CONSTITUTIVE MODELING OF ENGINEERING MATERIALS - THEORY AND COMPUTATION The Primer by Kenneth Runesson Lecture Notes, Dept. of Applied Mechanics, Chalmers University of Technology, Göteborg
Preface There seems to be an ever increasing demand in engineering practice for more realistic models as applied to metals as well as composites, ceramics, polymers and geological materials (such as soil and rock). Consequently, a vast amount of literature is available on the subject of nonlinear constitutive modeling, with strong emphasis on plasticity and damage. Such modeling efforts are parallelled by the development of numerical algorithms for use in Finite Element environment. For example, implicit (rather than explicit) integration techniques for plasticity problems are now predominant in commercial FE-codes. I am indebted to a great number of people who have contributed to the present volume: Mr. M. Enelund, Mr. L. Jacobsson, Mr. M. Johansson, Mr. L. Mähler and Mr. T. Svedberg, who are all graduate students at Chalmers Solid Mechanics, have read (parts of) the manuscript and struggled with the numerical examples. Mr. T. Ernby prepared some of the difficult figures. Ms. C. Johnsson, who is a graduate student in ancient Greek history at Göteborg University, quickly became an expert in handling equations in L A TEX. The contribution of each one is gratefully acknowledged. Göteborg in March 1996. Kenneth Runesson 2nd revised edition: I am grateful to Mr. Lars Jacobson and Mr. Magnus Johansson (in particular) for their help in revising parts of the manuscript. Göteborg in March 1997. Kenneth Runesson 3rd revised edition: Ms. EvaMari Runesson, who is a student in English at the University of Gothenburg (and also happens to be my daughter) did an excellent job in mastering L A TEXfor this edition.
iv Göteborg in March 1998. Kenneth Runesson 4th revised edition: Mr. Lars Jacobsson and Ms. EvaMari Runesson were of great help in typing the manuscript. Göteborg in January 1999. Kenneth Runesson 5th revised edition: Some small changes were made to improve the manuscript. Göteborg in January 2000. Kenneth Runesson 6th revised edition: Ms. Annicka Karlsson was of great help in revising the manuscript, mainly concerning the notation. Göteborg in January 2002 Kenneth Runesson 7th revised edition: The help by Mr. Mikkel Grymer in revising the manuscript is greatly acknowledged. Göteborg in March 2005 Kenneth Runesson
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Contents 1 CHARACTERISTICS OF ENGINEERING MATERIALS AND CON- STITUTIVE MODELING 1 1.1 General remarks on constitutive modeling.................. 1 1.1.1 Concept of a constitutive model.................... 1 1.1.2 The role of constitutive modeling................... 3 1.1.3 General constraints on constitutive models.............. 4 1.1.4 Approaches to constitutive modeling................. 5 1.2 Modeling of material failure Fracture................... 7 1.2.1 Continuum damage mechanics..................... 7 1.2.2 Fracture mechanics........................... 8 1.3 Common experimental test conditions..................... 8 1.4 Typical behavior of metals and alloys..................... 12 1.4.1 Plastic yielding Hardening and ductile fracture.......... 12 1.4.2 Constant loading Creep and relaxation.............. 13 1.4.3 Time-dependent loading Rate effect and damping........ 14 1.4.4 Cyclic loading and High-Cycle-Fatigue (HCF)............ 15 1.4.5 Cyclic loading and Low-Cycle-Fatigue (LCF)............. 16 1.4.6 Creep-fatigue and Relaxation-fatigue................. 20 1.5 Typical behavior of ceramics and cementitious composites......... 21 1.5.1 Monotonic loading Semi-brittle fracture............... 21
viii CONTENTS 1.5.2 Cyclic loading and fatigue....................... 22 1.5.3 Creep and relaxation.......................... 22 1.6 Typical behavior of granular materials.................... 22 1.6.1 Monotonic loading Basic features.................. 22 1.6.2 Constant loading Consolidation................... 23 1.6.3 Constant loading Creep and relaxation............... 23 2 THERMODYNAMICS A BRIEF SUMMARY 25 2.1 Free energy and constitutive relations..................... 25 2.1.1 General................................. 25 2.1.2 Stress-strain response relation..................... 26 2.1.3 Material classes............................. 27 3 VISCOELASTICITY 29 3.1 Introduction................................... 29 3.2 Prototype model: The Maxwell rheological model.............. 30 3.2.1 Thermodynamic basis Constitutive relation............ 30 3.2.2 Prescribed constant stress (pure creep)................ 32 3.2.3 Prescribed constant strain (pure relaxation)............. 32 3.3 Linear viscoelasticity Constitutive modeling................ 33 3.3.1 General characteristics......................... 33 3.3.2 Laplace-Carson transform....................... 35 3.3.3 Linear Standard Model (Generalized Maxwell Model)........ 37 3.3.4 Backward Euler method for linear standard model.......... 40 3.4 Linear viscoelasticity Structural analysis.................. 42 3.4.1 Structural behavior........................... 42 3.4.2 Solution strategies........................... 43 3.4.3 Analysis of truss Elastic analogy.................. 44
CONTENTS ix 3.4.4 Analysis of truss numerical integration.............. 47 3.4.5 Analysis of beam cross-section Elastic analogy.......... 49 3.4.6 Analysis of double-symmetric beam cross-section Numerical integration................................. 51 3.5 Nonlinear viscoelasticity Constitutive modeling.............. 53 3.5.1 General characteristics......................... 53 3.5.2 Norton creep law............................ 54 3.5.3 Backward Euler method for the Norton creep law.......... 56 3.6 Nonlinear viscoelasticity structural analysis................ 58 3.6.1 Structural behavior........................... 58 3.6.2 Analysis of truss............................ 58 3.6.3 Analysis of beam cross-section Stationary creep.......... 60 3.6.4 Analysis of double-symmetric beam cross-section Numerical integration................................. 63 3.6.5 Analysis of single-symmetric beam cross-section Numerical integration.................................. 64 3.7 Viscous damping and dynamic behavior.................... 67 3.7.1 Preliminaries.............................. 67 3.7.2 Forced vibration of discrete system.................. 68 3.7.3 Energy dissipation........................... 70 3.7.4 Evaluation of damping for the linear standard model........ 71 3.8 Appendix : Laplace - Carson transform.................... 75 4 PLASTICITY 77 4.1 Introduction................................... 77 4.2 Prototype rheological model for perfectly plastic behavior.......... 79 4.2.1 Thermodynamic basis Yield criterion............... 79 4.2.2 Plastic flow rule and elastic-plastic tangent relation......... 80
