Structural Components E-Book

• Herausgeber: John Wiley & Sons
• Kategorie: Fachliteratur
• Sprache: Englisch

Seitenzahl: 416

Veröffentlichungsjahr: 2013

Beschreibung

The mechanical tests presented in this book are essential for determining the basic properties of the materials used. Areas covered include elasticity, tensile and compression tests, hardness, endurance tests and dynamic tests.

Leseprobe

Table of Contents

Chapter 1. Introduction

Chapter 2. Constitutive Equations

2.1. Introduction

2.2. Fundamental concepts

2.3. Unified theory of viscoplasticity

2.4. Other types of modeling

2.5. Conclusion

2.6. Bibliography

Chapter 3. Measurement of Elastic Constants

3.1. Elastic constants

3.2. Quasi-static mechanical tests

3.3. Ultrasonic methods

3.4. Resonant methods

3.5. Modulus measurements of coatings

3.6. Bibliography

Chapter 4. Tensile and Compression Tests

4.1. Introduction

4.2. Description of the tensile test

4.3. Standard data

4.4. Determination of constitutive equations

4.5. Damage determination

4.6. Compression test

4.7. Conclusion

4.8. Notations

4.9. Bibliography

Chapter 5. Hardness Tests

5.1. Introduction

5.2. Standard hardness tests

5.3. Analytical approaches of hardness tests

5.4. Finite element analysis of hardness test

5.5. Conclusion

5.6. Appendices

5.7. Bibliography

Chapter 6. Fatigue Tests

6.1. Principles

6.2. High-cycle fatigue tests – endurance limit

6.3. Low-cycle fatigue tests

6.4. Measurement of the crack propagation rate in fatigue

6.5. Bibliography

Chapter 7. Impact Tests

7.1. Introduction

7.2. Some history [MAN 99, TOT 02]

7.3. Description of the impact test

7.4. Determination of transition curves

7.5. Transition temperature and upper shelf

7.6. Impact fracture energy-fracture toughness empirical correlations

7.7. Bibliography

Chapter 8. Fracture Toughness Measurement

8.1. Introduction

8.2. Fracture mechanics bases

8.3. Implementation of fracture toughness tests

8.4. Measurement of fracture toughness JIc

8.5. CTOD measurement

8.6. Conclusion

8.7. Notations

8.8. Bibliography

Chapter 9. Dynamic Tests

9.1. Introduction

9.2. Test methods relying on propagation techniques and on the Hopkinson bar

9.3. Dynamic fracture mechanics tests

9.4. Plate against plate test

9.5. Collision tests

9.6. Bibliography

Chapter 10. Notched Axi-symmetric Test Pieces

10.1. Introduction

10.2. Geometry and notations

10.3. Notch test piece testing

10.4. Specimen elastic analysis

10.5. Plastic analysis of specimens

10.6. Damage analysis

10.7. Viscoplasticity and creep damage

10.8. From notch to crack?

10.9. Bibliography

List of Authors

Index

First published in France in 2001 by Hermes Science Publications entitled: “Essais mécaniques et lois de comportement”

First published in Great Britain and the United States in 2008 by ISTE Ltd and John Wiley & Sons, Inc.

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Library of Congress Cataloging-in-Publication Data

Essais mécaniques et lois de comportement. English

  Structural components: mechanical tests and behavioral laws/edited by Dominique Francois.

 p. cm.

  Includes index.

  ISBN: 978-1-84821-015-8

 1. Materials--Mechanical properties. 2. Deformations (Mechanics) I. François, Dominique,

1948- II. Title.

  TA404.8.E87 2007

  620.1'123--dc22

2007021380

British Library Cataloguing-in-Publication Data

A CIP record for this book is available from the British Library

ISBN: 978-1-84821-015-8

Chapter 1

Introduction1

Structural components must resist various applied loadings. Their nature is very diverse: aggressive environments, temperature and mechanical loads. However, we will only consider this last area in this book. Engineers must design parts so their deformations under these loads remain acceptable and they are not damaged or broken. In many cases structures need to be as light as possible, in order to save materials on the one hand, but above all to decrease energy consumption on the other hand. It is thus important to optimize the shape of parts and to choose the materials they are made of so that they perform without excessive deformations and without loss of integrity. Following design, problems can arise in service, such as crack initiation, which require calculations of the stress and strain distributions. Finally, failure assessments also require such analyses. To reach these objectives, more and more sophisticated design tools are available for engineers, provided that they are suitably supplied with reliable data concerning the mechanical behavior of materials. This is the aim of mechanical testing.

