internal friction in metallic materials: a handbook

Springer Series in

materials science 90

Springer Series in

materials scienceEditors: R. Hull R.M. Osgood, Jr. J. Parisi H. Warlimont

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90 Internal Friction in Metallic Materials

By M.S. Blanter, I.S. Golovin,H. Neuhauser, and H.-R. Sinning

A Handbook

M.S. Blanter I.S. GolovinH. Neuhauser H.-R. Sinning

Internal Frictionin Metallic Materials

123

With 65 Figures and 53 Tables

A Handbook

Professor Dr. Mikhail S. BlanterMoscow State Universityof Instrumental Engineeringand Information ScienceStromynka 20, 107846, Moscow, RussiaE-mail: [email protected]

Professor Dr. Hartmut NeuhauserInstitut fur Physik der Kondensierten MaterieTechnische Universitat BraunschweigMendelssohnstr. 338106 Braunschweig, GermanyE-mail: [email protected]

Professor Dr. Igor S. Golovin

Institut fur WerkstoffeTechnische Universitat BraunschweigLanger Kamp 838106 Braunschweig, Germany

Series Editors:

Professor Robert HullUniversity of VirginiaDept. of Materials Science and EngineeringThornton HallCharlottesville, VA 22903-2442, USA

Professor R.M. Osgood, Jr.Microelectronics Science LaboratoryDepartment of Electrical EngineeringColumbia UniversitySeeley W. Mudd BuildingNew York, NY 10027, USA

Professor Jrgen ParisiUniversitat Oldenburg, Fachbereich PhysikAbt. Energie- und HalbleiterforschungCarl-von-Ossietzky-Strasse 91126129 Oldenburg, Germany

Professor Hans WarlimontInstitut fur Festkorper-und Werkstofforschung,Helmholtzstrasse 2001069 Dresden, Germany

ISSN 0933-033X

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ISBN-13 978-3-540-68757-3 Springer Berlin Heidelberg New York

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To our families

Preface

Internal friction and anelastic relaxation form the core of the mechanical spec-troscopy method, widely used in solid-state physics, physical metallurgy andmaterials science to study structural defects and their mobility, transport phe-nomena and phase transformations in solids. From the view-point of Mechan-ical Engineering, internal friction is responsible for the damping properties ofmaterials, including applications of high damping (vibration and noise reduc-tion) as well as of low damping (vibration sensors, high-precision instruments).

In many cases, the highly sensitive and selective spectra of internal friction(as a function of temperature, frequency, and amplitude of vibration) containunique microscopic information that cannot be obtained by other methods.On the other hand, owing to the large variety of phenomena, materials, andrelated microscopic models, a correct interpretation of measured internal fric-tion spectra is often dicult. An ecient use of mechanical spectroscopy maythen require both: (a) a systematic treatment of the dierent mechanisms ofinternal friction and anelastic relaxation, and (b) a comprehensive compila-tion of experimental data in order to facilitate the assignment of mechanismsto the observed phenomena.

Whereas the rst of these two approaches was developed since more thanhalf a century in several textbooks and monographs (e.g., Zener 1948, Krishtalet al. 1964, Nowick and Berry 1972, De Batist 1972, Schaller et al. 2001), thesecond requirement was met only by one Russian reference book (Blanter andPiguzov 1991), with no real equivalent in the international literature. Thepresent book, partly based on the Russian example, is intended to ll thisgap by providing readers with comprehensive information about publishedexperimental results on internal friction in metallic materials.

According to this objective, this handbook mainly consists of tables wheredetailed internal friction data are combined with specications of relax-ation mechanisms. The key to understand this very condensed informationis provided, besides appropriate lists of symbols and abbreviations, by theintroductory Chaps. 13: after the Introduction to Internal Friction in Chap. 1,dening and delimiting the subject and clarifying the terminology, the relevant

VIII Preface

internal friction mechanisms are briey reviewed in Chaps. 2 (AnelasticRelaxation) and 3 (Other Mechanisms). Although somewhat more space isobviously devoted to the former than to the latter, this part should not beunderstood as a systematic analysis of the physical sources of anelasticity anddamping; in that respect, the reader is referred e.g., to the above-mentionedtextbooks.

The data collection itself, as the main subject of the book, can be found inChaps. 4 and 5. The tables, generally in order of chemical composition, includethe main properties of all known relaxation peaks (like frequency, peak heightand temperature, activation parameters), the relaxation mechanisms as sug-gested by the original authors, and additional information about experimentalconditions. Other (e.g., hysteretic) damping phenomena, however, could notbe considered within the limited scope of this book, with very few exceptions.Chapter 4, which represents the main body of data on crystalline metals andalloys, is divided into subsections according to the group of the main metal-lic element in the periodic table, with alphabetic order within each subsec-tion. Chapter 5 contains several new types of metallic materials with specicstructures, which do not t well into the general scheme of Chap. 4. A shortsummary or specic explanations are included at the beginning of each table.

Although the authors made all eorts to be consistent in style throughoutthe book, some diculties in evaluating individual relaxation spectra led toslight deviations, concerning details of data presentation, between the dierentchapters and subsections. Since some of the data were evaluated from gures,the accuracy should generally be regarded with care; in cases of doubt, theoriginal papers should be consulted. Over 2000 references published until mid2006 were included, among which many earlier ones are still important be-cause certain alloys and eects are not covered by the more recent literature.Latest information, if missing in this book, might be found in three confer-ence proceedings published in the second half of 2006 (Mizubayashi et al.2006b, Igata and Takeuchi 2006, Darinskii and Magalas 2006), as well as inforthcoming continuations of these conference series.

This book is intended for students, researchers and engineers working insolid-state physics, materials science or mechanical engineering. From oneside, due to the relatively short summary of the basics of internal frictionin Chaps. 13, it may be helpful for nonspecialists and for beginners in theeld. From the other side, its probably most comprehensive data collectionever published on this topic should also be attractive for top specialists andexperienced researchers in mechanical spectroscopy and anelasticity of solids.

The authors acknowledge gratefully the help of Ms. Tatiana Sazonova withthe list of references, of Ms. Brigitte Brust with gures, and of Ms. SvetlanaGolovina with tables. We are also grateful to the Springer team, in particularDr. Claus Ascheron, Ms. Adelheid Duhm and Ms. Nandini Loganathan, forgood cooperation.

Moscow, Tula, Braunschweig Mikhail S. Blanter, Igor S. GolovinJanuary 2007 Hartmut Neuhauser, Hans-Rainer Sinning

Contents

1 Introduction to Internal Friction: Terms and Denitions . . . 11.1 General Phenomenon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Types of Mechanical Behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Anelastic Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.4 Thermal Activation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.5 Other Types of Internal Friction . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.6 Measurement of Internal Friction . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 Anelastic Relaxation Mechanisms of Internal Friction . . . . . . 112.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2 Point Defect Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2.1 The Snoek Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.2.2 Relaxation due to Foreign Interstitial Atoms

(C, N, O) in fcc and Hexagonal Metals . . . . . . . . . . . . . . . 282.2.3 The Zener Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.2.4 Anelastic Relaxation due to Hydrogen . . . . . . . . . . . . . . . 362.2.5 Other Kinds of Point-Defect Relaxation . . . . . . . . . . . . . . 48

2.3 Dislocation Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 502.3.1 Intrinsic Dislocation Relaxation Mechanisms:

Bordoni and NiblettWilks Peaks . . . . . . . . . . . . . . . . . . . 512.3.2 Coupling of Dislocations and Point Defects:

Hasiguti and SnoekKoster Peaks and Dislocation-Enhanced Snoek Eect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

2.3.3 Other Kinds of Dislocation Relaxation . . . . . . . . . . . . . . . 732.4 Interface Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

2.4.1 Grain Boundary Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . 782.4.2 Twin Boundary Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . 822.4.3 Nanocrystalline Metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

2.5 Thermoelastic Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 872.5.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

X Contents

2.5.2 Properties and Applications of ThermoelasticDamping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

2.6 Relaxation in Non-Crystalline and Complex Structures . . . . . . . 952.6.1 Amorphous Alloys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 972.6.2 Quasicrystals and Approximants . . . . . . . . . . . . . . . . . . . . 113

3 Other Mechanisms of Internal Friction . . . . . . . . . . . . . . . . . . . . . 1213.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1213.2 Internal Friction at Phase Transformations . . . . . . . . . . . . . . . . . 121

3.2.1 Martensitic Transformation . . . . . . . . . . . . . . . . . . . . . . . . . 1213.2.2 Polymorphic and Other Phase Transformations . . . . . . . 1293.2.3 Precipitation and Dissolution of a Second Phase . . . . . . . 133

3.3 Dislocation-Related Amplitude-Dependent InternalFriction (ADIF) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

3.4 Magneto-Mechanical Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1443.5 Mechanisms of Damping in High-Damping Materials . . . . . . . . . 148

4 Internal Friction Data of Crystalline Metalsand Alloys (Tables) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1574.1 Copper and Noble Metals and their Alloys . . . . . . . . . . . . . . . . . . 1584.2 Alkaline and Alkaline Earth Metals and their Alloys . . . . . . . . . 1894.3 Metals of the IIAVIIA Groups and their Alloys . . . . . . . . . . . . 1964.4 Metals of the IIIB Group, Rare Earth Metals and Actinides . . . 223

4.4.1 Rare Earth and Group IIIB Metals . . . . . . . . . . . . . . . . . . 2234.4.2 Actinides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235

4.5 Metals of the IVB Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2384.5.1 Titanium and its Alloys . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2384.5.2 Zirconium and its Alloys . . . . . . . . . . . . . . . . . . . . . . . . . . . 2634.5.3 Hafnium and its Alloys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275