x CONTENTS 4.2.3 Dissipation of energy.......................... 82 4.3 Prototype model for hardening plastic behavior............... 82 4.3.1 Thermodynamic basis Yield criterion............... 82 4.3.2 Plastic flow rule and elastic-plastic tangent relation......... 83 4.3.3 Dissipation of energy.......................... 86 4.4 Model for cyclic loading Mixed isotropic and kinematic hardening... 86 4.4.1 Thermodynamic basis Yield criterion............... 86 4.4.2 Associative flow and hardening rules Linear hardening...... 87 4.4.3 Characteristic response for linear hardening............. 88 4.4.4 Associative flow and nonassociative hardening rules Nonlinear hardening................................ 88 4.4.5 Characteristic response for nonlinear hardening........... 90 4.4.6 Backward Euler method for integration Linear hardening.... 96 4.5 Structural analysis............................... 101 4.5.1 Structural behavior Limit load analysis.............. 101 4.5.2 Analysis of truss Numerical integration.............. 101 4.5.3 Analysis of double-symmetric beam cross-section.......... 103 4.5.4 Analysis of single-symmetric beam cross-section Numerical integration.................................. 106 5 VISCOPLASTICITY 109 5.1 Introduction................................... 109 5.2 Prototype rheological model for perfectly viscoplastic behavior....... 110 5.2.1 Thermodynamic basis Quasistatic yield criterion......... 110 5.2.2 Viscoplastic flow rule Perzyna s formulation............ 111 5.2.3 Bingham model Perzyna s formulation............... 113 5.2.4 Norton model (creep law) Perfect viscoplasticity......... 114
CONTENTS xi 5.2.5 Limit behavior Viscoplastic regularization of rate-independent plasticity................................. 115 5.3 Prototype rheological model for hardening viscoplasticity.......... 115 5.3.1 Thermodynamic basis Quasistatic yield criterion......... 115 5.3.2 Viscoplastic flow and hardening rules Perzyna s formulation... 116 5.3.3 Bingham model Perzyna s formulation............... 117 5.3.4 Viscoplastic flow and hardening rules Duvaut-Lions formulation 118 5.3.5 Comparison of Perzyna s and Duvaut-Lions formulations...... 119 5.3.6 Bingham model Duvaut-Lions formulation............ 120 5.4 Model for cyclic loading Mixed isotropic and kinematic hardening... 121 5.4.1 Constitutive relations for linear hardening Perzyna s formulation 121 5.4.2 Backward Euler method for linear hardening Perzyna s formulation121 5.5 Structural analysis............................... 123 6 DAMAGE AND FRACTURE THEORY 125 6.1 Introduction to the modeling of damage................... 125 6.1.1 Concept of damage........................... 125 6.1.2 Physical nature of damage for different materials.......... 127 6.1.3 The concepts of effective stress and strain equivalence........ 128 6.2 Prototype model of damage coupled to elasticity............... 130 6.2.1 Thermodynamics Damage criterion................ 130 6.2.2 Damage law and tangent relations................... 133 6.3 Experimental measurement of damage.................... 135 7 DAMAGE COUPLED TO PLASTICITY 137 7.1 Prototype model for damage coupled to perfect plasticity.......... 137 7.1.1 Thermodynamic basis Yield and damage criterion........ 137 7.1.2 Plastic flow rule and damage law Constitutive relations..... 139
xii CONTENTS 7.1.3 Dissipation inequality.......................... 143 7.1.4 Dissipation of mechanical energy................... 144 7.2 Prototype model for damage coupled to hardening plasticity........ 148 7.2.1 Thermodynamics Yield and damage criterion........... 148 7.2.2 Dissipation rules Constitutive relations.............. 148 7.2.3 Dissipation of energy.......................... 151 7.3 Model for cyclic loading and fatigue Mixed linear isotropic and kinematic hardening.................................... 151 7.3.1 Constitutive relations for linear hardening.............. 151 7.3.2 Backward Euler algorithm for integration Linear hardening and uniaxial stress.............................. 152 8 DAMAGE COUPLED TO VISCOPLASTICITY 157 8.1 Prototype model for damage coupled to perfect viscoplasticity....... 157 8.1.1 Thermodynamic basis Quasistatic yield and damage criterion.. 157 8.1.2 Viscoplastic flow rule and damage law Perzyna s formulation.. 158 8.1.3 Norton model (creep Law) Perfect viscoplasticity......... 159 8.2 Prototype model for damage coupled to hardening viscoplasticity..... 162 8.2.1 Thermodynamic basis Quasistatic yield and damage criterion.. 162 8.2.2 Viscoplastic flow, hardening and damage rules Perzyna s formulation.................................. 163 8.3 Constitutive modeling of creep failure of metals and alloys......... 163 8.3.1 Modified damage law for tertiary creep................ 163 8.3.2 Typical results for creep at uniaxial stress.............. 164 9 FATIGUE PHENOMENON AND ANALYSIS 167 9.1 Background................................... 167 9.1.1 Nomenclature.............................. 167
CONTENTS xiii 9.1.2 Historical remarks........................... 168 9.1.3 Cyclic stress-strain relation...................... 169 9.2 Engineering approach to HCF and LCF based on stress-control....... 170 9.2.1 Basquin-relation............................. 170 9.2.2 Variable amplitude loading Palmgren-Miner rule......... 177 9.2.3 Multiaxial fatigue criteria based on stress............... 179 9.3 Engineering approach to LCF based on strain-control............ 181 9.3.1 Manson-Coffin relation......................... 181 9.3.2 Combined effects of creep and fatigue................. 183 9.3.3 Multiaxial fatigue criteria based on strain.............. 185 9.4 Life prediction strategies............................ 185 9.4.1 Coupled - decoupled approach..................... 185 9.4.2 Life prediction strategy based on the decoupled approach...... 187 9.5 Damage mechanics approach to LCF..................... 189 9.5.1 Simplified analysis of LCF Derivation of the Manson-Coffin and Basquin relations............................ 189 9.5.2 Rational approach to LCF - Damage coupled to plastic deformation 194 9.6 Damage mechanics approach to CLCF.................... 197 9.6.1 Simplified analysis of CLCF...................... 197 9.6.2 Damage coupled to viscoplastic deformation............. 203 9.7 Fracture mechanics approach to fatigue.................... 203 9.7.1 Preliminaries.............................. 203 9.7.2 Paris law for fatigue crack growth.................. 204 9.7.3 Variable amplitude loading....................... 207
