Internal friction in metallic materials

An efficient use of mechanical spectroscopy maythen require both: a a systematic treatment of the different mechanisms ofinternal friction and anelastic relaxation, and b a comprehensive co

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90 Internal Friction in Metallic Materials

By M.S Blanter, I.S Golovin,

H Neuh¨auser, and H.-R Sinning

A Handbook

M.S Blanter I.S Golovin

Moscow State University

of Instrumental Engineering

and Information Science

Stromynka 20, 107846, Moscow, Russia

E-mail: mike@blanter.msk.ru

Professor Dr Hartmut Neuh¨auser

Institut f¨ur Physik der Kondensierten Materie

Technische Universit¨at Braunschweig

Microelectronics Science Laboratory

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Professor Jürgen Parisi

Universit¨at Oldenburg, Fachbereich Physik Abt Energie- und Halbleiterforschung Carl-von-Ossietzky-Strasse 9–11

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Professor Hans Warlimont

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To our families

Internal friction and anelastic relaxation form the core of the mechanical 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).

spec-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 difficult An efficient use of mechanical spectroscopy maythen require both: (a) a systematic treatment of the different mechanisms ofinternal friction and anelastic relaxation, and (b) a comprehensive compila-tion of experimental data in order to facilitate the assignment of mechanisms

to the observed phenomena

Whereas the first of these two approaches was developed since more thanhalf a century in several textbooks and monographs (e.g., Zener 1948, Krishtal

et 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 fill 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 specifications of relax-ation mechanisms The key to understand this very condensed information

is provided, besides appropriate lists of symbols and abbreviations, by the

introductory Chaps 1–3: after the Introduction to Internal Friction in Chap 1,

defining and delimiting the subject and clarifying the terminology, the relevant

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 not

be 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 specificstructures, which do not fit well into the general scheme of Chap 4 A shortsummary or specific explanations are included at the beginning of each table.Although the authors made all efforts to be consistent in style throughoutthe book, some difficulties in evaluating individual relaxation spectra led toslight deviations, concerning details of data presentation, between the differentchapters and subsections Since some of the data were evaluated from figures,the accuracy should generally be regarded with care; in cases of doubt, theoriginal papers should be consulted Over 2000 references published until mid

2006 were included, among which many earlier ones are still important cause certain alloys and effects 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

be-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 friction

in Chaps 1–3, it may be helpful for nonspecialists and for beginners in thefield 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 figures, and of Ms SvetlanaGolovina with tables We are also grateful to the Springer team, in particular

Dr Claus Ascheron, Ms Adelheid Duhm and Ms Nandini Loganathan, forgood cooperation

Moscow, Tula, Braunschweig Mikhail S Blanter, Igor S Golovin

January 2007 Hartmut Neuh¨ auser, Hans-Rainer Sinning

1 Introduction to Internal Friction: Terms and Definitions 1

1.1 General Phenomenon 1

1.2 Types of Mechanical Behaviour 2

1.3 Anelastic Relaxation 3

1.4 Thermal Activation 5

1.5 Other Types of Internal Friction 7

1.6 Measurement of Internal Friction 8

2 Anelastic Relaxation Mechanisms of Internal Friction 11

2.1 Introduction 11

2.2 Point Defect Relaxation 11

2.2.1 The Snoek Relaxation 12

2.2.2 Relaxation due to Foreign Interstitial Atoms (C, N, O) in fcc and Hexagonal Metals 28

2.2.3 The Zener Relaxation 32

2.2.4 Anelastic Relaxation due to Hydrogen 36

2.2.5 Other Kinds of Point-Defect Relaxation 48

2.3 Dislocation Relaxation 50

2.3.1 Intrinsic Dislocation Relaxation Mechanisms: Bordoni and Niblett–Wilks Peaks 51

2.3.2 Coupling of Dislocations and Point Defects: Hasiguti and Snoek–K¨oster Peaks and Dislocation-Enhanced Snoek Effect 61

