Internal friction in metallic materials : a handbook 9783540687580, 3540687580, 9783540687573, 3540687572

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Springer Series in

materials science

90

Springer Series in

materials science Editors: R. Hull

R. M. Osgood, Jr.

J. Parisi

H. Warlimont

The Springer Series in Materials Science covers the complete spectrum of materials physics, including fundamental principles, physical properties, materials theory and design. Recognizing the increasing importance of materials science in future device technologies, the book titles in this series ref lect the state-of-the-art in understanding and controlling the structure and properties of all important classes of materials. 71 Dissipative Phenomena in Condensed Matter Some Applications By S. Dattagupta and S. Puri 72 Predictive Simulation of Semiconductor Processing Status and Challenges Editors: J. Dabrowski and E.R. Weber 73 SiC Power Materials Devices and Applications Editor: Z.C. Feng 74 Plastic Deformation in Nanocrystalline Materials By M.Yu. Gutkin and I.A. Ovid’ko 75 Wafer Bonding Applications and Technology Editors: M. Alexe and U. G¨osele 76 Spirally Anisotropic Composites By G.E. Freger, V.N. Kestelman, and D.G. Freger 77 Impurities Confined in Quantum Structures By P.O. Holtz and Q.X. Zhao

81 Metallopolymer Nanocomposites By A.D. Pomogailo and V.N. Kestelman 82 Plastics for Corrosion Inhibition By V.A. Goldade, L.S. Pinchuk, A.V. Makarevich and V.N. Kestelman 83 Spectroscopic Properties of Rare Earths in Optical Materials Editors: G. Liu and B. Jacquier 84 Hartree–Fock–Slater Method for Materials Science The DV–X Alpha Method for Design and Characterization of Materials Editors: H. Adachi, T. Mukoyama, and J. Kawai 85 Lifetime Spectroscopy A Method of Defect Characterization in Silicon for Photovoltaic Applications By S. Rein 86 Wide-Gap Chalcopyrites Editors: S. Siebentritt and U. Rau 87 Micro- and Nanostructured Glasses By D. H¨ulsenberg and A. Harnisch

78 Macromolecular Nanostructured Materials Editors: N. Ueyama and A. Harada

88 Introduction to Wave Scattering, Localization and Mesoscopic Phenomena By P. Sheng

79 Magnetism and Structure in Functional Materials Editors: A. Planes, L. Ma˜nosa, and A. Saxena

89 Magneto-Science Magnetic Field Effects on Materials: Fundamentals and Applications Editors: M. Yamaguchi and Y. Tanimoto

80 Micro- and Macro-Properties of Solids Thermal, Mechanical and Dielectric Properties By D.B. Sirdeshmukh, L. Sirdeshmukh, and K.G. Subhadra

90 Internal Friction in Metallic Materials A Handbook By M.S. Blanter, I.S. Golovin, H. Neuh¨auser, and H.-R. Sinning

M.S. Blanter I.S. Golovin H. Neuh¨auser H.-R. Sinning

Internal Friction in Metallic Materials A Handbook

With 65 Figures and 53 Tables

123

Professor Dr. Mikhail S. Blanter

Professor Dr. Igor S. Golovin

Moscow State University of Instrumental Engineering and Information Science Stromynka 20, 107846, Moscow, Russia E-mail: [email protected]

Physics of Metals and Materials Science Department Tula State University Lenin ave. 92, 300600 Tula, Russia E-mail: [email protected]

Professor Dr. Hartmut Neuh¨auser

Professor Dr. Hans-Rainer Sinning

Institut f¨ur Physik der Kondensierten Materie Technische Universit¨at Braunschweig Mendelssohnstr. 3 38106 Braunschweig, Germany E-mail: [email protected]

Institut f¨ur Werkstoffe Technische Universit¨at Braunschweig Langer Kamp 8 38106 Braunschweig, Germany E-mail: [email protected]

Series Editors:

Professor Robert Hull

Professor Jürgen Parisi

University of Virginia Dept. of Materials Science and Engineering Thornton Hall Charlottesville, VA 22903-2442, USA

Universit¨at Oldenburg, Fachbereich Physik Abt. Energie- und Halbleiterforschung Carl-von-Ossietzky-Strasse 9–11 26129 Oldenburg, Germany

Professor R. M. Osgood, Jr.

Professor Hans Warlimont

Microelectronics Science Laboratory Department of Electrical Engineering Columbia University Seeley W. Mudd Building New York, NY 10027, USA

Institut f¨ur Festk¨orperund Werkstofforschung, Helmholtzstrasse 20 01069 Dresden, Germany

ISSN 0933-033X ISBN-10 3-540-68757-2 Springer Berlin Heidelberg New York ISBN-13 978-3-540-68757-3 Springer Berlin Heidelberg New York Library of Congress Control Number: 2006938675 All rights reserved. No part of this book may be reproduced in any form, by photostat, microfilm, retrieval system, or any other means, without the written permission of Kodansha Ltd. (except in the case of brief quotation for criticism or review.) This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specif ically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microf ilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media. springeronline.com © Springer-Verlag Berlin Heidelberg 2007 The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specif ic statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. A X macro package Typesetting: Data prepared by SPi using a Springer LT E Cover: eStudio Calamar Steinen

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Preface

Internal friction and anelastic relaxation form the core of the mechanical spectroscopy method, widely used in solid-state physics, physical metallurgy and materials science to study structural defects and their mobility, transport phenomena and phase transformations in solids. From the view-point of Mechanical Engineering, internal friction is responsible for the damping properties of materials, including applications of high damping (vibration and noise reduction) 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) contain unique microscopic information that cannot be obtained by other methods. On the other hand, owing to the large variety of phenomena, materials, and related microscopic models, a correct interpretation of measured internal friction spectra is often difficult. An efficient use of mechanical spectroscopy may then require both: (a) a systematic treatment of the different mechanisms of internal friction and anelastic relaxation, and (b) a comprehensive compilation 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 than half 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), the second requirement was met only by one Russian reference book (Blanter and Piguzov 1991), with no real equivalent in the international literature. The present book, partly based on the Russian example, is intended to fill this gap by providing readers with comprehensive information about published experimental results on internal friction in metallic materials. According to this objective, this handbook mainly consists of tables where detailed internal friction data are combined with specifications of relaxation 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

VIII

Preface

internal friction mechanisms are briefly reviewed in Chaps. 2 (Anelastic Relaxation) and 3 (Other Mechanisms). Although somewhat more space is obviously devoted to the former than to the latter, this part should not be understood as a systematic analysis of the physical sources of anelasticity and damping; in that respect, the reader is referred e.g., to the above-mentioned textbooks. The data collection itself, as the main subject of the book, can be found in Chaps. 4 and 5. The tables, generally in order of chemical composition, include the main properties of all known relaxation peaks (like frequency, peak height and temperature, activation parameters), the relaxation mechanisms as suggested by the original authors, and additional information about experimental conditions. 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 and alloys, is divided into subsections according to the group of the main metallic element in the periodic table, with alphabetic order within each subsection. Chapter 5 contains several new types of metallic materials with specific structures, which do not fit well into the general scheme of Chap. 4. A short summary or specific explanations are included at the beginning of each table. Although the authors made all efforts to be consistent in style throughout the book, some difficulties in evaluating individual relaxation spectra led to slight deviations, concerning details of data presentation, between the different chapters and subsections. Since some of the data were evaluated from figures, the accuracy should generally be regarded with care; in cases of doubt, the original papers should be consulted. Over 2000 references published until mid 2006 were included, among which many earlier ones are still important because certain alloys and effects are not covered by the more recent literature. Latest information, if missing in this book, might be found in three conference proceedings published in the second half of 2006 (Mizubayashi et al. 2006b, Igata and Takeuchi 2006, Darinskii and Magalas 2006), as well as in forthcoming continuations of these conference series. This book is intended for students, researchers and engineers working in solid-state physics, materials science or mechanical engineering. From one side, 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 the field. From the other side, its probably most comprehensive data collection ever published on this topic should also be attractive for top specialists and experienced researchers in mechanical spectroscopy and anelasticity of solids. The authors acknowledge gratefully the help of Ms. Tatiana Sazonova with the list of references, of Ms. Brigitte Brust with figures, and of Ms. Svetlana Golovina with tables. We are also grateful to the Springer team, in particular Dr. Claus Ascheron, Ms. Adelheid Duhm and Ms. Nandini Loganathan, for good cooperation. Moscow, Tula, Braunschweig January 2007

Mikhail S. Blanter, Igor S. Golovin Hartmut Neuh¨ auser, Hans-Rainer Sinning

Contents

1

Introduction to Internal Friction: Terms and Definitions . . . 1.1 General Phenomenon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Types of Mechanical Behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Anelastic Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Thermal Activation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Other Types of Internal Friction . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Measurement of Internal Friction . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 2 3 5 7 8

2

Anelastic Relaxation Mechanisms of Internal Friction . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Point Defect Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 The Snoek Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Relaxation due to Foreign Interstitial Atoms (C, N, O) in fcc and Hexagonal Metals . . . . . . . . . . . . . . . 2.2.3 The Zener Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.4 Anelastic Relaxation due to Hydrogen . . . . . . . . . . . . . . . 2.2.5 Other Kinds of Point-Defect Relaxation . . . . . . . . . . . . . . 2.3 Dislocation Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Intrinsic Dislocation Relaxation Mechanisms: Bordoni and Niblett–Wilks Peaks . . . . . . . . . . . . . . . . . . . 2.3.2 Coupling of Dislocations and Point Defects: Hasiguti and Snoek–K¨ oster Peaks and DislocationEnhanced Snoek Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Other Kinds of Dislocation Relaxation . . . . . . . . . . . . . . . 2.4 Interface Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Grain Boundary Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Twin Boundary Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.3 Nanocrystalline Metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Thermoelastic Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11 11 11 12 28 32 36 48 50 51

61 73 77 78 82 83 87 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) ES ) which explains the observed rate of the GB relaxation.

82

2 Anelastic Relaxation Mechanisms of Internal Friction

Table 2.17. Parameters of grain-boundary maxima in some pure metals (f = 1 Hz) (after Ashmarin 1991) Me

peak

Tm (K)

Tm /Tmelt

H (kJ mol−1 )

Al

A C A B C A B C

480–610 795 473–638 703–820 925–1025 670–783 843–893 953–1075

0.5–0.65 0.85 0.35–0.47 0.52–0.60 0.68–0.76 0.39–0.45 0.49–0.52 0.55–0.62

117–160 294 113–168 189–210 202–263 185–294 217–246 260–328

Cu

Ni

A: low temperature peak; B: medium temperature peak; C: high temperature peak.

In pure metals three GB-induced damping maxima can exist (Ashmarin 1991): (a) a low-temperature peak with Tm ≈ (0.35–0.65)Tmelt (sometimes called “Kˆe peak”), (b) a medium-temperature peak with Tm ≈ (0.5–0.6)Tmelt associated with special grain boundaries but not observed in all metals, and (c) a high-temperature peak with Tm ≈ (0.55–0.85)Tmelt observed in coarsegrained samples. Examples of these peaks are displayed in Table 2.17. In low-concentrated substitutional solid solutions we may distinguish the lowtemperature peak (at the same temperature as in a pure metal) from an additional peak at higher temperature (so-called impurity grain-boundary peak), which might be connected with the aforementioned change in the ratecontrolling mechanism. The role of grain-boundary relaxation may become dominant in materials with extremely fine grains, where the GB regions constitute a substantial part of the total sample volume. These nanocrystalline materials (produced e.g., by extreme plastic deformation), with GB structures mostly far from equilibrium and particular mechanical properties, may require special model descriptions of deformation and anelastic relaxation beyond those mentioned earlier, and will be considered separately in Sect. 2.4.3. 2.4.2 Twin Boundary Relaxation A twin boundary is a very special type of grain boundary, separating two “twin” crystallites that are, with respect to their lattice, mirror images of each other (which is possible only at a well-defined misorientation angle). If the twin boundary is identical with the mirror plane, usually a low-indexed, close-packed crystallographic plane, it is called a coherent twin boundary. Since the twin crystals can be transformed into each other by a shear transformation parallel to the mirror plane, the formation of twins (“twinning”), which may occur under sufficiently high stress or during recrystallisation (in metals and alloys of low stacking-fault energy), represents an additional deformation mechanism. Also the perpendicular shift of a twin boundary (growth of one twin at the expense of the other) means a shear deformation.

2.4 Interface Relaxation

83

For anelastic relaxation and internal friction peaks to occur by stressinduced movement of twin boundaries, these boundaries must be sufficiently mobile. As crystallographic coherency exists across the twin interfaces, the relaxation mechanism cannot involve interfacial sliding (Nowick and Berry 1972). However, certain types of twin boundaries can be shifted as the result of movement of partial dislocations (Hirth and Lothe 1968); then, the corresponding dislocation mechanisms will be involved in twin boundary relaxation. Examples for the few existing experiments are those by Siefert and Worrell (1951) on Mn–12at%Cu, De Morton (1969) and Postnikov et al. (1968b, 1969, 1970) on In–Tl alloys. Twinning is most frequently accompanying diffusionless phase transformations (e.g., from cubic to tetragonal structure), which themselves involve a shear that can be accommodated by twinning in order to retain the external shape and to avoid high residual stresses in the sample. The high density of twin boundaries often produced in this case may give rise to large effects of anelastic relaxation and internal friction. The relation to martensitic transformations will be treated in Sect. 3.2.1. 2.4.3 Nanocrystalline Metals Subject of this section are polycrystals with ultra-fine grain sizes in the nanometer range. Such nanocrystalline materials form a special group of nanostructured materials (or “nanomaterials”) which also include other types of “nanosized” structures in one, two or three dimensions. Owing to the extremely rapid development of the field, a generally accepted terminology of nanomaterials has not yet been fully established. From the viewpoint of materials science, nanostructured materials may be classified into different groups according to the shape (dimensionality) and chemical composition of their constituent structural elements (Gleiter 1995, 2000); however, a less precise, synonymous use of terms like “nanocrystalline”, “nanostructured,” or “nanophase” is also found in the literature. There is also no commonly agreed grain size limit to define nanocrystalline materials. In the physical concept of highly disordered solids, it is the fraction of atoms situated in the cores of defects (grain boundaries, interfaces) which should be as high as possible. Under this viewpoint the grain size should be below 10 nm (Gleiter 1989, 2000), but also limiting values of 15, 20 or 30 nm have been mentioned. Engineers developing new materials, on the other hand, are sometimes using the prefix “nano” for length scales almost up to 1 µm, which generally lacks a physical justification. An application-oriented delimitation of “nanocrystalline” grain sizes should rather be linked to specific properties, which are expected to be different from those of conventional materials if dominated by the high density of grain boundaries. In some recent reviews, an upper limit of about 100 nm is introduced (Tjong and Chen 2004, Suryanarayana 2005), which seems to be a reasonable compromise.

84

2 Anelastic Relaxation Mechanisms of Internal Friction

Many different methods and techniques have been employed to produce nanocrystalline metallic materials (n-Me), like inert gas condensation, mechanical alloying, severe plastic deformation, devitrification of amorphous precursors and many others. They all have their specific advantages or drawbacks concerning the compositions, properties or shapes (e.g., porous or fully dense, bulk or thin film) of materials produced; for example, the amorphous route is well established to produce nanocrystalline soft magnetic alloys since the successful development of FINEMET (Yoshizawa et al. 1988). With respect to improved mechanical properties, severe plastic deformation (SPD) is one of the most important and widely used routes, as bulk and fully dense materials with ultra-fine grain (UFG) structure can be obtained in this way (Valiev et al. 2000, Mulyukov and Pshenichnyuk 2003). The surprisingly high temperature stability of such UFG structures has been attributed to a high GB diffusivity and low driving force of recrystallisation (Valiev 2002). In many cases it is not clear, however, whether there is a significant difference in GB diffusivity between the nanocrystalline and annealed states (Kolobov et al. 2001) or not (W¨ urschum et al. 2002). The small grain sizes of genuine n-Me lead to distinct changes in the mechanical properties including increased yield strength and hardness. A particular feature is the breakdown of the Hall–Petch relation at grain sizes around 20 nm or below. In this range, a decreasing grain size leads to anomalous softening, referred to as inverse Hall–Petch behaviour, which is associated with the operation of diffusion-controlled mechanisms combined with GB sliding (e.g., Schiøtz et al. 1998, 1999, Yamakov et al. 2002a,b, Van Swygenhoven 2002, Van Swygenhoven et al. 2003). The cross-over from normal to inverse Hall–Petch behaviour has been treated in a ‘two-phase’ model (Kim et al. 2000, Kim and Estrin 2005), in which the grain boundaries deform by a diffusion mechanism, and the grain interiors by a combination of dislocation glide and diffusion-controlled mechanisms. Anelastic grain-boundary relaxation (Kˆe 1999) is considered, in a recent theory of non-equilibrium GBs (Chuvildeev 2004), to be hardly detectable in UFG metals below a certain temperature (∼0.35Tmelt ), unless the dislocation density at the GBs is decreased. Alternatively, disclination concepts have also been discussed in connection with relaxation processes in n-Me (Romanov 2002, 2003). The conditions for GB relaxation are even less clear in multicomponent n-Me produced by the amorphous route, where the same factors which favour glass formation may also lead to stabilised and more densely packed GB structures, being less susceptible to relaxation. This latter type of n-Me (for which, to our knowledge, no systematic studies of GB relaxation exist) is not included in the following considerations, but will be mentioned further below in connection with the respective amorphous alloys (Sect. 2.6.1). Experimental studies of anelastic properties of n-Me were undertaken by several research groups using different mechanical spectroscopy techniques. The main results of some systematic and therefore reliable studies, including the materials studied and the main effects observed, are summarised briefly in

2.4 Interface Relaxation

85

Table 2.18. In most of these cases, the total temperature-dependent internal friction Q−1 (T ) can be written as Q−1 (T ) = Qb−1 (T ) + ΣQ−1 r (T ), Q−1 b (T )

(2.46)

ΣQ−1 r (T )

where the terms represent a “background” of and internal friction and a superposition of different anelastic relaxation peaks, respectively. The first term, closely related to composition and microstructure of the respective alloy phases as well as to the dislocation structure, can depend on the annealing time, as well on the amplitude and frequency of the imposed oscillatory strain. This contribution was reported to be substantial enough to consider nanostructured Cu (Mulyukov and Pshenichnyuk 2003) and Mg (reinforced by different microparticles, Trojanova et al. 2004) as high-damping materials (see Sect. 3.5) even for low vibration amplitudes. The second term, which may contain contributions not only from GB relaxation but also from almost all relaxation mechanisms related to dislocations or point defects, is time-independent but frequency-dependent and can often be described by the Debye equation. Because of the lack of a combined study of nanostructured metals by different mechanical spectroscopy techniques, varying as many experimental parameters (frequency, temperature, amplitude, annealing conditions) as possible, it is not easy to distinguish between “pure” anelastic relaxation mechanisms (most important: GB relaxation) and irreversible mechanisms of structural relaxation, which are in most cases due to changes in the density and distribution of dislocations. Summarising the pertinent results published in the literature (partly presented in Table 2.18), one can draw the following conclusions: – Almost all UFG and nanostructured metals (except those produced from the amorphous state, see above) exhibit an IF peak (very roughly with an activation energy of about 1 eV), which is not often found in well annealed (coarse-grained) metals. – The nature of this peak is still not entirely clear: some authors report a thermally activated, reversible anelastic GB relaxation consistent with the Kˆe approach (Kˆe 1999), while others attribute the effect to irreversible structural changes like recovery and recrystallisation, connected with short-range GB diffusion in non-equilibrium GB. – Internal friction can generally be correlated with superplastic properties and thus can be used for determining the optimum temperature for superplastic deformation. – Structural changes in severely deformed metals occur already around ambient temperature, as indicated by a group of low-temperature IF peaks observed after high-pressure torsion in Ti, Mg and several Fe-based alloys, which is extremely sensitive to heating. – In some SPD-processed metals like Cu, a high damping capacity was reported in a broad range of strain amplitude and temperature, however, has not been reproduced all published works (see Sect. 3.5).

86

2 Anelastic Relaxation Mechanisms of Internal Friction Table 2.18. Mechanical spectroscopy studies of UFG metallic materials

materials

∗

mechanical spectroscopy – short summary

references

Pd

1

Weller et al. (1991)

Cu: 99.997% 99.98%

2

Au 99.99%

3

Al, Ni

4

Cu (99.98%) Ni (99.98%)

2

Cu, Fe18Cr9Ni

5

TDIF, 1–5 Hz. Several IF peaks. IF peak H = 62.7 kJ mol−1 (reordering phenomena) TDIF at ∼10 Hz and ∼100 kHz; TD- and ADIF ∼5 MHz. IF peak ∼420 K: reversible dynamic GB rearrangement TDIF, 300–500 Hz: Bordoni peak ∼120 K, dislocation peak ∼460 K, GB peak ∼750 K. TDIF, 1–30 Hz , 0.01–200 Hz, 0.1–10 kHz. Relaxation IF peak 159 kJ mol−1 (Al: 475 K 1 Hz, GB relaxation) TDIF, 1–7 Hz and ∼1 kHz (time-dependent IF). Irreversible changes in the structure TDIF, ∼2.5 Hz, ADIF, ∼35 Hz. High damping; IF peaks at 54 and 475 K

Mg

6

Mg alloys: Mg–6Zn Mg–9Al Fe–25Ni

2

7

Fe–0.8C

5

Fe–25Al

5

Ti grad2

5

∗

TDIF, 0.5, 5, 50 Hz. Relaxation peaks at ∼70 K (116 kJ mol−1 due to dislocations) and at ∼620 K due to GB TDIF, ∼10 Hz. Irreversible IF peak 530–570 K, ∼87 kJ mol−1 enhanced GB diffusion TDIF, 0.5–5 Hz. IF peaks due to martensitic transformation TDIF, 1–2 kHz. Irreversible IF peak ∼550 K: recovery TDIF, 0.5–2 kHz. IF peaks (150–300 K) due to dislocations and self-interstitial atoms; unstable with respect to heating. TDIF, ∼2 kHz. IF (Hasiguti) peak ∼210 K: dislocations and self-interstitial atoms; possibly hydrogen-related effect at ∼410 K.

Akhmadeev et al. (1993)

Okuda et al. (1994) Bonetti and Pasquini (1999) Gryaznov et al. (1999) Mulyukov and Pshenichnyuk (2003) Trojanova et al. (2004)

Chuvildeev et al. (2004a) Wang et al. (2004a) Ivanisenko et al. 2004 Golovin et al. (2006a,c)

Golovin et al. (2006a)

Fabrication methods: (1) evaporation, condensation, compaction; (2) equal channel angular pressing (ECAP); (3) gas deposition; (4) mall milling; (5) high-pressure torsion; (6) ball milling, compaction, hot extrusion; (7) consolidation.

2.5 Thermoelastic Relaxation

87

2.5 Thermoelastic Relaxation 2.5.1 Theory Physical Principle In every solid, there exists a fundamental thermoelastic coupling between the thermal and mechanical states (e.g., between stress and temperature fields), with the thermal expansion coefficient α as the coupling constant. The best known phenomenon of thermoelastic coupling is thermal expansion, as the response of the mechanical state to an applied change in temperature. Conversely, fast adiabatic (i.e., isentropic) changes of the dilatational stress component result in (small) temperature changes, known as the thermoelastic effect. If such stress variations are spatially inhomogeneous – either externally according to the mode of loading (e.g., bending) or internally in a material with heterogeneous mechanical properties – temperature gradients are produced which can then relax by irreversible heat flow (thermoelastic relaxation), causing entropy production and dissipation of mechanical energy. The resulting thermoelastic damping 1 – not to be confused with damping due to “thermoelastic” martensite2 (Sect. 3.2.1) – represents the most fundamental among all mechanical damping mechanisms, since it does not require any defects but exists in all solids with non-zero thermal expansion, even in the most perfect crystals. Assuming that the mean free path of the phonons is small compared to the length scale of the stress inhomogeneities, which is generally the case except for very low temperatures and high frequencies, the heat flow during thermoelastic relaxation can be described as a classical diffusion process. Biot (1956) pointed out that it is the entropy which satisfies the diffusion equation. Zener’s Theory Thermoelastic damping is known since the late 1930s, when Zener was the first to give both a detailed theory (Zener 1937, 1938b) and a collection of related experimental results (Zener et al. 1938, Randall et al. 1939). The theory was developed in scalar (one-dimensional) form mainly for the transversal vibration of homogeneous reeds and wires, but some other cases like spherical cavities or polycrystals with randomly oriented crystallites were also considered 1

2

As a fundamental thermodynamic phenomenon, thermoelastic damping is sometimes also referred to as “thermodynamic damping” (e.g., Panteliou and Dimarogonas 1997, 2000). Other authors have called it “elastothermodynamic” (Bishop and Kinra 1995, 1997; Kinra and Bishop 1996) because the cause is “elastic” and the effect is both “thermal” and “dynamic” (i.e., time-dependent). A martensitic transformation is called “thermoelastic” if its thermal hysteresis and transformation energy is relatively small, comparable in magnitude with usual elastic strain energies. This alternative use of the term “thermoelastic” has nothing to do with thermoelastic coupling considered here.

