Internal Friction of Materials

INTERNAL FRICTION OF MATERIALS

i

ii

INTERNAL FRICTION OF MATERIALS Anton Puškár Transport and Communications Technical University Zilina, Slovak Republic ^

CAMBRIDGE INTERNATIONAL SCIENCE PUBLISHING iii

Published by

Cambridge International Science Publishing 7 Meadow Walk, Great Abington, Cambridge CB1 6AZ, UK http://www.demon.co.uk/cambsci/homepage.htm

First published May 2001

© Anton Puškár © Cambridge International Science Publishing

Conditions of sale All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher.

British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library

ISBN 1 898326 509

Production Irina Stupak Printed by PWP Acrolith Printing Ltd, Hertford, England

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PREFACE The complete absence of books characterising the internal friction of materials, its external and internal aspects and the application of measurements in various scientific and technical areas, especially in physical metallurgy and threshold states of materials, has been the impetus for the author to write this book. Without understanding the principle and mechanisms of anelasticity and the effect of various factors on internal friction, together with the application of methods of reproducible internal friction measurements, it is not possible to solve the problems of the application of these measurements as direct or indirect methods for the evaluation of the structural stability of alloys, problems of cyclic microplasticity and deeper understanding of processes associated with the response of materials to single or repeated loading. In addition to the original systematisation of the possibilities of using internal friction measurements in various sciences, the book presents the latest theories and results together with practical approaches to the measurement and evaluation of the resultant relationships Anton Puškár

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vi

CONTENTS

1

AIMS OF INTERNAL FRICTION MEASUREMENTS ....... 1

2

NATURE AND MECHANISMS OF ANELASTICITY ......... 5 ELASTICITY CHARACTERISTICS........................................... 5 Effect on elasticity characteristics .................................................. 13 Elasticity characteristics of structural materials ............................. 27 Elasticity characteristics of composite materials ............................ 37 MANIFESTATION OF ANELASTICITY ................................. 43 Delay of deformation in relation to stress ....................................... 44 Internal friction ............................................................................... 50 Mechanisms of energy scattering in the material ............................ 53 DEFECT OF THE YOUNG MODULUS ................................... 62

2.1 2.1.1 2.1.2 2.1.3 2.2 2.2.1 2.2.2 2.2.3 2.3

3 3.1. 3.1.1 3.1.2 3.1.3 3.1.4 3.1.5 3.2. 3.2.1 3.2.2 3.3 3.3.1 3.3.2 3.3.3 3.3.4 3.4 3.5 3.6

4 4.1

FACTORS AFFECTING ANELASTICITY OF MATERIALS ............................................................................... 79 INTERNAL FRICTION BACKGROUND ................................. 80 The substructural and structural state of material .......................... 81 Vacancy mechanism ........................................................................ 82 Diffusion-viscous mechanism ......................................................... 84 Dislocation mechanisms ................................................................. 85 The relaxation mechanism .............................................................. 87 EFFECT OF TEMPERATURE ON INTERNAL FRICTION .. 87 Mechanisms associated with point defects ..................................... 94 Dislocation relaxation mechanisms ................................................ 97 EFFECT OF STRAIN AMPLITUDE ....................................... 104 The Granato–Lücke spring model ................................................ 105 Thermal activation ........................................................................ 107 Internal friction with slight dependence on strain amplitude......... 109 Plastic internal friction ................................................................. 114 EFFECT OF LOADING FREQUENCY .................................. 114 EFFECT OF LOADING TIME ................................................. 121 EFFECT OF MAGNETIC AND ELECTRIC FIELDS ........... 128 MEASUREMENTS OF INTERNAL FRICTION AND THE DEFECT OF THE YOUNG MODULUS ............................. 133 APPARATUS AND EQUIPMENT ........................................... 133 vii

4.2 4.2.1 4.2.2 4.2.3 4.3 4.3.1 4.3.2 4.3.3

5 5.1 5.1.1 5.1.2 5.2 5.2.1 5.2.2 5.2.3 5.2.4 5.3 5.3.1 5.3.2 5.3.3 5.4 5.4.1 5.4.2 5.4.3 5.4.4 5.4.5 5.4.6

6 6.1 6.1.1 6.1.2 6.2 6.2.1

EXPERIMENTAL MEASUREMENTS AND EVALUATION ..... ...................................................................................................... 143 Infrasound methods ...................................................................... 144 Sonic and ultrasound methods ...................................................... 151 Hypersonic methods ..................................................................... 167 PROCESSING THE RESULTS OF MEASUREMENTS AND INACCURACY ........................................................................... 169 Inaccuracy caused by the design of equipment ............................. 169 Inaccuracies of the measurement method ..................................... 172 Errors in processing the measurement results ............................... 174 STRUCTURAL INSTABILITY OF ALLOYS .................... 181 DIFFUSION MOBILITY OF ATOMS ..................................... 181 Interstitial solid solutions .............................................................. 183 Substitutional and solid solutions ................................................. 195 RELAXATION OF DISLOCATIONS ...................................... 197 Low-temperature peaks ................................................................ 197 Snoek and Köster relaxation ......................................................... 198 Phenomena associated with martensitic transformation in steel ... 204 Migration of solute atoms in the region with dislocations ............ 205 RELAXATION AT GRAIN BOUNDARIES ............................ 213 Pure metals ................................................................................... 215 Solid solutions .............................................................................. 216 Relaxation models ........................................................................ 217 ANALYTICAL PROCESSING OF THE RESULTS OF MEASUREMENTS .................................................................... 219 Solubility boundaries .................................................................... 219 Activation energy and diffusion coefficient .................................. 221 Breakdown of the solid solution ................................................... 223 Intercrystalline adsorption ............................................................ 224 Transition of the material from ductile to brittle state .................. 227 Relaxation movement of microcracks ........................................... 230 CYCLIC MICROPLASTICITY ............................................ 234 CRITICAL STRAIN AMPLITUDES AND INTENSITY OF CHANGES OF CHARACTERISTICS ..................................... 237 Physical nature of the critical strain amplitude ............................. 239 Methods of evaluating critical amplitudes .................................... 245 CYCLIC MICROPLASTIC RESPONSE OF MATERIALS .. 248 Dislocation density and the activation volume of microplasticity . 248 viii

6.2.2

Condensation temperature of the atmospheres of solute elements 257

6.2.3

Deformation history ............................................................................... 262

6.2.4 Cyclic strain curve ........................................................................ 267 6.2.5 Temperature and cyclic microplasticity ........................................ 276 6.2.6 Magnetic field and microplasticity parameters ............................. 282 6.2.7 Saturation of cyclic microplasticity .............................................. 290 6.3 FATIGUE DAMAGE CUMULATION ..................................... 297 6.3.1 Hypothesis on relationship of Q–1 – ε and σa – Nf dependences ... 297 6.3.2 Deformation and energy criterion of fatigue life ........................... 300 6.3.3 Effect of loading frequency on fatigue limit ................................. 311 References ................................................................................................... 315 Index ...................................................................................................... 325

ix

x

SYMBOLS A Ap a B BHR b b c cikmn cm c0 cp D D ⊥, D ||

-

dz E

-

ED EN ER ES Eef ∆E/E e

-

F F⊥, F||

-

FR G ∆G H h I i K KS

-

k kI Lef Ln

-

coefficient of anisotropy approximate coefficient of anisotropy lattice spacing proportionality coefficient Blair, Hutchins and Rogers model Burgers vector fatigue life coefficient fatigue life exponent elasticity constant average concentration initial concentration heat capacity at constant pressure diffusion coefficient diffusion coefficient normal and parallel to the dislocation grain size modulus of elasticity (Young modulus) in tension or compression dynamic modulus of elasticity (Young modulus) non-relaxed modulus of elasticity relaxed modulus of elasticity modulus of elasticity effective modulus of elasticity defect of modulus of elasticity (Young modulus) temperature coefficient of the change of the modulus of elasticity coefficient of the shape of the hysteresis loop force acting normal and parallel in relation to the dislocation Finkel’stein-Rozin relaxation shear modulus of elasticity difference of moduli of elasticity activation enthalpy Planck’s constant magnetization current interstitial atom bulk modulus of elasticity mean capacity of absorption of energy in the microvolume Boltzmann’s constant coefficient of magnetomechanical bond effective length of the dislocation segment length of the pinned dislocation segment xi

Lp M

-

Nf n no P Qc Q–1 Q–1S Q–1SK Q–1m Q–1o

-

Q–1 p

-

Q–1 t R Re Rm SK ∆S s smnik ss T TF Th Tp Tt t V V+ v vd vl vt W Wt Wk ∆W Z1, Z2

-

α β γ

-

spacing of pinning points ratio of the extent of internal friction and the defect of the modulus of elasticity number of cycles to fracture coefficient of cyclic strain hardening density of geometrical inflections pressure activation energy for creep internal friction height of Snoek’s peak height of Snoek and Köster peak internal friction slightly depending on ε internal friction independent of ε, the so-called background internal friction strongly dependent on ε, so-called plastic friction internal friction strongly dependent on loading time degree of dynamic relaxation yield stress ultimate tensile strength Snoek and Köster relaxation entropy difference substitutional atom elasticity constant designation of a pair of substitutional atoms oscillation period thermal fluctuation relaxation homologous temperature ductile to brittle transition temperature melting point time volume activation volume vacancy dislocation velocity velocity of the longitudinal wave velocity of the transverse wave total energy supplied to the system energy consumed by material up to fracture half energy of the formation of a double kink energy scattered in the material during a cycle total power of the exciter and power required to overcome resistance in the exciter coefficient of intensity of damping widening of the peak of internal friction thermal conductivity coefficient xii

γkr1 γkr2 δ εac εap εc εd εe εi εmn εkr1 εkr2 εkr3 εp εt εd ΘD ΘE χ λ1 , λ2 µ υ ρ ρa ρd ρn ρp σ σa σC σf σ ik σK τ τr σε ϕ ϕo Ψ ω ∆ω

-

first critical strain amplitude second critical strain amplitude logarithmic decrement of damping total strain amplitude plastic strain amplitude total strain additional strain elastic strain fatigue ductility coefficient strain tensor first critical strain amplitude second critical strain amplitude third critical strain amplitude plastic strain strain at crack formation rate of change of additional strain Debye temperature Einstein temperature coefficient of proportionality parameters of the ellipsoid of deformation Poisson number Debye frequency specific density density of active dislocation sources dislocation density density of stationary dislocations density of mobile dislocations normal stress stress amplitude fatigue limit fatigue strength coefficient stress tensor physical yield limit shear stress relaxation time relaxation time at constant strain phase shift angle of deflection of the pendulum relative amount of scattered energy circular frequency half width of the resonance peak at half its height

xiii

xiv

Nature and Mechanisms of Anelasticity

1 AIMS OF INTERNAL FRICTION MEASUREMENTS The elasticity characteristics belong in the group of the important parameters of solids because they are often used in the analytical solution of the problems of deformation and failure. The elasticity values are used in all engineering calculations and the design of components, sections and whole structures. In the development and application of specific materials and high-strength components produced from them, it is necessary to consider the required rigidity and also the probability that a certain amount of energy will build up in the system during service. Actual solids are characterised by the scattering of mechanical energy in them, i.e. internal friction. This representation of the anelasticity of materials and their transition from elastic to anelastic, microplastic or even plastic response to external loading are the consequences of the effects of external loading and the activity of various mechanisms and sources of scattering of mechanical energy in the material which may be characterised by relaxation, dislocational, mechanical and magnetomechanical hysteresis. These mechanisms result in changes of the structure-sensitive properties of materials and also the structuresensitive component of the Young modulus which is still regarded erroneously in a number of publications as a material constant. The external factors, such as mechanical loading, temperature, the effect of the magnetic field, different frequency of the changes of loading, etc., lead to changes of the nature and mechanisms of the processes of scattering of mechanical energy in materials. The relationships between the changes of internal friction and the defect of the Young modulus with changes taking place in the material on the atomic level, on the level of a group of grains, in the volume of the loaded solid or in a group of solids, have already been confirmed and verified. 1

Internal Friction of Materials solid solutions diffusion

Solid state physics

thermal activ. param. phase transformations point defects dislocation structure grain boundaries

thermal strain

Damage

cyclic loading radiation hydrogen

damping capacity

Internal friction

Young's modulus

Threshold state of materials

micromechanical characteristics relaxation additional loading creep cracking resistance

quality of system noise in system

Vibroacoustics of system

amplitude-frequency spectrum of system structural damping aerodynamic damping flaw inspection vibrothermography vibrotechnologies

F ig .1.1 ig.1.1

The author of the book has developed an original system, Fig. 1.1, which, using the currently available data, shows the possibilities of the application of internal friction measurements in different sciences, from the atomic size up to entire structural complexes. The effect of the external factors on the occurrence of threshold 2

Nature Mechanisms Anelasticity Aims of and Internal FrictionofMeasurements

states of materials and components [1] is associated with continuous changes of the response of the material to their effect. This is reflected in changes of internal friction [2]. After compiling and verifying a physical model explaining the nature of changes of internal friction with the changes of the external factor it is possible to conclude that the internal friction measurements can be used as an indirect method of the monitoring of processes taking place in solids. The evaluation and quantification of the dynamics of changes in solid solutions, during diffusion and phase transformations, are often associated with the determination of the parameters of thermal activation, with the data on the point, line and area defects of the crystal lattice taken into account. The changes of temperature, static and cyclic deformation, and also radiation or the presence of hydrogen in the material are reflected in the degradation of the characteristics of the material as a result of damage cumulation. Accurate measurements of internal friction enable indirect quantification of this phenomenon. When evaluating the threshold state of materials and structures, it is also essential to quantify the damping capacity, the defect of the Young modulus, microplastic characteristics, relaxation and additional elasticity phenomena, etc. these processes are also accompanied by changes of internal friction so that the internal friction measurements can be used for examining the process and critical characteristics of specific materials. Vibroacoustic examination of a structure or machine under different service conditions, by evaluation of the acoustic quality, noise and amplitude–frequency spectrum makes it possible to propose measures for ensuring high reliability and safety of operation of the system, especially under resonance conditions. Design or aerodynamic damping of components, sections of the entire structure may be utilised here. The efficient selection of materials with the required damping capacity improves the functional behaviour and reliability of operation of the machine and decreases the ecological damage from vibrations and noise of machines. In technical practice, flaw inspection methods are used on a wide scale, but the possibilities of these methods have not as yet been exhausted. Vibrothermography is not yet used widely as a method for identification of the areas of preferential absorption and scattering of energy in a real solid. However, it represents a significant tool in the solution of problems of stressstrain heterogenities and concentrators in the solids, with one of the internal friction mechanisms playing the dominant role. Vibro-tech3

Internal Friction of Materials

nologies are gradually introduced into various production and transport applications where the level of internal friction in certain components may be very low whereas in others it may be high, depending on the system utilising vibrotechnology. Taking the actual scale of this subject, in this book, special attention is given only to some selected problems, associated mainly with physical metallurgy and threshold states.

4

Nature and Mechanisms of Anelasticity

2 NATURE AND MECHANISMS OF ANELASTICITY The solution of a set of problems associated with the nature, measurements, evaluation and application of information on the internal friction and the defect of the Young modulus of materials is based on a brief and functional characterisation of the elasticity parameters of structural monoliths and composite materials, and also on the evaluation of the effect of different internal and external factors on their magnitude. A special position is occupied by the effect of factors causing nonlinearity between stress and strain, i.e. anelastic behaviour of the materials. This includes the explanation of the phenomenon of deformation lagging behind stress, irreversible scattering of energy in materials and mechanisms by which the energy of vibrations is irreversibly scattered in the materials. These processes are also reflected in the level of the Young modulus and the occurrence of defects of the Young modulus, and this may influence the accuracy of calculations of permitted stresses in components or whole structures.

2.1 ELASTICITY CHARACTERISTICS Under the effect of a generally oriented force the solid can change its dimensions and shape. If the relative strain in a specific direction is denoted by the strain sensor mn and the force per unit crosssection, causing this strain, is denoted as the stress sensor ik, then s ik = cikmn e mn ,

(2.1)

e mn = smnik s ik ,

(2.2)

where c ikmn and s nmik are the elasticity constants. Strain tensor ε mn 5

Internal Friction of Materials

and stress tensor σ ik are the tensors of the second order. They can be described by nine pairs of strain components or nine pairs of stress components, and the unit volume is selected sufficiently small to ensure that the strain of stress unit is the same everywhere. Three pairs of components act in the direction of the x axis, i.e. σ xx in the direction normal to the wall of the cube; these pairs are oriented normal to the x axis. They include the normal stress which is positive for tension and negative for compression. The second pair of the components in the direction of the x axis acts on the walls oriented normal to the y axis. This pair is denoted σ xy and σ yx . The third pair of the components in the direction of the x axis acts on the walls normal to the z axis. This pair is denoted σ xz and σ zx . The second and third pair act in the given planes and tend to shift them mutually. They are denoted as tangential or shear stress. Since the elementary volume of the material does not rotate, the components must be in equilibrium, i.e. σ xz = σ zx , and they are denoted τ xz , and also σ xy = σ yz , denoted τ xy . Consequently, this gives a symmetric tensor of the second order with six components

s xx s ik = t xy t xz

t xy s yy t yz

t xz t yz . s zz

(2.3)

Every symmetric tensor of the second order has three main axes. Since the axes of the coordinates are regarded as synonymous with the axes of the tensor, only the main stress σ 1 is obtained. For an anisotropic medium, equation (2.1) can be expanded into a system of linear equations, i.e. for stresses σ xx , σ yy , σ zz , σ yz , σ xy , strains ε xx , ε yy , ε zz , γ yx , γ zx and γ xy , using the elasticity constants c ikmn , as given for the first of six rows in the form

s xx = c11e xx + c12 e yy + c13e zz + c14 g yz + c15 g zx + c16 g xy ,

(2.4)

where γ is shear. Similarly, equation (2.2) can be expanded utilising the elasticity constants s mnik , as given for the first of the six rows

e xx = c11s xx + s12 s yy + s13s zz + s14 t yz + s15 t zx + s16 t xy .

6

(2.5)

Nature and Mechanisms of Anelasticity

Generally, the elasticity constant c ikmn (and also s mnik ) has the form of a tensor with 39 components, where the first of the 6 rows is c 11 , c 12 , c 13 , c 14 , c 15 , c 16 (or s 11 , s 12 , s 13 , s 14 , s 15 , s 16 ). Proportionality is found between the elasticity constants c 12 = c 21 or s 11 = s 21 and, generally, c αβ = c βα , or s αβ = s βα . As a result of this symmetry, the crystallographic system with the lowest symmetry (triclinic) has only 21 independent components, and the number of independent components in the orthorhombic system decreases to 9, in the tetragonal and diagonal systems it decreases to 6, and in the hexagonal system to 5. For cubic crystals there are three elasticity constants, c 11 , c 12 and c 44 . The elasticity constants c ikmn and s nmik are linked by the defined relationships [3]. The physically determined elasticity constants for technical applications are complicated. For crystallographic systems with the symmetry higher than orthorhombic, the normal tensile stress in the direction of the x axis (σ xx) results in relative elongation ε xx and two reduction in areas ε yy and ε zz :

e xx =

s xx , E1

e yy = - m12 e xx ,

e zz = -m13e xx .

(2.6)

If we consider the normal stress in the direction of the y and z axes, we obtain three moduli in tension (Young modulus) and six Poisson numbers, of which only three are mutually independent, because

E1 E = 2, m12 m12

E2 E = 3 , m 23 m 32

E3 E = 1. m 31 m13

(2.7)

Shear stress τ yz in the yz plane causes shearing γ yz . Consequently, τ yz = G 1γ yz . Similarly, in the xz plane, where τ xz = G 2 γ xz and in the xy plane, where τ xy = G 3 γ xy . The orthorhombic crystal in the system of the technical elasticity moduli has only nine independent elasticity characteristics, i.e. three tensile (Young) moduli E, three shear (Coulomb) moduli g, and three Poisson numbers µ. For the cubic crystal, these are three characteristics (E, G, µ), and for an isotropic solid it is E, G, because

G=

E . 2 m +1

a f

(2.8) 7

Internal Friction of Materials

Equation (2.8) is also important because polycrystalline metals and alloys without a sharp texture behave as isotropic materials. The bulk Young modulus K is defined as isotropic pressure P divided by the relative volume change caused by this pressure

K=

E -P . = DV 3 1 - 2m V

a

f

(2.9)

The values of the elasticity moduli (E, G), the bulk Young modulus (K) and Poisson number (µ) for a number of metals are presented in Table 2.1. For rock and glass µ = 0.25, G = 0.4E, K = 2/3E, for metals µ = 0.33, G = 3/8E, K = E, and for elastomers µ = 0.5, G = 1/3E, and the K/E ratio is high. The Young modulus is closely linked with the velocity of propagation of sound in a metallic material. In the case of a longitudinal elastic wave, the velocity of propagation is nl =

E r

(2.10)

and in propagation of a transverse elastic wave, the velocity of propagation is nt =

G , r

(2.11)

where ρ is the specific density of the metallic material [4]. This phenomenon is utilised in accurate measurements of E and G, because measurements are taken without exchange of heat with the environment which enables also the determination of adiabatic elasticity moduli which differ from the elasticity moduli obtained under isothermal conditions (for example, in the tensile test). Table 2.1 gives values of v 1 and v t for several metals. The values of the elasticity constant make it possible to determine accurately the anisotropic factor of the elastic properties of metallic materials from the equation

8

Nature and Mechanisms of Anelasticity Ta b le 2.1

Elasticity moduli and other characteristics of several metals at 20°C

Typ e o f me ta l

E ( GP a )

G ( GP a )

µ

K ( GP a )

Vl (ms–1)

Vt (ms–1)

* Al Ca Ni Cu Pd Ag Ir Pt Au Pb

70.8 19.6 231.2 145.3 142.7 81.1 528.0 169.9 88.1 37.3

26.3 7.4 89.1 54.8 51.8 29.6 214.0 61.0 31.1 13.6

0.34 0.31 0.31 0.35 0.39 0.38 0.26 0.44 0.42 0.44

77.5 17.2 190.0 139.6 192.1 103.6 370.0 272.7 175.4 48.8

6355 – 5894 4726 4594 3686 – – 3361 2158

3126 – 3219 2298 1987 1677 – – 1239 860

10.5 7.2 4.6 131.2 279.7 223.2 1.3 107.0 330.0 190.0 393.7

4.0 2.7 1.7 48.3 102.0 86.9 0.47 39.2 11 9 . 7 71.1 153.0

0.36 0.32 0.35 0.36 0.28

11 . 8 8.3 4.0 154.3 196.5 173.1

0.39 0.30 0.35 0.29

159.0 282.5 194.3 308.1

5709 3078 – 6000 – 6064 – 5104 6649 4447 5319

2821 1434 – 2780 – 3325 – 2089 3512 2039 2843

*** Mg Ti Co Cd

44.8 11 4 . 4 220.9 65.5

17.6 43.3 84.5 24.6

0.28 0.36 0.32 0.30

34.5 107.2 190.3 62.2

5895 6263 5827 3130

3276 2922 3049 1663

**** In Sn

13.9 60.1

4.8 23.6

0.46 0.33

41.6 60.6

2459 3300

709 1649

** Li Na K V Cr Fe Rb Nb Mo Ta

Comment: * - cubic face centred; ** - cubic body centred; *** - hexagonal closedpacked; **** tetragonal

A=

2c44 , c11 - c12

(2.12)

where for an isotropic case A = 1. For some metals, the dependence of the mechanical properties on the loading direction is shown in Fig. 2.1. 9

Internal Friction of Materials

F ig .2.1. Directions of true stress at fracture of aluminium single crystal (a), elongation ig.2.1. of aluminium single crystal (b), Young modulus of aluminium single crystal (c), Young modulus of iron single crystal (d), shear modulus of elasticity of iron single crystal (e) and Young modulus of magnesium single crystal in tension (f).

Ta b le 2.2 Approximate (Ap) and accurate (A) anisotropy coefficients of elastic properties Typ e o f me ta l

E max in d ire c tio n < 111 > ( GP a )

E min in d ire c tio n <100> ( GP a )

Al Cu Fe W

7.7 19.4 29.0 40.0

6.4 6.8 13.5 40.0

Ap

G max in d ire c tio n <100> ( GP a )

G min in d ire c tio n < 111 > ( GP a )

Ap

Ap

1.175 2.870 2.150 1.000

2.9 7.7 11 . 8 15.5

2.5 3.1 6.1 15.5

1.13 2.50 1.93 1.00

1.2 3.3 2.4 1.0

The approximate coefficient of anisotropy of the elastic properties can be determined as the ratio of the maximum and minimum values of the elasticity moduli. Table 2.2 gives the accurate (A) and also approximate (A p ) anisotropy coefficients of the elastic properties of some metals. If E max (= E 111 ) and E min (= E 100 ) are available, it is possible to determine E in the direction characterised by the angles α, β, γ to the axes of the cube using the Weert’s equation in the form

FG H

1 1 1 =3 E E100 E111

IJ ccos K

2

h

a cos2 b + cos2 b cos 2 g + cos2 g cos 2 a .

(2.13)

The equation can also be used for polycrystalline materials with a texture, if the latter is expressed by two or more ideal orientations. 10

Nature and Mechanisms of Anelasticity

The elasticity moduli are associated with the characteristics determined by the force influence of interaction of the atoms in the crystal lattice linked with the thermal expansion coefficient, Debye temperature, sublimation temperature, melting point, etc. These considerations show that the elasticity moduli can be determined approximately, with an acceptable correlation factor, using the measured values of these characteristics. The relationship between the Young modulus E (or G) and the melting point of metal T m has the form

Tt = k1 A E ,

(2.14)

where k 1 = 5K in determination of E, and k 1 = 8.5K in determination of G, where K is the bulk Young modulus, and A is the proportionality coefficient. The relationship between the Young modulus, the volume coefficient of thermal expansion β and the relative molar heat capacity at constant pressure c p is determined by the equation

g 0 cp

K=

bVa

,

(2.15)

where γ 0 is a constant and V a is the molar volume. At room temperature and elevated temperatures T, the approximate validity of the following equation has been confirmed

E=

cp T ln t , βVa T

(2.16)

The Poisson number and constant γ 0 are linked by the equation

m=

2g0 - h , 3g 0 + h

(2.17)

where η = 1.5 for fcc metals, and η = 0.945 for bcc metals. Debye temperature Θ D is linked with the Young modulus by the equation

11

Internal Friction of Materials

F E IJ » 168 G H rA K 2

QD

2 1

1/ 6

,

(2.18)

where ρ is specific density, and A 1 is atomic density. The elasticity moduli can also be determined from accurately recorded results of tensile or torsion tests. However, the most accurate data are obtained by measuring the velocity of propagation of elastic waves v 1 or v t (equations 2.10 and 2.11). The resonance methods are effective and accurate (error 0.50.8%) in the determination of the elasticity moduli. However, it should be noted that the natural frequency of the longitudinal vibrations is an order of magnitude higher than the natural frequency of the bending vibrations. Increase of the loading frequency increases the intensity of relaxation processes. This is reflected in an increase of temperature and the associated decrease of the Young modulus. This results in a systematic error in the measurement of the elasticity moduli by the resonance method. This shortcoming can be eliminated using the pulsed methods of Young modulus measurements. These methods are based on the measurement of the velocity of propagation of a pulsed elastic wave in metal, and the wavelength is small in comparison with the dimensions of the solid. The Poisson number can be determined from X-ray diffraction measurements of the lattice parameters of the metallic material. The accuracy of the pulsed methods of measurement of the Young modulus is high (error is approximately 0.1%). However, these methods also have certain shortcomings. The most important problem is the fact that when measuring the velocity of propagation of a pulsed wave it is necessary to measure the Poisson number at a specific moment of time. Procedural problems do not enable measurements of the Young modulus to be taken at higher temperatures. The tabulated data on the elasticity moduli of metals and alloys are limited because they represent the characteristics at room temperature and do not describe the initial state of the material or its thermal-deformation history. This shortcoming is partially eliminated by a set of data [5] which contains the elasticity moduli for a large number of metals and alloys at elevated temperatures. The elasticity constants and also technical elasticity moduli are influenced by a large number of external and internal factors.

12

Nature and Mechanisms of Anelasticity

2.1.1 Effect on elasticity characteristics The effect of temperature on the elasticity characteristics is associated with the thermal expansion of the material, i.e. with the temperature dependence of the atomic spacing. Analysis of this problem has shown that the change of the elasticity moduli is not associated with absolute temperature, but is linked with homologous temperature T h = T/T m , where T is the temperature at which the Young modulus is determined. For the same homologous temperatures, the relative change of the elasticity characteristics for many metals is the same (Fig. 2.2). This relationship is linked with the identical homologous temperature dependence of the change of atomic spacings. Increase of temperature results in a decrease of E, G and K. The value of the Poisson number initially slowly decreases and then increases with a further increase of temperature; because of the different thermal strain history of the material, the dependence is more complicated. Decrease of temperature, like increase of pressure, results in the same change of the atomic spacing in the crystal lattice. This shows that the change of the Young modulus will be similar. The change of bulk Young modulus K at absolute temperature from 0 to T is described by the equation

DK g b T = , K0 3

E/E0

(2.19)

Th F ig .2.2. Dependence of relative values of the Young modulus of various metals on ig.2.2. homologous temperature, where E 0 is the Young modulus at 0 K. 13

Internal Friction of Materials

where

g=

1 dK K0 de

(2.20)

is the change of the Young modulus during deformation of the lattice by the value ε, K 0 is the bulk Young modulus at 0 K. Modulus K is proportional to the curvature of the relief of the potential energy of the crystal lattice in the area with the atom. Depending on the distance of the atom from the equilibrium position, the curvature of the potential relief decreases as a result of increase of ε. This shows that g < 0, and the value of the modulus decreases with increasing temperature. Consequently

∆K γ 0 gc pT . = 3Va K 0 K0

(2.21)

Equation (2.21) shows that the resultant value is strongly influenced by the value of c p . As in the case of equation (2.21), it is possible to write similar equations for the change of E or G. Like the temperature dependence of the heat capacity at constant pressure, the temperature dependence of the elasticity moduli can be divided into three ranges: low-temperature range, where T << Θ D , transition range, where Θ D ≤ T ≤ 0.5T t, and high-temperature range, where T > 0.5T t . In the low-temperature range, the coefficient of the effect of temperature on the change of the modulus e is proportional to t 0 g c p , and also proportional to (T/Θ D) 3. In the entire temperature range the dependence of the Young modulus on temperature has the shape K/ K 0 ~ T 4. Two cases can occur in the transition temperature range. If the Debye temperature for a specific metal or alloy is significantly lower than 0.5T t , then c p ≈ 3R, where R is the gas constant, and e = const. This shows that the modulus of elasticity increases proportionately with increasing temperature. When Θ D is close to 0 or higher than 0.5T t , the value of c p increases with increasing temperature and the dependence is ‘domed’ in the upward direction. The increase of temperature by one degree results in a decrease of the Young modulus by 0.02–0.04%, with the approximation sufficient for a wide range of the materials. 14

Nature and Mechanisms of Anelasticity

The selection of experimental dependences is very important; it is necessary to ensure that they describe with sufficient accuracy the changes of the elasticity moduli of the materials. For example, in Ref. 6, the temperature dependences of the Young modulus for VSt3 steel in the form E = (21.68 – 67×10 –4 T)×10 4 MPa, where T is in °C, were verified for the temperature range from –70 to +70°C. For metals with high melting points, for temperatures of up to 2000°C, the authors of Ref. 7 published the empirical dependences: for vanadium in the form E = (12.8 – 9.61×10 –4 T)×10 4 or G = (4.88 – 8.48 × 10 –4 T)×10 4 , for niobium E = (10 + 9.18×10 –4 T 4.11×10 –7 T 2 )×10 4 or G = (3.12 + 9.9×10 –5 T )×10 4 , and for tantalum E = (16.9–8.22×10 –4 T–1.66×10 –7 T 2 )×10 4 or G = (7.74–1.73× 10 –4 T)×10 4 , always in MPa. For tungsten, we can use the equations in the form E = E 0 [(T t T)1/T t ] 0.4 , or G = G 0 [(T t – T)1/T t ] 0.263 , E = E 0 [(T t – T)1/T t ] 0.463 , G = G 0 [(T t – T)1/T t ] 0.465 , where E 0 and G 0 are the moduli at 0 K. On the basis of analysis of the elastic characteristics of 40 alloys based on Fe, Ni, Cu and Al in the temperature range below 500 K, it was shown [8] that, with the exception of Invar alloys, the temperature dependence of the Young modulus is described quite accurately by the following equation

E = E0 −

η , ΘE eT − 1

(2.22)

here Θ E is the Einstein temperature, η/Θ E is the limiting value of the tangent dE/dT to the E(T) dependence. At elevated temperatures, above 0.5T m, the rate of decrease of the Young modulus rapidly increases, and the temperature at which the rapid decrease of the modulus starts is close to or identical with the temperature of the start of increase of the high-temperature background of internal friction. There are several hypotheses explaining the rapid decrease of the Young modulus in the high-temperature range. This may be caused by the nonlinear dependence of atomic forces on additional thermal strain. Some hypotheses are based on the assumption according to which this behaviour is the consequence of deformation due to dislocation movement. The hypotheses are supported by the assumption according to which the mobility of dislocations increases with increasing temperature. This is reflected in an increase of the contri15

Internal Friction of Materials

bution of dislocational anelasticity. The elasticity moduli also decrease with increasing internal friction. Measurements of the temperature dependence of the elasticity moduli in the temperature range (0.5–0.7) T m show that the activation energy of the change of the elasticity moduli is close to the activation energy for self-diffusion [9] which is close to the activation energy of the relaxation process at the grain boundaries in polycrystalline materials. In the temperature range close to the melting point (0.95–0.97) T m the Young modulus changes as a result of the temperature maximum on the temperature dependence of internal friction. Measurements of the Young modulus of Sn, Bi, Cd and Pb up to the melting point showed that in the vicinity of the melting point the Young modulus rapidly decreases, and a slight increase of the Young modulus is recorded only in the case of Sn at temperatures higher than 0.98T m [10]. This phenomenon was also reflected in the arrest of the decrease of the Young modulus of specimens of sintered iron with a tin filler in the vicinity of the melting point of Sn. Acceleration of decrease of the Young modulus with increasing temperature is also caused by relaxation processes taking place in the process of gradual increase of external stress. In the forties, Frenkel showed that a metal starts to melt when the Young modulus is 0. Theoretical calculation showed that the Young modulus at the melting point is 0.7–0.75 of the modulus at 0 K. The experimental dependences and also Fig. 2.2 show that with increase of temperature up to the melting point the Young modulus decreases by 40–60%. The difference between the theoretical calculations and the results of measurements confirms the effect of hightemperature relaxation. The form of the temperature dependence of the Young modulus at high temperatures may have a significant effect on the activation energy of creep [11]. In steady-state creep, the creep rate is determined by the equation

ε& = A2 σ

n

Q − e RT e

(2.23)

where A 2 is the proportionality constant, σ is the acting stress, n is the strain hardening coefficient, Q c is the activation energy for creep, and R is the gas constant. If we take into account the temperature dependence of the Young modulus, the equation has the 16

Nature and Mechanisms of Anelasticity

following form n

*

Qe  σ  − RT ε& = A3  e ,   E (T ) 

(2.24)

where A 3 is the proportionality constant and Q ∗c is the modified activation energy of creep which can be determined from the equation

Qc* = − R

d ln E (T ) δ ln ε& − nR 1 1 d  δ  T  T 

(2.25)

Consequently

∆Qc = Qc* − Qc =

nRT 2 dE . E dT

The change of the activation energy of creep may be quite considerable. For example, in the case of Inconel alloy at a temperature of 704°C, Q *c = 551.7 kJ mol , ∆Q = 80.9 kJ mol, and at 1037°C, Q = 251. 4 kJ⋅mol –1 and ∆Q c = 59.5 kJ⋅mol −1 . This example shows that in calculations it is important to take into account the information on the change of the Young modulus with increasing temperature. External pressure and internal stress also influence the level of the Young modulus. The increase of external pressure P results in increase of the Young modulus. Up to a pressure of 5 GPa, we can use the following dependence E = E 0 (1 + χ1 P ) ,

(2.26)

where E 0 is the Young modulus at the atmospheric pressure, and the value χ 1 varies from 10 –1 to 10 –2 GPa –1 . Specific values of χ 1 for various materials are presented in Ref. 5. The increase of the Young modulus with increasing hydrostatic pressure is caused by a decrease of the atomic spacing in the crystal 17

Internal Friction of Materials

F ig .2.3. Dependence of Young modulus on microstrain ∆ a / a for heat treated St3 ig.2.3. steel.

lattice. This hypothesis has been verified theoretically and also by measurements [12]. The dependence of the Young modulus on stress is general not only for the hydrostatic pressure conditions. The dependence of the Young modulus on the state of the structure of the hardened material, observed in a large number of experiments, is usually interpreted from the viewpoint of the magnitude, nature and distribution of internal stresses in the material [13]. Theoretical calculations of the dependence of the Young modulus on the microstrain of the crystal lattice ∆a/a were carried out by Levin [14]. The calculated and experimental results are presented in Fig. 2.3. Increasing microstrain decreases the Young modulus but the scatter of the measured Young modulus values increases. This is the result of the scatter of distribution of the internal stresses in the material. The results show that the Young moduli of the materials which contain internal stresses are in fact random quantities for which it is possible to obtain the corresponding dependence of dispersion S 2 on microstrain ∆a/a. In the case of the experimentally determined change of dispersion moduli S 2 it is necessary to take into account the dispersion caused by the measuring procedure and this value characterises the inaccuracy of determination of the elasticity moduli of the metals and alloys. 18

Nature and Mechanisms of Anelasticity

The change of the Young modulus with increasing pressure and temperature has the same basis, i.e. the change of the atomic spacing, thus yielding the equation

 ∂E   ∂E   ∂E    =  − 3α K   ,  ∂T C  ∂T V  ∂P T

(2.27)

where C, V and T indicate that the values are determined at constant concentration, volume, and temperature, and α is the coefficient of linear thermal expansion [15]. The quantity (∂E/∂T) V is included in the equation due to the dependence of the elasticity moduli of ferromagnetic materials on the degree of arrangement of magnetic domains. At temperatures lower than T c (Curie temperature) and for non-ferromagnetic materials, (∂E/∂T) V = 0. Consequently T

 ∂E  ET − ET0 = −3K   α (T ) dT .  ∂P T T0

∫

(2.28)

For pure iron and binary alloys of iron with cobalt, nickel, chromium and molybdenum at T << T c equation (2.28) describes satisfactorily the experimentally measured values. The dependence of the Young modulus on the type and severity of the stress state and, in particular, strain state is of principal importance not only for predicting the changes of the elastic materials in service but also for the development of suitable theoretical models of the strength and plasticity behaviour of metals and alloys. The level of the Young modulus is also influenced significantly by the strain. The addition of alloying elements to the main metal changes its elastic characteristics as a result of changes of the interatomic bonds in the alloys. This takes place in relation to the chemical interactions between foreign atoms as a result of the change of the concentration of ‘free’ electrons (electron factor) or as a result of the effect of lattice defects and, consequently, lattice parameters, which is the result of different atomic radii of the added and main metal (dimensional factor). In reality, these factors can affect the Young modulus in the same or opposite direction. Experiments show that the dependence of the change of the Young modulus on the con19

Internal Friction of Materials

centration of the alloying element (c) is the result of these factors acting in the same direction. This result can be expressed by the equation in the form

 ∂E   ∂ ln a  ∆E = ∆El − 3 K     c.  ∂P T  ∂c T

(2.29)

where ∆E e is the component of the change of the Young modulus as a result of the change of the number of electrons, a is the lattice spacing. The characteristic ∆E e has complicated form and depends on many influences, such as the change of the energy of positive ions in the electron ‘gas’, the change of the degree of overlapping of electron paths of the adjacent atoms, the zonal structure of the metal, etc. The dependence of the Young modulus on the concentration of the alloying elements is determined by the characteristics of the atoms of the alloying element and the main metal. Examples for binary systems with complete solubility of the metals in the solid state are presented in Fig. 2.4. For alloys of the metals which do not form intermetallic compounds together, the elasticity moduli are an additive property to a first approximation. For the majority of non-transitional and some transitional elements in the range of the mass concentration from 10 to 20% and for Cu–Pt, Mo–W, Cu–Ni and other alloys, the results obtained by Köster show that the concentration dependence of the Young modulus in the entire concentration range is almost linear. The contribution of the electron factor to the value of the Young modulus increases with increasing difference of the valency of the components of the alloying and the main metal (Fig. 2.4a). In most cases, the elasticity moduli of intermediate phases are higher than those of pure components. The experimental results obtained for the Cu–Sn and Cu–Zn systems show, Fig. 2.5, that higher elasticity moduli of the intermetallic phases influence the changes of the elasticity moduli in a wide concentration range. Compounds of metals and an intermetallic phase are characterised by high elasticity moduli. In the case of carbides, it should not be assumed that the Young modulus of carbon is E = 0.09 × 10 5 MPa. It is more accurate to consider the value for diamond, i.e. E = 1 × 10 5 MPa. For example, for titanium E = 1.1 × 10 5 MPa, and for TiC E = 0.46 × 10 5 MPa. For tungsten E = 3.96 × 10 5 20

Nature and Mechanisms of Anelasticity

wt.%

F ig .2.4. Effect of composition of alloys on their Young modulus. ig.2.4.

F ig .2.5. Change of the Young modulus of Cu–Zn and Cu–Sn in relation to composition ig.2.5. indicating the formation of intermetallic phases.