2.3.3 Other Kinds of Dislocation Relaxation 73

2.4 Interface Relaxation 77

2.4.1 Grain Boundary Relaxation 78

2.4.2 Twin Boundary Relaxation 82

2.4.3 Nanocrystalline Metals 83

2.5 Thermoelastic Relaxation 87

2.5.1 Theory 87

X Contents

2.5.2 Properties and Applications of Thermoelastic

Damping 90

2.6 Relaxation in Non-Crystalline and Complex Structures 95

2.6.1 Amorphous Alloys 97

2.6.2 Quasicrystals and Approximants 113

3 Other Mechanisms of Internal Friction 121

3.1 Introduction 121

3.2 Internal Friction at Phase Transformations 121

3.2.1 Martensitic Transformation 121

3.2.2 Polymorphic and Other Phase Transformations 129

3.2.3 Precipitation and Dissolution of a Second Phase 133

3.3 Dislocation-Related Amplitude-Dependent Internal Friction (ADIF) 136

3.4 Magneto-Mechanical Damping 144

3.5 Mechanisms of Damping in High-Damping Materials 148

4 Internal Friction Data of Crystalline Metals and Alloys (Tables) 157

4.1 Copper and Noble Metals and their Alloys 158

4.2 Alkaline and Alkaline Earth Metals and their Alloys 189

4.3 Metals of the IIA–VIIA Groups and their Alloys 196

4.4 Metals of the IIIB Group, Rare Earth Metals and Actinides 223

4.4.1 Rare Earth and Group IIIB Metals 223

4.4.2 Actinides 235

4.5 Metals of the IVB Group 238

4.5.1 Titanium and its Alloys 238

4.5.2 Zirconium and its Alloys 263

4.5.3 Hafnium and its Alloys 275

4.6 Metals of the VB Group 276

4.6.1 Vanadium and its Alloys 276

4.6.2 Niobium and its Alloys 287

4.6.3 Tantalum and its Alloys 321

4.7 Metals of the VIB Group 331

4.7.1 Chromium and its Alloys 331

4.7.2 Molybdenum and its Alloys 338

4.7.3 Tungsten and its Alloys 346

4.8 Metals of the VIIB group: Mn and Re 352

4.9 Iron and Iron-Based Alloys 356

4.9.1 Fe (“pure”) 357

4.9.2 Fe–Interstitial Atoms (C, H, N), Other Elements (As, B, Ce, La, P, S, Y) <1%, and Low Carbon Steels 360

4.9.3 Fe–(<3%)Me–(C, N) and Low Alloyed Steels (Me = Metal) 367

4.9.4 Fe–Al Alloys and Steels (Mainly bcc and bcc-Based) 372

4.9.5 Fe–Al-Based Ternary and Multi-Component Alloys

(e.g., Fe–Al–Cr, Fe–Al–Ge, Fe–Al–Si, etc.) 380

4.9.6 Fe–Co, –Ge, –Si, –Mo, –V, –W Alloys 385

4.9.7 Fe–Cr-Based Steels and Alloys 389

4.9.8 Fe–Mn-Based Steels and Alloys 397

4.9.9 Fe–Ni-Based Steels and Alloys 402

4.9.10 Other Fe-Based Multi-Component Alloys 408

4.10 Co, Ni and their Alloys 413

5 Internal Friction Data of Special Types of Metallic Materials (Tables) 423

5.1 Hydrogen-Absorbing Alloys 424

5.2 Metallic Glasses 439

5.3 Quasicrystals and Other Complex Alloys 449

References 453

Index 535

Irr(n) irradiated by neutrons

Irr(e) irradiated by electrons

Irr(p) irradiated by protons

Irr(d) irradiated by deuterons

Irr(γ) irradiated by γ-rays

Ufgr ultrafine-grained

SQC single quasicrystal (icosahedral phase)

QC (poly-) quasicrystalline (icosahedral phase)d-QC (poly-) quasicrystalline (decagonal phase)

Sput.th.f sputtered thin film

Evap.th.f evaporated thin film

Gas depos gas deposited thin film

DSR directional structural relaxation

AES activation energy spectrum

RRR residual resistivity ratio

Abbreviations of Relaxation Mechanisms and Internal Friction Peaks

S(H/O,N) hydrogen (H near O, N, etc.)

S(D/O,N) deuterium (D near O, N, etc.)

FR Finkelstein–Rosin relaxation due to:

IG(H) intercrystalline Gorsky effect of hydrogen

Z Zener (and Zener-type) relaxation due to:

Z(ord/disord) order/disordering

PD(O/sub) oxygen atom – substitutional atom

PD(O/dis) oxygen atom – dislocation

PD(O/M) peaks due to diffusion under stress of a foreign interstitialPD(N/M/M) atom near one or two metal atoms

Dislocations

DP(B1) dislocation (deformation) Niblett–Wilks peak (fcc)DP(B2) dislocation (deformation) Bordoni peak (fcc)

DP(α) dislocation (deformation)α-peak (bcc)

DP(β) dislocation (deformation)βpeak (bcc)

DP(δ) dislocation (deformation)δpeak (bcc)

DP(γ) dislocation (deformation)γ-peak (bcc)

DP(Pi) deformation Hasiguti peaks: i from 1 to 3

DP dislocation peaks in the range of intermediate and elevated

ODR overdamped dislocation resonance

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

Grain boundaries

GBI impurity grain boundary peak

GB(Me) impurity grain boundary peak due to MeGB(LT) GB peak in low temperature range

GB(IT) GB peak in intermediate temperature rangeGB(HT) GB peak in high temperature range

SB peak due to subgrain boundaries

Phase transitions

PhT peaks due to a phase transition, namely:

