CONSTITUTIVE MODELING OF ENGINEERING MATERIALS - THEORY...

CONSTITUTIVE MODELING OF ENGINEERING

MATERIALS - THEORY AND COMPUTATION

The Primer

by

Kenneth Runesson

Lecture Notes, Dept. of Applied Mechanics,

Chalmers University of Technology, Goteborg

Preface

There seems to be an ever increasing demand in engineering practice for more realistic

models as applied to metals as well as composites, ceramics, polymers and geological

materials (such as soil and rock). Consequently, a vast amount of literature is available

on the subject of “nonlinear constitutive modeling”, with strong emphasis on plastic-

ity and damage. Such modeling efforts are parallelled by the development of numerical

algorithms for use in Finite Element environment. For example, implicit (rather than ex-

plicit) integration techniques for plasticity problems are now predominant in commercial

FE-codes.

I am indebted to a great number of people who have contributed to the present volume:

Mr. M. Enelund, Mr. L. Jacobsson, Mr. M. Johansson, Mr. L. Mahler and Mr. T. Svedberg,

who are all graduate students at Chalmers Solid Mechanics, have read (parts of) the

manuscript and struggled with the numerical examples. Mr. T. Ernby prepared some of

the difficult figures. Ms. C. Johnsson, who is a graduate student in ancient Greek history

at Goteborg University, quickly became an expert in handling equations in LATEX. The

contribution of each one is gratefully acknowledged.

Goteborg in March 1996.

Kenneth Runesson

2nd revised edition:

I am grateful to Mr. Lars Jacobson and Mr. Magnus Johansson (in particular) for their

help in revising parts of the manuscript.


Kenneth Runesson

3rd revised edition:

Ms. EvaMari Runesson, who is a student in English at the University of Gothenburg (and

also happens to be my daughter) did an excellent job in mastering LATEXfor this edition.

iv


Kenneth Runesson

4th revised edition:

Mr. Lars Jacobsson and Ms. EvaMari Runesson were of great help in typing the manuscript.

Goteborg in January 1999.

Kenneth Runesson


Some small changes were made to improve the manuscript.

Goteborg in January 2000.

Kenneth Runesson


Ms. Annicka Karlsson was of great help in revising the manuscript, mainly concerning the

notation.

Goteborg in January 2002

Kenneth Runesson


The help by Mr. Mikkel Grymer in revising the manuscript is greatly acknowledged.

Goteborg in March 2005

Kenneth Runesson

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Contents

1 CHARACTERISTICS OF ENGINEERING MATERIALS AND CON-

STITUTIVE MODELING 1

1.1 General remarks on constitutive modeling . . . . . . . . . . . . . . . . . . 1

1.1.1 Concept of a constitutive model . . . . . . . . . . . . . . . . . . . . 1

1.1.2 The role of constitutive modeling . . . . . . . . . . . . . . . . . . . 3

1.1.3 General constraints on constitutive models . . . . . . . . . . . . . . 4

1.1.4 Approaches to constitutive modeling . . . . . . . . . . . . . . . . . 5

1.2 Modeling of material failure — Fracture . . . . . . . . . . . . . . . . . . . 7

1.2.1 Continuum damage mechanics . . . . . . . . . . . . . . . . . . . . . 7

1.2.2 Fracture mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.3 Common experimental test conditions . . . . . . . . . . . . . . . . . . . . . 8

1.4 Typical behavior of metals and alloys . . . . . . . . . . . . . . . . . . . . . 12

1.4.1 Plastic yielding — Hardening and ductile fracture . . . . . . . . . . 12

1.4.2 Constant loading — Creep and relaxation . . . . . . . . . . . . . . 13

1.4.3 Time-dependent loading — Rate effect and damping . . . . . . . . 14

1.4.4 Cyclic loading and High-Cycle-Fatigue (HCF) . . . . . . . . . . . . 15

1.4.5 Cyclic loading and Low-Cycle-Fatigue (LCF) . . . . . . . . . . . . . 16

1.4.6 Creep-fatigue and Relaxation-fatigue . . . . . . . . . . . . . . . . . 20

1.5 Typical behavior of ceramics and cementitious composites . . . . . . . . . 21

1.5.1 Monotonic loading – Semi-brittle fracture . . . . . . . . . . . . . . . 21

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1.5.2 Cyclic loading and fatigue . . . . . . . . . . . . . . . . . . . . . . . 22

1.5.3 Creep and relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . 22

1.6 Typical behavior of granular materials . . . . . . . . . . . . . . . . . . . . 22

1.6.1 Monotonic loading – Basic features . . . . . . . . . . . . . . . . . . 22

1.6.2 Constant loading – Consolidation . . . . . . . . . . . . . . . . . . . 23

1.6.3 Constant loading – Creep and relaxation . . . . . . . . . . . . . . . 23

2 THERMODYNAMICS — A BRIEF SUMMARY 25

2.1 Free energy and constitutive relations . . . . . . . . . . . . . . . . . . . . . 25

2.1.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.1.2 Stress-strain response relation . . . . . . . . . . . . . . . . . . . . . 26

2.1.3 Material classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3 VISCOELASTICITY 29

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.2 Prototype model: The Maxwell rheological model . . . . . . . . . . . . . . 30

3.2.1 Thermodynamic basis — Constitutive relation . . . . . . . . . . . . 30

3.2.2 Prescribed constant stress (pure creep) . . . . . . . . . . . . . . . . 32

3.2.3 Prescribed constant strain (pure relaxation) . . . . . . . . . . . . . 32

3.3 Linear viscoelasticity — Constitutive modeling . . . . . . . . . . . . . . . . 33

3.3.1 General characteristics . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.3.2 Laplace-Carson transform . . . . . . . . . . . . . . . . . . . . . . . 35

3.3.3 Linear Standard Model (Generalized Maxwell Model) . . . . . . . . 37

3.3.4 Backward Euler method for linear standard model . . . . . . . . . . 40

3.4 Linear viscoelasticity — Structural analysis . . . . . . . . . . . . . . . . . . 42

3.4.1 Structural behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.4.2 Solution strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.4.3 Analysis of truss — Elastic analogy . . . . . . . . . . . . . . . . . . 44

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3.4.4 Analysis of truss — numerical integration . . . . . . . . . . . . . . 47

3.4.5 Analysis of beam cross-section — Elastic analogy . . . . . . . . . . 49

3.4.6 Analysis of double-symmetric beam cross-section — Numerical in-

tegration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.5 Nonlinear viscoelasticity — Constitutive modeling . . . . . . . . . . . . . . 53

3.5.1 General characteristics . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.5.2 Norton creep law . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.5.3 Backward Euler method for the Norton creep law . . . . . . . . . . 56

3.6 Nonlinear viscoelasticity — structural analysis . . . . . . . . . . . . . . . . 58

3.6.1 Structural behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.6.2 Analysis of truss . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.6.3 Analysis of beam cross-section — Stationary creep . . . . . . . . . . 60

3.6.4 Analysis of double-symmetric beam cross-section — Numerical in-

tegration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.6.5 Analysis of single-symmetric beam cross-section — Numerical inte-

gration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

3.7 Viscous damping and dynamic behavior . . . . . . . . . . . . . . . . . . . . 67

3.7.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

3.7.2 Forced vibration of discrete system . . . . . . . . . . . . . . . . . . 68

3.7.3 Energy dissipation . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

3.7.4 Evaluation of damping for the linear standard model . . . . . . . . 71

3.8 Appendix : Laplace - Carson transform . . . . . . . . . . . . . . . . . . . . 75

4 PLASTICITY 77

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.2 Prototype rheological model for perfectly plastic behavior . . . . . . . . . . 79

4.2.1 Thermodynamic basis — Yield criterion . . . . . . . . . . . . . . . 79

4.2.2 Plastic flow rule and elastic-plastic tangent relation . . . . . . . . . 80

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4.2.3 Dissipation of energy . . . . . . . . . . . . . . . . . . . . . . . . . . 82

4.3 Prototype model for hardening plastic behavior . . . . . . . . . . . . . . . 82


4.3.2 Plastic flow rule and elastic-plastic tangent relation . . . . . . . . . 83


4.4 Model for cyclic loading — Mixed isotropic and kinematic hardening . . . 86


4.4.2 Associative flow and hardening rules — Linear hardening . . . . . . 87

4.4.3 Characteristic response for linear hardening . . . . . . . . . . . . . 88

4.4.4 Associative flow and nonassociative hardening rules — Nonlinear

hardening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

4.4.5 Characteristic response for nonlinear hardening . . . . . . . . . . . 90

4.4.6 Backward Euler method for integration — Linear hardening . . . . 96

4.5 Structural analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

4.5.1 Structural behavior — Limit load analysis . . . . . . . . . . . . . . 101

4.5.2 Analysis of truss — Numerical integration . . . . . . . . . . . . . . 101

4.5.3 Analysis of double-symmetric beam cross-section . . . . . . . . . . 103

4.5.4 Analysis of single-symmetric beam cross-section — Numerical inte-

gration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

5 VISCOPLASTICITY 109

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

5.2 Prototype rheological model for perfectly viscoplastic behavior . . . . . . . 110

5.2.1 Thermodynamic basis — Quasistatic yield criterion . . . . . . . . . 110

5.2.2 Viscoplastic flow rule — Perzyna’s formulation . . . . . . . . . . . . 111

5.2.3 Bingham model — Perzyna’s formulation . . . . . . . . . . . . . . . 113

5.2.4 Norton model (creep law) — Perfect viscoplasticity . . . . . . . . . 114

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5.2.5 Limit behavior — Viscoplastic regularization of rate-independent

plasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

5.3 Prototype rheological model for hardening viscoplasticity . . . . . . . . . . 115

5.3.1 Thermodynamic basis — Quasistatic yield criterion . . . . . . . . . 115

5.3.2 Viscoplastic flow and hardening rules — Perzyna’s formulation . . . 116

5.3.3 Bingham model — Perzyna’s formulation . . . . . . . . . . . . . . . 117

5.3.4 Viscoplastic flow and hardening rules — Duvaut-Lions’ formulation 118

5.3.5 Comparison of Perzyna’s and Duvaut-Lions’ formulations . . . . . . 119

5.3.6 Bingham model — Duvaut-Lions’ formulation . . . . . . . . . . . . 120

5.4 Model for cyclic loading — Mixed isotropic and kinematic hardening . . . 121

5.4.1 Constitutive relations for linear hardening — Perzyna’s formulation 121

5.4.2 Backward Euler method for linear hardening — Perzyna’s formulation121

5.5 Structural analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

6 DAMAGE AND FRACTURE THEORY 125

6.1 Introduction to the modeling of damage . . . . . . . . . . . . . . . . . . . 125

6.1.1 Concept of damage . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

6.1.2 Physical nature of damage for different materials . . . . . . . . . . 127

6.1.3 The concepts of effective stress and strain equivalence . . . . . . . . 128

6.2 Prototype model of damage coupled to elasticity . . . . . . . . . . . . . . . 130

6.2.1 Thermodynamics — Damage criterion . . . . . . . . . . . . . . . . 130

6.2.2 Damage law and tangent relations . . . . . . . . . . . . . . . . . . . 133

6.3 Experimental measurement of damage . . . . . . . . . . . . . . . . . . . . 135

7 DAMAGE COUPLED TO PLASTICITY 137

7.1 Prototype model for damage coupled to perfect plasticity . . . . . . . . . . 137

7.1.1 Thermodynamic basis — Yield and damage criterion . . . . . . . . 137

7.1.2 Plastic flow rule and damage law — Constitutive relations . . . . . 139

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7.1.3 Dissipation inequality . . . . . . . . . . . . . . . . . . . . . . . . . . 143

7.1.4 Dissipation of mechanical energy . . . . . . . . . . . . . . . . . . . 144

7.2 Prototype model for damage coupled to hardening plasticity . . . . . . . . 148

7.2.1 Thermodynamics — Yield and damage criterion . . . . . . . . . . . 148

7.2.2 Dissipation rules — Constitutive relations . . . . . . . . . . . . . . 148


7.3 Model for cyclic loading and fatigue — Mixed linear isotropic and kinematic

hardening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

7.3.1 Constitutive relations for linear hardening . . . . . . . . . . . . . . 151

7.3.2 Backward Euler algorithm for integration — Linear hardening and

uniaxial stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

8 DAMAGE COUPLED TO VISCOPLASTICITY 157

8.1 Prototype model for damage coupled to perfect viscoplasticity . . . . . . . 157

8.1.1 Thermodynamic basis — Quasistatic yield and damage criterion . . 157

8.1.2 Viscoplastic flow rule and damage law — Perzyna’s formulation . . 158

8.1.3 Norton model (creep Law) — Perfect viscoplasticity . . . . . . . . . 159

8.2 Prototype model for damage coupled to hardening viscoplasticity . . . . . 162

8.2.1 Thermodynamic basis — Quasistatic yield and damage criterion . . 162

8.2.2 Viscoplastic flow, hardening and damage rules — Perzyna’s formu-

lation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

8.3 Constitutive modeling of creep failure of metals and alloys . . . . . . . . . 163

8.3.1 Modified damage law for tertiary creep . . . . . . . . . . . . . . . . 163

8.3.2 Typical results for creep at uniaxial stress . . . . . . . . . . . . . . 164

9 FATIGUE — PHENOMENON AND ANALYSIS 167

9.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

9.1.1 Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

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9.1.2 Historical remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

9.1.3 Cyclic stress-strain relation . . . . . . . . . . . . . . . . . . . . . . 169

9.2 Engineering approach to HCF and LCF based on stress-control . . . . . . . 170

9.2.1 Basquin-relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

9.2.2 Variable amplitude loading — Palmgren-Miner rule . . . . . . . . . 177

9.2.3 Multiaxial fatigue criteria based on stress . . . . . . . . . . . . . . . 179

9.3 Engineering approach to LCF based on strain-control . . . . . . . . . . . . 181

9.3.1 Manson-Coffin relation . . . . . . . . . . . . . . . . . . . . . . . . . 181

9.3.2 Combined effects of creep and fatigue . . . . . . . . . . . . . . . . . 183

9.3.3 Multiaxial fatigue criteria based on strain . . . . . . . . . . . . . . 185

9.4 Life prediction strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

9.4.1 Coupled - decoupled approach . . . . . . . . . . . . . . . . . . . . . 185

9.4.2 Life prediction strategy based on the decoupled approach . . . . . . 187

9.5 Damage mechanics approach to LCF . . . . . . . . . . . . . . . . . . . . . 189

9.5.1 Simplified analysis of LCF — Derivation of the Manson-Coffin and

Basquin relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

9.5.2 Rational approach to LCF - Damage coupled to plastic deformation 194

9.6 Damage mechanics approach to CLCF . . . . . . . . . . . . . . . . . . . . 197

9.6.1 Simplified analysis of CLCF . . . . . . . . . . . . . . . . . . . . . . 197

9.6.2 Damage coupled to viscoplastic deformation . . . . . . . . . . . . . 203

9.7 Fracture mechanics approach to fatigue . . . . . . . . . . . . . . . . . . . . 203

9.7.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

9.7.2 Paris’ law for fatigue crack growth . . . . . . . . . . . . . . . . . . 204

9.7.3 Variable amplitude loading . . . . . . . . . . . . . . . . . . . . . . . 207

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Chapter 1

CHARACTERISTICS OF

ENGINEERING MATERIALS

AND CONSTITUTIVE

MODELING

In this chapter we give a brief introduction to the particular field within applied solid me-

chanics that deals with the establishment of constitutive models for engineering materials.

Some generally accepted constraints that must be imposed on constitutive models are dis-

cussed. Commonly occurring test conditions for obtaining results towards calibration and

validation are discussed briefly. Finally, the typical material (stress-strain) behavior of

the most important engineering materials (metals and alloys, cementitious composites,

granular materials) under various loading conditions is reviewed.

1.1 General remarks on constitutive modeling

1.1.1 Concept of a constitutive model

Common to all mechanical analysis of engineering materials and their behavior in struc-

tural components is the need for constitutive models that link the states of stress and

strain. From a mathematical viewpoint, the constitutive equations (that define the con-

stitutive model) are complementary equations to the balance and kinematic equations.

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21 CHARACTERISTICS OF ENGINEERING MATERIALS AND

CONSTITUTIVE MODELING

Taken together with the loading and boundary conditions, these are the sufficient, but

not always the necessary, equations in order to formulate a complete boundary value

problem, from which the motion of a given body can be calculated1.

It is clear that constitutive models may be very different for the various materials used in

engineering practice, such as metals and alloys, polymers, fiber composites (with polymer

or metal matrix), concrete and wood. However, to a large extent it is possible to employ

the same principles and concepts (and even the same terminology) in establishing con-

stitutive relations for these different materials, despite the fact that the physics behind

the macroscopical phenomena are entirely different. Indeed the characteristics of an en-

gineering material are determined by its microstructure, all the way down to its atomic

arrangement. Examples of microstructures (on the level below the macroscopic scale)

are shown in Figure 1.1. Crystalline and amorphous materials behave differently, as do

single crystals in comparison to polycrystalline materials. The mechanical properties are

often significantly affected by the temperature and by the loading rate. For example, the

ductility of a metal is reduced at low temperature and high loading rate.

Figure 1.1: Typical microstructure of (a) Steel (perlitic grain structure, eutectoid compo-

sition), (b) Concrete, (c) Wood.

1These are the necessary and sufficient conditions for any hyperstatic (statically indeterminate) struc-

ture, whereas it is not necessary to know the constitutive response to calculate the stresses in an isostatic

(statically determinate) structure.

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1.1 General remarks on constitutive modeling 3

1.1.2 The role of constitutive modeling

It is emphasized that constitutive models are just mathematical simplifications of a quite

complex physical behavior, and there is no such thing as an “exact” model. For example,

it is appropriate to claim that the behavior of steel can be represented by an elastic-

plastic model, but it does not make sense to claim that steel is elastic-plastic! In fact, it

is appropriate to model steel (and any other engineering material) in a number of ways

depending on the purpose and the required precision of the model predictions. Examples

of different purposes of the relevant model are given as follows:

• Structural analysis under working load: Linear elasticity

• Analysis of damped vibrations: Viscoelasticity

• Calculation of limit load: Rigid perfect plasticity

• Accurate calculation of permanent deformation after monotonic and cyclic loading:

Hardening elasto-plasticity

• Analysis of stationary creep and relaxation: Perfect (nonhardening) elasto-viscoplasticity

• Prediction of lifetime in high-cycle-fatigue: Damage coupled to elastic deformations

• Prediction of lifetime in low-cycle-fatigue: Damage coupled to plastic deformations

• Prediction of lifetime in creep and creep-fatigue: Damage coupled to viscoplastic

deformations

• Prediction of stability of a preexisting crack: Linear elasticity (from which singular

stress fields are derived for sharp cracks)

• Prediction of strain localization in shear bands and incipient material failure: Soft-

ening plasticity or damage coupled to plastic deformation

Most of the listed phenomena will be considered in some detail in this text. Clearly, the

task of the engineer is to choose a model that is sufficiently accurate, yet not unnecessarily

complex and computationally expensive. The questions that should be asked in regard

to the choice of a certain model are as follows:

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• Is the model relevant for describing the physical phenomena at hand?

• Does the model produce sufficiently accurate predictions for the given purpose?

• Is it possible to devise and implement a robust numerical algorithm (in a computer

code) to obtain a truly operational model?

1.1.3 General constraints on constitutive models

A list of constraints that must be placed on constitutive relations, that represent the

mechanical behavior of a continuous medium, is given below. Virtually all of these re-

quirements are intuitively obvious, although it is not trivial to express them properly in

mathematical language. Moreover, some constraints are important only in conjunction

with large deformations, say, at the modeling of material forming.

Principle of coordinate invariance

Constitutive relations, as well as other relations between physical entities, should not be

affected by arbitrary coordinate transformations.

This requirement is satisfied if proper tensorial relations are established.

Principle of determinism (or causality)

The stress in a given body is determined entirely by the history of the motion of the body,

i.e. it is not affected by the future events.

This requirement is always satisfied if intrinsically time-dependent relations are estab-

lished with time as (one of) the independent coordinate(s). It may be violated if relations

between Laplace or Fourier transformed variables are set up directly. For example, care

must be taken when internal damping relations (expressing energy dissipation) are pro-

posed in the “frequency domain”, as discussed by Crandall (1970).

Principle of material objectivity (or frame-indifference)

Constitutive relations must not be affected by arbitrary Rigid Body Motion (RBM) that

is superposed on the actual motion.

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1.1 General remarks on constitutive modeling 5

This requirement is most easily satisfied by employing objective tensor fields as the con-

stitutive variables. In particular, it is important to note that the ordinary time derivative

of common variables (stress, strain) is not objective. For example, the time rate of the

(Cauchy) stress tensor is not zero at RBM, even if the material does not “feel” any change

of stress, i.e. the stress components with respect to a corotating coordinate system do

not change. However, the non-zero time rate is merely a consequence of the rotation.

As a consequence, this time rate is not permissible in constitutive relations, at least not

for large material rotation. In small strain theory, which employs linear kinematics, the

requirement of objectivity can be ignored.

Constraints of material symmetry (or spatial covariance)

Response functions are unaffected by certain rotations of the chosen reference configu-

ration due to material symmetry. The most important special case is complete material

isotropy, which means that the response is equal in all directions or, more precisely, for

all possible spatial rotations of the chosen reference configuration. The precise definition

of symmetry is expressed mathematically in terms of the appropriate symmetry group (of

orthogonal transformations).

Second law of thermodynamics (or dissipation inequality)

The 2nd law of thermodynamics states that the production of internal entropy, or rate of

“material disorder”, must be non-negative. This statement is equivalent to the statement

that dissipation of energy is never negative.

This law, whose mathematical formulation is the Clausius-Duhem Inequality, is discussed

in Chapter 3. It is a cornerstone for the further developments in the present text. In

particular, its automatic satisfaction is a key feature of “standard dissipative materials”,

which are considered in various contexts subsequently.

1.1.4 Approaches to constitutive modeling

The conceptually different approaches to the derivation of macroscopical constitutive mod-

els may be defined as follows:

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Fundamental (or micromechanics) approach – Homogenization and computa-

tional multiscale modeling (CMM)

As indicated above, a complete understanding of the deformation and failure characteris-

tics requires the detailed knowledge of the microstructural processes. In the fundamental

approach this fact is acknowledged, and elementary constitutive relations are established

for the microstructural behavior (micromechanical modeling). A classical example is crys-

tal plasticity, in which relations between shear stress and shear slip are established for

single slip systems within the atomic lattice structure. A useful macroscopic model can

then obtained via averaging techniques (homogenization), which can sometimes be car-

ried out analytically, cf. Nemat-Nasser & Hori (1993). More generally, it is carried

out numerically with the aid of a Representative Volume Element (RVE), which must

be sufficiently large to admit statistical representations, yet small enough to represent

a ”point” from a continuum mechanics perspective. Sometimes the size of the RVE is

determined by periodicity of the microstructural arrangement.

A more powerful alternative to homogenization aimed at developing a macroscopic con-

stitutive model is to carry out a Computational Multiscale Modeling (CMM), whereby

the macroscopic constitutive model becomes obsolete. The response of the RVE is then

simulated as an integrated part of the macroscopic analysis of a given component, which

involves the global balance equations of mechanics. The macroscopic stress and strain

values are computed as averages (in some sense) of the corresponding microstructural

fields within the pertinent RVE in each spatial point subjected to the actual macroscopic

deformation. cf. Miehe (1996), Lilbacka et al. (2004), Grymer et al. (2006).

The fundamental approach to constitutive modeling is still less developed, although the

international activity is quite strong. Not only metals with ordered lattice structures are

considered, but also “disordered” media (soil, rock, etc.).

Phenomenological approach

The macroscopic model is established directly based on the observed characteristics from

elementary tests. The calibration is carried out mainly by comparison with experimental

results and/or with micromechanical predictions for well-defined boundary conditions on

the pertinent RVE, cf. the discussion above. Traditional models are sometimes simple

enough to admit the identification of the material parameter values one by one from

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1.2 Modeling of material failure — Fracture 7

well-defined elementary experiments. The obvious example is the observation of the yield

stress of mild steel from a tensile test. However, the general approach is to optimize the

predictive capability of the model in the calibration procedure. The objective function

to be minimized is a suitable measure (norm) of the difference between the predicted

response and the experimentally obtained data.

The arguments of the constitutive functions are observable variables (like stress, strain and

temperature) in addition to a sufficient number of nonobservable, or internal, variables

that represent the microstructural changes.

Statistical approach

Statistical “models” for describing material behavior are the least fundamental, in the

sense that they are normally established as response functions for specific loading and

environmental conditions. A variety of distributions can be used for describing the scatter

in strength data, whereby Weibull’s statistical theory is quite often used.

1.2 Modeling of material failure — Fracture

Two principally different views can be distinguished with respect to the analysis of ma-

terial failure. From a classical standpoint, these approaches are related to the fact that

a material may behave in a ductile or brittle fashion, depending on material composition,

aging, temperature, etc.

1.2.1 Continuum damage mechanics

The view of continuum damage mechanics is that the failure process starts with a gradual

deterioration of a continuously deforming material. After considerable inelastic deforma-

tion, due to the material ductility, the stress drops quite dramatically (in a displacement

controlled test) and deformations localize in a narrow zone (or band). This stage is defined

as the onset of fracture; cf. Figure 1.2. In many cases the localization is quite extreme

in the sense that a single macroscopic (discrete) crack starts to develop. Stresses can be

transferred across the crack until it is fully opened.

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σ

σ

σ

localizationzone=“neck”

microcracks=“damage”

(a) (b)

neckdevelops

brittle

ǫductile

Figure 1.2: Damage process (a) Localization (necking) in a bar of ductile material, (b)

Stress vs. strain characteristics.

1.2.2 Fracture mechanics

The view of fracture mechanics is that a macroscopic crack (or flaw) has already occured,

and the main task is to determine whether the crack will propagate or not. A crack

that propagates only when the externally applied load is increased is termed stable. No

consideration is then given to the process leading to the (preexisting) fully open crack.

The analysis of crack stability is usually based on the assumption that the behavior close

to the crack tip is linear elastic (Linear Fracture Mechanics), such that the stress field

singularity at the crack tip is determined from linear elasticity, cf. Figure 1.3. The

simplest crack stability criterion is the (empirical) Griffith criterion, by which the crack is

deemed stable if the pertinent stress intensity factor, that depends on the applied loading,

does not exceed a critical value. This concept can be extended to cyclic loading, e.g. in

the shape of a threshold level of stress in Paris’ law.

1.3 Common experimental test conditions

Phenomenological constitutive laws are calibrated with the aid of experimental data that

are obtained from well-defined laboratory tests. The idea is to design the test in such a way

that the specimen is subjected to homogeneous states of stress, strain and temperature.

A few common test conditions in practice are listed below:

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1.3 Common experimental test conditions 9

L

σ

u

u

macroscopiccrack

singularstress fieldat crack tipσ = ∞ (locally)

σ

area under σ − u curve =released fracture energy

brittle

(a) (b)

σ

Figure 1.3: Fracture process (a) Preexisting edge cracks, (b) Far-field stress vs. extension

characteristics. Note: ε = u/L is not well-defined as local measure of strain!

Uniaxial stress

A cylindrical bar is subjected to a state of uniform (axial) stress, which may be tensile or

compressive, as shown in Figure 1.4(a). The strain state is cylindrical, i.e. nonzero radial

and tangential normal strains normally appear. Either the axial stress or the axial strain

is controlled. This elementary test condition is common for most materials, at least those

possessing cohesion. By definition, cohesion materials have shear strength that prevails

when the mean (normal) stress is zero. Frictional materials, whose shear strength vanishes

when the mean stress is zero, can not be tested under the uniaxial stress condition without

precompaction.

Normal stress combined with shear

The conventional way of applying normal stresses, combined with shear stress, is to subject

a circular thin-walled tube to axial load, internal or external pressure, together with a

torsional moment, as shown in Figure 1.4(b). Since the wall thickness is small, the radial

stress varies approximately linearly through the thickness, and at the midplane of the tube

wall a well-defined triaxial stress state is obtained. Moreover, when torsion is applied,

the principal axes rotate due to additional shear stress. In the case there is no applied

pressure, a state of plane stress is obtained. This type of test is common for metals, but

has also been used for concrete and highly cohesive soil (such as clay).

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Conventional plane stress and plain strain

Cross-shaped plane specimens of metallic material may be subjected to biaxial (tensile or

compressive) loading under plane stress conditions. For soil and other granular materials,

a special biaxial apparatus (biaxial cell) is needed to ensure the appropriate out of plane

condition, in particular the plane strain condition. The principal stress directions can not

rotate.

Cylindrical stress and strain states

A cylindrical specimen is subjected to external radial pressure and axial compressive load,

as shown in Figure 1.4(c). This is a commonly used test condition for granular materials,

such as powder and soil, as well as for rock and concrete. Two usual test procedures are

denoted Conventional Triaxial Compression (CTC) and Conventional Triaxial Extension

(CTE). In the CTC-test, an isotropic state of stress is first applied during the socalled

consolidation phase. Then the radial (=circumferential) stress is held constant, while the

axial compressive loading is further increased. This compression may be either stress-

or strain-controlled. In the CTE-test, isotropic stress is first applied to consolidate the

sample in the same fashion as for the CTC-test. However, the axial stress is then kept

constant while the radial pressure is further increased.

In order to assess the principal difference between these two test conditions, we consider

the corresponding principal stresses σi < 0 (compression negative), where σ1 ≥ σ2 ≥ σ3.

Since σi = 0, the CTC-test is defined by σ1 = σ2 > σ3 and the axial stress is σ3, which is

the numerically largest principal stress. The CTE-test, on the other hand, is defined by

σ1 > σ2 = σ3 and the axial stress is now σ1. It is common to use these test results towards

the evaluation of a failure (or yield) criterion of the Mohr-Coulomb type, cf. Chapter 10.

True triaxial stress and strain states

Principal stresses can be applied independently in the cubical cell apparatus, as illustrated

in Figure 1.4(d). In practice, this is a quite complex device that has gained widespread

use for soil, rock and concrete.

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1.3 Common experimental test conditions 11

(a)

(c)

(b)

(d)

z

z

σr = σθ = −p

p

p

σ3

σ2

σ1

r

r

r θ

θ

θ

Figure 1.4: Stress and strain states in (a) Tensile test, (b) Normal load-torsion test of

thin-walled tube, (c) CTC- and CTE-tests, and (d) Cubical cell test.

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1.4 Typical behavior of metals and alloys

1.4.1 Plastic yielding — Hardening and ductile fracture

The basic behavior of a ductile metal is obtained under monotonic loading. Plastic yielding

will occur approximately at the same magnitude of stress in tension as in compression since

plastic slip is determined by the critical resolved shear stress along potential slip planes

(Schmid’s law). Yielding is independent of the magnitude of the mean stress, which

defines an ideal cohesive material. The further increase of stress beyond yielding is known

as hardening. Figure 1.5(a) shows the typical stress-strain relation in uniaxial tension at

monotonic loading of a hot-worked steel. The characteristic strength parameters are the

yield stress σy and the ultimate strength (peak stress) σu. Figure 1.5(b) shows the typical

yield surface in biaxial stress (approximately elliptical in reality for a polycrystalline

metal).

ǫǫuǫy

σ

σy

σy

σy

−σy

−σyσ1

σ2

σu

(a) (b)

Figure 1.5: (a) Stress-strain relation in uniaxial tension showing yielding, hardening and

ductile fracture. (b) Yield surface in biaxial stress.

The picture is complemented by unloading, followed by reversed loading which gives rise

to hysteresis loops, as shown in Figure 1.6. After significant straining the average un-

loading modulus will decrease significantly, which may be interpreted as a sign of internal

degradation (damage) and that fracture is approaching.

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1.4 Typical behavior of metals and alloys 13

ǫ

ǫ

A AA’ A’

B

t

σ

σy

EE E < E

(a) (b)

Figure 1.6: Response at loading/unloading showing eventual degradation (damage) and

ductile fracture

1.4.2 Constant loading — Creep and relaxation

The time-dependent response of a material at elevated temperature, after rapid initial

loading up to constant (nominal) stress, is denoted creep. For metals, viscous (creep)

behavior becomes important when the temperature exceeds, approximately, 30% of the

melting temperature. At this temperature, cavitation along the grain boundaries starts to

become an important deformation/failure mechanism. A typical creep curve, for constant

temperature, is shown in Figure 1.7(a). The recovery upon rapid unloading is also shown.

Three different stages of the creep process can be distinguished (in a classical description),

although the transition between them is, by no means, clear:

Transistent stage (or Primary stage, I)

The rate of creep is initially decreasing, which is a result of saturation of dislocations.

Stationary stage (or Secondary stage, II)

After the saturation level has been reached, the creep rate is rather constant. As will be

discussed later, the Norton creep law is traditionally adopted in this stage.

Creep failure stage (or Tertiary stage, III)

After certain creep deformation, the development of microstructural degradation (internal

damage) will result in an accelerated creep rate until failure occurs at time tR, which is

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the lifetime of the specimen. This process is strongly temperature dependent.

The time-dependent stress change after rapid loading, while the strain is held constant, is

denoted relaxation. The relaxation behavior is thus complementary to the creep behavior,

as shown in Figure 1.7(b).

σ

σ

σ0

σ0

A

A

AA

B

B

BB

t t

tt

ǫ

ǫ

ǫ0

ǫ0

(a) (b)

⇓⇓

tRI II III

rupture

recovery

creep

Figure 1.7: (a) Creep and (b) Relaxation curves.

1.4.3 Time-dependent loading — Rate effect and damping

Another aspect of viscous properties, besides creep, is the rate-dependence that is exhibited

in the stress-strain curve for certain materials. This is manifested by higher stiffness and

strength for larger loading rate, especially due to impact loading. In accordance with the

situation at creep, the rate-effect is more pronounced at elevated temperature. The typical

result at monotonic loading under prescribed strain rate for a rate-sensitive material is

shown in Figure 1.8. In a real structure the strain rate may vary considerably from the

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loaded region to other parts. Hence, it is important to model rate-effects in such a fashion

that the rate-independent situation is obtained merely as a special case.

ǫ

σ

ǫ = 0

ǫ > 0

ǫ = ∞

Figure 1.8: Rate effect on stress-strain relation.

Damping, in the sense that free vibrations of a structure will decay with time and even-

tually die out, can be explained as the result of energy dissipation in the material. If the

amount of damping is dependent on the frequency of the vibrations, then the damping is

of viscous character and can be modelled within the framework of viscoelasticity or vis-

coplasticity. If, on the other hand, the damping is independent on the frequency, then the

damping is commonly denoted as hysteretic and can be modelled within the framework

of rate-independent plasticity.

An alternative way of assessing damping and rate effects is to consider forced vibrations

due to a sustained harmonic load with given frequency. The structural response, in terms

of strain and stress, is then normally observed to be dependent on the frequency of the

exciting load, which points towards a rate-effect.

1.4.4 Cyclic loading and High-Cycle-Fatigue (HCF)

The usual way of testing cyclic and, eventually, fatigue behavior is to subject the speci-

men to a (slow) cyclic variation of stress or strain with constant amplitude. If the applied

load level is below the macroscopic yield stress, but above a certain threshold, the cyclic

response is in the elastic range and no macroscopic plastic deformation is observed. How-

ever, after many load cycles a reduction of the apparent elasticity modulus is noted. This

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degradation of the elastic stiffness is caused by microcracking and microslip due to local

stress-concentrations within the microstructure. The number of cycles to failure is very

high (NR > 100, 000), and the final fracture is brittle in character since failure is preceeded

by virtually no inelastic deformation, as shown in Figure 1.9.

ǫ

σ (MPa)

2 4 6 8

damage threshold200

100

0.2 × 10−2-0.2

σ

×105N

Figure 1.9: Result of HCF-test with constant strain amplitude.

1.4.5 Cyclic loading and Low-Cycle-Fatigue (LCF)

If the applied load level is high enough, the macroscopic yield stress will be exceeded, and

plastic strains will develop in each cycle. Not unlike the characteristics of a creep test,

three different stages of the deformation process may be distinguished:

Saturation Stage

Consider the early stage of cyclic loading with constant amplitude. The response is then

characterized as either cyclic hardening or cyclic softening. The typical behavior of cyclic

hardening is shown in Figure 1.10(a) for given strain amplitude (strain control) and in

Figure 1.10(b) for given stress amplitude (stress control). Cyclic hardening means that

the stress amplitude will initially increase in a few cycles to an asymptotic level in a strain

controlled test, whereas the strain amplitude will decrease in a stress controlled test. The

complementary behavior in the case of (initial) cyclic softening is shown in Figure 1.11(a)

and Figure 1.11(b). Hence, cyclic softening means that the stress amplitude will initially

decrease to an asymptotic level in a strain controlled test, whereas the strain amplitude

will increase in a stress controlled test.

In both stress- and strain-controlled cyclic loading, the ideal situation is that the respec-

tive strain or stress amplitude will shake-down quite rapidly to a stabilized stress-strain

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3

3

2

2

2

2

1

1

1

1

∆ǫ = constant

∆σ = constant

σmax

σmin

ǫmax

ǫmin

σmax = −σmin ⇒ σm = 0

ǫm 6= 0

(a)

(b)

σ

σ

σ

ǫ

ǫ

ǫ

Stabilized

Stabilized

t

t

Figure 1.10: Initial cyclic hardening as shown in (a) Strain control, (b) Stress control.