xiv CONTENTS
Chapter 1 CHARACTERISTICS OF ENGINEERING MATERIALS AND CONSTITUTIVE MODELING In this chapter we give a brief introduction to the particular field within applied solid mechanics that deals with the establishment of constitutive models for engineering materials. 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 the most important engineering materials (metals and alloys, cementitious composites, granular materials) under various loading conditions is reviewed. 1.1 General remarks on constitutive modeling 1.1.1 Concept of a constitutive model Common to all mechanical analysis of engineering materials and their behavior in structural components is the need for constitutive models that link the states of stress and strain. From a mathematical viewpoint, the constitutive equations (that define the constitutive model) are complementary equations to the balance and kinematic equations.
2 1 CHARACTERISTICS OF ENGINEERING MATERIALS AND CONSTITUTIVE MODELING Taken together with the loading and boundary conditions, these are the sufficient, but not always the necessary, equations in order to formulate a complete boundary value problem, from which the motion of a given body can be calculated 1. It is clear that constitutive models may be very different for the various materials used in engineering practice, such as metals and alloys, polymers, fiber composites (with polymer or metal matrix), concrete and wood. However, to a large extent it is possible to employ the same principles and concepts (and even the same terminology) in establishing constitutive relations for these different materials, despite the fact that the physics behind the macroscopical phenomena are entirely different. Indeed the characteristics of an engineering material are determined by its microstructure, all the way down to its atomic arrangement. Examples of microstructures (on the level below the macroscopic scale) are shown in Figure 1.1. Crystalline and amorphous materials behave differently, as do single crystals in comparison to polycrystalline materials. The mechanical properties are often significantly affected by the temperature and by the loading rate. For example, the ductility of a metal is reduced at low temperature and high loading rate. Figure 1.1: Typical microstructure of (a) Steel (perlitic grain structure, eutectoid composition), (b) Concrete, (c) Wood. 1 These are the necessary and sufficient conditions for any hyperstatic (statically indeterminate) structure, whereas it is not necessary to know the constitutive response to calculate the stresses in an isostatic (statically determinate) structure.
1.1 General remarks on constitutive modeling 3 1.1.2 The role of constitutive modeling It is emphasized that constitutive models are just mathematical simplifications of a quite complex physical behavior, and there is no such thing as an exact model. For example, it is appropriate to claim that the behavior of steel can be represented by an elasticplastic model, but it does not make sense to claim that steel is elastic-plastic! In fact, it is appropriate to model steel (and any other engineering material) in a number of ways depending on the purpose and the required precision of the model predictions. Examples of different purposes of the relevant model are given as follows: Structural analysis under working load: Linear elasticity Analysis of damped vibrations: Viscoelasticity Calculation of limit load: Rigid perfect plasticity Accurate calculation of permanent deformation after monotonic and cyclic loading: Hardening elasto-plasticity Analysis of stationary creep and relaxation: Perfect (nonhardening) elasto-viscoplasticity Prediction of lifetime in high-cycle-fatigue: Damage coupled to elastic deformations Prediction of lifetime in low-cycle-fatigue: Damage coupled to plastic deformations Prediction of lifetime in creep and creep-fatigue: Damage coupled to viscoplastic deformations Prediction of stability of a preexisting crack: Linear elasticity (from which singular stress fields are derived for sharp cracks) Prediction of strain localization in shear bands and incipient material failure: Softening plasticity or damage coupled to plastic deformation Most of the listed phenomena will be considered in some detail in this text. Clearly, the task of the engineer is to choose a model that is sufficiently accurate, yet not unnecessarily complex and computationally expensive. The questions that should be asked in regard to the choice of a certain model are as follows:
4 1 CHARACTERISTICS OF ENGINEERING MATERIALS AND CONSTITUTIVE MODELING Is the model relevant for describing the physical phenomena at hand? Does the model produce sufficiently accurate predictions for the given purpose? Is it possible to devise and implement a robust numerical algorithm (in a computer code) to obtain a truly operational model? 1.1.3 General constraints on constitutive models A list of constraints that must be placed on constitutive relations, that represent the mechanical behavior of a continuous medium, is given below. Virtually all of these requirements are intuitively obvious, although it is not trivial to express them properly in mathematical language. Moreover, some constraints are important only in conjunction with large deformations, say, at the modeling of material forming. Principle of coordinate invariance Constitutive relations, as well as other relations between physical entities, should not be affected by arbitrary coordinate transformations. This requirement is satisfied if proper tensorial relations are established. Principle of determinism (or causality) The stress in a given body is determined entirely by the history of the motion of the body, i.e. it is not affected by the future events. This requirement is always satisfied if intrinsically time-dependent relations are established with time as (one of) the independent coordinate(s). It may be violated if relations between Laplace or Fourier transformed variables are set up directly. For example, care must be taken when internal damping relations (expressing energy dissipation) are proposed in the frequency domain, as discussed by Crandall (1970). Principle of material objectivity (or frame-indifference) Constitutive relations must not be affected by arbitrary Rigid Body Motion (RBM) that is superposed on the actual motion.