The design of a part requires the knowledge of the relations between applied loads and deformations, as well as the limits not to be exceeded at the risk of damage or fracture. This will be the case, for example, with the stiffness of a spring and with its yield load. More generally, these relations involve the stresses and the strains, and more precisely the two corresponding tensors. Solving the problem is achieved by integrating the three stress equilibrium equations together with the boundary conditions. As there are six unknowns, the six components of the stress tensor, there is a lack of equations to reach a solution. They are provided precisely by the constitutive equations between the six components of the stress tensor and the six components of the strain tensor, which, it must not be forgotten, derive from the displacement field, which itself includes three components. The problem thus involves nine unknowns: the six components of the stress tensor and the three components of the displacement, and nine equations: the three equilibrium equations and the six constitutive equations.

These constitutive equations, as a general rule, involve not only the stresses and the strains at a given time, but also the elapsed time. Without even considering the aging effects, the strain state preserves the memory of past deformations. It is only for an elastic material, for which there is no residual deformation after unloading, that the past does not operate. The constitutive laws are empirical laws, which need to be determined experimentally. Nevertheless, various theoretical considerations, which are deduced from the knowledge of deformation mechanisms, enable us to formulate hypotheses regarding the structure of the constitutive equations. If, for example, it is permissible to assume that the material is isotropic, the stress and strain invariants can only appear in those constitutive equations, as the orientation of the coordinates has no influence.

Influence of past time can appear explicitly in the formulation of constitutive equations. More generally, it appears implicitly through the derivatives of the components of the stress and strain tensors, and the strain rate only in most cases. In this way, the influence of the loading path, i.e. the evolution of the eigen stresses and directions at a given point, can be introduced before being followed by the present instant. The cyclic character of the loads needs to be introduced, namely in the study of fatigue. The number of variables then becomes large: amplitude and mean level of stresses, frequency, loading paths. The constitutive equations also depend on temperature. In this way, their most general formulations are rather complex and involve a very large number of parameters. Their identification can then require a significant number of tests.

A major difficulty immediately appears in the determination of constitutive equations: the deformation of any test specimen under a given load can be calculated only if those equations are known, which assumes that the problem is solved. It is only for very simple shapes for which the stress and strain fields are homogeneous that direct determination is possible. This is the case for the gage of a tensile specimen, in an approximate way for the wall of a thin tube in torsion or under internal pressure. It is nevertheless necessary to perform more complicated tests to explore various three-dimensional stress and strain states. In that case, it is absolutely vital to couple experimentation with theoretical hypotheses concerning the form of the constitutive equations and with calculations.

As underlined above, the task of an engineer is far from being achieved when he has determined the deformation of a part; he still needs, and this is often the main part of the work, to fix the limits not to be exceeded, in term of stresses and strains, so as to avoid excessive yielding or worse fracture, or in order to reach a given life. It is then required to express these conditions at the level of the basic components of the stress and strain tensors. More or less solid hypotheses allow us to do this using simple tests results: for example, the Von Mises yield criterion depending on the yield strength as measured in a tensile test, and the Goodman diagram for the endurance limit as determined from rotary bending tests results and from the ultimate tensile stress. Otherwise, transposition at the level of parts of tests results of not so easy an interpretation, such as Charpy or fracture toughness KIctests, needs to be possible. The development of underlying theories is out of the question here. The reader will need a sufficient knowledge of resistance of materials, solid mechanics and fracture mechanics, concepts which are developed in other books.

This book is also not intended to cover the entire scope of tests which are performed routinely for the complete determination of constitutive equations in their full generality. Specific tests keep being designed for the determination of a particular parameter. On the contrary, the emphasis will be placed on the most widely used tests, those which are common practice in industrial laboratories, or otherwise, those which can be obtained from various specialized laboratories, without new adjustments and at reasonable costs. A number of them provide access directly to the parameters of the constitutive equations, whereas others do so only indirectly: for instance, hardness tests. Nevertheless, they will be discussed inasmuch that they are of current use, although we will not go into all the information, which they are able to provide. A number of tests are useful not only for the determination of the constitutive equations but also for the determination of the damage and fracture limits. Some essentially deal with these last aspects.

The majority of the tests to be discussed are normalized. Of course, it is important to conform rigorously to the standards, in order to avoid any dispute between client and supplier, and also to achieve the most meaningful results as possible. The provisions of standards result partly from compromises based on considerations that are barely scientific, but also on sound theoretical considerations and on round-robins. It is not always easy to understand the reason for which a particular requirement is imposed, and its importance is sometimes underestimated, particularly when difficulties are found in the literal application of procedures. Actually, as far as possible, explanations for the reasons behind the introduction of various provisions in a standard will be given, so that their scope can be suitably appreciated, and the validity limits of these tests can be understood. However, the purpose of this book is not to provide an exhaustive description of the various tests when they are standardized. For this, it will be more useful to refer directly to the standards.