4.6 Metals of the VB Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2764.6.1 Vanadium and its Alloys . . . . . . . . . . . . . . . . . . . . . . . . . . . 2764.6.2 Niobium and its Alloys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2874.6.3 Tantalum and its Alloys . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321

4.7 Metals of the VIB Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3314.7.1 Chromium and its Alloys . . . . . . . . . . . . . . . . . . . . . . . . . . . 3314.7.2 Molybdenum and its Alloys . . . . . . . . . . . . . . . . . . . . . . . . . 3384.7.3 Tungsten and its Alloys . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346

4.8 Metals of the VIIB group: Mn and Re . . . . . . . . . . . . . . . . . . . . . . 3524.9 Iron and Iron-Based Alloys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356

4.9.1 Fe (pure) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3574.9.2 FeInterstitial Atoms (C, H, N), Other Elements (As,

B, Ce, La, P, S, Y)

Contents XI

4.9.5 FeAl-Based Ternary and Multi-Component Alloys(e.g., FeAlCr, FeAlGe, FeAlSi, etc.) . . . . . . . . . . . . 380

4.9.6 FeCo, Ge, Si, Mo, V, W Alloys . . . . . . . . . . . . . . . . 3854.9.7 FeCr-Based Steels and Alloys . . . . . . . . . . . . . . . . . . . . . . 3894.9.8 FeMn-Based Steels and Alloys . . . . . . . . . . . . . . . . . . . . . 3974.9.9 FeNi-Based Steels and Alloys . . . . . . . . . . . . . . . . . . . . . . 4024.9.10 Other Fe-Based Multi-Component Alloys . . . . . . . . . . . . . 408

4.10 Co, Ni and their Alloys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413

5 Internal Friction Data of Special Types of MetallicMaterials (Tables) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4235.1 Hydrogen-Absorbing Alloys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4245.2 Metallic Glasses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4395.3 Quasicrystals and Other Complex Alloys . . . . . . . . . . . . . . . . . . . 449

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535

List of Abbreviations

General Abbreviations

IF internal frictionADIF amplitude-dependent internal frictionTDIF temperature-dependent internal frictionMS mechanical spectroscopyLT bgr. low-temperature backgroundHT bgr. high-temperature background

HT heat treatmentQuen. quenchedAnnl. annealedTemp. temperedRecr. recrystallisedPrec.tr. precipitation treatedMsp. melt-spunSint. sintered

CW cold workedHot roll. hot rolledFatig. fatiguedIrr(n) irradiated by neutronsIrr(e) irradiated by electronsIrr(p) irradiated by protonsIrr(d) irradiated by deuteronsIrr() irradiated by -rays

SC single crystalPC polycrystalGB grain boundary

XIV List of Abbreviations

Ufgr. ultrane-grainednc. nanocrystallineam. amorphousSQC single quasicrystal (icosahedral phase)QC (poly-) quasicrystalline (icosahedral phase)d-QC (poly-) quasicrystalline (decagonal phase)n-QC nanoquasicrystalSput.th.f. sputtered thin lmEvap.th.f. evaporated thin lmGas depos. gas deposited thin lmFM ferromagnetAFM antiferromagnetPM paramagnetSR structural relaxationDSR directional structural relaxationSRO short-range order(ed)AES activation energy spectrumRRR residual resistivity ratio

Abbreviations of Relaxation Mechanisms and InternalFriction Peaks

Point defects

S Snoek (and Snoek-type) relaxation due to:S(O) oxygenS(C) carbonS(N) nitrogenS(H) hydrogenS(H/O,N) hydrogen (H near O, N, etc.)S(D/O,N) deuterium (D near O, N, etc.)

FR FinkelsteinRosin relaxation due to:FR(C) carbonFR(N) nitrogen

G Gorsky relaxation due to:G(H) hydrogenG(D) deuteriumIG(H) intercrystalline Gorsky eect of hydrogen

Z Zener (and Zener-type) relaxation due to:Z(Cu) copperZ(ord/disord) order/disorderingZ(H) hydrogen

List of Abbreviations XV

PD relaxation due to point defects:PD(O) oxygenPD(N) carbonPD(C) nitrogenPD(O/vac) reorientation of complexes a oxygen atom vacancyPD(N/int) nitrogen atom self-interstitial atomPD(O/sub) oxygen atom substitutional atomPD(O/dis) oxygen atom dislocationPD(O/M) peaks due to diusion under stress of a foreign interstitialPD(N/M/M) atom near one or two metal atomsAR atomic reorientation

Dislocations

DP(B1) dislocation (deformation) NiblettWilks peak (fcc)DP(B2) dislocation (deformation) Bordoni peak (fcc)DP() dislocation (deformation) -peak (bcc)DP() dislocation (deformation) peak (bcc)DP() dislocation (deformation) peak (bcc)DP() dislocation (deformation) -peak (bcc)DP(Pi) deformation Hasiguti peaks: i from 1 to 3DP dislocation peaks in the range of intermediate and elevatedDPi temperatures; i 1ODR overdamped dislocation resonance

DP(am.) relaxation peak due to dislocation-like defects in amor-phous structures

SK SnoekKoster peak, due to:SK(O) oxygenSK(C) carbonSK(N) nitrogenSK(H) hydrogenSK(D) deuterium

DES dislocation-enhanced Snoek peak, due to:DES(O) oxygenDES(N) nitrogenDES(C) carbonDEFR dislocation-enhanced FinkelsteinRosin relaxation

XVI List of Abbreviations

Grain boundaries

GB grain boundary peakGBI impurity grain boundary peakGB(Me) impurity grain boundary peak due to MeGB(LT) GB peak in low temperature rangeGB(IT) GB peak in intermediate temperature rangeGB(HT) GB peak in high temperature rangeSB peak due to subgrain boundaries

Phase transitions

PhT peaks due to a phase transition, namely:PhT() precipitation or dissolutionPhT() phases precipitation or dissolutionPhT(hydr) hydrides precipitation or dissolutionPhT(deut) deuterides precipitation or dissolutionPhT( ) peaks due to transitionPhT(mag) magnetic phase transitionPhT(FMPM) ferro- to paramagnetic magnetic phase transitionPhT(melt) meltingPhT(cr) crystallizationPhT(ord) orderingPhT(polym) polymorphic transitionPhT(mart) martensite phase transitionPhT(super) transition to superconducting condition

Main symbols (selected)

a lattice parameterC concentrationD diusion coecientE Youngs modulusE/E storage/loss modulusEU/ER unrelaxed/relaxed modulusf frequencyG shear modulus (G, G, GU , GR accordingly)H activation energy/enthalpy of relaxationk Boltzmanns constantQ quality factorQ1 loss factor, internal frictionQ1m height of internal friction peakQ1b internal friction background

List of Abbreviations XVII

R universal gas constantt timeT temperatureTm temperature of IF peakTmelt melting temperature attenuation coecient of ultrasonic waves relaxation strength logarithmic decrement strain0 strain amplitude viscosity stress0 stress amplitude relaxation time0 limit relaxation time (pre-exponential factor)/ tan loss angle / loss tangent specic damping capacity circular frequency

Other, less frequently used or more specic symbols are explained directlyin the text. If the same symbol is used in dierent meanings which sometimescould not be avoided it is ensured that the correct meaning is clear fromthe respective context.

1Introduction to Internal Friction:Terms and Definitions

In this chapter, the reader is introduced into the terminology and nomencla-ture used in this book. The main subjects are dened and classied from thephenomenological point of view, and the related theoretical background andexperimental techniques are reviewed briey. The microscopic mechanisms,which are also included in the data collections as an important part of infor-mation, will then be introduced in Chaps. 2 and 3.

1.1 General Phenomenon

The phenomenon of internal friction most generally dened as the dissipa-tion of mechanical energy inside a gaseous, liquid or solid medium is basicallydierent from friction in the tribological sense, i.e., the resistance againstthe motion of two solid surfaces relative to each other (external friction). Ina solid material exposed to a time-dependent load within the elastic defor-mation range only this case is considered in this book internal frictionusually means energy dissipation connected with deviations from Hookes law,as manifested by some stressstrain hysteresis in the case of cyclic loading. Thecorresponding energy absorption W during one cycle, divided by the maxi-mum elastic stored energy W during that cycle, denes the specic dampingcapacity = W/W , or the loss factor W/2W , as the most general mea-sures of internal friction, for which no further assumptions are required (butfor most metallic materials this hysteresis is rather small, i.e., 1). Thereciprocal loss factor is also called the quality factor Q = 2W/W (Lazan1968), so that internal friction or damping can generally be written as

Q1 = W/2W = /2. (1.1)

It should be mentioned, however, that for some of these terms deviatingdenitions exist, which will be discussed later in this introductory chapter.Also the nomenclature (usage of names and symbols) is not always clear,

2 1 Introduction to Internal Friction: Terms and Denitions

t

t

t

t

Applied stress

= E (Hooke)

= (Newton)

(instantaneous plasticity)

elastic viscous plasticStrain response:

.

Fig. 1.1. Fundamental types of mechanical behaviour: response of strain (t) to aconstant stress of nite duration with abrupt loading and unloading

due to dierent traditions that have developed in the related scientic andtechnical disciplines. One example is the symbol which is reserved herefor the viscosity (see Fig. 1.1) according to its common use in physics, uidmechanics and materials science (including glasses and polymers), but whichhas a second meaning as loss factor (or loss coecient, Lazan 1968) in struc-tural engineering and part of technical mechanics. In this book internal frictionis according to the tradition of materials science, physical metallurgy andsolid-state physics from which mechanical spectroscopy has emerged gener-ally denoted as Q1.

