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
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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.
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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
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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
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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-
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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.
Vol I March 21, 2006
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
Vol I March 21, 2006
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Vol I March 21, 2006
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
1.3.5 Coordinate transformation . . . . . . . . . . . . . . . . . . . . . . . 9
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
1.4.4 Coordinate transformation . . . . . . . . . . . . . . . . . . . . . . . 12
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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
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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
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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
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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
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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
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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
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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
8.4.3 Examples of response simulations . . . . . . . . . . . . . . . . . . . 214
8.4.4 Examples of response simulations . . . . . . . . . . . . . . . . . . . 217
8.4.5 Appendix: Constitutive relations for the uniaxial stress state . . . . 217
8.5 Prototype model: The isotropic (nonlinear) Norton model . . . . . . . . . 219
8.5.1 The constitutive relations . . . . . . . . . . . . . . . . . . . . . . . 219
8.5.2 The constitutive integrator . . . . . . . . . . . . . . . . . . . . . . . 220
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8.5.3 Examples of response simulations . . . . . . . . . . . . . . . . . . . 223
8.5.4 Appendix: Constitutive relations for the uniaxial stress state . . . . 227
8.6 The constitutive framework - Nonlinear viscoelasticity with multiple dissi-
pative mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230
8.6.1 Free energy, thermodynamic forces and rate equations . . . . . . . . 230
8.6.2 Linear model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
8.7 Prototype model - The isotropic Linear Standard Viscoelasticity model . . 232
8.7.1 The constitutive relations . . . . . . . . . . . . . . . . . . . . . . . 232
8.7.2 The constitutive integrator . . . . . . . . . . . . . . . . . . . . . . . 234
8.7.3 Examples of response simulations . . . . . . . . . . . . . . . . . . . 235
9 THERMO-(VISCO)ELASTICITY 237
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237
9.2 The constitutive framework - Nonlinear thermo-viscoelasticity . . . . . . . 238
9.2.1 Free energy, thermodynamic forces and rate equations . . . . . . . . 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
9.3.1 Backward Euler method . . . . . . . . . . . . . . . . . . . . . . . . 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
9.4.1 The constitutive relations . . . . . . . . . . . . . . . . . . . . . . . 243
9.4.2 The constitutive integrator . . . . . . . . . . . . . . . . . . . . . . . 245
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
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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
10.3.1 Backward Euler method . . . . . . . . . . . . . . . . . . . . . . . . 256
10.3.2 Backward Euler method – Constrained minimization problem . . . 259
10.3.3 ATS-tensor for BE-rule . . . . . . . . . . . . . . . . . . . . . . . . . 260
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
10.4.1 The constitutive relations . . . . . . . . . . . . . . . . . . . . . . . 265
10.4.2 The constitutive integrator . . . . . . . . . . . . . . . . . . . . . . . 267
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
10.5.4 Continuum tangent relations . . . . . . . . . . . . . . . . . . . . . . 280
10.5.5 Significance of hardening versus softening . . . . . . . . . . . . . . . 282
10.5.6 Significance of total mechanical dissipation versus
“plastic dissipation” . . . . . . . . . . . . . . . . . . . . . . . . . . 284
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10.6 The constitutive integrator - Hardening
plasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284
10.6.1 Backward Euler method . . . . . . . . . . . . . . . . . . . . . . . . 284
10.6.2 Backward Euler method – Constrained minimization problem . . . 286
10.6.3 ATS-tensor for BE-rule . . . . . . . . . . . . . . . . . . . . . . . . . 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
10.7.1 The constitutive relations . . . . . . . . . . . . . . . . . . . . . . . 290
10.7.2 The constitutive integrator . . . . . . . . . . . . . . . . . . . . . . . 295