MPa, and for WC E = 6.7 × 10 5 MPa. These approaches are of information nature. In the systems with superlattices, the magnitude of the Young modulus depends on the state of the structure, for example, the val21

Internal Friction of Materials

F ig .2.6. Change of the elasticity moduli in relation to the concentration of the solid ig.2.6. solution.

ues for the disordered state of an alloy are presented in Fig. 2.6. After the formation of a superlattice, there are significant changes in relation to the initial condition. The dependence of the elasticity moduli on the composition of the alloying for some transitional elements is complicated becaus d-electrons take part in the formation of bonds in these metals. Cs, Ti, Mg and Fe are characterised by unstable electron configurations, reflected in allotropic transformations. Extrema on the E(c) dependences may also form when the transitional metal is the main component (Fe–Al, Ti–Al) or it is also an addition (Cu–Ti), or both (Ti–Mo). The complicated nature of the changes of binding forces in alloying of transition metals has so far prevented the construction of a unified mechanism of the effect of foreign atoms on the elastic characteristics of the transition metals. In ferrous alloys, it is possible to evaluate the effect of the electron factor on the change of the Young modulus in alloying with different elements [14]. The electron factor of the transition metals of the same period as iron (chromium, vanadium, cobalt, nickel, manganese) is almost 0 and, consequently, all changes of the elasticity moduli in alloying are associated with the dimensional factor. The atoms of the transitional elements from higher periods are charac22

Nature and Mechanisms of Anelasticity

F ig .2.7. Dependence of the Young modulus of iron-based alloys on the content of ig.2.7. different additions (molar %).

terised by non-zero electron factor values. The values of the electron factor of the elements from the same period (for example, Ru and Rh or Re, Ir, Pt) are the same and increase with increase of the number of the period. The dependence of the Young modulus on composition for ironbased alloys is complicated (Fig. 2.7 and 2.8). The most significant changes of the E(c) dependence are recorded at small amounts of the additions. This phenomenon is associated with the fact that the formation of the solid solutions of the transitional metals is accompanied by an increase of the number of electrons when interaction takes place between s and d electrons. This increases the Young modulus. The increase of the concentration of the alloying element is accompanied by an increase of the strength of the effect of the dimensional factor which increases the distance between the atoms. The determination of the changes of the Young modulus and the accompanying change of the parameters of the crystal lattice with the change of the composition of the alloy has made it possible to conclude that, after dissolving in iron, the alloying elements form additional bonds and the effect becomes stronger with the distance of the added element from iron in the periodic system of elements. The temperature dependence of the Young modulus for binary 23

Internal Friction of Materials

Fig .2.8. Effect of temperature on the dependence of the Young modulus on the temperature ig.2.8. of an Fe–Cr alloy.

alloys is characterised by the temperature coefficient which depends only slightly on the concentration of the added element. Consequently, the E(T) dependences with increasing content of the alloying element are almost parallel. The increase of the carbon content of iron decreases the elasticity characteristics. In annealed steels, the largest increase of the Young modulus is recorded at low carbon concentrations (to 0.2 wt.%). In quenched steels, the E(c) dependence decreases, even though the increase of the carbon content accelerates the increase of the Young modulus in comparison with the annealed condition. The dependence of the elasticity characteristics on composition in the case of the Fe–Ni system is also influenced by prior tensile deformation and annealing (Fig. 2.9). In the steels, carbon is present in most cases in cementite Fe 3 C and, consequently, the evaluated dependences E(c) are examined mainly from the viewpoint of the effect of different factors on cementite. Cementite is a ferromagnetic phase with T C ≈ 210°C. Since Armco iron has a conventional, almost linear dependence E(T), the dependence for carbon steels at T ≈ 200°C is characterised by a small delay which changes to a local maximum with the increase of the carbon content of the alloy, for example in white cast iron. The 24

Nature and Mechanisms of Anelasticity

wt.%

F ig .2.9 ig.2.9 .2.9. Dependence of the Young modulus and Poisson number on the composition of ferrous and nickel alloys after cold tensile deformation (symbol 1) and after annealing (symbol 2).

form of the previously plotted curves also enables extrapolation of the changes of the E(T) dependence for cementite. Examination of the titanium alloys with different iron and aluminium content [16] showed higher elasticity moduli and the increase of the Debye temperature with increasing content of the alloying elements. If the different electron structures of the Al and Fe atoms are considered, it may be concluded that aluminium has a significant effect on the nature of atomic interactions in titanium alloys. Increase of the aluminium content increases the force characteristics of the atomic bonds; this may be one of the reasons for the increase of the ultimate tensile strength when alloying titanium with aluminium from 480 MPa (for example, titanium alloy VT10) up to 1000 MPa (for example, titanium alloy VT5). Investigations of the elasticity characteristics of alloys of titanium with molybdenum, vanadium and niobium shows that the formation of extrema on the E(c) curves may be attributed to the start and finish of transformation of the α-phase to β-phase, and in the case of the titanium alloys with these elements, the dependences are quantitatively similar indicating the same nature of the mutual effect of titanium alloys with these elements. The additions decrease the elasticity moduli of the alloys with the α-phase and also with 25

Internal Friction of Materials

wt.% V

wt.% Mo

F ig .2.10. Dependence of the Young modulus of alloys of Ti with Mo (a) and with ig.2.10. V (b) after rapid cooling (1) and tempering (2).

α + β phases. Transition to the single-phase region (β) is associated with the minimum on the E(c) dependence. Alloying of the single-phase (β) solid solutions results in a monotonic increase of the elasticity moduli up to the values typical of the pure components. Figure 2.10 shows that α-Ti is characterised by higher values of the Young modulus in comparison with β-Ti. Differences are also evident in alloying. Whilst in the case of α-Ti the Young modulus decreases, in the case of β-Ti the Young modulus increases with increasing concentration of the additions. Rapid cooling of the alloy greatly changes the dependence of the Young modulus on concentration, Fig. 2.10. The rapid decrease of the Young modulus in the range of low concentration is associated with the formation of supersaturated α′- and α″-solid solutions. In the case of the Ti–Mo alloy this takes place up to 6 wt.% Mo. The subsequent increase of the E(c) dependence is caused by the formation of the martensitic phase ω which formed as a result of the transformation of β to ω. The maximum of the E(c) dependence corresponds to the highest volume content of the β-phase. The second minimum of the E(c) dependence can be explained by the increase of the content of the β-phase. This results in an increase of 26

Nature and Mechanisms of Anelasticity

the volume fraction of the metastable cubic volume-centred structure with less dense packing and, consequently, in a decrease of the Young modulus. In the region where there is only the β-phase, the increase of the Young modulus takes place as a result of the increase of the strength of the effect of alloying elements on the change of the interatomic forces. In the region of the single-phase β-alloys there are no significant differences in the values of the elasticity moduli for the annealed and quenched condition. The differences in the elasticity moduli are characterised by different sensitivity to the composition of the alloy. In the Ti–Zn alloy, the E(c) dependence is represented by a smooth curve with the minimum at c = 50 wt.%. In this range, K and µ change nonmonotonically in relation to composition. In alloys where the components form a mechanical mixture, specifically Al–Be, Al–Si, Bi–Sn, Fe–Fe 3C, etc., the dependence of the Young modulus on composition is linear. The Young modulus also depends on the distribution of the inclusions which is reflected in the upper and lower boundary of the elasticity moduli of the mechanical mixtures. Calculations show that at concentrations c < 40 wt.%, the elasticity moduli are the additive characteristics of the phases present. At a large amount of the second phase in the alloy, the calculated and measured values of the elasticity moduli differ. Ordering in the alloys also influences the elasticity moduli because the effect of the electron factor is very strong. Ordering increases the interatomic forces, the electron factor has the positive orientation and increases the level of the Young modulus. The dependence of the Young modulus on temperature in ordered alloys at temperatures below Kurnakov temperature T k is characterised by a constant value of the temperature factor of the change of the Young modulus e. With the increase of temperature to the range T ≈ T k , the dependence E(T) shows an anomaly in the form of a jump or a sudden decrease of the Young modulus to the value corresponding to the difference of the elasticity moduli of the ordered and disordered state of the alloy (Fig. 2.11). 2.1.2 Elasticity characteristics of structural materials In most cases, multiphase and multicomponent alloys with different chemical composition are used in technical practice. The complicated phase composition and the differences in the degree of metastability of structural components result in large differences in extent of the utilisation of the processes of structural and phase 27

Internal Friction of Materials

F ig .2.11. Effect of temperature on the Young modulus of ordered CuZn and Cu 3 Au ig.2.11. alloys.

changes, ageing, recovery, etc. in the application of various materials. Analysis of the effect of these factors on the elasticity characteristics is complicated owing to the fact that factors, determining the actual value of the Young modulus, are also important. These processes determine the effective value of the interatomic forces by the change of the concentration of the alloying elements in the solid solutions, the change of internal stresses at coherent phase boundaries, actual changes of the elasticity moduli of the independent phases, the increase of the number of structural defects, etc. (the first group of factors). The second group of factors includes the processes of anelasticity; this is reflected in the structural sensitivity of the elasticity moduli. The change of the structure and phases of the alloys in quenching, tempering, ageing, etc. results in a change of the level of the elasticity moduli; this change is large in comparison with the values given in the tables. Consequently, it is necessary to have not only the values from various tables or material specifications, but also data on the specific structural state and the relationships of their changes in the expected working conditions of the material. The sensitivity of the elasticity moduli of the material to the change of the structure and phases is the reason for the qualitatively new thermal hysteresis of the moduli. In this case, the curves E(T) in heating and cooling differ. Hysteresis is found only in a specific 28

Nature and Mechanisms of Anelasticity

F ig .2.12. Effect of temperature on the Young modulus of steel 40 (1) and white ig.2.12. cast iron (2).

temperature range (Fig. 2.12, curves 2). Figure 2.12 shows the changes of the Young modulus in relation to temperature for a steel with 0.4 wt.% C (curves 1) and for white cast iron with 2.8 wt.% C and 1.4 wt.% Si (curves 2). Both materials were normalised in the initial condition. In the temperature region below the eutectoid transformation temperature (curves 1) the dependence E(T) is slightly nonlinear. In the eutectoid transformation temperature range there is a large decrease of the elasticity moduli associated with the start of dissolution of cementite particles. Completion of the α→γ transformation results in a drop in the rate of decrease of the E(T) dependence. The large increase of E at temperatures of 1000–1050 °C is interpreted by the equalisation of the composition of the phases as a result of the dissolution of the solutes built up at the grain boundaries in the disappearance of the old grain boundaries during the growth of austenite grains. With the reversed change of temperature the extent of relaxation at the grain boundaries decreases and the dependence E(T) is shifted higher in comparison with gradual heating. The curves E(T) in heating and cooling become identical below the eutectoid transformation temperature. 29

Internal Friction of Materials

The special feature of the E(T) dependence of cast iron is the presence of a wide temperature range in which thermal hysteresis forms. The form of the E(T) dependence depends on the process of breakdown of cementite and ferrite formation. The most significant changes are recorded in the temperature range 680–850°C. In heating, the E(T) curve shows two distinctive steps associated with the start (710–720°C) and finish (810–840°C) of eutectoid transformation in white cast iron. The first stage corresponds to the transformation of pearlite to austenite which is part of ledeburite, and also to the compressive stress in the transformation of ferrite to austenite as a result of the differences in the coefficient of thermal expansion of eutectic cementite and pearlite. The second stage corresponds to the completion of the breakdown of pearlite at the original austenite grain boundaries where extensive relaxation of a large part of the residual stresses could have taken place during the tests. The measurements of internal friction with the change of temperature showed that every step on the E(T) curve corresponds to the internal friction maximum. The relaxation mechanism is associated with the change of the susceptibility of the grain boundaries to stress relaxation with the formation and growth of new phase particles on them. The elasticity characteristics of the materials change in relation to the method and conditions of heat treatment. The final results are the consequence of the mutual effect of the factors which increase or decrease the Young modulus. In quenching of carbon steels, the volume fraction of cementite in the structure of steel decreases. The Young modulus of cementite is lower. Therefore, this process increases the Young modulus. In addition, quenching is accompanied by martensitic transformation resulting in supersaturation of the dislocation density. This process decreases the Young modulus. After adding up the processes, the quenching of steels and cast irons decreases the Young modulus. The magnitude of the decrease depends on the carbon content, Fig. 2.13. The increase of the carbon content to 0.45 wt.% increases the relative change ∆E E1

 Ez − Ez  =  Ez  

of the steel after quenching from 900°C. A further increase of the carbon content has no longer any effect on the value of ∆E/E 1 . This can be explained by the formation of the effective values of the Young modulus with another influence. Increasing carbon content of 30

Nature and Mechanisms of Anelasticity

wt.%C

Fig .2.13 ig.2.13 .2.13. Dependence of the Young modulus of steel on carbon content after quenching from 900°C (1) or 1050°° C.

the steel decreases the temperature at which the martensitic transformation is completed. For steels with a carbon content higher than 0.5 wt.%, the temperature of completion of the martensitic transformation is lower than 0°C so that the structure of the quenched steel also contains retained austenite. The Young modulus of austenite is higher than that of ferrite, and this compensates the decrease of the actual values of the Young modulus as a result of the effect of the previously mentioned factors. The increase of quenching temperature to 1050°C changes the nature of the changes of the Young modulus in relation to the carbon content of the steel (Fig. 2.13, curve 2). Therefore, the increase of E/E 1 may be caused by increase of the grain size, resulting in a decrease of E and, consequently, increase of ∆E/E 1 . In the case of steels with a high carbon content, the decrease of E may also be reflected in the formation of various defects of the structure during quenching. Quenching temperature of the steel affects the Young modulus. The largest decrease of the Young modulus of the carbon steels is recorded at the quenching temperatures higher than 740–750°C. This is accompanied by a large and rapid increase of the value of ∆E/ 31

Internal Friction of Materials

E 1 in the temperature range to 900°C, and the increase of ∆E/E 1 is smaller in steels with a higher carbon content (U10A) than in the steels with a lower carbon content (U8A). At temperatures higher than 900°C the change of ∆E/E 1 in the case of U8A steel stabilises, but in U10A steel the change continues to increase. This may be associated with the formation of microcracks and also a decrease of E z when quenching from high temperatures. Quenching affects the temperature dependence of the Young modulus [17]. Figure 2.14 shows that the Young modulus of quenched U8A steel changes non-monotonically during heating. In the temperature ranges 50–150°C, 300–400°C, and 500–700°C, the temperature coefficient of the change of the Young modulus decreases. This indicates the increase of the rate of decrease of the Young modulus, and at temperatures of 50–300°C and 400–500°C there is a large increase of e. The maxima on the e(T) curves are distributed in the temperature ranges corresponding to the development of various transformations during tempering of steel. In the temperature range 50–150°C, where the value of e decreases, the ε-carbide, coherent with the mother phase, appears. The regions of internal stresses are characterised by

F ig .2.14. Temperature dependence of E / E and of e for USA steel after quenching ig.2.14. from 800 °C (a) and tempering at 150° (b), 300 (c) and 450°C (d). 32

Nature and Mechanisms of Anelasticity

a decrease of the values of the Young modulus and by the increase of the rate of decrease of the Young modulus in this temperature range. The increase of the degree of precipitation of the particles of the ε-carbide is accompanied by a decrease of the extent of distortion of the crystal lattice. This is also reflected in the increase of the Young modulus to the values corresponding to the given temperature. The rate of decrease of the Young modulus slows down with increasing temperature and this decreases the value of e. Prior tempering at 150°C results in the precipitation of the εcarbides and in the removal of distortions of the crystal lattice associated with the formation of this carbide. Consequently, in subsequent heating at 50–150°C the height of the maximum on the e(T) dependence decreases. The large increase of the value of parameter e in the temperature range 300–400°C is associated with the formation of cementite. Prior tempering at 450°C results in the precipitation of cementite and the removal of distortions of the crystal lattice associated with the formation of cementite. Subsequent heating in this temperature range is accompanied by a continuous change of the Young modulus. The values of the Young modulus of the quenched steel in the temperature range 200–300°C and the decrease of the values of e in this range are associated with the breakdown of retained austenite. The experiments with the changes of the retained austenite of quenched steels showed that the increase of the retained austenite content results in a decrease of e in the temperature range 200300°C. This phenomenon can be explained by the fact that the retained austenite is in the stressed state. It’s breakdown decreases the level of internal stresses thus decreasing the rate of decrease of the Young modulus with increasing temperature. The precipitation of carbon in the solid solution of alpha iron also contributes to the process. When the retained austenite content of the quenched steel is increased, the change of the Young modulus with increasing temperature is more marked. The decrease of the temperature coefficient of the change of the Young modulus e in the temperature range 50– 150°C with increasing quenching temperature is explained by the increase of the number of coherent particles of the ε-carbide as a result of the increase of the number of nucleation areas of the carbide. The alloying of steels increases the structural sensitivity of their Young modulus, especially after heat treatment. This is caused by the increase of the metastability of the structural components. There 33

Internal Friction of Materials

F ig .2.15. Temperature dependence of the Young modulus of 70S2ChA steel after ig.2.15. quenching from 900°C in oil (a) and tempering for 1 h at 300 ° (b) and 400 °C (c).

are also changes in the form of the E(T) dependence, reflected in addition changes of the curves of the temperature hysteresis of the elasticity moduli associated with the development of phase transformations in the material. The dependence of the Young modulus on temperature temperature in alloyed chromium steels is shown in Fig. 2.15. At temperatures of 150–200°C and 350–500°C, the elasticity moduli of the quenched steels increase. The decrease of the rate of increase of the Young modulus in tempering in the temperature range 150–200°C is associated with the breakdown of martensite and the formation of ε-carbide. The second ‘plateau’ in the temperature range 350–500°C is attributed to the transformation of carbides. The Young modulus increases as a result of tempering the quenched steel at temperatures higher than 500°C, especially in alloyed steels. This is associated with the fact that the addition of alloying elements, such as Mo, Cr, V reduces the rate of the breakdown of the supersaturated solid solution and increases the temperature range in which this process takes place. The extension of the breakdown time of martensite and the formation of pearlite increase the Young modulus after tempering at high temperatures. The increase of the chromium content in the steel increases the intensity 34

Nature and Mechanisms of Anelasticity

of this phenomenon. The increase of the Young modulus at temperatures higher than 500°C is also influenced by the process of coalescence of the carbide particles. This is reflected in a higher degree of equilibrium of the alloy. Whilst the quenched U8A steel is characterised by a monotonic decrease of the value of E with increasing temperature, the 70S2Kh2 and 60S3KhMVA steels, quenched from a temperature of 900°C, show an increase of the Young modulus by 1.2–1.5% in a specific temperature range. If the temperature in subsequent tempering is increased, the intensity of this defect decreases, Fig. 2.15c. In the alloy quenched steels in the temperature range 200–500°C the values of the dependence E(T) are lower than for the quenched carbon steels, although outside this temperature range there are no differences in the values of the E(T) dependence. This is explained by the shift of the region of transition of the metastable to stable carbide of the cementite type. The process of breakdown of the supersaturated solid solution of carbon in α-iron is accompanied by the formation and growth of finelly dispersed particles of metastable carbides, coherent with the matrix. This process is associated with the formation of high internal stresses which contribute to a further decrease of the Young modulus. The formation of the stable carbides with the disruption of coherence of the carbides and the matrix decreases the efficiency and number of defects of the crystal lattice and E increases. Silicon may also exert an effect; this element changes the nature of interatomic forces as a result of a change of the fraction of covalent bonding in the alloys. The increase of the elasticity moduli of the materials with structural inhomogeneity in the transition from the metastable to stable phase is probably a general phenomenon which takes place not only in steels because similar behaviour was found in Cu–Be, Cu–Ti–Cr and Cu–Ni–Sn systems containing metastable intermetallic phases. With the change of temperature, the Young modulus is greatly affected by combined heat treatment and plastic deformation. If under normal conditions the change of the Young modulus after plastic deformation is not high, for example, in U8 steel it does not exceed 2.5%, in the case of alloyed steels these changes are considerably larger. For alloy steel 1Cr18Ni9Ti after quenching from 900°C and 40% deformation, the change of the Young modulus ∆E/E is 10– 11%. In nickel martensitic steels after deformation and ageing, the decrease of E in comparison with the condition of the steel after tempering at high temperature is as high as 14%. Evaluation of the elasticity characteristics of 36NKhTYu alloy steel after quenching 35

Internal Friction of Materials

and ageing showed that the decrease of the Young modulus of the material is associated with the change of the texture during heating to the quenching temperature as a result of recrystallisation. After ageing the Young modulus increases. This phenomenon is controlled by the process of short-range ordering and by the formation of nuclei of the γ′-phase with a size of 1–1.5 nm. Continuation of ageing results in a small increase of the Young modulus. This is explained by the anisotropy of the elasticity characteristics of the particles of the γ′-phase during their growth. The elasticity moduli are influenced not only by the type and number of the particles of the strengthening phase but also by the uniformity of distribution of this phase in the volume of the material. For example, the Young modulus of a high-speed tool steel decrease in the case of the nonuniform distribution of the carbide phase, or the moduli of elasticity of steels for stamping dies decrease as a result of the formation of small areas with free ferrite. This increases the structural homogeneity of the material. The effect of the main metallic mass and the shape of graphite on the change of the Young modulus with increasing temperature for grey and high-strength cast iron was investigated in Ref. 18. The anomalies of the E(T) dependence are caused mainly by the number, dimensions and shape of graphite and also, to a lesser degree, whether the main metallic mass is ferritic or pearlitic. The method of measuring the E(T) dependence can also be used in examination of the kinetics of arrangement of the silicon atoms in the main the metallic mass of ferritic grey cast iron. The decrease of the Young modulus after quenching when the solid solution is disordered, and also the anomalous increase of the temperature dependence of the Young modulus in the temperature range 350–550°C are explained by the breakdown of the superlattice. Evaluation of the recovery of the elasticity moduli after quenching grey cast iron from different temperatures made it possible to determine the critical value (approximately 775°C) separating the characteristic changes of the elasticity moduli. In quenching from temperatures in the range from Curie temperature to 775°C, the solid solutions of iron and silicon in the matrix are characterised by the disordered distribution of the solute atoms resulting in a decrease of the elasticity moduli. Quenching from 775°C and higher temperatures results in the rapid decrease of the Young modulus as a result of more extensive defects in the structure formed in the process of quenching from higher temperatures. 36

Nature and Mechanisms of Anelasticity

2.1.3 Elasticity characteristics of composite materials Significant changes of the elasticity moduli are caused by the relaxation of stresses in materials with high structural inhomogeneity, especially in metals with pores. The effect of porosity on the level of the elasticity moduli is also observed in the case of many metals in the cast condition (alloys with a high melting point, sintered systems, composites, etc.). The Young modulus of the material with porosity p, denoted E (p) , can be determined from the empirical equation in the form

E( p ) = E(0) (1 − p ) , m

(2.30)

where m is the experimental factor. Increase of porosity is accompanied by a decrease of the Poisson number; this is often ignored when comparing the mechanical properties of dense and porous materials. In fact, the µ–p dependence is almost the same as that for the dense material. Frantsevich et al. [5] recommend the following equation

E p = E0

1 − p2/3 , 1 + ap b

(2.31)

where a, b are the experimental coefficients taking into account the effect of stress concentration in the vicinity of non-spherical pores. The elastic characteristics of the material, representing a mechanical mixture of certain types of particles with different deformation, can be evaluated from the deterministic and statistical viewpoint. In the first case, the mechanical model of a composite material is represented by a solid in which the interaction of the components depends on their mutual displacement. The equation of motion is derived for the displacement of the matrix and the filler. The equation contains factors and variables which depend on the heterogeneity parameters of the material. Some models solve the propagation of planar waves when: 1. The stress tensor of one component depends on the strain tensors of other components and on the tensors of the partial initial volume strains; 2. A force pulse is transferred from one component to another; this pulse depends on the uniaxial displacement and the deformation of the volumes of components. 37

Internal Friction of Materials

In the second case, in the statistical theory of random fields in the mechanics of inhomogeneous deformation, it is necessary to solve a large number of problems of the elastic equilibrium of inhomogeneous media. This is carried out on the basis of the statistical characteristics of the stress and strain states of the solids. In inhomogeneous media, the mean stress at a specific point of the solid depends not only on the deformation at this point but also on the deformation at all other points of the solid. A special situation arises in the intermediate layer which in the homogeneous media is represented by, for example, the surface of the solid, but for inhomogeneous media this region is found more often. The composite materials based on metals include systems with dispersion strengthening and materials reinforced with particles or fibres. In all cases, the material contains a metallic or alloy matrix with other phases, particles of fibres with different distribution [19]. Usually, the second phase is regarded as stronger and more rigid than the matrix. The elasticity characteristics of a two-component composite can be described, for the upper limit, using the equation E h = E1c1 + E 2 c2

(2.32)

and for the lower limit

Es =

E1 E2 , E2 c1 + E1c2

(2.33)

where E 1 , E 2 are the Young moduli of the filler and the matrix, c 1 , c 2 are the volume fractions of the components. The expression for E h corresponds to the case with the same deformation in the two components, and the expression for E s for the same stresses. The difference between E h and E s decreases if calculations are carried out using the variance principles, the method of the theory of random processes, etc. The calculated and experimentally determined elasticity moduli for powder composites based on iron, molybdenum and tungsten with the content of an copper–iron binder to 35% are compared in Table 2.3. The deviation of the experimental values of the Young modulus to higher values is explained by the fact that the expressions for E h and E s do not take into account factors such as the interaction of 38

Nature and Mechanisms of Anelasticity Ta b le 2.3 Calculated and experimental values of the Young modulus of composite materials with different matrix

% o f fille r

C a lc ula te d va lue o f E (1 0 5 MP a ) Eh

Es

Exp e rime nts o f va lue s o f E (1 0 5 MP a ) a t te mp e ra ture s o f (º C )

Em

Eν

20

200

400

600

800

* 5 25 30

1.875 1.875 1.875

1.768 1.768 1.768

1.930 1.808 1.779

– – –

1.950 1.840 1.813

1.80 1.77 1.73

1.70 1.61 1.56

1.60 1.33 1.34

1.48 – –

** 20 25 32 36

2.687 2.687 2.687 2.687

2.213 2.213 2.213 2.213

2.549 2.446 2.246 2.128

2.496 2.246 1.824 1.386

2.381 2.130 1.921 1.740

2.380 2.130 1.920 1.725

2.33 2.06 1.90 1.69

2.33 1.96 1.86 1.65

2.16 1.87 1.78 1.59

*** 25 38

3.137 –

2.410 –

– –

– –

2.855 2.170

2.73 2.05

2.73 2.05

2.65 2.00

2.55 1.95

Comment: * - iron; ** molybdenum; *** - tungsten

the elastic fields of the inclusions and the matrix, the local nonuniformity of the stress state, the elastic properties of the components, etc. More accurate calculations of the elastic characteristics of the composites can be carried out taking into account the statistical nature of the distribution of heterogenities, the mutual effect of the components and correlations between their elastic fields. The theory of isotropic deformation of elastic solids with random heterogenities, using the elasticity factors ν and λ, gives

Em =

ν (3λ + 2ν ) . λ+ν

(2.34)

The method of calculating E m is suitable for composites with a small volume fraction of the filler. For the volume fraction of the filler larger than 20% we can use the following equation

Eν = E0

M (c ) cck (η − 1)

(ck − 1)(η − 1) + M (c )ck

,

(2.35) 39

Internal Friction of Materials

where E 0 is the Young (elasticity) modulus of the matrix material, M (c) is the function which takes into account the mutual effect of the microinclusions, c is the volume fraction of the filler, c k is the critical concentration of the filler at which E = E 1, i.e. the Young modulus of the filler, η(= E/E 0 ). The data presented in Table 2.3 show that the values obtained by calculations using equation (2.35) are in good agreement with the experimentally determined values for the case in which the volume fraction of the filler is larger than 20%, whereas equation (2.34) is suitable for volume fractions of the filler smaller than 20% and also for materials with a small difference of the elasticity moduli of the matrix and the filler. The experimental results show that the values of the Young modulus decrease with increasing volume fraction of the filler. As the Young modulus of the binder decreases, the rate of decrease of the Young modulus increases. At higher temperatures this relationship is sometimes not fulfilled. The experimental results obtained for powder composites, produced by liquid-phase sintering (iron–copper, tungsten–copper, tungsten–copper–nickel, tungsten-iron–nickel, molybdenum–silver, nickel–silver, etc.) show that the dependence of the elasticity moduli on the volume fraction deviates from the lower values determined in accordance with the rules of mixing. The dispersion-strengthening particles affect the composite when they restrict the deformation of the matrix. Usually, the Young modulus of such a system is lower than that indicated by equation (2.32). Calculations carried out using equation (2.32) and the experimentally obtained data are compared in Fig. 2.16. For all examined systems, the experiments show positive deviations of the Young modulus values from the calculated values, indicating the constriction and limited deformation of the matrix. In Ref. 20 and 21, the method of finite elements was used to determine the algorithms for predicting the elasticity moduli of homogeneous isotropic two-phase materials with any geometry of the phases forming the composite. The authors analysed the stress fields in the vicinity of the individual particles and groups of the particles in a continuous matrix, and also examples of the application of the algorithm for determining the elasticity characteristics of the porous and sintered materials. The strength of the reinforcing fibres in the composites is often very high but the cross-section of the components may be unsuitable because it does not ensure the required rigidity and, consequently, the stability of the structure. In some cases, the high rela40

Nature and Mechanisms of Anelasticity

F ig .2.16. Effect of the volume fraction of dispersion-strengthening particles in a ig.2.16. composite on the relative change of experimentally and theoretically determined elasticity moduli.

tive strength of the composites is not utilised because the energy of elastic deformation (σ 2 /E) increases in direct proportion to the square of stress or yield limit. All reliability parameters decrease with increasing energy of elastic deformation. The parameters affecting the actual service efficiency of materials greatly differ from those of real structures. For example, at R m = 5000 MPa, the energy of elastic deformation decreases by an order of magnitude and the rigidity parameters are 3–4 times lower than the required boundary values. One of the methods of solving these problem is to produce alloys where not only the tensile strength but also Young modulus increases. When retaining the characteristic determined by the ratio E/σρ n , where ρ is the specific density and 1>n>1/3. For materials used in aviation industry with ρ = 1.1–9 g⋅cm –3 , n = 0.5. The composites characterised by a high Young modulus and low specific density have advantages not only in comparison with steel but also magnesium and aluminium alloys. The behaviour of the composites outside the elastic range depends on whether the strengthening particles of fibres are deformed. Solid surfaces of the inclusions restrict deformation of the softer matrix under loading. When the hydrostatic component becomes 3–4 times 41

Internal Friction of Materials

higher than the yield limit of the matrix, failure takes place. When this stress is insufficient for particles to be deformed, failure propagates through the matrix. This description of the elasticity characteristics is also valid for laminated composites [22]. To solve the boundary problems of the elasticity theory, it is necessary to know the geometrical parameters of the phase, the distribution and cross-section of the fibres. Irregular distribution of the fibres in the cross-section of the component greatly complicates the calculations. If the geometry of the phases and approximation of the stress fields are taken into account, it is possible to find simple methods of joining the elements of the composite. The main problem in determination of the elastic characteristics of the composites with fibres is the evaluation of the moduli by the variance calculation procedures. The modulus of elasticity along a solid in the direction of reinforcing fibres can be determined as an additive characteristic, whereas the values in the direction normal to the direction of the strengthening fibres greatly differ from the values determined from the decrease rule. Reinforcement of the matrix with fibres greatly increases the Young modulus in the direction normal to the direction of fibres. However, the increase of the Young modulus of the fibre results in a situation in which the increase of the transverse modulus of the composite is not significant and the solution approaches the values determined for infinitely rigid fibres. The ratio of the longitudinal and transverse modulus of normal elasticity at higher values of the Young modulus of the fibres is small and this restricts the application of fibres with high elasticity moduli in the composites. The increase of the rigidity of the system in the transverse direction can be achieved by selecting the orientation of fibres in different layers which obviously decreases the rigidity of the composite in the axis of the component. A suitable example is a laminated composite in which the orientation of the fibres in the individual layers deviates by the selected angle. The properties of this composite are similar to those with the isotropic characteristics, and the values of the Young modulus fit in the group of the values determined in the longitudinal and transverse direction in relation to the distribution of the fibres. The elasticity properties of the component of the composite characterise to a certain extent the conditions of failure of components made of composite materials. The fibre-reinforced composite remains sound if 42

Nature and Mechanisms of Anelasticity

σ≤

3.25 η F

(1 − µ ) E d 2

,

(2.36)

ν ν

where η is the proportionality factor, F is the friction factor, µ is the Poisson number, E v is the Young modulus, and d v is the fibre diameter. The magnitude of the tensile stresses formed in the fibre is low and, consequently, the critical value of the tensile stresses σ is obtained in the case of long fibres (for example, for boron fibres in and aluminium matrix the length of the fibres is 0.65–0.70 m). Equation (2.36) characterises a simple condition for the process of single-component plastic deformation of the matrix in elastic deformation of the fibre. The complicated nature of the process of failure of fibrereinforced composites under cyclic loading [22] is shown in Fig. 2.17. The composites consist of a matrix made of and aluminium alloys and strengthening fibres of molybdenum or tungsten, with a different volume fraction of the fibres in the component. Puskar [23] describes the characteristic stages of failure in the longitudinal loading of components in tension and compression. Initially, transverse cracks form and propagate (Fig. 2.17), and this is followed by the formation and propagation of longitudinal cracks (Fig. 2.17b) in the matrix. With loading, the entire cross-section fails by fatigue (Fig. 2.17c) or the matrix disintegrates and fall out, initially from the surface and in later stages from the space between the fibres (Fig. 2.17d). The formation of a specific stage is determined mainly by the type and dimensions of the reinforcing fibre and its volume fraction in the component. The author of [23] assumes that the mechanism of failure of composites is based on the significant difference of the elasticity moduli of the strengthening fibres and the matrix material during propagation of an elastoplastic wave in the component. 2.2 MANIFESTATION OF ANELASTICITY Elastic deformation is characterised by the complete reversibility of the Hooke law which is fulfilled only when the loading rate is very low and the level of acting stress does not cause any changes in the density and distribution of lattice defects or in the distribution of magnetic moments.

43

Internal Friction of Materials

F ig .2.17. Stages of damage and failure of a composite material. ig.2.17.

2.2.1 Delay of deformation in relation to stress The Hooke law characterises the time relationship between stress σ and strain ε. The total elastic strain of the solid ε c is the sum of instantaneous elastic strain ε e and additional (quasi-inelastic) strain ε d whose equilibrium value is obtained only after stress σ has been operating for some time (Fig. 2.18). The time required to obtain the equilibrium value ε d is determined by the processes associated with the redistribution of atoms, magnetic moments and temperature in the material. The redistribution of the atoms in the interstitial solid solutions under loading can be illustrated on an example of the fcc lattice (for example, α-iron) with interstitial atoms (for example, carbon). The interstitial atoms can be displaced to the positions (0, 1/2, 1/2) and (1/2, 1/4,0). In the first case, the atomic spacing is 0.90 nm, in the second case it is 0.36 nm. The radius of the carbon atom is 0.8 nm; this results in non-symmetric deformation of the lattice during the formation of a solid solution. The first position is characterised by a potential well whose depth is smaller than that in the second position. The application of the criteria of the minimum deformation energy for the given positions shows that the first position is significantly more stable than the second position with respect to the 44

Nature and Mechanisms of Anelasticity

Fig .2.18. Time dependence of the change of strain when loading the solid with constant ig.2.18. stress (a) and the change of stress when loading the solid with constant strain (b).

positioning of the interstitial atom. If there is no external stress, the interstitial atoms can travel to the positions in the direction of the x, y and z axes. Loading with a stress along some axis, for example z, increases the space in the faces of the cube parallel with this axis in comparison with the faces of the cube parallel with the axes x and y. The distribution of the interstitial atoms in the positions of the z axis is therefore preferred. This results in tetragonality of the lattice in the z axis. Consequently, additional deformation takes place and its magnitude increases with loading time. In fcc lattices, the largest void is found at the point of intersection of the body diagonals. The interstitial atom in this void causes cubic stretching of the lattice. Tetragonality maybe the result of the presence of a pair of interstitial atoms placed in two adjacent lattices. Therefore, the redistribution of the atoms in the FCC lattice is observed that lower concentrations of the interstitial atoms in comparison with the FCC lattice. The pair of the interstitial atoms is reoriented in space under the effect of external stress resulting in additional deformation which increases with time. The time for the establishment of the new distribution of the interstitial atoms (relaxation time τ r) is the function of the frequency of transitions of the interstitial atoms from one position to another. In this case, it is determined by diffusion equilibrium. 45

Internal Friction of Materials

The redistribution of the magnetic characteristics is the reflection of the relationship between the mechanical and magnetic properties of ferromagnetic materials, especially magnetostriction, i.e. the change of the geometrical dimensions of the ferromagnetic materials in the direction of the external magnetic field. The phenomenon is reversible, which means that mechanical loading results in the displacement of Bloch walls in these materials. The phenomenon can be suppressed using a sufficiently strong external magnetic field. The redistribution of temperatures can be illustrated by, for example, bending of a beam resting on two supports. When bending is adiabatic, for example, at a high rate, the flexure is proportional to the force and is reflected in the compression of the upper fibres and stretching of the lower fibres. The compressed area is heated whereas the stretched area is cooled. A temperature gradient forms in the cross-section of the beam and causes additional deformation whose magnitude depends on time. The time for obtaining the same temperature is determined by thermal conductivity, heat capacity and density. With longitudinal and torsional methods of loading the intensity of the phenomenon is low, but for bend loading it can reach a significant value with the scattering of the mechanical energy in the material. These examples indicate the occurrence of additional deformation, in addition to primary deformation, especially under repeated loading, which is the reason for the lower values of the dynamic elasticity moduli in comparison with the values determined by static methods. The deformation process of the actual component is linked with time by means of additional deformation which may be reflected immediately after loading or after a certain period of time, and the change of the magnitude of additional deformation is controlled by the exponential law and described by the relaxation equation. Movement towards uniform deformation becomes faster with the increase of the initial deviation of the characteristics. There may also be cases in which additional deformation copies the course of damped vibrations. Consequently, vibrations may change to resonance under the effect of the external force. Additional deformation may depend on stress by a directly proportional dependence ε d = cσ, by means of a certain function (ε d = f(σ)) or hysteresis. From the viewpoint of time, we can determine immediately f(t) = const, determine gradually f(t) = e –(t/γ), or in damped or resonance manner determine f(t) = e –βt e iωt , where c is the proportionality constant, σ is stress, t is time, τ and β the characteristics of the material and ω is circular frequency. 46

Nature and Mechanisms of Anelasticity

These examples of the dependence of additional deformation on stress and time can be combined. Three combinations are important for practice: relaxation processes (combination of the relationships ε d = cσ and f(t) = e –βt e iωt ), resonance processes (combination of relationships ε d = cσ and f(t) = e –βt e (t/γ) ), and mechanical hysteresis (combination of hysteresis and f(t) = const). The relaxation processes are described in Fig. 2.18. At time t = 0, stress σ is generated in the material and its magnitude is maintained constant. The deformation, corresponding to this stress, is not detected immediately. Elastic strain ε c forms immediately but the total value of strain ε c is obtained only after a certain period of time. The rate of approach to the value ε c is

ε& =

1 ( ε c − ε ε ). τσ

(2.37)

The value τ σ is the time required to obtain ε c under the effect of constant stress. The dependences of σ and ε in loading and unloading (Fig. 2.19) show that during a deformation process the specimen is loaded with a constant stress for some period of time. The tangent of the angle inclination of the ON line is the Young modulus of the material of the specimen in the stage in which the total deformation has not yet been realised. This modulus corresponds to the adiabatic deformation process and is referred to as the adiabatic or non-relaxed Young modulus E N . The slope of the OR line determines the modulus of elasticity of the material of the specimen when total deformation has taken place and relaxation processes have occurred in the specimen. In the conditions with slow deformation (isothermal loading), during the time longer than the relaxation time we obtain the isothermal or relaxed Young modulus E R which is lower than E N . A different approach to the phenomenon can also be used. The strain ε c is generated in the specimen at a specific time and is maintained constant. It is necessary to decrease the stress, and the rate of this decrease increasea with increase of the difference of the stress and equilibrium values σ 0 (Fig. 2.18b), therefore

σ=

1 (σ0 − σ ). τε

(2.38)

47

Internal Friction of Materials

Fig .2.19. Stress–strain dependence under the effect of connstant stress (a) and constant ig.2.19. strain (b).

The values τ ε is the relaxation time of stress at constant strain. The dependence between σ and ε is in Fig. 2.19b. It shows that the non-relaxed and relaxed elasticity moduli must also be differentiated in this case. Since σ D = E R ε c and from equations (2.37) and (2.39) we obtain the values of σ 0 and ε c , the relaxation equation has the following form

σ + τεσ = ER (ε + τσε& ).

(2.39)

Solids fulfilling this condition are referred to as standard linear solids. The evaluation of additional strain by static methods is difficult because these strains are low and the relaxation time is short. It is therefore necessary to use repeated loading ε in which strain lags behind applied stress σ and the phase shift is ϕ (Fig. 2.20a). During a single loading cycle in the σ−ε coordinates we obtain a characteristic hysteresis loop (Fig. 2.20b). Its area corresponds to the energy scattered in the material during a single loading cycle (∆W). Its axis (line OD) has, however, a different slope in comparison with the one corresponding to the non-relaxed or relaxed Young modulus. Consequently, the tangent of the angle of the OD line characterises the dynamic Young modulus E D . For the metals, we can use the equation in the form 48

Nature and Mechanisms of Anelasticity

F ig .2.20 ig.2.20 .2.20. Delay of strain behind stress (a) and the hysteresis loop (b).