One of the drawbacks of normalized tests is that they often require the entry of only a few specific pieces of data: for example, engineering yield strength, ultimate tensile strength, elongation at fracture. Merely settling with those, a large part of the information that could be extracted from the tests is lost for the determination of constitutive equations. This impoverishment can paradoxically be increased by the computerized entry of data, when what is not explicitly required is deleted. Paper records do not have this drawback. It must not be forgotten either that the specimen themselves, which were used in the test, constitute results to be carefully processed by various inspections such as dimensional measurements and macro and micro-fractographies. One of the aims of this book is to provide the means to exploit as completely as possible the whole test dataset.

Following this introduction, a chapter will be devoted to constitutive equations. Without entering in the entire justification of the forms which they take up, it will allow us to understand their structure and to envisage the parameters to be determined, which are more or less numerous according to the complexity of the problem to be solved. Then, various mechanical tests will be discussed in turn, beginning with those, which are mostly used for the determination of the parameters of these constitutive equations and finishing with those which are specifically intended for the study of damage and fracture.

1 Chapter written by Dominique FRANÇOIS.

Chapter 2

Constitutive Equations1

2.1. Introduction

The constitutive law of the material is an essential ingredient in any structural design analysis. It provides the indispensable relation between strains and stresses, a linear relation in the case of elastic analyses (Hooke’s law), and a much more complex non-linear relation in inelastic analyses, involving time and additional internal variables.

This book is limited to the traditional continuous medium approach, i.e. the representative volume element (RVE) of the material is considered under quasi-uniform macroscopic strain or stress. This continuous medium hypothesis amounts to neglecting the local heterogenity of stresses and strains within the RVE, by working on averaged quantities, the effects of the heterogenities operating only indirectly through a certain number of internal variables. Furthermore, within the framework of the local state method of thermo-mechanics of continuous media, it is assumed that the state of a material point (or of its immediate neighborhood in the sense of RVE) is independent of the state of the neighboring material point, and that the stress or strain gradients do not operate in the constitutive equations. This hypothesis is obviously questioned in recent theories on the mechanics of generalized continuous media, which will not be addressed here.

This entire study is limited to quasi-static movements, which are considered to be sufficiently slow, within the framework of small perturbations (small strains, less than 20% for example). Furthermore, the indicated laws will be formulated without introducing the influence of temperature (though it can be very significant in some cases). In other words, in accordance with common practice in the determination of the constitutive equations of solid materials, a constant temperature (uniform in the RVE) will be assumed. The influence of the temperature will operate only through the evolution of the material parameters defining the constitutive equations. Nothing will be said about the linear elasticity law, three-dimensional Hooke’s law (see Chapter 3), or about viscoelasticity, whether linear or not. Thus, the book focuses more directly on metallic materials, with elastoplastic or elasto-viscoplastic behaviors, even if, in a certain manner, viscoelasticity, i.e. the influence of viscosity on elasticity, could be modeled based on a viscoplastic model. Therefore, among the effects to be considered will be: irreversible deformation, or plastic deformation, associated hardening phenomena, and the effects of time, whether they occur via the influence of the loading rate or through slow temporal evolutions of the various variables (time recovery for instance). Aging phenomena (associated with possible modifications of the metallurgical structure) and the effects of the damage will be mentioned only briefly. The anelasticity of metals (very low viscous hysteresis in the so-called domain of elasticity), which corresponds to reversible movements of dislocations, will not be discussed either. It may be observed immediately that summarizing constitutive laws in a single chapter will not allow us to be exhaustive at all in terms of the presentation of the various theories, the various specific models, their advantages/drawbacks, or to give all the necessary details. We only hope to provide the indispensable general elements as well as the main types of modeling. The interested reader should refer to more complete specialized books [FRA 91, KRA 96, LEM 85b, MIL 87].

2.2. Fundamental concepts

2.2.1. Domain of elasticity

Before discussing inelastic phenomena, it is appropriate to remind the existence (more or less real) of a domain of elasticity, typically the domain of the stress space (Figure 2.1) within which stress changes cause only reversible strains (no plasticity).