1.1 General remarks on constitutive modeling 5 This requirement is most easily satisfied by employing objective tensor fields as the constitutive variables. In particular, it is important to note that the ordinary time derivative of common variables (stress, strain) is not objective. For example, the time rate of the (Cauchy) stress tensor is not zero at RBM, even if the material does not feel any change of stress, i.e. the stress components with respect to a corotating coordinate system do not change. However, the non-zero time rate is merely a consequence of the rotation. As a consequence, this time rate is not permissible in constitutive relations, at least not for large material rotation. In small strain theory, which employs linear kinematics, the requirement of objectivity can be ignored. Constraints of material symmetry (or spatial covariance) Response functions are unaffected by certain rotations of the chosen reference configuration due to material symmetry. The most important special case is complete material isotropy, which means that the response is equal in all directions or, more precisely, for all possible spatial rotations of the chosen reference configuration. The precise definition of symmetry is expressed mathematically in terms of the appropriate symmetry group (of orthogonal transformations). Second law of thermodynamics (or dissipation inequality) The 2nd law of thermodynamics states that the production of internal entropy, or rate of material disorder, must be non-negative. This statement is equivalent to the statement that dissipation of energy is never negative. This law, whose mathematical formulation is the Clausius-Duhem Inequality, is discussed in Chapter 3. It is a cornerstone for the further developments in the present text. In particular, its automatic satisfaction is a key feature of standard dissipative materials, which are considered in various contexts subsequently. 1.1.4 Approaches to constitutive modeling The conceptually different approaches to the derivation of macroscopical constitutive models may be defined as follows:
6 1 CHARACTERISTICS OF ENGINEERING MATERIALS AND CONSTITUTIVE MODELING Fundamental (or micromechanics) approach Homogenization and computational multiscale modeling (CMM) As indicated above, a complete understanding of the deformation and failure characteristics requires the detailed knowledge of the microstructural processes. In the fundamental approach this fact is acknowledged, and elementary constitutive relations are established for the microstructural behavior (micromechanical modeling). A classical example is crystal plasticity, in which relations between shear stress and shear slip are established for single slip systems within the atomic lattice structure. A useful macroscopic model can then obtained via averaging techniques (homogenization), which can sometimes be carried out analytically, cf. Nemat-Nasser & Hori (1993). More generally, it is carried out numerically with the aid of a Representative Volume Element (RVE), which must be sufficiently large to admit statistical representations, yet small enough to represent a point from a continuum mechanics perspective. Sometimes the size of the RVE is determined by periodicity of the microstructural arrangement. A more powerful alternative to homogenization aimed at developing a macroscopic constitutive model is to carry out a Computational Multiscale Modeling (CMM), whereby the macroscopic constitutive model becomes obsolete. The response of the RVE is then simulated as an integrated part of the macroscopic analysis of a given component, which involves the global balance equations of mechanics. The macroscopic stress and strain values are computed as averages (in some sense) of the corresponding microstructural fields within the pertinent RVE in each spatial point subjected to the actual macroscopic deformation. cf. Miehe (1996), Lilbacka et al. (2004), Grymer et al. (2006). The fundamental approach to constitutive modeling is still less developed, although the international activity is quite strong. Not only metals with ordered lattice structures are considered, but also disordered media (soil, rock, etc.). Phenomenological approach The macroscopic model is established directly based on the observed characteristics from elementary tests. The calibration is carried out mainly by comparison with experimental results and/or with micromechanical predictions for well-defined boundary conditions on the pertinent RVE, cf. the discussion above. Traditional models are sometimes simple enough to admit the identification of the material parameter values one by one from
1.2 Modeling of material failure Fracture 7 well-defined elementary experiments. The obvious example is the observation of the yield stress of mild steel from a tensile test. However, the general approach is to optimize the predictive capability of the model in the calibration procedure. The objective function to be minimized is a suitable measure (norm) of the difference between the predicted response and the experimentally obtained data. The arguments of the constitutive functions are observable variables (like stress, strain and temperature) in addition to a sufficient number of nonobservable, or internal, variables that represent the microstructural changes. Statistical approach Statistical models for describing material behavior are the least fundamental, in the sense that they are normally established as response functions for specific loading and environmental conditions. A variety of distributions can be used for describing the scatter in strength data, whereby Weibull s statistical theory is quite often used. 1.2 Modeling of material failure Fracture Two principally different views can be distinguished with respect to the analysis of material failure. From a classical standpoint, these approaches are related to the fact that a material may behave in a ductile or brittle fashion, depending on material composition, aging, temperature, etc. 1.2.1 Continuum damage mechanics The view of continuum damage mechanics is that the failure process starts with a gradual deterioration of a continuously deforming material. After considerable inelastic deformation, due to the material ductility, the stress drops quite dramatically (in a displacement controlled test) and deformations localize in a narrow zone (or band). This stage is defined as the onset of fracture; cf. Figure 1.2. In many cases the localization is quite extreme in the sense that a single macroscopic (discrete) crack starts to develop. Stresses can be transferred across the crack until it is fully opened.