R   , Ed = EN 1 − 2 2   1 + ω τr 

(2.40)

where R is the degree of dynamic relaxation

R=

EN − ER , EN

ω is the circular loading frequency and τ r is the characteristic process time. The Young modulus determined by the dynamic method is lower than that the determined by the static methods, i.e. E D < E R . The relative difference is approximately 1% and is not associated with the measurement error. The lower value of E D is caused by the fact that repeated loading is accompanied by higher elastic deformation in the solid in comparison with the same stress under static loading. Another aspect of the phenomenon (E D < E R) is that alternated loading by the same stress as in static loading decreases the deformation resistance of the material. The phase shift ϕ is the function of the loading frequency. At low frequencies (ω→∞) the relaxation process can take place and ∆W is low (Fig. 2.21). Consequently, E D → E R. At high frequencies (ω → ∞), the relaxation process does 49

Internal Friction of Materials

F ig .2.21. Frequency dependence of the change of the Young modulus and energy ig.2.21. scattered during a single cycle.

not manage to take place, there is no additional deformation and E D → E A and ∆W are low (Fig. 2.21). For an intermediate case in which ωτ r = 1 E D = (E N + E R)/2 and ∆W is high (Fig. 2.21). In cases with the effect of external factors (for example, the magnitude of acting stress) or internal factors (for example, the resonance of vibrations of dislocations segments with the frequency of external loading) the area of the hysteresis loop ∆W increases with increasing number of loading cycles (Fig. 2.22). Consequently, the dynamic Young modulus gradually decreases as a result of repeated deformation and this is reflected in the fact that the selected level of stress leads to higher deformation of the material. 2.2.2 Internal friction Evaluation of the scatter of the energy inside metal is often used in the direct experiments, for example, when measuring the dynamic hysteresis loop, internal friction in the region in which it is dependent on the strain amplitude, etc. Evaluation of the energy losses in a single loading cycle ∆W in long-term loading characterises the kinetics of fatigue damage cumulation. The temperature, frequency, time and amplitude dependences of internal friction provide a large amount of important information on the mechanisms of microplasticity [24] or elastic characteristics, the defect of the Young modulus, the degree of relaxation of the stresses in the examined material, etc. 50

Nature and Mechanisms of Anelasticity

F ig .2.22. Schematic representation of changes of hysteresis loops. ig.2.22.

Internal friction is the property of the solid characterising its capacity to scatter irreversibly the energy of mechanical vibrations [25, 26]. In resonance methods, the internal friction of the material is determined from the width of a peak or depression on the curve of the amplitude of the deviation from the loading frequency at a constant amplitude of vibrations [27]. The amount of the energy scattered in the material is measured using the quantity

Q −1 =

∆ω , 3ω0

(2.41)

where ∆ω is the half width of the resonance peak at half its height, and ω 0 is the circular resonance frequency of vibrations of the specimen. Taking equation (2.41) into account, we obtain

Q −1 =

R ωτr . 1 + ω2 τr2

(2.42)

At ωτ r = 1, internal friction is maximum, Q –1max = R/2. Internal friction is also characterised by the relative amount of 51

Internal Friction of Materials

energy scattered in a single load cycle Ψ, which is determined from the area of the hysteresis loop ∆W and from the total energy supply to the system W, corresponding to the maximum strain in the cycle in which ∆W was determined, therefore

ψ=

∆W . W

(2.43)

In dynamic measurements

Q −1 =

ψ . 2π

(2.44)

The logarithmic decrement of vibrations δ is determined by the equation

δ = ln

zn , z n +1

(2.45)

where z n and z n+1 is the amplitude of the n-th and n + 1 cycle of damped vibrations of the solid. The numerical values δ are equal to the relative scattering of energy (irreversible change of the energy of vibrations to heat) during a vibration cycle. When δ << 1, the following equation is valid

ψ = 2δ.

(2.46)

For small phase shift values of σ and ε (Fig. 2.20) tan ϕ ≈ ϕ = (1/2π)(∆W/W). The characteristics Q –1 , Ψ, δ, ϕ are linked by the equations

Q −1 = tan ϕ ≈ ϕ =

ψ δ ∆W . = = 2 π π 2 πW

(2.47)

The amount of energy absorbed by the solid can also be measured using other criteria, such as the increment of the vibrations, the coefficient of absorption of the acoustic wave, etc. The values of Q –1 , ϕ, δ, Ψ etc. can be defined for any stress 52

Nature and Mechanisms of Anelasticity

state. The difference is that in the case of inhomogeneous stress the characteristics are identical in the entire volume. For inhomogeneous stress this holds only when Q –1 , Ψ, δ, ϕ, etc., are independent of the strain amplitude. Otherwise, these characteristics represent the mean values of these quantities in the entire volume of the solid. Equation (2.47) is valid only for low values of internal friction. At high values of Q –1 (and, consequently, of Ψ, δ, ϕ, etc.) the internal friction depends on the internal friction mechanisms and must be determined separately for every case. The measurements of ϕ and δ depend not only on the frequency and amplitude of vibrations but also on the type of loading. The values were determined in transverse and longitudinal vibrations and depend on, for example, the relaxation mechanism. The difference of the Young modulus is determined by the equation

∆G =

3 ∆ E. 2 (1 + µ )

(2.48)

For the majority of metals and alloys µ = 0.27–0.37 which shows that the changes of ∆G are 10–18% higher than the changes of ∆E. It is therefore more efficient to determine the value of ∆G than ∆E.

2.2.3 Mechanisms of energy scattering in the material Internal friction is an integral representation of the activity of the individual components which depend or do not depend on frequency. The frequency-dependent components include the thermoelastic phenomenon, the movement of valency electrons, the viscosity of grain boundaries, the movement of interstitial atoms in interstitial solid solutions, the movement of interstitial atoms in substitutional solid solutions, the change of the orientation of paired defects, and dislocation relaxation. The thermoelastic phenomenon can be observed when loading solids produced from single crystals or polycrystalline materials. If a test bar is loaded with a sufficiently high rate, the bar is extended by the value 0a' (Fig. 2.23). Accurate measurements show that the elongation of the bar is accompanied by its cooling. The extended and partially cooled bar equalises its temperature to ambient temperature and, consequently, it is extended by the value a'b'. This extension is referred to as the direct elastic after-effect and its rate 53

Internal Friction of Materials

F ig .2.23. Diagram illustrating the thermoelastic phenomenon. ig.2.23.

is low. As a result of compression the bar is again heated in relation to the environment and a reverse process takes place. In repeated loading of the material, this phenomenon causes the scatter of supplied energy. Zener calculated that the scatter depends on the dilation coefficient, heat conductivity, specific mass and the Young modulus of the examined material. The thermoelastic phenomenon is strongest in the frequency range from 1 Hz to 1 kHz. The frequency at which damping is maximum is equal to D/D 2z, where D is the selfdiffusion coefficient and d z is the mean the grain size of the polycrystalline material. The movement of valency electrons, associated with external loading, can be observed especially at low temperatures (several Kelvins) and at a high frequency of changes of the orientation of loading of the solid. At a relatively high rate of compression of the material, for example, in the x direction, the velocity of movement of the valency electrons in this direction increases. This changes the original, for example, the spherical Fermi level to an ellipsoid. The new shape of the Fermi level is obtained after a certain period of time and the stress in compression, required for elastic deformation, gradually decreases. Part of the supplied vibrational energy is consumed for these changes. The viscosity of the grain boundaries is characterised by microplastic deformation which already forms under the effect of low shear stresses and is associated with the displacement of the atoms of the grain boundaries, even though the deformation of the grains is only elastic. This leads to hysteresis phenomena. When a polycrystal is subjected to repeated loading at low temperature, the energy scatter is small (Fig. 2.24) because the time available is too short for atom displacement. In loading in the medium temperature 54

Nature and Mechanisms of Anelasticity

range there is a significant scatter of the supplied energy explained by the self diffusion of atoms along the grain boundaries. The movement of interstitial atoms in the interstitial solid solutions can be evaluated most efficiently in α-iron which contains carbon and nitrogen, already at room temperature and a frequency of approximately 1 Hz. According to Snoek, if steel is not subjected to external loading, the carbon or nitrogen atoms are in octahedral positions, Fig. 2.25. When stress is applied, the atoms are transferred to stretched interstitial positions from compressed positions. The length of the jump in repeated loading is x = (a√2)/2, where a is the lattice spacing. The time t required to travel the distance x is determined by the equation x 2 = D⋅t, where D is the coefficient of diffusion in α-iron. For a diffusion process, time τ 1, required for a single jump, is determined by the equation

τ1 = τ0 e

−

q kT

(2.49)

.

F ig .2.24. Temperature dependence of internal friction for polycrystalline aluminium ig.2.24. (1) and single crystal aluminium (2) at a frequency of vibrations of 0.8 Hz. 55

Internal Friction of Materials

F ig .2.25. Positions for the distribution of interstitial atoms in the cubic body-centred ig.2.25. lattice: 1) atoms of the main metal; 2) octahedral positions; 3) tetrahedral positions.

where τ 0 is the physical constant, k is the Boltzmann constant, T is absolute temperature, and q is the size of the energy barrier which the carbon atom must overcome when jumping from the interstitial position to the adjacent one. When the time t, given by the frequency of external stress, is many times higher than τ 1 for the given temperature, the carbon atoms manage to move under the effect of external stress, without deformation lagging behind stress. When t << τ 1, the carbon atoms remain in their sites and there is no delay (ϕ). Only at t ≈ τ 1 , the interstitial atoms exert time-dependent resistance to deformation resulting in a delay and, consequently, damping, Fig. 2.26. In this case, x 2 = Dτ 1 . Consequently

a2

2 2

= D τ1.

The maximum of internal friction Q –1 is directly proportional to the concentration of the interstitial element in the solid solution. The increase of the frequency of vibrations displaces the maxi56

Nature and Mechanisms of Anelasticity

F ig .2.26. Temperature dependence of internal friction for pure iron with 0.027 wt.% ig.2.26. of nitrogen.

mum of Q –1 to higher temperatures, without any change of the height of the peak. Similar maxima of Q –1 were detected for the bcc metals with interstitial atoms (for example, nitrogen in iron, nitrogen in tantalum, oxygen and nitrogen in vanadium and niobium, etc.). Snoek friction enables the diffusion coefficient to be measured at room temperature. It makes it possible to evaluate the change of the carbon content of the solid solution; this is highly advantageous in investigations of precipitation processes. The movement of interstitial atoms in fcc lattices, being an analogue of the previous description, was observed but the internal friction resulting from the movement of these atoms was extremely small because the deformation of the lattice by the interstitial atoms is small. This also applies to vacancies or substitutional atoms placed in the nodal sites of the lattice. The change of the orientation of pair defects contributes to the formation of internal friction. The effect of the external forces changes the orientation of the individual pairs of defects (pairs of vacancies, substitutional and interstitial atoms, substitutional atom and vacancy, in the close vicinity). It is assumed that a pair of substitutional atoms is situated in the cubic lattice. A pair of defects causes deformation of the close vicinity and results in 57

Internal Friction of Materials

F ig .2.27. Illustration of the change of a pair of defects under the effect of a pair ig.2.27. of defects.

tetragonality of the lattice. If the metal is not subjected to the effect of external forces, the individual pairs of the defects are oriented randomly according to the orientation of the individual crystals, Fig. 2.27. After applying an external force, the orientation of the pair of the defects changes. When the pairs of the defects are formed by atoms larger than the atoms of the main lattice, under the effect of the external force they are oriented in the direction of the external force. This may be used to explain the maximum of damping for Cu–Zn alloy (Zeener), Ag–Zn (Nowick, Fig. 2.28) and other alloys. The extent of damping is proportional to the number of defect pairs which can change the orientation. The relaxation of dislocations (Bordoni phenomenon) is associated with the change of stress in the dislocation line. The decrease of stress is determined by the movement of small kinks of the dislocations pinned at impurities or other dislocations (Manson). The dislocation line can be shortened as a result of vibrations of the dislocations and depends on the shape of the dislocation line and the level of local stress (Seeger). The formation of the maximum of internal friction at a specific temperature depends strongly on the frequency of repeated stress because the phenomenon is of the resonance nature. Dislocational frequency-dependent friction forms at megahertz frequencies. In the case of kilohertz frequencies, it is pos58

Nature and Mechanisms of Anelasticity

F ig .2.28. Temperature dependence of internal friction of Ag–Zn single crystals with ig.2.28. the (110) orientation containing: 1) 3.7 wt.% Zn; 2) 10.6 wt.% Zn; 3) 16.5 wt.% Zn; 4) 26.2 wt.% Zn; 5) 30.25 wt.% Zn.

sible to ignore the dislocational frequency-dependent friction. Information on the operation of the individual mechanisms of scattering of mechanical energy in the material at different frequencies is presented in Fig. 2.29. The formation of internal friction peaks, their height and frequency affiliation are a function of the type of material, its composition, substructural state and, in particular, temperature during loading. The frequency-independent processes include the magnetomechanical phenomenon and the amplitude-dependent vibration of dislocation segments. The magnetomechanical phenomenon is the result of magnetic hysteresis in ferromagnetic materials and is associated with the movement of Bloch walls. The effect of this component of internal friction can be greatly suppressed by applying a sufficiently strong external field to the loaded solid. For example, a magnetic flux with an intensity of 2⋅10 4 A⋅m –1 greatly decreases the contribution of this phenomenon in iron and mild steel. The amplitude-dependent vibration of the segments of the dislocations can be described using the model of vibration of an an59

Internal Friction of Materials

F ig .2.29. Change of internal friction in relation to loading frequency the orientation ig.2.29. indication of maxima at 20°C for different energy scattering mechanisms: 1) change of orientation of pair defects; 2) viscosity of grain boundaries; 3) viscosity of twin boundaries; 4) movement of interstitial atoms; 5) thermoelastic phenomenon; 6) exchange of thermal energy between grains.

chored spring, in accordance with the Granato–Lücke theory [28], supplemented by other authors. We shall assume a random dislocation distribution in a crystal of a real metal. Some parts of the dislocations L are pinned, for example, due to the presence of clusters of point defects K at the dislocations, Fig. 2.30. Selecting a dislocation segment and gradually increasing the amplitude of external stress σ a , after reaching some value σ a the segments with the length L p will bend (Fig. 2.30b) in areas where they are not blocked. As a result of tensile loading, the pinning points can move in the dislocation line, depending on the origin. After reaching the critical value σ a , when 2r = L p (2.30c), the individual segments with the original length L p can combine in a single segment with length L n (Fig. 2.30d). The increase of stress σ results in a linear increase of strain ε disl (section ABC in Fig. 2.31) up to the stress resulting in unpinning of the dislocation from the pinning points (section CD). Another increase of stress increases the strain in accordance with the DG curve, up to the formation of a new dislocation from a source with the distance of the pinning points L n . After unstressing, the strain 60

Nature and Mechanisms of Anelasticity

σa = 0

increasing σa→

F ig .2.30. Diagram describing the vibration of parts of pinned dislocations and the ig.2.30. formation of new dislocations.

decreases along the GDA line, i.e. in the direction differing from that during loading. In repeated loading the curve passes along the points ABEF and FEA. The area defined by these curves is proportional to the absorbed scattered energy and its size depends on the number of points from which the dislocation became unpinned. The process of unpinning of the dislocation and generation of new dislocations is shown in Fig. 2.30e. After a further increase of the stress amplitude up to some critical value, the dislocations segment with the initial length L n can therefore generate dislocation kinks

F ig .2.31. Dependence of stress on dislocational strain. ig.2.31. 61

Internal Friction of Materials

and renew the activity of the dislocations source (Fig. 2.30f) which leads in the final analysis to cyclic microplastic deformation. These processes, starting with the vibration of the segments of the dislocations pinned at the points K up to the generation of new dislocations, depends on the magnitude of the stress amplitude σ a or the strain amplitude ε and provide different contributions to the extent of internal friction. Because of the importance of the effect of the stress amplitude on the extent of internal friction in the examination of substructural, structural and deformation characteristics of materials and the derivation of relationships of fatigue damage cumulation in materials, the problem is examined in detail, especially in section 3.2 and chapter 6 of this book. Hysteresis is observed to various degrees in all materials. The main reason for it is the irreversible movement of the dislocations in the stress field, i.e. microplastic deformation. The hysteresis curve in the ideal case is independent of time and depends only on the magnitude of stress. Consequently, the Young modulus and internal friction are independent of frequency but depend on the strain amplitude. The slope of the hysteresis loop and its area do not change in relation to the rate of repetition of loading. The shape of the hysteresis loop is influenced mainly by the level of the maximum stress. Figure 2.32 shows schematically the frequency (A) and amplitude (B) dependence of the Young modulus and internal friction for the case of relaxation (I), resonance (II) and hysteresis (III). The third case is of special importance for engineering. Strain amplitude-dependent internal friction, formed at high strain amplitudes, makes it possible to suppress the development of dangerous resonance states of components, sections and whole structures. 2.3 Defect of the Young modulus The above results show that the Young modulus is not a physical constant but it is a characteristic influenced by a large number of factors. In practice, the concept of the ideal Young modulus can be used only for an isotropic, single-phase and defect-free metallic material, deformed in the elastic stress range. When loading a real solid, total strain ε contains the elastic strain ε e and additional plastic strain ε p, which depends on the magnitude of acting stress and also loading rate, Fig. 2.33. The occurrence of the additional, plastic deformation changes the Young 62

Nature and Mechanisms of Anelasticity

F ig .2.32. Changes of the Young modulus and internal friction in relation to frequency ig.2.32. (A) and strain amplitude (B) for relaxation (I), resonance (II) and hysteresis processes (III).

modulus of the material which is characterised in this case by the effective value, for example, E ef . In static measurements of E ef , we can use the following equation

Eef =

σ . εe + ε p

(2.50)

When measuring the elastic characteristics of materials we determine their effective values. The difference between the ideal and effective modulus of elasticity (Young moduus) is small but, in some cases, for example, in measurements of the elasticity moduli by static methods at stresses of (0.7–0.8) R e the difference may reach 40–50% [29]. When evaluating the effective values of the elasticity moduli, it is useful to take into account information on the structural sensitivity of the elasticity moduli. The effective values of the elasticity moduli are useful not only for engineering calculations but they also represent a source of valuable information on the nature and mechanisms of the additional, plastic component of the total strain. In this case, it is recommended 63

Internal Friction of Materials

F ig .2.33. Composition of total strain during loading of material. ig.2.33.

to use a dimensionless parameter, the so-called defect of the Young modulus, defined by the equation

∆E E − Eef = , E E

(2.51)

where E is the ideal or initial value of the Young modulus. The relationship of the defect of the Young modulus with ε p depends on the loading conditions. Figure 2.33 shows that in static loading

∆E d ε p = . E dε In repeated loading with the stress σ = σ 0 exp (i ω t) the resultant strain contains the elastic component of strain ε e and the plastic component of strain ε p , i.e. ε = ε e + ε p . The inelastic, plastic component of strain can be divided into the following parts: ε p = ε′p – iε″p . The phase of the first part (ε′ p ) is identical with the external loading phase. The phase of the second part (ε″p ) is displaced by 90° in relation to the external loading phase. The occurrence of ε p is determined by the scatter of the elastic energy of vibrations in 64

Nature and Mechanisms of Anelasticity

the material. Different mechanisms of the transformation of the energy of elastic deformation to thermal energy are included in the integral name: internal friction. Consequently

Q

−1

=

ε**p εe

.

(2.52)

The formation of ε p results in a change of the elastic properties of the material which are characterised in this case by the dynamic Young modulus

ED =

σ . εe + ε′p

(2.53)

The difference between the values of the static and dynamic elasticity moduli is associated with the fact that the real solid is not characterised by ideal elasticity. The value of E D depends on the type, nature and mechanisms of anelasticity of the material and, consequently, it is the equivalent of E ef , determined in static loading. For the case of repeated loading we have

∆E ε′p = . E εe

(2.54)

The effective elasticity moduli are the result of the effect of several factors which can be conventionally divided into two groups. The first group includes the factors which take into account the phase composition, structural state, the level of micro- and macrostresses, etc. This classification has a number of problems because both groups overlap. It is useful to distinguish between the nonlinear and nonelastic behaviour of the material. The anelastic behaviour the material results in the formation of a hysteresis loop. The nonlinear nature of the σ–ε dependence results in the formation of E ef and ∆E/E with the change of ε. The anelasticity phenomena, associated with the formation of the hysteresis loops, indicate the nonlinear behaviour of the material. 65

Internal Friction of Materials

F ig .2.34 ig.2.34 .2.34. Form of the hysteresis loop of the material with ∆ E / E = 0 (a) and ∆ E / E ≠ 0 (b)

However, the occurrence of the defect of the Young modulus is not direct confirmation of the anelastic behaviour of the material. The evaluation of the contour of the hysteresis loop makes it possible to determine the parameter characterising the amount of scattered energy, i.e. the area of the hysteresis loop ∆W. The level of the defect of the Young modulus is determined by the middle line of the hysteresis loop [30]. In the symmetric hysteresis loop (Fig. 2.34a) the middle line is straight. Here, the defect of the Young modulus is equal to 0. If ∆E/ E ≠ 0, the loading curve must be nonlinear, Fig. 2.34b. Consequently

2W , ε2

ED =

(2.55)

where

z bg

ε0

W = σ ε dε. 0

The nonlinear form of the middle line of the hysteresis loop is the result of the effect of two types of factors. They are factors whose effect results only in the formation of the defect of the Young modu66

Nature and Mechanisms of Anelasticity

lus, and factors causing both the defect of the Young modulus and anelastic scattering of energy in the material. Additional deformation is immediately evident and depends only on the magnitude of acting stress. The type of anelastic processes, related to the second group of the factors, is determined by the type of the dependence of the additional plastic deformation on the acting stress and loading time. In anelastic processes of the relaxation type, the defect of the Young modulus, referred to as the degree of relaxation, is determined by the equation

∆E E N − E R = . E EN

(2.56)

The magnetomechanical mechanism of relaxation in ferromagnetic materials can cause the formation of the defect of the Young modulus depending on saturation. The value ∆E is the tensor of the fifth order and depends on the mutual orientation of the main axes of the strain tensor, crystallographic axes and the saturation vector. The formation of the ∆E effect is associated with the change of the domain structure of the ferromagnetic material during loading. It is reflected in additional, anelastic deformation. The increase of saturation from zero to the value I results in a change [31] of the Young modulus in accordance with the equation

 2b  K( I ) − K(0) = Vf + g  2 − 1 , a 

(2.57)

where

f =

∂2F , ∂ 2V

F is the free energy of the ferromagnetic material

g=

∂F , ∂V

a, b are the coefficients in the Bridgman equation 67

Internal Friction of Materials

∆V = aP + 6 P 2 , V where P is pressure and V is volume. The change of the Young modulus ∆E of the ferromagnetic materials ∆E = E S − E ,

(2.58)

where E S and E are the elasticity moduli of the material in the saturated and initial conditions. The value of the ∆E phenomenon is high in materials with high magnetostriction, with moderate magnetic crystallographic anisotropy and a low level of internal stresses. The characteristic ∆E/E S reaches almost 20%, for example, in the case of annealed nickel. One of the manifestations of the ∆E phenomenon is the difference between the values of the elasticity moduli measured by dynamic methods whilst maintaining a constant value of the external magnetic field (E H ) and saturation (E S ), and

k12 =

ES − E H , EH

(2.59)

where k 1 is the coefficient of the magnetomechanical bond. When the temperature is increased to the range around Curie temperature, the material changes to the paramagnetic state, the ∆E phenomenon disappears and normal changes of the elasticity moduli in relation to temperature are recorded above T C (Fig. 2.35). In the saturated condition, the dependence of the Young modulus on temperature has the conventional form. However, for certain alloys (for example, Elinvar) the effect of the magnetic field with the saturated value results in anomalies in the E – T dependence. Iron with 45 wt.% Ni in a magnetic field with an intensity of 0.8 × 10 4 A m –1 shows values independent of temperature from 0 to 470°C. The movement of the dislocations under the effect of the external stress is the reason for the dislocation or anelasticity of the materials. The existence of a large variety of dislocation structures and of specific features of the interaction of dislocations with point 68

Nature and Mechanisms of Anelasticity

F ig .2.35. Temperature dependence of the Young modulus of polycrystalline nickel ig.2.35. for different saturated magnetic fields (10 4 Am –1 ): 1) 0; 2) 0.05; 3) 0.08; 4) 0.33; 5) 0.85; 6) 4.8 (complete saturation).

defects and atoms of the alloying elements in alloys requires the development of various mechanisms describing dislocation anelasticity. These mechanisms have been described in greater detail in Ref. 2, 32. Additional strain ε p, caused by dislocation anelasticity, is proportional to the total length of moving dislocations and the mean value of displacement of the dislocations from the initial (equilibrium) position. The controlling factor is the change of the local characteristics of interaction of the dislocations and the atoms of the alloying elements situated in the atmospheres around the dislocations in the solid solution. In cases in which the dislocation does not interact with other dislocations, the movement of a single unpinned dislocation in the ideal, pure material is of the viscous type. The rheological properties of the elastic–viscous bond are important in this case. The hysteresis loops of mechanical hysteresis of such a material are in fact ellipses which grow symmetrically with increasing stress (Fig. 2.36a). In actual metallic materials, the movement of the individual dislocations is associated with overcoming barriers of various type. The magnitude of stress, required for the start of movement of the dislocations, is associated with the considerations of the dry fric69

Internal Friction of Materials

F ig .2.36. Hysteresis loops in operation of different mechanisms of energy scattering ig.2.36. in material.

tion mechanism. The nonlinear form of the σ – ε curve and opening of the hysteresis loop takes place only at σ > σ t, where σ t is the stress for dry friction (Fig. 2.36b). Here, the defect of the Young modulus and internal friction depend on the amplitude of acting stress. The form of the curve in Fig. 2.36b is identical with the deformation diagram for the mechanism of dragging the atmospheres of alloying elements by the dislocations temporarily arrested and blocked on a system of the disordered obstacles. σ t has its own physical meaning for every case. The mechanisms of dislocation anelasticity are based on the physical models of two types. They are the models of detachment of dislocations which are probable a modification of the dry friction model. In these models, friction does not take place in all points of the space, but only in microvolumes with the effect of solute atoms. The hysteresis loop for this case is in Fig. 2.36c. The second type of model links the displacement of the elastic stress field, which surrounds the dislocation during its movement, with special features of the redistribution of the atmosphere of solute atoms. Redistribution is regarded as a diffusion process in a changing inhomogeneous stress field around the dislocation and is described on the basis of the considerations of the elastic–viscous nature of bonding. 70

Nature and Mechanisms of Anelasticity

Qualitatively similar results were obtained for the case of interaction of the dislocations by the mechanism of dislocation anelasticity. They are based on the assumption on the conservation of the dislocation density of the type of dislocation configuration. Hysteresis anelasticity, formed by the movement of dislocations, results in a decrease of the elasticity moduli in the form

∆E E d εdisl . , = E dσ

(2.60)

where ε disl is the dislocational strain. Equation (2.60) characterises the defect of the Young modulus in the stress range preceding the development of microplastic deformation. Taking into account the model of vibrations of a spring according to Granato and Lücke [33], the defect of the Young modulus is determined by the equation

∆E 6Ω = 2 ρd L2 , E π

(2.61)

where Ω is the orientation factor, ρ d is the dislocation density, L is the effective length of the dislocation determined by the equation 1/L = 1/L p + 1/L n . The redistribution of dislocations and also dislocation multiplication processes contributes to the anelasticity of the material. For these processes to take place, it must be σ > σ t (Fig. 2.36d). The increase of the proportion of mutual interaction of the dislocations at a sufficiently high external stress results in a more distinctive nonlinear form of the σ−ε curve and in a significant increase of the defect of the Young modulus. In the microplastic range, the dependence of the defect of the Young modulus on the stress amplitude or strain (e) amplitude has the form

∆E = Ae a , E

(2.62)

where A and a are the parameters that depend on the type and conditions of loading. The mechanisms of dislocational anelasticity are characterised by 71

Internal Friction of Materials

a wide spectrum of the relaxation times. All models are characterised by the directly proportional relationship of the defect of the Young modulus with the density of moving dislocations. In annealed materials, the defect of the Young modulus, based on dislocation behaviour, is only several tenths of a per cent. At higher temperatures, the anelasticity mechanism at the grain boundaries is also important. It is associated mainly with the viscous slip in the vicinity of the grain boundaries. Calculations of the defect of the Young modulus for this type of relaxation show that it is strongly influenced by the Poisson number and may represent several tens of per cent, as confirmed for alpha brass. The anelastic nature of deformation of the material during cycling loading results in the occurrence of two mutually linked phenomena, internal friction and the defect of the Young modulus. In accordance with the equations (2.52) and (2.54), we can write that

M =

Q −1 ε p ′′ , = ∆E ε p′ E

(2.63)

The main source of information on the anelasticity mechanisms and the values of the parameters controlling these mechanisms is still the measurement of internal friction. On the basis of a large number of measurements and publications of internal friction it was assumed that the defect of the Young modulus is a characteristic of secondary importance which does not provide any further information. The latest results show that internal friction and the defect of the Young modulus are not identical anelasticity characteristics, because internal friction evaluates the size of the hysteresis loop and the latter the nonlinearity of the central line of the loop. Under the combined effect of a large number of external factors and reactions of the material in the form of activation of various anelasticity mechanisms, it may be seen that each of them provides a separate contribution to the formation of hysteresis loops observed in practice. Detailed information on the behaviour of the material under the effect of external alternating loading may be obtained by combined measurements with evaluation of the internal friction and the defect of the Young modulus. These characteristics (Q –1 , ∆E/E) are measured by two methods, i.e. as the isothermal dependences of the defect of the Young modu72

Nature and Mechanisms of Anelasticity

lus on strain amplitude or as the dependence of the defect of the Young modulus on temperature at a constant stress or strain amplitude. In the first case, the defect of the Young modulus is determined from the equation

f2− f2 ∆E =A r 2 x , E fr

(2.64)

where f r and f x is the natural frequency of vibrations of the specimen, determined at the minimum stress amplitude and at the selected stress amplitude. In the second case, the defect of the Young modulus is determined from equation (2.51), and E ef has the meaning of the dynamic Young modulus at temperature T determined from equation (2.22). The numerical values of the parameters in equation (2.22) are determined by the approximation of the temperature dependence of the elasticity moduli from the region of lower temperatures where the effect of the evaluated mechanisms of anelasticity can be ignored. The relationship between Q –1 and ∆E/E (equation 2.63) is determined by the type of the mechanism of the scattering of mechanical energy in the material. For the case of the effect of the relaxation and resonance mechanisms, internal friction has only the temperature-frequency dependence. The internal friction, caused by the hysteresis mechanism, depends on the strain amplitude and remains almost constant with increasing loading frequency. Taking the results into account, equation (2.63) changes for different mechanisms of scattering of mechanical energy. When the value of the fraction M in equation (2.63) is 0.5, the internal friction peak characterises a specific relaxation time, or the peak forms by the superimposition of several relaxation maxima. In the second case, the value of the defect of the Young modulus is determined with the accuracy equal to the accuracy of determination of the sum of the degrees of relaxation of every relaxation process. The total height of the peak is smaller than the sum of the heights of every separate maximum. In the case of superimposition of several relaxation maxima, the value of ratio M in equation (2.63) is lower than 0.5. If thermal relaxation takes place in the material, the defect of the Young modulus is determined by the difference of the adiabatic and isothermal elasticity moduli. The effect of the resonance mechanisms of internal scatter of 73

Internal Friction of Materials

energy can be explained on the example of reverse movement of the dislocations in a viscous medium (Granato–Lücke model). Additional deformation occurs by displacement of the dislocations in the external stress field ε p = ρ d bl, where ρ d is dislocation density, b is the Burgers vector, and l is the mean displacement of the dislocations in the slip plane determined from the equation

l=

1 Lp

Lp

∫ l ( x ) dl,

(2.65)

0

where L p is the length of the dislocation segment between the pinning points. The x axis is oriented in the direction of dislocation movement. For the model of the elastic string, the equation of movement at kilohertz and megahertz frequencies has the form

B

∂l ∂ 2l + χ 2 2 = σ b eiωt , ∂t ∂x

(2.66)

where B is the characteristic of viscous friction, χ 2 is the linear energy of the dislocations. Consequently

Q−1 =

A1 ω B ρd L4p χ2

,

(2.67)

∆E = A2 ρd L2p , E

(2.68)

where A 1 ≈ A 2 ≈ 1/6 is a coefficient which depends on the value of χ 2. The equations (2.67) and (2.16) show that the values of Q –1 and ∆E/E for the same model characterise different aspects of the process. The area of the hysteresis loop and the formation of Q –1 are determined by the viscous movement of the dislocations and, consequently, depend on the loading frequency and viscous friction coefficient B. The defect of the Young modulus is determined by the

74

Nature and Mechanisms of Anelasticity

effective force of linear stretching of the dislocation. For the given resonance mechanism of anelasticity, the ratio M from equation (2.63) is proportional to

B ω L2p χ2

.

For copper at room temperature, a frequency of 100 kHz and L p = 5 × 10 –3 mm M = 0.04. Simultaneous measurements of Q –1 and ∆E/E using equations (2.67) and (2.16) have been described only in a very small number of cases. The available studies indicate that for alloys the ratio M from equation (2.63) is in the range 0.05–0.1 when the content of the atoms of the alloying elements is increased from 0.5 up to 5 wt.%, but when the prior strain is increased to 30%, the value of the ratio rapidly decreases. Increase of the carbon concentration results in a decrease of L p in proportion to C –0.6 , and the value of L p is 10–40 times higher than the calculated values of the mean spacing of the atoms of the alloying elements on the dislocations. Areas of dislocation pinning in concentrated solid solutions are not represented by the individual atoms of the alloying elements but by their clusters containing several tens of these atoms, as shown for solid solutions based on aluminium and iron [34]. For the hysteresis type of anelasticity of the material, the relationship between Q –1 and ∆E/E depends on the relationship between anelastic strain ε p and external stress σ. When the anelasticity of the material is linear under the effect of external stress, both components ε′p and ε″p depend in the same manner on the external stress σ. Consequently, the ratio determined by equation (2.63) does not depend on stress amplitude σ nor on loading frequency. Two mechanisms of dislocational anelasticity have been developed. For the first case, the hysteresis of dislocations is the result of cyclic break away of the dislocations from the atmospheres of the solute atoms. The ratio M, determined by equation (2.63), for this case is a constant and its value varies from ~ 0.3 (for the theoretical model according to Granato and Lücke) up to ~ 1 for the mechanisms of thermal activation of release of dislocations from the atmospheres. In these models, the maximum value of dislocational strain is restricted by the tensile force in the dislocation. Increase of the ex75

Internal Friction of Materials

ternal stress amplitude increases the magnitude of bowout of the dislocation and the level of dislocational strain is influenced by interactions of the dislocations with internal barriers, for example, with stress fields from a dislocation forrest, mosaic boundaries, etc. Subsequently, increase of the stress amplitude should result in a decrease of the ratio determined by equation (2.63). The second type of mechanisms is based on the evaluation of viscous movement of the dislocations in the lattice with uniformly distributed solute atoms. For this approach, the dependence of the speed of dislocations v d on the value of acting stress σ has the form m

σ Vd = V0   ,  σ1 

(2.69)

where v 0, σ 1, m are experimental constants. For this model [35] with the change of m from 1 to higher values, the ratio determined by equation (2.63) is in the range 0.85–0.93. In simultaneous measurements of Q –1 and ∆E/E on single crystals of copper, zinc, and on solid solutions of these elements with the content of solute elements to 0.01 wt.%, the ratio determined by equation (2.63) in the range of dependence of Q –1 on strain amplitude is 0.5–1.0. For a copper alloy with 35 wt.% Zn, the ratio is from 0.06 to 0.25. Under the effect of higher strain amplitude the material behaves as a nonlinear solid. This is reflected in the value of the ratio determined by equation (2.63), because it becomes dependent on the strain amplitude. Two types of dependence can be observed in this case. In the case of a purely hysteresis mechanism of anelasticity, the increase of nonlinearity of the σ−ε dependence results in a directly proportional decrease of the value of the ratio determined by equation (2.63), as shown in Fig. 2.37a. The second type of the dependence of M on stress amplitude γ appears when the viscous movement of fresh unpinned dislocations in the crystal lattice dominates at γ < γ cr2 . Increase of the level of microplasticity increases the density of fresh dislocations and this lead to an increase of the value of coefficient B which depends on the frequency of intersection of the dislocations moving in non-complanar slip planes. In this case, the value of M is proportional to the parameter B and will increase with increasing ε. This is observed for, for example, copper, Fig. 76

Nature and Mechanisms of Anelasticity

F ig .2.37. Dependence of σ, ∆ G / G and r on the strain amplitude of annealed materials ig.2.37. at a frequency of 2 Hz.

2.37. At γ < γ cr2 M is constant. The change of M with increasing γ with both dependences added up together is shown in Fig. 2.37c. This shows that the M–ε dependence is highly sensitive to the development of nonlinear anelasticity. Consequently, measurements of the M–ε dependence can be used to determine the range of linear and microplastic anelasticity. If microplastic processes take place in the material, it can be seen that the dependence of ∆G/G on γ is irreversible. Evaluation of the defect of the Young modulus with the change of temperature and at a specific value of the strain amplitude showed that the values of ∆E/E decrease to lower strain amplitude and are associated with the processes of thermal activation of dislocation movement. Consequently − ∆E = Ae E

H −σV kT

,

(2.70)

where A is the proportionality constant, H is the effective value of 77

Internal Friction of Materials

activation enthalpy, and V is the activation volume. It is obvious that the elasticity characteristics and anelasticity of the material are influenced by a large number of external and internal factors.

78

Factors Affecting Anelasticity of Materials

3 FACTORS AFFECTING ANELASTICITY OF MATERIALS The resultant value of the extent of internal friction and the defect of the Young modulus is the integral representation of frequencydependent and frequency-independent mechanisms of scattering of mechanical energy in materials and also of many other processes, phenomena and mechanisms which can operate in different materials under the effect of different external influences. In this chapter, attention will be given to explaining the physical nature of internal friction which is independent of the strain amplitude, the so-called internal friction background Q –1 and also the effect of temperature for materials with different structures and substructures. Since the problem of cyclic microplasticity is examined in Chapter 6, in this chapter, we shall present only the main data on the effect of the strain amplitude on internal friction and the defect of the elasticity (Young) modulus. However, special attention will be given to the effect of loading frequency, loading time, the number of load cycles and the effect of the magnetic field in ferromagnetic materials. Each section shows that, in addition to understanding the nature and effects of the individual influences, it is also important to obtain information on the accuracy of measurement of Q –1 or ∆E/E on the change of these factors. Many of the effects are not unique and often overlap, act together or against each other. This is also explained by the differences in the effect of temperature on internal friction and possibly the defect of the Young modulus, because other mechanisms are determined by the response of the material to repeated loading and this reaction is significantly influenced by temperature in loading.

79

Internal Friction of Materials

3.1. INTERNAL FRICTION BACKGROUND When measuring the internal friction of materials in the area in which there are no phase transformations and in the range of the strain amplitudes in which the internal friction does not depend on the strain amplitude, the internal friction background is recorded as the reflection of the integral mechanisms of scattering of mechanical energy in the material. The quantification and analysis of the internal friction background has been the subject of special attention in the development of new structurally stable creep-resisting materials [25,28]. Consequently, it is necessary to evaluate theoretical models and activity of the mechanisms in a wide temperature range [36]. There are three stages of the dependence of the internal friction background Q –1 on temperature: 1) slight, approximately linear 0 dependence Q –1 – T at temperatures lower than (0.5–0.6)T m ; 2) 0 the experimental dependence of the type Q –1 at temperatures higher 0 than 0.6T m ; 3) the stage of decrease of the growth of the internal friction background at temperatures close to the melting point. In the first stage of the dependence Q –1 (T) we can use the 0 theory proposed by Granato and Lucke [28] in the form

Q0−1 ~ ρL4

and

∆M ~ ρL2 M

(3.1)

where ρ is the density of the vibrating dislocations, and the meaning of L was explained in the section 2/2/3. The temperature deis associated with the function of distribution of pendence of Q –1 0 dislocations on the basis of their orientation [37] and with the type of dislocation clusters [13]. The length subdivision of the dislocation segments has no significant effect on Q –1 [39]. 0 At present, special attention is given to the next two stages of the Q –1 (T) dependence. The general relationships were evaluated 0 for the first time by the author of Ref. 40 and then in Ref. 26, where it was shown that the internal friction background correlates with the creep strength of the materials, i.e. as the creep strength of the alloy increases, the value of Q –1 at the examined tempera0 ture decreases. The results showed a relationship between the flow stress and creep rate with the value of internal friction [41]. The internal friction background is influenced by many factors.

80

Factors Affecting Anelasticity of Materials

Fig. 3.1. Dependence of internal friction on temperature for copper filament crystals (a) silicon filament crystals (b). a: 1) initial condition, 2) after plastic deformation at room temperature, 3) polycrystalline specimens. b: 1) initial condition, 2) after 0.6% deformation, 3) after 1.1% deformation, 4) after 1.5% deformation, 5) after anneling at 800°C/1 h.