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2 Anelastic Relaxation Mechanisms of Internal Friction

by Zener. The simplest and best described case is certainly that of alternating transverse thermal currents (Nowick and Berry 1972) between the compressed and dilated sides of a homogeneous and isotropic, rectangular beam, vibrating in flexure with the frequency f . The thermoelastic damping of such a beam is in good approximation given by Q−1 (f, T ) = ∆T

f · f0 f 2 + f02

(2.47)

with the relaxation strength ∆T = α2 EU T /ρCp

(2.48)

f0 = πλ/2h2 ρCp ,

(2.49)

and the peak frequency where α is the linear thermal expansion coefficient, EU the unrelaxed Young’s modulus, ρ the density, Cp the specific heat capacity at constant pressure (or stress)3 , λ the thermal conductivity and h the thickness of the beam (i.e., the distance over which heat flow occurs). Equation (2.47) has the same functional form as (1.8) and represents a Debye peak as a function of frequency, with a single relaxation time τT = 1/2πf0 = h2 /π 2 Dth

(2.50)

where Dth = λ/ρCp is also called the thermal diffusivity. The analogy between (2.50) for the thermoelastic and (2.17) for the Gorsky relaxation, respectively, reflects the more general analogy between “thermal” and “atomic” diffusion already pointed out by Zener (1948). In the same way, the intercrystalline Gorsky effect introduced in (2.18) and (2.19) is analogous to the case of intercrystalline thermal currents (Zener 1948, Nowick and Berry 1972) with ∆IT = R(3α)2 KU T /ρCp τIT = d2 /3π 2 Dth ,

(2.51) (2.52)

where R is an elastic anisotropy factor (see Zener 1938b for an estimate for cubic metals with randomly oriented crystallites), 3α denotes the volumetric expansion coefficient, KU the unrelaxed bulk modulus and d the dominating grain size in the polycrystal. Despite this analogy between atomic and thermal diffusion, it should be noted that the Arrhenius relation of thermal activation, (1.9), only holds for 3

Here we understand Cp per unit mass as found in most data collections; if considered per unit volume as in Zener’s original equations, the density ρ does not appear in these equations. Instead of Cp , the symbol Cσ (for constant stress) has also been used in the literature. If, on the other hand, Cp or Cσ is replaced by Cν or Cε (at constant volume or strain), a small error in ∆T (of the order of ∆T 2 ) is introduced (Lifshitz and Roukes 2000).

2.5 Thermoelastic Relaxation

89

the former but not for the latter having a comparatively weak temperature dependence. Thus, unlike the Gorsky relaxation, the thermoelastic Debye peak is found only as a function of frequency but not of temperature. Instead, both ∆T /T in (2.48) and f0 in (2.49) are only weakly temperature-dependent (at least above the Debye temperature), so that thermoelastic damping is nearly proportional to the temperature. Another possibility of thermoelastic damping is related to longitudinal thermal currents between the “hills” and “valleys” of longitudinal elastic waves. In this case, treated in some detail by L¨ ucke (1956), the relaxation time is itself frequency-dependent as ω −2 because the thermal diffusion distance is given by half the wavelength, which means that in contrast to the normal case adiabatic conditions are expected here in the low-frequency (!) limit. With expected peak frequencies in the GHz range or even higher, longitudinal thermoelastic damping is usually negligibly small (Nowick and Berry 1972). Advanced Theories More extended and fundamental, three-dimensional and mathematically more rigorous treatments can be found in many later theoretical papers (e.g., Biot 1956, Alblas 1961, 1981, Chadwick 1962a,b, Lord and Shulman 1967b, Kinra and Milligan 1994, Lifshitz and Roukes 2000, Norris and Photiadis 2005). However, although the general thermoelastic equations and also some specific solutions (most often for the transversely vibrating Euler–Bernoulli beam) are now well known, it is up to the present date still difficult to calculate the thermoelastic damping explicitly for more complex cases beyond those already treated by Zener. An exact solution for the thin Euler–Bernoulli beam was given by Lifshitz and Roukes (2000), who also showed that Zener’s approximation is valid within 2% in most of the relevant frequency range, except for the highfrequency side of the peak far above f0 where the deviations grow up to a 20% underestimation in the limit f → ∞. Therefore, the still widely spread use of Zener’s (2.47)–(2.49) is sufficiently accurate for many practical purposes, at least in the classical case of transverse thermal currents during flexural vibration of homogeneous samples. The analysis of Kinra and Milligan (1994) formed the basis for further model calculations of thermoelastic damping also in heterogeneous structures like fibre- or particle-reinforced composites (Milligan and Kinra 1995, Bishop and Kinra 1995), hollow spherical inclusions (Kinra and Bishop 1997), laminated composite beams (Bishop and Kinra 1993, 1997; Srikar 2005b) or some specific cases of pores and cracks (Kinra and Bishop 1996, Panteliou and Dimarogonas 1997, 2000; Panteliou et al. 2001). In the special case of flexural resonators made of polycrystals (e.g., of silicon) with particularly low thermal conductivity across the grain boundaries compared to that in the crystals,

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2 Anelastic Relaxation Mechanisms of Internal Friction

a preliminary fast equilibration of the transverse thermal currents is possible inside the grains, which has been called intracrystalline thermoelastic damping (Srikar and Senturia 2002). Another branch of theories is devoted to resonators with more complex external shape, usually in form of planar structures made of thin, flat plates vibrating predominantly either in flexure or in torsion. Although the thermoelastic loss should be zero in case of pure shear, it is important to note that even the nominally torsional vibration modes almost always contain some flexural component which can produce significant thermoelastic damping. To solve this problem, a flexural “modal participation factor” (MPF) has been defined as the fraction of potential elastic energy stored in flexure (Photiadis et al. 2002, Houston et al. 2002, 2004). Assuming classical transverse thermal currents for this flexural component, the thermoelastic damping of any particular vibration mode is then obtained by multiplying the MPF with the classical result for the flexural beam e.g., from Zener’s theory. The MPF itself can be calculated by integrating the curvature tensor of the vibration mode over the volume of the sample, provided the displacement field of the mode is known (Norris and Photiadis 2005). The problem then mainly consists of determining the mode shape, e.g., with the help of finite element modelling and/or advanced experimental techniques like laser-Doppler vibrometry (Liu et al. 2001). 2.5.2 Properties and Applications of Thermoelastic Damping To judge the practical importance of thermoelastic damping in a given material, we have to consider primarily the magnitude of the transverse relaxation strength ∆T and the related peak frequency f0 according to (2.48) and (2.49). A detailed compilation of room-temperature relaxation strengths, including results of four data collections from the literature as well as re-calculated data using (2.48), is given in Table 2.19 for many pure metals and also a limited number of non-metallic materials. It is typical that in Table 2.19 the ∆T values taken from different sources never match exactly. This scatter may come from unspecified microstructural influences (defects, textures) causing some variation mainly in the possibly anisotropic quantities α and E, among which deviations in α have a particularly strong effect due to the quadratic dependence in (2.48). For our own re-calculations of ∆T , the underlying basic data were checked for reliability by comparing different sources wherever possible. Ideally, the data in Table 2.19 refer to random polycrystals at least in case of metals. Exceptions are Si and Ge where single crystal values are given, according to the [100]-oriented wafers from which most of the respective resonators are fabricated. The second practically important quantity is the peak frequency f0 or, vice versa, the sample thickness h(f0 ) which belongs to a pre-selected peak frequency according to (2.49). With thermal diffusivities Dth usually in the

6.0 32.1 6.4 25 (Weast 1973)

4.8

Mo 329

100 (Srikar 2005b) 11 528 45 10280

5323 7310 22650 1738

7874

251

321 233 131 1025

449

0.88

0.63 2.0 2.2 4.7

2.5

0.86

0.45 3.1 2.35 4.8–5.4

2.2–2.6

4.6 1.4 10 3.4 3.0–3.7

Ge In Ir Mg

211

3.3 1.44 7.1 2.8 3.1

11.8

1820 122 231 421 384

1850 9780 8650 8900 8920

Fe

287 32 50 209 130

11.3 13.4 30.8 13.0 16.5

4.6–5.1

Be Bi Cd Co Cu

129

19300 1.9

4.7

2.4–3.5

1.7–2.2

78

904

3.6

14.2

2700

235

∆T Lit × 103

Au

70

10490

Cp ρ ∆T calc −3 −1 −1 (kg m ) (J kg K ) × 103

2.7–3.7

23.1

Al

83

E (GPa)

Al2 O3

18.9

α (10−6 K−1 )

Ag

material

Zener (1948), Riehemann (1996), Srikar (2005b) Zener (1948), Kinra and Milligan (1994), Riehemann (1996), Srikar (2005b) Kinra and Milligan (1994), Srikar (2005b) Zener (1948), Kinra and Milligan (1994), Riehemann (1996), Srikar (2005b) Zener (1948) Zener (1948) Zener (1948) Riehemann (1996) Zener (1948), Riehemann (1996), Srikar 2005b Zener (1948), Riehemann (1996), Srikar 2005b Srikar 2005b Riehemann (1996) Riehemann (1996) Zener (1948), Kinra and Milligan (1994), Riehemann (1996) Riehemann (1996)

reference for ∆T Lit

Table 2.19. Thermoelastic relaxation strengths ∆T of pure metals and some other selected materials at 300 K: ∆T calc calculated from the intrinsic properties α, E, ρ and Cp , and ∆T Lit taken directly from the literature

2.5 Thermoelastic Relaxation 91

4.5

30.2

5.7

TiC W

Zn

Zr

7140 6510

68

19 250

7310 16 650 4507

11 340 12 023 21 090 12 450 6697 2330

8570 8908

ρ (kg m−3 )

108

411

50 186 116

16 121 168 275 55 160 (Srikar 2005b)

105 200

E (GPa)

278

388

132

227 140 522

127 244 133 243 207 712

265 445

0.37

10.7

0.98

4.4 0.95 1.1

2.8 1.72 1.39 1.8 1.44 0.2

0.74 2.7

Cp ∆T calc −1 −1 (J kg K ) × 103

0.68

5.8–18(!)

1.4 0.8–1.3

0.22 0.003 4.0–4.8 0.3 0.8–1.2

2.5–2.8 2.0–2.5 1.5 0.7 1.5–1.8 0.19 0.35–0.6

0.71 2.6–2.9

∆T Lit × 103

Riehemann (1996) Zener (1948), Riehemann (1996), Srikar (2005b) Zener (1948), Riehemann (1996) Zener (1948), Riehemann (1996) Zener (1948) Zener (1948) Zener (1948), Riehemann (1996) Srikar (2005b) Kinra and Milligan (1994), Srikar (2005b) Srikar (2005b) Srikar (2005b) Zener (1948), Riehemann (1996) Zener (1948) Kinra and Milligan (1994), Riehemann (1996),Srikar (2005b) Kinra and Milligan (1994) Zener (1948), Riehemann (1996), Srikar (2005b) Zener (1948), Riehemann (1996), Srikar (2005b) Riehemann (1996)

reference for ∆T Lit

References: unless noted otherwise, the intrinsic properties α, E, ρ and Cp were taken from WebElements [http://www. webelements.com/].

22.0 6.3 8.6

28.9 11.8 8.8 8.2 11.0 2.6

Pb Pd Pt Rh Sb Si SiC

Si3 N4 SiO2 Sn Ta Ti

7.3 13.4

α (10−6 K−1 )

Nb Ni

material

Table 2.19. Continued

92 2 Anelastic Relaxation Mechanisms of Internal Friction

2.5 Thermoelastic Relaxation

93

range of 10−6 to 10−4 m2 s−1 , samples have to be prepared mostly with thicknesses between 0.05 and 0.5 mm in order to have maximum thermoelastic damping at 1 kHz. Metallic Materials In Table 2.19 the strongest effect is predicted for Zn with a relaxation strength as high as 0.01 (according to ∆T calc ) and a maximum thermoelastic damping Qm −1 = ∆T /2 ≈ 0.005, followed by Cd, Al, Mg, Sn; but also for Ag, Be, Co, Fe, Ni and Pb the thermoelastic loss factor at room temperature can exceed 10−3 . Although such values are easily observable and practically significant, the interest in thermoelastic damping of metals has as yet been rather limited from both the fundamental and applied sides, and systematic experimental studies are very scarce. On the fundamental side, mechanical spectroscopy is usually concerned with thermally activated relaxation peaks, measured as a function of temperature to study defects and transformations in solids. Thermoelastic damping is then noticed mainly as a linear background to be subtracted, but very rarely studied for its own sake. This has also experimental reasons: to trace out the full peak after (2.47), flexural frequency and sample thickness have to be mutually adjusted and varied accordingly, e.g., over at least two orders of magnitude in frequency, which requires more effort than just varying the temperature on a single sample. In addition, to observe the pure thermoelastic losses, other kinds of damping have to be suppressed effectively e.g., by suitable alloying. Only in the early days – before many other mechanisms were known – thermoelastic damping in metals was a subject of intense study as a main source of energy dissipation. The probably still most careful measurements of the thermoelastic relaxation peak come from that time, like the study of Bennewitz and R¨otger (1938) on German silver, and in particular that of Berry (1955) on α-brass which gave an impressively exact confirmation of Zener’s theory of transverse thermal currents without any adjustable parameters (see also Nowick and Berry 1972). Based on this fundamental work, the height and position of the thermoelastic peak were occasionally used later to determine coefficients of thermal expansion and conductivity, respectively, e.g., for some metallic glasses (Berry 1978, Sinning et al. 1988) or commercial Al and Mg alloys (G¨ oken and Riehemann 2002). As an example, Fig. 2.33 shows the annealing-induced shift of the thermoelastic Debye peak, according to an increase in thermal conductivity from 7 over 11 to 17 W mK−1 , due to structural relaxation and subsequent crystallisation of an amorphous Ni alloy (Sinning et al. 1988). In this case, the measurement temperature had been lowered to 170 K to reduce the amount of other damping contributions, partly still visible in Fig. 2.33 on the low-frequency side of the peak for the as-quenched state; therefore, the thermoelastic peaks in Fig. 2.33 are almost a factor of two smaller than they would be at room temperature.

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2 Anelastic Relaxation Mechanisms of Internal Friction

Fig. 2.33. Frequency-dependent internal friction of a rapidly quenched, meltspun Ni78 Si8 B14 ribbon (thickness h = 0.05 mm) at T = 170 K after different annealing treatments (the solid lines are fits to (2.47)): (a) as-quenched amorphous state, f0 = 1200 Hz; (b) after 2 h at 618 K (structurally relaxed amorphous state), f0 = 1730 Hz; (c) crystallised, f0 = 2750 Hz (Sinning et al. 1988)

Also worth mentioning in this context is the early work of Randall et al. (1939) on α-brass with systematically varied grain sizes, which seems still to represent the only known example of a reliable observation of intercrystalline thermal currents. On the side of application, the main problem is that damping due to transverse thermal currents is available only in a relatively narrow frequency range around f0 , depending on the geometry of the respective structural component. On the other hand, it might be possible in certain cases to adjust the geometrical dimensions or the thermal conductivity (by alloying) according to the technical requirements of damping properties. Much more interesting from the applied viewpoint are those thermoelastic damping contributions that occur in heterogeneous metallic materials like composites or porous metals. Three types of effects may be expected from such heterogeneities: 1. The introduction of new internal length scales, in addition to the sample dimensions, will distribute the dissipation processes over a much wider frequency range. This effect has been discussed qualitatively for metallic foams (Golovin and Sinning 2003b, 2004). 2. Thermoelastic damping will no longer be confined to flexural vibrations but will occur also in other deformation modes. 3. Additional heterogeneities cause additional temperature gradients and thus additional dissipation processes, i.e., more damping will be produced. This is the most promising but also least understood aspect: in fact, model calculations for specific arrays of pores (Panteliou and Dimarogonas 1997, 2000) have predicted a strong increase of thermoelastic damping with porosity, up to much higher values than in the case of classical transverse thermal currents; but the consequences for real materials are not yet clear. There is a strong need for theoretical as well as experimental

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95

research in this field, which then might open new perspectives towards the development of heterogeneous metallic materials with tailored properties of thermoelastic damping. Applications in Microsystems The recently renewed interest in thermoelastic damping is, in its main part, not related to the aforementioned perspectives of metallic materials but has a completely different reason: the rapid development of micro- and nanoelectromechanical systems (MEMS and NEMS) which include silicon-based micromechanical resonators as central elements. Irrespective of the specific application (e.g., force sensors, accelerometers, bolometers, magnetometers, high-frequency mechanical filters or ultrafast actuators), the performance of the micromechanical system (e.g., sensor sensitivity) critically depends on the quality factor Q of the resonator which should be as high (i.e., the damping Q−1 as low) as possible. That is, contrary to the metallic case discussed earlier, the aim is here not to produce damping but to avoid it. If in the most perfect silicon resonators all defect-induced sources of dissipation are removed, the thermoelastic damping remains and can be influenced only by a proper geometrical design and fabrication of the resonator. Especially with more complex-shaped resonators like single- or double-paddle oscillators (Kleiman et al. 1985, Liu et al. 2001, Houston et al. 2004) attempting quality factors as high as 108 , or in case of layered structures including metallic or ceramic coatings (Srikar 2005b), this is not a trivial task. Since most of the recent theoretical progress on thermoelastic damping since about 2000 (see above) was without doubt strongly motivated by the needs of MEMS and NEMS, we have briefly sketched these important new developments here – although their basic material, silicon, is as a semiconductor not included in the main data collections of this book. Finally, it should be mentioned as well that thermoelastic damping is also an important factor limiting the ultimate sensitivity of interferometric gravitational wave detectors (Black et al. 2004).

2.6 Relaxation in Non-Crystalline and Complex Structures With the important exception of the universal thermoelastic damping treated in the preceding section, most mechanisms of anelastic relaxation comprise the motion of defects interacting with an applied stress. According to the classical understanding of defects as structural imperfections in (periodic) crystals, such relaxation mechanisms are traditionally defined for crystalline solids (Nowick and Berry 1972). This classical line was also followed in the Sects. 2.2–2.4 on point defects, dislocations and interfaces, where the respective microscopic processes of relaxation were introduced for the crystalline

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2 Anelastic Relaxation Mechanisms of Internal Friction

case. An extension of such defect-related mechanisms to non-crystalline structures is not obvious, except for some special cases like interstitial diffusion jumps of hydrogen atoms (if not coupled with the motion of matrix defects, see Sect. 2.2.4). In this context, the term “non-crystalline” is traditionally understood as opposed to periodic crystals, which then includes both amorphous solids and quasicrystals. To some extent this is still common practice (and practically useful), although it deviates from crystallographically correct terminology. In proper crystallographic terms, quasicrystals are in fact crystals in the wider sense of quasiperiodic crystals, which include both periodic and aperiodic, long-range ordered structures (Lifshitz 2003). From the viewpoint of anelastic relaxation of metals, on the other hand, quasicrystals and amorphous structures have many things in common, at least in case of icosahedral short-range order (cf. Sect. 2.2.4). There is a borderline, however, between common periodic crystals (in most practical cases with relatively simple crystal structures) on the one side, and other metallic structure types – amorphous alloys, quasicrystals and to some extent even structurally complex periodic crystals with giant unit cells (Urban and Feuerbacher 2004) – on the other side: in the former case, most defect-related mechanisms are quantitatively well understood and classified within the systematic and wellfounded concepts of “anelastic relaxation in crystalline solids” (Nowick and Berry 1972; at that time “crystals” were always understood as periodic crystals), whereas in the latter case many details of the theoretical concepts have still to be developed. In principle we may distinguish roughly, in relation to the classical relaxation processes in crystalline solids, between three types of relaxation mechanisms in “non-crystalline” structures (in the above “traditional” meaning including quasicrystals): (a) Mechanisms which are independent of the structure type and exist in the same way in crystalline as well as in non-crystalline structures, with only numerical differences. Examples are thermoelastic damping and the Gorsky effect (at least in the basic form of transverse thermal or atomic diffusion currents), where relaxation strength and time may vary according to the values of the respective parameters, but all essential characteristics of the relaxation remain unchanged. (b) Mechanisms which are modified by the structure type, i.e., which are based on the same principle but with some conceptual differences calling for a modified or extended theoretical treatment. Examples are the Snoek-type relaxation in the generalised form as introduced for hydrogen in Sect. 2.2.4, or a hypothetical dislocation relaxation in an amorphous structure which can only be treated using a more general dislocation concept (independent of a crystal lattice). (c) Mechanisms which are specifically found in “non-crystalline” but not in (simple) crystalline structures. Examples are cooperative processes of

2.6 Relaxation in Non-Crystalline and Complex Structures

97

directional structural relaxation or viscous flow (e.g., near the glass transition) in metallic glasses, or some types of relaxation related to phasons in quasicrystals. While there is no reason to mention again type (a), we will focus in the following on mechanisms of types (b) and in particular (c) which can not always be differentiated clearly from each other. The aim is to give an introduction into those aspects of anelastic/viscoelastic relaxation in amorphous (Sect. 2.6.1) and quasicrystalline (Sect. 2.6.2) structures that have not yet been considered in the previous parts of this chapter. 2.6.1 Amorphous Alloys The most important aspect to be considered in amorphous alloys, also called metallic glasses, is the relation between structural and mechanical relaxation which are closely connected. To discuss this relation, it is first necessary to know the a-priori different definitions and characteristics of both kinds of relaxation. Since mechanical (anelastic or viscoelastic) relaxation has already been introduced in Chap. 1, a brief introduction into structural relaxation will be given here. Structural Relaxation In the literal sense, any time-dependent equilibration of the atomic structure of condensed matter, after any kind of external perturbation, may be called “structural relaxation” (SR). This may in principle include production, annihilation and rearrangement of defects in crystals (like equilibration of thermal vacancies after changes in temperature, or recovery and recrystallisation after plastic deformation or irradiation), and even certain cases of phase transformations. However, it is more common to use the name “structural relaxation” more specifically for continuous changes of amorphous structures – in particular in glass-forming systems – which are not so easily expressed in terms of defect concentrations but rather appear as integral modifications of the whole structure. For instance, temperature changes generally give rise to SR due to the temperature dependence of amorphous structures in (stable or metastable) equilibrium. The Glass Physics Approach Understanding SR in glass-forming systems is the key to understand “glass” per se, i.e., the formation and nature of glasses and the glass transition below which SR is largely frozen. According to many renowned experts, this is still the most challenging unsolved problem in condensed matter physics. The difficult task of summarising the state of knowledge in this complex field was

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2 Anelastic Relaxation Mechanisms of Internal Friction

tackled by Angell et al. (2000), by posing detailed key questions and reviewing “the best answers available” as given by experts and specialists in about 500 references. The subject was divided into four parts, i.e., three temperature domains A–C with respect to the glass transition temperature Tg , and a fourth part D dealing with “short time dynamics” which can be skipped here. The main emphasis in the review by Angell et al. (2000) is put on the hightemperature domain A of the (supercooled) viscous liquid at T > Tg where the system is ergodic (i.e., its properties have no history dependence). Important items to be understood are the temperature dependences of transport properties and relaxation times, e.g., in form of the Vogel–Fulcher–Tammann (VFT) equation and deviations from it, as well as non-exponential relaxation functions of the form exp[−(t/τ )β ] with 0 < β < 1 (Kohlrausch–Williams– Watts (KWW) or stretched exponential function, which was given a physical meaning e.g., by Ngai’s coupling model of cooperative many-body molecular dynamics (Ngai et al. 1991, Ngai 2000)). The VFT equation, e.g., for the viscosity η, can be written as η = η0 exp[D∗ T0 /(T − T0 )],

(2.53)

with the so-called fragility parameter D∗ and VFT temperature T0 , which are coupled with respect to the glass transition according to Tg /T0 = 1 + D∗ / ln(ηg /η0 ) ≈ 1 + D∗ /39,

(2.54)

where ηg and η0 represent the viscosities at T = Tg and T → ∞, respectively (Angell 1995). The fragility parameter D∗ is used to distinguish between “strong” liquids or glasses with large D∗ and almost Arrhenius-like behaviour (which would be exact for D∗ = ∞ implying T0 = 0), and “fragile” ones with small D∗ , a pronounced curvature in a Tg -scaled Arrhenius plot, and a very rapid breakdown of shear resistance on heating directly above Tg . A similar temperature dependence is also found for the relaxation time τ , which in this range A is so short that the structure can generally be considered to be in a “relaxed” state of internal equilibrium. The low-temperature domain C of the “truly glassy” state (T ≪ Tg ), on the opposite side, can be defined as the range where the cooperative SR of the viscous liquid (also called “main”, “primary” or “α” relaxation) is completely frozen. Here the properties change essentially reversibly with temperature (as they do in range A) but now depend strongly on history, i.e., on the initial time-temperature path on which the system was frozen. Relaxation in this glassy range is possible only by decoupled, localised motion of easily mobile species (also called “secondary” relaxations4 ). 4

These secondary relaxations are sometimes classified further as β, γ, δ, . . . relaxations, which is more appropriate for polymers where the stepwise freezing of various local degrees of freedom may be associated with specific molecular groups, than for anorganic or metallic systems.