1.2 Types of Mechanical Behaviour

Before characterising the types and sources of internal friction in more detail,we have to consider the phenomenology of mechanical behaviour; for this pur-pose, the use of mechanical (or rheological) models is very helpful. Elementsof such models are deduced from fundamental types of mechanical behaviourof solids and liquids like those shown in Fig. 1.1; most important as linearelements are the spring and the dashpot which denote, respectively, an ideal(Hookean) elastic solid with stiness or modulus E, and an ideal (New-tonian) viscous liquid with viscosity (for non-linear models used to describeplasticity, see e.g. Palmov 1998, Fantozzi 2001). Combinations of springs anddashpots generally dene viscoelastic behaviour (Palmov 1998), in particularlinear viscoelasticity since the related constitutive equations are linear (forconvenience we consider uniaxial deformation and scalar quantities, but thegeneralisation to the tensor form is straightforward).

Within this denition, the simplest case of linear viscoelasticity is repre-sented by a Maxwell model, i.e., a spring and a dashpot in series. On theother hand, the respective parallel combination (VoigtKelvin model) is un-realistic because of innite instantaneous stiness. Whereas in principle anynumber of springs and dashpots can be combined, we have to distinguish

1.3 Anelastic Relaxation 3

tt

RU

RU

(a) (b)

Fig. 1.2. Examples of viscoelastic mechanical models, with the same applied stressas in Fig. 1.1: (a) completely recoverable three-parameter models (standard anelas-tic solid); (b) partially recoverable four-parameter model

Table 1.1. Dierent existing terminologies for the distinction between recoverableand non-recoverable types of (linear) viscoelasticity

recoverable non-recoverable reference (example)

anelastic viscoelastic Nowick and Berry 1972viscoelastic viscoplastic Fantozzi 2001, Rosler et al. 2003viscoelastic elastoviscous Meyer and Guicking 1974viscoelastic solid viscoelastic liquid Ferry 1970

between models with a continuous chain of springs resulting in a completelyrecoverable strain (or more precisely, a unique equilibrium relationship be-tween stress and strain, Fig. 1.2a), and those with a single dashpot in seriesshowing a permanent deformation after unloading (absence of a stressstrainequilibrium, Fig. 1.2b).

The terminology of this latter distinction (which can also be interpretedas a borderline between solids and liquids) is again not consistent throughoutthe scientic literature: a list of a few related denitions is given in Table 1.1.In the present book, a completely recoverable behaviour is named anelastic(Zener 1948, Krishtal et al. 1964, Lazan 1968, Nowick and Berry 1972, Lakes1999) because this term appears to be the most clearly dened one. The cor-responding approach to internal equilibrium, after an external perturbation,is known as anelastic relaxation. Non-recoverable components like in Fig. 1.2bmay then be called viscous or viscoplastic (but are of minor importance inthis book), whereas viscoelasticity may include both recoverable and non-recoverable behaviour (Lazan 1968, Ferry 1970, Palmov 1998, Lakes 1999).

1.3 Anelastic Relaxation

Anelastic relaxation, as the main source of internal friction considered in thisbook, is seen in Fig. 1.2a both as a saturating creep strain (t) after loading,with unrelaxed and relaxed values U and R, and as a decaying elastic


after-eect after unloading, and may also be observed as stress relaxationin case of a constant applied strain. It is characterised by a relaxationstrength = (R U)/U1 which can be written in dierent ways e.g. as = (EU ER)/ER using a time-dependent modulus E(t) dened for stressrelaxation (generalised Hookes law) and by a distribution of relaxationtimes . In the simplest possible case, the so-called standard anelastic solid(Nowick and Berry 1972) or standard linear solid (Zener 1948, Fantozzi 2001)dened by the two equivalent three-parameter models in Fig. 1.2a, the time-dependent changes are of the form et/ with a single relaxation time (either for constant stress or for constant strain, with = = for 1).

More important in the context of this handbook, than this quasi-staticbehaviour of an anelastic solid, is that one under a cyclic applied stress orstrain. In the linear theory of anelasticity (Nowick and Berry 1972), usingthe mathematically convenient complex notation (with complex quantitiesmarked by an asterisk) for sinusoidally varying stress and strain,

= 0eit and = 0ei(t) = ( i )eit, (1.2)

several dynamic response functions are dened as a function of the circularfrequency like, for instance, a complex modulus

E() = / = E()ei() = E() + iE(). (1.3)

The real quantities E(), E() and E() are called absolute dynamic mod-ulus, storage modulus and loss modulus, respectively; the phase lag betweenstress and strain is also known as the loss angle.2 The real parts of (1.2) formthe parametric equations of an ellipse as stressstrain hysteresis loop (notonly in the anelastic case but generally for linear viscoelasticity), so that thecalculation of the dissipated energy W shows that in this case the loss tan-gent tan is identical with the more generally dened loss factor introducedearlier (Fantozzi 2001):

Q1 = W/2W = tan = E/E = /. (1.4)

The dynamic response functions of the standard anelastic solid are given bythe well-known Debye equations (rst derived in 1929 by Debye for the caseof dielectric relaxation) which can be found in detail in many textbooks andmonographs (e.g. Zener 1948, Nowick and Berry 1972, Fantozzi 2001). TheDebye equations can be written in dierent ways, e.g.

1 This specic use of the alone-standing symbol for the relaxation strength, fol-lowing the common practice in the literature on anelastic relaxation, should notbe confused with its general meaning as a dierence sign in combinations like W .

2 Sometimes the loss angle is dened as internal friction (Nowick and Berry1972, Fantozzi 2001), implying that internal friction would exist only in linearviscoelastic materials. Since there is no physical reason for such a restriction, weprefer the more general use of this term introduced earlier.

1.4 Thermal Activation 5

012 1 2log

Q1()Q1(T)

ER

EU

ER(T) EU(T) /2E(T)

Tm T

1.144

E() (b)(a)

Fig. 1.3. Dynamic modulus E and internal friction Q1 of the standard anelasticsolid: (a) as a function of frequency on a log scale; (b) as a function of temper-ature at constant frequency. In the latter case, the relaxation-induced step in E(T )is superimposed on the intrinsic temperature dependence of EU(T ) and ER(T )

E() = ER

(1 +

22

1 + 22

)= EU

(1

1 + 1

1 + 22

)(1.5)

and

Q1() =

1 +

1 + 2

, (1.6)

where EU, ER, , and have the same meaning as in the quasi-staticcase considered above. In the case of 1, these equations simplify to

E() = E() = ER

(1 +

22

1 + 22

)= EU

(1

1 + 22

)(1.7)

andQ1() =

1 + 22. (1.8)

The resulting Debye peak Q1(), as shown in Fig. 1.3a, is characterised bya well-dened shape and width (1.144 at half-maximum on a log10 scale)with a damping maximum Qm1 = /2 at = 1. The asymptotic behaviourfor 0 and implies that at these limits the loss angle vanishesand the elliptic stressstrain hysteresis loop degenerates to a straight line(purely elastic behaviour with slopes EU for 1 and ER for 1), sothat losses are detectable only in a certain range around = 1 (dynamichysteresis).

1.4 Thermal Activation

Most of the known mechanisms of anelastic relaxation, to be described later inChap. 2, have their origin in the thermally activated motion of various kindsof defects. In this case, a reciprocal Arrhenius equation


= 0 exp (H/kT ) (1.9)

can be assumed for the relaxation time which now represents a reciprocaljump frequency of the defects ( = 1) to overcome the energy barrier H atthe temperature T . Inserting (1.9) into (1.8), the Debye peak is now obtainedalso as a function of temperature at constant oscillation frequency f = /2(Fig. 1.3b),

Q1(T ) =

2sech

H

k

(1T 1

Tm

), (1.10)

where the peak (or maximum) temperature Tm is dened by the condition = 1, and sech x = (cosh x)1 = 2/(ex + ex). On the reciprocal tem-perature scale (not shown in Fig. 1.3) the Debye peak is symmetric witha half-width of 2.635 k/H. The thermal activation parameters of the relax-ation process, i.e., the (eective) activation enthalpy (apparent activationenergy) H and the limit relaxation time (reciprocal attempt frequency)3

0 = 01, are usually determined from the shift of the peak temperature Tmwhen changing the vibration frequency f , according to

ln(

f2f1

)=

H

k

(1

Tm1 1

Tm2

). (1.11)

This is even possible for loss peaks which are much broader than a Debye peak,which means that the underlying physical process includes a whole spectruminstead of a single value of relaxation times; in that case, average activationparameters are obtained (for more details on the analysis of relaxation spectraand peak deconvolution, see Nowick and Berry 1972, San Juan 2001). In fact,more or less broadened relaxation peaks are the experimental rule rather thanthe exception, so that most data on H and 0 given in this book, usuallydetermined empirically from the frequency shift of Tm after (1.11), are in thissense some average values.

From this latter, experimental viewpoint, it is necessary to point out thatthe separation of H and 0, important for clarifying the relaxation mecha-nisms as well as for predicting the peak position at arbitrary frequencies, isinevitably connected with a loss in precision compared to directly measureddata like Tm. A reliable evaluation of H and 0 and even more of discreteor continuous spectra in these quantities is highly sensitive to the quality ofthe experiments, and mainly requires the variation of frequency over a rangeas broad as possible. Such evaluated activation parameters may therefore bequestionable in cases of scatter in the primary data or too small frequencyvariation, which can only partially be estimated from the information givenin the tables of Chaps. 4 and 5. Since it has not been possible in this bookto classify the quality of the literature data accordingly, the reader should be3 Sometimes the symbol (indicating the limit T ) is used instead of 0. Onthe other hand, the use of 0 is consistent with other common quantities like D0in the related diusion equation D = D0 exp(H/kT ).

1.5 Other Types of Internal Friction 7

aware that there may be strong variations especially in the reliability of theactivation parameters H and 0. If doubts remain even after consulting theoriginal papers, it is recommended to use the primary experimental data likeTm rather than H and 0.