3.1.1. The substructural and structural state of material The intergral value of Q –1 is influenced significantly by the mecha0 nisms determined by the presence of dislocations [42,43]. For example, examination of whisker crystals showed that there is a relationship between the value of Q –1 and dislocation density, Fig. 0 3.1. Internal friction reacts sensitively to the increase of dislocation density in whisker crystals [44], i.e. Q –1 increases with in0 creasing temperature and the second and third stage of the Q –1 0 dependence is displaced to lower temperatures. Perfect filament crystals also show increase of Q –1 , in the vicinity of the melting 0 point [45], i.e. when the vacancy concentration rapidly increases. On the basis of the capacity to stabilise the internal friction background of iron, the alloying elements can be placed in the following sequence: Ni, Co, Cu, Cr, Mo. Increase of the content of a carbide-forming element in iron (for example, molybdenum) shifts the high-temperature section to the range of high temperatures. Polymorphous transformations are characterised by sudden changes of the internal friction background. For example, such a sudden change is well-known to take place in the transformation of alpha iron to gamma iron [46,47]. In pure metals in the form of single or polycrystals at temperatures close to the melting point there are anomalies of the changes of the internal friction background (Fig. 3.2a) [48], i.e. with increasing melting point, the internal friction background is saturated or decreases. Similar behaviour has 81

Internal Friction of Materials

Fig. 3.2. Temperature dependence of internal friction. a) aluminium, magnesium and copper; b) beta tin, where: 1) initial heating; 2) after heating at 310°C/1 h, cooling, 3) initial heating of stabilised beta tin, 4) after heating at 310°C/1 h, cooling, 5) heating after holding for 1 h at 410°C.

been observed for many metals (Al, Mg, Cu [48], Ta [49], Pt [49], etc). According to Ref. 49, these phenomena at temperatures close to the melting point can be interpreted by diffusion mechanisms. After melting of the metals which have closely packed lattices in the solid state, the liquid state is characterised by the occurrence of a structural relaxation mechanism of energy absorption [50] so that it is possible to obtain, for example, for tin, the relationship between the structure in the solid and liquid state. The hightemperature gamma phase is obtained by cooling the melt at 300°C, the stable beta phase is obtained from the melt at 410–450°C and the ß phase with the nuclei of the gamma phase can be obtained from the melt at 320°C (Fig. 3.2b). The coefficient of absorption of sound α(T), caused by structural relaxation, has a background nature in molten metals and changes with temperature in accordance with the change of the ratio T 3 /v 3 , where v is the velocity of sound. These considerations are justified when the short-range order is retained in melting. 3.1.2. Vacancy mechanism The models used to explain the value of the internal friction background differ. The vacancy mechanism is based on the consideration according to which the intensity of the background is controlled by the equilibrium concentration of the vacancies migrating in the 82

Factors Affecting Anelasticity of Materials

field of repeated loading [51]. The formation of the equilibrium vacancy concentration requires a certain period of time (relaxation time). The relaxation of vacancy concentration causes nonuniform deformation leading to the scattering of the energy of mechanical vibrations, i.e. the internal friction background. This can be expressed by the equation

Q0−1 =

U f (σ ) − RT e T

(3.2)

where f(σ) is the function of the magnitude of repeated loading σ, U is the energy required for the formation of 1 mole of vacancies. The most frequently used equation for calculating the activation energy of the internal friction background according to Schoeck [52] has the form

Q0−1

 U  = const ωe RT   

− n1

,

(3.3)

where ω is the circular frequency of oscillations, n 1 is the structural constant. To determine this constant, it is necessary to obtain the dependence of internal friction on frequency since

ln n1 =

Q0−1 ( ω1 )

Q0−1 ( ω2 ) . ω2 ln ω1

(3.4)

The experimentally determined values of n 1 for annealed, polycrystalline samples of metals are in the range from 0.17 to 0.38 [53], and for copper single crystal n 1 = 0.65 [49]. At the these values of n 1 we can obtain the activation energy of the internal friction background U, corresponding to the energy of activation of self-diffusion in pure metals. The phenomenological description of the Schoeck model was published in Ref. 54. Here the internal friction background is treated as the superimposition of relaxations with a wide time spectrum. 83

Internal Friction of Materials

3.1.3.Diffusion-viscous mechanism In real solids, processes determined by directional flows of point defects can also take place [55]. The sources and sinks of these flows can be inclusions and various interfaces. If we use the mechanisms of diffusion-conditioned deformation for interpreting the internal friction background at infra-sound frequencies (ω<<10 2 s –1 ), we obtain the asymptotic form of the dependence [56]

const  = 2  ω e RT d z T  U

Q0−1

−1

  , 

(3.5)

where d g is the mean grain size. At high frequencies (ω << 10 2 s – ) we can use the equation in the form

1

U ef const  RT −1  ωe Q0 = d zT  

   

1 − , 2

(3.6)

where U ef is the effective energy of bonding, numerically equal to the sublimation energy. The dependence of the internal friction background on the grain size (decrease of Q –1 with increasing grain 0 size) has been determined in many studies [57]. When characterising the internal friction background, it is also possible to use the mechanisms of diffusion flow with the moving dislocations. Anelastic strain, caused by the displacement of dislocations, is ε n (T) = –bρξ(T), where ρ = 1/dh is the dislocation density in the crystal, d is the distance between the dislocations in the dislocation wall. If we obtain the conventional dependence of the internal friction background on temperature and loading frequency, this dependence is similar to the Schoeck characteristic with the exponent n 1 = 1 at low and n 1 = 1/2 at high loading frequencies [48]. The activation energy of the internal friction background in models of unlimited dislocation walls and in the model of blocks for polycrystals is identical at the corresponding loading frequencies. Excellent agreement between the theoretical and experimental data is obtained for the model of displacement of the dislocation at the grain boundaries [58]. This procedure can be used to obtain the analytical dependence of frequency of temperature 84

Factors Affecting Anelasticity of Materials

Fig. 3.3. Displacement of a dislocation segment in the stress field (a–e), where the solid lines indicate the position of the dislocation, broken lines the initial position, 1 is the position of the point defect at the start of displacement, 2 is the position of the point defect in the vibration period T, and a) t = 0, b) t = T/4, c) t = T/2, d) t = 3T/4, e) t = T. The diagram of relaxation of the dislocation in the stress field (f), where the solid lines represent the positions with maximum energy, broken lines those with minimum energy, A and B are the areas of pinning of dislocations with length L, H is the height of the Peierls barrier.

[59]. For example, at a frequency of several hertz and at T > (0.3– 0.4)T m , for the fcc metals, the dependence has the following form U  −1 RT Q0 ~  ω e  

  

−

m 2

,

(3.7)

where U is the activation energy of bulk self-diffusion. The magnitude of m is determined by comparing the distribution function f(N) and the experimentally determined histograms of the distribution of the grain boundaries N(ϑ) on the basis of the angle of misorientation ϑ of the crystals [60]. For copper m ~ 0.43–0.78. The model of displacement of the dislocations at the grain boundaries for describing the internal friction background has restrictions at specific temperatures because of the recovery of the structure as a result of the formation of grain boundaries [48]. 3.1.4 Dislocation mechanisms In accordance with the calculations based on the model of the oscillating side, at low frequencies the internal friction background is negligible. However, the experimental results show that the value 85

Internal Friction of Materials

of Q –1 in the Hertz frequency range is relatively high [61]. This 0 phenomenon can be explained of the basis of the assumption that there is a possibility of small displacements of pinned impurities behind the moving dislocations (Fig.3.3a–e) [62]. Internal friction is determined by two components. The first component is the directly proportional and the other one indirectly proportional to frequency. With decreasing loading frequency the significance of the first component decreases and that of the second component increases. The diffusibility of the blocking areas determines the exponential dependence of the internal friction background on temperature in the form

Q0−1 ~

U

− D D = 0 e RT , ωRT ωRT

(3.8)

where D is the coefficient diffusion of point defects, U is the activation energy of diffusion. The contribution from overcoming Peierls and Nabarro barriers can be taken into account utilising the model of relaxation of dislocations [62]. In the Mason model, the dislocations are parallel with one of the directions occupied most densely by the atoms in the crystal (Fig. 3.3f) and the length of the initial double kink on the dislocation is equal to the entire length of the dislocations segment. The scatter of energy, associated with the high-temperature background of internal friction, forms during the thermally activated separation of adjacent double kinks from their pinning point. Since the number of separations is described by the Boltzmann distribution, the following equation is valid 1/ 2

 2τ  Q0−1 =   G

νt N 0 Lef 2π

e

−

U RT

,

(3.9)

where τ(T) is the critical shear stress required for overcoming a Peierls–Nabarro barrier, v t is the velocity of sound during transverse vibrations, N 0 is the number of atoms in 1 cm 3 of materials, L ef is the effective length of the dislocation segment, U is the energy of activation of separation from the solute atom. The thermally-activated release of the dislocations with the kinks from the pinning points increases their mean length which increases with 86

Factors Affecting Anelasticity of Materials

increasing temperature. The internal friction background increases with increase in temperature in proportion to T 2 [63]. 3.1.5. The relaxation mechanism The general result, according to which the magnitude of internal friction is determined by the structural defects in metals, has been utilised in the relaxation mechanism of energy scattering during vibrations of solids [64]. The collective excitation of the crystal lattice with increasing temperature, the so-called relaxons, are areas with the disrupted arrangement of the particles which form in the areas of occurrence of structural defects of the crystal lattice. Table 3.1 gives the values of the internal friction background and relaxons of some metals at m min = 7, where U is the actual and U ef = nU is the effective activation energy of the internal friction background according to Schoeck [52]. For all evaluated metals, the ratio U min /U is approximately the same (~0.2, with the exception of lead). The value of exponent n for cadmium, aluminium and copper is also close to 0.2. Comparison of these parameters confirms the possibility of operation of the relaxation mechanism. The formation of a general theory of the internal friction background is still an open area with many complications.

3.2. EFFECT OF TEMPERATURE ON INTERNAL FRICTION The measurements of internal friction at different temperatures provide a large amount of information because temperature, combined with other external factors of the threshold states [1], has varying effects in different mechanisms of the scattering of mechanical energy in metallic materials. Many examples have been presented and others will be given in the following chapters of the book. Table 3.1 Parameters of the relaxation structure and internal friction background in metals

M e ta l

q · 1 0 20 (J)

U 0 min· 1 0 20 (J)

U 0 min/U

U ef· 1 0 20 (J )

n = U ef /U

Cd Pb Al Ag Au Cu

1.04 0.80 1.79 1.87 2 . 11 2.17

2.49 2.24 4.96 5.28 5.93 6.08

0.2 0.13 0.22 0.18 0.21 0.18

2.08 6.72 4.80 – – 8.01

0.17 0.33 0.20 – – 0.25

87

Internal Friction of Materials

The total relaxation maxima of this type can be found by, for example, the superimposition of the Snoek maximum, determined by the presence of carbon and nitrogen in iron. In some cases, the relaxation maxima of different nature can overlap if they are distributed closely to each other on the temperature axis. In both cases, the evaluation of the individual relaxation mechanisms starts with the division of the overall curve Q 0–1 (T) into the individual internal friction maxima caused by the effect of various mechanisms of mechanical relaxation. This task is similar to the case of distribution of the profile of X-ray diffraction into individual components. Like in X-ray diffraction, satisfactory solutions are obtained only when certain conditions, relating to the distribution of maxima on the temperature axis and the height of the partial peaks of internal friction, are fulfilled. An advantageous feature is that when dividing the spectrum of the maxima of internal friction, the specific relaxation time τ = τ 0 e H/RT holds for the given maximum and the analytical description of the profile of the relaxation maximum of internal friction Q r–1 (T) = Q –10 (T) – Q –1 (T) is available in the form t H −1 Qr−1 (T ) = Qmax sech   R

1 1   −  ,  T Tmax  

(3.10)

where Q –1max is the maximum height of the internal friction maximum (half of the relaxation step of the process) obtained at the maximum temperature T max . In the occurrence of independently acting relaxation processes, the experimentally determined internal friction at any temperature is the sum of the partial contributions of the individual relaxation processes in accordance with the equation

Qr−1 (T ) =

∑ Q (T ). −1 ri

(3.11)

i

Since the partial relaxation process is characterised by only one relaxation time, equation (3.11) has the following form

Qr−1 (T ) =

∑Q

−1 max i

i

H  1 1 sech  i  −  R  T Tmax 88

  , 

(3.12)

Factors Affecting Anelasticity of Materials

where summation is carried out for all partial relaxation mechanisms involved. When dividing the overall relationship Q –1 (T) into partial processes, we can use graphical and analytical methods. The graphical method is used when combining several relaxation processes determined by the single relaxation mechanisms. In this case, it is necessary to process the Q –1r (T) curve in a hyperbolic coordinate system, i.e.

arsech

Qr−1 1 = f  . −1 Qmax T 

(3.13)

In this system of coordinates, the internal friction curve that describes the single relaxation maximum is a straight line. The point of intersection of the straight line with the temperature axis indicates maximum temperature T max . The slope of the straight line makes it possible to determine the activation enthalpy of the process. In the case of purely relaxation processes, the straight line is shifted when the loading frequency is changed. The graphical method can be used most efficiently in cases in which the maxima are divided far away from each other on the temperature axis. Consequently, on the overall curve Q –1 (T) it is possible to separate the branches of the maxima in which the course of the Q r–1 (T) curves is determined by the effect of only one relaxation process. The graphical procedure depends on the relationship between the heights of the maxima. If the height of one maximum is many times greater than that of another maximum, –1 then the parameter Q max in the equation is replaced by the height of the total maximum at the maximum point (Fig. 3.4a). Transformation to the hyperbolic coordinate system (Fig. 3.4b) makes it possible to determine the linear section. If this section is analysed, it is possible to determine the activation parameter of the relaxation process which forms the highest internal friction maximum. The application of the determined parameters makes it possible to express the analytical form of the higher maximum. The smaller maximum of internal friction is determined by subtraction from the evaluated maximum. In most cases, it is necessary to use the analytical methods based on the approximation of the Q r–1 (T) dependence by the equations of the type (3.12), using the method of least squares. The calculation programmes of the break-up of the internal friction 89

Internal Friction of Materials

Fig. 3.4. Temperature dependence of internal friction of Fe–N–2 at% Mn alloy in conventional (a) and hyperbolic coordinates (b), broken lines show the main peaks of the dependence.

spectrum into individual components are based on the assumption on the ideal Debye shape of the individual partial relaxation maxima [65]. The method of dividing the Q –1r (T) can be greatly simplified if we use the apriori data on the values of the activation enthalpy of the temperature position of the maxima, even though only for a part of the individual relaxation processes. The width of the examined partial maxima of internal friction, especially the width of these maxima, often exceeds the width of the ideal Debye maximum, as indicated by the narrow spectra of relaxation times or, in most cases, the specific distribution of the activation enthalpies of the partial relaxation process. The expansion of the observed relaxation maxima β can be evaluated in accordance with Ref. 66. If there is no expansion (β = 0), the Debye maximum has the ideal shape, and after transforming the data to the reciprocal coordinate values of temperatures (1/T), the maximum is slightly asymmetric. Analysis of the Snoek maximum 90

Factors Affecting Anelasticity of Materials

Fig. 3.5. Temperature dependence of internal friction on Ti–33 at.% Mo alloy after annealing for 1 h at 1200°C. The dotted lines represent the main peaks, the solid curve is the theoretical curve.

by the calculation method, for titanium alloys with 20–40 mol% of niobium or molybdenum, taking into account the expansion of the partial maxima, was carried out in Ref. 67. The Snoek maximum was divided into five partial maxima (Fig. 3.5), associated with the occurrence of oxygen atoms in the lattice distributed in five different energy positions. Using chromium steels as an example, it is possible to evaluate the thermodynamic activity of carbon in alloyed ferrite where microregions with a different chromium concentration can form at the nodal points of the crystal lattice [68]. The graphical and analytical division of the partial relaxation maxima makes it also possible to determine the activation parameters of the relaxation parameters, especially activation enthalpy H. In some cases, the parameters H, τ 0 and Q –1max are already obtained when dividing the dependence into partial maxima. In other cases, the activation parameters of the processes are determined independently. Usually, it is necessary to use two groups of methods, based on analysis of the shape or temperature at which the relaxation maxima occur. The temperature at which a relaxation maximum occurs and the activation parameters of the relaxation process are linked by the equation:

91

Internal Friction of Materials

ωτ0 e

H = 1, RT

(3.14)

Q r–1 which makes it possible to determine experimentally H and τ 0 . If we change the loading frequency of the specimens from ω 1 to ω 2 , the relaxation maximum Q –1r (T) is displaced from the temperature T max1 to T max2 . Consequently, H and τ 0 can be determined from the equations

H =R

ω2 , ω1

(3.15)

H Tmax1 + Tmax 2 1 − ln ( ω1ω2 ). 2 R Tmax1Tmax 2

(3.16)

Tmax1Tmax 2 Tmax 2 − Tmax1

ln τ0 = −

ln

If it is required to use equation (3.15) with sufficiently high accuracy, frequency ω must be reduced by several orders of magnitude. When it is difficult to fulfil the condition of the change of frequency and the shift of the temperature maximum is 20–40°C, the relative error of determination of the enthalpy of activation in the sense of equation (3.15) is up to 20%. The accuracy of measuring the temperature shift T max2 – T max1 can be increased by evaluating the position of the maximum on the basis of transformation of the curves to hyperbolic co-ordinates and by applying linear regression to the resultant dependence. Two obstacles may occur in this procedure: final expansion of the maximum of the dependences Q r–1 (T) or the occurrence of the temperature dependence Q –1 .In max this case, the method gives average values of activation enthalpy. The activation enthalpy of the relaxation processes can be determined efficiently using the equation proposed by Wert and Marx in the form

H = RTmax ln

kTmax + Tmax ∆S , hω

(3.17)

where h is Planck’s constant, ∆S is activation entropy, ω is loading frequency. 92

Factors Affecting Anelasticity of Materials

Activation entropy is 40–20 J⋅mol –1 K –1 . Usually, the values ∆S = 10–12 J mol –1 K –1 are used. The equation proposed by Wert and Marx is used mainly owing to the fact that it is very simple. When using this equation, it must be taken into account that the boundary conditions must be fulfilled. When deriving equation (3.17), it was assumed that the value τ 0 in equation (3.14) is the same for all relaxation processes and equals 10 –13 s, which corresponds to the frequency of Debye oscillations of a single atom υ D= 1/τ 0 . There are a large number of processes in which τ 0 greatly differs from this value. For example, in the mechanism of dislocation relaxation τ 0 ≈ 10 – 9 –10 –10 s. Equation (3.17) can be used in cases in which the relaxation process is associated with the thermal activation of displacement of the individual atoms to interatomic distances. The shape of the relaxation maximum can be used in evaluating the activation parameters of relaxation. The activation enthalpy can be determined utilising the low- and high-temperature sections of the maximum of internal friction Q –1r (T). From equation (3.14) and (3.23) we obtain the following equation for the low-temperature section of the maximum (T < T max )

Qr−1 (T ) ≈ e

−

H RT

(3.18)

and for the high-temperature section of the maximum (T > T max ) the equation H

Qr−1 ≈ e RT .

(3.19)

Equation (3.18) shows that H can be determined from the angle of inclination of the low-temperature and high-temperature sections of the maximum of the dependence Q –1r (T) in the co-ordinates ln Q r–1 – 1/T. This procedure is suitable only for distinctive maxima whose broadening is close to zero. Activation enthalpy H can also be determined from the width of the maximum of the dependence Q r–1 (T). Usually, the width of the maximum is determined at the position Q –1max /2. The temperatures corresponding to this level of Q r–1 (T) in accordance with equation (3.23) fulfil the condition

93

Internal Friction of Materials

H

ωτ 0 e RT 1+

2H ω 2 τ02 e RT

=

1 . 4

(3.20)

The roots of equation (3.20) fulfil the equation, ωτ 0e H/RT= 2 ± √3 which shows that

H = 2.63R

T1T2 , T2 − T1

(3.21)

where T 1 and T 2 are temperatures at which Q –1r = Q –1max /2. The value H determined using equation (3.21) is accurate when the process is characterised by a single relaxation time. If we have the spectrum of relaxation times, the effective value of H, obtained using equation (3.21), will be lower. The analysed methods of calculations of the activation enthalpy of relaxation processes show that the most suitable method is the one based on analysis of the temperature shift of the maximum of the dependence Q –1r (T). Other methods can be used under certain conditions when the process is characterised by a single relaxation time and there is no expansion of the maximum (β = 0), but when the mechanism of the relaxation process is clear. The individual mechanisms of the relaxation processes are described in subsequent parts of this chapter. 3.2.1 Mechanisms associated with point defects In substitutional solid solutions, there is relaxation associated with the change of the relative position of the atoms of the alloying element and the main metal under the effect of external load. The maximum of Q –1 was reported for the first time by Zeener [69] in the relaxation spectrum of α-brass and later by Nowick [70] in the spectrum of the Ag–Zn alloys (see Fig.2.28). In this case, the maximum is more distinctive because there is a larger difference between the radii of the atoms of silver and zinc than between the atoms of copper and zinc. At low concentrations of the alloying elements the height of the maximum is directly proportional to the square of the concentration of the dissolved element. For solid solutions with a high content of the added metal, where the atoms of the alloying elements form complicated complexes rather than iso94

Factors Affecting Anelasticity of Materials

lated pairs, the proposed relaxation mechanism takes into account the changes of the short-range order after loading the body. The kinetics of Zeener’s relaxation is determined by the spatial migration of the atoms of the solid solution. This results in a change of the number of pairs of atoms with different orientation, or the parameters of short-range order. The same mechanism is the basis of the diffusion phenomena and, consequently, the kinetic characteristics of relaxation (relaxation time) are similar to the diffusion parameters in appropriate alloys. Experimental examination of Q –1 in ternary and multicomponent alloys showed that Zeener ’s relaxation processes also take place here [71]. The magnitude of relaxation is several times higher and the relaxation maximum is 1.5–2 times higher than in the binary alloys of the components forming the ternary alloy. The presence of alloying elements in interstitial solid solutions based on the cubic space-centred lattice results in a more complicated relaxation spectrum of Snoek’s mechanism and in the development of other relaxation processes [72], as shown in Table 3.2. Fe–C–metal alloys show high-temperature and also additional low-temperature maxima. The increase of the difference of the size of the atoms of iron and the alloying elements at comparable concentrations results in an increase of the temperature difference of the additional maxima and Snoek’s maximum. The magnitude of relaxation of the additional maxima of both types for interstitial solid solutions with carbon or oxygen does not exceed 1×10 –3 – 5×10 –3 . The maximum proposed by Finkel’shtein and Rozin is relatively wide and in certain cases is divided into two closely spaced peaks [73]. The height of the maximum increases with increasing concentration of the interstitial atoms. At concentrations lower than critical concentrations c cr the dependence of the height of the maximum on carbon content is linear. The shape of the concentration dependence of the height of the maximum is the same for all alloys in which this phenomenon has been observed. The value differs from 0.1 to 0.25 wt.% of carbon or nitrogen. In solid solutions with a face-centred cubic lattice or a hexagonal close-packed lattice the octahedral and tetrahedral planes, in which the interstitial atoms can be distributed, have the same symmetry as the lattice itself. In these materials, Snoek’s relaxation process does not occur. Despite this, the point defects (interstitial atoms) are the reason for a special relaxation phenomenon referred to as the Finkel’shtein and Rozin phenomenon. This maxi95

Internal Friction of Materials Table 3.2 Characteristics of the additional maximum in alloyed ferrite

Allo ying e le me nt ( wt . % )

Lo a d ing fre q ue nc y (HZ)

Te mp e ra ture o f S no e k ma ximum (K )

Te mp e ra ture o f fo rma tio n o f a d d itio na l ma ximum (K )

Ac tiva tio n e ne rgy o f a d d itio na l ma ximum (k J. mo l–1)

1

297

1 1 1 0.8

296 296 296 283

280 308 320 348 360 323

69* 81.6* 84.0** 92.0** 95.0** 81.0

450 0.2 950 950 1 960 0.65 950 950 960 945 970 1 1 1 1 1

383 299 388 390 312 393 – 389 389 394 386 396 – – – – –

348 291 473 427 334 431 353 453 362 360 363 473 460 510 535 550 560

80.5

F e - N S yste m 0 . 5 – 2 Mn 0.5 C r 0 . 5 Mo 0.5 V 1 . 0 Al F e – C S yste m 0.49 – 5.15 V 1.02 – 3.43 Mo 0 . 8 6 Mo 1.65 Si 3 Si 3.46 Si 0 . 1 – 0 . 2 2 Ti 0.9 C o 4.53 C o 4.4 W 3.9 N i 3.9 C r 8 Cr 15 C r 20 C r 25 C r 30 C r

105.5 92.2 87.0 93.1 133.0 101.3 78.6 77.2 78.9 102.5 – – – 134 136

Comment: * - calculated from the shift of the frequency of temperature maximum; ** - calculated from maximum temperature at τ 0 = 10 –11.5 s

mum was detected for the first time in an austenitic chromium– nickel alloy in the vicinity of 300°C at a loading frequency of 1 Hz and later in different austenitic steels and alloys based on the face– centred cubic lattice which also contained nitrogen or carbon. The principle of Finkel’shtein and Rozin phenomenon is explained by Werner’s model, based on the diffusion rotation of a pair of interstitial atoms of nitrogen or carbon under the effect of external stress. This pair if interstitial atoms can be stable when it is oriented along 〈110〉, 〈112〉, 〈130〉 and distributed in the third to fifth 96

Factors Affecting Anelasticity of Materials

co-ordination sphere. The pairs result in tetragonal distortion of the lattice. During rotation of the pair in relation to the axis of tensile loading of the specimen this causes mechanical relaxation to take place. The increase of the level of internal stresses by quenching, cold deformation, ageing, etc., causes significant broadening and increases the height of the maximum [74]. For the Finkel’shtein– Rozin phenomenon to take place, the material must contain lattice defects. For example, after cold deformation, the high-temperature part of the Finkel’shtein–Rozin maximum shows an additional maximum whose height depends on the level of internal stresses and their heterogeneity. For a steel with 24 wt.% Ni, 0.5 wt.% Mn and 0.5 wt.% C, the maximum height of the additional maximum is obtained when plastic deformation results in the formation of a homogeneous dislocation structure. When deformation leads to the formation of a cellular structure, the height of the maximum rapidly decreases and the additional maximum is not detected in the fine-grained recrystallised material. 3.2.2 Dislocation relaxation mechanisms The effect of low strain amplitudes characterised by internal friction irrespective of the strain amplitude results in the operation of different mechanisms of relaxation of dislocation segments and entire dislocation substructures. The activation energy of the processes, controlling the movement of dislocations in the crystal lattice, is not higher than 2.4×10 –19 J. All processes of dislocation relaxation under repeated loading from Hertz to kilohertz frequencies occur at temperatures lower than 500–600 K, i.e., below the condensation temperature of Cottrell atmospheres T C . At temperature lower than T C, every dislocation in annealed material is surrounded by the atmospheres of solute and alloying elements. The dislocations blocked in this manner are stationary and do not contribute to the process of dislocation relaxation. For mobile dislocations to appear in this material, the amplitude of repeated loading must be higher than the microscopic limit of elasticity. In the region in which internal friction is independent of the strain amplitude, the dislocation relaxation processes can develop as a result of the movement of fresh dislocations formed during previous plastic deformation of the material, or during phase transformations accompanied by strengthening. Since strengthening is a condition for dislocation relaxation, the appropriate maximum on the 97

Internal Friction of Materials

Q r–1 (T) curve is regarded as the deformation maximum or the maximum of cold deformation. Bordoni’s relaxation is the main process of dislocation relaxation in pure metals. At kilohertz frequencies, the Bordoni maxima on the Q r–1 (T) dependence are found at low temperatures (~100 K). The phenomenon is found in single crystals and polycrystalline materials, but always after prior plastic deformation. Efficiently annealed metals do not show Bordoni’s relaxation. The height of the maximum depends on the magnitude of prior plastic deformation ε p . The increase of ε p to approximately 3% results in a non-monotonic increase of the maximum. A further increase of ε p has no longer any effect on the height of the maximum (Fig.3.6). Bordoni’s maximum is resistant to annealing and disappears only after complete recrystallisation (for copper at 500°C). In heat treatment (Fig.3.7) and also when the content of the alloying elements is increased, the height of the maximum decreases, despite the fact that the temperature position of the maximum does not depend on the concentration or type of alloying atoms. With the change of the frequency of oscillations the temperature position of the maximum changes, but the change of strain amplitude has the form of a maximum. The activation energy of the relaxation proc-

Fig. 3.6. Bordoni’s peak of polycrystalline copper after plastic deformation of: a) 0.1%, b) 0.5%, c) 2.2%, d) 8.4%. Loading frequency 1.1 kHz. 98

Factors Affecting Anelasticity of Materials

Fig. 3.7. Effect of annealing temperature of polycrystalline copper on the height of Bordoni’s peak: 1) after 8.4% cold deformation, 2) after annealing for 1 h at 180°C, 3) after annealing for 1 h at 350°C. Loading frequency 1.1 kHz.

ess for all evaluated metals varies from 8 to 20 kJ⋅mol –1 and τ 0 varies from 10 –10 to 10 –12 s. A special feature of Bordoni’s relaxation is that the height of the maximum is an order of magnitude higher that in the case of the relaxation maximum with a single relaxation time. Bordoni’s relaxation is explained by the mechanism of dislocation relaxation based on examination of the thermally-activated formation of pairs of kinks on screw dislocations situated in the directions parallel to the direction of the closest packing, in positions with the minimum Peierls potential energy. Analysis shows that the enthalpy of activation of Bordoni’s relaxation is H = 2W k , where 2W k is the energy of formation of a double kink on the dislocation. Although the model is in good agreement with the observed phenomena, the width of the maximum is several times smaller than that indicated by the model. However, if we consider the displacement of the dislocation in the field of external repeated loading or thermal or geometrical kinks that are on dislocations and are not situated in potential wells, the theory is in good agreement with the observations. The width of the maximum is explained by the distribution of the length l of dislocation segments N (l). Consequently,

99

Internal Friction of Materials

Q −1 =

An0 3 ωτ l N (l ) dl , kT l 1 + ω2 τ2

∫

(3.22)

where n 0 is the density of geometrical kinks without the presence of external stress, τ = 1/πD, where D is the coefficient of diffusion of kinks of a longitudinal dislocation, A is the proportionality coefficient. At temperatures lower than the room temperature, fcc metals show other relaxation phenomena after plastic deformation. These phenomena are explained by the movement of kinks on the dislocations [2]. To explain low-temperature maxima that are close to Bordoni’s maximum, Hasighuti proposed a concept according to which in certain cases relaxation can be caused by braking of diffusing dislocation kinks by point defects, especially vacancies. The recorded maxima are characterised by high activation energy in comparison with that for Bordoni’s maximum. Dislocation relaxation in deformed alloys has many other manifestations. The internal friction maximum of deformed iron with nitrogen atoms was reported for the first time by Snoek in 1941 at a temperature of 200°C and a loading frequency of 0.2 Hz. The studies by Snoek, Kêo, Köster and others show that the occurrence of the maximum at 200°C requires the presence of a small number of atoms of C and N and also prior plastic deformation. In Fe–C and Fe–N systems at loading frequencies of 0.2–0.1 Hz is recorded at 200–250°C and are referred to as the Snoek–Köster maximum (S–K relaxation). Later, relaxations identical with S–K relaxation were observed in different metals with the fcc lattice (Ta, V, Mo), when the solid solution contained the atoms of C, N, O, H [75], and also in substitutional solutions and interstitial solutions based on the fcc lattice [76] and the hcp lattice [77]. This shows that the Snoek– Koster relaxation is a general phenomenon in the case of interaction of dislocation with atoms forming atmospheres around the dislocations that formed during plastic deformation. The activation energy of the deformation maxima is always higher than the activation energy of the volume diffusion of the appropriate alloying element. For example, the activation energy of S–K relaxation in alloys of iron with atoms of C and N is 127–168 kJ⋅mol –1 and the frequency factor τ −10 = 10 14 s –1 . When the level of prior strain is increased, the height of the S–K maximum increases by 50–70% 100

Factors Affecting Anelasticity of Materials

to saturation which depends on ε p . In unidirectional deformation, the height of the S–K maximum is Q –1 ≈ 10 –2 ε p 1/2 , and in thermoS–K –1 –2 . mechanical treatment it is Q S–K ≈ 10 ε p The increase of the prior strain changes the temperature position of the S–K maximum. For example, for iron with 10 –3 mol.% C after ε p = 2% at room temperature T S–K = 557 K, but after ε p = 10% it decreases so that T S–K = 545 K. The height of the S–K maximum depends on the type of deformation and is determined not only by the density of fresh dislocations but also by their type and mutual distribution. The decrease of temperature during deformation from 200 to –70°C reduces the height of the S–K maximum. When ε p = const, the highest height of the maximum after deformation is recorded at temperatures around 200°C. The effect of the grain size on the height of the S–K maximum and its temperature position is only very slight. However, the height of the S–K maximum depends strongly on the content of interstitial atoms in the solid solution. When ε p = const, the height of the S–K maximum increases with increasing content of the interstitial atoms, initially linearly and later rapidly increases to saturation whose level increases with increasing ε p . S–K relaxation in iron with nitrogen is characterised by a certain anomaly because the deformation maximum is recorded at a low nitrogen content of the solid solution insufficient for the formation of a measurable Snoek’s

Fig. 3.8. Dependence of the height of the Snoeck–Köster maximum of Armco iron on its heating rate. 101

wt.%

Internal Friction of Materials

Fig. 3.9. Effect of Cr content on the temperature at which Snoek’s maximum (1) and Snoek–Köster maximum (upper curve) of Fe–C alloy is observed. The temperatures at which the maxima are detected are indicated (2 Hz).

maximum. The temperature of S–K relaxation initially increases with increasing content of the interstitial atoms and, subsequently, after stabilisation of the height of the maximum, it no longer changes. The concentration, corresponding to the maximum height of the S–K maximum, for nitrogen atoms is on the whole 9–10 times higher than in the case of carbon, and the maximum corresponding to the nitrogen is 4–5 times higher than the maximum corresponding to the Fe–C system. The height of the S–K maximum is strongly influenced by the heating rate when measuring the changes of internal friction in relation to temperature (Fig.3.8). Sarrak and Golovin generalised the results of evaluation of the effect of the chromium content of iron on the temperature position of the S–K maximum and of the mixed maximum formed after quenching a steel from 1250°C followed by 12% cold deformation (Fig.3.9). The S–K phenomenon forms as a result of phase hardening during martensitic transformation. The presence of retained austenite has only an indirect effect on the parameters of the maximum because it is manifested in quenching in the form of Finkel’shtein–Rozin relaxation. Certain special features are observed in interpretation of this phenomenon. It is assumed that the control102

Factors Affecting Anelasticity of Materials

Fig. 3.10. Snoek maximum (lower curve) and deformation maximum (upper curve) of Fe–C alloy with temperatures at which maxima are recorded (2 Hz).

ling role in the formation of the S–K maximum in phase hardening is played by the dislocation density and the local distribution of the clusters of carbon atoms in the vicinity of the dislocation kernel where the arrangement process may take place. The interaction of dislocations with the clusters of carbon atoms is used to explain the S–K relaxation in quenched or deformed steel [78]. The distribution of carbon in the vicinity of structural defects causes heterogeneity of internal stresses in martensite. In addition to the maxima, recorded at 200°C in the iron–carbon and iron–nitrogen systems, there are also other maxima resulting from the deformation of the material. For example, in the Fe–N system, cold deformation results in the formation of a small maximum at 29°C, i.e. several degrees higher than the Snoek maximum (T S = 22°C, f = 1 Hz). The maximum is found after quenching from the temperatures higher than the temperature of the ferrite to austenite transformation, and also after cooling from temperatures corresponding to ferrite and subsequent plastic deformation ε p = 10 –4 –10 –3 . When the strain amplitude increases from 2.8 × 10 –4 to 5.7 × 10 –4 , this 29°C maximum increases, whereas the Snoek maximum decreases. The phenomenon is explained by the interaction of dislocations with the nitrogen atoms. A similar maximum was also found by Breshers in the Fe–C system. When the carbon content was ~5 × 10 –6 mol%, a small Snoek maximum is overlapped in the deformed material by a high deformation maximum which is displaced by 7°C to higher temperatures, Fig. 3.10. The deformation maxima of internal friction recorded for alloys 103

Internal Friction of Materials

of iron with carbon or nitrogen can also be detected at temperatures below 75 and 135°C at a loading frequency of 1 Hz [79]. The maximum was recorded immediately after deformation during the first measurement, but after cooling or during the second measurement it was no longer recorded. At temperatures higher than 200– 250°C, the mobility of the atoms of carbon and nitrogen is already quite high, the atmospheres around the dislocations are dispersed and the intensity of interaction of the interstitial atoms with the dislocations is insufficient for the formation of deformation maxima. In fcc metals, such as Ta and Nb with an oxygen impurity, deformation maxima are recorded in the temperature range 287–470°C or 272–452°C at a loading frequency of 1 Hz [75]. The interpretation of the Snoek and Köster relaxation, proposed by Shoeck and improved by Seeger and also Seeger and Hirth, is described in greater detail in Ref. 14. Further examples of the effect on internal friction are in fact presented in every other chapter of this book.

3.3 EFFECT OF STRAIN AMPLITUDE The dependence of internal friction on strain amplitude or alternating stress in external loading is interesting. This dependence is the result of the effect of various mechanisms, such as the relaxation mechanisms associated with point structural defects of the crystal lattice, relaxation and resonance dislocation mechanisms, the relaxation mechanisms associated with the boundaries of the blocks and grains, inelastic phenomena during phase transformations, the magnetomechanical relaxation mechanisms, the mechanical–thermal relaxation mechanisms, and others. The overall value of Q –1 is the result of the joint effect of a set of inelasticity mechanisms operating in the given ranges of temperature, frequency and amplitude of acting stresses. In many cases, Q –1 = Q 0– 1 + Q r–1, where Q 0– 1 is the internal friction background, Q r–1 is the internal friction associated with the effect of the relaxation mechanisms. A special feature of relaxation in materials is that the relaxation maximum of internal friction Q r–1 can be described in most cases by relaxation time τ t , using the equation

Qr−1 =

∆E ωτ , E 1 + ω2 τr2

(3.23) 104

Factors Affecting Anelasticity of Materials

where ω is the frequency of external loading, and ∆E/E is 10 –3 – 10 –1 . An exception is represented by inelastic phenomena associated with phase transformations and magnetomechanical phenomena. 3.3.1 The Granato–Lücke spring model The studies and investigations of the effect of strain amplitude on the extent of internal friction are based on the spring model, proposed by Granato and Lücke [28]. The initial experiments already showed good agreement between the theoretical calculations and experimental data, and further work has confirmed assumptions, although the experiments have been carried out under different conditions. The original theory is applicable to different metals with a specific dislocation structure at 0 K. The criticism of the model is associated with the development of detailed physical considerations regarding the effect of bows on the displacement of dislocations, with the new results obtained regarding the movement of point defects anchoring the dislocations, and with the data on thermal activation and microplasticity. However, the theory can also be applied to materials with a high content of solute elements [80], at temperatures higher than the temperature of Bordoni relaxation [81], etc. The model of the vibrating spring (Fig. 2.30), which already takes into account the tensile force in the dislocation, and viscous and inertia forces, was proposed by Köhler [82]. The model also links the scatter of mechanical energy with unpinning of the dislocations from the pinning points. Granato and Lücke [28] proposed a quantitative theory of the movement of dislocations based on the dependence of internal friction on strain amplitude. The calculations are based on the following assumptions: 1. All dislocation segments of the dislocation network have the same length L n , 2. L n > > L p 3. When stress is zero, the distribution of the dislocation segments with length l is disordered, and is expressed by the Köhler distribution in the form

N (l ) dl =

 l  ρ exp  −  dl , L2p  Lp 

(3.24)

105

Internal Friction of Materials

where N(l) dl is the number of segments with the length from l to l + dl, L p is the mean length of the segment, ρ is the density of the dislocations released from the pinning points. Consequently C2

C1 −ε0 Q = 1/2 πε0 −1

n

∑X

(3.25)

i

i=1

where

C1 =

Ω∆0ρL3n C2 Lp

(3.26)

C2 =

k ηb , Lp

(3.27)

here Ω is the orientation factor, ε 0 is the strain amplitude, ∆ 0 is:

∆0 =

8G b 2 , π2 S

G is the shear modulus of elasticity, S is the tensile force in the dislocation, k is the Boltzmann constant, η is the Cottrell parameter, b is the Burgers vector. Calculations can be carried out under the condition that σ 0 /r << 1, where σ 0 is the amplitude of the stress in external loading, r is the stress required for separating the dislocation with the length L p . Shortcomings, such as, for example, the large calculated distance of movement of the dislocations after unpinning, are explained by the inaccuracy of the boundary conditions. At low stress amplitudes, equation (3.25) is in good agreement with the experimental results. The condition L n >> L p shows that theory is not suitable for pure metals and this has been confirmed in a large number of investigations. Theory is in excellent agreement with the experiments on commercial purity metals at temperatures higher than the condensation temperature of the Cottrell atmospheres T on the dislocations. 106

Factors Affecting Anelasticity of Materials

Theory is supplemented by the dynamic losses in unpinning of the dislocations from the solute atoms [83] by the assumption according to which the movement of the dislocation, unpinned from the pinning points, is determined not only by linear stretching but also by the interaction of the dislocation with the solute atoms present in the solute solution, in the area in which the dislocation moves. Statistical analysis of the process of unpinning of the dislocation from the pinning points with different binding forces for medium stress amplitudes gives Q –1 ≈ σ 2 [84]. If equation (3.24) is replaced by the distribution in accordance with the equation N (l ) = c....li−1/ 2

we obtain the value Q –1 close to that determined using equation (3.24). 3.3.2 Thermal activation In the field of repeated stresses, structural defects, especially dislocations, change their position and configuration ‘in a jump’. Energy scatter in the formation of steps by thermal activation is determined from the following procedure [85]. If N(t) is the number of defects (dislocations) which have ‘jumped’ from position x to position y during time t, then

dN (t ) =  Nox − N (t )ν  x Px −  N0 y + N (t ) ν y Py , dt

(3.28)

where N ox (N oy ) is the number of defects at position x (or y) at time t, x (ν y ) is the frequency of vibrations of the defects at position x (or y), P x (P y ) is the probability of jump from position x to position y (or from position y to position x). The probability of a thermally activated jump

 U (σ )  P = exp  − .  kT    If we consider the deformation of the crystal ε d , we can determine 107

Internal Friction of Materials

Fig. 3.11. Model of separation (a), dependence of the energy of the double loop on the distance s between the dislocation and the pinning point (b) and stress (c), where 1 is linear energy, 2 Cottrell energy and 3 is the strain energy; m - low stress, v - high stress,

Q−1 =

Ñ∫ ϕ(σ, l ) N (t ) d σ ,

(3.29)

2πW

where ∆W =

Ñ∫ ε d σ; ε d

d

= ϕ (σ, 1) N (t ).