Figure 2.1. (a) Domain of elasticity in the stress space); (b) definition of the plastic strain and conventional yield limit

In general, this domain is defined in a 6-component space (the six components of the symmetric stress tensor). By convention, it is assumed that the boundary of the domain of elasticity is defined by (where is the stress tensor). If f < 0, the stress state is inside the domain. Of course, the definition of the yield limit is often subjective, or conventional, defined by the loss of linearity of the stress-strain response (through an offset), low for a correct characterization, 10–4 for example, higher for a conventional definition (for example, 0.2% for the yield strength Rp0.2 which would correspond to an irreversible strain of 0.2%). The notion of macroscopic plastic strain already appears, which supposes, in small perturbations, the partition of the total strain (Figure 2.1b):

[2.1]

where is the linear elastic strain (proportional to by Hooke’s law) and the irreversible strain or plastic strain (residual strain after an almost instantaneous unloading). Remember that this is a macroscopic definition, of continuous medium type. This plastic strain can be negligible macroscopically and not so at the local scale (within some misoriented grains for instance). If the material is assumed to be isotropic, i.e. without any preferred material direction, it is known that the domain of elasticity can be expressed using only the three invariants of the stress tensor. The criterion that is most commonly used and which provides a very good approximation for a large number of materials is the Von Mises criterion:

[2.2]

where denotes the deviator of the stress tensor and is the first invariant, or hydrostatic pressure. is the equivalent Von Mises stress or second stress invariant. The third invariant does not operate in this criterion. A parenthesis on the notations: the symbol “.” between two tensors denotes the product contracted once with Einstein’s summation, represents the square of the tensor ); the symbol “:” denotes the product contracted twice (for example, scalar

[2.3]

where is a fourth-order tensor respecting material symmetries. If the material is orthotropic (for example, rolled sheet, composites, etc.), the tensor displays only 9 independent constants (in fact 8 as, giving directions, it can be normalized, and most often, assuming iso-volume, reduces to only 7 constants). Figure 2.2 gives the example of the orthotropic Hill criterion determined for a Zircalloy alloy, and the comparison with the Von Mises criterion.

2.2.2. Hardening

Hardening is the manifestation of the evolution of the domain of elasticity with prior plastic strain. Normally, after tensile loading, an increase of the yield strength can be observed: a new plastic flow will occur only at a stress level close to the one previously applied for pre-deforming. In the same manner, during a tensile test, to increase the plastic strain, an increase of the stress is necessary. This phenomenon is associated with the inhomogenities of the material existing at various scales. For instance, for a polycrystal, some misoriented grains will be deformed plastically and others will not. This inhomogenity of plastic strain at the micro-scale can be accommodated only by elastic strains, thus (internal) stresses, which results in global hardening. A very simple convincing model about the source of this effect consists of an assembly of skids and springs associated in parallel. Each of the branches is elastic-plastic (without hardening) but their parallel functioning causes the kinematic hardening effect. At lower scales, the initial inter-granular inhomogenities (inclusions, defects, etc.), as well as those produced by the displacements of the dislocations, the increase in the density of dislocations and the creation of substructures (walls, cells, twins, etc.), will also contribute to the hardening by globally similar effects (here this can be a combination of isotropic hardening and kinematic hardening).

Figure 2.2. Example of a plastically anisotropic material (Hill criterion)

The terms isotropic hardening and kinematic hardening have been mentioned. What do they mean exactly?

Isotropic hardening refers to an increase of the yield limit identical in all the directions (in particular, an initially isotropic material remains isotropic). In the stress space, this corresponds to an expansion, or proportional transformation (illustrated in Figure 2.3 for an initially isotropic material). Just one scalar parameter is sufficient to express this change, the increase R of the yield strength:

[2.4]

It can be shown that the state variable associated with the (additional) stress R is the cumulated plastic strain p, defined by integration of the rate modulus:

[2.5] in the isotropic case.

Thus, p is the length of the plastic strain path (in the plastic strain space). Some theories instead consider the cumulated plastic work

Figure 2.3. Isotropic hardening: schematic representation in the stress space and under traction-compression

Kinematic hardening refers to a translation of the domain of elasticity in the stress space. Thus, the increase of the yield strength in traction after a tensile pre-deformation is associated with a decrease (in modulus) of the yield strength in compression, this effect being commonly called the Bauschinger effect. To express kinematic hardening, it suffices to use a second order tensor, , defining the position of the new center of the domain of elasticity:

[2.6]

Figure 2.4 illustrates this transformation in the case of an initially isotropic material and with a linear kinematic hardening.

Figure 2.4. Kinematic hardening: schematic representation in the stress space and under traction-compression

Of course, hardening can have more complex manifestations. Apart from the two basic types described above, the evolution of a fourth-order tensor (the tensor) or even of tensors of a greater order could be introduced to create distortions of the surface. For practical purposes, the combination of kinematic hardening and isotropic hardening only is used. The rest of the chapter is restricted to the case of an initially isotropic material (with , the fourth order deviatoric identity tensor, such that , so that:

[2.7]

It may be noticed that and R are often called internal stresses. They correspond to the macroscopic manifestation of the existence of internal stresses (or residual stresses) at the microscopic scale, but, by definition, the mean value of those is zero. In fact, and R effectively transfer the effects of the microscopic internal stresses to the macroscopic scale.