1.5 Other Types of Internal Friction

Although the data collections in this book are focussed on anelastic relax-ation, the main characteristics of other types of internal friction should alsobe considered briey. It is useful to know about these characteristics not onlyfor separating anelastic and other contributions when superimposed on eachother, but also from the viewpoint of understanding the anelastic relaxationphenomena and mechanisms themselves.

Viscous damping, in the sense of non-recoverable linear viscoelasticity in-troduced earlier, can appear in quite dierent forms depending on the loadingconditions and quantities considered. In contrast to anelasticity, all relaxedquantities are now either zero or innite, and some parameters like or are meaningless; on the other hand, stress relaxation and the modulus-typeresponse functions are quite analogous in both cases. For the Maxwell modelas the simplest case, the dynamic functions E() and E() have the sameform as for the standard anelastic solid, e.g. a Debye peak in E() withrelaxation time and peak height EU/2; however, the related loss tangenttan = ()1 does not show any peak but goes to innity in the low-frequency or high-temperature limit. An example of such viscous damping isthe so-called relaxation of metallic glasses (see Sect. 2.6.1).

Non-linear damping, i.e., internal friction beyond linear viscoelasticity,can be described by mechanical models containing specialised non-linearelements (Palmov 1998, Fantozzi 2001) in addition to springs and dashpots;such models are beyond the scope of this introduction, however. Generally,non-linearity always means that the loss factor becomes amplitude-dependent ,whereas in linear viscoelasticity the related quantities in (1.4) do not dependon the amplitude of a sinusoidal vibration.

The amplitude dependence is usually connected with a static hysteresiscomponent, i.e., a virtually frequency-independent contribution to the stressstrain hysteresis loop which does not vanish in the limit 0. In simpliedterms, such idealised type of non-linear, amplitude-dependent and frequency-independent behaviour is often named hysteretic, as opposed to relaxation(i.e., linear, amplitude-independent and frequency-dependent); although incertain cases of non-linear relaxation this may be an oversimplication. Forhigh-damping applications, hysteretic damping is generally preferred overrelaxation because of its weak frequency dependence.

In contrast to linear viscoelasticity with always elliptically-shaped dynamichysteresis loops (cf. (1.2)), non-linear damping may be based on many dierent


types and shapes of static hysteresis loops (see De Batist 2001 for some exam-ples); the underlying mechanical behaviour may be of microplastic, pseudo-elastic or other type depending on the related microscopic mechanisms. Sincetan is not well dened in these cases, non-linear internal friction should beexpressed using more general denitions like the specic damping capacity in (1.1).

A few aspects of amplitude-dependent internal friction (ADIF) areaddressed later in Chaps. 3.33.5. However, in spite of its importance forhigh-damping materials, ADIF is generally not included in the data collec-tions of this book, with very few exceptions. A main reason besides themere quantity of data which would by far exceed the volume of the book isthat the amplitude dependence can be very sensitive to the microstructuralstate of the sample. This makes it more dicult to collect suciently detailedinformation to compare dierent ADIF studies to each other, but also tocondense this information, if available, in the form of tables.

1.6 Measurement of Internal Friction

With ascending frequency, the experimental techniques of mechanical spec-troscopy are generally divided into four groups: quasi-static, subresonance,resonance and wave-propagation (pulse-echo) methods. While measuring dif-ferent quantities and response functions, they all can be used to determineinternal friction of metallic materials, preferably under vacuum to avoidunwanted aerodynamic losses. More details about the following techniques,which can be mentioned only very briey in this introduction, can be foundin the books by Nowick and Berry (1972), Lakes (1999), Schaller et al. (2001),and related references.

Quasi-static tests can be performed using conventional mechanical testingequipment in two dierent ways: (1) in a quasi-static relaxation experiment(as creep/elastic after-eect (t) at constant stress like in Fig. 1.2, or as stressrelaxation (t) at constant strain), or (2) in a cyclic measurement of the stressstrain hysteresis (), e.g. at an alternating constant strain rate d/dt.

The relaxation experiment (1) is suitable to study linear relaxationprocesses if the loading or unloading time of the testing machine is smallcompared to the relaxation time of interest. Measured are quasi-static re-sponse functions including quantities like and , from which dynamicproperties like internal friction can be calculated (Nowick and Berry 1972;cf. (1.8)). On the contrary, the cyclic test (2) is useful for obtaining thefrequency-independent component of ADIF directly from (1.1) using thedirectly measured area W of the static hysteresis loop.

Whereas quasi-static hysteresis can in principle be measured with arbitrarytime functions of stress and strain, the remaining three dynamic methodsideally work with sinusoidal (harmonic) vibrations or waves with a well-denedfrequency = 2f and a wavelength for the related elastic waves. They

1.6 Measurement of Internal Friction 9

can be distinguished by means of the relation between and the length l ofthe sample.

Subresonant experiments, with l and no external inertia attachedto the sample, are working in forced vibration far below the resonance fre-quency of the system. The directly measured quantity is the phase lag (lossangle) between stress and strain, from which internal friction is deter-mined according to (1.4). Commercial instruments of this type (dynamicmechanical analyser), mostly working in bending mode, are widely usedfor polymers with a generally higher viscoelastic damping level. For met-als, the low-frequency forced torsion pendulum is generally preferred becauseof its higher sensitivity. The main advantage of this technique is the pos-sibility to perform isothermal experiments in a very large and continuousfrequency range (104 to almost 102 Hz), which may be important in case oftemperature-dependent structural changes (Rivie`re 2001b).

Resonant experiments form the largest and oldest group of mechanicalspectroscopy methods, with the greatest variety of special techniques, andmay be divided into subgroups where resonance either refers to the eigen-vibrations of the sample ( l) or to a larger system ( l, with an externalinertia attached to the sample). The most important resonance techniques torsion pendulum, vibrating-reed (bending vibration of at samples), compositeoscillator (longitudinal vibration of rods), and resonant ultrasound spec-troscopy of rectangular parallelepiped samples (Leisure 2004) altogetherspan a frequency range from about 101 to more than 106 Hz.

Internal friction can be determined from resonant experiments in dierentways. One possibility is the direct determination of W and W by a care-ful analysis of the relative magnitudes of the input and output signals of thestationary resonant vibration which, however, needs a high stabilisation andcalibration eort. More widely used are two other methods, called resonantbandwidth and free decay, respectively. In the rst case, the width of theresonance peak at the resonance frequency r is measured using forced vibra-tions with constant excitation. If 1 and 2 denote the frequencies on bothsides of the peak where the oscillation amplitude falls to 1/

2 of its maximum

value (half-power or 3 dB points), the internal friction is given by4

Q1 = (2 1)/r. (1.12)4 In fact, (1.1) and (1.12) are both found in the literature as denitions of thequality factor Q. In special cases of electrical circuits from which the concept ofthe quality factor is adopted, both expressions have been shown to be equivalent;possible dierences at higher damping are less important for electrical networkswhich are normally designed for low damping. The related mechanical problemwas analysed by Graesser and Wong (1992), using a linear complex springmodel with frequency-independent loss factor. In this case agreement within 1%is found between both denitions for Q1 < 0.28, whereas beyond that range thedeviations grow rapidly up to a limit of Q1 =

2 at tan = 1, above which Q1

after (1.12) is no longer dened. Hence, if Q1 is used for high damping values,the chosen denition should be indicated clearly.


The second, perhaps still most widely spread method uses free damped vibra-tions after turning o the excitation. The measured quantity is the logarithmicdecrement dened as

= ln(An/An+1), (1.13)

where An and An+1 are the vibration amplitudes in two successive cycles.There are dierent ways of determining depending on the damping level ofthe sample, the quality of the signal, and details of measurement techniqueand data processing. Internal friction is usually determined by

Q1 = / (1.14)

as a well-known low-damping approximation. The deviations at high dampingdepend again on the exact response of the material; related results from theliterature are restricted to special cases (sometimes also questionable) and willnot be given here. In the perhaps most careful analysis available, Graesserand Wong (1992) give Q1 < 0.2 as range of validity, within 1% deviation,for (1.14) in case of the complex spring model.

In wave-propagation experiments, short high-frequency pulses ( l; fre-quency about 106109 Hz or even more) are sent through the sample. Theattenuation coecient

= d(ln u(x))/dx = /, (1.15)

where u(x) is the envelope of the wave during propagation in x direction,corresponds to the logarithmic decrement that would be expected for arelated free vibration. Hence, the internal friction is

Q1 = / = / (1.16)

with limitations analogous to those mentioned earlier for (1.14).In addition to these experimental standard methods, there are also

some highly specialised, combined techniques like acoustic coupling or scan-ning local acceleration microscopy (Gremaud et al. 2001a,b). Furthermore,advanced microfabrication technology and the development of micro- ornanoelectromechanical systems (MEMS/NEMS) open new applications forclassical resonance techniques using miniaturised resonators (Yasumura et al.2000) or thin lms on specially designed, complex-shaped oscillators (Liu andPohl 1998, Harms et al. 1999).

2Anelastic Relaxation Mechanismsof Internal Friction

2.1 Introduction

In this chapter the main mechanisms are considered which produce anelasticdamping as dened in Chap. 1. They are associated with the diusive motionunder stress of point defects (Sect. 2.2), the motion of dislocations or partsof them (Sect. 2.3), and the motion of grain boundaries or other interfaces(Sect. 2.4, where a section about nanocrystalline materials is added). Thefundamental thermoelastic relaxation, always present in internal friction exp-eriments at least as a background, is treated in Sect. 2.5. Specic features ofanelastic and viscoelastic relaxation in non-crystalline metallic structures, ifnot included in the earlier sections, are nally considered in Sect. 2.6.