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3

∆σ = constant

σmax

σmin

ǫmax = −ǫmin ⇒ ǫm = 0

∆ǫ = constant

σm 6= 0

ǫmax

ǫmin

(a)

(b)

1

1

1

1

2

2

2

2

Stabilized

Stabilized

t

t

σ

σ

σ

ǫ

ǫ

ǫ

Figure 1.11: Initial cyclic softening as shown in (a) Strain control, and (b) Stress control.

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hysteresis loop, as shown in Figure 1.12(a). This loop is symmetrical in tension and com-

pression in the ideal situation. In reality, the stabilized (shake-down) amplitudes on the

tension and compression sides may not be symmetrical, even if the applied cyclic action is

symmetrical, as shown in Figure 1.10 and Figure 1.11. Denoting the stabilized maximum

values by σmax and ǫmax, and the minimum values by σmin and ǫmin, we define the cyclic

mean stress and cyclic mean strain, respectively, as

σm =1

2[σmax + σmin], ǫm =

1

2[ǫmax + ǫmin] (1.1)

where it is noted that algebraic values are used. We thus conclude that, in general at

the saturation level, σm 6= 0 in the strain-controlled test, whereas ǫm 6= 0 in the stress-

controlled test.

However, it is also possible that stabilization does not occur at all (or is very slow). For

prescribed constant stress amplitude, this lack of stabilization is evident as “ever increas-

ing” plastic strain, or ratchetting, which is shown in Figure 1.12(b). Such ratchetting can

be expected when σm 6= 0, in particular.

σσ

ǫǫ

Shakedown

1

1

2

2

∆ǫr

Ratchetting strain(a) (b)

Figure 1.12: Phenomena of (a) Shakedown and (b) Ratchetting.

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Fatigue failure stage

After certain amount of plastic deformation has accumulated in the hysteretic loops after

saturation, damage starts to develop. This damage development will eventually, say

after 1,000-10,000 cycles, result in cyclic softening until failure occurs. Hence, LCF is

characterized by relatively small values of NR, which is the number of cycles to failure

(NR < 10, 000).

The characteristics of LCF are observed in the strain-controlled as well as in the stress-

controlled environment. The elastic unloading modulus is continually decreasing in such

a way that the hysteresis-loops become more and more “flattened”. As a result, the

stress amplitude gradually decreases in the strain-controlled test, as shown in Figure 1.13,

whereas the strain-amplitude grows in an uncontrolled fashion in the stress controlled test.

Moreover, further ratchetting may be obtained due to the fact that the rate of damage

development is smaller in compression than in tension (and it is assumed to be zero in

the figure).

stabilized(saturation) σ (MPa)

−0.2 0.2 × 10−2

100200300

ǫ

σ

damage threshold

500 1000 1500 2000N

Figure 1.13: Result of LCF-test with constant strain amplitude.

1.4.6 Creep-fatigue and Relaxation-fatigue

Creep-fatigue is obtained when the stress varies in a cyclic fashion with a predefined hold-

time within each cycle. The failure is caused by the combined action of creep deformation

and deterioration of the stiffness due to LCF. This phenomenon is of particular importance

at the design of jet engines and other gas turbines, which operate under high temperature.

Relaxation-fatigue is the counterpart of creep-fatigue when the strain is allowed to vary

in a cyclic fashion with predefined hold-time.

Sometimes, the notion thermal fatigue refers to the situation where the stress/strain vari-

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1.5 Typical behavior of ceramics and cementitious composites 21

ation is due to cyclic temperature change. The effect becomes more pronounced for high

degree of static indeterminacy (when stresses are larger). Clearly, it is not possible to

control the stress or strain amplitude when the temperature is varied. The most gen-

eral loading situation is denoted thermomechanical fatigue, in which case a component is

subjected to cyclic variation of the mechanical load as well as the temperature.

1.5 Typical behavior of ceramics and cementitious

composites

1.5.1 Monotonic loading – Semi-brittle fracture

At monotonic loading, cementitious materials (such as concrete) show nearly linear elastic

response at small load levels. However, the type of failure is entirely different in tension

and compression. Tensile failure will occur in a quite brittle (quasi-brittle) manner at

the tensile strength, σ = σtu, whereby a macroscopic crack starts to develop and is

fully open when the stress has dropped to zero. The corresponding post-peak stress-

strain relationship is not well-defined, cf. the discussion in Section 2.2. This response is

depicted in Figure 1.14(a). Compressive failure, on the other hand, will occur in a ductile

manner after the compressive strength, σ = −σcu, has been reached. The stress drop in

the post-peak regime represents gradual crushing of the microstructure. Typically, the

ratio σtu/σcu is of the order 0.1. Quite often the response in compression close to failure is

modelled as elastic-plastic, whereby the yield criterion is strongly mean-stress dependant.

In order to compensate for the low tensile strength cementitious materials must in practice

be reinforced by steel bars, glass-fiber bars or distributed ductile fibers.

A typical failure criterion is that of Mohr-Coulomb (which is discussed in further detail

in Chapter 10). This criterion is shown in Figure 1.14(b) for biaxial stress states.

Structural failure in massive concrete structures can be very dramatic due to the large

amount of elastic energy that is stored in a large volume at the point of cracking. An

example of a major disaster was the failure of the Sleipner oil platform outside Stavanger,

Norway in 19XX.

Remark: Mean stress dependent yielding and failure is typical for different granular and

particulate materials, e.g. soil and powders. Another example is (graphitic) grey-cast

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replacemen

(a)(a)

σ

ǫσ1

σ2

σtu

σtu

σtu

−σcu

−σcu

−σcu

Figure 1.14: (a) Stress-strain relation in uniaxial tension and compression. (b) Failure

surface according to Mohr-Coulomb for biaxial stress states (plane stress).

iron, for which the ratio of yield stress in tension and compression, σty/σcy, is of the order

3. 2

1.5.2 Cyclic loading and fatigue

1.5.3 Creep and relaxation

Creep phenomena in cementitious materials can be characterized similarly to those of

metals.

1.6 Typical behavior of granular materials

1.6.1 Monotonic loading – Basic features

Granular materials, such as soil and (ceramic and metal) powders, show frictional charac-

teristics. A purely frictional material, such as sand, gravel or fragmented rock (ballast),

can sustain shear only in the presence of compressive normal stress between the particles.

Moreover, in a purely frictional material the tensile strength is zero (σtu = 0). Many fine-

grained materials, such as clay and powders that have been subjected to precompaction,

show combined frictional and cohesive characteristics, which means that the material can

sustain some shear stress even without any normal stress. In particular, this is the case for

clayey soils and for rock and concrete. Hence, frictional/cohesive features can be trans-

lated into mean-stress dependent failure criteria, cf. the discussion in Section 2.5, and

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1.6 Typical behavior of granular materials 23

it can be concluded that soils, powders and concrete do, in fact, have much in common

when it comes to the modeling of failure characteristics.

The inelastic deformations of granular materials contain a volumetric component (con-

trary to the case for most metals). The deformation is dilatant at dense initial packing,

whereas it is contractant at loose initial packing. Dilatant behavior is associated with

softening response (negative hardening), whereas contractant response is accompanied

by hardening. In reality there may be a significant elastic-plastic coupling in the sense

that the inelastic volume change affects the elastic moduli. The mechanical response is

normally tested in a triaxial stress apparatus under cylindrical stress conditions, cf. the

CTC-and CTE-conditions discussed in Section 2.3.

Remark: In metals it necessary to account for evolving porosity close to failure, whereby

the yielding characteristics resemble those of a powder compact. 2

1.6.2 Constant loading – Consolidation

Natural fine-grained soils (in particular clay) show a more complex mechanical response

due to the presence of fluid (water and air) in the open pores. The resulting hydro-

mechanical interaction of such poro-mechanical materials introduces time-dependent de-

formation at constant applied load. Such a time-delayed deformation process is denoted

consolidation in soil mechanics (which must not be confused with creep due to viscous

character of the solid particles). Basically, consolidation is the process of ”squeezing a

sponge filled with water”. Oil reservoirs constitute a complex geological system of solid,

liquid (oil/water) and gas in a mixture state.

In the extreme case the permeability is so small that virtually no seepeage of fluid can

take place in the pore system, which is termed undrained condition. In the special case of

water-saturated pores, such an undrained state corresponds to overall incompressibility

of the granular material.

1.6.3 Constant loading – Creep and relaxation

In addition to consolidation, fine-grained soils show creep under constant loading. Such

creep, which is sometimes denoted ”secondary consolidation”, can be observed experi-

mentally in the triaxial apparatus or in the oedometer (which imposes a state of uniaxial

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strain).

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Chapter 2

THERMODYNAMICS — A BRIEF

SUMMARY

A rigorous treatment of the thermodynamic background to the constitutive models of solid

materials is beyond the scope of this introduction and brief summary. In this chapter, we

shall only introduce the necessary concepts and relations as ad hoc statements. Moreover,

we shall restrict the treatment to isothermal response, i.e. thermal effects are ignored. As a

consequence, in the case that thermal effects must be taken into account, the temperature

is treated merely as a parameter.

2.1 Free energy and constitutive relations

2.1.1 General

The free energy per unit volume of a dissipative material is defined as Ψ(ǫ, kα), where ǫ is

the (macroscopic) strain, whereas kα constitute a finite set of, say N , internal variables

that represent irreversible microstructural processes in the material. A typical example

(that we shall consider later in more detail) is the plastic deformation that is caused by

dislocations of crystal planes in a metal.

From Ψ(ǫ, kα) we may calculate the stress σ and the socalled dissipative stresses κα (that

are energy-conjugated to kα) from the constitutive equations:

σ =∂Ψ

∂ǫ, κα

def= −

∂Ψ

∂kα, α = 1, 2, . . . , N (2.1)

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26 2 THERMODYNAMICS — A BRIEF SUMMARY

Remark: The relations (2.1) are consequences of the 2. law of thermodynamics (which

is not proven here) and are sometimes known as Coleman’s relations. 2

Hence, for given values of the state variables ǫ and kα, we may always calculate the

dependent state variables σ and κα from the relation (2.1). In order to link values of

kα to the observable variable ǫ, further constitutive relations must be established. Such

relations are expressed as rate equations of the form

kα = fα(ǫ, kα; ǫ), α = 1, 2, . . . , N (2.2)

The functions fα(ǫ, kα; ǫ) must be chosen in such a fashion that the dissipation inequality

D =N∑

α=1

καkα ≥ 0 (2.3)

is satisfied for all possibles values of the strain rate ǫ.

Remark: The inequality (2.3), which is known as the Clausius-Duhem inequality, is also

a consequence of the 2. law of thermodynamics. 2

In conclusion, any material model that satisfies the relations (2.1) to (2.3) is consistent

with fundamental thermodynamic requirements.

2.1.2 Stress-strain response relation

Strain control

A strain-driven solution strategy is the natural approach in finite element codes based on

the displacement method. By integrating (2.2) for given strain history, i.e. ǫ(t) is known,

and given initial values kα(0) = 0, we may calculate kα(t). It is then straightforward to

calculate σ(t) from (2.1)1, at any time when the arguments ǫ(t) and kα(t) are known.

Stress control

When the stress history, σ(t), is known, we assume that it is possible to invert (2.1)1, to

obtain ǫ = ǫ(σ, kα). This expression can then be inserted into (2.2), which can now be

integrated for kα(t).

Remark: In practice it is customary to use the strain-driven algorithm even in this case;

however, it is necessary to carry out iterations in order to compute ǫ(t) such that the

prescribed value σ(t) is obtained from (2.1)1. 2

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2.1 Free energy and constitutive relations 27

2.1.3 Material classes

Important subclasses of the general dissipative material may be identified depending on

the particular choice of fα in (2.2).

Elastic material response

Elastic (non-dissipative) material response is obtained if kα does not occur as arguments

in Ψ , whereby Ψ(ǫ) represents the strain energy. Clearly, nonlinear elastic response is

obtained whenever Ψ is not a quadratic function in ǫ.

Viscous dissipative material response

Viscous, or rate-dependent, material response is defined by the special form of (2.2):

kα = fα(ǫ, kα) (2.4)

where fα are bounded state functions. Examples of model classes are viscoelasticity and

viscoplasticity, which are discussed in greater detail in Chapter 3 and 5.

Since fα is bounded, there will be insignificant (zero) change of kα during a “step loading”

in time of ǫ, i.e. for very rapid change of ǫ. This means that we obtain elastic response

(that is defined by constant kα) during such a loading.

Nonviscous dissipative material response

Nonviscous, or rate-independent, material response is defined by the special case of (2.2):

kα = f (ǫ)α (ǫ, kα)ǫ (2.5)

where f(ǫ)α are bounded state functions. Since the rate equations (2.5) are linear in ǫ,

the corresponding material response is often termed “incrementally linear”. The most

important example is plasticity, which is treated in Chapter 4.

That the material is rate-independent may be illustrated by its indifference to a “change

of clock”. To show this, we assume that a different “clock”, i.e. another time-scale, is

introduced as the strictly monotonic function s(t), i.e. s(t) > 0. It is then possible to

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28 2 THERMODYNAMICS — A BRIEF SUMMARY

invert s(t) to give t = t(s), from which we can obtain kα(s)def= kα(t(s)) and ǫ(s)

def= ǫ(t(s)).

Upon using the chain rule, e.g. ǫ(t) = (dǫ/ds)s(t), it follows from (2.5) that

[dkα

ds− f (ǫ)

α (ǫ(s), kα(s))dǫ

ds

]

s(t) = 0 ∀t (2.6)

which givesdkα

ds= f (ǫ)

α (ǫ(s), kα(s))dǫ

ds(2.7)

Upon integrating (2.7), we note that the same value of kα is obtained for given value of s

independent of the function s(t). Hence, the response is not dependent on the “clock” or

real time but only on the history ǫ(s). Hence, the speed of the straining process (in real

time) is of no relevance for the solution of (2.5), as shown schematically in Figure 2.1.

s

s

t1 t2

s (t)1s (t)2

t s

kα

kα

s

Figure 2.1: Illustration of rate-independent material behavior.

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Chapter 3

VISCOELASTICITY

In this chapter, we outline the elements of linear, as well as nonlinear, viscoelasticity. The

Maxwell model is taken as the prototype model. Both (Laplace) transform technique and

numerical integration are described for handling the time-dependence of the constitutive

relations. Structural analysis of a truss and a beam cross-section is outlined. Finally, the

modeling of damping (in dynamic analysis) using viscoelasticity is discussed.

3.1 Introduction

The theory of viscoelasticity is used to model time-dependent response of a variety of

materials at elevated temperature (typical examples are metals and polymers) as well

as at ambient temperature (a typical example is fine-grained soil). The response of a

class of fluids, which include biological fluids, such as blood, and melted metals can

be predicted well by quite complex nonlinear viscoelastic models; hence, such fluids are

denoted viscoelastic.

The time-dependent deformation due to constant stress (creep) or the time-dependent

stress due to constant deformation (relaxation) are two “dual” cases of particular interest.

More generally, it is often possible to use a viscoelastic model to describe how the stress-

strain response is affected by the rate at which the control variables (stress or strain) are

applied, at least if the temperature is sufficiently high. By definition, viscoelastic materials

do not possess a truly elastic region, i.e. the response is never fully recoverable at finite

rate of loading. Viscoelasticty is sometimes used to model socalled viscous damping, as

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30 3 VISCOELASTICITY

opposed to hysteretic damping, which is a measure of the frequency-dependent dissipation

of energy during a period of harmonic loading. Hysteretic damping, on the other hand,

is frequency-independent and can be predicted by rate-independent dissipative models.

Linear viscoelasticity is normally described in the literature in terms of rheological models,

that give a direct physical “feeling” for the response under uniaxial stress condition.

Names of such rheological models are taken from scientists such as Maxwell, Kelvin,

Voigt, Burgers, etc. For any linear viscoelasticity model it is possible to define creep

and relaxation functions1. Their general characeristics are discussed below, and explicit

functional expressions are given for the Maxwell model.

The difference between nonlinear viscoelasticity and viscoplasticity is somewhat diffuse,

since viscoplasticity models possess a quasistatic yield surface (enclosing the truly elastic

region) which can be allowed to shrink to a point at the origin of stress space. The

distinction made here is that viscoelastic models do not allow hardening such that elastic

regions can develop with time. Hence, the model by Bodner & Partom (1975) does

not qualify as a viscoelasticity model; it is rather a special case of a viscoplasticity model.

The most well-known nonlinear law, expressing staionary creep rate, is that of Norton in

the 1930’s. Other relevant names are Odqvist (in the 1940’s) and Spencer & Boyle (in

the 1970’s). Finally, we mention that improved damping characteristics can be obtained

if the (conventinal) first order time-derivative in the evolution equations for the internal

variables is replaced by a socalled fractional time-derivative.

3.2 Prototype model: The Maxwell rheological model

3.2.1 Thermodynamic basis — Constitutive relation

One of the simplest rheological models featuring combined viscous and elastic response

is the Maxwell model, as depicted in Figure 3.1. It is characterized by a spring (with

elasticity modulus E) serially connected to a dashpot (with viscosity coefficient µ). Since

the dashpot represents a dissipative element, its strain will conveniently be treated as an

internal variable, that is subsequently denoted ǫv.

1Relaxation functions do not exist for all rheological models.

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3.2 Prototype model: The Maxwell rheological model 31

σE

vε

µσ

εε − ve =ε

Figure 3.1: Maxwell model representing viscoelastic material.

The simplest choice of the free energy that gives the desired constitutive behavior is

Ψ =1

2E(ǫ − ǫv)2 (3.1)

from which we obtain

σ =∂Ψ

∂ǫ= E(ǫ − ǫv) (3.2)

This constitutive equation can readily be derived from the rheological model in Figure 3.1.

The appropriate rate equation that determines the development of the internal variable

(viscous strain) is given as

ǫv =1

µσv (3.3)

If we introduce the dissipative stress σv associated with ǫv from the constitutive equation

σv = −∂Ψ

∂ǫv= E(ǫ − ǫv) ≡ σ (3.4)

we may use this identity to express the rate equation (3.3) as

ǫv =1

µσ =

E

µ(ǫ − ǫv) (3.5)

With (3.3), we note that

D = σvǫv = σǫv =1

µσ2 ≥ 0 (3.6)

and, hence, the CDI is satisfied.

By eliminating ǫv in (3.5), we obtain the linear differential equation

σ +1

t∗σ = Eǫ (3.7)

where t∗ = µ/E is the natural relaxation time. It appears that ǫ(t) may be solved for

prescribed σ(t). Alternatively, σ(t) may be solved for prescribed ǫ(t).

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3.2.2 Prescribed constant stress (pure creep)

Assume that the stress σ0 is applied suddenly at t = t0, whereafter it is held constant as

time elapses, i.e. σ(t) can be written as

σ(t) = σ0H(t) (3.8)

where H(t) is Heaviside’s function defined as H(t) = 0 when t < 0, and H(t) = 1 when

t ≥ 0. Since ǫv(0) = 0, we obtain from (3.5)

ǫv(t) =σ0

E

t

t∗(3.9)

Combining this expression with the constitutive expression for σ in (3.2), we obtain

ǫ(t) = C(t)σ0, with C(t) =1

E

(

1 +t

t∗

)

(3.10)

where C(t) is the creep function for the Maxwell model, which is depicted in Figure 3.2(a).

3.2.3 Prescribed constant strain (pure relaxation)

We may, instead of prescribed stress, assume that the strain ǫ0 is applied suddenly at

t = t0, whereafter it is held constant in time, i.e. ǫ(t) is written as

ǫ(t) = ǫ0H(t) (3.11)

From (3.5), we may solve for ǫv(t) as

ǫv(t) = ǫ0

(

1 − e−t

t∗

)

(3.12)

where it was used, again, that ǫv(0) = 0. Combining this expression with the constitutive

equation for σ in (3.2), we obtain

σ(t) = R(t)ǫ0, with R(t) = Ee−t

t∗ (3.13)

where R(t) is the relaxation function 2 for the Maxwell model which is depicted in Fig-

ure 3.2(b).

Remark: It is noted that the strain σ0/E and the stress Eǫ0 represent the instantaneous

elastic response preceding the time-dependent creep and relaxation processes, respectively.

2

2It is noted that R(t) 6= 1C(t) . However, such a simple inversion is possible for the corresponding

Laplace transforms, which is discussed below.

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3.3 Linear viscoelasticity — Constitutive modeling 33

C

EE

1

(a) (b)

R

tt∗

tt∗

1E

Figure 3.2: (a) Creep function, (b) Relaxation function for Maxwell material.

3.3 Linear viscoelasticity — Constitutive modeling

3.3.1 General characteristics

That creep and relaxation functions exist can be taken as the definition of a linear vis-

coelastic material response. Hence,

σ(t) = σ0H(t) ⇒ ǫ(t) = C(t)σ0 (3.14)

as shown in Figure 3.3(a,b).

(a) (b)

σ

σ0

ǫC(t)σ0

tt

⇒

Figure 3.3: (a) Constant stress loading, (b) Creep response due to linear viscoelastic

behavior.

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Viscoelastic solids and viscoelastic fluids

This behavior may also be described in terms of straight isochrone curves, as shown in

Figure 3.4(a,b) for a (viscoelastic) solid and a (viscoelastic) fluid, respectively. These are

defined by the following properties of the creep function:

C(t) < ∞, t → ∞ ⇒ solid

C(t) = ∞, t → ∞ ⇒ fluid (3.15)

For example, according to this definition, the Maxwell model represents a viscoelastic

fluid, whereas the Kelvin model (which is discussed below) represents a viscoelastic solid.

ε

t1 > 0

t 2 > t 1

→t ∞

( a )

t = 0σ0

ε

t1 > 0

t 2 > t 1

→t ∞

( b )

t = 0σ0

Figure 3.4: Straight isochrone curves for, (a) Solid and, (b) Fluid behavior, which are

characteristic for linear viscoelastic models.

Hereditary integrals for prescribed stress (generalized creep)

It is possible to obtain the total strain response for a prescribed stress history σ(t) by

using the expression (3.10) and the principle of superposition. By applying the small

stress amplitude dσ(t′) at time t′ ≤ t, we obtain the strain response

dǫ(t, t′) = C(t − t′)dσ(t′) = C(t − t′)dσ

dt′(t′)dt′, t′ ≤ t (3.16)

This expression may be integrated in the form of a hereditary (or convolution) integral

ǫ(t) =

∫ t

0

dǫ(t, t′) dt′ =

∫ t

0

C(t − t′)dσ

dt′(t′)dt′ or ǫ = C ∗

dσ

dt(3.17)

where the star (∗) denotes convolution product.

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Remark: The expression in (3.17) is valid also for non-differentiable functions σ(t) if the

derivative dσ/dt is taken in the sense of a distribution. For example,

σ(t) = σ0H(t − t0) ⇒dσ

dt(t) = σ0δ(t − t0) (3.18)

where δ(t) is the Dirac delta distribution. Formal use of (3.17) then gives

ǫ(t) =

∫ t

0

C(t − t′)σ0δ(t′ − t0)dt′ = C(t − t0)σ0 (3.19)

which is precisely the definition of the creep function. If, in particular, σ0 is applied at

t = 0 due to rapid initial loading, then the expression in (3.10) is retrieved. Henceforth,

we shall therefore always assume that σ(0) = 0 and ǫ(0) = 0 with possible step values

applied at t = 0+. 2

Hereditary integrals for prescribed strain (generalized relaxation)

The dual formulation of (3.17) for the total stress response due to a prescribed strain

history ǫ(t) is obtained in an analogous fashion by using the relaxation function. We

obtain

σ(t) =

∫ t

0

R(t − t′)dǫ

dt′(t′)dt′ or σ = R ∗

dǫ

dt(3.20)

3.3.2 Laplace-Carson transform

The analysis of linear viscoelastic materials can be reduced to the analysis of linear elastic

materials by means of symbolic calculus in the form of the Laplace-Carson transform. For

the present purpose, we consider functions f(t), which are piecewise differentiable and

which vanish identically for t ≤ 0. The transform is defined as (f)∗(s):

(f)∗(s) = s

∫ ∞

0

f(t)e−stdt (3.21)

A table of transforms for frequently occuring functions are given in the Appendix of this

chapter.

A very useful property is given in the following theorem:

Theorem: The transforms of convolution products are given as

(ǫ)∗(s)def=

(

C ∗dσ

dt

)∗

(s) = (C)∗ (s)(σ)∗(s) (3.22)

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(σ)∗(s)def=

(

R ∗dǫ

dt

)∗

(s) = (R)∗(s)(ǫ)∗(s) (3.23)

Proof: Homework!

From (3.22) and (3.23) follows that

(C)∗ (s)(R)∗(s) = 1 (3.24)

whereby we have shown that

C ∗dR

dt= R ∗

dC

dt= 1, t > 0 (3.25)

Example: Maxwell model

We may check explicitly that the results in (3.24) and (3.25) are valid for the Maxwell

model. For this model we derived the creep and relaxation functions

C(t) =1

E

(

1 +t

t∗

)

, R(t) = Ee−t

t∗ (3.26)

With the requirement that C(t) = R(t) = 0 for t < 0, both functions have a finite jump

at t = 0.

Using the tabulated transforms in the Appendix, we obtain

(C)∗ (s) =1

E

(

1 +1

t∗s

)

, (R)∗(s) = Et∗s

t∗s + 1(3.27)

First, we conclude that

(C)∗ (s)(R)∗(s) = 1 (3.28)

Secondly, we obtain

C ∗dR

dt= C(t)R(0) +

∫ t

0

C(t − t′)dR

dt′(t′)dt′ =

(

1 +t

t∗

)

· 1 +

∫ t

0

(

1 +t − t′

t∗

)(

−1

t∗e−

t′

t∗

)

dt′ = 1 (3.29)

and

R ∗dC

dt= R(t)C(0) +

∫ t

0

R(t − t′)dC

dt′(t′)dt′ =

e−t

t∗ +

∫ t

0

e−t−t′

t∗1

t∗dt′ = 1 (3.30)

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3.3.3 Linear Standard Model (Generalized Maxwell Model)

As a prototype of a general linear viscoelastic material, one may take the Linear Standard

Viscoelastic Model, whose rheological design is shown in Figure 3.5.

σ σ

µ1

µ2

µα

µN

E1

E2

Eα

EN

αεε − vαvε

Figure 3.5: Linear Standard Model.

The model consists of N Maxwell chains coupled in parallel. The expression for Ψ , that

was given for the Maxwell model, is extended in the following obvious fashion:

Ψ =1

2

N∑

α=1

Eα(ǫ − ǫvα)2 (3.31)


σ =∂Ψ

∂ǫ=

N∑

α=1

σα with σαdef= Eα(ǫ − ǫv

α) (3.32)

Hence, σ can be expressed as

σ = E(∞)ǫ −

N∑

α=1

Eαǫvα with E(∞) =

N∑

α=1

Eα (3.33)

Remark: It appears that E(∞), as defined in (3.33)2, is the elastic stiffness of the model at

infinite loading rate, in which case no viscous strain will develop in the dashpots (ǫvα = 0).

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Moreover, the considered Linear Standard Model can represent solid behavior only if it

has non-zero elastic stiffness E(0) at zero loading rate. Clearly, this is the case only if one

of the dashpots disappear, e.g. µN = ∞ which gives E(0) = EN . These situations will be

elaborated further in Section 3.7 in conjunction with dynamic behavior. 2

The rate equations for the N internal variables ǫvα are chosen as

ǫvα =

1

µασα, α = 1, 2, . . . , N (3.34)

and it appears that

D ≡N∑

α=1

σαǫvα =

N∑

α=1

1

µα

(σα)2 (3.35)

Since all µα > 0, it is clear that D > 0, i.e. the CDI is satisfied also for this general

model.

Upon substituting σα from the constitutive equations (3.32)2 into (3.34), we obtain the

uncoupled rate equations

ǫvα =

1

t∗α(ǫ − ǫv

α), α = 1, 2, . . . , N with t∗α =µα

Eα

(3.36)

We have now established the complete set of constitutive equations from which the creep

and relaxation functions can be derived.

It turns out to be convenient to start with the relaxation function R(t). By applying the

Laplace-Carson transform to (3.36), with the initial values ǫvα(0) = 0, we obtain

(ǫvα)∗ =

1

t∗αs + 1ǫ∗ (3.37)

which may be combined with (3.32)2 to give

(σα)∗ =Eαt∗αs

t∗αs + 1ǫ∗ (3.38)

Finally, upon combining this result with (3.32)1 gives

(σ)∗ = (R)∗(ǫ)∗ with (R)∗ =

N∑

α=1

Eαt∗αs

t∗αs + 1(3.39)

where the last expression has a meaning if Eα < ∞, α = 1, 2, . . . , N . The relaxation

function R(t) is thus obtained as the Prony series

R(t) =N∑

α=1

Eαe−t

t∗α (3.40)

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It appears readily that R(0) = E(∞), whereas R(∞) = 0 if t∗α < ∞ for all α.

Remark: If, in the α:th Maxwell chain, its spring stiffness is infinite (Eα = ∞), then

R(t) does not exist. 2

We may now invert (R)∗ in (3.39) to obtain (C)∗. Hence,

(ǫ)∗ = (C)∗(σ)∗, (C)∗ =1

(R)∗=

(N∑

α=1

Eαt∗αs

t∗αs + 1

)−1

(3.41)

It appears that no simple expression of C(t) can be found, except in special cases. Below

we (re)consider the Maxwell and Kelvin models.

Maxwell model as special case of linear standard model

The Maxwell model (which represents a fluid) is obtained from the Linear Standard Model

by the special choice of one single chain with

E1 ≡ E and t∗1 ≡ t∗ (3.42)

We then obtain directly from (3.39) and (3.41) the expressions

(R)∗ =Et∗s

t∗s + 1, (C)∗ =

1

E

(

1 +1

t∗s

)

(3.43)

which gives

R(t) = Ee−t

t∗ , C(t) =1

E

(

1 +t

t∗

)

(3.44)

We note that E(∞) = E, whereas E(0) = 0.

Kelvin model as special case of linear standard model

The Kelvin model (which represents a solid), as shown in Figure 3.6(a), is obtained from

the Linear Standard Model by the choice of two chains with

E1 = ∞, E2 = E and µ1 = µ , µ2 = ∞ (3.45)

and is shown in Figure 3.6. This is a more “tricky” model, since (R)∗ can not be obtained

directly from (3.39). Rather, we consider Figure 3.6 to obtain

σ∗ = σ∗1 + σ∗

2 = µsǫ∗ + Eǫ∗ = E(1 + t∗s)ǫ∗ with t∗

def=

µ

E(3.46)

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which gives

(R)∗ = E(1 + t∗s), (C)∗ =1

E

(1

1 + t∗s

)

(3.47)

Remark: The result in (3.47)2 can be obtained directly from (3.41). 2

We conclude that R(t) does not exist, whereas

C(t) =1

E

(

1 − e−t

t∗

)

(3.48)

Moreover, we note that E(∞) = ∞, whereas E(0) = E.

Three-parameter model as special case of linear standard model

The Three-parameter model (which is also called the Standard Solid in the literature),

shown in Figure 3.6(b), is the simplest special case of the Linear Standard Model.3 It is

obtained by the choice of two chains with

E1 = E(∞) − E(0), E2 = E(0) and t∗1 = t∗, t∗2 = ∞ (3.49)

From (3.39), the expression for (R)∗ is given as

(R)∗ =(E(∞) − E(0))t∗s

1 + t∗s+ E(0) (3.50)

and from (3.40) we obtain

R(t) = (E(∞) − E(0))e− t

t∗ + E(0) (3.51)

We note that E(0) is the value of R(t) at very slow loading, i.e. for t = ∞. Likewise, E(∞)

is the value of R(t) at very rapid loading corresponding to t = 0.

Homework: Find the explicit expression for C(t). 2

3.3.4 Backward Euler method for linear standard model

When it is difficult to find the inverse from Laplace transform space back to the time-

domain, the constitutive relations may instead be integrated in a step-by-step fashion.

Here we shall apply the Backward Euler (BE) method to the Linear Standard Model in

the case of prescribed strain history.

3It is too simple to represent realistic solid behavior.

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E2 = E µ2 ∞=

µ1 = µE1 = ∞

σ σ

( )

( )

E2 µ2 ∞=

µ1 = µ

E1σ σ

( )

(a)

(b)

Figure 3.6: (a) Kelvin model, (b) Three-parameter model.

Applying BE to the rate equations (3.34), we first obtain

n+1ǫvα = nǫv

α +∆t

µα

n+1σvα (3.52)

which may be combined with (3.32)2 to give

n+1σα = nσvα + Ev

α∆ǫ, 0σα = 0 (3.53)

where

nσvα = aα(∆t) nσα, Ev

α = aα(∆t)Eα with aα(∆t) =

(

1 +∆t

t∗α

)−1

(3.54)

The (total) stress n+1σ is then obtained from (3.32)1 as

n+1σ =N∑

α=1

n+1σα = nσv + Ev∆ǫ (3.55)

where

nσv =

N∑

α=1

nσvα, Ev =

N∑

α=1

Evα (3.56)

Remark: Pure elastic response is obtained when t∗α → ∞, in which case we obtain from

(3.54), (3.55) and (3.33):

n+1σ = nσ + E(∞)∆ǫ 2 (3.57)

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Maxwell model

In the special case of the Maxwell model, which is obtained if we set N = 1 with E1 ≡ E

and t∗1 ≡ t∗, we obtain from (3.53) and (3.55)

n+1σ = a(∆t)n+1σtr with a(∆t) =

(

1 +∆t

t∗

)−1

(3.58)

where n+1σtr is the “elastic trial stress” defined as

n+1σtr = nσ + E∆ǫ (3.59)

3.4 Linear viscoelasticity — Structural analysis

3.4.1 Structural behavior

The behavior of a linear viscoelastic structure will depend on its statical (in)det-

erminacy. Here, we shall outline some general features, which will be illustrated later in

conjunction with the discussion of truss structures.

Isostatic structures

For an isostatic structure, the stresses are (by definition) uniquely determined by the

applied load. For the important special case when the load is constant in time (after initial

step loading), the stresses are also constant. Hence, the entire structure is subjected to a

state of pure creep. If, in addition, the creep functions of every material point are affine,

i.e. a common relative creep function C(t) can be identified, then C(t) applies also to the

entire structural response.

Hyperstatic structures

For a hyperstatic structure, the stresses are not uniquely determined by the applied load

but depend on the material properties. Hence, the stresses will generally redistribute

with time even for a load that is constant in time. This phenomenon is denoted structural

relaxation. The exception is, again, the situation when a common relative creep function

C(t) for each material point can be identified. Then the structural relaxation is zero (for

initial step loading), and C(t) applies to the entire structural response.

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3.4 Linear viscoelasticity — Structural analysis 43

3.4.2 Solution strategies

Solution based on elastic analogy in transform space

It follows from the relations (3.22) and (3.23) that linear viscoelasticity becomes quite

analogous to linear elasticity upon Laplace-Carson transformation. As a consequence,

it is always possible (in theory) to take advantage of this fact when solving structural

problems. The equivalent linear elastic problem in transform space is then solved (an-

alytically or numerically), which is followed by inversion back to the time-domain. For

real structures, such inverse transformation must generally be carried out numerically. A

powerful algorithm was developed by Talbot (1979). The entire procedure is illustrated

in Figure 3.7.

Solution

Viscoelasticsolution

Viscoelasticproblem

Equivalent elasticproblem

Elasticsolution

Laplace - Ctransform

Inversetransform

Figure 3.7: Solution procedure for viscoelastic problem based on elastic analogy.

Solution based on numerical integration

The “elastic analogy” strategy outlined above may be efficient when the response is sought

only at one point (or a few points) in time. In most cases, however, it is desirable to know

the entire time response, which makes it more advantageous to employ a step-by-step

procedure in time. This means that the constitutive relations are solved by numerical

integration.

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3.4.3 Analysis of truss — Elastic analogy

Here we shall briefly repeat the crucial relations and steps in the (finite element) analysis

of a truss structure built of uniform bars of linearly elastic material. The truss structure

under consideration, which is depicted in Figure 3.8, is subjected to time dependent

loads collected in the load vector P (t). The energy-conjugated displacement components,

collected in p(t), are sought.

P1

P22

bar element}p

1p

node

normal force N

elongation n

length L

i

i

i

Figure 3.8: Truss structure (with used notation).

Each uniform bar is assumed to have the length Li, cross-section area Ai and relaxation

function Ri(t). Its elongation and normal force are denoted ni and Ni respectively. After

applying the Laplace-Carson transform, we obtain the following constitutive relations for

the i:th bar (in complete analogy with the corresponding linear elastic bar):

(ni)∗ =

Li

(Ri)∗Ai(Ni)

∗ or (Ni)∗ =

(Ri)∗Ai

Li(ni)

∗ (3.60)

We may collect these relations for all bars in the matrix relations

(n)∗ = (F e)∗(N)∗ or (N)∗ = (Se)

∗(n)∗ (3.61)

where (F e)∗ and (Se)

∗ are the diagonal compliance and stiffness matrices, respectively,

in transform space of the element assembly (denoted by subindex “e”). More explicitly,

Vol 0 March 7, 2006


we may rewrite, for example, the flexibility relation in (3.61)1 as

(n1)∗

(n2)∗

·

=

L1

(R1)∗A10 ·

0 L2

(R2)∗A2·

· · ·

(N1)∗

(N2)∗

·

(3.62)

Next, we shall outline the structural analysis pertinent to a statically determinate (iso-

static) and statically indeterminate (hyperstatic) truss, respectively.

Analysis of isostatic truss

From equilibrium of the joints (nodes) we obtain the relations

(P )∗ = AT(N)∗ (3.63)

which in the case of an isostatic truss has the unique solution

(N)∗ = B(P )∗, B = (AT)−1 (3.64)

It is noted that the matrix AT represents only geometric relations. (In fact, A is the

natural transformation, that is defined below for the hyperstatic truss.)