2.6 Relaxation in Non-Crystalline and Complex Structures

99

In the intermediate temperature domain B near and not too far below the glass transition (T
Tg ), primary SR must be considered explicitly as it occurs continuously on all experimental time scales, but without reaching equilibrium except for long annealing times. This is the most difficult range in which structure and properties depend on both history and actual time during the measurement. A first approach relies on the “principle of thermorheological and structural simplicity” (Angell et al. 2000) which relates the molecular or atomic mobility to the structural departure from equilibrium, as described by a single parameter like the so-called fictive temperature Tf . As depicted in Fig. 2.34, the fictive temperature can be found by projecting the actual value of a certain property p (like volume, enthalpy, entropy, etc.) on the equilibrium curve for the liquid extrapolated from range A, using the slope ∂p/∂T from the frozen range C. Structural relaxation in range B can then be described as a relaxation of Tf , in the simplest case according to T˙f = (T − Tf )/τ

(2.55)

with limiting conditions Tf = T in range A and Tf = const. in range C, respectively. The relaxation time τ now depends on both T and Tf , as expressed first by Tool (1946) τ (T, Tf ) = τ0 exp[xA/kT + (1 − x)A/kTf )],

(2.56)

where x is a dimensionless “non-linearity parameter” (0 < x < 1, typically x ≈ 0.5), and A is an activation energy (J¨ackle 1986, Angell et al. 2000).

Fig. 2.34. Definition of the fictive temperature Tf in different relaxing or frozen glassy states: (1) during and (2) after rapid cooling, (3) during slow cooling, (4) during heating after slow cooling. Indicated are also the temperature ranges A–C (Angell et al. 2000; see text). For frozen states like in case (2), Tf may be considered identical with Tg for a given heating or cooling rate

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In this simple form the fictive temperature concept has been useful for modelling relaxation in the difficult temperature range B; however, some ambiguity remains as regards which property p is chosen, and also the non-exponentiality (KWW function), found here as well, is not accounted for. The latter point is addressed by more advanced concepts like that of “hierarchically constrained dynamics”, considering elementary atomic relaxation events to occur not in parallel but in series (Palmer et al. 1984). The link in relaxation dynamics between ranges A and B is also underlined by correlations between the parameters β, D∗ , A and x (Angell et al. 2000). Up to this point, the synopsis of SR under the viewpoint of glass physics applies to all kinds of glasses (polymers, metals, oxides), necessarily neglecting more specific aspects in these different classes of materials. In particular, for certain characteristics of SR in metallic glasses, some different viewpoints exist independently in the traditions of solid-state physics and materials science rather than of glass physics. SR in Metallic Glasses An obvious difference, as compared to non-metallic glasses, is that in metallic glasses SR has long been noticed mainly as a strong irreversible (irrecoverable) effect deep in the solid range (T ≪ Tg ) existing even at room temperature, rather than as a phenomenon originating in the reversible properties of the undercooled melt above Tg as introduced earlier. This is a consequence of the high cooling rates used during production, especially in case of rapidly quenched “conventional” metallic glasses being in a highly unstable state far from equilibrium (high Tf ). The undercooled melt, on the other side, is more difficult to study and has been totally inaccessible before the development of “bulk” metallic glasses which, although first prepared by Chen (1974), became popular not before the 1990s (see Wang et al. 2004b for a review). On this historical background, some conceptually restricted usage of the term “structural relaxation” has partly developed for metallic glasses, regarding SR as being absent in the state of metastable equilibrium above Tg (e.g., Fursova and Khonik 2000) as observed macroscopically. This would however unnecessarily exclude from the term those fast dynamic processes in the viscous liquid which are needed to maintain equilibrium (e.g., during temperature changes), and which in glass physics just form the core of SR, being only slowed down below Tg . To avoid this obvious inconsistency, in this chapter we use “structural relaxation” in its general physical meaning and only speak of different “types” or “components” of SR if necessary. It was shown long ago that the irreversible type of SR in metallic glasses, e.g., during annealing of a rapidly quenched Pd–Si glass, can increase viscosity by five orders of magnitude (Taub and Spaepen 1979, 1980), indicating enhanced atomic mobility in the initial unrelaxed state. In other words: this irreversible SR, affecting virtually all physical and mechanical properties p

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(Cahn 1983), cannot be a “secondary” relaxation in the frozen temperature range C but should be considered as a primary one in range B, kinetically extended to lower temperatures. At this point it seems surprising that at temperatures so far below Tg , there is also a reversible (recoverable) component of SR being even faster than the irreversible one, as observed e.g., for Young’s modulus (Kurˇsumovi´c et al. 1980, Scott and Kurˇsumovi´c 1982) or enthalpy (Scott 1981, Sommer et al. 1985, G¨ orlitz and Ruppersberg 1985), but hardly for density or volume (Cahn et al. 1984, Sinning et al. 1985). This (selective) low-temperature reversible SR component, to be distinguished from reversible behaviour at the glass transition, is difficult to understand in terms of fictive temperature or primary/secondary relaxations, but at least roughly consistent with an earlier hypothesis by Egami (1978) relating reversible and irreversible SR, respectively, to changes in chemical and topological (or geometrical, Egami 1983) short-range order. The (also non-exponential) kinetics of such “solid-state” SR phenomena in metallic glasses, extensively studied in both “conventional” and “bulk” metallic glasses during the past three decades, have been widely analysed in terms of an activation energy spectrum (AES) model, introduced by Gibbs et al. (1983) on the basis of earlier work by Primak (1955), and subjected to some later extensions and modifications. This model is based on a wide non-equilibrium distribution of Debye-type relaxation events, which during annealing is gradually cut down from the low-energy side. While mathematically equivalent to the use of a KWW function, the physics behind this model seems to be more consistent with the idea of independent “relaxation centres” (see later), instead of the picture of true cooperative motion associated with a KWW function. For a microscopic understanding of SR in metallic glasses, the oldest and maybe still most widely spread concept is that of free volume, which was introduced by Cohen and Turnbull (1959) and worked out later by van den Beukel and coworkers, incorporating also Egami’s distinction between topological and chemical short-range order (e.g., van den Beukel 1993 and references therein). Alternative concepts were added more recently, for example based on interstitialcy theory (describing an amorphous solid as a crystal containing a few per cent of self-interstitials; e.g., Granato 1992, 1994, 2002; Granato and Khonik 2004), or on the theory of local topological fluctuations (of atomic bonds and atomic-level stresses; Egami 2006). As SR is closely related to diffusion, much can be learned from the recent progress in understanding diffusion mechanisms in metallic glasses (Faupel et al. 2003), which generally revealed highly collective atomic processes (contrary to crystalline metals): according to molecular dynamics simulation supported by critical experiments, atomic migration mainly occurs in thermally activated displacement chains or rings. Being rather local at low temperature, these chains grow in size and concentration with increasing temperature until they finally merge into flow.

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Relation Between Structural and Mechanical Relaxation Any structural relaxation – whatever the exact microscopic mechanism is – must involve atomic movements directed to lower the Gibbs free energy under the acting external perturbation, generally including anisotropic atomiclevel distortions oriented in different directions (like the above displacement chains). If the external perturbation is isotropic, e.g., in case of a purely thermal deviation from equilibrium, such local anisotropies may be averaged out so that only a macroscopically isotropic volume change is observed. In the presence of a mechanical stress, however, the distribution of the local events may become asymmetric producing a net distortion in the direction of energetically favoured orientations, i.e., a mechanical relaxation due to a directional structural relaxation (DSR). In this generality, and using the widest meaning of SR which in principle applies to crystalline structures as well (see above), every mechanical relaxation mechanism based on the motion of defects, including all cases considered in Sect. 2.2–2.4, might be called a DSR: under this viewpoint, DSR forms a very general principle of mechanical relaxation which of course also applies to amorphous structures. Thus, the connection between structural and mechanical relaxation is generally a rather close and direct one. More specifically, the different types and temperature ranges of SR in glassforming systems must be considered. In the range of the primary α relaxation around the glass transition, the same cooperative atomic motions cause both viscous flow and SR (i.e., “SR occurs by viscous flow”), so that relaxation time and viscosity can directly be converted into each other (for which, in spite of non-exponential relaxation, often a simple Maxwell model with τε = η/EU is used, cf. Chap. 1). Therefore, in the range where a mechanical (e.g., internal friction) measurement is dominated by viscous flow, the result directly reflects the structural α relaxation. There is a superabundant number of (mechanical and other) studies of the α relaxation over wide frequency and temperature ranges in more stable non-metallic glass formers, whereas in metallic systems the α relaxation is accessible only under favourable conditions using the best bulk metallic glass formers and low frequencies (see later). The situation is less clear in metallic glasses at temperatures further below Tg down to about 400 K where the above-mentioned, specific types of irreversible and reversible SR are found, mechanical relaxation is at least partly anelastic (recoverable) in nature (Berry 1978), but plastic deformation still occurs mainly by homogeneous flow. By assuming spatially separated structural “relaxation centres” represented by two-well systems, Kosilov, Khonik and coworkers developed a specific DSR model which applies in this range not only to mechanical relaxation but to mechanical properties in general (e.g., Kosilov and Khonik 1993; Khonik 2000, 2003 and references therein). The relaxation centres (two-well systems) were divided into irreversible (highly asymmetric) and reversible (rather symmetric) ones, the former being responsible for mainly viscoplastic low-frequency internal

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friction, plastic flow and even for reversible strain recovery (Csach et al. 2001), whereas the latter cause anelastic processes seen at higher frequencies (Khonik 1996, Eggers et al. 2006). At still lower temperatures where plastic deformation of metallic glasses is known to change to a highly localised shear band mode, the primary SR is eventually frozen (range C in Fig. 2.34, in many cases below about 400 K). If speaking of “DSR” in this range at all, this can only mean “secondary” relaxations of special, easily mobile species, like those of interstitially dissolved hydrogen which have already been treated in Sect. 2.2.4. However, since such anelastic processes in metallic glasses – classified as “type (b)” in the introduction to non-crystalline structures at the beginning of this section – have more in common with crystalline structures than primary DSR, a true solidstate picture with a clear distinction of the relaxing defect might be more appropriate in this low-temperature range than the more general viewpoint of DSR. Internal Friction Phenomena in Metallic Glasses General Aspects Amorphous alloys have to be produced with the help of some non-equilibrium procedure (like rapid cooling from the melt, mechanical alloying, various kinds of deposition, etc.), during which the formation of the thermodynamically stable crystalline state is kinetically hindered. Therefore, all amorphous alloys crystallise when heated into a temperature range with sufficient atomic mobility, which is always connected with a maximum of internal friction at a temperature close to the onset of crystallisation (Fig. 2.35). In fact this “crystallisation peak”, with a position usually depending on heating rate but not on frequency (e.g., Zhang et al. 2002), is not a true relaxation peak but a transitory effect. It basically reflects the irreversible transition from the high and monotonically increasing IF in the glassy amorphous phase to a much lower damping level in the crystalline state, but can be a quite complex-shaped superposition of many different effects in the frequent case of a multiple-step crystallisation process. Once passed during heating, the crystallisation peak completely disappears during subsequent cooling or during a second heating run. It has been used in some cases to study details of the crystallisation process including kinetics and activation energies (Sinning and Haessner 1985, Klosek et al. 1989, Nicolaus et al. 1992). In contrast to the high damping level at the onset of crystallisation, the internal friction in metallic glasses is generally low at room temperature and below, and at acoustic (vibrating-reed) frequencies often reduced to the thermoelastic background (see Sect. 2.5) if no special low-temperature effects are there (see later). The temperature dependence of IF is rather weak up to about 400–500 K, where a stronger, often exponential increase sets in which

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Fig. 2.35. Comparison of the low- and high-frequency IF behaviour, at a heating rate of 0.3 K min−1 , for two Ni-based glasses with (Ni60 Pd20 P20 ) and without (Ni78 Si8 B14 ) a glass transition before crystallisation. (1) Ni60 Pd20 P20 , 0.08 Hz; (2) Ni60 Pd20 P20 , 450 Hz; (3) Ni78 Si8 B14 , 0.095 Hz; (4) Ni78 Si8 B14 , 400 Hz. The maxima of all curves (at ∼600 K for Ni60 Pd20 P20 and ∼700 K for Ni78 Si8 B14 ) correspond to the onset of partial (primary) crystallisation followed by further transformations (Sinning and Haessner 1988a)

continues up to crystallisation. At all temperatures damping is higher at 0.1 Hz than at acoustic frequencies, indicating a broad spectrum of additional low-frequency processes. In this context, two main groups of metallic glasses have to be distinguished: those which crystallise from the solid state before reaching the glass transition, and those which first show a glass transition and then crystallise from the undercooled melt (which largely corresponds to the distinction between “conventional” and “bulk” metallic glasses, except for a few intermediate cases like CuTi showing a Tg in a torsion pendulum at 0.3–0.5 Hz without being a bulk glass former (Moorthy et al. 1994)). Glass Transition and α Relaxation As shown in Fig. 2.35 for a still moderate example, the occurrence of a glass transition has a dramatic effect on the height of the crystallisation peak at low frequencies which easily exceeds tan φ = 1, while the high-frequency IF peak remains unaffected and shows about the same height (tan φ < 0.1) as without a glass transition. The reason for this dramatic low-frequency IF increase, seen in Fig. 2.35 as the strong upward bend of curve 1 at Tg which is missing for the “conventional” metallic glass (curve 3), is the onset of dominating viscous damping Qv −1 due to the α relaxation (described as Qv −1 (T ) = EU /ωη(T ) using a Maxwell model). It has been shown that this viscous onset,

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shifting to higher temperature with increasing frequency, is located just at the dynamic glass transition (assuming ηg = 1012 N s m−2 ) if the frequency is around 0.1 Hz; under certain conditions, it could be used for determining Tg at heating rates much lower than possible with the common DSC technique (Sinning and Haessner 1986, 1987, 1988b; Sinning 1991a, 1993a). It is important to note, however, that such maxima in the loss factor tan φ (or Q−1 ) remain always transitory “crystallisation peaks” as mentioned earlier, even in presence of a glass transition: there is no “glass transition peak” or “ α relaxation peak” in tan φ in metallic (or more generally in low molecular weight) glasses, contrary to occasional misinterpretations in the literature. The glass transition alone, without the intervention of crystallisation, produces an α relaxation peak only in the loss modulus E ′′ (or G′′ in case of shear) but not in tan φ = E ′′ /E ′ which would in this case grow infinitely as E ′ goes to zero in the supercooled liquid. The typical situation, producing a “peak” in tan φ, is depicted in Fig. 2.36 for Zr65 Al7.5 Cu27.5 (a moderate bulk glass former not very different from Ni60 Pd20 P20 in Fig. 2.35): whereas the loss modulus E ′′ shows two separate peaks, being identified with the α relaxation and with losses during crystallisation, respectively (Rambousky et al. 1995), the single maximum in tan φ does not reflect these two peaks. It is rather dominated by the behaviour of the storage modulus E ′ in the denominator, which falls down in the supercooled liquid above Tg by more than one order of magnitude, to a sharp minimum that is solely determined by the onset of crystallisation (note the different, logarithmic and linear scales for the moduli and tan φ, respectively). Therefore, only the rising part of the damping “peak” may be associated with the α relaxation.

Fig. 2.36. Storage modulus E ′ , loss modulus E ′′ and damping tan φ of as-quenched amorphous Zr65 Al7.5 Cu27.5 , measured at 1 Hz during heating with 10 K min−1 using a dynamic mechanical analyser (Rambousky et al. 1995). Tg denotes the onset of the calorimetric glass transition

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For studying the α relaxation by mechanical spectroscopy, it is therefore more appropriate to look at E ′ and E ′′ (or G′ and G′′ ) separately, rather than just considering internal friction. To trace out the full α relaxation peak in the loss modulus as a function of either temperature or frequency, it is important to have a wide supercooled liquid range, i.e., to use the best bulk metallic glasses available. Meanwhile such studies have been performed on several more advanced Zr- and Pd-based bulk glasses (e.g., Schr¨ oter et al. 1998, Pelletier and Van de Moort`ele 2002a, Pelletier et al. 2002b, Lee et al. 2003a, Wen et al. 2004); an example is shown in Fig. 2.37. The results follow the time-temperature superposition principle, well known from non-metallic glass formers: all curves fall on a master curve when shifted by a temperaturedependent relaxation time which usually obeys the VFT equation. Occasional low-temperature shoulders of the α peak in E ′′ are sometimes interpreted as a β relaxation (Pelletier and Van de Moort`ele 2002a). Contrary to the loss modulus, the loss compliance J ′′ does not show an α relaxation peak either, but monotonically falls (like tan φ) with increasing frequency or decreasing temperature. An analysis of its frequency dependence, with an exponent typically changing from −1 at low to −1/3 at high frequencies, may be used to separate Newtonian viscous flow from “relaxation” components and to discuss related models (Schr¨ oter et al. 1998). In addition, the interest in more specific questions of glassy dynamics in this range (which are beyond the scope of this chapter), calling for advanced or extended experimental conditions, has triggered some remarkable new experimental developments in mechanical spectroscopy: for instance, a non-resonant

Fig. 2.37. The α relaxation of the Zr46.75 Ti8.25 Cu7.5 Ni10 Be27.5 bulk metallic glass studied by dynamic mechanical analysis. (a) Temperature dependence during heating with 1 K min−1 at different frequencies; (b) isothermal frequency dependence and fit to the KWW equation with an exponent β = 0.5 (solid lines) at different temperatures (Wen et al. 2004)

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vibrating-reed technique with an extremely wide frequency range (Lippok 2000), or a special “double-paddle oscillator” for studying thin films (Liu and Pohl 1998) applied to glassy alloys at high temperatures and high frequencies (R¨osner et al. 2003, 2004). The results of such fundamental studies on bulk metallic glasses generally confirm the main characteristics of the α relaxation in the high-temperature domain A as briefly outlined above, known from non-metallic glass formers: in this respect, the underlying physics appears to be the same for quite different classes of glass-forming systems. Intermediate Temperature Range This is the classical range of materials science in which the study of internal friction in metallic glasses began (Chen et al. 1971), and where most of our knowledge is still based on results obtained on rapidly quenched samples (although in this range there seems to be no big difference to bulk alloys; Berlev et al. 2003, Eggers et al. 2006). The main feature is here the exponential increase of IF with temperature mentioned earlier, e.g., in form of the (on the logarithmic scale) linear rise of curves 2–4 in Fig. 2.35 towards the maximum. The following main characteristics have been reported for this rising part of the IF spectrum: 1. It is reduced in its lower part (or shifted to higher temperature) by irreversible structural relaxation. For instance, if an as-quenched sample is heated with a constant rate to successively increasing temperatures (Fig. 2.38), in each heating run the IF is reduced compared to the previous one, in close correlation to an irreversible increase of Young’s modulus or resonance frequency (seen more clearly in isothermal experiments; Morito and Egami 1984a, Neuh¨ auser et al. 1990). Such an annealing behaviour is often analysed in terms of the AES model mentioned earlier, resulting in a broad spectrum for irreversible structural (not mechanical) relaxation ranging from about 100 to 200 kJ mol−1 in case of Fig. 2.38. 2. Using appropriate isothermal anneals, Morito and Egami 1984b and Bothe (1985) have been able to cycle the IF spectrum reversibly (Fig. 2.39), proving an effect of reversible SR as well. 3. At frequencies about 1 Hz and below, the IF of as-quenched samples depends on the heating rate (Bobrov et al. 1996, Yoshinari et al. 1996b). 4. After a stabilising anneal and subtraction of the thermoelastic background Q−1 B , the IF at constant frequency often shows a straight line in an Arrhenius plot (Fig. 2.40), i.e., it is of the empirical form −A/kT . Q−1 − Q−1 B ∝e

(2.57)

The slope parameter A increases with annealing (Berry 1978). 5. The IF increase shifts to higher temperatures at higher frequencies, i.e., it is a thermally activated relaxation effect (Fig. 2.40). The apparent activation enthalpies H, taken from cuts at constant damping according to

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Fig. 2.38. Variation of (a) frequency and (b) damping of a vibrating amorphous Ni78 Si8 B14 reed during heating-cooling cycles with 1 K min−1 . The vertical arrows indicate the onsets of structural relaxation and crystallisation, respectively, (Neuh¨ auser et al. 1990)

ln(f2 /f1 ) = (H/k)(T1−1 − T2−1 ),

(2.58)

vary between 115 and 250 kJ mol−1 for different Pd- and Fe-based glasses and annealing treatments (with sometimes unphysically high attempt frequencies τ0−1 ), and are in most cases much higher than the related slope parameters A (22–125 kJ mol−1 ) (Soshiroda et al. 1976, Berry 1978, Neuh¨ auser et al. 1990). The problem of this discrepancy could be solved by Kr¨ uger et al. (1993) by showing theoretically that the ratio A/H is identical with the KWW exponent β obtained under quasi-static conditions, which also reduces the attempt frequency to reasonable values of the order of the Debye frequency. Including such A/H ratios, experimental KWW exponents of metallic glasses may span a very wide range between e.g., 0.23 for Fe75 P15 C10 (Berry 1978) and 0.67 for Zr65 Al7.5 Cu27.5 (Weiss et al. 1996). 6. At a fixed temperature, the thermally activated damping Q−1 − Q−1 B should exhibit a frequency dependence proportional to 1/f β in case of a KWW law (Kr¨ uger et al. 1993), or to ln (1/f ) from the empirical equations (2.57) and (2.58) corresponding to Fig. 2.40, as based mainly on results at acoustic and lower frequencies down to about 1 Hz. However, when extending frequency into the infralow range down to 3 mHz,

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Fig. 2.39. Damping of amorphous Fe32 Ni36 Cr14 P12 B6 at 0.4 Hz and 2.5 K min−1 . (a) as quenched; (b) annealed at 300◦ C for 72 h; (c) annealed further at 375◦ C for 3 min; (d) additionally annealed at 270◦ C for 50 h (Morito and Egami 1984b)

Fig. 2.40. Thermally activated IF of amorphous Fe80 B20 (expressed as logarithmic decrement δ = πQ−1 ), measured under a saturating magnetic field to suppress magnetoelastic damping (Berry 1978)

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Fig. 2.41. Internal friction of amorphous Co70 Fe5 Si15 B10 at f = 0.003–300 Hz and T = 500–700 K as a function of the vibration period. The inset shows the “highfrequency” range (f > 0.1 Hz) on an enlarged scale (Fursova and Khonik 2000)

Fursova and Khonik (2000) have found in an as-quenched Co-based glass that below about 0.1 Hz the damping varies linearly with the period of vibration (Fig. 2.41, obtained from isothermal cuts of linear heating curves at 3.3 K min−1 and different frequencies), i.e., that in this low-frequency range IF is dominated by a component proportional to 1/f rather than to ln (1/f ) or 1/f β . According to these results and some additional isothermal experiments (e.g., Morito 1983, Bothe and Neuh¨ auser 1983, Bobrov et al. 1996, Fursova and Khonik 2002a), at least four or five components of IF in metallic glasses may be distinguished in this intermediate temperature range: (a) the thermoelastic background, (b) thermally activated IF in the “structurally relaxed” state with KWW kinetics, (c) a relatively small component which follows reversible SR and (d) a larger component (mainly in as-quenched glasses) which is gradually eliminated by irreversible SR, and which probably splits up into a high- and a low-frequency part, the latter having transient character (Yoshinari et al. 1996b). A coarse scheme of “primary” and “secondary,” or “α” and “β” relaxations seems rather inadequate for such subtle distinctions. As an additional unexplained anomaly, an isothermal re-increase of low-frequency IF at 0.1 Hz, after the decay of component (d), was observed in as-quenched Pd40 Ni40 P20 but not in other cases like Pd77.5 Ag6 Si16.5 at temperatures approaching the glass transition (Sinning 1991a, 1993a).

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To understand this IF behaviour in relation to structural relaxation, the question whether the atomic rearrangements which contribute to anelasticity and those which change it are identical or not, has already been posed by Morito and Egami (1984b). Exactly this identity (but not restricted to anelasticity) is assumed in the DSR model mentioned earlier, where IF is considered to be proportional to the rate rather than to the degree of SR (Khonik 1996, Fursova and Khonik 2000) – like in the case of the α relaxation above Tg where, however, the “degree” of SR is without meaning because the system is in metastable equilibrium. A similar interpretation was given by Yoshinari et al. (1996b) for the “transient” heating-rate dependent IF component in Fe33 Zr67 . According to the DSR model, damping should be viscoplastic in nature as described by a Maxwell body, and under linear heating conditions proportional to T˙ /f (Khonik 2000). This is confirmed by the low-frequency behaviour in Fig. 2.41, and by the observed heating-rate dependence. Therefore, the DSR model seems to describe some extension of the viscous α relaxation to highly irreversible non-equilibrium conditions at lower temperatures, and is apparently consistent with low-frequency IF of as-quenched metallic glasses. As regards the underlying microscopic mechanisms of internal friction, the above-mentioned progress with respect to diffusion mechanisms (Faupel et al. 2003) seems to show where to go for a better understanding of mechanical relaxation as well. The main problem will be to bring together this kind of knowledge about possible atomic rearrangements with the respective microscopic aspects (e.g., coupling/cooperativity versus separated relaxation centres) of the different relaxation models. Low-Temperature Effects True low-temperature anelastic relaxation peaks5 – i.e., true “secondary relaxations” in the sense of decoupled, localised processes in the language of glass physics – have been found in metallic glasses mainly in two cases: after absorption of hydrogen and after cold plastic deformation 6 . Since the former case has already been treated in Sect. 2.2.4, we will now focus on the latter one. IF peaks due to plastic deformation are usually located below room temperature, in form of quite different spectra ranging from well-defined single 5

6

“Low temperature” means here the range from room temperature down to about 50–100 K. Effects at liquid helium temperature and below, like tunneling and low-energy excitations, are not considered. There are some reports of similar IF peaks in metallic glasses without a specific treatment, especially in the earlier literature, where either no hypothesis about their origin or some speculations about other relaxation mechanisms were given (e.g., K¨ unzi et al. 1979, Sinning et al. 1988). Most of them have been identified later, or may be suspected, to originate either from hydrogen (as an impurity, in alloys containing metals like Zr or Y with a high affinity to hydrogen), or from unintentional deformation (due to gripping or clamping in the sample holders).

