CONSTITUTIVE MODELING OF ENGINEERING MATERIALS - THEORY AND COMPUTATION Volume I General Concepts and Inelasticity

Transcription

1 CONSTITUTIVE MODELING OF ENGINEERING MATERIALS - THEORY AND COMPUTATION Volume I General Concepts and Inelasticity by Kenneth Runesson, Paul Steinmann, Magnus Ekh and Andreas Menzel

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3 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 techniques within a setting of geometrically linear theory. Rate-independent as well as ratedependent 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 response. 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

4 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 constitutive models used in engineering practice. Presenting ideas underlying numerical procedures for integrating the evolution equations 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 obtained 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 mechanics that deals with the establishment of constitutive models for engineering materials.

5 v Some generally accepted constraints that must be imposed on constitutive models are discussed. 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 material 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 consequence 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 balance 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 iterations. 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 Dissipative Material. Both rate-independent and rate-dependent response are considered (under the assumption of isothermal conditions). Extension is then made to non-associative structure (whereby the normality property is lost). Finally, issues of controllability, stability, 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 incremental 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

6 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 incremental 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 numerical 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

7 vii the celebrated postulate of Maximum Plastic Dissipation, which is the fundamental basis of classical plasticity. This postulate infers the normality rule, and it provides general loading criteria (in terms of the complementary Kuhn-Tucker conditions) for any choice of control variables. To illustrate the developments, the von Mises yield criterion with mixed isotropic and kinematic hardening is investigated in detail as a prototype model. The chapter is concluded with a review of classical isotropic yield (and failure) criteria. In Chapter 11 we elaborate on advanced concepts of plasticity, that are often necessary to account for in order to provide a realistic model. Here, we limit the list of such concepts to: non-associative flow and hardening rules, non-smooth yield surface, and anisotropic yield surface. A prototype model (or class of models) is selected to illustrate each concept. In particular, we discuss the Cam-Clay family of yield surfaces, developed for granular materials, as a proponent of plasticity models that display quite general (non-associative) hardening. In Chapter 12 we discuss elastic-viscoplastic material response, which is characterized by the presence of rate-dependent dissipation mechanisms when the stress exceeds the yield stress. Viscoplasticity is shown to be the regularization of rate-independent plasticity in the sense that the flow and hardening rules are obtained from a penalty formulation of the MPD-principle (in the spirit of Perzyna s viscoplasticity concept). To illustrate the developments, the Bingham/Norton model with mixed hardening is investigated in detail as a prototype model. In Chapter 13 we discuss elastic-viscoplastic response with thermomechanical coupling. A generic format of the rate equations is presented for simple hardening of the quasistatic yield surface, whereby Perzyna s viscoplasticity concept is adopted. The continuum tangent 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 considered. 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-

8 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 damage 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 membrane 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.

9 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 Mähler and Dr. Thomas Svedberg, who are present and former graduate students at Chalmers, have read (parts of) the manuscript and struggled with the numerical simulations. Ms. EvaMari Runesson, an English and History student at Göteborg University (and who also happens to be the daughter the first author), quickly became an expert in L A TEX. 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 including running the response simulations pertinent to the various prototype models. The contribution of both is gratefully acknowledged. Göteborg and Kaiserslautern in January Kenneth Runesson and Paul Steinmann

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11 Contents 1 TENSOR CALCULUS TOOLBOX Introduction Preliminaries about style and notation Symbolic and component notation Differential operators Elementary algebra of vectors Component representations Scalar product and length Coordinate transformation Elementary algebra of 2nd order tensors Component representations Scalar product(s) Symmetry and skew-symmetry Special tensors Coordinate transformation Elementary algebra of 4th order tensors Component representation Symmetry and skew-symmetry Special tensors Coordinate transformation

12 xii CONTENTS Appendix: Voigt-matrix representation of 4th order tensor transformation Permutation tensor (symbol) and its usage Spectral properties and invariants of a symmetric 2nd order tensor Principal values - Spectral decomposition Basic invariants Principal invariants - Cayley-Hamilton s theorem Octahedral invariants of the stress and strain tensors Derivatives of a 2nd order tensor Derivatives of invariants, etc Representation of eigendyads Representation theorems Coordinate transformation vs. vector rotation Scalar-valued isotropic tensor functions of one argument Scalar-valued isotropic tensor functions of two arguments Scalar-valued isotropic tensor functions of three arguments Symmetric tensor-valued isotropic tensor functions of one argument Symmetric tensor-valued isotropic tensor function of two arguments 31 2 CHARACTERISTICS OF ENGINEERING MATERIALS AND CON- STITUTIVE MODELING General remarks on constitutive modeling Concept of a constitutive model The role of constitutive modeling General constraints on constitutive models Approaches to constitutive modeling Modeling of material failure Fracture Continuum damage mechanics