From virtual work considerations (Clebsch’s theorem), we may establish the kinematically

dual relation of (3.63) as:

(p)∗ = BT(n)∗ (3.65)

which may be combined with (3.61) and (3.64) to give the structural flexibility relation

in transform space:

(p)∗ = BT(F e)∗B(P )∗ = (F )∗(P )∗ with (F )∗ = BT(F e)

∗B (3.66)

as shown in Figure 3.9. Considering (3.64), we immediately see that this relation can be

backtransformed in trivial fashion to the time-domain as

N (t) = BP (t) (3.67)

which is unaffected by the viscoelastic properties. In particular, we obtain

P (t) = P 0H(t) ⇒ N(t) = N 0H(t) with N 0 = BP 0 (3.68)

where N 0 is the instantaneous normal force that is uniquely determined (from equilib-

rium) by the suddenly applied load P 0 at t = 0.

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As to the structural displacements p(t), it appears from (3.66) that no simple inversion

can be obtained in general. However, in the special case that the relaxation functions are

affine, i.e.

(Ri)∗(s) = Ei(R)∗(s), i = 1, 2, . . . (3.69)

where ¯(R)∗(s) is the transform of a common relative relaxation function R(t), then we

may conclude from (3.62) and (3.66) that

(p)∗ =1

(R)∗F (P )∗ = F (C)∗(P )∗ with (C)∗ =

1

(R)∗(3.70)

where F represents the elastic structural flexibility activated for a suddenly applied load.

For example,

P (t) = P 0H(t) ⇒ (p)∗ = (C)∗p0 with p0 = FP 0 ⇒ p(t) = C(t)p0 (3.71)

where p0 is the initial elastic response due to the suddenly applied load P 0 at t = 0.

Figure 3.9: Transformation diagram for structural analysis based on elastic analogy (Inner

loop is well-defined only for isostatic structures).

Analysis of hyperstatic truss

In the case of a hyperstatic truss the equilibrium equations have no unique solution

(N)∗. The dual kinematic relation is obtained from virtual work considerations (Clebsch’s

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theorem) as

(n)∗ = A(p)∗ (3.72)

We may combine this relation with (3.63) and (3.61)2 to give the structural stiffness

relation in transform space:

(P )∗ = AT(Se)∗A(p)∗ = (S)∗(p)∗ with (S)∗ = AT(Se)

∗A ⇒ (p)∗ = ((S)∗)−1(P )∗

(3.73)

as shown in Figure 3.9. When (p)∗ has been calculated, we may obtain (N)∗ from (3.61)2,

(3.72) and (3.73) as

(N)∗ = (Se)∗A((S)∗)−1(P )∗ (3.74)

No simple inversion back to the time domain can be obtained in general. The exception

is the case when the relaxation functions are affine, as defined in (3.69). We then obtain

(Se)∗ = Se(R)∗ and (S)∗ = S(R)∗ (3.75)

whereby

(p)∗ = S−1(C)∗(P )∗ and (N)∗ = SeAS−1(P )∗ ⇒ N(t) = SeAS−1P (t) (3.76)

and it is noted that N(t) is unaffected by the viscoelastic properties. In particular, we

obtain

P (t) = P 0H(t) ⇒ p(t) = C(t)p0, N(t) = N 0 (3.77)

where p0 is the initial elastic response and N 0 is the corresponding normal force due to

the suddenly applied load P 0 at t = 0.

Remark: In the general situation we obtain for a step loading:

(N)∗ = (Se)∗A((S)∗)−1P 0 ⇒ N 6= 0 (3.78)

and it follows that there will be a time-dependent redistribution of stresses even for the

constant load P 0. This phenomenon is termed structural relaxation. 2

3.4.4 Analysis of truss — numerical integration

We shall consider the Linear Standard Model. From (3.55) we obtain the normal forcen+1Ni

def= Ni in the i:th bar:

Ni = nNvi +

Evi Ai

Li∆ni (3.79)

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This relation can be written in matrix form for all bars as

N = nN v + Sve ∆n (3.80)

where

Sve =

Ev1A1

L10 ·

0Ev

2A2

L2·

· · ·

(3.81)

The pertinent equilibrium and kinematic relations for the truss structure are given as

P = AT N and ∆n = A ∆p (3.82)

By combining these relations with the constitutive relation (3.79), we obtain the structural

stiffness relation

P = AT nN v + Sv ∆p with Sv = ATSveA (3.83)

where Sv is the “algorithmic” structural stiffness matrix. We may now solve for ∆p from

(3.83) to obtain the updated displacement vector n+1pdef= p as

p = np + ∆p with ∆p = (Sv)−1( P − AT nN v) (3.84)

A few special loading situations will be considered next:

Step loading

If ∆t = 0, or t∗ = ∞ (for all α), then Sv = Se (the elastic structural stiffness matrix).

We then obtain from (3.84)

∆p = (Se)−1∆P with ∆P = P − nP (3.85)

This is purely elastic response due to a step loading ∆P , that is applied instantaneously

at the time t = tn.

Creep after initial step loading

The load is applied suddenly at t = 0, whereafter it is held constant, i.e.

P (t) = P 0H(t), 0P = P (0) = P 0 (3.86)

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This gives the algorithm:

p = np + ∆p with ∆p = (Sv)−1(P 0 − AT nN v), n ≥ 0 (3.87)

where the initial solution is given by

0p = (Se)−1P 0 and 0N = SeeA

0p (3.88)

3.4.5 Analysis of beam cross-section — Elastic analogy

A beam cross-section is intrinsically statically indeterminate if the curvature κc is taken as

the “displacement” and the section moment M is taken as the “load”. The structural be-

havior of the truss (discussed above) then carries over to the beam cross-section to a large

extent. For example, stress redistribution across the height of the beam is to be expected,

unless a common relative creep function C(t) can be identified for the entire cross-section.

This is, in particular, the situation when the elastic modulus has an arbitrary distribution,

while the relaxation times t∗α are the same, as will be shown below.

For simplicity, we consider only double-symmetric cross-sections and pure bending.4 A

typical cross-section for a sandwich-structure is shown in Figure 3.10. A soft core with

(small) t(1)∗ is “sandwiched” between hard surface layers with (large) t

(2)∗ .

Using the elastic analogy, we may derive the stiffness relation in the Laplace transform

variables as

(M)∗ = (S)∗(κc)∗ with (S)∗ =

∫

A

(R)∗(z)z2dA (3.89)

and the stress transform (σ)∗(z) is obtained as

(σ)∗(z) = (R)∗(z)(ǫ)∗(z) = (R)∗(z)(κc)∗z =

(R)∗(z)(M)∗

(S)∗z (3.90)

with (κc)∗ calculated from (3.89) for given (M)∗.

In the special case that the relaxation function R(t) has an affine distribution in z, as

defined in (3.69), i.e.

(R)∗(z, s) = E(z)(R)∗(s) (3.91)

then we obtain from (3.89)

(M)∗ = Se(κc)∗(R)∗ with Se =

∫

A

E(z)z2dA (3.92)

4This means that the neutral axis is always identical with the symmetry axis, from which the coordinate

z is introduced.

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b

( a ) ( c )( b )

hh1

t = 0σ, t = ∞σ,z

Figure 3.10: (a) Typical cross-section of sandwich beam. Stress distribution for Maxwell

material in the core for (b) t = 0 and (c) t = ∞.

and from (3.88)

(σ)∗(z) =E(z)(M)∗

Sez ⇒ σ(z, t) =

E(z)M(t)

Sez (3.93)

In this case it is clear that the distribution of σ over the cross-section is unaffected by the

viscoelastic properties. In particular, we obtain

M(t) = M0H(t) ⇒ κc(t) = C(t) κc,0 and σ(z, t) = σ0(z) (3.94)

where κc,0 is the initial elastic curvature in response to the suddenly applied moment at

t = 0, and σ0(z) is the initial (elastic) stress distribution:

κc,0 =M0

Se, σ0(z) =

E(z)M0

Sez (3.95)

Now, returning to the sandwich beam, we conclude that

(S)∗ = (R1)∗I1 + (R2)

∗I2, I1 =bh3

1

12, I2 =

b(h3 − h31)

12(3.96)

where Ri(t), i = 1, 2, are the relevant relaxation functions for the inner core and surface

layers, respectively. The moments of inertia Ii, i = 1, 2, are defined w.r.t. the neutral

axis, and we note that I = I1 + I2.

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Example: Sandwich beam with Maxwell material

Consider the case when the core of the sandwich beam responds as a Maxwell-model,

whereas the surface layers are elastic (t(2)∗ = ∞). For simplicity, the (unrealistic) assump-

tion is made that the E-modulus is the same, i.e. E1 = E2 = E. Since the relaxation

function for the Maxwell model is given by its transform

(R)∗(s) = Et∗s

t∗s + 1(3.97)

we obtain

(S)∗ = EI1t∗s

t∗s + 1+ EI2 ⇒ ((S)∗)−1 =

1

EI2

[

1 −

(

1 −I2

I

)t∗s

t∗s + I2I

]

(3.98)

We have phrased ((S)∗)−1 in a form that provides directly for the inverse transform. With

M0 applied at t = t0, we obtain

κc(t) =M0

EI2

[

1 −

(

1 −I2

I

)

e−I2I

tt∗

]

(3.99)

with the limiting values

κc(0) =M0

EI, κc(t) =

M0

EI2when t = ∞ (3.100)

To work out the corresponding stress-distribution with time is left as homework, (cf.

Figure 3.10(b,c)). Consider the cases I2 → 0 (homogenous Maxwell model) and I2 → I

(homogeneous elasticity).

3.4.6 Analysis of double-symmetric beam cross-section — Nu-

merical integration

Numerical integration in time and space (over the beam cross-section) is a versatile and

effective technique, which is general w.r.t. the choice of the material properties. For

this purpose, we consider the double-symmetric cross-section as shown in Figure 3.11.

The cross-section is subdivided into 2nint lamellas of width bi and thickness ∆zi, where

subindex int stands for “integration”.

The moment n+1Mdef= M at time t = tn+1 is given as

M =

∫ h/2

−h/2

σ(z)zb(z)dz ≈ 2

nint∑

i=1

σizi∆Ai with ∆Ai = bi∆zi (3.101)

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symm

z i

∆ z i

h2

h2

z

b i

Figure 3.11: Double-symmetric cross-section with lamella at z = zi.

where n+1σidef= σi is the updated stress value in the i:th lamella.5 Again, we shall consider

the Linear Standard Model, for which σ was given in (3.55). We thus use the relation

σ(z) = nσv(z) + Ev(z)∆ǫ(z) with ∆ǫ(z) = z∆κc (3.102)

This expression is inserted into (3.101), which gives the equation

M = nMv + Sv∆κc (3.103)

where

nMv =

∫ h/2

−h/2

nσv(z)zb(z)dz = 2

nint∑

i=1

nσvi zi∆Ai (3.104)

and

Sv =

∫ h/2

−h/2

Ev(z)z2b(z)dz = 2

nint∑

i=1

Evi Ii with Ii = z2

i ∆Ai (3.105)

from which ∆κc is solved to give the updated curvature n+1κcdef= κc

κc = nκc + ∆κc with ∆κc = (Sv)−1(M − nMv) (3.106)

The following special situations are considered:

5Subindex i (referring to the i:th lamella) must not be confused with subindex α in (3.53), which

refers to the α:th Maxwell chain in the Linear Standard Model.

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3.5 Nonlinear viscoelasticity — Constitutive modeling 53

Step loading

If ∆t = 0, or t∗α = ∞ (for all α), then Sv = Se (the fully elastic response). It then follows

that nMv = nM and

∆κc = (Se)−1∆M with ∆M = M − nM (3.107)

Creep after initial step loading

Consider the step loading at t = 0:

M(t) = M0H(t) and 0M = M(0) = M0 (3.108)

which gives

κc = nκc + ∆κc with ∆κc = (Sv)−1(M0 −nMv), n ≥ 0 (3.109)

where the initial solution is given by

0κc = (Se)−1M0 and 0σ(z) = E0(z)κcz (3.110)

3.5 Nonlinear viscoelasticity — Constitutive model-

ing

3.5.1 General characteristics

For a nonlinear viscoelastic material, the creep curves are nonlinear functions of the

initially applied stress σ0, as indicated in Figure 3.12. In practice, the major effort has been

spent on modeling only the stationary creep stage, which is also indicated in Figure 3.12.

(In order to be able to model both the transient and stationary stages with a unified

model, one has to resort to hardening viscoplasticity, as discussed in Chapter 5.)

It is clear that ǫ = ǫ(σ0, t), where the dependence on t may be of quite general character,

i.e.

ǫ(t) 6= C(t)σ0 (3.111)

Such behavior is characterized by nonlinear isochrone curves for the creep behavior, as

shown in Figure 3.13.

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σ

σ0

t

( b )( a )

t

ε

Figure 3.12: (a) Constant stress loading, (b) Creep response due to nonlinear viscoelastic

behavior.

ε

σ0t = 0

t 1 > 0

t 2 > t 1

Figure 3.13: Nonlinear isochrone curves.

3.5.2 Norton creep law

The most commonly used creep law for the modeling of stationary creep in metals, which

is due to Norton (1929), is obtained by generalizing the Maxwell model. This may be

done in such a fashion that the rate law for the viscous strain is a simple power law in

the stress:

ǫv =1

τ∗

(|σ|

σc

)nc σ

|σ|(3.112)

where σc is the creep modulus, nc is the creep exponent and τ∗ is a relaxation time (whose

value is normally chosen constant). The value of σc will depend on the choice of nc.

In addition, σc is strongly temperature-dependent and σc → 0 when the temperature

approaches the melting point (T → Ts). The creep exponent nc is less sensitive to tem-

perature increase, but nc → 1 when T → Ts. Corresponding values of σc and nc can be

found in, for example, Hult (1984), p 106.

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Remark: From (3.112) follows that the same creep rate is modelled in tension and

compression (for the same magnitude of stress). 2

Upon combining (3.112) with Hookes law in (3.2), we obtain the governing constitutive

equation

σ +E

τ∗

(|σ|

σc

)nc σ

|σ|= Eǫ (3.113)

Remark: Equations (3.112) and (3.113) represent the conventional choice of parameters

based on the a priori chosen value of τ∗. Another possibility is to conform directly to the

Maxwell model by postulating the creep law

ǫv =1

t∗

(|σ|

E

)nc σ

|σ|(3.114)

where E is the (static) value of the elasticity modulus. We then have the relation

t∗ = τ∗

(σc

E

)nc

(3.115)

in which case t∗ → 0 when σc → 0 for high temperature. We may then rewrite (3.113) as

σ +E

t∗

(|σ|

E

)nc σ

|σ|= Eǫ (3.116)

which format is used subsequently. The Maxwell model s readlily retrieved when nc = 1

2

Creep

In the creep situation, when σ = σ0 > 0 is held constant (after rapid loading), we obtain

the solution of (3.113), with the notation in (3.115), as

ǫ(t) =1

t∗

(σ0

E

)nc

, ǫ(t) =σ0

E+(σ0

E

)nc t

t∗, t > 0 (3.117)

which is characteristic for stationary (or stage II) creep, as shown in Figure 3.14(a).

Relaxation

In the relaxation situation, when ǫ = ǫ0 > 0 is held constant (after rapid loading), we

obtain from (3.113) the problem

σ +E

t∗

( σ

E

)nc

= 0, σ(0) = σ0 = Eǫ0 (3.118)

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(a) (b)

ǫ σ

σ0

σ0E 1

(σ0E

)nc

tt∗

tt∗

nc = 1

nc > 1

Figure 3.14: (a) Creep and (b) Relaxation curves for a Norton-material

whose solution is

σ(t) =

σ0 e−t

t∗ , nc = 1[

σ−(nc−1)0 + (nc−1)

Enc−1tt∗

]− 1nc−1

, nc 6= 1(3.119)

It appears that σ → 0 when t → ∞. Typical results are shown in Figure 3.14(b).

3.5.3 Backward Euler method for the Norton creep law

In practice, most calculations require numerical integration of the constitutive relation in

(3.116). Applying the Backward Euler rule, we obtain

n+1σ = n+1σtr − Eµn+1σ

|n+1σ|with µ =

∆t

t∗

(|n+1σ|

E

)nc

(3.120)

where n+1σtr is the “elastic trial stress” given in (3.59). We may rearrange in (3.120) to

obtain (

1 +Eµ

|n+1σ|

)

︸︷︷︸

>0

n+1σ = n+1σtr (3.121)

which shows that sign(n+1σ) = sign(n+1σtr). Now, taking the absolute value of both sides

of (3.121) gives

σe + Eµ = σtre with σe

def= |σ|, σtr

edef= |σtr|, n+1σ

def= σ (3.122)

and it is noted that σe ≤ σtre since µ ≥ 0. Upon introducing the function

η(σe) =(σe

E

)nc

⇒ µ =∆t

t∗η(σe) (3.123)

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we may combine (3.122), (3.123) to obtain a set of equations from which σe and µ can

be solved. Two different iterative schemes will be discussed below. After convergence of

such a scheme it is possible to obtain n+1σ as

n+1σ =

{

σe if n+1σtr > 0

−σe if n+1σtr < 0(3.124)

Solution method 1: Fixed point iterations

For k = 0, 1, . . . solve in sequence

σ(k+1)e = σtr

e − Eµ(k), µ(k+1) =∆t

t∗η(σ(k+1)

e ) (3.125)

A suitable starting guess is

µ(0) =∆t

t∗η(|nσ|) (3.126)

Iterations are stopped when |σ(k+1)e − σ

(k)e | and |µ(k+1) − µ(k)| are sufficiently small.

A few iterations are normally sufficient. If very small time steps ∆t are taken, it is possible

to avoid iteration entirely and to accept the semi-explicit expression that is defined for

k = 0, i.e. we may setn+1σ ≈ n+1σ(1) (3.127)

Solution method 2: Newton iterations

Recall (3.122)and (3.123) as the set of equations

Rσ(σe, µ) = σe − σtre + Eµ = 0

Rµ(σe, µ) = η(σe) −t∗∆t

µ = 0(3.128)

It is noted that nonlinearity is embedded in η(σe) only. We may rephrase (3.128) in matrix

form as

R(X) = 0 with Rdef=

[

Rσ

Rµ

]

, Xdef=

[

σe

µ

]

(3.129)

Applying Newton iterations to solve R(X) = 0, we need the Jacobian J

Jdef=

∂R

∂X=

[∂Rσ

∂σe

∂Rσ

∂µ∂Rµ

∂σe

∂Rµ

∂µ

]

=

[

1 E

η′(σe) − t∗∆t

]

(3.130)

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with

η′(σe)def=

dη

dσe

(σe) =nc

E

(σe

E

)nc−1

(3.131)

The inverse of J can be obtained explicitly as

J−1 =1

ha

[t∗∆t

1η′

E 1η′

1 − 1η′

]

, hadef= E +

t∗∆t

1

η′(3.132)

The Newton scheme becomes: For k = 0, 1, . . . compute

X(k+1) = X(k) + dX with dX = −(J (k)) −1R(k) (3.133)

which is stopped when |R(k)| is sufficiently small.

Remark: The characteristics of the Norton law is embedded in µ. It is clear that any

other nonlinear viscoelastic law can be phrased in the format (3.120) by using the appro-

priate expression for µ. 2

3.6 Nonlinear viscoelasticity — structural analysis

3.6.1 Structural behavior

Like in the case of linear viscoelasticity, the structural response will depend on the statical

(in)determinacy. Structural relaxation will always take place (even if all truss members

have the same material properties) for a hyperstatic structure. In practice, most engi-

neering analyses for design purposes deal only with the stationary state that is achieved

a long time after a time-independent structural load has been applied to the structure.

Because of the inherent material nonlinearity, the analysis of the transient structural

behavior must be based on numerical integration of the constitutive relations. Iterations

are needed to find the incremental displacements in each time step.

3.6.2 Analysis of truss

We recall the expression in (3.120) for the updated stress pertinent n+1σidef= σi to the

Norton law:

σi = ai σtri with a = 1 −

Eµ

|σtr|and µ

def=

∆t

t∗

(|σ|

E

)nc

(3.134)

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3.6 Nonlinear viscoelasticity — structural analysis 59

The normal force in the i:th bar is then given as

Ni = Ai σi (3.135)

and the normal forces are collected in the column matrix N(∆n).

Remark: In the case of linear elasticity, we have a = 1. Moreover, the Maxwell model is

retrieved at the choice nc = 1, whereby we obtain

σ = σtr

(

1 −∆t

t∗

|σ|

|σtr|

)

; a =

(

1 +∆t

t∗

)−1

(3.136)

which is the expression for a given in (3.58)2. 2

The equilibrium and kinematic relations for the truss are recalled as

P = AT N and ∆n = A ∆p (3.137)

Iterations are required in order to find the solution ∆p from (3.137)1. For given loadn+1P

def= P , we may devise the following Newton procedure (k being the iteration count):

∆p(k+1) = ∆p(k) + δp

where δp is the solution of the linear set of equations:

Sv(k)a δp = −(P − AT N (k)) with Sv

a =∂(AT N)

∂(∆p)(3.138)

and with ∆p(0) chosen as the converged value of ∆p in the previous timestep.

In (3.138), we introduced the algorithmic tangent stiffness (ATS) matrix Sva, which is a

nonlinear function of the incremental solution ∆p for a given timestep.

Lemma: The ATS-matrix Sva is given as

Sva = ATSv

eaA with Svea = diag

[A1E

va1

L1,

A2Eva2

L2, . . .

]

(3.139)

where, for each bar, the algorithmic tangent stiffness modulus Eva is defined as

Eva =

E

1 + nc∆tt∗

(|σ|E

)nc−1 2 (3.140)

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Proof: From the definition in (3.138) follows that

Sva = AT ∂(N )

∂(∆n)A = AT ∂(∆N )

∂(∆n)A (3.141)

where it was used that d(∆n) = A d(∆p). Moreover,

∂(∆N )

∂(∆n)= diag

[d(∆N1)

d(∆n1),

d(∆N2)

d(∆n2), . . .

]

(3.142)

For each bar, we obtain

d(∆N)

d(∆n)= A

d(∆σ)

d(∆ǫ)

d(∆ǫ)

d(∆n)=

AEva

Lwith Ev

a =d(∆σ)

d(∆ǫ)(3.143)

where it was used that d(∆ǫ) = d(∆n)/L. An explicit expression for Eva pertinent to the

Norton law is obtained upon differentiating the relation (3.120). We obtain

d(∆σ) = E d(∆ǫ) − E d(µ)σ

|σ|(3.144)

with

d(µ) = nc∆t

t∗

1

E

(|σ|

E

)nc−1σ

|σ|d(∆σ) (3.145)

Upon rearranging terms in this expression, we obtain

d(∆σ) =E

1 + nc∆tt∗

(|σ|E

)nc−1 d(∆ǫ) (3.146)

which gives Eva in (3.140). 2

The crudest approximation of Sva is obtained if Ev

a is replaced by the elastic modulus E,

which means that Sva is replaced by the elastic stiffness matrix Se.

3.6.3 Analysis of beam cross-section — Stationary creep

Consider the situation when a moment M0 is applied suddenly at t = 0. Because of the

structural relaxation for t > 0, the stress distribution will change from the linear elastic

(initially) to a nonlinear one in a fashion that is shown schematically in Figure 3.15

until a steady state is achieved after long time. This situation of stationary creep is, in

Vol 0 March 7, 2006


Figure 3.15: (a) Non-stationary creep of Norton material, (b) Stationary stress distribu-

tion in double-symmetric cross-section.

practice, the only one that can be analyzed analytically. Again, only double-symmetric

cross-sections are considered (for simplicity).

Since no elastic strain develops at the steady state, we have from (3.111)

ǫ → ǫ∞ =1

t∗

(|σ∞|

E

)nc σ∞

|σ∞|, ǫ∞(z) = κc∞z (3.147)

where we used the notation ǫ∞ and σ∞ for the steady state values at t = ∞. From (3.147),

we conclude that

|ǫ∞| =1

t∗

(|σ∞|

E

)nc

⇒ |σ∞| = E|z|1

nc (t∗κc∞)1

nc (3.148)

which may be inserted into (3.147) to give

σ∞ = E(t∗κc∞)1

nc |z|1

nc−1z (3.149)

Upon inserting (3.149) in the expression for the moment, we obtain

M0 =

∫

A

σ∞zdA = E(t∗κc∞)1

nc In with In =

∫

A

|z|1

nc+1dA (3.150)

Finally, we may solve for κc∞ from (3.150) and insert into (3.149) to obtain the stress

distribution from the generalization of the Navier formula (for elastic response) as follows:

σ∞ =M0

In|z|

1nc

−1z (3.151)

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Remark: In is a generalized moment of inertia, and the usual moment of inertia is define

as I = I1 (for nc = 1). In this latter case it appears that the classical linear stress

distribution, which is pertinent to linear elastic response, is retrieved. 2

Remark: The stress distribution in (3.151) is the same as for the nonlinear elastic Bach

material defined by the stress-strain law

ǫ = ǫ0

(|σ|

σ0

)nc σ

|σ|; σ =

M

In|z|

1nc

−1 z (3.152)

which is depicted in Figure 3.16. The reader should show this as homework! 2

CC

Figure 3.16: Nonlinear elastic stress-strain law characterizing a Bach material.

In the case of a cross-section with height h, the maximum stress (for z = h/2) at t =

∞ (σmax∞ ) and at t = 0 (σmax

0 ) are given as

σmax∞ =

M0

In

(h

2

) 1nc

and σmax0 =

M0

I1

h

2;

σmax∞

σmax0

=I1

In

(h

2

) 1nc

−1

(3.153)

Example: Rectangular cross-section

For a rectangular cross-section, we obtain

In =2b

1nc

+ 2

(h

2

) 1nc

+2

⇒σmax∞

σmax0

≤1nc

+ 2

3(3.154)

and it follows that2

3≤

σmax∞

σmax0

≤ 1 (3.155)

The lower bound is obtained for nc = ∞, which corresponds to a rectangular stress-

distribution. The typical distributions are shown in Figure 3.17.

Remark: The rectangular stress-distribution in Figure 3.17 is also obtained for rigid-

perfectly plastic response, as discussed in Chapter 4. 2

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C C C

Figure 3.17: Stress distributions in beam cross-section at stationary creep for different

values of the creep exponent nc.

3.6.4 Analysis of double-symmetric beam cross-section — Nu-

merical integration

In order to trace the transient behavior, it is necessary to resort to numerical integration.

This is carried out in much the same fashion as for linear viscoelasticity, that was outlined

in Subsection 3.4.6. The main difference is that iterations will now be required, since the

problem is nonlinear.

For a discretized double-symmetric cross-section we thus obtain the moment n+1Mdef= M

at time t = tn+1 as

M =

∫ h/2

−h/2

σ(z)zb(z)dz ≈ 2

nint∑

i=1

σizi∆Ai with ∆Ai = bi∆zi (3.156)

The updated stress n+1σdef= σ was given in (3.121), i.e

σ = σtr − Eµσtr

|σtr|(3.157)

where

σtr = nσ + E∆ǫ with ∆ǫ = z∆κc (3.158)

As discussed previously, σ must usually be calculated in an iterative fashion for given σtr.

Iterations are used to find the solution ∆κc from (3.156). For a given moment M0 in a

creep situation, we may devise the Newton procedure

∆κ(k+1)c = ∆κ(k)

c + δκc (3.159)

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where δκc is obtained from

δκc = −(Sv(k)a )−1( M (k) − M0) with Sv

a =d(∆M)

d(∆κc)(3.160)

and with ∆κ(0)c chosen as the converged value of ∆κc in the previous timestep.

The algorithmic tangent bending stiffness Sva is obtained as follows: Assuming that σ =

σ(∆ǫ(∆κc)), we obtain, upon differentiation

d(∆σ) =d(∆σ)

d(∆ǫ)d(∆ǫ) = Ev

a zd(∆κc) (3.161)

where the expression of Eva pertinent to the Norton law was given in (3.140). With (3.141),

we then obtain

d(∆M) =

∫ h/2

−h/2

d(∆σ)zb(z)dz =

(∫ h/2

−h/2

Eva (z)z2b(z)dz

)

d(∆κc) (3.162)

Hence, we have shown that

Sva =

∫ h/2

−h/2

Eva (z)z2b(z)dz = 2

nint∑

i=1

(Eva )iIi with Ii = z2

i ∆Ai (3.163)

Figure 3.18 shows the stress relaxation with time for the rectangular cross-section sub-

jected to the moment M0 applied at t = 0. (The creep exponent is nc = 2.)

3.6.5 Analysis of single-symmetric beam cross-section — Nu-

merical integration

Let us, next, consider a single-symmetric cross-section, as shown in Figure 3.19. To

achieve greatest possible flexibility of the formulation, we assume that the y-axis is located

arbitrarily at distance h− and h+ from the upper edge and lower edge, respectively. The

deformation of the cross-section is now defined by the axial strain ξ at the level z = 0 (in

addition to the curvature κc. We thus compute the sectional normal force n+1Ndef= N and

the moment n+1Mdef= M as

N =

∫ h+

−h−

σ(z)b(z)dz =

nint∑

i=1

σi∆Ai (3.164)

M =

∫ h+

−h−

σ(z)zb(z)dz =

nint∑

i=1

σizi∆Ai (3.165)

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0

0

stress

z−co

ordi

nate

σ0max

h2

h2

t = 0

t = ∞

Figure 3.18: Stress relaxation with time in rectangular cross-section for the creep exponent

value nc = 2.

where nint is now the total number of integration points across the height of the beam

cross-section. The updated stress is still given by (3.157) and (3.158), where ∆ǫi is now

defined by

∆ǫi = ∆ξ + zi∆κc (3.166)

Hence, for given N = N0 and M = M0, we solve for ∆ξ and ∆κc from the system

N(∆ξ, ∆κc) = N0

M(∆ξ, ∆κc) = M0 (3.167)

Newton iterations give

∆ξ(k+1) = ∆ξ(k) + δξ, ∆κ(k+1)c = ∆κ(k)

c + δκc (3.168)

where δξ and δκc are obtained from the linear set of equations

[

Sv(k)a,NN S

v(k)a,NM

Sv(k)a,NM S

v(k)a,MM

][

δξ

δκc

]

= −

[

N (k) − N0

M (k) − M0

]

(3.169)

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The algorithmic stiffness moduli in (3.169) are obtained as follows: First, we obtain

d(∆σ) = Evad(∆ǫ) = Ev

ad(∆ξ) + Evazd(∆κc) (3.170)

Linearizing (3.167), we obtain

d(∆N) =

∫ h+

−h−

d(∆σ)b(z)dz =

(∫ h+

−h−

Eva (z)b(z)dz

)

d(∆ξ) +

(∫ h+

−h−

Eva (z)zb(z)dz

)

d(∆κc) (3.171)

d(∆M) =

∫ h+

−h−

d(∆σ)zb(z)dz =

(∫ h+

−h−

Eva (z)zb(z)dz

)

d(∆ξ) +

(∫ h+

−h−

Eva (z)z2b(z)dz

)

d(∆κc)(3.172)


Sva,NN =

∫ h+

−h−

Eva (z)b(z)dz =

nint∑

i=1

(Eva )i∆Ai (3.173)

Sva,NM =

∫ h+

−h−

Eva (z)zb(z)dz =

nint∑

i=1

(Eva )izi∆Ai (3.174)

Sva,MM =

∫ h+

−h−

Eva (z)z2b(z)dz =

nint∑

i=1

(Eva )iIi (3.175)

Assume now that the y-axis is located at the center of gravity, i.e.

∫ h+

−h−

zb(z)dz = 0 (3.176)

The simplest algorithm is obtained if we choose Eva = E, by which

Sva,NN = EA, Sv

a,NM = 0, Sva,MM = EI (3.177)

Hence, we obtain from (3.169)

δξ =1

EA(N0 −N (k)), δκc =

1

EI(M0 −M (k)) (3.178)

Remark: The expressions in (3.178) define the incremental change at step loading, for

which the response becomes purely elastic. 2

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3.7 Viscous damping and dynamic behavior 67

h−

h+

y

z

z

ǫ

κc

ξ

ǫ(z) = ξ + κc · z

σ

t > 0

t = 0 N0

M0

Figure 3.19: Single-symmetric cross-section.

3.7 Viscous damping and dynamic behavior

3.7.1 Preliminaries

The viscoelastic response can be assessed, and the used model calibrated, from obser-

vations made under quasistatic or dynamic conditions. Creep and relaxation are typical

quasistatic phenomena (where inertial forces can be ignored), whereas rapid loading gives

rise to inertial forces that must be accounted for. In the latter case, the viscoelastic dis-

sipation of energy results in damping of the response (as compared to the purely elastic

response). In the literature, the Kelvin model has traditionally been used to describe

the damped behavior of vibrating structures. This is usually denoted viscous6 damping,

which represents a pronounced dependence of the rate of loading (as we shall see later),

whereas the other extreme that the damping is not dependent on the rate of loading is

termed hysteretic damping. In conclusion, we may state that creep and damping are “two

sides of the same coin”.

The viscoelastic characteristics may be observed from free and forced vibrations. Free

vibrations occur after an initial disturbance of the static state of equilibrium and will

die out eventually. Forced vibrations, on the other hand, represent the stationary (har-

6Since all realistic models are generically of viscous character, we shall use this term henceforth without

specific reference to the Kelvin model.

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monic) motion due to (harmonic) loading. This is the only loading situation considered

henceforth. Moreover, we restrict our consideration to linear viscoelastic response.

3.7.2 Forced vibration of discrete system

Consider the uniform bar in Figure 3.20 with its mass ρL lumped to one end. The forced

vibrations are caused by the applied harmonic force

f(t) = fa cos(ωt) = Re{fF(t)} with fF(t) = fa eiωt (3.179)

where fa is the amplitude and ω is the angular frequency with which the load excites the

system7. Assuming uniform strain ǫ (= u/L) in the bar, we obtain the equation of motion

as

mǫ + σ = Re{faeiωt} with m = ρL2 (3.180)

It is convenient to solve (3.180) with f(t) replaced by the equivalent Fourier component

fF to obtain the corresponding ǫF and σF. Since the actual load is f(t) = Re{fF(t)}, we

obtain ǫ(t) = Re{ǫF(t)}, etc. We thus obtain from (3.180)

−mω2(ǫ)∗ + (σ)∗ = (f)∗ with (f)∗ = fa (3.181)

Upon introducing (σ)∗ = (R)∗(ǫ)∗ into (3.150), we solve for (ǫ)∗ to obtain

(ǫ)∗ =fa

(R)∗ − mω2= ǫae

iϕǫ; ǫF(t) = ǫae

i(ωt+ϕǫ) (3.182)

7For any variable u(t), we define its complex Fourier transform (u)∗(ω)

(u)∗(ω) = iω

∫∞

0

u(t)e−iωtdt = (u)∗R(ω) + i(u)∗I (ω)

where (u)∗R and (u)∗I are real-valued functions. The corresponding Fourier component uF is defined as

uF = (u)∗eiωt = uaei(ωt+ϕu)

where ua is the (real) amplitude and ϕu is the phase angle given from

ua =√

((u)∗R)2 + ((u)∗I )2, tan ϕu =

(u)∗I(u)∗R

It appears that the Fourier transform is identical to the Laplace transform upon setting s = iω. A more

explicit discussion of the use of complex variable technique for damped vibrations (including the Fourier

transform) is found in Akesson (1992).

Vol 0 March 7, 2006


L

u

ρLcos ωtfa

Figure 3.20: Uniform bar of viscoelastic material with lumped mass.

where

ǫa =fa

(R)∗R

([

1 − (ω

ωref)2

]2

+ η2

)− 12

, tan ϕǫ = −η

1 − ( ωωref

)2(3.183)

In order to obtain the expressions in (3.183), we introduced the following representation

for (R)∗:

(R)∗ = (R)∗R(1 + iη) = RaeiϕE with η =

(R)∗I(R)∗R

(3.184)

where η(ω) is the loss factor. Moreover, we introduced the reference frequency ωref(ω)

from the definition

ωref =

((R)∗Rm

) 12

(3.185)

and the amplitude Ra(ω) as

Ra = (R)∗R(1 + η2

) 12 , tanϕE = η (3.186)

We shall define E(0) = Ra(0) and E(∞) = Ra(∞) as the apparent elastic moduli for slow

(ω = 0) and rapid (ω = ∞) loading, respectively.

Remark: For a general viscoelastic model, both (R)∗R and (R)∗I are frequency-dependent.

Hence, it is concluded that ωref = ωref(ω), and the subindex “ref” might seem awkward.

However, we shall see later that ωref becomes the undamped eigenfrequency ω0 > 0 for

the classical Kelvin model. 2

With (3.182) and (3.184), we may now obtain (σ)∗ as

(σ)∗ = (R)∗(ǫ)∗ = Raǫaei(ϕǫ+ϕE) ≡ σae

iϕσ (3.187)

from which we conclude that

σa = Raǫa, ϕσ = ϕǫ + ϕE (3.188)

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Moreover, using (3.183) and (3.186), we obtain from (3.188) that

σa = fa

(

1 + η2

[1 − ( ωωref

)2]2 + η2

) 12

(3.189)

The results in (3.182) and (3.187) are shown schematically in Figure 3.21.

(R)

(R)

Figure 3.21: Complex representation of Fourier transforms (ǫ)∗ and (σ)∗ due to the loading

(f)∗ = fa cos ωt.