Thermomechanically activated separation of the dislocation (Teutonico model) describes the situation of two dislocation segments pinned in the centre between a point defect [86]. The total energy of the system consists of the energy of the bowed dislocation U 1 , the energy of elastic interaction U 2 and the work carried out by applied stress U 3. The diagram of the model is in Fig. 3.11a, b. The effect of the magnitude of acting stress is shown in Fig. 3.11c. Under the effect of very small (4) or very high amplitudes (4) there is only one energy minimum, which means that there is only one stable position of the segments. For the mean values of the stress amplitude (5) we record two stable positions, separated by an energy barrier. Under the effect of low stresses, there is no unpinning and the increase of stress results in the formation of a second energy minimum and unpinning may take place. This results in the disappearance of the first energy minimum and the dislocation is mechanically unpinned from the pinning points. 108

Factors Affecting Anelasticity of Materials

The stress of thermal–mechanical unpinning can be expressed as the stress of mechanical unpinning using the equation

  kT ν C ′ 1/ 2  σ = σkr 1 −  ln 1   . ω     U 0 

(3.30)

The model is constructed for a temperature of 0 K. If we consider the effect of temperature, it can be seen that the internal friction is proportional to

 σ  exp  − kr  ,  σ0  where σ cr is determined by equation (3.30) and σ 0 is the external stress amplitude. The development of the Teutonico model in Ref. 87 resulted in the determination of the conditions of catastrophic unpinning of the dislocation pinned by several point defects. The conclusions show that catastrophic unpinning takes place in a narrow stress range (~σ t1 , where the difference between σ t1 and σ kr increases with increasing temperature), and catastrophic repeated pinning also takes place in a narrow the stress range (~σ t2 , where σ t2 << σ t1 ). These results support the equations of the Granato–Lücke model, when σ kr is replaced by σ t1 . 3.3.3 Internal friction with slight dependence on strain amplitude Lücke et al. [88] analysed the conditions of thermomechanical separation for a dislocation segment pinned in the centre of the length by a point defect. It is assumed that the value of f changes exponentially from f 0 to the equilibrium value

−1

 ν U −Ub  . f ∞ = 1 + 2 exp a kT   ν1

(3.31)

109

Internal Friction of Materials

Fig. 3.12. Diagrams of the dependence of individual quantities in thermomechanical separation of the dislocation.

At low temperatures and high frequencies, f ∞ (σ t2 ) and f ∞ (σ t1 ) approach 0 and 1. The changes of f ∞ are described by a stepped function. Figure 3.12 presents the dependences σ – ε d and σ d , f ∞ , σ in relation to t for the given conditions. The investigations carried out in Ref. 88 shows that the internal friction, dependent on the stress amplitude, is

Qm−1

= ρL

3

σ3r1

 σt  exp  − 1  . σ0  σ0 

(3.32)

Friedel [89] derived equations forming a link between internal friction, frequency and temperature, assuming a certain number ρ of dislocations with length L n uniformly pinned by point defects, located at the distance L p , where L n >> L p >> b. The change of energy in relation to the magnitude of stress is expressed by the equation

110

Factors Affecting Anelasticity of Materials

U (σ ) = WM − σ b dL p ,

(3.33)

where W M is the total energy of the point defect–dislocation bond, d is the distance of the effective bond which is (1–3) b. Repeated pinning of the dislocation was ignored. For the given conditions

Qm−1 =

 WM − σ0b dL p  ρ L4n ν 0b exp  − . 2 kT 24πL p  

(3.34)

The equation is valid when

σ0 < σkr =

WM . b dLp

(3.35)

Unpinning takes place at σ 0 ≥ σ kr . Therefore 2

ρ L3  σ  = n  kr  . 12π  σ0 

Qm−1

(3.36)

at σ 0 < σ kr changes with the change of temThe magnitude of Q –1 m perature and frequency. Unpinning from the first pinning point takes place at a low frequency in comparison with unpinning from the last pinning points, so that the following conditions must be satisfied

exp

ε

σ b dLp

Lp b

2kT

> 1.

(3.37)

> 10 −2.

(3.38)

For a high-purity material, the conditions (3.37) and (3.38) are fulfilled when L p > 10 2 b, ε > 10 –4 , and the average concentration of the solutes is c 0 < 10 –5 wt.%. When calculating Q m–1, Koiva and Hasigutti [90] used the follow111

Internal Friction of Materials

Fig. 3.13. Positions of dislocation lines in the process of overcoming a point defect, where a is the equilibrium position when overcoming the obstacle, b shows approach to the concentrated force, c shows the binding energy for the case of repulsion (w > 0) and attraction (w < 0). 1) stable position in front of the obstacle, 2) unstable position behind the obstacle (saddle of the curve); 3) stable position after overcoming the obstacle.

ing equation instead of (3.28)

dN (t ) dt

 U − σν  =  N0 − N (t ) ν1 exp − 0 . kT  

(3.39)

Approximate calculations were carried out for the case in which σ 0 ν/kT < 1, and they obtained the equation

Qm−1 = N0

L3n 12π

σ0 ν    f (α ) + kT h (α ) .  

(3.40)

An interesting result of the solution is that it predicts a maximum on the dependence of internal friction on strain amplitude with 112

Factors Affecting Anelasticity of Materials

the change of temperature. The level of stress influences the functions f(α), h(α) and the shape of the maximum. The presented data show that the model proposed by Lücke et al. is suitable for low temperatures and high frequencies, whereas the model proposed by Friedel is suitable for pure materials, etc. The results are generalised in Ref. 91. The model proposed by Indenbon and Chernov [92] analyzes the thermal fluctuation overcoming of an elastic field of a point defect by a dislocation, by introducing transitional configuration of the dislocation, prior to the establishment of the equilibrium position. The dislocation segment is situated on the slip plane x, y (Fig. 3.13), and is pinned at the ends at points (±l, 0, 0). Under the effect of external stress σ 0 and thermal fluctuation, the segment is separated from the point defect C(0, 0, z 0). The equilibrium positions are calculated in a computer. With a probability close to 1, unpinning takes place at the energy

 ν  U(σ0 ) = kT ln  ξ 0  ,  ν

(3.41)

where ν 0 is the frequency of vibrations of the dislocation segment, ν is the frequency of external loading. The temperature dependence of the segment length L p fulfils the equation

 ∂L p   ∂T

k ln (ν 0 / ν )  . =  (∂U / ∂L p )(σ )

(3.42)

0

With increasing temperature, the shortest segments are the first to be unpinned and segments with the length being a multiple of L p form at relatively low stresses. The first stage is characterised by the increase of the number of segments contributing to the dislocation hysteresis, and the second stage is reflected in a decrease of the strength of the effect of each segment. Consequently, internal friction

Qm−1

∞

2

σ  (T , σ0 ) = Ln  n  N (l ) dl,  σ0  Lmin

∫

(3.43) 113

Internal Friction of Materials

where 2L n is the length of the dislocation segment, σ n is the stress at which the thermal fluctuation unpinning of the segment from the defect takes place, σ 0 is the external stress amplitude. The value L min = L M (σ 0 , T). The effect of thermal activation can also be evaluated by solving the equations of vibrations of a solid in analysis of functions for additional strain ε d [93]. Into the vibration equation we substitute the rate of change of the additional strain determined by the movement of the dislocation  U 0 − σν   U 0 + σν    ∆S   ε d = b ∆ F ρ 0 ν1 exp    exp  −  − exp  −  , kT kT  k     

(3.44)

where ∆F is the area through which the dislocation travels, ∆s is the change of entropy accompanying the displacement of point defects. From the resultant equations it is possible to determine the characteristics of internal friction and the defect of the Young modulus. At relatively high strain amplitudes, microplastic deformation can take place in certain areas of the material; this is a source of significant scattering of the energy in the material. 3.3.4 Plastic internal friction In accordance with the schematic representation in Fig. 2.30, the effect of a certain amplitude of external stress results in the generation of dislocations, Fig. 2.30, and cyclic microplastic deformation of the material takes place. In order to describe the given process uniformly and in considerable detail, and take into account the importance of this response of the material to repeated loading, this part of the dependence Q –1 (ε) is discussed in Chapter 6 of this book. 3.4 EFFECT OF LOADING FREQUENCY The Granato–Lücke theory [28] is suitable for examining the effect of frequency on internal friction and also for predicting many processes taking place in solids. However, there are two complicating features. Calculations of the dislocations strain at stresses (10 –7 –10 –8 ) G in the MHz frequency range with the value δ = 10 –3 and at a dislocation density of ρ = 10 10 m –2 indicate that the segments vibrate with an amplitude of 10 –7 –10 –8 mm, which is smaller than the 114

Factors Affecting Anelasticity of Materials

atomic spacing. This small displacement can no longer be described by the tensile force and the Burgers vector. It has been confirmed that the dislocations start to move under the effect of the Peierls stress which is lower than (10 –7 –10 –8 )G. The model of the vibrating spring does not take into account the existence of Peierls barriers and does not explain the possibility of overcoming these barriers in measurement of Q –1 in the range of low strain amplitudes. The theory of internal friction caused by dislocations does not consider the possibility of thermally activated release of the dislocation segments. It is well-known that already at room temperature, the frequency of unpinning of the dislocation segments as a result of temperature variations amounts to hundredths of a megahertz. In the MHz frequency range, a change of external stress is accompanied by a multiple, thermally-activated separation of the segments. Despite the fact that these complications are important, it is quite surprising that the theory of the vibrating spring describes sufficiently the resultant experimental dependences. The explanation is based on taking into account other phenomena, for example the concept and mechanism of movement of bows on the dislocations during overcoming barriers according to Shockley [94]. The statistical and dynamic theory of bows assumes that a dislocation in the given slip plane is oriented preferentially in the direction with the highest density of the atoms and, consequently, causes bows to form. At temperatures higher than 0 K, the dislocation lines are unstable, form pairs of bows of the reversed sign and are periodically displaced from one potential well to another. The energy barrier, overcome by movement of the bow along the dislocation line in the direction of dense distribution of the atoms (so-called Peierls stress of the second kind) at stresses of 10 –4 G and the width of the bow of 100b is 10 –7 G, which corresponds to the stress amplitudes used in internal friction measurements. Evaluation of the strain amplitude during movement of bows showed that its value is 1/n pb times higher than the amplitude used in the model of the vibrating spring (n p is the number of bows on a dislocation segment with length l). The theory of resonance dislocational internal friction solves the movement of bows under the effect of repeated external loading if the following conditions are fulfilled: –the pinning points of the dislocation line are not located in the same potential well and, consequently, the dislocations contain geometrical bows; 115

Internal Friction of Materials

–below the temperature corresponding to the Bordoni peaks, n p is constant and does not change during the loading cycle; –the activation energy of movement of the bows along the dislocation line is relatively small in comparison with the energy of formation of double bows; – with increasing excitation amplitude used in measurement of –1 Q the dislocations situated in the same potential well do not contribute to anelastic deformation. Consequently, the defect of the Young modulus is 2

 ω 1−   ∆G  ω0  , =Z 2 G   ω 2  2 1 −    + (ωτ )   ω0  

(3.45)

and internal friction

Q−1 = Z

ωτ 2

  ω 2  2 1 −    + (ωτ)   ω0  

, (3.46)

where

Z=

ω02

8Ωρl 2 Gb 2 aB , τ= , 4 m0 ω02 π Sl

(3.47)

π4 aSl sin ϕ Gb 2β γkT , Sl = . ln n p + = 2 4π q sin ϕ m0 l

(3.48)

In these equations, Ω is the orientation factor, γ is the size factor equal to 5 at a high angle ϕ and equal to 1 at small angle ϕ. Relaxation time

116

Factors Affecting Anelasticity of Materials

τ=

aBl 2 . π 2 γkT

Resonance frequency ω 0 decreases with decreasing angle ϕ and approaches a constant value

π2 γ kT . m0l 2 Stretching S 1 contains the quantity

γ kT , a sin ϕ where a is the lattice parameter. At low frequencies, ωτ < 1, the above equations have the following form

∆G 8ΩρG b 2l 2 , = G π4 Dl

Q −1 =

(3.49)

8Ωρ Gb 2 1 Bl 4 ω π6 Sl2 sin ϕ

(3.50)

and at high frequencies, ωτ > 1, the form of these equations is

∆G 8ωπ G b 2 Sl sin 2 ϕ = , G l2B2 ω2 Q −1 =

(3.51)

8ΩπGb 2 sin ϕ , ωB π2

(3.52)

where the meaning of the symbols is the same as in the case of equations (3.45)–(3.48). 117

Internal Friction of Materials

Fig.3.14. Dependence of the relative friction decrement on relative frequency at different values of the friction constant (B) for the delta distribution of the length of dislocation segments.

The internal friction associated with the movement of bows along the dislocation lines increases in direct proportion to loading frequency and the value l 4 when approaching the resonance maximum (ωτ = 1). After passing through resonance, the values start to decrease in inverse proportion to loading frequency. The movement of bows on the dislocations can be evaluated by internal friction measurements in the case of small angles ϕ that with the magnitude of acting stresses below 10 –6 G. Figure 3.14 shows the dependence of normalised friction on normalised frequency for different values of the friction characteristic B, plotted using equations (3.45) and (3.46). It can be seen that friction is of resonance nature, and the resonance frequency is controlled by the values of S 1 , B and l, and also by the distribution of the lengths of the dislocation segments in the material. For the case shown in Fig. 3.40 we use the delta distribution. The ‘quality’ of the specimen decreases with increasing B, the resonance peak becomes wider and is displaced to lower frequencies. For the case of damped resonance (ω/ω 0 ) 2 << 1 we have 118

o

Factors Affecting Anelasticity of Materials Table 3.3 Resonance frequencies ω 0 and position of internal friction maxima ω max M e ta l

ϖ 0 (MHz)

ϖ max (MHz)

Ag Al Au Cu Ni Pb Zn Ge

467 868 352 628 784 210 586 640

76.0 65.0 79.0 92.0 135.0 24.5 69.0 51.0

Z  ∆G  −1   = Qmax = . 2  G  max

(3.53)

The frequency position of the maximum

ω max

π 2 S l m ω02 = = , B Bl 2

(3.54)

makes it possible to determine, together with the value of Q –1 , the max parameters ρl 2 /S 1 , S/l 2 B and ρ/B and specify these values further when the values of S 1 and ρ are known. Lücke [95] determined the resonance frequency of the dislocations ω 0 and the frequency position of the ω max of internal friction max for several metals at l = 10 –3 mm and B = 5 × 10 –4 dyne s/ cm 2 (Table 3.3). At frequencies lower (ωτ < 1) and higher (ωτ > 1) than the peak, internal friction is proportional to l 4Bω or ω –1 .

Fig.3.15. Displacement of dislocations under the effect of stress: a) displacement at low (1) and high (2) friction; b) the case of high friction with oscillations not in phase (1) or in phase (2); c) change of situation after adding another pinning point (as b). 119

Internal Friction of Materials

In the range of low (sonic) frequencies the dislocation friction is low, as indicated by, for example, experiments with measurement of Q –1 at 1 Hz and 10 kHz. The value of Q –1 consists of two components, where the first component is directly proportional and the second one indirectly proportional to frequency. With decrease of frequency, the importance of the first and second component increases. The spring theory predicts the anomalous decrease of the velocity of sound after irradiation with neutrons or low plastic deformation [96]. The magnitude of internal friction and of the defect of the Young modulus are influenced by various parts of dislocation deformation that are in phase ε f or not in phase ε f0 with the induced stresses, i.e. Q –1 = ε f0 /ε e , ∆G/G = ε f /ε e , where ε 0 is the elastic strain. The increase of loading frequency increases the viscous friction force and also the value ε 0 (Fig. 3.15), and decreases ε f . This increase the velocity of sound and friction (normal phenomenon). In the limiting case ω→∞, ε f = 0, the wave propagates as in a medium without dislocations. In the presence of other pinning points, formed as a result of neutron irradiation or low deformation, the value of ε f increases (Fig. 3.15c) and ε f0 decreases. This results in a decrease of the velocity of sound and a decrease of friction (anomalous phenomenon). At a large number of pinning points, the dislocation component, which vibrates in the phase with the acting stress, is completely blocked thus increasing the velocity of sound. The main shortcoming of the spring theory is the small displacement, i.e. the amplitude of dislocation vibrations. For example, at a stress of 10 –7 G this amplitude is higher than the amplitude of thermal vibrations of the lattice at room temperature only when l ≥ 10 5 b, at a purity of the material better than 99.999 at.%. However, if it’s taken into account that the spring model is equivalent to the model of movement of bows with a high density, these shortcomings are eliminated. The equivalence of the models was confirmed by Suzuki and Elbaum [97] at temperatures corresponding to Bordoni’s peak and at a strain amplitude sufficient for the formation of a single double bow on a dislocation. These considerations also show that the mechanisms of internal scattering of energy, regarded as frequency-independent, can be influenced by frequency, although not directly but by means of a set of mechanisms that are sensitive to frequency and form, for example, a large part of the internal friction background. 120

Factors Affecting Anelasticity of Materials

3.5 EFFECT OF LOADING TIME The anelasticity characteristics of the material depend on loading time and, consequently, the number of load cycles, in the range of strain amplitudes that do not yet cause microplastic reactions. The change of internal friction and the defect of the Young modulus, caused by extending loading time, makes it possible to determine the parameters of diffusion of atoms along dislocations, the characteristics of stability and long-term strength, the processes of fatigue damage cumulation in materials, etc. The time dependence of internal friction is the result of diffusion of solute atoms in a stress field. The solute atoms form clusters and, consequently, dislocation segments are released from the pinning area and the extent of friction increases. In addition to diffusion displacement of the atoms along the dislocations, these atoms also diffuse from the dislocations into the volume of the lattice. The type of atoms placement mechanisms that operates in a specific case depends on the magnitude of acting stress, loading frequency, temperature, concentration of the solute atoms at dislocations, bonds between the solute atoms and the dislocation, and on the mutual effect of the solute atoms between each other. External loading results in bending of dislocation segments (Fig. 3.16). The tensile force in the dislocations results in the formation of forces acting on the solute atom. These forces can be divided into mutually normal components. Consequently

Fig.3.16. Diagram of the effect of forces on the pinned atom: a) length of dislocation segments A = B; b) A < B. 121

Internal Friction of Materials

F⊥ =

σb σ2b 2 2 Lp + l p +1 ) , F|| = Lp − L2p +1 , ( 2 8C

(

)

(3.55)

where σ is the acting stress, b is the Burgers vector, C is the stretching of the dislocation line, L p and L p+1 are the lengths of the dislocation segments on each side of the pinned atom. Force F || is independent of the orientation of external loading and is always directed from larger to shorter dislocation segment. Force F ⊥ changes its sense in accordance with the sense of applied stress. Both forces (F ⊥ and F || ) can cause movement of the blocking solute atoms. In the case of small bows of dislocation segments, the following equation can be used

F|| F⊥

=

Gb ( L p − L p +1 ) 4C

,

(3.56)

It can be seen that F || is many times smaller than F ⊥ . The movement of the pinning points in the direction normal to the dislocation line should be examined under static loading. With repeated loading, the redistribution of the pinning atoms depends on the relationship between the relaxation time and the frequency of changes of external loading. For the simplest model of the dislocation segment with a single pinning point, situated between two pinning points, the relaxation time of movement of a pinned atom together with the dislocation at the stress acting in the direction normal to and parallel to the initial position is

τ⊥ =

L1 L2 4C ; τ|| = , D⊥ D|| 2 2 C ( L1 + L2 ) ( L1 + L2 )b σ kT kT

(3.57)

where D ⊥ and D || are the diffusion coefficients of the atoms of the solute in the direction normal and parallel to the dislocation line, L 1, L 2 are the lengths of the adjacent dislocation segments. The displacement of the pinning atoms is interpreted by the diffusion under the effect of the external force and, consequently, it is possible to determine the diffusion coefficients along the dislo122

Factors Affecting Anelasticity of Materials

cations from the internal friction measurements at the start of recovery of the material. A more efficient procedure is the one using the curves of saturation of changes because the measurement of the course is more accurate, the distance between the solute atoms is sufficiently large and the interaction of the atoms can be ignored. The curve of saturation of the changes Q –1 can be de(τ) scribed by the equation −

2τ τ1

1− e Q(−τ1) = Q(−σ1) +  Q(−01) − Q(−σ1)  , −2 τ 0   τ1 1− e

(3.58)

, Q –1 is the internal friction under the effect of stress where Q –1 (τ) (0) σ and without it, τ is time, τ 0 is the duration of saturation of the changes, and τ 1 is the time related to the loading frequency. The unpinning of the dislocations from the solute atoms and their new distribution in the atmosphere is described in detail in the analysis of the Q –1 – ε dependences. In this section, it is possible to examine the case in which a solute atom moves together with a dislocation. Force F ⊥ (Fig. 3.16) has the meaning corresponding to the meaning of external loading. The solute atom does not travel far

Fig. 3.17. Region of separation (A), diffusion with activation energy of 0.32 × 10 –19 J (B), diffusion with the activation energy of 0.8 × 10 –19 J (C) and the position of pinning (D) of the dislocation in relating to pinning atoms in the stress–temperature diagram. 123

Internal Friction of Materials

away from the dislocation. When the applied stress carries out work higher than the binding energy of the solute atom and the dislocation, the dislocation separates from the pinning atoms. With increasing temperatures the jump time of the solute atom τ ⊥ decreases and the solute atom and the dislocation can move together. The σ – T dependence (Fig. 3.17) shows the region of unpinning of the dislocation (A) and the region of displacement of the solute atoms (B) together with the dislocation. The boundaries of the regions are a function of temperature, loading frequency, binding energy of the solute atoms and the dislocations, and the activation energy of diffusion. It should be noted that due to rigid pinning (C) the probability of joint movement of the solute atoms with the dislocations increases with increasing temperature. This model explains the temperature dependence of the critical strain amplitude ε cr where the value of ε cr decreases with increasing temperature. Examination of the Q –1 – γ dependence for copper with zinc and germanium shows that the change of the temperature dependence of γ cr occurs at a temperature of ~170°C. This temperature denotes the start of displacement of pinning atoms together with the dislocations, with the average displacement being 4b. Consequently, D ⊥ can be determined. The joint displacement of the atoms of the solute with the dislocation may not be reflected only in the time dependence of internal friction, because the direction of diffusion of the pinned atom during transverse movement changes during cycling loading. Displacement may also influence the function of the distribution of the solute atoms along the dislocations, with a decrease of the probability of formation of clusters of the atoms. In addition, the deformation, caused by the displacement of dislocations increases, and there are also losses associated with the frequency of jumps of the atoms of the solute with the frequency of external loading. Continuous recording has a large number of advantages in comparison with discrete recording where vibrations must be periodically decreased when the saturation branch forms, and when the unstressing branch appears it is necessary to excite the specimens for measurement of Q –1 . Disregarding the short measurement time, this results in errors which are especially evident at short relaxation times. Since the diffusion coefficient of the solute atoms along the dislocations causes a change of Q –1 , the time dependence of Q –1 is reflected in certain temperature and frequency ranges. At low temperature, the mobility of the solute atoms is low and the long-term 124

Factors Affecting Anelasticity of Materials

Fig. 3.18. Temperature dependence of internal friction (A) for aluminium A999 at different frequencies: 1) 1.38 kHz, 2) 2.41 kHz in instantaneous measurements; 3 and 4) the same frequency and after saturation lasting 3 min. b) temperature dependence of friction decrement. Table 3.4 Temperature ranges of the time dependence of internal friction

M e ta l

C o nte nt o f so lute s ( wt . % )

Te mp e ra ture ra nge (K )

Te mp e ra ture ra nge re la te d to me lting p o int

Lo a d ing fre q ue nc y

Cu Ni Al Fe

0.01 0.01 0.01 0.01

370–540 630–720 250–400 500–720

0.28–0.40 0.36–0.42 0.30–0.47 0.28–0.40

0.7 0.9 0.9–2.5 0.9

effect of loading does not lead to any significant changes of Q –1 . At high temperatures, relaxation takes place already during the period of increasing loading and it is difficult to determine the time dependence of the changes of Q –1 , because the atoms managed to return to the initial position at the end of the loading cycle. Figure 3.18a shows the temperature dependence of Q –1 of pure aluminium, recorded at the start of saturation and after loading for three minutes. Figure 3.18b shows the increase of the friction decrement ∆δ on temperature. The temperature ranges in which Q –1 depends on temperature and also the homologous temperatures of the boundaries of the ranges for different materials are presented in Table 3.4. It can be seen that for all evaluated materials, the temperature ranges with the time dependence of the changes of Q –1 are on average close to temperature (0.3–0.45)T t . In these ranges, the relaxation stability of the metals is reduced. Measurements of the time dependence of the changes of Q –1 can be utilised efficiently in the quantitative evaluation of the relaxa125

Internal Friction of Materials

r

→

Fig. 3.19. Dependence of the increase of the decrement of oscillations on the radius of the solute atom.

tion stability and stability of the dimensions of metallic solids operating under repeated loading. Measurements should be taken using specimens with the natural frequency, the same as the frequency of the source of the vibrations. Static and repeated loading results in the redistribution and clustering of the solute atoms as a result of their displacement along the dislocations which is the initial stage of low-temperature creep, and the dimensions of the solid change by ~0.1%. The temperature ranges of occurrence of this phenomenon should be avoided in service of components. Loading frequency also influences the occurrence of temperature ranges in which changes of Q –1 depend on time. The decrease of the loading frequency results in a decrease of temperature at which the time dependence of the changes of internal friction is evident. The time dependence of the changes of Q –1 is recorded in the region in which Q –1 is independent and also dependent on the strain amplitude. Long-term loading results in a decrease of the critical strain amplitude and increases internal friction. The largest increase of Q –1 is at stresses corresponding to the critical amplitude, but with increasing ε the increase of Q –1 becomes smaller. In the entire strain amplitude range up to the microplasticity range the increase of time increases internal friction (see Chapter 6). The radius of the solute atoms also influences the increase of the logarithmic decrement. The largest radius of the solute atom is related to the largest change of ∆δ (Fig. 3.19). Evidently, the spacing of the solute atoms on the dislocations is also important in this case. Prior plastic deformation, accompanied by ageing, influences the increase of the logarithmic decrement when examining the time de126

127

bn

bn

Q–1h

Q

–1

Q

Irreversib le mo vement o f d o main b o und aries

Reversib le p ro c e s s e s o f ro tatio n o f the vecto r

Reversib le mo vEment o f d o main b o und aries

Magnetic elastic static hysterisis

M a c ro e d d y currents

M a c ro e d d y currents

C hange o f general magnetisatio n

Q–1a

–1

Mechanical internal frictio n

Reaso n fo r internal frictio n

P arameter

f < > f0

f <>f 0

f <>f 1

F req uency range

~f ~f –1

~f f –1

~

~f ~f –1/2

Limiting c a se

f /f 0 1 + (f + f 0 )2

f/f 0 1 + (f + f 0 )2

f/f 1 1 + (f + f 1 )2

General d ep end ence

F req uency f

Ind ep end ent to f » 1 0 6 Hz

f0»ρ /2 5 πXRD2

f0»ρ /9 6 πXaD2

f 1»ρ /4 π X R2

Limiting freq uency

Q–1bn= 0 at I = Is

Q–1a= 0 at I = 0 and at I = Is

Magnetisatio n I

– F, at Q–1h= 0 at F < < F s is I= Is– 1 /F 2 at F=F2

Ind ep end ent

Ind ep end ent

Amp litud e F

Dep end ence o f Q–1 o n

–

Do es no t d ep end

Dep end ence

S hap e and d imensio n o f sp ecimens

Table 3.5 Friction mechanisms in ferromagnetic materials (σ – specific electrical resistance, R – the radius of the specimen, X a – the initial magnetisation determined by the movement of domain boundaries, X R – reversible magnetization, D – size of domains, σ s – stress of magnetic elastic saturation)

Factors Affecting Anelasticity of Materials

Internal Friction of Materials

pendence of the changes of Q –1. In this case, the dislocation density increases and the number of the solute atoms situated along the unit length of the dislocation in the solid solutions decreases. This is reflected in an increase of ∆δ. Of special importance is the examination of the changes of Q –1 and ∆E/E in relation to the time of repeated loading in the range of cyclic microplasticity and in the part of the Q –1 – ε dependence in which Q –1 is an evident function of the loading time (Chapter 6).

3.6 EFFECT OF MAGNETIC AND ELECTRIC FIELDS In ferromagnetic materials, a large part of external mechanical energy can be dispersed as a result of changes of general magnetisation, reversible movement of the domain boundaries, the reversible process of movement of the vector of magnetisation in domains and irreversible displacement of the domains boundaries; eddy microcurrents or magnetic–elastic static hysteresis are also important [98]. Table 3.5 summarises the most important effects on the extent of internal friction, with special attention to the effect of the magnetic field. The scatter of energy, associated with eddy micro- and macrocurrents, is independent of the strain amplitude and is directly proportional to loading frequency. However, magnetomechanical friction is more important; in this case, the domain structure of the ferromagnetic materials is characterised by different orientation of the vectors of magnetisation of the adjacent domains separated by Bloch walls. When the orien-

Fig. 3.20. Deformation curve for ferromagnetic materials. 128

Factors Affecting Anelasticity of Materials

tation of external loading is changed, the magnetisation vector also changes and the domain boundaries are displaced. The formation of elastic strain ε c is accompanied by magnetostriction deformation λ, caused by the change of the orientation of vectors of local magnetisation; the total deformation of the material is ε = ε c + λ. Unstressing is followed by the deformation of the stepped boundary of the domains. The effect of cyclic loading is accompanied by continuous displacement of the domain boundaries in two mutually normal directions with the frequency identical with the frequency of external loading. This phenomenon results in additional magnetomechanical scatter of energy in the ferromagnetic materials, the ∆E effect appears and appropriate changes take place in the hysteresis loops (Fig. 3.20). When the ferromagnetic material is placed in a saturated magnetic field (H = H s ), the entire material has the properties of a single magnetic domain and the deformation curve is linear. Consequently, the friction characteristics of the ferromagnetic material are low. The scattering of energy in the ferromagnetic materials depends on the initial condition of the material and test temperature, in relation to Curie temperature. The loss of energy in the material is independent of the frequency of vibrations up to 10 5 – 10 6 Hz, i.e. to the frequencies at which the duration of the Barkhausen jumps is considerably shorter than the period of elastic vibrations [99].

Fig. 3.21. Main shapes of the curves of the dependence of the amount of scattered energy on stress amplitude in ferromagnetic materials. 129

Internal Friction of Materials

The scatter of energy by magnetic–elastic hysteresis is associated with the amplitude of the acting cyclic stress through the equation ∆W = Dσ n , where n ≤ 3. Under the effect of a low stress amplitude n = 3, and with increasing amplitude the value of n decreases to 2 and at the maximum stress amplitude n = 0 at magnetic saturation. Similarly, at low stress amplitude, the magnetic component of internal scattering of the energy Ψ h ≈ 6, and the slope of the Ψ h (σ) curve decreases. The function Ψ h (σ) is characterised by a curve with a maximum (for example, for nickel and alloys of iron with nickel). The dependence of Ψ h on the stress amplitude has the form of an S-shaped curve; this has been confirmed by experiments for iron and carbon steels. With increasing amplitude of the vibrations the magnetic−elastic hysteresis loop of soft ferromagnetic materials gradually changes from oval to bent shape; this is associated with the formation of the ∆G effect [100]. Kekalo [98] classified the main types of the dependence of internal friction on the strain amplitude for ferromagnetic materials, Fig. 3.21. The curves with a high maximum, Fig. 3.21a, are recorded in cases when τ m = τ max << τ < τ d , where τ d is the minimum shear stress at which the energy losses, associated with dislocation processes, are recorded. This may be observed in iron, nickel, cobalt, low-carbon and low-alloy steels, and also many ferrous alloys. The dependence of internal friction on the stress or strain amplitude is often different (Fig. 3.21b–d) in comparison with that observed in nickel, iron, gadolinium, maraging steels, carbon alloys, and also other alloys. For these cases τ m = τ max ≈ τ d < τ. At τ d > τ max we obtain the case shown in Fig. 3.21d. The curves with the form shown in Fig. 3.21d are recorded for slightly different materials, i.e. the magnetomechanical losses are not completely suppressed. The form of the dependence shown in Fig. 3.21e is recorded in two cases, i.e. when τ << τ max , or in materials with increased magnetic hardness (τ max ≈ τ el ). The form of the curve in Fig. 3.21f is associated with pinning of domains by different obstacles, for example, after stabilisation of the domain structure in the ordered condition, and this is reflected in the formation of an additional maximum. The effect of the structure on the form of the dependence of internal friction on the stress or strain amplitude from the viewpoint of magnetomechanical phenomena is quite clear. All factors, decreasing the speed of movement of the domain boundaries, de130

Factors Affecting Anelasticity of Materials

Fig.3.22. Dependence of internal friction on strain amplitude of Co and its alloys: 1) Co; 2) 2% Re; 3) 10% Re; 4) 1% Zr; 5) 8% Zr; 6) 1% Nb; 7) 10% Nb; 8) 8% V; 9) 7% Fe; 10) 5.65% Mn; 11) 10% Mn; 12) 5% Cr; 13) 15% Cr; 14) 25% Cr; 15) 65% Ni; 16) 10% Mo; 17) 2% Ti (wt.%).

Fig. 3.23. Dependence of internal friction on the strain amplitude of the alloy at 23 kHz. 1) without the magnetic field, 2) using the electric field with 3 × 10 4 V m –1 , 3) after application of the magnetic field with an intensity of 1.1 × 10 4 V m –1.

crease the height of the maximum and increase the stress τ max . The form of the dependences is also influenced by the measurement conditions, such as temperature, application of a constant or variable magnetic field, electric current, static loading, etc. the magnitude of internal friction of cobalt and cobalt alloys is significantly influence by the ε → α polymorphous transformation. Increasing content of the alloying element decreases the amount of the low-temperature ε-phase with a hexagonal close-packed lattice and the temperature of the ε → α transformation increases. This is also reflected in the dependence of internal friction on strain 131

Internal Friction of Materials

amplitude, Fig. 3.22. The magnitude of internal friction of the alloys under the effect of a specific strain amplitude increases with a decrease of the content of alloying elements. If it is taken into account that the transformation in pure cobalt is not completed, the magnitude of internal friction in a wide strain amplitude range depends on the amount of the high-temperature phase present in the alloys. Annealing at temperatures in the vicinity of the martensitic transformation temperature decreases the residual stresses and increases the friction capacity of cobalt [101]. Measurements taken on amorphous alloys of iron, chromium and nickel [102], produced in the form of ribbons 0.02 mm thick and 2.5 mm wide, by cooling from a melt at a speed of 10 6 K s –1 , with a tensile strength of approximately 2400 MPa show, Fig. 3.23, that the level of their internal friction is influenced not only by the strength of the magnetic field but also by the application of a unidirectional electric field. The main relationships recorded for crystalline materials are also valid in the amorphous materials. Iron alloys are also sensitive to the strength of the external magnetic field and, consequently, the magnetomechanical component of internal friction contributes significantly to the level of their internal friction. The overall effect depends not only on the chemical composition of the alloy (this results in the changes of the amount and distribution of the ferromagnetic phase in the alloy), but also on heat treatment which may greatly change the fraction of the ferromagnetic phases in the alloys of a specific composition. Further results of measurements in this area are published in Chapter 6. In concluding this chapter, it should be noted that the specific level of internal friction and of the defect of the Young modulus is a function of a large number of factors having different effects on the mechanisms of scattering of mechanical energy in the material. It has also been found that in efficiently prepared experiments, internal friction and the defect of the Young modulus reflect the substructure, structure, prior deformation, thermal, deformation– thermal and other treatments. This means that the measurements of Q –1 or ∆E/E provide additional data on the material and on changes taking place in it.

132

Measurements of Internal Friction and the Defect of the Young Modulus

4 MEASUREMENTS OF INTERNAL FRICTION AND THE DEFECT OF THE YOUNG MODULUS The requirements on equipment and devices for measuring internal friction and elasticity moduli different [1,103]. For example, the determination of an independent characteristic, such as the internal scattering of energy, i.e. damping capacity, is carried out with different force effects on actual solids. For many actual structural components it is important to know a wide amplitude and frequency dependence of oscillations so that it is important to use a suitable test system. On the other hand, the tasks of physical metallurgy and threshold states require exact experiments in different directions (for example, in analysis of the relaxation spectrum of internal friction in solids it is necessary to regulate temperature from the liquid helium temperature up to temperatures close to the melting point, different frequencies at low strain amplitude). In the following section, we discuss the fundamentals of experimental procedures and equipment for measuring internal friction and the defect of the Young modulus and also examine the possibilities of automating these methods. 4.1 APPARATUS AND EQUIPMENT The basis of the classification of the methods of measuring internal friction and elasticity moduli is the general approach to the construction of scientific devices and metrology (Fig. 4.1). The main structural feature of apparatus and equipment is the excitation section, the measurements section and the section for recording vibrations of the specimens or test system. These sections are characterised by defined parameters and represent independent elements of testing equipment. Frequency is the criterion used in the 133

134

Selfexcitation

External independent

Rotating

Transverse

Longitudinal

Running

Standing

Hypersound

Excitation method

oscillations

Type of

Elastic waves

Measured characteristics

Relative

Absolute

Ultrasound

Frequency range Indirect

Measurement method Direct

Supplied power Calorimetric

Sound

Hysteresis loop

Measurements

Oscillation damping Oscillation increment Resonance curve Phase delay

Excitation

Ultrasound Differential

Infrasound

Recording output

Recording method

Recording

ELASTIC AND ANELASTIC PROPERTIES

Analogue x–y rec. device

Computer

Coding

Programming

Digital print

Mechanical

Piezoelectric

Electromagnetic

Electrostatic

Visual

Internal Friction of Materials

F ig ig.. 4.1. Classification of the test methods, measurements and recording of internal friction and the defect of the Young modulus.

Measurements of Internal Friction and the Defect of the Young Modulus

conventional approach for determination of the group of the excitation section. Using frequency, the systems are divided into infrasonic, sonic, ultrasonic and hypersonic. The frequency range in the test characterises different structural principles of the construction of equipment and provides different temperature and time changes of the elasticity effects in metals. The frequency criterion has physical meaning when the appropriate frequency range is determined in relation to the dimensions of the specimens a and the velocity of propagation of the wave v in the material [103]. Consequently, the range of low frequencies is represented by the case in which ω = 2π f << 2πv/a, the range of medium frequencies ω = 2πv/a, and the range of high frequencies ω > > 2πv/a. Another important classification criterion for the excitation of vibrations in the measurements of internal friction and elasticity moduli, reflecting the nature of propagation of the elastic waves in the material, is the nature of elastic waves. These waves may be standing or running. The standing waves are excited in a system whose part is the specimens (for example, a torsion pendulum) or directly in the specimens (for example, resonance waveguide). Running waves are excited in the surface of the specimens (for example, Rayleigh waves) or in its volume (for example, the pulsed method). The form of the vibrations of the specimens is determined by the method of excitation of vibrations. There are longitudinal, transverse and torsional vibrations. The nature of the vibrations determines not only the special features of the structure of equipment but also the measured characteristics of the materials (for example, normal or shear Young modulus). According to the excitation method, there are various units (Fig. 4.2), i.e. mechanical, electromagnetic, magnetostriction, piezoelectric, laser, etc. Consequently, the excitation methods can be combined on the basis of the controlling criterion, i.e. external independent and self-excited (Fig. 4.1). The classification of the circuit of the measurement unit is determined by two factors. It is the aim of measurements and the measurements method. When measuring the elastic or inelastic characteristics of metals, there may be different tasks, e.g. determination of the absolute values or relative changes of the characteristics, associated with the change of the state of the evaluated systems, or with differences between the determined states. In the first case, the high accuracy of measurements is controlling, whereas high sensitivity is the controlling factor in the second case. The classification of the measurement methods is more complicated [2, 103, 104]. The direct methods include the evaluation of the 135

Internal Friction of Materials

F ig ig.. 4.2. Methods of excitation (V) and recording (P is the receiver) of internal damping and the defect of the Young modulus (1): a) mechanical (2 – hammer, 3 – microscope), b) capacitance (4 – electrode of the condenser), c) magnetostriction (5 – magnetostriction device), d) electromagnetic (6 – magnetic layer), e) piezoelectric (7 – piezocrystal), f) eddy current.

