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CONSTITUTIVE MODELING OF ENGINEERING

MATERIALS - THEORY AND COMPUTATION

Volume I General Concepts and Inelasticity

by

Kenneth Runesson, Paul Steinmann, Magnus Ekh and Andreas Menzel

Preface

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

mathematical models that can be used for describing and simulating the material response

of metals as well as composites, ceramics, polymers and geological materials (such as soil

and rock) under a variety of loading and environmental conditions. Consequently, a vast

amount of literature is available on the subject of “nonlinear constitutive modeling”, with

strong emphasis on plasticity and damage. Such modeling efforts are parallelled by the

development of numerical algorithms for use in the Finite Element environment. For

example, implicit (rather than explicit) integration techniques for plasticity problems are

now predominant in commercial FE-codes.

In the present book (comprising three volumes), we set out to give a coherent treatise of

the assumptions and concepts underlying the development of commonly used constitutive

models that involve ”dissipative mechanisms”. The archetypes of such ”mechanisms” are

those of inelasticity, viscosity and damage, which may combined in a quite general fashion

to realistically mimic complex macroscopic nonlinear and time-dependent response of a

large variety of engineeering materials. The pertinent constitutive relations are based

heavily on thermodynamics, in particular on the second law expressed as the constraint

of non-negative dissipation.

Volume I presents the general concepts of cconstitutive modeling and computational tech-

niques within a setting of geometrically linear theory. Rate-independent as well as rate-

dependent inelastic response are considered in a quite unified fashion. We consider only

phenomenological (macroscopic) models, although frequent refernces are made to the fact

that it is the microstructure of any given material that determines its macroscopic re-

sponse.

Volume II presents concepts and models for describing material failure at various scales,

including localized failure in narrow bands. Issues of damage mechanics, crack mechanics

Vol I March 21, 2006

iv

and fatigue are covered. Higher order continuum models, that involve a material length

scale, for modeling size effects are also covered. Although most of the emphasis is on

macroscopic models, we also discuss microstructural modeling, homogenization technique

and multiscale modeling strategies.

Volume III extends selected concepts and models of the two first volumes to geometrically

nonlinear theory.

We may thus summarize what the book is essentially about:

• Conveying concepts underlying the most important classes of macroscopic constitu-

tive models used in engineering practice.

• Presenting ideas underlying numerical procedures for integrating the evolution equa-

tions that are part of the constitutive framework.

To achieve greater clarity about our intentions, we also indicate what the book is not

about:

• Listing elaborate and ”fancy” models used in engineering practice, which are ob-

tained by a more less obvious combination of features of the considered archetype

models.

• Calibrating models to realistic data as obtained from experiments and working out

numerical solutions to real-world problems.

One must always bear in mind that a constitutive model (like any other mathematical

model in science and technology) may be useful, but it is never correct!

The present Volume I is outlined as follows:

Chapter 1 contains ”the tensor calculus toolbox” in as much as it summarizes the used

notation and elementary vector and tensor algebra and calculus. Throughout the book we

adopt symbolic (coordinate free) notation; however, index notation is exploited at times

for clarity. In this introductory chapter we also give useful formulas and results that can

not easily be found in standard text-books on continuum mechanics. A typical example

is the Simo-Serrin formualae for closed-form spectral representations.

In Chapter 2 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.


v

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

metals and alloys under various loading conditions is reviewed.

In Chapter 3 we present the basic relations of continuum thermodynamics for solid ma-

terial behavior. Constitutive relations are established for the most general situation of

non-isothermal behavior. Particular emphasis is placed on the dissipation inequality as

derived from the 2nd law of thermodynamics. That this inequality is satisfied will be

used as a criterion on thermodynamic admissibility, that will be referred to repeatedly

throughout this book. The different thermodynamic potentials are exploited as a conse-

quence of the fact that one has a freedom in choosing the independent state variables (as

arguments of the potentials). Finally, the archetypes of dissipative materials are discussed

in a generic context.

In Chapter 4 we present the continuous variational format (in space) of the relevant bal-

ance laws for the fully coupled thermomechanical problem, whereby the primary unknown

fields are the displacement, velocity, and temperature fields. Special cases are: Isothermal

format, isometric format (rigid heat conductor) and adiabatic format. The corresponding

discrete formats in time and space are established based on the fully implicit (Backward

Euler) method in the time domain and a finite element discretization in space. Finally,

it is shown how to solve the resulting nonlinear incremental relations using Newton iter-

ations. The relevant matrices involved in Newton iterations are obtained upon consistent

linearization of the incremental relations for the chosen time integration method.

In Chapter 5, we outline the fundamental ideas that define the ”canonical constitutive

framework” for dissipative material response. An important subclass is the Standard Dis-

sipative Material. Both rate-independent and rate-dependent response are considered (un-

der the assumption of isothermal conditions). Extension is then made to non-associative

structure (whereby the normality property is lost). Finally, issues of controllability, sta-

bility, and uniqueness for the rate-independent response are discussed.

In Chapter 6 we present a generic algorithm for the integration of the constitutive relations

under complete strain control (strain-driven format), which results in a ”local” incremen-

tal problem. This integration algorithm is based on the Backward Euler (BE) method.

The (iterative) strategy to handle prescribed stress components is outlined, whereby the

”core-algorithm” based on the strain-driven format is employed. The constitutive driver


vi

CONSTLAB c©, written in MATLAB, is based on this strategy. A generic format of the

Algorithmic Tangent Stiffness (ATS) tensor, arizing from the linearization of the incre-

mental stress-strain relation, is given. In particular, the ATS-tensor is used in Newton

iterations when the stresses are prescribed. We also discuss a startegy to handle mixed

stress and starin control, that adopts the strain-controlled format as the ”core-algorithm”.

Finally, we discuss issues related to model calibration.

In Chapter 7 we consider elastic response, which represents the conceptually simplest class

of material behavior. No dissipative mechanism is involved, i.e. the free energy does not

depend on any internal variables. Starting with the prototype model of linear elasticity,

we then extend the discussion to the general nonlinear (hyperelastic) format. Certain

widespread classes of nonlinear material response, including the total deformation format

of plasticity, can be obtained as special cases of the general theory. We then turn to the

general anisotropic response, which is represented using structure tensors of 2nd order.

The special cases of orthotropy and transverse isotropy are evaluated both in the symbolic

format and the Voigt matrix format.

In Chapter 8 we discuss viscoelastic material response, which is characterized by the

presence of rate-dependent dissipative mechanisms for any level of stress. A generic format

of the rate equations is presented for a rather large class of nonlinear viscoelasticity

models based on a single ”dissipative mechanism”. Specializations are introduced in

various respects; in particular to achieve the linear prototype model (that extends the

rheological model of Maxwell type to multiaxial stress and strain conditions). It is shown

how thermodynamic admissibility is satisfied for the most general anisotropic response.

Extension of the theory is also made to situations where multiple dissipation mechanisms

are included (like for the Linear Standard Viscoelastity model). The simple Maxwell

model is chosen as the prototype model for numerical investigation.

In Chapter 9 we discuss viscoelastic material response with thermomechanical coupling.

A generic format of the rate equations is presented for the class of nonlinear viscoelasticity

models based on a single ”dissipative mechanism”, that were discussed in the previous

Chapter. The simple Maxwell/Fourier model is chosen as the prototype model for numer-

ical investigation.

In Chapter 10 we discuss elastic-plastic material response, which is characterized by the

presence of rate-independent dissipation mechanisms when the stress exceeds a certain

threshold value (yield stress). The thermodynamic basis is presented in conjunction with


vii

the celebrated postulate of Maximum Plastic Dissipation, which is the fundamental basis

of classical plasticity. This postulate infers the normality rule, and it provides general

loading criteria (in terms of the complementary Kuhn-Tucker conditions) for any choice

of control variables. To illustrate the developments, the von Mises yield criterion with

mixed isotropic and kinematic hardening is investigated in detail as a prototype model.

The chapter is concluded with a review of classical isotropic yield (and failure) criteria.

In Chapter 11 we elaborate on ”advanced concepts” of plasticity, that are often necessary

to account for in order to provide a realistic model. Here, we limit the list of such concepts

to: non-associative flow and hardening rules, non-smooth yield surface, and anisotropic

yield surface. A prototype model (or class of models) is selected to illustrate each concept.

In particular, we discuss the Cam-Clay family of yield surfaces, developed for granular

materials, as a proponent of plasticity models that display quite general (non-associative)

hardening.

In Chapter 12 we discuss elastic-viscoplastic material response, which is characterized by

the presence of rate-dependent dissipation mechanisms when the stress exceeds the yield

stress. Viscoplasticity is shown to be the regularization of rate-independent plasticity in

the sense that the flow and hardening rules are obtained from a penalty formulation of

the MPD-principle (in the spirit of Perzyna’s viscoplasticity concept). To illustrate the

developments, the Bingham/Norton model with mixed hardening is investigated in detail

as a prototype model.

In Chapter 13 we discuss elastic-viscoplastic response with thermomechanical coupling.

A generic format of the rate equations is presented for simple hardening of the quasistatic

yield surface, whereby Perzyna’s viscoplasticity concept is adopted. The continuum tan-

gent formulation pertinent to the rate-independent limit is outlined. To illustrate the

developments, the Bingham/Fourier model with mixed hardening and thermal softening

is investigated in detail as a prototype model.

In Chapter 14 we outline the fundamental ideas behind Continuum Damage Mechanics, as

a direct application of the Nonstandard Dissipative Materials. Both scalar and tensorial

damage (giving rise to isotropic as well as anisotropic incremental response) are consid-

ered. We adopt the concept of strain energy equivalence as the basis for the proposed

model framework, and we introduce the concepts of effective configuration and integrity.

The (scalar or tensorial) integrity measure is used as argument in the free energy. The

Microcrack-Closure-Reopening effect, due to different behavior in tension and compres-


viii

sion, is discussed. We limit the discussion to rate-independent, elastic-damaged response

under isothermal conditions.

In Chapter 15 we discuss the coupling of damage to elastic-plastic material response,

whereby the concepts and algorithms introduced in the previous chapter are used. Both

scalar and tensorial damage are considered. With the introduction of a tensorial dam-

age measure in the yield criterion, this criterion will inevitably become anisotropic on

the nominal configuration. The von Mises yield criterion with isotropic hardening (as

formulated on the effective configuartion) is chosen as the prototype model.

Undoubtedly, the course material is best digested with the aid of computer simulations

that show the predictive capability/performance of the various models/algorithms. To

this end, the description of each prototype model is complemented by such illustrative

predictions for homogeneous as well as non-homogeneous states (a FE-discretized mem-

brane in plane stress). Moreover, a separate problem book containing suggested computer

assignments, that represent extensions and variations of those already included in the text.

The only necessary prerequisites for a good understanding of the subject matter are basic

courses in solid mechanics and numerical analysis, while it is helpful to have taken an

introductory course on finite elements. We therefore believe that the material in this

book is well suited for an advanced undergraduate course as well as for an introductory

graduate course on constitutive relations. A more advanced course should include the

same type of material as applied to nonlinear kinematics.


Acknowledgements

We are indebted to a great number of people who have contributed to the making of

this book. In particular, we would like to thank Dr. Magnus Ekh, Assistant Professor

at the Department of Solid Mechanics, Chalmers University, who has read the entire

manuscript and who is the mastermind behind the computer software CONSTLAB. We

are also indebted to Mr. Andreas Menzel, Ph.D. student at the Department of Technical

Mechanics, University of Kaiserslautern, who contributed greatly at the late stages of the

preparation of the book. Many others have contributed to the book at its various stages

from Lecture Notes at Chalmers University up to its present form: Mr. Lars Jacobsson,

Dr. Lennart Mahler and Dr. Thomas Svedberg, who are present and former graduate stu-

dents at Chalmers, have read (parts of) the manuscript and struggled with the numerical

simulations.

Ms. EvaMari Runesson, an English and History student at Goteborg University (and who

also happens to be the daughter the first author), quickly became an expert in LATEX.

At the final stage of the book Ms. Annicka Karlsson did a great job in preparing figures,

organising the manuscript and, as part of her M.Sc. theses, developing CONSTLAB in-

cluding running the response simulations pertinent to the various prototype models. The

contribution of both is gratefully acknowledged.

Goteborg and Kaiserslautern in January 2002.

Kenneth Runesson and Paul Steinmann


x


Contents

1 TENSOR CALCULUS TOOLBOX 1

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Preliminaries about style and notation . . . . . . . . . . . . . . . . 1

1.1.2 Symbolic and component notation . . . . . . . . . . . . . . . . . . . 2

1.1.3 Differential operators . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Elementary algebra of vectors . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2.1 Component representations . . . . . . . . . . . . . . . . . . . . . . 3

1.2.2 Scalar product and length . . . . . . . . . . . . . . . . . . . . . . . 4

1.2.3 Coordinate transformation . . . . . . . . . . . . . . . . . . . . . . . 5

1.3 Elementary algebra of 2nd order tensors . . . . . . . . . . . . . . . . . . . 5

1.3.1 Component representations . . . . . . . . . . . . . . . . . . . . . . 5

1.3.2 Scalar product(s) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.3.3 Symmetry and skew-symmetry . . . . . . . . . . . . . . . . . . . . 7

1.3.4 Special tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8


1.4 Elementary algebra of 4th order tensors . . . . . . . . . . . . . . . . . . . . 9

1.4.1 Component representation . . . . . . . . . . . . . . . . . . . . . . . 9

1.4.2 Symmetry and skew-symmetry . . . . . . . . . . . . . . . . . . . . 10

1.4.3 Special tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10



xii CONTENTS

1.4.5 Appendix: Voigt-matrix representation of 4th order tensor trans-

formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.5 Permutation tensor (symbol) and its usage . . . . . . . . . . . . . . . . . . 14

1.6 Spectral properties and invariants of a symmetric 2nd order tensor . . . . . 16

1.6.1 Principal values - Spectral decomposition . . . . . . . . . . . . . . . 16

1.6.2 Basic invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

1.6.3 Principal invariants - Cayley-Hamilton’s theorem . . . . . . . . . . 19

1.6.4 Octahedral invariants of the stress and strain tensors . . . . . . . . 21

1.6.5 Derivatives of a 2nd order tensor . . . . . . . . . . . . . . . . . . . 22

1.6.6 Derivatives of invariants, etc. . . . . . . . . . . . . . . . . . . . . . 24

1.6.7 Representation of eigendyads . . . . . . . . . . . . . . . . . . . . . 25

1.7 Representation theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

1.7.1 Coordinate transformation vs. vector rotation . . . . . . . . . . . . 27

1.7.2 Scalar-valued isotropic tensor functions of one argument . . . . . . 29

1.7.3 Scalar-valued isotropic tensor functions of two arguments . . . . . . 29

1.7.4 Scalar-valued isotropic tensor functions of three arguments . . . . . 30

1.7.5 Symmetric tensor-valued isotropic tensor functions of one argument 30

1.7.6 Symmetric tensor-valued isotropic tensor function of two arguments 31

2 CHARACTERISTICS OF ENGINEERING MATERIALS AND CON-

STITUTIVE MODELING 33

2.1 General remarks on constitutive modeling . . . . . . . . . . . . . . . . . . 33

2.1.1 Concept of a constitutive model . . . . . . . . . . . . . . . . . . . . 33

2.1.2 The role of constitutive modeling . . . . . . . . . . . . . . . . . . . 35

2.1.3 General constraints on constitutive models . . . . . . . . . . . . . . 36

2.1.4 Approaches to constitutive modeling . . . . . . . . . . . . . . . . . 37

2.2 Modeling of material failure — Fracture . . . . . . . . . . . . . . . . . . . 39

2.2.1 Continuum damage mechanics . . . . . . . . . . . . . . . . . . . . . 39


CONTENTS xiii

2.2.2 Fracture mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2.3 Common experimental test conditions . . . . . . . . . . . . . . . . . . . . . 40

2.4 Typical behavior of metals and alloys . . . . . . . . . . . . . . . . . . . . . 44

2.4.1 Plastic yielding — Hardening and ductile fracture . . . . . . . . . . 44

2.4.2 Constant loading — Creep and relaxation . . . . . . . . . . . . . . 45

2.4.3 Time-dependent loading — Rate effect and damping . . . . . . . . 46

2.4.4 Cyclic loading and High-Cycle-Fatigue (HCF) . . . . . . . . . . . . 47

2.4.5 Cyclic loading and Low-Cycle-Fatigue (LCF) . . . . . . . . . . . . . 48

2.4.6 Creep-fatigue and Relaxation-fatigue . . . . . . . . . . . . . . . . . 52

2.5 Typical behavior of ceramics and cementitious composites . . . . . . . . . 53

2.5.1 Monotonic loading – Semi-brittle fracture . . . . . . . . . . . . . . . 53

2.5.2 Cyclic loading and fatigue . . . . . . . . . . . . . . . . . . . . . . . 54

2.5.3 Creep and relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . 54

2.6 Typical behavior of granular materials . . . . . . . . . . . . . . . . . . . . 54

2.6.1 Monotonic loading – Basic features . . . . . . . . . . . . . . . . . . 54

2.6.2 Constant loading – Consolidation . . . . . . . . . . . . . . . . . . . 55

2.6.3 Constant loading – Creep and relaxation . . . . . . . . . . . . . . . 55

3 INTRODUCTION TO CONTINUUM THERMODYNAMICS 57

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.1.1 Motivation and literature overview . . . . . . . . . . . . . . . . . . 57

3.1.2 The role of continuum thermodynamics . . . . . . . . . . . . . . . . 59

3.1.3 Thermodynamic system . . . . . . . . . . . . . . . . . . . . . . . . 59

3.1.4 Thermodynamic state variables . . . . . . . . . . . . . . . . . . . . 60

3.1.5 Thermodynamic processes . . . . . . . . . . . . . . . . . . . . . . . 62

3.2 Mechanical balance laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.2.1 Global and local formats of the momentum balance law – Equilib-

rium equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63


xiv CONTENTS

3.2.2 Global and local formats of the moment of the momentum balance

law - Symmetry of stress . . . . . . . . . . . . . . . . . . . . . . . . 64

3.3 Energy balance – The first law of thermodynamics . . . . . . . . . . . . . . 65

3.3.1 Kinetic and internal energy . . . . . . . . . . . . . . . . . . . . . . 65

3.3.2 Global and local formats of the energy equation . . . . . . . . . . . 65

3.4 Entropy inequality – The second law of thermodynamics . . . . . . . . . . 67

3.4.1 Entropy - Motivation from statistical mechanics . . . . . . . . . . . 67

3.4.2 Global and local formats of the entropy inequality . . . . . . . . . . 68

3.4.3 Basic constitutive relations . . . . . . . . . . . . . . . . . . . . . . . 70

3.5 Choice of independent state variables - Thermodynamic potentials . . . . . 73

3.5.1 Legendre-Fenchel transformations . . . . . . . . . . . . . . . . . . . 73

3.5.2 Format based on internal energy . . . . . . . . . . . . . . . . . . . . 74

3.5.3 Format based on enthalpy . . . . . . . . . . . . . . . . . . . . . . . 74

3.5.4 Format based on (Helmholtz’) free energy . . . . . . . . . . . . . . 75

3.5.5 Format based on (Gibbs’) free enthalpy . . . . . . . . . . . . . . . . 76

3.5.6 Evaluation of thermodynamic processes . . . . . . . . . . . . . . . . 78

3.5.7 Strain and stress energy for reversible system . . . . . . . . . . . . 79

3.5.8 Tangent stiffness and compliance relations at prescibed temperature

- General situation . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

3.5.9 Tangent stiffness and compliance relations at prescribed tempera-

ture - Adiabatic case . . . . . . . . . . . . . . . . . . . . . . . . . . 82

3.6 The archetypes of dissipative materials . . . . . . . . . . . . . . . . . . . . 82

3.6.1 Generic constitutive relations . . . . . . . . . . . . . . . . . . . . . 82

3.6.2 Inviscid (rate-independent) response . . . . . . . . . . . . . . . . . 83

3.6.3 Viscous (rate-dependent) response . . . . . . . . . . . . . . . . . . . 84

3.7 Appendix: Legendre transformations . . . . . . . . . . . . . . . . . . . . . 84

3.8 Questions and problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88


CONTENTS xv

4 SPACE-TIME DISCRETIZED FORMATS OF THERMOMECHANI-

CAL RELATIONS 89

4.1 The continuous formats of continuum thermodynamics . . . . . . . . . . . 90

4.1.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

4.1.2 The fully coupled format . . . . . . . . . . . . . . . . . . . . . . . . 92

4.1.3 The isothermal format . . . . . . . . . . . . . . . . . . . . . . . . . 94

4.1.4 The isometric format (rigid heat conductor) . . . . . . . . . . . . . 94

4.1.5 The thermomechanically decoupled format . . . . . . . . . . . . . . 95

4.1.6 The adiabatic format . . . . . . . . . . . . . . . . . . . . . . . . . . 96

4.1.7 The eisentropic format . . . . . . . . . . . . . . . . . . . . . . . . . 96

4.2 The discrete formats of continuum thermodynamics . . . . . . . . . . . . . 97

4.2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

4.2.2 The fully coupled format . . . . . . . . . . . . . . . . . . . . . . . . 98

4.2.3 The fully coupled format - reduced version . . . . . . . . . . . . . . 103

4.2.4 The isothermal format . . . . . . . . . . . . . . . . . . . . . . . . . 104

4.2.5 The isometric format . . . . . . . . . . . . . . . . . . . . . . . . . . 104

4.2.6 The adiabatic and eisentropic formats . . . . . . . . . . . . . . . . . 105

4.3 Global solution algorithm - Newton iterations . . . . . . . . . . . . . . . . 106

4.3.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

4.3.2 Iterative solution for the fully coupled format . . . . . . . . . . . . 107

4.3.3 Iterative solution for the isothermal format . . . . . . . . . . . . . . 110

4.3.4 Iterative solution for the isometric format . . . . . . . . . . . . . . 111

4.3.5 Iterative solution for the adiabatic and eisentropic formats . . . . . 111

5 THE CANONICAL CONSTITUTIVE FRAMEWORK FOR DISSIPA-

TIVE MATERIALS 113

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

5.2 Common features of a canonical constitutive framework . . . . . . . . . . . 114


xvi CONTENTS

5.2.1 Free energy and thermodynamic stresses . . . . . . . . . . . . . . . 114

5.2.2 Concepts of dissipation surface and elastic region . . . . . . . . . . 115

5.3 Associative structure - Postulate of Maximum Dissipation (MD) . . . . . . 116

5.3.1 Rate-independent models - Exact format of MD . . . . . . . . . . . 116

5.3.2 Rate-independent models – Alternative formats of the constitutive

equations pertinent to the exact format of MD . . . . . . . . . . . . 118

5.3.3 Rate-independent models - Continuum tangent operators . . . . . . 119

5.3.4 Rate-dependent models - Penalized enforcement of MD . . . . . . . 123

5.3.5 Rate-dependent models – Alternative formats of the constitutive

equations pertinent to the penalized enforcement of MD . . . . . . 125

5.4 Non-associative structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

5.4.1 Rate-independent models . . . . . . . . . . . . . . . . . . . . . . . . 126

5.4.2 Rate-independent models - Continuum tangent operators . . . . . . 127

5.5 Issues of controllability, stability and uniqueness . . . . . . . . . . . . . . . 128

5.5.1 Controllability of rate-independent response for strain and stress

control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

5.5.2 Limit points for rate-independent response - Spectral properties of

CTS-tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

5.5.3 Second order work - Spectral properties of symmetric part of CTS-

tensor - Hill-stability . . . . . . . . . . . . . . . . . . . . . . . . . . 135

5.5.4 Uniqueness of boundary value problem . . . . . . . . . . . . . . . . 138

6 THE CONSTITUTIVE INTEGRATOR 143

6.1 Introduction - The concept of a Constitutive Laboratory . . . . . . . . . . 143

6.1.1 Response functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

6.1.2 Constitutive Laboratory code CONSTLAB c© . . . . . . . . . . . . 147

6.2 Integrator - Backward Euler rule for the rate-independent canonical format 151

6.2.1 Incremental format . . . . . . . . . . . . . . . . . . . . . . . . . . . 151


CONTENTS xvii

6.2.2 The Algorithmic Tangent Stiffness (ATS) tensor . . . . . . . . . . . 154

6.2.3 The ATS-tensor - Alternative derivation . . . . . . . . . . . . . . . 157

6.3 Integrator - Backward Euler rule of the rate-dependent canonical format . 158

6.3.1 Incremental format . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

6.3.2 The ATS-tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

6.4 Iterator - Newton algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 160

6.4.1 Isothermal format . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

6.4.2 Adiabatic format . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

6.5 Numerical differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

6.6 Appendix: Total derivative of an implicit function . . . . . . . . . . . . . . 163

7 ELASTICITY 165

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

7.1.1 General characteristics of nonlinear elasticity . . . . . . . . . . . . . 165

7.1.2 Material symmetry - Isotropy . . . . . . . . . . . . . . . . . . . . . 167

7.1.3 Appendix: Voigt-matrix representation of tangent relations . . . . . 168

7.2 Constitutive relations - Isotropic nonlinear elasticity . . . . . . . . . . . . . 169

7.2.1 Generic format of free energy . . . . . . . . . . . . . . . . . . . . . 169

7.2.2 Generic format of Continuum Tangent Stiffness tensor . . . . . . . 170

7.2.3 Volumetric/deviatoric decomposition of the free energy . . . . . . . 171

7.2.4 Deformation theory of plasticity . . . . . . . . . . . . . . . . . . . . 174

7.3 Prototype model: Hooke’s model of isotropic linear elasticity . . . . . . . . 177

7.3.1 Constitutive relations . . . . . . . . . . . . . . . . . . . . . . . . . . 177

7.3.2 Examples of response simulations . . . . . . . . . . . . . . . . . . . 181

7.4 Constitutive framework - Anisotropic nonlinear elasticity . . . . . . . . . . 182

7.4.1 Generic format of the free energy - Symmetry classes . . . . . . . . 182

7.4.2 Representation of anisotropy with structure tensors . . . . . . . . . 188


xviii CONTENTS

7.4.3 Kelvin-modes and spectral decomposition of the tangent stiffness

tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

7.5 Constitutive framework - Anisotropic linear elasticity . . . . . . . . . . . . 193

7.5.1 Orthogonal symmetry . . . . . . . . . . . . . . . . . . . . . . . . . 193

7.5.2 Tetragonal symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . 194

7.5.3 Transverse isotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

7.5.4 Cubic symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198

7.5.5 Isotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

8 VISCOELASTICITY 201

8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

8.2 The constitutive framework - Nonlinear viscoelasticity . . . . . . . . . . . . 203

8.2.1 Free energy, thermodynamic forces and rate equations . . . . . . . . 203

8.2.2 Linear viscoelasticity - Creep and relaxation functions . . . . . . . . 204

8.3 The constitutive integrator - Nonlinear viscoelasticity . . . . . . . . . . . . 206

8.3.1 Backward Euler method . . . . . . . . . . . . . . . . . . . . . . . . 206

8.3.2 ATS-tensor for BE-rule . . . . . . . . . . . . . . . . . . . . . . . . . 208

8.3.3 Backward Euler rule for linear elasticity . . . . . . . . . . . . . . . 209

8.3.4 Backward Euler rule for linear viscoelasticity . . . . . . . . . . . . . 210

8.4 Prototype model: The isotropic (linear) Maxwell model . . . . . . . . . . . 210

8.4.1 The constitutive relations . . . . . . . . . . . . . . . . . . . . . . . 210

8.4.2 The constitutive integrator . . . . . . . . . . . . . . . . . . . . . . . 213



8.4.5 Appendix: Constitutive relations for the uniaxial stress state . . . . 217

8.5 Prototype model: The isotropic (nonlinear) Norton model . . . . . . . . . 219




CONTENTS xix



8.6 The constitutive framework - Nonlinear viscoelasticity with multiple dissi-

pative mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230


8.6.2 Linear model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231

8.7 Prototype model - The isotropic Linear Standard Viscoelasticity model . . 232




9 THERMO-(VISCO)ELASTICITY 237

9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237

9.2 The constitutive framework - Nonlinear thermo-viscoelasticity . . . . . . . 238


9.2.2 The energy equation . . . . . . . . . . . . . . . . . . . . . . . . . . 239

9.2.3 The locally adiabatic case . . . . . . . . . . . . . . . . . . . . . . . 240

9.3 The constitutive integrator - Nonlinear thermo-viscoelasticity . . . . . . . . 240


9.3.2 ATS-tensor and other algorithmic quantities for the BE-rule . . . . 241

9.4 Prototype model: The isotropic (linear) Maxwell-Fourier model . . . . . . 243



9.4.3 The constitutive integrator - Adiabatic condition . . . . . . . . . . 246

9.4.4 Examples of response simulations (adiabatic case) . . . . . . . . . . 246

10 PLASTICITY - BASIC CONCEPTS 247

10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247


xx CONTENTS

10.1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247

10.1.2 Literature overview . . . . . . . . . . . . . . . . . . . . . . . . . . . 249

10.2 The constitutive framework - Perfect plasticity . . . . . . . . . . . . . . . . 250

10.2.1 Free energy and thermodynamic stresses . . . . . . . . . . . . . . . 250

10.2.2 Associative structure - Postulate of Maximum Dissipation . . . . . 251

10.2.3 Continuum tangent relations . . . . . . . . . . . . . . . . . . . . . . 254

10.3 The constitutive integrator - Perfect plasticity . . . . . . . . . . . . . . . . 256


10.3.2 Backward Euler method – Constrained minimization problem . . . 259


10.3.4 Backward Euler method for linear elasticity - Solution in stress space261

10.3.5 Concept of Closest-Point-Projection for linear elasticity . . . . . . . 263

10.4 Prototype model: Hooke elasticity and von Mises yield surface . . . . . . . 265



10.4.3 Examples of response computations . . . . . . . . . . . . . . . . . . 268

10.4.4 Appendix I: Constitutive relations for the uniaxial stress state . . . 269

10.4.5 Appendix II: Voigt format of prototype model . . . . . . . . . . . . 273

10.5 The constitutive framework - Hardening plasticity . . . . . . . . . . . . . . 277

10.5.1 Free energy and thermodynamic forces . . . . . . . . . . . . . . . . 277

10.5.2 Representation of hardening - Constraints and classification . . . . 277

10.5.3 Associative structure - Postulate of Maximum Dissipation . . . . . 279


10.5.5 Significance of hardening versus softening . . . . . . . . . . . . . . . 282

10.5.6 Significance of total mechanical dissipation versus

“plastic dissipation” . . . . . . . . . . . . . . . . . . . . . . . . . . 284


CONTENTS xxi

10.6 The constitutive integrator - Hardening

plasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284


10.6.2 Backward Euler method – Constrained minimization problem . . . 286


10.6.4 Backward Euler method for linear elasticity and linear hardening . 288

10.6.5 Concept of Closest-Point-Projection for linear elasticity and linear

hardening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290

10.7 Prototype model: Hooke elasticity and von Mises yield surface with linear

mixed hardening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290





10.8 Classical isotropic yield criteria . . . . . . . . . . . . . . . . . . . . . . . . 311

10.8.1 Basic concepts - Cohesive and frictional character . . . . . . . . . . 311

10.8.2 Isotropic yield criteria - General characteristics . . . . . . . . . . . 312

10.8.3 The Tresca criterion . . . . . . . . . . . . . . . . . . . . . . . . . . 318

10.8.4 The von Mises criterion . . . . . . . . . . . . . . . . . . . . . . . . 319

10.8.5 Hosford’s yield criterion . . . . . . . . . . . . . . . . . . . . . . . . 322

10.8.6 The Mohr criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . 323

10.8.7 The Mohr-Coulomb criterion . . . . . . . . . . . . . . . . . . . . . . 325

10.8.8 The Drucker-Prager criterion . . . . . . . . . . . . . . . . . . . . . 328

10.8.9 Appendix: Geometric invariants in principal stress space . . . . . . 329

10.9 The constitutive integrator for a special class: Isotropic linear elasticity

and isotropic yield criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . 334

10.9.1 Backward Euler method - Preliminaries . . . . . . . . . . . . . . . . 334


xxii CONTENTS

10.9.2 Backward Euler method for two-invariant yield surfaces (indepen-

dent of the Lode angle) . . . . . . . . . . . . . . . . . . . . . . . . . 335

10.9.3 Backward Euler method for three-invariant yield surfaces . . . . . . 337

10.9.4 ATS-tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341

10.10Prototype model: Hooke elasticity and Hosford’s family of yield surfaces . 342

10.10.1The constitutive relations . . . . . . . . . . . . . . . . . . . . . . . 342

10.10.2The constitutive integrator . . . . . . . . . . . . . . . . . . . . . . . 343

10.10.3Examples of response simulations . . . . . . . . . . . . . . . . . . . 344

10.11Questions and problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345

11 PLASTICITY - ADVANCED CONCEPTS 347

11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347

11.2 The constitutive framework – Nonassociative structure . . . . . . . . . . . 349


11.2.2 Non-associative flow and hardening rules . . . . . . . . . . . . . . . 349

11.2.3 Continuum tangent relations (for smooth yield surface) . . . . . . . 350

11.2.4 Non-associative hardening - Special choice . . . . . . . . . . . . . . 351

11.3 The constitutive integrator – Nonassociative structure . . . . . . . . . . . . 352


11.3.2 ATS-tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353

11.3.3 Closest-Point-Projection for linear elasticity and linear hardening . 353

11.3.4 Volumetric non-associativity – Isotropic elasticity and plasticity . . 353

11.4 Prototype model: Hooke elasticity and von Mises yield surface with non-

linear mixed (saturation) hardening . . . . . . . . . . . . . . . . . . . . . . 355




11.5 Prototype model: Hosford yield surface and von Mises plastic potential . . 368


CONTENTS xxiii

11.6 Prototype model: Parabolic Drucker-Prager yield surface . . . . . . . . . . 368




11.7 Non-smooth yield surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 371

11.7.1 Associative flow rule - Koiter’s rule . . . . . . . . . . . . . . . . . . 371

11.7.2 Continuum tangent relations for non-smooth yield surface . . . . . 372

11.7.3 Backward Euler method - CPPM for linear elasticity . . . . . . . . 377

11.7.4 ATS-tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381

11.8 Prototype model: Linear Drucker-Prager yield surface . . . . . . . . . . . . 381



11.9 Anisotropic yield surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 389

11.9.1 Oriented materials - Representation of anisotropy with structure

tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389

11.9.2 Orthotropy - Restriction to quadratic forms . . . . . . . . . . . . . 389

11.9.3 Transverse isotropy - Restriction to quadratic forms . . . . . . . . . 392

11.9.4 Hill’s yield criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . 394

11.10Prototype model: Transversely isotropic elasticity and Hill’s yield criterion 399

11.10.1The constitutive relations . . . . . . . . . . . . . . . . . . . . . . . 399

11.10.2The constitutive integrator . . . . . . . . . . . . . . . . . . . . . . . 400

11.10.3Examples of response simulations . . . . . . . . . . . . . . . . . . . 401