3.7.3 Energy dissipation

During a period T of loading (from t = t0 to t = t0 + T ), the internal dissipation of

mechanical energy is given as

Wdiss =

∫ t0+T

t0

σǫdt =

∫ t0+T

t0

Re{σF}Re{ǫF}dt (3.190)

where

σF = σaei(ωt+ϕǫ+ϕE), ǫF = iωǫF = ǫaωei(ωt+ϕǫ+

π2) (3.191)

Upon inserting (3.191) into (3.190), we obtain

Wdiss = ωσaǫa

∫ t0+T

t0

sin(ωt + ϕǫ) cos(ωt + ϕǫ + ϕE)dt = πσaǫa sin ϕE (3.192)

Vol 0 March 7, 2006


where it was used that ωT = 2π. However, since sin ϕE = η/√

1 + η2 we obtain with

(3.186) and (3.188)

Wdiss = πη(R)∗Rǫ2a = π(R)∗I ǫ

2a (3.193)

3.7.4 Evaluation of damping for the linear standard model

With the choice t∗N = ∞ in the Linear Standard Model (by which solid behavior is

represented), we obtain from (3.39)

(R)∗(ω) =

N−1∑

α=1

Eαt∗αiω

1 + t∗αiω+ EN = (R)∗R + i(R)∗I (3.194)

where

(R)∗R =N−1∑

α=1

Eα(t∗αω)2

1 + (t∗αω)2+ EN , (R)∗I =

N−1∑

α=1

Eαt∗αω

1 + (t∗αω)2(3.195)

We note that

(R)∗R(0) = EN ≡ E(0), (R)∗R(∞) =

N∑

α=1

Eα ≡ E(∞) (3.196)

and

(R)∗I (0) = (R)∗I (∞) = 0 ; η(0) = η(∞) = 0 (3.197)

which confirms the notation

Ra(0) = E(0) and Ra(∞) = E(∞) (3.198)

It also follows from (3.185) and (3.195)1 that

ω2ref =

N−1∑

α=1

ω2α(t∗αω)2

1 + (t∗αω)2+ ω2

(0) with ω2α =

Eα

m, ω2

(0) =EN

m≡

E(0)

m(3.199)

which gives

ωref(0) = ω(0) =

(E(0)

m

) 12

, ωref(∞) = ω(∞) =

(N−1∑

α=1

ω2α + ω2

(0)

) 12

=

(E(∞)

m

) 12

(3.200)

The typical behavior of η(ω) is shown in Figure 3.22, which also shows the critical fre-

quency defined as

ωηcr = arg[max η(ω)] (3.201)

Vol 0 March 7, 2006


0 1 2 3 4 5 6 7 80

0.5

1

1.5

2

2.5

3

ωt /* (ω t )*crη

η(ω

)/η m

ax

Kelvin

Linear Standard Solid (3-parameter)

Maxwell

Figure 3.22: Frequency-dependence of loss factor η for the Three-parameter model (for

E(∞)/E(0) = 4), the Kelvin model and the Maxwell model.

From (3.183), we obtain the static response by setting ω = 0, which gives

ǫa(0) = ǫstatica =

fa

E(0)

(3.202)

and we may obtain the dynamic amplification factor A(ω) as the ratio

A(ω) ≡ǫa

ǫstatica

=E(0)

(R)∗R

[

1 −

(ω

ωref

)2]2

+ η2

− 12

(3.203)

It appears that A(0) = 1 and A(∞) = 0, and we show the characteristic behavior of A(ω)

in Figure 3.23, where ωAcr denotes the damped resonance frequency defined as

ωAcr = arg[maxA(ω)] (3.204)

Vol 0 March 7, 2006


0 20 40 60 80 100 120 140 160 180 2000

0.2

0.4

0.6

0.8

1

1.2

ω [rad/s]

A(ω

)

E /E = 4/1 (3-parameter)(0)(∞)

E /E → ∞ (Kelvin)(0)(∞)

Figure 3.23: Frequency-dependence of dynamic amplification factor A for a bar with

lumped mass and with the Three-parameter model and the Kelvin model. Chosen material

parameters (representing a polymer) are: E(0) = 2.5 MPa, ρ = 1000 kg/m3, L = 1 m,

t∗ = 0.025 sec.

Damping of the Kelvin model

The Kelvin model is defined by

(R)∗(ω) = E(1 + it∗ω) ; (R)∗R = E, η = t∗ω (3.205)

We also note that

Ra = E[1 + (t∗ω)2

] 12

; E(0) = E, E(∞) = ∞ (3.206)

Remark: As pointed out previously, the situation E(∞) = ∞, i.e. infinite stiffness at

very rapid loading, represents an extreme situation of solid behavior. 2

We thus obtain the quite unrealistic result that η grows linearly with ω without bound,

which is indicated in Figure 3.22. This behavior characterizes the classical notion of

viscous damping. Moreover, we conclude that ωref = ω0 =√

E/m, and we obtain from

Vol 0 March 7, 2006


(3.203) that

A(ω) =

[

1 −

(ω

ω0

)2]2

+ (t∗ω)2

− 12

(3.207)

Remark: In the standard expression of A(ω) in the literature, it is common to introduce

the critical value of damping for which no free damped vibrations can occur. This critical

value corresponds to (t∗)cr = 2/ω0. 2

Damping of the Maxwell model

The Maxwell model is defined by

(R)∗(ω) =Et∗iω

1 + t∗iω; (R)∗R =

E(t∗ω)2

1 + (t∗ω)2, η =

1

t∗ω(3.208)

which gives

Ra =E(t∗ω)2

1 + (t∗ω)2

(

1 +1

t∗ω

) 12

; E(0) = 0, E(∞) = E (3.209)

Hence, we obtain the peculiar result that η is unlimited when ω = 0, as indicated in

Figure 3.22. Since the static solution is unbounded, the definition of A(ω) does not make

any sense.

Damping of the three-parameter model

With the choice of parameters defined in (3.49), we obtain directly from (3.195) that

(R)∗R =E(∞)(t∗ω)2 + E(0)

1 + (t∗ω)2, η =

(E(∞) − E(0))t∗ω

E(∞)(t∗ω)2 + E(0)

(3.210)

We obtain

ωηcr =

1

t∗

(E(0)

E(∞)

) 12

and ηmax =1

2

[(E(0)

E(∞)

)− 12

−

(E(0)

E(∞)

) 12

]

(3.211)

Moreover, the dynamic amplification factor A(ω) becomes

A(ω) =E(0)[1 + (t∗ω)2]

E(∞)(t∗ω)2 + E(0)

[

1 −

(ω

ωref

)2]2

+ η2

− 12

(3.212)

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3.8 Appendix : Laplace - Carson transform 75

with η(ω) given in (3.210)2 and ωref defined by

ω2ref =

ω2(∞)(t∗ω)2 + ω2

(0)

1 + (t∗ω)2(3.213)

We refrain from calculating ωAcr and Amax explicitly.

3.8 Appendix : Laplace - Carson transform

In the table below, the Laplace-Carson transform of some elementary (useful) functions

are listed. It is assumed that f(t) = g(t) = 0 when t = 0.

Function f(t), g(t) Transform (f)∗(s), (g)∗(s)

dnfdtn

(t) sn(f)∗(s)

H(t) 1

H(t − t′), t′ > 0 e−st′

f(t − t′)H(t − t′), t′ > 0 (f)∗(s)e−st′

dfdt

(t) ∗ g(t) (f)∗(s)(g)∗(s)

tn, n ≥ 0 n!sn (= 1

sfor n = 1; = 1 for n = 0)

e−at ss+a

1 − e−at as+a

cos ωt s2

s2+ω2

sin ωt sωs2+ω2

f(t)e−at (f)∗(a + s) ss+a

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The reader should verify this table as homework!

Vol 0 March 7, 2006

Chapter 4

PLASTICITY

In this chapter we discuss rate-independent elastic-plastic material response, which is

the classical theory for metallic materials showing a marked yield stress at ambient tem-

perature. Both the perfectly plastic and the hardening prototype models are discussed.

Numerical integration of the constitutive equations is described. Structural analysis of a

truss and a beam cross-section is outlined.

4.1 Introduction

The macroscopic theory of plasticity is probably the most important (and celebrated) the-

ory of inelastic response of engineering materials, when judged from its widespread use in

commercial FE-codes. The word “plastic” is a transliteration of the ancient Greek verb

that means to “shape” or “form”. Plasticity theory is traditionally associated with the

irreversible deformation of metals, viz. low-carbon steel, for which the inelastic deforma-

tion occurs mainly as distortion (shear), whereas the inelastic volume change is normally

negligible. However, plasticity theory has also won widespread use in the modeling of

non-metallic ductile materials, such as certain polymers and fine-grained soil (e.g. clay).

For these highly porous materials, the inelastic deformation has both distortional and

volumetric components.

The conceptual background of plastic (and viscoplastic) deformation in metals is plastic

slip along crystal planes in the direction of the largest resolved shear stress, or Schmid-

stress, and this slip is caused by the motion of dislocations of atom planes. In a perfect

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78 4 PLASTICITY

crystal structure the plastic slip results in a macroscopic shear deformation without other

distortion of the lattice structure itself. This deformation is superposed by elastic defor-

mation, as illustrated in Figure 4.1.

ǫp ǫe

Figure 4.1: Microstructure of single crystal showing plastic deformation followed by elastic

deformation.

However, most metals are polycrystalline materials. This means that grains with differ-

ent crystallographic orientations and lattice structure (that represents different thermo-

dynamic phases) are interacting in the mesostructure, cf. Figure 4.2. If the distribution

Figure 4.2: Mesostructure of grains interacting via grain boundaries and possessing dif-

ferent crystal orientations.

of crystal orientations is statistically uniform, i.e. each orientation is equally probable

within a Representative Volume Element, then the resulting macroscopic response can

be expected to be isotropic. This is the ideal situation which is hardly encountered in

practice. Plastic (and elastic) anisotropy are induced by the manufacturing process, e.g.

elongation of grains in the rolling direction for metal sheet products.

Vol 0 March 7, 2006

4.2 Prototype rheological model for perfectly plastic behavior 79

4.2 Prototype rheological model for perfectly plastic

behavior

4.2.1 Thermodynamic basis — Yield criterion

Perfectly plastic behavior may be represented by the prototype model shown in Figure 4.3.

The frictional-plastic slider is inactive as long as |σ| < σy, where σy is the yield stress.

=0 (L)

σσ

σ E

ε =ε−ε εe p p

y

σ

ε ε

−σ

σ

ε

p e

y

y

(a) (b)

⋅

σ=Eε (U)⋅ ⋅

(L)

(U)

Figure 4.3: (a) Prototype model for elastic-(perfectly)-plastic material, (b) Stress-strain

relationship.

As the single internal variable we take the plastic strain ǫp, and the expression for the

free energy is chosen as

Ψ =1

2E(ǫe)2 =

1

2E(ǫ − ǫp)2 (4.1)

where ǫe = ǫ− ǫp is the elastic strain of the Hookean spring with modulus of elasticity E.

We then obtain the constitutive equation for the stress as

σ =∂Ψ

∂ǫ= E(ǫ − ǫp) (4.2)

and for the dissipative stress, that is conjugated to ǫp, as

σp = −∂Ψ

∂ǫp= E(ǫ − ǫp) ≡ σ (4.3)

The yield criterion is Φ = 0, where Φ is chosen as

Φ(σ) = |σ| − σy (4.4)

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80 4 PLASTICITY

Since the magnitude of stress can never exceed the yield stress (in this simple prototype

model), it follows that the admissible stress range is defined as those stresses for which

Φ ≤ 0.

4.2.2 Plastic flow rule and elastic-plastic tangent relation

It is assumed that no plastic strain will be produced when Φ < 0, i.e. when |σ| < σy.

The material response is then elastic and |σ| < σy thus defines the elastic stress range.

However, when Φ = 0 plastic strain may be produced. The constitutive rate equation for

ǫp is then postulated as the associative flow rule:

ǫp = λ∂Φ

∂σ= λ

σ

|σ|(4.5)

where the plastic (Lagrangian) multiplier λ is a non-negative scalar variable. Combining

(4.5) with Hooke’s law expressed in (4.2), we obtain the differential equation for the stress

as

σ = Eǫ − λEσ

|σ|(4.6)

The problem formulation is complemented by the so-called elastic-plastic loading criteria.

It follows from the aforesaid that the general format of the loading criteria is

λ ≥ 0, Φ(σ) ≤ 0, λΦ(σ) = 0 (4.7)

Elastic-plastic tangent stiffness relation

Let us consider a plastic state defined by Φ(σ) = 0. Since Φ ≥ 0 is not admissible,

due to the constraint Φ = 0, the plastic multiplier λ is determined from the consistency

condition Φ ≤ 0:

Φ =∂Φ

∂σσ ≤ 0 (4.8)


Φ =σ

|σ|E

(

ǫ − λσ

|σ|

)

=σ

|σ|Eǫ − Eλ ≤ 0 (4.9)

Plastic loading (L) is defined by the situation λ > 0 and Φ = 0, in which case we may

solve for λ from (4.9) to obtain

λ =σ

|σ|ǫ (4.10)

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4.2 Prototype rheological model for perfectly plastic behavior 81

It follows that this is a valid solution only when (σ/|σ|)ǫ > 0, which is the appropriate

loading criterion, that must be satisfied in order for plastic strain to evolve.

Elastic unloading (U) is defined by λ = 0 and Φ ≤ 0, which is obtained whenever

(σ/|σ|)ǫ ≤ 0.

Upon inserting the expression for λ given in (4.10), into (4.5), we obtain the rate equation

for the internal variable ǫp in terms of the control variable ǫ as

ǫp = ǫ (L), ǫp = 0 (U) (4.11)

By inserting this result in (4.6), we obtain the corresponding tangent stiffness relation as

σ = 0 (L), σ = Eǫ (U) (4.12)

This result may be summarized as follows:

When |σ| < σy, then we obtain ǫp = 0 and σ = Eǫ corresponding to elastic response.

When the yield criterion is satisfied, i.e. when |σ| = σy, two different situations are

possible:

The first situation is characterized by ǫ and σ having the same sign, which gives plastic

loading (L). The solution is then ǫp = ǫ and σ = 0, which can be expected for perfectly

plastic behavior (as shown in Figure 4.3(b)) for which the tangent stiffness is zero.

The second situation is characterized by ǫ and σ having opposite signs, which gives elastic

unloading (U). The solution is then defined by ǫp = 0 and σ = Eǫ, which corresponds to

elastic response, as shown in Figure 4.3(b).

Remark: The expressions for Ψ in (4.1) and Φ in (4.4) are not the only possible ones.

For example, we may introduce two internal variables (ǫp and k) and set

Ψ =1

2E(ǫ − ǫp)2 − σyk (4.13)

Φ = |σp| − κ (4.14)

However, σ is still given by (4.2), while σp and κ (that are the energy conjugate variables

to ǫp and k) are given as

σp = −∂Ψ

∂ǫp= E(ǫ − ǫp) ≡ σ (4.15)

κ = −∂Ψ

∂k= σy (4.16)

and it is realized that the resulting model is equivalent to the original one (which is simpler

since it contains only one internal variable). 2

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82 4 PLASTICITY

4.2.3 Dissipation of energy

The dissipation function D was defined in (2.3). For the formulation defined by (4.1) and

(4.4), we obtain for Φ = 0, i.e. when |σ| = σy, the expression

D = σǫp = λ|σ| = λσy ≥ 0 (4.17)

Remark: It is interesting to note that D will not get the same value in the model

formulation that was defined in the previous Remark. For this formulation, defined by

(4.13) and (4.14), we obtain

D = σǫp + κk = λσy − λσy = 0 (4.18)

where it was used that

k = λ∂Φ

∂κ= −λ (4.19)

For both formulations it is concluded that D ≥ 0 and, hence, the dissipation inequality

is satisfied. 2

4.3 Prototype model for hardening plastic behavior


Hardening plastic behavior is represented by the prototype model shown in Figure 4.4.

The frictional-plastic slider is now increasing its resistance due to the amount of slip

developed. More specifically, the excess stress over the initial yield stress is due to the

“hardening spring” with stiffness H that is related to the plastic strain. Upon unloading

and reloading, the slider will thus become inactive until the stress has resumed the previous

level during loading, i.e. as long as |σ| < σy + H|ǫp|, where H > 0 is the (constant)

hardening modulus. This behavior is typical for hardening plasticity.

Apart from ǫp, we now introduce the (isotropic) hardening variable k, such that the free

energy density is expressed as

Ψ =1

2E(ǫ − ǫp)2 +

1

2Hk2 (4.20)

From Coleman’s equations we still obtain

σ =∂Ψ

∂ǫ= E(ǫ − ǫp) (4.21)

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4.3 Prototype model for hardening plastic behavior 83

E

σ σ

H

σ

ε ε

σ

σ

σ=Hε

σ=Eε

ε

e p

y

y

p. .

. .

(a) (b)

Figure 4.4: Prototype model for elastic-hardening-plastic material, (a) Rheological model,

(b) Stress-strain relationship.

σp = −∂Ψ

∂ǫp≡ σ (4.22)

whereas the dissipative stress κ, associated with k, is given as

κ = −∂Ψ

∂k= −Hk (4.23)

The yield function is now defined as

Φ(σ, κ) = |σ| − σy − κ (4.24)

In the literature, κ is frequently denoted the “drag stress”.

4.3.2 Plastic flow rule and elastic-plastic tangent relation

Inelastic deformation can be produced when Φ = 0. The associative flow and hardening

rules are then defined as

ǫp = λ∂Φ

∂σ= λ

σ

|σ|(4.25)

k = λ∂Φ

∂κ= −λ (4.26)

where the plastic multiplier is still defined by the loading criteria given in (4.7). The

pair (ǫp, k) can be perceived as the outward pointing normal from the cone defined by

Vol 0 March 7, 2006

84 4 PLASTICITY

0

=p.

=-p.

=0 =0

yy- y- y

( , )=0

( , ). .p

(a) (b)

λ λ

ǫ

ǫ ǫ

σ

σσσ σ

σσ

κ

κ

κ

E

Φ

ΦΦ

Figure 4.5: (a) Associative flow rule for perfect plasticity, (b) Associative flow and hard-

ening rules for hardening plasticity.

Φ(σ, κ) = 0, as shown in Figure 4.5(b). (As a comparison, the simpler situation of an

associated flow rule in perfect plasticity is depicted in Figure 4.5(a).)

By combining (4.25) with (4.21), we still obtain the differential equation for the stress in

(4.6). A rate equation for κ may also be obtained by differentiating (4.23), and combining

with (4.26). We thus obtain the set of equations

Eǫ = Eǫ − λEσ

|σ|(4.27)

κ = Hλ (4.28)

which are subjected to the loading criteria

λ ≥ 0, Φ(σ, κ) ≤ 0, λΦ(σ, κ) = 0 (4.29)

Remark: The loading criteria (4.29) are known as the Kuhn-Tucker complementary

conditions, which stem from constrained (convex) minimization. In fact, the associative

flow and hardening rules can be derived from a minimization principle. [The theoretical

basis is beyond the scope of this introductory treatment.] 2

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4.3 Prototype model for hardening plastic behavior 85

Elastic-plastic tangent stiffness relation

Consider a plastic state defined as Φ = 0. We then have the constraint Φ ≤ 0, which

gives

Φ =∂Φ

∂σσ +

∂Φ

∂κκ ≤ 0 (4.30)

Upon inserting (4.27) and (4.28) into (4.30), we obtain

σ

|σ|Eǫ − hλ ≤ 0 (4.31)

where h is defined as the generalized plastic hardening modulus

hdef= E + H (4.32)

Plastic loading (L), is defined as λ > 0 and Φ = 0. Since h > 0, this situation is at hand

whenever (σ/|σ|)ǫ > 0, and we obtain λ from (4.31) as

λ =E

h

σ

|σ|ǫ (4.33)

On the other hand, elastic unloading (U), which is defined by λ = 0 and Φ ≤ 0, is obtained

whenever (σ/|σ|)ǫ ≤ 0. It is noted that the criteria for loading/unloading are exactly the

same as for perfect-plasticity.

We may now obtain

ǫp =E

hǫ (L), ǫp = 0 (U) (4.34)

and

σ = Eepǫ (L), σ = Eǫ (U) (4.35)

where Eep is the elastic-plastic tangent stiffness modulus defined as

Eep =

(

1 −E

h

)

E =E

hH (4.36)

With a generalization of the hardening concept, the following situations are distinguished:

Hardening : H > 0 ⇒ Eep > 0, (κ > 0) (4.37)

Perfect plasticity : H = 0 ⇒ Eep = 0, (κ = 0) (4.38)

Softening : H < 0 ⇒ Eep < 0, (κ < 0) (4.39)

It also follows that

H = ∞ ⇒ Eep = E (elastic) (4.40)

H = −E (h = 0) ⇒ Eep = −∞ (infinitely brittle) (4.41)

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86 4 PLASTICITY


When Φ = 0, where Φ is given by (4.24), we obtain (by definition of D) with (4.25) and

(4.26) that the dissipation of energy is given as

D = σǫp + κk = λ(|σ| − κ) = λσy ≥ 0 (4.42)

The portion of D that relates to “dissipation of mechanical energy” is denoted Dp and is

defined as

Dp = σǫp = λ|σ| ≥ D (4.43)

It is noted that (in this particular case) the mechanical dissipation is larger than the total

dissipation.

4.4 Model for cyclic loading — Mixed isotropic and

kinematic hardening


In order to describe the behavior of metals at cyclic loading in a realistic fashion, it is

common to employ mixed isotropic and kinematic hardening. We then define the free

energy as

Ψ =1

2E(ǫ − ǫp)2 +

1

2rHk2 +

1

2(1 − r)Ha2 (4.44)

where k represents isotropic hardening, whereas a represents kinematic hardening. The

parameter r, with 0 ≤ r ≤ 1, controls the relation between isotropic and kinematic

hardening:

• r = 0: purely kinematic hardening

• r = 1: purely isotropic hardening

The conjugated dissipative stresses are given as:

κ = −∂Ψ

∂k= −rHk (4.45)

α = −∂Ψ

∂a= −(1 − r)Ha (4.46)

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4.4 Model for cyclic loading — Mixed isotropic and kinematic hardening 87

where κ is the “drag-stress” due to the isotropic portion of hardening, whereas α is the

“back-stress” due to the kinematic portion of hardening.

For the considered uniaxial stress state, the yield criterion is now defined as

Φ = |σred| − σy − κ = 0 with σred = σ − α (4.47)

where σred is denoted the reduced stress, cf. Figure 4.6.

Remark: Occasionally, we introduce the notation σe = |σ|, where σe is the equivalent

stress. Likewise, σrede = |σred| is the equivalent reduced stress, where the notion “reduced”

refers to the translation in stress space due to the backstress α. 2

α

κ κΦ(σ, κ, α) = 0 Φ(σred, κ) = 0

σ σred

Figure 4.6: Illustration of mixed isotropic and kinematic hardening in stress space.

4.4.2 Associative flow and hardening rules — Linear hardening

The associative flow and (linear) hardening rules are then defined as

ǫp = λ∂Φ

∂σ= λ

σred

|σred|(4.48)

k = λ∂Φ

∂κ= −λ (4.49)

a = λ∂Φ

∂α= −λ

σred

|σred|= −ǫp (4.50)

We may now combine these relations with (4.21), (4.45) and (4.46). The pertinent differ-

ential equations are then obtained as

σ = E(ǫ − ǫp) = Eǫ − Eλσred

|σred|(4.51)

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88 4 PLASTICITY

κ = rHλ = rHǫp (4.52)

α = (1 − r)Hλσred

|σred|= (1 − r)Hǫp (4.53)

With the proper initial conditions, these equations taken together with the loading criteria

of type (4.29) define the elastic-plastic constitutive relations for the considered material

model.

4.4.3 Characteristic response for linear hardening

The characteristic response is illustrated for pure isotropic, pure kinematic and mixed

hardening, respectively, in Figure 4.7. By introducing kinematic hardening, it is possible

to pick up the Bauschinger effect, i.e. that the yield stress in compression, upon reversed

loading from tension, is smaller than it was in tension.

Remark: This reduction in compressive yield strength should not be confused with

the softening phenomenon, which means that the yield strength is reduced in tension

(compression) whilst the material is actually loaded in tension (compression). 2

4.4.4 Associative flow and nonassociative hardening rules —

Nonlinear hardening

In order to obtain a more realistic response in cyclic loading, we should resort to nonlinear

laws of hardening. For example, it is of value to model the asymptotic case of perfect

plasticity for large plastic strains, which corresponds to saturation of dislocations. As

to the specific choice of flow and hardening rules, Φ can not be used as the potential

function (for the hardening rules). Instead, we introduce the plastic potential Φ∗ 6= Φ of

the following form:

Φ∗ = Φ +κ2

2κ∞

+α2

2α∞

(4.54)

In this fashion, the flow rule will still be of the associative type, while the hardening rules

for k and a are both non-associative. The positive constants κ∞ and α∞ are saturation

values of the drag-stress κ and the back-stress α, respectively. We thus obtain the flow

and hardening rules

ǫp = λ∂Φ∗

∂σ= λ

σred

|σred|(4.55)

Vol 0 March 7, 2006


Figure 4.7: Uniaxial stress versus plastic strain characteristics for (a) Linear isotropic

hardening, (b) Linear kinematic hardening, (c) Mixed linear isotropic and linear kinematic

hardening.

Vol 0 March 7, 2006

90 4 PLASTICITY

k = λ∂Φ∗

∂κ= −λ

(

1 −κ

κ∞

)

(4.56)

a = λ∂Φ∗

∂α= −λ

(σred

|σred|−

α

α∞

)

= −ǫp + λα

α∞(4.57)

The pertinent differential equations are now obtained as the slightly adjusted versions of

those in (4.51) to (4.53) as:

σ = E(ǫ − ǫp) = Eǫ − Eλσred

|σred|(4.58)

κ = rHλ

(

1 −κ

κ∞

)

(4.59)

α = (1 − r)Hλ

(σred

|σred|−

α

α∞

)

= (1 − r)H

(

ǫp − |ǫp|α

α∞

)

(4.60)

The characteristic response is shown schematically in Figure 4.8.

Figure 4.8: Uniaxial stress versus plastic strain characteristics for mixed nonlinear

isotropic and kinematic hardening.

4.4.5 Characteristic response for nonlinear hardening

Next we consider the cases of loading with increasing plastic strain, ǫp > 0, and reversed

loading with decreasing plastic strain, ǫp < 0, in further detail. At this analysis we denote

the assumed initial state by ǫp0 , κ0 and α0. First we note that ǫp = λ when ǫp > 0, whereas

ǫp = −λ when ǫp < 0, which follows from (4.55) upon observing that λ ≥ 0.

Vol 0 March 7, 2006


Loading

The solution of (4.59) at loading, i.e. when ǫp > 0, is

κ = κ∞ − (κ∞ − κ0)e−

rH(ǫp−ǫp0)

κ∞ (4.61)

Similarly, the solution of (4.60) becomes

α = α∞ − (α∞ − α0)e−

(1−r)H(ǫp−ǫp0)

α∞ (4.62)

In the case of monotonic loading with the initial conditions κ0 = α0 = 0 when ǫp0 = 0, we

obtain from (4.61) and (4.62) the solutions

κ = κ∞

(

1 − e−rHǫp

κ∞

)

, α = α∞

(

1 − e−(1−r)Hǫp

α∞

)

(4.63)

Upon differentiating (4.63), we conclude that

dκ

dǫp= rH and

dα

dǫp= (1 − r)H when ǫp = 0 (4.64)

i.e. the initial hardening values are the same as those of linear hardening. It is also clear

that the behavior tends to the perfectly plastic one when ǫp → ∞, i.e.

σ = σy + α∞ + κ∞ when ǫp = ∞ (4.65)

Remark: Linear hardening characteristics are obtained by simply setting κ∞ = α∞ = ∞.

2

Reversed loading

The solution of (4.59) at reversed loading, i.e. when ǫp < 0, is

κ = κ∞ − (κ∞ − κ0)erH(ǫp−ǫ

p0)

κ∞ (4.66)

Similarly, the solution of (4.60) becomes

α = −α∞ + (α∞ + α0)e(1−r)H(ǫp−ǫ

p0)

α∞ (4.67)

From the solutions (4.61), (4.62), (4.66) and (4.67) we obtain simply that

κ = κ∞ and |α| = α∞ when ǫp = ∞ and ǫp = −∞ (4.68)

Vol 0 March 7, 2006

92 4 PLASTICITY

Hence, in this situation we may use the triangle inequality to conclude that

|σ| ≤ |σ − α| + |α| ≤ σy + α∞ + κ∞ (4.69)

where (4.47) and (4.68) were used. The criterion

|σ| − σy − α∞ − κ∞ = 0 (4.70)

represents the “limit criterion” that can never be violated for any amount of accumulated

plastic strain.

Shake-down and ratchetting (prescribed stress)

We shall next consider the situation of cyclic loading with constant stress amplitude, and

we intend to investigate whether shake-down or ratchetting is predicted by the nonlinear

mixed hardening model. To this end we consider the extremes of pure isotropic hardening

and pure kinematic hardening, respectively. However, we shall refer to Figure 4.9 that

depicts the typical behavior in reversed loading followed by renewed loading when both

isotropic and kinematic hardening effects are present.

For isotropic hardening (r = 1), we obtain shake-down upon renewed loading if the

material yielded plastically at reversed loading. This may be seen formally as follows:

During reversed loading from 1’ to 2 it is obvious that κ will increase from the value κ1

to the value κ2. The question is whether the maximum stress σmax at 3 will be reached

before yielding takes place. This is indeed the case, since

σmax = σy + κ1 ≤ σy + κ2 (4.71)

Hence, the state 2’ is never reached and immediate shake-down is obtained.

For kinematic hardening (r = 0), we obtain constant ratchetting in each cycle for constant

stress amplitude, which may be shown as follows: During reversed loading from 1’ to 2,

we may use Eqn. (4.67) to calculate the reduction of α from the value α1 to the value α2

according to the expression

α2 = −α∞ + (α∞ + α1)eH(ǫ

p2−ǫ

p1)

α∞ (4.72)

The value of α remains constant upon renewed elastic loading up to 2’, whereafter renewed

plastic loading takes place. Eqn. (4.62) now gives the increase of α from the value α2 to

Vol 0 March 7, 2006


Figure 4.9: Results for uniaxial stress showing characteristic behavior of stress and back-

stress at reversed and renewed loading for nonlinear mixed hardening model.

the value α3 according to the expression

α3 = α∞ − (α∞ − α2)e−

H(ǫp3−ǫ

p2)

α∞ (4.73)

Upon combining (4.72) and (4.73), while introducing the plastic range ǫp3 − ǫp

2 = ∆ǫp, we

obtain

∆ǫp =α∞

Hln

(α∞

2 − α22

(α∞ − α3)(α∞ + α1)

)

(4.74)

Now, since the yield criterion is satisfied at 1 as well as at 3, we obtain (since κ = 0)

α1 = α3 = σmax − σy (4.75)

where it was used that σmax > α1. Moreover, since the yield criterion is satisfied also at

2, and σmin < α2, we obtain

α2 = σmin + σy (4.76)

Upon inserting the values of α1, α2 and α3 from (4.75) and (4.76) into (4.74), we obtain

∆ǫp =α∞

Hln

(α∞

2 − (σmin + σy)2

α∞2 − (σmax − σy)2

)

(4.77)

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94 4 PLASTICITY

For given value of σmax, it follows from (4.77) that

∆ǫp is maximum when σmin = −σy (4.78)

∆ǫp is minimum (= 0) when σmin = −σmax

Hence, no ratchetting is obtained when the cyclic stress variation is symmetric in tension

and compression.

In the general situation of mixed hardening we expect reduced ratchetting, that gradually

decreases with the accumulation of plastic strain. However, since the analytic solution

becomes quite complex in this case, the pertinent model behavior is most conveniently

assessed from numerical integration of the constitutive equations.

Shake-down (prescribed strain)

In the case of prescribed cyclic strain, we investigate whether shake-down or growing

stress amplitude is obtained. To start with, it was concluded that shake-down is obtained

immediately in the case of linear kinematic hardening, whereas the stress amplitude will

continue to grow till the response becomes cyclically elastic in the case of linear isotropic

hardening. Using numerical integration, we may confirm this behavior in Figure 4.10(a)

and Figure 4.10(e). A more realistic picture of the shake-down behavior is obtained

if nonlinear hardening is assumed, which is demonstrated for kinematic and isotropic

hardening in Figure 4.10(b) and Figure 4.10(f), respectively. Common for all results in

Figure 4.10 are the input data:

σy

E= 0.001,

H

E= 0.1, 0 ≤ r ≤ 1, κ∞ = α∞ = aσy (4.79)

ǫ ∈ [−2ǫy, 2ǫy], where ǫy =σy

E

where a is a parameter that controls the amount of nonlinear hardening. Linear hardening

is defined by a = ∞, whereas the chosen nonlinear hardening model is characterized by

the choice a = 0.25. Kinematic hardening is defined by r = 0, whereas isotropic hardening

is defined by r = 1.

Vol 0 March 7, 2006


−2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5

x 10−3

−2

−1.5

−1

−0.5

0

0.5

1

a1.5

2

E=1000⋅σy , α

∞=κ

∞=∞, H=E/10, r=0

strain

stre

ss

−2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5

x 10−3

−2

−1.5

−1

−0.5

0

0.5

1

b1.5

2

strain

stre

ss

E=1000⋅σy , α

∞=κ

∞=σ

y/4, H=E/10, r=0

−2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5

x 10−3

−2

−1.5

−1

−0.5

0

0.5

1

c1.5

2

strain

stre

ss

E=1000⋅σy , α

∞=κ

∞=∞, H=E/10, r=0.5

−2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5

x 10−3

−2

−1.5

−1

−0.5

0

0.5

1

d1.5

2

strain

stre

ss

E=1000⋅σy , α

∞=κ

∞=σ

y/4, H=E/10, r=0.5

−2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5

x 10−3

−2

−1.5

−1

−0.5

0

0.5

1

e1.5

2

strain

stre

ss

E=1000⋅σy , α

∞=κ

∞=∞, H=E/10, r=1

−2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5

x 10−3

−2

−1.5

−1

−0.5

0

0.5

1

f1.5

2

strain

stre

ss

E=1000⋅σy , α

∞=κ

∞=σ

y/4, H=E/10, r=1

Figure 4.10: Predicted results from cyclic straining of linear and nonlinear hardening

models. Results for linear hardening are shown in (a,c,e), whereas results for nonlinear

hardening are shown in (b,d,f).

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96 4 PLASTICITY

4.4.6 Backward Euler method for integration — Linear harden-

ing

Applying the fully implicit (Backward Euler) integration rule to (4.51), we obtain

n+1σ − nσ

∆t= E

(∆ǫ

∆t−

µ

∆t

n+1σred

n+1σrede

)

with n+1σrede = |n+1σred|, µ

def= λ∆t (4.80)

which can be rewritten as

n+1σ = n+1σtr −Eµ

n+1σrede

n+1σred (4.81)

where n+1σtr is the “elastic trial” stress defined by

n+1σtr = nσ + E∆ǫ (4.82)

We shall use implicit integration of (4.52) and (4.53) as well, whereby we obtain

n+1κ = nκ + rHµ (4.83)

n+1α = nα +(1 − r)Hµ

n+1σrede

n+1σred (4.84)

In principle, µ can be solved from (4.81), (4.83) and (4.84) together with the loading

conditions

µ ≥ 0, n+1Φ ≤ 0, µ n+1Φ = 0 (4.85)

Upon combining (4.81) and (4.84), we obtain(

1 +[E + (1 − r)H ]µ

n+1σrede

)

n+1σred = n+1σred,tr with n+1σred,tr = n+1σtr − nα (4.86)

Taking the absolute values of both sides, we obtain

n+1σrede = n+1σred,tr

e − [E + (1 − r)H ]µ with n+1σred,tre = |n+1σred,tr| (4.87)

and it is noted that n+1σred,tre is a known quantity when ∆ǫ has been prescribed. Now,

we may introduce the updated stresses from (4.83) and (4.87) into (4.47) to obtain the

updated yield function

n+1Φ = n+1σrede − σy −

n+1κ = n+1Φtr − hµ ≤ 0 (4.88)

where we have introduced the “trial” value of the yield function

n+1Φtr = n+1σred,tre − σy −

nκ (4.89)

and h is (still) the generalized plastic modulus defined as

h = E + H (4.90)

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Plastic loading or elastic unloading

Whether plastic loading or elastic unloading is at hand can be determined with use of the

complementary conditions in (4.85). We distinguish between the following conditions:

Loading (L) is defined by n+1Φ = 0 and µ > 0, which gives the solution

µ =n+1Φtr

h> 0 when n+1Φtr > 0 (4.91)

Unloading (U), on the other hand, is defined by n+1Φ ≤ 0 and µ = 0, which obviously

gives

µ = 0 when n+1Φtr < 0 (4.92)

Updated solution

When µ has been determined, it is possible to compute the updated values of all state

variables of interest. First, we obtain from (4.86) that

n+1σ − n+1α =n+1σred

e

n+1σred,tre

n+1σred,tr (4.93)

which may be inserted into (4.81) and (4.84) to give the updated solution

n+1σ = n+1σtr −Eµ

n+1σred,tre

n+1σred,tr = c1n+1σtr + (1 − c1)

nα (4.94)

n+1κ = nκ + rHµ (4.95)

n+1α = nα +(1 − r)Hµn+1σred,tr

e

n+1σred,tr = c2n+1σtr + (1 − c2)

nα (4.96)

where

c1 = 1 −Eµ

n+1σred,tre

, c2 =(1 − r)Hµn+1σred,tr

e

(4.97)

In the very simplest situation of perfect plasticity, defined by H = 0 and nκ = nα= 0, we

obtain from (4.91), in the case of (L), that

µ =1

E(|n+1σtr| − σy)

which may be inserted to (4.94) to give

n+1σ = σy

n+1σtr

|n+1σtr|

Remark: In the present case of linear hardening, it appears that the crucial equation

(4.88) is linear in µ. In fact, it can be shown that this linearity carries over to the

multiaxial situation. 2

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98 4 PLASTICITY

Summary of algorithm

We summarize the solution algorithm in Box 4.1.