PhT(α) αprecipitation or dissolution

PhT(θ) θphases precipitation or dissolution

PhT(hydr) hydrides precipitation or dissolution

PhT(deut) deuterides precipitation or dissolution

PhT(α→β) peaks due toα→β transition

PhT(mag) magnetic phase transition

PhT(FM–PM) ferro- to paramagnetic magnetic phase transitionPhT(melt) melting

PhT(cr) crystallization

PhT(ord) ordering

PhT(polym) polymorphic transition

PhT(mart) martensite phase transition

PhT(super) transition to superconducting condition

Main symbols (selected)

G shear modulus (G
, G

, G U , G R accordingly)

H activation energy/enthalpy of relaxation

Q −1 loss factor, internal friction

Q −1m height of internal friction peak

Q −1b internal friction background

List of Abbreviations XVII

Tm temperature of IF peak

Tmelt melting temperature

α attenuation coefficient of ultrasonic waves

τ0 limit relaxation time (pre-exponential factor)

φ/ tan φ loss angle / loss tangent

Other, less frequently used or more specific symbols are explained directly

in the text If the same symbol is used in different meanings – which sometimescould not be avoided – it is ensured that the correct meaning is clear fromthe respective context

Introduction to Internal Friction:

Terms and Definitions

In this chapter, the reader is introduced into the terminology and ture used in this book The main subjects are defined and classified from the

nomencla-phenomenological point of view, and the related theoretical background and experimental techniques are reviewed briefly The microscopic mechanisms,

which are also included in the data collections as an important part of mation, will then be introduced in Chaps 2 and 3

infor-1.1 General Phenomenon

The phenomenon of internal friction – most generally defined as the

dissipa-tion of mechanical energy inside a gaseous, liquid or solid medium – is basicallydifferent from “friction” in the tribological sense, i.e., the resistance againstthe motion of two solid surfaces relative to each other (“external friction”) In

a solid material exposed to a time-dependent load within the “elastic”

defor-mation range – only this case is considered in this book – “internal friction”usually means energy dissipation connected with deviations from Hooke’s law,

as manifested by some stress–strain hysteresis in the case of cyclic loading The

corresponding energy absorption ∆W during one cycle, divided by the mum elastic stored energy W during that cycle, defines the specific damping capacity Ψ = ∆W/W , or the loss factor ∆W/2πW , as the most general mea-

maxi-sures of internal friction, for which no further assumptions are required (but

for most metallic materials this hysteresis is rather small, i.e., Ψ  1) The reciprocal loss factor is also called the quality factor Q = 2πW/∆W (Lazan

1968), so that “internal friction” or “damping” can generally be written as

It should be mentioned, however, that for some of these terms deviatingdefinitions 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 Definitions

t Applied stress σ

σ = E ε (Hooke)

σ = η ε (Newton)

(instantaneous plasticity)

Strain response:

.

constant stress of finite duration with abrupt loading and unloading

due to different traditions that have developed in the related scientific and

technical disciplines One example is the symbol η which is reserved here for the viscosity (see Fig 1.1) according to its common use in physics, fluid

mechanics and materials science (including glasses and polymers), but which

has a second meaning as loss factor (or loss coefficient, Lazan 1968) in

struc-tural engineering and part of technical mechanics In this book internal friction

is – according to the tradition of materials science, physical metallurgy and

solid-state physics from which mechanical spectroscopy has emerged – ally denoted as Q −1

gener-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 pose, the use of mechanical (or rheological) models is very helpful Elements

pur-of such models are deduced from fundamental types pur-of mechanical behaviour

of 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 stiffness or “modulus” E, and an ideal tonian) viscous liquid with viscosity η (for non-linear models used to describe

(New-plasticity, see e.g Palmov 1998, Fantozzi 2001) Combinations of springs anddashpots generally define viscoelastic behaviour (Palmov 1998), in particular

linear viscoelasticity since the related constitutive equations are linear (for

convenience we consider uniaxial deformation and scalar quantities, but thegeneralisation to the tensor form is straightforward)

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

repre-t t

ε

ε R

ε U ε

ε R −ε U

Fig 1.2 Examples of viscoelastic mechanical models, with the same applied stress

as in Fig 1.1: (a) completely recoverable three-parameter models (“standard tic solid”); (b) partially recoverable four-parameter model

anelas-Table 1.1 Different existing terminologies for the distinction between recoverable

and non-recoverable types of (linear) viscoelasticity

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 stress–strainequilibrium, Fig 1.2b)

The terminology of this latter distinction (which can also be interpreted

as a borderline between solids and liquids) is again not consistent throughoutthe scientific literature: a list of a few related definitions 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 defined one The cor-responding approach to internal equilibrium, after an external perturbation,

is known as anelastic relaxation Non-recoverable components like in Fig 1.2b

may 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 this

book, is seen in Fig 1.2a both as a saturating “creep” strain ε(t) after loading, with “unrelaxed” and “relaxed” values εU and εR, and as a decaying “elastic

4 1 Introduction to Internal Friction: Terms and Definitions

after-effect” after unloading, and may also be observed as stress relaxation

in case of a constant applied strain It is characterised by a relaxation strength ∆ = (εR− εU)/εU1 which can be written in different ways – e.g as

∆ = (EU− ER)/ER using a time-dependent modulus E(t) defined for stress relaxation (generalised Hooke’s law) – and by a distribution of relaxation times τ In the simplest possible case, the so-called standard anelastic solid (Nowick and Berry 1972) or standard linear solid (Zener 1948, Fantozzi 2001)

defined by the two equivalent three-parameter models in Fig 1.2a, the dependent changes are of the form e−t/τ with a single relaxation time τ (either

time-τσ for constant stress or τ ε for constant strain, with τ σ = τ ε = τ for ∆  1).