13 CONTENTS xiii Fracture mechanics Common experimental test conditions Typical behavior of metals and alloys Plastic yielding Hardening and ductile fracture Constant loading Creep and relaxation Time-dependent loading Rate effect and damping Cyclic loading and High-Cycle-Fatigue (HCF) Cyclic loading and Low-Cycle-Fatigue (LCF) Creep-fatigue and Relaxation-fatigue Typical behavior of ceramics and cementitious composites Monotonic loading Semi-brittle fracture Cyclic loading and fatigue Creep and relaxation Typical behavior of granular materials Monotonic loading Basic features Constant loading Consolidation Constant loading Creep and relaxation INTRODUCTION TO CONTINUUM THERMODYNAMICS Introduction Motivation and literature overview The role of continuum thermodynamics Thermodynamic system Thermodynamic state variables Thermodynamic processes Mechanical balance laws Global and local formats of the momentum balance law Equilibrium equation

14 xiv CONTENTS Global and local formats of the moment of the momentum balance law - Symmetry of stress Energy balance The first law of thermodynamics Kinetic and internal energy Global and local formats of the energy equation Entropy inequality The second law of thermodynamics Entropy - Motivation from statistical mechanics Global and local formats of the entropy inequality Basic constitutive relations Choice of independent state variables - Thermodynamic potentials Legendre-Fenchel transformations Format based on internal energy Format based on enthalpy Format based on (Helmholtz ) free energy Format based on (Gibbs ) free enthalpy Evaluation of thermodynamic processes Strain and stress energy for reversible system Tangent stiffness and compliance relations at prescibed temperature - General situation Tangent stiffness and compliance relations at prescribed temperature - Adiabatic case The archetypes of dissipative materials Generic constitutive relations Inviscid (rate-independent) response Viscous (rate-dependent) response Appendix: Legendre transformations Questions and problems

15 CONTENTS xv 4 SPACE-TIME DISCRETIZED FORMATS OF THERMOMECHANI- CAL RELATIONS The continuous formats of continuum thermodynamics Preliminaries The fully coupled format The isothermal format The isometric format (rigid heat conductor) The thermomechanically decoupled format The adiabatic format The eisentropic format The discrete formats of continuum thermodynamics Preliminaries The fully coupled format The fully coupled format - reduced version The isothermal format The isometric format The adiabatic and eisentropic formats Global solution algorithm - Newton iterations Preliminaries Iterative solution for the fully coupled format Iterative solution for the isothermal format Iterative solution for the isometric format Iterative solution for the adiabatic and eisentropic formats THE CANONICAL CONSTITUTIVE FRAMEWORK FOR DISSIPA- TIVE MATERIALS Introduction Common features of a canonical constitutive framework

16 xvi CONTENTS Free energy and thermodynamic stresses Concepts of dissipation surface and elastic region Associative structure - Postulate of Maximum Dissipation (MD) Rate-independent models - Exact format of MD Rate-independent models Alternative formats of the constitutive equations pertinent to the exact format of MD Rate-independent models - Continuum tangent operators Rate-dependent models - Penalized enforcement of MD Rate-dependent models Alternative formats of the constitutive equations pertinent to the penalized enforcement of MD Non-associative structure Rate-independent models Rate-independent models - Continuum tangent operators Issues of controllability, stability and uniqueness Controllability of rate-independent response for strain and stress control Limit points for rate-independent response - Spectral properties of CTS-tensor Second order work - Spectral properties of symmetric part of CTStensor - Hill-stability Uniqueness of boundary value problem THE CONSTITUTIVE INTEGRATOR Introduction - The concept of a Constitutive Laboratory Response functions Constitutive Laboratory code CONSTLAB c Integrator - Backward Euler rule for the rate-independent canonical format Incremental format

17 CONTENTS xvii The Algorithmic Tangent Stiffness (ATS) tensor The ATS-tensor - Alternative derivation Integrator - Backward Euler rule of the rate-dependent canonical format Incremental format The ATS-tensor Iterator - Newton algorithm Isothermal format Adiabatic format Numerical differentiation Appendix: Total derivative of an implicit function ELASTICITY Introduction General characteristics of nonlinear elasticity Material symmetry - Isotropy Appendix: Voigt-matrix representation of tangent relations Constitutive relations - Isotropic nonlinear elasticity Generic format of free energy Generic format of Continuum Tangent Stiffness tensor Volumetric/deviatoric decomposition of the free energy Deformation theory of plasticity Prototype model: Hooke s model of isotropic linear elasticity Constitutive relations Examples of response simulations Constitutive framework - Anisotropic nonlinear elasticity Generic format of the free energy - Symmetry classes Representation of anisotropy with structure tensors