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relaxation peaks over more irregular ones up to broad, multiple-peak spectra; a frequent case is a main peak at about 200–300 K and a smaller side peak around 100 K. The dependence on the degree of deformation can be very different and is as yet hardly predictable: whereas Pd77.5 Cu6 Si16.5 showed a monotonically increasing damping from 23% to 79% deformation (Zolotukhin et al. 1985), in most other cases (Ni60 Nb40 , Cu50 Ti50 , Ni78 Si8 B14 ) quite the opposite trend was found: a relatively sharp, though scattered maximum of the effect at only about 2–3% plastic strain followed by a strong decrease towards stronger deformation (Zolotukhin et al. 1989; Khonik 1994, 1996; Khonik and Spivak 1996). Intermediate rolling degrees around 20% were successful to produce damping peaks in Co33 Zr67 (Winter et al. 1996) but ineffective in both Zr52.5 Cu17.9 Ni14.6 Al10 Ti5 and Pd40 Cu30 Ni10 P20 (Eggers et al. 2007). In addition, the effect usually shows a pronounced amplitude dependence, and is suppressed by annealing or irradiation. An example is given in Fig. 2.42, indicating a mixture of hysteretic and relaxation components (Khonik 1996). It is obvious that the origin of this deformation-induced IF must lie in the local structural changes (i.e., “defects”) produced in the shear bands which carry the highly localised plastic deformation of metallic glasses at low temperatures. A microscopic understanding of the relaxation processes must therefore be based on an understanding of the shear-band deformation itself. Some main ideas have been based on free volume (Spaepen 1977, Argon 1979, Steif et al. 1982, Wright et al. 2003, Kanungo et al. 2004), polycluster (Bakai 1994, Bakai et al. 1997) or dislocation concepts (Gilman 1973, 1975; Li 1978); however, as this is an own extensive topic since about 30–40 years, it cannot be discussed here. Since several characteristic features of the deformation-induced IF peaks

Fig. 2.42. Low-temperature IF peaks in amorphous Ni78 Si8 B14 after cold rolling (f ≈ 290 Hz): (a) temperature dependence at constant strain amplitude (7 · 10−6 ) and different rolling degrees; (b) amplitude dependence at constant rolling degree (3.6%) and different temperatures; the almost amplitude-independent behaviour of the “as cast” undeformed sample is shown for comparison (Khonik 1996)

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in metallic glasses (not to be listed here in detail) have been found analogous to those of dislocations in crystals, most authors of the above damping studies favour a dislocation concept, i.e., they assume that in metallic glasses deformation occurs by non-crystalline types of dislocations or dislocation-like defects (Bobrov and Khonik 1995, Khonik 2003 and further references). An idea by Khonik and Spivak (1996) that even the IF peak produced by hydrogen in metallic glasses (Sect. 2.2.4) – observed in almost the same temperature range as the deformation peak – could be an indirect, hydrogenationinduced deformation effect, rather than a Snoek-type relaxation, gave rise to a longer controversy and further critical experiments (Khonik 1996, Winter et al. 1996, Takeuchi et al. 2004, Sinning 2006a, Eggers et al. 2007). However, apart from the more fundamental considerations on the Snoek-type mechanism in a wider structural context given in Sect. 2.2.4, it turns out that the properties of both types of IF peaks in metallic glasses (only to mention annealing behaviour and amplitude dependence) are plainly too different to be explained by the same mechanism. There remain some earlier, less detailed observations of smaller IF peaks around or above room temperature, preceding the exponential IF increase towards the glass transition (e.g., Yoon and Eisenberg 1978, Hettwer and Haessner 1982, Deng and Argon 1986), which appear neither to be related to hydrogen nor to plastic deformation, so that the question for their origin must remain open. Beyond anelastic relaxation, in ferromagnetic metallic glasses also magnetomechanical damping (and a “giant” ∆E effect) is observed, which is generally analogous to the respective phenomenon in crystalline materials (cf. Sect. 3.4), but modified by the absence of magnetocrystalline anisotropy in the amorphous structure (Berry 1978, Kobelev et al. 1987, Lu et al. 1990). 2.6.2 Quasicrystals and Approximants Quasiperiodic Order and the Role of Phasons Similarly as structural relaxation played the key role in the earlier discussion of anelasticity and viscoelasticity in amorphous alloys, so do the phasons in case of quasicrystals. Phasons are specific perturbations of the quasiperiodic long-range order that defines this novel class of solid-state structures, which was discovered by Shechtman et al. (1984). Quasiperiodic order is characterised by sharp Bragg diffraction spots in spite of the absence of translational periodicity. There are different types of quasicrystals with quasiperiodic order in one, two or three dimensions (and periodic order in the remaining directions); the most important are the decagonal (2D) and the icosahedral (3D) ones. Mathematically, quasiperiodic order can be constructed as a cut through a periodic hyper-structure in a higher-dimensional space, like a 3D cut through a 6D periodic hyperlattice in case of icosahedral quasicrystals (Duneau and

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2 Anelastic Relaxation Mechanisms of Internal Friction

Fig. 2.43. Simplified scheme of phason defects in quasicrystals: (a) hyperlattice construction of a 1D quasicrystal with a phason flip produced by an orthogonal shift of the cut; (b) a 2D phason flip; (c) 2D simulation of shear deformation with a phason wall in the glide plane behind the dislocation (after Mikulla et al. 1995)

Katz 1985, Elser 1985). The construction is symbolised in Fig. 2.43a for a simple, 1D example: the “quasilattice points” in the “real” 1D space are determined by the cuts through “atomic hypersurfaces” in the orthogonal space around each point of the 2D hyperlattice. In this hyperspace, additional degrees of freedom exist orthogonal to physical space, which give rise to deviations from perfect order (from the “ideal” cut) in form of dynamic excitations (“phasons” as opposed to phonons in physical space) as well as static “phason defects”. The latter are violating certain “matching rules” in real space (instead of perturbing translational symmetry as defects in crystals do), and are connected with correlated rearrangements of certain local atomic configurations. Since these rearrangements also change the elastic energy (“phason strain”; G¨ ahler et al. 2003), it should in principle be possible to study their dynamics by mechanical spectroscopy. In the most elementary case (so-called “phason flips”; Fig. 2.43b), the situation may be similar to that of point defects in regular crystals forming “elastic dipoles” (Weller and Damson 2003). However, as most “conventional” defects known from regular crystals also exist in quasicrystals, contributions to mechanical relaxation may be expected from these conventional defects as well as from the presence and motion of phasons. As concerns plastic deformation, quasicrystals are usually brittle at room temperature, while shear deformation by dislocation glide is possible at high temperatures. An important difference from crystals is that dislocation glide does not restore perfect order, but a “phason wall” (a special type of stacking fault) is left behind each gliding dislocation (Fig. 2.43c). Such coupling between conventional defects (dislocations) and phasons might also give rise to relaxation effects.

2.6 Relaxation in Non-Crystalline and Complex Structures

115

Survey of Mechanical Spectroscopy Studies Although mechanical spectroscopy was used from the very beginning to characterise the quasicrystalline state, the number of related studies in this young field is still limited. As seen in the semi-chronological, comprehensive list in Table 2.20, the first papers appeared in the field of low-temperature physics. The next steps were influenced by the development of sample preparation, from ultrafine-grained melt-spun ribbons over coarse-grained polyquasicrystals up to large-size single quasicrystals grown by advanced techniques (Feuerbacher et al. 2003): with this increase in available crystal size, fundamental studies on single quasicrystals became possible first for elastic constants and later for anelastic relaxation which generally needs longer samples. On the other hand the Ti- and Zr-based quasicrystals, related to more applied aspects like hydrogen storage (Stroud et al. 1996) or precipitation strengthening (Xing et al. 1999, Inoue et al. 2000), offer access to the other extreme case of nano-quasicrystalline structures close to the amorphous limit (Nicula et al. 2000, Jianu et al. 2004). With a few exceptions mentioned later, most mechanical spectroscopy studies were performed on the icosahedral (3D) type of quasicrystalline structures. Table 2.20. Topics introduced into the study of quasicrystals by mechanical spectroscopy (left column: year of first publication) year

subject

reference

1987

low-temperature (0.01–100 K) acoustics and relaxation elastic constants of Al-based single quasicrystals (ultrasonic techniques) anelastic relaxation in Al-based poly-quasicrystals anelastic relaxation in Al-based single quasicrystals

Birge et al. (1987), VanCleve et al. (1990) Reynolds et al. (1990); Sathish et al. (1991); Amazit et al. (1992); Spoor et al. (1995) Okumura et al. (1994), Damson et al. (2000a) Feuerbacher et al. (1996), Weller et al. (1996), Damson et al. (2000a,b), Weller and Damson (2003) Foster et al. (1999b)

1990

1994 1996

1999 2000

2002 2004

elastic moduli of Ti-based (poly-) quasicrystals relaxation in Ti/Zr-based quasicrystals, including hydrogen-induced damping peaks quasicrystalline coatings and composites with quasicrystals nano-quasicrystalline states and amorphous-quasicrystalline transitions

Foster et al. (2000), Scarfone and Sinning (2000); Sinning and Scarfone (2002); Sinning et al. (2003, 2004a) Fikar et al. (2002, 2004b) Sinning et al. (2004b,c)

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2 Anelastic Relaxation Mechanisms of Internal Friction

With increasing temperature (in total covering a range from about 0.01 to 1000 K), the following relaxation and IF phenomena can roughly be distinguished: – A low-temperature damping plateau at T < 1 K, – A linear damping increase with temperature (typical range 5–100 K), – A hydrogen-induced relaxation peak in H-absorbing Zr/Ti-based quasicrystals (see Sect. 2.2.4), – An intrinsic relaxation peak in Al-based quasicrystals around 400 K, – A high-temperature relaxation peak in Al–Pd–Mn single quasicrystals at about 900 K, – A strongly (e.g., exponentially) increasing high-temperature background. The low-temperature effects, not to be considered here in detail, were mainly explained by atomic tunneling systems resulting from structural disorder, assumed to be of phasonic type and hence correlated with phason strain; the following linear increase may be of mainly thermoelastic origin. Since the hydrogen peak has been treated separately in Sect. 2.2.4, the following considerations will focus on the remaining three points (i.e., intrinsic relaxation peaks and high-temperature background), before briefly coming back to a specific aspect of the hydrogen peak. Internal Friction Phenomena in Quasicrystals Intrinsic Effects The as yet most careful studies of the intrinsic dynamic losses in quasicrystals have undoubtedly been performed by Weller and coworkers, including also high-quality icosahedral Al70.3 Pd21.5 Mn8.2 single quasicrystals (see references in Table 2.20). Two well-defined, thermally activated relaxation peaks (denoted “A” and “B” by the authors, see curves 1 and 2 in Fig. 2.44) were observed near 400 and 900 K. Thanks to a frequency variation by seven orders of magnitude (10−3 –104 Hz, using forced and resonant torsion pendula as well as a resonant bar apparatus), the apparent activation energies and pre-exponential frequency factors could be determined rather accurately as 95 kJ mol−1 /2·1015 s−1 for peak A and 386 kJ mol−1 /3·1024 s−1 for peak B, respectively. These values were considered as indicating a point-defect mechanism for peak A, but a collective motion of a large number of atoms for the high-temperature peak B (Damson et al. 2000a, 2000b; Weller and Damson 2003). Phasons are considered to contribute to both types of mechanisms, in form of (A) isolated phason flips and (B) extended phason formation and migration connected with the movement of dislocations. However, a clear distinction between e.g., phason flips and ordinary vacancy jumps, in case of the point-defect mechanism of peak A, is not possible at the present stage. Obviously, this point-defect mechanism is not confined to single quasicrystals,

2.6 Relaxation in Non-Crystalline and Complex Structures

117

Fig. 2.44. Examples of high-temperature IF spectra of icosahedral quasicrystals. Curves 1, 2: Al70.3 Pd21.5 Mn8.2 single quasicrystals with peaks “A” and “B” (curve 1: frequency 3 Hz, curve 2: 2 kHz; Weller and Damson 2003); curves 3, 4: microcrystalline Ti41 Zr42 Ni17 (curve 3: 360 Hz, curve 4: 3.6 kHz; Sinning et al. 2003)

since a similar peak had previously been observed in icosahedral Al75 Cu15 V10 polycrystals (Okumura et al. 1994). A related peak, with somewhat higher activation energies (135–210 kJ mol−1 ) but otherwise similar characteristics, was also found in Al–Ni–Co alloys with decagonal quasicrystals (Damson et al. 2000a, Weller and Damson 2003). On the other hand, as exemplified by curve 3 in Fig. 2.44, intrinsic relaxation peaks like peak A are absent in all Ti- and Zr-based icosahedral quasicrystals studied as yet (Sinning et al. 2003, 2004a, 2004b). As concerns peak B which was seen only in the icosahedral Al–Pd–Mn single quasicrystals, one should not exclude that it exists in icosahedral polycrystals as well, but such measurements at sufficiently high temperatures (sometimes prevented, like in the Ti–Zr–Ni system, by the formation of crystalline phases) are not found in the literature. Therefore, results like curves 3 and 4 in Fig. 2.44 are often treated as a monotonic high-temperature background; the frequency shift between these two curves, taken very roughly at a damping level of Q−1 = 0.01, corresponds to activation parameters of 320 kJ mol−1 /1025 s−1 comparable to those of peak B. A high-temperature background also exists in decagonal Al–Ni–Co up to 1000 K or even 1200 K; in that case, however, there is no such huge difference in activation parameters (especially as concerns the pre-exponential factors) like that between peaks A and B in Al–Pd–Mn. Consequently, the mechanisms of the “hightemperature damping background” may well be different between icosahedral and decagonal quasicrystals, and in the latter case mainly associated with volume diffusion (Weller and Damson 2003). Another interesting observation has been made in two bulk metallic glasses, which form quasicrystals as the first devitrification product from the undercooled melt above Tg : different from all previous observations in comparable cases, the damping level above room temperature was in this case found

118

2 Anelastic Relaxation Mechanisms of Internal Friction

Fig. 2.45. High-temperature damping increase in two bulk metallic glasses forming icosahedral quasicrystals from the undercooled melt (2 K min−1 , 330–350 Hz). Solid lines Zr69.5 Cu12 Ni11 Al7.5 , dashed lines Zr61.6 Ti8.7 Nb2.7 Cu15 Ni12 ; thin lines structurally relaxed amorphous, and thick ones quasicrystalline states (Sinning 2006b)

higher in the fine-grained quasicrystalline than in the initial amorphous state (Fig. 2.45). This seems to indicate that for building up long-range quasiperiodic order (though only over grain diameters of some tens of nanometres), defects or free volume must be produced in the relatively densely packed, structurally relaxed bulk metallic glass, which then become part of the fine polyquasicrystalline structure and give rise to enhanced anelastic or viscoelastic relaxation (Sinning 2006b). Hydrogen as a Probe It was already mentioned briefly earlier in Sect. 2.2.4 that H-induced IF peaks can be used as a local structural probe, e.g., in case of Fig. 2.14 to detect differences in short-range order between amorphous alloys. We should add here that of course the same is true for H-absorbing quasicrystals, i.e., that the position (like in Fig. 2.15) and shape of a Snoek-type hydrogen peak contain information about the local atomic structures in which the interstitial H atoms are moving. Prominent examples of such local atomic structures are in icosahedral quasicrystals the so-called “Bergman” and “Mackay” clusters (e.g., Kim et al. 1998), which seem to be distinguishable by the width of the hydrogen relaxation peak: Sinning (2006b) found evidence for a wider, composed peak (i.e., a stronger splitting of interstitial site or saddle point energies) for the Mackay clusters (in Ti65 Zr10 Fe25 ) compared to that one observed for the Bergman clusters (e.g., in Ti41 Zr42 Ni17 ). If, in the latter case, the sharpness of the mechanical loss peak can be related to the degree of icosahedral order in the local clusters, then this local order seems to be practically the same in amorphous and nano-quasicrystalline structures up to grains sizes of about 20 nm (in spite of a strong Young’s

2.6 Relaxation in Non-Crystalline and Complex Structures

119

modulus increase up to 50% between these two states), but distinctly improved at grain sizes beyond about 100 nm (Sinning 2006b). Approximants and Complex Alloy Phases Approximants are periodic crystals that “approximate” quasicrystals in the sense that, despite different long-range order, their structures are locally the same, i.e., they contain the same atomic clusters or structural building units as their quasiperiodic counterparts. In the higher-dimensional description mentioned earlier (Fig. 2.43a), they are called “rational approximants” as represented by rational coordinates of the cut through the hyperlattice, contrary to irrational ones for quasiperiodic structures. Due to the possibility of classical crystallographic structure determination, approximants are the main key for analysing the structure of quasicrystals. Unfortunately we are not aware of systematic mechanical spectroscopy work on approximants, except for one ultrasonic study of the hydrogen peak in the Ti–Zr–Ni W phase (a large-unit-cell “1/1 bcc approximant” containing Bergman clusters, Kim et al. 1998) by Foster et al. (2000) which, however, allows only for a limited comparison with the respective effect in the quasicrystal because of different H concentrations. Apart from their value as approximants to study quasicrystals, the so-called “structurally complex alloy phases” – defined as intermetallic phases with giant unit cells (Urban and Feuerbacher 2004) – are very fascinating to be studied in their own right, because they contain novel types of defects neither found in quasicrystals nor in simpler metallic structures, which may also produce interesting new physical properties. The most prominent of such defects are the so-called “metadislocations” (Klein et al. 1999) carrying plastic deformation, which would be a very promising new class of defects to be studied by mechanical spectroscopy.

3 Other Mechanisms of Internal Friction

3.1 Introduction In this chapter those internal friction phenomena are described which are not only associated with an anelastic mechanism, but which are mainly due to some hysteretic deformation occurring during stress cycling. Such hysteresis may originate from several kinds of phase transformations, like polymorphic or martensitic transformations or precipitation/dissolution of second phases (Sect. 3.2), from dislocation motion (Sect. 3.3), or from magnetostriction in ferromagnets (Sect. 3.4). Some of these mechanisms may lead to exceptionally high damping values defining so-called “Hidamets” (high-damping materials), which are considered in Sect. 3.5.

3.2 Internal Friction at Phase Transformations If a phase transformation is connected with a volume change or a shear deformation, it may oscillate under the influence of an alternating stress so that a transformation-induced, alternating strain is produced in addition to the elastic strain. There is usually a phase lag between stress and strain in such cases, causing a hysteresis and an energy loss measured as internal friction. The most important types of such stress-active phase transformations are considered briefly in this section, together with some typical damping effects (see Benoit 2001d for a review). 3.2.1 Martensitic Transformation A martensitic transformation (MT) is a special type of phase transformation, characterised by the following features (Cohen et al. 1979): – It is a diffusionless transformation between a high-temperature “austenite” and a low-temperature “martensite” phase; even if diffusion occurs it is not essential for the transformation

122

3 Other Mechanisms of Internal Friction

– It involves a lattice distortion, which consists mainly of a deviatoric but not a dilatational component – The kinetics and morphology are dominated by the strain energy There are three types of MT: athermal, isothermal and thermoelastic.1 Another distinction refers to ferrous and non-ferrous martensite. Athermal in this sense refers to the way how the martensite phase is growing upon cooling, e.g., by a sudden, jump-like (“athermal”) formation of martensite plates of a certain size which cannot grow further due to immobile interfaces, so that a further transformation forth or back requires new nucleation events. This athermal MT, characterised by large values of all relevant energy and strain parameters and high hardness in the martensitic state, is mainly found in interstitial Fe-based alloys including steels (ferrous martensite). Because of its large transformation hysteresis, it is of little interest from the viewpoint of internal friction and will not be considered further. The isothermal, i.e., time-dependent MT, discovered by Kurdyumov and Maksimova (1948, 1950), is a relatively rare type of MT; its effect on internal friction will shortly be discussed at the end of this chapter. Finally, it is the thermoelastic MT (Kurdyumov and Khandros 1949) in (mostly) non-ferrous materials which is of high importance for internal friction. With e.g., small transformation strains, energies, hysteresis and easily mobile interfaces, it is in many respects opposite to the athermal MT. The amount of research papers on the thermoelastic MT is huge, also because many of these alloys exhibit a shape-memory effect, and it is difficult to provide the reader even with a list of corresponding review papers. Under the aspect of IF and damping capacity, recent reviews by Van Humbeeck (2001, 2003), San Juan and P´erez-S´aez (2001) and San Juan and N´ o (2003), are recommended for deeper reading and have been used here. In the following short overview, we can only give an outline of the main damping mechanisms related to the thermoelastic MT, and mention the main groups of alloys in which these mechanisms occur. Damping Mechanisms Usually the total internal friction Q−1 tot associated with a phase transition consists of three components (Bidaux et al. 1989): −1 −1 −1 Q−1 tot = QT r + QP T + Qint .

(3.1)

The first two components belong directly to the phase transition in form of ˙ a transient part Q−1 T r depending on a finite heating or cooling rate T = 0, 1

Here the term “thermoelastic” refers to the comparability between thermal transformation and elastic strain energies, which has nothing to do with the fundamental thermoelastic coupling considered in Sect. 2.5.