More important in the context of this handbook, than this quasi-static

behaviour of an anelastic solid, is that one under a cyclic applied stress or

strain 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,

σ ∗ = σ0eiωt and ε ∗ = ε0ei(ωt−φ) = (ε
− i ε

)eiωt , (1.2)several dynamic response functions are defined as a function of the circular

frequency ω like, for instance, a complex modulus

E ∗ (ω) = σ ∗ /ε ∗ = E(ω)e iφ(ω) = E
(ω) + iE

(ω). (1.3)

The real quantities E(ω), E
(ω) and E

(ω) are called absolute dynamic ulus, storage modulus and loss modulus, respectively; the phase lag φ between stress and strain is also known as the loss angle.2 The real parts of (1.2) formthe parametric equations of an ellipse as stress–strain hysteresis loop (notonly in the anelastic case but generally for linear viscoelasticity), so that the

mod-calculation of the dissipated energy ∆W shows that in this case the loss gent tan φ is identical with the more generally defined loss factor introduced

tan-earlier (Fantozzi 2001):

Q −1 = ∆W/2πW = tan φ = E

/E
= ε

/ε
(1.4)The dynamic response functions of the standard anelastic solid are given bythe well-known Debye equations (first derived in 1929 by Debye for the case

of 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 different ways, e.g

1

fol-lowing the common practice in the literature on anelastic relaxation, should not

1972, 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

Q −1 (ω) = √ ∆

1 + ∆

ω √ τστε

where EU, ER, ∆, τ σ and τ ε have the same meaning as in the quasi-static

case considered above In the case of ∆  1, these equations simplify to

and

Q −1 (ω) = ∆ ωτ

The resulting Debye peak Q −1 (ω), as shown in Fig 1.3a, is characterised by

a well-defined shape and width (1.144 at half-maximum on a log10ωτ scale) with a damping maximum Qm−1 = ∆/2 at ωτ = 1 The asymptotic behaviour for ωτ → 0 and ωτ → ∞ implies that at these limits the loss angle φ vanishes

and the elliptic stress–strain hysteresis loop degenerates to a straight line

(purely elastic behaviour with slopes EU for ωτ  1 and ERfor ωτ  1), so that losses are detectable only in a certain range around ωτ = 1 (“dynamic”

6 1 Introduction to Internal Friction: Terms and Definitions

can be assumed for the relaxation time which now represents a reciprocal

jump frequency of the defects (τ = ν −1 ) to overcome the energy barrier H at the temperature T Inserting (1.9) into (1.8), the Debye peak is now obtained also as a function of temperature at constant oscillation frequency f = ω/2π

(Fig 1.3b),

Q −1 (T ) = ∆

2 sech

H k

1

T − 1Tm



where the peak (or maximum) temperature Tm is defined by the condition

ωτ = 1, and sech x = (cosh x) −1 = 2/(e x+ e−x) On the reciprocal perature scale (not shown in Fig 1.3) the Debye peak is symmetric with

tem-a htem-alf-width of 2.635 k/H The thermtem-al tem-activtem-ation ptem-artem-ameters of the reltem-ax-

relax-ation process, i.e., the (effective) activrelax-ation enthalpy (“apparent activrelax-ation

energy”) H and the “limit relaxation time” (reciprocal “attempt frequency”)3

τ0= ν0−1 , are usually determined from the shift of the peak temperature Tm

when changing the vibration frequency f , according to

1

instead of a single value of relaxation times; in that case, average activation

parameters 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 than

the exception, so that most data on H and τ0 given in this book, usually

determined empirically from the frequency shift of Tmafter (1.11), are in thissense some average values

From this latter, experimental viewpoint, it is necessary to point out that

the separation of H and τ0, important for clarifying the relaxation nisms as well as for predicting the peak position at arbitrary frequencies, isinevitably connected with a loss in precision compared to directly measured

mecha-data like Tm A reliable evaluation of H and τ0 – and even more of discrete

or continuous spectra in these quantities – is highly sensitive to the quality ofthe experiments, and mainly requires the variation of frequency over a range

as 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 given

in the tables of Chaps 4 and 5 Since it has not been possible in this book

to classify the quality of the literature data accordingly, the reader should be

3

aware that there may be strong variations especially in the reliability of the

activation parameters H and τ0 If doubts remain even after consulting theoriginal papers, it is recommended to use the primary experimental data like