18 xviii CONTENTS Kelvin-modes and spectral decomposition of the tangent stiffness tensor Constitutive framework - Anisotropic linear elasticity Orthogonal symmetry Tetragonal symmetry Transverse isotropy Cubic symmetry Isotropy VISCOELASTICITY Introduction The constitutive framework - Nonlinear viscoelasticity Free energy, thermodynamic forces and rate equations Linear viscoelasticity - Creep and relaxation functions The constitutive integrator - Nonlinear viscoelasticity Backward Euler method ATS-tensor for BE-rule Backward Euler rule for linear elasticity Backward Euler rule for linear viscoelasticity Prototype model: The isotropic (linear) Maxwell model The constitutive relations The constitutive integrator Examples of response simulations Examples of response simulations Appendix: Constitutive relations for the uniaxial stress state Prototype model: The isotropic (nonlinear) Norton model The constitutive relations The constitutive integrator

19 CONTENTS xix Examples of response simulations Appendix: Constitutive relations for the uniaxial stress state The constitutive framework - Nonlinear viscoelasticity with multiple dissipative mechanisms Free energy, thermodynamic forces and rate equations Linear model Prototype model - The isotropic Linear Standard Viscoelasticity model The constitutive relations The constitutive integrator Examples of response simulations THERMO-(VISCO)ELASTICITY Introduction The constitutive framework - Nonlinear thermo-viscoelasticity Free energy, thermodynamic forces and rate equations The energy equation The locally adiabatic case The constitutive integrator - Nonlinear thermo-viscoelasticity Backward Euler method ATS-tensor and other algorithmic quantities for the BE-rule Prototype model: The isotropic (linear) Maxwell-Fourier model The constitutive relations The constitutive integrator The constitutive integrator - Adiabatic condition Examples of response simulations (adiabatic case) PLASTICITY - BASIC CONCEPTS Introduction

20 xx CONTENTS Motivation Literature overview The constitutive framework - Perfect plasticity Free energy and thermodynamic stresses Associative structure - Postulate of Maximum Dissipation Continuum tangent relations The constitutive integrator - Perfect plasticity Backward Euler method Backward Euler method Constrained minimization problem ATS-tensor for BE-rule Backward Euler method for linear elasticity - Solution in stress space Concept of Closest-Point-Projection for linear elasticity Prototype model: Hooke elasticity and von Mises yield surface The constitutive relations The constitutive integrator Examples of response computations Appendix I: Constitutive relations for the uniaxial stress state Appendix II: Voigt format of prototype model The constitutive framework - Hardening plasticity Free energy and thermodynamic forces Representation of hardening - Constraints and classification Associative structure - Postulate of Maximum Dissipation Continuum tangent relations Significance of hardening versus softening Significance of total mechanical dissipation versus plastic dissipation

21 CONTENTS xxi 10.6 The constitutive integrator - Hardening plasticity Backward Euler method Backward Euler method Constrained minimization problem ATS-tensor for BE-rule Backward Euler method for linear elasticity and linear hardening Concept of Closest-Point-Projection for linear elasticity and linear hardening Prototype model: Hooke elasticity and von Mises yield surface with linear mixed hardening The constitutive relations The constitutive integrator Examples of response simulations Appendix: Constitutive relations for the uniaxial stress state Classical isotropic yield criteria Basic concepts - Cohesive and frictional character Isotropic yield criteria - General characteristics The Tresca criterion The von Mises criterion Hosford s yield criterion The Mohr criterion The Mohr-Coulomb criterion The Drucker-Prager criterion Appendix: Geometric invariants in principal stress space The constitutive integrator for a special class: Isotropic linear elasticity and isotropic yield criteria Backward Euler method - Preliminaries

22 xxii CONTENTS Backward Euler method for two-invariant yield surfaces (independent of the Lode angle) Backward Euler method for three-invariant yield surfaces ATS-tensor Prototype model: Hooke elasticity and Hosford s family of yield surfaces The constitutive relations The constitutive integrator Examples of response simulations Questions and problems PLASTICITY - ADVANCED CONCEPTS Introduction The constitutive framework Nonassociative structure Free energy and thermodynamic forces Non-associative flow and hardening rules Continuum tangent relations (for smooth yield surface) Non-associative hardening - Special choice The constitutive integrator Nonassociative structure Backward Euler method ATS-tensor Closest-Point-Projection for linear elasticity and linear hardening Volumetric non-associativity Isotropic elasticity and plasticity Prototype model: Hooke elasticity and von Mises yield surface with nonlinear mixed (saturation) hardening The constitutive relations The constitutive integrator Examples of response simulations Prototype model: Hosford yield surface and von Mises plastic potential.. 368