12 PLASTICITY - MORE ADVANCED CONCEPTS 405

12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405

12.2 The constitutive framework – Plasticity with non-simple hardening . . . . 406


12.2.2 Non-associative flow and hardening rules . . . . . . . . . . . . . . . 406


xxiv CONTENTS

12.2.3 Continuum tangent relations (for smooth yield surface) . . . . . . . 407

12.2.4 Controllability under strain and stress control . . . . . . . . . . . . 408

12.2.5 Material failure and stability – General . . . . . . . . . . . . . . . . 411

12.2.6 Hill’s and Drucker’s criteria of material stability . . . . . . . . . . . 415

12.2.7 The constitutive integrator – BE-rule . . . . . . . . . . . . . . . . . 416

12.2.8 ATS-tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417

12.3 Prototype model: Cam-Clay family of yield surfaces . . . . . . . . . . . . . 417

12.3.1 Porosity measures — relative density . . . . . . . . . . . . . . . . . 417

12.3.2 Generic relations in Critical State Soil Mechanics . . . . . . . . . . 419

12.3.3 The generic Cam-Clay family of yield surfaces . . . . . . . . . . . . 423




12.4 Prototype model: Gurson model family of yield surfaces . . . . . . . . . . 432




13 VISCOPLASTICITY 435

13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435

13.2 The constitutive framework – Perzyna format . . . . . . . . . . . . . . . . 436


13.2.2 Penalty formulation of the Postulate of Maximum Plastic Dissipa-

tion – Rate equations . . . . . . . . . . . . . . . . . . . . . . . . . . 437

13.2.3 Plasticity as the limit situation . . . . . . . . . . . . . . . . . . . . 439

13.2.4 Elasticity as the limit situation . . . . . . . . . . . . . . . . . . . . 440

13.2.5 Creep and relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . 441

13.2.6 Generalized rate laws – Concept of dynamic yield surface . . . . . . 442


CONTENTS xxv

13.3 The constitutive integrator – Perzyna format . . . . . . . . . . . . . . . . . 443



13.3.3 Backward Euler method for linear elasticity and linear hardening . 447

13.3.4 Concept of Closest-Point-Projection for linear elasticity and linear

hardening - “Quasi-projection” property . . . . . . . . . . . . . . . 447

13.4 Prototype model: Hooke elasticity and Bingham viscoplasticity with linear

mixed hardening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449





13.5 The constitutive framework – Duvaut-Lions’ format . . . . . . . . . . . . . 455

13.5.1 Flow and hardening rules . . . . . . . . . . . . . . . . . . . . . . . . 455

13.5.2 Thermodynamic abmissibility . . . . . . . . . . . . . . . . . . . . . 457

13.6 The constitutive integrator – Duvaut-Lions’ format . . . . . . . . . . . . . 457

13.6.1 Backward Euler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457


13.7 Prototype model: Hooke elasticity and Bingham-type viscoplasticity with

linear mixed hardening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 460




14 THERMO-(VISCO)PLASTICITY 463

14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463

14.2 The constitutive framework for thermo-viscoplasticity – Perzyna format . . 464



xxvi CONTENTS

14.2.2 The energy equation . . . . . . . . . . . . . . . . . . . . . . . . . . 465

14.2.3 The locally adiabatic case . . . . . . . . . . . . . . . . . . . . . . . 466

14.3 The constitutive framework of rate-independent thermo-plasticity . . . . . 466

14.3.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 466


14.3.3 Continuum tangent relations - The locally adiabatic case . . . . . . 469

14.4 The constitutive integrator – Perzyna format . . . . . . . . . . . . . . . . . 470


14.4.2 ATS-tensor and other algorithmic quantities for the BE-rule . . . . 471

14.5 Prototype model: Hooke elasticity and Bingham (visco)plasticity with lin-

ear mixed hardening and thermal softening . . . . . . . . . . . . . . . . . . 472


14.5.2 The constitutive relations for the rate-independent response . . . . 473

14.5.3 The constitutive equations for dynamic yield surface and temperature-

dependent quasistatic yield surface . . . . . . . . . . . . . . . . . . 474



15 CALIBRATION OF CONSTITUTIVE MODELS 477

15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477

15.1.1 Definition of parameter identification problem . . . . . . . . . . . . 477

15.1.2 Experimental data – Testing and measurements . . . . . . . . . . . 479

15.1.3 Parameter identification from direct identification . . . . . . . . . . 480

15.2 Calibration via optimization - Least squares format . . . . . . . . . . . . . 484

15.2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484

15.2.2 Choice of state-, control- and response variable . . . . . . . . . . . . 484

15.2.3 Objective function . . . . . . . . . . . . . . . . . . . . . . . . . . . 486

15.2.4 Evaluation of the objective function . . . . . . . . . . . . . . . . . . 487


CONTENTS xxvii

15.2.5 Quality of the solution . . . . . . . . . . . . . . . . . . . . . . . . . 488

15.2.6 Prototype example: Norton’s model . . . . . . . . . . . . . . . . . . 491

15.3 Optimization strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492

15.3.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492

15.3.2 Gradient-free methods . . . . . . . . . . . . . . . . . . . . . . . . . 493

15.3.3 Gradient-based methods . . . . . . . . . . . . . . . . . . . . . . . . 493

15.4 Sensitivity assessment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493

15.4.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493

15.4.2 Monte Carlo method . . . . . . . . . . . . . . . . . . . . . . . . . . 493

15.4.3 Perturbation method . . . . . . . . . . . . . . . . . . . . . . . . . . 493

15.4.4 Correlation matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . 494

15.4.5 Influence from discretization errors . . . . . . . . . . . . . . . . . . 494

15.5 Self-adjoint format for calibration . . . . . . . . . . . . . . . . . . . . . . . 494

15.5.1 Optimality condition . . . . . . . . . . . . . . . . . . . . . . . . . . 494

15.5.2 Gradient-based methods for optimization . . . . . . . . . . . . . . . 494

15.5.3 Sensitivity assessment . . . . . . . . . . . . . . . . . . . . . . . . . 494


xxviii CONTENTS


Chapter 1

TENSOR CALCULUS TOOLBOX

const201.tex

In this introductory chapter we introduce the commonly used notation and summarize

some basic definitions and results from vector and tensor calculus. We also give some

useful formulas and results that can not easily be found in standard text-books on con-

tinuum mechanics and constitutive theory. A typical example is Serrin’s formula. Most

relations are given without rigorous proofs. In essence, this chapter will serve as reference.

For further reading on tensor calculus, we refer to the abundant literature on continuum

mechanics, e.g. Malvern (1969), Gurtin (1981), Holzapfel (2000).

1.1 Introduction

1.1.1 Preliminaries about style and notation

As a rule, the theoretical developments are presented in the traditional direct style (com-

mon in physics), which means that the final result comes after a sequence of derivations

given in consecutive order. However, at times the indirect style (common in mathemat-

ics) is adopted, whereby theorems are preceding proofs. Independently of the presentation

style, some important results are placed within a frame. A box (2) is used to mark the

end of a Theorem, Proof, Remark, etc.

A meager italic character is used to denote scalars, e.g. a, A, whereas boldface italic char-

acters are used to denote tensors of 1st order (vectors), 2nd order, or 3rd order, e.g. a, A.


2 1 TENSOR CALCULUS TOOLBOX

Boldface sanserif characters denote 4th order tensors, e.g. A. Sets are denoted by black-

board characters, e.g. R. Configurations of a body, i.e. a set of points in Euclidean space,

are denoted by calligraphic characters, e.g. B. A meager character with an “underscore”

denotes matrix, e.g. a and A. A superimposed dot denotes (material) time derivative, e.g.

udef= du/dt.

Italic characters are used to denote a running index, whereas roman characters are used

to denote a fixed index, e.g. εpij .

Regular brackets are used for functional arguments, whereas square brackets are used to

separate expressions, e.g. f(x) = 2 [x[1 + x]]−1. Curly brackets are used to denote sets,

e.g. E = {x| φ(x) ≤ 0}.Throughout the chapter (and the whole book) we shall consider only Cartesian coordi-

nates, unless otherwise is explicitly stated.

1.1.2 Symbolic and component notation

Einstein’s summation convention is used for indices, e.g. for vectors (1st order tensors)

we have the representation in terms of Cartesian components and unit base vectors ei

v = vieidef=

3∑

i=1

viei , w = wαeαdef=

2∑

α=1

wαeα (1.1)

Latin letters are used for a running index ranging from 1 to 3 (corresponding to a repre-

sentation of v in E3), whereas Greek letters are used when the running index ranges from

1 to 2 (corresponding to a representation of w in E2)1. Indeed, Cartesian coordinates are

used if not otherwise is stated explicitly. For a 2nd order tensor A and an n:th order

tensor T we thus have the component representation

A = Aijei ⊗ ej, T = Ti1i2...inei1 ⊗ ei2 ⊗ ...ein (1.2)

where ⊗ denotes the open product symbol. The tensor components are normally denoted

by a meager character (such as Aij). However, in order to avoid possible sources of

confusion and provide maximal transparency, we sometimes use the notation (A)ij instead

1Note that Greek letters are also used to label matrix elements, e.g. A = [Aαβ ], when these elements

do not represent the components of a 2nd order tensor. In such a case, the index range is defined by the

context.


1.2 Elementary algebra of vectors 3

of Aij . This notation is useful in expressions such as

At = (At)ijei ⊗ ej = (A)jiei ⊗ ej, A · B = (A · B)ijei ⊗ ej (1.3)

As a matter of policy, we avoid explicit component representations, if possible. We rather

use symbolic notation, which is generally valid regardless of the coordinate system (the

choice of which is a matter of taste and convenience), cf. Malvern (1969).

Square brackets are used to define the matrix contents. As a special case, matrix notation

is used to represent components. Examples are

u = [ui] =

[

u1

u2

]

, ε = [εij ] =

[

ε11 ε12

ε21 ε22

]

(1.4)

1.1.3 Differential operators

Differential operators are defined in terms of the gradient (vector) operator ∇, which can

operate both “forward” and “backward” on a tensor field as follows:

∇ ⊗ [•] = ei ⊗∂[•]∂xi

, [•] ⊗ ∇ =∂[•]∂xi

⊗ ei (1.5)

The dot product (divergence) and the cross product (rotation or curl) are defined as

∇ · [•] = ei ·∂[•]∂xi

, ∇ × [•] = ei ×∂[•]∂xi

(1.6)

In particular, if a is a scalar field, then ∇a = (∂a/∂xi)ei.

1.2 Elementary algebra of vectors

1.2.1 Component representations

Repeating (1.1), we conclude that any vector v (1st order tensor) can be represented in

an arbitrary Cartesian coordinate system as follows:

v = viei (1.7)

Remark: Adopting more general non-Cartesian coordinates, we have the two possible

representations

v = vigi = vigi (1.8)



PSfrag replacementsg2

g2

g1

e1

g1

v

e2

Figure 1.1: Co- and contravariant base vectors. Cartesian base vectors.

where vi, gi are the covariant components and base vectors, respectively, whereas vi, gi are

the corresponding contravariant quantities. The covariant base vectors gi are “natural”

in the sense that they are tangential to the coordinate lines (which are lines in space along

which the two other coordinates have constant values). The contravariant vectors gi are

defined as mutually orthogonal to gi, i.e.

gi · gj = δ·ji with δ·ji =

{

1 if i = j

0 if i 6= j(1.9)

where δ·ji represents the mixed co-contravariant components of the metric (or unit) tensor

I. Usually, δ·ji is known as the Kronecker delta symbol.

The scalar product of gi and gj constitute the covariant components gij of the metric

tensor. Likewise, the scalar product of gi and gj constitute the contravariant components

gij of the metric tensor, i.e.

gi · gj = gij = gji, gi · gj = gij = gji (1.10)

So much for general coordinates. 2

1.2.2 Scalar product and length

With u = uiei and v = vjej, we obtain

u · v = [uiei] · [vjej] = uivjδij = uivi (1.11)


1.3 Elementary algebra of 2nd order tensors 5

The length (Euclidean norm) of u, denoted |u|, is defined as

|u| = [u · u]1/2 = [uiui]1/2 = [[u1]

2 + [u2]2 + [u3]

2]1/2 (1.12)

1.2.3 Coordinate transformation

We shall consider the effect of coordinate transformation between two Cartesian coor-

dinate systems with base vectors that are denoted ei and e′i respectively. The rela-

tions between these are defined by a linear transformation. Upon performing a scalar

multiplication of the ansatz e′i = Mijej with ek and using ej · ek = δjk, we obtain

e′i · ek = Mijej · ek = Mik. Hence, we summarize

e′i = Mijej, Mij = e′

i · ej = cos(e′i, ej) (1.13)

Moreover, we make the ansatz u = u′ie′i to obtain

e′k · u = e′

k · [uiei] = ui[e′k · ei] = uiMki

= e′k · [u′ie′

i] = u′i[e′k · e′

i] = u′iδki = u′k (1.14)

and we conclude that

u = uiei = u′ie′i, with u′i = Mijuj (1.15)

In matrix form, the component relation (1.15)2 reads

u′ = M u with M = [Mij ] (1.16)

1.3 Elementary algebra of 2nd order tensors

1.3.1 Component representations

The simplest form of a 2nd order tensor T is a dyad, which is defined as the open (or

dyadic) product of two vectors u and v:

T = u ⊗ vdef= [uiei] ⊗ [vjej] = uivjei ⊗ ej (1.17)

where ⊗ is the “open product” symbol. The products ei ⊗ ej, which are denoted base

dyads, form the basis of the product space E3×E

3 (in the same way that ei form the basis

of E3). Clearly, the dyad in (1.17) is only a special case of the general representation

T = Tijei ⊗ ej (1.18)



The matrix representation of, say e1 ⊗ e2, w.r.t its own basis is

[

(e1 ⊗ e2)ij

]

=

1

0

0

[

0 1 0]

=

0 1 0

0 0 0

0 0 0

(1.19)

Hence, the matrix format of the general dyad u ⊗ v is

[

(u ⊗ v)ij

]

=

u1

u2

u3

[

v1 v2 v3

]

=

u1v1 u1v2 u1v3

u2v1 u2v2 u2v3

u3v1 u3v2 u3v3

(1.20)

whereas the matrix format of the general tensor T is

T = [Tij ] =

T11 T12 T13

T21 T22 T23

T31 T32 T33

(1.21)

Any 2nd order tensor defines a linear mapping of E3 onto E

3, since

T · u = [Tijei ⊗ ej] · [ukek] = Tijuk[ej · ek]ei

= Tijukδjkei = Tijujei = viei = v (1.22)

where we introduced the vector v with components vi = Tijuj . In matrix form, this

component relation reads

v = Tu with T = [Tij ] (1.23)

The transpose (or dual) of T , denoted T t, is defined via the relation v · T t = T · v for

any choice of v. This identity can be expressed as

vi(Tt)ij = Tjivi ∀ vi ⇒ (T t)ij = Tji (1.24)

In summary,

T t = (T t)ijei ⊗ ej = Tjiei ⊗ ej (1.25)

The matrix format of T t is

T t = [Tji] =

T11 T21 T31

T12 T22 T32

T13 T23 T33

(1.26)


1.3 Elementary algebra of 2nd order tensors 7

1.3.2 Scalar product(s)

The single scalar product implies contraction with one index (as used above). Double

scalar products of two dyads ei ⊗ ej and ek ⊗ el are of two kinds:

[ei ⊗ ej] · ·[ek ⊗ el]def= [ej · ek][ei · el] = δjkδil (1.27)

[ei ⊗ ej] : [ek ⊗ el]def= [ei · ek][ej · el] = δikδjl (1.28)

Hence, for two second order tensors T and U , we obtain

T · ·U = TijUklδjkδil = TikUki = tr(T · U ) (1.29)

T : U = TijUklδikδjl = TklUkl = Tkl(Ut)lk = tr(T · UT) (1.30)

If T and U are symmetrical, i.e. if T T = T and UT = U , then T · ·U = T : U .

The length (Euclidean norm) of a second order tensor is defined as

|T | = [T : T ]1/2 = [TijTij ]1/2 (1.31)

We note the following rule:

T : [u ⊗ v] = u · T · v = uiTijvj = uTTv (1.32)

Finally, we note that the single scalar product T ·S defines a linear mapping from E3×E

3

onto E3 × E

3, since

T · U = [Tijei ⊗ ej] · [Uklek ⊗ el] = TijUkl[ej · ek]ei ⊗ el

= TijUklδjkei ⊗ el = TikUklei ⊗ el = Vilei ⊗ el = V (1.33)

where we introduced the second order tensor V with components Vij = TikUkj. In matrix

form, this component relation reads

V = TU with V = [Vij ], T = [Tij ], U = [Uij] (1.34)

1.3.3 Symmetry and skew-symmetry

The symmetric part of T , denoted T sym, and the skew-symmetric part of T , denoted

T skw, are defined as follows:

T sym =1

2[T + T t], T skw =

1

2[T − T t] (1.35)

T is symmetrical when T sym = T (and T skw = 0), i.e. when T = T t. In component form,

Tij = Tji. T is skew-symmetrical when T skw = T (and T sym = 0), i.e. when T = −T t.

In component form, Tij = −Tji, which in particular infers that Tij = 0 for i = j.



1.3.4 Special tensors

The 2nd order identity (or unit) tensor I is defined by the identity I · u = u for any

vector u, which gives the component representation

I = δijei ⊗ ej (1.36)

whose matrix format is

I = [δij ] =

1 0 0

0 1 0

0 0 1

(1.37)

The deviator of a symmetric tensor T , denoted T dev, is defined as

T devdef= T − 1

3[I : T ]I = T − 1

3TvolI with Tvol

def= I : T (= Tkk) (1.38)

and it follows that I : T dev = 0.

The spherical part of T , denoted T sph, is defined as

T sph = T − T dev =1

3TvolI =

1

3[I ⊗ I] : T (1.39)

The inverse of a nonsingular tensor T , denoted T −1, is defined by the identity T ·T −1 = I.

In component form, Tik(T−1)kj = δij .

Assume that T is a rank-one update of I. Its inverse can be computed explicitly according

to the Sherman-Morrison formula:

T = I + αu ⊗ v ; T −1 = I − α

1 + αu · v · u ⊗ v (1.40)

where u,v are arbitrary vectors and α is an arbitrary scalar such that α 6= −1/[u · v] (so

that T is non-singular).

The result in (1.40) is shown by making the ansatz T −1 = I + βu ⊗ v, carrying out the

multiplications involved in T · T −1 = I, and identifying components.

A straightforward generalization of the formula in (1.40) is the following:

T = U + αu ⊗ v ; T −1 = U−1 − α

1 + αv · U−1 · uU−1 · u ⊗ v · U−1 (1.41)

where it is assumed that U is a non-singular tensor. Show this as homework!

Hint: Express T = U · T with T = I + αU−1 · u ⊗ v, such that T −1 = T−1 · U−1, and

use (1.40). 2


1.4 Elementary algebra of 4th order tensors 9


Upon making the ansatz T = T ′ije

′i ⊗ e′

j, we obtain

e′k · T · e′

l = e′k · [Tijei ⊗ ej] · e′

l = [e′k · ei]Tij [ej · e′

l] = MkiTijMlj

= e′k · [T ′

ije′i ⊗ e′

j] · e′l = [e′

k · e′i]T

′ij [e

′j · e′

l] = δkiT′ijδjl = T ′

kl (1.42)

Hence, we conclude that

T = Tijei ⊗ ej = T ′ije

′i ⊗ e′

j with T ′ij = MikTklMjl (1.43)

In matrix form, the component relation (1.43)2 reads

T ′ = M T MT (1.44)

1.4 Elementary algebra of 4th order tensors

1.4.1 Component representation

The simplest form of a 4th order tensor A is a quad, which is defined as the open product

of two 2nd order tensors T and U , i.e.

A = T ⊗ U = [Tijei ⊗ ej] ⊗ [Uklek ⊗ el] = TijUklei ⊗ ej ⊗ ek ⊗ el (1.45)

The products ei ⊗ ej ⊗ ek ⊗ el, which are denoted the base quadrads, form the basis of

the product space E3 × E

3 × E3 × E

3. The expression in (1.45) is, clearly, only a special

case of the general representation of a 4th order tensor

A = Aijklei ⊗ ej ⊗ ek ⊗ el (1.46)

Any 4th order tensor defines a linear mapping from E3 × E

3 to E3 × E

3, since

A : T = [Aijklei ⊗ ej ⊗ ek ⊗ el] : [Tmnem ⊗ en] = AijklTmn[ek · em][el · en]ei ⊗ ej

= AijklTmnδkmδlnei ⊗ ej = AijklTklei ⊗ ej = Uijei ⊗ ej = U (1.47)

where we introduced the tensor U with components Uij = AijklTkl.

Useful notations are the “overline open product” ⊗ and the “underline open product” ⊗,

which are defined via the component representations

T ⊗Udef= TikUjlei ⊗ ej ⊗ ek ⊗ el, T⊗U

def= TilUjkei ⊗ ej ⊗ ek ⊗ el (1.48)



Useful rules, that involve the open product symbols, for 2nd order tensors U ,V and W

are:

[U ⊗ V ] : W = U [W : V ] , W : [U ⊗ V ] = [U : W ] V (1.49)

[U⊗V ] : W = U · W · V t, W : [U⊗V ] = U t · W · V (1.50)

[U⊗V ] : W = U · W t · V t, W : [U⊗V ] =[U t · W · V

]t= V t · W t · U (1.51)

1.4.2 Symmetry and skew-symmetry

The major transpose of a 4th order tensor A is defined as

AT = Aklijei ⊗ ej ⊗ ek ⊗ el (1.52)

i.e. the transpose is associated with a “major shift” of indices. The major-symmetric

part of A, denoted ASYM, and the major-skew-symmetric part of A, denoted A

SKW, are

defined as follows:

ASYM =

1

2[A + A

T], ASKW =

1

2[A − A

T] (1.53)

A possesses major symmetry if ASYM = A (and A

SKW = 0), i.e. when A = AT. In

component form, Aijkl = Aklij. A possesses major skew-symmetry when ASKW = A (and

ASYM = 0), i.e. when A = −A

T. In component form, Aijkl = −Aklij, which (in particular)

infers that Aijkl = 0 for ij = kl.

Moreover, A possesses 1st and 2nd minor symmetry if Aijkl = Ajikl and Aijkl = Aijlk,

respectively. Likewise, A possesses 1st and 2nd minor skew-symmetry if Aijkl = −Ajikl

and Aijkl = −Aijlk, respectively.

1.4.3 Special tensors

The 4th order identity, or unit, tensor I is defined by the identity I : T = T for any 2nd

order tensor T . This gives the component representation

Idef= I⊗I = δikδjlei ⊗ ej ⊗ ek ⊗ el (1.54)

It follows that I possesses major symmetry, since (I)klij = δkiδlj = δikδjl = (I)ijkl, i.e.

I = IT.



The minor transpose of I, denoted It, is defined by the identity I

t : T = T t for any 2nd

order tensor T . This gives the component representation

It def

= I⊗I = δilδjkei ⊗ ej ⊗ ek ⊗ el (1.55)

Remark: It represents a transpose with respect to both the two first and the two last

indices, i.e. (It)ijkl = (I)jikl = (I)ijlk. 2

Even It retains major symmetry, since (It)klij = δkjδli = δilδjk = (It)ijkl, i.e. I

t = [It]T.

The minor-symmetric part of I, denoted Isym, and the minor-skew-symmetric part of I,

denoted Iskw, are defined as follows:

Isym def

=1

2[I + I

t] =1

2[I⊗I + I⊗I] =

1

2[δikδjl + δilδjk] (1.56)

Iskw def

=1

2[I − I

t] =1

2[I⊗I − I⊗I] =

1

2[δikδjl + δilδjk] (1.57)

Remark: We note that Isym possesses both 1st and 2nd minor symmetries (in addition

to major symmetry). Likewise, Iskw possesses both 1st and 2nd minor skew-symmetries

(while still possessing major symmetry). Show this as homework! 2

It also follows that Isym : T = T sym and I

skw : T = T skw for any 2nd order tensor T .

We define the deviator projection tensors from the 4th order identity tensors as follows:

Idevdef= I − 1

3I⊗I, I

symdev = I

sym − 1

3I⊗I, I

skwdev = I

skw − 1

3I⊗I (1.58)

which infers that Idev : T = T dev, Isymdev : T = T

symdev and I

skwdev : T = T skw

dev . It follows that

Idev : I = Isymdev : I = I

skwdev : I = 0.

Sometimes the spherical projection operator Isph is introduced:

Isph = I − Idev =1

3I⊗I (1.59)

It appears that Isph : T dev = 0 for any 2nd order tensor T , and that Isph : I = I.

The inverse of a non-singular tensor A, denoted A−1, is defined by the identity A : A

−1 = I.

In component form, Aijmn(A−1)mnkl = Iijkl.

Assume that A is a rank-one update of I. Its inverse can be computed explicitly using

the Sherman-Morrison formula:

A = I + αT ⊗ U ; A−1 = I − α

1 + αT : UT ⊗ U (1.60)



where T ,U are arbitrary 2nd order tensors and α is an arbitrary scalar such that α 6=−1/[T : U ] (so that A is non-singular).

A straight-forward generalization of the formula in (1.60) is the following:

A = B + αT ⊗ U ; A−1 = B

−1 − α

1 + αU : B−1 : T

B−1 : T ⊗ U : B

−1 (1.61)

where it is assumed that B is a non-singular tensor.


Upon making the ansatz A = A′ijkle

′i ⊗ e′

j ⊗ e′k ⊗ e′

l, we obtain

[e′m ⊗ e′

n] : A : [e′p ⊗ e′

q] = [e′m ⊗ e′

n] : [Aijklei ⊗ ej ⊗ ek ⊗ el] : [e′p ⊗ e′

q]

= [e′m · ei][e

′n · ej]Aijkl[ek · e′

p][el · e′q]

= MmiMnjAijklMpkMql

= [e′m ⊗ e′

n] : [A′ijkle

′i ⊗ e′

j ⊗ e′k ⊗ e′

l] : [e′p ⊗ e′

q]

= δmiδnjA′ijklδkpδlq = A′

mnpq (1.62)

Hence, we conclude that

A = Aijklei ⊗ ej ⊗ ek ⊗ el = A′ijkle

′i ⊗ e′

j ⊗ e′k ⊗ e′

l (1.63)

with

A′ijkl = MimMjnAmnpqMkpMlq (1.64)

Remark: The component relation (1.64) can not be simply expressed in matrix format.

2

1.4.5 Appendix: Voigt-matrix representation of 4th order ten-

sor transformation

Consider the linear transformation

σ = E : ε (1.65)

where σ and ε are symmetric 2nd order tensors2 and E is a 4th order tensor possessing

minor (but not necessarily major) symmetry. Due to the symmetry of σ and ε, they

2The mechanical interpretation of σ and ε are stress and strain, respectively.



have 6 independent components in, say, a given Cartesian coordinate system. Hence, it

is possible to consider (1.65) as a linear transformation from the 6-dimensional Euclidean

vector space onto itself.

The Voigt-matrix format is defined by the representation

σ =

σ11

σ22

σ33

σ23

σ13

σ12

, ε =

ε11

ε22

ε33

γ23

γ13

γ12

with γij = 2εij , i 6= j (1.66)

Remark: In terms of stresses and strains, γij are the engineering “shear strain” compo-

nents. 2

In this way the matrices σ and ε become energy-conjugated in the sense that σ : ε = σTε.

Due to the minor symmetry of E, it is possible to establish the component representation

of (1.65) in the Voigt-matrix format

σ = Eε (1.67)

which can be expanded as

σ11

σ22

σ33

σ23

σ13

σ12

=

E1111 E1122 E1133 E1123 E1113 E1112

E2211 E2222 E2233 E2223 E2213 E2212

E3311 E3322 E3333 E3323 E3313 E3312

E2311 E2322 E2333 E2323 E2313 E2312

E1311 E1322 E1333 E1323 E1313 E1312

E1211 E1222 E1233 E1223 E1213 E1212

ε11

ε22

ε33

γ23

γ13

γ12

(1.68)

It is noted that E is symmetrical if E possess major symmetry. Furthermore, if E is

invertible so that Cdef= E

−1 exists, then the matrix Cdef= E−1 exists.

Remark: The 9 × 9-matrix corresponding to the full 9-dimensional representation of

(1.65) is not invertible. Show this as homework! 2

The Voigt-matrix format of the inverse relation is

ε = Cσ (1.69)



which can be expanded as

ε11

ε22

ε33

γ23

γ13

γ12

=

C1111 C1122 C1133 2C1123 2C1113 2C1112

C2211 C2222 C2233 2C2223 2C2213 2C2212

C3311 C3322 C3333 2C3323 2C3313 2C3312

2C2311 2C2322 2C2333 4C2323 4C2313 4C2312

2C1311 2C1322 2C1333 4C1323 4C1313 4C1312

2C1211 2C1222 2C1233 4C1223 4C1213 4C1212

σ11

σ22

σ33

σ23

σ13

σ12

(1.70)

For example, we note that

(C)11 = C1111 , (C)14 = 2C1123 , (C)44 = 4C2323 (1.71)

Remark: Another possibility to define ε and σ, although less common in practice, is

σ =

σ11

σ22

σ33√2 σ23√2 σ13√2 σ12

, ε =

ε11

ε22

ε33√2 ε23√2 ε13√2 ε12

(1.72)

This choice leads to other definitions of E and C in terms of the tensor components of E

and C. Establish the relevant expressions of E and C in this case as homework! 2

1.5 Permutation tensor (symbol) and its usage

The permutation tensor e is the 3rd order tensor

e = eijkei ⊗ ej ⊗ ek (1.73)

where eijk is the permutation symbol defined as

eijk =

1 if (i, j, k) is a cyclic permutation of (1, 2, 3)

−1 if (i, j, k) is an anti-cyclic permutation of (1, 2, 3)

0 if (i, j, k) is no permutation of (1, 2, 3)

(1.74)

Remark: That (i, j, k) is a permutation means that i, j, k are all distinct. 2


1.5 Permutation tensor (symbol) and its usage 15

Remark: The permutation tensor is the only isotropic 3rd order tensor, i.e. the compo-

nents are invariants w.r.t. any Cartesian coordinate change. 2

The following relations hold:

e : e = 2I, e · e = 2Iskw (1.75)

e is useful in expressing the vector product of two vectors u and w, which is another

vector symbolically denoted u × v. This vector can be expanded as

u × v = [u2v3 − u3v2]e1 + [u3v1 − u1v3]e2 + [u1v2 − u2v1]e3 (1.76)

Using e, we can express u × v as

u × v = e : [u ⊗ v] (1.77)

Show this as homework!

Likewise, e is useful in expressing the triple product [uvw]def= [u × v] · w, which is the

scalar quantity

[uvw] = [u2v3 − u3v2]w1 + [u3v1 − u1v3]w2 + [u1v2 − u2v1]w3 (1.78)

Using e, we can express [uvw] as

[uvw] = u · e : [v ⊗ w] (1.79)

Remark: [uvw] is invariant for any cyclic permutation of (u, v, w). 2

Finally, we define the axial vector w

w = axl(W )def= −1

2e : W or [wi] =

W32

W13

W21

(1.80)

where W is a skew-symmetric 2nd order tensor. Conversely, we may define the skew-

symmetric tensor W as

W = spn(w)def= −e · w or [Wij ] =

0 −w3 w2

w3 0 −w1

−w2 w1 0

(1.81)

for arbitrary w.



1.6 Spectral properties and invariants of a symmetric

2nd order tensor

1.6.1 Principal values - Spectral decomposition

Consider a 2nd order symmetric tensor A (such as the stress σ or the strain ε) with the

components Aij in a Cartesian coordinate system with base vectors ei. Eigenvalues Ai

and eigenvectors ei, for i = 1, 2, 3, are defined from

A · ei = Aiei (no summation on i) (1.82)

In component form we obtain

A11 A12 A13

A21 A22 A23

A31 A32 A33

(ei)1

(ei)2

(ei)3

= Ai

(ei)1

(ei)2

(ei)3

, i = 1, 2, 3 (1.83)

or

Akl(ei)l = Ai(ei)k (no summation on i) (1.84)

where (ei)j is the component of ei with respect to ej. For simplicity, we shall assume

henceforth that ei are unit vectors, i.e. |ei| = 1. Since Aij is a symmetric matrix, it

follows that Ai are real.

Since (1.82) is valid regardless of the chosen coordinate system to represent components,

given in (1.83), it follows that both Ai and ei are invariants, i.e. they do not change at

coordinate transformation. In the following, Ai will be denoted spectral invariants, and

all other invariants can be expressed in terms of the spectral invariants.

Let us next make a coordinate transformation to the Cartesian coordinate system associ-

ated with the principal base vectors ei. The components of A in this system are denoted

Aij. The coordinate transformation is defined by

ei = Mijej, Mij = cos (ei, ej) = (ei)j (1.85)

from which it follows that

Aij = MikAklMjl = (ei)kAkl(ej)l (1.86)

Inserting (1.84) into (1.86), we obtain the component identity

Aij = (ei)kAj(ej)k = Aj ei · ej︸︷︷︸

δij

= Aiδij (no summation on i) (1.87)


1.6 Spectral properties and invariants of a symmetric 2nd order tensor 17

or, more explicitly,

A11 A12 A13

A21 A22 A23

A31 A32 A33

=

A1 0 0

0 A2 0

0 0 A3

(1.88)

As an example of spectral invariants, we may consider the stress tensor σ with principal

stresses σi, i = 1, 2, 3, as shown in Figure 1.2.PSfrag replacements

σ2

σ1

σ22

σ12

σ11

e1

e1

e2

e2

Figure 1.2: Transformation to principal coordinates of the stress tensor

Spectral representation

Upon introducing the eigendyad midef= ei ⊗ ei, it follows directly from (1.88) that A can

be represented by the spectral decomposition:

A =3∑

i=1

Aimi (= A1e1 ⊗ e1 + A2e2 ⊗ e2 + A3e3 ⊗ e3) (1.89)

We may also check that A is represented correctly w.r.t. the arbitrary Cartesian basis ei.

To this end, we first conclude from (1.85) that mi has the dyadic expansion:

mi = ei ⊗ ei = MikMilek ⊗ el (no summation on i) (1.90)

Now, from (1.89) follows that

Akl = MikAijMjl =3∑

i=1

AiMikMil (1.91)



Upon combining (1.89) with (1.90) and (1.91), we obtain

A =3∑

i=1

AiMikMilek ⊗ el = Aklek ⊗ el = Aijei ⊗ ej (1.92)

Remark: The component representation of mi w.r.t. the principal basis ei is:

[(m1)ij] =

1 0 0

0 0 0

0 0 0

, [(m2)ij] =

0 0 0

0 1 0

0 0 0

, [(m3)ij] =

0 0 0

0 0 0

0 0 1

(1.93)

whereas the component representation w.r.t. ei is:

[(mk)ij] = [(ek)i(ek)j] = [MkiMkj] =

Mk1Mk1 Mk1Mk2 Mk1Mk3



2 (1.94)

It is possible to generalize (1.89) to

An =3∑

i=1

[Ai]nmi and f(A)

def=

3∑

i=1

f(Ai)mi (1.95)

where f is an isotropic tensor-valued function of A. In particular, we may choose n = 0

to obtain

I =3∑

i=1

mi (1.96)

1.6.2 Basic invariants

The basic invariants of A are denoted i1, i2, i3, and are defined as 3

i1 = I : A = tr(A) =3∑

i=1

Ai (1.97)

i2 = I : A2 = tr(A2) = |A|2 =3∑

i=1

[Ai]2 (1.98)

i3 = I : A3 = tr(A3) =3∑

i=1

[Ai]3 (1.99)

3Cartesian coordinates: i1 = Akk, i2 = (A2)kk = AkiAik = |A|2 = A : A, i3 = (A3)kk =

AkiAijAjk.



In order to obtain the last equality in (1.98) to (1.99), we conveniently use the spectral

decomposition in (1.89).

Alternatively, we may define invariants of the deviator Adev via the decomposition

A = Adev +1

3i1I (1.100)

or, in principal coordinates,

(Adev)i = Ai − Am with Am =1

3i1 (1.101)

Remark: It follows that Adev and A have the same principal directions (since they differ

by the isotropic tensor AmI). 2

We then obtain

i1 = I : [Adev +1

3i1I] = j1 + i1 with j1 = I : Adev = 0 (1.102)

i2 = I :

[

[Adev]2 +

1

9[i1]

2I

]

= j2 +1

3[i1]

2 with j2 = I : [Adev]2 (1.103)

i3 = I :

[

[Adev]3 +

1

27[i1]

2I

]

= j3 +1

9[i1]

3 with j3 = I : [Adev]3 (1.104)

Hence, we may choose i1, j2, j3 as an alternative set of basic invariants.