Tmrather than H and τ0

1.5 Other Types of Internal Friction

Although the data collections in this book are focussed on anelastic ation, the main characteristics of other types of internal friction should also

relax-be considered briefly 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 different forms depending on the loadingconditions and quantities considered In contrast to anelasticity, all “relaxed”

quantities are now either zero or infinite, and some parameters like ∆ or τ σ

are meaningless; on the other hand, stress relaxation and the modulus-typeresponse functions are quite analogous in both cases For the Maxwell model

as the simplest case, the dynamic functions E
(ω) and E

(ω) have the same form as for the standard anelastic solid, e.g a “Debye peak” in E

(ω) with relaxation time τ ε and peak height EU/2; however, the related loss tangent tan φ = (ωτ ε)−1 does not show any peak but goes to infinity in the low-frequency or high-temperature limit An example of such viscous damping is

the so-called “α 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 depend

on 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 stress–

strain hysteresis loop which does not vanish in the limit ω → 0 In simplified

terms, such idealised type of non-linear, amplitude-dependent and 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 oversimplification Forhigh-damping applications, “hysteretic” damping is generally preferred overrelaxation because of its weak frequency dependence

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

8 1 Introduction to Internal Friction: Terms and Definitions

types and shapes of static hysteresis loops (see De Batist 2001 for some ples); the underlying mechanical behaviour may be of microplastic, pseudo-elastic or other type depending on the related microscopic mechanisms Since

exam-tan φ is not well defined in these cases, non-linear internal friction should be expressed using more general definitions like the specific damping capacity Ψ

in (1.1)

A few aspects of amplitude-dependent internal friction (ADIF) areaddressed later in Chaps 3.3–3.5 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 difficult to collect sufficiently detailedinformation to compare different 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 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 briefly in this introduction, can be found

spec-in 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 testing

equipment in two different ways: (1) in a quasi-static relaxation experiment

(as creep/elastic after-effect ε(t) at constant stress like in Fig 1.2, or as stress relaxation σ(t) at constant strain), or (2) in a cyclic measurement of the stress– strain hysteresis σ(ε), e.g at an alternating constant strain rate ±dε/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 τ , from which dynamic

properties 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 the

directly measured area ∆W of the “static” hysteresis loop.

Whereas quasi-static hysteresis can in principle be measured with arbitrary

time functions of stress and strain, the remaining three dynamic methods

ideally work with sinusoidal (harmonic) vibrations or waves with a well-defined

frequency ω = 2πf and a wavelength λ for the related elastic waves They

can be distinguished by means of the relation between λ and the length l of

the sample

Subresonant experiments, with λ  l and no external inertia attached

to the sample, are working in forced vibration far below the resonance quency of the system The directly measured quantity is the phase lag (loss

fre-angle) φ 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 because

of its higher sensitivity The main advantage of this technique is the sibility to perform isothermal experiments in a very large and continuousfrequency range (10−4 to almost 102Hz), which may be important in case oftemperature-dependent structural changes (Rivi`ere 2001b)

pos-Resonant experiments form the largest and oldest group of mechanical

spectroscopy methods, with the greatest variety of special techniques, andmay be divided into subgroups where resonance either refers to the eigen-

vibrations of the sample (λ ≈ l) or to a larger system (λ  l, with an external

inertia attached to the sample) The most important resonance techniques –

torsion pendulum, vibrating-reed (bending vibration of flat samples), composite oscillator (longitudinal vibration of rods), and resonant ultrasound spec- troscopy of rectangular parallelepiped samples (Leisure 2004) – altogether

span a frequency range from about 10−1 to more than 106Hz

Internal friction can be determined from resonant experiments in different

ways 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 effort More widely used are two other methods, called “resonantbandwidth” and “free decay”, respectively In the first case, the width of the

resonance peak at the resonance frequency ω ris measured using forced

vibra-tions with constant excitation If ω1 and ω2 denote the frequencies on both

sides of the peak where the oscillation amplitude falls to 1/ √

2 of its maximumvalue (“half-power” or “3 dB” points), the internal friction is given by4

Q −1 = (ω2− ω1)/ω r. (1.12)

4

In fact, (1.1) and (1.12) are both found in the literature as definitions of the

the quality factor is adopted, both expressions have been shown to be equivalent;possible differences at higher damping are less important for electrical networks

which are normally designed for low damping The related mechanical problem

was analysed by Graesser and Wong (1992), using a linear “complex spring”model with frequency-independent loss factor In this case agreement within 1%

the chosen definition should be indicated clearly

10 1 Introduction to Internal Friction: Terms and Definitions

The second, perhaps still most widely spread method uses free damped tions after turning off the excitation The measured quantity is the logarithmic

vibra-decrement δ defined as

where A n and A n+1 are the vibration amplitudes in two successive cycles

There are different ways of determining δ depending on the damping level of

the sample, the quality of the signal, and details of measurement techniqueand data processing Internal friction is usually determined by