23 CONTENTS xxiii 11.6 Prototype model: Parabolic Drucker-Prager yield surface The constitutive relations The constitutive integrator Examples of response simulations Non-smooth yield surfaces Associative flow rule - Koiter s rule Continuum tangent relations for non-smooth yield surface Backward Euler method - CPPM for linear elasticity ATS-tensor Prototype model: Linear Drucker-Prager yield surface The constitutive relations The constitutive integrator Anisotropic yield surfaces Oriented materials - Representation of anisotropy with structure tensors Orthotropy - Restriction to quadratic forms Transverse isotropy - Restriction to quadratic forms Hill s yield criterion Prototype model: Transversely isotropic elasticity and Hill s yield criterion The constitutive relations The constitutive integrator Examples of response simulations PLASTICITY - MORE ADVANCED CONCEPTS Introduction The constitutive framework Plasticity with non-simple hardening Free energy and thermodynamic forces Non-associative flow and hardening rules

24 xxiv CONTENTS Continuum tangent relations (for smooth yield surface) Controllability under strain and stress control Material failure and stability General Hill s and Drucker s criteria of material stability The constitutive integrator BE-rule ATS-tensor Prototype model: Cam-Clay family of yield surfaces Porosity measures relative density Generic relations in Critical State Soil Mechanics The generic Cam-Clay family of yield surfaces The constitutive relations The constitutive integrator Examples of response simulations Prototype model: Gurson model family of yield surfaces The constitutive relations The constitutive integrator Examples of response simulations VISCOPLASTICITY Introduction The constitutive framework Perzyna format Free energy and thermodynamic forces Penalty formulation of the Postulate of Maximum Plastic Dissipation Rate equations Plasticity as the limit situation Elasticity as the limit situation Creep and relaxation Generalized rate laws Concept of dynamic yield surface

25 CONTENTS xxv 13.3 The constitutive integrator Perzyna format Backward Euler method ATS-tensor for BE-rule Backward Euler method for linear elasticity and linear hardening Concept of Closest-Point-Projection for linear elasticity and linear hardening - Quasi-projection property Prototype model: Hooke elasticity and Bingham viscoplasticity with linear mixed hardening The constitutive relations The constitutive integrator Examples of response simulations Appendix: Constitutive relations for the uniaxial stress state The constitutive framework Duvaut-Lions format Flow and hardening rules Thermodynamic abmissibility The constitutive integrator Duvaut-Lions format Backward Euler ATS-tensor for BE-rule Prototype model: Hooke elasticity and Bingham-type viscoplasticity with linear mixed hardening The constitutive relations The constitutive integrator Examples of response simulations THERMO-(VISCO)PLASTICITY Introduction The constitutive framework for thermo-viscoplasticity Perzyna format Free energy, thermodynamic forces and rate equations

26 xxvi CONTENTS The energy equation The locally adiabatic case The constitutive framework of rate-independent thermo-plasticity Preliminaries Continuum tangent relations Continuum tangent relations - The locally adiabatic case The constitutive integrator Perzyna format Backward Euler method ATS-tensor and other algorithmic quantities for the BE-rule Prototype model: Hooke elasticity and Bingham (visco)plasticity with linear mixed hardening and thermal softening The constitutive relations The constitutive relations for the rate-independent response The constitutive equations for dynamic yield surface and temperaturedependent quasistatic yield surface The constitutive integrator Examples of response simulations CALIBRATION OF CONSTITUTIVE MODELS Introduction Definition of parameter identification problem Experimental data Testing and measurements Parameter identification from direct identification Calibration via optimization - Least squares format Preliminaries Choice of state-, control- and response variable Objective function Evaluation of the objective function

27 CONTENTS xxvii Quality of the solution Prototype example: Norton s model Optimization strategies Preliminaries Gradient-free methods Gradient-based methods Sensitivity assessment Preliminaries Monte Carlo method Perturbation method Correlation matrix Influence from discretization errors Self-adjoint format for calibration Optimality condition Gradient-based methods for optimization Sensitivity assessment

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29 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 continuum 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 Preliminaries about style and notation As a rule, the theoretical developments are presented in the traditional direct style (common 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 mathematics) is adopted, whereby theorems are preceding proofs. Independently of the presentation style, some important results are placed within a frame. A box ( ) 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 characters are used to denote tensors of 1st order (vectors), 2nd order, or 3rd order, e.g. a, A.