F ig ig.. 4.3. Dynamic hysteresis loop: a) away from resonance (ω / ω r < 1), b) in resonance (ω / ω r = 1).

static and dynamic hysteresis loops. The method is based on the measurement of stress and resultant strain σ in the process of gradual loading ε and unloading of the specimen. The internal scatter of energy is characterised by the equation (2.43). In repeated loading, it is possible to record simultaneously stress 136

Measurements of Internal Friction and the Defect of the Young Modulus

and strain in the form of a hysteresis loop. Figure 4.3 shows examples of a dynamic hysteresis loop away from resonance, Fig. 4.3a, and at resonance, Fig. 4.3b. The energy measurement method is based on evaluating the difference of the electrical or mechanical output of the excitation device, required for maintaining the selected amplitude of the vibrations of the specimen. The relative scatter of energy in the material of the evaluated specimen is determined from the equation

Ψ=

Z1 − Z 2 , fW

(4.1)

where Z 1 is the total output of the excitation device, Z 2 is the output required to overcome resistance in the excitation device with the replacement of the output scattered in the circuit of the vibrating system, f is the frequency of the resultant vibrations, and W is the potential energy of the deformed specimen, corresponding to the amplitude of the resultant vibrations. The thermal method is based on the measurement of thermal energy U generated in the specimen during cyclic deformation with frequency f during time t. Energy ∆W scattered in the material during a single load cycle is determined by the equation

∆W =

U . tf

(4.2)

The method is integral and can be used in the case of the homogeneous stress state of the specimen, without connected masses. Indirect methods of internal friction measurements, Fig. 4.1, also include the method of damping vibrations and other methods. In this method, the specimen is the elastic part of the vibrating system. Internal friction is characterised by the logarithmic decrement of vibrations δ. Taking into account the decrease of the energy of the system during a vibration cycle, it is possible to determine the difference of the energies corresponding to the adjacent maxima A t , A t+∆t . We introduce the concept of the logarithmic decrement of vibrations in the form δ = Q –1 . Consequently,

137

Internal Friction of Materials

δ = ln

At = β∆t . At + ∆t

(4.3)

The method of increasing vibrations eliminates the shortcoming of the previous method because the latter requires initial deformation of the specimen with the maximum amplitude which may cause changes in the material and inaccuracies so that the evaluated state of the specimen is not the initial state. The method of increasing vibrations, i.e. the increment of vibrations ν, utilises resonance and the constant magnitude of the excitation force. The increment of the vibrations

ν=2

At + ∆t − At ∆At . = At + ∆t + At Ats

(4.4)

where A ts is the mean amplitude. The increment and decrement of the vibrations, corresponding to the amplitude A ts , are connected by the equation [104] in the form

δ = δ0

A0 − ν, Ats

(4.5)

here δ 0 is the decrement of the vibrations, corresponding to the maximum amplitude at resonance A 0 . The increment of the vibrations is determined from the recording of the vibrations of the increasing resonance vibrations and the values of δ 0 , A 0 are determined from the results of processing the recording of damping of the vibrations. The resonance vibrations are excited in the self-excitation regime. The method of the resonance curve determines the width of the resonant maximum or resonance valley in relation to the frequency dependence of the vibrations, at a constant excitation force (Fig. 4.4). Internal friction is determined from the equation Q −1 =

∆ ω 0.7 fr

and

Q −1 =

∆ ω 0.5 , 3 fr

(4.6)

i.e. from the width of the resonance maximum at the height of 0.7 138

Measurements of Internal Friction and the Defect of the Young Modulus

F ig ig.. 4.4. Resonance peak of the frequency dependence of deviation µ 0 .

or 0.5 of the maximum, where f r is the internal frequency of vibrations of the system without internal friction. The equations (4.6) are valid for the determination of internal friction in the area in which the internal friction is independent of the strain amplitude. For other measurement ranges, the equations must be modified. Internal friction can also be determined from the width of the resonance ‘well’ [104] using the equation

Q −1 = Kβ

∆ωβ fr

,

(4.7)

here K β is the coefficient of proportionality. Equation (4.7) can be used for any dependence of Q –1 on the strain amplitude. Internal friction can also be determined, using the method of the resonance curve, from the following equation [4,105]

Q −1 = K1

∆ω3db , ωr

(4.8)

where K 1 is the proportionality factor which takes into account the effective mass of the resonance system, ∆ω 3db is the three decibell deviation of resonance frequency and ω r is the resonance frequency of the system. The phase shift method is based on the time delay between the acting stress and resultant strain, denoted by angle ϕ. Consequently 139

Internal Friction of Materials

 ω2 Ψ = 2π 1 − fr 

  tan ϕ, 

(4.9)

where ω is the frequency of the excitation force, f r is the internal frequency of the system. Since ϕ ≈ 1, then

Ψ = 2 π tan ϕ.

(4.10)

For the linear system of the hysteresis type, it is possible to find the relationship of the decrement of the vibrations with the phase shift ϕ in the form

δ=

π  f r20 − ω2  tan ϕ,  f r2 

(4.11)

where f r0 is the resonance or internal frequency of the vibrations of the system, taking into account the dependence of internal friction on strain amplitude. In the ultrasound method, a wave is passed through the specimen. The velocity of the wave and its damping in the material are examined. The wave equation has the following form

x  A = A0 e −αx cos  t −  ,  ν

(4.12)

where α is the damping factor, A and A 0 is the actual and initial amplitude of the wave, x is the coordinate in the direction of wave propagation. The damping factor α is determined from the scatter of energy of the sound wave at its length λ. The value ∆W/W is linked with α by the equation

∆W Ax − Ax +λ = = 1 − e −2αλ , W Ax

(

)

(4.13)

and at low values of α and λ by the equation

140

Measurements of Internal Friction and the Defect of the Young Modulus

∆W πν ≈ 2αλ = 4α . ω W

(4.14)

Elasticity (Young) moduli The elasticity moduli are measured using bar-shaped specimens whose length is significantly greater than the transverse dimensions. In the case of low damping, we have

ν=

M ρ −1 ,

where M is the characteristic of the Young modulus (E or G). Consequently

νl =

E and ν t = ρ

G , ρ

(4.15)

where v 1 and v t are the velocities of propagation of the longitudinal and transverse wave. It is also possible to measure the internal frequency of longitudinal (f l ) or torsional (f t ) vibrations of the bar [106]. Consequently

fl =

n 2L

E ρ

 n 2 π2µ 2 I 1 − 2 SL2 

 , 

(4.16)

and

ft =

n G , 2L ρ

where n is the number of harmonic vibrations, L is the length of the bar, ρ is specific density, µ is Poisson’s number, S is the cross sectional area, and I is the moment of inertia of the specimen in relation to the longitudinal axis (for the square cross section I = πS 2 / 6, for the circular cross section with radius r I = πr 2 /2). The second term in the brackets in equation (4.16) is the correction coefficient according to Rayleigh, which takes into account the effect of 141

Internal Friction of Materials

transverse vibrations on the frequency of longitudinal vibrations. The correction coefficient for the specimens with L/2r > 3 made of a cubic lattice material is less than 1% and can be ignored. In complicated systems with inertia masses (for example, a torsional pendulum with a moving weight), the moment of inertia can be determined from the equations derived in Ref. 107, but in most cases, it is recommended use the dependences G − f 2 with changes of temperature. Usually, measurements are taken at the internal or resonance frequency which correspond to the main harmonic vibrations of the specimen. The main requirements in measurements of the elasticity moduli are described by the standards [14]. The recommended dynamic methods (strain rates 10 3 –10 4 s –1 ) for the determination of the elasticity moduli are based on the measurements of the resonance frequency of induced vibrations of the specimen in the form of a bar with a constant cross section. The resonance methods enable rapid and relatively accurate (relative error 0.5–0.8%) determination of the Young modulus. When using the pulsed methods, the elastic strain range is 10 6 – 10 8 s –1 . In these methods, it is necessary to determine the velocity of propagation of the elastic wave through a bar where the wavelength is small in comparison with the dimensions of the specimen. The elasticity moduli are calculated using equation (4.15). The pulsed methods of measurements of the Young modulus are characterised by high accuracy because the measurement error is ~0.1%. However, the extent of application of these methods is limited, especially at high temperatures of the material. The values of the elasticity moduli for many materials and alloys are presented in the literature (for example, Ref. 5,14,108). Let us return to the diagram in Fig. 4.1. The sections for recording the vibrations differ, depending on their construction and the methods of obtaining information. For the most extensively used experimental machines, the methods of recording vibrations can be divided into visual, mechanical, piezoelectric, electromagnetic, electrostatic and others. The methods of recording data from the measurement sections are discrete or analogue. The results of measurements can be recorded in a coordinate-recording device or on strips of digital recording systems. After transformation, coding and processing, the results are entered into a computer for further processing and analysis. According to the level and state of automation, the experimental procedures of measurements of internal friction and the defect of the 142

Measurements of Internal Friction and the Defect of the Young Modulus

Young modulus can be classified into four groups: non-automated systems, semi-automated systems, automatic systems, where the processes of measurement and recording data in the analogue or digital form are automated, and fully automated systems, controlled by a computer, with computer processing of the results. The general principles for the construction of the circuits of the automated systems with continuous recording of internal friction and the defect of Young modulus are presented in Ref. 109. The main requirements on the systems for measuring internal friction and the defect of the Young modulus is to ensure high accuracy and sensitivity of equipment. The relative error of measurements of internal friction and the defect of the Young modulus in the most effective systems is: for the torsional pendulum ±2.5%, for resonance bars ±0.3–0.1%, and for the pulsed methods ±10–0.01%. The construction of sensitive equipment and systems is based on their automation and resolution capacity. In the case of the automated systems, the resolution power of the measurements of internal friction is sufficiently high: for the torsional pendulum ±0.03%, for the resonance systems up to 10 %, and for the method with the running wave up to 10 %. When comparing characteristics, it is very important to ensure the high resolution power of the systems for measuring internal friction and the defect of the Young modulus [106]. The design of the devices and systems can differ greatly. The mechanical section must fulfil special requirements (for example, the sections of connection of the specimen, the section of excitation of vibrations, etc.). It is always necessary to optimise the test conditions in order to eliminate unnecessary scattering of the energy in the measurement section. The introduction of measurements of internal friction into laboratories in plants requires reliable systems are reliable and fast measurements. In the case of scientific investigations, special attachments can be used. 4.2 EXPERIMENTAL MEASUREMENTS AND EVALUATION In the last 20–30 years, the extent of application of the methods of measuring internal friction and the defect of the Young modulus has expanded greatly. A large amount of experience has been accumulated in the application of devices and systems in solid-state physics, in evaluation of damage and threshold state of materials (see Fig. 1.1). We shall use the conventional division of the methods of measurement of internal friction and the defect of the Young modulus 143

Internal Friction of Materials

when using the frequency criterion for: infrasound (10 –4 –10 2 Hz), sonic (10 2 –10 4 Hz), ultrasonic (10 5 –10 8 Hz) and hypersonic (10 9 – 10 11 Hz). The conventional nature of the division also results from the region of sonic and low frequency oscillations but has a certain physical substantiation [26]. 4.2.1 Infrasound methods The methods of the static and dynamic hysteresis loops are used in examination of the characteristic the internal friction and the defect of the Young modulus of materials at stresses close to or higher than the fatigue limit. The sensitivity and accuracy of this method is determined by the error of measurement of deformation and by the efficiency of the measuring devices (strain gauges, dynamometers). The range of possible measurements of energy scattering in efficient test systems of the type Instron and Strainset is greatly limited [110]. The principal diagram of the system for measurement of internal friction in metals using strain gauges is shown in Fig. 4.5 [111]. A pair of sensors 4 is attached to the specimen, two other pairs of sensors 5 and 9 are attached to the dynamometer. The signal from the sensors 4 is directly proportional to the deformation of the

F ig ig.. 4.5. Diagram of equipment for measuring internal friction using strain gauges: 1) oscilloscope; 2, 12) frequency-selective filters; 3) amplifier; 4,5,9) strain gauges; 6) switch; 7,8) resistors; 10) two-channel amplifier of strain gauge data; 11) compensator of the phase shift of the signal. 144

Measurements of Internal Friction and the Defect of the Young Modulus

specimen, passes through the strain gauge bridge, the amplifier and the filter to the horizontal axis of the oscilloscope. The signal from the sensors 9, proportional to the stress, passes through the strain gauge bridge, the reverse change of the phase shift and the filter to the vertical axis of the oscilloscope. The beam of the electrons on the screen of the oscilloscope describes the loop in the stress–strain coordinates. The application of accurate strain gauges (nonlinearity 2.1 × 10 –4 %) makes it possible to increase greatly the sensitivity of measurements. In evaluation of materials with Ψ ≥ 0.001 the error does not exceed 10%, Efficient connection of the amplifier makes it possible to subtract the elastic strain from the total strain of the specimen and, consequently, increase the sensitivity of the system [110]. This system is universal and is utilised in fatigue systems loading the specimen by pull–push, bending or torsional loading. At higher loading frequencies, the method of the dynamic hysteresis loop is used for recording the diagrams of cyclic deformation [111]. The torsional pendulum is used in the low-frequency range and is the most efficient method of measurement of the internal friction dependent on the strain amplitude, and in measurements of the internal friction dependent on temperature. The shear Young modulus is determined by the square of frequency f 2 of damped vibrations (G ~ M * f, where M * is the moment of inertia of the system). There are three methods of using the torsional pendulum: direct, reversed, and combined (Fig. 4.6). A suitable example of the design of measuring heads in the

F ig ig.. 4.6. Torsional pendulum systems: a) direct, b) reversed, c) combined (1 – specimen, 2 – rod of the moving clamping jaw, 3) inertia mass for testing the specimen, 4 – clamping jaws, 5 – rod of the fixed jaw, 6 – suspension, 7 – counterweight). 145

Internal Friction of Materials

arrangement for the measurements of the damping of vibrations and internal vibrations is the RKM-TPI torsion pendulum. In the first variant [112], internal friction and the square of frequency were measured visually or semiautomatically using a counter of vibrations. The second variant [113] is based on the automatic recording of the results of measurements, with continuous drawing of the curve of the temperature dependence of internal friction and the dependence of internal friction on the strain amplitude with discrete recording of the data. The system makes it possible to use specimens with a length of 50–120 mm of circular or right-angled cross section. The frequency range is 0.2–100 Hz, the temperature range up to 2500°C. The strain amplitude range is from 10 –6 to 10 –3 , and the internal friction background is smaller than 10 –4 units of internal friction. Measurements can be taken in a unidirectional magnetic field with an intensity of 4×10 4 A×m –1 . The RKM-TPI system can be fitted with clamping heads with the section for deformation of the specimen during measurements [114] by superplastic loading [115], and by an attachment for the effect of corrosive media. Other systems based on the torsional pendulum of different design are also available [116,117]. Improvement of design is associated with the widening of the working characteristics of systems, such as the range of strain amplitude and frequency of vibrations, temperature [118,119], modification of the measuring heads [120, 121], increase of sensitivity for the application of strain (for example, from 10 –11 ) [122]. Some systems based on the torsional pendulum also make it possible to carry out tests with bending vibrations, in addition to torsional vibrations [123]. It is important to note low-frequency combined pendulum systems with the original design of the suspension [123]. When using the pendulum, excitation is carried out by the mutual effect of the magnetic field of coils distributed on a suspension system, with a permanent magnet [124]. To prevent deformation of the specimen as a result of the weight of the rest of the system, a container with a damping fluid is used. However, systems not using fluid dampers are also available [124]. Centring of the vibrating system is improved by a system of rollers [126] or using precision dimples [125]. Good results have been obtained with the system in which the ferromagnetic cylinder of the suspension is centered in holes of two toroidal electromagnets with the core in the coils thus forming an orthogonal source of alternating current [127]. The suspension also 146

Measurements of Internal Friction and the Defect of the Young Modulus

contains a square magnet which interacts with four electromagnets powered by direct current. If the voltage to the electromagnet is changed, the moment of inertia and frequency of vibrations of the system change. Compensation of the mass of the suspension system and the formation of a certain mechanical stress in the specimen can be achieved using lever or spring systems [128]. In measurements of the characteristics of brittle material and glass, it is necessary to ensure rigid contact of the specimen with the clamping heads. In one of the interesting solutions, the lower clamping jaw has the form of a vessel and the upper has the form of a seed (the specimen is pulled from the alloy and this is followed by the measurements), or contact is made by means of eutectics with a low melting point [129]. In the basic models of the torsional pendulum, the change of frequencies achieved by moving a weight on the one-sided lever of the inertia system, or by changing the number of weights. Improvement have been made in the automatic displacement of the weights along a two-sided lever, by regulating the number of discs in the inertia mass on the vertical shafts, or by changing the number of weights– permanent magnets on a shafts as a result of their contact with electric coils. Another solution is the mutually effect of the magnetic fields of the inertia mass in the form of a permanent magnetic with a stationary electric coil [127]. Expansion of the frequency range is possible in the case of the torsional pendulum by using the regime of forced vibrations [113]. The automation of the low-frequency system for measurements of internal friction and the defect of the Young modulus is based on the application of photocells. The diagram of recording vibrations for this principle is shown in Fig. 4.7a. The light beam from the mirror, secured to the inertia system, falls on photo resistances and the signal from the passes, after amplification, to a recording device. The distribution of the photoresistances in a specific sequence makes it possible to obtain recordings in the logarithmic coordinates. Of considerable importance for automation was the concept of determination of the decrement of vibrations from the decrease of the speed of vibrations (time periods) during the passage of the beam (Fig. 4.7b [130] which greatly decrease the duration of a single measurement in the strain amplitude range 10 –4 – 10 –2 and the frequency range 0.2 – 20 Hz. The method of measurement of the duration of passage of the light trace was gradually greatly improved as a result of the application of photocells in the goniometric head [116]. 147

Internal Friction of Materials

F ig ig.. 4.7. The block diagram of equipment: a) (1 – mirror, 2 – light beam, 3 – the light source, 4 – photoresistance, 5 – amplifier, 6,7 – the recording device with attachments); b) (1 – mirror, 2 – light source, 3 – light beam, 4 – grating, 5 – photoamplifier, 6 – the device for measuring the transfer of vibrations, 7 – the device for measuring time during the passage of vibrations); c) (1 – mirror, 2 – the light source, 3 – the input Schmidt trigger, 4 – the frequency divider, 5 – chronometer, 6 – printer, 7- photocells, 8 – the output Schmidt trigger); d) (1 – pendulum, 2 – mirror, 3 – the light source, 4 – shut-off attachment, 5 – the source of the signal for arresting the cycle counter, 6 – the unit for stopping the counting of cycles, 7 – the counter of pulses, 8 – the unit for the double number of the pulses, 9 – indicating device, 10-the device for selecting the number of pulses, 11 – the unit for the double number of the pulses, 12 – photodiodes, 13 – triggering photographic equipment, 14 – the source of the starting signal, 15 – the unit for starting the counter, 16 – printer).

Accurate measurements of the time periods in the frequency range of operation of the pendulum from 0.3 to 3 Hz are taken using silicon photodiodes [117]. The central part of the illuminating scale contains photodiodes distributed at a distance of 10 mm from the centre of the scale [Fig. 4.7c]. During the gradual passage of the beam through the photodiodes, Schmidt triggers convert the electric signal to square pulses which are transferred to a frequency divider and then to a timer. The data for determining the logarithmic decrement of friction

1 t  σ = ln  n  n  t0 

(4.17) 148

Measurements of Internal Friction and the Defect of the Young Modulus

are automatically recorded in a printer. In equation (4.17), n is the number of vibrations, t 0 is the duration of passage of the light beam through a constant base at the start of counting, t n is the identical time after n vibrations. Another modification is shown in Fig. 4.7d. The light beam, reflected from the mirror, falls on the triggering photograph equipment. The signals from the starting and stopping mechanism are change to square pulses which are then transferred to the indicating unit. The equipment for starting and stopping the counter is connected to a chronometer. The pulses are then transferred to the control system, consisting of the selection device for the number of vibrations (n = 1, 2, 4, 8,...) and from a double counter forming the signal for stopping the counter in accordance with the selected programme. The device for stopping the counter issues a signal for stopping the chronometer which records the duration of passage of the light beam from the triggering to stopping device after the given number of vibrations of the system. Equipment is also fitted with a device for the generation of high-power alternating voltage. The modified form makes it possible to determine automatically the dependence of internal friction on temperature, strain amplitude or time, in the form of tables or graphical dependences [131,132]. When applying transverse or bending vibrations, it is possible to use specimens of simple shape and different dimensions, a simple excitation method and a simple method of recording vibrations, a wide frequency range and also a wide range of external loading of the specimens. As in the case of the torsional pendulum, the bending vibrations do not load the entire cross section of the specimen and this impairs the conditions of obtaining the actual internal friction characteristics. The systems with a cantilever specimen are also used [124, 133]. Equipment with or without connected mass is available. The second method is used when it is required to generate various stresses in the specimen. The principle of this method is shown in Fig. 4.8. The vibrations are excited by the electromagnetic method. Recording from contact-free shadow observation of the vibrations is carried out using a photographic sensor [134]. Four forms of vibrations of the specimen are used. In the case of non-ferromagnetic materials, thin sheets of a ferromagnetic material are secured to the specimen. The logarithmic decrement of damping is determined by the method of damped vibrations or resonance vibrations. The systems suspended at the nodes of vibrations are used for the maximum equalisation of the stress along the specimen under the 149

Internal Friction of Materials

F ig ig.. 4.8. Block diagram of equipment (a) and the shape of pulsed of the cantilever specimen (b): 1) amplifier, 2) relay, 3) sound generator, 4) oscilloscope, 5) screw, 6) measuring discs, 7) base plate, 8) wedge, 9) variable mass, 10) indicating device, 11) electromagnet, 12) specimen, 13) microscope, 14) spring.

F ig ig.. 4.9. Diagram of the oscillating system and optical recording system. 1) spring, 2) inertia mass, 3) prismatic specimen, 4) mirror, 5) light source, 6) recorder.

effect of bending vibrations. On the basis of the principle shown in Fig. 4.9, the Institute of Strength of Materials of the Academy of Sciences of the Ukraine has developed a set of systems [104]. The prismatic specimen 3 is suspended on thin springs 1. The inertia mass 2 is connected to the specimen. The mass can be connected using threads, clamps, wedges, etc. Vibration systems consisting of two specimens and inertia mass are also available [135]. 150

Measurements of Internal Friction and the Defect of the Young Modulus

A system with a U-shaped specimen (‘tuning fork’) utilises the magnetoelectric excitation to resonance at the natural frequency or stationary excitation [136]. The frequency of the exciting vibrations is changed automatically and the signal, directly proportional to the amplitude of the vibrations, is recorded. The second method is based on the maintenance of a specific frequency, closed to the resonance frequency, with the change of the temperature of the specimen [104]. The results show that the method of measuring the supplied power is most promising in continuous measurements of internal friction with the change of temperature, amplitude or loading time [113]. The methods based on longitudinal infrasound vibrations are used mostly in evaluation of materials with high elasticity [26]. For special purposes, it is possible to use an attachment with cryogenic or higher temperature for the examination of the characteristics under static or cyclic deformation, under the effect of Xray or ion radiation, for testing whisker crystals, thin foils and highly brittle materials [2,26,104,107]. Experimental results show that in addition to the measurements of internal friction and Young modulus, it is also possible to evaluate other physical properties [137]. 4.2.2 Sonic and ultrasound methods The design and application of equipment for this frequency range have been described in a number of studies [2,26,106,138]. The design of equipment does not contain any special inertia systems connected with the specimen. The main groups of the measuring equipment are: –equipment with the specimen secured at nodes by the contact method [139], –equipment in which specimens are secured at nodes by a contactless method [147], –equipment with one specimen secured at the node of the vibrations [141], –cantilever-type equipment [142]. When the specimens are secured by the contact method, it is not necessary to use any wires or supports connecting the specimens with the sensor and receiver of the vibrations in the vicinity of the node of the vibrations. The receiver and the sensor are far away from the specimen and, consequently, they can be protected from the effect of temperature, magnetic or other energy field. It should be remembered that additional energy scattering takes place in the area 151

Internal Friction of Materials

of contact of the specimen with the filament. For example, when using filaments 0.1 mm thick and in the case of the minimum distance to the nodes of the vibrations, the scattering of energy in the contact zone is less than ≤ 3% when the level of the background of the internal friction of equipment is Q –1 < 10 –5 . This systematic error is almost completely eliminated in the systems without contact where the correction of the filaments or supports to the specimen is carried out exactly at the nodes of the vibrations. For example, in the presence of bending vibrations, the contact is realised in the form of needles. The displacement of the sensors of excitation or the change of the method of securing the specimens make it possible to produce not only basic vibrations but also higher harmonic vibrations. A shortcoming of these systems is the low level of the strain amplitude (ε ≤ 10 –5 ) and in the case of contactless connection also the proximity of the measuring unit to the zone of external influences. In both the contact and contactless variant of equipment with single connection at the node of the vibrations it is possible to excite several types of vibrations. All these systems can be fully automated. Equipment for cantilever securing of the specimen is used in evaluation of thin sheets, foils, filaments and wires. A special position is occupied by the vibrators consisting of stepped and exponential concentrators because they make it possible to apply a wide range of strain amplitudes [105,143]. The measurements of vibrations and of their changes using piezoceramic sheets connected by long waveguides with the specimens are carried out to evaluate the relative change of internal friction because energy is lost in contact areas. In electrostatic excitation, one of the ends of the bar is subjected to the effect of a periodic force formed as a result of electrostatic attraction between the conducting end of the bar and the stationary electrode (Fig. 4.10). In excitation of the bending vibrations the electrodes are situated in the area with the maximum amplitude of the vibrations. The electrodynamic method of excitation is associated with the mutual effect of the conducting metal strip, deposited on the surface of a non-conducting specimen, with the magnetic field. This results in the formation of an alternating force causing vibrations in the specimen. These methods are suitable for materials with low internal friction in a wide frequency range. The effect of the electrostatic force between capacitor plates, where one of the plates is represented by the part of the surface of the specimen, makes it possible to excite longitudinal, bending or 152

Measurements of Internal Friction and the Defect of the Young Modulus

F ig ig.. 4.10. Diagram of electrostatic excitation of longitudinal (a) and bending (b) vibrations in specimen 3: 1) power supply from the oscillator; 2) recorded signal travels to the receiver, recording and printing.

torsional vibrations in the specimen. In this excitation method, the bond between the mechanical and electrical system is weak. Consequently, the effect of the electrical circuit can be ignored. To eliminate aerodynamic losses, the working space is evacuated. Elastic vibrations in the specimen can also form as a result of Coulomb forces [144] between the face of the specimen and the electrode which is parallel to the face of the specimen. The second electrode (the receiver on the other end of the specimen) changes the mechanical vibrations to an electrical signal. The modified method requires only one electrode [145]. The frequency modulation of the loading and measuring signal is utilised in this case. Equipment with electrostatic excitation and reception of information on vibrations is being used at the A.F. Joffe Physical-Technical Institute [146]. One of the first automatic systems of this type was built at the Academy of Sciences of Georgia in the former USSR [147]. This equipment makes it possible to carry out continuous measurement of the temperature dependence of internal friction. The equipment is based on the method of supplied power of the method of damping of vibrations. The electrostatic method of excitation ensures good acoustic contact between the specimen and the electromechanical converter. This impairs the measurement results when using the piezovibrator or the pulsed method, especially at high temperatures. Equipment of this type has been used for a large number of studies of the properties of semiconductors, metallic and dielectric materials [106]. Various methods are used in electromagnetic citation. These methods are based on the excitation of longitudinal, bending or torsional vibrations using conventional electromagnetic converters (Fig. 4.11). In the first variant, Fig. 4.11a, longitudinal vibrations are excited using thin sheets connected to the ends of the specimen. Disadvantages of the method are the displacement of the discs in the 153

Internal Friction of Materials

F ig ig.. 4.11. Equipment with electromagnetic excitation of oscillations. a: 1 – specimen, 2 – ferromagnetic plates, 3 – emitter, 4 – receiver, 5 – oscillator, 6 – amplifier and oscilloscope; b: 1 – specimen, 2 – elastic suspension, 3 – emitter, 4 – receiver, 5 – oscillator, 6 – amplifier and oscilloscope, 7 – furnace or cryogenic unit.

fixing area and the fact that direct heating of the specimens is not possible. Therefore, a new system was developed, Fig. 4.11b, where thin wire suspension is used for inducing and receiving electromagnetic vibrations. On the one side, the suspension is connected to the membrane of the electromagnetic converters and on the other side at the nodes of the vibrations (or the ends of the specimen). The suspension is outside the furnace and the entire electrical circuit is at room temperature [139]. The circuit for automatic equipment with electromagnetic converters for the continuous recording of internal friction and Young modulus at normal or torsional vibrations is shown in Fig. 4.12 [139]. The parameters of this variant are as follows: frequency range 0.5–3 kHz, the strain amplitude range from 10 –7 to 10 –4 , temperature range from –196 to 600°C, vacuum 0.13 N⋅m –2 , the background of equipment is ≤10 –5 unit of internal friction, the sensitivity of equipment to the relative change of internal friction is 10 –4 %. On the basis of the circuit with the electromagnetic excitation of vibrations and suspension of the specimens on filaments, S.A. Golovin and A.A. Morozyuk (Tula Polytechnic Institute) constructed systems based on the differential measurement method. This system makes it possible to record differences of the signals from the reference specimen and the measured specimen. This greatly improved the accuracy of the method. Equipment of company Elastomat is also used widely. These systems include electromagnetic and piezoelectric exciters in different temperature ranges of measurement of internal friction and Young modulus (–190°C to 200°C, or 20 to 154

Measurements of Internal Friction and the Defect of the Young Modulus

F ig ig.. 4.12. Block diagram of automatic equipment with electromagnetic excitation: 1 – heating (cooling) system, 2) specimen, 3) sensor of vibrations, 4) amplifier, 5 – filter, 6 – potentiometer, 7 – power amplifier, 8 – exciter of vibrations, 9 – comparison unit, 10 – servoamplifier, 11 – servomotor, 12 – source of stabilised voltage, 13 – potentiometer, 14 – recording device, 15 – thermocouple, 16 – recording device, 17 – device for recording data from the potentiometer, 18 – voltmeter, 19 – oscilloscope, 20 – frequency meter, 21 – discriminator, 22 – pulse counter, 23 – oscillator, 24 – amplitude limiter – frequency-dependent link – sensor unit.

1000°C, for the 1.015 model), using longitudinal, transverse and torsional vibrations on the basic or higher harmonic vibrations. Automatic systems with the excitation of internal vibrations were described in Ref. 148 and 149. A large amount of data on the properties of metals and structural materials have been obtained on the systems with electromagnetic excitation of vibrations [2,26,104]. A method of piezoelectric excitation of vibrations in specimens for determination of the elastic characteristics of materials was proposed by S.L. Quimbi. At present, this method is used in the vibrator proposed by J. Marx [150]. A specimen whose section is the same as that of the piezoceramic specimen is connected to one of the ends of the piezoceramic specimen (Fig. 4.13). The piezoceramic specimen is secured at the node of elastic displacements formed after supplying an alternating electric field to the electrodes. Measurement of the resonant frequency, the quality of the piezoceramic material and the piezoceramic–specimen system makes it possible to calculate the internal friction characteristics and the Young modulus. At high temperatures, it is recommended to use a triple vibra155

Internal Friction of Materials

F ig ig.. 4.13 4.13. Diagram of piezoelectric excitation: 1) piezocrystal, 2) specimen, 3) oscillator, 4,5) voltmetres, 6,7) resistances.

tor consisting of piezoceramics, a ceramic bar, and the specimen. Various types of a compound piezoceramic vibrator are possible: 1) piezoceramics with a specimen of arbitrary length; 2) piezoceramics loaded simultaneously with two specimens of any or identical length; 3) piezoceramics with the specimen whose length corresponds to the resonance frequency of the piezoceramics. The third case is most useful because the nodal plane of the vibrator is identical with the nodal plane of the piezoceramics. The damping decrement of the specimen δ 0 is determined by the decrement and mass m of the elements of the compound vibrator [151]. For a double vibrator

δ0 = δ 2 +

m1 (δ2 − δ1 ) , m0

(4.18)

for a triple vibrator

δ 0 = δ3 +

m m1 (δ3 − δ1 ) + e (δ3 − δ1 ), m0 m0

(4.19)

where m e is the mass of the reference bar, δ 0 and δ 1 or m 0 and m 1 are the decrement of vibrations or the mass of the specimen and the piezoceramics, δ 2 is the decrement of the vibrations of the piezoceramics–bar–specimen system. For the natural frequency of the specimen in the vibrator

f0 = f 2 +

m1 m0

( f 2 − f1 ),

(4.20) 156

Measurements of Internal Friction and the Defect of the Young Modulus

and for a triple vibrator

f0 = f 2 +

m1 m0

( f3 − f1 ) +

me ( f3 − fe ) , m0

(4.21)

here f e is the resonance frequency of the reference bar. Crystals of alkali halides have been used successfully in the evaluation of the effect of the amplitude vibrations on internal friction and the defect of the Young modulus [152]. A compound vibrator, consisting of the specimen, a duralumin bar and piezoceramics, is placed in a cryostat. The equipment has the following characteristics: temperature range from 4.2 to 300 K, working frequency 50– 125 kHz, strain amplitude from 10 –7 to 10 –4 , with the error of measurement of the decrement of vibrations of approximately 10%. A suitable example of the development of acoustic systems with piezoceramics is the automatic system developed at the Joffe Physical-Technical Institute of the Russian Academy of Sciences. The problem was solved on two planes: analogue automation [153], and the variant using a computer and digital equipment [154]. Examples of the application of this type of equipment for polycrystalline magnesium have been published in Ref. 106. Step vibrators with magnetostriction or piezoceramic exciters (Fig. 4.14) can be utilised in the quantification of internal friction. A fully automatic system of measurement of internal friction and

F ig ig.. 4.14. Vibrators with a ceramic converter and stepped concentrator (a) and with an exponential concentrator (b): 1) supply of electric signal, 2) earthed electrode, 3) sensing electrode (dimensions in mm). 157

Internal Friction of Materials

the defect of the Young modulus was developed at the Technical University at Zilina (Slovakia) under the supervision of the author of the book [505]. The measurement of internal friction Q –1 by the method of determination of the quality of the resonance system and the defect of the modulus of elasticity ∆E/E by the method of determination of the displacement of the resonance frequency of the system is carried out in a resonant system consisting of a piezoceramic converter 1, a titanium attachment 2 and the specimen 3 (Fig. 4.15). All parts of the resonant system (1, 2, 3) fulfil the resonance condition according to which their length is half the wavelength of damping in the given material at resonance frequency f 1. The internal friction of the entire system Q –11,2,3 or of the system without the specimen Q –1 , is determined by the equation 1,2

 ∆f  −1 Q1,2,3 =  r 3dB  ,  f r 1,2,3 or  ∆f  −1 Q1,2 =  r 3 dB  ,  f r 1,2

(4.22)

where ∆f r3dB is the 3 dB deviation of the resonance peak. If voltage

F ig ig.. 4.15. Diagram of equipment for automatic measurement of internal friction and defect of the Young modulus (Technical University, Zilina). 158

Measurements of Internal Friction and the Defect of the Young Modulus

U n with frequency f r is supplied to the piezocrystal, a standing wave forms in the resonant system and the piezocrystal generates, on different planes, the voltage recorded as U s . This voltage is directly proportional to the strain amplitude in the central part of the specimen in accordance with the equation ε = k1 U s ,

(4.23)

where k 1 is a constant for the given system with the specimen. The internal friction of the entire resonant system is also determined by the equation −1 Q1,2,3 =

U n K1,2,3 U s fr

,

(4.24)

where K 1,2,3 is the characteristic of the system, determined from the equation

K1,2,3 = ∆f r 3dB

Us . Un

(4.25)

The quantity ∆f r3dB is determined at U s /√2 by increasing the frequency of supplied current gradually until we record U s max at f t . Subsequently, the system is detuned to value f rx at which U s = U s max /√2. The increase of frequency above f r also results in a decrease of U s max to the value U s max /√2 at which the frequency of the system is f ry . Consequently

f r 3 dB = f ry − f rx .

(4.26)

is determined from the The internal friction of the specimen Q –1 3 equation −1 −1 , Q3−1 = k2 Q1,2,3 − k3Q1,2

(4.27)

where k 2 and k 3 are the constants calculated from the dimensions, shape and properties of the connected specimens and their material (156). 159

Internal Friction of Materials

The defect of the Young modulus of the specimen ∆E/E is determined from the equation

∆E m1,2,3 f r1 − f r , = E m3 fr

(4.28)

where m 1,2,3 and m 1,2 at the calculated effective masses of the entire system without the specimen, f r1 is the resonance frequency of the entire system, determined using the value of ε differing from that used in the determination of f r . For the currently available range of ε from 5×10 –6 to 7×10 –4 it is sufficient to measure at least 30 points at different values of ε from the given range. For every point it is necessary to determine U n , U s , f r . Special attention must be given to at least tenfold repetition of the measurement for the determination of ∆f r3dB (equation 4.26), because this value is included in the calculations of Q –11,2,3 . In the manual setting of the values of U n in the range from 100 mV to 1000 mV on a dial-type electronic voltmeter, with the manual setting of the frequency of the resonance peak with the accuracy of 1 Hz in the frequency meter, and with subjective determination of the value of U s on a dial-type electronic millivoltmeter with the manual search for the value of f rx of f rt with the accuracy of ±1 Hz, subjective determination of U s at U s max /√2, taking into account the manual inputting of the initial data, interoperational values and the output values, we obtain that at f r = 22.5 kHz and Q 3– 1 = 10 –4 the measurement error is approximately 1% and at ∆E/E = 10 –3 it is ~0.1%. The measurement time of every experimental point is no longer than 60 seconds, i.e. for a set of 60 measurements, together with the adjustment time of the measurements, it is no less than 3600 s, with high concentration of the operator. Puskar, Palcek and Houba [155] constructed equipment in which these disadvantages are removed. The decrease of the measurement time and improvement of the accuracy of measurement of internal friction and the defect of the Young modulus are based on the complete automation of the selection of power supply, measurement, evaluation and graphical expression of the results of measurement using a microcomputer. The resonant system 1 + 2 + 3, or 1 + 2 + 7 + 3 (Fig. 4.15) is powered by the oscillator 11, controlled by a programmable generator with the attenuator 10, activated using the interface 7 from the microcomputer 4. The magnitude of U n is 160

Measurements of Internal Friction and the Defect of the Young Modulus

monitored using the multi-meter 8. The data on the magnitude of U n are entered through the interface 7 into the operating memory of the computer 4. When a standing wave forms in the resonant system the resonance frequency f t is measured and the signal is transferred to through the interface 7 to the operating memory of the microcomputer 4. Subsequently, voltage U s is generated in the piezoceramic material of the converter 1, and the magnitude of this voltage is measured with the multimeter 9 and the signal travels through the interface 7 to the operating memory of the microcomputer 4. The measurement algorithm is presented in the flow chart in Fig. 4.16. After completing measurements at a selected point, the microcomputer 4 issues a signal which, after passing through the interface 7, defines the next measurement point for the programmable generator with the alternator 10. The operating memory of the microcomputer 4 records data of all measurements. On the basis of the programme,

Density of specimen ?: H Size of specimen?:L1,L2,D1,D2 Speed of sound?: v Frequency range?:t 1,t2 εmax?: εmax Initial value U?:U 0

Interpolation U(i) Calcul. U s max, ε, f r, Q –1, dE/E

+ +

Interpolation U(i) Calcul. Us max, f r, frx ,∆f3dB Calcul. ε, Q–1, k2, k3, m1, m2

Printing of results Graph

F ig ig.. 4.16. Flow chart showing measurement and evaluation of internal friction and the defect of the Young modulus in equipment VTP-A (Technical University, Zilina). 161

Internal Friction of Materials

it transfers the data to the disk storage 5 and the signal for printing the selected data and the graphical dependences in the printer 6. For the case of a sandwich converter 13, Fig. 4.15, consisting of the piezoceramic plates 14, adhesion bonded to the metallic sections 15, the algorithm, Fig. 4.16, is the same. Using the extension waveguide 17 with the length nλ/2, where n is an integer and λ is the wavelength of the waves in the material of the waveguide, specimen 3 can be placed in a furnace or a cryogenic unit with programmable temperature change, in a unidirectional magnetic field with the programmed change of the intensity, in vacuum which can be varied using a programme, or in the testing space of a nuclear reactor where the intensity of neutron radiation can be varied in accordance with a programme, or it can be placed in other energy fields which are either constant or can be varied in accordance with a programme. The data on the parameters in the unit 18 are processed by the unit for evaluating the defect 19 and, subsequently, the data are transferred through the interface 7 into the microcomputer 4 which then controls the parameters in unit 18. The system reduces the measurement time of a single point to 4 s which means that 30 points can be measured within 600 s, including the setting time. The system also increases the accuracy of measurement of U n , U s , f r , ∆f r3dB , f r1 at a frequency of 22.5 kHz so that at Q 3– 1 = 10 –4 the measurement error is approximately 10 percent. The system described in the invention eliminates the subjective error of measurement caused by the operator. As a result of eliminating manual calculations and drawing graphical dependences in the selected coordinates, the time required for obtaining reproducible data on Q 3– 1 and ∆E/E is reduced further. The application of the programmable generator with the alternator 10 and digital multimeters 8 and 9 makes it possible to set more sensitively the value of U n and measure appropriate values, as well as select U n max = 1500 V thus expanding the strain amplitude range from 1.2×10 –7 to 1×10 –3 . The programmed control of the measurements enables data to be obtained on the time dependences of Q 3– 1 and ∆E/E from 4 s to the selected time. Magnetostriction vibrators for a frequency of 3–20 kHz using longitudinal vibrations have also been developed at the Institute of Strength of Materials of the Academy of Sciences of Ukraine [104, 157]. The evaluation of internal friction by the calorimetric method at a frequency of ~ 20 kHz, Fig. 4.17a, can also be carried out using the magnetostriction vibrator (1). The magnetostriction phenomenon 162

Measurements of Internal Friction and the Defect of the Young Modulus

F ig ig.. 4.17. Diagram of calorimetric (a) and resonance (b) methods: a: 1) magneto striction vibrator, 2) concentrator, 3) specimen, 4) calorimeter filling, 5) thermocouple, 6) casing of the calorimeter, 7) microscope, 8) mixer, 9) rubber membrane, 10) base plate, 11) collar, 12) cooling of vibrator; b: 1) stabilized power source, 2) capacitance sensor, 3) detector, 4) electronic potentiometer, 5) frequency meter, 6) oscillator, 7) amplifier, 9) vibrator, 10) concentrator, 11) specimen.

is based on the observation which shows that the magnetostriction material (Fe, Co, Ni, and alloys of these metals) changes its dimensions with the change of the magnetic field in which it is placed. Concentration of the energy in a small cross section is carried out using the concentrator 2 of the conical or stepped type which changes the magnetic field. The length of the vibrator and of the concentrator corresponds to half the wavelength of the waves in the material of the vibrator and the concentrator at a frequency of f r . Specimen 3 is attached to the concentrators; the natural frequency of the specimen correspond to f r . The areas where the mechanical joints of the system are located are not loaded with mechanical stress. The electric signal from the generator, connected to amplifier, is supplied to the winding of the vibrator. The latter must be cooled. Consequently, the vibrator is placed in the vessel 12 with running water. The heat dissipated in the material can be measured using the calorimeter 4 filled with water, containing a thermometer and a mixer 8. The relative scattered energy is calculated from the equation 163

Internal Friction of Materials

Ψ=

4 E m ∆T , V σ02 ft

(4.29)

where E is the Young modulus of the material of the specimen, m is the mass of water in the calorimeter, ∆T is the change of the water temperature in the calorimeter during time t of cycling loading, V is the volume of the working section of the specimen, σ 0 is the stress amplitude in the specimen, and f is the frequency of the vibrations. The method has been used in evaluation of the fatigue properties of structural steels and alloys. Recording of the resonance curve, Fig. 4.17b, for a cyclically deformed specimen is based on the excitation of longitudinal vibrations of the system which contains the specimen 11, concentrator 10 and vibrator 9. The excitation of vibrations with a frequency of 3–10 kHz is determined by the coil of the magnetostriction vibrator receiving alternating current from the amplifier 7 and direct current for the magnetisation of the vibrator. The system is powered by an ultrasound oscillator operating in the independent excitation regime. The amplitude of vibrations of the specimen can be measured using the electronic potentiometer 4. The amplitude of the vibrations of the specimen is recorded by the capacitance sensor 2 powered by the stabilised source 1. The alternating voltage, whose magnitude is proportional to the amplitude of vibrations of the end of the specimen, is transferred from the sensor to the amplifier and the detector 3. The voltage is then further amplified and direct voltage is measured. The voltage is then transferred to the potentiometer. When the movement of paper in the recording device corresponds to the frequency and the movement of the pen to the amplitude of the vibrations, we obtain a recording of the resonance curve. Puskar [158] constructed and tested equipment for the visualisation of hysteresis loops at kilohertz loading frequencies. Consequently, it was possible to evaluate the cyclic strain curves of different materials. The pulsed excitation of the vibrations is based on the moving wave generated in the converter, Fig. 4.18. The pulse from the highfrequency generator with the frequency corresponding to the resonance frequency of the plate of the transducer based on the principle of the inversed piezoelectric phenomenon then induces vibrations in the ceramics or a crystal. The vibrations are transferred through a thin transition layer to the specimen in the direction normal to its 164

Measurements of Internal Friction and the Defect of the Young Modulus

F ig ig.. 4.18. Placing of the transducer on the specimen. 1,2) silver coatings with the transducer between them, 3) transition layer, 4) specimen.

surface. A moving ultrasound wave forms. After interrupting the electric signal passing into the transducer, the ultrasound waves are reflected many times between the walls of the specimen up to complete attenuation of the vibrations. The time to complete attenuation is determined using the transducer, Fig. 4.18, which now operates as a receiver. In the transmission method, one side of the specimen contains the transducer operating as an emitter and the other side the transducer acting as a receiver. The amplitude and analysis of echo signals on the screen of the oscilloscope which is synchronised with the frequency of the pulses makes it possible to determine the internal friction. When the sensitivity of the oscilloscope is sufficiently high, it is possible to determine the delay of various echo signals and, consequently, the speed of propagation of elastic or waves in the material can be determined. The relative decrease of the amplitude of the reflected signals enables the coefficient of absorption of ultrasound in the evaluated specimen to be determined. The method of connection of the reflected echo signals is based on the phenomenon according to which a group of reflected signals enters an oscilloscope synchronised with the frequency of the emitted signals from the oscillator. When the repetition frequency τ s of the synchronised pulses is matched with the measurement time τ the echo signals, i.e. at τ s = τ∆n, where ∆n is the difference between the number of reflections of the combined echo signals, the screen of the oscilloscope shows the image of superimposed pulses of different orders n. The value of τ s can be determined with high accuracy using and electronic frequency meter. The main inaccuracy of the method is associated with the visual placement of the signals on each other on the screen of the oscilloscope which makes it possible to record the change of acoustic delay from 0.01 to 0.05 µs. The accuracy of the method is 0.2–0.5%. 165

Internal Friction of Materials

F ig ig.. 4.19. Diagram of propagation of an acoustic pulse in the examined medium.