Damping phenomena which are mainly due to non-linear hysteretic mech-anisms will be treated in Chap. 3 even if they also contain some anelasticcomponent.

2.2 Point Defect Relaxation

Point defect relaxation generally means an anelastic relaxation caused by adiusive redistribution of point defects under the action of an applied stress(diusion under stress). This necessarily requires an elastic interaction bet-ween the applied stress and the distortions of a crystal lattice (or possibly of anon-crystalline matrix) created by the point defects, so that under the actionof the external stress the internal equilibrium distribution of the defects ischanged and a driving force for a directed diusion is produced.

One such possibility, already predicted by Gorsky (1935), is the movementof interstitial atoms from compressed to dilated regions in an inhomogeneousstress eld, i.e., a long-range diusion driven by the hydrostatic stress compo-nent. Since this Gorsky relaxation has been observed in fact only for hydrogenas the most mobile interstitial species, it will be introduced later in connectionwith H-induced anelasticity (Sect. 2.2.4).

12 2 Anelastic Relaxation Mechanisms of Internal Friction

The other possibility, named reorientation, is related to the anisotropyof both the applied stresses and the defect-induced distortions. Comparedto the Gorsky relaxation, reorientation processes have much more practicalimportance for two reasons: (a) they apply to a much larger variety of pointdefects and their clusters, and (b) they require only short-range diusion,ideally over atomic distances, so that the relaxation times are much shorterand more likely to cause internal friction of elastic vibrations in practicallyrelevant frequency ranges.

However, not all point defects in metals are subject to a reorientationmechanism: some of them may cause damping, while others may not. Thisability depends on specic symmetry relations and on the direction of theoscillating applied stress. The main condition is that the symmetry of thelocal elastic distortions, caused by the defects in the crystal lattice, is lowerthan the symmetry of the lattice itself; when specied in crystallographicterms, this is known as so-called selection rules for anelasticity (Table 2.1).

The temperature of anelastic relaxation (i.e., of an internal friction peak)is determined by the activation energy of diusion of the point defect andby the frequency of vibrations. The relaxation strength is determined by theconcentration of defects and by the strength of the individual, defect-induceddistortions. Such a distortion eld, also called an elastic dipole because of itsanisotropic character (Kroner 1958, Nowick and Berry 1972), is described bythe -tensor with the components

(p)ij = ij/Cp, (2.1)

where ij are the components of the strain tensor and Cp is the partial defectconcentration in a specic orientation p(p = 1 . . . nd; nd = number of possible,crystallographically equivalent defect orientations). The -tensor is symmet-ric, i.e., ij(p) = ji(p), and in the coordinate system of the 3 principal axesalso diagonal:

=

1 0 00 2 0

0 0 3

(2.2)

The principal values 1, 2, 3 are the same for all orientations p. Theproperties of the -tensor for various defect symmetries are summarised inTable 2.2. If elastic dipoles present in a crystal have dierent -tensors (dier-ent orientations), then they interact with the applied stress eld in a dierentway. This leads to the reorientation of defects by local atomic jumps in theexternal stress eld and to anelasticity.

2.2.1 The Snoek Relaxation

The classical Snoek relaxation, a mechanism described rst by Snoek (1941)to explain the damping due to C in -Fe, is an anelastic relaxation caused by

2.2 Point Defect Relaxation 13

Table

2.1.Se le c

t ion

r ulesforv a

r iousde fe c

t sand

c rys tals y

s te m

s( N

o wick

and

Be r

r y1972)

c rys tal

sym

met

ryc o

mplianc e

de fe c

ts y

mm

e tr y

t et ragonal

trigonal

or thor h

ombic

monoclin

ict riclinic

100

110

100

110

Cubic

S11S12

10

21

21

2S44

01

01

12

3

tetragonal

[001]

100

110

S11S12

0

10

11

01

S66

0

01

10

11

S44

0

00

01

12

hex

agonal

[001]

[100]

S11S12

0

12

12

S44

0

00

12

trigonal

S11S12,S44,S14

0

12

Adash

mea

nsth

eabse

nce

ofth

edefec

tin

thecr

ystal;

zero

mea

nsth

atth

ere

laxation

ofth

eco

mpliance

ispro

hib

ited

and

1,2

or3

mea

nsth

eex

isting

ofre

laxation

ofdi

eren

tty

pes


Table 2.2. The -tensor (Nowick and Berry 1972)

defectsymmetry

principalvalues

principal axes number ofindependent components

cubic 1 = 2 = 3 arbitrary 1

tetragonal, 1 = 2 = 3 axis 1 along major symmetry axis 2hexagonal, or orand trigonal 1 = 2 = 3 axis 3 along major symmetry axisorthorhombic 1 = 2 = 3 along the three symmetry axes 3monoclinic 1 = 2 = 3 axis 1 or 3 along symmetry axis 4triclinic 1 = 2 = 3 unrelated to crystal axes 6

heavy foreign interstitial atoms (IA) in the body-centred cubic (bcc) metals.It is observed in O, N, C interstitial solid solutions in metals belonging to thegroups VB (V, Nb, Ta) and VIB (Cr, Mo, W), and also in -Fe. The Snoek-type relaxation, i.e., the same mechanism extended to the case of alloys, isobserved in many bcc dilute and concentrated substitutional alloys where theadditional interaction between interstitial and substitutional atoms inuencesthe relaxation parameters (see 2.2.1).1

Relaxation Mechanism

The O, N and C atoms are located in the octahedral interstices of the bcc metallattice (Dijkstra 1947; Nowick and Berry 1972; Weller 2001). The two metalatoms nearest to the interstice (distance about a/2) move aside by a/10; thedisplacement of the other four atoms, located in a distance of a/

2, is about

one order of magnitude smaller (Blanter and Khachaturyan 1978) (Fig. 2.1.).The resulting lattice distortions (elastic dipoles) are oriented mainly along oneaxis (x, y or z) and have a tetragonal symmetry. According to the selectionrules (Table 2.1), a defect with tetragonal symmetry in the cubic lattice causesrelaxation of the elastic compliance (S11S12) and therefore must cause energylosses.

The three types of interstices corresponding to the three lattice directions(x, y, z) form three sublattices (numbered p = 1, 2, 3). In the absence of exter-nal stresses, the dissolved IA are distributed uniformly among the intersticesin all three sublattices: the related occupation probabilities n1, n2 and n3 areequal to each other. By applying a tensile stress along one cubic crystal axis(e.g., X in Fig. 2.1), the arrangement of dissolved atoms in the octahedralinterstices of the sublattice with the number p = 1 becomes energeticallymore favourable than those with p = 2 or 3. Therefore, the dissolved IA will

1 A more general case of Snoek-type relaxation, in a very large variety of alloystructures, is found for hydrogen as the lightest foreign IA (see Sect. 2.2.4).


ZY

X

Fig. 2.1. Octahedral interstices in the bcc crystal lattice: large circles are metalatoms; small circles, squares and triangles are interstices of the sublattices p = 1, 2and 3, respectively

diuse from sublattices 2 and 3 into 1, and n1 will be higher than n2and n3. When the stress sign changes the reverse process sets in, and underthe action of alternating periodic stresses this diusion under stress of IAcauses periodic variations of the occupation numbers n1 to n3.

This change in the distribution of IA among the sublattices of octahedralinterstices causes an anelastic deformation of the crystal associated with achange in lattice spacings along the three main crystal axes:

ax = a0[1 + 1(n2 + n3) + 2n1]ay = a0[1 + 1(n1 + n3) + 2n2] (2.3)az = a0[1 + 1(n1 + n2) + 2n3],

where 1 and 2 are two components of the -tensor not equal to each other(Table 2.2); a0 is the lattice parameter of the pure metal. The dierence|2 1| determines the elastic dipole strength. Methods for determining1 and 2 are described later.

The relaxation time of this Snoek relaxation process is associated withthe diusion of IA on the octahedral interstices, D = D0 exp[H/(RT )],where D0 is the pre-exponential factor and H is the activation energy ofdiusion of IA (Nowick and Berry 1972):

= a20/(36 D), (2.4)0 = a20/(36 D0). (2.5)

Therefore, the activation energy of the Snoek relaxation is equal to the acti-vation energy H of the IA diusion. The Snoek peak temperature Tm thenfollows from (2.4) for 2f = 1:

Tm = H/{R ln[a20f/(18D0)]}, (2.6)where f is the imposed frequency of mechanical vibrations.

16 2 Anelastic Relaxation Mechanisms of Internal FrictionTable

2.3.Snoek

relaxation

para

met

ers

system

Tm

(K)

(f=

1Hz)

H(k

Jm

ol

1)

0(1

0

15s)

Qm

1

104

(per

one

at.%

ofdisso

lved

ato

ms,

polycr

ystal,

at

T=

Tm)

refere

nce

Cr

C435

2620a

Zem

skij

and

Spasskij

(1966),

Golovin

etal.

(1997)

Cr

N429

114.4

1.43

1880

Weller(2

001),

Klein

(1967)

-F

eC

314

83.7

1.89

2150

Weller(2

001),

Blante

rand

Piguzo

v(1

991)

-F

eN

300

78.8

2.38

2000

Weller(2

001),

Blante

rand

Piguzo

v(1

991)

MoC

579596

165

2170a

Yam

aneand

Masu

moto

(1981),

Shch

elkonogov

etal.

(1968)

MoN

498

125

11

2140a

Weller(2

001)

MoO

483515

130140

1340a

Yam

aneand

Masu

moto

(1981),

Gra

ndin

iet

al.

(1996a)

NbC

514

137.5

660a

Weller(2

001)

NbN

562

151.0

1.22

480

Weller(2

001),

Blante

r(1

978),

Weller

etal.