1. Given ∆ǫ −→ ∆σtr −→ n+1σtr = nσ + E∆ǫ

2. Check L/U

If n+1Φtr ≤ 0, then µ = 0

else µ =n+1Φtr

h> 0, h = E + H

3. Update solution

n+1σ = c1n+1σtr + (1 − c1)

nα, c1 = 1 −Eµ

n+1σred,tre

n+1κ = nκ + rHµ

n+1α = c2n+1σtr + (1 − c2)

nα, c2 =(1 − r)Hµn+1σred,tr

e

Box 4.1: Solution algorithm for linear mixed hardening in plasticity.

Closest-point-projection-method (CPPM)

It appears that we may rewrite (4.81), (4.83) and (4.84) in a slightly more general fashion

asn+1σ = n+1σtr − µE

n+1(

∂Φ

∂σ

)

(4.98)

n+1κ = nκ − µrHn+1(

∂Φ

∂κ

)

(4.99)

n+1α = nα − µ(1 − r)Hn+1(

∂Φ

∂α

)

(4.100)

where we directly used the characteristics of the flow and the hardening rules. We also

repeat, for completeness, the loading conditions (4.85)

µ ≥ 0, n+1Φ ≤ 0, µn+1Φ = 0 (4.101)

Let us now introduce the convex set E of plastically admissible dissipative stresses as

E = {(σ, κ, α) | Φ(σ, κ, α) ≤ 0} (4.102)

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where we (implicitly) used that σ is the dissipative stress that is energy conjugated to ǫp.

We are now in the position to establish the following important theorem:

Theorem: The updated solution (n+1σ, n+1κ, n+1α) of (4.98) to (4.101) is also the solution

of the convex minimization problem as follows:

(n+1σ, n+1κ, n+1α) = arg

[

Min

(σ, κ, α) ∈ EΠ (σ, κ, α)

]

(4.103)

where Π is defined as

Π (σ, κ, α) =1

2

[1

E(n+1σtr − σ)2 +

1

rH(nκ − κ)2 +

1

(1 − r)H(nα − α)2

]

(4.104)

which has a unique solution when H ≥ 0. It thus appears that Eqns. (4.98) to (4.101)

represent the Kuhn-Tucker problem corresponding to the minimization problem (4.103).

Proof: The KT-conditions of (4.103) are also the stationary conditions of the Lagrangian

function

Λ(σ, κ, α, µ) = Π (σ, κ, α) + µ n+1Φ(σ, κ, α) (4.105)

in the following sense:∂Λ

∂σ= 0,

∂Λ

∂κ= 0,

∂Λ

∂α= 0 (4.106)

and∂Λ

∂µ≤ 0, µ

∂Λ

∂µ= 0, µ ≥ 0 (4.107)

It can readily be checked that the conditions (4.106) and (4.107) are identical to (4.98)

to (4.101).

The updated solution is illustrated in Figure 4.11 for the special case that r = 1, which

means that there is no kinematic hardening. Considering, for example, the situation thatn+1σtr > 0, we may combine (4.94) and (4.95) to the “vector” equation in the (σ, κ)-space

as follows:

(n+1σ, n+1κ) = (n+1σtr, nκ) − (E,−H)µ (4.108)

which relation is illustrated in Figure 4.11(b). We have thus established that (n+1σtr, n+1κ, n+1α)

is the projection of (n+1σtr, nκ, nα) onto the convex set E in the particular metric defined

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100 4 PLASTICITY

Figure 4.11: Updated solution (n+1σ, n+1κ) for isotropic hardening, (a) Restriction to

σ − ǫ- relation, (b) Space of dissipative stresses (σ, κ).

by the norms defined in the quadratic form Π . This is the reason why the BE-method

applied to the plasticity problem is also known in the literature as the Closest-Point-

Projection-Method. The projection will also be denoted as the operator CPPM in the

mapping

(n+1σ, n+1κ, n+1α) = CPPM{n+1σtr, nκ, nα; E, rH, (1− r)H} (4.109)

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4.5 Structural analysis 101

where the arguments E, rH and (1 − r)H define the projection metric.

4.5 Structural analysis

4.5.1 Structural behavior — Limit load analysis

Like for viscoelasticity, the behavior of an elastic-plastic structure will depend on its

statical (in)determinacy. We shall then first outline some general features, which will be

illustrated later in conjunction with the discussion of truss structures.

Isostatic structures

We recall that, for an isostatic structure, the stress distribution is uniquely determined

by the applied load. Considering, for example, a truss with bars of perfectly plastic

material, we conclude that the maximum load bearing capacity can simply be calculated

as the load for which the normal stresses in the most severely stressed bar has reached

the yield point. At this load level, the structural tangent stiffness matrix is zero and the

truss behaves like a mechanism, i.e. it can not sustain any further load increase without

excessive geometrical distortion. The considered load is thus the limit load.

Hyperstatic structures

We recall from Subsection 3.4.1 that, for a hyperstatic structure, the stress distribution

is not uniquely defined by the load but is determined also by the material properties. For

example, for the truss with perfectly plastic material in the bars, the statical redundancy

is reduced during the load increase each time a new bar is yielding plastically until the

structure (truss) has become isostatical. For further load increase, the discussion of the

isostatic truss (above) applies so that the limit load is achieved when a mechanism has

been obtained.

4.5.2 Analysis of truss — Numerical integration

We shall analyze the truss in Figure 3.8 in the same fashion as for nonlinear viscoelastic

response, and we refer to Figure 3.8 for the introduced definitions and notation. Hence, the

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102 4 PLASTICITY

truss is subjected to time-independent1 loads collected in P (t), and the energy-conjugated

displacement components p(t) are sought.

We recall the expression in (4.94) for the updated stress n+1σidef= σi in the i:th bar

according to linear mixed isotropic and kinematic hardening:

σi = (c1)i σtri + (1 − (c1)i)

nαi, i = 1, 2, . . . (4.110)

where

σtri = nσi + Ei(∆ǫ)i with (∆ǫ)i =

(∆n)i

Li

(4.111)

In analogy with (3.136), we now obtain the corresponding normal forces n+1Nidef= Ni as

Ni = Ai σtri (∆ǫ) (4.112)

which are collected in the column matrix N(∆n). The equilibrium and kinematic rela-

tions for the truss are still those of (3.138), i.e.

P = AT N and ∆n = A ∆p (4.113)

and iterations are required in order to find the solution ∆p from (4.113)1. For given loadn+1P , we may device the following Newton procedure (k being the iteration count):

∆p(k+1) = ∆p(k) + δp

where δp is the solution of the linear set of equations:

Sep(k)a δp = −(P − AT N (k)) with Sep

a =∂(AT N)

∂(∆p)(4.114)

and with ∆p(0) chosen as the converged value of ∆p in the previous timestep. It thus fol-

lows that Sepa is the ATS-matrix for elastic-plastic behavior, which is a nonlinear function

of the incremental solution ∆p for a given timestep.

Lemma: The ATS-matrix Sepa is given as

Sepa = ATSep

eaA with Sepea = diag

[A1(E

epa )1

L1,

A2(Eepa )2

L2, . . .

]

(4.115)

1The variable t is a “time-like” parameter, which may be chosen as the real time (although this is not

necessary). Since the material response is rate-independent, it is concluded that real time is irrelevant.

Vol 0 March 7, 2006


where, for each bar, the algorithmic tangent stiffness modulus Eepa is defined as d(∆σ)/d(∆ǫ).

Proof: From the definition in (4.114) follows that

Sepa = AT ∂(N )

∂(∆n)A = AT∂(∆N )

∂(∆n)A (4.116)

where it was used that d(∆n) = A d(∆p). Moreover,

∂(∆N )

∂(∆n)= diag

[d(∆N1)

d(∆n1),

d(∆N2)

d(∆n2), . . .

]

(4.117)

For each bar, we obtain

d(∆N)

d(∆n)= A

d(∆σ)

d(∆ǫ)

d(∆ǫ)

d(∆n)=

AEepa

Lwith Eep

a =d(∆σ)

d(∆ǫ)(4.118)

where it was used that d(∆ǫ) = d(∆n)/L. 2

Remark: For linear mixed hardening Eepa ≡ Eep, where Eep = EH/h is the tangent

stiffness modulus defined already in (4.36). Show this as homework! 2

4.5.3 Analysis of double-symmetric beam cross-section

Perfect plasticity — Limit moment

In accordance with the discussion for a general structure, the cross-section moment for

which the most stressed fiber starts to yield plastically is denoted the elastic limit mo-

ment Mel. If the material is assumed to be perfectly plastic, then the limit moment Ml

corresponds to plastic yielding in the whole cross-section, i.e. |σ| = σy everywhere.

Subsequently, we consider (for the sake of simplicity) a double-symmetric cross-section.

Figure 4.12 shows the situation in the elastic stage, the elastic-plastic stage and the (fully)

plastic stage, respectively.

It follows trivially that

Mel = Wσy with W =2I

h(4.119)

When the plastic zone has advanced to the location defined by z = zy, the corresponding

stress distribution is defined as

σ =

{

σyzzy

, 0 ≤ |z| ≤ zy

σyz|z|

, zy ≤ |z| ≤ h2

(4.120)

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104 4 PLASTICITY

h2

h2

|σ|=σ |σ|=σ|σ|<σ

z

z z =0

y y

y

y

y

y

(a) (b) (c)

Figure 4.12: Progressive yielding in double-symmetric cross-section for elastic-perfectly

plastic response.

For a given moment M ≥ Mel, we may calculate z = zy from the condition

M = 2

∫ h/2

0

σ(z)zb(z)dz =2σy

zy

∫ zy

0

z2b(z)dz + 2σy

∫ h/2

zy

|z|b(z)dz

=σy

zyIy + σy(Z − Zy) (4.121)

where we introduced the notation

Iy = 2

∫ zy

0

z2b(z)dz, Zy = 2

∫ zy

0

|z|b(z)dz and Z = 2

∫ h/2

0

|z|b(z)dz (4.122)

In the fully plastic regime, i.e. when zy = 0, we have Iy = 0 and Zy = 0, which inserted

into (4.121) gives the limit moment

Ml = Zσy (4.123)

The plastic shape factor a is defined as

a =Ml

Mel

=Z

W(4.124)

It is also possible, as an alternative, to express M as a function of the curvature in a slight

reformulation of (4.121). First, we introduce κc,el as the curvature corresponding to Mel.

In this situation the yield strain is achieved at z = h/2. Hence we obtain

ǫy = κczy = κc,elh

2⇒ zy =

h

2

(κc

κc,el

)−1

(4.125)

Vol 0 March 7, 2006


by which we may rewrite (4.121) as

M

(κc

κc,el

)

= 2σy

h

κc

κc,el

Iy

(κc

κc,el

)

+ σy

[

Z − Zy

(κc

κc,el

)]

,κc

κc,el

≥ 1 (4.126)

or, in the more general form, as

M

(κc

κc,el

)

= MelM

(κc

κc,el

)

with M(1) = 1 and M(∞) = a (4.127)

where M is a non-dimensional function, such that 1 ≤ M ≤ a.

Special case: For a rectangular cross-section we obtain

W =bh2

6, Z =

bh2

4and a = 1.5 (4.128)

Moreover, we obtain the relation

M

(κc

κc,el

)

= 1.5

[

1 −1

3

(κc

κc,el

)−2]

(4.129)

This relation is shown in Figure 4.13 together with typical relations for other common

cross-sectional shapes. 2

Figure 4.13: Relation between moment and curvature for various cross-sections.

Hardening plasticity

When the material is hardening in a general fashion, it is expedient to use numerical

integration along the lines set out in Subsection 3.6.4 for the Norton material. The

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106 4 PLASTICITY

problem becomes incrementally nonlinear in general (for nonlinear hardening), in which

case iterations must be used.

For a discretized double-symmetric cross-section we thus obtain the moment n+1Mdef= M

as

M =

∫ h/2

−h/2

σ(z)zb(z)dz ≈ 2

nint∑

i=1

σizi(∆A)i with (∆A)i = bi(∆z)i (4.130)

In the case of linear hardening, n+1σdef= σ was defined in (4.94) as

σ = c1 σtr + (1 − c1)nα (4.131)

where

σtr = nσ + E∆ǫ with ∆ǫ = ∆κcz (4.132)

The Newton-type iteration procedure to calculate ∆κc for a given moment M at t = tn+1

reads

∆κ(k+1)c = ∆κ(k)

c + δκc (4.133)

δκc = −(Sepa )−1(M − M (k)) with Sep

a =d(∆M)

d(∆κc)(4.134)

and with ∆κ(0)c chosen as the converged value of ∆κc in the previous timestep. Hence, we

conclude that Sepa becomes

Sepa =

∫ h/2

−h/2

Eepa (z)z2b(z)dz = 2

nint∑

i=1

(Eepa )iIi with Ii = z2

i (∆A)i (4.135)

Remark: To find the algorithmic bending stiffness Sepa is straightforward for the case

of linear hardening, since it will coincide with the tangent bending stiffness Sep in this

particular case. 2

4.5.4 Analysis of single-symmetric beam cross-section — Nu-

merical integration

Next, we consider a single-symmetric cross-section, as shown in Figure 4.14. We compute

(like in the case of nonlinear viscoelasticity discussed in Subsection 3.6.5) the normal force

N and the moment M as

N =

∫ h+

−h−

σ(z)b(z)dz =

nint∑

i=1

σi∆Ai (4.136)

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M =

∫ h+

−h−

σ(z)zb(z)dz =

nint∑

i=1

σizi∆Ai (4.137)

which must equilibrate the prescribed normal force N and moment M . We may compute

σ = σ(∆ǫ(z)) from (4.131) with

∆ǫ(z) = ∆ξ + z∆κc (4.138)

Hence, we solve ∆ξ and ∆κc from the system

N(∆ξ, ∆κc) = N (4.139)

M(∆ξ, ∆κc) = M (4.140)

h−

h+

y

z

z

ǫ

κc

ξ

ǫ(z) = ξ + κc · z

σ

t > 0

t = 0 N

M

Figure 4.14: Single-symmetric cross-section.

Newton iterations give

∆ξ(k+1) = ∆ξ(k) + δξ, ∆κ(k+1)c = ∆κ(k)

c + δκc (4.141)

where δξ and δκc are obtained from the linear set of equations[

Sep(k)a,NN S

ep(k)a,NM

Sep(k)a,NM S

ep(k)a,MM

][

δξ

δκc

]

= −

[

N (k) − N

M (k) − M

]

(4.142)

The algorithmic tangent stiffness moduli are obtained precisely as in the case of nonlinear

viscoelasticity (discussed in Subsection 3.6.5):

Sepa,NN =

∫ h+

−h−

Eepa (z)b(z)dz =

nint∑

i=1

(Eepa )i∆Ai (4.143)

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108 4 PLASTICITY

Sepa,NM =

∫ h+

−h−

Eepa (z)zb(z)dz =

nint∑

i=1

(Eepa )izi∆Ai (4.144)

Sepa,MM =

∫ h+

−h−

Eepa (z)z2b(z)dz =

nint∑

i=1

(Eepa )iIi (4.145)

Moreover, by assuming that the y-axis is located at the center of gravity and choosing

Eepa = E, we obtain

Sepa,NN = EA, Sep

a,NM = 0, Sepa,MM = EI (4.146)

and we obtain from (4.142):

δξ =1

EA(N − N (k)), δκc =

1

EI(M − M (k)) (4.147)

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Chapter 5

VISCOPLASTICITY

In this chapter, we extend the concept of plastic behavior to rate-dependent material

response, by which certain creep and relaxation phenomena of metallic materials at ele-

vated temperature can be modelled. Both the perfectly viscoplastic and the hardening

prototype models are discussed. Numerical integration of the constitutive equations is

described.

5.1 Introduction

The idea of “overstress”, i.e. there exists a threshold value of stress that must be exceeded

before time-dependent inelastic deformation will take place, was invented in the 1920’s

by Bingham for metals (while employing von Mises quasistatic yield surface), and it was

generalized by Hohenemser & Prager to also include hardening of the quasistatic yield

surface. However, this is only a special case of the specific formulation of viscoplasticity of

Perzyna, which applies to general quasistatic yield criteria. In this way, it is possible to

treat the time-dependent response of cohesive, as well as frictional, materials in a unified

fashion. For example, viscoplasticity based on the Perzyna concept has been used with

considerable success to model creep (secondary consolidation) of soft soils, such as clay.

Very sophisticated models for simulating the rate-dependent response of metals, including

classical creep and relaxation situations, have been developed. Models that are able to

represent complex features such as time-recovery of back-stress, strain-range memorization

at cyclic loading, etc., have been proposed.

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110 5 VISCOPLASTICITY

A special class of “viscoplastic” models do not possess any quasistatic yield surface at

all (which corresponds to zero threshold stress), i.e. the models are essentially of the

nonlinear viscoelasticity type, although they are termed viscoplastic in the literature. It

thus appears that viscoplasticity has much in common with nonlinear viscoelasticity; the

main difference being the concept of an elastic region in stress space (like in plasticity).

For example, it is possible to retrieve the simple Maxwell viscoelastic model as a special

case of viscoplasticity.

In recent years the format of viscoplasticity put forward by Duvaut & Lions has been

advocated. As it turns out, this formulation is the natural extension of the underlying

rate-independent plasticity problem when it is set in a incremental format (after integra-

tion using the Backward Euler rule). In fact, the Perzyna and Duvaut-Lions versions of

plasticity can be obtained as special subclasses within a more general framework.

5.2 Prototype rheological model for perfectly viscoplas-

tic behavior

5.2.1 Thermodynamic basis — Quasistatic yield criterion

As the prototype for perfectly viscoplastic behavior, we consider the rheological model

in Figure 5.1, which is often denoted the Bingham model. Like for rate-independent

plasticity, the plastic slider is inactive as long as |σ| < σy, where σy is the quasistatic

yield stress. This indicates that σy is the proper yield stress only at very slow loading.

Hence, the major difference in comparison to (rate-independent) plasticity is that the

stress is allowed to exceed the (quasistatic) yield stress, i.e. |σ| > σy, since stress can be

transferred to the viscous dashpot when the frictional resistance of the slider has been

exhausted. As the single internal variable we take the viscoplastic strain ǫp in the slider

and the dashpot, and the expression for the free energy is chosen (like in the case of

plasticity) as

Ψ =1

2E(ǫe)2 =

1

2E(ǫ − ǫp)2 (5.1)

where ǫe = ǫ− ǫp is the elastic strain of the Hookean spring with modulus of elasticity E.

We then obtain the constitutive equation for the stress as

σ =∂Ψ

∂ǫ= E(ǫ − ǫp) (5.2)

Vol 0 March 7, 2006

5.2 Prototype rheological model for perfectly viscoplastic behavior 111

t ,η(Φ)

ε ε

E

σ σ

σ

e p

y

*

Figure 5.1: Prototype model for elastic-viscoplastic material.

and for the dissipative stress, that is conjugated to ǫp, as

σp = −∂Ψ

∂ǫp= E(ǫ − ǫp) ≡ σ (5.3)

The quasistatic yield criterion is Φ = 0, where Φ is chosen as

Φ = |σ| − σy (5.4)

5.2.2 Viscoplastic flow rule — Perzyna’s formulation

In a formulation suggested by Perzyna (1966), it is assumed that no viscoplastic strain

will be produced when |σ| ≤ σy, i.e. when Φ ≤ 0, in which case the material response is

elastic. Note that this behavior is different from that of inviscid plasticity, for which elastic

response is unconditional only when |σ| < σy, whereas plastic strains can develop when

|σ| = σy. However, viscoplastic strain may be produced due to the “spillover” of stress

from the slider to the dashpot whenever |σ| > σy. As a consequence, the constitutive rate

equation for ǫp is postulated as

ǫp =1

t∗η(Φ)

∂Φ

∂σ=

1

t∗η(Φ)

σ

|σ|(5.5)

where t∗ is the natural relaxation time, and η(Φ) is a non-dimensional strictly monotoni-

cally increasing overstress function with the properties

η(Φ) > 0 when Φ > 0, η(Φ) = 0 when Φ ≤ 0

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It appears that we can write

λ =1

t∗η(Φ) ≥ 0 (5.6)

where λ now plays the role of the plastic multiplier in plasticity. However, in the present

theory of viscoplasticity it should be noted that λ is a state function. Hence, there is

no need to introduce any consistency condition (to ensure that the yield criterion is not

violated) like in plasticity.

By combining (5.2) and (5.5), we may obtain the constitutive differential equation in σ

and ǫ:

σ = E(ǫ − ǫp) = Eǫ −E

t∗η(Φ)

σ

|σ|(5.7)

or

σ +E

t∗η(Φ)

σ

|σ|= Eǫ (5.8)

Like for viscoelasticity, it is possible to solve for ǫ(t) when σ(t) is a prescribed function

or, alternatively, we may solve for σ(t) when ǫ(t) is prescribed. However, because of the

nonlinearity we must resort to numerical integration schemes (like in the case of nonlinear

viscoelasticity).

Creep

In the particular case when σ = σ0 > 0 is held constant (after rapid loading), we obtain

the solution of (5.8) as

ǫ(t) =σ0

E+ η(Φ0)

t

t∗(5.9)

which may be compared with the expression in (3.108) pertinent to the Norton law.

Relaxation

When ǫ(t) is a given function, we may solve for σ(t) directly from (5.8). As long as the

stress is small enough to satisfy σ ≤ σy, then (5.8) has the trivial elastic solution σ = Eǫ.

Let us next consider the situation when the yield stress has been exceeded. In the relax-

ation situation, when ǫ = ǫ0 > 0 is held constant (after rapid loading), then the stress

must be solved from

σ +E

t∗η(Φ) = 0, σ(0) = σ0 = Eǫ0 > 0 (5.10)

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5.2 Prototype rheological model for perfectly viscoplastic behavior 113

It appears from (5.10) that the stress is decreasing monotonically until the stage has been

reached (after long time) when Φ = 0. This holds independently of the parameter values

E and t∗ as well as of the explicit choice of the overstress function η(Φ).

5.2.3 Bingham model — Perzyna’s formulation

The simplest possible model, that features viscoplastic behavior, is the Bingham model.

This model is defined by the choice

η(Φ) =〈Φ〉

E=

〈|σ| − σy〉

E(5.11)

where 〈·〉 is the McCauley bracket defined as

〈Φ〉 =

{

Φ if Φ > 0

0 if Φ ≤ 0(5.12)

Inserting this expression in (5.8), we obtain the constitutive equation

σ +1

t∗〈|σ| − σy〉

σ

|σ|= Eǫ (5.13)

Creep

In the creep situation, when σ = σ0 > 0, we obtain the solution of (5.13) as

ǫ(t) =σ0

E+

〈σ0 − σy〉

E

t

t∗(5.14)

This solution is shown in Figure 5.2(a).

Relaxation

In the case of relaxation, when ǫ = ǫ0 > 0, (5.13) reduces to the problem

σ +1

t∗〈σ − σy〉 = 0, σ(0) = σ0 (5.15)

which has the solution

σ = σ0 − 〈σ0 − σy〉(

1 − e−t

t∗

)

(5.16)

This solution is shown in Figure 5.2(b).

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1

σ -σ E

0 y(2)

σ(2)0

E

σ(1)0

E

σ(2)0 >σy

σ(1)0 <σy

σ(2)0

σ(1)0 <σy

σ(2)0 >σy

σ(1)0

ε σ

tt*

tt*

(a) (b)

Figure 5.2: (a) Creep and (b) Relaxation curves for a Bingham-material.

Special case: A trivially simple case is defined by the choice σy = 0 and

η(Φ) =Φ

E=

|σ|

E(5.17)

Upon introducing the viscosity parameter µ = Et∗, we may insert (5.17) into (5.13) to

obtain

σ +E

µσ = Eǫ (5.18)

This is, clearly, the Maxwell model of viscoelasticity. 2

5.2.4 Norton model (creep law) — Perfect viscoplasticity

A common choice of η(Φ), that defines the generalized Norton viscoplastic law, is given

as

η(Φ) =

(〈Φ〉

E

)nc

=

(〈|σ| − σy〉

E

)nc

and t∗ = τ(σc

E

)nc

(5.19)

where σc is the creep modulus, τ is the relaxation time and nc is the creep exponent (as

introduced in Chapter 3). The classical Norton creep law is retrieved from (5.19) when

σy = 0, in which case we may insert (5.19) into (5.5) and (5.8) to obtain

ǫp =1

t∗

(|σ|

E

)nc σ

|σ|, σ +

E

t∗

(|σ|

E

)nc σ

|σ|= Eǫ (5.20)

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5.3 Prototype rheological model for hardening viscoplasticity 115

5.2.5 Limit behavior — Viscoplastic regularization of rate-independent

plasticity

It is of some interest to assess the behavior of the prototype model when t∗ = 0 and

t∗ = ∞, respectively:

In the case that t∗ = 0, it can be shown from (5.8) that η(Φ) = 0, i.e. Φ = 0, which

means that the plastic consistency condition is satisfied at all times independent of the

loading. Hence, this viscoplastic model coincides with the plasticity model when t∗ = 0.

This is the reason why the elastic-viscoplastic model can be considered as a regularization

of the corresponding elastic-plastic model.

In the case that t∗ = ∞, it follows directly from (5.8) that σ = Eǫ, i.e. we obtain the

elasticity solution. From the model point of view, this case corresponds to a rigid dashpot

(or slider with infinite yield stress).

5.3 Prototype rheological model for hardening vis-

coplasticity

5.3.1 Thermodynamic basis — Quasistatic yield criterion

In analogy with the concept of hardening in rate-independent plasticity, we may introduce

hardening of the quasistatic yield surface, as shown in the prototype model in Figure 5.3.

The resistance of the frictional-plastic slider is increasing due to the amount of slip de-

veloped, which will reduce the stress that can be transferred to the viscous dashpot.

Like in the case of (linear) hardening plasticity, we propose

Ψ =1

2E(ǫ − ǫp)2 +

1

2Hk2 (5.21)

which gives, once more, the constitutive relations

σ =∂Ψ

∂ǫ= E(ǫ − ǫp) = σp (5.22)

κ = −∂Ψ

∂k= −Hk (5.23)

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t ,η(Φ)

ε ε

E σ σ

σ

e p

y

*

H

Figure 5.3: Prototype model for elastic-hardening-viscoplastic material.

The quasistatic yield function is now defined as

Φ(σ, κ) = |σ| − σy − κ (5.24)

5.3.2 Viscoplastic flow and hardening rules — Perzyna’s formu-

lation

Associative flow and hardening rules are defined as

ǫp =1

t∗η(Φ)

∂Φ

∂σ=

1

t∗η(Φ)

σ

|σ|(5.25)

k =1

t∗η(Φ)

∂Φ

∂κ= −

1

t∗η(Φ) (5.26)

By combining (5.25) and (5.26) with Hooke’s law (5.22) and with (5.23), we obtain the

pertinent differential equations for σ and κ:

σ +E

t∗η(Φ)

σ

|σ|= Eǫ (5.27)

κ −H

t∗η(Φ) = 0 (5.28)

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Like in the case of perfect viscoplasticity, we may solve for ǫ(t) and κ(t) when σ(t) is

a prescribed function (creep), or we may solve for σ(t) and κ(t) when ǫ(t) is prescribed

(relaxation). Solutions must in practice be obtained using numerical integration, except

in the particularly simple case when η is a linear function, which defines the Bingham

model. This model is considered next.

5.3.3 Bingham model — Perzyna’s formulation

Choosing η(Φ) as in (5.11), we obtain from (5.27) and (5.28) the set of governing equations:

σ +1

t∗〈|σ| − σy − κ〉

σ

|σ|= Eǫ (5.29)

κ −H

Et∗〈|σ| − σy − κ〉 = 0 (5.30)

Creep

Consider the creep situation, when σ = σ0 > 0 is held constant (after rapid loading).

Upon using the initial conditions ǫ(0) = σ0/E and κ(0) = 0, we obtain the solutions of

(5.29) and (5.30) as

ǫ(t) =σ0

E+

〈σ0 − σy〉

H

(

1 − e−HE

tt∗

)

(5.31)

κ(t) = 〈σ0 − σy〉(

1 − e−HE

tt∗

)

(5.32)

It is noted that ǫ(t) → 0 when t → ∞. Hence, this model is less suitable for describing

real creep behavior, although it can be used to describe the primary stage (stage I) for

small times. In order to mimic the transient as well as the stationary stages (stages I and

II), we must resort to nonlinear hardening characteristics.

Relaxation

In the case of relaxation, when ǫ = ǫ0 > 0, we obtain from (5.29) and (5.30) the problem

σ +1

t∗〈σ − σy − κ〉 = 0, σ(0) = σ0 = Eǫ0 (5.33)

κ −H

Et∗〈σ − σy − κ〉 = 0, κ(0) = 0 (5.34)

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which has the solution

σ(t) = σ0 −E

h〈σ0 − σy〉

(

1 − e−ht

Et∗

)

(5.35)

κ(t) =H

h〈σ0 − σy〉

(

1 − e−ht

Et∗

)

(5.36)

where h is defined as h = E + H .

Homework: Verify the creep and relaxation solutions stated above! 2

The solutions for ǫ(t) in (5.31) and σ(t) in (5.35) are illustrated in Figure 5.4.

σ

yσ

tt*

σ0

σ0(E/h) + (H/h)yσ

σ0

tt*

1

σ0 yσ-

E

Eσ0 yσ-

H

Stage I

ε

(a) (b)

Figure 5.4: (a) Creep and (b) Relaxation solution for linear hardening of the Bingham

material when σ0 > σy.

5.3.4 Viscoplastic flow and hardening rules — Duvaut-Lions’

formulation

As an alternative to the formulation of Perzyna, we may formulate the flow and hardening

rules (that are pertinent to the quasistatic yield surface) in the spirit of a formulation

first suggested by Duvaut and Lions (1972). With a slight generalization of their

formulation we propose

ǫp =1

t∗ηD(ρ)

1

E(σ − σs) (5.37)

k =1

t∗ηD(ρ)

1

H(κ − κs) (5.38)

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where ρ = |σ − σs| and where ηD(ρ) (subscript “D” refers to the formulation of Duvaut-

Lions) is a non-dimensional strictly monotonically increasing overstress function such that

ηD(ρ) ≥ 1,dηD

dρ(ρ) > 0 for ρ > 0 (5.39)

Remark: Both Φ and ρ are state functions, i.e. Φ = Φ(σ, κ) and ρ = ρ(σ, κ). In general

ηD(ρ) 6= η(Φ) for a given state. As will be shown later, the two formulations of Perzyna

and Duvaut-Lions are generally not equivalent. 2

The pair (σs, κs) is quasistatically admissible in the sense that it is defined as the CPPM-

projection of the current state (σ, κ) onto the convex set E defined as

E = {(σ, κ)|Φ(σ, κ) ≤ 0} (5.40)

i.e. (σs, κs) is the solution of the constrained minimization problem

(σs, κs) = arg

{

Min(σ,κ)∈E

[1

E(σ − σ)2 +

1

H(κ − κ)2

]}

(5.41)

Since E is a convex set, the solution (σs, κs) is unique in the case of strict hardening, i.e.

when H > 0.

Remark: The similarity between the rate-independent solution (n+1σ, n+1κ) in Subsec-

tion 4.4.6 and the present static solution (σs, κs) is striking. 2

We may now combine (5.37) and (5.38) with (5.22) and (5.23) to obtain the constitutive

differential equations for σ and κ as follows:

σ +1

t∗ηD(ρ)(σ − σs) = Eǫ (5.42)

κ +1

t∗ηD(ρ)(κ − κs) = 0 (5.43)

5.3.5 Comparison of Perzyna’s and Duvaut-Lions’ formulations

Since E and H are assumed to be constants, it may be shown that the optimality condi-

tions corresponding to the constrained minimization problem in (5.41) are:

σs = σ − λsE

(∂Φ

∂σ

)s

= σ − λsEσs

|σs|(5.44)

κs = κ − λsH

(∂Φ

∂κ

)s

= κ + λsH (5.45)

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where the Lagrangian multiplier λs can be calculated if (5.44) and (5.45) are subjected

to the complementary conditions

λs ≥ 0, Φs ≤ 0, λsΦs = 0 (5.46)

In fact, in the present case it is possible to calculate λs explicitly (see Chapter 4) as

λs =〈Φ〉

E + H(5.47)

Upon inserting the solutions (5.44) and (5.45) into (5.42) and (5.43), we obtain

ǫp =1

t∗ηD(ρ)λs σs

|σs|(5.48)

k = −1

t∗ηD(ρ)λs (5.49)

Since it is obvious that sign(σ) = sign(σs), we conclude that the two formulations of

Perzyna and Duvaut-Lions are identical if

Homework: Verify this result! 2

5.3.6 Bingham model — Duvaut-Lions’ formulation

The Bingham model is defined by choosing η(Φ) as in (5.11). Upon inserting this ex-

pression in (5.49), we retrieve the same model within the framework of Duvaut-Lions’

formulation by choosing

ηD(ρ) = 1 +H

E(5.50)

This expression can be inserted into (5.42) and (5.43) to give the constitutive rate equa-

tions

σ +1

t∗

(

1 +H

E

)

(σ − σs) = Eǫ (5.51)

κ +1

t∗

(

1 +H

E

)

(κ − κs) = 0 (5.52)

Remark: The original form of the Duvaut-Lions’ formulation, cf. Simo & Hughes

(1988), seems to be

σ +1

t∗(σ − σs) = Eǫ (5.53)

κ +1

t∗(κ − κs) = 0 (5.54)

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rather than (5.52) and (5.53). From the discussion above, it should be clear that the

relations (5.54) and (5.55) are equivalent to those of Perzyna only in the special case

when H = 0, in which case ηD(ρ) = 1. 2

5.4 Model for cyclic loading — Mixed isotropic and

kinematic hardening

5.4.1 Constitutive relations for linear hardening — Perzyna’s

formulation

In the case of uniaxial stress, the relevant quasistatic yield function was given in (4.47)

as:

Φ = |σred| − σy − κ, σred = σ − α (5.55)

and the constitutive differential equations according to the Perzyna formulation become

σ +E

t∗η(Φ)

σred

|σred|= Eǫ (5.56)

κ −rH

t∗η(Φ) = 0 (5.57)

α −(1 − r)H

t∗η(Φ)

σred

|σred|= 0 (5.58)

These equations should be compared with those of rate-independent plasticity given in

(4.51) to (4.53), and it turns out that they are quite similar.

5.4.2 Backward Euler method for linear hardening — Perzyna’s

formulation

When the Backward Euler method is used to integrate Eqns. (5.56) to (5.58), we obtain

(formally) the same expressions for the updated stresses as in rate-independent plasticity

that were given in (4.94) to (4.97). The expressions are repeated here for completeness:

n+1σ = c1n+1σtr + (1 − c1)

nα (5.59)

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n+1κ = nκ + rHµ (5.60)

n+1α = c2n+1σtr + (1 − c2)

nα (5.61)

where

c1 = 1 −Eµ

n+1σred,tre

, c2 =(1 − r)Hµn+1σred,tr

e

(5.62)

The multiplier µ is still given as the solution of the equation

y(µ) = η(

n+1Φ(µ))−

t∗∆t

µ = 0 (5.63)

where n+1Φ(µ) is given as

n+1Φ(µ) =n+1 Φtr − hepµ with hep = E + H (5.64)

Hence, (5.63) may be rewritten more explicitely as

η(n+1Φtr − hepµ) =t∗∆t

µ (5.65)

Let us next consider the simple model of Bingham, defined in (5.11). From (5.65) we

obtain

η(Φ) =< Φ >

E=⇒ <n+1 Φtr − hepµ > =

Et∗∆t

µ (5.66)

In the case of loading (L), defined as n+1Φ ≥ 0 and µ ≥ 0, we obtain from (5.66)

µ =n+1Φtr

hevp> 0 when n+1Φtr > 0 (5.67)

where hevp, defined as

hevp = hep +Et∗∆t

(5.68)

is the viscoplastic “enhancement” of hep.

Summary of algorithm

We summarize the solution algorithm in Box 5.1.

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1. Given ∆ǫ −→ ∆σtr −→ n+1σtr = nσ + E∆ǫ

2. Check L/U

If n+1Φtr ≤ 0, then µ = 0

else µ =n+1Φtr

hevp> 0

3. Update solution

n+1σ = c1n+1σtr + (1 − c1)

nα, c1 = 1 −Eµ

n+1σred,tre

n+1κ = nκ + rHµ

n+1α = c2n+1σtr + (1 − c2)

nα, c2 =(1 − r)Hµn+1σred,tr

e

Box 5.1: Solution algorithm for linear mixed hardening in the Perzyna formulation of viscoplasticity.