Remark: We define the trace of Am, for m = 1, 2, . . ., by

im = I : Am = tr(Am) =3∑

i=1

[Ai]m (1.105)

However, im for i ≥ 4 are not independent since they can be expressed in terms of i1, i2, i3

using the Cayley-Hamilton’s theorem, see below. 2

1.6.3 Principal invariants - Cayley-Hamilton’s theorem

Consider next the characteristic equation for determining Ai expressed in the principal

coordinates, i.e.

det(Aij − Aδij) = 0 (1.106)

which may be expanded as

[A1 − A][A2 − A][A3 − A] = 0 or A3 − I1A2 − I2A− I3 = 0 (1.107)



The characteristic invariants I1, I2 and I3 are defined as

I1 = i1 = A1 + A2 + A3

I2 =1

2

[i2 − [i1]

2]

= −[A2A3 + A3A1 + A1A2] (1.108)

I3 = det(A) = A1A2A3

Since I1, I2 and I3 are coefficients of the characteristic equation, it is clear that they are

indeed invariants.

Since (1.107)2 must be satisfied by each Ai, we obtain the equation

A13 0 0

0 A23 0

0 0 A33

− I1

A12 0 0

0 A22 0

0 0 A32

− I2

A1 0 0

0 A2 0

0 0 A3

− I3

1 0 0

0 1 0

0 0 1

= 0

(1.109)

which is the representation in the principal coordinates of the tensor equation

A3 − I1A2 − I2A − I3I = 0 (1.110)

This is the Cayley-Hamilton theorem, which can be used in order to obtain an alternative

expression for I3. Taking the trace of (1.110), we obtain

i3 − I1i2 − I2i1 − 3I3 = 0 (1.111)

where it was used that trI = 3. Solving for I3 from (1.111), we obtain

I3 =1

3[i3 − I1i2 − I2i1] =

1

6[2i3 − 3i2i2 + [i1]

3] (1.112)

Alternatively, the principal invariants may be expressed in terms of the deviator Adev. As

the set of independent generic invariants we may choose I1, J2 and J3 defined as 4

I1 = i1 = A1 + A2 + A3

J2 =1

2j2 =

1

2

[[(Adev)1]

2 + [(Adev)2]2 + [(Adev)3]

2]

(1.113)

J3 = det(Adev) =1

3j3 =

1

3

[[(Adev)1]

3 + [(Adev)2]3 + [(Adev)3]

3]

The expressions for J2 and J3 follow from (1.108) and (1.112) when it is used that J1 =

j1 = tr(Adev) = 0.

4Cartesian components: J2 = 12 (Adev)ij(Adev)ij , J3 = 1

3 (Adev)ik(Adev)kj(Adev)ji.



1.6.4 Octahedral invariants of the stress and strain tensors

The traction vector (stress vector) s on a given plane defined by the normal n can be

expressed as

s = σ · n = [σdev + σmI] · n = σdev · n + σmn (1.114)

The normal stress σ is defined, with (1.114), as

σdef= s · n = n · σ · n = n · σdev · n + σm (1.115)

whereas the shear stress τ is defined, with (1.114) and (1.115), as

τ 2 def= |s|2 − σ2 = n · σ2 · n − σ2 = n · σ2

dev · n − [n · σdev · n]2 (1.116)

The octahedral plane is the physical plane that is defined by the normal vector n with

components ni = 1/√

3 with respect to the principal coordinate axes ei. This plane is

shown in Figure 1.3.

PSfrag replacements e1

e2

e3

tn

s

st = τoctt

sn = σoctn

n = 1√3[1, 1, 1]

Figure 1.3: Octahedral plane in physical space defined by [ni] = [1, 1, 1]/√

3 in principal

coordinates

The octahedral normal stress σoct and the octahedral shear stress τoct are magnitudes of

the normal and tangential components of the traction vector s = σ · n on the octahedral

plane. First we note that n · σdev · n = 0. Hence, we may use (1.115) and (1.116) to

express σoct and τoct as

σoctdef= s · n = σm =

1

3[σ1 + σ2 + σ3] =

1

3i1 (1.117)



τ 2oct

def= |s|2 − σ2

oct = n · σ2 · n − σ2oct =

1

3

[[σdev,1]

2 + [σdev,2]2 + [σdev,3]

2]

=1

3j2 (1.118)

The octahedral stresses are thus related to the mean stress σm and the equivalent stress

σe (associated with the von Mises yield criterion) as follows:

σoct = σm, τoct =

√2

3σe =

1√3|σdev| with σe =

√

3

2|σdev| (1.119)

Remark: In the literature on granular materials, it is common to use the notation

p = −σm and q = σe. Hence, the pairs (σoct, τoct), (σm, σe), (ξ, ρ) and (p, q) are all equiv-

alent pairs of invariants, since they differ only by scaling factors. 2

In the special case of a uniaxial stress state (σ1 = σ, σ2 = σ3 = 0) we obtain q = σe = σ.

We may also define, in similar fashion, the octahedral normal strain εoct and the octahedral

shear strain γoct as follows:

εoctdef= n · ε · n =

1

3[ε1 + ε2 + ε3] =

1

3i1 (1.120)

γ2oct

def= n · ε2 · n − ε2oct =

1

3[ε21 + ε22 + ε23] − ε2oct (1.121)

The octahedral strains are related to the volumetric strain εvol and the equivalent strain

εe as follows:

εoct =1

3εvol, γoct =

1√2εe =

1√3|εdev| with εe =

√

2

3|εdev| (1.122)

1.6.5 Derivatives of a 2nd order tensor

For a 2nd order symmetric tensor A, we have the derivatives

∂A

∂A= I

sym ,∂Asph

∂A= Isph ,

∂Adev

∂A= I

symdev (1.123)

∂Ani

∂A= nAn−1

i mi ,∂An : I

∂A= nAn−1 (1.124)

In order to show (1.124)1, we first use the chain rule to obtain

∂Ani

∂A= nAn−1

i

∂Ai

∂A(1.125)



We then use the definition of Ai (and ei), i.e.

[A − AiI] · ei = 0 , i = 1, 2, 3 (in 3D) (1.126)

and differentiate this equation to obtain

[dA − dAiI] · ei + [A − AiI] · dei = 0 (1.127)

Upon premultiplying (1.127) with ei and using (1.126), we obtain

dAi = ei · dA · ei = [ei ⊗ ei] : dA = mi : dA ;∂Ai

∂A= mi (1.128)

In order to show (1.124)2, it suffices to use (1.124)1 together with

An : I =3∑

i=1

Ani ,

3∑

i=1

Ani mi = An (1.129)

to obtain∂An : I

∂A=

3∑

i=1

nAn−1i mi = nAn−1 (1.130)

We shall also need the following derivatives:

∂A2

∂A= 2PAI =

1

2[A⊗I + A⊗I + I⊗A + I⊗A] (1.131)

∂A−1

∂A= −PA−1A−1 = −1

2[A−1⊗A−1 + A−1⊗A−1] (1.132)

where we introduced the symmetric 4th order projection tensor PAB, which is composed

of the two symmetric 2nd order tensors A and B:

PAB =1

4[A⊗B + A⊗B + B⊗A + B⊗A] (1.133)

Remark: It appears that I = PII. 2

In order to show (1.131), we first use the chain rule to obtain

dA2 = A · dA + dA · A = A · dA · I + I · dA · A= [A⊗I + I⊗A] : dA =

1

2[A⊗I + A⊗I + I⊗A + I⊗A] : dA

= 2PAI : dA (1.134)

where we used the symmetry of A in order to obtain the next to last equality.



To show (1.132), we first differentiate the identity A · A−1 = I to obtain

dA−1 = −A−1 · dA · A−1 = −[A−1⊗A−1] : dA

= −1

2[A−1⊗A−1 + A−1⊗A−1] : dA

= −PA−1A−1 : dA (1.135)

where we, once again, used the symmetry of A (or rather A−1) to obtain the next to last

equality.

Finally, we note the useful result

∂|Adev|∂A

=Adev

|Adev|(1.136)

Since j2 = I : [Adev]2 = |Adev|2 and j2 = Adev : Adev, we have

dj2 = 2|Adev|∂|Adev|∂A

: dA and dj2 = 2Adev : dAdev = 2Adev : dA (1.137)

Eliminating between (1.137)1 and (1.137)2 gives (1.136).

Recalling the definition of equivalent stress σe in (1.119), we thus obtain

∂σe

∂σ=

√

3

2

σdev

|σdev|=

3

2σe

σdev (1.138)

1.6.6 Derivatives of invariants, etc.

For the generic 2nd order symmetric tensor A, we summarize the spectral, basic and

principal invariants as the sets

Is = {A1, A2, A3}, Is,dev = {(Adev)1, (Adev)2, (Adev)3} (1.139)

Ib = {i1, i2, i3}, Ib,dev = {i1, j2, j3} (1.140)

Ip = {I1, I2, I3}, Ip,dev = {I1, J2, J3} (1.141)

Subsequently, we shall need the derivatives of all these invariants w.r.t. A. They are

listed as follows:∂Ai

∂A= mi (1.142)

∂i1∂A

= I,∂i2∂A

= 2A,∂i3∂A

= 3A2,∂j2∂A

= 2Adev,∂j3∂A

= 3[Adev]2 − j2I (1.143)



∂I1∂A

= I,∂I2∂A

= A− I1I,∂I3∂A

= A2 − I1A− I2I,∂J2

∂A= Adev,

∂J3

∂A= [Adev]

2 − 2

3J2I

(1.144)

Alternative results for ∂I3/∂A and ∂J3/∂A are

∂I3∂A

= I3A−1,

∂J3

∂A= J3[Adev]

−1 +1

3J2I (1.145)

We first note that (1.142) is the special case of (1.124)1 when n = 1.

The expressions in (1.143) are obtained upon using the definitions of i1, j2, j3 in (1.97),

(1.103)2, and (1.104)2, the spectral representation of A in (1.95) and the result in (1.142).

The expressions in (1.144) are obtained by using the identities in (1.108) and the results

in (1.143). To show the results in (1.145) is left as homework for the reader. Hint: Use

the Cayley-Hamilton theorem in (1.110).

1.6.7 Representation of eigendyads

The eigendyads mi can be given a remarkably simple explicit representation that does

not require the knowledge of ei. This representation is known as Serrin’s formula, cf.

Ting (1985):

Lemma: If Ai are distinct, i.e. A1 6= A2 6= A3, then

mi =1

di

3∏

l=1/i

[A − AlI]

=1

di

[A2 − [I1 − Ai]A + I3[Ai]

−1I]

=Ai

di

[A − [I1 − Ai]I + I3[Ai]

−1A−1]

(1.146)

where di is defined as

didef=

3∏

l=1/i

[Ai − Al] = [Ai − Aj][Ai − Ak] = 2[Ai]2 − I1Ai + I3[Ai]

−1 (1.147)

and where the set of indices i, j, k are an even permutation of 1,2,3. That (Ai, Aj, Ak)

must be distinct for (1.146) to be valid follows directly from the necessary condition that

di 6= 0 in (1.147).



Proof: From the spectral decompositions of A and I, we conclude that

A − AiI =3∑

l=1

[Al − Ai]ml , i = 1, 2, 3 (1.148)

Upon using (1.148), we now obtain

3∏

l=1/i

[A − AlI] = [A − AjI] · [A − AkI]

=3∑

l=1

[Al − AjI]ml ·3∑

m=1

[Am − Ak]mk =3∑

l=1

[Al − Aj][Al − Ak]ml

= [Ai − Aj][Ai − Ak]mi =∏

l=1/i

[Ai − Al]mi = dimi

(1.149)

by which

mi =1

di

3∏

l=1/i

[A − AlI] (1.150)

Moreover, we may expand

3∏

l=1/i

[A − AlI] = [A − AjI] · [A − AkI] = A2 − AkA − AjA + AjAkI

= A2 − [I1 − Ai]A + I3[Ai]−1I (1.151)

and we have proved the first two rows of (1.146). Finally, premultiplying this expression

with A−1, while using (1.82), we obtain the alternative expression

mi =Ai

di

[A − [I1 − Ai]I + I3[Ai]

−1A−1]

(1.152)

which completes the proof. 2

We shall also need the quadrad Mi, defined as

Midef=∂mi

∂A(1.153)

The following representations of Mi are essentially due to Simo & Taylor (1992):


1.7 Representation theorems 27

Lemma: In the case that Ai are distinct, i.e. A1 6= A2 6= A3, then

Mi =Ai

di

[Isym − I ⊗ I − I3[Ai]

−1[PA−1A−1 − A−1 ⊗ A−1] + I ⊗ mi + mi ⊗ I

−I3[Ai]−2[A−1 ⊗ mi + mi ⊗ A−1] + 2[I3[Ai]

−3 − 1]mi ⊗ mi

](1.154)

Proof: This expression is associated with the second expression for mi in (1.146). The

result follows directly upon using the derivatives of the invariants Ai, I1 and I3 and the

derivatives of A2 and A−1, as given in the previous Subsections. 2

We remark that if two, or all, principal values Ai are equal, then (1.146) and (1.154) must

be replaced by other expressions accordingly, as shown by Simo (1991). In practice it is

possible to use (1.146) and (1.154) even in such a case, if Ai are perturbed by a small

amount so that they become distinct. However, extreme care must be exercised in order

not to run into numerical difficulties.

Finally in this Subsection, we note the following identities:

A−1 =3∑

i=1

[Ai]−1mi ⇒ ∂A−1

∂A= −PA−1A−1 =

3∑

i=1

[−[Ai]

−2mi ⊗ mi + [Ai]−1

Mi

]

(1.155)

I =3∑

i=1

mi ⇒ 0 =∂I

∂A=

3∑

i=1

Mi (1.156)

A =3∑

i=1

Aimi ⇒ ∂A

∂A= I

sym =3∑

i=1

[mi ⊗ mi + AiMi] (1.157)

1.7 Representation theorems

1.7.1 Coordinate transformation vs. vector rotation

Consider a coordinate transformation e′i = Mikek and a vector u = uiei. A new vector

u′ is obtained by rotation of u if it can be written as u′ = uie′i, i.e. the rotation leaves

the components of u unchanged w.r.t. a corotating coordinate system. We obtain

u′ def= uie

′i = uiMikek = [Mlkek ⊗ el] · [uiei] = Q · u (1.158)

where

Q = Qijei ⊗ ej with Qij = Mji = e′j · ei (1.159)



The rotation tensor Q can, alternatively, be expressed as

Q = [e′j · ei]ei ⊗ ej = e′

j · [ei ⊗ ei] ⊗ ej = e′j · I ⊗ ej = e′

j ⊗ ej (1.160)

where it was used that I = δikei ⊗ ek = ei ⊗ ei.

Remark: In matrix form, (1.159)2 reads Q = MT. Hence, (1.158)2 can be rewritten in

terms of the components on matrix form as

u′ = Qu = MTu (1.161)

As compared with the coordinate transformation in (1.16) we note the following important

distinction:

• In (1.16): u′ are the components of a vector u in new coordinates defined by rotating

the base vectors ei to e′i = Qt · ei, whereby u are the components of u w.r.t ei.

• In (1.161): u′ are the components w.r.t. ei of a new vector u′ obtained by rotating

the vector u to u′ = Q · u, whereby u are the components of u w.r.t. ei.

Coordinate transformation and vector rotation must not be confused! 2

It follows directly that Q is orthonormal, since

Qt · Q = MikMjkei ⊗ ej = δijei ⊗ ej = I ⇒ Qt = Q−1 (1.162)

Likewise, for a given tensor T the rotated tensor T ′ is obtained as

T ′ def= Tije

′i ⊗ e′

j = TijMikek ⊗Mjlel = MikTijMjlek ⊗ el = Q · T · Qt (1.163)

It is noted that the rotation leaves the length unchanged, i.e. |T ′| = |T |. Verify this

property as homework.

Remark: As an alternative, we may define the 4th order rotation tensor Qdef= Q⊗Q5,

by which T ′ = Q : T . It appears that Q possesses minor, but not major, symmetry. 2

We now turn to representation theorems, which are given without proof. The interested

reader is referred to Spencer (1980).

5Cartesian coordinates (Q)abcd = QacQbd



1.7.2 Scalar-valued isotropic tensor functions of one argument

A scalar-valued function Φ(A) of one symmetric 2nd order tensor argument A is isotropic

(or invariant) if it satisfies the invariance condition

Φ(A) = Φ(Q · A · Qt) ∀Q ∈ SO(3) (1.164)

where SO(3) is the set of all possible proper rotations (proper meaning that det(Q = 1).

The form in which A can occur as argument of Φ is called the representation, which in

the present case is defined by the following representation theorem:

Φ(A) = Φ(I(A)) (1.165)

where the irreducible set of invariants I(A) can be chosen as any of those previously

discussed in Section 1.6. For example, we may choose the basic invariants defined as Ib:

I(A) = {i1, i2, i3} = {I : A, I : A2, I : A3} (1.166)

It is common in many applications, such as yield and failure functions, to choose the

spectral invariant set Is. For this choice it can be shown that Φ is symmetrical in its

arguments, i.e.

Φ(A1, A2, A3) = Φ(Ai, Aj, Ak) (1.167)

for any permutation (i, j, k) of (1,2,3).

1.7.3 Scalar-valued isotropic tensor functions of two arguments

A scalar-valued function Φ(A1,A2) of two symmetric 2nd order tensor arguments A1 and

A2 is isotropic if it satisfies the condition

Φ(A1,A2) = Φ(Q · A1 · Qt,Q · A2 · Qt) ∀Q ∈ SO(3) (1.168)

The corresponding representation theorem is:

Φ(A1,A2) = Φ(I(A1), I(A2), I(A1,A2)) (1.169)

where the functional basis is composed of the following irreducible sets of invariants:

I(Ai) = {I : Ai, I : [Ai]2, I : [Ai]

3}, i = 1, 2 (1.170)

I(A1,A2) = {A1 : A2, A1 : [A2]2, [A1]

2 : A2, [A1]2 : [A2]

2} (1.171)



1.7.4 Scalar-valued isotropic tensor functions of three arguments

A scalar-valued function Φ(A1,A2,A3) of three symmetric 2nd order tensor arguments

A1, A2 and A3 is isotropic if it satisfies the condition

Φ(A1,A2,A3) = Φ(Q · A1 · Qt,Q · A2 · Qt,Q · A3 · Qt) ∀Q ∈ SO(3) (1.172)

The corresponding representation theorem is:

Φ(A1,A2,A3) = Φ(I(A1), I(A2), I(A3), I(A1,A2), I(A2,A3), I(A3,A1), I(A1,A2,A3))

(1.173)

where the functional basis is composed of the following irreducible sets of invariants

I(Ai) = {I : Ai, I : [Ai]2, I : [Ai]

3}, i = 1, 2, 3 (1.174)

I(Ai,Aj) = {Ai : Aj, Ai : [Aj]2, [Ai]

2 : Aj, [Ai]2 : [Aj]

2}, i = 1, 2, 3, i 6= j (1.175)

I(A1,A2,A3) = {I : [A1 · A2 · A3]} (1.176)

1.7.5 Symmetric tensor-valued isotropic tensor functions of one

argument

A tensor valued function, of symmetric 2nd order, T (A) of one symmetric 2nd order

tensor argument A is isotropic if it satisfies the invariant condition

T (A) = Qt · T (Q · A · Qt) · Q ∀Q ∈ SO(3) (1.177)

The corresponding representation theorem is, due to Rivlin and Ericksen (19),

T (A) =3∑

a=1

φa(I(A))Ga (1.178)

where the eigenbasis tensors (2nd order symmetric tensors) Ga belong to the irreducible

set of generators

Ga ∈ {G(0),G(A)} (1.179)

The irreducible set of generators are

G(0) = {I} (1.180)

G(A) = {A,A2} (1.181)

The irreducible set of invariants are the same as for the scalar-valued isotropic tensor

function in (1.166), e.g..

I(A) = {I : A, I : A2, I : A3} (1.182)



1.7.6 Symmetric tensor-valued isotropic tensor function of two

arguments

A tensor-valued function, of symmetric 2nd order, T (A1,A2) of two symmetric 2nd order

tensor arguments A1 and A2 is isotropic if it satisfies the condition

T (A1,A2) = Qt · T (Q · A1 · Qt,Q · A2 · Qt) · Q ∀Q ∈ SO(3) (1.183)

The corresponding representation theorem is

T (A1,A2) =8∑

a=1

φa(I(A1), I(A2), I(A1,A2))Ga (1.184)

where the eigenbasis tensors Ga belong to the irreducible set of eigenvectors

Ga ∈ {G(0),G(A1),G(A2),G(A1,A2)} (1.185)

The irreducible set of generators are

G(0) = {I} (1.186)

G(Ai) = {Ai, [Ai]2}, i = 1, 2 (1.187)

G(A1,A2) = {A1 · A2 + A2 · A1, [A1 · A2 · A1], [A2 · A1 · A2]} (1.188)

The irreducible set of invariants are the same as for the scalar-valued isotropic tensor

function in (1.170) and (1.171), e.g.

I(Ai) = {I : Ai, I : [Ai]2, I : [Ai]

3}, i = 1, 2 (1.189)

I(A1,A2) = {A1 : A2, A1 : [A2]2, [A1]

2 : A2, [A1]2 : [A2]

2} (1.190)


Chapter 7

ELASTICITY

In this Chapter we consider elastic response, which represents the conceptually simplest

class of material behavior. No dissipative mechanism is involved, i.e. the free energy

does not depend on any internal variables. Starting with the prototype model of linear

elasticity, we then extend the discussion to the general nonlinear (hyperelastic) format.

Certain widespread classes of nonlinear material response, including the total deformation

format of plasticity, can be obtained as special cases of the general theory. We then

turn to the general anisotropic response, which is represented using structure tensors of

2nd order. The special cases of orthogonal symmetry (orthotropy), trigonal symmetry

(transverse isotropy) and cubic symmetry are evaluated both in the symbolic format and

the Voigt matrix format.

7.1 Introduction

7.1.1 General characteristics of nonlinear elasticity

The elementary prototype for an elastic material is the Hookean spring. An elastic ma-

terial is defined by the absence of internal variables in the constitutive equations. Linear

elasticity is defined by a quadratic function Ψ in terms of the strain ε (under isothermal

conditions):

Ψ(ε) =1

2ε : E

e : ε with Ψ(ε)def= ρψ(ε) (7.1)


166 7 ELASTICITY

Here, Ee is the 4th order constant elastic stiffness modulus tensor, which possesses both

(1st and 2nd) minor and major symmetry. Moreover, Ee is positive definite, defined as

ε : Ee : ε > 0 ∀ε 6= 0. We thus obtain

σ =∂Ψ

∂ε= E

e : ε (7.2)

Hyperelasticity (nonlinear elastic response) is defined by the direct generalization of the

free energy from a quadratic form in ε for linear elasticity to a strictly convex, but

otherwise general, function Ψ(ε). We thus obtain the stress σ(ε) as a nonlinear function

of ε from the constitutive relation

σ =∂Ψ

∂ε(7.3)

The rate format (or linearized format), corresponding to the total format in (7.3), is

defined as follows:

σ = Ee : ε ; E

e =∂σ

∂ε=

∂2Ψ

∂ε ⊗ ∂ε(7.4)

where Ee(ε) is the Elastic Continuum Tangent Stiffness (ECTS) tensor pertinent to elastic

response1.

Remark: Ee must be positive definite in order for Ψ to be strictly convex. If the domain

of definition is unbounded (in strain space), then it follows that a natural stress state

exists, i.e. it is certainly possible to “shift the origin” of the strain space in such a fashion

that σ = 0 for ε = 0. 2

The following consequences follow from the simple potential character of Ψ:

• Path-independence: Ψ serves as a strain energy potential. Hence, for two different

strain states ε0 and ε1, we have

Ψ(ε1) − Ψ(ε0) =

∫ ε1

ε0

∂Ψ

∂ε′: dε′ =

∫ ε1

ε0

σ(ε′) : dε′ (7.5)

This difference in strain energy does not depend on the strain path that is traced

at loading, as shown in Figure 7.1.

• No dissipation for closed path:

ε1 = ε0 ⇒ Ψ(ε1) = Ψ(ε0) ⇒∮

σ(ε′) : dε′ = 0 (7.6)

1Cartesian components: (Ee)abcd = ∂2Ψ∂εab∂εcd

.


7.1 Introduction 167

PSfrag replacements

ε2

ε1

ε0[σ0]

ε1[σ1]

Figure 7.1: Path-independent strain energy.

• Major symmetry of ECTS-tensor: Since ε and σ are symmetric tensors, i.e. εT = ε

and σT = σ, it is concluded that Ee possesses both 1st and 2nd minor symmetries2.

As a consequence, the number of independent entries of Ee is reduced from 92 = 81 to

62 = 36. Moreover, since Ψ is a potential for σ it follows that Ee must satisfy major

symmetry defined as [Ee]T = Ee 3. As a consequence, the number of independent

entries of Ee is further reduced from 36 to 21.

7.1.2 Material symmetry - Isotropy

The functional relation σ(ε), as obtained from (7.3), is subjected to certain restrictions

due to existing material symmetry conditions. Symmetry is the invariance of the con-

stitutive relations under a given set of rotations of the reference configuration. For a

hyperelastic material response this invariance means that Ψ remains unchanged under

the rotation, which condition will reduce the number of independent entries in Ee. For

example, in the case of linear elasticity, when Ee is a constant tensor, this reduction is

directly reflected by a reduced number of (constant) material parameters, which will be

shown explicitly below. The most restrictive form of symmetry is isotropy, in which case

Ψ remains unchanged for all possible rotations. In other words, in the case of isotropy the

material properties are the same in all directions.

2Cartesian coordinates: (Ee)jikl = (Ee)ijlk = (Ee)ijkl3Cartesian coordinates: (Ee)klij = (Ee)ijkl


168 7 ELASTICITY

These rather abstract statements will be substantiated subsequently.

7.1.3 Appendix: Voigt-matrix representation of tangent rela-

tions

The CTS-tensor Ee can be interpreted as a symmetric linear transformation from the

6-dimensional Euclidean space on itself in the general case of 3-dimensional stress and

strain states, whereby the Cartesian components of ε and σ span 6-dimensional vector

spaces, cf. Subsection 1.4.5.

Let us consider linear elasticity, whereby Ee is a constant tensor and σ = E

e : ε. In the

6-dimensional vector space, we may then represent this relation as the component relation

σ = Eeε in the Voigt-matrix notation:

σ11

σ22

σ33

σ23

σ13

σ12

=

Ee1111 Ee

1122 · · · Ee1112

Ee1122

· ·· ·· ·

Ee1112 · · · · Ee

1212

ε11

ε22

ε33

γ23

γ13

γ12

(7.7)

where γij = 2εij , i 6= j, are the engineering “shear strain” components. The compliance

relation ε = Ce : σ, with C

e def= [Ee]−1, is represented as ε = Ceσ in matrix form:

ε11

ε22

ε33

γ23

γ13

γ12

=

Ce1111 Ce

1122 · · · 2Ce1112

Ce1122

···

2Ce1112 · · · · 4Ce

1212

σ11

σ22

σ33

σ23

σ13

σ12

(7.8)

From the symmetry properties it follows that, in the most general case of complete

anisotropy, Ee and C

e are defined by 21 independent parameters.

Dimensional reduction is obtained in the two special cases of plane strain and plane stress:


7.2 Constitutive relations - Isotropic nonlinear elasticity 169

Special case: Plane strain

The condition ε33 = ε23 = ε13 = 0 gives the special case of the general stiffness relation

(7.7):

σ11

σ22

σ12

=

Ee1111 Ee

1122 Ee1112

Ee1122 Ee

2222 Ee2212

Ee1112 Ee

2212 Ee1212

ε11

ε22

γ12

(7.9)

The transversal (out-of-plane) stresses are given as

σ33

σ23

σ13

=

Ee1133 Ee

2233 Ee3312

Ee1123 Ee

2223 Ee1223

Ee1113 Ee

2213 Ee1213

ε11

ε22

γ12

(7.10)

Special case: Plane stress

The condition σ33 = σ23 = σ13 = 0 gives the special case of the general compliance relation

(7.8):

ε11

ε22

γ12

=

Ce1111 Ce

1122 2Ce1112

Ce1122 Ce

2222 2Ce2212

2Ce1112 2Ce

2212 4Ce1212

σ11

σ22

σ12

(7.11)

The transversal (out-of-plane) stresses are given as

ε33

γ23

γ13

=

Ce1133 Ce

2233 2Ce3312

2Ce1123 2Ce

2223 4Ce1223

2Ce1113 2Ce

2213 4Ce1213

σ11

σ22

σ12

(7.12)

7.2 Constitutive relations - Isotropic nonlinear elas-

ticity

7.2.1 Generic format of free energy

In the case of complete isotropy, then Ψ(ε) belongs to the class of scalar-valued isotropic

(or invariant) functions of the single argument ε. This means that we may choose any

irreducible set of invariants (of ε) as the arguments of Ψ, e.g. Is , Ib or Ip (see Chapter


170 7 ELASTICITY

1). Using the results regarding the derivatives of the invariants taken w.r.t. ε, we may

use the chain rule to obtain the following results:

For Ψ(Is(ε)) → Ψ(ε1, ε2, ε3):

σ =3∑

i=1

σim(ε)i with σi =

∂Ψ

∂εi(7.13)

For Ψ(Ib(ε)) → Ψ(i1, i2, i3):

σ =∂Ψ

∂i1I + 2

∂Ψ

∂i2ε + 3

∂Ψ

∂i3ε2 (7.14)

For Ψ(Ip(ε)) → Ψ(I1, I2, I3):

σ =∂Ψ

∂I1I +

∂Ψ

∂I2[ε − I1I] +

∂Ψ

∂I3[ε2 − I1ε + I2I]

=

[∂Ψ

∂I1− I1

∂Ψ

∂I2+ I2

∂Ψ

∂I3

]

I +

[∂Ψ

∂I2− I1

∂Ψ

∂I3

]

ε +∂Ψ

∂I3ε2 (7.15)

The expressions for σ in (7.14) and (7.15) can be summarized as

σ = Φ0I + Φ1ε + Φ2ε2 =

3∑

i=1

σim(ε)i with σi = Φ0 + Φ1εi + Φ2[εi]

2 (7.16)

which shows the formal equivalence between (7.13), (7.14) and (7.15). The appropriate

expressions of Φi are defined by comparison with (7.14) and (7.15). Clearly, Φi are scalar

isotropic invariant functions of ε.

7.2.2 Generic format of Continuum Tangent Stiffness tensor

The CTS-tensor Ee can be obtained explicitly for any choice of invariant representation:

For Ψ(ε1, ε2, ε3):

Ee =

3∑

i=1

3∑

j=1

∂2Ψ

∂εi∂εjm

(ε)i ⊗ m

(ε)j +

3∑

i=1

σiM(ε)i (7.17)

where the fully symmetrical 4th order tensor M(ε)i

def= ∂m

(ε)i /∂ε was given in (1.110). It is

noted that∂2Ψ

∂εi∂εj=∂σi

∂εj=∂σj

∂εi(7.18)



For Ψ(i1, i2, i3):

Ee = I ⊗

[∂2Ψ

∂i1∂i1I + 2

∂2Ψ

∂i1∂i2ε + 3

∂2Ψ

∂i1∂i3ε2

]

+

ε ⊗[

2∂2Ψ

∂i2∂i1I + 4

∂2Ψ

∂i2∂i2ε + 6

∂2Ψ

∂i2∂i3ε2

]

+

ε2 ⊗[

3∂2Ψ

∂i3∂i1I + 6

∂2Ψ

∂i2∂i3ε + 9

∂2Ψ

∂i3∂i3ε2

]

+

Φ1Isym + 2Φ2PεI (7.19)

where

PεI =1

4[ε⊗I + ε⊗I + I⊗ε + I⊗ε] (7.20)

The corresponding expression of Ee for Ψ(I1, I2, I3) is left as homework to the interested

reader.

It turns out that Ee for both Ψ(i1, i2, i3) and Ψ(I1, I2, I3) can be summarized as follows:

Ee = ϕ00I ⊗ I + ϕ01[I ⊗ ε + ε ⊗ I] + ϕ02[I ⊗ ε2 + ε2 ⊗ I] +

ϕ11ε ⊗ ε + ϕ12[ε ⊗ ε2 + ε2 ⊗ ε] + ϕ22ε2 ⊗ ε2 +

Φ1Isym + 2Φ2PεI (7.21)

where the coefficients ϕij are scalar invariant functions, e.g. ϕij(i1, i2, i3).

Remark: The major symmetry of Ee is clearly visible in (7.21). 2

7.2.3 Volumetric/deviatoric decomposition of the free energy

We shall next consider the special case that it is possible to decompose Ψ additively into

purely volumetric and deviatoric parts, as follows:

Ψ(i1, i2, i3) = Ψvol(i1) + Ψdev(j2, j3) (7.22)

This gives

σ =dΨvol

di1I + 2

∂Ψdev

∂j2εdev +

∂Ψdev

∂j3

[3[εdev]

2 − j2I]

(7.23)

whereby the mean stress σm is obtained as

σmdev=

1

3I : σ =

dΨvol

di1(7.24)


172 7 ELASTICITY

whereas the deviator stress σdev becomes

σdev = σ − σmI = 2∂Ψdev

∂j2εdev +

∂Ψdev

∂j3

[3[εdev]

2 − j2I]

(7.25)

Hence, a complete decoupling of the volumetric and the deviatoric responses has been

obtained. We may introduce further simplification by dropping the influence of the third

invariant, i.e. ∂Ψdev/∂j3 = 0, by which (7.24, 7.25) give

σm =dΨvol

di1, σdev = 2

dΨdev

dj2εdev (7.26)

A particularly useful representation of Ψ is that which employs the octahedral strains εoct

and γoct, defined as

εoct =1

3i1, γoct =

1√3[j2]

12 ⇒ ∂εoct

∂ε=

1

3I,

∂γoct

∂ε=

1

3

εdev

γoct

(7.27)

Clearly, εoct and γoct can be chosen to replace i1 and j2, respectively, as arguments in Ψ.

Hence, assuming Ψvol (εoct) and Ψdev (γoct), we obtain from (7.26) using the chain rule of

differentiation, the secant relations:

σoct =1

3

dΨvol

dεoct

= 3Ks(εoct)εoct with Ks(εoct)def=

1

9εoct

dΨvol

dεoct

(7.28)

σdev =1

3γoct

dΨdev

dγoct

εdev = 2Gs(γoct)εdev with Gs(γoct)def=

1

6γoct

dΨdev

dγoct

(7.29)

where Ks(εoct) and Gs(γoct) are the secant moduli that are completely uncoupled. We

may rewrite (7.28) and (7.29) in terms of the octahedral stresses as

σoct = fσ(εoct) = 3Ks(εoct)εoct, τoct = fτ (γoct) = 2Gs(γoct)γoct (7.30)

Upon combining (7.28) and (7.29), we obtain

σ = 2Gsεdev + 3KsεoctI = Ees : ε (7.31)

where Ees is the elastic secant stiffness (ESS) tensor, defined as

Ees = 2GsI

symdev +KsI ⊗ I (7.32)

It appears that σ can be expressed in a format that completely resembles isotropic linear

elasticity if only K and G are replaced by the secant moduli Ks and Gs, respectively.



We shall now introduce the tangential moduli Gt and Kt by differentiating (7.30), i.e.