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, Graesser

and Wong (1992) give Q −1 < 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 (λ  l;

fre-quency about 106–109 Hz or even more) are sent through the sample Theattenuation coefficient

with limitations analogous to those mentioned earlier for (1.14)

In addition to these experimental standard methods, there are alsosome 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 films on specially designed, complex-shaped oscillators (Liu andPohl 1998, Harms et al 1999)

Anelastic Relaxation Mechanisms

of Internal Friction

2.1 Introduction

In this chapter the main mechanisms are considered which produce anelasticdamping as defined in Chap 1 They are associated with the diffusive motionunder stress of point defects (Sect 2.2), the motion of dislocations or parts

of 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 Specific features ofanelastic and viscoelastic relaxation in non-crystalline metallic structures, ifnot included in the earlier sections, are finally 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 adiffusive redistribution of point defects under the action of an applied stress(“diffusion 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 action

of the external stress the internal equilibrium distribution of the defects ischanged and a driving force for a directed diffusion is produced

One such possibility, already predicted by Gorsky (1935), is the movement

of interstitial atoms from compressed to dilated regions in an inhomogeneousstress field, i.e., a long-range diffusion driven by the hydrostatic stress compo-

nent Since this Gorsky relaxation has been observed in fact only for hydrogen

as 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 anisotropy

of both the applied stresses and the defect-induced distortions Compared

to 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 diffusion,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 specific 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 specified 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 diffusion of the point defect and

by the frequency of vibrations The relaxation strength is determined by theconcentration of defects and by the strength of the individual, defect-induced

distortions Such a distortion field, also called an elastic dipole because of its

anisotropic character (Kr¨oner 1958, Nowick and Berry 1972), is described by

λ (p)ij = ∂εij/∂Cp, (2.1)

where εij are the components of the strain tensor and Cp is the partial defect

concentration in a specific orientation p(p = 1 nd; nd= number of possible,crystallographically equivalent defect orientations) The λ-tensor is symmet-

ric, i.e., λij(p) = λji(p), and in the coordinate system of the 3 principal axesalso diagonal:

The principal values λ1, λ2, λ3 are the same for all orientations p The

properties of the λ-tensor for various defect symmetries are summarised in

Table 2.2 If elastic dipoles present in a crystal have differentλ-tensors

(differ-ent ori(differ-entations), then they interact with the applied stress field in a differ(differ-entway This leads to the reorientation of defects by local atomic jumps in theexternal stress field and to anelasticity

2.2.1 The Snoek Relaxation

The classical Snoek relaxation, a mechanism described first by Snoek (1941)

to explain the damping due to C inα-Fe, is an anelastic relaxation caused by

14 2 Anelastic Relaxation Mechanisms of Internal Friction

“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 inα-Fe The Snoek- type relaxation, i.e., the same mechanism extended to the case of alloys, is

observed in many bcc dilute and concentrated substitutional alloys where theadditional interaction between interstitial and substitutional atoms influencesthe relaxation parameters (see 2.2.1).1

axis (x, y or z) and have a tetragonal symmetry According to the selection

rules (Table 2.1), a defect with tetragonal symmetry in the cubic lattice causes

relaxation of the elastic compliance (S11–S12) 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 interstices

in all three sublattices: the related occupation probabilities n1, n2and n3areequal 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 octahedral

interstices of the sublattice with the number p = 1 becomes energetically more 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)

Z Y X

Fig 2.1 Octahedral interstices in the bcc crystal lattice: large circles are metal

atoms; small circles, squares and triangles are interstices of the sublattices p = 1, 2

and 3, respectively

diffuse from sublattices “2” and “3” into “1”, and n1 will be higher than n2

and n3 When the stress sign changes the reverse process sets in, and underthe action of alternating periodic stresses this “diffusion under stress” of IA

causes 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 + λ1(n2+ n3) + λ2n1]

ay = a0[1 + λ1(n1+ n3) + λ2n2] (2.3)

az = a0[1 + λ1(n1+ n2) + λ2n3], where λ1 and λ2 are two components of theλ-tensor not equal to each other

(Table 2.2); a0 is the lattice parameter of the pure metal The difference

|λ2− λ1| determines the “elastic dipole strength” Methods for determining

λ1 and λ2 are described later

The relaxation time τ of this Snoek relaxation process is associated with the diffusion of IA on the octahedral interstices, D = D0exp[−H/(RT )], where D0 is the pre-exponential factor and H is the activation energy of

diffusion of IA (Nowick and Berry 1972):

Therefore, the activation energy of the Snoek relaxation is equal to the

acti-vation energy H of the IA diffusion The Snoek peak temperature Tm thenfollows from (2.4) for 2πf · τ = 1:

Tm= H/{R ln[πa2

where f is the imposed frequency of mechanical vibrations.