5.5 Structural analysis

The structural analysis is carried out in exactly the same fashion as for rate-independent

plasticity. For example, the analysis of a truss is the same as in Subsection 4.5.2 if only

the ATS-matrix Sepa is replaced with Sevp

a . We shall then need the algorithmic stiffness

Eevpa , defined by the relation

Eevpa =

d(∆σ)

d(∆ǫ)(5.69)

It is then remarked that the corresponding CTS-relations in viscoplasticity (which would

be denoted Eevp) does not exist!

Remark: For linear mixed hardening of Perzyna viscoplasticity, we obtain

Eevpa = E

(

1 −E

hevp

)

(5.70)

In the special case of when t∗ = 0 (rate-independent response), we obtain

Eevpa → Eep

a = E

(

1 −E

hep

)

=EH

hep, hep = E + H (5.71)

Show this as homework! Hint: Use (5.59), (5.62)1 and (5.67). 2

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Chapter 6

DAMAGE AND FRACTURE

THEORY

In this chapter we introduce the concept of distributed (or continuum) damage. A model

for damage coupled to elastic deformation is presented. We also touch upon the issue of

how to experimentally quantify damage.

6.1 Introduction to the modeling of damage

6.1.1 Concept of damage

Close to the state of failure the microstructure of any engineering material will start to

disintegrate or “break up”. The physical nature of this deterioration is, of course, not the

same for different materials. For example, in a metal the deformation accelerates due to

the simultaneous propagation of microcracks and growth of microcavities. These defects

initiate due to severe stress concentrations in the neighborhood of in-situ inclusions and

interfaces, e.g. along the grain boundaries. Eventually, these microcracks and microvoids

coalesce to form an incipient macroscopical crack. From the material point of view, failure

has already occurred at this point, which may be well in advance of the stage where the

crack can be observed.

The process of successive material degradation may be modelled by damage theory. New

(internal) damage variables are introduced to represent the degree to which the material

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126 6 DAMAGE AND FRACTURE THEORY

has degraded. Clearly, it would be desirable to use this concept throughout the deforma-

tion process up to and including the localized mode of failure that represents a partially

open crack. Then the validity of continuum theory could be extended up to complete fail-

ure. However, as pointed out already in Chapter 1, the traditional approach is to employ

classical linear fracture mechanics in order to assess the possibility for further propagation

of the (pre)existing crack. The main purpose is then to establish whether the crack will

propagate in a stable or unstable manner at increased loading, and the latter situation is

viewed as structural failure.

The development of damage may be linked to the development of inelastic (plastic, creep,

et.) strain, or it may be governed by a separate criterion (that is analogous to the yield

criterion). The first approach is feasible for the modeling of ductile failure and creep

failure, as demonstrated in Chapters 7 and 8, whereas the latter is useful in conjunction

with brittle behavior, which is shown more explicitly in this Chapter. The following

classification of damage-related phenomena is commonly made:

• Brittle damage (also denoted elastic damage): Only elastic strains occur (in the

intact material), and damage develops with the total strain after a certain threshold

of strain has been exceeded.

• Ductile damage: Damage develops with plastic strains after a threshold of plastic

strain has been exceeded. Plastic strains and damage may localize at incipient

failure, cf. Figure 1.2.

• Creep damage: Damage dominates the tertiary creep phase, and is enhanced at

elevated temperature. Intergranular decohesion is pronounced, cf. Figure 1.7.

• Low-cycle fatigue: Ductile failure terminates the cyclic loading process, cf. Fig-

ure 1.13.

• High-cycle fatigue: Brittle failure terminates the cyclic loading process, cf. Fig-

ure 1.9.

An important feature of damage is that it leads to strain-softening of the stress-strain

relationship, i.e. the stress drops to zero more or less rapidly (depending on the rate of

damage development) after the ultimate stress is traversed. This behavior may be taken

as material instability.

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6.1 Introduction to the modeling of damage 127

A major borderline can be drawn between the models in which damage is coupled to

the total strain (elastic damage) and those models in which damage is coupled to the

inelastic part of the strain (plastic and viscoplastic damgage including low-cycle-fatigue

and creep). The principal behavior is sketched in Figure 6.1.

Figure 6.1: Undamaged and damaged (a) Elastic material, (b) Elastic-plastic material.

6.1.2 Physical nature of damage for different materials

The debonding mechanisms, that are represented as damage, are different in metals and

alloys, polymers, composites, ceramics, concrete and wood. It should be noted that

damage is not the same thing as deformation, although it may be coupled to deformation.

Let us, for example, consider a metal. Plastic slip, which is enhanced by a high dislocation

density, occurs without debonding. However, dislocation movements can be stopped by

an inclusion in the lattice in such a way that plastic deformation can not continue without

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braking the atomic bonds, i.e. resulting in damage. In this way a microcrack is created.

Since the number of atomic bonds (which is a measure of the elastic stiffness) decreases

with damage development, it follows that the elastic modulus decreases with damage.

Hence, the development of damage in a metal is coupled to the development of inelastic

strain, but it is the reduced elasticity modulus that is a measure of damage, as will be

shown below.

The physical nature of damage for a variety of engineering materials are listed as follows:

• Metals and alloys: Nucleation of microcracks at inclusions within the grains and

the matrix. Debonding between grains and matrix along weakened interfaces.

• Polymers: Brakeage of bonds between long chains of molecules.

• Fiber composites: Debonding between fibers and the polymeric or metallic ma-

trix.

• Concrete: Decohesion between aggregates (stones) and cement paste. Debonding

between reinforcement steel and cement paste.

• Wood: Debonding along cellular walls.

6.1.3 The concepts of effective stress and strain equivalence

Consider a unit area of the material which has been damaged so that the area portion

d is completely broken and can not sustain any stress. The scalar damage variable d is

the simplest form of representing this situation. It is clear that d = 1 corresponds to a

completely deteriorated material.

Remark: The material will reach its ultimate strength at a certain finite value dF , which

may be taken as the failure value. For larger damage, the stress-strain relation shows

softening. 2

The view of damage taken above leads to the Equivalent Strain Principle: The material

is considered as if it would consist of parallel fibers subjected to the same strain. At a

certain state the area portion d of the fibers are broken, and the remaining portion 1− d

is intact, as shown in Figure 6.2. For given applied nominal stress σ, the reduction in

stress-carrying area leads to a corresponding increase of stress in the undamaged portion.

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6.1 Introduction to the modeling of damage 129

This is the intrinsic or effective stress σ, which notion was first introduced by Kachanov

(1958). A simple equilibrium equation for the unit area of material (see Figure 6.2) gives

σ =σ

1 − d(6.1)

Figure 6.2: Elementary model of elastic damage based on the Equivalent Strain Principle.

Remark: This is a very special case of a two-phase material with parallel coupling of

the two phases, where the extreme view is taken that one of the phases has completely

deteriorated. 2

A consequence of the Equivalent Strain Principle is that any constitutive functional rela-

tionship expressing the behavior of undamaged material will also be valid for the damaged

material if only the nominal stress is replaced by the effective stress. For example, assum-

ing that the virgin (undamaged) material obeys the constitutive law

σ = f(ǫ, kd) (6.2)

then the damaged material is characterized simply by

σ = f(ǫ, kd) (6.3)

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6.2 Prototype model of damage coupled to elasticity

6.2.1 Thermodynamics — Damage criterion

The simplest form of damage coupled to elastic behavior is obtained by defining Ψ =

Ψ(ǫ, d) as

Ψ =1

2(1 − d)Eǫ2 (6.4)

where the damage variable (0 ≤ d ≤ 1) represents the dissipative mechanism, whereas E is

the constant modulus of elasticity that represents linear elastic response of the undamaged

material.

Coleman’s equations applied to Ψ as defined in (6.4) gives the nominal stress σ as

σ =∂Ψ

∂ǫ= (1 − d)Eǫ = (1 − d)σ (6.5)

where (6.1) was used. It thus follows that the effective stress is given as

σ = Eǫ (6.6)

which illustrates the simplicity of the notion of effective stress (as alluded to in the Remark

above).

The secant modulus of the damaged material, denoted E, is defined from

σ = Eǫ (6.7)

A comparison of (6.5), (6.6) and (6.7) shows that

E = Eσ

σ= (1 − d)E (6.8)

which gives the direct interpretation of d as

d = 1 −E

E(6.9)

Remark: The relation (6.9) is the basis of a simple and straightforward method for

experimentally determining the amount of damage, which subject is further discussed in

a subsequent subsection. 2

We may also obtain the dissipative “force” δ that is energy conjugate to d as

δ = −∂Ψ

∂d=

1

2Eǫ2 =

1

2Eσ2 (6.10)

which is the effective elastic stress energy stored in the material. In order to obtain the

last equality in (6.10), we used (6.6).

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6.2 Prototype model of damage coupled to elasticity 131

Damage energy release rate

Let us introduce Gibbs’ free energy Ψ via the Legendre transformation

Ψ(σ, d) = σǫ − Ψ =1

2E(1 − d)σ2 (6.11)

We then obtain readily that

∂Ψ

∂d=

1

2E(1 − d)2σ2 =

1

2Eσ2 = δ

(

= −∂Ψ

∂d

)

(6.12)

Since ∂Ψ/∂d is the release rate of elastic stress energy during damage growth at constant

nominal stress, the dissipative force δ is sometimes denoted the damage energy release

rate. This definition is analogous to the definition of the fracture energy release rate Gf

in fracture mechanics, cf. Chapter 9.

Damage criterion

Similarly to the situation in plasticity, we may introduce a damage criterion (rather than

a yield criterion) expressed as Φ = 0, where Φ(δ, d) is the damage function. In the case

that Φ < 0 the response is elastic, whereas the case Φ = 0 admits the possibility for

development of damage. A quite general class of damage criteria, that couples damage to

the total “elastic” strain, may be expressed in the form

Φ = δ − g(d) (6.13)

where g(d) is a monotonically increasing function such that

g(0) = 0, g(1) = gmax (6.14)

For reasons that will be evident later, we shall here consider the specific choice

g(d) = gmax[1 − (1 − d)m]23 , gmax =

E

2

(6S

Em

) 23

(6.15)

where S is the damage modulus (dimension N/m2) and m is the damage exponent, both

of which govern the rate of damage development.

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Stress-strain relation

Since we consider, at the moment, a situation of uniaxial stress, it is possible to derive

the stress-strain relationship directly from (6.13) under the assumption of inelastic loading

(for which the proper condition will be evaluated below). At loading the damage criterion

is satisfied, i.e. Φ = 0, which gives (after some elaboration, where the definition of δ is

used):

d = 1 −

(

1 −Em

6S|ǫ|3) 1

m

(6.16)

Upon using the failure condition that d = 1 when ǫ = ǫf , we may rewrite (6.16) as

d = 1 −

(

1 −

(|ǫ|

ǫf

)3) 1

m

with ǫ3f =

6S

Em(6.17)

This expression may now be inserted into (6.5) to give

σ

Eǫf

=ǫ

ǫf

(

1 −

(|ǫ|

ǫf

)3) 1

m

(6.18)

The ultimate stress σ = σu is obtained for ǫ = ǫu from the condition dσ(ǫ)/dǫ = 0 at

ǫ = ǫu, which gives

ǫu

ǫf=

(m

m + 3

) 13

(6.19)

σu

Eǫf=

(m

m + 3

) 13(

3

m + 3

) 1m

(6.20)

du = 1 −

(3

m + 3

) 1m

(6.21)

In the special case that m = 1, we obtain a model that has turned out to be reasonable for

modeling the uniaxial compression for concrete, cf. Lemaitre and Chaboche (1990).

Typical results are shown in Figure 6.3. In the extreme situation when m = ∞, we obtain

from (6.19)-(6.21) thatǫu

ǫf= 1,

σu

Eǫf= 1, du = 0 (6.22)

This evidently corresponds to perfectly brittle response preceeded by only elastic defor-

mations. In the model above it is possible to introduce a threshold strain ǫ below which

no damage will develop. The resulting behavior is shown schematically in Figure 6.4. (To

derive the pertinent expressions is left as an exercise for the reader.)

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6.2 Prototype model of damage coupled to elasticity 133

(a) (b)

σEǫf

m = 1

m = 1

m = 2

m = 3

2 3 ǫǫf

ǫǫf

d

1

1

1

1

Figure 6.3: Characteristic behavior of elastic-damage model.

6.2.2 Damage law and tangent relations

The damage law is given formally as the associated law

d = λ∂Φ

∂δ= λ (6.23)

Whenever the damage criterion Φ = 0 is satisfied, it is possible to determine the multiplier

λ from the complementary conditions

λ ≥ 0, Φ ≤ 0, λΦ = 0 (6.24)

We may use the definition of δ and the damage law in (6.23) to obtain

Φ =∂Φ

∂δδ +

∂Φ

∂dd = Eǫǫ +

∂Φ

∂dλ ≤ 0 (6.25)

In order to conform the formulation with the elastic-plastic behavior (discussed in Chapter

4), we may rewrite (6.25) as

Φtr − hλ ≤ 0 (6.26)

where we have introduced the “elastic” rate Φtr and the inelastic (damage) modulus h as

Φtr = Eǫǫ (6.27)

h = −∂Φ

∂d=

∂g

∂d=

2m

3gmax[1 − (1 − d)m]−

13 (1 − d)m−1 > 0 (6.28)

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1

1

1

1

(a) (b)

σEǫf

m = 1m = 1

m = 2

m = 3

2 3

ǫǫf

ǫǫf

d

ǫǫf

ǫǫf

Figure 6.4: Characteristic behavior of elastic-damage model with strain threshold.

Damage (loading) occurs when Φtr > 0, or

ǫǫ > 0 (6.29)

in which situation we obtain the solution for λ from (6.26) as

λ =Eǫ

hǫ (6.30)

Upon inserting (6.30) into (6.23), we may express the damage law as

d =Eǫ

hǫ =

σ

hǫ =

E|ǫ|

h

σ

|σ|ǫ =

δ

S(1 − d)m−1

σ

|σ|ǫ (6.31)

The last expression of (6.31), which is obtained using the definition of h in (6.28) and

noting that E|ǫ| = (2δE)1/2, is quite interesting, since it reminds strongly about the

formulation of the damage law that is frequently used in the context of damage coupled

to plasticity, as will be discussed in some detail in Chapter 7.

Remark: The damage law in (6.31) can be taken as the basic law, rather than the

postulated damage criterion in (6.15). In fact, integration of the damage law will give

precisely the criterion in (6.15) and, as shown above, the stress-strain behavior that is

valid for loading. 2

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6.3 Experimental measurement of damage 135

It is now a simple matter to obtain the tangent relation by first differentiating (6.5) to

obtain

σ + σd = (1 − d)Eǫ (6.32)

and then inserting the damage law in (6.32) to finally obtain the relation

σ =

(

(1 − d)E −1

hσ2

)

ǫ =

(

(1 − d)E −δ|σ|

S(1 − d)m−1

)

ǫ (6.33)

Finally, we conclude that elastic unloading without development of damage will take place

if

ǫǫ ≤ 0 (6.34)

We thus obtain the tangent relation

σ = (1 − d)Eǫ (6.35)

where the value d = d remains constant during unloading. It is simple to integrate (6.35)

to obtain the unloading relation

σ

Eǫf= (1 − d)

ǫ

ǫf(6.36)

It is noted that the unloading stress-strain path returns straight to the origin, i.e. (6.36)

represents a secant relation. This extreme situation is quite unrealistic for most materials,

but has nevertheless been used in modeling. For concrete, this behavior was denoted

elastic-fracturing by Dougill (1976). The more realistic behavior, that there is some

irreversible strain after unloading, is discussed in the following Chapters.

6.3 Experimental measurement of damage

A review of the more significant methods to measure damage in the special case of uniaxial

stress is given as follows:

Degradation of elasticity

As stated above, damage can be measured from the unloading modulus at unloading,

following monotonic loading, as the relation

d = 1 −E

E(6.37)

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where E is the “undamaged” modulus, whereas E is the “damaged”, or effective, modulus

defined in (6.7). Accurate measurements require that the damage is uniformly distributed

in the bar specimen, i.e. the strain must be measured before localization occurs (or in

the localized zone).

Degradation of ultrasonic wave velocity

If the stress state is uniaxial, there is a simple relation between the (axial) wave velocity

vL in the bar and its elasticity modulus. Provided that the density ρ is unaffected by the

damage, we have the relations

vL2 =

E

ρ, v2

L =E

ρ(6.38)

Using this expression in (6.8), we obtain the alternative measure of damage as

d = 1 −E

E= 1 −

v2L

vL2

(6.39)

Stress amplitude drop at fatigue and creep failure

Estimates of damage from fatigue and creep failure tests are discussed in Chapters 10 and

8.

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Chapter 7

DAMAGE COUPLED TO

PLASTICITY

In order to describe ductile fracture under monotonic loading and fatigue failure under

cyclic loading, it is necessary to couple damage to the development of plastic deformation.

(The physical explanation for this assumption was given in Chapter 6). In particular, low-

cycle fatigue (LCF) is the consequence of damage development under cyclic loading, which

is discussed in further detail in Chapter 9.

7.1 Prototype model for damage coupled to perfect

plasticity

7.1.1 Thermodynamic basis — Yield and damage criterion

The presence of material damage means that part of the material has lost its bearing

capacity, as illustrated schematically for the prototype model in Figure 7.1.

As a result, the free energy is reduced, and we may consider a prototype model based on

the following expression for Ψ(ǫ, ǫp, d):

Ψ =1

2(1 − d)E(ǫ − ǫp)2 (7.1)

from which Coleman’s equations give

σ =∂Ψ

∂ǫ= (1 − d)E(ǫ − ǫp) = (1 − d)σ with σ =

σ

1 − d= E(ǫ − ǫp) (7.2)

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138 7 DAMAGE COUPLED TO PLASTICITY

Figure 7.1: Prototype model of elastic-plastic damage (based on the Equivalent Strain

Principle).

σp = −∂Ψ

∂ǫp= (1 − d)E(ǫ − ǫp) ≡ σ (7.3)

δ = −∂Ψ

∂d=

1

2E(ǫ − ǫp)2 =

1

2Eσ2 (7.4)

It is noted that the expression δ = −∂Ψ/∂d for the damage force is exactly the same as

for damage coupled to “elastic deformation”, that was discussed in the previous Chapter.

We now introduce the yield function Φ = Φ(σ)

Φ(σ) = |σ| − σy (7.5)

which corresponds to a perfectly plastic behavior of the undamaged material. The plastic

potential Φ∗ = Φ∗(σ, δ, d) is proposed as

Φ∗ = Φ +δ2

2S(1 − d)m6= Φ (7.6)

where S is a material constant; dim(S)=dim(σ), and m is an exponent that governs the

rate of damage evolution (which will be evident below).

Remark: Since the second term of Φ∗ in (7.6) is quadratic in δ, the dissipation inequal-

ity is always satisfied. Hence, the chosen model is admissible from a thermodynamics

viewpoint, and it is termed thermodynamically consistent. 2

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7.1 Prototype model for damage coupled to perfect plasticity 139

7.1.2 Plastic flow rule and damage law — Constitutive relations

The constitutive rate equations for ǫp and d are obtained as the associative flow rule and

the non-associative damage rule as:

ǫp = λ∂Φ∗

∂σ=

λ

1 − d

∂Φ∗

∂σ=

λ

1 − d

σ

|σ|(7.7)

d = λ∂Φ∗

∂δ=

λ

(1 − d)m

σ2

2ES=

σ2

2ES(1 − d)m−1

σ

|σ|ǫp ≥ 0 (7.8)

Remark: The last expression of (7.8), which was obtained upon using (7.7), is completely

equivalent to that obtained for elastic coupling in (6.30); the only difference is that ǫ is

now replaced by its plastic part ǫp. 2

From (7.7), we obtain

λ = (1 − d)ǫpe with ǫp

e = |ǫp| (7.9)

Combining (7.7) with Hooke’s law in (7.2), we obtain

˙σ = E(ǫ − ǫp) = E

(

ǫ −λ

1 − d

σ

|σ|

)

= Eǫ −λ

1 − dE

σ

|σ|(7.10)

Together with (7.8), this equation is solved when subjected to the loading criteria (com-

plementarity conditions):

λ ≥ 0, Φ(σ) ≤ 0, λΦ(σ) = 0 (7.11)

Tangent stiffness relation

We may use the consistency condition Φ ≤ 0 in a plastic state (Φ = 0) to derive

Φ =∂Φ

∂σ˙σ ≤ 0 (7.12)


Φ =σ

|σ|Eǫ − hλ ≤ 0 (7.13)

where the plastic/damage modulus h is given as

h =E

1 − d(7.14)

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Plastic/damage loading (L) is obtained when Φtr > 0, where

Φtr =σ

|σ|Eǫ (7.15)

which is precisely the same loading criterion as for plasticity without damage. At loading

(L) we thus obtain λ as

λ =Φtr

h= (1 − d)

σ

|σ|ǫ > 0 (L) (7.16)

and we may insert this expression into (7.7) and (7.8) with (7.4) to obtain

ǫp = ǫ (L) (7.17)

d =|σ|σ

2ES(1 − d)m−1ǫ (L) (7.18)

Finally, we obtain the tangent stiffness relation by combining (7.2), (7.10), (7.17) and

(7.18):

σ = (1 − d) ˙σ − σd = −|σ|3

2ES(1 − d)m−1ǫ (L) (7.19)

Remark: We note that ˙σ = 0 due to perfectly plastic response of the undamaged mate-

rial. Moreover, the strain ǫ can be considered as a fixed parameter during the process of

unloading. 2

Stress-strain relation

In this simple case it is possible to integrate (7.18) and (7.19). By using the yield criterion

(7.5), i.e. |σ| = σy at loading, we first obtain tangent relations

d =σy

2

2ES(1 − d)m−1sign (σ)ǫ (7.20)

σ = −σy

3

2ES(1 − d)m−1ǫ (7.21)

Next, we solve (7.20) and (7.21) for two different load cases representing tensile and

compressive states.

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Load case I

Assume that σ = σy > 0, ǫ > 0 (Φtr > 0). The initial conditions d = 0 and σ = σy, when

ǫ = ǫy, give the solutions of (7.20) and (7.21):

d = 1 −

[

1 −σy

2m

2ES(ǫ − ǫy)

] 1m

, ǫ ≥ ǫy (7.22)

σ = σy

[

1 −σy

2m

2ES(ǫ − ǫy)

] 1m

, ǫ ≥ ǫy (7.23)

Load case II

Assume that σ = −σy < 0, ǫ < 0 (Φtr > 0). The initial conditions d = 0 and σ = −σy,

when ǫ = −ǫy, give the solutions of (7.20) and (7.21):

d = 1 −

[

1 +σy

2m

2ES(ǫ + ǫy)

] 1m

, ǫ ≤ −ǫy (7.24)

σ = −σy

[

1 +σy

2m

2ES(ǫ + ǫy)

] 1m

, ǫ ≤ −ǫy (7.25)

Alternatively, we may eliminate the parameter S in terms of the fracture strain ǫf . For

example, in Load Case I we obtain d = 1 when ǫ = ǫf , which gives

S

m=

σy2

2E(ǫf − ǫy) (7.26)

Remark: It is possible to express the damage modulus S as

S =ms

2σyǫ

2y, s =

ǫf

ǫy

− 1 (7.27)

where s is a non-dimensional ductility measure. For metals, s is in the range 100-200 for

ductile damage, while s may be as small as 1-2 for brittle damage. 2

Upon introducing the expression (7.27) into (7.22) and (7.23), we obtain

d = 1 −

(ǫf − ǫ

ǫf − ǫy

) 1m

, ǫ ≥ ǫy (7.28)

σ = σy

(ǫf − ǫ

ǫf − ǫy

) 1m

, ǫ ≥ ǫy (7.29)

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Let us next consider elastic unloading (E), which takes place while the damage is constant,

d = d. In this particular case, the damage value d corresponds to the strain ǫ via the

relation (7.28). Hence, the stress rate is given by the rate law

σ = (1 − d)Eǫ =

(ǫf − ǫ

ǫf − ǫy

) 1m

Eǫ (7.30)

Characteristic stress-strain and damage-strain relationships are depicted in Figure 7.2 for

the choice m > 1. (The values m < 1 are less realistic). The influence of the value of m

on the characteristics of these relationships is demonstrated in Figure 7.3. Bilinear curves

are obtained in the special case that m = 1.

Figure 7.2: (a) Typical stress-strain and (b) Damage-strain relationships (for m > 1)

showing loading and unloading characteristics.

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Let us consider the behavior when ǫ → ǫf . From (7.27) and (7.28) we obtain

dd

dǫ=

1

m(ǫf − ǫy)

(ǫf − ǫ

ǫf − ǫy

) 1m−1

≥ 0 (7.31)

dσ

dǫ= −

σy

m(ǫf − ǫy)

(ǫf − ǫ

ǫf − ǫy

) 1m−1

≤ 0 (7.32)

It is clear that both dd/dǫ and dσ/dǫ tend to zero when ǫ → ǫf in the case m < 1, whereas

they tend to infinity in the case m > 1. The latter behavior is more in accordance with

experimental evidence.

Figure 7.3: (a) Stress-strain and (b) Damage-strain relationships in loading showing the

influence of the exponent m.

7.1.3 Dissipation inequality

The rate of dissipation D is given as

D = σǫp + δd = σλ

1 − d

σ

|σ|+ λ

δ

S(1 − d)m(7.33)

= λ

(

σy +δ2

S(1 − d)m

)

≥ 0

where it was used that |σ| = σy at plastic loading. Hence, the model is thermodynamically

consistent.

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7.1.4 Dissipation of mechanical energy

The rate of work W and the rate of plastic work Dp are defined as

Wdef= σǫ , Dp def

= σǫp = λσy ≥ 0 (7.34)

We shall also introduce the fracture energy release gf (energy per unit volume of material),

defined as

gf =

∫ ∞

0

Wdt (7.35)

where it is assumed that σ = 0 when t = ∞ (like in the prototype model above). Hence, gf

is the necessary “consumed energy” in order to reduce the stress to zero, which corresponds

to the fully fractured state. Thereby we consider the prototype model based on perfect

plasticity coupled to damage as well as the alternative model of softening plasticity without

damage. The latter model mimics the same (nonlinear) softening characteristics as does

the damage-based model when m < 1.

In order to unify the developments in an explicit fashion, we rederive the results of the

damage-based model while using stress control, i.e., σ is the control variable. The same

strategy is then used for the plasticity model. Moreover, we shall directly use the condition

σ > 0 in pure tension.

Perfect plasticity coupled to damage

The model is defined by

Φ = |σ| − σy , Φ∗ = Φ +δ2

2S(1 − d)m, δ =

|σ|2

2E=

σ2y

2E(7.36)

which gives

ǫp =λ

1 − d, d = λ

σ2y

2ES(1 − d)m(7.37)

The consistency condition (at loading) reads

Φ = ˙σ = 0 (7.38)

However,

σ =σ

1 − d; ˙σ =

σ

1 − d+

σ

1 − dd =

σ

1 − d+

σy

1 − dd (7.39)

Vol 0 March 7, 2006


Inserting (7.37) into (7.39) gives

λ =1

hσ with h = −

σ3y

2ES(1 − d)m(7.40)

Now, upon inserting (7.40) in (7.37)2, or directly from (7.39), we obtain

d = −σ

σy; d = 1 −

σ

σy(7.41)

where it was used that d = 0 when σ = σy.

Remark: The result in (7.41) could have been obtained directly from (7.20) and (7.21)

for σ > 0, which are derived under the assumption of strain control. 2

We may combine the result in (7.40) with (7.41)2 and insert into (7.37)1 to obtain

ǫp = −2ESσm−1

(σy)m+2σ ; Dp = σǫp = −

2ESσm

(σy)m+2σ (7.42)

As to W , we first conclude that

ǫ = ǫe + ǫp , ǫe =σ

E(7.43)

However, since ˙σ = 0 from (7.38), we obtain ǫe = 0 and ǫ = ǫp, which gives W = Dp.

Remark: In the general situation of plastic hardening when ˙σ 6= 0, then ǫe 6= 0 and

W 6= Dp. 2

Hence, upon integration of (7.42)2, we obtain

gf = gef + gp

f =σ2

y

2E−

2ES

(σy)m+2

∫ 0

σy

σm dσ =σ2

y

2E+

2ES

(m + 1)σy≃

2ES

(m + 1)σy(7.44)

and we have obtained one equation for determining the unknown parameters S and m.

The remaining equation is obtained from an experimentally given value of the fracture

strain ǫf (or ductility measure s, as defined in (7.27)). To this end, we integrate (7.42)1

as follows:

ǫf = ǫef + ǫp

f =σy

E−

2ES

(σy)m+2

∫ 0

σy

σm−1 dσ =σy

E+

2ES

mσ2y

≃2ES

mσ2y

if ǫy ≪ ǫf (7.45)

Remark: The relation (7.45)1 is identical to (7.27) expressed in terms of the ducility

measure s. 2

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We may now solve for S and m from (7.44) and (7.45)2 to obtain

S =σygf

2E

(

1 −gf

ǫfσy

)−1

, m =gf

ǫfσy

(

1 −gf

ǫfσy

)−1

(7.46)

It appears that a downwards convex stress vs. strain relation, defined by m < 1, requires

thatgf

ǫfσy<

1

2(7.47)

Softening plasticity

In the case of elastic-plastic response (with linear elasticity), we have

ǫ =σ

E+ ǫp

; gf =

∫ ∞

0

σσ

Edt +

∫ ∞

0

Dpdt =

∫ ∞

0

Dpdt (7.48)

where it was used that the elastic part of the strain does not contribute to the dissipation

of energy in a closed stress cycle (when stress is increased to the peak stress and then

reduced to zero during the fracture process).

To represent the same type of behavior as the damage-based model, we propose the

nonlinear softening model defined by

Φ = |σ| − σy − κ , Φ∗ = Φ +|κ|m

mσm−1y

(7.49)

which gives

ǫp = λ , k = −λ

[

1 −

(|κ|

σy

)m−1]

(7.50)

Hence, we obtain

κ = −H0k = H0λ

[

1 −

(|κ|

σy

)m−1]

with H0 < 0 (7.51)

σ = E(ǫ − ǫp) = E(ǫ − λ) (7.52)

The consistency relation (at loading) reads

Φ = σ − κ = 0 (7.53)

from which

κ = σ ; κ = σ − σy < 0 (7.54)

Vol 0 March 7, 2006


where it was used that κ = 0 when σ = σy. Moreover, by inserting (7.51) and (7.52) into

(7.53), we obtain

λ =1

hσ with h = H0

[

1 −

(

1 −σ

σy

)m−1]

(7.55)

Now, upon combining (7.55) with (7.50)1, we obtain

ǫp =1

hσ ; Dp =

σ

H0

[

1 −(

1 − σσy

)m−1] σ (7.56)

Hence, we obtain (after variable substitution):

gf = −σ2

y

H

∫ 1

0

1 − x

1 − xm−1dx (7.57)

Let us consider two special cases:

Case 1: m = ∞ corresponds to linear softening and gives

gf = −σ2

y

2H(7.58)

Case 2: m = 2 corresponds to the (classical) nonlinear saturation hardening, cf. Sec-

tion 4.4.4, and gives

gf = −σ2

y

H(7.59)

These two cases are depicted in Figure 7.4.

εfεy

σ

ε

m=2σy m=∞

−H

1

Figure 7.4: Softening plasticity relation.

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7.2 Prototype model for damage coupled to harden-

ing plasticity

7.2.1 Thermodynamics — Yield and damage criterion

Damage coupled to (isotropic) hardening plasticity is defined by the free energy density

Ψ =1

2(1 − d)E(ǫ − ǫp)2 +

1

2Hk2 (7.60)

From Coleman’s equations, we obtain

σ =∂Ψ

∂ǫ= (1 − d)σ, σp ≡ σ (7.61)

κ = −∂Ψ

∂k= −Hk (7.62)

δ = −∂Ψ

∂d=

1

2E(ǫ − ǫp)2 =

1

2Eσ2 (7.63)

The yield criterion is given as

Φ(σ, κ) = |σ| − σy − κ (7.64)

whereas, still, the plastic potential is defined as in (7.6), i.e.

Φ∗ = Φ +δ2

2S(1 − d)m(7.65)

7.2.2 Dissipation rules — Constitutive relations

The constitutive rate equations for the internal variables are given as

ǫp = λ∂Φ∗

∂σ=

λ

1 − d

∂Φ

∂σ=

λ

1 − d

σ

|σ|(7.66)

k = λ∂Φ∗

∂κ= λ

∂Φ

∂κ= −λ (7.67)

d = λ∂Φ∗

∂δ=

λ

(1 − d)m

δ

S=

λ

(1 − d)m

σ2

2ES(7.68)

It is noted that the flow and hardening rules are associated with Φ, whereas this is not

the case for the damage rule. Upon inserting (7.66) and (7.67) into (7.61) and (7.62), we

obtain the constitutive rate equations

˙σ = Eǫ −λ

1 − dE

σ

|σ|(7.69)

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7.2 Prototype model for damage coupled to hardening plasticity 149

κ = Hλ (7.70)

Together with (7.64), these equations can be solved when they are subjected to the loading

condition

λ ≥ 0, Φ(σ, κ) ≤ 0, λΦ(σ, κ) = 0 (7.71)

It remains to establish the tangent stiffness relation.

Tangent stiffness relation

From the consistency condition Φ ≤ 0, valid at the plastic state (Φ = 0), we obtain

Φ =∂Φ

∂σ˙σ +

∂Φ

∂κκ ≤ 0 (7.72)

Analogously with the situation of perfect plasticity, we obtain

Φ = Φtr − hλ ≤ 0 (7.73)

where Φtr is still given (like in the case of perfect plasticity) as

Φtr =σ

|σ|Eǫ (7.74)

whereas h is now given as

h =E

1 − d+ H > 0 (7.75)

Plastic/damage loading (L) is obtained when Φtr > 0, in which situation (7.73) gives

λ =Φtr

h=

E

h

σ

|σ|ǫ > 0 (L) (7.76)

which expression may be inserted into (7.66) and (7.68) to give

ǫp =E

(1 − d)hǫ (L) (7.77)

d =|σ|σ

2hS(1 − d)mǫ (L) (7.78)

Finally, we obtain the tangent stiffness relation by inserting (7.68) and (7.69) into (7.61)

to give

σ = (1 − d) ˙σ − σd = Eepǫ (L) (7.79)

where

Eep =(1 − d)E

h

[

H −|σ|3

2ES(1 − d)m+1

]

(7.80)

When H = 0, we readily retrieve the tangent stiffness relation in (7.19).

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Transition of hardening to softening

It is of some interest to investigate under which conditions the response is hardening.

Firstly, right at the onset of yielding (when d = κ = 0), we obtain the initial value Eep0

from (7.80) as

Eep0 =

E

E + H

(

H −σy

3

2ES

)

=E

E + H(H − Hcr) with Hcr =

σy3

2ES(7.81)

and it follows that

Eep0 > 0 if H > Hcr (7.82)

For a particular state during the development of damage and plastic deformation, we

obtain (for given H > Hcr) that

Eep > 0 if d < dcr with dcr = 1 −

[(σy + κcr)

3

2ESH

] 1m+1

(7.83)

where it is noted that κ = κcr depends on the history of damage development (that must

be calculated while the constitutive relations are integrated numerically), as illustrated

in Figure 7.5.

Figure 7.5: Typical stress-strain response curves depending on the hardening.

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7.3 Model for cyclic loading and fatigue — Mixed linear isotropic and kinematichardening 151


As to the dissipation of energy, we obtain

D = σǫp + κk + δd = λ

(

σy +δ2

S(1 − d)m

)

≥ 0 (7.84)

where it was used that |σ| − κ = σy at plastic loading. Hence, the model is thermody-

namically consistent.

7.3 Model for cyclic loading and fatigue — Mixed

linear isotropic and kinematic hardening

7.3.1 Constitutive relations for linear hardening

The pertinent constitutive equations for damage coupled to linear mixed isotropic and

kinematic hardening plasticity in the case of uniaxial stress are summarized as follows:

The yield function of von Mises, accounting for damage, is defined by

Φ = σrede − σy − κ with σred

e = |σred|, σred = σ − α (7.85)

The constitutive equations become

˙σ = Eǫ −λ

1 − dE

σred

σrede

and σ =σ

1 − d(7.86)

κ = λrH (7.87)

α = λ(1 − r)H

(σred

σrede

)

(7.88)

d = λσ2

2ES(1 − d)m(7.89)

which can be solved when they are subjected to the complementary conditions

λ ≥ 0, Φ ≤ 0, λΦ = 0 (7.90)

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7.3.2 Backward Euler algorithm for integration — Linear hard-

ening and uniaxial stress

Integration of damage law

Upon applying the Backward Euler (or fully implicit) rule for integrating the damage law

in (7.89), we obtain the relation

n+1d = nd + µn+1σ2

2ES(1 − n+1d)m(7.91)

We are thus seeking the solution d = n+1d of the equation

y(d) = d − nd − µ(d)n+1σ(d)2

2ES(1− d)m= 0 (7.92)

A convenient and robust (although not necessarily the most efficient) iterative technique

to obtain the root of y(d) = 0, if such a root exists, is based on a bisecting procedure within

the interval 0 ≤ d < 1, which will be discussed in detail below. For each iterative value of

d = n+1d in this procedure, it is necessary to calculate the appropriate values of µ(d) andn+1σ(d). This can be done in a fashion that is almost identical to the solution procedure for

plasticity without damage (as described in the next subsubsection). The major difference

is that nominal stress quantities are replaced with the effective correspondents.