σoct = 3Kt(εoct)εoct with Kt = Ks +dKs

dεoct

εoct (7.33)

τoct = 2Gt(εoct)γoct with Gt = Gs +dGs

dγoct

γoct (7.34)

The ETS-tensor becomes

Ee = E

es + 2γoct

dGs

dγoct

εdev

|εdev|⊗ εdev

|εdev|+ εoct

dKs

dεoct

I ⊗ I

= 2GsIsymdev + 2[Gt −Gs]

εdev

|εdev|⊗ εdev

|εdev|+KtI ⊗ I (7.35)

The presented socalled “K-G model” is attractive because of its conceptual simplicity and

its ease of calibration. Rewriting (7.30) generically as

σoct = fσ(εoct), τoct = fτ (γoct) (7.36)

we may calibrate the model in pure isotropic tension/compression and pure shear upon

assuming suitable functions fσ and fτ . A variety of such functions have been suggested

in the literature, and we list only a few below4:

• Power law, Ludwik (?)

f(ε) = σ0

[ |ε|ε0

]nε

|ε| (7.37)

• Modified power law

f(ε) =

Eε for ε <σ0

E

σ0

[ |ε|ε0

]nε

|ε| for ε ≥ σ0

E

(7.38)

• Inverse Ramberg-Osgood law, Goldberg-Richard (1963)

f(ε) = σ0

[

1 +

[ |ε|ε0

]n]− 1n ε

ε0(7.39)

• Logarithmic law, Larsson et al. (1999)

f(ε) = σ0 ln

(

1 +|ε|ε0

)ε

|ε| (7.40)

A more comprehensive list was presented by Willam (?). The principal behavior of the

octahedral stress-strain relations (response functions) is shown in Figure 7.2.

4The expressions are given for uniaxial stress. This means, for example, that E is a generic modulus

that is replaced by 3K for fσ and 2G for fτ , where K and G are suitable parameters.


174 7 ELASTICITY

PSfrag replacements

σoct

εoct

τoct

γoct

1

3Ks

3Kt

1

2Gt1

1

11

12Gs

fσ(εoct)

fτ (γoct)

Figure 7.2: Octahedral response functions for volumetric/deviatoric decomposition. (a)

Octahedral normal stress, (b) Octahedral shear stress.

7.2.4 Deformation theory of plasticity

The octahedral formulation of hyperelasticity (based on decoupling of the volumetric and

shear responses) encompasses, as a special case, the deformation theory of plasticity. This

formulation, which is attributed to Hencky (1923), is based on the assumption that

the volumetric response is linear, whereas the shear response is nonlinear. Moreover, the

nonlinearity in shear is based on the von Mises yield criterion and a uniaxial stress-strain

relation. The generic equations are

σoct = 3Kεoct, τoct = 2Gs(γoct)γoct (7.41)

where K is the (constant) bulk modulus, whereas Gs(γoct) is the nonlinear shear modulus.

In order to derive the desired relation for Gs(γoct) it is necessary to introduce the assump-

tion of “elastic-plastic” decomposition

εdev = εedev + ε

pdev (7.42)

where

εedev =

1

2Gσdev (7.43)

εpdev =

0 if τoct <√

23σy(0)

µσdev if τoct =√

23σy(k) ≥

√2

3σy(0)

(7.44)



PSfrag replacementsσy(0)

σy

k

Figure 7.3: Yield stress function

Here we have introduced the “plastic multiplier” µ ≥ 0 and the “equivalent plastic strain”

k, defined as

k =

√

2

3|εp

dev| =√

2 γpoct with γp

octdef=

1√3|εp

dev| (7.45)

Moreover, σy(k) is the current yield stress (under uniaxial stress), which is assumed to be

a known function from a monotonic tensile (or compression) test, cf. Figure 7.3. We may

combine (7.42), (7.43) and (7.44) to obtain

σdev =2G

1 + 2Gµε ⇒ τoct = 2Gsγoct with Gs =

G

1 + 2Gµ(7.46)

From (7.44)2 and (7.45) we obtain

µ(k) =k√2τoct

=3k

2σy(k)(7.47)

However, our aim is to express µ (or rather Gs) in terms of γoct. To this end we use

the “yield criterion” in (7.44)1 and combine with (7.46)2 and (7.47) to obtain an implicit

equation in k for given γoct, from which it is (formally) possible to solve for k = g(γoct).

Hence, we obtain from (7.47)

µ(γoct) =3g(γoct)

2(σy ◦ g)(γoct)(7.48)

and Gs(γoct) is given from (7.46).

Finally, we obtain the secant relationship

σ = 2Gs(γoct)εdev +KεvolI = Ees : ε (7.49)


176 7 ELASTICITY

where Ees was given in (7.32). Moreover, the ETS-tensor E

e was given in (7.35). In this

case it is slightly simplified due to the linear volumetric response:

σ = Ee : ε with E

e = Ees + 2[Gt −Gs]

εdev

| εdev | ⊗εdev

| εdev | (7.50)

Remark: Although we have introduced a “plastic” strain εp, it can be shown that the

dissipation is zero such that the model indeed qualifies as elastic. To this end, we (tem-

porarily) introduce the extended free energy

Ψ(i1, j2, k) = Ψvol(i1) + Ψdev(je2) + Ψhar(k) (7.51)

where the “hardening” portion Ψhar is chosen such that

∂Ψhar

∂k= σy(κ) (7.52)

The dissipation rate then becomes

D def= −∂Ψdev

∂εp: εp − ∂Ψhar

∂kk = σdev : ε

pdev − σy(k)k (7.53)

where it was used that

∂Ψdev

∂εp= −∂Ψdev

∂ε= −∂Ψdev

∂je2

∂je2

∂ε= −2Gεe

dev = −σdev (7.54)

Now, upon differentiating (7.52)1, and combining with (7.51)2, we obtain

k =2µ

3kσdev : ε

pdev (7.55)

Finally, combining (7.55) with (7.47), we obtain from (7.53) that D = 0. 2

Remark: When this model is used in practice, it has been common to specify a criterion

for “loading versus unloading” in order to resemble the characteristics of a truly elastic-

plastic response. Loading (L) means that the tangent expression in (7.50) is valid, whereas

unloading (U) means that the response is purely elastic as defined by Hooke’s law:

σ = Eelin : ε with E

elin = 2GI

symdev +KI ⊗ I (7.56)

It is difficult to find a natural loading criterion such that the derived response is obtained

at unloading. For example, we may rephrase (7.50) as

σ = Ees : ε +

1

j2[Gt −Gs]εdev

d

dt[j2] (7.57)


7.3 Prototype model: Hooke’s model of isotropic linear elasticity 177

and choose the sign of ddt

[j2] as the loading criterion such that

d

dt[j2] > 0 (L)

≤ 0 (U) (7.58)

However, the response at (U) would then become

σ = Ees : ε 6= E

elin : ε (7.59)

This principal deficiency of the total deformation theory of plasticity makes it less ap-

pealing in comparison with the conventional incremental theory of plasticity, which is

discussed in Chapter 10. 2

7.3 Prototype model: Hooke’s model of isotropic lin-

ear elasticity

7.3.1 Constitutive relations

We shall now derive Hooke’s law from the general expression of Ψ by introducing the

assumption of linearity in the stress-strain relationship. As the point of departure, we

take the expression for σ in (7.14), i.e.

σ =∂Ψ

∂i1I + 2

∂Ψ

∂i2ε + 3

∂Ψ

∂i3ε2 (7.60)

The assumption of linearity gives

∂Ψ

∂i1= Li1,

∂Ψ

∂i2= G,

∂Ψ

∂i3= 0 (7.61)

where G and L are the usual Lame’s constants. This gives, with ε = εdev + 13εvolI,

σ = 2Gε + Li1I = 2Gε + LεvolI = 2Gεdev +KεvolI = Ee : ε (7.62)

where K is the bulk (compression) modulus. The following relations hold:

G =E

2[1 + ν], L =

Eν

[1 + ν][1 − 2ν], K = L+

2

3G =

E

3[1 − 2ν](7.63)

We may express Ee as

Ee = 2GI

sym + LI ⊗ I = 2GIsymdev +KI ⊗ I (7.64)


178 7 ELASTICITY

and its inverse, the elastic compliance tensor Ce, as

Ce def

= [Ee]−1 =1

2GIsym − L

2G[2G+ 3L]I ⊗ I =

1

2GIsymdev +

1

9KI ⊗ I (7.65)

Remark: It is possible to obtain Ce from E

e upon using the Sherman-Morrison

formula. Show this as homework! 2

It follows that Ψ(i1, i2) can be represented as

Ψ(i1, i2) =1

2ε : E

e : ε = Gi2 +1

2L[i1]

2 = Gj2 +1

2K[i1]

2 (7.66)

where we used the relations i2 = j2 + 13[i1]

2 and K = L+ 23G.

Uniqueness of boundary value problems in elasticity problems requires that Ee is positive

definite or, equivalently, Ψ is a convex function. It follows directly from (7.66) that

convexity of Ψ requires that G > 0 and K > 0. These conditions place the following

restrictions on E and ν:

E > 0, −1 < ν <1

2(7.67)

Remark: Since Ψ(i1, i2) does not depend on i3 for isotropic linear response, it is concluded

that this material response represents volumetric/deviatoric decoupling. Moreover, the

response is completely defined by two material constants (any two out of K, L, G, E and

ν). 2



Voigt-matrix representation

From (7.64), we obtain the Voigt-matrix representation of Ee as follows

Ee =

2G+ L L L 0 0 0

L 2G+ L L 0 0 0

L L 2G+ L 0 0 0

0 0 0 G 0 0

0 0 0 0 G 0

0 0 0 0 0 G

=E

[1 + ν][1 − 2ν]

1 − ν ν ν 0 0 0

ν 1 − ν ν 0 0 0

ν ν 1 − ν 0 0 0

0 0 0 1−2ν2

0 0

0 0 0 0 1−2ν2

0

0 0 0 0 0 1−2ν2

(7.68)

whereas (7.65) gives the Voigt-matrix representation of Ce as

Ce =1

2G[2G+ 3L]

2[G+ L] −L −L 0 0 0

−L 2[G+ L] −L 0 0 0

−L −L 2[G+ L] 0 0 0

0 0 0 2[2G+ 3L] 0 0

0 0 0 0 2[2G+ 3L] 0

0 0 0 0 0 2[2G+ 3L]

=1

E

1 −ν −ν 0 0 0

−ν 1 −ν 0 0 0

−ν −ν 1 0 0 0

0 0 0 2[1 + ν] 0 0

0 0 0 0 2[1 + ν] 0

0 0 0 0 0 2[1 + ν]

(7.69)

In establishing (7.68) and (7.69) we observed that, for example,

I1212 =1

2⇒ Ee

1212 = G , Ce1212 =

1

4G(7.70)


180 7 ELASTICITY

In the special case of simple shear in the e1e2-plane, we have

σ12 = Ee1212γ12 = Gγ12 , γ12 = 4Ce

1212σ12 =1

Gσ12 (7.71)

The two special cases of plane strain and plane stress are defined as follows:

Special case: Plane strain

Upon setting ε33 = ε23 = ε13 = 0 in (7.68), we obtain

σ11

σ22

σ12

=

2G+ L L 0

L 2G+ L 0

0 0 G

ε11

ε22

γ12

=E

[1 + ν][1 − 2ν]

1 − ν ν 0

ν 1 − ν 0

0 0 1−2ν2

ε11

ε22

γ12

(7.72)

and

σ33

σ23

σ13

=

L L 2G+ L

0 0 0

0 0 0

ε11

ε22

γ12

=E

[1 + ν][1 − 2ν]

ν ν 1 − ν

0 0 0

0 0 0

ε11

ε22

γ12

(7.73)

which are special cases of (7.9) and (7.10).

Special case: Plane stress

Upon setting σ33 = σ23 = σ13 = 0 in (7.69), we obtain

ε11

ε22

γ12

=

1

2G[2G+ 3L]

2[G+ L] −L 0

−L 2[G+ L] 0

0 0 2[2G+ 3L]

σ11

σ22

σ12

=1

E

1 −ν 0

−ν 1 0

0 0 2[1 + ν]

σ11

σ22

σ12

(7.74)



and

ε33

ε23

γ13

=

1

2G[2G+ 3L]

−L −L 2[G+ L]

0 0 0

0 0 0

σ11

σ22

σ12

=1

E

−ν −ν 1

0 0 0

0 0 0

σ11

σ22

σ12

(7.75)

which are special cases of (7.11) and (7.12).

Upon inverting (7.74), we obtain

σ11

σ22

σ12

=

G

2G+ L

4[G+ L] 2L 0

2L 4[G+ L] 0

0 0 2G+ L

ε11

ε22

γ12

=E

1 − ν2

1 ν 0

ν 1 0

0 0 1−ν2

ε11

ε22

γ12

(7.76)

7.3.2 Examples of response simulations

In the subsequent numerical examples, we choose the material data

E = 205 GPa , ν = 0.3

which are typical elastic data for a carbon steel.

Numerical example 1: Uniaxial stress

The common tensile test under uniaxial stress is considered, cf. Figure 7.4. The prescribed

strain and stress components are:

ε11(t) = 5t ∗ 10−4

σ22 = σ33 = 0 , σij = 0 for i 6= j(7.77)

In Figure 7.5 the response is plotted in terms of σ11 versus ε11. The constant slope is

equal to the elasticity modulus E. Since the response is rate-independent, the actual rate

of loading is irrelevant for the stress-strain response.


182 7 ELASTICITY

PSfrag replacements

x1

x2

x3

ε11 6= 0

Figure 7.4: Tensile test under uniaxial stress

Numerical example 2: Biaxial strain with plane stress

As a second numerical example, a loading case defined by biaxial strain in the x1x2-

plane and zero stress in the x3-direction (plane stress) is considered, cf. Figure 7.6. The

prescribed strain and stress components are thus:

ε11(t) = ε22(t) = 5t ∗ 10−4 , ε12(t) = 0

σ33 = 0 , σ13 = σ23 = 0(7.78)

In Figure 7.7 the response is plotted in terms of σ11 = σ22 versus ε11. It is noted that

the slope (stiffness) of the relation σ11 vs. ε11 is higher than in Figure 7.5 (that represents

uniaxial stress).

7.4 Constitutive framework - Anisotropic nonlinear

elasticity

7.4.1 Generic format of the free energy - Symmetry classes

Anisotropic response is obtained when the material is oriented, i.e. it has one (or more)

privileged direction(s). For example, a quite common situation is that the material is

oriented in only one single direction defined by the unit vector a. This case of transverse

isotropy is discussed further below. In practise, certain types of material symmetry can


7.4 Constitutive framework - Anisotropic nonlinear elasticity 183

0 0.01 0.02 0.03 0.04 0.050

2

4

6

8

10

12Uniaxial stress

ε11

σ 11 [G

Pa]

Figure 7.5: Stress-strain behavior for uniaxial stress. Isotropic linear elasticity.

PSfrag replacements

x1

x2

x3

ε11 6= 0

ε22 6= 0

σ33 = 0

Figure 7.6: Biaxial strain in x1x2-plane and zero out-of-plane stress (plane stress).


184 7 ELASTICITY

0 0.01 0.02 0.03 0.04 0.050

5

10

15Biaxial strain, plane stress

ε11

σ 11=

σ 22 [G

Pa]

Figure 7.7: Stress-strain behavior for biaxial strain and plane stress. Isotropic linear

elasticity.



be identified, which reduces the complexity (and generality) of the free energy Ψ as

compared to the most general case of anisotropy. The pertinent symmetry conditions are

often derived directly from the crystal micro-structure in a metal; however, they can also

relate to the manufacturing and deformation processes, etc.

The hierarchy of important symmetry classes (with their crystallographic equivalence

within parantheses) are shown in Figure 7.8 and discussed in some further detail subse-

quently.

PSfrag replacements

Ortogonal symmetry = orthotropy

(parallelepipedic crystal)

Tetragonal symmetry

(tetragonal crystal)

Transverse isotropy

(hexagonal crystal)

Cubic symmetry

(cubic crystal)

Isotropy

Figure 7.8: Hierarchy of common symmetry conditions.

Orthogonal symmetry

Orthogonal symmetry (orthotropy) is defined by the situation that there exist three or-

thogonal principal material directions ai, i = 1, 2, 3, such that normal stresses in the


186 7 ELASTICITY

principal directions will give rise only to normal strains, and that a shear stress in a

particular principal plane will cause shear deformation only in this plane. However, the

elastic moduli associated with each principal direction and plane are all different. This

is illustrated with the parallel-epipedic grid in the cube in Figure 7.9a. In terms of a

crystal microstructure, this situation corresponds to lattice dimensions l1 6= l2 6= l3, cf.

Figure 7.9b. Examples of engineering significance where orthogonal symmetry can be

PSfrag replacements

a3 a2

a1

(a) (b)

[001]

[010]

[100]

l1

l3

l2

Figure 7.9: Representation of orthogonal symmetry.

expected:

• monocrystalline high-strength alloys (inherent anisotropy)

• texture in sheet metal due to large plastic deformations from rolling (induced

anisotropy)

• oriented microfracture (=damage) in metals close to failure (induced anisotropy)

• composites built up of orthogonal plys with uniaxial parallel fibers embedded in

isotropic matrix (inherent anisotropy)

Tetragonal and cubic symmetries

Tetragonal symmetry is defined as the special case when two principal material directions

are equivalent, e.g. those defined by a2 and a3. In terms of the crystallographic symmetry,

this situation corresponds to the condition l1 6= l2 = l3 (cf. Figure 7.9b).



In the case of cubic symmetry, all the principal material directions and planes are equiv-

alent. In terms of the crystal structure, this means that l1 = l2 = l3.

Transverse isotropy

Transverse isotropy is defined by the special case when one plane is isotropic, e.g. that

spanned by a2 and a3. In other words, not only the directions a2 and a3, but all directions

that can be spanned by a2 and a3, are equivalent. This “higher” form of symmetry is

illustrated with the “pipe-grid” in Figure 7.10 , where the “pipes” have random cross-

sectional shape and size.

PSfrag replacements

a3 a2

a1

Figure 7.10: Representation of transverse isotropy.

Examples of engineering significance when tetragonal symmetry can be expected:

• uniaxial parallel fibers embedded in isotropic matrix (inherent anisotropy)

• stratification (=layering) in sedimented soils and rocks (inherent anisotropy)

• fabric in biological materials, such as muscular tissue, wood, etc. (inherent anisotropy)

• texture across the thickness of sheet metal due to large plastic deformation from

rolling (induced anisotropy)

• drawing of wire (induced anisotropy)

• oriented microfracture (=damage) in metals close to failure when the stress state is

cylindrical (induced anisotropy)


188 7 ELASTICITY

Isotropy

Isotropic response can be obtained as a special case of either transverse isotropy or cubic

symmetry, cf. Figure 7.8. It is defined by the situation when all possible directions and

planes in the material are completely equivalent. Isotropy is, typically, associated with

the behavior of a polycrystalline metal with random shape and size of the grains.

Remark: For composite materials, the macroscopic response can be obtained upon

adding the free energy contributions for each constituent material and then using the

strain equivalence principle (parallel coupling of the constituents). For example, the com-

posite may be composed of plys of a certain thickness and with individual symmetry

properties. It is then plausible to add the respective free energies in proportion to the

relative thickness of the plys, as illustrated in Figure 7.11.

Ψ(ε) =1

h

K∑

k=1

hkΨk, h =K∑

k=1

hk (7.79)

where the thickness of each ply is hk, k = 1, 2, ...K. 2

PSfrag replacements

θ1

Ψ1(ε)

θ2

Ψk(ε)

+ ...

Ψ(ε)

+ ... =

Figure 7.11: General anisotropic composite built up from transversely isotropic plys.

7.4.2 Representation of anisotropy with structure tensors

To introduce the concept of structure tensors, let us consider the situation when the

material is oriented only in the direction a. In other words, the material is assumed to

possess a “layered structure”, and a is normal to the isotropic layers. It is then convenient

to introduce the associated structure tensor Adef= a⊗a and use the representation theorem

for scalar functions of two tensors, cf. Chapter 1. It is recalled that this theorem states

that the arguments of Ψ can always be chosen as a suitable integrity basis consisting of



the irreducible invariants I(ε), I(A) and I(ε,A). However, since A is a fixed tensor with

the properties A = A2 = A3 and A : I = 1, we conclude that we are left with only five

such irreducible invariants that form the integry basis: {i1, i2, i3, i1(A), i2(A)}, where the

single invariants i1, i2 and i3 were defined in (1.97) to (1.99), whereas the mixed invariants

i1(A) and i2(A) are defined as5:

i1(A)def= A : ε = a · ε · a = εn(a), i2(A)

def= A : ε2 = a · ε2 · a (7.80)

Here, the mechanical interpretation of εn(a) is the normal strain in the “fiber direction”,

cf. Figure 7.12. In the chosen coordinate system we have

PSfrag replacements

θ

a

x1

x2

Figure 7.12: Example of transverse isotropy: Uniaxial fibers in isotropic matrix.

(a)i =

cos θ

sin θ

0

, (A)ij =

[cos θ]2 cos θ sin θ 0

cos θ sin θ [sin θ]2 0

0 0 0

(7.81)

Remark: Although formally an argument of Ψ, A is a parameter tensor that represents

the material characteristics and should not be confused with the thermodynamic variable

ε. 2

More orientations can be introduced, via unit vectors ai, i = 1, 2, . . ., in order to achieve

more general anisotropic response. For example, orthotropic response can be described

by introducing two orthogonal vectors a1 and a2, forming the dyads A1def= a1 ⊗ a1 and

A2def= a2⊗a2, and introducing an extended integrity basis as the independent arguments

of Ψ. This technique will be exploited below in the context of linear elasticity.

5The unambiguous definition is i(ε)k(A)

def= A : εk. However, to avoid unnecessary notation, we introduce

the notation ik(A)def= i

(ε)k(A).


190 7 ELASTICITY

7.4.3 Kelvin-modes and spectral decomposition of the tangent

stiffness tensor

Consider the eigenvalue problem

Ee : ϕ = λiϕ, i = 1, 2, ..., 6 (3D-case) (7.82)

Since Ee is symmetrical and positive definite, the eigenvalues λi are all real and positive.

Henceforth, we shall assume that the 2nd order eigentensors ϕi form an orthonormal

basis, i.e. ϕi : ϕj = δij for i, j = 1, 2, ..., 6.

Due to the variations in material symmetry properties of the elastic response, not all λi

are distinct. We assume that there are nmode ≤ 6 distinct eigenvalues λj, j = 1, 2, ...nmode,

with multiplicity Kj, and we introduce the sets Kj , each one containing Kj indices in the

range 1 to 6. We thus have∑nmode

j=1 Kj = 6. Associated with each distinct λj, we define

the projection operator

Pj =∑

k∈Kj

ϕk ⊗ ϕk (7.83)

so that the spectral decomposition of Ee becomes

Ee =

nmode∑

i=1

λiPi, [Ee]−1 =

nmode∑

i=1

[λi]−1

Pi (7.84)

Remark: The projection operators have the properties

Pi : Pj = δijPi, i, j = 1, 2, ..., nmode ;

nmode∑

i=1

Pi = Isym (7.85)

Show this as homework! 2

We can now identify the Kelvin-modes εi of strain as follows:

εidef= Pi : ε, ε =

nmode∑

i=1

εi (7.86)

and the corresponding Kelvin-modes σi of stress as

σi = λiεi (no sum) ; σi = Pi : σ, σ =

nmode∑

i=1

σi (7.87)

The relation (7.87) is obtained upon using the relation σ = Ee : ε and inserting (7.84)

with (7.86).



A direct representation of the projection operators Pi can be given in the spirit of Serrin’s

formula as follows:

Pi =

∑nmode/ik=1

[E

e − λkI]

∑nmode/ik=1

[λi − λk

] , i = 1, 2, ..., nmode (7.88)

which was demonstrated by Luehr & Rubin (). 2

In the next subsection, we shall consider anisotropy in the simplest case of linear response,

and we derive the stress-strain relation for various symmetry classes discussed above. In

particular, we shall identify the number of elastic constants, ncons, and the number of

distinct eigenvalues, nmode, corresponding to the Kelvin modes. We also give (in a few

cases) the explicit expressions for the eigenvalues λi and the corresponding projection

operators Pi. To this end, we first list the most typical Kelvin modes:

Typical Kelvin modes

Dilatation mode, Figure 7.13(a):

ϕd =1√3

[a1 ⊗ a1 + a2 ⊗ a2 + a3 ⊗ a3] =I√3

(7.89)

Isochoric extension modes, Figure 7.13(b):

ϕe1 =

1√6[2a1 ⊗ a1 − a2 ⊗ a2 − a3 ⊗ a3] (7.90)

ϕe2 =

1√6[2a2 ⊗ a2 − a3 ⊗ a3 − a1 ⊗ a1] (7.91)

ϕe3 =

1√6[2a3 ⊗ a3 − a1 ⊗ a1 − a2 ⊗ a2] (7.92)

Isochoric pure shear modes, Figure 7.13(c):

ϕps1 =

1√2[a2 ⊗ a2 − a3 ⊗ a3] (7.93)

ϕps2 =

1√2[a3 ⊗ a3 − a1 ⊗ a1] (7.94)

ϕps3 =

1√2[a1 ⊗ a1 − a2 ⊗ a2] (7.95)

Isochoric simple shear modes, Figure 7.13(d):

ϕss1 =

1√2[a2 ⊗ a3 + a3 ⊗ a2] (7.96)


192 7 ELASTICITY

ϕss2 =

1√2[a3 ⊗ a1 + a1 ⊗ a3] (7.97)

ϕss3 =

1√2[a1 ⊗ a2 + a2 ⊗ a1] (7.98)

Non-isochoric extension modes:

ϕne1 = ? (7.99)

ϕne2 = ? (7.100)

ϕne3 = ? (7.101)

where the constant α depends on the actual symmetry case, cf. below.PSfrag replacements

1 2

3

PSfrag replacements

1

2

3

(a) (b)PSfrag replacements

1

2

3

PSfrag replacements

1

2

3

(c) (d)

Figure 7.13: Typical Kelvin modes. (a) Dilatation mode, (b) Isochoric extension modes,

(c) Isochoric shear modes, (d) Isochoric simple shear modes.

It is possible to establish certain characteristics (linear dependence, orthogonality, etc.)

for the various types of Kelvin modes. For example, the isochoric extension and pure

shear modes ϕei and ϕ

psi , respectively, are linearly dependent since

ϕe1 + ϕe

2 + ϕe3 = 0 , ϕ

ps1 + ϕ

ps2 + ϕ

ps3 = 0 (7.102)

whereas the isochoric simple shear modes ϕssi are linearly independent. Moreover, the

isochoric extension, pure shear and simple shear modes are deviatoric, since

I : ϕei = 0 , I : ϕ

psi = 0 , I : ϕss

i = 0 , i = 1, 2, 3 (7.103)


7.5 Constitutive framework - Anisotropic linear elasticity 193

7.5 Constitutive framework - Anisotropic linear elas-

ticity

7.5.1 Orthogonal symmetry

In this Subsection we start out by considering the general case of orthogonal symmetry

when Ψ depends on I(ε), I(A1), I(A2), I(ε,A1), I(ε,A2), and I(ε,A1,A2). However,

because of the properties of A1, A2 (in particular orthogonality), we conclude that the

only independent invariants to be included in the integrity basis are the seven invariants

{i1, i2, i3} and {i1(Ai), i2(Ai)}, i = 1, 2, where i1(Ai) and i2(Ai) are defined as

i1(Ai)def= Ai : ε = ai · ε · ai = εn(ai), i2(Ai)

def= Ai : ε2 = ai · ε2 · ai (7.104)

Subsequently, we shall adopt the restricted form of Ψ where the dependence on i3 has been

dropped. In this case, the integrity basis is restricted to the invariants {i1, i2, i1(A1), i2(A1),

i1(A2), i2(A2)} However, upon introducing the orthogonal vectors a1,a2,a3, where a3 =

a1 × a2 and noting that I = A1 + A2 + A3, we conclude that

i1 = I : ε =3∑

i=1

i1(Ai), i2 = I : ε2 =3∑

i=1

i2(Ai) (7.105)

and it is thus possible to choose the integrity basis as the new set of irreducible invariants

{i1(A1), i2(A1), i1(A2), i2(A2), i1(A3), i2(A3)}.From the choice Ψ(i1(A1), i2(A1), i1(A2), i2(A2), i1(A3), i2(A3)), we obtain

σ =3∑

i=1

[∂Ψ

∂i1(Ai)

Ai + 2∂Ψ

∂i2(Ai)

[ε · Ai]sym

]

(7.106)

We shall next restrict to linear response, which imposes the restrictions

∂Ψ

∂i1(A1)

= φ11i1(A1) + φ12i1(A2) + φ13i1(A3) (7.107)

∂Ψ

∂i1(A2)

= φ12i1(A1) + φ22i1(A2) + φ23i1(A3) (7.108)

∂Ψ

∂i1(A3)

= φ13i1(A1) + φ23i1(A2) + φ33i1(A3) (7.109)

2∂Ψ

∂i2(A1)

= φ4 , 2∂Ψ

∂i2(A2)

= φ5 , 2∂Ψ

∂i2(A3)

= φ6 (7.110)


194 7 ELASTICITY

where is was used that φij = φji, and that the coefficients φ are all constants. It thus

appears that the general orthotropic response is defined by 9 independent (constant)

parameters, i.e. ncons = 9 in this case. Hence, we may express σ as

σ =3∑

i=1

3∑

j=1

φiji1(Aj)Ai +3∑

i=1

φ3+i[ε · Ai]sym = E

e : ε (7.111)

where Ee is the constant ECTS-tensor

Ee =

3∑

i=1

3∑

j=1

φijAi ⊗ Aj +3∑

i=1

φ3+iAi (7.112)

and Ai is the symmetric 4th order tensor defined as

Ai = PAiI =1

4[Ai⊗I + Ai⊗I + I⊗Ai + I⊗Ai] (7.113)

The 9 constants φ11, φ12, φ13, φ22, φ23, φ33, φ4, φ5 and φ6 are interpreted in a more explicit

manner upon resorting to the Voigt matrix format and orienting the coordinate system

such that ai = ei, i = 1, 2, 3. In this coordinate system, we obtain the elastic stiffness

matrix (in 3D) as follows:

Ee =

φ11 + φ4 φ12 φ13 0 0 0

φ12 φ22 + φ5 φ23 0 0 0

φ13 φ23 φ33 + φ6 0 0 0

0 0 0 14[φ5 + φ6] 0 0

0 0 0 0 14[φ4 + φ6] 0

0 0 0 0 0 14[φ4 + φ5]

(7.114)

Show this as homework! Please note the ordering of the shear components as defined in

Subsection 7.1.3.

As to the spectral properties (and Kelvin modes), we merely note that nmode = 6 in this

case, i.e. all eigenvalues are distinct.

7.5.2 Tetragonal symmetry

Tetragonal symmetry is retrieved from the case of orthogonal symmetry, if it is assumed

that the directions x2 and x3 are equivalent. From (7.108, 7.109) this leads to the restric-

tions

φ12 = φ13 , φ22 = φ33 , φ5 = φ6 (7.115)



by which Ee in (7.112) takes the form

Ee = φ11A1 ⊗ A1 + φ12 [A1 ⊗ [A2 + A3] + [A2 + A3] ⊗ A1]

+ φ22[A2 ⊗ A2 + A3 ⊗ A3] + φ23[A2 ⊗ A3 + A3 ⊗ A2] + φ4A1 + φ5[A2 + A3] (7.116)

The Voigt-format of (7.116)is obtained from (7.114) and (7.115) as follows:

Ee =

φ11 + φ4 φ12 φ12 0 0 0

φ12 φ22 + φ5 φ23 0 0 0

φ12 φ23 φ22 + φ5 0 0 0

0 0 0 12φ5 0 0

0 0 0 0 14[φ4 + φ5] 0

0 0 0 0 0 14[φ4 + φ5]

(7.117)

We conclude that ncons = 6 in this case. As to the spectral properties (and Kelvin modes),

we merely note that nmode = 5.

7.5.3 Transverse isotropy

Transverse isotropy is obtained as the special case of tetragonal symmetry when the

plane spanned by e2 and e3 is isotropic. This leads to the additional condition that

φ23 = φ22[= φ33], as compared to (7.115). Show this as homework!

As a result, Ee in (7.116) is further simplified and becomes

Ee = φ11A1 ⊗ A1 + φ12[A1 ⊗ [A2 + A3] + [A2 + A3] ⊗ A1]

+φ22[A2 + A3] ⊗ [A2 + A3] + φ4A1 + φ5[A2 + A3] (7.118)

Now, setting A1 = A, A1 = A and using the identities A2+A3 = I−A, A2+A3 = Isym−A

we may rephrase (7.118) as follows:

Ee = φ22I ⊗ I + φ5I

sym + [φ12 − φ22][I ⊗ A + A ⊗ I[

+[φ11 − 2φ12 + φ22]A ⊗ A + [φ4 − φ5]A (7.119)

Remark: It is possible to deduce this expression directly from the general expression for

the free energy upon introducing the single structure tensor A and consider the integrity

basis consisting of I(ε), I(A) and I(ε,A). If the dependence on i3 is dropped, we are left


196 7 ELASTICITY

with the four independent invariants {i1, i2, i1(A), i2(A)} to be used as the arguments of Ψ.

2

The Voigt-format of (7.119) is obtained directly from (7.117) with the extra condition

φ23 = φ22 as follows:

Ee =

φ11 + φ4 φ12 φ12 0 0 0

φ12 φ22 + φ5 φ22 0 0 0

φ12 φ22 φ22 + φ5 0 0 0

0 0 0 12φ5 0 0

0 0 0 0 14[φ4 + φ5] 0

0 0 0 0 0 14[φ4 + φ5]

(7.120)

We conclude that ncons = 5.

Using a more conventional notation, we may rephrase (7.120) as

Ee =

M‖ L‖ L‖ 0 0 0

L‖ 2G⊥ + L⊥ L⊥ 0 0 0

L‖ L⊥ 2G⊥ + L⊥ 0 0 0

0 0 0 G⊥ 0 0

0 0 0 0 G‖ 0

0 0 0 0 0 G‖

(7.121)

where a quantity with subscript ⊥ is associated with the isotropic planes x2x3, whereas the

subscript ‖ relates to the anisotropic planes x1x2 and x1x3. Apart from the usual Lame’s

constants L and G, we have introduced the uniaxial strain modulus M. Identification of

coefficients between (7.120) and (7.121) gives the relations

φ11 + φ4 = M‖, φ12 = L‖, φ22 + φ5 = 2G⊥ + L⊥, φ22 = L⊥,

1

4[φ4 + φ5] = G‖,

1

2φ5 = G⊥

which has the solution

φ11 = M‖ − 4G‖ + 2G⊥, φ12 = L‖, φ22 = L⊥, φ4 = 4G‖ − 2G⊥, φ5 = 2G⊥ (7.122)

Hence, we may express Ee in (7.119) as

Ee = L⊥I ⊗ I + 2G⊥I

sym + [L‖ − L⊥][I ⊗ A + A ⊗ I]

+[M‖ − 4G‖ + 2G⊥ − 2L‖ + L⊥]A ⊗ A + 4[G‖ −G⊥]A (7.123)



Special case: Fiber-reinforced elastic composite

We shall consider the case of a fiber-reinforced elastic composite, whereby the matrix is

assumed to be isotropic. To simplify matters, we assume that the fibers can only carry

longitudinal stresses corresponding to the axial stiffness Ef per unit cross-sectional area,

i.e. their transverse and shear stiffness contributions are negligible.