30 2 1 TENSOR CALCULUS TOOLBOX Boldface sanserif characters denote 4th order tensors, e.g. A. Sets are denoted by blackboard 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. u def = du/dt. Italic characters are used to denote a running index, whereas roman characters are used to denote a fixed index, e.g. ɛ p ij. 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 coordinates, unless otherwise is explicitly stated 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 e i v = v i e i def = 3 v i e i, i=1 w = w α e α def = 2 w α e α (1.1) α=1 Latin letters are used for a running index ranging from 1 to 3 (corresponding to a representation of v in E 3 ), whereas Greek letters are used when the running index ranges from 1 to 2 (corresponding to a representation of w in E 2 ) 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 = A ij e i e j, T = T i1 i 2...i n e i1 e i2...e in (1.2) where denotes the open product symbol. The tensor components are normally denoted by a meager character (such as A ij ). However, in order to avoid possible sources of confusion and provide maximal transparency, we sometimes use the notation (A) ij instead 1 Note that Greek letters are also used to label matrix elements, e.g. A = [A αβ ], 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.

31 1.2 Elementary algebra of vectors 3 of A ij. This notation is useful in expressions such as A t = (A t ) ij e i e j = (A) ji e i e j, A B = (A B) ij e i e j (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 1 ɛ 11 ɛ 12 u = [u i ] =, ɛ = [ɛ ij ] = (1.4) u 2 ɛ 21 ɛ 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: [ ] = e i [ ] x i, [ ] = [ ] x i e i (1.5) The dot product (divergence) and the cross product (rotation or curl) are defined as [ ] = e i [ ] x i, In particular, if a is a scalar field, then a = ( a/ x i )e i. [ ] = e i [ ] x i (1.6) 1.2 Elementary algebra of vectors 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 = v i e i (1.7) Remark: Adopting more general non-cartesian coordinates, we have the two possible representations v = v i g i = v i g i (1.8)

32 4 1 TENSOR CALCULUS TOOLBOX PSfrag replacements g2 v e 2 g 2 g 1 g 1 e 1 Figure 1.1: Co- and contravariant base vectors. Cartesian base vectors. where v i, g i are the covariant components and base vectors, respectively, whereas v i, g i are the corresponding contravariant quantities. The covariant base vectors g i 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 g i are defined as mutually orthogonal to g i, i.e. g i g j = δ j i with δ j i = { 1 if i = j 0 if i j (1.9) where δ j i represents the mixed co-contravariant components of the metric (or unit) tensor I. Usually, δ j i is known as the Kronecker delta symbol. The scalar product of g i and g j constitute the covariant components g ij of the metric tensor. Likewise, the scalar product of g i and g j constitute the contravariant components g ij of the metric tensor, i.e. So much for general coordinates. g i g j = g ij = g ji, g i g j = g ij = g ji (1.10) Scalar product and length With u = u i e i and v = v j e j, we obtain u v = [u i e i ] [v j e j ] = u i v j δ ij = u i v i (1.11)

33 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 = [u i u i ] 1/2 = [[u 1 ] 2 + [u 2 ] 2 + [u 3 ] 2 ] 1/2 (1.12) Coordinate transformation We shall consider the effect of coordinate transformation between two Cartesian coordinate systems with base vectors that are denoted e i and e i respectively. The relations between these are defined by a linear transformation. Upon performing a scalar multiplication of the ansatz e i = M ij e j with e k and using e j e k = δ jk, we obtain e i e k = M ij e j e k = M ik. Hence, we summarize e i = M ij e j, M ij = e i e j = cos(e i, e j ) (1.13) Moreover, we make the ansatz u = u ie i to obtain e k u = e k [u i e i ] = u i [e k e i ] = u i M ki = e k [u ie i] = u i[e k e i] = u iδ ki = u k (1.14) and we conclude that u = u i e i = u ie i, with u i = M ij u j (1.15) In matrix form, the component relation (1.15) 2 reads u = M u with M = [M ij ] (1.16) 1.3 Elementary algebra of 2nd order tensors 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 v def = [u i e i ] [v j e j ] = u i v j e i e j (1.17) where is the open product symbol. The products e i e j, which are denoted base dyads, form the basis of the product space E 3 E 3 (in the same way that e i form the basis of E 3 ). Clearly, the dyad in (1.17) is only a special case of the general representation T = T ij e i e j (1.18)