermoviscoplasticity 13. 6. 4 Constant Axes of Elastic Anisotropy 13. 6. 5 Closing Remark
References
Index
XIII
541 542 545 548 549 549 550 552 558 560
561
575
Notation
a,A,a,a,A, ...
a,A, .. .
a,A, .. .
ak , ak , bk1 , bk1 , bk1 , bk1 ' C k1 , ...
a+b
aa
o a·b, akbk , akbk , ...
axb
a is) b , akb1 , akb1, akb1
A+B aA
o AT
trA,Akk,A/, ...
detA
AD
AB, AklB1m' AklB1m , ...
Av, Aklv1 , Ak1 vI' Ak1vl , ...
A· B, AklBkl , Akl Bkl , Ak1Bkl , ...
Indices, scalars, constants
Vectors
Tensors
Components, matrices
Addition of vectors
Product of scalar and vector
Zero vector
Scalar product of vectors
Vector product of vectors
Tensor product of vectors
Addition of tensors
Product of scalar and tensor
Zero tensor
Transpose of A
Trace of A
Determinant of A
Deviator of A
Product (composition) of tensors
Product of vector and tensor
Scalar product of tensors
XVI
IN
IR IE
V 'lI'
o
• o
•
f:A----B x .......... y = f(x)
1 Cf. BRONSTEIN et al. [1997], p. 50.
Identity tensor
Set of natural numbers
Set of real numbers
Euclidean space
Vector space
Tangent space
Set of second order tensors
Set of symmetric tensors
Set of antisymmetric tensors
Set of unimodular tensors
Set of orthogonal tensors
Notation
Set of orthogonal tensors with positive determinant
End of a definition
End of a natural law, axiom or principle
End of a theorem
End of a proof
lim f(x) = A ;= 0 1
x ...... o xn
lim f(x) = 0 x ...... o xn
Map from set A into set B
Transformation from x E A to y E B