10.7.3 Examples of response simulations . . . . . . . . . . . . . . . . . . . 301
10.7.4 Appendix: Constitutive relations for the uniaxial stress state . . . . 305
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
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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
11.2.1 Free energy and thermodynamic forces . . . . . . . . . . . . . . . . 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
11.3.1 Backward Euler method . . . . . . . . . . . . . . . . . . . . . . . . 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
11.4.1 The constitutive relations . . . . . . . . . . . . . . . . . . . . . . . 355
11.4.2 The constitutive integrator . . . . . . . . . . . . . . . . . . . . . . . 359
11.4.3 Examples of response simulations . . . . . . . . . . . . . . . . . . . 363
11.5 Prototype model: Hosford yield surface and von Mises plastic potential . . 368
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11.6 Prototype model: Parabolic Drucker-Prager yield surface . . . . . . . . . . 368
11.6.1 The constitutive relations . . . . . . . . . . . . . . . . . . . . . . . 368
11.6.2 The constitutive integrator . . . . . . . . . . . . . . . . . . . . . . . 370
11.6.3 Examples of response simulations . . . . . . . . . . . . . . . . . . . 371
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
11.8.1 The constitutive relations . . . . . . . . . . . . . . . . . . . . . . . 381
11.8.2 The constitutive integrator . . . . . . . . . . . . . . . . . . . . . . . 385
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
12.2.1 Free energy and thermodynamic forces . . . . . . . . . . . . . . . . 406
12.2.2 Non-associative flow and hardening rules . . . . . . . . . . . . . . . 406
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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
12.3.4 The constitutive relations . . . . . . . . . . . . . . . . . . . . . . . 425
12.3.5 The constitutive integrator . . . . . . . . . . . . . . . . . . . . . . . 429
12.3.6 Examples of response simulations . . . . . . . . . . . . . . . . . . . 432
12.4 Prototype model: Gurson model family of yield surfaces . . . . . . . . . . 432
12.4.1 The constitutive relations . . . . . . . . . . . . . . . . . . . . . . . 432
12.4.2 The constitutive integrator . . . . . . . . . . . . . . . . . . . . . . . 432
12.4.3 Examples of response simulations . . . . . . . . . . . . . . . . . . . 432
13 VISCOPLASTICITY 435
13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435
13.2 The constitutive framework – Perzyna format . . . . . . . . . . . . . . . . 436
13.2.1 Free energy and thermodynamic forces . . . . . . . . . . . . . . . . 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
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13.3 The constitutive integrator – Perzyna format . . . . . . . . . . . . . . . . . 443
13.3.1 Backward Euler method . . . . . . . . . . . . . . . . . . . . . . . . 443
13.3.2 ATS-tensor for BE-rule . . . . . . . . . . . . . . . . . . . . . . . . . 445
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
13.4.1 The constitutive relations . . . . . . . . . . . . . . . . . . . . . . . 449
13.4.2 The constitutive integrator . . . . . . . . . . . . . . . . . . . . . . . 449
13.4.3 Examples of response simulations . . . . . . . . . . . . . . . . . . . 451
13.4.4 Appendix: Constitutive relations for the uniaxial stress state . . . . 451
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
13.6.2 ATS-tensor for BE-rule . . . . . . . . . . . . . . . . . . . . . . . . . 459
13.7 Prototype model: Hooke elasticity and Bingham-type viscoplasticity with
linear mixed hardening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 460
13.7.1 The constitutive relations . . . . . . . . . . . . . . . . . . . . . . . 460
13.7.2 The constitutive integrator . . . . . . . . . . . . . . . . . . . . . . . 461
13.7.3 Examples of response simulations . . . . . . . . . . . . . . . . . . . 461
14 THERMO-(VISCO)PLASTICITY 463
14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463
14.2 The constitutive framework for thermo-viscoplasticity – Perzyna format . . 464
14.2.1 Free energy, thermodynamic forces and rate equations . . . . . . . . 464
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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
14.3.2 Continuum tangent relations . . . . . . . . . . . . . . . . . . . . . . 467
14.3.3 Continuum tangent relations - The locally adiabatic case . . . . . . 469
14.4 The constitutive integrator – Perzyna format . . . . . . . . . . . . . . . . . 470
14.4.1 Backward Euler method . . . . . . . . . . . . . . . . . . . . . . . . 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
14.5.1 The constitutive relations . . . . . . . . . . . . . . . . . . . . . . . 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
14.5.4 The constitutive integrator . . . . . . . . . . . . . . . . . . . . . . . 476
14.5.5 Examples of response simulations . . . . . . . . . . . . . . . . . . . 476
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
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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
Vol I March 21, 2006
xxviii CONTENTS
Vol I March 21, 2006
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.