16 2 Anelastic Relaxation Mechanisms of Internal Friction

Some peak temperatures Tm, determined experimentally for f = 1 Hz, are

listed in Table 2.3 Since the respective diffusion characteristics differ

signifi-cantly, the Tmvalues are also rather different If different IA are dissolved inthe same metal, their Snoek peaks sometimes overlap, as seen for C and O in

V, 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 been

recorded, the values of D0 (and thus τ0) are rather similar Therefore, there

is a reliable linear dependence of Tm on H (Fig 2.3), which corresponds to

τ0 = 2.08 · 10 −15 s (quite similar for all solutions) and Tm (K) = 3.765 H

(kJ mol−1) (Weller 1985)

Orientation Dependence

The relaxation strength ∆ and the peak maximum Qm−1 depend on the

dir-ection of the stress applied to the crystal lattice according to the seldir-ectionrules and the Snoek relaxation mechanism described earlier This directiondetermines the change in the energy of a dissolved IA in the interstices ofdifferent sublattices if an external stress is applied This effect is described by

the orientation parameter Γ :

Γ = cos α1· cos α2+ cos α1· cos α3+ cos α2· cos α3 (2.7)

where α1, α2, α3 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)

Qm−1 = ηC0· V · (λ2− λ1)2· F (Γ ) · M/(RTm) (2.8)

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

18 2 Anelastic Relaxation Mechanisms of Internal Friction

where C0 is the atomic fraction of interstitial atoms in the solution, V is

the volume of one mole of the host metal In case of torsional vibration, the

parameter η = 2/3, and M = G, F (Γ ) = Γ ; for flexural vibration, η = 1/9,

M = E, F (Γ ) = 1–3Γ This results in different orientation dependences for

different kinds of deformation

In case of flexure the extending and compressing stresses both are applied

along the [100] direction, cos α1= 1, cos α2= cos α3= 0, Γ = 0, thus the value

Qm−1 is maximal (Table 2.4 and Fig 2.4) The most prominent differences

observed are the ones between the IA energies in the octahedral interstices of

sublattices p = 1 and p = 2, 3 with the maximum number of atoms passing

from one sublattice to another If the stress is applied along 1 =

cos α2 = cos α3 = 1/ √

3, Γ = 3 and Qm−1 = 0 In this case, octahedral

interstices of all three sublattices are deformed similarly and “diffusion understress” is absent In case of the 1 = cos α2 = 1/ √

2,

cos α3= 0, Γ = 1/3 and Qm−1 has an intermediate value Similar orientation

dependences are observed for the longitudinal vibrations For torsionial and

relaxations in several bcc metals (Weller 1985)

Table 2.4 Orientation dependence of the carbon Snoek peak height of iron

monocrystals (Ino and Inokuti 1972)

Fig 2.4 Orientation dependence of the carbon Snoek maximum for iron single

110 Type of vibrations: 1, 3, 5: flexure, f = 1.15÷1.18 Hz; 2, 4, 6: torsion, f = 3.25–

3.46 Hz

transversal vibrations, the

effect, the

For polycrystalline samples, averaging over all grain orientations gives

Γ ≈ 0.2 (Nowick and Berry 1972) From (2.8) one can obtain for torsional

vibrations

Qm−1 = (0.4G/3) · [C0V · (λ2− λ1)2/(RTm)], (2.9)and for flexural vibrations

Qm−1 = (0.4E/9) · [C0V · (λ2− λ1)2/(RTm)]. (2.10)

For metals, the Poisson ratio µ ≈ 0.3 (Livshiz et al 1980) and therefore

G = 0.5·E/(µ+1) ≈ 0.4·E, and then the height of the Snoek peak almost does

not 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 leads

to a dependence of Qm−1 on the type of oscillations and on the direction of

applied stress (Fig 2.5)

Concentration Dependence

A linear dependence of the Snoek peak height on the dissolved element tration follows from (2.8) It is more prominent for a larger difference|λ2−λ1|,

concen-i.e., for a higher asymmetry of distortions caused by the dissolved IA Such

a linear dependence for Fe–C and Fe–N is shown in Fig 2.6, for Nb–O in

20 2 Anelastic Relaxation Mechanisms of Internal Friction

Fig 2.5 Snoek carbon peaks for iron samples cut parallel to the rolling direction

Fig 2.6 Concentration dependence (wt%) of the carbon (1) and nitrogen (2) Snoek

Fig 2.7 The Fe samples were quenched from temperatures, at which the total

C or N concentration in iron was within solubility in theα-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 thanα-Fe (up

to 0.5–1 at.% at room temperature), the linear dependence is observed up to

∼0.3 at.% (Weller et al 1981c; Heulin 1985a): in the system Nb–O up to 0.35

at.%, and in the system Nb–N 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 (Schulze

et al 1981)