The intrinsic constitutive relation for the matrix material is

σ = Ee,iso : ε with E

e,iso = 2GIsym + LI ⊗ I (7.124)

whereas the intrinsic constitutive relation for the fibers is

σ = Eef : ε with E

ef = EfA ⊗ A (7.125)

From (7.125) we conclude that σ = Efεn(a)A, where εn(a) = A : ε. Again, choosing the

coordinate system such that a = e1, we obtain

σ11 = Efε11, σij = 0 for i, j 6= 1 (7.126)

Letting the (dilute) volume concentration of fibers be cf , we may use the “strain equiva-

lence principle” by adding stiffness contributions from (7.124) and (7.125) to obtain the

homogenized constitutive law:

σ =[[1 − cf ]E

e,iso + cfEfA ⊗ A : ε = Ee]

: ε (7.127)

whereby

Ee = [1 − cf ]E

e,iso + cfEfA ⊗ A = [1 − cf ][2GIsym + LI ⊗ I] + cfEfA ⊗ A (7.128)

Upon comparing this expression with (7.123), we conclude that it corresponds to the

choice

G‖ = G⊥ = [1 − cf ]G, L‖ = L⊥ = [1 − cf ]L

M‖ = [1 − cf ][2G+ L] + cfEf = kM with (7.129)

kdef= 1 − cf +

cfEf

2G+ L, M

def= 2G+ L


198 7 ELASTICITY

Spectral properties – Kelvin modes

The 4 distinct eigenvalues (nmode = 4) are

λ1 =1

2

[

M‖ + 2G⊥ + 2L⊥ +√

[M‖ − 2G⊥ − 2L⊥]2 + 8[L‖]2]

, K1 = 1 (7.130)

λ2 = 2G⊥ , K2 = 2 (7.131)

λ3 =1

2

[

M‖ + 2G⊥ + 2L⊥ −√

[M‖ − 2G⊥ − 2L⊥]2 + 8[L‖]2]

, K3 = 1 (7.132)

λ4 = 2G⊥ , K4 = 2 (7.133)

corresponding to the projection operators

P1 = ϕne1 ⊗ ϕne

1 with α =2L‖

M‖ − λ1

(7.134)

P2 = ϕps1 ⊗ ϕ

ps1 + ϕss

1 ⊗ ϕss1 (7.135)

P3 = ϕne1 ⊗ ϕne

1 with α =2L‖

M‖ − λ3

(7.136)

P4 = ϕss2 ⊗ ϕss

2 + ϕss3 ⊗ ϕss

3 (7.137)

7.5.4 Cubic symmetry

Cubic symmetry is obtained as the special case of ortotropy (or tetragonal symmetry) by

introducing the restriction that all directions x1, x2 and x3 represent the same response.

Considering (7.114), we then introduce the following conditions:

φ11 + φ4 = φ22 + φ5 = φ33 + φ6

φ12 = φ13 = φ23def= L (7.138)

φ4 + φ5 = φ5 + φ6 = φ4 + φ6

The solution to (7.138) is

φ11 = φ22 = φ33def= L′, φ4 = φ5 = φ6

def= 2G (7.139)

As a result, Ee takes the form

Ee =L′

3∑

i=1

Ai ⊗ Ai + 2GIsym

+ L[A2 ⊗ A3 + A1 ⊗ A3 + A1 ⊗ A2 + A3 ⊗ A2 + A3 ⊗ A1 + A2 ⊗ A1]

=LI ⊗ I + 2GIsym + [L′ − L]

3∑

i=1

Ai ⊗ Ai

(7.140)




The Voigt format of Ee becomes

Ee =

M L L 0 0 0

L M L 0 0 0

L L M 0 0 0

0 0 0 G 0 0

0 0 0 0 G 0

0 0 0 0 0 G

(7.141)

where Mdef= L′ + 2G. We conclude that ncons = 3.

Remark: Comparing (7.141) with (7.68) pertinent to complete isotropy, we note that

isotropic response is readily retrieved when L′ = L. 2



λ1 = M + 2L , K1 = 1 (7.142)

λ2 = M − L , K2 = 2 (7.143)

λ3 = 2G , K3 = 3 (7.144)


P1 = ϕd ⊗ ϕd =1

3I ⊗ I = Ivol (7.145)

P2 = ϕei ⊗ ϕe

i + ϕpsi ⊗ ϕ

psi , i = 1, 2 or 3 (7.146)

P3 = ϕss1 ⊗ ϕss

1 + ϕss2 ⊗ ϕss

2 + ϕss3 ⊗ ϕss

3 (7.147)

It is noted that

P2 + P3 = Isym − Ivol

def= I

symdev (7.148)

7.5.5 Isotropy

The case of isotropic linear elastic response was discussed in some depth in Section 7.3, and

it is included here for completeness only. It is concluded that isotropy can be obtained

as the simplest special case of either transverse isotropy upon setting L‖ = L⊥def= L,


200 7 ELASTICITY

G‖ = G⊥def= G and M‖

def= L+ 2G in (7.123) or from cubic symmetry by setting L′ = L in

(7.140). Hence, in this case ncons = 2.



λ1 = 3K , K1 = 1 (7.149)

λ2 = 2G , K2 = 5 (7.150)


P1 = Ivol (7.151)

P2 = Isymdev (7.152)


Chapter 10

PLASTICITY - BASIC CONCEPTS

In this chapter we discuss elastic-plastic material response, which is characterized by the

presence of rate-independent dissipation mechanisms when the stress exceeds a certain

threshold value (yield stress). The thermodynamic basis is presented in conjunction with

the celebrated postulate of Maximum Plastic Dissipation, which is the fundamental basis

of classical plasticity. This postulate infers the normality rule, and it provides general

loading criteria (in terms of the complementary Kuhn-Tucker conditions) for any choice

of control variables. To illustrate the developments, the von Mises yield criterion with

mixed isotropic and kinematic hardening is investigated in detail as a prototype model.

The chapter is concluded with a review of classical isotropic yield (and failure) criteria.

10.1 Introduction

10.1.1 Motivation

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).


248 10 PLASTICITY - BASIC CONCEPTS

For these highly porous materials, the inelastic deformation has both distortional and

volumetric components, cf. Chapter 11.

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

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 10.1. However, most metals are polycrystalline materials.

PSfrag replacements

εp εe

Figure 10.1: Microstructure of single crystal showing plastic deformation followed by

elastic deformation.

This means that grains with different crystallographic orientations and lattice structure

(that represents different thermodynamic phases) are interacting in the mesostructure, cf.

Figure 10.2. If the distribution of crystal orientations is statistically uniform, i.e. each ori-

Figure 10.2: Mesostructure of grains interacting via grain boundaries and possessing

different crystal orientations.

entation 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


10.1 Introduction 249

manufacturing process, e.g. elongation of grains in the rolling direction for metal sheet

products. Anisotropic yield criteria are discussed in Chapter 11.

10.1.2 Literature overview

Early contributions to the macroscopic (mathematical) theory of plasticity were given by

Huber, von Mises, Prandtl, Hencky and Reuss from 1900 up to the 1920’s. The

concept of isotropic hardening was introduced by Odqvist (1933), whereas the concept

of kinematic hardening is due to Melan (1938) and Prager (1947), who coined the

term “kinematic” hardening. For metals the main characteristics of a macroscopic theory

are the pressure-independent yield criterion and isochoric plastic deformation, which were

observed by Bridgman (1923).

It is not our intension to compete with the abundance of existing text-books in describing

the classical concepts of plasticiy. We merely select the three books by Lubliner (1990),

Lemaitre & Chaboche (1990), Maugin (1992) and Haupt (2000), to which the

reader is referred for comprehensive treatments of the macroscopic modeling issues. A

number of books and volumes dealing with the material science aspects of inelasticity,

including computational issues, have been published in recent years. Examples are those

of Phillips (xxx), Teodosiu (ed.) (xxx), and Kocks, Tome, & Wenke (eds.)

(xxx).

As to literature dealing with the numerical technique, viz. the integration of evolution

equations and iteration of the resulting nonlinear systems, we consider the most compre-

hensive and modern treatment to be that of Simo & Hughes (xxx). A good account of

the issues associated with reliability and robustness for solution of the incremental con-

stitutive problem is given by Armero & Perez-Foguet (xxx). We remark that there

is a fundamental difference between the CTS- and ATS-tensor. The first publication, to

our knowledge, of the explicit form of the ATS-tensor for the Backward Euler method

applied to von Mises plasticity was given by Runesson & Booker (1982).



10.2 The constitutive framework - Perfect plasticity

10.2.1 Free energy and thermodynamic stresses

A prototype model for nonviscous, or rate-independent, plasticity is characterized by

rate-independent dissipation of energy in the “frictional slider”; i.e. the internal variables

change their values, and energy is dissipated, at the same rate as the loading is applied.

As a result, rate-independent inelastic stress-strain behavior is obtained.

The strain is decomposed additively into elastic and plastic parts, such that the elastic

strain is defined as 1

εe(ε, εp) = ε − εp (10.1)

where the plastic strain εp is an internal variable. We thus represent the free energy as

Ψ(ε, εp)def= Ψ(εe), which represents the elastic response. The constitutive equation for

the stress is then obtained from (1.65) as

σ =∂Ψ

∂εe(10.2)

which is the same result as for viscoelasticity.

The thermodynamic, or dissipative, stress σp, which is energy-conjugated to εp, becomes

σp = − ∂Ψ

∂εp=∂Ψ

∂εe= σ (10.3)

and the dissipation inequality becomes 2

D = σ : εp ≥ 0 (10.4)

The classical formulation of plasticity is based on the existence of a yield surface in stress

space. Inside this surface the states are entirely elastic, which means that the response

is reversible for any change of the state. We thus assume the existence of a convex yield

1The additive strain decomposition can be justified from a kinematic viewpoint only in the (present)

context of small strain theory.2Dred

mech is replaced by D to abbreviate notation.


10.2 The constitutive framework - Perfect plasticity 251

function3 Φ (σ) such that plastically admissible states are contained in the convex set4E:

E = {σ | Φ (σ) ≤ 0} (10.5)

that is assumed to contain the origin, i.e. Φ(0) ≤ 0. We also define the elastic region as

the open set defined by the interior E of E.

Remark: The more general situation is that plastic strains develop for any thermo-

dynamic state, provided that a certain loading criterion is satisfied. The loading cri-

terion is expressed in terms of the rate of change of the chosen control variable. In

the literature such a nonclassical situation is termed plasticity without yield surface, cf.

Lubliner (1972). 2

10.2.2 Associative structure - Postulate of Maximum Dissipa-

tion

The classical properties associated with plasticity theory can be derived from the pos-

tulate of Maximum Dissipation (abbreviated MD-postulate), Hill (1950), whereby the

maximization is performed over the admissible space E of dissipative stresses. (Although

the consequences of this postulate were discussed in a more general context in Section 5.3,

the arguments are repeated here for completeness.) We first recall that the dissipation

function can be expressed as

D (σ) = σ : εp, σ ∈ E (10.6)

where D is (for the moment) considered to be a function of the (nominal) stresses σ for

given values εp.

MD-postulate: The actual value of σ satisfies the constrained maximum of D such that

σ = arg[max D (σ∗) , ∀σ∗ ∈ E

](10.7)

3A function F (x) : Rn → R is convex iff, for any pair x1,x2, the following inequality holds:

F (αx1 + [1 − α]x2) ≤ αF (x1) + [1 − α]F (x2) ∀α ∈ [0, 1]

4A set K is convex iff

x1,x2 ∈ K ⇒ αx1 + [1 − α]x2 ∈ K ∀α ∈ [0, 1]



Equivalently, the solution σ satisfies the variational inequality:

D(σ) −D (σ∗) ≥ 0, ∀σ∗ ∈ E 2 (10.8)

The formulations (10.7) and (10.8) are completely equivalent, but they will be discussed

separately below.

We remark that the modern mathematical treatment of plasticity is heavily based on

the MD-postulate and techniques in convex analysis, e.g. Duvaut & Lions (1972),

Johnson (?) (?), Moreau (1976), Temam (1980).

Consequences of the MD-postulate are:

• The dissipation inequality is automatically satisfied

• The flow rule is of the associative type in the space of (dissipative) stresses

These properties were demonstrated in Chapter 5 in the more general context of the canon-

ical framework for dissipative materials. Here, we shall only elaborate on the associative

flow rule from a few different viewpoints.

Variational inequality formulation: Associative flow rule

Let us reformulate (10.8) more explicitly as

[σ − σ∗] : εp ≥ 0, ∀σ∗ ∈ E (10.9)

and consider a given state σ ∈ E. It is then possible to conclude from (10.9) that the

corresponding εp must be directed in the outward “normal” direction to the yield surface.

This normal direction is unique if the yield surface is smooth, whereas it is possible for

εp to vary within a “normal cone” if the yield surface has a sharp corner. Both situations

are illustrated in Figure 10.3(a,b). Moreover, it follows that εp = 0 is the only possibility

when the state is elastic, i.e. when Φ(σ) < 0, which is shown in Figure 10.3(c). Finally,

Figure 10.3(d) shows the physically unrealistic situation that E is non-convex. This must

be rejected, since it would imply that the only solution of (10.9) is εp = 0, even when

Φ = 0.

In the particular case of a smooth yield surface, the findings above can be conveniently

summarized as the associative flow rule (normality rule):

εp = λ∂Φ(σ)

∂σ(10.10)


10.2 The constitutive framework - Perfect plasticity 253PSfrag replacements

σ − σ∗

σ∗

σ

(a)

εp ∼ ν

σ∗

σ

εp

(b)

(c) (d)

σ∗

σ

σ − σ∗

σ − σ∗

σ∗σ∗

σ∗

σ

σ

σσ

σ − σ∗

σ − σ∗

σ − σ∗

σ − σ∗

Figure 10.3: MD-postulate in stress space for (a) Φ = 0, convex smooth yield surface, (b)

Φ = 0, convex non-smooth yield surface, (c) Φ < 0, convex smooth or non-smooth yield

surface, (d) Φ = 0, non-convex (smooth) yield surface

where the plastic multiplier λ is determined by the complementarity conditions

λ ≥ 0, Φ(σ) ≤ 0, λΦ(σ) = 0 (10.11)

Optimality conditions (Kuhn-Tucker problem): Associative flow rule

An alternative way of deriving the flow rule is provided by establishing the optimality

conditions that are pertinent to the maximization problem (10.7). These optimality con-

ditions are known as the Kuhn-Tucker problem. To be more explicit, we consider the case

of a smooth yield surface and introduce the Lagrangian function L defined by

L(σ∗, λ∗) = −D(σ∗) + λ∗Φ(σ∗) (10.12)



It follows from classical optimization theory that the solution of (10.7) also defines the

unconstrained saddle-point of L(σ∗, λ∗). The optimality conditions are

∂L(σ, λ)

∂σ= −εp + λ

∂Φ(σ)

∂σ= 0 (10.13)

λ ≥ 0, Φ ≤ 0, λΦ = 0 (10.14)

and it is seen that (10.13) is equivalent to (10.10). The Kuhn-Tucker complementary

conditions (10.14) are the general loading conditions, that hold regardless of the choice

of control variables, and thus infer that λ ≥ 0 when Φ = 0, whereas λ = 0 when Φ < 0.

(The corresponding formulations of (10.10) to (10.11) for a non-smooth yield surface are

more technical and are postponed to the next chapter).

Next, we summarize and elaborate the findings above (for the special case of a smooth

yield surface): As shown above, the constitutive rate equation for εp may be expressed as

the associative plastic flow rule:

εp = λν with νdef=∂Φ

∂σ(10.15)

By differentiating σ in (10.2) and invoking (10.15), we obtain the constitutive rate equa-

tion for σ as follows:

σ = Ee : [ε − λν] = E

e : ε − λEe : ν with Ee def

=∂2Ψ

∂εe ⊗ ∂εe(10.16)

In the case that the elastic response is linear, then (10.16) is particularly useful as the

basis for implicit integration, which will be discussed in Section 10.3.

10.2.3 Continuum tangent relations

In the special case that Φ = 0, i.e. the stress satisfies the yield criterion and the state is

currently “plastic”, then we obtain (since Φ ≤ 0 when Φ = 0) the consistency condition:

λ ≥ 0, Φ ≤ 0, λΦ = 0 (10.17)

and we distinguish between the two main loading situations:

• Φ = 0, λ > 0 plastic loading (L)

• Φ ≤ 0, λ = 0 elastic unloading (U)


10.2 The constitutive framework - Perfect plasticity 255

Using the constitutive rate equation (10.16) combined with the consistency condition

(10.17), we may derive the pertinent tangent relations. Upon differentiating Φ, we first

obtain

Φ = ν : σ ≤ 0 (10.18)

Inserting (10.16) into (10.18), we obtain

Φ = Φtr − hλ ≤ 0 (10.19)

where we introduced the loading function Φtr, defined as

Φtr = ν : Ee : ε (10.20)

and the plastic modulus h, defined as

h = ν : Ee : ν > 0 (10.21)

That h > 0 follows from the fact that Ee is assumed to be positive definite. Hence, it is

concluded from the general discussion in Section 5.3 that a unique solution of λ exists for

any given value of Φtr, and this solution is defined as follows:

If Φtr > 0, then we must have plastic loading (L), defined by λ > 0 and Φ = 0, and from

(10.19) we obtain the solution

λ =1

hΦtr > 0 (10.22)

If Φe ≤ 0, then we must have elastic unloading (U), defined by λ = 0 and Φ ≤ 0, and

from (10.19) we obtain

Φ = Φtr ≤ 0 (10.23)

The special situation Φe = 0 is sometimes denoted neutral loading. However, for all

practical purposes this case need not be distinguished from other situations of elastic

unloading, since the response is, indeed, elastic.

Remark: The loading function in (10.20) is valid for strain control, in which case ε

is prescribed. This is the natural choice in terms of the formulation of boundary value

problems and their solution by the displacement-based finite element method. However, it

is possible to express the loading function for stress control, in which case σ is prescribed.

Such a choice is useful when a mixed finite element method is adopted. In the sequel we

shall consider strain control (unless otherwise is stated explicitly). 2



Consider now the case of plastic loading (L). By inserting (10.22) with (10.20) into (10.15),

we obtain

εp =Φtr

hν =

1

hν ⊗ ν : E

e : ε (10.24)

It now remains to combine (10.22) with (10.16) to obtain the tangent stiffness relation

σ = Eep : ε with E

ep def= E

e − 1

hE

e : ν ⊗ ν : Ee (10.25)

where Eep is the elastic-plastic Continuum Tangent Stiffness (CTS) tensor.

Considering the components of Eep, we note that [Eep]ijkl possess “major symmetry” in

the indices ij respective kl (because of the use of an associative flow rule for εp).

In the case of elastic unloading (U), the incremental response is clearly elastic, since we

have λ = 0. Hence, we conclude that εp = 0 and, therefore

σ = Ee : ε (10.26)

10.3 The constitutive integrator - Perfect plasticity

10.3.1 Backward Euler method

Here we shall consider only the fully implicit Backward Euler method, which is the most

commonly used method as of today. Since σ = σ(εe), it is convenient to compute εe

as the primary unknown; thus we shall follow the strategy outlined in Section 8.3. The

pertinent constitutive equations for a smooth yield surface are then:

εe = ε − λν(σ(εe)) (10.27)

λ ≥ 0, Φ(σ(εe)) ≤ 0, λΦ(σ(εe)) = 0 (10.28)

We consider, in turn, the two cases of elastic unloading (U) and plastic loading (L).

Elastic unloading

In the case of elastic unloading (U), the integration of (10.27) and (10.28) becomes trivial,

since λ = 0. Hence, we obtain simply ∆εp = 0 and

εe = εe,tr with εe,tr def= n−1εe + ∆ε , σ = σtr with σtr def

= σ(εe,tr) (10.29)


10.3 The constitutive integrator - Perfect plasticity 257

where σtr is commonly denoted the “trial stress” in the literature. The condition for

validity of (10.29) is that Φtr def= Φ(σtr) ≤ 0.

Plastic loading

In the case of plastic loading (L), defined by Φ(σtr) > 0, the Backward Euler (BE) applied

to (10.27) yields the incremental problem (µ = ∆tλ)

εe = εe,tr − µν (σ(εe)) (10.30)

Φ(σ(εe)) = 0 (10.31)

The local incremental problem (10.30) and (10.31) can be rewritten as follows:

R\ε(ε

e, µ) = εe − εe,tr + µν(σ(εe)) = 0 (10.32)

R\µ(εe) = Φ(σ(εe)) = 0 (10.33)

or

R\(X) = 0 with X =

[

εe

µ

]

, R\ =

[

R\ε

R\µ

]

(10.34)

This is the most general format and can be employed whether the elastic and plastic prop-

erties are isotropic or anisotropic, and regardless of the explicit choice of the coordinate

system. In particular, we may choose Cartesian coordinates. For given ε = ε(tn), it is

possible to solve for X from (10.34)1 in an iterative fashion (in the general situation). It

is convenient to use Newton iterations, whereby we use the associated Jacobian matrix

J \ of R\(X), defined formally as

J \ =

Isym + µN : E

e ν

ν : Ee 0

(10.35)

where N was given in (8.27) as the Hessian of Φ:

Ndef=

∂2Φ

∂σ ⊗ ∂σ(10.36)

In the Newton procedure, we may instead introduce the scaled “out-of-balance” forces

Rε = Ee : R\

ε, Rµ = R\µ (10.37)



which are associated with the symmetric Jacobian matrix J defined as

J =

Ee + µEe : N : E

eE

e : ν

ν : Ee 0

=

Ee : [Ee

a]−1 : E

eE

e : ν

ν : Ee 0

(10.38)

where Eea was defined already in (8.31) as

Eea

def=[[Ee]−1 + µN

]−1(10.39)

It is possible to compute J−1 explicitly as

J−1 =1

ha

ha[Ee]−1 : Ea : [Ee]−1 [Ee]−1 : E

ea : ν

ν : Eea : [Ee]−1 −1

(10.40)

where

Eadef= E

ea −

1

ha

Eea : ν ⊗ ν : E

ea (10.41)

ha = ν : Eea : ν (10.42)

Remark: With the exception of a slight modification of ha, these expressions are identical

to those pertinent to nonlinear viscoelasticity, discussed in Section 8.3. 2

The iteration procedure is summarized in Box 10.1.



1. Given the values X (k) in iteration k, then compute J (k)

2. Calculate improved solution

X(k+1) = X(k) + δX with δX = −[J (k)]−1R(k)

3. Calculate “unbalanced stress” R(k+1) = R(X(k+1))

4. Check convergence

If |δX| < TOL and |R(k+1)| < TOL, then stop

else goto 1 and continue iteration.

Box 10.1: Newton iterations in state space

10.3.2 Backward Euler method – Constrained minimization prob-

lem

It is possible to obtain εe, which is (part of) the solution of (10.29) and (10.30, 10.31),

from the solution of a constrained minimization problem in stress space. To this end, we

first introduce the free enthalphy (or complementary free energy) Ψ∗(σ) via the Legendre

transformation

Ψ∗(σ) = supεe

[σ : εe − Ψ(εe)] (10.43)

which, in particular, gives

εe =∂Ψ∗

∂σ= εe(σ) (10.44)

Show this as homework! Hint: Compare with 3.5.5.

Let us now consider the solution σ of the constrained minimization problem

σ = arg

[

minσ∗ ∈E

[Ψ∗(σ∗) − σ∗ : εe,tr

]]

(10.45)

for given, fixed, value εe,tr. By using the Lagrange multiplier method, it is possible to

establish the Kuhn-Tucker conditions that are equivalent to the constrained minimization

problem (10.45). To this end, we introduce the Lagrangian function L, defined as

L(σ∗, µ∗) = Ψ∗(σ∗) − σ∗ : εe,tr + µ∗Φ(σ∗) (10.46)



whose gradient in the (σ, µ)-space is

∂L(σ∗, µ∗)

∂σ∗ =∂Ψ∗(σ∗)

∂σ∗ − εe,tr + µ∗ν(σ∗) (10.47)

∂L(σ∗, µ∗)

∂µ∗ = Φ(σ∗) (10.48)

The corresponding KT-conditions, that define the saddle-point (σ, µ) of L, are:

∂L∂σ∗ (σ, µ) =

∂Ψ∗

∂σ∗ (σ) − εe,tr + µν(σ) = 0 (10.49)

µ ≥ 0, Φ(σ) ≤ 0, µΦ(σ) = 0 (10.50)

Because of the relation (10.44), it appears that (10.49, 10.50) are precisely equivalent to

the relations (10.29) and (10.30, 10.31).

Remark: It is emphasized that solving directly for σ and µ from (10.49, 10.50) would

require that Ψ∗(σ) is first computed/known. Normally, Ψ(εe) is given a priori which

means that the formulation in terms of εe and µ, as shown in Subsection 10.3.1, is the

more convenient one in practice. The exception is linear elasticity, in which case the

difference becomes trivial. 2

It remains to check that σ∗ = σ, obtained as the solution of (10.49) and (10.50), does in

fact represent a minimum of F(σ∗)def= Ψ∗(σ∗) − σ∗ : εe,tr. We obtain

∂2F(σ)

∂σ∗ ⊗ ∂σ∗ =∂2Ψ∗(σ)

∂σ∗ ⊗ ∂σ∗ = Ce (10.51)

and since Ce is positive definite, it is clear that F is min at σ∗ = σ.

10.3.3 ATS-tensor for BE-rule

The derivation of the ATS-tensor can be carried out in a fashion that is completely iden-

tical to that of viscoelasticity in Chapter 8. Hence, we conclude that Ea in (10.41) is the

pertinent explicit expression of the ATS-tensor, which is repeated here for completeness:

dσ = Eepa : dε (10.52)

where

Eepa

def= E

ea −

1

ha

Eea : ν ⊗ ν : E

ea (10.53)



The similarity in mathematical structure of Eepa and the corresponding CTS-tensor E

ep is

striking. The formal difference is the subscript “a”, that indicates “algorithmic”. In fact,

Eep is obtained as

limµ→0

Eepa = E

ep (10.54)

This identity follows readily when µ = 0, in which case we obtain Eea = E

e and ha = h.

10.3.4 Backward Euler method for linear elasticity - Solution in

stress space

In the case of linear elasticity, defined by σ = Ee : εe, where E

e is the constant elastic

stiffness tensor, then we may rewrite (10.27) and (10.28) as follows:

σ = Ee : ε − λEe : ν(σ) (10.55)

λ ≥ 0, Φ(σ) ≤ 0, λΦ(σ) = 0 (10.56)

Hence, the solution is conveniently sought in stress space rather than in the space of elastic

strains (as in the generic case).

The two cases of elastic unloading (U) and plastic loading (L) are now distinguished as

follows:

Elastic unloading

In the case of elastic unloading (U), then (10.29) is replaced by

σ = σtr with σtr def= E

e : εe,tr = n−1σ + Ee : ∆ε (10.57)

where it was used that εe,tr = n−1εe + ∆ε and n−1σ = Ee : n−1εe. This expression for

σe,tr is the classical one for the trial stress.

Plastic loading

In the case of plastic loading (L), the local incremental problem (10.32) and (10.33) is

replaced by

R\σ(σ, µ) = σ − σtr + µEe : ν(σ) = 0 (10.58)

R\µ = Φ(σ) = 0 (10.59)


262 10 PLASTICITY - BASIC CONCEPTSPSfrag replacements

E

σ11

σ12

Φ(σ) = 0

E

σ

ν

µEe : ν

nσ

4σtr

σtr

Figure 10.4: Predictor-corrector algorithm in stress space (perfect plasticity) pertinent to

the Backward Euler method in the case of linear elasticity.

or

R\(X) = 0 with X =

[

σ

µ

]

, R\ =

[

R\σ

R\µ

]

(10.60)

The elastic predictor-plastic corrector characteristics of the BE-method are illustrated in

Figure 10.4.

The Jacobian matrix J \ of R\(X) is given as

J \ =

Isym + µEe : N E

e : ν

ν 0

(10.61)

In this case a symmetric Jacobian J is obtained upon introducing the scaled “out-of-

balance” forces

Rσ = [Ee]−1 : R\σ, Rµ = R\

µ (10.62)

which gives

J =

[Ee]−1 + µN ν

ν 0

=

[Eea]

−1 ν

ν 0

(10.63)

where Eea is, again, the AES-tensor defined by (10.39). Moreover, it is possible to compute



J−1 explicitly as

J−1 =1

ha

haEa Eea : ν

ν : Eea −1

(10.64)

Remark: The ATS-tensor was derived based on the generic incremental format in Sub-

section 10.3.1. However, it is simple to show that a similar derivation based on the present

format (whenever this format is found appropriate) would give the same result. 2

10.3.5 Concept of Closest-Point-Projection for linear elasticity

In the case of linear elasticity, the solution σ = nσ, as defined by the relations (10.57) and

(10.58, 10.59), can be obtained as a projection in complementary elastic energy norm. This

follows quite trivially from the result in Subsection 10.3.2: In the case of linear elasticity

we obtain

Ψ∗(σ) =1

2σ : [Ee]−1 : σ, σtr = E

e : εe,tr (10.65)

and (10.45) thus becomes

σ = arg

[

minσ∗ ∈E

[1

2σ∗ : [Ee]−1 : σ∗ − σ∗ : [Ee]−1 : σtr

]]

= arg

[

minσ∗ ∈E

Ψ∗(σ∗ − σtr)

]

(10.66)

To obtain the last identity, we added the constant quantity σtr : [Ee]−1 : σ∗/2. In other

words, σ is the projection of σtr onto the convex set E in the particular metric defined

by the norm Ψ∗. This is the reason why the BE-method applied to the plasticity problem

is also known as the Closest-Point-Projection-Method (CPPM). More schematically, the

CPPM is defined as the mapping

σ = CPPM(σtr;Ee

)(10.67)

where Ee indicates the projection metric.

Theorem: The minimization problem in (10.66) is also equivalent to the variational

inequality

[σ − σ∗] : [Ee]−1 : [σtr − σ] ≥ 0 ∀σ∗ ∈ E (10.68)

Proof: Since Ee is positive definite, we introduce the decomposition

Ee = E

e[ : E

e[ with E

e[

def= [Ee]

12 (10.69)



and we define the transformed stress σ[ and transformed set E[ as follows:

σ[ = [Ee[ ]−1 : σ ⇒ σ = E

e[ : σ[, E[ = {σ[ | σ ∈ E} (10.70)

We may now rewrite (10.68) as

[σ[ − σ∗[ ] : [σtr

[ − σ[] ≥ 0 ∀σ∗[ ∈ E[ (10.71)

and it follows directly that the solution σ[ of (10.71) is also the Euclidean projection of

σtr[ onto E[, i.e. we have

σ[ = arg

[

minσ∗

[∈E[

|σtr[ − σ∗

[ |2]

(10.72)

Now, upon back-transforming to the original stress space, we realize that

|σtr[ − σ∗

[ |2 = 2Ψ∗(σtr − σ∗) (10.73)

and it follows that (10.72) is equivalent to (10.66). 2

A geometric illustration of the inequality (10.68) and the corresponding constrained min-

imization problem is shown in Figure 10.5. In particular, it is shown that the Euclidean

projection is retrieved in the transformed stress space for a general yield surface. Later

we shall consider important special yield surfaces for which the Euclidean projection is

obtained even in the nominal stress space.

In the case of elastic unloading, i.e. when σtr ∈ E, then the solution of (10.68) or (10.66)

is σ = σtr, which is the identity projection.

We remark that (10.68) and, hence, (10.66) are valid formulations even in the general

situation when the yield criterion is non-smooth. How to efficiently carry out the mini-

mization in such a situation is discussed later.


10.4 Prototype model: Hooke elasticity and von Mises yield surface 265

PSfrag replacements

Eb

Eb ={σb | σ ∈ E

}

σ[

σ[ − σ∗[

σtr[

σ∗[

σtr[ − σ[

Figure 10.5: Closest-Point-Projection-Method for perfect plasticity (CPPM) in trans-

formed stress space defined by σ[ = [Ee[ ]−1 : σ where E

e[

def= [Ee]1/2.

10.4 Prototype model: Hooke elasticity and von Mises

yield surface

10.4.1 The constitutive relations

As the prototype model we consider linear isotropic elasticity in conjunction with the von

Mises yield surface. Using

σ = Ee : εe with E

e = 2GIsymdev +KI ⊗ I (10.74)

we may split σ as follows:

σ = σdev + σmI (10.75)

where

σdev = 2Gεedev, σm = Kεevol (10.76)

The von Mises yield function is defined as

Φ = σe − σy (10.77)

where σedef=√

32|σdev| is the equivalent stress, and σy is the initial yield stress.



Remark: Quite frequently, σe is denoted the “effective” stress. However, this notation

will be reserved for the stress quantity that accounts for damage, cf. Chapter 11. 2

The associative flow rule is given as

εp def=∂Φ

∂σ= λν with ν =

∂σe

∂σ=

3

2σe

σdev (10.78)

Squaring both sides of (10.78), we may deduce that

λ = ep with ep def=

√

2

3|εp| (10.79)

where ep is the rate of accumulated plastic strain. Moreover, (10.78) shows that the plastic

strain is purely deviatoric, i.e. the material displays plastic incompressibility or

tr (εp) = εpkk

def= εpvol = 0 (10.80)

Remark: In the general case ep def=√

2/3|εpdev|. However, since εp = ε

pdev for the present

model, the expression (10.79) is sufficient. 2

Using Ee in (10.74) together with ν in (10.78)2, we may now express the rate equation

for σ more explicitly as

σdev = 2Gεdev − λ3G

σe

σdev, σm = Kεvol (10.81)

and it is concluded that the inelastic response is confined to the deviator stress.

Tangent relations

In a “plastic” state, defined as Φ = 0, the tangent relations can be established. Upon

inserting the pertinent relations above into the expression for h in (10.21), we obtain

h = 3G (10.82)

and the loading function Φtr becomes

Φtr =3G

σe

σdev : ε ⇒ λ =1

σe

σdev : ε (10.83)

Finally, we obtain the CTS-tensor Eep in (10.25) as

Eep = E

e − 3G

[σy]2σdev ⊗ σdev (10.84)



Dissipation inequality

It may be of interest to check explicitly that the dissipation D (or the rate of internal

production of entropy) is non-negative:

D = σ : εp = λσ : ν = λσe = λσy > 0 (10.85)

where it was used that Φ = 0, i.e. that σe = σy.

10.4.2 The constitutive integrator

The incremental relations due to implicit integration are obtained quite analogously to

those of the isotropic Norton model in the context of nonlinear elasticity. However, since

the elastic properties are linear and isotropic, we shall give a slightly different derivation

based on the “stress format”, as outlined for the generic case in Subsection 10.3.3. We

thus obtain from (10.58)

σdev = σtrdev −

3Gµ

σe

σdev, σm = σtrm (10.86)

where the (elastic) trial stress σtr is split into the deviatoric and mean parts

σtrdev = n−1σdev + 2G∆εdev, σtr

m = n−1σm +K∆εvol (10.87)

where µ is determined from the incremental complementary conditions. Hence, at loading

(L) we compute σe and µ from the following relations

R\σ(σe, µ) = σe − σtr

e + 3Gµ = 0 (10.88)

R\µ(σe) = σe − σy = 0 (10.89)

where the condition for loading (L) is that Φtr ≥ 0, where

Φtr def= σtr

e − σy, σtre

def=

√

3

2|σtr

dev| (10.90)

It is simple to solve for µ and σe directly from the linear equations (10.88, 10.89) to obtain

µ =σtr

e − σy

3G, σe = σy (10.91)

and we may compute the updated stress σ as

σ = cσtrdev + σtr

mI with c = c(µ) = 1 − 3Gµ

σtre

=σy

σtre

≤ 1 (10.92)

where we used (10.88) to obtain the scalar c. This “radial return” property of σdev from

σtrdev is illustrated in Figure 10.6.