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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.
Vol I March 21, 2006
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)
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4 1 TENSOR CALCULUS TOOLBOX
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)
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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)
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6 1 TENSOR CALCULUS TOOLBOX
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)
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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.
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8 1 TENSOR CALCULUS TOOLBOX
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
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1.4 Elementary algebra of 4th order tensors 9
1.3.5 Coordinate transformation
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)
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10 1 TENSOR CALCULUS TOOLBOX
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.
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1.4 Elementary algebra of 4th order tensors 11
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)
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12 1 TENSOR CALCULUS TOOLBOX
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.
1.4.4 Coordinate transformation
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.
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1.4 Elementary algebra of 4th order tensors 13
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)
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14 1 TENSOR CALCULUS TOOLBOX
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
Vol I March 21, 2006
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.
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16 1 TENSOR CALCULUS TOOLBOX
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)
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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)
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18 1 TENSOR CALCULUS TOOLBOX
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
Mk2Mk1 Mk2Mk2 Mk2Mk3
Mk3Mk1 Mk3Mk2 Mk3Mk3
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.
Vol I March 21, 2006
1.6 Spectral properties and invariants of a symmetric 2nd order tensor 19
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)
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20 1 TENSOR CALCULUS TOOLBOX
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.
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1.6 Spectral properties and invariants of a symmetric 2nd order tensor 21
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)
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22 1 TENSOR CALCULUS TOOLBOX
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)
Vol I March 21, 2006
1.6 Spectral properties and invariants of a symmetric 2nd order tensor 23
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.
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24 1 TENSOR CALCULUS TOOLBOX
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)
Vol I March 21, 2006
1.6 Spectral properties and invariants of a symmetric 2nd order tensor 25
∂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).
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26 1 TENSOR CALCULUS TOOLBOX
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):
Vol I March 21, 2006
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)
Vol I March 21, 2006
28 1 TENSOR CALCULUS TOOLBOX
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
Vol I March 21, 2006
1.7 Representation theorems 29
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)
Vol I March 21, 2006
30 1 TENSOR CALCULUS TOOLBOX
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)
Vol I March 21, 2006
1.7 Representation theorems 31
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)
Vol I March 21, 2006
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)
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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
.
Vol I March 21, 2006
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
Vol I March 21, 2006
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:
Vol I March 21, 2006
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
Vol I March 21, 2006
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)
Vol I March 21, 2006
7.2 Constitutive relations - Isotropic nonlinear elasticity 171
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)
Vol I March 21, 2006
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.
Vol I March 21, 2006
7.2 Constitutive relations - Isotropic nonlinear elasticity 173
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.
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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)
Vol I March 21, 2006
7.2 Constitutive relations - Isotropic nonlinear elasticity 175
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)
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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)
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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)
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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
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7.3 Prototype model: Hooke’s model of isotropic linear elasticity 179
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)
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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)
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7.3 Prototype model: Hooke’s model of isotropic linear elasticity 181
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.
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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
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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).
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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.
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7.4 Constitutive framework - Anisotropic nonlinear elasticity 185
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
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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).
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7.4 Constitutive framework - Anisotropic nonlinear elasticity 187
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)
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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
Vol I March 21, 2006
7.4 Constitutive framework - Anisotropic nonlinear elasticity 189
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).
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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).
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7.4 Constitutive framework - Anisotropic nonlinear elasticity 191
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)
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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)
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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)
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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)
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7.5 Constitutive framework - Anisotropic linear elasticity 195
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
Vol I March 21, 2006
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)
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7.5 Constitutive framework - Anisotropic linear elasticity 197
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
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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)
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7.5 Constitutive framework - Anisotropic linear elasticity 199
Show this as homework!
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
Spectral properties – Kelvin modes
The 3 distinct eigenvalues (nmode = 3) are
λ1 = M + 2L , K1 = 1 (7.142)
λ2 = M − L , K2 = 2 (7.143)
λ3 = 2G , K3 = 3 (7.144)
corresponding to the projection operators
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,
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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.