3.2 Internal Friction at Phase Transformations

123

Fig. 3.1. Scheme of the three contributions to IF during a martensitic phase transformation: the transitory term IFTr , the phase transition (or isothermal) term IFPT and the intrinsic term IFInt (From San Juan and P´erez-S´ aez 2001)

and a rate-independent, non-transient part Q−1 P T found even under isothermal conditions. The third, intrinsic part −1 −1 Q−1 int (T ) = V Qmart (T ) + (1 − V )Qβ (T )

(3.2)

represents the intrinsic IF in both co-existing phases (the high-temperature (β) and martensite phases in case of the MT), taking into account the temperature-dependent volume fraction V of martensite. As sketched schematically in Fig. 3.1, the three contributions sum up to form an IF peak, with often very high damping capacity in case of a thermoelastic MT (see Sect. 3.5); this peak is usually accompanied by a minimum in the temperature-dependent elastic modulus. The experimental methods to separate these components have been discussed in detail by San Juan and N´ o (2003). −1 For the non-transient phase-transformation damping QP T , different mechanisms (related e.g., to interface or dislocation movements) are discussed in the literature with respect to different materials (Dejonghe et al. 1976, Mercier and Melton 1976, Clapp 1979, Koshimizu 1981, Kustov et al. 1995). Two basic approaches can be distinguished (Van Humbeeck 2001): (1) Q−1 P T is a result of coexistence of macroscopic amounts of martensite and austenite phases, or (2) Q−1 P T results from a “pre-transformation” state with sub-microscopical nuclei of the martensite phase. These approaches assume different dependencies of Q−1 P T on frequency and amplitude of vibrations, as studied by Kustov et al. (1995). The transient damping Q−1 T r , being the dominating part if the frequency is in the Hz range, is basically attributed to the transformation rate ∂n/∂t, where n is the amount of transformed material (Scheil and M¨ uller 1956). The related

124

3 Other Mechanisms of Internal Friction

anelastic strain dεan = k(∂n/∂t)dt, which determines Q−1 T r , comes from the lattice deformation when the material is transformed. Several approaches have been developed on this background: 1. Belko et al. (1969a) introduced a model of thermally activated formation and growth of nuclei of the martensitic phase. 2. Delorme and Gobin (1973a,b) suggested a “transformation plasticity” approach with the transformation deformation dεp = kσdn to be linearly dependent on the transformed volume fraction and on the applied stress σ, leading to k dn T˙ . (3.3) Q−1 Tr = J dT ω 3. Dejonghe et al. (1976) added a stress-dependent term to the transformation rate, and the possibility of retransformation during one cycle, to obtain


3 σC ∂n 2 k ∂n T˙ −1 QT r = + σ0 1− , (3.4) J ∂T ω 3π ∂σ σ0 where σC is the critical stress necessary to re-orientate already existing martensite variants or to nucleate new ones. 4. Similarly, the influence of the stress amplitude σ0 on the transformation rate was taken into account by Gremaud et al. (1987b) by ∂n ∂n = · (T˙ + ασ), ˙ ∂t ∂T where α is the Clausius–Clapeyron factor (α = υεt /∆S with the molar volume υ, the transformation strain εt and the transformation entropy ∆S), which also agrees better with experiments showing that Q−1 T r does not linearly depend on T˙ /ω. 5. Stoiber introduced a “fragmentation” parameter x(n) which represents additional anelastic deformations due to the migration of intervariant boundaries (Stoiber 1993, Stoiber et al. 1994):   ˙ T ∂n ·f . (3.5) Q−1 T r = k · x(n) · ∂t ω · σ0 More models have been discussed by San Juan and P´erez-S´aez (2001). Important Factors ˙ Q−1 tot increases with heating or cooling rate T via the corresponding changes −1 ˙ in QT r , known as the so-called “T effect”. At T˙ = 0, Q−1 T r = 0; in that case, some frequency dependence was measured in the low-frequency range (Pelosin and Rivi´ere 1997). If T˙ = 0, Q−1 T r is inversely proportional to the frequency;

3.2 Internal Friction at Phase Transformations

125

Fig. 3.2. Dependence of IF on oscillation amplitude (ε0 ) measured in Cu–Zn–Al martensite according to Koshimizu et al. (1979). Domain C: ε0 < 10−6 ; domain B: ε0 < 10−5 ; domain A: ε0 > 10−5

on the other hand, in the martensitic phase the (intrinsic) IF is frequencyindependent apart from some relaxation peaks. Three amplitude domains can be distinguished in martensite, as suggested by experimental data on Cu–Zn–Al (Koshimizu et al. 1979, see Fig. 3.2) and ucke Au–Cd (Zhu et al. 1983). At low amplitudes (10−7 −10−6 ), the Granato–L¨ mechanism can be accepted (Granato and L¨ ucke 1956a, 1956b; see Sect. 3.3). No clear amplitude dependence is observed between 10−6 and about 10−5 , whereas above that range the amplitude dependence is clearly seen again. For many different alloys, the critical onset values of the second amplitudedependent range are found between 5×10−6 and 2×10−5 . Several models were applied to this latter range, i.e., by Granato and L¨ ucke (1956a,b), Takahashi (1956) and Peguin et al. (1967). At constant amplitude, an additional time dependence of damping (either decreasing with time or going through a maximum, following structural changes in the material) was observed in NiTi and CuAlZn e.g., by Mercier et al. (1979), Morin et al. (1985, 1987b), Van Humbeeck and Delaey (1982, 1984) and Van Humbeeck et al. (1985). In several Mn–Cu based alloys, a magnetic field increases the damping in the premartensitic range, above the IF peak temperature at the MT, during cooling but not during heating (Markova 2004, Markova et al. 2004). This contribution is characterised by the area SQ between the TDIF curves measured with and without magnetic field, which is proportional to the magnetic field applied (Fig. 3.3). The effect is explained by weak ferromagnetism in antiferromagnets due to spinodal decomposition and an inhomogeneous distribution of Mn atoms. Some Selected Materials (see also Van Humbeeck 2001) Ni–Ti-Based Alloys Ni50 Ti50 -based alloys with an approximate content of Ni between 48 and 52 at% are known as shape-memory alloys (Saburi 1989, 1998; Wayman

126

3 Other Mechanisms of Internal Friction SQ arb.un.

1200

80

Mn-Cu-Ni

Q-1, 10-4

60

Ms

5

800

Mn-Cu-Cr-Ni Mn-Cu-Ni Mn-Cu-Cr Mn-Cu

400

4 0

40

3

1

2 H[104A/m]

20

2 1

0 80

110

140

170

200

230

T,⬚C

Fig. 3.3. Influence of a magnetic field on TDIF at 1 Hz for Mn–Cu–Ni, with (1) H = 0; (2) H = 0.6 · 104 A m−1 ; (3) H = 1.3 · 104 A m−1 ; (4) H = 1.9 · 104 A m−1 ; (5) H = 2.5 · 104 A m−1 . Inset: Influence of magnetic field on the damping in the premartensitic range (area SQ , see text) in the Mn–Cu based alloys Mn80 Cu20 , Mn80 Cu17 Ni3 , Mn80 Cu17 Cr3 and Mn80 Cu14 Ni3 Cr3 (from Markova et al. 2004)

1989; Markova et al. 1996; Golovin et al. 1997b; Ilyin et al. 1998; Ilchuck and Moravic 1998; Coluzzi et al. 2000; Golyandin et al. 2000; Biscarini et al. 2003a; Igata et al. 2003a; Yoshida and Yoshida 2003; Straube et al. 2004; Mazzolai et al. 2004). During cooling in the temperature range between 150 and −40◦ C (depending on the Ni content), the high-temperature B2 phase is transformed into the B19′ monoclinic martensite. In some cases the martensitic transformation is preceded by formation of the rhombohedral R-phase, followed by the R-to-B19′ transformation (an example is given in Fig. 3.4.). At heating only the reverse MT takes place. Adding a third element, i.e., replacing some Ni or Ti atoms, helps to adapt Ni–Ti alloys to different applications by influencing the strength, ductility and parameters of the MT (Eckelmeyer 1976, Honma et al. 1979, HuismanKleinherenbrink 1991, Kachin 1989, Kachin et al. 1995): Cu decreases and Nb increases the transformation hysteresis; Fe, Cr, Co, Al lower and Hf, Zr, Pd, Pt, Au rise the transformation temperatures; Mo, W, O, C strengthens the matrix, H increases the total damping around room temperature. The effect of hydrogen on internal friction in Ni–Ti alloys (Coluzzi et al. 2006a,b) is discussed in Sects. 2.2.4 and 3.5. Cu-Based Alloys These are Cu–Zn-, Cu–Al- and Cu–Sn-based alloys with the structure of the Hume–Rothery β phase and fcc Cu–Mn-based alloys, which all show the martensitic transformation.

3.2 Internal Friction at Phase Transformations Q−1,10−4 B19’

← R ← B2

450

127

0.9

f cooling

f,

heating

Hz

Q−1 300

cooling

0.8

heating 150

1 K / min 0.7

0 200

250

300

350

T, K

Fig. 3.4. Temperature dependencies of IF (Q−1 ) and resonance frequency (f) at cooling and heating for a Ti–50.6at%Ni alloy after annealing at 575◦ C (Golovin et al. 1997b)

Most of the former alloys have disordered bcc structures at high temperatures which order to the B2, D03 or L21 structures at lower temperatures. They are often used as Hidamets (Sect. 3.5) because of their very high damping, and are also shape-memory alloys (Favstov et al. 1980, Favstov 1984, Kustov et al. 1996a,b, Shen et al. 1996, Coluzzi et al. 1996, Deborde et al. 1996, Kustov et al. 2000, Mazzolai et al. 2000a, Covarel et al. 2000, Van Humbeeck 2001, Kustov et al. 2006). Damping in fcc Cu–Mn alloys is very sensitive to heat treatment and carbon content (Birchon et al. 1968, Sugimoto et al. 1973, Men’shikov et al. 1975, Vintaykin et al. 1978, Sugimoto 1981, Kˆe et al. 1987a, Udovenko et al. 1990, Laddha and Van Aken 1997, Markova 1998, 2002, 2004, Kostrubiec et al. 2003, Pelosin and Rivi´ere 2004, Recarte et al. 2004). These alloys can lose about 50% of their damping capacity during natural ageing even at room temperature. This effect can be prevented by additional alloying. Fe-Based Alloys The thermoelastic (“non-ferrous”) type of MT is also found in some Fe-based alloys. Depending on composition, the austenite phase (γ) can be transferred into three kinds of martensite (α′ (bcc), ε(hcp) and fct martensite), accompanied by the shape-memory effect (Delaey 1995, Gu et al. 1994, Maki 1998). Most attention has been paid to alloys with the γ → ε transformation like Fe–Mn, Fe–Mn–Cr, Fe–Mn–Si, Fe–Mn–Al or Fe–Cr–Ni (Sato et al. 1982a,b; Volinova et al. 1987; Volinova and Medov 1998; Sato 1989; Robinson and McCormick 1990; Lee et al. 2003a, 2004; Okada et al. 2003; Igata et al. 2003a, 2004; Wan et al. 2006; Sawaguchi et al. 2006; Dong et al. 2006). The amount

128

3 Other Mechanisms of Internal Friction

of ε martensite plays the most important role if high damping is required (see Sect. 3.5), the specific damping capacity is about 20–25% (see also two symposium proceedings on High Damping Materials: J. All. Comp. 355 (2003) and Key Eng. Mater. 319 (2006)). Isothermal Martensitic Transformation Martensitic transformations of the ferrous type occur as athermal barrierless transformations when a certain driving force is achieved, or with the help of a thermally activated process at lower driving force. Type and kinetics of the MT in a certain Fe-based alloy depend on the structure and on the state of the high-temperature austenite phase, and may be either athermal or isothermal (Kurdyumov 1949a, Kurdyumov and Maksimova 1948, 1950, 1951; Maksimova 1999). In the latter case special attention is paid to the interrelated development of two subsystems in the crystals: the formation and growth of interphase boundaries, and the migration of lattice imperfections (Roitburd 1978; Roytburd 1995, Wang and Khachaturyan 1997). The redistribution of defects in the lattice may affect the mode of MT substantially (Olson and Cohen 1976, Clapp 1995, Golovin et al. 1999). The IF method was first applied by Rodrigues and Prioul (1985, 1986) to study the isothermal MT in Fe–Ni–Mo alloys which, exhibiting different kinetics and types of the γ → α transformation, give a prominent example as regards the effects of composition and initial morphology on the transformation. By measuring IF during cooling in the ranges of athermal and isothermal MT, both by low- and high-frequency techniques, the influence of the dislocation-impurity interaction on the MT was studied in several Fe–Ni, Fe–Ni–Mo and Fe–Ni–Cr steels, and estimations of the activation energy for the isothermal MT were given (Golovin et al. 1994, 1995, 2000; Golovin and Golovin 1996). It was thus shown that IF is sensitive to the isothermal MT and permits to follow its kinetics. Several IF peaks occur in connection with the isothermal MT: the FR relaxation peak in austenite, a transformation peak including transient and intrinsic components, and a peak in martensite; these peaks give information on the behaviour of interstitials and dislocations before, during and after the diffusionless transformation. From this viewpoint, two states of Fe–Ni–Mo austenite may be distinguished with respect to their effect on the MT (see Fig. 3.5): (a) A state with saturated atmospheres of interstitial atoms around dislocations (weak amplitude dependence of IF and existence of an FR peak, see Sect. 2.2.2, indicating pinned dislocations and supersaturation of the fcc solid solution with C, respectively), resulting in an athermal MT during cooling. (b) A state with non-saturated dislocation atmospheres (stronger amplitude dependence and absence of an FR peak), connected with an isothermal MT component.

3.2 Internal Friction at Phase Transformations (a)

0.50

(b)

0.45

QFR

0.40 2 0.35 0.30

0 0.02

0.04

0.06

0.08

0.10

C, %

0

MSiso

−50

MSath

MS

4

−1

QFR−1 (104)

tgαADIF

tgαADIF

0.00

129

−100 −150 0.00

0.02

0.04

0.06

0.08

0.10

C, %

Fig. 3.5. Effect of the carbon content in Fe–24Ni–5Mo on (a) the amplitude dependence of IF (slope tgαADIF of IF taken in the linear part) and the height Q−1 FR of the FR peak and (b) the start temperatures (in ◦ C) for the athermal and isothermal martensitic transformations

The correlation between dislocation pinning (suppression of ADIF), stress relaxation in austenite and the martensite points (MS.iso , MS.ath ), i.e., the type of MT kinetics, shows the importance of dislocation–interstitial interaction. Dislocation pinning by interstitials in austenite therefore leads to an athermal MT and to a suppression of the isothermal MT component. The activation energies of the isothermal MT, 15–35 kJ mol−1 in Fe–Cr– Ni–Mo and 3–12 kJ mol−1 in Fe–Ni–Mo compared to 135–145 kJ mol−1 for the C diffusion in austenite, cannot be explained by diffusion processes but seem to be consistent with a dislocation-assisted nucleation of isothermal martensite (Golovin et al. 1996b, 2000). The binding energy between dislocations and interstitials in Fe–Ni–Mo alloys is found to be ≈10 kJ mol−1 . 3.2.2 Polymorphic and Other Phase Transformations Polymorphic Transformations In the vicinity of polymorphic phase transition temperatures, high maxima or jumps of IF may occur (Figs. 3.6–3.7). For example, maxima were observed in Zr (Garber et al. 1976, Boyarskij 1986), Co (Selle and Focke 1969a; Boyarskij et al. 1986; Bidaux et al. 1985, 1987), Ti (Wegielnik and Chomka 1975, Semashko et al. 1988), La (Dashkovskij and Savitskij 1961, Pan et al. 1985), Ce (Sharshakov et al. 1977, Postnikov et al. 1975a, Korshunov et al. 1981), Nd (Maltseva et al. 1980), Tl (Mordyuk 1963, Maltseva and Ivlev 1974), Fe (Dashkovskij et al. 1960a), Np (Selle and Rechtien 1969b), whereas jumps were found in U and Pu (Selle and Focke 1969a). According to Selle and Focke (1969a), the mechanisms of allotropic transformations can be classified by comparing the IF of both phases directly above and below the transformation temperature: considerably different values indicate a diffusive transformation, and approximately equal ones a shear

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3 Other Mechanisms of Internal Friction

Fig. 3.6. Changes in internal friction at f = 1 Hz of (a) uranium and (b) plutonium during polymorphic transformations. Solid lines: heating; dashed lines: cooling (Selle and Focke 1969a)

transformation. For instance, in U all transformations are diffusive (Fig. 3.6a); in Pu α → β, δ → δ′ and δ′ → ε are shear transformations, while β → γ and γ → δ are diffusive ones (Fig. 3.6b). IF maxima at polymorphic transformations have the following features: 1. They have a non-Debye shape and are often narrow. 2. They do not shift in temperature with changes in the measurement frequency, that is, they are not thermally activated. 3. They display a temperature hysteresis: on heating, they occur at higher temperatures than on cooling, and also the respective peak heights may differ significantly (Fig. 3.7). Magnetic Transformations Maxima of mechanical losses, as well as steps due to different IF in the different phases, may appear at magnetic phase transitions as well. For example, Fig. 3.8a shows the maxima of ultrasonic attenuation found in dysprosium, where transitions exist both from paramagnetic to antiferromagnetic (N´eel temperature, 174 K) and from antiferromagnetic to ferromagnetic states (Curie temperature, 84 K). These non-relaxation maxima are located at somewhat lower temperatures than the phase transitions themselves, as predicted by Landau and Khalatnikov (1954), due to the effect of stresses on the magnetic ordering processes in the spin system and related fluctuation processes. This kind of internal friction is essential only at high frequencies in a narrow temperature interval close to the respective phase transition. Another kind of IF behaviour is shown in Fig. 3.8b for terbium single crystals (Shubin et al. 1985), where transitions occur from FM to AFM at 214–221 K (θ1 ), and from AFM to PM at 225–230 K (θ2 ). At θ1 and θ2 , respectively, a bending and a sharp drop in Q−1 can be seen with increasing

3.2 Internal Friction at Phase Transformations

131

Fig. 3.7. Changes in internal friction of neptunium during the β → γ and γ → β polymorphic transformations (f ≈ 1.5 Hz; Selle and Rechtien 1969b)

Fig. 3.8. Temperature-dependent effects in rare-earth metals: (a) longitudinal (α1 ) and transversal (αt ) ultrasonic attenuation in a Dy polycrystal (f = 10 MHz; Rosen 1968c); (b) IF and elastic modulus of a Tb single crystal (f = 1 kHz; Shubin et al. 1985)

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3 Other Mechanisms of Internal Friction

temperature. The difference between these two examples is due to the fact that Dy was studied at a frequency of 10 MHz, but Tb at about 1 kHz. At 1.3 Hz the maximum at the PM → AFM transition in Dy is absent, too, and a stepwise increase in IF is observed (Sharshakov et al. 1978a). In chromium, a non-relaxation IF peak at the N´eel temperature (TN = 310-312 K) was first reported by Fine et al. (1951) and then confirmed by many authors (see Table 4.7.1). Another magnetic IF effect due to a spinflip transition at Tsf was found by De Morton (1963) and Street (1963), and again confirmed by many authors (Table 4.7.1). Between the N´eel and spinflip temperatures the spin-density waves are transversely polarised, i.e., the polarisation vector P is perpendicular to the wave vector Q (TSDW); below TSF the polarisation is longitudinal (LSDW) and P is parallel to Q (Dubiel 2003); a related measurement is shown in Fig. 3.9. The IF peak at the N´eel temperature can be asymmetric or even double-shaped, which may be explained by the influence of internal stresses of different origin (Pal-Val et al. 1989, Golovina and Golovin 2004). In steels an IF peak due to a magnetic transformation in Fe3 C was detected by Sudnik et al. (1974), and confirmed by Rokhmanov and Sirenko (1993). Melting In the vicinity of the melting point at T ≈ (0.94–0.99)Tmelt, , internal friction maxima are observed due to grain boundary melting (Postnikov et al. 1963,

Fig. 3.9. Temperature dependence of Q−1 (bigger grey filled circles) and f (smaller black points) for chromium near the N´eel point (≈ 311 K) and spin-flip transition (≈ 120 K) (Golovina and Golovin 2004). The dotted lines show the influence of quenching from 1100◦ C on the peak near the N´eel point. The intervals of magnetic transformations are shown in the bottom (after Dubiel 2003)

3.2 Internal Friction at Phase Transformations

133

Table 3.1. Internal friction peaks due to grain boundary melting metal f (Hz) Tm (K) Tm /Tmelt Qm −1

Bi

Cd

Pb

Sn

548 538 0.99 0.07

636 573 0.97 0.02

524 563 0.94 0.01

707 493 0.98 0.05

Drapkin et al. 1980, Drapkin and Kononenko 1981; see Table 3.1), which are in fact pseudo-maxima on the high-temperature background due to a decrease in internal friction. There have been some attempts to use this effect for highdamping applications (see Sect. 3.5). Superconducting Transition The transition to the superconducting state is accompanied by the evolution of hysteresis maxima, e.g., in Nb (Postnikov et al. 1975b; Pal-Val et al. 1993) or Ta (Miloshenko and Shukhalov 1976), or by a drop in the internal friction curve (e.g., Nb–H; Wang et al. 1984). In perfect single crystals, it is necessary to introduce defects by preliminary plastic deformation for the maxima to appear. 3.2.3 Precipitation and Dissolution of a Second Phase A special group of first-order phase transitions produces, by means of nucleation and growth, a two-phase mixture consisting of precipitates in a host matrix, in general with different compositions and structures (Hardy and Teal 1954, Kelly and Nicholson 1963). If an external stress induces some change in shape and distribution of these precipitates (by affecting volume, elastic constants, transformation temperatures, and/or different defect mobilities), this will result in internal friction. The high sensitivity of IF for any structural change can be used to monitor changes in the precipitated state; however, owing to the wide variety of possible reasons for the IF signal, the identification of the relevant mechanisms is usually a difficult task and requires support by other methods (e.g., DSC, TEM, X-ray diffraction, thermoelectric power), which detect the microstructural changes in a more direct way. For the simple case of an isotropic material, Krivoglaz (1960, 1962) proposed a theory for anelastic relaxation of the bulk modulus in such a phase mixture, controlled by the time for the phase transformation to occur. In case of two components, this time is governed by the difference in composition between precipitate and matrix as well as by diffusion, resulting in a relaxation time τσ (affected by the external stress σ) τσ =

r03 , 2Dx2

(3.6)

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3 Other Mechanisms of Internal Friction

where r0 is the radius of the particles (assumed to be spheres), and x2 is the volume fraction of the second (particle) phase. The first-order transition (nucleation and growth) implies the occurrence of a temperature hysteresis in sequential heating and cooling experiments, connected with dissolution and re-precipitation of the particles. As a characteristic feature of (3.6), important for identifying the nature of the resulting damping peak, the peak position on the temperature scale is independent of the vibration frequency, which only affects the peak height. Many examples can be found in the literature, as reviewed by Nowick and Berry (1972) and more recently by Schaller (2001b); a typical one is shown in Fig. 3.10. In this case (and also in systems like Al–Cu or Cu–Ag), precipitation from the homogeneous solid solution – produced by quenching from a sufficiently high temperature – starts with the formation of coherent solute clusters or Guinier–Preston (GP) zones, which grow while the surrounding matrix is depleted from the solute. In order to reduce strain and surface energy, the system may transform via metastable configurations into incoherent precipitates which finally attain their equilibrium structure and shape. Such transformations can be followed by mechanical spectroscopy in different ways: besides the IF signals found at the transformation itself like in Fig. 3.10, one can also monitor the height of the Zener peak (cf. Sect. 2.2.4), which decreases during precipitation as the number of reorientable, free solute pairs is diminished. Another possibility is to follow the decrease of the dislocation hysteresis peak due to the blocking of dislocation motion by the evolving precipitates (Schaller 2001b). The signals associated directly with the precipitation reactions, as in Fig. 3.10, may consist of one or several new relaxation peaks corresponding to

Fig. 3.10. Damping spectrum (Q−1 ) and elastic modulus (∝ f 2 , f = frequency) of Al–20wt%Ag, (1) after quenching, and (2) after 10 min annealing at 520 K. Peak P1 corresponds to the formation of GP zones from solid solution, and P2 to the transformation of the GP zones into metastable γ ′′ precipitates. These transform during the annealing treatment into stable γ ′ precipitates which produce peak P3 . (From Schaller 2001b)

3.2 Internal Friction at Phase Transformations

135

the various configurations of precipitates, superimposed on the increasing high-temperature IF background (cf. section “Damping Background at Elevated Temperatures” in Sect. 2.3.3). These peaks may arise from the anelastic strain due to stress-induced shape changes of the precipitates themselves (Damask and Nowick 1955). As such shape changes occur by dissolution on one side and precipitation on the other side of the particle, it will be governed by solute diffusion in the interface region, thus the activation energy of the process is expected to be somewhat less than that for solute diffusion in the bulk. In more detail, according to Schoeck and Bisogni (1969) and Schoeck (1969), the movement of partial dislocations around the precipitates may be involved, in agreement with electron microscopic observations, with dragging of solutes by the moving dislocations (cf. Sect. 2.3). Miner et al. (1969) propose ledge motion along the edges of the precipitate particles, and Entwistle et al. (1978) consider the relaxation of atomic groups within individual clusters involving vacancies, thus a point-defect process (Sect. 2.2). Even more spectacular IF effects are observed in case of discontinuous precipitation which starts at favourable places in the sample, such as grain boundaries or dislocations, subsequently spreading out into the interior, e.g., in Al–Zn (Nowick 1951). As the same mechanism may produce both the hightemperature background and some characteristic relaxation peak, these are difficult to separate (cf. Sect. 2.3.3). The reason for internal friction again lies in processes within the interface between precipitate particles and matrix, e.g., of the types considered in Sect. 2.4.1. Some features of the IF mechanisms due to precipitates have been compiled in the review by Schaller (2001b), and can be divided into (a) Interface relaxation, considered by Schoeck (1969) who distinguished between volume changes of coherent precipitates, and modulus changes due to both coherent and incoherent particles. While the first two involve elastic strain energy due to the imposed stress, the last one comprises shear along the interface if the local stress exceeds some critical stress value. This implies that only incoherent and semi-coherent precipitates should give a hysteresis relaxation peak, while the coherent ones only contribute to the background damping. This has been verified with the systems Al–Cu (Hanauer et al. 1972), Cu–Co (Mondino and Schoeck 1971) and Cu–Si (Mondino and Gugelmeier 1980). However, the situation may be more complex, because in Cu–Fe the coherent precipitates do produce a peak which disappears when they loose their coherency (Pelletier et al. 1975). In another approach by Kohen et al. (1975), an anisotropic interface mobility of the precipitates is suggested to cause relaxation, in particular if this interface mobility is high. (b) Relaxation within the precipitates, e.g., in Al–Ag alloys with semicoherent, hcp γ ′ (Ag2 Al) precipitates, which can be distinguished from interface relaxation by its stability against overageing (Ostwald ripening, with strongly varying volume-to-surface ratio; Merlin et al. 1978,

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Schaller 1980). This was corroborated by a test with a sample consisting of the pure γ ′ phase (Al–85at%Ag, Schaller and Benoit 1980). In this case the relaxation is explained by a reorientation of elastic dipoles within the γ ′ precipitate, i.e., a Zener effect. (c) Internal friction, of only partly anelastic relaxation character, by diffusion-controlled movement of interfaces at sufficiently elevated temperatures beyond the Zener effect. Corresponding “precipitate peaks” near the solvus have been observed by Kiss et al. (1986) in Al–Ag. The common experiments performed with gradually increasing temperature (e.g., with T˙ = dT /dt = const) may exhibit peaks near the transition temperature which do not reflect the true precipitation kinetics because of ongoing transformation during the temperature rise. This problem can be avoided in isothermal experiments with variation of frequency where the microstructure has been stabilised by previous annealing. For example, this method permitted to detect a new (not thermally activated) relaxation peak due to θ precipitates in the 2024 Al alloy (Rivi`ere and Pelosin 2000), and was also applied by Ogi et al. (2000) to study the Snoek relaxation in a Cuprecipitated alloy steel. Internal friction due to phase precipitation and dissolution has been investigated best for the cases of hydrides and deuterides in metals of the groups IV and V, as well as in Pd (“hydride precipitation peak”, see also Sect. 2.2.4): V–D (Cannelli and Mazzolai 1971, 1973; Yoshinari et al. 1977, 1978), V–H (Cannelli and Mazzolai 1970; Koiwa and Shibata 1980a,b; Yoshinari and Koiwa 1982a; Cannelli et al. 1983, 1984b), Nb–D (Buck et al. 1971), Nb–H (Wert et al. 1970; Yoshinari and Koiwa 1982a,b; Yoshinari et al. 1985), Ta–D (Cannelli and Cantelli 1974, 1975), Ta–H (Cannelli and Mazzolai 1969, Yoshinari et al. 1980; Yoshinari and Koiwa 1982a; Li et al. 1985), Ti–D (Numakura and Koiwa 1985; Kato et al. 1988), Ti–H (K¨ oster et al. 1956a; Tung and Sommer 1974; Ritchie and Sprungmann 1982; Gong et al. 1990), Zr–D (Provenzano et al. 1974; Numakura et al. 1988), Zr–H (Provenzano et al. 1974; Ritchie and Sprungmann 1983; Numakura et al. 1988; Pan and Puls 2000), Pd–H (Yoshinari and Koiwa 1987). During cooling, internal friction begins to increase abruptly at the precipitation temperature, before a non-relaxation IF peak appears below this temperature. This peak has a λ form with two com ˙ /f . T ∝ ponents, i.e., an equilibrium value and a transient damping Q−1 Tr