ATS-tensor

As to the pertinent ATS-tensor Ea, it can be established in a fashion that is identical

to that of the (nonlinear) viscoelastic Norton model discussed in Section 8.5. Upon

introducing the simplification that ha = 3G and that σe = σy at loading, we obtain

Ea = Ee − 2GbQ − 3G

[σy]2σdev ⊗ σdev with b =

3Gµ

σy

[

1 +3Gµ

σy

]−1

(10.93)

where Q is the projection tensor

Qdef= I

symdev − 3

2[σe]2σdev ⊗ σdev (10.94)

that was used already in Chapter 8 in the context of the isotropic Norton model (as the

prototype model of nonlinear viscoelasticity). We note that Ea can be rephrased as

Ea = Eep − 2GbQ (10.95)

where Eep is the CTS-tensor defined in (10.84). For vanishing timestep size, µ = 0, we

obtain b = 0 and, hence, Ea = Eep in such a case.

PSfrag replacements

nσdevσdev = c1σ

trdev

Φ = σe − σy = 0

∆σtrdev σtr

dev

σ1

σ2σ3

Figure 10.6: “Radial return” in deviatoric stress space when the Backward Euler rule is

applied to the von Mises yield criterion in the case of perfect plasticity.

10.4.3 Examples of response computations

To be completed.



10.4.4 Appendix I: Constitutive relations for the uniaxial stress

state

The a priori assumptions of stress and strain states pertinent to the uniaxial stress state

are given in Box 10.2. The developments leading to the tangent stiffness relation of

interest (in the axial direction) are given in Box 10.3, whereby the general multiaxial

tangent relations are exploited.

• Stress tensor:

[σ]ij = σ

1

0

0

; [σdev]ij =

1

3σ

2

−1

−1

, σm =

1

3σ (i)

• Strain tensor:

[ε]ij =

ε

ε⊥

ε⊥

; [εdev]ij =

1

3[ε− ε⊥]

2

−1

−1

, εvol = ε+ 2ε⊥

(ii)

• Stress invariants:

σe = |σ| (iii)

Box 10.2: Characterization of uniaxial stress state.



• Constitutive relations for stress, from (10.81) and uniaxial stress constraint:

{

σ = [L+ 2G]εe + Lεe⊥ + Lεe⊥ (a)

0 = Lεe + [L+ 2G]εe⊥ + Lεe⊥ (b)

where

L = K − 2G

3=

Eν

[1 − 2ν][1 + ν], G =

E

2[1 + ν], K =

E

3[1 − 2ν]

Solution of (a, b): εe⊥ = −νεe ⇒

σ = Eεe = E[ε− εp] (iv)

• Evolution equation for internal variable (flow rule)

[ν]ij =1

2

σ

|σ|

2

−1

−1

⇒ εp = λ

σ

|σ| (v)

• Constitutive relations (iv), (v) and (10.56)

σ = Eε− Eλσ

σe

(vi)

λ ≥ 0 , Φ ≤ 0 , λΦ = 0 (vii)

• Yield condition:

Φ = |σ| − σy ≤ 0 (viii)



• Tangent relation at loading (L):

Φtr = 2Gσ

|σ| [ε− ε⊥] (ix)

h = 3G (x)

⇒λ =

2

3

σ

|σ| [ε− ε⊥] (xi)

• Plastic strain rate at loading (L):

[εp]ij = [εpdev]ij = λ[ν]ij =

1

3[ε− ε⊥]

2

−1

−1

(xii)

• Elastic strain rate at loading (L)

[εe]ij = [ε]ij − [εp]ij =

ε

ε⊥

ε⊥

− 1

3[ε− ε⊥]

2

−1

−1

=1

3[ε+ 2ε⊥]

1

1

1

def=

εe

εe⊥εe⊥

(xiii)

Condition: εe⊥ = −νεe + (xiii) ;

1

3[ε+ 2ε⊥] = −ν [ε+ 2ε⊥]

; ε⊥ = −1

2ε ⇒ εe = 0 (xiv)



• Resulting rates at loading (L):

[ε]ij = [εp]ij =ε

2

2

−1

−1

, εp = ε (xv)

[εe]ij = 0 ·

1

−ν−ν

, εe = 0 (xvi)

• Tangent relations:

[σ]ij = 0 ·

1

0

0

, σ = 0 (L) (xvii)

[σ]ij = Eε

1

0

0

, σ = Eε (U) (xviii)

Box 10.3: Constitutive relations for prototype perfect plasticity model (uniaxial stress

state).



• Constitutive relations:

σ = Eεe = Eε− Eλσ

σe

(xix)

λ ≥ 0 , Φ ≤ 0 , λΦ = 0 , Φdef= σe − σy (xx)

• Backward Euler applied to (xix, xx) at (L), Φtr > 0:

R\σ(σe, µ) = σe − σtr

e + Eµ = 0 (xxi)

R\µ(σe) = σe − σy = 0 (xxii)

where

σtre

def=∣∣σtr∣∣ , Φtr def

= σtre − σy , σtr def

= n−1σ + E∆ε

Solution:

µ =σtr

e − σy

E, σe = σy ⇒ (xxiii)

σ = c(µ)σtr , c(µ) = 1 − Eµ

σtre

=σy

σtre

≤ 1 (xxiv)

• Backward Euler applied to (xix, xx) at (U), Φtr ≤ 0:

µ = 0 (xxv)

σ = σtr (xxvi)

Box 10.4: Constitutive integrator for prototype perfect plasticity model (uniaxial stress

state).

10.4.5 Appendix II: Voigt format of prototype model

In the Voigt format, the split of stress into deviatoric and spherical parts reads

σ = σdev + σmI (10.97)



where we introduced the (matrix) notation

σ =

σ11

σ22

σ33

σ23

σ13

σ12

, σmdef=

1

3(σ11 + σ22 + σ33), I =

1

1

1

0

0

0

(10.98)

by which the deviator (column) vector becomes

σdevdef=

σdev,11

σdev,22

σdev,33

σdev,23

σdev,13

σdev,12

= σ − σmI =

13(2σ11 − σ22 − σ33)

13(−σ11 + 2σ22 − σ33)

13(−σ11 − σ22 + 2σ33)

σ23

σ13

σ12

(10.99)

In principal coordinates, we obtain the component (column) vector σ as

σdef=

σ1

σ2

σ3

0

0

0

, σmdef=

1

3(σ1 + σ2 + σ3) (10.100)

by which

σdev = σ − σmI =

13(2σ1 − σ2 − σ3)

13(−σ1 + 2σ2 − σ3)

13(−σ1 − σ2 + 2σ3)

0

0

0

, σmdef=

1

3(σ1 + σ2 + σ3) (10.101)



From (1.119) 3 and (10.101) follows that the “equivalent stress” σe can be expressed as

σe =

√

3

2

[(σdev,11)

2 + (σdev,22)2 + (σdev,33)

2 + 2(σdev,23)2 + 2(σdev,13)

2 + 2(σdev,12)2] 1

2

=1√2

[(σ2 − σ3)

2 + (σ1 − σ3)2 + (σ1 − σ2)

2] 1

2 (10.102)

Let us now introduce the “odd” column vector σ of components w.r.t. the chosen Carte-

sian coordinates as

σdef=

σ11

σ22

σ33

2σ23

2σ13

2σ12

, σdev = σ − σmI (10.103)

Hence, the only difference to σ is the factor 2 in the shear components, and we conclude,

upon comparison with the expression for ε in (1.66), that σdev has a “strain–like” structure.

It is now possible to express σe and the gradient ν as

σe =

(3

2σT

devσdev

) 12

, νdef=∂σe

∂σ=

3

2σe

σdev (10.104)

(Show this as homework!).

We are now in the position to summarize the relevant constitutive relations in Voigt

format, that are pertinent to the prototype model: Hooke’s law for isotropic linear elastic

response reads

σ = Eeεe = Ee(ε− εp) (10.105)

where Ee was given by (7.68)

Ee =

2G+ L L L 0 0 0

L 2G+ L L 0 0 0

L L 2G+ L 0 0 0

0 0 0 G 0 0

0 0 0 0 G 0

0 0 0 0 0 G

(10.106)



From the relations

σ = 2Gεe + LεevI, σm = Kεev with K =1

3(2G+ 3L) (10.107)

we obtain

σdevdef= σ − σmI = 2Gεe − (K − L

︸︷︷︸23G

)εevI = 2Gεedev (10.108)

Moreover, we can rewrite (10.108) and (10.107) 2 as

σdev = 2G(εdev − εpdev), σm = K(εv − εpv) (10.109)

which corresponds to (10.76).

Now, the plastic flow rule can be expressed as

εp = λ∂φ

∂σ= λ

3

2σe

σdev (10.110)

Finally, the constitutive rate equations become

˙σdev = 2Gεdev − λ3G

σe

σdev, σm = Kεv (10.111)

λ ≥ 0, Φ ≤ 0, λΦ = 0 (10.112)

As to the integration of the constitutive relations, we obtain from (10.111)

σdev = σtrdev −

3Gµ

σe

σdev, σm = σtrm (10.113)

where

σtrdev

def= n−1σdev + 2G∆εdev, σtr

m = n−1σm +K∆εv (10.114)

which correspond to (10.86) with (10.87). Further rearrangement in (10.113)1 gives

(

1 +3Gµ

σe

)

︸︷︷︸

=k>0

σdev = σtrdev ⇒ kσdev = σtr

dev (10.115)

Upon multiplying the LHS of (10.115)1 with kσdev and RHS by σtrdev, we obtain kσe = σtr

e

or

σe − σtre + 3Gµ = 0 (10.116)


10.5 The constitutive framework - Hardening plasticity 277

which is precisely (10.88). Hence, the solution for µ in (10.91) follows and, finally, we

obtain

σ = cσtrdev

︸︷︷︸

σdev

+σtrmI → σ, σdev (10.117)

where the scalar c was given in (10.92)2.

10.5 The constitutive framework - Hardening plas-

ticity

10.5.1 Free energy and thermodynamic forces

The motion of the current yield surface Φ = 0 in stress space with plastic deformation is

denoted hardening. Such hardening may result in expansion (without change of shape),

translation (without change of size or shape), or distortion of the current yield surface.

The corresponding hardening modes are commonly denoted isotropic, kinematic and dis-

tortional hardening, respectively, which will be given a more precise meaning later.5

To quantify the hardening response, we introduce a set of additional variables, which are

tensors of even order (scalars, 2nd order tensors and 4th order tensors). The components

of these hardening variables are collected in (the column vector) k, which is added to the

arguments of Ψ. Hence, the free energy is represented as Ψ(εe, k).

The constitutive equations are now given as

σ =∂Ψ

∂εe, κ = −∂Ψ

∂k(10.118)

where κ are the hardening stresses. The dissipation inequality becomes

D = σ : εp + κTk ≥ 0 (10.119)

5More adequate notions would be expansional (instead of isotropic) and translational (instead of

kinematic), but the classical expressions are used here.



10.5.2 Representation of hardening - Constraints and classifica-

tion

A quite general type of hardening representation is defined by functional relationships of

the format Φ(σ, κ; k), i.e. both k and κ are used to represent the hardening. The question

is whether this is possible? Since hardening is defined as the “motion of the yield surface

in stress space due to plastic deformation”, it is clear that κ is eligible as an argument of

Φ only if κ = κ(k), i.e. κ may not depend on elastic strains εe. It then follows that

∂κ

∂εe=∂σ

∂k= 0 (10.120)

Moreover, this situation can occur only if it is possible to decompose Ψ additively in elastic

and plastic parts:

Ψ(εe, k) = Ψe(εe) + Ψp(k) ; σ = σ(εe) , κ = κ(k) (10.121)

On the other hand, if we assume the representation Φ(σ; k), then it is possible to retain

the general format of Ψ(εe, k) that gives σ = σ(εe, k). Clearly, it is also possible to adopt

(10.121) in such a case.

We are now in the position to make the following classification of hardening:

• Simple hardening: Φ(σ(εe), κ(k)) or Φ(σ(εe), κ(k); k). It is possible to choose the

flow and hardening rules such that the resulting model is contained in the category

of Standard Dissipative Materials and Generalized Standard Dissipative Materials,

respectively.

• Non-simple hardening: Φ(σ(εe); k) or Φ(σ(εe, k); k). It is not possible to choose

the hardening rules such that the resulting model is contained in any category of

standard materials.

We remark that most models in engineering practice employ the framework of simple

hardening. However, sometimes it is desirable to take into account that the elastic stiffness

is affected by the development of hardening. Such elastic-plastic coupling can obviously

only be modeled within the framework of non-simple hardening. A pertinent example is

the densification of porous materials (such as soils and powders), which affects the yield

surface as well as the elastic moduli.



Henceforth in this Chapter, we shall (for simplicity) employ the most basic format of Φ

within the framework of simple hardening. We thus consider yield functions Φ (σ, κ),

which are assumed to be convex in the space of dissipative stresses (σ, κ). The corre-

sponding convex set of plastically admissible states E is defined as

E = {σ, κ | Φ (σ, κ) ≤ 0} (10.122)

that is (still) assumed to contain the origin, i.e. Φ(0, 0) ≤ 0. Moreover, we define the

admissible region in stress space as the subset of E, denoted Eκ, such that

Eκ = {σ | Φ (σ, κ) ≤ 0, κ fixed } (10.123)

. .PSfrag replacements

σ2

σ1

Φ(σ, 0) = 0

Φ(σ, κ) = 0

plastic state

elastic state

Figure 10.7: Current yield surfaces in stress space as a result of hardening.

10.5.3 Associative structure - Postulate of Maximum Dissipa-

tion

It is possible to generalize the MD-postulate in straight-forward fashion to the space of

dissipative stresses. In this case the dissipation function is expressed as

D (σ, κ) = σ : εp + [κ]Tk , (σ, κ) ∈ E (10.124)

where D is now a function of (σ, κ) for given values of (εp, k). Hence, the MD-principle

is formulated as the constrained maximum of D such that

(σ, κ) = arg[max D (σ∗, κ∗) , ∀ (σ∗, κ∗) ∈ E

](10.125)



or in terms of the variational inequality:

D (σ, κ) −D (σ∗, κ∗) ≥ 0, ∀ (σ∗, κ∗) ∈ E (10.126)

Generalizing the arguments for perfect plasticity, we conclude that the pair (εp, k) must

satisfy the normality condition in the space spanned by (σ, κ). Correspondingly, for a

smooth yield surface, the Lagrangian function L is now defined as

L(σ∗, κ∗, λ∗) = −D(σ∗, κ∗) + λ∗Φ(σ∗, κ∗) (10.127)

and the optimality conditions are

∂L(σ, κ, λ)

∂σ= −εp + λ

∂Φ(σ, κ)

∂σ= 0 (10.128)

∂L(σ, κ, λ)

∂κ= −k + λ

∂Φ(σ, κ)

∂κ= 0 (10.129)

λ ≥ 0, Φ(σ, κ) ≤ 0, λΦ(σ, κ) = 0 (10.130)

which give rise to the associative plastic flow and hardening rules:

εp = λν, k = λζ with ζdef=∂Φ

∂κ(10.131)

We may now differentiate σ and κ in (10.118) and invoke (10.131) to obtain the tangent

relations

σ = Ee : ε − λEe : ν with E

e(εe) =∂2Ψ

∂εe ⊗ ∂εe(10.132)

κ = −λHζ with H =∂2Ψ

∂k∂kT(10.133)

where H is the symmetric matrix of hardening moduli. If H is positive definite, then

the response is characterized as strictly hardening. If, on the other hand, H is negative

definite, then the response is strictly softening. In all other cases, the hardening/softening

characteristics are not unique but depend on the actual loading situation. The significance

of these characteristics will be discussed later.

10.5.4 Continuum tangent relations

Even in the case of hardening the consistency conditions (10.17) hold, and the conditions

defining plastic and elastic loading, respectively, are the same as those already derived



for perfect plasticity. For the sake of completeness, we shall next (re)derive the pertinent

tangent stiffness relations, while pointing out the additional contributions due to harden-

ing. Hence, we consider the situation that the current state is “plastic”, i.e. that Φ = 0.

Upon differentiating Φ, we now obtain

Φ = ν : σ + ζTκ ≤ 0 (10.134)

Inserting (10.132) and (10.133) into (10.134), we still obtain

Φ = Φtr − hλ ≤ 0 (10.135)

The loading function Φtr is still defined as

Φtr def= ν : E

e : ε (10.136)

whereas the plastic modulus h is defined as

hdef= ν : E

e : ν + H with Hdef= ζTHζ (10.137)

We shall denote H the effective hardening modulus.

It is recalled that a necessary and sufficient condition for uniqueness of response, i.e. that

a unique value of λ exists for any given value of Φtr, is that h > 0. Provided this condition

is satisfied, the two relevant loading situations are distinguished as follows:

If Φtr > 0, then we must have plastic loading (L), defined by λ > 0 and Φ = 0, and from

(10.135) we obtain the solution

λ =1

hΦtr > 0 (10.138)

If Φtr ≤ 0, then we must have elastic unloading (U), defined by λ = 0 and Φ ≤ 0, and

from (10.135) we obtain

Φ = Φtr ≤ 0 (10.139)

Remark: The condition h > 0 is a condition of controllability of loading. This refers

to the proper formulation of the constitutive model, and it must not be confused with

uniqueness of the solution to a given boundary value problem. 2

We shall now summarize the tangent relations. Consider the case of plastic loading (L).

By inserting (10.138) into (10.131), we obtain

εp =Φtr

hν =

1

hν ⊗ ν : E

e : ε, k =Φtr

hζ =

1

hζν : E

e : ε (10.140)



The structure of the tangent stiffness relation becomes identical to that of perfect plas-

ticity, i.e.

σ = Eep : ε with E

ep def= E

e − 1

hE

e : ν ⊗ ν : Ee (10.141)

In the case of elastic unloading (U), we obtain

εp = 0, k = 0 (10.142)

and, therefore,

σ = Ee : ε (10.143)

10.5.5 Significance of hardening versus softening

So far we have introduced the generic notion of “hardening” without assessing its physical

significance. This is best done under stress control, i.e. for given σ, since hardening

represents the “motion in stress space” of the yield surface due to inelastic deformation.

To this end we reconsider (10.134) at stress control and under the condition of plastic

loading (L), i.e.

Φ = ν : σ − Hλ = 0, λ > 0 (10.144)

Depending on the sign of H, we define the three different situations w.r.t. the direction

of σ:

ν : σ > 0 if H > 0 hardening response

= 0 = 0 limit state (perfectly plastic response)

< 0 < 0 softening response

(10.145)

and these conditions have a simple geometric interpretation as shown in Figure 10.6.

Hardening now obviously means that the projection of the yield surface in stress space

must expand (locally), whereas softening means that it must shrink (locally).

In the case of elastic unloading (U), then

Φ = ν : σ < 0 (λ = 0) (10.146)

Obviously, both when the response is softening during plastic loading and when it is

purely elastic, the stress rate is directed inwards the current yield surface. Hence, the

sign of ν : σ can not be used as a diagnostic measure of (L) versus (U) when softening is

involved.



= =PSfrag replacementsσ for H > 0, ν : σ > 0

σ for H = 0, ν : σ = 0

σ for H < 0, ν : σ < 0

σtrE

e : ν

Φ(σ, κ) = 0

Figure 10.8: Tangential response for hardening, perfectly plastic and softening behavior.



10.5.6 Significance of total mechanical dissipation versus

“plastic dissipation”

Not withstanding the fact that D ≥ 0, it appears that the rate of plastic work Dp def= σ : εp,

which is commonly denoted “plastic dissipation”, can be negative. Indeed, this is the

situation for sufficient amount of kinematic hardening, which situation is demonstrated

in Figure 10.9. In fact, this figure shows that it is possible to guarantee that Dp ≥ 0 only

as long as the origin of the stress space is located inside the current yield surface Φ = 0,

i.e. 0 ∈ ∂Eκ. It is evident from Figure 10.9 that this is not always the case.

PSfrag replacements

εp

σσ

σεp

εp

Φ(σ, κ) = 0

Φ(σ, κ) = 0, κ fixed

1

2

Figure 10.9: Two positions in stress space of the current yield surface representing situa-

tions where the “plastic dissipation” (1) is always positive, (2) may be negative.

10.6 The constitutive integrator - Hardening

plasticity

10.6.1 Backward Euler method

We recall the generic constitutive relations in terms of the flow and hardening rules for a

smooth yield surface

εe = ε − λν(σ(εe), κ(k)) (10.147)

k = λζ(σ(εe), κ(k)) (10.148)


10.6 The constitutive integrator - Hardeningplasticity 285

λ ≥ 0, Φ(σ(εe), κ(k)) ≤ 0, λΦ(σ(εe), κ(k)) = 0 (10.149)

Applying the Backward Euler rule to the set of equations (10.147) to (10.149), we then

obtain the following incremental equations (µ = ∆tλ)

εe = εe,tr − µν(σ(εe), κ(k)) (10.150)

k = n−1k + µζ(σ(εe), κ(k)) (10.151)

µ ≥ 0, Φ(σ(εe), κ(k)) ≤ 0, µΦ(σ(εe), κ(k)) = 0 (10.152)

In the case of plastic loading (L), we may replace (10.150) to (10.152) by

R\ε(ε

e, k, µ) = εe − εe,tr + µν(σ(εe), κ(k)) = 0 (10.153)

R\k(ε

e, k, µ) = k − n−1k − µζ(σ(εe), κ(k)) = 0 (10.154)

R\µ(εe, k) = Φ(σ(εe), κ(k)) = 0 (10.155)

or

R\(X) = 0 with X =

εe

k

µ

, R

\ =

R\ε

R\k

R\µ

(10.156)

For given ε(tn) it is possible to solve for X from (10.156) in an iterative fashion along

the same lines as for perfect plasticity (discussed in Section 10.3). We thus immediately

introduce the scaled “out-of-balance” forces

Rε = Ee : R\

ε, Rk = HR\k, Rµ = R\

µ (10.157)

which are associated with the symmetric Jacobian matrix J defined as

J =

Ee + µEe : N : E

e −µEe : NTκH E

e : ν

−µHNκ : Ee H + µHZH −Hζ

ν : Ee −ζTH 0

(10.158)

Here, we have introduced the notation

Nκdef=

∂2Φ

∂κ∂σ, Z

def=

∂2Φ

∂κ∂κT(10.159)



10.6.2 Backward Euler method – Constrained minimization prob-

lem

10.6.3 ATS-tensor for BE-rule

In order to compute the ATS- tensor Ea we need

dR\ε|X = −dε, dR\

k|X = 0, dR\µ|X = 0 (10.160)

by which we may obtain the “scaled” differentials

dRε|X = −Ee : dε, dRk|X = 0, dRµ|X = 0 (10.161)

We need to express dεe in terms of dε, which is obtained from the solution of

JdX = −dR|X (10.162)

which with (10.158) and (10.161) can be expanded as

Ee + µEe : N : E

e −µEe : NTκH E

e : ν

−µHNκ : Ee H + µHZH −Hζ

ν : Ee −ζTH 0

: dεe

dk

dµ

=

Ee : dε

0

0

(10.163)

By solving for dεe from (10.163), it is possible to derive an expression of Ea that preserves

the tensorial structure in a quite explicit fashion. Our aim is now to eliminate dk and dµ

from the equations in (10.163) in such a fashion that dεe can be expressed in terms of dε.

The result is given in the following theorem:

Theorem: The ATS-tensor Ea can be expressed as

Ea = Eepa

def= E

ea −

1

ha

Eea : νa ⊗ νa : E

ea (10.164)

where the following notation was introduced:



Eea is the Algorithmic Elastic Stiffness (AES) tensor defined as 6

Eea =

[[Ee]−1 + µN − µ2NT

κHaNκ

]−1(10.165)

Moreover, νa is the algorithmic gradient of Φ defined as

νadef= ν − µNT

κHaζ (10.166)

whereas ha is the algorithmic plastic modulus defined as

hadef= νa : E

ea : νa + Ha with Ha

def= ζTHaζ (10.167)

Finally, the algorithmic hardening matrix Ha is defined as

Hadef=[H−1 + µZ

]−1(10.168)

Proof: From the second equation in (10.163) we may first solve for dk to obtain

Hdk = Haζdµ+ µHaNκ : Ee : dεe (10.169)

which may be inserted into the first equation of (10.163) to yield, after arrangement of

terms, the expression

dσ = Ee : dεe = E

ea : [dε − νadµ] (10.170)

Now, upon back-substituting (10.170) into (10.169), we obtain

Hdk =[Haζ − µHaNκ : E

ea : νa

]dµ+ µHaNκ : E

ea : dε (10.171)

It now remains to compute dµ from the third equation in (10.163), which represents

the consistency condition dΦ = 0. Upon inserting (10.170) and (10.171) into the third

equation, we may solve for dµ in terms of dε as follows:

dµ =1

ha

νa : Eea : dε (10.172)

Finally, upon inserting (10.172) into (10.170), we obtain the desired expression Eepa given

in (10.164). 2

6The matrix multiplication invoking Nκ should be understood as follows:

NTκHaNκ =

∑

α,β

(Ha)αβ

∂2Φ

∂κα∂σ⊗

∂2Φ

∂σ∂κβ



Like in the case of perfect plasticity, we conclude that the corresponding CTS-tensor Eep

is obtained as the special case

limµ→0

Eepa = E

ep (10.173)

This identity follows readily when µ = 0, in which case we obtain Eea = E

e, νa = ν and

ha = h.

Remark: In the special case that ν = ν(σ) and ζ = ζ(κ), we obtain the simplified

expressions:

Eea =

[[Ee]−1 + µN

]−1, νa = ν (10.174)

A trivial example is perfect plasticity (when also H = 0). 2

Finally, we may derive the following algorithmic tangent relation directly from (10.168)

and (10.169):

dκ = F a : dε with F adef= − 1

ha

Haζaνa : E

ea (10.175)

where we introduced the algorithmic gradient ζa

as follows:

ζa

= ζ − µNκ : Eea : νa (10.176)

10.6.4 Backward Euler method for linear elasticity and linear

hardening

In the case of linear elasticity, defined by σ = Ee : εe, and linear (simple) hardening,

defined by κ = −Hk (when Ee and H are constant), then we may rewrite (10.147) to

(10.149) as follows:

σ = Ee : ε − λEeν(σ, κ) (10.177)

κ = −λHζ(σ, κ) (10.178)

λ ≥ 0, Φ(σ, κ) ≤ 0, λΦ(σ, κ) = 0 (10.179)

Hence, the solution is sought directly in the generalized stress space of σ and κ.

In the case of plastic loading (L), the system (10.153) to (10.155) is now replaced by

R\σ(σ, κ, µ) = σ − σtr + µE : ν(σ, κ) = 0 (10.180)

R\κ(σ, κ, µ) = κ− n−1κ+ µHζ(σ, κ) = 0 (10.181)



R\µ(σ, κ) = Φ(σ, κ) = 0 (10.182)

or

R\(X) = 0 with X =

σ

κ

µ

, R

\ =

R\σ

R\κ

R\µ

(10.183)

The scaled “out-of-balance” forces

Rσ = [Ee]−1 : R\σ, Rκ = H−1R\

κ, Rµ = R\µ (10.184)

are associated with the symmetric Jacobian matrix J that is now defined as

J =

[Ee]−1 + µN µNTκ ν

µNκ H−1 + µZ ζ

ν ζT 0

(10.185)

Remark: The scaled format is obviously relevant only if H is nonsingular. For example,

if H is semi-definite (corresponding to perfectly plastic response at a certain loading

direction), it is not possible to use κ as the unknown variable. However, k can still be

computed. Hence, the generic algorithm in 10.6.1 is also more general. 2

Special case: Formulation in reduced stress space

In some cases it is possible to solve for σ and κ, for given ∆ε, in terms of µ from (10.179)

and (10.180), respectively. The yield criterion (10.181) can then be expressed as

ϕ(µ)def= Φ (σ(µ), κ(µ)) = 0 (10.186)

whereby the local problem (10.182) is reduced to one single scalar (nonlinear) equation

in µ. Hence, in the Newton iterations we obtain

R\ = ϕ(µ), X = µ and J \ def=

dϕ

dµ(10.187)

When µ has been computed, we may find σ(µ) and κ(µ). The ATS-tensor can still be

computed according to the general structure in Subsection 10.6.2.



10.6.5 Concept of Closest-Point-Projection for linear elasticity

and linear hardening

The Closest-Point-Projection property discussed in Subsection 10.3.5 for linear elasticity

in conjunction with perfect plasticity can be generalized to the case of linear hardening,

whereby the updated values (σ, κ) can be interpreted as the projection of (σtr, κtr), with

κtr def= n−1κ, in a special metric. To this end, we first generalize the complementary elastic

energy Ψ∗ as follows:

Ψ∗(σ, κ) =1

2σ : [Ee]−1 : σ +

1

2κTH−1κ (10.188)

while making the significant assumption that H is positive definite, i.e. the material re-

sponse in strictly hardening. It is then simple to show (by inspection, as homework) that

the solution (σ, κ) to the system (10.180) to (10.182) is also the solution of the constrained

convex minimization problem

(σ, κ) = arg

[

min(σ∗,κ∗)∈E

Ψ∗(σ∗ − σtr, κ∗ − κtr)

]

(10.189)

which, again, defines the Closest-Point-Projection-Method (CPPM). More schematically,

the CPPM is defined as the mapping

(σ, κ) = CPPM(σtr, κtr;Ee, H

)(10.190)

where (Ee, H) indicates the projection metric.

10.7 Prototype model: Hooke elasticity and von Mises

yield surface with linear mixed hardening


The most widely used model within metal plasticity employs mixed isotropic and kine-

matic hardening of the von Mises’ yield surface. This model employs the same free energy

as in the case of perfect plasticity, i.e.

σ = Ee : εe with E

e = 2GIsymdev +KI ⊗ I ; (10.191)


10.7 Prototype model: Hooke elasticity and von Mises yield surface with linearmixed hardening 291

σ = σdev + σmI (10.192)

where

σdev = 2Gεedev, σm = Kεevol (10.193)

In addition, the free energy representing hardening is chosen as follows:

Ψp =1

2rHk2 +

1

2[1 − r]Ha2

e with ae =

√

2

3|adev| (10.194)

where k and a are isotropic and kinematic hardening variables, respectively. Moreover, H

is the constant hardening modulus of the uniaxial stress-strain curve (which is assumed

to be bilinear), whereas r is a parameter that controls the relation between isotropic and

kinematic hardening: r = 0 represents purely kinematic hardening, and r = 1 represents

purely isotropic hardening.

The von Mises yield function with mixed hardening is defined as

Φ = σrede − σy − κ, σred

e =

√

3

2|σred

dev| with σred def= σ − α (10.195)

where σrede is the (reduced) equivalent stress, σy is the initial yield stress, κ is the “drag-

stress” due to isotropic hardening, and α is the “back-stress” due to kinematic hardening.

With (10.194) we obtain the dissipative stresses

κ = −∂Ψp

∂k= −rHk, α = −∂Ψp

∂a= −2

3[1 − r]Hadev (10.196)

from which we obtain

Hdef= −

∂κ

∂k

∂α

∂k∂κ

∂a

∂α

∂a

=

[

rH 0

0 23[1 − r]HI

symdev

]

(10.197)

Associative flow and hardening rules - Prager’s rule of kinematic hardening

The associative flow and hardening rules for the considered model are given as

εp = λν with νdef=∂Φ

∂σ=

3

2σrede

σreddev (10.198)

k = λζκ with ζκdef=∂Φ

∂κ= −1 (10.199)



a = λζα with ζαdef=

∂Φ

∂α= − 3

2σrede

σreddev = −ν (10.200)

or

εp = λ3

2σrede

σreddev, k = −λ, a = −λ 3

2σrede

σreddev = −εp (10.201)

The associative kinematic hardening rule in (10.201) is originally due to Prager (1955).

Within the present thermodynamic framework, this hardening rule has been given the

more precise meaning that it ensures a non-negative dissipation (since it is formulated as

an associative rule).

Using Ee in (10.191) together with ν in (10.198), we may express the rate equation for σ

more explicitly as

σdev = 2Gεdev − λ3G

σrede

σreddev, σm = Kεvol (10.202)

We may also combine (10.196) with (10.201) to obtain the differential equations for the

hardening stresses:

κ = λrH (10.203)

αdev = λ[1 − r]H

σrede

σreddev, αm = 0 (10.204)

which are subjected to the homogeneous initial conditions κ(0) = 0 and α(0) = 0.

The characteristic response is illustrated for purely isotropic, purely kinematic and mixed

hardening, respectively, in Figures 10.10, 10.11 and 10.12.

Remark: By the introduction of kinematic hardening, it is possible to simulate the

Bauschinger effect, i.e. that the yield stress in compression, upon reversed loading from

tension, is smaller than it was in tension. 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

Tangent relations

In a “plastic” state, defined as Φ = 0, the tangent relations can be established. Upon

inserting the pertinent relations above into the expression for h in (10.137), we obtain

h = 3G+ H with H = H (10.205)



PSfrag replacements

σ

σdev

σ3

σ2

Φ = 0

σ1

σy

2[σy+κ]

εp

H1

r = 1

εp

reversed loadingσy+κ

σy+κ

σy+κ

Figure 10.10: Linear isotropic hardening of von Mises yield surface, (a) Hardening in

deviator stress space, (b) Uniaxial stress versus plastic strain characteristics.

The loading function Φtr in (10.136) becomes

Φtr =3G

σrede

σreddev : ε ⇒ λ =

3G

hσrede

σreddev : ε (10.206)

From (10.201) and (10.206), we obtain the constitutive evolution equations for the isotropic

and kinematic hardening variables as:

εp =9G

2h[σrede ]2

σreddev ⊗ σred

dev : ε (10.207)

k = − 3G

hσrede

σreddev : ε (10.208)

a = −εp = − 9G

2h[σrede ]2

σreddev ⊗ σred

dev : ε (10.209)

Finally, we obtain the CTS-tensor Eep in (10.141) as

Eep = E

e − 9G2

h[σrede ]2

σreddev ⊗ σred

dev (10.210)



PSfrag replacements

σ2

σ1σ3

σdev

Φ = 0

σ

σy

σyσy

2σy

εp

r = 0

α

reversed loading

shake-down

εp

H1

αdev

Figure 10.11: Linear kinematic hardening of von Mises yield surface, (a) Hardening in

deviator stress space, (b) Uniaxial stress versus plastic strain characteristics.

Dissipation inequality

It may be of interest to check explicitly that the dissipation D (or the rate of internal

production of entropy) is non-negative:

D def= σ : εp + κk + α : a = λ [σ : ν + κζκ + α : ζα]

= λ[σrede − κ] = λσy > 0 (10.211)

where it was used that Φ = 0, i.e. that σrede − κ = σy.

Significance of hardening/softening

In order to illustrate the significance of hardening and softening, as discussed in Subsection

10.5.4, we consider the scalar product ν : σ. From H = H, we obtain from (10.144) in

the case of (L):

ν : σ = Hλ (10.212)

which result is illustrated in Figure 10.13.



PSfrag replacements

σy+κ

εp2[σy+κ]

σ2

εp

σ1σ3

σdev

αdev

σy

α(1 − r)H

H

1

1

1

0< r < 1

reversed loadingno shake-down

Figure 10.12: Linear mixed isotropic and kinematic hardening of von Mises yield surface,

a) Hardening in deviator stress space, (b) Uniaxial stress versus plastic strain character-

istics.