Spectral properties – Kelvin modes
The 2 distinct eigenvalues (nmode = 2) are
λ1 = 3K , K1 = 1 (7.149)
λ2 = 2G , K2 = 5 (7.150)
corresponding to the projection operators
P1 = Ivol (7.151)
P2 = Isymdev (7.152)
Vol I March 21, 2006
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).
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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
Vol I March 21, 2006
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).
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250 10 PLASTICITY - BASIC CONCEPTS
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.
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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]
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252 10 PLASTICITY - BASIC CONCEPTS
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)
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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)
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254 10 PLASTICITY - BASIC CONCEPTS
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)
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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
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256 10 PLASTICITY - BASIC CONCEPTS
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)
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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)
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258 10 PLASTICITY - BASIC CONCEPTS
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.
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10.3 The constitutive integrator - Perfect plasticity 259
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)
Vol I March 21, 2006
260 10 PLASTICITY - BASIC CONCEPTS
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)
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10.3 The constitutive integrator - Perfect plasticity 261
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)
Vol I March 21, 2006
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
Vol I March 21, 2006
10.3 The constitutive integrator - Perfect plasticity 263
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)
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264 10 PLASTICITY - BASIC CONCEPTS
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.
Vol I March 21, 2006
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.
Vol I March 21, 2006
266 10 PLASTICITY - BASIC CONCEPTS
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)
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10.4 Prototype model: Hooke elasticity and von Mises yield surface 267
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.
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268 10 PLASTICITY - BASIC CONCEPTS
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.
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10.4 Prototype model: Hooke elasticity and von Mises yield surface 269
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.
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270 10 PLASTICITY - BASIC CONCEPTS
• 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)
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10.4 Prototype model: Hooke elasticity and von Mises yield surface 271
• 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)
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272 10 PLASTICITY - BASIC CONCEPTS
• 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).
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10.4 Prototype model: Hooke elasticity and von Mises yield surface 273
• 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)
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274 10 PLASTICITY - BASIC CONCEPTS
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)
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10.4 Prototype model: Hooke elasticity and von Mises yield surface 275
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)
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276 10 PLASTICITY - BASIC CONCEPTS
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)
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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.
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278 10 PLASTICITY - BASIC CONCEPTS
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.
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10.5 The constitutive framework - Hardening plasticity 279
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)
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280 10 PLASTICITY - BASIC CONCEPTS
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
Vol I March 21, 2006
10.5 The constitutive framework - Hardening plasticity 281
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)
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282 10 PLASTICITY - BASIC CONCEPTS
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.
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10.5 The constitutive framework - Hardening plasticity 283
= =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.
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284 10 PLASTICITY - BASIC CONCEPTS
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)
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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)
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286 10 PLASTICITY - BASIC CONCEPTS
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:
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10.6 The constitutive integrator - Hardeningplasticity 287
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
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288 10 PLASTICITY - BASIC CONCEPTS
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)
Vol I March 21, 2006
10.6 The constitutive integrator - Hardeningplasticity 289
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.
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290 10 PLASTICITY - BASIC CONCEPTS
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
10.7.1 The constitutive relations
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)
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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)
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292 10 PLASTICITY - BASIC CONCEPTS
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)
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10.7 Prototype model: Hooke elasticity and von Mises yield surface with linearmixed hardening 293
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)
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294 10 PLASTICITY - BASIC CONCEPTS
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.
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10.7 Prototype model: Hooke elasticity and von Mises yield surface with linearmixed hardening 295
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.
10.7.2 The constitutive integrator
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)
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296 10 PLASTICITY - BASIC CONCEPTS
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)
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10.7 Prototype model: Hooke elasticity and von Mises yield surface with linearmixed hardening 297
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:
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298 10 PLASTICITY - BASIC CONCEPTS
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)
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10.7 Prototype model: Hooke elasticity and von Mises yield surface with linearmixed hardening 299
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.
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300 10 PLASTICITY - BASIC CONCEPTS
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)
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10.7 Prototype model: Hooke elasticity and von Mises yield surface with linearmixed hardening 301
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).
10.7.3 Examples of response simulations
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.
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302 10 PLASTICITY - BASIC CONCEPTS
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.
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10.7 Prototype model: Hooke elasticity and von Mises yield surface with linearmixed hardening 303
−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
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304 10 PLASTICITY - BASIC CONCEPTS
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.