The pulse-phase method of measurement of ultrasound is based on the comparison of the phases of ultrasound waves passed through different paths in the examined material, Fig. 4.19, in accordance with Ref. 159. The piezoconverter 1 emits the ultrasound signal 3 into the specimen 2. At the moment of incidence of the reflected signal on the piezoconverter 1 (during 2τ) another reference signal is supplied (4) and, consequently, interference of the signals 3 and 4 takes place in the specimen. The complete attenuation of the vibrations taking place in the profile occurs when

ωn τ +

ϕn π = ( 2 n − 1) , 2 2

(4.30)

where ω n is the circular frequency at attenuation of vibrations, ϕ is the phase shift of the n-th reflected wave, n is a positive integer. The meaning of (2n–1) is determined from two levels for ω n and ω n+1 . In the vicinity of the resonance frequency of the converter where the thickness of the transition layer is smaller than 5 µm, the compensation condition can be simplified to the following form

 ωn Z1  − ,  ∆ω Z2 

( 2n − 1) = 2 

(4.31)

where Z 1 and Z 2 are the acoustic resistances of the piezoceramics and the specimen. Since the time τ is real only at the natural frequency of the converter ω 0 , we use frequencies ω n and ω n+1 closer to ω 0 so that ω n < ω 0 < ω n+1 . Consequently

166

Measurements of Internal Friction and the Defect of the Young Modulus

τ1 =

( 2 n − 1) 2 π 4ωn

and

τ2 =

( 2 n + 1) 2 π . 4 ω n +1

(4.32)

Linear interpolation of τ 1 and τ 2 to frequency ω 0 makes it possible to determine the actual time of propagation of the wave in the specimen τ and, consequently, the phase velocity of ultrasound. The relative error of the method is approximately 0.01%. The authors of Ref. 160 described ultrasound equipment for the measurement of the velocity of sound in solids by the pulse-phase method. The sensitivity to the change of the velocity of sound is 10 –6. This method makes it possible to measure the elasticity moduli of the third order from constants characterising the anharmonic properties of the solid and accompanying phenomena [161], examine anharmonic behaviour of the dislocation structure (162] and other phenomena [163]. The compensation phase method is based on comparison of the phases from two high-frequency signals which have passed through different media, a specimen and a reference liquid [164]. The system has the form of a balanced (with respect to 0) acoustic bridge. It can record very small relative changes of the velocity of sound in the specimen (~10 –6), caused by the effect of external factors. The method with the pulsed, continuous or combined wave emission regime can be used. The acoustic bridge can also be of the self-balancing type. The phase difference formed when the balance of the bridge is disrupted results in a change of the amplitude of the shifted ultrasound signals and, consequently, the frequency of the oscillator is automatically adjusted up to complete compensation of the phase shift. Similarly, small changes of the velocity of ultrasound in the specimen are transformed to the change of the frequency of the reference oscillator; this change can be carried out with a high accuracy. Other methods of the excitation of vibrations, for example, the eddy current method [165], the laser pulse method [166], and others are also available. 4.2.3 Hypersonic methods Microwave ultra-acoustics is a new direction in the solid state physics. The generation of acoustic waves with a frequency higher than 10 9 Hz is difficult in the excitation of higher harmonic vibrations of piezoceramic sheets. However, some authors have managed to measure the absorption of hypersound up to 2000 Hz, for exam167

Internal Friction of Materials

ple, Ref. 167. In fact, high-frequency microwaves are similar to thermal phonons, with the exception of the non-coherent nature of the phonons. For example, the characteristic frequency of thermal phonons of the lattice at 1 K is approximately 20 GHz. Using an X-cut of quartz, place in a superhigh-frequency resonator, Baranovskii [168] observed the generation of waves with a frequency of 10–20 GHz. The diagram of the experimental setup is shown in Fig. 4.20. A quartz horn with the X-cut (or Y-cut) with parallel faces and the diameter considerably smaller than the length of the electromagnetic wave but greater than the length of the acoustic wave was placed in the superhigh-frequency resonator and subjected to the effect of radiowave pulses with a frequency of 10–20 GHz; the duration of the effect of the pulses was approximately 1 µs. Every elementary volume of the piezocrystal, situated in the zone of the effect of superhigh frequency, generates an acoustic wave with the direction normal to the face of the crystal. After leaving the resonator, a planar hypersonic wave is generated in the horn and it can be transferred to the examined object. The echo signal is received at the reversed effect of excitation of the resonator by the acoustic wave. The excitation of shear waves with a frequency of 1 GHz as a result of the ferromagnetic resonance in a thin layer of nickel deposited on the quartz surface was described in Ref. 169. The application of the piezoelectric phenomenon for the generation

F ig ig.. 4.20. Principle of the hypersonic method utilising the piezoelectric phenomenon: 1) resonator for superhigh frequency; 2) ultrasound pulse with a length of 1 µs; 3) crystal; 4) transition layer; 5) specimen; 6) propagation of ultrasound waves; 7) shape of the echo signal on the oscilloscope. 168

Measurements of Internal Friction and the Defect of the Young Modulus

of hypersound in quartz was discussed in Ref. 170, where a layer of Permalloy was used at the end of the horn to make it operate as a receiving transducer. The horn was placed in the super high-frequency resonator and vibrations with a frequency of 8.9 GHz were produced. The development of this method was impaired by difficulties in producing high-intensity magnetic fields. In most cases, hypersonic measurements are taken at low temperatures in order to reduce the extent of damping of the examined material.

4.3 PROCESSING THE RESULTS OF MEASUREMENTS AND INACCURACY The resultant values of the internal friction and the defect of the Young modulus as a basis for evaluating the physical processes taking place in the metals and alloys are influenced by the design of testing equipment and also by the methods used to determine the characteristics. 4.3.1 Inaccuracy caused by the design of equipment Mechanical friction The main requirement when designing testing equipment is to ensure minimum losses in the system. The losses maybe caused by mechanical friction in clamping sections, by the friction of moving components with the environment and by the dispersion of the energy of vibrations in the materials of the components of the system. It is assumed that the losses in the clamping sections are small and can be ignored, and the mass of the specimen is many times greater than the mass in the area of clamping. Using a simple model of a vibrating system, the authors of Ref. 170 determined the ratio of the damping decrement of the entire system δ s to the decrement of the specimen δ v , i.e. δ s /δ v in relation to the ratio of the inertia moments of the fixing section M 1 and the specimen M 2 , i.e. M 1 /M 2 , for three constant values of the elasticity characteristics (q 1/q 2), Fig. 4.21a. The ratio δ s /δ v is selected as 10 3 , which corresponds to the experimental observations. For steel specimens q 1 /q 2 = 5. The effect of the fixing section on the determined value internal friction is determined from an example. The effect of the design factors on the optimisation of low-frequency systems of the type of portional pendulum was evaluated in Ref. 171. The optimisation criterion was the background of internal friction at room temperature. The factor of the effect is determined by the ratio of the rigidity of securing the specimen and the 169

Internal Friction of Materials

F ig ig.. 4.21. Calculated (1–3) and experimental (4) dependences log (σ s, σ v ) on log (M 1 /M 2 ) (a) and nomogram for selecting structural members of equipment with the minimum level of losses in the system (b).

rigidity of the bar moving the specimen (K t /K v , where K = (πd 4 G)/ (32l), d is diameter, l is length), and by the ratio of the rigidity of the suspension wire to the rigidity of the specimen. The tests were carried out on specimens of aluminium, stainless steel and brass. The diameter of the securing bars was 3, 3.5 or 6 mm. The wires were produced from tungsten, nickel–chromium alloy or capron of different diameters. The zero level and the variation range were selected for all combinations. Processing of the results has provided a regression equation including the coefficient of the mutual paired effect. Verification confirmed the accuracy of the model. This was followed by constructing nomograms for selecting design parts of equipment in order to ensure the minimum losses in the system (Fig. 4.21b). Reference measurements were taken on specimens of 25KhNMFA steel. Securing sections made of brass and the suspension bars with a diameter 5 mm were selected from the nomograms. Measurements of the internal friction background showed a decrease of 25% in comparison with the case in which the normal parts of the measuring system were used (clamping sections made of stainless steel, suspension bars with a diameter 3 mm). It should be noted that this improvement of the efficiency of measuring equipment has made it possible to obtain information on the interaction of dislocations with solute atoms. For the equipment based on the principle of torsional pendulum 170

Measurements of Internal Friction and the Defect of the Young Modulus

the ratio K t /K v ≥ 100 is sufficient for obtaining accurate data on the internal friction of materials. Aerodynamic losses are a function of the density of the medium in which measurements are taken, and also of the shape of moving components and the rate of changes of their temperature. Taking into account the fact that the distance of different parts of the system from the axis of rotation differs, and the speeds of these parts are also different, it is necessary to solve a problem of the nonstationary movement of a solid in a viscous medium. In accordance with the calculations carried out by Yu.V. Piguzov [107], the aerodynamic losses by friction with the environment can be expressed by the equation

δa =

8π2αρ S R3ϕ0 , 3K ν T 2

(4.33)

here α is the resistance coefficient whose value at low speeds is equal to the value of the Reynolds number (Re), ρ is the density of the surrounding environment, S is the cross sectional area of the bar, normal to the direction of the vector of the speed of movement v, R is the half length of the inertia arm, ϕ 0 is the maximum angle of rotation of the torsional pendulum, K v is the rigidity of the specimen, and T is the period of vibrations. The results of the calculation show that the losses of this type decrease with increase of the vibration frequency and increase in proportion with the increase of the strain amplitude. This gives the nonlinear dependence of the aerodynamic losses on the strain amplitude in the specimens [172]. It should be noted that these conclusions are valid up to a frequency of 100 Hz. The calculations of losses for high loading frequencies should be supplemented by the effect of the resistance coefficient α and by the contribution of solids with a complicated shape. The aerodynamic losses can be significantly suppressed is the active section of the measuring system is placed in vacuum. The magnitude of the losses decreases asymptotically with decrease of the pressure of the environment, down to the linear dependence at a pressure of 1.33 N⋅m –2 [107].

171

Internal Friction of Materials

4.3.2 Inaccuracies of the measurement method The magnitude of the error of internal friction measurements depends of the selected testing method and the sensitivity of measuring equipment. The method of damped vibrations has a relative error based on the measurement and expressed by the equation derived in Ref. 107

∆Q −1 =

1 (∆Ap + ∆Ak ),  Ap  ln    Ak 

(4.34)

where A p , A k is the initial and final amplitude of vibrations, respectively; ∆A p is the dependence of the magnitude of the relative error ∆Q –1 /Q –1 on the value of the ratio A p /A k. The magnitude of the error when determining this ratio is 3–4%. The logarithmic decrement is the measure of internal friction from Q –1 ~ 10 –1 . A different measurement method must be used at higher values. When evaluating the recording of damped vibrations, the inaccuracies are determined by the accuracy of determination of the range of vibrations (2A i ). At the value of the absolute inaccuracy ±h of measurements of the i-th range of the vibrations, the relative error of calculation of internal friction is determined by the equation [104]

Ai hi+ n − hi Ai+ n −1 . ∆Q =  Ai   Ai hi + hi+ n  Ain 1 − +  1 +  2 Ai +n   Ai +n   Ai +n

(4.35)

The required value 2A i , at which these calculations can be carried out with the error not exceeding the maximum error of the damped vibrations on photographic material and in determination of the range by a slide rule, is h = 0.2–0.3 mm. This means that for internal friction calculations with the inaccuracy smaller than 5% the required difference of the ranges is 8–12 mm. The application of automatic discriminators passing signals proportional to the amplitude of the damped vibrations in the range A i to A i+n and also of counting devices for the determination of the 172

Measurements of Internal Friction and the Defect of the Young Modulus

value of n in the given range or of timers of time periods for n vibrations enables the accuracy of measurements to be increased up to 2.5%. The resonance curve, used for the determination of internal friction, is also characterised by some inaccuracies when processing the results. The inaccuracies are caused by the determination of the frequency of the half width of the curve ∆ω r (ω r is the resonance frequency of the vibrations) from the equation ∆ Q − 1 = ∆ ( ∆ω ) − ∆ω r ,

(4.36)

where ∆(∆ω) is the inaccuracy of determination of the half width of the resonance curve, ∆ω r is the relative inaccuracy of determination of the resonance frequency of the vibrations. The systematic analysis of the inaccuracies in low-frequency experiments showed [124] that at strain amplitude lower than 10 –6 the relative error of measurements as a result of the inaccuracy ∆(∆ω) is ~ 10% and rapidly decreases with increasing strain amplitude. Figure 4.22b shows the inaccuracy of the measurement of internal friction determined by different methods (1 – the method of damped vibrations, 2 – the method of the resonance curve) in relation to the extent of internal friction in the specimen. The resonance method is used in equipment with kilohertz frequencies in measurements of the Young modulus. For example, when using cylindrical specimens, the inaccuracy of determination of the Young modulus in tension is 0.8%, and in torsion it is 0.5%. The inaccuracies of the determination of the Young modulus were evaluated methodically using the instructions in Ref. 173. The method of supplied power depends, as regards the measurement accuracy, on the accuracy and stability of operation of the automatic control equipment, the selection of suitable devices for measurements, the stability of the power source, the adaptability of the feedback system, etc. For the equipment of this type, the relative inaccuracy of internal friction measurements can be determined from the equation

∆ Q −1 = ∆ Z + ∆ C + ∆ N ,

(4.37)

where ∆Z is the error caused by the instability of the amplification factor of the feedback, ∆C is the calibration error, ∆N = ∆Z + ∆N e , 173

Internal Friction of Materials

F ig ig.. 4.22. Dependence of the relative error of measurement of internal friction on the ratio of the initial and final amplitude (a) and internal friction (b).

is the reference voltage. The relative inaccuracy for the equipment assembled on the basis of the principle of the supplied power is approximately 0.5%. The accuracy of the absolute measurements of internal friction is determined by the accuracy of calibration and the scale of the devices for determination of the data. Evaluation of the inaccuracies of internal friction measurements is described in instructions published in the former Soviet Union in Ref. 174. Phase and ultrasound methods require consideration of many special inaccuracies. For example, when using the echo method, there may be inaccuracies caused by non-monochromatic pulses, diffraction phenomena, non-parallel form of the ends of the specimen, the effect of side walls, losses in the transition layer, etc. When analysing inaccuracies, reference should be made to specialised literature [2, 151].

4.3.3 Errors in processing the measurement results When plotting various dependences of internal friction, for example on temperature, there are sources of additional errors caused by thermal instability [107]

∆Q −1 = 0.4 H

∆T , RT 2 q −1

(4.38)

174

Measurements of Internal Friction and the Defect of the Young Modulus

where H is activation enthalpy, ∆T is the absolute error of determination of the temperature peak, q –1 = Q –1 /Q –1max Q –1 is the internal friction at temperature T, Q –1max is the internal friction at T max , and R is the gas constant. The overlapping of the relaxation maxima, observed in the experiments, may exert different effects on the temperature dependence of internal friction. In Ref. 171, the authors presented models and results of experiments carried out in the area of overlapping of the maxima of the relaxation processes, where the following effects are recorded: 1 – superimposition of two relaxation Snoek peaks in iron with different types of solutes, 2 – superimposition of three high– temperature diffusion processes in vanadium, molybdenum, or tungsten, containing solute atoms. In the first case, it is possible to separate the maxima of the temperature-dependent internal friction, where T max = T max – T max1 , and the height of the peaks is 1:0.5 (Fig.4.23a). Figure 4.23b shows the dependence of the resolution of the spectrum for temperature– dependent internal friction (the relationship of the height of the smaller peak and internal friction in the saddle area Q –1s ) on the relationship of the calculated maxima. The resolution power of the spectrum becomes lower with increasing difference in the sizes of the peaks, for example, those corresponding to the concentration of the interstitial impurities in the solid solution. When the ratio of the heights of the peaks is 1:0.8 or 1:1, the general curve shows a plateau. For complicated shapes of the maximum, for example, in the high-temperature region, the resolution power of the spectrum is

100

1.8

80

1.6

60

1.4

40

1.2

−1 Qmax Qs−1

2.0

Q–1× 10–4

120

20

1.0

0 0

40 80 120

0 1

T, °C

2 3

−1 Qmax 1

4 5 6

−1 Qmax 2

F ig ig.. 4.23. Temperature dependence of internal friction ( T m 1 = 313 K, T m 2 = 343 K) for computer-calculated two maxima (a) and the dependence of the resolution of the spectrum on the ratio of the height of the maxima (b). 175

Internal Friction of Materials

considerably lower. This is associated with the superimposition of the high-temperature background of internal friction and appropriate maxima. The systematic calculations carried out in computers show that the decrease of the resolution power of the spectrum is often caused by lower efficiency of the devices and equipment used. The errors in processing the results of measurements of the dependence of internal friction on strain amplitude were evaluated in detailed in the Laboratory of Metals Physics and Strength of the Tula Polytechnic Institute [175]. Figure 4.24a shows the dependence of the decrement of damping on the amplitude of the deviation of the arm of a torsional pendulum for steel 22KhNMFA, obtained from 25 parallel measurements carried out in a RKM-TPI torsional pendulum [112]. Figure 4.24b shows the dispersion of determination of the mean range of the value of the decrement S 2. The scatter of the results of the measurement increases with increasing amplitude. The coefficient of variation v of determination of the decrement of vibrations is divided into two parts by a horizontal broken line (Fig. 4.24b). Above the line v > 5%, below the line v < 5%. A similar result can also be obtained for the alloys, for example, alloys of copper, iron, etc. The application of semiautomatic and automatic systems almost doubles the range of amplitude in which the condition of identical measurement accuracy is fulfilled. In many pure materials and alloys, the dependence of internal

F ig ig.. 4.24. Dependence of the decrement of oscillations δ of annealed 25Kh2NMFA steel on the amplitude of oscillations (a) and the dependence of the variance factor v and dispersion S 2 on the amplitude of oscillations (b). 176

Measurements of Internal Friction and the Defect of the Young Modulus

friction on amplitude is regarded as a linear dependence with the slope determined by the tangent of the angle of slope of the dependence of the part of the decrement of the vibrations in relation to the axis of the amplitude (tan α ≈ α). It is recommended to determine the parameter α by the least squares method which makes it possible to determine markedly not only the value of α but also the dispersion and probability range of the appropriate quantity. When the aim of the experiments is the examination of damping mechanisms, the linearisation hypothesis cannot be accepted but it should be carefully evaluated because otherwise it would be possible to loose a large number of valuable information and the evaluated values of the dispersion and the probability range will be inaccurate. If measurements are taken in the evaluation of kinetic, time or other processes and if it is required to describe the intensity of changes in the dependence of internal friction on the strain amplitude and the visual examination results lead, in the first approximation, to a linear dependence, it is then possible to treat the dependence of internal friction on amplitude as linear. The verification of whether internal friction depends in a linear manner on amplitude is necessary when dealing with a new material for the experimentator and also in cases in which there is no original information on the nature of the dependence of internal friction on strain amplitude. Verification of the linearity hypothesis is based on comparing the dispersion corresponding to the scatter of the mean values of the decrement of the vibrations, in relation to the empirically determined regression line; this makes it possible to determine whether the model is adequate and the dispersion, determined by the error in repetition of the method, can also be determined. The non-adequacy dispersion S 2n can be efficiently calculated from the equation 2   m    n y1  1  m 2  i =1  2 − Sn =  n yi − m m − 2  i =1 n  i =1 

∑

∑

∑

2  m    n ( xi − x ) yi    i =1  ,  m 2 − n ( xi x )   i =1 

∑

∑

(4.39)

where n is the number of parallel measurements of the dependence of internal friction on amplitude taken on the same specimen, or the number of specimens used for measuring the given dependence in the 177

Internal Friction of Materials

selected amplitude range, m is the number of measurement points on the dependence, y i is the mean value of the decrement of the vibrations at the given amplitude, determined from a series of n repeated measurements, y i is the value of the decrement of the vibrations at the given amplitude, calculated from the regression equation, x is the mean value of the amplitude, x i is the actual amplitude. The dispersion of repeatability is determined by the equation m

S 02 =

n

∑∑ y

m

ij

i =1 j =1

  

−n

∑y i =1

 ni  − m i =1  m

−2 i

,

(4.40)

∑

where y i is the change of the decrements selected m times from each curve at n parallel measurements of the dependence of internal friction on amplitude. If repeated measurements are not taken (this is often the case in measurements of the dependence of internal friction on amplitude), the nonadequacy dispersion S 2n can be calculated. In this case, the experimentator plots the results graphically and evaluates them visually. Consequently, the dispersion can be determined using equation (4.39), if the value of α is known; in determination of the probability range, the number of degrees of freedom is selected m = 2. Verification of the linearity hypothesis using this procedure can be carried out only if the dispersion of internal friction in all amplitude ranges is the same. When the dispersion of determination of the decrement of the vibrations depends on the strain amplitude (this is the most frequent case in the measurements), it is also necessary to consider differences in the accuracy of the measurements [176,177]. Special groups of experiments are then prepared for evaluating the repeatability of the method, and the number of parallel measurements in each group is more than 20. The input data are initially verified whether the accuracy is the same, and further statistical processing is carried out using the amplitude range in which the dispersions are equal. The results of measurements of the dependence of internal friction on amplitude, presented in the following part, fulfilled the linearity hypothesis. 178

Measurements of Internal Friction and the Defect of the Young Modulus

The internal friction background of iron, determined on 25 different specimens subjected to tensile deformation of 5% for the formation of an unstable structural state, is characterised by a variance factor of 5%. The results indicate that the measurements of the background are highly reproducible, despite the fact that in many studies it has been shown that the results are highly sensitive to the structural condition. The variance factor in measurements of background on a single specimen with 25 parallel measurements, carried out without dismantling the specimen, was approximately 3% (v = 2.97%). The accuracy of determination of the internal friction background is sufficiently high, and the extent of damping in the first case was ~ 28×10 –4 and in the second case 8×10 –4 . The intensity of damping α after annealing of the specimen was evaluated by measurements. The measurement of the dependence of internal friction on the strain amplitude was taken by two methods. In the first method, internal friction was measured at increasing strain amplitude, in the range in which the linearity hypothesis was valid. The amplitude was always increased by 5 mm and then decreased by 5 mm. In the second method, 20 parallel measurements were taken at the extreme values of the amplitude range, always after 10 measurements at every extremum. The measurement time in both cases was the same. This was followed by standard statistical processing of the results to determine the dispersion and the variance coefficient within the individual groups. In the first case, S 20 = 1.747, in the second case S 20 = 0.903. Comparison of the dispersion carried out using the F criterion showed that the difference in the dispersion was negligible so that the same accuracy was used in all cases. In the first case, v = 3.4%, in the second case v = 2.4%. Since the material has a linear dependence of the decrement on amplitude, the accuracy of determination of α is 2–4%, i.e. slightly higher than the accuracy of determination of the internal friction background. Similar measurements were taken on specimens after 5% tensile deformation. Subsequently, v = 30% in a batch of 20 specimens. This shows that the variance coefficient in determination of internal friction increases when the specimens are in the unstable structural condition. The critical strain amplitude can be obtained simply if we have the experimentally determined regression equation, obtained by the method of least squares, and if we know the level of the internal friction background and the entire course of the dependence of internal friction on strain amplitude. The background is substituted into the regression equation and this gives the value of the first 179

Internal Friction of Materials

critical strain amplitude ε kr1 . The variance factor for ε kr1 , determined from a series of 20 parallel measurements, taken on specimens prior to tensile deformation, was approximately 15%. If the specimens were not annealed prior to measurements, the measurement error ε kr1 was not higher than 5% and for ε kr2 it was 9%. To conclude this chapter, it should be stressed that the efficient selection of the equipment and devices, careful execution of the measurements and evaluation of the results in accordance with the required procedures necessitate a critical review and determination of the measurement accuracy.

180

Structural Instability of Alloys

5 STRUCTURAL INSTABILITY OF ALLOYS The thermal and frequency dependence of internal friction as a result of relaxation is associated with the mean time of movement of atoms in the equilibrium position of the crystal lattice τ and relaxation time τ r . The relaxation time is determined by the nature of processes and is a material characteristic. In this chapter, we present several mechanisms of the relaxation of internal friction used in solving the problems of physical metallurgy and threshold states of materials. 5.1 DIFFUSION MOBILITY OF ATOMS Internal friction measurements make it possible to determine the diffusion characteristics of point defects, their thermal activation, formation of different pairs of point defects and their redistribution under the effect of external stresses. A point defect produces local elastic strains in the crystal and an elastic dipole, which is a tensor of the second order, forms. On the basis of the group theory, Nowick and Berry [66] produced tables which make it possible to determine inelastic interactions for elastic dipoles and the formation of peaks on the Q –1 T or Q –1 f dependence. The best known inelastic interaction is the one determined by the presence of an interstitial atom i in the cubic body-centred lattice. This atom forms tetragonal symmetry and can be located in octahedral or tetrahedral voids. The difference of these tetragonal dipoles can be determined from the absolute value of the shape factor (λ 1 – λ 2) of the deformation ellipsoid [66]. It has been confirmed that the interstitial atoms in iron are distributed in octahedral voids; this is also confirmed by the interpretation of the Snoek maximum. In the cubic fcc and hcp lattices, the interstitial atoms do not form these anelastic formations [66]. A single substitutional atom 181

Internal Friction of Materials

s or vacancy in fcc, bcc and hcp lattices does not form the anelasticity state, because a defect results in distortion whose symmetry is the same as that of the crystal. However, in the formation of the s – s or s – v pair, the anelastic phenomenon may occur.For example, the s–s pair in the bcc lattice forms a tetragonal and in the fcc lattice an orthorhombic dipole with the orientation 〈110〉, and under the effect of external stress they can be mutually displaced. Zener relaxation and the associated occurence of a kink on the curve of the internal friction dependence in the single crystal of α-brass, caused by the initial displacement of a pair of atoms, is a general feature associated with the dissolution of atoms in the substitutional solid solutions. The temperature of formation of the maximum of internal friction and the activation parameters of the relaxation process are connected together by the expression ωτ 0 exp (H/RT), which is also used for the determination of H and τ 0. If the frequency is changed from ω 1 to ω 2 , the internal friction peak is displaced on the temperature axis from T max1 to T max2 . Consequently

H =R

Tmax1Tmax 2 ω ln 2 , Tmax 2 − Tmax1 ω1

ln τ 0 = −

(5.1)

b

g

H Tmax1 + Tmax 2 1 − ln ω 1ω 2 . 2 R Tmax 2Tmax1 2

(5.2)

The calculation of activation enthalpy H with sufficient accuracy requires the changes of ω by several orders of magnitude. When the shift of the maximum with the change of the frequency of oscillations is 20–40°C, the relative error in determining H is approximately 20%. Determination of τ 0 from equation (5.2) using the effective value of H requires analysis of the physical nature of the relaxation process. Otherwise, the physical meaning may be lost. Wert and Marx proposed a suitable equation for further determination of H in the form

H = RTmax ln

kTmax + ∆S , hω

(5.3)

182

Structural Instability of Alloys

where R is the gas constant, k is Boltzmann constant, h is the reduced Planck constant, ∆S is the activation entropy (~10–12 J⋅mol –1 ⋅K –1 ). Equation (5.3) can be used only when the relaxation process is associated with the thermal activation of displacement of the individual atoms by the distance equal to the atomic spacing. Even in this case, it should be expected that there will be a systematic error in determination of H, caused by the difference of the relaxation process and the ideal model according to Debye. The diffusion coefficient, determined by internal friction measurements, does not depend on the thermodynamic factor. Its value can be determined from the equation

a2 D=α , τ

(5.4)

where a is the spacing of the adjacent atoms, α is the geometrical factor with the value 1/24 for the bcc lattice and 1/12 for the fcc lattice, where τ = 3/2τ r for the interstitial solid solution, or τ = τ r for the substitutional solid solution. Basically, it is possible to evaluate not only the mobility of individual atoms and pairs but also of larger clusters of the atoms capable of migrating in the crystal in different dimensions and with different speeds [2]. Determination of the value of D using equation (5.4) and the value of H using equation (5.1) or (5.3) makes it possible to determine the value of D 0 in the equation D = D 0 exp (–H/RT). When comparing the effective coefficient of diffusion from the displacement of the mass, determined from internal friction measurements, the thermodynamic activity of the element is characterised by the correlation factor [2]. 5.1.1 Interstitial solid solutions The fcc, bcc and hcp lattices contain octahedral voids (OV) and tetrahedral voids (TV). The dimensions of these voids in the fcc and hcp lattices are 0.41r for (OV) and 0.225r for (TV). The size of these voids in the bcc lattice is 0.154r and 0.291r, and in the case of the OV it is the minimum spacing between two adjacent atoms in the lattice. The void is asymmetric and the spacing of the four atoms, forming this void, is 0.631r. In evaluation of the internal friction of interstitial solid solutions attention is given especially to the orientation polarization of the paraelastic medium. This polari183

Internal Friction of Materials

zation forms when the introduction of a defect results in disruption of the symmetry which is considerably lower than the symmetry of the lattice. This phenomenon is based on the tendency of the dipole to orient in the stress field in such a manner that its energy is minimum. The symmetry group of the tensor of the dipole is the subgroup of the symmetric group of the crystal, and the following inequality must be fulfilled

nt =

hk ≥ 1, ht

(5.5)

where h k and h t are the numbers of operations of symmetry for the crystal and for the tensor, n t is the number of possible operations of the dipole in the lattice. The condition m t ≥ 1 is essential but insufficient for relaxation to take place. This model makes it possible to determine the boundary of application of internal friction measurements for a homogeneous solid solution [178]. The shape factor varies in the range from 0.01 to 1.0. If it is taken into account that the sensitivity of internal friction measurements is 10 –4 to 10 –5 , it can be seen that it is possible to record changes of the solid concentration range from 10 –4 to 10 –3 . If the measurement conditions are not favourable, this range varies from 0.1% to 1.0%. The shape factor is related with solubility and increases with decreasing solubility. When evaluating the relative change of concentration, the sensitivity of the method, based on internal friction measurements, is a constant value in a wide concentration range. The Snoek relaxation is the best known phenomenon caused by the diffusion of interstitial atoms in the bcc lattice. Snoek [179] observed this phenomenon in iron at a frequency of approximately 1 Hz at temperatures close to room temperature. The shape of the peak is similar to the Debye dependence (Fig. 5.1). The formation of the peak is associated with the migration of interstitial atoms (carbon and nitrogen) and of the effect of the external stress in octahedral positions. In the diffusion of oxygen, nitrogen and carbon in iron, this mechanism has been confirmed by experiments. For the metals with the bcc lattices, the interstitial atoms can migrate in the OV and also TV [708]. These types of migration are also referred to as Snoek’s relaxation. This relaxation is controlled by the thermally activated jumps of atoms. The relaxation time can be determined using the Arrhenius 184

Structural Instability of Alloys

F ig ig.. 5.1. Dependence of internal friction on temperature for tantalum with 0.013 wt.% C at two loading frequencies.

equation in the form τ r = τ 0 exp (H/kT). Internal friction measurements are usually taken at a constant frequency. If we disregard temperature changes ∆(T) and ∆(T) at ωτ = 1 and T = T max , we can use the equation

    ∆ H −1  , Q (T ) = sech  1 2  1  k  −    T Tmax  

(5.6)

where ∆ is the degree of relaxation of the elasticity modulus (at ωτ = 1 Q –1max = ∆/2). In the exact procedure, the experimentally determined Snoek maximum does not correspond to the position of the peak at ωτ = 1 and corresponds to the condition (ωτ) 2 = (1 + kT/H)(1 – kT/H), because ∆ ~ 1/T [180]. Therefore, 185

Internal Friction of Materials Ta b le 5.1 Parameters of Snoek relaxation in bcc metals and interstitial solute atoms

S yste m

H· 1 0 1 0 (J)

τ 0· 1 0 15 (s )

T max a t 1 Hz (K )

N b–O N b–N Ta – O Ta – N V– O V– N C r– N Mo – N F e–C F e–C F e–N F e–N F e–N N b–C Ta – C V– C

1.486 0.01 2.520 0.03 1.769 0.01 2.664 0.03 2.064 2.512 1.904 2.080 1.331 1.294 1.275 1.217 1.273 2.256 2.768 1.952

2.65 0.8 1.22 0.8 8.56 2.5 3.64 2.0 0.96 0.51 1.44 11 5.77 13.6 4.66 18.0 5.72 2.08 20.8 2.08

422 562 419 615 458 544 429 530 312 312 296 296 298 512 627 443

c0 ν0 (δα ) F ( g ) 2

∆=α

kT M −1 ( g )

,

(5.7)

where c 0 is the concentration of the interstitial atoms in the solid solution, v 0 is the atomic volume, δλ = λ 1 – λ 2 is the resistance of the dipoles (the shape factor), g is the geometrical factor of the main stress axis, α = 4/3, M = G, F(g) = g (in torsion), but α = 2/9, M = E, F(g) = (1 – 3) g (in tensile loading). The quantity τ 0 = ω 0 [∆S/kT], where ∆S is the activation entropy, ω 0 = 6υ (υ is the Debye frequency). According to Wert and Marx [181], one can confirm, with good approximation, the linear relationship of the temperature of occurrence of the Snoek maximum τ max and activation enthalpy H using the equation

ω  H = Tmax k ln  0  + Tmax ∆S = −Tmax k ln ( ωτ0 ).  ω

(5.8)

Table 5.1 shows the generalised data on the Snoek maximum (H, τ 0) in the interstitial solid solutions obtained by measurements of the temperature dependence of internal friction [180]. The published re186

Structural Instability of Alloys

F ig ig.. 5.2. Dependence of activation enthalpy on temperature of formation of Snoek relaxation peak.

sults contain only data on the experiments in which the number of interstitial atoms and the resultant dependence of the height of the Snoek peak on the solute content were determined by analysis. According to Weller [180], the value of the relaxation parameters corresponds to the experiments with displacement as a result of the change of frequency (for example, for tantalum H = 1.92 × 10 –19 J and τ 0 = 7 × 10 –5 s [182]). Using equation (5.8) at a low loading frequency (~1 Hz), we can calculate the temperatures of formation of the Snoek maximum and plot the H(T) graphic dependence shown in Fig. 5.2. After extrapolation for the case H = 0, T = 0, we have H 2 = 4.4 × 10 –23 ⋅T max (J). If H = 0 at T max = 0, H 1 = 4.5 × 10 –23 ⋅T max – 8.8 × 10 –21 (J), and this value corresponds to τ 0 = 2.08 × 10 –15 s = const. The solubility of carbon in niobium, tantalum and vanadium is very low. This complicates experiments because of the need to use superpure metals and dope the material with low concentrations of carbon [180]. Snoek peaks are recorded at 525 K (ω = 2.1 Hz) in the Nb – C system, at 642 (2.2 Hz) in the Ta – C system, and at 454 K (1.93 Hz) in the V – C system. Details on the temperature at which the Snoek maximum is recorded are obtained for nitrogen and oxygen in, for example, the V – N system (T max = 540 K), V – O (T max = 455 K), Nb – N (T max = 559 K), Nb – O (T max = 418 K), Ta – N (T max = 616 K) and Ta – O (T max = 415 K). The data on the Snoek peaks for the metals of group VIa are pre187

Internal Friction of Materials Ta b le 5.2 Parameters of Snoek relaxation for metals of group VIa P eak

P a ra me te r

C r– C

Exp e rime nta l

T max (K ) f (Hz)

413- 415 0.7–1.35

C a lc ula te d

T max (K ) H· 1 0 1 9 (J)

417 1.84

C r– O

582 2.57

Mo – C

Mo – O

W–C

600 –1

475 –1

653÷683 0.4÷1

580 2.56

488 2.16

708 3.12

W–O

W–N

378 1.66

450÷560 1.99÷2.48

sented in Table 5.2. Difficulties in experiments are formed not only by the low solubility of interstitial atoms (C, N, O), but also by defects of the actual spectrum of the dependence of internal friction and temperature as a result of the formation of Snoek and Köster peaks (production of alloys maybe accompanied by deformation, e.g. annealing in a gas atmosphere followed by rapid cooling). The possibility of the formation of a Snoek peak as a result of the migration of boron atoms in the solid solution of α-iron has been discussed in the technical literature. Experiments yielded peaks on the Q –1(T) dependence [183], widening of the initial peak caused by carbon migration (T max = 30°C at 1 Hz) [184], and also the formation of unstable peaks (T max = 13°C at 1–5 Hz) [185]. Later, systematic experiments [186] showed that the addition of boron to iron does not result in the formation of a new peak caused by the migration of boron to the interstitial positions in α-iron. No widening of the peak, associated with carbon migration, was detected. For the interstitial solid solutions of metals with the bcc lattice, the diffusion coefficient at T = T max is determined by the equation

D=

a2 , 36 τr

(5.9)

where

τr =

1 ωmax

=

1 . 2π f max

Measurements of the diffusibility of atoms with the evaluation of Q –1 are in agreement with the results obtained by other methods, Fig. 5.3a. The set of points at higher temperatures corresponds to 188