Applying the Backward Euler rule to integrate the evolution equations for σ, κ and α in

(10.202) to (10.204), we obtain

σdev = σtrdev −

3Gµ

σrede

σreddev, σm = σtr

m (10.213)

κ = n−1κ+ rH∆λ (10.214)

αdev = n−1αdev +[1 − r]Hµ

σrede

σreddev, αm = 0 (10.215)

According to the basic recipe, given in Chapter 6, the unknown variables in the local

solution vector X are σdev, κ, αdev(= α) and µ. However, we shall aim at a formulation

that includes only the scalar quantities σrede , κ and µ (in analogy with the situation for

perfect plasticity). In order to achieve this goal, we compute σreddev by subtracting (10.215)

from (10.213) to obtain

σreddev = σ

red,trdev − 3Gµ[1 + a]

σrede

σreddev, a

def=

[1 − r]H

3G(10.216)

where we introduced

σred,trdev

def= σtr

dev − n−1αdev (10.217)



PSfrag replacements

σedev

σ3 σ1

σdev

νσ2

σdev for H > 0σdev for H=0

σdev for H < 0

Figure 10.13: Tangential response for hardening, perfectly plastic and softening behavior

for the von Mises yield criterion.

In a strain-driven algorithm the trial value σred,trdev is computable a priori (at each new

timestep). Rearranging in (10.216), we obtain

[

1 +3Gµ

σrede

[1 + a]

]

σreddev = σ

red,trdev (10.218)

We now conclude that σreddev is the “radial return” from σ

red,trdev . Taking the norm of both

sides of (10.218), we obtain

σrede + 3Gµ[1 + a] = σred,tr

e with σred,tre

def=

√

3

2|σred,tr

dev | (10.219)

and, again, it appears that σrede ≤ σred,tr

e .

We are now in the position to establish the pertinent incremental constitutive relations

at loading (L) as follows:

R\σ(σred

e , µ) = σrede − σred,tr

e + 3Gµ[1 + a] = 0 (10.220)

R\κ(κ, µ) = κ− n−1κ− rHµ = 0 (10.221)

R\µ(σred

e , κ) = σrede − σy − κ = 0 (10.222)

The condition for loading is (still) that Φtr ≥ 0, where

Φtr def= σred,tr

e − σy − n−1κ (10.223)



Even in this case it is concluded that the equations (10.220) to (10.222) are linear. Upon

elimination, we obtain the solution

µ =Φtr

hwith h

def= 3G+H (10.224)

Using this value, we may compute

σrede = σred,tr

e − 3Gµ[1 + a] (10.225)

κ = n−1κ+ rHµ (10.226)

It now remains to compute the updated values of σdev, κ and α = αdev in terms of the

“trial” values σred,trdev , σtr

m, n−1κ and n−1α. From (10.215)1 and (10.216)2, we first obtain

σdevdef= σred

dev + αdev =

[

1 +3Gaµ

σrede

]

σreddev + n−1αdev (10.227)

However, using the radial return property

σreddev =

σrede

σred,tre

σred,trdev (10.228)

and inserting this expression in (10.227), while also using the solution (10.225), we obtain

σdev =

[

1 − 3Gµ

σred,tre

]

σred,trdev + n−1αdev (10.229)

Likewise, we may introduce (10.228) into (10.215) to obtain

αdev = n−1αdev +3Gaµ

σred,tre

σred,trdev (10.230)

Summarizing, we arrive at the updated values

σdev = c1σred,trdev + n−1αdev, σm = σtr

m (10.231)

κ = n−1κ+ rHµ (10.232)

αdev = c2σred,trdev + n−1αdev, αm = 0 (10.233)

where

c1 = 1 − 3Gµ

σred,tre

, c2 =3Gaµ

σred,tre

(10.234)

Finally, we consider separately the extreme situations of purely isotropic hardening (r = 1)

and purely kinematic hardening (r = 0). The pertinent solutions are as follows:



Special case: Linear pure isotropic hardening

From the general solution above, we obtain with r = 1:

σdev = c1σtrdev, σm = σtr

m (10.235)

κ = n−1κ+Hµ (10.236)

where

c1 = 1 − 3Gµ

σtre

(10.237)

This solution is shown in Figure 10.14.

It is possible to combine (10.235) and (10.236) to define the “vector” equation in (σe, κ)-

space as

(σe, κ) = (σtre ,

n−1κ) − (3G,−H)µ (10.238)

which is identical to the uniaxial stress situation if the elastic modulus E is replaced by

3G. 2

Special case: Linear pure kinematic hardening

From the general solution above, we obtain with r = 0

σdev = c1σred,tr + n−1αdev, σm = σtr

m (10.239)

αdev = c2σred,trdev + n−1αdev, αm = 0 (10.240)

where

c1 = 1 − 3Gµ

σred,tre

, c2 =Hµ

σred,tre

(10.241)

This solution is shown in Figure 10.15. 2

ATS-tensor

Linear mixed hardening belongs to the class of associative hardening rules, for which the

pertinent general expression of the ATS-tensor Ea was given in (10.164). We now obtain

N =3

2σrede

Q with Qdef= I

symdev − 3

2[σrede ]2

σreddev ⊗ σred

dev (10.242)



PSfrag replacements

Φ = 0

n−1Φ = 0

σdev

σtrdev

σ1

σ2σ3

Figure 10.14: Stress projection for CPPM applied to the von Mises criterion with linear

isotropic hardening.

PSfrag replacements

σ3

σ1

σ2

n−1Φ = 0

Φ = 0

σdev

σtrdev

αdev

n−1αdev

Figure 10.15: Stress projection for CPPM applied to the von Mises criterion with linear

kinematic hardening.



thus generalizing the expression in (8.117) to account for kinematic hardening. Still,

Q is an idempotent, singular projection tensor. The 2nd order eigentensors of Q, that

correspond to zero eigenvalue, are σreddev and I.

We also derive

Nκdef=

∂ν

∂κ∂ν

∂α

=

0

− 3

σrede

Q

(10.243)

Zdef=

∂ζκ∂κ

∂ζα

∂κ∂ζκ∂α

∂ζα

∂α

=

0 0

03

2σrede

Q

(10.244)

By combining the expression the expression for the hardening matrix H in (10.197) with

Z in (10.244), we obtain

Hadef=

[

(Ha)κκ 0

0 (Ha)αα

]

=

[

rH 0

0 23[1 − r]H

[

Isym − c2

1+c2Q

]

]

(10.245)

where the non-dimensional scalar c2 was defined in (10.234). Moreover, to obtain the

expressions for (Ha)αα in (10.245)2, we used the Sherman-Morrison formula.

The next task is to calculate the AES-tensor Eea. We start by observing that

NTκHaNκ =

∂ν

∂α: (Ha)αα :

∂ν

∂α=

3[1 − r]H

2[1 + c2][σrede ]2

Q (10.246)

where (10.243) and (10.245) were used. We may then obtain Eea (after some algebraic

manipulations) as

Eea = E

e − 2GbQ with b =

3Gµσrede

1 + 3Gµσrede

[1 + a], a

def=

[1 − r]H

3G(10.247)

We also obtain

νa = ν =3

2σrede

σreddev ; E

ea : νa =

3G

σrede

σreddev (10.248)

As to the algorithmic plastic moduli Ha and ha, we obtain Ha = H and ha(= h) = 3G+H.

Finally, we are in the position to establish Ea explicitly as

Ea = Ee − 2GbQ − 9G2

h[σrede ]2

σreddev ⊗ σred

dev (10.249)



When ∆ε = 0 we obtain µ = 0 and, consequently, b = 0. Hence, when ∆ε = 0 we may

readily retrieve the identity Ea = Eep upon comparing with the pertinent expression of

the CTS-tensor Eep given in (10.144).


The material parameters are chosen as

E = 200 MPa , ν = 0.3 , σy = 543 MPa

The values of H and r are allowed to vary in the different numerical examples below.

Numerical example 1: Uniaxial stress – Monotonic loading

As a first example of the present prototype model the tensile test under uniaxial stress is

(re)considered. The prescribed strain and stress components are:

ε11(t) = 5t ∗ 10−4

σ22 = σ33 = 0 , σij = 0 for i 6= j(10.250)

In Figure 10.16 the response is plotted in terms of σ11 versus ε11 for different values of

H. It is noted that the result is independent on the choice of r ∈ [0, 1] for this particular

loading case.

Numerical example 2: Uniaxial stress – Cyclic loading

In order to get illustrative curves for the difference between isotropic and kinematic hard-

ening we run an example with cyclic loading for r = 0, 0.5, 1, and H = 6.5 GPa. The

result is plotted in Figure 10.17.

Numerical example 3: Biaxial strain, plane stress

The second example for this prototype model is biaxial strain and plane stress, which can

be seen in Figure 10.18. Here, the prescribed stresses and strains are listed in equation

(10.251).

ε11(t) = ε22(t) = 5t ∗ 10−4 , εij = 0 , where i 6= j

σ33 = 0(10.251)

The stress-strain response is plotted in Figure 10.19 for different values of H.



0 0.01 0.02 0.03 0.04 0.050

0.2

0.4

0.6

0.8

1Uniaxial stress

ε11

σ 11 [G

Pa]

H = 8 GPa

H = 6.5 GPa

H = 5 GPa

Figure 10.16: Stress-strain behavior for different hardening under the condition of uniaxial

stress. Hooke elasticity and von Mises yield criterion with linear hardening.



−0.025 −0.02 −0.015 −0.01 −0.005 0 0.005 0.01 0.015 0.02 0.025−3000

−2000

−1000

0

1000

2000

3000Uniaxial stress, cyclic loading

ε11

σ 11 [M

Pa]

−0.025 −0.02 −0.015 −0.01 −0.005 0 0.005 0.01 0.015 0.02 0.025−3000

−2000

−1000

0

1000

2000

3000

ε11

σ 11 [M

Pa]

−0.025 −0.02 −0.015 −0.01 −0.005 0 0.005 0.01 0.015 0.02 0.025−3000

−2000

−1000

0

1000

2000

3000

ε11

σ 11 [M

Pa]PSfrag replacements

r = 0.5

r = 1

r = 0

(a)

(b)

(c)

Figure 10.17: Cyclic loading



PSfrag replacements

x1

x2

x3

ε11 6= 0

ε22 6= 0

σ33 = 0

Figure 10.18: Biaxial strain in plane x1x2 and plane stress

0 0.01 0.02 0.03 0.04 0.050

0.2

0.4

0.6

0.8

1

1.2

1.4Biaxial strain, plane stress

ε11

= ε22

σ 11 =

σ22

[GP

a]

H = 5 GPa

H = 6.5 GPa

H = 8 GPa

Figure 10.19: Stress-strain behavior for different hardening under the condition of bi-

axial strain with plane stress. Hooke elasticity and von Mises yield criteria with linear

hardening.



10.7.4 Appendix: Constitutive relations for the uniaxial stress

state

The a priori assumptions of stress and strain states pertinent to the uniaxial stress state are

given in Box 10.5. The developments leading to the tangent stiffness relation of interest

(in the axial direction) are given in Box 10.6, whereby the general multiaxial tangent

relations are exploited. The uniaxial stress-strain relation is shown in Figure 10.20.

PSfrag replacements

εp

1

H1+ H

E

σy

σ

ε

1

σyσy

σσ

H1

1

1

E

Figure 10.20: Uniaxial stress-strain relation for linear hardening.



• Stress tensor:

[σ]ij = σ

1

0

0

; [σdev]ij =

1

3σ

2

−1

−1

, σm =

1

3σ (i)

• Strain tensor:

[ε]ij =

ε

ε⊥

ε⊥

; [εdev]ij =

1

3[ε− ε⊥]

2

−1

−1

, εvol = ε+ 2ε⊥

(ii)

• Back stress tensor (α11def= 2α/3):

[α]ij = [αdev]ij =1

3α

2

−1

−1

(iii)

• Deviatoric reduced stress tensor:

[σreddev]ij =

1

3σred

2

−1

−1

with σred def

= σ − α (iv)

• Stress invariants:

σe = |σ| , αe = |α| , σrede = |σred| (v)

Box 10.5: Characterization of uniaxial stress state.



• Constitutive relation for stress, from (10.191) and uniaxial stress constraint:

{

σ = [L+ 2G]εe + Lεe⊥ + Lεe⊥ (a)

0 = Lεe + [L+ 2G]εe⊥ + Lεe⊥ (b)

where

L = K − 2G

3=

Eν

[1 − 2ν][1 + ν], G =

E

2[1 + ν], K =

E

3[1 − 2ν]

Solution of (a, b): εe⊥ = −νεe ⇒

σ = Eεe = E[ε− εp] (vi)

• Evolution equations for internal variables

[ν]ij =1

2

σred

|σred|

2

−1

−1

⇒ εp = λ

σred

|σred| (vii)

ζκ = −1 ⇒ k = −λ (viii)

[ζα]ij = −[ν]ij = −1

2

σred

|σred|

2

−1

−1

⇒ a = −λ σ

red

|σred| (ix)

• Constitutive relations (vi),(vii), (10.202), (10.203) and (10.129)

σ = Eε− Eλσred

|σred| (x)

κ = rHλ (xi)

α = [1 − r]Hλσred

|σred| (xii)

λ ≥ 0 , Φ ≤ 0 , λΦ = 0 (xiii)



• Yield condition:

Φ = |σred| − [σy + κ] ≤ 0 (xiv)

• Tangent relation at loading (L):

Φtr = 2Gσred

|σred| [ε− ε⊥] (xv)

h = 3G+ H (xvi)

⇒λ =

2G

h

σred

|σred| [ε− ε⊥] (xvii)

• Plastic strain rate at loading (L):

[εp]ij = [εpdev]ij = λ[ν]ij =

G

h[ε− ε⊥]

2

−1

−1

(xviii)

• Elastic strain rate at loading (L):

[εe]ij = [ε]ij − [εp]ij =

ε

ε⊥

ε⊥

− G

h[ε− ε⊥]

2

−1

−1

=

[

1 − 2G

h

]

ε+2G

hε⊥

G

hε+

[

1 − G

h

]

ε⊥

G

hε+

[

1 − G

h

]

ε⊥

def=

εe

εe⊥εe⊥

(xix)



Condition: εe⊥ = −νεe + (xix) ;

G

hε+

[

1 − G

h

]

ε⊥ = −ν[[

1 − 2G

h

]

ε+2G

hε⊥

]

; ε⊥ =Gh[2ν − 1] − ν

Gh[2ν − 1] + 1

ε ⇒ εe =1 − 3G

hGh[2ν − 1] + 1

ε =H

E +Hε (xx)

• Resulting rates at loading (L):

[ε]ij = ε

1

k

k

with k

def=

Gh[2ν − 1] − ν

Gh[2ν − 1] + 1

(xxi)

[εp]ij =E

2[E + H]ε

2

−1

−1

, εp =

E

E +Hε (xxii)

[εe]ij =H

E + Hε

1

−ν−ν

, εe =

H

E +Hε (xxiii)

• Resulting tangent stiffness relations:

[σ]ij =EH

E + Hε

1

0

0

, σ = Eεe =

EH

E +Hε[= Hεp

](L) (xxiv)

[σ]ij = Eε

1

0

0

, σ = Eε (U) (xxv)

• Comparison with uniaxial stress-strain curve, defined by σ = Hεp, H = H, cf

Figure 10.20.

Box 10.6: Prototype linear mixed hardening model (uniaxial stress state).



Remark: The condition for controllability of the loading process is E + H > 0, i.e.

H > −E, which is a constraint on the amount of softening that can be tolerated. This

issue is further discussed in Chapter 11. 2

• Summary of constitutive relations:

σ = Eε− Eλσred

σrede

(xxvi)

κ = rHλ (xxvii)

α = [1 − r]Hλσred

σrede

(xxviii)

λ ≥ 0 , Φ ≤ 0 , λΦ = 0 , Φdef= σred

e − [σy + κ] (xxix)

• Backward Euler applied to (xxvi) to (xxix) at (L), Φtr > 0, after rearrangement:

R\σ(σred

e , µ) = σrede − σred,tr

e + Eµ[1 + a] = 0 , adef=

[1 − r]H

E(xxx)

R\κ(κ, µ) = κ− n−1κ− rHµ = 0 (xxxi)

R\µ(σred

e , κ) = σrede − σy − κ = 0 (xxxii)

where

Φtr def= σred,tr

e −σy− n−1κ , σred,tre

def= |σred,tr| , σred,tr = n−1σ− n−1α+E∆ε (xxxiii)


10.8 Classical isotropic yield criteria 311

Solution:

µ =Φtr

hwith h

def= E +H (xxxiv)

σ = c1(µ)σred,tr + n−1α , c1(µ) = 1 − Eµ

σred,tre

(xxxv)

κ = n−1κ+ rHµ (xxxvi)

α = c2(µ)σred,tr + n−1α , c2(µ) =Eaµ

σred,tre

(xxxvii)

• Backward Euler applied to (xxvi) to (xxix) at (U), Φtr ≤ 0:

µ = 0 (xxxviii)

σ = σtr = n−1σ + E∆ε (xxxix)

κ = n−1κ (xl)

α = n−1α (xli)

Box 10.7: Constitutive integrator for prototype hardening plasticity model (uniaxial

stress state).

10.8 Classical isotropic yield criteria

10.8.1 Basic concepts - Cohesive and frictional character

The classical yield criteria of von Mises (1913) and Tresca (1864) are isotropic and

do not involve the mean-stress. When the mean stress does not effect the yielding char-

acteristics, these are denoted cohesive. Moreover, for a cohesive material it is normally

sufficient to assume plastic incompressibility, i.e. the volumetric part of the plastic strain

is zero. Cohesion essentially relates to the shear yield strength in crystal planes in met-

als. From a conceptual point of view, Tresca’s criterion may be thought of as a direct

generalization of slip due to the resolved shear stress on a predefined slip plane with the

assumption that the slip plane is determined in the principal triad. This gives rise to a

non-smooth yield surface, which makes the numerical treatment more difficult. Whereas



the von Mises criterion is defined by one single stress invariant (2nd invariant of σdev),

Tresca’s criterion requires two invariants (2nd and 3rd invariants of σdev), cf. below.

Many materials of engineering importance are highly sensitive to the applied mean stress

(1st invariant of σ): Soil, rock, concrete, ceramics and powder are typical examples.

When the mean stress has a significant effect on the yielding characteristics, these are

denoted frictional. Truly frictional materials, such as dry granular materials, have no

cohesion at all. A common feature of these materials is that they are porous (to a varying

degree). Frictional materials show significant plastic compressibility, i.e. the volumetric

part of the plastic strain is non-zero (dilatant or contractant behavior). Plasticity theory

for frictional materials is based on the failure theory by Coulomb in the 1770’s and by

Mohr around 1900.

Although pressure-dependence is most significant for non-metallic porous materials (as

described above), the mean stress may affect the yield strength in shear deformation even

for metallic materials as a result of the microstructural composition: Gray-cast iron is one

important example. Under intense straining, in particular at elevated temperature and

close to failure, steel and other metallic alloys may develop voids and microcracks to the

extent that the mean stress will have an influence on the process of plastic deformation.

As a consequence, also the elastic properties will be affected due to such a development

of material damage.

10.8.2 Isotropic yield criteria - General characteristics

That the yield criterion is isotropic means that Φ is a scalar invariant function of σ.

Possible arguments of Φ are then invariants of σ as follows:

• Spectral invariants, i.e. principal stresses, Is(σ) = {σ1, σ2, σ3}

• Basic invariants Ib,dev(σ) = {i1, j2, j3} or principal invariants Ip,dev(σ) = {I1, J2, J3}

• Geometric invariants {ζ, ρ, θ} or {p, q, θ} or {σm, σe, θ} or {σoct, τoct, θ}

The definition of these invariants (and how they are interrelated) are given in Subsection

10.8.9 (Appendix). Here, we only note that q = σe is the equivalent stress which is

associated with the von Mises criterion.



A typical cross-section of an isotropic yield surface in the deviatoric plane is shown in

Figure 10.21, whereby the stress space diagonal is pointing towards the viewer. ThePSfrag replacements

Φ(ξ, ρ, θ) = 0 or Φ(σm, σe, θ) = 0

σ1

σ2σ3

θ = 0◦

θθ = 60◦ρt ρy(θ)

ρc

Figure 10.21: Typical deviatoric cross-section of isotropic yield surface.

different meridian planes of special interest are defined in Subsection 10.8.9 (Appendix):

Tensile meridian (θ = 0o), compressive meridian (θ = 60o) and shear meridian (θ = 30o).

Meridian symmetry planes

Without loss of generality, it may be assumed that σ1 ≥ σ2 ≥ σ3, which means that the

only region of interest in the deviator planes is the sector defined by 0 ≤ θ ≤ 60o, as shown

in Figure 10.21. This is equivalent to the conclusion that the tensile and compressive

meridian planes (θ = 0o and θ = 60o) are symmetry planes.

To show this, we first conclude that the value of Φ is uniquely defined by the values of J2

and J3 (for given I1). We then consider two different positions on the deviatoric trace of

the yield surface defined by different θ- values, denoted θ(1) and θ(2). Since these values

must correspond to the same value of J2 and J3, it follows from (10.253) that

cos 3θ(1) = cos 3θ(2); θ(1) = ±θ(2) + n · 120o, n = 0, 1, . . . (10.253)

In particular, we obtain from (10.253) with u = 0 that θ(1) = −θ(2), which shows symmetry

w.r.t. θ = 0. Setting θ(1) = −θ(2) + 120o with θ(1) = 60o + α(1) and θ(2) = 60o + α(2), we

obtain 60o + α(1) = 60o − α(2) which shows symmetry w.r.t θ = 60o.



Mean stress independence

In metal plasticity, it was shown already by Bridgman (1923) that the mean stress σm

does not significantly affect yielding, i.e. we may write Φ = Φ(σe, θ). This means that the

yield surface is a cylinder along the ζ -axis in stress space. This property may be taken as

the definition of purely cohesive response. Moreover, for metals the uniaxial yield strength

is the same in tension as in compression. We may also assume that the “directional yield

stress” in the deviatoric plane is the same upon reversal of direction, which is expressed

as the condition Φ(σdev) = Φ(-σdev) for any σ. Taken together, the conditions above

infer that the yield surface is symmetrical w.r.t. the plane θ = 30o, so that the sector of

interest in the deviator planes has been reduced to 0 ≤ θ ≤ 30o. This is shown as follows:

Let the stress σ(1) be associated with J(1)2 and J

(1)3 , whereas σ(2) is associated with J

(2)2 and

J(2)3 . From the condition σ

(1)dev = -σ

(2)dev, we conclude that J

(1)2 = J

(2)2 , whereas J

(1)3 = −J (2)

3 .

Hence, for the corresponding values θ(1) and θ(2), we obtain

cos 3θ(1) = − cos 3θ(2); θ(1) = ±θ(2) + 60o + n · 120o, n = 0, 1, . . . (10.254)

Now, setting θ(1) = −θ(2) + 60o (for n = 0) with θ(1) = 30o +α(1) and θ(2) = 30o +α(2), we

obtain 30o + α(1) = 30o − α(2), which shows the symmetry w.r.t. θ = 30o.

Gradient of isotropic yield function - Normal to yield surface

Let us consider a regular, i.e. smooth, portion of the isotropic yield surface Φ = 0. The

gradient of Φ will be given subsequently in terms of different representations.

Φ = Φ(σ1, σ2, σ3), where (for convenience) σ1 ≥ σ2 ≥ σ3:

νdef=∂Φ

∂σ=

3∑

i=1

νimi with νi =∂Φ(σ1, σ2, σ3)

∂σi

(10.255)

where mi are the eigendyads of σ (and ν). It is possible to express mi explicitly in terms

of σ and σi via the Simo-Serrin’s formula (for distinct σi)7, as shown in Chapter 1:

mi =σi

di

[σ − [I1 − σi]I + I3[σi]−1σ−1], i = 1, 2, 3 (10.256)

7In the case they are not all distinct, it is still convenient to employ the “general” formula in (10.256)

upon numerical “perturbation” of the identical values by a very small amount.



where the scalars di are given as

di = 2[σi]2 − I1σi + I3[σi]

−1 (10.257)

Φ = Φ(σm, σe, θ):

ν = a1I + a2σdev + a3[σdev]2 (10.258)

where the coefficients ai(σm, σe, θ), which are scalar invariant functions, are given as

a1 =1

3

∂Φ

∂σm

+1

σe sin 3θ

∂Φ

∂θ=

1

3

∂Φ

∂σm

− 1

σe cos 3θ

∂Φ

∂θ(10.259)

a2 =3

2σe

[∂Φ

∂σe

+1

σe tan 3θ

∂Φ

∂θ

]

=3

2σe

[∂Φ

∂σe

− 1

σe cot 3θ

∂Φ

∂θ

]

(10.260)

a3 = − 9

2σ3e sin 3θ

∂Φ

∂θ= − 9

2σ3e cos 3θ

∂Φ

∂θ(10.261)

with θ = θ − 30o and the range of application 0 < θ < 60o (−30o < θ < 30o).

From (10.258) follows that

νdev = −2

9σ2

ea3I + a2σdev + a3[σdev]2, νvol = 3

[

a1 +2

9σ2

ea3

]

=∂Φ

∂σm

(10.262)

Remark: Since ν and σ are coaxial tensors for an isotropic yield criterion, it follows that

εp and σ are coaxial for an associative flow rule. However, coaxiality may hold also for

other yield criteria and does not imply associativity. 2

Rate of plastic work - Equivalent strain

The rate of plastic work Dp can always be decomposed into deviatoric and volumetric

parts as follows:

Dp def= σ : εp = Dp

dev + Dpvol with Dp

dev = σdev : εpdev, Dp

vol = σmεpvol (10.263)

It follows that εpvol is the strain quantity that is the energy conjugate to σm for any flow

rule.

Next, we define εpe as the strain quantity that is the energy conjugate to σe in the sense of

σeεpe = σdev : ε

pdev

def= σdev : ep

; Dp = σeεpe + σmε

pvol (10.264)



where we have introduced the purely deviatoric quantity ep as the portion of εp that is

proportional to σdev, i.e. ep is defined as

ep = (epdev =)

σdev ⊗ σdev

|σdev|2: εp

; ep =3σdev

2σe

εpe (10.265)

as shown in Figure 10.22. From (10.265)2, we obtain the definition

εpe =

√

2

3|ep| (10.266)

which is the rate of equivalent plastic strain for an arbitrary flow rule.

PSfrag replacements

εp

ep

Φ = 0

σdev

σ3 σ2

σ1

Figure 10.22: Direction of ep, which is used to define the equivalent plastic strain.

Remark: The expression in (10.266) is the classical definition of equivalent strain in the

case of von Mises yield criterion, whereby ep ≡ εp. In the more general situation when

Φ = Φ(σm, σe), then an associative flow rule gives ep = εpdev. However, in the most general

case the proper definition of εp depends explicitly on the actual yield criterion. 2

Let us consider the representation Φ = Φ(σm, σe, θ). For an associative flow rule, we may

now use (10.261) and (10.262) to obtain the identities (Show this as homework!):

εpe = λσdev : νdev

σe

= λ∂Φ

∂σe

, εpvol = λνvol = λ∂Φ

∂σm

(10.267)

This shows that εpvol and εpe are components of the normal to the yield surface Φ(σm, σe, θ)

in the restriction to the (σm, σe)-space, as shown in Figure 10.23.



PSfrag replacements

(σm, σe)

σe, εpe

(εpvol, εpe ) = λ

(∂Φ∂σm

, ∂Φ∂σe

)

Φ(σm, σe, θ) = 0

σm, εpvol

Figure 10.23: Isotropic yield surface in (σm, σe)-space.



We may now combine the result in (10.267) with (10.263) to obtain

Dp = λ

[

σm∂Φ

∂σm

+ σe∂Φ

∂σe

]

(10.268)

Remark: It is of interest to note that ∂Φ∂θ

does not contribute to the rate of plastic

dissipation. 2

Remark: In the geomechanics literature, it is common to use the notation q = σe and

p = −σm, which gives

εpe = λ∂Φ

∂q, εpvol = −λ∂Φ

∂p(10.269)

and

Dp = λ

[

p∂Φ

∂p+ q

∂Φ

∂q

]

(10.270)

where it was tacitly used that Φ = Φ(p, q, θ). 2

10.8.3 The Tresca criterion

According to the Tresca yield criterion the material yields plastically when the maximum

shear stress reaches the shear yield stress τy, i.e.

Φ(σ1, σ2, σ3) = |τ |max − τy =1

2[σ1 − σ3] − τy (10.271)

where it is assumed that σ1 ≥ σ2 ≥ σ3 and, hence, |τ |max = [σ1 − σ3]/2; cf. Figure 10.24.

It is noted that this criterion is mean stress independent, since σ1−σ3 = (σdev)1− (σdev)3.

The deviatoric cross-section of the Tresca yield surface is shown in Figure 10.25a, whereas

the biaxial section (for plane stress) is shown in Figure 10.25b8. Using the expression for

(σdev)i, i = 1, 2, 3 in (10.321) to (10.323), we may express the yield function in (10.271)

in terms of invariants as

Φ(σe, θ) =1

3σe [cos θ − cos (θ − 240o)] − τy, 0o ≤ θ ≤ 60o (10.272)

This expression can be simplified by the substitution θ = 30o + θ, which gives

Φ(σe, θ) =1√3σe cos θ − τy, −30o ≤ θ ≤ 30o (10.273)

8In this case x3 is taken as the out-of-plane principal direction, i.e. σ1 ≥ σ2 represent the in-plane

stresses, whereas σ3 = 0.



PSfrag replacements

σ1

σ2

σ3 σ

τy

τ

Φ = 0

Figure 10.24: Tresca’s yield criterion expressed as the ’maximum shear stress’ criterion

in the Mohr-representation.

From this expression it is simple to obtain the yield values of σe for θ = −30o, θ = 0o,

θ = 30o, corresponding to the tensile, shear and compressive meridians respectively:

σe = 2τy for θ = ±30◦ (10.274)

and

σe =√

3 τy for θ = 0 (10.275)

From (10.274) it is concluded that σy = 2τy according to Tresca’s criterion (since σe = σy

in uniaxial stress).

10.8.4 The von Mises criterion

The von Mises criterion is most commonly expressed in terms of the uniaxial yield stress

σy, i.e.

Φ(σe) = σe − σy (10.276)

In particular, we obtain the same yield value of σe(= σy) for θ = −30◦, θ = 0◦ and

θ = 30◦. It is also concluded that σy =√

3 τy according to the von Mises criterion (since

σe =√

3 τ in pure shear).

The von Mises yield criterion can be compared with the Tresca criterion as follows:



PSfrag replacements

σ1

σ2

σ3

ρ =√

2τy

ρ = 2√

23τy

2τy

−2τy

−2τy

−2τy

σ2

σ2

σ1

σ1

2τy

2τy

(a)

30◦

(b)

Figure 10.25: Tresca’s yield surface (a) Deviatoric stress plane, (b) Biaxial stress plane

(σ3 = 0).

• von Mises’ yield surface circumscribes Tresca’s yield surface if yielding takes place

for both in uniaxial stress (when θ = ±30◦), i.e. σ(M)y = σ

(T)y . This means that

σ(M)y =

√3 τ (M)

y = σ(T)y = 2τ (T)

y ⇒ τ (M)y =

2√3τ (T)y (10.277)

• von Mises’ yield surface inscribes Tresca’s yield surface if yielding takes place for

both in pure shear (when θ = 0◦), i.e. τ(M)y = τ

(T)y . This means that

τ (M)y =

1√3σ(M)

y = τ (T)y =

1

2τ (T)y ⇒ σ(M)

y =

√3

2σ(T)

y (10.278)

In terms of principal stresses, we may express σe as

σe =1√2

[[σ2 − σ3]

2 + [σ1 − σ3]2 + [σ1 − σ2]

2]1/2

(10.279)

The deviatoric cross-section of the von Mises criterion is shown in Figure 10.26a, whereas

the biaxial section (when σ3 = 0) is shown in Figure 10.26b. For σ3 = 0, we obtain the

simpler expression of σe:

σe = [σ21 + σ2

2 − σ1σ2]1/2 (10.280)

Remark: The von Mises criterion is sometimes termed the “deviatoric work criterion”,

since it corresponds to plastic yielding when the deviatoric portion Ψedev of the stored



PSfrag replacements

√3τ

(T)y

σ1−σ3

σ2−σ3

2τ(T)y

−2τy

−2τy

−2τy

σ1

σ2σ3

v.Mises (circumscribed)

v.Mises (inscribed)

Tresca

2τ(T)y

2τ(T)y

(a) (b)

30◦

Figure 10.26: von Mises yield surface (a) Deviatoric stress plane, (b) Biaxial stress plane

elastic energy reaches a critical amount. For linear elastic behavior we have

Ψedev =

1

2σdev : εdev (10.281)

and upon introducing isotropic elasticity via εdev = 12G

σdev, we obtain

Ψedev =

1

4G|σdev|2 =

1

6Gσ2

e =1

6Gσ2

y 2 (10.282)

The von Mises yield surface is sometimes interpreted merely as a smooth approximation

of the Tresca yield surface. However, since the von Mises yield surface has a very simple

geometric shape in stress space (a circular cylinder), it may be preferable from the view-

point of numerical manipulation as compared to the Tresca surface, which has “nasty”

corners. Generally speaking, smooth surfaces are simpler than those with irregularities.

This is particularly true in the context of integrating the resulting constitutive relations.

A generalized von Mises criterion

It is possible to generalize (10.276) by including the effect of θ as follows:

Φ(σe, θ) = σeg(θ) − σc (10.283)

where g(θ) should be chosen conveniently. An example is the expression by Willam-

Warnke (?), that was originally suggested for concrete:

g(θ) =4[1 − e2] cos2 θ + [2e− 1]2

2[1 − e2] cos θ + [2e− 1] [4[1 − e2] cos2 θ + 5e2 − 4e]1/2(10.284)



where e is an “excentricity” parameter such that 12< e ≤ 1. We see that

• g(0o) = 1e

for the tensile meridian, σe = σt = eσc

• g(60o) = 1 for the compressive meridian, σe = σc

The deviatoric cross-section is shown in Figure 10.27a, whereas g(θ) in (10.284) is shown

in Figure 10.27b.

0.0 20.0 40.0 60.01.0

1.2

1.4

1.6

1.8

2.0

e = 0.5e = 0.6e = 0.7e = 0.8e = 0.9e = 1.0

PSfrag replacementsσ1

σ2σ3

ρt

ρc

elliptic shape

(a) (b)

θ(◦)

g(θ

)

Figure 10.27: Generalized von Mises yield surface (a) Deviatoric stress plane, (b) Function

g(θ).

10.8.5 Hosford’s yield criterion

The yield criterion proposed by Hosford (1972) can be expressed in its most fundamental

format as

Φ = σHe (σ1, σ2, σ3) − σy (10.285)

where we introduced the “generalized effective stress” σHe as follows:

σHe

def=

1

[2]12k

[[σ2 − σ3]

2k + [σ1 − σ3]2k + [σ1 − σ2]

2k] 1

2k (10.286)

and where k = 1, 2, ... is taken as an integer. A family of yield surfaces Φ = 0 is thus

obtained, defined by the specific choice of k. All yield surfaces coincide at uniaxial loading,

defined by the yield stress σy.



Let us now introduce the functions yij(θ) as follows:

y12(θ)def= cos θ − cos(θ − 120o) =

√3

2cos θ − 3

2sin θ

y23(θ)def= cos(θ − 120o) − cos(θ − 240o) =

√3

2cos θ +

3

2sin θ (10.287)

y13(θ)def= cos θ − cos(θ − 240o) =

√3 cos θ

where we used the substitution θ = 30o + θ. Upon using the expressions for (σdev)i,

i = 1, 2, 3, in (10.324) to (10.326), we can rephrase σHe in (10.286) as follows:

σHe (σe, θ) =

2

3[2]12k

σe

[[y23(θ)]

2k + [y13(θ)]2k + [y12(θ)]

2k] 1

2k (10.288)

Special case: von Mises yield function

Upon setting k = 1 in (10.286) or (10.288), we obtain σHe = σe and it appears that the

von Mises yield function is retrieved. Show this as homework! In order to obtain von

Mises’ yield function from (10.288), show first that [y23(θ)]2 + [y13(θ)]

2 + [y12(θ)]2 = 9/2.