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10.7 Prototype model: Hooke elasticity and von Mises yield surface with linearmixed hardening 305
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.
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306 10 PLASTICITY - BASIC CONCEPTS
• 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.
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10.7 Prototype model: Hooke elasticity and von Mises yield surface with linearmixed hardening 307
• 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)
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308 10 PLASTICITY - BASIC CONCEPTS
• 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)
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10.7 Prototype model: Hooke elasticity and von Mises yield surface with linearmixed hardening 309
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).
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310 10 PLASTICITY - BASIC CONCEPTS
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)
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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
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312 10 PLASTICITY - BASIC CONCEPTS
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.
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10.8 Classical isotropic yield criteria 313
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.
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314 10 PLASTICITY - BASIC CONCEPTS
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.
Vol I March 21, 2006
10.8 Classical isotropic yield criteria 315
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)
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316 10 PLASTICITY - BASIC CONCEPTS
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.
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10.8 Classical isotropic yield criteria 317
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.
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318 10 PLASTICITY - BASIC CONCEPTS
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.
Vol I March 21, 2006
10.8 Classical isotropic yield criteria 319
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:
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320 10 PLASTICITY - BASIC CONCEPTS
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
Vol I March 21, 2006
10.8 Classical isotropic yield criteria 321
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)
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322 10 PLASTICITY - BASIC CONCEPTS
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.
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10.8 Classical isotropic yield criteria 323
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)
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324 10 PLASTICITY - BASIC CONCEPTS
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.
Vol I March 21, 2006
10.8 Classical isotropic yield criteria 325
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)
Vol I March 21, 2006
326 10 PLASTICITY - BASIC CONCEPTS
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)
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10.8 Classical isotropic yield criteria 327
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.
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328 10 PLASTICITY - BASIC CONCEPTS
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.
Vol I March 21, 2006
10.8 Classical isotropic yield criteria 329
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).
Vol I March 21, 2006
330 10 PLASTICITY - BASIC CONCEPTS
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)
Vol I March 21, 2006
10.8 Classical isotropic yield criteria 331
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)
Show this as homework!
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
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332 10 PLASTICITY - BASIC CONCEPTS
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)
σ2 =1√3ξ + (σdev)2 with (σdev)2 =
√
2
3ρ cos(θ − 120o) (10.322)
σ3 =1√3ξ + (σdev)3 with (σdev)3 =
√
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.
Vol I March 21, 2006
10.8 Classical isotropic yield criteria 333
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.
Vol I March 21, 2006
334 10 PLASTICITY - BASIC CONCEPTS
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)
Vol I March 21, 2006
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)
Vol I March 21, 2006
336 10 PLASTICITY - BASIC CONCEPTS
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)
Vol I March 21, 2006
10.9 The constitutive integrator for a special class: Isotropic linear elasticity andisotropic yield criteria 337
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
µ ∂2Φ∂σm∂σe
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)
Vol I March 21, 2006
338 10 PLASTICITY - BASIC CONCEPTS
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.
Vol I March 21, 2006
10.9 The constitutive integrator for a special class: Isotropic linear elasticity andisotropic yield criteria 339
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)
Vol I March 21, 2006
340 10 PLASTICITY - BASIC CONCEPTS
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∂σe
µ ∂2Φ∂σm∂θ
∂Φ∂σm
µ ∂2Φ∂σm∂σe
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.
Vol I March 21, 2006
10.9 The constitutive integrator for a special class: Isotropic linear elasticity andisotropic yield criteria 341
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.
Vol I March 21, 2006
342 10 PLASTICITY - BASIC CONCEPTS
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
10.10.1 The constitutive relations
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)
Vol I March 21, 2006
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)
10.10.2 The constitutive integrator
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)
Vol I March 21, 2006
344 10 PLASTICITY - BASIC CONCEPTS
10.10.3 Examples of response simulations
Perfect plasticity, variation of parameter k. Extremes: k = 1 (von Mises), k large (Tresca).
To be completed.
Vol I March 21, 2006
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.
Vol I March 21, 2006
346 10 PLASTICITY - BASIC CONCEPTS
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?
Vol I March 21, 2006
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