3.3 Dislocation-Related Amplitude-Dependent Internal Friction (ADIF) In Sect. 2.3, those dislocation-related relaxation mechanisms were considered which can be regarded as purely anelastic, i.e., as linear and amplitudeindependent, and which prevail at very low vibration amplitudes. However, with increasing amplitude of applied stress or strain, the dislocations may

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

137

break away from their pinning points, move over considerable distances, and even start to multiply (Hull and Bacon 1984, Hirth and Lothe 1968), resulting in an increasing amplitude dependence of internal friction caused by non-linear relaxation and/or hysteretic phenomena (Gremaud 1987). Although the data part of this book (Chaps. 4 and 5) generally does not include ADIF data, a few aspects of ADIF will be treated here because their knowledge is important for separating the different damping components and for identifying the relevant relaxation mechanisms. ADIF studies are extensively applied to investigate the microstructure and its evolution with deformation and heat treatments in various materials. As an example, Fig. 3.11 shows typical ADIF curves of an Al–Mg alloy (Schwarz 1985). Starting from a plateau value at low amplitudes (note the logarithmic scale!), the damping rises exponentially if a certain amplitude is exceeded, indicating some critical stress necessary to overcome obstacles for dislocation motion (see also, e.g., Nishino et al. 2004, Trojanova et al. 2004, G¨ oken et al. 2005). An extended and thorough review of important results, measuring techniques and theoretical treatments of dislocation mechanisms can be found in the paper by Gremaud (2001), who considers five categories of dislocation mechanisms leading to different kinds and amounts of ADIF: (a) Relaxation models which include drag mechanisms acting continuously along the line of the moving dislocation, such as phonon and electron

Fig. 3.11. Amplitude dependence of the decrement δ(= π · Q−1 ) for Al–0.1at%Mg at 128 K before (a) and after (b) plastic deformation (bending with surface strain of ≈ 0.02). (From Schwarz 1985)

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3 Other Mechanisms of Internal Friction

drag; these are relevant at very high frequencies and show at most a slight amplitude dependence, if any. (b) Drag resulting from the diffusion of point obstacles distributed on the dislocation segments (i.e., point defects moving together with the bowing dislocation segment at sufficiently high temperature, as in the cases of Hasiguti or Snoek–K¨ oster peaks), where in addition to what has been said in Sect. 2.3, also a slight amplitude dependence may occur. (c) Dissipation due to breakaway of the dislocation from immobile point obstacles segregated at the dislocation line, yielding a pronounced ADIF from the resulting hysteresis of the stress–strain curve (Fig. 3.12a). As the dislocation will be repinned by the same point obstacles during its reverse motion, the damping is of quasi-static hysteresis type. In the kHz range, the breakaway time is negligible and the shaded area in Fig. 3.12a corresponds to the dissipated power per stress cycle, as first treated by Granato and L¨ ucke (1956a). The related anelastic strain is taken as

Fig. 3.12. Dislocation models (dislocation loops and obstacles) and stress-strain hysteresis loops for (a) dislocation breakaway from obstacles along the dislocation line (case (c)) and (b) dislocation motion across a random array of point obstacles in the slip plane (case (c)). The shaded area indicates the energy loss during a full vibration cycle; the different symbols are explained in the text. In (a) the situation with a stress less than the Frank–Read stress for dislocation multiplication is shown. (From Gremaud 2001)

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

εd = Λ b y m

139

(3.7)

with the dislocation density Λ and the mean effective shift ym of the dislocation averaged over its length. Taking the distribution of segment lengths l between pinning points in a loop of length L as f(l) = 2 ) exp(−l/lm ) with an average segment length lm (Koehler 1952, (Λ/lm Granato and L¨ ucke 1956a, Gremaud 2001), a damping contribution of
 K 2 Ec Kt Λ exp − (3.8) Q−1 = ε0 lm ε 0 is found, with Kt = 4Ω(1–G)L3 /(π4 lm ) (depending on the crystal orientation Ω) and K2 = ∆G/G, where Ec is the dislocation–obstacle interaction energy and ε0 the applied strain amplitude. Although this theory has been devised for T = 0 (athermal behaviour), it compares well with experiments at T > 0, where the breakaway from pinning points occurs with a thermally activated waiting time (relaxation time) τ = τ0 exp(Ec /kT ) (Teutonico et al. 1964, Granato and L¨ ucke 1981, L¨ ucke and Granato 1981). This leads to a similar expression as (3.8) with a slight additional frequency dependence of Q−1 , corresponding to an intermediate mode between relaxation and hysteresis. According to (3.8), ADIF may be used to extract information on dislocation density as well as on obstacle strength and concentration by means of the so-called Granato–L¨ ucke plot of log(ε0 Q−1 ) vs. 1/ε0 , which ideally gives a straight line with slope and intercept determined by the mean segment length lm and by the dislocation density Λ, respectively. Equation (3.8) may also be written as
 ∆ σ0cr σ0cr −1 Q ≈ · exp − (3.9) π σ0 σ0 with the critical yield stress σ0cr = f0cr /b¯l = nf0cr /bL, where σ0 , f0cr , L and l are the applied stress amplitude, the critical force on the obstacle at breakaway, the dislocation loop length between anchoring points, and the segment length between n point obstacles along the dislocation, respectively. The relaxation strength ∆ is found to be 2

Λb2 L ∆= 12Γ J

(3.10)

with the dislocation line tension Γ and the unrelaxed compliance J. The dependence of Q−1 on temperature and frequency enters through σ0cr which is mainly controlled by thermally activated processes depending on temperature and deformation rate. (d) Damping losses resulting from thermally activated motion of dislocations through a random array of point obstacles, providing a hysteresis different from that in case (c) (cf. Fig. 3.12).

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3 Other Mechanisms of Internal Friction

Neglecting first thermal activation, the dependence of damping on the stress amplitude σ0 becomes (Gremaud 2001)
 4∆ σ0cr σ0cr −1 Q = · (3.11) 1− if σ0 > σ0cr π σ0 σ0 and Q−1 = 0 if σ0 < σ0cr . Thus damping again depends indirectly on temperature and frequency and shows a maximum at σ0 = 2σ0cr . Taking now thermal activation into account, different dependencies are found in the various ranges of temperature and obstacle strength (see Gremaud 2001). With a transition stress σtr = 4∆Gm /h2 b depending on the maximum interaction energy ∆Gm between dislocation and point defect and on the average area h2 = 2d¯l of the glide plane per point defect (with distances d and ¯l between neighbouring point defects normal and along the average dislocation line, respectively), the cases σ0cr (T ) > σtr (Friedel’s (1964) statistics, in the low-temperature range) and σ0cr (T ) < σtr (Mott’s statistics (see Labusch 1970, Schwarz and Labusch 1978), in the higher temperature range may be distinguished. The corresponding Granato–L¨ ucke (G–L) plot of ln(σ0 Q−1 ) vs. 1/σ0 will now be positively curved (instead of the straight line for case (c)), as shown in the elucidating experiments by Schwarz (1981), Schwarz and Funk (1983), Fig. 3.13.

Fig. 3.13. Granato–L¨ ucke plots of ADIF in Al–0.1at%Mg at 300 K: (A) measured directly following a 10 min high-amplitude straining, (B) after several static anneals of 1, 5 and 10 min. (From Schwarz and Funk 1983)

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

141

The stress amplitude necessary to keep a constant ADIF value is found to provide a direct measure of the critical yield stress σ0cr , which thus can be studied by mechanical spectroscopy with the advantage that microstructural changes are largely avoided (Lebedev 1992, Lebedev and Ivanov 1993, Lebedev and Pilecki 1995), contrary to usual deformation experiments. An analysis of applicability of the G–L model to study the structure of different materials has shown the necessity to specify better in which interval of amplitudes the dislocation hysteresis plays the dominating role, i.e., in which amplitude intervals the G–L model can describe experimental data reliably (Krishtal et al. 1964). For that reason, critical amplitudes of deformation restricting this interval were introduced, by writing the ADIF (here as a function of strain amplitude, Q−1 (ε0 )) as (Golovin 1968):   m−2 C2 (ε0 − εcr1 ) ϕ2 (ε0 ) −1 exp , + ϕ3 (ε0 ) Q (ε0 ) = ϕ1 (ε0 ) + m ε0 − εcr1 ε0 − εcr1 εcr1 (3.12) where εcr1 and εcr2 are the first and second critical amplitudes. The resulting IF is amplitude-independent below εcr1 , is reversible and nearly linearly increasing with amplitude (dislocation hysteresis) between εcr1 and εcr2 , and becomes irreversible above εcr2 where the hysteretic loop is not closed, but new dislocations can be generated and microdeformation takes place. ϕ1 , ϕ2 , ϕ3 , C2 and m are numerical parameters. The study of damping mechanisms in different amplitude and frequency domains was carried out for materials with different types of structures (Beresnev and Sarrak 1966, Golovin et al. 1979, Levin et al. 1980, Lebedev and Kustov 1987, Dudarev 1988), which helped to understand microdeformation mechanisms and to determine their influence on possible fatigue damage at ε0 > εcr2 (Golovin and Puˇsk´ar 1980). With increasing temperature the point obstacles, which up to now have been considered as immobile, may attain some diffusional mobility which then leads to a transition from depinning to dragging. While pure dragging was considered in case (b), the transition region provides special difficulties (cf. Gremaud 2001). An anomalous behaviour of relaxation peaks (a maximum in the amplitude dependence of the peak height) was discovered by Kˆe (1950) in Al–Cu and Al–Mg alloys and discussed in terms of diffusion of solute atoms in the dislocation core (Kˆe 1985a, Fang and Kˆe 1985, 1996, Kˆe et al. 1987a, Fang 1997; Fang and Wang 2000a). (e) Finally, the hysteretic damping resulting from an athermal long-range interaction with point obstacles distributed in the bulk has recently been discussed and treated by Gremaud and Kustov (1999), and Kustov et al. (1999, 2006), who investigated this type of ADIF by detailed

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3 Other Mechanisms of Internal Friction

Fig. 3.14. Strain amplitude dependence of the decrement measured for a Cu– 7.6at%Ni single crystal at various temperatures (f = 100 kHz). The domains (I) (II) and (III) are explained in the text. (From Gremaud and Kustov 1999)

measurements in a wide temperature range in solid solutions of Cu–Ni alloys (example in Fig. 3.14). Their model includes short-range (localised) obstacles (i.e., solute atoms in planes adjacent to the dislocation glide plane), long-range (diffuse) obstacles (i.e., weak obstacles in some distance from the glide plane), and unbreakable pinners (i.e., nodes in the dislocation network). Two critical stresses σ0cr1 and σ0cr2 exist for the diffuse (athermal) and localised (thermally activated) forces, respectively, of which only the latter depend on temperature and frequency. This model predicts, in agreement with the measurements in Fig. 3.14, three domains: In domain (I) at low temperatures, neither σ0cr1 nor σ0cr2 are exceeded by the external stress, so that low-amplitude ADIF due to the dislocation motion over diffuse forces remains purely athermal. In domain II at more elevated temperatures and moderate amplitudes, σ0cr2 is exceeded but not σ0cr1 , resulting in a gradual increase of ADIF. In domain III at still higher amplitudes both critical stresses are exceeded, resulting in a steep increase of internal friction (Gremaud and Kustov 1999). These mechanisms also explain the so-called peaking effect during irradiation (Caro and Mondino 1981a, 1982), i.e., the development of an IF maximum as a function of irradiation time. Damping Effects in Fatigue Internal friction connected with fatigue is not only an extreme case of hysteretic damping, but has also been applied extensively as a method to study fatigue. Without discussing this in detail, readers should at least be provided with some important references. The high sensitivity of IF to monitor the accumulation of cyclic microdeformation has been pointed out by many authors (e.g., Lazan 1961, Koca´ nda 1972, Puˇsk´ar and Golovin 1985, Klesnil and Lukas 1992, Blanter et al. 1994,

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

143

(a)

(b) Fig. 3.15. Internal friction (torsion pendulum 0.3 Hz, surface strain < 10−5 ) of polycrystalline Cu wire after ultrasonic fatigue with 20 kHz: (a) after N = 4 · 105 cycles at different amplitudes ε0 , compared to an annealed sample (650◦ C/2 h); (b) after different cycle numbers N at constant ε0 = 3.5 · 10−4 . (From Bajons and Weiss 1971)

Puˇsk´ar 2001, Terentev 2002). For example, some fatigue-induced changes of temperature-dependent, low-frequency IF spectra are shown in Fig. 3.15. As concerns amplitude dependence, the relation between “critical” amplitudes (with respect to IF and related modulus changes) and the fatigue characteristics (W¨ohler curves, microstructures) has been discussed by Puˇsk´ar and Golovin (1985), Klesnil and Lukas (1992) and Vincent and Foug`eres (2001). It was shown that a clear correlation exists between fatigue lifetime and certain intervals in the ADIF curve, so that the fatigue behaviour at different amplitudes can be predicted from ADIF measurements. However, for further analysis (Puˇsk´ar 2001), a serious difficulty is the inhomogeneity of the dislocation structure.

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3 Other Mechanisms of Internal Friction

Such application of mechanical spectroscopy to fatigue problems is possible in different ways and with different objectives: (a) By applying deformation- or energy-based empirical criteria of fatigue life. The resistance to fatigue is known to depend on the area of the hysteresis loop (Feltner–Morrow criterion, see Morrow 1960, Feltner and Morrow 1961), and on the value of anelastic deformation which can also be evaluated from the hysteresis loop shape (Coffin–Manson criterion, see Coffin 1954 and Manson 1953). Some modifications of these criteria, including their applicability to different types of metallic materials in cases of low and high cycle fatigue, have been the subject of different studies (Langer 1962, Troschenko 1980, Troschenko et al. 1985, Puˇsk´ar and Golovin 1985, Puˇsk´ar 2001, Vincent and Foug`eres 2001). (b) By studying the mechanisms of deformation and damage accumulation in materials subjected to cyclic loading. Several models of interpreting dislocation-related IF during fatigue damage accumulation have been proposed (Peguin et al. 1967, Burdett 1971, Jon et al. 1976, Golovin et al. 1979), and analysed with respect to their applicability to different materials (Jon et al. 1976, Puˇsk´ar and Golovin 1985, Puˇsk´ar 2001). The similarity between parameters of micromechanical tests and of ADIF was pointed out by Golovin (1968), Browne (1972), Koca´ nda (1972), Golovin and Puˇsk´ar (1992). Ultrasonic tests after (Bajons and Weiss 1971) or during fatigue cycling (Hirao et al. 2000) were used to study the kinetics of the process, to find out a dislocation model for fatigue development (Vincent and Foug`eres 2001), and to control the material state under cycling (Schenk et al. 1960, Golovin et al. 1975). Additional aspects of IF originating from microcracks and other types of damage were considered e.g., by Golovin et al. (1975), Carre˜ no-Morelli (2001), Robby (2001), Schaller (2001a), G¨ oken et al. (2004), Golovin et al. (2004a) and Arkhipov et al. (2005, 2006).

3.4 Magneto-Mechanical Damping Applying a stress to a ferromagnetic material causes a variation of magnetisation due to the magneto-elastic coupling (magnetostriction), which results in the so-called “∆E effect” (i.e., an apparent reduction of Young’s modulus below the purely elastic value found in the magnetically saturated state), and also in a related dissipation of mechanical energy during loading/unloading or in case of vibration. The latter effect can give rise to a strong magneto-mechanical damping with stress-dependent and stress-independent components, used for application in high-damping materials (Chap. 3.5). Whereas the basics of magnetisation phenomena are treated in many standard textbooks (e.g., Bozorth 1951, Chikazumi 1966, Jiles 1998), more specific information on the corresponding internal friction effects can be

3.4 Magneto-Mechanical Damping

145

found in reviews by e.g., Smith and Birchak (1968, 1970), Berry and Pritchet (1975), Degauque and Astie (1980), Astie et al. (1981), Augustyniak (1994), Kekalo and Potemkin (1969), Kekalo et al. (1970), Kekalo (1973, 1986), Kekalo and Samarin (1989), Stolyarov (1991), Riehemann (1996), Blythe et al. (2000). The following short summary is partly based on an introduction into magneto-mechanical damping given recently by Degauque (2001). Considering five main contributions to the total energy of a ferromagnet without an external field (exchange energy Wex , magnetocrystalline anisotropy energy WK , magneto-elastic or magnetostrictive energy Wλ , magnetostatic energy Wm , energy of magnetic domain walls WW ), four main mechanisms of magneto-mechanical damping may be defined: – – – –

Magnetoelastic hysteresis damping (Qh −1 ) Macroeddy-current damping (Qa −1 ) Microeddy-current damping (Qµ −1 ) Damping at magnetic transformations (QPhT −1 ) (e.g., at Curie and N´eel temperatures, spin-flip transitions etc.; see Sect. 3.2.2).

Therefore, the total magnetomechanical damping QM −1 in ferromagnets can be considered as a sum of these components: QM −1 (ε, f, T ) = Qh −1 (ε, f, T ) + Qa −1 (f, T ) + Qµ −1 (f, T ) + QPh.T −1 where, contrary to Qa −1 and Qµ −1 , the hysteretic contribution Qh −1 depends on the strain amplitude ε. Macroeddy-Current Macroeddy-current amplitude independent damping (Qa −1 ) is a result of the eddy currents induced temporarily in an electrically conductive sample as a response to a stress-induced change dB/dσ in the total magnetic induction B. For the case of a partially magnetised rod of radius R vibrating with a periodic stress σ of frequency f, Qa −1 has been given by Bozorth (1951) in the low(Lf)- and high(Hf)-frequency limits, respectively, as Q−1 a(Lf ) =

π 4




dB dσ

2

R2 Ef ρ

and Q−1 a(Hf ) =

1 3/2

8π 2 µr




dB dσ

2


R2 f ρ

−1/2

,

(3.13)

where ρ is the electrical resistivity and µr the reversible magnetic permeability. Zener (1938a) also considered the damping due to macroeddy currents by summation of single Debye peaks. Some experimental examples given by Berry and Pritchet (1975, 1976, 1978a) have been discussed by Degauque (2001). Microeddy-Current Microeddy-current amplitude independent damping (Qµ −1 ) is due to the microscopic eddy currents caused by local changes of the magnetisation arising during the stress-induced displacement of magnetic domain walls. Unlike Qa −1

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3 Other Mechanisms of Internal Friction

which requires substantial magnetisation to be effective, Q−1 µ has its largest value near the demagnetised state (M/MS < 0.3, Cangana et al. 1971). Depending on the frequency range and on the type of motion assumed for the domain walls, different equations have been suggested for Q−1 µ , like Q−1 µ =A

El2 MS2 · f, µρσi2

(3.14)

for reversible motion of rigid domain walls at relatively “low” frequency, where µ is the magnetic permeability, MS the saturating magnetisation, and A a parameter depending on the level of internal stresses σi with wavelength l (Bozorth 1951, Williams et al. 1950) or Q−1 µ =

mω/β ′ GN a2 ¯l 2 · 2, 12Ww 1 + (mω/β ′ )

(3.15)

for reversible motion of flexible domain walls considered as elastic membranes pinned by dislocations (Latiff and Fiorre 1974). Here G is the shear modulus, N a2 the area per unit volume of domain walls with effective mass m, ¯l the average distance between pinning dislocations, and β ′ a viscous damping parameter. A similar expression was introduced by Mason (1951, 1958) in form of a Debye-type law Q−1 µ =

9µi λs E f /fr · , 20πMS 1 + (f /fr )2

with

fr =

πρ , 2 24µi Dw

(3.16)

where µi is the initial magnetic permeability, λS the saturation magnetostriction, and DW the domain size. Values for Armco iron (DW ≈ 3×10−3 cm, ρ ≈ 10−5 Ωcm, µi ≈ 800) give rise to a peak at fr ≈ 180 kHz; for the same material, taking the frequency value f in Hz, Coronel and Beshers (1998) proposed the relation f 3 . (3.17) Q−1 µ = 3 · 10 · 3.2 · 1010 + f 2 Magnetoelastic Hysteresis Damping Magnetoelastic hysteresis damping (Q−1 h ) is the most important and powerful type of damping in ferromagnets, so that the terms “magnetomechanical” or “magneto-elastic” are sometimes narrowed down to denote only this hysteretic, amplitude-dependent damping component. It is due to the stress-induced motion of non-180◦ domain walls (while the 180◦ walls are stress-insensitive), including irreversible Barkhausen jumps beyond a critical stress τcr . At low applied shear stresses τ or shear strains γ (i.e., in the “Rayleigh region” of the magnetisation curve), the hysteretic energy dissipated is (Kornetski 1938, Bozorth 1951, Degauque 2001): ∆Wh =

4 dG−1 3 ∆Wh τ , so that Q−1 . h = 3 dτ 2πW

(3.18)

3.4 Magneto-Mechanical Damping

147

As the vibration energy W varies with τ 2 , this implies a linear amplitude dependence of internal friction. For higher stresses, however, the hysteretic losses ∆Wh no longer increase with τ 3 but with a stress-dependent exponent 0 < n < 3 (Sumner and Entwistle 1959), and finally reach a saturation level; consequently, Q−1 h shows a maximum as a function of stress or strain amplitude. Relating the saturation value of ∆Wh to the magnitude of the “effective” internal stresses opposed to the movement of the domain walls, Degauque (2001) pointed out that the position and height of the amplitudedependent damping maximum of Q−1 h can be used to determine the level of these internal stresses, and found a value of 7 MPa for the case of high-purity iron if carefully recrystallised. Magneto-elastic ADIF curves measured with rising and falling shear amplitudes γ, respectively, do not necessarily coincide but can be different according to the details of magnetisation and demagnetisation. This behaviour may also be influenced by microplastic deformation processes, depending on the relative magnitudes of the peak position γh. max , the highest amplitude γ0 used in the tests, and the amplitude γcr2 of the onset of microplastic deformation (Golovin 1993, Degauque 2001; cf. Sect. 3.3). Moreover, these effects depend on temperature, time and static stress applied. The temperature dependence of Q−1 h , although always approaching zero at the Curie point, may vary from material to material (sometimes even with rather different literature data for similar materials), as discussed e.g., by Kekalo (1973) and Degauque (2001). Finally, a frequency dependence of Q−1 h is not expected as long as the time for an irreversible domain jump, related to the relaxation time of the microeddy currents, can be neglected: thus, Q−1 h should decline at frequencies where the microeddy current peak appears (Nowick and Berry 1972). To summarise, the different dependencies of magneto-elastic hysteresis damping on temperature, amplitude, frequency and magnetic field are given schematically, in simplified form, in Fig. 3.16. The total damping Qtot −1 of ferromagnetic high-damping materials (Sect. 3.5) can often be described approximately as the sum of magnetomechanical damping (QM −1 , mainly represented by the hysteretic component

Fig. 3.16. Schematic representation of the dependence of magneto-mechanical damping on temperature, strain amplitude, frequency and magnetic field (adapted from De Batist 1983)

148

3 Other Mechanisms of Internal Friction

Qh −1 ), and dislocation damping (Qd −1 ) as an efficient nonmagnetic damping source: Qtot −1 (ε, f, T ) ≈ QM −1 (ε, f, T ) + Qd −1 (ε, f, T ). Kekalo (1973) summarised the six most typical ADIF curves for different materials of this type, depending on the amplitude values for the maximum of Qh −1 (εh. max ) and for the beginning of irreversible motion of dislocations (εcr2 ), respectively (see Sect. 3.3). For separating the magnetic and nonmagnetic contributions, one may apply a saturating magnetic field to suppress the magnetic losses. Some examples of the competing ADIF contributions QM −1 and Qd −1 in Fe– Cr and Fe–Al alloys, measured at different temperatures, have been reported by Golovin (1994), Golovin et al. (2004b). A particular situation is met in some Fe-based metallic glasses (cf. Sect. 2.6) with also rather high magnetostriction values but small anisotropy energies (due to the absence of a crystal lattice), so that domain wall motion is supplemented and partially replaced by reversible and irreversible rotation processes of magnetisation. Besides some related modifications of the “normal” magneto-elastic damping and ∆E effects, a special “giant” ∆E effect has been found for a particular domain structure, achieved by a specific annealing treatment in a magnetic field (Kobelev et al. 1987; cf. Degauque 2001). Finally, the aforementioned damping QPhT −1 at magnetic transformations should also be noticed, in addition to the three kinds of magneto-mechanical damping (Qa −1 , Qµ −1 , Qh −1 ) described earlier. Found e.g., at kHz frequencies in a narrow interval around Curie (TC ) or N´eel (TN ) points, QPhT −1 may result from the influence of an external stress on the processes of magnetic ordering (Stolyarov 1991). Such effects have already been discussed in more detail in Chap. 3.2.2.