It can also be noticed that what has been discussed above is presented within the framework of time-independent plasticity, for which the stress state cannot move out of the domain of elasticity (f ≤ 0). The expression of the viscoplasticity effect will be discussed below.

2.2.3. Normality rules

Hill’s maximum work principle, not stated here, implies that in the case of plastic flow, the direction of the rate of plastic strain will be normal to the plastic load surface at the current stress point. In the case of associated plasticity, the (current) boundary of the domain of elasticity f ≤ 0 is the same as the load surface, and consequently:

[2.8]

[2.9]

Figure 2.5. Schematic representation of the domain of elasticity and the viscoplasticity equipotential surfaces

Thus, any stress state can be decomposed in the following manner, in which the function will be deduced by inverting the relation

[2.10]

We may note that consistency between the choice of the Von Mises criterion, independent of the first invariant and the isovolume of the plastic strain (the latter, produced by slips of the atomic planes, at least in metallic materials, is achieved at constant volume). In fact, according to [2.2] and [2.9], we obtain:

[2.11]

2.3. Unified theory of viscoplasticity

To simplify, the presentation will fall directly in the scheme of viscoplasticity. The case of time-independent plasticity will be deduced simply in the limit. A relatively general form of constitutive laws will first be given, then the most common specific choices for the viscosity function and for isotropic and kinematic hardening will be considered. Thereafter, the case of time-independent plasticity will be covered and a few indications on the determination, based on experiments, of the parameters of the equations will conclude.

2.3.1. General form of the constitutive law

First of all, it can be observed that the constitutive law can be stated in the general formal framework of thermodynamics of continuous media. This subject will not be covered here. The interested reader can refer to [CHA 96, GER 73, HAL 75], for example.

The expression of the viscoplastic constitutive law essentially comprises two aspects:

– choice of the viscosity function (see section 2.3.2), or choice of the viscoplastic potential Ω, which will operate in the expression of the plastic strain rate (its dependence on the viscous stress) via the normality rule expressed above [2.9];

– choice of hardening laws for the whole internal variables. These are provisionally denoted by aj, (j =1, 2, …, N); they can be scalar or tensorial. The proposed general form deals with a strain hardening term, a dynamic recovery term and a static recovery term:

[2.12]

[2.13]

2.3.2. Choice of viscosity law

This relation between the viscous part of the stress and the modulus of the plastic strain rate is most often highly non-linear. Thus, in a large range of rates, it can be approximated by a power function:

[2.14]

[2.15]

[2.16]

[2.17]

In all these expressions, the exponent n is highly temperature-dependent, the viscosity phenomenon being thermally activated (n becomes low at high temperatures).

Another, not incompatible, manner of defining the viscosity function is given by the Zener-Hollomon formulation [ZEN 44], which combines the influence of temperature and the influence of velocity in a single master curve. This approach consists of writing:

[2.18]

where Z is a unique monotonic function to be defined and where θ(T) and σo(T) are two functions of the temperature to be defined. The advantage of this formulation, illustrated in Figure 2.7 [FRE 93], is to avoid the high non-linearity of the power function in which the exponent is highly temperature-dependent. The function Z being defined for a large number of decades in rate (24 for instance), the role of the function θ(T) is then to shift by normalization the useful rate field (in practice, limited to 6 to 8 decades in rate). The equivalent exponent (the slope of the Z function in the bilogarithmic diagram) thus moves from a very low value in a certain area of the curve (low values of to a very high value in the opposite area (high values of ).

Figure 2.6. Viscous stress – plastic strain rate relationship: 316L stainless steel (17-12 SPH) at 600°C

2.3.3. Isotropic hardening laws

Considering the expression of the modulus of the strain rate [2.14], by replacing f with [2.7], we obtain:

[2.19]

Figure 2.7. Behavior in stationary creep: (a) data on aluminum; (b) data on copper (reproduced from the International Journal of Plasticity, Volume 9, A. Freed & K.P. Walker, with the permission of Elsevier Science Ltd, Kidlington, UK)

Three possibilities to express an isotropic hardening are noted:

– through the variable R, by increasing the size of the domain of elasticity;

– by increasing the drag stress D;

– by a coupling with the law of evolution of the kinematic variable.

In the first two cases, which alone are considered here, it suffices to define the biunivocal function of the dependence between R (or D) and the state variable of the isotropic hardening, namely, the cumulated plastic strain p (possibly the cumulated plastic work Wp).

