Continuum Mechanics and Theory of Materials

Springer-Verlag Berlin Heidelberg GmbH

Advanced Texts in Physics

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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.

Softcover reprint of the hardcover I st edition 2000

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To Vivien, Jonas and Viola

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