3.5 Mechanisms of Damping in High-Damping Materials High-damping metallic materials, often abbreviated as “Hidamets” or HDM (Birchon 1964, James 1969), are used in various practical applications like, for instance, reducing noise and vibration, preventing fatigue problems or increasing the quality of cutting tools. Contrary to this wide application, there is some confusion as concerns the physical mechanisms which can effectively produce high damping, including the definition of “Hidamets” themselves. Here we will briefly consider “engineering” and “physical” viewpoints on metallic materials with high intrinsic (“passive”) damping. Questions of active vibration control and resonance damping due to the shape or mass of a construction, as well as damping in non-metallic materials, are not subject of this chapter. One specific engineering viewpoint was summarised in a panel discussion at the International HDM Symposium, Tokyo 2002 (Igata et al. 2003b): “. . . the definition of Hidamets . . . is only possible when a particular application is specified”. This concept is directed to the practical needs for damping under specific operating conditions, but asks neither for the physical origin

3.5 Mechanisms of Damping in High-Damping Materials

149

of “high” damping nor for a more general definition of “Hidamets” as engineering materials. Consequently, under this viewpoint a material may already be called a HDM if, for one particular application, the special operating conditions (frequency, temperature) match the damping maximum of a narrow relaxation peak, while under deviating conditions the damping of the same material can be much lower. It is clear that this viewpoint, although reasonable for the specific engineering application considered, does not lead to an unambiguous classification of Hidamets. Therefore, for such a classification it has been suggested to specify the physical mechanisms which may reliably cause “true” high-damping properties; or in the words of Schaller (2003): “HDM . . . should exhibit high damping over a wide frequency and temperature range. In metals, only a limited number of dissipative mechanisms . . . can be used to achieve such performance. Point defect relaxations are not useful in this application because, generally, they give rise to relaxation peaks located in narrow frequency and temperature domains. A damping mechanism is active over a wide frequency range only if it is not thermally activated”. This latter, “physical” viewpoint seems more appropriate to clarify the physical (or structural) background for high damping, i.e., to define which materials should be called Hidamets (Golovin 2006a). Important steps into this direction were done in many earlier publications, to mention only a few of them: James (1969), Sugimoto (1974), De Batist (1983, 1994), Schaller and Benoit (1985), Golovin et al. (1987), Ritchie and Pan (1991b), Graesser and Wong (1992), Schaller (2001a). It is obvious that HDM must possess not only a stable and powerful source of damping but also a certain level of strength. From the viewpoints of physics as well as of engineering, therefore, the damping capacity for different materials should be estimated under homologous conditions, like THD = n · Tmelt , σHD = m · σ0.2 (with n, m < 1, e.g., m = 0.1 (James 1969) in case of anelasticity), relating the operating conditions (temperature THD and stress amplitude σHD ) for high damping to the melting temperature Tmelt and yield stress σ0.2 , respectively. On the other hand, for many engineering applications it is desirable to consider damping at room temperature, which leads to a somewhat different engineering classification of Hidamets (James 1969, Sugimoto 1974) than comparing different materials under strictly homologous temperature conditions e.g., with n = 0.3 (Golovin and Arkhangelskij 1966). Before turning to the mechanisms of high damping, a short historical survey of the problem might be useful to explain the importance of the “engineering” approach. As often found in history, the first HDM were used by people without deep knowledge about structure and physical mechanisms. This is especially true for cast iron which has been used for centuries in countless applications, e.g., in tools and processes like forging, pressing, stamping or extruding, in order to protect people and machinery (for the effect of vibration frequencies on the human body see Frolov and Furman 1990): although

150

3 Other Mechanisms of Internal Friction

being a cheap material, grey cast iron effectively reduces vibration and noise at plants, fabrics, and workshops (Golovin et al. 1980, Millet et al. 1985). In the next, more recent stage, a specific damping index (SDI) was defined in order to achieve a better comparability of different materials. The SDI is the specific damping capacity Ψ measured by means of a torsion pendulum at 1 Hz, under a surface shear-stress amplitude of one tenth of the 0.2% tensile yield stress, thus denoted as Ψ0.1σY S or more shortly as Ψ0.1 . By comparing many different materials, definitions of low-, medium- and high-damping materials were given by Ψ0.1 < 1%, 1% < Ψ0.1 < 10%, and Ψ0.1 > 10%, respectively (James 1969). Mg-based, Cu–Zn–Al, Mn–Cu and Ni–Ti alloys take the first places in such a classification. Already at this stage, the problem of standardisation and unification of tests of damping capacity became very important, not to say crucial, and has not been solved up to our days. As already mentioned in the Introduction (Chap. 1), measures of damping cannot be properly transformed into each other if the damping is very high, because the relative deviations increase almost exponentially with increasing damping values (Rivi`ere 2003). Moreover, different measuring techniques (free decay, resonance and sub-resonance methods, torsion, bending etc.) have different limitations and accuracies in case of high damping. Several helpful conclusions and recommendations on the interrelation of damping measures for materials with high damping capacity, and on the methods of measuring high damping, can be found in the articles by Golovin et al. (1987), and Graesser and Wong (1992). A problem of comparability of damping values arises in the experimental measurements on equivalent samples performed by several laboratories both in Western Europe (organised by Rivi`ere) and in Soviet Union (organised by Blanter, Golovin and Sarrak), where a lack of agreement was reported. Considering the SDI alone is not sufficient for selecting materials for applications requiring high damping at a certain strength level. Consequently, the next stage was to consider the damping capacity in connection with other mechanical properties. Plotting the SDI Ψ0.1 against the ultimate tensile strength σUTS (Sugimoto 1974, 1975; Fig. 3.17), high-damping materials were characterised by a parameter α = Ψ0.1 × σUTS > 100, or even > 1000, where Ψ0.1 and σUTS are given in % and kg mm−2 , respectively (or by α > 1000 or even 10000 if SI units are used). Other authors suggested to use the logarithmic decrement δ0.1 instead of Ψ0.1 ≈ 2δ0.1 (in case of freely decaying vibrations), and the yield stress σYS instead of the ultimate tensile strength, or its ratio to the density ρ in applications where weight is important: (δ0.1 × σYS )/ρ (Skvortsov et al. 1987, Golovin and Golovin 1989, Golovin and Sinning 2003a). As another combination of properties, the loss coefficient Q−1 vs. Young’s modulus E was proposed by Waterman and Ashby (1991). Most HDM are, with respect to their strength, exceptions from the common rule of thumb that damping of metals is inversely proportional to the yield stress. Such decoupling of damping and strength implies that the

3.5 Mechanisms of Damping in High-Damping Materials

151

Fig. 3.17. Sugimoto plot: specific damping index vs. tensile strength for different metallic materials

mechanism responsible for damping should not be identical to that one controlling mechanical properties like the yield stress. One can find in the literature of the last century a large number of different mechanisms proposed for high damping, and many different solutions for particular practical applications. Some high-damping alloys are well known in industry, e.g., Sonoston and Incramute (Mn–Cu based), Proteus (Cu–Zn–Al), Nitinol (Ni–Ti), Maximum (Mg–Zr based), Nivco (Ni–Co based), Silentalloy (Fe–Cr–Al based), Gentalloy (Fe–Mo–W), Trancalloy, Vacosil, etc. However, most of these Hidamets remain rather expensive materials. As an important step forward to bring more physical clarity into the problem of Hidamets (polymers and ceramics as non-metallic materials are not discussed in this book), a useful as well as simple structure-based approach was introduced by Kondratev et al. (1986), and discussed in detail by Golovin and Golovin (1989, 1990)). By considering the main “structural units” responsible for high damping (instead of the detailed physical mechanisms), all Hidamets were consistently divided into four sub-groups: 1. 2. 3. 4.

Hidamets Hidamets Hidamets Hidamets

with with with with

highly heterogeneous structure thermoelastic martensite magnetic domains easily moveable dislocations

152

3 Other Mechanisms of Internal Friction

Within these subgroups damping can be rather high, and stable with respect to varying test conditions (T , f, σ). It can be optimised with respect to other properties (strength, stiffness, etc.) by choosing a proper heat treatment or chemical composition, and shows as a rule of thumb a monotonically increasing amplitude dependence. On the basis of new developments, however, the addition of a fifth group was recently suggested by Golovin (in the aforementioned panel discussion, Igata et al. 2003b): 5. Hidamets with extremely high hydrogen concentration. This additional group mainly consists of some amorphous or crystalline, Zr- or Ti-based alloys capable of absorbing a high amount of hydrogen. The related relaxation peaks are really high, situated not far away from room temperature, and much broader that a Debye peak (see also Sect. 2.2.4), which at least partly compensates the disadvantages of a thermally activated mechanism for damping applications. These alloys are attractive from the practical viewpoint and may be collected in one group from the physical background, i.e., easily activated hydrogen atom jumps in metallic materials. An additional advantage of this mechanism is that the high damping is amplitude-independent and occurs already at very low vibration amplitudes. In the following, these five groups of Hidamets are characterised briefly with respect to their main damping mechanisms and limitations. A certain instability of the high damping values against heat treatment is a limiting factor in almost all cases. 1. Hidamets with highly heterogeneous structure. In these natural or artificial composites (see e.g., Schaller 2003, Skvortsov 2004a), the main damping mechanism is local plastic deformation of soft phases and at interphase boundaries. Examples for “natural” composites are cast iron (where the main contribution arises from dislocations in graphite), lead bronze or pseudo-alloys (e.g., Fe–Cu, Fe–Pb, Al–Zn, Al–Sn, Ti–Pb, Ti–Sn). An unlimited variety of additional opportunities and advantages is provided in case of artificial composites (including cellular metals, see later) to combine damping and other properties of metallic materials, and also to vary the properties in different directions. 2. Hidamets with thermoelastic martensite (e.g., Van Humbeeck 1996, 2003, San Juan and N´ o 2003, Lee et al. 2003a, Igata et al. 2003a, Wuttig 2003). Here the damping is mainly due to the hysteretic movement of interphase (e.g., martensite–austenite) or twin boundaries, stacking faults or dislocations in martensite (see also Sect. 3.2.2). These materials exhibit the second highest values of Ψ0.1 after pure Mg, and in some alloys (e.g., Cu– Zn–Al) damping can be as high as in Mg if tested in a proper temperature interval. Common examples are Ni–Ti, Cu–Mn, Cu–Zn–Al, Cu–Al–Ni, In–Tl and (α + β) Ti-based alloys, but also Fe–Mn-based alloys with ε martensite are included in this group. The main limitation is the martensitic transformation temperature, as the damping in the parent austenitic

3.5 Mechanisms of Damping in High-Damping Materials

153

phase is low. Important factors are also heat treatment and the frequency of vibration, with generally lower damping at high frequencies. 3. Hidamets with magnetic domains (e.g., Kekalo 1986, Coronel and Beshers 1998, Skvortsov 2004b, Udovenko and Chudakov 2006). The main mechanism is magneto-mechanical hysteresis, causing a pronounced peak of ADIF at ε ∼ 10−4 (see also Sect. 3.4). Materials are Fe, Ni, Co and their alloys, like Fe–Cr, Fe–Al, Fe–Cr–Al, Fe–Co, Ni–Co or Co–Zr. The main limitation is the external magnetic field, and heat treatment may also have a strong influence. While the absolute values of Ψ0.1 are lower than those of Mg, Cu–Mn or Ni–Ti, the advantage is that some Fe-based Hidamets of this type are relatively cheap and have good mechanical characteristics. 4. Hidamets with easily moveable dislocations (e.g., Sugimoto et al. 1977, Favstov 1984, Nishiyama et al. 2003, Schaller 2003, Trojanova et al. 2003, Mielczarek et al. 2006, Trojanova et al. 2006). The main mechanism is dislocation movement; see also Sects. 2.3 and 3.3. Materials are Mg-based alloys (with Zr, Ni, Si etc.) which probably exhibit the highest values of Ψ0.1 , and some austenitic steels. Heat treatment is an important factor, and low weight an additional advantage. The main limitation is the low yield stress in pure Mg, as both damping and strength are controlled by the same mechanism, i.e., dislocation mobility. The problem can be solved at least partially by creating structures of natural composites as suggested by different authors (Table 3.2).

Table 3.2. Damping in some Mg alloys (group 4) (a) James (1969), Driz and Rokhlin (1983), Favstov (1984) alloys (group 4)

σ0.2 (MPa)

Ψ0.1 (%)

21 61 45 52 120 130 190 280

60 48 60 52 4 0.27 6.5 0.2

Mg 99.9% Mg 99.9% Mg–0.6Zr Mg–0.7Si Mg–9.8Al–0.2Mn Mg–9Al–2Zr–0.2Mn Mg–3Al–1Zn–0.4Mn Mg–3Th–1.2Mn

state as cast CW as cast as cast as cast as cast CW CW

(b) Schaller (2003) alloys (group 4) Mg–5Ni Mg–10Ni Mg–15Ni Mg–22.6Ni (eutectic) Mg–6Al–3Zn (AZ63)

σ0.2 (MPa)

Ψ (%) (ε = 10−5 )

68 117 163 170 113

4.7 4.6 3.8 1.5 0.1

Q−1 × 103 (ε = 10−5 ) 7.5 7.4 6.1 2.4 0.2

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3 Other Mechanisms of Internal Friction

5. Hidamets with extremely high hydrogen concentration. Depending on the alloy structure (amorphous, quasicrystalline or crystalline), the main damping mechanism may be either a strongly broadened, pure Snoek- or Zener-type hydrogen reorientation or a coupled process involving both H diffusion and motion of matrix defects, like twin boundaries in the presence of a martensitic transformation (cf. Sect. 2.2.4). Materials include bulk metallic glasses (mostly Zr-based; see e.g., Mizubayashi et al. 2003, 2004b, Hasegawa et al. 2003, Yagi et al. 2004, Sinning 2006a) as well as NiTi(Cu) shape-memory alloys (Biscarini et al. 1999a, 2003a,b; Mazzolai et al. 2003, Coluzzi et al. 2006). Here the temperature range of effective damping may be limited either by the vibration frequency in case of relaxation peaks (often situated slightly below room temperature at 1 kHz), or (as in group 2 earlier) by the martensitic transformation temperature in case of H-enhanced “transformation” peaks. Another limiting factor is the necessity to introduce hydrogen into the alloys and to keep it inside. Although this structure-based distinction of five groups has clearly improved the classification of Hidamets, it may be subject to further modification during the continuing development of novel materials. As an example we may think of nanostructured metals: it has been reported that nanostructured Cu (Mulyukov and Pshenichnyuk 2003) and Al (Koizumi et al. 2003), produced by severe plastic deformation (SPD), can exhibit relatively high damping due to a contribution of grain boundary dislocations (in combination with a significant increase in yield strength), as compared to the undeformed state. At present these results remain questionable in view of other publications where an increase in damping due to SPD was not observed. In addition, several IF peaks at different temperatures are enhanced by SPD (cf. Sect. 2.4.3 and Table 2.18). Highly porous cellular metallic materials (CMM) form another type of relatively new, rapidly developing engineering materials, attractive from the viewpoint of low weight. In the above classification CMM may belong to the first group (“highly heterogeneous structure”), although the variety of possible damping mechanisms in CMM is reduced as compared to “true” composites: in CMM like metallic foams, sponges or porous (sintered) metals, most damping contributions come from the same basic mechanisms as in dense metals. It is the combination of the existing damping mechanisms with the mechanical properties and the extremely low density which makes these materials interesting: the capability of damping mechanical vibrations, when estimated by complex parameters which also consider strength, stiffness and weight, demonstrates some advantages of CMM compared to the dense materials they are made from (Golovin and Sinning 2003a, 2004). Studied in a wide range of vibration amplitudes (Han et al. 1997, Liu et al. 1998a,b, 2000, Golovin and Sinning 2002, 2003a, 2004), three amplitude ranges of IF in CMM can be distinguished in a similar way as

3.5 Mechanisms of Damping in High-Damping Materials

155

considered earlier for dislocation-related ADIF (Sect. 3.3): an amplitudeindependent range with linear relaxation (including thermoelastic damping), a first amplitude-dependent range with reversible hysteresis effects, and a second one with irreversible phenomena connected with fatigue problems. The critical point in CMM is the quantitative modification of the different, known damping mechanisms determining the extension of and the transitions between these ranges, which has as yet been treated only in relatively simple models (Golovin et al. 2004a, Arkhipov et al. 2005, 2006) and remains still an open problem for real cellular structures. Whereas from this viewpoint the direct applicability of CMM as Hidamets is unclear, Al foams have already found related, dissipative applications for sound and energy absorption (when crashed) in cases where weight minimisation is demanded (Gibson and Ashby 1997, Banhart et al. 2001). Absorption coefficients for airborne sound as high as 90% have been found even for some closed-cell foams (Gibson and Ashby 1997, Ashby et al. 2000), although other authors report lower values (Han et al. 1998, Kovacik and Simancik 2002). Open-cell sponges with interconnecting pores are in principle more appropriate for this kind of aerodynamic losses in the surrounding atmosphere which are, however, beyond “internal” friction and hence outside the topic of this book. To sum up, there are two main viewpoints on high-damping materials. The first, “engineering” approach regards the damping of vibrations or noise under the specific conditions of a particular application; any material which is able to give the desirable damping at the chosen conditions – whatever the mechanism of damping is – is a high-damping material under this viewpoint. Within the second, “physical” approach, high-damping metals are defined, instead of varying application conditions, on the background of the most powerful damping mechanisms acting in the material, as resulting from its structure and physical properties.

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

In the following tables we attempt to collect the experimental material about internal friction produced by anelastic relaxation in crystalline metals and alloys, as published until mid 2006. If not specified otherwise, the data refer to temperature-dependent mechanical loss peaks, including (a) the measured primary quantities vibration frequency (f), temperature (Tm ) and height (Qm −1 ) of the damping maximum, (b) the related activation enthalpies (H) and preexponential times (τ0 ) if available, and (c) the specimen conditions and probable relaxation mechanisms. Comments and further parameters, if necessary, are added directly on the spot. Each table starts with a brief summary of important features of internal friction in the considered group of metals and alloys, and of the observed relaxation mechanisms. The data collection is strictly confined to metallic materials, i.e., data on ceramics, semiconductors or polymers are not included even in cases where the same mechanisms occur as in metals. Experimental results refer exclusively to dynamic measurements (sub-resonance and resonance vibrations as well as ultrasonic waves), whereas quasi-static creep, stress relaxation or elastic after-effect data are not considered. That means, slow phenomena like viscous flow or Gorsky effects are included only if studied, with dynamic methods, in form of damping of vibrations. The selection of data is further restricted to those studies in which the information on relaxation has really been evaluated. So, for instance, a big volume of work is intentionally omitted that concentrates on amplitudedependent internal friction phenomena (see some indications on data in this field in Sect. 3.3). Another limiting area is that of low-temperature physics (at liquid helium temperatures and below): such data are mentioned only occasionally as part of a broader characterisation of a material, whereas exclusive low-temperature studies e.g., on tunneling systems or spin glasses are generally omitted as well. In some cases, earlier data collections have been used and are referred to, rather than citing the original papers. Of course, the present collection cannot

158

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

be fully complete and exhaustive, and the authors apologise for not having detected all the relevant publications. It should be noted that the data and mechanisms given in the tables are those provided by the authors of the cited papers; no judgment as to the accuracy of values or plausibility of mechanisms has been attempted from the view of present knowledge, except for some rare cases in which later papers question the proposed mechanism. The task to judge each value would be formidable and even impossible to fulfill with the available information. For the same reason we have to renounce indications of error limits to the values given in the tables; in cases of doubt, primary data like Tm are generally more reliable than evaluated parameters like H or τ0 . For more information the reader has to consult the original papers referred to. The order, in which the data for the different metals and alloys are presented, largely follows the groups of metals in the Periodic Table of the Elements, with alphabetical order within each individual table (cf. the headings given in the table of contents). In each case the values are presented first for the pure metal, and then for binary, ternary and more complex alloys in succession. The compositions are given either in weight or in atomic percent as specified at the beginning of each table, owing to the fact that in the literature different styles are prevailing for different metals (depending on their different use in research and application). If subscripts are used for the composition (as often in multi-component alloys and intermetallic compounds), they always mean atomic concentrations or stoichiometries. Entries in the tables belonging to different alloys, different qualities (e.g. purities) of elemental metals, or in some cases groups of alloys with certain concentration ranges, are separated by thin horizontal lines. In general, the entries in each column, within such a block between the horizontal lines, apply also to the following lines if these are empty. On the contrary, a hyphen in a cell means that the preceding entry does not apply to this cell. For some special groups of multi-component alloys with particular structures or properties (hydrogen-absorbing alloys, metallic glasses, quasicrystals), which do not fit well into the above ordering scheme, the data are compiled in the Tables of Chap. 5 (see also the introduction to this table).

4.1 Copper and Noble Metals and their Alloys Copper with its alloys is one of the most thoroughly investigated metals. Thus, nearly all kinds of known relaxation effects have been found in the group of Cu and noble metals. The following internal friction effects have been observed: In pure metals: – Point defect related peaks (PD), in particular after irradiation, in Ag, Cu;

4.1 Copper and Noble Metals and their Alloys

159

– Intrinsic dislocation related relaxations (DP), namely Bordoni (B1 ) and Niblett–Wilks (B2 ) peaks in Ag, Au, Cu, Pd, Pt; Hasiguti peaks (P1 , . . . , P4 ) and other not specified DP, in Ag, Au, Cu; – Low temperature background (probably dislocation related) in Ag, Au, . . . ; – High temperature background (related to dislocations, diffusion, recovery or recrystallization), in Au, Cu – Grain boundary related relaxations (GB), sometimes involving impurities or particles, often associated with grain boundary sliding, sometimes with special boundaries like twin boundaries, sub-boundaries, in Ag, Au, Cu, Ir – Zener-type relaxations with divacancies in Au – Overdamped dislocation resonance (MHz range) in Cu, etc. In solid solutions: – – – –

Point defect related relaxations (PD), in Ag-based solid solutions; Snoek–K¨ oster relaxations (SK) (involving dislocations) in Pd–H Zener peaks (Z), in Ag-, Au-, Cu-, Pd-based solid solutions Peaks due to phase transformations of various kinds, including ordering, martensitic transformation – Grain boundary (GB) and interface relaxations, associated with diffusional stress relaxation at interfaces between matrix and precipitates. In pure Ag and Cu, as well as in Cu–Zn and Cu–Ni–Zn alloys, thermoelastic damping by transverse and intercrystalline thermal currents has been observed.

f(Hz)

240

1.5

8 · 103 –8 · 105

1000

0.7 1.5

1

20

600

0.2

composition

Ag Ag

Ag(99.999)

Ag(99.8)

Ag(99.99)

Ag(99.999) Ag(99.99)

Ag(99.999)

Ag(99.9)

Ag(99.999)

Ag(99.999)

45–85 55 850

173 200 245 61 523 630 436 629 280

410 580 80

293

mechanism state of specimen

8–108 6 40

60 185 60 38 39 30

55 190 25

147

172 235

21 35.5 44 9.6

99 167 12

1 · 10−9

3.8 · 10−22

5 · 10−10 5 · 10−13 5 · 10−13 1.6 · 10−12

2.5 · 10−13

GB, jogs

DP(B1 )

DP(P1 ) DP(P2 ) DP(P3 ) DP(B2 ) GB GB GB GBI DP/interstitials

GB GBI DP(B2 )

PC, CW at 77 K SC, CW PC bamboo structure

annl.

CW annl.

CW

CW

annl.

30 incl. bgr. first report of IF peaks as transverse wire, diam. a function of frequency thermal currents 1.01 mm

Tm (K) Qm −1 (10−4 ) H(kJ mol−1 ) τ0 (s)

Table 4.1. Copper and Noble Metals and their Alloys (at.% if not specified differently)

Okuda (1963a,b,c) Postnikov et al. (1963) Cordea and Spretnak (1966) Pstroko´ nski and Chomka (1982) Mecs and Nowick (1969) Roberts and Barrand (1969b)

Bennewitz and R¨ otger (1936), after Zener et al. (1938) Pearson and Rotherham (1956) Bordoni (1960), Bordoni et al. (1960a,b) Hasiguti et al. (1962)

reference

160 4 Internal Friction Data of Crystalline Metals and Alloys (Tables)

Ag (collected data from literature) Ag(99.995)

666

(fl = flexural, to = torsional vibration)

40

740 937

456

1

Ag(99.999)

15–60

45–70

∼1 ∼1

60

∼103

Ag(99.9999)

10–6

521

∼103 ∼103 ∼103

Ag(99.95) Ag(99.998) Ag(99.995) +Cu(50) ppm Ag(99.995) +Cu(150) ppm Ag(99.999)

1

18 88 23 40

3.5

(fl) (to) (fl) (to)

93.6 – 99.7 – 149 116 8.7 8.7–12.5

–

164 –

495 533 500

1

Ag(99.999) 13.5 30 9.5

435 305–785 85 526 67–170 109 588 188–595 166 741 256–961 193 Qm −1 dependent on grain size 456 35 96

0.3

Ag(99.999)

3 · 10−12 – 10−9 – 10−11 3 · 10−8 2 · 10−12 10−12 − 10−13

DP

10−12

DP DP DP(B1 ) DP(B2 ) PD(int)

DP

DP

DB(B)

DP(B)

GB

2.3 · 10−11 4.9 · 10−12 2.1 · 10−15 1.5 · 10−14

irr(e)

CW, annl., recr. CW, annl., recr. SC, PC

CW, annl., recr.

annl.