[2.20]

One possibility is to assume the two evolutions to be proportional. The R(p) function alone then needs to be defined and from it can be deduced:

[2.21]

By the decomposition of the equivalent Von Mises stress (in the case without kinematic hardening,the different roles of the two types of isotropic hardening can be noticed:

[2.22]

In the first case, with R, the increase of the domain of elasticity will take place identically irrespective of the strain rate; in the second case, the increase in D will result in an increase of σeqwhich will be all the more significant the higher the strain rate.

[2.23]

This multiplicative form of hardening is very easy to determine [LEM 71] and provides good results in a relatively large range, at least for quasi-proportional monotonic loadings.

2.3.4. Kinematic hardening laws

Kinematic hardening being a quite general manifestation, at least in the domain of small strains, it will be needed to resort to the corresponding models as soon as it is desirable to express correctly either non-proportional monotonic loadings (variation of the direction of the loading or variation of relationships between independent loadings, or thermo-mechanical loadings, etc.) or cyclic loadings.

The most common kinematic hardening models are indicated here in an increasing order of complexity. More advanced models for the expression of particular effects will be discussed in section 2.4.3. At present, strain hardening only will be dealt with, the effects of time recovery being discussed in section 2.3.6.

The simplest model is Prager’s non-linear kinematic hardening [PRA 49], in which the evolution of the kinematic variable(called back-stress) is collinear to the evolution of the plastic strain. Thus:

[2.24]

The associated linearity of the stress-strain response (Figure 2.4b) is rarely observed (except perhaps in the significant strains regime). The model initially proposed by Armstrong and Frederick [ARM 66], introducing a feedback term, called dynamic recovery, gives a better description:

[2.25]

The feedback term is collinear to(as in general equation [2.12]) and proportional to the modulus of plastic strain rate. The evolution of,instead of being linear, is thus exponential for a monotonic uni-axial loading, with saturation (suppression of the hardening) for a value C/γ. In fact, the integration of [2.25] depending on εp, for uni-axial loading, gives:

[2.26]

where vXo and εpoare the values of X and εpat the start of the loading branch considered.

[2.27]

A better approximation consists of adding several models such as [2.25], with significantly different spring constants γi(5 to 20 factors between each of them):

[2.28]

which helps to express a more extended strain range, ensuring a better description of the smooth transition between elasticity and the beginning of the plastic flow. Figure 2.8 shows the significant improvement obtained in the case where only two variables are superimposed, one being linear, with γ20. Other more complex combinations can be used [CAI 95] instead of [2.28], but they do not provide an analytical uni-axial expression.

Figure 2.8. Cyclic curves of a few materials and interpretation using the non-linear kinematic hardening model

2.3.5. Cyclic hardening and softening

[2.29]

Note that in the case of cyclic softening, c < 0 can be chosen. Note also that the drag stress D can be used instead of the threshold stress R, or the two combined, or a coupling between kinematic hardening and isotropic hardening can even be introduced [MAR 79], with a function ϕ (p) to be defined:

[2.30]

2.3.6. Static recovery

The recovery of hardening over time, whether kinematic or isotropic, will generally occur at high temperatures. These mechanisms, thermally activated, are described macroscopically by relations such as [2.12]. Thus, for kinematic hardening, power functions for instance will be used in the feedback term operating according to time:

[2.31]

where mi, τi, Midepend on the material and temperature. In practice, MiCi / γiwill be chosen and the time constant τiwill be highly time-dependent.

For the restoration of isotropic hardening, any function can be used, for instance [CHA 89b, NOU 83]:

[2.32]

2.3.7. Time-independent limit case

So far the case of viscoplasticity has been considered, with a part of the stress that is rate-dependent (relations in section 2.3.2). When the temperature is sufficiently low, the viscosity effect can be ignored. For some applications, even at high temperatures, the time-independent plasticity scheme can be preferred.

Figure 2.9. Modeling of cyclic relaxation tests on 316L stainless steel (17-12 SPH) at 600°C: σ max: maximal stress at the stabilized cycle; σ rel : stress after relaxation; tH : hold time

For this, it is only necessary to make the viscous stress tend towards zero, by decreasing the value of the drag stress (D → 0) in an expression like [2.14] or [2.22]. It necessarily follows that σv→ 0 and the criterion f ≤0 will be automatically met. Of course, in an expression like [2.14], an indetermination (0/0) is reached, but it can be eliminated by a consistency condition f in the case of plastic flow (is indeterminate in pure elasticity, i.e. if f < 0). The formal treatment of time-independent plasticity is a little more complex than that of viscoplasticity, introducing a loading-unloading condition and additional difficulties when the material has negative hardening. These aspects will not be discussed here.