8 1 CHARACTERISTICS OF ENGINEERING MATERIALS AND CONSTITUTIVE MODELING σ microcracks = damage σ neck develops localization zone= neck brittle ǫ ductile (a) σ (b) Figure 1.2: Damage process (a) Localization (necking) in a bar of ductile material, (b) Stress vs. strain characteristics. 1.2.2 Fracture mechanics The view of fracture mechanics is that a macroscopic crack (or flaw) has already occured, and the main task is to determine whether the crack will propagate or not. A crack that propagates only when the externally applied load is increased is termed stable. No consideration is then given to the process leading to the (preexisting) fully open crack. The analysis of crack stability is usually based on the assumption that the behavior close to the crack tip is linear elastic (Linear Fracture Mechanics), such that the stress field singularity at the crack tip is determined from linear elasticity, cf. Figure 1.3. The simplest crack stability criterion is the (empirical) Griffith criterion, by which the crack is deemed stable if the pertinent stress intensity factor, that depends on the applied loading, does not exceed a critical value. This concept can be extended to cyclic loading, e.g. in the shape of a threshold level of stress in Paris law. 1.3 Common experimental test conditions Phenomenological constitutive laws are calibrated with the aid of experimental data that are obtained from well-defined laboratory tests. The idea is to design the test in such a way that the specimen is subjected to homogeneous states of stress, strain and temperature. A few common test conditions in practice are listed below:
1.3 Common experimental test conditions 9 L σ macroscopic crack u singular stress field at crack tip σ = (locally) σ brittle area under σ u curve = released fracture energy u σ (a) (b) Figure 1.3: Fracture process (a) Preexisting edge cracks, (b) Far-field stress vs. extension characteristics. Note: ε = u/l is not well-defined as local measure of strain! Uniaxial stress A cylindrical bar is subjected to a state of uniform (axial) stress, which may be tensile or compressive, as shown in Figure 1.4(a). The strain state is cylindrical, i.e. nonzero radial and tangential normal strains normally appear. Either the axial stress or the axial strain is controlled. This elementary test condition is common for most materials, at least those possessing cohesion. By definition, cohesion materials have shear strength that prevails when the mean (normal) stress is zero. Frictional materials, whose shear strength vanishes when the mean stress is zero, can not be tested under the uniaxial stress condition without precompaction. Normal stress combined with shear The conventional way of applying normal stresses, combined with shear stress, is to subject a circular thin-walled tube to axial load, internal or external pressure, together with a torsional moment, as shown in Figure 1.4(b). Since the wall thickness is small, the radial stress varies approximately linearly through the thickness, and at the midplane of the tube wall a well-defined triaxial stress state is obtained. Moreover, when torsion is applied, the principal axes rotate due to additional shear stress. In the case there is no applied pressure, a state of plane stress is obtained. This type of test is common for metals, but has also been used for concrete and highly cohesive soil (such as clay).
10 1 CHARACTERISTICS OF ENGINEERING MATERIALS AND CONSTITUTIVE MODELING Conventional plane stress and plain strain Cross-shaped plane specimens of metallic material may be subjected to biaxial (tensile or compressive) loading under plane stress conditions. For soil and other granular materials, a special biaxial apparatus (biaxial cell) is needed to ensure the appropriate out of plane condition, in particular the plane strain condition. The principal stress directions can not rotate. Cylindrical stress and strain states A cylindrical specimen is subjected to external radial pressure and axial compressive load, as shown in Figure 1.4(c). This is a commonly used test condition for granular materials, such as powder and soil, as well as for rock and concrete. Two usual test procedures are denoted Conventional Triaxial Compression (CTC) and Conventional Triaxial Extension (CTE). In the CTC-test, an isotropic state of stress is first applied during the socalled consolidation phase. Then the radial (=circumferential) stress is held constant, while the axial compressive loading is further increased. This compression may be either stressor strain-controlled. In the CTE-test, isotropic stress is first applied to consolidate the sample in the same fashion as for the CTC-test. However, the axial stress is then kept constant while the radial pressure is further increased. In order to assess the principal difference between these two test conditions, we consider the corresponding principal stresses σ i < 0 (compression negative), where σ 1 σ 2 σ 3. Since σ i = 0, the CTC-test is defined by σ 1 = σ 2 > σ 3 and the axial stress is σ 3, which is the numerically largest principal stress. The CTE-test, on the other hand, is defined by σ 1 > σ 2 = σ 3 and the axial stress is now σ 1. It is common to use these test results towards the evaluation of a failure (or yield) criterion of the Mohr-Coulomb type, cf. Chapter 10. True triaxial stress and strain states Principal stresses can be applied independently in the cubical cell apparatus, as illustrated in Figure 1.4(d). In practice, this is a quite complex device that has gained widespread use for soil, rock and concrete.