(1981b)

NbO

422

111

2.65

490

Weller(2

001),

Blante

r(1

978)

TaC

626

160.6

1030a

Weller(2

001)

TaN

615

160.1

3.6

730

Weller(2

001),

Blante

r(1

978)

TaO

420

106.2

8.55

740

Weller(2

001),

Blante

r(1

978)

VC

443

114.7117.3

1070a

Blante

r(1

978),

Welleret

al.

(1985)

VN

544

151.0

0.51

800

Weller(2

001),

Blante

r(1

978),

Bora

tto

and

Ree

d-H

ill(1

977)

VO

458

124.0

1.0

800

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Some peak temperatures Tm, determined experimentally for f = 1Hz, arelisted in Table 2.3. Since the respective diusion characteristics dier signi-cantly, the Tm values are also rather dierent. If dierent IA are dissolved inthe same metal, their Snoek peaks sometimes overlap, as seen for C and O inV, for O and N in Mo or for C and N in Ta, Cr and Fe, respectively; in othercases, each IA gives its own Snoek peak (Fig. 2.2).

For all interstitial solid solutions in which the Snoek relaxation has beenrecorded, the values of D0 (and thus 0) are rather similar. Therefore, thereis a reliable linear dependence of Tm on H (Fig. 2.3), which corresponds to0 = 2.08 1015 s (quite similar for all solutions) and Tm (K) = 3.765H(kJmol1) (Weller 1985).

Orientation Dependence

The relaxation strength and the peak maximum Qm1 depend on the dir-ection of the stress applied to the crystal lattice according to the selectionrules and the Snoek relaxation mechanism described earlier. This directiondetermines the change in the energy of a dissolved IA in the interstices ofdierent sublattices if an external stress is applied. This eect is described bythe orientation parameter :

= cos1 cos2 + cos1 cos3 + cos2 cos3 (2.7)where 1, 2, 3 are the angles formed by the applied stress and the cubeaxes [100], [010] and [001], respectively (Nowick and Berry 1972).

According to (Swartz et al. 1968; Nowick and Berry 1972; Weller 1985)

Qm1 = C0 V (2 1)2 F ( ) M/(RTm) (2.8)

Fig. 2.2. The Snoek peaks of O and N in Nb (Grandini et al. 2005)


where C0 is the atomic fraction of interstitial atoms in the solution, V isthe volume of one mole of the host metal. In case of torsional vibration, theparameter = 2/3, and M = G, F ( ) = ; for exural vibration, = 1/9,M = E, F ( ) = 13 . This results in dierent orientation dependences fordierent kinds of deformation.

In case of exure the extending and compressing stresses both are appliedalong the [100] direction, cos1 = 1, cos2 = cos3 = 0, = 0, thus the valueQm

1 is maximal (Table 2.4 and Fig. 2.4). The most prominent dierencesobserved are the ones between the IA energies in the octahedral interstices ofsublattices p = 1 and p = 2, 3 with the maximum number of atoms passingfrom one sublattice to another. If the stress is applied along 111, cos1 =cos2 = cos3 = 1/

3, = 3 and Qm1 = 0. In this case, octahedral

interstices of all three sublattices are deformed similarly and diusion understress is absent. In case of the 110 direction, cos1 = cos2 = 1/

2,

cos3 = 0, = 1/3 and Qm1 has an intermediate value. Similar orientationdependences are observed for the longitudinal vibrations. For torsionial and

Fig. 2.3. Plot of the activation energy H versus peak temperature Tm for Snoekrelaxations in several bcc metals (Weller 1985)

Table 2.4. Orientation dependence of the carbon Snoek peak height of ironmonocrystals (Ino and Inokuti 1972)

crystal axis Qm1 104 carbon concentration,

104 wt%torsion exure

100 1.54 65.3 70 10110 26.7 28.3 65 10111 58.1 2.7 65 10polycrystal 40 34.1 62 10


Fig. 2.4. Orientation dependence of the carbon Snoek maximum for iron singlecrystals (Ino and Inokuti 1972). Crystal orientations: 1, 6: 100; 2, 5: 111; 3, 4:110. Type of vibrations: 1, 3, 5: exure, f = 1.151.18Hz; 2, 4, 6: torsion, f = 3.253.46Hz

transversal vibrations, the 111 direction of applied stress gives the maximaleect, the 100 direction the minimal eect, and 110 an intermediate one.

For polycrystalline samples, averaging over all grain orientations gives 0.2 (Nowick and Berry 1972). From (2.8) one can obtain for torsionalvibrations

Qm1 = (0.4G/3) [C0V (2 1)2/(RTm)], (2.9)

and for exural vibrations

Qm1 = (0.4E/9) [C0V (2 1)2/(RTm)]. (2.10)

For metals, the Poisson ratio 0.3 (Livshiz et al. 1980) and thereforeG = 0.5E/(+1) 0.4E, and then the height of the Snoek peak almost doesnot depend on the types of oscillation for texture-less polycrystalline samples(Table 2.4) as it follows from (2.9) and (2.10). The presence of a preferredcrystallographic orientation of grains in a polycrystal, a texture, also leadsto a dependence of Qm1 on the type of oscillations and on the direction ofapplied stress (Fig. 2.5).

Concentration Dependence

A linear dependence of the Snoek peak height on the dissolved element concen-tration follows from (2.8). It is more prominent for a larger dierence |21|,i.e., for a higher asymmetry of distortions caused by the dissolved IA. Sucha linear dependence for FeC and FeN is shown in Fig. 2.6, for NbO in


Fig. 2.5. Snoek carbon peaks for iron samples cut parallel to the rolling direction(0), at 45 and at 90 to the rolling direction. f = 1Hz (Magalas et al. 1996)

Fig. 2.6. Concentration dependence (wt%) of the carbon (1) and nitrogen (2) Snoekpeak heights for iron. f = 1Hz (Lenz and Dahl 1974)

Fig. 2.7. The Fe samples were quenched from temperatures, at which the totalC or N concentration in iron was within solubility in the -solid solution. If asupersaturated solution decomposes, the height of the peak is determined bythe concentration of interstitials remaining in solid solution. In the metals ofthe VB group (V, Nb, Ta) having higher solubility of O and N than -Fe (upto 0.51 at.% at room temperature), the linear dependence is observed up to0.3 at.% (Weller et al. 1981c; Heulin 1985a): in the system NbO up to 0.35at.%, and in the system NbN up to 0.25 at.% (Ahmad and Szkopiak 1972).In high-purity metals, this boundary may be higher (Weller 2001).


Fig. 2.7. Variation of the Snoek peak height in Nb with oxygen content (Schulzeet al. 1981)

A deviation from the linear dependence, with a weaker increase in Qm1

with rising concentration, occurs at higher concentrations of solute IA as aresult of the formation of groups of dissolved atoms, e.g., pairs or triplets,due to IA interaction. The IF peaks caused by reorientation of these groupsunder stress are characterised by higher activation energies and occur at highertemperatures than the main Snoek peak. The Snoek peak broadens at hightemperatures (Powers and Doyle 1956; Gibala and Wert 1966a, 1966c; Ahmadand Szkopiak 1970). If the concentration of dissolved IA increases, a signicantpart does not contribute to the main Snoek peak as isolated atoms, thus Qm1

increases less strongly with concentration. The existence of pairs and tripletsof IA is questioned in some papers (Weller et al. 1981b, 1985), while otherpublications (Cost and Stanley 1985; Heulin 1985a; Gibala 1985) conrm theformation of such complexes. The inuence of IA concentration on the Snoekpeak shape can be explained by IA long-range interaction. It was analysedby simulation of IA short-range order and its inuence on relaxation (Welleret al. (1992); Haneczok et al. 1992, 1993; Blanter and Fradkov 1992; Haneczok1998; Blanter and Magalas 2003).

The slope of Qm1 as a function of C0 (increase per unit concentrationof IA) is determined mainly by the value (1 2), i.e., by the level of dis-tortions created in the crystal lattice by a dissolved IA. The values 1 and


2 may be determined by two methods (Nowick and Berry 1972; Blanter andKhachaturyan 1978; Khachaturyan 1983):

(a) By the dependence of the lattice parameter on concentration in long-rangeordered interstitial solid solution with a known atomic structure accordingto (2.3), as it is done for C and N in iron using the martensite:

1 = a10 dax/dn1, (2.11)2 = a10 daz/dn2, (2.12)

where z is the tetragonal axis.(b) The dierence (1 2) can be determined by the concentration depen-

dence of Qm1 from (2.8), and the value (21+2) by the lattice parame-ter dependence on concentration in the non-ordered solution. Accordingto such a method, 1 and 2 were determined for O and N in V, Nb, Ta(Blanter and Khachaturyan 1978) (Table 2.5).

For solutions of C in V, Nb, Ta and for solutions of N and O in Cr, Mo, W,there are no reliable experimental data for 1 and 2 because of low solubility.It was shown (Blanter 1985) that the tetragonality factor = 1/2 0.1for all interstitial solid solutions in octahedral interstices of the bcc lattice.The 2 value is higher if the dierence between the sizes of the dissolved IAand the octahedral interstice is bigger, 2 is lower in case of stronger chemicalinteraction between the dissolved IA and host metal atoms. The calculatedvalues 1 and 2 for C in V, Nb, Ta and C, O, N in Cr, Mo, W are given inTable 2.5.

The level of distortions produced by dissolved IA in a bcc lattice (2)increases in the sequence O N C, and while moving upwards along everyperiodic table group in the sequence Ta Nb V and W Mo Cr foreach dissolved element. This is due to the increase in size incompatibility,caused in the rst case by the increase in the dissolved atom diameter and inthe second case by the decrease in metal lattice spacings, and to weakening ofthe metalmetalloid chemical interaction. In the VIB group metals, distortionsare more pronounced than in the VB group metals because of lower latticeparameters and weaker chemical interaction.