The laws of monotonic or cyclic viscoplasticity, with the associated hardening models, simply degenerate into the time-independent case, without any other change than the dimension of the pure domain of elasticity (see section 2.3.8). We may mention the special case of isotropic hardening, for which relation [2.22] becomes:

[2.33]

Quite often in the applications, the relation R(p) can be considered as defined point by point, based on the expression σk + R (εp ), which is equivalent in the uni-axial case. This function is then directly drawn from the experimental tensile curve. Quite often it can be assimilated into a power function:

[2.34]

2.3.8. Methods of determination

The identification of unified viscoplasticity models, combining isotropic hardening, kinematics, viscosity and recovery effects, may be relatively difficult. Here, a step-by-step determination approach, which has often proven applicable, is proposed.

2.3.8.1. Determination of hardening laws within independent time-scheme

It is assumed that monotonous and cyclic uni-axial tests are available, for example low cycle fatigue tests up to the stabilized cycle, with σ - εprecordings (see Chapter 6). It is also assumed that these tests are achieved for rather constantand relatively high (for example rates. Starting from the cyclic curve, considering that Cte at cycle maxima, the following relation will be identified; this will be valid after cyclic hardening or softening effects:

[2.35]

Subsequently, the determination of (time-independent) hardening laws is completed according to the available data in monotonous tensile loading, or possibly in compression loading (see Chapter 4); the corresponding experimental curve must be expressed by:

[2.36]

The isotropic hardening rate coefficient g will be supplied by the number of cycles necessary to saturate the cyclic hardening or softening with fixed amplitude Δεp/2: 2gNΔεp ≈ 5 is a good criterion for exponential function saturation. A more precise manner is that of tracing out the succession of normalized maxima (σmax(N) - σmax(0))/ (σmax (Nsat ) - σmax(0)) according to p ≈2NΔεpfor some low cycle fatigue tests. An iterative treatment of the set of these data, with some readjustments, provides k*, Ci, γi, c, g (and function ϕ(p)).

2.3.8.2. Determination of the viscosity law

The data in the variable strain rate domains between, say, 10-6 s-1 and 10-2 s-1, is now used in order to determine the viscosity law, for example exponent n, constant K, and final value k of the true domain of elasticity. The following necessity of readjustment is noted:

[2.37]

[2.38]

between the already determined version in the time-independent approximation (with a rate pretty much equal to ) and the complete version, taking into account the choice [2.21] for the isotropic hardening associated with the evolution of the drag. If monotonous or cyclic relaxation tests are achieved, the determination of n and K will be greatly facilitated by the possible use of a graphic determination method [LEM 85b]. Certain iterations are necessary in order to reach a satisfactory solution (in all these data processing, the parameters determined in stage 1 are taken into account).

2.3.8.3. Determination of static recovery effects

Available data is used with a very low strain rate in creep tests or in long-term relaxation tests. As illustrated in Figure 2.6 for the 316L steel, the influence of the recovery mechanism appears to be directly visible through the appreciable reduction of the stress supported for a given strain rate. By successive approximations, while all the other parameters remain fixed, the static recovery parameters of the envisaged models (mi, τi , Qr , mr , γr ) are relatively easily obtained. If specific recovery tests are available, these effects and the corresponding parameters will be much more directly measured. For example, such tests consist of normal cyclic loading, until stabilization, then a partial unloading and a hold time, in temperature, of significant duration (100 hours, for example), then cyclic loading again. Recovery must be achieved at a sufficiently low, but possibly non-zero, strain or stress level, chosen in such a manner that partial recovery of the plastic deformation cannot be produced. The comparison and identification of responses before and after recovery thus furnish, quite directly, the values of the sought parameters.

2.3.9. Other unified approaches

For the past 20 years, numerous unified theories have been developed [BRU 94, DEL 88, FRE 88, KRE 86, MIL 76, ROB 83, WAL 81]. They have almost the same ingredients in common, the essential differences being the choice of expressions of functions, for example the viscosity function. The kinematic hardening form, with a hardening term, a dynamic recovery term and a static recovery term, is almost a constant feature, save for Bodner’s approach [CHA 89c], who rejects, despite experimental evidence, the very notion of kinematic hardening. Other minor differences may be signaled:

– Robinson’s model [ROB 83] presupposes a non-linear relation between the kinematic hardening module and the kinematic variable itself (power function). This procedure, which implies a biunivocal relationship between the plastic strain and the hardening variable, requires a change of the origin to be made at each cycle change, by producing discontinuities in the tangent moduli, which is of delicate numerical treatment;

– Krempl’s model [KRE 86] considers a direct hardening term (the first term of relation [2.12]) proportional to the total strain rate, instead of the plastic strain rate. Although it presents a certain advantage for the description of ratcheting phenomena, this approach is considered to be inadmissible from the thermodynamic point of view [CHA 93, 96];

– Miller’s model [SCH 81] involves a dragging phenomenon by the atoms in solution in the law of isotropic hardening, a phenomenon which corresponds to dynamic aging and to the Portevin-Le Chatelier effects;

– other ways of expressing non-linear kinematic hardening have been proposed, involving the simultaneous use of two surfaces (plastic load surface and yield surface). Let us mention the Dafalias and Popov model [DAF 76] in a time-independent plasticity context and McDowell’s work [MDO 90] in a more general framework.