1.3 Common experimental test conditions 11 z θ r θ r p (a) z (b) σ 2 θ r σ r = σ θ = p σ 1 p σ 3 (c) (d) Figure 1.4: Stress and strain states in (a) Tensile test, (b) Normal load-torsion test of thin-walled tube, (c) CTC- and CTE-tests, and (d) Cubical cell test.
12 1 CHARACTERISTICS OF ENGINEERING MATERIALS AND CONSTITUTIVE MODELING 1.4 Typical behavior of metals and alloys 1.4.1 Plastic yielding Hardening and ductile fracture The basic behavior of a ductile metal is obtained under monotonic loading. Plastic yielding will occur approximately at the same magnitude of stress in tension as in compression since plastic slip is determined by the critical resolved shear stress along potential slip planes (Schmid s law). Yielding is independent of the magnitude of the mean stress, which defines an ideal cohesive material. The further increase of stress beyond yielding is known as hardening. Figure 1.5(a) shows the typical stress-strain relation in uniaxial tension at monotonic loading of a hot-worked steel. The characteristic strength parameters are the yield stress σ y and the ultimate strength (peak stress) σ u. Figure 1.5(b) shows the typical yield surface in biaxial stress (approximately elliptical in reality for a polycrystalline metal). σ u σ y σ σ 2 σ y σ y σ 1 ǫ y ǫ u ǫ σ y σ y (a) (b) Figure 1.5: (a) Stress-strain relation in uniaxial tension showing yielding, hardening and ductile fracture. (b) Yield surface in biaxial stress. The picture is complemented by unloading, followed by reversed loading which gives rise to hysteresis loops, as shown in Figure 1.6. After significant straining the average unloading modulus will decrease significantly, which may be interpreted as a sign of internal degradation (damage) and that fracture is approaching.
1.4 Typical behavior of metals and alloys 13 ǫ σ A A A A B σ y E E Ê < E (a) t (b) ǫ Figure 1.6: Response at loading/unloading showing eventual degradation (damage) and ductile fracture 1.4.2 Constant loading Creep and relaxation The time-dependent response of a material at elevated temperature, after rapid initial loading up to constant (nominal) stress, is denoted creep. For metals, viscous (creep) behavior becomes important when the temperature exceeds, approximately, 30% of the melting temperature. At this temperature, cavitation along the grain boundaries starts to become an important deformation/failure mechanism. A typical creep curve, for constant temperature, is shown in Figure 1.7(a). The recovery upon rapid unloading is also shown. Three different stages of the creep process can be distinguished (in a classical description), although the transition between them is, by no means, clear: Transistent stage (or Primary stage, I) The rate of creep is initially decreasing, which is a result of saturation of dislocations. Stationary stage (or Secondary stage, II) After the saturation level has been reached, the creep rate is rather constant. As will be discussed later, the Norton creep law is traditionally adopted in this stage. Creep failure stage (or Tertiary stage, III) After certain creep deformation, the development of microstructural degradation (internal damage) will result in an accelerated creep rate until failure occurs at time t R, which is
14 1 CHARACTERISTICS OF ENGINEERING MATERIALS AND CONSTITUTIVE MODELING the lifetime of the specimen. This process is strongly temperature dependent. The time-dependent stress change after rapid loading, while the strain is held constant, is denoted relaxation. The relaxation behavior is thus complementary to the creep behavior, as shown in Figure 1.7(b). σ ǫ σ 0 A ǫ 0 A B t B t ǫ creep A rupture σ ǫ 0 B recovery σ 0 A t R I II III t t (a) (b) Figure 1.7: (a) Creep and (b) Relaxation curves. B 1.4.3 Time-dependent loading Rate effect and damping Another aspect of viscous properties, besides creep, is the rate-dependence that is exhibited in the stress-strain curve for certain materials. This is manifested by higher stiffness and strength for larger loading rate, especially due to impact loading. In accordance with the situation at creep, the rate-effect is more pronounced at elevated temperature. The typical result at monotonic loading under prescribed strain rate for a rate-sensitive material is shown in Figure 1.8. In a real structure the strain rate may vary considerably from the