Integration of stress (for given damage)

Following the procedure outlined in subsection 4.4.6, we arrive at the constitutive equation

n+1Φ(µ) = n+1σrede − σy −

n+1κ = n+1Φtr − n+1hµ ≤ 0 (7.93)

where n+1Φtr is given as

n+1Φtr = n+1σred,tre −σy−

nκ with n+1σred,tre = |n+1σred,tr| , n+1σred,tr = n+1σtr−nα (7.94)

The generated plastic modulus n+1h is slightly modified as compared to (4.90), i.e. it is

defined asn+1h =

E

1 − n+1d+ H (7.95)

where we note the occurrence of the “effective elastic modulus” E/(1 − n+1d).

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Loading is defined as in (4.91), that is

n+1Φ(0) = n+1Φtr(0) = n+1Φtr > 0 (7.96)

where n+1Φtr was defined in (7.94). From (7.93), we obtain the solution

µ =n+1Φtr

n+1h(7.97)

Remark: The loading condition need to be checked (within each load step) only at the

first iteration on the damage variable when d = nd. 2

The updated state becomes

n+1σ = c1n+1σtr + (1 − c1)

nα (7.98)

n+1κ = nκ + rHµ (7.99)

n+1α = c2n+1σtr + (1 − c2)

nα (7.100)

where the coefficients c1 and c2 are given as

c1 = 1 −Eµ

n+1σred,tre (1 − n+1d)

, c2 = aα(1 − r)Hµn+1σred,tr

e

(7.101)

Computational algorithm

The complete computational algorithm for a time increment ∆t is given in Box 7.1 and

in Figure 7.6.

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1. For given state (nǫ, nσ, nκ, nα, nd), compute the updated strain

and “effective elastic trial stress”

n+1ǫ = nǫ + ∆ǫ, n+1σtr = nσ + E∆ǫ

2. Check loading/unloading of updated solution (n+1σ, n+1κ, n+1α).

If n+1Φtr ≤ 0, then elastic unloading

n+1σ = n+1σtr, n+1κ = nκ, n+1α = nα, n+1d = nd

Stop

elseif n+1Φtr > 0, then plastic loading

3. Set d(0) max = 1, dmin = nd and n+1d = nd; then start iteration

4. For given value n+1d(k) in iteration k, calculate updated values

dmax and dmin as follows:

Integration ;n+1σ(k), µ(k)

; y(k)

If y(k) ≥ 0 then dmax = n+1d(k)

elseif y(k) < 0 then dmin = n+1d(k)

5. Bisect the interval [dmin, dmax] to obtain

n+1d(k+1) =1

2(dmin + dmax)

6. Check convergence

If | n+1d(k+1) − n+1d(k)| < tol then Stop

If k > maxiter, then reduce ∆t and goto 1

else goto 4 and continue iteration

Box 7.1: Algorithm within time increment.

Remark: As to the bisecting procedure, we remark that µ > 0 always, which means that

y(nd) < 0. Hence, one of the two situations depicted in Figure 7.6 is the relevant one.

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Figure 7.6: Function y(d) used in bisectioning algorithm. (a) Solution exists, (b) Solution

does not exist since ∆t is too large.

Moreover, failure is detected at the actual state whenever

∆d

∆t> large value (= 10000) or d ≥ dcr (= 0.99) (7.102)

The first criterion in (7.102) corresponds to loss of controllability in the sense that d grows

in an unlimited fashion with time. 2

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Chapter 8

DAMAGE COUPLED TO

VISCOPLASTICITY

In order to describe creep rupture under constant or moderately varying load as well as

creep-fatigue phenomena under cyclic loading, it is necessary to couple damage to the

development of viscoplastic deformation.

8.1 Prototype model for damage coupled to perfect

viscoplasticity

8.1.1 Thermodynamic basis — Quasistatic yield and damage

criterion

Like in the case of damage coupled to rate-independent plasticity (that was discussed in

Chapter 7), we consider the simplest prototype model based on the following expression

for Ψ(ǫ, ǫp, d):

Ψ =1

2(1 − d)E(ǫ − ǫp)2 (8.1)

Hence, Coleman’s equations are the same as those of damage coupled to rate-independent

plasticity:

σ =∂Ψ

∂ǫ= (1 − d)E(ǫ − ǫp) = (1 − d)σ with σ =

σ

1 − d≡ E(ǫ − ǫp) (8.2)

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158 8 DAMAGE COUPLED TO VISCOPLASTICITY

σp = −∂Ψ

∂ǫp= (1 − d)E(ǫ − ǫp) ≡ σ (8.3)

δ = −∂Ψ

∂d=

1

2E(ǫ − ǫp)2 =

1

2Eσ2 (8.4)

The perfectly plastic quasistatic yield function Φ = Φ(σ) is chosen as

Φ = |σ| − σy (8.5)

which corresponds to a perfectly viscoplastic behavior of the undamaged material. The

plastic potential Φ∗ = Φ∗(σ, δ, d) is proposed as

Φ∗ = Φ +δ2

2S(1 − d)m6= Φ (8.6)

where S is a material constant; dim(S) = dim(σ), and m is an exponent that governs

the rate of damage evolution. All these expressions coincide with their rate-independent

counterparts.

8.1.2 Viscoplastic flow rule and damage law — Perzyna’s for-

mulation

Analogously to the situation of damage coupled to rate-independent plasticity, we propose

ǫp =1

t∗η(Φ)

∂Φ∗

∂σ=

1

t∗(1 − d)η(Φ)

∂Φ∗

∂σ=

1

t∗(1 − d)η(Φ)

σ

|σ|(8.7)

d =1

t∗η(Φ)

∂Φ∗

∂δ=

1

t∗

η(Φ)

(1 − d)m

σ2

2ES≥ 0 (8.8)

where t∗ is (still) the natural relaxation time, and η(Φ) is a non-dimensional monotonically

increasing overstress function (that were both introduced in Chapter 5).

By combining (8.2) and (8.7), we now obtain the differential equations in σ and ǫ:

˙σ = E(ǫ − ǫp) = Eǫ −E

t∗(1 − d)η(Φ)

σ

|σ|(8.9)

or˙σ +

E

t∗(1 − d)η(Φ)

σ

|σ|= Eǫ (8.10)

We may introduce the definition of σ to derive the alternative expression

σ + σd +E

t∗η(Φ)

σ

|σ|= Eǫ with E = (1 − d)E (8.11)

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8.1 Prototype model for damage coupled to perfect viscoplasticity 159

8.1.3 Norton model (creep Law) — Perfect viscoplasticity

Like in the case of viscoplasticity without damage, cf. Chapter 5, we define the generalized

Norton viscoplastic law by the following choice of η(Φ):

η(Φ) =

(〈Φ〉

E

)nc

=

(〈|σ| − σy〉

E

)nc

(8.12)

We shall only consider the classical Norton creep law, which is retrieved from (8.12) when

σy = 0, in which case we may insert (8.12) into (8.7), (8.8) and (8.11) to obtain

ǫp =1

t∗(1 − d)

(|σ|

E

)nc σ

|σ|, and d =

1

t∗

1

(1 − d)m

(|σ|

E

)nc σ2

2ES≥ 0 (8.13)

σ + σd +E

t∗

(|σ|

E

)nc σ

|σ|= Eǫ (8.14)

Creep (stage III)

It is assumed that the stress σ = σ0 > 0 is applied suddenly, whereafter creep occurs for

a constant stress. This process is defined by the initial value problem

d =1

t∗

1

(1 − d)m+nc+2

(σ0

E

)nc σ02

2ES, d(0) = 0 (8.15)

ǫ =σ0d

E(1 − d)2+

1

t∗

1

(1 − d)nc+1

(σ0

E

)nc

, ǫ(0) =σ0

E(8.16)

It is possible to solve (8.15) first, which gives

d(t) = 1 − p(t)1

m+nc+3 (8.17)

where we have introduced the function

p(t) = 1 −(m + nc + 3)σ0

2

2ES

(σ0

E

)nc t

t∗; 0 < p(t) ≤ 1 (8.18)

Upon introducing the solution for d(t) given in (8.17) into (8.16), we obtain, after some

elaboration, the solutions for ǫ(t) and ǫ(t) as

ǫ(t) =1

t∗

(σ0

E

)nc[

σ03

2E2Sp(t)−

m+nc+4m+nc+3 + p(t)−

nc+1m+nc+3

]

≈1

t∗

(σ0

E

)nc

p(t)−nc+1

m+nc+3 (8.19)

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ǫ(t) =σ0

Ep(t)−

1m+nc+3 +

2ES

(m + 2)σ02

(

1 − p(t)m+2

m+nc+3

)

(8.20)

≈σ0

E+

2ES

(m + 2)σ02

(

1 − p(t)m+2

m+nc+2

)

The first term of the exact solution in (8.19) is due to the change of elastic stiffness when

the stress is held constant, i.e. σ = σ0. It appears that this term can be neglected in

comparison with the viscoplastic contribution only when the following criterion is satisfied:

k ≡σ0

3

2E2S<< 1 (8.21)

For realistic values (σ0 ≈ 200MPa, E = 200GPa, S ≈ 1MPa), we obtain k ≈ 0.0001,

which means that the indicated approximation is justified except when t is close to tR.

The approximation is frequently adopted in the literature, cf. Lemaitre (1992).

Remark: According to the exact solution, ǫ → ∞ when t → tR, whereas the introduced

approximation gives a finite value of ǫ when t = tR. 2

In the particular case that S = ∞, we obtain

p(t) = 1 ; d(t) = 0 (8.22)

Hence, we retrieve the solution without damage (given in Chapter 6):

ǫ(t) =1

t∗

(σ0

E

)nc

, ǫ(t) =σ0

E+(σ0

E

)nc t

t∗(8.23)

Let us next consider the situation when t = 0, at which moment we have p(0) = 1, and

(8.19) gives

ǫ(0) =1

t∗

(σ0

E

)nc(

σ03

2E2S+ 1

)

(8.24)

where the first term in this expression is due to damage development.

Creep rupture

From (8.15) and (8.16) we note that ǫ → ∞ and d → ∞ when d → 1. This situation

is achieved after the (finite) rupture time tR, which can be calculated from the condition

d(tR) = 1. This condition is, obviously, equivalent to the condition p(tR) = 0, and from

(8.18) we then obtaintRt∗

=2ES

(m + nc + 3)σ02

(σ0

E

)−nc

(8.25)

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8.1 Prototype model for damage coupled to perfect viscoplasticity 161

It is noted that the life-time is reduced by an increase of the stress level. Moreover, the

rate of damage development is increased by reducing the modulus S and raising the value

of the exponent m in the damage law. As a result the life time is reduced. The typical

behavior is shown in Figure 8.1.

Figure 8.1: Creep rupture behavior for the Norton-material in terms of (a) Creep strain

rate, (b) Creep strain.

Taking the logarithm of both sides of (8.25), we obtain the alternative expression

lntRt∗

= ln2S

(m + nc + 3)E− (nc + 2) ln

σ0

E(8.26)

This relation, which shows the influence of the creep exponent nc, is depicted in Figure 8.2.

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Figure 8.2: Creep rupture time tR versus applied stress σ0.

8.2 Prototype model for damage coupled to harden-

ing viscoplasticity

8.2.1 Thermodynamic basis — Quasistatic yield and damage

criterion

Isotropic hardening of the quasistatic yield surface can be included in a fashion that is

similar to viscoplasticity without damage. We thus propose

Ψ =1

2(1 − d)E(ǫ − ǫp)2 +

1

2Hk2 (8.27)

and Coleman’s equations give

σ =∂Ψ

∂ǫ= (1 − d)σ, σ = E(ǫ − ǫp), σp ≡ σ (8.28)

κ = −∂Ψ

∂k= −Hk (8.29)

δ = −∂Ψ

∂d=

1

2Eσ2 (8.30)

The quasistatic yield function accounting for isotropic hardening is given as

Φ(σ, κ) = |σ| − σy − κ (8.31)

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8.3 Constitutive modeling of creep failure of metals and alloys 163

while the plastic potential Φ∗(σ, κ, δ, d) is still given by (8.6).

8.2.2 Viscoplastic flow, hardening and damage rules — Perzyna’s

formulation

The flow, hardening and damage rules are proposed as

ǫp =1

t∗η(Φ)

∂Φ∗

∂σ=

1

t∗(1 − d)η(Φ)

∂Φ∗

∂σ=

1

t∗(1 − d)η(Φ)

σ

|σ|(8.32)

k =1

t∗η(Φ)

∂Φ∗

∂κ= −

1

t∗η(Φ) (8.33)

d =1

t∗η(Φ)

∂Φ∗

∂δ=

1

t∗η(Φ)

σ2

2ES(1 − d)m≥ 0 (8.34)

By combining (8.32) and (8.33) with (8.28) and (8.29), we obtain the constitutive rate

equations

˙σ +E

t∗(1 − d)η(Φ)

σ

|σ|= Eǫ (8.35)

κ −H

t∗η(Φ) = 0 (8.36)

Together with (8.34), these equations are sufficient to determine the time dependent

stress-strain relation under various loading conditions. In practice, it is necessary to

use numerical integration methods to solve even this (moderately complex) initial value

problem.

8.3 Constitutive modeling of creep failure of metals

and alloys

8.3.1 Modified damage law for tertiary creep

The general characteristics of the commonly observed behavior of metals and alloys under

creep conditions was discussed already in Chapter 1. The subsequent comments refer to

Figure 8.3.

After the transient (primary) phase, corresponding to saturation hardening of the qua-

sistatic yield surface, the behavior is virtually “perfectly viscoplastic” up to a certain level

Vol 0 March 7, 2006


of accumulated creep strain ǫpe where damage starts to develop. This (secondary) phase of

stationary creep is ideally assumed to take place with a constant quasistatic yield surface,

although the transition from the transient to the stationary stage is smooth. In practice,

it seems reasonable to employ the nonlinear hardening model without damage (discussed

in Chapter 5) for these two first stages.

In order to model the tertiary stage of the creep process, which terminates in creep failure

after the time tR, it is necessary to modify the damage law given in (8.34) in such a fashion

that ǫpe will serve as the threshold value, below which no damage develops. This value is

achieved after the the time t. It should be noted that t is not a material constant, since

it is dependent on the stress level. The damage law in (8.34) is thus reformulated as:

d =1

t∗η(Φ)

σ2

2ES(1 − d)mH(ǫp

e − ǫpe ) (8.37)

where H(x) denotes the Heaviside function defined by

H(x) = 1 if x ≥ 0, H(x) = 0 if x < 0 (8.38)

Due to the development of damage, the effective stress moves away from the quasistatic

yield surface (as shown in Figure 8.3), which results in increasing creep rate. Hence, it is

possible to unify the typical behavior of all creep stages within one single model concept.

8.3.2 Typical results for creep at uniaxial stress

In order to increase the model capability, it is possible to introduce (nonlinear) saturation

hardening. The model described by Ekh (2000) employs this feature and a modified

damage law of the type in (8.37), and we shall adopt this model to predict the uniaxial

behavior in creep. The following data were used for the calculations:

σy

E= 0.001,

H

E= 0.02, r = 1, κ∞ = 0.25σy, m = 2, σ0 = 2σy (8.39)

where κ∞ is the saturation value of isotropic hardening. The results are shown in Fig-

ure 8.4.

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8.3 Constitutive modeling of creep failure of metals and alloys 165

σ

σ1

2σ=σ fixed

σ

σ1

2σ=σ fixed

σ

σ1

2

σ

I Saturation hardening

II Perfect viscoplasticity(stationary creep)

III Softening

II III

t

t

t

tR

tR

tR

ε

ε

I II III

I II III

.

fixedσ

t II t I

t II t I

t II t I

σ

Φ = 0, t = tI

Φ = 0, t = 0

IΦ = 0, t ≤ t ≤ t

II

(saturated hardening)

, t > tIIσ

IIΦ = 0, t ≤ t ≤ t

R

σ-σy

Φ I

Figure 8.3: Characteristics of the creep stages I, II, and III.

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0 20 40 60 80

5

10

15

20

25

30

t / t*

εε / y

(a)

Figure 8.4: (a) Strain versus time, (b) Hardening stress and damage versus time.

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Chapter 9

FATIGUE — PHENOMENON

AND ANALYSIS

The fatigue phenomenon, which is the most common reason for failure of engineering

components, is treated in this chapter. Classical approaches (which are based on empirical

relations) are reviewed. A damage mechanics approach is then discussed, whereby the

famous Manson-Coffin relation is derived. Finally, the growth of a macroscopic fatigue

crack, using the fracture mechanics approach, is discussed.

9.1 Background

9.1.1 Nomenclature

As alluded to already in the Chapter 1, fatigue is the accepted term for the damage and

eventual failure of a material that is subjected to cyclic variation of the loading. The basic

situation is defined as mechanical fatigue at ambient temperature, which occurs as a result

of rate-independent material behavior for cyclic variation of the externally applied loads

only. At elevated temperature, depending on the rate of loading, creep deformation may

take place in combination with mechanical LCF. This gives rise to creep-fatigue (CLCF),

which is of particular importance in hot engine parts, such as in jet turbines. Thermal

fatigue is encountered when the stress change is caused by cyclic variation of temperature

in a statically indeterminate structure. A typical situation is start-stop of an engine. The

combined effect of cyclic mechanical and thermal loading is denoted thermomechanical

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168 9 FATIGUE — PHENOMENON AND ANALYSIS

fatigue.

Another fatigue phenomenon is the socalled corrosion fatigue. This is rather the gradual

deterioration of the microstructure due to chemically aggressive or embrittling environ-

ment.

From the microstructural point of view, the progression of fatigue for a metal may be

classified as follows:

1. Saturation of dislocations after which nucleation and creation of microscopic voids

and cracks take place, often at the grain boundaries (saturation phase, N cycles).

2. Growth of the microscopic flaws to complete coalescence and initiation of a macro-

scopic crack (damage phase, Nd cycles).

3. Propagation of the macroscopic crack until its growth becomes unstable and struc-

tural failure occurs (fracture phase, Nf cycles).

The crack growth can be accelerated by stress-corrosion at the grain boundaries and by

the presence of embrittling substance, such as hydrogen and chromium.

Remark: The total fatigue life is the sum of the cycles in the damage and fracture phases.

However, in the engineering approach discussed in this chapter, no distinction is made

between the different phases. 2

9.1.2 Historical remarks

The most famous investigations of fatigue failure (in railroad axles) were carried out

by Wohler (1860), although it was not the first study of fatigue. Based on various

experimental findings, an empirical relation between stress amplitude and the number of

cycles to failure was proposed 50 years later. Of importance for the understanding of

LCF is also the discovery by Bauschinger that the current yield stress upon reversed

loading is smaller. From a Swedish perspective, we may mention the concept of “linear

damage accumulation” for assessing the combined effects of a series of block-loadings with

different, but constant, amplitudes, which was proposed by Palmgren (1924).

A design criterion for LCF was established quite late. Coffin (1954) and Manson

(1954) independently proposed an empirical relation between plastic strain amplitude

and number of cycles to failure.

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9.1 Background 169

It is wideley accepted that the early phase of fatigue is characterized by slow growth of

damage. However, a rational theory was suggested rather recently by French scientists, see

Lemaitre (1992). Based on the concept of an evolution law for continuum damage in

addition to plastic deformation, the Basquin and Coffin-Manson relations can be derived,

which is also shown below in Section 9.2. The analysis is still based on simplifying

assumptions. For completely arbitrary variable amplitude loadings, it is necessary to

adopt a more complete constitutive theory in order to predict the fatigue life. This is also

shown later in this Chapter for LCF. A corresponding theory for HCF has been suggested

by Ottosen (1995).

After the initiation of a macroscopic fatigue crack, its propagation can be considered

within the realm of fracture mechanics. Paris & al. (1961) proposed an evolution law

for the fatigue crack such that is advancement per load cycle is related to the amplitude

of the stress intensity factor at constant amplitude loading.

9.1.3 Cyclic stress-strain relation

The basic mode of fatigue testing involves a loading program with constant amplitude

stress or strain cycles. Under rather ideal conditions the cyclic response will result in

stabilized hysteretic loops after a number of cycles. This is particularly true for smooth-

surfaced specimens without notches and excessive surface roughness (that act as stress

concentrators), whereby material deterioration does not start until after quite many load

cycles. Under the further idealized assumption that the stabilized hysteretic loops are

symmetrical, it is sometimes useful (in conjunction with fatigue life predictions) to re-

late the strain amplitude ǫa to the stress amplitude σa of the stabilized loops, as shown

in Figure 9.1. It is noted that this stabilized cyclic stress-strain curve, can be located

above (cyclic hardening) or below (cyclic softening) the corresponding monotonic stress-

strain curve. Generally speaking, well-annealed metals and alloys exhibit cyclic hardening,

whereas work-hardened materials undergo cyclic softening.

In the case that plastic strains are produced during a typical cycle, then a commonly used

idealization of the stabilized curve is the power-law expression

ǫa = ǫea + ǫp

a =σa

E+

(σa

σ′cyc

) 1ncyc

(9.1)

Remark: The expression (9.1) is valid only if the yield stress was exceeded in the first

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cycle, which pertains to LCF. If the macroscopic stress is purely elastic, which pertains

to HCF, then the expression (9.1) has no meaning. 2

As to the parameters, E is the elasticity modulus, whereas σ′cyc and ncyc are material

parameters to be determined via a suitable curve-fitting procedure. In the literature

σ′cyc is denoted the cyclic strength coefficient, whereas ncyc is the cyclic strain hardening

coefficient. For most metals, ncyc is in the range 0.1-0.2.

Remark: Do not confuse the cyclic strain hardening modulus ncyc with the creep expo-

nent in the Norton law c. 2

σa

ε a

σ

ε

σa

σa

ε aε a

Figure 9.1: Stabilized cyclic stress-strain relation.

9.2 Engineering approach to HCF and LCF based on

stress-control

9.2.1 Basquin-relation

In the approach for assessing the fatigue life that was suggested by Wohler (1860),

smooth (unnotched) test specimens are fatigue-tested in plane bending, rotating bending

or uniaxial tension-compression. The most basic loading modes are defined as reversed

stress (σm = 0) and pulsating stress (σm = σa), which are shown in Figure 9.2.

Let us first consider the case of fully reversed cyclic loading, in which case σm = 0. The

typical result of fatigue tests are shown in Figure 9.3, where σa is plotted against NR (the

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9.2 Engineering approach to HCF and LCF based on stress-control 171

σa

( b )

= 0σ m

− σa

σ

t

2σa

( c )

σaσ m =σ

t

σa

σ max

σ m

σ

σ min

( a )

t

2σa

Figure 9.2: (a) Constant stress amplitude cyclic loading, (b) Reversed stress loading, (c)

Pulsating stress loading.

number of cycles to failure). Such curves are known as S-N-curves, or Wohler-curves.

A simple empirical expression for the Wohler-curves is the following socalled Basquin-

relation:

σa = σ′f(2NR)b (9.2)

where σ′f is the fatigue strength coefficient, which (as a rule of thumb) can be taken as

the fracture strength as observed in a monotonic tensile test. Moreover, b is the socalled

Basquin exponent, whose value is in the range −0.005 to −0.012 for most metals and

alloys.

Remark: Eqn. (9.2) is supposed to be valid for both HCF and LCF, although the

physical mechanisms of damage are different. For example, in LCF plastic strains will

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σa

( a ) ( b )

flσ > 0

= 0flσ

flσ

ln σa

RNln σa = ln f′σ ln+ b ( )2

RNln ( )2

b < 0

1

RN

Figure 9.3: (a) Wohler-curves (S-N-curves) for material with σfl > 0 and with σfl = 0, (b)

Basquin relation (σfl = 0).

develop according to the cyclic stress-strain relation in (9.1). 2

Some materials can sustain “infinite” number of cycles if only σa is below a certain thresh-

old level. This threshold amplitude σfl is denoted the fatigue limit, or endurance limit.

For most materials σfl is in the range 35 % to 50 % of σu, where σu is the ultimate tensile

strength. Therefore, we introduce the modified Basquin-relation:

σa − σfl = σ′f(2NR)b (9.3)

However, for many high strength steels and aluminum alloys, no fatigue limit exists, i.e.

σfl = 0.

It is emphasized that NR represents the total fatigue life, i.e. NR includes the initial

damage phase as well as the subsequent fracture phase (as already discussed in Chapter

1). The damage portion may vary from 0 % for structures containing severe stress con-

centrations or surface defects to 80 % in very smooth surfaced, defect-free materials of

high purity.

Effect of mid-stress. Haigh-diagram

The mid stress σm has a significant influence on the fatigue life such that the Wohler

curves are lowered for increasing σm, as shown in Figure 9.4(a). In other words, for

given life NR, it appears that σa is reduced with increasing σm. The resulting socalled

Haigh-diagram is shown in Figure 9.4(b).

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σa

σ m

( a )

RN

(1) (2)σ m σ m<

(2)σ m

σap

σ y σu

representspulsating load

σa

σar

σmr = 0 σ mp = σap

( b )

Figure 9.4: (a) Wohler-curves for different constant σm, (b) Typical Haigh-diagram show-

ing possible range of σa versus σm for given life NR.

A reasonable approximation for ductile alloys is the socalled Gerber-relation

σa = σar

[

1 −

(σm

σu

)2]

(9.4)

where σar is the amplitude for reversed loading (σmr = 0). From this relation we may

calculate (σa =)σap, which is the pertinent amplitude for pulsating loading, by setting

(σm =)σmp = σap.

For hand calculation purposes, the relation may be further simplified by a bilinear re-

lationship, as shown in Figure 9.5(a). An alternative, but equivalent, representation is

given by the socalled Goodman-diagram, Figure 9.5(b), which shows the stress range

(σm + σa, σm − σa) as a function of σm.

Effect of stress concentration at HCF

Stress concentrations at holes and notches will have an influence on the fatigue life, al-

though in a manner that is different from that of stress concentrations for monotonic

loading. In the case of monotonic loading, we recall that it is possible to relate the local

stress concentration to the far-field stress by the stress concentration factor Kt, which is

calculated for the appropriate constitutive relation. For example, for a hole in an infinite

plate with isotropic elastic material behavior, we obtain Kt = 3.

In order to account for the effect of stress concentrations on the fatigue life, we introduce

(admittedly in a somewhat ad-hoc manner) the fatigue notch factor Kf via the empirical

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2σap

− σar

( b )

σ m +−σ = σa

( a )

σap

σu

σa

σar

σa σ m=

σ m

σap

σu

σar

σ m

σu

σ σ m=

σap

Figure 9.5: (a) Haigh-diagram and (b) Goodman-diagram suitable for design.

relation

Kf = 1 + q(Kt − 1), 0 ≤ q ≤ 1, Kf ≤ Kt (9.5)

where q is the socalled notch sensitivity index. For a notch with root radius ρ, it has been

suggested that q is calculated according to the empirical relation:

q =

(

1 +

√

An

ρ

)−1

(9.6)

where An is a constant that depends on the strength and ductility of the material.

(An ∼0.025–0.25 mm for steel, where the lower limit is for very high strength steel,

whereas the upper limit is for well-annealed steel.) In brief, we note that An = 0 (q = 1)

for very brittle material response, while An is large for ductile materials corresponding to

a smaller value of q. Obviously, the larger value of An, the smaller sensitivity to a notch.

It is noted that

ρ = 0 ⇒ q = 0, ρ = ∞ ⇒ q = 1 (9.7)

In practice, the stress concentration factor Kt, and therefore also Kf , are always calculated

under the assumption that the material is isotropic elastic. This is valid with good

approximation only for HCF, in which situation Kf is not affected by the actual stress

level.

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The design procedure is then as follows: Use the relations for the nominal (far-field) stress

by making the substitutions

σm → Ktσm, σa → Kfσa (9.8)

Then use these values (with the appropriate safety factors) in the design situation.

Effect of stress concentration at LCF

LCF will occur in the inelastic regime, whereby the stress concentration due to a notch

will be less than that for the elastic response. Moreover, the corresponding concentration

of strain at the notch root will be different than that of stress. To this end, we introduce

the notation

σ∗a = Kσσa, ǫ∗a = Kǫǫa = Kǫ

σa

E, Kσ ≤ Kǫ, Kσ ≤ Kf (9.9)

where Kσ and Kǫ are the stress and strain concentration factors, respectively, and where

σa and ǫa are the nominal (far-field) stress and strain values, which are calculated under

the assumption of elastic response. Clearly, if the material was elastic (HCF), then it

follows trivially that Kσ = Kǫ = Kf , which is illustrated in Figure 9.6 (dotted line).

For LCF, the actual values of Kσ (and Kǫ) are obtained from the following approximate

method, denoted Neuber’s method: It is assumed that the stress and strain states at the

notch satisfy the socalled Neuber hyperbola, which is satisfied by the (fictitious) elastic

state (Kfǫa, Kfσa), as shown in Figure 9.6. This means that the Neuber hyperbola has

the equation

σǫ =K2

f σ2a

Ewith σa = Eǫa (9.10)

Let us next assume that the actual stabilized cyclic stress-strain curve is given by the

function

σa = f(ǫa) (9.11)

For each stress level σa, it is assumed that the state (ǫ∗a, σ∗a) at the notch satisfies the

hyperbola (9.10), in addition to the cyclic relation (9.11). We thus obtain

σ∗a = f(ǫ∗a) ⇒ Kσσa = f

(

Kǫσa

E

)

(9.12)

Moreover, introducing (ǫ∗a, σ∗a) from (9.9) into (9.10) gives the relation between Kσ and

Kǫ:

KσKǫ = K2f (9.13)

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σa

elastic

σ = f ( )ε

const. ( Neuber hyperbola )σ ε =

K f σa

ε aK f

Kσσa

K ε ε aε a

σa

ε a

a a

Figure 9.6: Use of Neuber hyperbola for calculation of Kσ and Kǫ.

Combining (9.12) and (9.13), we may solve for Kσ (and then for Kǫ) for given σa from

the relation

Kσσa = f

(K2

f

KσEσa

)

(9.14)

Finally, the design procedure is defined as follows: Use the relations for the far-field stress

by making the substitutions

σm → Ktσm, σa → Kσσa (9.15)

Design — Safety

So far, we have given explicit consideration to the influence of mid stress and stress

concentration, which both reduce the fatigue strength. However, in reality there are a

number of other effects that reduce the fatigue strength, but which are difficult to quantify:

• Non-smooth surface (with micro-notches)

• Large size of specimen (with greater probability of defects and impurities)

• Corrosive environment (including high temperature)

To account for these effects the calculated stress is amplified by a suitable load factor, or

safety factor, which may be taken differently depending on whether the safety is related

to σm, σa, or a combination of σm and σa. Usually, the following situations are considered:

Vol 0 March 7, 2006


• σm = const ⇒ Design stresses: (Ktσm, FaKσσa)

• σa = const ⇒ Design stresses: (FmKtσm, Kσσa)

• σa

σm= const ⇒ Design stresses: (FamKtσm, FamKσσa)

This is shown schematically in Figure 9.7. Note that Kσ = Kf in the case of HCF.

9.2.2 Variable amplitude loading — Palmgren-Miner rule

In practice, engineering components are invariably subjected to cyclic loading programs

with varying stress amplitude and mean stress. Quite often it can be assumed that

the loading program is composed of a sequence of blocks with constant amplitude. For

this situation, the Palmgren-Miner linear damage accumulation rule, first proposed by

Palmgren (1924), is widely used. The amount of damage d accumulated is written as

the sum

d =

n∑

i=1

∆Ni

NR,i(9.16)

where ∆Ni is the number of cycles in the i:th loading block with constant amplitude σa,i,

and NR,i is the fatigue life at σa,i, as shown in Figure 9.8. The total number of blocks in

the sequence is n. Fatigue failure then corresponds to d = 1.

The following assumptions are implicit in (9.16):

• The same amount of damage is caused by each cycle for a given amplitude (inde-

pendent of whether it occurs at the beginning or end of the fatigue life).

• The order of the blocks of different amplitude does not affect the damage develop-

ment and the total fatigue life.

Neither of these assumptions are particularly realistic. Later in this chapter, we examine

in further detail the assumptions above in conjunction with the derivation of fatigue-life

curves based on a rational damage mechanics approach. Despite its weak theoretical basis,

the Palmgren-Miner rule is widely used in a variety of contexts in fatigue life analysis.

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σa

( a )

σu

σar

σ m

( )K t σ m , K f σa

( ),K t σ m FaK f σa


( b )

σar

σuσ m

( )FmK t σ m , K f σa

σa

σar

σu

( c )


( )Fam K t σ m Fam K f σa,

σ m

σa

Figure 9.7: Design diagram for (a) Constant σm, (b) Constant σa, (c) Constant σa/σm.

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σ

∆ N1 ∆ N2

NR

σ 1a σ 2a

Figure 9.8: Loading consisting of sequence of blocks with constant stress amplitude (at

reversed cyclic loading).

9.2.3 Multiaxial fatigue criteria based on stress

So far we have discussed fatigue at uniaxial stress. However, in most situations in en-

gineering practice, the stress state is multiaxial, and it is not obvious how to generalize

the analysis for uniaxial stress to this more general situation. In order to simplify the

discussion, we shall first assume that all components of the stress tensor are periodic with

the same frequency (period T ). The only difference is the phase angle, i.e. we can write

σij(t) = (σm)ij + (σa)ij A(ωt + θij), i, j = 1, 2, 3 (9.17)

where (σm)ij is the mid stress and (σa)ij is the amplitude defined as

(σm)ij =1

2

(

max0≤t≤T

σij(t) + min0≤t≤T

σij(t)

)

(σa)ij =1

2

(

max0≤t≤T

σij(t) − min0≤t≤T

σij(t)

)

(9.18)

Moreover, A(ωt + θij) is a periodic function with a unit amplitude, |A(ωt + θij)| ≤ 1,

where ω is the angular frequency and θij is the phase angle for the stress component σij .

For an isotropic material, we may consider only the principal stresses, i.e.

σi(t) = (σm)i + (σa)iA(ωt + θi), i = 1, 2, 3 (9.19)

A quite-straightforward extension of the uniaxial fatigue analysis is the Sines criterion,

which is defined as follows: The equivalent stress (according to von Mises) of σa(t) is

Vol 0 March 7, 2006


defined as

σa,e(t) =

(

3

2

3∑

i=1

(σ′a)

2i A

2(ωt + θi)

)1/2

(9.20)

where a prime denotes deviator, i.e. σ′i = σi −

13(σ1 + σ2 + σ3).

The maximum equivalent stress over one period T is defined as

σa = max0≤t≤T

σa,e(t) = σa,e(tmax) (9.21)

which is illustrated in Figure 9.9.

σa e, ( )t

maxωt

ωt= 0t

σa

Figure 9.9: Illustration of maximum equivalent stress σa.

Remark: When all stress components vary cyclically in-phase, i.e. θi = 0 for i = 1, 2, 3,

we obtain from (9.20) and (9.21) that

σa,ph =

(

3

2

3∑

i=1

(σ′a)

2i

)1/2

(9.22)

where it was used that max |δ| = 1. In all other situations than in-phase changes, we have

σa ≤ σa,ph 2 (9.23)

The equivalent mid stress σm is defined in precisely the same way as σa, but simply by

replacing (σ′a)i with (σ′

m)i in (9.22). Hence, we obtain

σm = σm,e with σm,e =

(

3

2

3∑

i=1

(σ′m)2

i

)1/2

(9.24)

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9.3 Engineering approach to LCF based on strain-control 181

In the special case of uniaxial stress, the usual expressions

σa = σa, σm = σm with σa = (σa)1, σm = (σm)1 (9.25)

are retrieved.

The simplest way of extending the analysis for the uniaxial stress state to the multiaxial

stress state is to replace σa by σa and σm by σm and to use the Haigh diagram and the

Basquin equation

σa − σfl = σ′f(2NR)b (9.26)

9.3 Engineering approach to LCF based on strain-

control

9.3.1 Manson-Coffin relation

Theoretically, the stress-based approach discussed above is applicable to HCF as well as

to LCF. However, in the case of LCF (for high stress levels producing significant plastic

strain) the fatigue behavior is more accurately described in terms of strain-controlled

tests. Like for stress-controlled loading, the simplest constant amplitude tests are defined

by reversed strain (ǫm = 0) and pulsating strain (ǫm = ǫa).

Let us consider the case of fully reversed cyclic strain, i.e. ǫm = 0. Like for stress

controlled tests, ǫa is plotted against NR, as shown in Figure 9.10. These curves are often

summarized in the form of the Manson-Coffin relation

ǫa = ǫea + ǫp

a =σ′

f

E(2NR)b + ǫ′f(2NR)c (9.27)

where σ′f is the fatigue strength coefficient (introduced for stress controlled fatigue) and

ǫ′f is denoted the fatigue ductility coefficient. As a rule of thumb, ǫ′f can be taken as the

fracture ductility as observed in a monotonic tensile test. Moreover, b is (still) the Basquin

exponent, whereas c is an auxiliary exponent. Considering (9.1), we may conclude that

b and c are not independent, since they can be related to the cyclic hardening coefficient

ncyc as follows: By identifying the elastic and plastic parts of strain in (9.1) and (9.27),

we obtain directly the relation

b = c ncyc and ǫ′f =

(σ′

f

σ′cyc

) 1ncyc

(9.28)

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For example, for b = −0.12, which is a reasonable value for a ductile material, and

ncyc = 0.2, we obtain c = −0.6. These values of b and c are quite often adopted for repre-

senting the universal Manson-Coffin relation, since it has turned out that these exponent

values are remarkably insensitive to variations of the material composition (for the same

temperature). However, all material coefficients are strongly temperature-dependent.

RNln ( )21

ε alnversusε aln RNln ( )2

eε aln = bE

f′σ+ RNln ( )2

= ′ε f cpε aln ln + RNln ( )2

c

b

1

1

2

1

2

ln

Figure 9.10: Manson-Coffin relation for LCF.