2

Special case: Tresca’s yield function

Upon setting k = ∞ in (10.286) or (10.288), we obtain

σHe = σ1 − σ2 and σH

e =2√3σe cos θ (10.289)

respectively. A comparison of these expressions with (10.271) and (10.273) immediately

shows that the Tresca yield criterion has been retrieved. 2

The Hosford family of yield surfaces for selected values of the exponent k are shown in

Figure 10.28.

10.8.6 The Mohr criterion

Mohr’s classical hypothesis of failure states that failure (and plastic slip) will take place on

planes in the body for which, at a certain stress state, a given combination of the normal

and shear stresses reaches a critical value. This failure criterion is thus expressed as

τ = g(σ) (10.290)



Figure 10.28: Hosford’s family of yield surfaces (a) Deviatoric stress plane, (b) Biaxial

stress plane.

where σ and τ are the magnitudes of normal and shear9 stresses on the plane with normal

n

σ = n · σ · n, τ =[n · [σ2 − σ · n ⊗ n · σ] · n

]1/2(10.291)

Remark: On the octahedral plane, which is the special plane defined by n = [e1 + e2 +

e3]/√

3, or ni = 1/√

3, i = 1, 2, 3, in the principal (stress) coordinate system, we obtain

σ = σoct and τ = τoct. 2

The criterion (10.290) is most simply shown in a Mohr-circle diagram; as shown in Figure

10.29. The normal to the failure plane at the point of failure is given as

nf = cosψf e1 + sinψf e3 (10.292)

and inserting this expression into (10.291), we can express the stresses σf and τf at the

point of contact between the Mohr-envelope and the largest Mohr-circle in terms of the

major and minor principal stresses σ1 and σ3. It then appears that the intermediate stress

σ2 is without any importance:

σf = σ1 cos2 ψf + σ3 sin2 ψf , τf =1

2[σ1 − σ3] sin 2ψf (10.293)

Let us now introduce the angle of internal friction φ such that

dg

dσ= − tanφ = −µ(σ) (10.294)

Hence, µ is the current coefficient of internal friction, which is mean stress dependent (in

general). Directly from Figure 10.29, we obtain Mohr’s slip plane solution at failure,

9In the context of crystal plasticity, τ is the Schmid stress, or “resolved shear stress”, on a given slip

plane.



PSfrag replacements

σ1

σ3

σ2σ

τ = g(σ)

(σf , τf)

2Ψf

(a) (b)

τf σf

σ1σ1

σ3

σ3

Ψf

x1

x

x

x

x3Tangent point

Failure plane

τ

xx xx

Figure 10.29: Mohr’s failure criterion

defined by φ = φf , as

φf = 90o − 2ψf ⇒ ψf = 45o − 1

2φf (10.295)

Combining (10.290) and (10.293) with (10.294), we obtain a relation expressed in the

principal stresses σ1 and σ3, which is the Mohr criterion.

10.8.7 The Mohr-Coulomb criterion

A special case of the Mohr criterion is the Mohr-Coulomb criterion, which is defined by

a constant value of the angle of internal friction, φ. This implies that

τ = g(σ) = c− σ tanφ ⇒ τ + σ tanφ = c (10.296)

where c is the internal cohesion, as shown in Figure 10.30. From this figure follows that

the yield criterion can be expressed in terms of σ1 and σ3. The contact stresses are

σf =1

2[σ1 + σ3] +

1

2[σ1 − σ3] sinφ, τf =

1

2[σ1 − σ3] cosφ (10.297)

which can be combined with (10.290) to yield

Φ(σ1, σ2, σ3) =1

2[σ1 − σ3] +

1

2[σ1 + σ3] sinφ− c · cosφ (10.298)



PSfrag replacements

σ1 σσ3

2Ψf

(σf , τf)

φ

τ

c

x

Figure 10.30: Mohr-Coulomb’s failure (yield) criterion

Remark: The special case when φ = 0 defines the Tresca criterion. A comparison with

(10.271) shows that c = τy in this case. 2

Let us next consider a few alternative formulations of (10.298). Firstly, we may express

sinφ and c · cosφ in terms of the uniaxial yield stress in tension (σ1 = σt, σ3 = 0) and

compression (σ1 = 0, σ3 = −σc) to obtain

sinφ =σc − σt

σc + σt

, c · cosφ =σcσt

σc + σt

(10.299)

whereby (10.298) is formulated as

Φ(σ1, σ2, σ3) =1

2[σ1 − σ3] +

1

2[σ1 + σ3]

σc − σt

σc + σt

− σcσt

σc + σt

(10.300)

Remark: We may obtain the Rankine criterion, or the principal stress criterion, by

setting σc → ∞ in (10.300). This gives

Φ(σ1) = σ1 − σt (10.301)

From (10.298) we see that this is equivalent to setting φ = 90o and c · cosφ = σt. 2

It is possible to give an alternative expression of Φ in (10.298) in terms of the geometric

stress invariants p, q and θ. To this end, we may use the expression for σi, i = 1, 2, 3, in

(10.321) to (10.323) in order to obtain

Φ(p, q, θ) = −p sinφ+1√3q cos θ − 1

3q sin θ sinφ− c · cosφ (10.302)



for −30o ≤ θ ≤ 30o, where θ is (still) defined as θ = θ − 30o.

The deviatoric cross-section of the Mohr-Coulomb yield surface is shown in Figure 10.31a,

whereas the biaxial section (for plane stress) is shown in Figure 10.31b.10

PSfrag replacements

ρt/ρc = e

σ1

σ3 σ2

ρt

ρc

(φ = π/2)

(0 < φ < π/2)

Tresca

Mohr-CoulombRankine

60◦

σ2/σt

σ1/σt

−[1 + sinφ]/[1 − sinφ]

Tresca

Tresca

Rankine

Rankine

M-C

(a) (b)

−11

Figure 10.31: Comparison of yield surfaces obtained as special cases of the Mohr-Coulomb

yield surface. (a) Deviatoric stress plane, (b) Biaxial stress plane (σ3 = 0).

The tensile and compressive meridians are defined by θ = −30o and θ = 30o, respectively,

which inserted into (10.302) gives

qt =2[p sinφ+ c · cosφ]

1 + 13sinφ

, qc =2[p sinφ+ c · cosφ]

1 − 13sinφ

(10.303)

The ratio of qt and qc is given as

e =qtqc

=%t

%c

=1 − 1

3sinφ

1 + 13sinφ

,1

2< e ≤ 1 (10.304)

where the lower bound e = 1/2 is obtained for φ = 90o (the Rankine criterion), whereas

the upper bound e = 1 is obtained for φ = 0 (the Tresca criterion).

10In this case x3 is taken as the out-of-plane principal direction, i.e. σ1 ≥ σ2 represent the in-plane

stresses, whereas σ3 = 0.



Finally, let us consider the shear meridian, defined by θ = 0, which yields

qs =√

3[p sinφ+ c · cosφ] (10.305)

10.8.8 The Drucker-Prager criterion

A general expression for the Drucker-Prager yield function (sometimes known as the

Extended von Mises yield function) is

Φ(σm, σe) = σe + µσm − κ (10.306)

where µ represents the angle of internal friction and κ is a cohesion parameter. The

corresponding yield surface may be considered as a smooth approximation of the Mohr-

Coulomb surface in the shape of a circular cone. Of course, such an approximation is not

unique. The least conservative choice is to circumscribe the Mohr-Coulomb surface by the

cylindrical cone, which corresponds to equating the yield strength along the compressive

meridian θ = 30o. The most conservative choice is to inscribe the cylindrical cone into

the Mohr-Coulomb surface. Other cones, which intersect the Mohr-Coulomb surface, are

obtained by equating the yield stresses along the tensile meridian (θ = −30o) and the

shear meridian (θ = 0o). The various deviatoric cross-sections are shown in Figure 10.32.

PSfrag replacements

σ1

σ2σ3

Compressive D-P, µc

Tensile D-P, µt

Inscribed D-P, µi

(a)

Φ = 0σe

κ

σm

(b)

µ1

Figure 10.32: Drucker-Prager’s yield surface. (a) Deviatoric stress plane, (b) Meridian

stress plane.



It is straightforward to calculate µc, µt and µs and the corresponding k-values merely by

inserting the proper value of θ in (10.302). For example, let us obtain µc and κc (for the

circumscribed cone). We then set θ = 30o to obtain qc according to (10.303)2, and the

criterion q = qc now gives (10.306) with

µc =2 sinφ

1 − 13sinφ

, κc =2 cosφ

1 − 13sinφ

(10.307)

To obtain µi and κi (for the inscribed cone), we must first calculate the value θi that

defines the tangent point of the inscribed cone with the Mohr-Coulomb surface. We

obtain

tan θi = −√

3[1 − e]

1 + e= − 1√

3sinφ (10.308)

where the last equality was obtained via (10.304). Inserting (10.308) into (10.302) we

obtain, for a given p -value, the pertinent value of qi, which gives

µi =sinφ

1√3cos θi − 1

3sin θi

, κi =c · cosφ

1√3cos θi − 1

3sin θi

(10.309)

where θi is defined in (10.308).

The Willam-Warnke (generalized Drucker-Prager) yield criterion

Analogous to the generalization of the von Mises yield function defined in (10.276), we

may generalize (10.306) to include the dependence on θ as

Φ(σm, σe, θ) = σeg(θ) + µσm − κ (10.310)

where, for example, g(θ) can be chosen as in (10.284).

10.8.9 Appendix: Geometric invariants in principal stress space

Consider the three-dimensional vector space of principal stresses spanned by the Cartesian

axes σi. The stress state is then represented by the vector OP = (σ1, σ2, σ3) in this space11, as shown in Figure 10.33. Polar coordinates (ξ, ρ, θ), which are sometimes known as

the Haigh-Westergaard coordinates, are now introduced as follows: ξ is the coordinate

11This representation of the principal stress space as a vector space should not be confused with the

Cartesian space spanned by the axes (x1, x2, x3) or principal axes (x1, x2, x3).



along the stress space diagonal (isotropic axis), ρ is the radius vector that is orthogonal

to the space diagonal (in the deviatoric plane), whereas θ is the clockwise angle in the

deviatoric plane from the σ1 -axis to the radius vector. This angle is occasionally denoted

the Lode angle.

Sometimes the following notation is used:

• π-plane (deviator plane), spanned by (ρ, θ) for given value of ξ.

• Meridian plane, spanned by (ξ, ρ) for given value of θ

PSfrag replacements

σ1

σ3

σ2

isotropic axis

ρ

P(σ1, σ2, σ3)

O

(a)

θ

P

ρρ

σ1

σ1

σ3σ3

σ2

σ2120o

(b)

N

ξMM1

M2M3

Figure 10.33: (a) Principal stress space with polar coordinates and ξ and ρ; (b) Deviatoric

plane showing ρ and θ.

With the normal vector OM = (1, 1, 1)/√

3 directed along the spatial diagonal, we may

express the coordinate ξ as

ξ = OM · OP =1√3[σ1 + σ2 + σ3] =

1√3i1 =

1√3I1 =

√3σm (10.311)

which shows that ξ is equivalent to I1 or (σm).

From Figure 10.33 we may deduce that ON represents the stress deviator σdev since

NP = OP − ON = (σ1, σ2, σ3) − σm(1, 1, 1) ≡ ((σdev)1, (σdev)2, (σdev)3) (10.312)



It then follows that ρ represents the invariant J2 since

ρ2 = |NP |2 = [(σdev)1]2 + [(σdev)2]

2 + [(σdev)3]2 = j2 = 2J2 = |σdev|2 =

2

3[σe]

2 (10.313)

It remains to define θ in terms of the generic invariants. Let OM i, i = 1, 2, 3, be unit

vectors in the deviatoric plane along the axes, as shown in Figure 10.33b:

OM 1 =1√6(2,−1,−1), OM 2 =

1√6(−1, 2,−1), OM 3 =

1√6(−1,−1, 2) (10.314)

We obtain the identity

ρ cos θ = OM 1 · NP =1√6

[2(σdev)1 − (σdev)2 − (σdev,3)] =

√

3

2(σdev)1 (10.315)

where the last equality follows by adding the quantity tr[σdev] = (σdev)1 + (σdev)2 +

(σdev)3 = 0. Similarly, from Figure 10.33b, we obtain

ρ cos(120o − θ) = ρ cos(θ − 120o) = OM 2 · NP =

√

3

2(σdev)2 (10.316)

ρ cos(240o − θ) = ρ cos(θ − 240o) = OM 3 · NP =

√

3

2(σdev)3 (10.317)

Next we shall make use of the trigonometric identity:

cos 3θ =4

3

[cos3 θ + cos3(θ − 120o) + cos3(θ − 240o)

](10.318)


Together with (10.315), (10.316), (10.317), and (1.139), equation (10.318) gives:

ρ3 cos 3θ =4

3

[√

3

2

]2[[(σdev)1]

3 + [(σdev)2]3 + [(σdev)3]

3]

= 3√

6J3 (10.319)

Finally, we may rewrite (10.319) as:

cos 3θ =3√

3

2

J3

[J2]3/2=

√6

j3

[j2]3/2

(10.320)

Remark: Because the angular argument of (10.320) is 3θ, it appears that the range of

interest is 0 ≤ θ ≤ 60o. 2



When the invariants (ξ, ρ, θ) are known, then σi may be calculated from the expressions

in (10.312), and (10.315) to (10.317):

σ1 =1√3ξ + (σdev)1 with (σdev)1 =

√

2

3ρ cos θ (10.321)


√

2

3ρ cos(θ − 120o) (10.322)


√

2

3ρ cos(θ − 240o) (10.323)

where it was used that σi = σm + (σdev)i, i = 1, 2, 3.

We may, alternatively, rephrase these expressions in terms of θ, defined from the substi-

tution θ = 30o + θ, which gives

(σdev)1 =

√

2

3ρ

[√

3

2cos θ − 1

2sin θ

]

(10.324)

(σdev)2 =

√

2

3ρ sin θ (10.325)

(σdev)3 =

√

2

3ρ

[

−√

3

2cos θ − 1

2sin θ

]

(10.326)

for −30o ≤ θ ≤ 30o.

Remark: The principal stresses are ordered so that, in fact, σ1 ≥ σ2 ≥ σ3 with σ1 = σ2

for θ = 30o and σ2 = σ3 for θ = −30o. Show this as homework, with the aid of (10.324)

to (10.326). 2

Special case: Isotropic stress

Consider a stress state defined as σ1 = σ2 = σ3 = −p, where p is the pressure. This

isotropic stress state is defined as σ = −pI, and it is directed along the isotropic axis in

Figure 10.33a.



Special case: Uniaxial tension and compression

Uniaxial tension is defined as12

σ1 = σ > 0, σ2 = σ3 = 0 ⇒ σ = σm1 (10.327)

and

σm =1

3σ , σe = |σ1| = σ (10.328)

The tensile meridian is defined by all stress states that can be obtained by superimposing

an isotropic stress on any given uniaxial tensile stress, i.e. which are characterized by

σ1 > σ2 = σ3.

This gives

J2 =1

3[σ1 − σ3]

2, J3 =2

27[σ1 − σ3]

3; cos 3θ = 1 or θ = 0o (10.329)

Similarly, uniaxial compression is defined by

σ1 = σ2 = 0, σ3 = σ < 0 ⇒ σ = σm3 (10.330)

and

σm =1

3σ , σe = |σ1| = −σ (10.331)

The compressive meridian is defined as all those stress states that can be obtained by

superimposing an isotropic stress on any given uniaxial compressive stress, i.e. which are

characterized by σ1 = σ2 > σ3. This gives

J2 =1

3[σ1 − σ3]

2, J3 = − 2

27[σ1 − σ3]

3; cos 3θ = −1 or θ = 60o (10.332)

Special case: Pure shear

A state of pure shear is defined as (for τ > 0)

σ1 = τ, σ2 = 0, σ3 = −τ ⇒ σ = τ [m1 − m3] (10.333)

and

σm = 0 , σe =√

3 τ (10.334)

12midef= ei ⊗ ei is the dyad associated with the principal (stress) direction ei, i = 1, 2, 3.



The shear meridian is defined as all those stress states that can be obtained by superim-

posing an isotropic stress onto any given state of pure shear. Let such an isotropic state

be σ = k[m1 + m2 + m3], which gives the compounded stress

σ1 > k > σ3 with k(= σ2) =1

2[σ1 + σ3] (10.335)

This gives

J3 = 0 ; cos 3θ = 0 or θ = 30o (10.336)

To obtain the given θ-values, we used the expression for θ in (10.320). The various

meridians are depicted in Figure 10.21.

10.9 The constitutive integrator for a special class:

Isotropic linear elasticity and isotropic yield cri-

teria

10.9.1 Backward Euler method - Preliminaries

We shall assume linear isotropic elasticity, in which case it is recalled that the elastic

stiffness modulus tensor Ee and its inverse are given by

Ee = 2GI

symdev +KI ⊗ I, [Ee]−1 =

1

2GIsymdev +

1

9KI ⊗ I (10.337)

Upon inserting (10.337) into the basic incremental relation (10.58), we may decompose

(10.58) in terms of the deviatoric and spherical portions of σ as follows:

σ = σdev + σmI (10.338)

with

σdev = σtrdev − 2Gµνdev, σm = σtr

m −Kµνvol (10.339)

where the trial values are given as

σtrdev

def= 2Gε

e,trdev = n−1σdev + 2G∆εdev, σtr

mdef= Kεe,trvol = n−1σm +K∆εvol (10.340)

We shall now consider isotropic yield criteria, and we choose the representation Φ =

Φ(σm, σe, θ). According to (10.262) we then have

νdev = −2

9σ2

ea3I + a2σdev + a3[σdev]2, νvol = 3

[

a1 +2

9σ2

ea3

]

=∂Φ

∂σm

(10.341)


10.9 The constitutive integrator for a special class: Isotropic linear elasticity andisotropic yield criteria 335

where the coefficients ai(σm, σe, θ) were given in (10.259) to (10.261).

Next, we note the following important property for isotropic yield criteria:

Lemma: If Φ is an isotropic function of σ, then σ and σtr are coaxial tensors, i.e. they

have the same principal directions.

Proof: For isotropic Φ, we recall the expression for νdev in (10.341):

νdev = −2

9σ2

ea3I + a2σdev + a3[σdev]2 (10.342)

We may thus write (10.339)1 as:

σtrdev = a′1I + a′2σdev + a′3[σdev]

2 (10.343)

where a′i(σm, σe, θ), i = 1, 2, 3, are a new set of scalars. Choosing the principal directions

of σ, we immediately obtain from (10.343) that

(σtrdev)i = a′1 + a′2(σdev)i + a′3([σdev]

2)i (10.344)

which shows that σe and σ are coaxial. 2

In order to be as explicit as possible, we shall next consider the special case that the

dependence of the third invariant is ignored, i.e. Φ = Φ(σm, σe), before treating the

general situation defined by Φ = Φ(σm, σe, θ). The reason for distinguishing these two

cases is that the latter one requires significantly more technical derivations, which are

best carried out using the projection property of the CPPM.

10.9.2 Backward Euler method for two-invariant yield surfaces

(independent of the Lode angle)

In the case that Φ = Φ(σm, σe), then the expression for ν in (10.341) can be simplified as

follows:

νdev =3

2σe

∂Φ

∂σe

σdev, νvol =∂Φ

∂σm

(10.345)

Combining (10.345) with (10.339)1, we obtain

σdev =

[

1 +3Gµ

σe

∂Φ

∂σe

]−1

σtrdev ; σ =

σe

σtre

σtrdev + σmI (10.346)



PSfrag replacements

σtrdev

σdev = kσtrdev

Φ(σm, σe) = 0 , σm fixed

σ1

σ2σ3

R(σm)

Figure 10.34: Radial return property in deviator space for the yield surface Φ(σm, σe) = 0.

We have thus shown the important property: σdev is proportional to σtrdev or, in other

words, σdev is obtained from σtrdev by “radial return” to the yield surface. This finding

generalizes the property already discovered for the von Mises criterion. In fact, for any

fixed σm then Φ = Φ(σm, σe) = 0 always represents a circular shape in the deviatoric

planes, which is illustrated in Figure 10.34. A trivial example is the von Mises yield

surface, defined by

σe − σy = 0 ; σ =σy

σtre

σtrdev + σtr

mI (10.347)

In the case of plastic loading (L), we also conclude from (10.339) and (10.345) that σm,

σe and µ are the solutions of the following set of equations

R\m = σm − σtr

m + µK∂Φ

∂σm

(σm, σe) = 0 (10.348)

R\e = σe − σtr

e + µ3G∂Φ

∂σe

(σm, σe) = 0 (10.349)

R\µ = Φ(σm, σe) = 0 (10.350)



or

R\(X) = 0 with X =

σm

σe

µ

, R

\ =

R\m

R\e

R\µ

(10.351)

Remark: It is possible to arrive at the equations (10.348) to (10.350) via the CPPM-

projection property, whereby these equations are the KT-conditions of the associated

constrained minimization problem in complementary elastic energy metric. This is shown

below in the context of the (more general) three-invariant formulation. 2

The Jacobian of R\(X) is given as

J \ =

1 + µK ∂2Φ(∂σm)2

µK ∂2Φ∂σm∂σe

K ∂Φ∂σm

µ3G ∂2Φ∂σm∂σe

1 + µ3G ∂2Φ(∂σe)2

3G ∂Φ∂σe

∂Φ∂σm

∂Φ∂σe

0

(10.352)

The scaled version is defined by

Rm =1

Kb

R\m, Re =

1

3GR\

e, Rµ = R\µ (10.353)

which is associated with the symmetric Jacobian J defined as

J =

1K

+ µ ∂2Φ(∂σm)2

µ ∂2Φ∂σm∂σe

∂Φ∂σm


13G

+ µ ∂2Φ(∂σe)2

∂Φ∂σe

∂Φ∂σm

∂Φ∂σe

0

(10.354)

10.9.3 Backward Euler method for three-invariant yield surfaces

In the most general case that Φ = Φ(σm, σe, θ), the derivation of the appropriate incre-

mental relations is considerably more technical. In fact, the pertinent relations are most

easily derived by using the CPPM-property. Hence, we note that σ is obtained, in the

case of plastic loading, as

σ = arg

[

minΦ(σ∗)=0

Ψ∗(σtr − σ∗)

]

(10.355)



where the complementary elastic energy norm may be expressed as

Ψ∗(σtr − σ∗) =1

4G|σtr

dev − σ∗dev|2 +

1

2K[σtr

m − σ∗m]2 (10.356)

Remark: For given, i.e. fixed, value of σm, it follows directly from (10.356) that σdev is

obtained as the truly Euclidian projection of σtrdev onto the deviatoric trace of the yield

surface (π-plane). This property is illustrated in Figure 10.35.

PSfrag replacements

σtrdev

Φ (σm, σe, θ ) = 0, σm fixed

σdev

θtr

θ

σ1

σ2σ3

Figure 10.35: Euclidean projection in the deviator subspace for given value of σm.

We need to express the deviator term in (10.356) in terms of the chosen invariants. For

any two stresses σ(1)dev and σ

(2)dev, we have

|σ(1)dev − σ

(2)dev|2 = [|σ(1)

dev| − |σ(2)dev|]2 + 2[|σ(1)

dev||σ(2)dev| − σ

(1)dev : σ

(2)dev]

= [ρ(1) − ρ(2)]2 + 2[ρ(1)ρ(2) − σ(1)dev : σ

(2)dev] (10.357)

where the notation ρ = |σdev| was used as an alternative representation of the 2nd stress

invariant.



Since σtr and the solution σ are coaxial, we may restrict our attention to those σ(1) and

σ(2) which are coaxial. Upon using the result in the Appendix we obtain (after some

elaboration) the following relation:

σ(1)dev : σ

(2)dev = (σ

(1)dev)1(σ

(2)dev)1 + (σ

(1)dev)2(σ

(2)dev)2 + (σ

(1)dev)3(σ

(2)dev)3

=2

3ρ(1)ρ(2)

[cos θ(1) cos θ(2) + cos

(θ(1) − 120o

)cos(θ(2) − 120o

)

+ cos(θ(1) − 240o

)cos(θ(2) − 240o

)]

= ρ(1)ρ(2) cos(θ(1) − θ(2)) (10.358)

Remark: It is possible to derive the expression in (10.358) from purely geometric con-

siderations in the deviator plane of the principal stress space. The reader should show

this as homework! 2

Upon inserting (10.357) with (10.358) into (10.356), we obtain

Ψ∗(σtr − σ∗)def= Ψ∗(σ∗

m, σ∗e , θ

∗; σtrm, σ

tre , θ

tr)

def=

1

6G[σtr

e − σ∗e ]

2 +2

6Gσtr

e σ∗e [1 − cos(θtr − θ∗)] +

1

2K[σtr

m − σ∗m]2

(10.359)

The interesting result has thus been obtained that the minimization in (10.355) can be

carried out entirely in terms of the stress invariants. Hence, in the case of plastic loading,

it appears that (σm, σe, θ) is the solution of the constrained minimization problem

(σm, σe, θ) = arg

[

minΦ(σ∗

m,σ∗

e ,θ∗)=0Ψ∗(σ∗

m, σ∗e , θ

∗; σtrm, σ

tre , θ

tr)

]

(10.360)

The Lagrangian multiplier method may be used for transforming the problem (10.360)

into an unconstrained minimization problem. We thus seek the solution (σm, σe, θ;µ) that

corresponds to a saddle point of the Lagrangian function L, defined as

L(σ∗m, σ

∗e , θ

∗, µ∗) = Ψ∗(σ∗m, σ

∗e , θ

∗; σtrm, σ

tre , θ

tr) + µ∗Φ(σ∗m, σ

∗e , θ

∗) (10.361)

The extremal conditions of (10.361) are

R\m = σm − σtr

m +Kµ∂Φ

∂σm

(σm, σe, θ) = 0 (10.362)

R\e = σe − σtr

e cos(θ − θtr) + 3Gµ∂Φ

∂σe

(σm, σe, θ) = 0 (10.363)



R\θ = σeσ

tre sin(θ − θtr) + 3Gµ

∂Φ

∂θ(σm, σe, θ) = 0 (10.364)

R\µ = Φ(σm, σe, θ) = 0 (10.365)

or

R\(X) = 0 with X =

σm

σe

θ

µ

, R\ =

R\m

R\e

R\θ

R\µ

(10.366)

The scaled version is defined as

Rm =1

KR\

m, Re =1

3GR\

e, Rθ =1

3GR\

θ, Rµ = R\µ (10.367)

which is associated with the symmetric Jacobian

J =

1K

+ µ ∂2Φ(∂σm)2


µ ∂2Φ∂σm∂θ

∂Φ∂σm


13G

+ µ ∂2Φ(∂σe)2

σtre

3Gsin(θ − θtr) + µ ∂2Φ

∂σe∂θ∂Φ∂σe

µ ∂2Φ∂σm∂θ

σtre

3Gsin(θ − θtr) + µ ∂2Φ

∂σe∂θσeσtr

e

3Gcos(θ − θtr) + µ∂2Φ

∂θ2∂Φ∂θ

∂Φ∂σm

∂Φ∂σe

∂Φ∂θ

0

(10.368)

Remark: When Φ is only weakly dependent on θ, then it may be computationally ad-

vantageous to split the (Newton) iterations for solving (10.362) to (10.365) in a two-level

strategy, whereby (10.362),(10.363) and (10.365) are solved on the lower iteration level

for given θ. A new value of θ is then computed on the higher iteration level upon using

the values of σm, σe and µ from the lower level iteration. 2

When the invariants σm, σe and θ have been determined (via iterations in general), it

remains to calculate the Cartesian components of the updated solution σ. We shall

then use the fact that σtr and σ are coaxial tensors. If the principal directions of σtr are

defined by the unit vectors e1, e2 and e3 (with components in a given Cartesian coordinate

system), we may use the spectral decomposition to express σtr and σ as

σtr =3∑

i=1

σtri mi, σ =

3∑

i=1

σi mi with midef= ei ⊗ ei (10.369)

It is noted that the dyads mi can be computed from the Serrin formula given in Chapter

1.



The principal values σi, i = 1, 2, 3, can be computed from the (now known) values

σm, σe, θ, as shown in Appendix:

σi = (σdev)i + σm with (σdev)i =3

2σe cos (θ − [i− 1]120o) , i = 1, 2, 3 (10.370)

Finally, σ is given in terms of its Cartesian components from the spectral representation

in (10.369)2.

Special case: Loading along tensile and compressive meridians

In the special case that the trial state is located along one of the meridians defined by

θ = 0o (tensile meridian) or θ = 60o (compressive meridian), then it follows directly from

symmetry arguments in Figure 10.35 that the updated stress solution must also be located

along the same meridian, i.e. θ = θtr. 2

Special case: Two-invariant yield surface (no dependence on the Lode angle)

In the special case that Φ = Φ(σm, σe), then ∂Φ/∂θ = 0 and it follows directly from

(10.364) that R\θ = 0 is satisfied only if θ = θtr. Hence, the radial return property in the

deviatoric hyperplane follows. 2

Special case: Two-invariant yield surface (no dependence on the mean stress)

In the special case that Φ = Φ(σe, θ), then ∂Φ/∂σm = 0 and it follows from (10.362) that

R\m = 0 is satisfied only if σm = σtr

m. The remaining pertinent equations are those in

(10.363), (10.364), which relate only to the deviatoric plane for σm = σtrm. 2

10.9.4 ATS-tensor

In the general situation of a three-invariant yield surface, we may most conveniently use

the relations for the ATS-tensor Ea, that were outlined in Subsection 10.3.3. It is noted

that the main technical difficulty lies in obtaining Eea, which requires the inversion of a

tensor. As it turns out, it may not be a simple matter to obtain a closed-form solution of

Eea in the general case.



However, in the case of a two-invariant formulation, it is simple to entirely avoid the

computation of Eea. To this end, we use the generic expression

dX(ε) = −J−1dR|X(ε) (10.371)

where dR|X is defined by its components

dRm|X = −I : dε, dRe|X = −σtrdev

σtre

: dε, dRµ|X = 0 (10.372)

We then obtain from (10.371):

dσm = (J−1)mmI : dε + (J−1)meσtr

dev

σtre

: dε, dσe = (J−1)meI : dε + (J−1)eeσtr

dev

σtre

: dε

(10.373)

where (J−1)mm, (J−1)me and (J−1)ee are submatrices of J−1.

We may now differentiate the expression for σ in (10.346)2, while using the chain rule, to

obtain

Ea = 2Gσe

σtre

Isymdev +

[(J−1)ee

[σtre ]2

− 3Gσe

[σtre ]3

]

σtrdev ⊗ σtr

dev

−(J−1)me

σtre

[σtr

dev ⊗ I + I ⊗ σtrdev

]+ (J−1)mmI ⊗ I (10.374)

It appears readily that Ea possesses major (and, of course, minor) symmetry.

10.10 Prototype model: Hooke elasticity and Hos-

ford’s family of yield surfaces


As the prototype model including the two stress invariants σe and θ, we consider linear

isotropic elasticity in conjunction with the Hosford family of yield surfaces, defined in

Subsection 10.8.5:

Φ(σe, θ) = σHe (σe, θ) − σy (10.375)

with

σHe (σe, θ) =

2

3[2]12k

σezk(θ) (10.376)


10.10 Prototype model: Hooke elasticity and Hosford’s family of yield surfaces343

zk(θ) =[[y23(θ)]

2k + [y13(θ)]2k + [y12(θ)]

2k] 1

2k (10.377)

where k is an integer parameter (k ≥ 1).

The flow rule is given as

εp = λν with ν = νdev = −2

9[σe]

2a3I + a2σdev + a3[σdev]2 (10.378)

where the coefficients a2(σe, θ) and a3(σe, θ) were given in (10.260, 10.261):

a2 =3

2σe

[∂Φ

∂σe

− 1

σe cot 3θ

∂Φ

∂θ

]

, a3 = − 9

2[σe]3 cos 3θ

∂Φ

∂θ(10.379)

In this case we have∂Φ

∂σe

=2

3[2]12k

zk(θ) (10.380)

∂Φ

∂θ=

2

3[2]12k

zk(θ)1−2k

[

[y23]2k−1 dy23

dθ+ [y13]

2k−1 dy13

dθ+ [y12]

2k−1 dy12

dθ

]

(10.381)

and

dy12

dθ= −

√3

2sin θ− 3

2cos θ,

dy23

dθ= −

√3

2sin θ +

3

2cos θ,

dy13

dθ= −

√3 sin θ (10.382)


The pertinent relations for the local constitutive problem is defined by

R\e = σe − σtr

e cos(θ − θtr) + 3Gµ∂Φ

∂σe

(σe, θ) = 0

R\θ = σe − σtr

e sin(θ − θtr) + 3Gµ∂Φ

∂θ(σe, θ) = 0 (10.383)

R\µ = Φ(σe, θ) = 0

whose solution is obtained directly upon following the “recepy” in Subsection 10.9.3.

Moreover, the ATS-tensor was given in Subsection 10.9.4 for the generic 3-invariant for-

mat. However, it is simplified due to the independence on σm. Since (J−1)mm = K, and

(J−1)me = 0, we obtain

Ea = 2Gσe

σtre

Isymdev +

[(J−1)ee

[σtre ]2

− 3Gσe

[σtre ]3

]

σtrdev ⊗ σtr

dev +KI ⊗ I (10.384)




Perfect plasticity, variation of parameter k. Extremes: k = 1 (von Mises), k large (Tresca).

To be completed.


10.11 Questions and problems 345

10.11 Questions and problems

1. In the case of perfect plasticity, show that the MD-postulate is a sufficient, but not

necessary, condition for the dissipation inequality to be satisfied.

2. Consider the prototype model for perfectly plastic response in Section 10.4. (a)

Derive the expression for the flow rule in (10.78) while using the projection tensor

Isymdev defined in (1.58). (b) Show that the volumetric response is purely elastic.

3. Carry out the explicit derivations leading to the expression for the elastic-plastic

tangent stiffness tensor Eep in (10.84). Moreover, derive explicit expressions for all

components (Eep)ijkl in terms of the elastic parameters G, K, the equivalent stress

σe and the components σij. Moreover, give the tangent stiffness relation in the Voigt

matrix format.

4. The “radial return” property of the BE-rule applied to the prototype model for

perfectly plastic response in Section 10.4 is a celebrated “geometrical” property in

stress space. What are the essential ingredients in the model that brings about this

property?

5. A “hybrid” integration rule for plastcity is to adopt the Forward Euler rule for the

flow rule, while the loading conditions are still established at the updated state (at

the end of the current timestep). Apply this method to the prototype model in

Section 10.4 and compare the result with that obtained from the BE-rule. Show

that it is possible that the solution for the updated stress does not exist.

6. What are the main conseqences of assuming “simple hardening”?

7. In the case of hardening, continuum tangent relations pertinent to strain control

are derived in Subsection 10.5.4. Establish the corresponding tangent compliance

relations for stress control, i.e. assuming that σ is the controlling quantity. In

particular, establish Cep in the relation

ε = Cep : σ with C

ep def= C

e +1

Hν ⊗ ν

Hence, establish the condition for uniqueness of the response under stress control,

and discuss the consequences for softening behavior.



8. The concept of Closest-Point-Projection is a property of the BE-rule in the case

of linear elasticity and linear hardening, as discussed in Subsection 10.6.4. Show

explicitly that this property holds for both loading (L) and unloading (U). Why is

it necessary to require strict hardening in this context?


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