Inserting values |21| from Table 2.5 into (2.8), the values Qm1 per 1at.% of dissolved IA were calculated for some solid solutions in the metals ofthe VB and VIB groups (Blanter 1989); these values are also given in Table 2.3.There are no reliable experimental data on the Snoek peak in most of thesemetals mainly due to the low solubility of IA. The height of the IF peakscaused by O and N are close to each other, while those for C are signicantlyhigher. The Snoek maxima are higher in the VIB group metals and in -Fethan in the VB group metals at similar concentrations due to higher crystallattice distortions by dissolved IA.


Table 2.5. The -tensor components for interstitial atoms in octahedral intersticesof the bcc metals

metal IA 1 2 2 1 method reference-Fe C 0.09 0.86 0.95 1 Roberts (1953)

0.10 0.89 0.99 1 Cheng et al. (1990)0.83 2 Weller (2001)

N 0.07 0.83 0.90 1 Bell and Owen (1967)0.06 0.92 0.98 1 Cheng et al. (1990)

1.0 2 Weller (2001)V C 0.08 0.78 0.86 4 Blanter (1985)

N 0.14 0.69 0.83 3 Blanter and Khachaturyan (1978)O 0.10 0.66 0.76 3 Blanter and Khachaturyan (1978)

Nb C 0.07 0.68 0.75 4 Blanter (1985)N 0.05 0.60 0.65 3 Blanter and Khachaturyan (1978)

0.65 2 Weller (2001)O 0.06 0.50 0.56 3 Blanter and Khachaturyan (1978)

0.61 2 Weller (2001)Ta C 0.07 0.67 0.74 4 Blanter (1985)

N 0.05 0.56 0.61 3 Blanter and Khachaturyan (1978)0.70 2 Weller (2001)

O 0.04 0.47 0.51 3 Blanter and Khachaturyan (1978)0.67 2 Weller (2001)

Cr C 0.09 0.85 0.94 4 Blanter (1985), Blanter and Gladilin (1985)N 0.07 0.69 0.76 4 Blanter (1985), Blanter and Gladilin (1985)O 0.06 0.63 0.69 4 Blanter (1985), Blanter and Gladilin (1985)

Mo C 0.08 0.76 0.84 4 Blanter (1985), Blanter and Gladilin (1985)N 0.07 0.67 0.74 4 Blanter (1985), Blanter and Gladilin (1985)O 0.06 0.55 0.61 4 Blanter (1985), Blanter and Gladilin (1985)

W C 0.08 0.76 0.84 4 Blanter (1985), Blanter and Gladilin (1985)N 0.06 0.64 0.70 4 Blanter (1985), Blanter and Gladilin (1985)O 0.05 0.52 0.57 4 Blanter (1985), Blanter and Gladilin (1985)

Methods: The values 1 and 2 or their dierence were determined: 1 by thedependence of the lattice parameters on concentration in the martensite; 2 by theQm

1 concentration dependence of high-purity monocrystals; 3 by the Qm1 andthe lattice parameter concentration dependence and 4 by calculation

Temperature Dependence

The height of the Snoek peak Qm1 is proportional to 1/Tm (2.8), andtherefore an increase in vibration frequency is associated with an increaseof Tm and decrease of Qm1 for the same concentration of IA. In some casesQm

1 (TmT0)1 (Nowick and Berry 1972), where T0 is the ordering tem-perature of IA in a solid solution. However, T0 Tm for low concentrationof IA, and the value T0 can be omitted. The relation Qm1 1/Tm is welldocumented for Ta 0.074 at.% O (Fig. 2.8; Weller et al. 1981b).


Fig. 2.8. Temperature dependence of the oxygen Snoek peak height in Ta with0.074 at.% (Weller et al. 1981b)

Grain Size Dependence

With decreasing grain size of a polycrystal, the Snoek peak height decreases.The coecient k in the formula Qm1 = kC0, where C0 is the concentrationin wt%, is a function of grain size (Ahmad and Szkopiak 1972):

1/k = 1.85 + 0.28 d1/2 for NbO1/k = 2.14 + 0.3 d1/2 for NbN, (2.13)

where d is the mean grain size in mm. The value k increases by a factor oftwo with an increase of grain size from d 0.04mm to d 1mm. The valuesof the coecient k for C and N in iron are given in Table 2.6 (Ferro-Miloneand Mezzetti 1975).

There are three reasons for the decrease in Snoek peak height with decreas-ing grain size: (1) dierent textures in ne- and coarse-grained specimens; (2)adsorption of dissolved IA on the grain boundaries which reduces the solidsolution concentration of IA (Nowick and Berry 1972; Ahmad and Szkopiak1972); (3) peculiarities of the stress distribution in crystals with dierentgrain sizes (Ferro-Milone and Mezzetti 1975). The data for Qm1 in Table 2.3are obtained mainly for coarse-grained polycrystalline specimens without anywell dened texture. However, in some papers on the Snoek relaxation theexistence of texture was not checked and the grain size was not given.

Inuence of Alloying Elements:The Snoek-Type Relaxation in Ternary Alloys (MeSAIA)

Substitutional atoms (SA) in a host lattice must inuence the parameters ofthe original Snoek relaxation via SAIA interaction, i.e., by a change in the


Table 2.6. Inuence of grain size on the Snoek maximum height in -Fe (Ferro-Milone and Mezzetti 1975)

interstitial grain size (mm) concentration Cm (wt%) k = Qm1/Cm,

1/wt%

C 0.0150.040 0.02 0.50.0150.050 0.031 0.530.070.20 0.02 0.70.52.0 0.023 1.14

N 0.0150.025 0.0060.045 0.740.05 0.032 0.780.07 0.017 0.830.15 0.032 0.90.28 0.017 0.940.40 0.032 10.70 0.017 1.12

0.52.0 0.0060.045 1.05

activation energy of the relaxation, and by a change of the value (1 2),i.e., the relaxation strength. In spite of more than 50 years of study sincethe pioneering works of Wert (1950, 1952), Dijkstra and Sladek (1953) onalloyed iron, the experimental situation about the inuence of SA on theSnoek relaxation has remained not clear enough. Some SA may not aect theposition and height of the Snoek peak, others may reduce the peak height,lead to the appearance of additional peaks at higher temperature besides theheight reduction, or suppress the Snoek peak and produce new peaks at highertemperatures (e.g., Krishtal et al. 1964; Saitoh et al. 2004). One can also ndcontradicting results in the literature.

The Snoek-type relaxation in alloys can be explained in many cases on thebasis of the SAIA interaction in the crystal lattice, which leads to a changein the diusion activation energy of dissolved IA located near the relativelyimmobile SA. The recent status of research shows that in most cases twocategories of alloys are distinguished with respect to Snoek relaxation: dilute(or non-concentrated) alloys, where SA can be considered as randomly dis-tributed, non-interacting with each other and sometimes giving an additionalcontribution to the Snoek peak, and concentrated alloys, where SA alwaysaect the IA jumps in the host lattice: SA may be distributed either ran-domly or in some order on a superlattice. Finally, the type of SAIA interac-tion, which may vary from a short-range type in case of a strong chemicalinteraction to a long-range elastic interaction, aects the Snoek relaxation.The critical concentration at which alloys should be considered as concen-trated depends on the range of SAIA interaction in the host lattice and isdierent for dierent alloys.


The Snoek-Type Relaxation in Dilute Alloys

Most experimental studies were done using dilute alloys: in most cases theSA concentration in dilute alloys is assumed to be


occurs by up to 2% Co, Mn, Si, Mn, P (Saitoh et al. 2004) and even 3 at.%Ge, Co (Golovin and Golovina 2006) in FeC. In some papers an additionalSnoek-type peak was reported at higher or lower temperatures and explainedby IASA and SAIASA interactions (e.g., FeBC, FeCo(4.5wt%)C,FeSi(3wt%)C (Golovin 1978)). Additional peaks were observed in FeN al-loyed by Mn (Ritchie and Rawlings 1967), Cu and Ni (Ogi et al. 2000, 2001),etc. Peaks at higher temperatures were observed in NbO and NbN alloyedby 1 at.% Cr, Zr or Hf, while addition of Mo to these alloys leads only to adecrease in the peak height (Szkopiak and Smith 1975). The theory of the in-uence of SAIA interaction on the Snoek relaxation has been given by Koiwa(1971a,b,c,d, 1972) and Numakura and Koiwa (1996) and has been proved inmany experimental papers mainly for iron-based alloys.

The Snoek-Type Relaxation in Concentrated Alloys

There is no well-established theory as yet for the case of higher SA concentra-tion in a host lattice: dierent approaches were used to describe the situationsin dierent alloys. Roughly, one can distinguish between two methods, con-sidering the total IF spectrum either as a sum of several Debye peaks, orexplaining it by distributions in either activation energy or relaxation time,e.g., Golovin (2000). The summing of Debye peaks to interpret a total IFspectrum works better in case of short range IASA interaction, i.e., for well-dened positions of IA in a solid solution. Some examples of such an inter-pretation can be found for FeSiC (Krishtal 1965), FeCrC (Golovin et al.1987a, 1992), FeAlC (Golovin and Golovina 2006), NbMoO and NbVO(Kushnareva and Snezhko 1994) alloys. In case of long-range IASA interac-tion the activation energies cannot be described by only few values, and theassumption of a continuous distribution is more reasonable. This distributionleads to a signicant broadening of the Snoek peak (e.g., in FeAlC (Tanaka1971; Golovin et al. 1998a) and FeCrC (Golovin et al. 1997a)). The Snoek-type relaxation has been well studied for the FeAl system in a wide rangeof Al concentrations (Fischbach 1962; Hren 1963; Janiche et al. 1966; Tanaka1971; Golovin et al. 1998c, 2004b; Strahl 2006; see Fig. 2.9b). The Snoek-typepeak in FeAl alloys (including the Fe3Al intermetallic compound) is broad-ened and shifted to higher temperatures. At an Al concentration of 1012 at.%,the original FeC component of the Snoek-type peak vanishes (i.e., no Al freesurroundings of a jumping C atom exist any more). The Snoek-type peakalso drastically decreases in presence of the carbide-forming elements Ti, Nb(Golovin et al. 2004c, 2005a) and Zr, Ta (Golovin et al. 2006b), where the C in-terstitials get xed in carbides. Ordering in FeAl decreases the peak widths(Tanaka 1971; Golovin et al. 1998a,c; Pozdova 2001, Pozdova and Golovin2003) because only a few well-dened CAl distances occur. High hydrogenconcentration inuences the oxygen diusion and therefore the oxygen Snoekpeak in Nb and NbTi: the relaxation strength decreases and the activationenergy increases with hydrogen content (Schmidt and Wipf 1993; Golovinet al. 1996a, 1998b), see also Sect. 2.2.4.