CW, annl., recr.

Fantozzi et al. (1982) B¨ orner and Robrock (1985)

Stadelmann and Benoit (1977) Stadelmann and Benoit (1977) Rivi`ere et al. (1981)

Isor´e et al. (1976)

Povolo (1975)

Woirgard et al. (1974)

4.1 Copper and Noble Metals and their Alloys 161

1

5 · 103

0.8–1

Ag(99.999)

Ag(99.999)

Ag(99.999)

(10

40–85

1 1

Ag–Cd(29) Ag–Cd(39) 503

591–541

1

Ag–Cd(21–35.3)

∼100

2.7–20

Qm

1 2 400 130 50 10–100 500 13 230 280–390 40 390

430–400

40 60 0.4–150

Tm (K)

solutions 40 40 0.36 733 0.36 613 0.36 623 1 637–771 0.36 733 1.5 510 708 Ag–Cd(3.5–15.8) 1.5 725–703 Ag–Cd(32.4) 1.5 533 640 Ag–Cd(32) 0.6 493

Ag binary solid Ag–Al(0.1) Ag–Au(0.1) Ag–Au(42.1) Ag–Au(58.5) Ag–Au(68) Ag–Au(25–80) Ag–Au(42) Ag–Cd(0.88)

f(Hz)

composition

−1 −4

152.3 123

152–140

– 183 – 176 165.3 – 170 184–189 – 159 147

85

12.5

(9–0.7)·10−15

Z Z

Z

annl. 600◦ C

irr(e) irr(e) PC various grain size

nc Gas depos.

Turner and Williams (1962) Shtrakhman and Piguzov (1964) Mills (1971)

Turner and Williams (1963) Pearson and Rotherham (1956)

B¨ orner and Robrock (1985) Turner and Williams Jr. (1960)

GB

10−11

PD(int) PD(int) GB GB GB Z GB GB GBI GBI GB GBI Z

N´ o and San Juan (1993) Xiao Liu et al. (1999) Y. Wang et al. (2001)

DP(B1 ) DP(B2 ) LT bgr.

7 · 10−13 sput.th.f.

reference

mechanism state of specimen

) H(kJ mol−1 ) τ0 (s)

Table 4.1. Continued

162 4 Internal Friction Data of Crystalline Metals and Alloys (Tables)

0.4–2.5

Ag–Cd(50)

580–536 500

1

388

1 1

10−3 –10−1

1.6

1.5

0.4–1.8

0.7–30

Ag3 Ga

Ag–Ge(0.1)

Ag–In(7.5–15.6) Ag–In(9.6–17.9)

Ag–In(10.8–18.1)

Ag–In(16)

Ag–In(1–16)

Ag–La(50)

Ag–Mg(12–25)

483–543 723–628 123–450 640

543

44

533–638

613

450–530 730 590–570 25 40 35–75

193

Ag–Cu(0.1)

Ag–Cu(0.1)

1

Ag–Cd(45)

6–15 320–350 20 100

145

100

3–30

0.7

2500

75–80 1 10 0.2–1.7

700

– 172–188 – 70 176–151

147–140

159–133 153–130.5

135

10−16.7 – 10−14.3

10−12.5

Z Z

GB GBI

Z

Z

Z Z

PD

PhT(ξ ↔ ξ ′ )

quen.

PhT(β′ ↔ ξ) PhT(ξ ↔ β) PhT(Θ) PD(int) PD(int) PD

annl. 600◦ C

annl. 600◦ C

irr(n)

irr(n)

irr(e)

quen.

PhT(β → β′ )

orner and B¨ Robrock (1985) Kobiyama and Takamura (1985) Artemenko et al. (1975) Kobiyama and Takamura (1985) Mills (1971) Finkelshtein and Shtrakhman (1964) Williams and Turner (1968) Childs and LeClaire (1954) Pearson and Rotherham (1956) Artemenko et al. (1975) Banerjee and Millis (1976)

Sharshakov et al. (1978b) Artemenko et al. (1975)

4.1 Copper and Noble Metals and their Alloys 163

274

440

1.5

10−3 –10−1

Ag–Si(0.2)

Ag–Sn(0.05)

Ag–Sn(0.93)

Ag–Sn(8.1) 40 38–74

Ag–Zn(0.05)

470 610 550

72

Ag–Zn(0.1)

255

510

10−3 –10−1

Ag–Sb(6.3) 34–92

38–44

393

Ag–Pb(0.1)

423 523 48–74

365

0.4–1.8

Ag–Mg(50)

Ag–Sb(0.1)

20

Ag–Mg(25.3)

270 255 255

40

20

Ag–Mg(23.2)

0.2

1

6 350 25

0.2

0.5

25

0.3

1

10 20 40 5 5 60 0.1

Tm (K) Qm

Ag–Pt(0.1)

f(Hz)

composition

−1

(10

−4

– 172 131.8

136

– 77

mechanism state of specimen

PD(Zn/int)

PD(int)

GB GBI Z

PD(Sn/int)

PD(Si/int)

Z

PD(Sb/int)

PD(int)

irr(n)

irr(e)

annl. 600◦ C

irr(n)

irr(n)

irr(n)

irr(e)

No peaks DP/interstitials, ordered in unde- impurities disord. formed ordered state disord. annl. 600◦ C PhT(ξ → β′ ) Z PD(Pb/int) irr(n)

) H(kJ mol−1 ) τ0 (s)

Table 4.1. Continued

Artemenko et al. (1975) Kobiyama and Takamura (1985) B¨ orner and Robrock (1985) Kobiyama and Takamura (1985) Williams and Turner (1968) Kobiyama and Takamura (1985) Kobiyama and Takamura (1985) Pearson and Rotherham (1956) Williams and Turner (1968) B¨ orner and Robrock (1985) Kobiyama and Takamura (1985)

Pstroko´ nski and Chomka (1982)

reference

164 4 Internal Friction Data of Crystalline Metals and Alloys (Tables)

398 513 573–600 713

1

1 0.7 1.6 · 104 0.5

Au(99.9998)

Au(99.99)

Au(99.88)

Au(99.999) 43 65 77

115

511 677 603

570–533

1

Au Au (collected data from literature)

533–505

0.7

Ag–Zn(15–30) Ag ternary alloys Ag–Al(14.3–25)– Mn(14.3–25) Ag5 MnAl

510 763–633

0.6

Ag–Zn(17)

533

Ag–Zn(3.7–30.25)

0.7

Ag–Zn(15)

25 – 30

20

220 160

50

100

90–130

60–70

10–490

300

120

9.6 18.4 18.3

15.2

6.5 15.2 −19.3 144.3 242.7 141.4

150–139

142–136

140

PhT(β) PhT(γ2 ) Z PhT(α ↔ β)

Z

Z

Z

Z

PC, CW

CW

DP(B1 ) DP(B2 )

5 · 10−13 1 · 10−13

CW, annl.

CW, annl.

quen. 780◦ C

annl. 780◦ C

2.3 · 10−11 DP(B2 )

DP(B1 ) 3 · 10−10 1.5 · 10−13 DP(B2 ) 1.4 · 10−11 GB GB GB

10−14.6 – 10−14.4

10−14.6

Z

Marsh and Hall (1953) oster et al. K¨ (1956b) Bordoni (1960), Bordoni et al. (1960a,b) Okuda (1963a,b,c)

Fantozzi et al. (1982)

Sharshakov et al. (1981)

Putilin (1985b)

Pirson and Wert (1962) Kobiyama and Takamura (1985) Seraphim and Nowick (1961) Nowick (1952) 4.1 Copper and Noble Metals and their Alloys 165

1

1

0.5

1.3 · 104

Au(99.999)

Au(99.99995)

Au(99.999)

Au(99.999)

45 77 200–210

Au(99.9999)

1

113

10

15 · 103

120 200. . . 190 63 113 93–118 – 36 – 68 273

130 190 220 160 230 290 210 673

Tm (K)

Au(99.99)

Au(99.9999)

4

Au(99.999)

104–5 7 · 104

f(Hz)

composition

20 30 50

3 2. . . 45 1.3 1.4 – – 0.05 – 0.05

10 100 30 10 10 20 2 500

6.5 19.3 11.6 – 14.5 – 11.6 62.7

57.7 435

21.4 32.9 34.7

2.5 · 10−10 1.3 · 10−13 1.6 · 10−10 – 4 · 10−12 – 2.5 · 10−10

5 · 10−10 – 3 · 10−10

Qm −1 (10−4 ) H(kJ mol−1 ) τ0 (s)

Table 4.1. Continued

DP(B1 ) DP(B2 ) DP(P1 )

DP(B)

DP(P1 ) DP(P2 ) DP(B1 ) DP(B2 ) DP(B2 ) DP(B1 ) DP(B2 ) DP(B1 ) DP(B2 ) Z(divac)

Z(divac) GB sliding

DP(P1 ) DP(P2 ) DP(P3 ) DP DP

mechanism

CW

quen. 1000◦ C

SC 111 Franklin and Birnbaum (1971) Stadelmann and Benoit (1977) Bonjour and Benoit (1979)

CW annl. PC, CW, irr(n) Grandchamp (1971) SC 100 SC 110

De Morton and Leak (1967) Benoit et al. (1970)

Neumann (1966)

quen. 700◦ C quen. 800◦ C quen. 1100◦ C annl. 900◦ C

Okuda and Hasiguti (1963)

reference

CW

state of specimen

166 4 Internal Friction Data of Crystalline Metals and Alloys (Tables)

1

1 270–470

5 · 103 340 145

Au(99.99)

Au(99.999)

Au(99.9999)

Au

0.3–3 · 103

4 · 103

Au

Au(99.9999)+ Pt(10–1000) ppm Au(99.9999)

Au(99.9999)+ 4 · 103 Cu(10–1000) ppm Au(99.9999)+ 1 Pt(10–500) ppm

Au(99.9999)+ Cu(10–500) ppm

60, 90

110 30, 50

60

30 50 400–550 0.8–3

70 370 –

20–10 30–13 24–12

45–50 78–83 105–115 60 100 180 40 80 120–130 210 430–450 0.5–100

23–2

19–4 28–6 40–17/8

47–530 80–90 200– 175/215 105–115

∼19

18.3

7 · 10−13

double peak

GB

GB

DP(B1 ) DP(B2 ) DP(B) DP(P1 ) recovery LT bgr.

DP

CW (10 K)

DP(B1 ) DP(B2 ) DP(B)

Okuda et al. (1994)

N´ o and San Juan (1993)

Baur and Benoit (1986)

Bonjour and Benoit (1979)

Bonjour and Benoit (1979)

Xiao Liu et al. (1999) nc. Gas depos. Tanimoto et al. Ribbon (2004a) nc. Gas depos. Tanimoto et al. Ribbon, irr(e), (2004b) irr(p)

Sput.th.f.

Gas depos. thin film

Study of IF background

CW (300 K)

CW (300 K)

CW (10 K)

DP(B)

DP(B1 ) DP(B2 ) DP(Pi )

4.1 Copper and Noble Metals and their Alloys 167

f(Hz)

1 1 1 ∼0.5 0.7 0.2–1

0.7

Au–Fe(5–7)

Au–Fe(10–27) Au–Ni(7.7–90.8) Au–Ni(30) Au–Zn(15)

Au–Zn(50)

Au ternary alloys Au–Ag(42)–Zn(15) 0.5

1

Au3 Cu (Au–Cu(25))

Au–Cu(21–42)− Zn(15–17)

0.75

Au3 Cu

Au binary solid solutions Au–Cd(46.1–52.5) ∼1; 5 · 104 ; 5 · 106 Au–Cu(10–90) 1

composition

573–653

533

533–563

2500–700

550

50

320

200 300 400 12–85 75–60 175–75

250–100 >20 700–3800 5–20

448–523 600–665 825–1102 170–230 490 625 760 638–653 808–837 663 670–925 ∼653 523

40–160

146

140

– 190 240 159 177.8 151 182–251 88.3 218

– 115–165 202–342

PhT(ord)

Z

PhT(ord) + vac

PhT(ord) Z GB Z PhT(Fe) Z Z Z(vac) Z

GB Z GBI anom. peak

PhT(mart)

Qm −1 (10−4 ) H(kJ mol−1 ) τ0 (s) mechanism

300–360

Tm (K)

Table 4.1. Continued

annl.

quen.

quen., CW, annl. quen. only quen., annl.

state of specimen

Pirson and Wert (1962) Pirson and Wert (1962)

Ang et al. (1955) Cost (1965b) Pirson and Wert (1962) Mukherjee (1966)

Mynard and Leak (1970)

Iwasaki (1986c)

Y. Wang et al. (1981) Maltseva et al. (1963); De Morton and Leak (1967) Iwasaki (1981a)

reference

168 4 Internal Friction Data of Crystalline Metals and Alloys (Tables)

2 · 104 –6 · 106

(0.5–10)106

1.09 · 104

1

1.5 1 1 5 600

1.2

Cu (techn. purity, (99.7))

Cu(99.999)

Cu(99.999)

Cu(99.999)

Cu(99.999)

Cu(99.999)

Cu(99.987)

Cu(99.999)

293

900

Cu Cu

488 573 30 70 190 483

149 166 240 22

80. . .100 60 135 38 79

40

613

Au–Cu(63)–Zn(17) 0.5 Z Pirson and Wert (1962)

23 incl. IF peak as a function of transverse cold roll. 0.45 mm Zener et al. (1938) frequency bgr. thermal currents DP(B1 ) Bordoni (1960), ≈5 4 1 · 10−9 PC, CW, annl. Bordoni et al. DP(B2 ) ≈30 11.8 1.2 · 10−11 (1959, 1960a) DP(B1 ) SC 4.8 Alers and 5 · 1013 DP(B2 ) 10.9 1 · 10−12 Thompson (1961) 4.2 V¨ olkl et al. (1965), DP(B1 ) SC, CW Niblett and Wilks 11.7 DP(B2 ) (1956), Sack (1962), Okuda (1963a,b) DP(P1 ) CW Koiwa and Hasiguti 26 31 10−12 DP(P2 ) (1963) – 34 1.7 · 10−12 DP(P3 ) – 41.5 1.2 · 10−10 DP(B1 ) CW Okuda (1963b,c) 0.3 4.3 2.5 · 10−11 11.6 DP(B2 ) 157 GB Peters et al. (1964) 156.9 GB annl. DP(B1 ) quen., CW Mecs and Nowick DP(B2 ) (1965) DP(P) GB CW, annl. Cordea and 165 132 2 · 10−15 Spretnak (1966)

130

4.1 Copper and Noble Metals and their Alloys 169

∼1

Cu(99.9999)

0.6

0.3

6 · 106 1 · 104 1

4 4 4 4 0.6 0.6

Cu(99.999)

Cu (electrolytic purity)

Cu(99.999)

Cu(99.999)

Cu(99.999)

1.6

Cu(99.999)

1 Cu (spectroscopical purity

f(Hz)

composition 689 1008 570 820 970 150 173 225 1110 660 730 140 225 240 165 105 – 550 690 590 715 835 950 540 670 – 30 50 200 30 60 30 10 10 70

145 70 200 24 38 11 650 60 10

– 16.4

435 169.5 154 290 445 30 63 100 201 – – 10−11.2 – –

10−13.3 10−12 –10−21 10−21.4

Tm (K) Qm −1 (10−4 ) H(kJ mol−1 ) τ0 (s)

GB twin boundaries GB DP(P1 ) DP(P2 ) DP(P3 ) GB(HT), jogs DP(P2 ) DP(P3 ) DP(divac) disl.(vac) disl.(int) disl.(int) DP(B1 ) – DP(P1 )? DP(P2 ) DP(P1 ) DP(P2 ) DP(P3 ) DP(P4 ) DP(P1 ) DP(P2 )

GB sliding

mechanism

Table 4.1. Continued

Woirgard (1974)

Besson et al. (1971)

Bajons and Weiss (1971)

Roberts and Barrand (1969b)

Iwasaki (1981b)

De Morton and Leak (1966, 1967) Williams and Leak (1967b)

reference

SC, CW, annl. 900◦ C

SC, CW, annl. Woirgard et al. (1975) 600◦ C

PC

SC, CW

fatig.

bamboo structure

CW

annl. 400◦ C

state of specimen

170 4 Internal Friction Data of Crystalline Metals and Alloys (Tables)

293

293

13 (1–5) · 107 (1–9) · 109 107 –8 · 107 107 –2 · 108 2

103 –2 · 104 200–400

Cu

Cu(99.999)

Cu + Au(50) ppm

Cu(RRR=1100) +Fe(100) ppm (RRR=15) Cu(RRR=1100)

Cu(RRR=2200)

Cu(99.999)

Cu(99.999)

Cu(99.9999) 32 65

170 220 270

273–50

12

Cu(99.999) Cu(99.999)

740 910 550 30 50 170 25 60 343 353 135–165

0.6 0.6 1 1.5

10.6 –9.6

3.8 13.5

92.6

2 4

30 5 5

γ dose-dependent

50 60 – 50 70 20 17 17

9.4 · 10−13 16 · 10−13

1 · 10−9 1 · 10−12

10−10

DP(P2 ) DP(P3 ) DP(P4 ) peaking effect DP(B1 ) DP(B2 )

ODR

DP(P3 ) DO(P4 ) DP DP(B1 ) DP(B2 ) DP(P2 ) DP(B1 ) DP(B2 ) DP(B) DP(B) DP(B)

SC, PC, irr(γ, e)

SC 111 compr., irr(γ) SC, CW, annl., irr(γ) intern. oxidised

internally oxidised

SC 110 SC 100

SC, CW

CW

Caro and Mondino (1981a,b) Lauzier et al. (1981)

Marx et al. (1981)

Lenz et al. (1981)

Schmidt et al. (1981b)

Soifer and Shteinberg (1978) Farman and Niblett (1980) Schulz and Lenz (1981) Schmidt et al. (1981a)

Povolo (1975) Lauzier et al. (1975)

4.1 Copper and Noble Metals and their Alloys 171

1 (1–5) · 107 15 15 5 · 106 5 · 107

Cu

Cu(RRR=1500)

Cu(99.997)–nc Cu(99.98)–nc Cu(99.98)–nc Cu(RRR=1500)

Cu(99.95)

Cu(99.99) 1

1

Cu(99.999)

Cu (collected data from literature)

f(Hz)

composition

383 453 433 170 155

140 155 220 516 708 926 30 56 128–173

556 732 868

Tm (K) Qm

−1

(10

−4

132 181 237 – 13.5

8.3 17.4

91.7 168 250 3.8–4.8

mechanism

5 · 10−13 − DP(B1 ) 10−9 DP(B2 ) 10−15 − 2 · 10−10 DP(P1 ) DP(P2 ) DP(P3 ) GB(LT) GB(IT) GB(HT) DP(B1 ) DP(B2 ) 7 · 10−13

τ0 (s) )H (kJ mol−1 )

Table 4.1. Continued

Moreno-Gobbi and Eiras (1994)

SC, CW 3% SC, CW 10%

ufgr, nc

Moreno-Gobbi and Eiras (1993) Soifer (1994)

N´ o and San Juan (1993)

Ashmarin et al. (1990)

Quenet and Gremaud (1989)

Fantozzi et al. (1982)

Rivi`ere et al. (1981)

reference

SC, CW

CW, annl. CW CW, annl.

CW

SC, PC

state of specimen

172 4 Internal Friction Data of Crystalline Metals and Alloys (Tables)

5

0.2

0.6

0.7

3

160

Cu(99.99)

Cu(99.99) N-implanted

Cu(99.99)

Cu(99.95)–nc

Cu(99.99) Cu(99.99999) Cu (zone refined)

(1–5) · 107

200

Cu(99.999)

Cu(99.99) Cu(99.99999) Cu (zone refined) Cu(99.999) 5–100

20–60 100–200

99.9) 6–9

Tm (K)

f(Hz)

composition

τ0 (s) Qm −1 (10−4 ) H (kJ mol−1 )

Table 4.1. Continued

DP(Ag) PD

PD

DP(disl. in cell walls) DP(P) by oxidation, vacancies relax. due to H

DP(B) DP(B) DP(B)

LT bgr.

mechanism

SC, CW

CW

irr(n)

SC 111 CW electrochemical oxidation CW, recov. El.chem. H loading

SC, CW

SC, CW

PC, fatig.

sput.th.f.

state of specimen

Kobiyama and Takamura (1985) Iwasaki et al. (1980) Tonn et al. (1983)

Aaltonen et al. (2004)

Jagodzinski et al. (2000a)

Xiao Liu et al. (1999) Hirao et al. (2000) Moreno-Gobbi et al. (2000) Moreno-Gobbi and Eiras (2000) Eiras (2000)

reference

174 4 Internal Friction Data of Crystalline Metals and Alloys (Tables)

145

1.5 · 107 1

1

1.1 1 ∼2 0.66 0.1 0.1 0.1 0.1 1–16

Cu–Al(2–10)

Cu–Al(3.1–7.3)

Cu–Al(9.3–11.4)

Cu–Al(10)

Cu–Al(10.1) (wt%)

Cu–Al(15) (wt%) Cu–Al(16.8)

Cu–Al(3.1) Cu–Al(7.3) Cu–Al(11.4) Cu–Al(14) Cu–Al(3–19)

1–20 3 peaks

689 618 – – 640–700

553 545 213 633 4 7 18 27 ∼120

35

903–1068 1213– 1325 1195–992 1328– 1124 700–850 (heating)

823 250 200–143

1 1.6 1.6

Cu–Ag(0.71) Cu–Ag(2–15) Cu–Al(0.01–1)

166 170 180 192 166–183

174.9

192–220 154–192

192–145 160–125

24

193

1.6 · 10−14 1.7 · 10−14 1.5 · 10−15 1.2 · 10−16 1.6 · 10−14 − 2 · 10−16

Z Z Z Z Z

Z

PD(vac) ordering.

PhTβ′ → β

SC

quen., annl. slow cooling CW

quen. 950◦ C

SC, CW

10−10 –10−11 10−7 –10−8

CW, annl. CW, annl. ADIF CW SC, CW

DP(B)

GB DP(solute)

10−11 –10−9 10−6

6 · 10−52

Rivi`ere and Gadaud (1996)

Sharshakov et al. (1978c) Lebienvenu et al. (1981b) Iseki et al. (1977) Li and Nowick (1956) Belanui et al. (1993)

Belhes et al. (1985)

Kayano et al. (1967) Belhes et al. (1985)

Peters et al. (1964) Kong et al. (1981a) Ishii (1994)

4.1 Copper and Noble Metals and their Alloys 175

f(Hz)

0.1

1

1

837

∼1 1 1 1 0.35

2.5–52 6–114 8 · 104

1.6

1

composition

Cu–CuAlO

Cu–Au(0–100)

Cu3 Au

Cu–Au(0.02)

Cu–Au(1.5) Cu–Au(12–25) Cu–Au(18) Cu–Au(25) Cu–Au(5)

Cu–Au(18) Cu3 Au Cu3 Au (Cu–Au(25))

Cu3 Au

Cu–B(0.38)

680 800 560 730

675–752 704–741 662

223 670–690 780 770

290–400 400 – – 653 873–930 12–87

Tm (K)

173 188

4 · 10−15 3.2 · 10−16

– 6.9 · 10−10 5 · 10−24 –3 · 10−14

τ0 (s)

DP(Au) Z Z Z HT bgr.

– Z GB PD

Rel.around particles

mechanism

≈200 Z ≈300 +PhT(ord) (jump in Q−1 from 2 to 4 · 10−4 on heating when passing the order/disorder (Kurnakov) ordering temperature) 90 Z 700 241 GB GB 370 80 GB(?)

10–80 80 250

0.4

Qm −1 (10−4 ) H (kJ mol−1 ) 2–10 – 70–80 86.8 436–168

Table 4.1. Continued

annl. 730◦ C

quen., annl.

SC

Ashmarin et al. (1985b)

Povolo and Hermida (1996b) Povolo and Hermida (1996a) Miura and Maruyama (1985) Iwasaki (1986b)

PC

CW PC SC

Kobiyama and Takamura (1985) Iwasaki (1978) Povolo and Armas (1983)

Ngantcha and Rivi`ere (2000) De Morton and Leak (1967)

reference

irr(n)

PC, int.oxid.

state of specimen

176 4 Internal Friction Data of Crystalline Metals and Alloys (Tables)

646 ∼1 2 ∼1 ∼1 1

∼1 0.2

0.2 1299 1–4

Cu–Be(1)

Cu–Co Cu–Fe(0.07–0.1)

Cu–Fe(0.5–10)

Cu–Fe(