1.4 Typical behavior of metals and alloys 15 loaded region to other parts. Hence, it is important to model rate-effects in such a fashion that the rate-independent situation is obtained merely as a special case. σ ǫ = ǫ > 0 ǫ = 0 ǫ Figure 1.8: Rate effect on stress-strain relation. Damping, in the sense that free vibrations of a structure will decay with time and eventually die out, can be explained as the result of energy dissipation in the material. If the amount of damping is dependent on the frequency of the vibrations, then the damping is of viscous character and can be modelled within the framework of viscoelasticity or viscoplasticity. If, on the other hand, the damping is independent on the frequency, then the damping is commonly denoted as hysteretic and can be modelled within the framework of rate-independent plasticity. An alternative way of assessing damping and rate effects is to consider forced vibrations due to a sustained harmonic load with given frequency. The structural response, in terms of strain and stress, is then normally observed to be dependent on the frequency of the exciting load, which points towards a rate-effect. 1.4.4 Cyclic loading and High-Cycle-Fatigue (HCF) The usual way of testing cyclic and, eventually, fatigue behavior is to subject the specimen to a (slow) cyclic variation of stress or strain with constant amplitude. If the applied load level is below the macroscopic yield stress, but above a certain threshold, the cyclic response is in the elastic range and no macroscopic plastic deformation is observed. However, after many load cycles a reduction of the apparent elasticity modulus is noted. This
16 1 CHARACTERISTICS OF ENGINEERING MATERIALS AND CONSTITUTIVE MODELING degradation of the elastic stiffness is caused by microcracking and microslip due to local stress-concentrations within the microstructure. The number of cycles to failure is very high (N R > 100, 000), and the final fracture is brittle in character since failure is preceeded by virtually no inelastic deformation, as shown in Figure 1.9. σ (MPa) σ -0.2 200 100 ǫ 0.2 10 2 damage threshold 2 4 6 8 10 5N Figure 1.9: Result of HCF-test with constant strain amplitude. 1.4.5 Cyclic loading and Low-Cycle-Fatigue (LCF) If the applied load level is high enough, the macroscopic yield stress will be exceeded, and plastic strains will develop in each cycle. Not unlike the characteristics of a creep test, three different stages of the deformation process may be distinguished: Saturation Stage Consider the early stage of cyclic loading with constant amplitude. The response is then characterized as either cyclic hardening or cyclic softening. The typical behavior of cyclic hardening is shown in Figure 1.10(a) for given strain amplitude (strain control) and in Figure 1.10(b) for given stress amplitude (stress control). Cyclic hardening means that the stress amplitude will initially increase in a few cycles to an asymptotic level in a strain controlled test, whereas the strain amplitude will decrease in a stress controlled test. The complementary behavior in the case of (initial) cyclic softening is shown in Figure 1.11(a) and Figure 1.11(b). Hence, cyclic softening means that the stress amplitude will initially decrease to an asymptotic level in a strain controlled test, whereas the strain amplitude will increase in a stress controlled test. In both stress- and strain-controlled cyclic loading, the ideal situation is that the respective strain or stress amplitude will shake-down quite rapidly to a stabilized stress-strain
1.4 Typical behavior of metals and alloys 17 σ ǫ = constant σ max σ 3 2 1 t ǫ ǫ σ min Stabilized σ max = σ min σ m = 0 (a) 1 2 σ σ = constant ǫ max 3 2 1 ǫ min t Stabilized ǫ ǫ m 0 1 2 (b) Figure 1.10: Initial cyclic hardening as shown in (a) Strain control, (b) Stress control.
18 1 CHARACTERISTICS OF ENGINEERING MATERIALS AND CONSTITUTIVE MODELING σ ǫ = constant σ max σ 1 2 ǫ σ = constant σ min σ m 0 Stabilized t (a) 2 1 σ ǫ ǫ max 1 2 3 t ǫ ǫ min 2 1 Stabilized (b) ǫ max = ǫ min ǫ m = 0 Figure 1.11: Initial cyclic softening as shown in (a) Strain control, and (b) Stress control.
1.4 Typical behavior of metals and alloys 19 hysteresis loop, as shown in Figure 1.12(a). This loop is symmetrical in tension and compression in the ideal situation. In reality, the stabilized (shake-down) amplitudes on the tension and compression sides may not be symmetrical, even if the applied cyclic action is symmetrical, as shown in Figure 1.10 and Figure 1.11. Denoting the stabilized maximum values by σ max and ǫ max, and the minimum values by σ min and ǫ min, we define the cyclic mean stress and cyclic mean strain, respectively, as σ m = 1 2 [σ max + σ min ], ǫ m = 1 2 [ǫ max + ǫ min ] (1.1) where it is noted that algebraic values are used. We thus conclude that, in general at the saturation level, σ m 0 in the strain-controlled test, whereas ǫ m 0 in the stresscontrolled test. However, it is also possible that stabilization does not occur at all (or is very slow). For prescribed constant stress amplitude, this lack of stabilization is evident as ever increasing plastic strain, or ratchetting, which is shown in Figure 1.12(b). Such ratchetting can be expected when σ m 0, in particular. σ σ 1 2 ǫ ǫ Shakedown 1 2 ǫ r (a) (b) Ratchetting strain Figure 1.12: Phenomena of (a) Shakedown and (b) Ratchetting.