A deviation from the linear dependence, with a weaker increase in Qm−1

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 significant

part does not contribute to the main Snoek peak as isolated atoms, thus Qm−1

increases less strongly with concentration The existence of pairs and triplets

of IA is questioned in some papers (Weller et al 1981b, 1985), while otherpublications (Cost and Stanley 1985; Heulin 1985a; Gibala 1985) confirm theformation of such complexes The influence of IA concentration on the Snoekpeak shape can be explained by IA long-range interaction It was analysed

by simulation of IA short-range order and its influence on relaxation (Weller

et al (1992); Haneczok et al 1992, 1993; Blanter and Fradkov 1992; Haneczok1998; Blanter and Magalas 2003)

The slope of Qm−1 as a function of C0 (increase per unit concentration

of IA) is determined mainly by the value (λ1− λ2), i.e., by the level of

dis-tortions created in the crystal lattice by a dissolved IA The values λ1 and

22 2 Anelastic Relaxation Mechanisms of Internal Friction

λ2may 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 according

to (2.3), as it is done for C and N in iron using the martensite:

λ1= a −10 dax/dn1, (2.11)

λ2= a −10 daz/dn2, (2.12)

where z is the tetragonal axis.

(b) The difference (λ1− λ2) can be determined by the concentration

depen-dence of Qm−1 from (2.8), and the value (2λ1+ λ2) by the lattice ter dependence on concentration in the non-ordered solution According

parame-to such a method, λ1 and λ2 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 λ1and λ2because of low solubility

It was shown (Blanter 1985) that the tetragonality factor ξ = λ1/λ2≈ −0.1

for all interstitial solid solutions in octahedral interstices of the bcc lattice

The λ2 value is higher if the difference between the sizes of the dissolved IA

and the octahedral interstice is bigger, λ2is lower in case of stronger chemicalinteraction between the dissolved IA and host metal atoms The calculated

values λ1 and λ2for 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 (λ2)increases in the sequence O→ N → C, and while moving upwards along every

periodic table group in the sequence Ta→ Nb → V and W → Mo → Cr for

each dissolved element This is due to the increase in size incompatibility,caused in the first case by the increase in the dissolved atom diameter and inthe second case by the decrease in metal lattice spacings, and to weakening ofthe metal–metalloid 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|λ2− λ1| from Table 2.5 into (2.8), the values Qm−1per 1

at.% 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 significantlyhigher The Snoek maxima are higher in the VIB group metals and in α-Fethan in the VB group metals at similar concentrations due to higher crystallattice distortions by dissolved IA

Table 2.5 Theλ-tensor components for interstitial atoms in octahedral interstices

N −0.14 0.69 0.83 3 Blanter and Khachaturyan (1978)

O −0.10 0.66 0.76 3 Blanter and Khachaturyan (1978)

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 λ1 and λ2 or their difference were determined: 1 – by thedependence of the lattice parameters on concentration in the martensite; 2 – by the

the lattice parameter concentration dependence and 4 – by calculation

Temperature Dependence

The height of the Snoek peak Qm−1 is proportional to 1/Tm (2.8), andtherefore an increase in vibration frequency is associated with an increase

of Tm and decrease of Qm−1 for the same concentration of IA In some cases

Qm−1 ∝ (Tm− T0)−1 (Nowick and Berry 1972), where T0is the ordering

tem-perature of IA in a solid solution However, T0  Tm for low concentration

of IA, and the value T0 can be omitted The relation Qm−1 ∝ 1/Tm is welldocumented for Ta – 0.074 at.% O (Fig 2.8; Weller et al 1981b)

24 2 Anelastic Relaxation Mechanisms of Internal Friction

Fig 2.8 Temperature dependence of the oxygen Snoek peak height in Ta with

0.074 at.% (Weller et al 1981b)

Grain Size Dependence

With decreasing grain size of a polycrystal, the Snoek peak height decreases

The coefficient k in the formula Qm−1 = kC0, where C0 is the concentration

in wt%, is a function of grain size (Ahmad and Szkopiak 1972):

decreas-grain sizes (Ferro-Milone and Mezzetti 1975) The data for Qm−1in Table 2.3

are obtained mainly for coarse-grained polycrystalline specimens without anywell defined texture However, in some papers on the Snoek relaxation theexistence of texture was not checked and the grain size was not given

Influence of Alloying Elements:

The Snoek-Type Relaxation in Ternary Alloys (Me–SA–IA)

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

Table 2.6 Influence of grain size on the Snoek maximum height in α-Fe Milone and Mezzetti 1975)

The Snoek-type relaxation in alloys can be explained in many cases on thebasis of the SA–IA interaction in the crystal lattice, which leads to a change

in the diffusion 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 alwaysaffect 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 SA–IA interac-tion, which may vary from a short-range type in case of a strong ‘chemical’interaction to a long-range ‘elastic’ interaction, affects the Snoek relaxation.The critical concentration at which alloys should be considered as concen-trated depends on the range of SA–IA interaction in the host lattice and isdifferent for different alloys