2.4. Other types of modeling

2.4.1. Plasticity-creep partition

This is the oldest way of describing plasticity and viscosity phenomena simultaneously by adding two independent inelastic deformations. Equation [2.1] is then replaced by:

[2.39]

First of all it is observed that the plastic strain of the preceding sections contained both plasticity and creep effects in a unified manner. On the contrary, here it is considered that they are dissociated and, in general, independent. The description of the evolutions of the two inelastic deformations will thus be made:

– by means of time-independent plasticity theory for with a normality rule such as [2.8], and in association with the hardening laws that are convenient to the envisaged type of application: isotropic hardening for the applications under quasi-monotonous loading, and kinematic hardening or combinations of the two for the applications under cyclic loading, or when non-proportional multi-axial effects can operate. Without explaining them further, they can be formally written:

[2.40]

The interest of the dissociation between plasticity and creep is that of enabling a simple determination of the material parameters starting either from the monotonous tensile curve, or from the cyclic curve (stage 1 of section 2.3.8);

– by means of a creep-type law for , incorporating primary creep and secondary creep, or in an integrated form such that:

[2.41]

or in a differential form, which is more correct because it implies strain hardening:

[2.42]

for which it is also possible to again use a multiplicative-type hardening law such as [2.23], replacing pby εceq. It is also possible to adopt a form similar to [2.12], with additional hardening variables (which combine isotropic and kinematic hardenings):

[2.43]

This form of evolution law combines strain hardening and static recovery (time effect, which is important in creep). In addition, in this case, regardless of the chosen form of hardening law, the dissociation with plasticity allows a quite simple determination, starting either from pure creep tests or from relaxation tests.

This method by means of partition of inelastic deformation has been currently used, up to the point of experimental evidence, which has been reported on numerous occasions, consisting of an obvious association between plastic and creep deformation, by means of associated hardening effects. Figure 2.10 shows schematically the type of observation made on a high rate tensile test, interrupted by a long-duration creep period (constant stress). Clearly there is a quasi-immediate lapse of the creep period and experimental evidence regarding the fact that hardening is correlated with the sum εp+ εc(and not defined in an independent manner with εpor εc).This type of observation, like many others pointing to a similar conclusion, has led to the development of non-unified approaches, but with coupled hardening effects [CAI 95, CON 89], by writing, for example:

[2.44]

[2.45]

with all types of possible variations. These approaches are relatively seldom used because they present complications or difficulties of determination that are analogous to those of the unified theories.

Figure 2.10. Schematic results of tensile-creep-tensile tests in comparison with a constant strain rate tensile test

2.4.2. Methods by means of micro-macro transposition

These methods consist of resorting to basic laws of crystalline plasticity, by inscribing directly in the modeling the various slip systems that can be activated for the various grain directions under consideration within the RVE of the polycrystal. Figure 2.11 gives the functional diagram of such an approach, being limited (formally) to a situation of imposed macroscopic stress (diagram realized on a time increment); the output is the macroscopic plastic strain The method involves the intervention of two localization stages and two averaging stages:

macro ↔ grain “g” ↔ slip system

Here Cailletaud’s [CAI 92] and Pilvin’s [PIL 94] formulation is followed; it is easy enough to use, but at the same time is sufficiently rigorous (even if it maintains a marked phenomenological character).

Figure 2.11. Functional diagram of a model for polycrystalline materials with a constitutive law written on the slip systems of each grain (on average)

The passage from the macro level to that of the (average) stress in each grain is achieved by the following Kröner-type [KRÖ 61] localization rule, but corrected, valid for a polycrystal with grains of identical nature and with macroscopically isotropic elasticity:

[2.46]

where µ is the shear modulus of elasticity, a an adjustment parameter, β ga state variable for each grain, which is analogous to the average plastic strain of the grain and the corresponding average value:

[2.47]

where cgis the concentration corresponding to each direction under consideration. Kröner’s elastic localization rule, which is known to be too rigid [ZAO 93], would consist of replacing with and with The originality of Cailletaud’s and Pilvin’s approach is that of adapting this rule, so that it is quasi-elastic in the regime of low plastic deformations, in order to tend towards a tangent rule for the higher strains. This is supplied by the following evolution law for , which is very similar to a non-linear kinematic hardening law (combined to linear hardening):

[2.48]

Dand δ