20 1 CHARACTERISTICS OF ENGINEERING MATERIALS AND CONSTITUTIVE MODELING Fatigue failure stage After certain amount of plastic deformation has accumulated in the hysteretic loops after saturation, damage starts to develop. This damage development will eventually, say after 1,000-10,000 cycles, result in cyclic softening until failure occurs. Hence, LCF is characterized by relatively small values of N R, which is the number of cycles to failure (N R < 10, 000). The characteristics of LCF are observed in the strain-controlled as well as in the stresscontrolled environment. The elastic unloading modulus is continually decreasing in such a way that the hysteresis-loops become more and more flattened. As a result, the stress amplitude gradually decreases in the strain-controlled test, as shown in Figure 1.13, whereas the strain-amplitude grows in an uncontrolled fashion in the stress controlled test. Moreover, further ratchetting may be obtained due to the fact that the rate of damage development is smaller in compression than in tension (and it is assumed to be zero in the figure). stabilized (saturation) σ (MPa) 300 200 100 ǫ 0.2 0.2 10 2 σ damage threshold 500 1000 1500 2000 N Figure 1.13: Result of LCF-test with constant strain amplitude. 1.4.6 Creep-fatigue and Relaxation-fatigue Creep-fatigue is obtained when the stress varies in a cyclic fashion with a predefined holdtime within each cycle. The failure is caused by the combined action of creep deformation and deterioration of the stiffness due to LCF. This phenomenon is of particular importance at the design of jet engines and other gas turbines, which operate under high temperature. Relaxation-fatigue is the counterpart of creep-fatigue when the strain is allowed to vary in a cyclic fashion with predefined hold-time. Sometimes, the notion thermal fatigue refers to the situation where the stress/strain vari-
1.5 Typical behavior of ceramics and cementitious composites 21 ation is due to cyclic temperature change. The effect becomes more pronounced for high degree of static indeterminacy (when stresses are larger). Clearly, it is not possible to control the stress or strain amplitude when the temperature is varied. The most general loading situation is denoted thermomechanical fatigue, in which case a component is subjected to cyclic variation of the mechanical load as well as the temperature. 1.5 Typical behavior of ceramics and cementitious composites 1.5.1 Monotonic loading Semi-brittle fracture At monotonic loading, cementitious materials (such as concrete) show nearly linear elastic response at small load levels. However, the type of failure is entirely different in tension and compression. Tensile failure will occur in a quite brittle (quasi-brittle) manner at the tensile strength, σ = σ tu, whereby a macroscopic crack starts to develop and is fully open when the stress has dropped to zero. The corresponding post-peak stressstrain relationship is not well-defined, cf. the discussion in Section 2.2. This response is depicted in Figure 1.14(a). Compressive failure, on the other hand, will occur in a ductile manner after the compressive strength, σ = σ cu, has been reached. The stress drop in the post-peak regime represents gradual crushing of the microstructure. Typically, the ratio σ tu /σ cu is of the order 0.1. Quite often the response in compression close to failure is modelled as elastic-plastic, whereby the yield criterion is strongly mean-stress dependant. In order to compensate for the low tensile strength cementitious materials must in practice be reinforced by steel bars, glass-fiber bars or distributed ductile fibers. A typical failure criterion is that of Mohr-Coulomb (which is discussed in further detail in Chapter 10). This criterion is shown in Figure 1.14(b) for biaxial stress states. Structural failure in massive concrete structures can be very dramatic due to the large amount of elastic energy that is stored in a large volume at the point of cracking. An example of a major disaster was the failure of the Sleipner oil platform outside Stavanger, Norway in 19XX. Remark: Mean stress dependent yielding and failure is typical for different granular and particulate materials, e.g. soil and powders. Another example is (graphitic) grey-cast
22 replacemen 1 CHARACTERISTICS OF ENGINEERING MATERIALS AND CONSTITUTIVE MODELING σ tu σ ǫ σ cu σ tu σ 2 σ tu σ 1 (a) σ cu (a) σ cu Figure 1.14: (a) Stress-strain relation in uniaxial tension and compression. (b) Failure surface according to Mohr-Coulomb for biaxial stress states (plane stress). iron, for which the ratio of yield stress in tension and compression, σ ty /σ cy, is of the order 3. 1.5.2 Cyclic loading and fatigue 1.5.3 Creep and relaxation Creep phenomena in cementitious materials can be characterized similarly to those of metals. 1.6 Typical behavior of granular materials 1.6.1 Monotonic loading Basic features Granular materials, such as soil and (ceramic and metal) powders, show frictional characteristics. A purely frictional material, such as sand, gravel or fragmented rock (ballast), can sustain shear only in the presence of compressive normal stress between the particles. Moreover, in a purely frictional material the tensile strength is zero (σ tu = 0). Many finegrained materials, such as clay and powders that have been subjected to precompaction, show combined frictional and cohesive characteristics, which means that the material can sustain some shear stress even without any normal stress. In particular, this is the case for clayey soils and for rock and concrete. Hence, frictional/cohesive features can be translated into mean-stress dependent failure criteria, cf. the discussion in Section 2.5, and
1.6 Typical behavior of granular materials 23 it can be concluded that soils, powders and concrete do, in fact, have much in common when it comes to the modeling of failure characteristics. The inelastic deformations of granular materials contain a volumetric component (contrary to the case for most metals). The deformation is dilatant at dense initial packing, whereas it is contractant at loose initial packing. Dilatant behavior is associated with softening response (negative hardening), whereas contractant response is accompanied by hardening. In reality there may be a significant elastic-plastic coupling in the sense that the inelastic volume change affects the elastic moduli. The mechanical response is normally tested in a triaxial stress apparatus under cylindrical stress conditions, cf. the CTC-and CTE-conditions discussed in Section 2.3. Remark: In metals it necessary to account for evolving porosity close to failure, whereby the yielding characteristics resemble those of a powder compact. 1.6.2 Constant loading Consolidation Natural fine-grained soils (in particular clay) show a more complex mechanical response due to the presence of fluid (water and air) in the open pores. The resulting hydromechanical interaction of such poro-mechanical materials introduces time-dependent deformation at constant applied load. Such a time-delayed deformation process is denoted consolidation in soil mechanics (which must not be confused with creep due to viscous character of the solid particles). Basically, consolidation is the process of squeezing a sponge filled with water. Oil reservoirs constitute a complex geological system of solid, liquid (oil/water) and gas in a mixture state. In the extreme case the permeability is so small that virtually no seepeage of fluid can take place in the pore system, which is termed undrained condition. In the special case of water-saturated pores, such an undrained state corresponds to overall incompressibility of the granular material. 1.6.3 Constant loading Creep and relaxation In addition to consolidation, fine-grained soils show creep under constant loading. Such creep, which is sometimes denoted secondary consolidation, can be observed experimentally in the triaxial apparatus or in the oedometer (which imposes a state of uniaxial
24 1 CHARACTERISTICS OF ENGINEERING MATERIALS AND CONSTITUTIVE MODELING strain).