From (9.28), and with values of b and c given above, it may be concluded that ǫpa is

the dominant term for short fatigue lives (small values of NR), whereas ǫea is the more

significant term for long fatigue lives (large values of NR).

Effect of mid-strain

A prescribed tensile mid strain ǫm reduces the fatigue life (in much the same fashion

as does a mid stress σm for stress-controlled fatigue tests.) This will lead to mid stress

relaxation, i.e. σm → 0, which is a gradual reduction of mid stress with cycling. Such mid

stress relaxation under strain-controlled fatigue loading is the counterpart of ratchetting

(or cyclic creep), that takes place under stress-controlled loading.

Effect of strain concentration

The effect of a strain concentration at LCF is treated in the same fashion as the case of

stress-controlled testing, that was discussed in Section 9.2. The only difference is that Kǫ

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9.3 Engineering approach to LCF based on strain-control 183

is now the quantity of interest (rather than Kσ). We thus obtain from (9.12)

σ∗a = f(ǫ∗a) ; KσEǫa = f(Kǫǫa) (9.29)

Upon eliminating Kσ via the use of (9.13), we obtain

K2f

Kǫ

Eǫa = f(Kǫǫa) (9.30)

from which Kǫ can be solved for given ǫa.

Remark: This result is the same as if Kσ is calculated first from (9.14), whereafter Kǫ

is given by (9.13). 2

The design procedure is defined, finally, by the substitutions

ǫm → Ktǫm, ǫa → Kǫǫa (9.31)

in the relations for the far-field strain.

9.3.2 Combined effects of creep and fatigue

The simplest loading program that involves the combined effects of LCF and creep, so-

called creep-fatigue, is defined as cyclic loading with a prescribed constant strain ampli-

tude that has a certain duration (hold time) th in each cycle. This is shown schematically

in Figure 9.11(a). A typical stabilized cyclic stress-strain relationship (which is valid prior

to excessive development of damage) is shown in Figure 9.11(b). The hold time th cor-

responds to the portion ǫpca of the total inelastic strain amplitude ǫp

a, whereas ǫpfa is the

time-independent LCF-portion of ǫpa , as also shown in Figure 9.11(b).

In order for creep effects to become significant, the temperature must be sufficiently high.

In the simplest case, the temperature is constant in time, whereby the failure is denoted

creep-fatigue (CLCF). If, on the other hand, the mechanical loading is held constant in

time while the temperature is varying cyclically with the hold time th, then we speak of

thermal-fatigue (TLCF). In the more general situation both the mechanical and thermal

loadings will vary in a cyclic fashion, which is denoted thermomechanical-fatigue. This

variation may be in-phase or out-of-phase, while the same frequency is retained. In the

most general case, the mechanical loading and the temperature will not vary with the

same frequency.

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σa

σ

2ε apc

creep fatigue

1

2 3

4

5 6

1

2 3

4

56

2 pfε a

pε

σ

t

t h

( a ) ( b )

Figure 9.11: Creep-fatigue, (a) Loading program, (b) Stabilized stress-strain cycle.

Let us consider the case of thermomechanical fatigue with in-phase variation of temper-

ature. It is then common to extend the Palmgren-Miner rule to combine the effects of

creep and fatigue as follows:NR

N fR

+tRtcR

= 1 (9.32)

Upon introducing the relations

tR = NRth, tcR = N cRth (9.33)

we may rephrase (9.32) as

NR

N fR

+NR

N cR

= 1, or NR = N fcR =

(1

N fR

+1

N cR

)−1

(9.34)

This relation is known as the Taira rule of linear creep-fatigue damage accumulation. The

similarity with the Palmgren-Miner rule is striking. In fact, it is possible to extend the

rule to the case of sequential loading with piecewise constant strain amplitudes to obtain

n∑

i=1

(

∆Ni

N fR,i

+∆Ni

N cR,i

)

= 1, orn∑

i=1

∆Ni

N fcR,i

= 1, N fcR,i =

(

1

N fR,i

+1

N cR,i

)−1

(9.35)

It must be realized that the theoretical basis for the assumption about linear damage

accumulation is very weak. Hence, it can not be expected that the agreement with real

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9.4 Life prediction strategies 185

behavior is good. Nor does the simplified rule necessarily give acceptable agreement with

a more rigorous analytical prediction. In fact, it is known that creep-fatigue interaction is

quite nonlinear in such a fashion that, unfortunately, (9.34) may overestimate the number

of cycles to failure. Nevertheless, since a more accurate analysis is quite involved, the Taira

rule has been used in practice, at least as a “rule of thumb”.

9.3.3 Multiaxial fatigue criteria based on strain

The most straight-forward generalization of the procedure at strain-control to the case

of multiaxial strain is to introduce the equivalent strain (according to von Mises) of the

strain amplitude as

ǫa,e(t) =

(

2

3

3∑

i=1

(ǫ′a)2i A(ωt + θi)

)1/2

(9.36)

where a prime denotes deviator, i.e. ǫ′i = ǫi −13(ǫ1 + ǫ2 + ǫ3).

The maximum equivalent strain over the period T is defined as

ǫa = max0≤t≤T

ǫa,e(t) (9.37)

which expression is next used in the Manson-Coffin relation to give

ǫa =σ′

f

E(2NR)b + ǫ′f(2NR)c (9.38)

Remark: Because of mid stress relaxation, it is often claimed that the corresponding

mid strain ǫm has a negligible effect on the fatigue life. 2

9.4 Life prediction strategies

9.4.1 Coupled - decoupled approach

All fatigue analyses aim at predicting the lifetime of a component subjected to cyclic (me-

chanical and/or thermal) loading. To this end, a number of different prediction strategies

are possible, corresponding to the acceptable level of complexity and realism of analysis.

Comprehensive reviews are found in Skelton (ed.) (1987) and Riedel (1987). In

most strategies, the concept of a loading (stress or strain) cycle plays an important role.

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In the classical ”engineering approach” to fatigue analysis, as discussed in Sections 9.3

and 9.4, the loading cycles are assumed to have an ideal shape in time1, e.g. sinusoidal

variation, which is repeated without any change until failure occurs. It is clear that such

an ideal variation may only remotely remind about the real loading. For example, a tur-

bine blade in a jet engine is subjected to a quite complex thermomechanical loading cycle

during a single flight.

The engineering approach represents the least sophisticated type of analysis, and it is

based directly on component tests. These tests are carried under stress or strain control,

cf. the historical remarks above.

The most complete type of (thermomechanical) fatigue analysis employs a constitutive

model that accounts for the gradual deterioration (damage) in each cycle, while at the

same time reflecting the redistribution of stresses due to the change of stiffness brought

about by the damage accumulation. Hence, a suitable model should include all the nec-

essary nonlinear and time-dependent effects. Typically, the LCF-analysis may be based

on damage coupled to plasticity (as discussed in Chapter 7), whereas CLCF-analysis may

conveniently be based on damage coupled to viscoplasticity (as discussed in Chapter 8).

In fact, LCF is only a special case of CLCF, and this fact should also be reflected by the

adopted model.2

As to the issue of devicing a rational strategy for lifetime prediction based on damage

mechanics, two different approaches are possible:

• Coupled approach: The constitutive model is used as an integral part of the

pertinent boundary value problem for each given combination of loading and envi-

ronmental condition. However, such an approach may be computationally overly

demanding in practice (at least in 3D) due to the need to integrate the constitutive

relations over a large number of ”cycles”. On the other hand, a definite advantage

of the coupled approach is that it is not necessary to introduce the very concept of

a ”cycle”. This concept does not have any relevance from a computational point of

view, but may still be convenient as a measure of the lifetime.

• Decoupled approach: The concept of ”decoupling” of the ”global” and ”local”

behavior, cf. Lemaitre & Chaboche (1990) opens up for a variety of rational

1For rate-independent response, the real time may be replaced with any time-like loading parameter.2For didactic purposes the analysis of LCF and CLCF are treated separately in this Chapter.

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9.4 Life prediction strategies 187

lifetime prediction strategies. This concept relies heavily on the presumed existence

of a ”stabilized cyclic response”, which is predicted without any provision for ma-

terial deterioration (in terms of damage accumulation or crack propagation). The

analysis of damage accumulation is then carried out in a post-processing step, based

on the assumption that the stabilized response is relevant up to failure.

9.4.2 Life prediction strategy based on the decoupled approach

A rational strategy for lifetime prediction should consist of the following three main steps:

Step 1: Parameterizing the loading

Upon parameterizing the applied mechanical and thermal loading, we may characterize

each loading as a point in the design parameter space P = (p1, p2, . . .). For a turbine

blade the loading may conveniently be defined by the angular frequency ω(t) and the

temperature θ(t) in the surrounding fluid. Typically, for in-phase thermomechanical cyclic

loading, the relevant parameters are the working frequency amplitude ωa, the temperature

amplitude θa, and the hold time th in each cycle (with duration tc), which is shown in

Figure 9.12. This gives P = (ωa, θa, th). Corresponding to each point in P, there is a

critical number of cycles to failure, NR (obtained in Step 3 below). Hence, the failure

state is defined by a surface in the space (P, N).

Step 2: Representative stabilized cyclic response

Compute the stress, strain and temperature fields (in space and time) during a few cycles

of the (thermomechanical) fatigue process until reasonably stabilized cyclic behavior has

been obtained. In this step, a (more or less) elaborate material law without damage is em-

ployed, and it is calibrated to fit the stabilized response. The underlying key assumption

is that the damage development will not significantly affect the stiffness variation with

time (until close to global failure), such that the computed stress and strain fields will

be representative throughout the major part of the lifetime. In particular, if the design

of the component under consideration is such that the stress field is (almost) statically

determinate, it is obvious that the stress field should be insensitive to the approximation

employed in the constitutive relations. The corresponding strains may, however, be quite

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tc

th

t

θa , ωa

Figure 9.12: Parameterization of in-phase loading on a turbine blade.

sensitive to the choice of the constitutive model.

Step 3: Damage calculation and lifetime prediction

This is the key step in the strategy. First, the critical regions (or points) for the damage

accumulation analysis are selected. A critical location may, for example, be characterized

by maximum effective stress or maximum damage stress (elastic energy density release

rate), as defined below. For given stress and temperature histories3 from Step 2, compute

the damage accumulation. The aim is to obtain the lifetime NR satisfying

d(NRtc) = dR ≤ 1 (9.39)

where dR is the (chosen) critical damage value, and where tc is the duration of a single

cycle. The value dR may be selected to represent a maximal allowable surface density

of microcracks. Alternatively, dR is computed as the result of a localization analysis to

determine the onset of a macroscopic crack.

As to the computation of damage accumulation, the following two different approaches

are considered:

3The reason for choosing the stress field (rather than the strain field) as input was discussed above.

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9.5 Damage mechanics approach to LCF 189

• No degradation (due to damage) or relaxation (due to viscous behavior) of the stress

level is assumed to take place during a single stabilized cycle. In the CLCF-situation,

the inelastic strain in the stabilized cycle, for given fixed stress amplitude, is par-

tioned into elastic-plastic and creep strains. This is denoted ”Simplified Analysis”

in what follows.

• The appropriate constitutive relation is used to compute the damage accumulation,

as well as possible creep strain, during a stabilized cycle. The stress history, as

obtained in Step 2, and the pertinent temperature history are used as data to this

calculation.

9.5 Damage mechanics approach to LCF

9.5.1 Simplified analysis of LCF — Derivation of the Manson-

Coffin and Basquin relations

Assumptions

A simplified analysis of LCF in the case of fully reversed strain and stress loadings, which

leads to the Mansion-Coffin and Basquin relations, is outlined below. The necessary

simplifying assumptions are:

• Stabilization under cyclic hardening occurs rapidly so that only stabilized cycles

need to be considered. These loops are assumed to correspond to perfectly plastic

response (in terms of the effective stress).

• The stabilized cyclic hardening relation is given by (9.1) for prescribed strain as well

as prescribed stress amplitude. Most importantly, ǫpa is connected to the effective

stress amplitude σa via the Ramberg - Osgood relation in (9.1).

• The change of damage is associated with increasing number of cycles; i.e. the

damage variable is taken as a constant during each cycle.

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Prescribed strain amplitude and reversed loading

We first consider cyclic loading with constant strain amplitude ǫa and fully reversed cyclic

strain.

Accounting for damage, we can rewrite the cyclic hardening relation (9.1) as

ǫa =σa

E+ ǫp

a , ǫpa =

(σa

σ′cyc

) 1ncyc

(9.40)

Now, we may solve for σa in terms of ǫpa and insert this expression into (9.40), i.e.

σa = σ′cyc (ǫp

a)ncyc

; ǫa =σ′

cyc

E(ǫp

a)ncyc + ǫp

a (9.41)

Since ǫa remains a constant amount of strain during the cyclic process, it follows from

(9.40) that σa is constant. Hence, ǫpa is also constant.

The change of accumulated strain ǫpe after each cycle is 4ǫp

a (corresponding to 4 quarter-

cycles), i.e.dǫp

e

dN= 4ǫp

a (9.42)

Hence, after N cycles, the accumulated strain is

ǫpe (N) = 4ǫp

aN (9.43)

The threshold value N for the initiation of damage is obtained from (9.43) simply as

N =ǫpe

4ǫpa

(9.44)

Let us now consider the damage law, given as

d = ǫpe

σ2

2ES(1 − d)m−1(9.45)

Since we have assumed perfect plasticity and no damage development during an individual

cycle, we may formally integrate (9.45) during a single cycle to obtain the rate of change

of damage per cycle as

dd

dN=

σ2a

2ES(1 − d)m−14ǫp

a = kF1

(1 − d)m−1, m ≥ 1 (9.46)

where the coefficient kF is given as

kF(ǫpa) =

2(σ′cyc)

2(ǫpa)

2ncyc+1

ES(9.47)

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In order to obtain this expression we used (9.41)1.

With the initial condition that N = N when d = 0, we may integrate (9.46) to obtain the

relation

kF(ǫpa) =

fm(d)

N − N(9.48)

where we introduced the function

fm(d) =1

m[1 − (1 − d)m] (9.49)

Alternatively, we may use (9.47) and combine with (9.48)to obtain

ǫpa = Cǫ(d)(N − N)

− 12ncyc+1 (9.50)

where

Cǫ(d) =

(ESfm(d)

2(σ′cyc)

2

) 12ncyc+1

(9.51)

It is noted that fm(0) = 0. The number of cycles to failure is NR corresponding to the

situation when d = dR. From (9.50) we obtain

ǫpa = CǫR(NR − N)

− 12ncyc+1 with CǫR = Cǫ(dR) (9.52)

By combining (9.41) with (9.52), we obtain the Manson-Coffin relation

ǫa =σ′

cyc

E(CǫR)ncyc(NR − N)

−ncyc

2ncyc+1 + CǫR(NR − N)− 1

2ncyc+1 (9.53)

Now, in order to compare this predicted relation with the ad-hoc relation (9.27), we restrict

to the case that N = 0. Firstly, we note that

c = −1

2ncyc + 1and b = −

ncyc

2ncyc + 1= c ncyc (9.54)

which is precisely the relation (9.28). Secondly, we may express σ′f in terms of ǫ′f as follows:

ǫ′f = 2−cCǫR ⇒ σ′f = 2−bσ′

cyc(CǫR)ncyc = σ′cyc(ǫ

′f)

ncyc (9.55)

where (9.54) was used. Hence, we conclude that the two expressions of the Manson-Coffin

relation in (9.27) and (9.53) are completely equivalent.

Remark: It would be possible to plot fatigue curves for values of d < dR by replacing

CǫR with Cǫ(d). Since Cǫ(d) ≤ CǫR, we note that such curves are located below the fatigue

curve defined by CǫR. 2

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We may also solve for d as a function of N − N from the expressions in (9.50) and (9.52).

The pertinent relationship becomes

d = 1 −

(

1 −N − N

NR − Nmfm(dR)

) 1m

, N ≥ N (9.56)

which is shown in Figure 9.13 in the special case that dR = 1. It is noted that the ratio

N/NR decreases with ǫa for given value of ǫpe . In fact, from (9.44) and (9.52) it is obtained

thatN

NR

=1

1 + c(ǫpa)2ncyc+2

with c =4

ǫpe (CǫR)2ncyc+1

(9.57)

The interested reader should show this as homework!

Figure 9.13: Damage development for LCF for constant amplitude loading.

Prescribed stress amplitude and reversed loading

It is possible (although less usual in practice) to carry out LCF-tests with constant stress

amplitude σa. The relationship in (9.40) for stabilized cyclic loops is still employed. The

threshold value N is now obtained, by using (9.40) into (9.44), as

N =(σ′

cyc)1

ncyc ǫpe

4(σa)1

ncyc

(9.58)

where it was also used that d = 0 when N ≤ N .

Let us, again, consider the damage law in (9.46), which can be expressed in terms of the

stress amplitude asdd

dN= kF

1

(1 − d)m+1+ 1

ncyc

, m ≥ 1 (9.59)

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where the coefficient kF is now defined as

kF(σa) =2(σa)

2+ 1ncyc

ES(σ′cyc)

1ncyc

(9.60)

Upon integration of (9.59), we obtain

kF(σa) =gm(d)

N − N(9.61)

where we introduced the function

gm(d) = fm+2+ 1ncyc

(d) (9.62)

Alternatively, we may combine (9.60) and (9.61) to obtain

σa(d) = Cσ(d)(N − N)−

ncyc2ncyc+1 (9.63)

where

Cσ(d) =

(

ES(σ′cyc)

1ncyc gm(d)

2

) ncyc2ncyc+1

(9.64)

The number of cycles to failure is obtained from (9.63) according to the Basquin relation

σa = CσR(NR − N)−

ncyc2ncyc+1 with CσR = Cσ(dR) (9.65)

Hence, the Basquin relation may be derived in a fashion that is similar to the Manson-

Coffin relation.

The relation between d and N is now obtained from (9.63) and (9.65) as

d = 1 −

(

1 −N − N

NR − N(m + 2 +

1

ncyc)gm(dR)

) ncyc1+(m+2)ncyc

, N ≥ N (9.66)

which is quite analogous to the relation (9.56) for prescribed strain amplitude.

Damage accumulation for variable amplitude — Validity of Palmgren-Miner

rule

Let us consider how damage accumulates if the cyclic loading history consists of n blocks

with constant amplitude, as was shown schematically in Figure 9.8. Since d is a nonlinear

Vol 0 March 7, 2006


function in N in general, it is clear that it is not possible to establish any simple accumu-

lation rule for the composite loading. Hence, we restrict our attention to the simple case

of linear accumulation of damage, which is obtained when the exponent is unity in (9.56)

or (9.66). For example, in the case of prescribed strain amplitude, defined by (9.56), we

must require that m = 1. For simplicity, we shall also assume that dR = 1.

Let each constant amplitude loading block of duration ∆Ni cycles be associated with the

fatigue life NR,i and with the threshold value Ni (if it was acting alone). The contribution

to the damage from each such sequence is then obtained from (9.56) or (9.66) as

∆d1 =∆N1 − N1

NR,1 − N1

∆di =∆Ni

NR,i − Ni

, 2 ≤ i ≤ n (9.67)

It was tacitly assumed that the first applied sequence always will have a duration that

exceeds the corresponding threshold value N1. At failure we must have

1 =

n∑

i=1

∆di =

n∑

i=1

∆Ni

NR,i − Ni

−N1

NR,1 − N1

(9.68)

and we obtain the accumulation rulen∑

i=1

NR,i

NR,i − Ni

∆Ni

NR,i

=NR,1

NR,1 − N1

(9.69)

A particularly simple situation is obtained if we set

Ni

NR,i

= constant, i = 1, 2, . . . , n (9.70)

since (9.69) is then reduced ton∑

i=1

∆Ni

NR,i

= 1 (9.71)

We have thus obtained the Palmgren-Miner rule of linear damage accumulation. However,

due to the severe restrictions imposed in order to motivate its validity, the Palmgren-Miner

rule should always be used with suspicion.

9.5.2 Rational approach to LCF - Damage coupled to plastic

deformation

After saturation hardening of a smooth-surfaced specimen, the behavior can be considered

as cyclically “perfectly plastic” up to a certain threshold level of accumulated plastic

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strain, where damage starts to develop due to nucleation of microcracks. For this first

phase of the deformation process, it is possible to employ the nonlinear hardening concept

without damage, as described in detail in Subsection 4.3. In the absence of damage

development, it is possible to show that the hardening (drag) stress κ can be expressed

for a completely arbitrary loading path, as

κ = κ∞

(

1 − e−rHǫ

pe

K∞

)

(9.72)

It is then possible to choose ǫpe = ǫp

e to correspond to a certain degree of hardening

saturation, defined as κ = κ = xκ∞, where x is a given constant. Upon inserting this

condition into (9.72), we obtain

ǫpe = −

κ∞

rHln(1 − x) (9.73)

For example, x = 0.9 ⇒ ǫpe = 2.30κ∞

rH.

In order to model the second phase of LCF, it is thus necessary to modify the damage

law given in (7.41) in such a fashion that ǫpe will serve as the threshold value, below which

no damage develops. This value corresponds to the number N of load cycles. (It should

be noted that N is not a material constant; for instance, it depends on the prescribed

stress/strain amplitude.) The damage law is thus reformulated as:

d = λδ

S(1 − d)mH(ǫp

e − ǫpe ) = ǫp

e

δ

S(1 − d)m−1H(ǫp

e − ǫpe ) with δ =

σ2

2E(9.74)

where H(x) denotes the Heaviside function defined by

H(x) = 1 if x ≥ 0, H(x) = 0 if x < 0 (9.75)

This behavior is illustrated in Figure 9.14 for the case of uniaxial monotonic loading.

The corresponding scenario at cyclic loading with prescribed constant strain amplitude

leading to LCF is shown in Figure 9.15.

It may be remarked that the shape of the strain-cycles (in time) is completely irrelevant,

since it was assumed that the material is rate-independent.

The (expected) cyclic behavior is shown in Figure 9.16, which was obtained for the fol-

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Figure 9.14: Delayed development of damage at uniaxial loading defined by the threshold

value ǫp.

lowing data:

σy

E= 0.001,

H

E= 0.02, r = 0.5, κ∞ = α∞ = 0.25σy, (9.76)

S

E= 0.5 × 10−6, m = 2,

ǫ ∈ [−2ǫy, 2ǫy] , where ǫy =σy

E,

ǫpe =

20σy

E,

Remark: In order to take into account the effect of deactivation of damage under com-

pressive stress (often referred to as the Microcrack-Closure-Reopening effect), the results

in Figure 9.16 were computed with the (refined) effective stress relation

σ =3 − (1 + 2H(σ))d

3(1 − d)(1 −H(σ)d)σ =

1 −H(−σ)d3

1 − dσ (9.77)

Hence, we obtain

σ =σ

1 − dwhen σ > 0

=(1 − d

3)σ

1 − dwhen σ ≤ 0 (9.78)

The precise rationale behind the choice in (9.77) requires a 3D analysis. 2

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9.6 Damage mechanics approach to CLCF 197

σ

ε a

ε a

N1 2

ε

RNN

N

Figure 9.15: LCF for constant strain amplitude loading with threshold for damage devel-

opment.

Repeating the analysis for different strain amplitudes, we may construct a typical Manson-

Coffin relation, as shown in Figure 9.17. It is noted that no damage can occur if the

strain amplitude is less than the yield strain. Hence, the yield strain may be taken as the

theoretical “fatigue-limit” in the context of LCF.

9.6 Damage mechanics approach to CLCF

9.6.1 Simplified analysis of CLCF

Assumptions

A simplified analysis of CLCF in the case of fully reversed stress loading under constant

temperature is outlined below. The necessary simplifying assumptions are (cf. those in

Subsection 9.5.1):

• Stabilization under cyclic hardening occurs rapidly, and the stabilized loops core-

spond to perfectly plastic response (in terms of the effective stress).

• The inelastic strain during a cycle is decomposed into one portion from fatigue, ǫpfa ,

and another from creep, ǫpca . The rate-independent part (ǫpf

a ) is connected to the

effective stress amplitude (σa) via the Ramberg-Osgood relation in (9.1)1.4

4Such a decomposition is not consistent with the ”unified” approach, in which ǫpfa and ǫpc

a are insepa-

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100

101

102

0

0.2

0.4

0.6

0.8

1

1.2

N

stre

ss a

mpl

itude

/ yi

eld

stre

ss

100

101

102

0

0.2

0.4

0.6

0.8

1

N

dam

age

Figure 9.16: Predicted LCF-result for prescribed strain amplitude, (a) σa versus N and

(b) d versus N .

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101

102

103

100

101

N

stra

in a

mpl

itude

/ yi

eld

stra

in

Figure 9.17: Predicted Manson-Coffin relation.

• During the hold time, th, Norton creep is assumed. Assuming constant temperature

and ignoring any Microcrack-Closure-Reopening effects, the creep strain ǫpca is equal

in tension and compression.

• The change of damage is associated with increasing number of cycles, i.e. the

damage variable is taken as constant during each cycle.

Prescribed stress amplitude and reversed loading

Because of the assumptions above, we may integrate the Norton-type creep law during a

quarter of the hold time to obtain

ǫpa = ǫpf

a + ǫpca =

(σa

σ′cyc

) 1ncyc

+th4t∗

(σa

E

)nc

(9.79)

where t∗ is the relaxation time, and nc is the classical Norton-creep coefficient.

rable from a conceptual viewpoint and their sum ǫpa is rather treated as an internal variable.

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The damage law is still given in (9.45), i.e.

d = ǫpe

σ2

2ES(1 − d)m−1(9.80)

Upon integrating (9.80) during one single cycle, while observing (9.79), we obtain

dd

dN= 4ǫp

a

σ2a

2ES(1 − d)m−1= kF

1

(1 − d)m+1+ 1

ncyc

+ kC1

(1 − d)m+1+nc(9.81)

where the coefficents kF and kC are given as

kF(σa) =2(σa)

2+ 1ncyc

ES(σ′cyc)

1ncyc

, kC(σa) =tht∗

(σa)2+nc

(E)nc+1S(9.82)

The critical number to failure, NR, is obtained by integrating (9.82), i.e. NR is obtained

from the relation

∫ dcr

0

dd

kF(σa)(1 − d)−m−1− 1

ncyc + kC(σa)(1 − d)−m−1−nc

= NR − N (9.83)

which must be solved by numerical integration in the general situation (for given ampli-

tude σa) due to the coupling between the fatigue (kF) and creep (kC) effects. However,

analytical solutions are easily obtained when decoupling is assumed. For given NR, cor-

responding to given dR, we may compute kF = kF,R when kC = 0 and kC = kC,R when

kF = 0. Evaluating (9.83) in these two special cases, we obtain

kF,R =fm+2+ 1

ncyc(dR)

NR − N, kC,R =

fm+2+nc(dR)

NR − N(9.84)

These values may be inserted into (9.83) to give

f(kF, kC)def=

∫ dR

0

dd

kFfm+2+ 1ncyc

(dR)(1 − d)−m−1− 1

ncyc + kCfm+2+nc(dR)(1 − d)−m−1−nc

= 1

(9.85)

where we introduced the non-dimensional variables

kF =kF

kF,R, kC =

kC

kC,R(9.86)

Remark: By definition, kF = 1, kC = 0 and kF = 0, kC = 1 are solutions of (9.85). 2

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Independent on the value NR, there is a unique relation between kF and kC, satisfying

(9.85), that can be depicted in the interaction diagram in Figure 9.18. In the same

diagram, we show the approximation

kF + kC = 1 (9.87)

which is the result of assumed independent damage accumulation form fatigue and creep,

respectively. This is Taira’s rule of linear fatigue-creep interaction (as discussed in Sub-

section 9.3.2).

Remark: For the special case that nc = 1/ncyc, the coupling between fatigue and creep

disappears and Taira’s rule is recovered. 2

It is also possible to construct Wohler (or S -N) curves, for given values of th (which

serves as a parameter). For each choice of σa, we first compute kF and kC from (9.81),

and inserting the result in (9.82) we may compute NR. Typically, these curves look like

in Figure 9.19.

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

kF

kC

f (kF,k

C)=1

kF+k

C=1 (Tairas rule)’

Figure 9.18: CLCF interaction diagram for nonlinear and linear interaction of fatigue and

creep. (dR = 1, m = 2, ncyc = 0.2, nc = 8).

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0 2 4 6 8 10

x 104

250

300

350

400

σa

Ncrit

th

= 0

th

> 0

Figure 9.19: Wohler curves for given holdtime th. (dR = 1, m = 2, ncyc = 0.2, nc = 8, E =

200e3, σcyc = 3000, t∗ = 1e − 14).

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9.7 Fracture mechanics approach to fatigue 203

9.6.2 Damage coupled to viscoplastic deformation

9.7 Fracture mechanics approach to fatigue

9.7.1 Preliminaries

As discussed in Subsection 9.1.1, the fracture phase follows the damage phase of the

fatigue life upon the occurrence of a macroscopic crack. It is then important to predict

the number of cycles from the initiation of the crack via the process of stable crack

propagation up to the final structural failure. Assuming that the crack length is a(N),

such failure is defined by unstable crack propagation, i.e.

da

dN→ ∞ when N → NR (9.88)

Remark: It is usual in the fracture mechanics approach to fatigue to assume that the

crack has a given initial length irrespective of the preceding loading. In this way, no link

exists to the preceding damage phase of the fatigue life. 2

The typical crack growth behavior for constant far-field stress amplitude loading, that

can be observed experimentally, is shown in Figure 9.20.

N

( 2 )σa σa( 1 ) ( 2 )σa<

a

a i 1

dadN

Figure 9.20: Typical crack growth behavior in constant far-field stress amplitude loading.

Remark: The successive advancement of the crack front in each cycle leads to the typical

striations in the fractured surface (which can be observed experimentally after complete

failure). 2

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The rate of crack propagation is influenced by a variety of factors (apart from the far

field stress amplitude σa that is applied). These are the mid stress σm, the cyclic load

frequency, the temperature and other environmental conditions.

9.7.2 Paris’ law for fatigue crack growth

The following empirical relation for the fatigue crack growth rate was suggested by Paris

& al. (1961):da

dN= C ′(2Ka)

mf (9.89)

where C ′ and mf are scaling parameters, whereas Ka is the stress intensity factor ampli-

tude. This amplitude can be calculated from σa according to equation (10.10) with the

appropriate geometric configuration factor f . If nothing else is stated, tensile fatigue is

considered, i.e. Ka refers to mode I. To be more precise, we should write KIa. Clearly,

KIIa and KIIIa may be introduced in a similar fashion to characterize fatigue crack growth

in mode II and mode III, respectively.

Remark: The relation (9.89) is somewhat awkward, since the actual dimension of C ′

depends on the value of the exponent mf . Typically, mf ≈ 2 − 4 for ductile alloys. 2

It is noted that the mid stress σm has a significant influence on the crack growth, since

the ”driving force” for crack growth is essentially the tensile stress range. In the ideal

situation of elastic material response, the crack remains fully open along its entire length

if σmin ≥ 0, i.e. when σm ≥ σa. For σmin ≤ 0, the crack faces are in contact and transmit

compressive stresses. The pertinent modification of the Paris’ law to accommodate such

crack closure effects is discussed below.

Effect of mid-stress — Crack closure

The effect of σm stems from the fact that the crack has a tendency to close at unloading

already for a tensile stress, which effect is due to plastic yielding. It is obvious that if σm

is large, then the tendency of crack-closure is smaller. In order to quantify this effect, we

use the stress ratio R = σmin/σmax = Kmin/Kmax.

The far-field stress σ for which the crack starts to close upon stress reversal in a cycle

corresponds to the value Kopen. Clearly, in the ideal situation that the crack closes at

σ = 0, we would have Kopen = 0. However, experimental investigations have shown that

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the crack starts to close already for σ > 0. Hence, stress reversal below Kopen > 0 will

not contribute to crack growth. A simple empirical expression is

Kopen = ϕ(R)Kmax with ϕ(R) =

{

0.25 + 0.5R + 0.25R2 , −1 ≤ R ≤ 1

0 , R ≤ −1(9.90)

which is shown in Figure 9.21.

1−1

= ( )RϕKmax

Kopen

Kmax

minK= R

R

Figure 9.21: Crack closure function for the interval −∞ ≤ R ≤ 1.

Remark: Since Kmin = RKmax, we conclude that Kopen ≥ Kmin with ϕ(R) given in

(9.90). 2

The strategy is now to replace Ka in Paris’ law (9.89), which is valid for a specific value

of R, with the effective stress intensity factor amplitude Keffa as follows:

Keffa =

1

2(Kmax − Kopen) =

1

2(1 − ϕ(R))Kmax =

1 − ϕ(R)

1 − RKa (9.91)

Moreover, C ′ is replaced by the universal coefficient C. Hence, instead of (9.89), we obtain

the expression

da

dN= C(2Keff

a )mf = CR(2Ka)mf with CR = C

(1 − ϕ(R)

1 − R

)mf

(9.92)

Remark: It is clear that C ′ introduced in (9.89) is identical to CR in (9.92). 2

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Validity of Paris’ law — Lower and upper limits

A plot of da/dN against Ka in log-scale reveals that this relationship is roughly linear

only for intermediate values of Ka. In fact, there is a lower limit Kal below which virtually

no crack propagation takes place. On the other hand, there is an upper limit Kau, above

which the crack growth rate becomes very large. These limit values may be defined by

the corresponding values

alKIc ≤ Kmax ≤ KIc (9.93)

where al is a scalar which for metals are in the order of 0.01. In terms of Ka = (1 −

R)Kmax/2, we thus obtain

Kal = al(1 − R)

2KIc, Kau =

1 − R

2KIc (9.94)

This behavior is shown in Figure 9.22.

1

( )dadN

ln

m

K aln

<R( 1 ) ( 2 )

R( 2 )

R

alK( 1 )

auK( 1 )

valid range forParis’ law

Figure 9.22: Range of validity for Paris’ law.

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Fatigue life calculations

The fatigue life Nf = NR−(N +Nd) is calculated by integrating Paris’ law (9.92) from the

initial crack length ai up to the failure crack length af . The geometric configuration factor

f is thereby treated as a constant in order to simplify the integration. (For example, we

may choose the value of f for the initial crack length ai in order to avoid an iterative

procedure.) We thus obtain

Nf =

1CRf2(2σa)2π

ln af

aiif mf = 2

2(mf−2)CRfmf (2σa)mf πmf/2

[

a−(mf

2−1)

i − a−(mf

2−1)

f

]

if mf ≥ 3

(9.95)

9.7.3 Variable amplitude loading

Like in the “engineering approach” to fatigue analysis, it is necessary to cope with situ-

ations when the loading is irregular and has variable amplitude. Such loading programs

may, in the simplest case, be composed of a sequence of “blocks” of constant amplitude

stress (or strain), as discussed in Subsection 9.2.2. In the more general situation, quite

arbitrary spectrum loads (with random character) are encountered. Vibrations in vehicles

due to uneven road surface is one typical example.

For the practical analysis, a few basic approaches may be identified, and they all involve

the use of Paris’ law.

Equivalent cycle methods — The Palmgren-Miner rule

When the loading program consists of a sequence of different blocks of constant amplitude,

it is common to use the Palmgren-Miner rule in much the same fashion as described in

Subsection 9.2.2. Paris’ law is used to calculate the appropriate life Nf,k for the k:th stress

amplitude. Each such calculation is done completely independently under the assumption

that the crack starts from its initial length ai. Clearly, this means that the influence of

the order, in which the loading blocks occur, is ignored.

In the more complex situation of random amplitude loading the main difficulty is to de-

termine what a “cycle” is. To this end, it is possible to use a variety of “cycle counting”

techniques, among which the most widely used is the socalled Rain-Flow-Count method;

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see e.g. Suresh (1991), p502. More recently, a method based on statistical repre-

sentations, the Level-Crossing method, was proposed by Holm and deMare (1992).

(Neither of these methods will be further discussed here.)

Cycle-by-cycle method

A quite basic method for an arbitrary loading program is to express the variation of

amplitude as a (continuous) function of N , i.e. we first determine Keffa (N). Paris’ law is

then used in straight-forward fashion as

da

dN= C(2Keff

a (N))mf (9.96)

In practice, this relation has to be integrated numerically.

Equivalent amplitude method

Sometimes the random loading program consists of a sequence of (nearly) identical blocks,

each of which consists of n cycles with different amplitudes, which is shown in Figure 9.23.

The strategy is then to define an equivalent stress intensity factor Keffa , that represents

the block loading, such that the crack advance caused by the n cycles in each block, with

amplitudes Keffa,k, k = 1, 2, . . . , n, is the same as the crack advance of n cycles of amplitudes

Keffa . Thereby, it is assumed that the effect of each cycle is independent. This can be

expressed as follows:

The crack advance per cycle of the original cycles is(

da

dN

)

k

= C(2Keffa,k)

mf k = 1, 2, . . . , n (9.97)

We thus obtain the relation

n∑

k=1

(da

dN

)

k

=n∑

k=1

C(2Keffa,k)

mf = nC(2Keffa )mf ⇒ 2Keff

a =

(

1

n

n∑

k=1

(2Keffa,k)

mf

) 1mf

(9.98)

Paris’ law is now used in straightforward fashion if only Keffa is replaced by Keff

a , i.e.

da

dN= C(2Keff

a )mf = C1

n

n∑

k=1

(2Keffa,k)

mf (9.99)

It is clear that also this approach is based on the assumption that it does not make any

difference in which order the different amplitudes occur.

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N

n cycles

σa

{Figure 9.23: Sequence of identical loading blocks used as basis for the Equivalent Ampli-

tude method.

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