2.2.2 Relaxation due to Foreign Interstitial Atoms (C, N, O)in fcc and Hexagonal Metals

fcc Metals

Interstitial solute atoms in face-centered cubic (fcc) metals occupy octahedralinterstitial sites having the same symmetry as the host lattice. For this reason,an isolated interstitial atom does not contribute to anelastic relaxation if astress is applied. In case they form complexes, the reorientation of atom pairsmay occur under an external stress. This eect was reported for the rst timeby Rosin and Finkelshtein (1953) for carbon in fcc steels with 25wt% Cr and20wt% Ni, and later conrmed for NiC and NiAlC alloys (Ke and Wang1955a, 1955b). Up to now, such internal friction peaks have been studied inmany austenitic steels (e.g., Ke and Tsien 1957, Golovin and Belkin 1965,1972, Golovin et al. 1966), in CoC, CoNiC, NiTiH, YbO, YbN alloys(Mah and Wert 1964, 1968; Numakura et al. 1995, 1996b, 2000; Yokoyamaet al. 1998), see Table 2.7.

In spite of the fact that the peaks in austenitic steels, known asFinkelshteinRosin peaks, have many common features with the peaks inbinary alloys (NiC, CoC), they also have some peculiarities and will bedescribed separately. The peaks in other fcc metals (no steels) are not calledFinkelshteinRosin peaks.

The main features of the FinkelshteinRosin relaxation can be describedas follows (Rosin and Finkelshtein 1953; Hausch et al. 1983; Ito et al. 1981,Ito and Tsukishima 1985; Ke et al. 1987a):

Table 2.7. Examples of relaxation processes due to interstitial atoms in fcc metals

alloy (wt%) f (Hz) Tm (K) H (kJmol1)

0 (s) atompairs

Ref.

FeC(0.5) 2000 648 125 1.2 1014 C-sub Teplov (1978, 1985)Ni(12)Mo(3)

FeCr(18) 1 625 142 1013 N-sub Banov et al. (1978a,b)Mn(4)N(0.4)

NiC 1 520 144 8 1016 CC Numakara et al. (2000)PdC 1 450 127 CC Yokoyama et al. (1998)

NiTiH 1 165 30.8 2 1011 HTi Numakura et al. (1995)YbN 1 393 105 N-sub Mah and Wert (1964)

YbO 0.5 413 117 O-sub Mah and Wert (1964)In some fcc metals H atoms occupy octahedral interstices as O, N, C; in such casesH relaxation peaks may be similar to those produced by the heavy interstitialatoms.


1. The FR peak position Tm (at f = 1Hz) in steels is within the temper-ature range 470570K and depends (not very strongly) on the chemicalcomposition of the steel.

2. An increase in the interstitial atom content leads to an increase in theFR peak height and a decrease in the peak temperature: for instance, ina manganese steel with 0.60.75wt% C, Tm 540K(f = 2Hz), and in asteel with 0.880.97wt% C, Tm is only 530K.

3. The activation energy is close to the diusion activation energy of car-bon (or nitrogen) in alloyed austenitic steel and decreases slightly withincreasing content of interstitial elements.

4. The peak height, Qm1 0.005 per atomic per cent carbon, is much lowerthan that of the Snoek relaxation, Qm1 0.2 per at.%. This is due tothe fact that C and N atoms disturb the bcc lattice of -Fe much morethan the fcc lattice of -Fe.

5. The FR peak is broader than a Debye peak with a single relaxation time.Sometimes, it can be decomposed into two peaks located close to eachother.

6. The peak height Qm1 is proportional to C2 at low C or N concentrationsC (


second-neighbour positions. For the NiC alloy Qm1 Cn, where n = 1.68although n = 2 would be expected from the CC pair reorientation model.This deviation is the result of a strong elastic interaction of C atoms in Ni,which aects the concentration dependence of CC pairs (Numakara et al.2000). In the PdC alloy, the elastic interaction of C atoms is weaker bec-ause of the larger Pd lattice parameter, thus n = 2 (Yokoyama et al. 1998;Numakara et al. 2000).

In the NiTiH alloy, the peak is attributed to the reorientation ofHTi pairs (Numakura et al. 1995). The orientation dependence of the peakheight suggests that the defect symmetry is 111 trigonal. One of the pos-sible congurations is the HTi pair in which hydrogen occupies the secondneighbour octahedral sites around the substitutional atom. Simulations of theshort range order for these alloys and internal friction spectra taking into acc-ount the long-range interaction of solute atoms (Blanter 1999) conrm thatthe HTi complexes can be responsible for the hydrogen peak in Ni withsubstitutional Ti atoms.

To summarise: the internal friction peaks in fcc interstitial solid solutionare caused by IAIA or IASA pairs.

Hexagonal Metals

Relaxation maxima due to diusion under stress of heavy interstitialatoms are observed in solid solutions of oxygen in Ti, Sc, Hf, Y, Zr, of nitrogenin Ti, Hf, Y and of carbon in Ti, Y (Table 2.8). Similar peaks from dissolvedlight interstitials (H, D) were recorded in the rare earth metals Lu, Scand Y, and attributed to a Zener or Zener-type relaxation (e.g., Vajda et al.1981, 1983, 1990, 1991a; Vajda 1994; Kappesser et al. 1993, 1996a, 1997;Trequattrini et al. 1999, 2000, 2004b). In Ti, Zr and Hf with much lower H orD solubility, related eects are found only in the hydride phase but not in solidsolution (see also Sects. 2.2.3 and 2.2.4). Here we discuss only the relaxationcaused by the heavy interstitial atoms.

In the literature, only very limited information can be found for most ofthe materials included in Table 2.8. The TiO and ZrO solid solutions havebeen most extensively studied. The existing information permits to summarisethe main features of the relaxation process as follows:

1. The peak shape is close to a Debye peak both for polycrystals (Fuller andMiller 1977) and single crystals (Ritchie et al. 1976), i.e., this is a processwith a single relaxation time.

2. The peak height increases linearly with increasing interstitial concentra-tion C in technical grade pure Zr (Gacougnolle et al. 1970, 1971) andTi (Bleasdale and Bacon 1976). In the high purity metals, Qm1 C2(Bertin et al. 1976; Namas 1978; Gacougnolle et al. 1975; Fuller and Miller1977). The peak is absent in Zr in the case of oxygen concentration lessthan 3 at.% (Gacougnolle et al. 1975). In Ti, Qm1 C if C < 0.3 at.%


Table 2.8. Parameters of relaxation processes due to heavy interstitial atoms inhexagonal metals

alloy (wt%) Tm or Tm(aver)a (K) H or H(aver)

(kJmol1)Qm

1b

HfZr(6)N 753 [a] 209 [a]

HfZr(6)O 753 [a] 243 [a]

ScO 430(f = 3.5 kHz) [b]520(f = 3.5 kHz) [b]

Qm1 = 7 105CO [b]

TiC 677 [c] 202 [c] Qm1 2.8 105 CC2 [c]

TiN 773 [l] 241 [l]

TiO (697754)\722[c,d,e,f,g,h,i]

(200242)\218[c,d,g,h,i,j]

Qm1 0.7 105 CO2 [c]

YC 500 [k] 117 [k]

YN 430 [k] 100 [k]

YO 388 [k] 90 [k]

YO 440(f = 1.8 kHz) [m] 67.3 [m]

ZrO (690704)\700[l,n,o,p,r]

(192219)\ 202[l,n,o,p,r]

Qm1 1.6 104 CO [n].

Qm1 (2.6 CO2 +

0.6 CO3) 106 [o],Qm

1 0.6 105 CO2 [r]a after recalculation to f = 1Hz, if the frequency is not specied.b concentrations CC, CN, CO are given in at.%Tm

(aver) and H(aver) are average values.

References: a-Bisogni et al. 1964; b-Trequattrini et al. 2004a; c-Miller and Browne1970; d-Bratina 1962; e-Browne 1972; f-Clauss et al. 1974; g-Bertin et al. 1976;h-Namas 1978; i-Wegielnik and Chomka 1979; j-Pratt et al. 1954; k-Borisov et al.1971; l-Miller 1962; m-Cannelli et al. 1996, 1997; n-Gacougnolle et al. 1970, 1971;o-Gacougnolle et al. 1975; p-Ritchie et al. 1976, 1977, r-Fuller and Miller 1977.

O and Qm1 C2 if C = 38 at.% O (Clauss et al. 1974). Qm1 is muchhigher in technical grade pure metals than in high purity ones, and alsomuch higher in TiO as compared to ZrO: the O atoms create higher dis-tortions in the Ti crystal lattice according to the concentration coecientsof lattice expansion in Ti and Zr (Blanter et al. 2002).

3. T

internal friction in metallic materials: a handbook

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