Structural Instability of Alloys

F ig ig.. 5.3. Temperature dependence of the coefficient of carbon diffusion in alpha iron. 1) low-temperature range; 2) high-temperature range; 3) experimentally determined curve; 4) calculated according to Wert and Marx.

the results obtained in other methods, for medium temperatures it corresponds to the method of measurement of Q –1 and at low temperatures to the elastic loading method. Processing of the results yielded the equation for carbon diffusion in α-iron in the following form

 84  DC = 0.02exp  −   RT 

(5.10)

and for nitrogen in α-iron

 76  DN = 0.003exp  − ,  RT 

(5.11)

where activation enthalpy is in kJ⋅mol –1 , and the dimension of the diffusion coefficient is cm 2 s –1 . Figure 5.3b shows a more accurate determination of the dependence D C (T). One of the curves was obtained on the basis of empiri189

Internal Friction of Materials

cal dependences, the other one was calculated according to Wert and Marx from the data on internal friction. If the slope of the straight line in Fig. 5.3a is modified in such manner as to describe the low– and high-temperature measurements, the straight line 4 in Fig. 5.3b reflects accurately the results of measurements of Snoek’s peak. The reason for the difference of the data obtained in the high-temperature range was described in detail in Ref. 2 and is based on the occurrence of an additional activation process, causing a deviation from the Arrhenius relationship, and on the effect of vacancies. The data on the diffusion characteristics in transition metals can be supplemented by analysis of the results presented in Table 5.1 and 5.2. Consequently Ta

Cr

Mo

W

D0 (c m2 s–1)

0.0018

0.009

0.0013

0.014

H (k J mo l–1)

104.6

109.2

160.0

182.2

The experimental data on the activation entropy ∆S e can be obtained from the equation D 0 = p αδ 2 υ exp (∆S/R), where p is the number of equivalent diffusion paths, α is the geometrical factor, δ is the magnitude of the jump of the atom, υ is the frequency of vibrations of the atoms. Wert confirmed good agreement of the values of ∆S e and ∆S theor for carbon and nitrogen in α-iron and oxygen and nitrogen in tantalum, as stated in Ref. 2. At present, a large number of data is available on the diffusion characteristics in interstitial solid solutions with the bcc lattice obtained from the parameters of the Snoek maximum [187]. The strength of the effect of interstitial atoms on the thermodynamic properties and the diffusion coefficient of the interstitial atoms in the ternary solid solutions based on niobium changes in the following sequence: Ta, Ti, W, V, Hf, Zr, Cr, which corresponds to the increase of the difference of the dimensions of the interstitial atoms and niobium and the energy of the deformation interaction of the s – i atoms [188]. The binding energy of the s – i complexes is higher than that of the i – i complexes. This results in complete absence of the i – i complexes in the ternary alloys at any temperature. Details of the Snoek relaxation in the concentrated solid solutions based on Ta, niobium and vanadium have been published in Weller’s study [180], and the data on the anisotropy of Snoek’s phenomenon 190

Structural Instability of Alloys

F ig ig.. 5.4 5.4. Model for explaining the Finkelstein–Rozin phenomenon (A is the iron atom, V the foreign atom, e.g. substitutional or interstitial atom or vacancy).

in niobium with oxygen and nitrogen are in Ref. 189. The relaxation according to Finkelstein and Rozin was observed in 1952 in an austenitic steel. The height of the peak is directly proportional to the carbon content of the solid solution and the activation energy of the peak is close to the activation energy of carbon diffusion [190]. These relationships were confirmed for nickel, Ni–Al alloys and a large number of austenitic steels [191,192]. Initial studies already showed the extensive possibilities of utilising the Finkelstein–Rozin (FR) phenomenon in the examination of dissociation of carbides in austenitic steels and also in analysis of phase transformations in austenitic–martensitic steels and metals with the fcc lattice. Kê et al showed that the formation of the FR peak is determined by the reversible movement of carbon atoms (Fig. 5.4), because the carbon atom occupies the vacancy V and forms a pair with the carbon atoms in the position 2 or 2'; possibly, two atoms are situated in the positions 2,2', and the vacancy is in the position V. The concentration dependence of the parameters of the Finkelstein–Rozin peak for different alloys is approximately the same [193]. At low concentrations, the height of the peak increases in proportion to c 2 , and at higher concentrations the height of the peak is proportional to c, where c is concentration. For manganesecontaining alloys, the parabolic region of the concentration dependence is less significant because the peak is detected only at concen191

Internal Friction of Materials

trations higher than the critical value c 0 . The transition from the parabolic to linear dependence is characterised by the concentration c k . Its level differs for different alloys (usually c k > c 0 ). When approaching the solubility limit, the concentration dependence starts to deviate from linear. Increase of the amount of the second phase may result in a decrease of the height of the Finkel’stein–Rozin peak with increase of the concentration of interstitial atoms. Verner [193] confirmed the absence of any strong effect of substitutional atoms and vacancies on the FR phenomenon. This makes it possible to understand the relaxation mechanism associated with the rotation of a pair of interstitial atoms under the effect of external stress. Evaluation of the probability of formation of these pairs, formed by the interstitial and substitutional atoms, was carried out using the data in Table 5.3 expressed in fractions of interaction in the coordination sphere. In all cases, repulsion is detected in the first two co-ordination spheres. This shows why pairs of the nearest neighTa b le 5.3 Energy of interaction of pairs of atoms in fcc alloys in relation to the first coordination sphere Inte ra c tio n o f p a irs o f a to ms Inte rstitia l Inte rstitia l a nd sub stitutio na l S ub stitutio na l

C o o rd ina tio n sp he re 1

2

3

4

5

6

+1.00

+1.10

–0.34

–0.33

–0.45

0.16

+1.00 +1.00

+1.10 +1.80

+0.90 +0.90

+0.60 +0.80

+0.70 +0.20

0.00 –0.10

bours do not form. In the case of interstitial solid solutions, pairs can form between the atoms of the third and fifth co-ordination sphere. On the basis of the results it may be assumed that relaxation in the interstitial solid solutions takes place as a result of the formation of pairs of interstitial atoms distributed in the third and fifth co-ordination sphere. Alternating loading results in the redistribution of the pair by the transition of the interstitial atom from one co-ordination sphere to another. According to Verner [193]:

192

Structural Instability of Alloys

2

−1 Qmax

 dλ  2  W  E  c exp   dN m  kTmax    , =   W   9 N 0 kTmax  24 c exp   + 1 k T max    

(5.12)

where

1  dλ  N0    dNm  is the change of the distortion of the lattice during rotation of a single pair from the direction parallel to the loading axis to the direction normal to this axis, N 0 is the number of voids in the unit volume, W is the energy of interaction of the interstitial atoms. The relationships for the vacancies are reversed and, in this case, pairs can form around the vacancies in the adjacent co-ordination spheres. In the case of the interstitial and substitutional atoms, attraction can take place from the sixth co-ordination sphere. However, this pair will not change its orientation and no phenomenon will be recorded. When the concentration of interstitial atoms is low, then 24 c exp (W/kT max ) << 1, we obtain that

 W  −1 Qmax = 6β c2 exp  ,  kTmax  where 2

1 E  dλ  1 . β=   9 N 0  dN m  kTmax At higher concentrations Q –1max = β/4c. The transition from the quadratic to linear concentration dependence is expressed less accurately by the equation 24c 0 exp (W/kT max ) ≈ 1. In substitutional solid solutions, which greatly differ by the atomic size from the solvent, the phenomenon of suppression of the process of formation of the in193

Internal Friction of Materials

terstitial pairs takes place until the concentration of the interstitial atoms reaches the value at which in the interstitial atoms are distributed in the fifth and lower coordination sphere. The critical concentration is approximately 0.1–0.25 wt.% C (or N), and the experimental value of c 0 is also in this range [178]. Relaxation in the hcp metals, observed on the Q –1 (T) dependence, has not been sufficiently quantified. More systematic analysis has been carried out in the case of solid solutions α-Ti. The maximum on the Q –1 (T) dependence in titanium is observed only in cases in which the distribution of the substitutional atoms in the solid solution [194] is such that the individual interstitial atoms situated in the octahedral voids cause distortion with the symmetry identical with that of the lattice and no dipoles appear. A dipole forms when the interstitial atom is in the vicinity of the substitutional atom. The height of the peak on the Q –1 (T) dependence increases with increase of the difference of the size of the atoms of titanium and the substitutional atom. At a constant content of the alloying substitutional element (for example, zirconium), the peak grows in direct proportion to the increase of the atomic concentration of the interstitial atoms (oxygen). In the case of a constant content of the interstitial atoms, the dependence of the height of the peak on the concentration of substitutional atoms is not unambiguous [178]. For example, in alloys of titanium with zirconium, the height of the peak of internal friction is maximum at low zirconium content, decreases with increasing zirconium content, passes through a minimum at 0.5 at% Zr (for an alloy with 1 at.% oxygen). The position of the minimum corresponds to the line situated in the ternary Ti–Zr–O diagram in the Zr–O relationship, corresponding to compound ZrO 2 . Measurements of the Q –1 (T) dependence and of the activation energy of the processes which enables the formation of the peak on the Q –1 (T) dependence at different zirconium content show in the Ti–Zr–O system the formation of an ordered structure, preceding precipitation of the new phase. Similar phenomena can be detected in the Mg–Cd substitutional solid solutions in the formation of the ordered phase Mg 3Cd in them [195] and also in the Ti–O–Al system for alloys with the ratio of the aluminium and oxygen content close to Al 2 O 3 . The peak of the dependence Q –1 (T) for these alloys is completely suppressed. The absence of data from the measurements of the Q –1 (T) dependence on single crystals of solid solutions with different concentration complicates the application of the rule of selection for determination of the symmetric of the dipole in the hcp lattice. Despite this 194

Structural Instability of Alloys

F ig ig.. 5.5. Temperature dependence of internal friction for brass ( f = 620 Hz).

fact, examination of the solid solutions by measurement of the Q –1 (T) dependence is highly promising and useful. 5.1.2 Substitutional and solid solutions At comparable concentrations of components of alloys of fcc, DCC and hcp metals and the formation of substitutional solid solutions we can observe relaxation caused by changes of the relative distribution of the atoms in the alloy under the effect of external loading. As mentioned previously, Zener detected, for 30% brass, a peak on the Q –1 (T) dependence which was attributed to the change of the orientation of the axes of the isolated pairs of adjacent atoms of solutes under shear loading (Fig. 5.5) [196]. At low solute concentration the height of the relaxation peak is directly proportional to the square of the concentration of the dissolved element. The theory of relaxation of the pair loses the meaning for higher concentrations of the dissolved element, where the assumption on the separated pair is not justified. For solid solutions with higher concentration, where the solute atoms form complexes, Le Claire and Lomer [197] proposed a model which takes into account the change of the shot-range order in alloys after loading the material (the model of ‘directional shot-range ordering’). Zener relaxation is caused by the spatial migration of the atoms 195

Internal Friction of Materials

of the solution and this results in changes of the number of pairs of atoms with different orientation or of the shot-range order parameters in the solution. Consequently, it is possible to examine, for example, the microscopic characteristics of the ordered alloys. The diffusion characteristics and thermodynamic activity of a large number of substitutional solid solutions have been measured by different authors [2]. For example, for the Ag–15.8–30.2 at.% Zn system, these characteristics were determined by Nowick, for the Al–0.01–0.5 at% Mg by Green and Pavlov, for the Ag–32 at% Cd and Au–32 at% Ag systems by Terner and Williams, etc.. The question why the Zener peak broadens has not as yet been answered. There are two extreme cases: a) there is a spectrum of the relaxation times, with each time depending on the temperature in accordance with the Arrhenius equation; b) the temperature dependence of the individual relaxation times differs from the Arrhenius dependence. Nowick and Berry [66] assumed the formation of spectral relaxation times characterised by the normal Gaussian distribution of the parameters

β = β0 +

βH , kT

(5.13)

where β 0 and β H are independent of temperature and determine the half width of distribution of ln τ 0 and H 1 . After calculating the dispersion of the distribution β c, the fluctuation of concentration in the alloy with the approximation according to Gorsky, Bragg and Williams and after relating the quantities β H and β c by the equation β H = (dH/dc m)β c , we obtain the equation

 1 dH  zW 1 βH = + −  n dcm  kT 2 cm (1 − cm ) 

−1/ 2

,

(5.14)

where n is the number of atoms in the region of concentrational heterogeneity, W is the energy of the ordered alloy, c m is the mean concentration of one of the components, H is the mean activation energy, z is the co-ordination number. Therefore, the spectrum of the relaxation times is justified by the fluctuation of the concentration in the alloys. 196

Structural Instability of Alloys

Zener relaxation is controlled by the kinetics of changes of shotrange ordering in the alloy [198]. The depth of the potential well is determined by the energy of interaction of the migrating atom with its closest neighbours. The height of the potential barrier does not depend on the atoms surrounding the potential well and is determined only by the type of migrating atom. This approach gives equations for the relaxation time of Zener relaxation. The value of the activation energy of Zener relaxation is between the values of the activation energy of diffusion of the components of the alloys. Many theoretical studies take into account the kinetics of changes of shot-range ordering for different cases of loading and deformation in fcc crystals [195], changes of shot-range ordering resulting from the effect of temperature [200] and also the correlation functions characterising the order around the vacancies [201]. The calculated and experimental values of the diffusion parameters are close. However, coefficients, not reflecting the physical nature of the process, are quoted in certain cases. 5.2 RELAXATION OF DISLOCATIONS The Q –1 (T) dependences of the metal than alloys show, even under the effect of very low stresses, anomalies in the changes of internal friction with changes of temperature. The activation energy of the processes controlling the movement of dislocations in the crystal for the maturity of material is not higher than 2.4 × 10 –19 K. The processes associated with dislocations relaxation in repeated loading of the material with the frequency in the hertz and kilohertz range take place at temperatures lower than 500–600 K, i.e. below the condensation temperature of Cottrell atmospheres T C. This means that in the range in which the internal friction depends on the state amplitude, all types of dislocation relaxation can take place only as a result of the movement of fresh, non-pinned dislocations formed during prior plastic deformation or phase hardening. The condition of prior plastic deformation from the processes to occur causes that these peaks are regarded as the strain maximum. 5.2.1 Low-temperature peaks The main phenomenon of dislocations relaxation in pure metals is the Bordoni relaxation [202] whose main features were described in section 3.2.2. In loading in the range of kilohertz frequencies the Bordoni peaks are recorded at low temperatures (~ 100 K). The Bordoni peak does not change during annealing up to the recrystallisation temperature. The increase of the content of the al197

Internal Friction of Materials

loying additions increases the height of the peak, even if the temperature at which this is recorded does not change. However, the change of the vibration frequency results in changes of the temperature at which the Bordoni peak is observed, but the shape of the peak remains unchanged. This peak is also slightly influenced by the strain amplitude. The activation energy of the relaxation process varies from 8 to 10 kJ×mole –1 and τ 0 = 10 –1 – 10 –12 s [203]. The characteristic feature of Bordoni relaxation is that the height of the peak is almost an order of magnitude higher than in the relaxation peaks with a single relaxation time. At temperatures lower than room temperature, the fcc metals subjected to plastic deformation undergo other relaxation phenomena, differing from Bordoni relaxation, even though their occurrence is explained on the basis of the assumption on the movement of bows in the dislocations. To explain the occurrence of peaks at low temperatures, but higher than the Bordoni peak, the author of [204] proposed an interpretation based on the assumption that in certain cases the relaxation may be the result of breaking of diffusing dislocation bows by point defects, especially vacancies. Different types of the defects then determine the formation of several peaks. 5.2.2 Snoek and Köster relaxation The characteristic peak of the Q –1 (T) dependence for deformed iron with the nitrogen content was detected by Snoek in 1941, at a temperature of 200 °C and at a loading frequency of the material of 0.2 Hz. Further experiment were carried out by Köster and many other authors [205, 206]. They investigated the main characteristics of the peak, the need for the presence of a small amount of nitrogen atoms in the solid solution and the need for prior deformation ε p . The relaxation nature of the process was already justified in this stage, despite the fact that the width of the peak is significantly greater than that of the Debye peak with a single relaxation time. The activation energy of relaxation in iron alloys containing carbon or nitrogen is 127–168 kJ⋅mol –1 , and the frequency factor is τ ≈ 10 14 s –1 (for iron with 10 at.% C after 2% cold deformation and T max S–K = 557 K and after 10% cold deformation T max S–K = 545 K). The temperature at which the S–K peak is observed in iron alloys at 0.2–1 Hz is approximately 200–250°C. Later, in subsequent investigations it was shown [207,208] that the peak with this characteristic is recorded in different bcc metals (iron, tantalum, vanadium, niobium, molybdenum, tungsten) if they contain the atoms of carbon, nitrogen, oxygen and hydrogen in the 198

Structural Instability of Alloys

F ig ig.. 5.6. Relationship of the height of Snoek and Köster peaks ( Q –1SK ) and Snoek peak ( Q –1 ) in ferrous alloys. S

solid solution. This peak is referred to as the Snoek–Köster maximum or peak (S–K peak). The main conditions for the formation of the S–K peak is the presence of interstitial atoms in the solid solution of the material, prior deformation of the material and the application of heat treatment under certain conditions [209]. Usually, it is concluded that the height of the S–K peak (Q –1 ) increases with the increase of the S–K content of interstitial atoms in the solid solution C 0 (Fig. 5.6). This hypothesis should be treated with care because the authors carried out deformation in the condition after strengthening by heat treatment, in order to determine C 0 from the measurement of the Snoek maximum. It is evident that this was reflected in the redistribution of the interstitial atoms between the precipitates and the atmospheres around the dislocations (Fig. 5.7a). The concentration range of the formation of the S–K peak was determined in practice [209] in alloys of iron with carbon using the Fibonachi method taking into account the actual state of the alloy prior to deformation. At ε p ≈ 20%, the dependence Q –1 (C 0 ) shows S–K an extremum in the range of the low initial carbon content in the solid solution (~9 × 10 –4 wt.% C). An analytical procedure was used to determine the optimum value of ε p for the maximum height of the S–K peak (for Armco iron ε p ≈ 55%, Fig. 5.7b). This shows that the 199

Internal Friction of Materials

0.001 wt.%C

1 F ig on strain (a), the duration of prior a geing at ig.. 5.7. Dependence of Q –SK 200 °C prior to 20% deformation of iron with 0.015 wt.% C (b), on temperature in heating (c) and on holding time at T = T max (d) for quenched and deformed (20%) specimens with 0.015 wt.% C.

dependence Q –1S–K ~ ε 1/2 is not valid generally [205]. The maximum p height of the S–K peak for iron with 0.015 wt.% C after 20% deformation of the specimens is obtained at 220°C after one hour. It should be taken into account that it is necessary to determine the condition of the atmosphere of solutes during the formation of the carbon S–K peak. There are differences in the effect of nitrogen and carbon on S–K relaxation. According to Snoek, during measurement of Q –1 in loading with hertz frequency of nitrogen-alloyed iron we obtain a high, complete peak. In the case of carburised iron, the height of the S–K peak is initially small and after heating the material to the temperature range T ≥ T max S–K this difference may increase [209]. Similar observation can also be made when heating the Fe–C alloy for short periods of time to 300°C. At the same concentration of the solid solution, the height of the Q –1S–K peak is approximately 4–5 times higher than in the case of the Fe–N alloy. The boundary concentration for obtaining the maximum value of Q –1S–K in nitrogenalloyed iron is approximately an order of magnitude higher, even 200

Structural Instability of Alloys

though the effective binding energy of C and N with the dislocations in ferrite is similar (~1.28 × 10 –19 J). The residual peak, observed at zero nitrogen content in the solid solution, was described in Ref. 210. Its occurrence is associated with the partial breakdown of nitrides during plastic deformation [211]. The characteristics of the S–K peak depend on the type of deformation and are determined not only by the density of fresh dislocations but also by their type and configuration [212]. In 1963, Shoeck [213] proposed the quantitative interpretation of the S–K peak. The model is based on the dragging of the atmospheres of solute atoms by dislocations moving under the effect of repeated external loading. The dislocations are regarded here as the model of a spring and the interaction of the dislocations with the solute atoms is interpreted as the unperturbed distribution of the solutes in the Cottrell atmospheres characterised by viscous movement. Consequently

QS−−1 K (T ) = A ρ lc2

∞

ωτz 5 f ( z ) dz, 2 2 4 + ω τ 1 z 0

∫

(5.15)

where A is the coefficient of proportionality, ρ is the density of moving dislocations taking part during the formation of the S–K peak, l c is the mean value of the length of the dislocation segment, f(z) is the function of distribution of the length of the dislocation segments l, z = l/l c , and the value τ=α

kT c d lc2 , G b3 D

(5.16)

where α is the proportionality factor, c d and D are the concentration and diffusion coefficient of interstitial atoms in the vicinity of the dislocation kernel. Equation (5.15) shows that the maximum of internal friction is characterised by the spectrum of relaxation times. The distribution of the lengths of the dislocation segments can be described according to Keller by the equation f(z) = e –z . Consequently, the integral in equation (5.15) in the Shoeck model has the maximum value of 2.2 at ωτ = 0.07 (in the Debye process ωτ = 1). The dependence Q –1 (T) in the Shoeck model is characterised by the dependence c d (T) 201

Internal Friction of Materials

and D(T), and D = D 0 exp(–Hd/kT), where H d is the activation energy of the interstitial atom in the vicinity of the dislocation. The dependence c d(T) depends on the selected model of the atmosphere around the dislocation. Several models of the S–K relaxation have been proposed for the solid solutions with lower and high concentration of solutes, since the distribution of the solute atoms in the atmosphere of the dislocations is modelled by the Boltzmann, Fermi or Dirac approach [214, 215, 216, 217]. The relaxation in the presence of hydrogen is still the subject of special interest [218]. The experiments show that the presence of hydrogen suppresses the low-temperature α and β peaks on the Q –1 (T) dependence, caused by the formation of a double bow simultaneously on the mixed and screw dislocations [215], with the formation of the S–K peak. S–K relaxation is the confirmed phenomenon of the Q –1 (T) dependence at temperatures of 80–170 K indicating the interaction of the hydrogen atoms with a dislocation. The mean value of the activation enthalpy of the relaxation process is H ≈ 31.8 s –1 . The results show that the values are comparable with the values resulting in the formation of S–K peaks in the Fe–C and Fe–N systems. Wirth [218] stressed the restrictions of the Shoeck model (Fig. 5.8), when describing the relaxation interaction of hydrogen with the dislocation under external loading. For example, the activation enthalpy H = 31.8 kJ⋅mol –1 does not correspond to the enthalpy of bonding of hydrogen with the dislocation (H H= 58.6 kJ⋅mol –1 ). An alternative model is based on the bending of the double bow of the dislocation atmosphere, formed by the hydrogen atoms under external loading (Fig. 5.8b). The relaxation time of the bent dislocation can be expressed by the equation

τ=

2l 2 , π2 B G b2

(5.17)

where l is the length of the dislocation segment, B is the dislocation mobility parameter. When the dislocation travels the distance x < 1, the following equation applies:

202

Structural Instability of Alloys

F ig .5.8. Models of bending of a dislocation with the atmosphere formed by hydrogen ig.5.8. atoms (a) and kinks formed in double bow (b): 1) without stress; 2) under stress; 3) 3,4) dislocations with a bow without hydrogen (3) and with hydrogen in dislocations (4); 5) bending of a kind with the formation of a double bow.

B=

 F*  2bDk exp  − k  , kT  kT 

(5.18)

When the dislocation moves through the distance x > 1

B=

 2F *  l Dk exp  − k  , kT  kT 

(5.19)

where D k is the diffusion coefficient of the bow, 2F *k is the free energy required for the formation of a double bow with the critical size. Since in the case of the schema in Fig. 5.8b–3, it is not possible to correlate the experimental data, we can use the schema of bending (Fig. 5.8, b–4) and obtain the value of the frequency factor irrespective of the value of H H τ0 =

2 l kT . π ν Gb 4

(5.20)

2

At T = 120 K υ = 10 13 s –1 and, therefore, τ 0 = 2.6 × 10 17 l/b, which is in agreement with the experimental observations, since l/b = 10 3 to 10 4 [219]. The model of bending of the double bow with the atmosphere generated by hydrogen indicates that the absorption of hydrogen suppresses the first peak (α) and displaces the second peak (β) of internal friction to lower temperatures. 203

Internal Friction of Materials

wt.%C

wt.%C

0.9 wt.%C

F ig ig.. 5.9. Dependence of the peak formed at 250°C ( Q –1 or Q –1SK ) of quenched max carbon steels on carbon content (a, b), quenching temperature (c) and subsequent tempering (d) at a carbon content (wt.%C) of: 1) 1.16; 2) 0,92; 3) 0.71; 4) 0.32; 5) 0.12 ( f = 1 Hz).

5.2.3 Phenomena associated with martensitic transformation in steel Since martensitic transformation took place in the steel, the dependence Q –1 (T) shows a peak similar to the S–K peak. The general relationships governing the formation of the peak recorded at 250°C, at a loading frequency of ~1 Hz, in the specimens of carbon steels subjected to quenching and tempering were presented in Ref. 220 and 221. The height of the peak increases with the increase of the carbon content of the steel and with increasing quenching temperature (Fig. 5.9), and T max is displaced higher temperature. The increase of the tempering temperature of steel decreases the height of the peak. Many authors interpret this peak on the basis of models used to explain the Snoek and Köster peaks. However, this relaxation process has certain different features. Alloying elements, influencing the bonding of dislocations and interstitial atoms and changing the breakdown of cementite during 204

Structural Instability of Alloys

heating, also change the parameters of the peak of the Q –1 (T) dependence. In steels alloyed with several elements there are also other additional peaks [222]. Special attention should be given to the data on the origin of the peak recorded at 160°C in quenched Fe–Ni–C materials. The authors of Ref. 222 showed that the formation of the peak is associated with the movement of twin boundaries containing moving carbon atoms, when the material is subjected to alternating external loading. Twinned martensite makes it possible to detect a peak at a temperature of 160°C on the Q –1 (T) dependence at a frequency of ~1 Hz in alloyed steels, dislocation martensite at 250°C, and mixed martensite shows both peaks on the Q –1 (T) dependence. The Snoek and Köster phenomenon and its analogy in the quenched carbon steels are only some of several interesting representations of the interaction of solute atoms with the dislocations. It can be expected that further work in this area will provide more detailed relationships between the value of internal friction and the diffusion parameters of the atoms, the movement of grain boundaries and behaviour of carbon atoms under external loading, especially in alloys with complicated microstructure and structure. 5.2.4 Migration of solute atoms in the region with dislocations It will be assumed that low temperatures are those at which T < T 0= –H/kln(fτ 0) and at these temperatures the solute atom does not move under the effect of external loading. High temperatures are those at which T > T 0 , which means that the atom of the solute at time ∆t = 1/f completes a certain number of jumps, i.e. diffuses to a certain distance (f is the frequency of changes of external loading). In the low-temperature range, all internal friction mechanisms can be divided into two large groups. The first group of the mechanisms is associated with the analysis of separation of the dislocation from the stationary atoms of the solutes forming the atmospheres around the dislocations (unpinning mechanism). The second group of the mechanisms is associated with the analysis of the relationships governing the movement of the dislocation in the slip plane in which the atoms of the solutes are distributed ('friction' mechanism). If the temperature is increased above T 0 , to the model is mentioned previously we can also add another group of mechanisms, in which anelasticity is manifested or depends strongly on the processes of diffusion displacement of the solid atoms during the period of external loading. If the material is subjected to mechanical stress 205

Internal Friction of Materials

τ m , it can be seen that each pinning point will be subjected to the effect of the force F(τ) caused by the bending of the adjacent dislocation segments (see also section 3.5). The force can be divided into the component F and F || , i.e. to the direction normal and parallel with the initial position of the dislocation line [223], determined by the equations

F⊥ =

τ2b2 2 2 l1 − l2 , 8m

(5.21)

F|| =

τb (l1 + l2 ), 2

(5.22)

(

)

where b and m are the Burgers vector and the linear elongation of the dislocation, l 1 and l 2 are the lengths of the dislocation segments of the dislocations in the vicinity of the pinning point. The existence of the forces F I and F II is the physical reason for the appearance of many anelastic processes (Table 5.4) at T > T 0 determined by the effect of the normal or parallel component of the force during movement of the atom of the solute in the region of the dislocation kernel. A special feature of these manifestations is that under the effect of external loading the dislocation lines do not separate from the atmospheres of the solute atoms, even though the atoms themselves can move along the dislocation line by the pipe diffusion mechanism. Thermally activated separation of the dislocations can take place as an independent process supplementing the phenomena caused by the diffusion displacement of the solute atoms. The phenomena of dislocation anelasticity, caused by the diffusion displacement of the atoms of the solutes in the atmospheres around the dislocations are controlled by the transverse and longitudinal mobility of the atoms of the solutes in the dislocation kernel. This model can be used in the quantification of the appropriate diffusion characteristics. A special feature of Table 5.4 is that it includes the mechanism of formation of the thermal–fluctuation relaxation peak of internal friction (TF relaxation), caused by the separation of the dislocations from the pinnig solid atoms. In this case, the separation of the dislocations segments is initiated by the thermal–fluctuation jump of the atom of the solute which pins the segments in the direction normal to the dislocation line. 206

207

Relaxation peak Q –1

Oscillation of Q –1

Time dependence of Q –1

Phenomenological models

Snoek and Köster peak

Microscopic models

Longitudinal movement

Dragging without unpinning

Peaks of Q –1 caused by dragging of solute atoms

Time dependence of Q –1

Phenomenological models

Mechanism of formation of peak of Q –1 due to unpinning of dislocation initiatied by thermallyactivated jumps of solute atoms

Dragging and thermally activated unpinning of dislocations from atmospheres of solute elements

Relaxation peak of Q –1 caused by reversed transverse jumps of atoms

Microscopic models

Transverse movement

Anelastic phenomena caused by movement of solute atoms in dislocation kernel

Ta b le 5.4 Systematization of anelastic phenomena

Structural Instability of Alloys

Internal Friction of Materials

Analysis of the model in which two dislocation segments l 1 and l 2 are pinned by the solute atoms under the effect of the defined force law during displacement of the dislocation [224] indicates that the width of the relaxation peak will be greater than the width of the Debye peak with a single relaxation time, although the broadening of the peak will not be very large. It is also expected that the increase of the stress amplitude will increase the peak with saturation. An anomaly of this dependence is that with the increase of mechanical stress the peak is displaced to higher temperatures on the temperature axis of the Q –1 (T) dependence. Analysis also shows that the increase of the content of the solute changes the degree of relaxation and the position on the temperature axis in the same manner as mechanical stress τ mo . At stress amplitudes of τ mo ≥ 10 –6 G and binding energies H = 0.8 – 16 × 10 –19 J, the activation parameters of the TF peak can be determined approximately from the equation

HTF = −kTmax ln ( 2π f κ3τm0 ) ,

(5.23)

where T max is the temperature at which the maximum of internal friction is recorded at the loading frequency f, where κ 3 is the coefficient of proportionality expressed in units. Many features of the TF peak are similar to those of the Snoek or Koster peaks; for this peak to form, the material must contain dislocations and a solid solution with a relatively low solute content. The temperature range in which the peak is observed is in the temperature range of recording the time dependence of internal friction. The relaxation peaks formed as a result of the TF mechanism were detected in solid solutions based on aluminium (Fig. 5.10) and copper [2] and in the Fe–C interstitials solid solutions [224]. The activation parameters of the peaks H TF and τ 0 of many solid solutions are presented in Table 5.5. The level of the activation energy of the diffusion of solute atoms along the dislocations H was determined by the method of the time dependence of internal friction. The value of τ 0 varies in the range 10 –12 to 10 –13 s –1 , which corresponds to the value 1/υd, where υd is the Debye frequency of the vibrations of a free atom. The value of Q –1 in the process of measurement of the time dependence of Q –1 is the function of temperature in measurements and reaches the highest value at temperatures where the TF is the peak of internal friction. 208

Structural Instability of Alloys

F ig .5.10. Effect of temperature and frequency on internal friction of Al–0.02 ig.5.10. wt.% C: a) effect of temperature on Q –1 at different frequencies: 1) f = 1.3 Hz, 2) 1820 Hz, 3) 2820 Hz, at strain amplitudes during measurement: 1) ε = 2.42 × 10 –5 , 2,3) 4×10 –6 ; b) frequency shift of the temperature at which the peak forms. Ta b le 5.5 Activation characteristics of different peaks for fcc and bcc metals

S yste m

Hτ · 1 0 1 9 (J)

τ · 1 0 12 (s )

HL · 1 0 1 9 (J)

Hτ/Hv

Al– C u Al– Mg F e–C

0.864 1.008 1.920

1.3 0.7 0.26

0.704 0.784 –

0.44 0.53 1.38

The processes of dragging of the atmospheres of the solute elements by the dislocations belong to the main mechanisms controlling the dynamics of build-up of microplastic deformation of solid solutions at elevated temperatures. Examination and quantification of the TF of internal friction makes it possible to determine the activation energy of cross diffusion and, under the given conditions, also the type of solute atoms inhibiting the movement of the dislocations. For the temperature range below τ 0 , the authors of Ref.225 proposed a model of dislocation internal friction based on the mechanism of the thermally activated unpinning of the dislocation from the pinning solute atoms. Blair, Hutchison and Rogers presented a model (BHR model) which is a modification of the internal friction mechanism, proposed by Teutonico, Granato and Lucke. The BHR model takes into account the general interaction law between the dislocation and the solute atoms for the frequently encountered case 209

Internal Friction of Materials

of the uniform distribution of the pinning centres along the dislocation line. Analysis of the replacement of the anelasticity mechanisms with increasing temperature or stress amplitude makes it possible to observe, in the T – τ m coordinates, several regions with different processes of the unpinning of dislocations from the pinning points (Fig. 5.11): – in the regions A and E, movement of the dislocations takes place as a result of simultaneous thermally activated unpinning from several solute atoms. In the regions B, C & D, the process of release of the entire dislocation line starts with the separation of the dislocation line from a single solute atom; – in the regions A & B, only a small part of the dislocations are separated as a result of the effect of alternating external stress. At high temperatures (regions C, D and E), almost all dislocations are separated in every loading cycle; – in region D, the activation energy of separation of the dislocation is constant. In other regions, one may expect the dependence of activation energy on stress amplitude. Critical temperature T k is linked with the binding energy of the solute atom with the dislocation. The value of H v is determined by the equation H v = kT k ln (δ 1/ω), where δ 1 is the effective frequency of oscillations of the dislocation segment. The BHR theory has been used in analysis of the redistribution of the solute atoms in the atmosphere around the dislocations dur-

F ig .5.11. Maps of mechanisms of thermally activated unpinning of dislocations ig.5.11. from solute atoms (a) and dislocation anelasticity of beryllium (b). Region 1 – thermally activated unpinning of dislocations from stationary solute atoms, 2) diffusion along dislocations, 3) diffusion dragging of solute atoms in the atmosphere by the dislocation. 210

Structural Instability of Alloys

ing their vibrations, by the mechanism of cross diffusion. The currently available results indicate that every diffusion process may result in the formation of a relaxation peak on the Q –1 (T) dependence. This hypothesis has been used to develop a single model of a dislocation pinned at a single point [226]. The process of scattering of energy is associated in this case with the diffusion dragging of the solute atoms by the dislocations, but no attention is given in the model to thermally activated unpinning of the dislocations. The displacement of the atom along the dislocation line by the pipe diffusion mechanism results in the formation of relaxation peak (peak α), with the relaxation time

τL =

(

lc2

Dα π2 + α 2

)

,

where

 H  DL = DLC exp  − L   kT  is the coefficient of diffusion along the dislocation,

α 2 = τ 02

G 2 lc3 . 4m k T

If the stress amplitude is increased, the peak α will be displaced to lower temperatures. The degree of relaxation of the peak will increase. The cross diffusion processes also lead to the formation of a relaxation peak (T peak), for which

τT =

lc kT , 2 DT m

where

211

Internal Friction of Materials

H  DT = DT0 exp  T   kT  is the coefficient of cross diffusion of the solute atoms in the dislocation kernel. Generally, the T peak is reflected in the superimposition of many elementary relaxation peaks and must be wider than the Debye peak with a single relaxation time. It is useful to draw attention to the anomalous dependence of the characteristics of the T peak on stress amplitude. If τ m is increased, the position T of the peak on the temperature axis Tmax does not change. Further increase of τ m already causes the displacement of the peak on the temperature axis to higher values. The development of diffusion processes in the dislocation kernel results in the formation, in the temperature–stress dependence (Fig. 5.11b), of two additional boundaries of regions of new dislocation relaxation mechanisms (2, 3). When examining the Q –1 (T) dependence for pure polycrystalline beryllium, Levin [227] observed three Q –1 maxima (Fig. 5.12). Analysis of the experiments showed that both high-temperature peaks formed as a result of the mechanism of pipe (L) and transverse (T) diffusion in the atmosphere of the solute elements. The peak P is the result of the thermally activated unpinning of the dislocations from the stationary solute atoms, in accordance with the BHR model. The calculation of the activation parameters of the L, T and P peaks yielded data on the characteristics of the dislocation complexes and solute atoms in beryllium (Table 5.6) and the results are used to construct a general temperature–stress diagram (Fig. 5.11b), expressing the operation of different dislocation anelasticity mechanisms, such as thermally-activated unpinning of the dislocation, the processes of diffusion along and normal to the diffusion line, and dragging of the dislocation by the solute atoms. The boundaries of the additional regions are determined by the equations 2πfT Tmax = 1 and 4πfT max L = 1. Three peaks, similar to P, L, T in Fig. 5.12, were also observed in zirconium [228]. Analysis of the data provides information on the temperature–stress diagram, thermally activated unpinning of the dislocations and diffusion-controlled dragging of the solute atoms (probably oxygen) in single crystals of zirconium and in polycrystalline zirconium.

212

Structural Instability of Alloys

F ig ig.. 5.12. Dependence of internal friction in heating (a) and cooling (b) and on the square of frequency (c) on temperature, with the strain amplitude τ/ G = 4× 10 –5 . Ta b le 5.6 Anelasticity parameters determined for internal friction peaks

P a ra me te r Hi · 1 0 1 9 ( J )

P eaks L

P 2.096

0.08

2.92

0.12

Hβ · 1 0 1 9 ( J )

2.096

lc

160 b

u = Hb/F m

1.5 b

T 3.072

0.224

5.3 RELAXATION AT GRAIN BOUNDARIES Measurement of internal friction associated with the relaxation at the grain boundaries of polycrystalline materials provides a large amount of information on the structure and properties of the grain boundaries. The Q –1 (T) dependence shows a peak which, after analysis, provides information on the kinetics of changes at the grain boundaries, taking under the effect of external loading; these changes depend on the structural condition of the polycrystalline 213

Internal Friction of Materials

material (mean grain size, the presence of a dislocation substructure, the distribution of the group of the grain boundaries on the basis of the angle of misorientation or the same orientation, etc.). This makes it possible to investigate and evaluate the interaction of the grain boundaries with other structural defects and also recombination of the grain boundaries under the effect of various internal and external influences on the material. A low content of solute atoms already has a significant effect on the nature of the peak associated with grain boundary relaxation due to the formation of the ss peak (the designation of the peak is taken from Ref. 229), which is the result of the interaction of the grain boundaries with the solid atoms. Analysis of the ss peak provides information on this interaction [230]. The peak, associated with the relaxation at the grain boundaries observed on the Q –1 (T)dependence, was described and analysed for the first time by Kê in 1947. The main data obtained in the measurement of the Q –1 (T) dependence in polycrystalline aluminium can be summarised as follows [231]: –at a loading frequency of ~1 Hz and a temperature of 300°C, the curves of the Q –1 (T) dependence for polycrystalline aluminium show a peak associated with the presence of the grain boundaries, because this peak does not form when measuring the Q –1 (T) dependence on single crystal aluminium; –the degree of relaxation of the peak is independent of the grain size d z in the limiting case when the grain size is smaller than the size of the specimen; –relaxation time τ is directly proportional to the grain size; –the activation enthalpy of the relaxation process is close to the activation enthalpy of self diffusion in aluminium and τ 0 ~ 10 –14 s; –the relaxation peak is considerably wider than the peak obtained for the conventional inelastic material. Similar relationships were also recorded for other metals, so that the observation made by Kê is universal. The interpretation of the relaxation of the grain boundaries and the associated peak on the Q –1 (T) dependence which, to simplify concentrations, will be referred to as GBIF, is based on the assumption on viscous sliding (accommodation) of the grains in their boundaries, formulated by Zeener. The results of a large number of experiments show that the phenomenon of grain boundary relaxation is more complicated because it depends strongly on the solutes manifested in the formation of the ss peak. 214

Structural Instability of Alloys

5.3.1 Pure metals The peaks on the Q –1 (T) dependences for pure silver, gold, and copper are considerably smaller and wider than in the case of aluminium. The relaxation spectrum is characterised by a complicated structure reflected in doubling of the peak after high-temperature annealing. The first peak corresponds to the Kê peak. The second peak is recorded at higher temperatures. The high-temperature maximum is associated with grain boundary relaxation, because it is not found in single crystals. In copper and nickel there is also a third peak at transition temperatures. This peak is associated with the formation of annealing twins [232]. The relationship between ∆ and γ (where γ is the stacking fault energy) in the case of gold, copper, nickel and aluminium is linear [233]. The width of splitting

r=

G b2 . 24 πγ

The results show that the activation enthalpy H and quantity r are linked by the dependence similar to the dependence of ∆ on r (Fig. 5.13). These dependences have been confirmed for many metals with different crystal lattices [66]. However, other data are also available. For example, in the case of copper [234] the height of the peak increases with a decrease of the grain size and the width of the peak decreases. For 99.999% copper, the value of ∆ is independent of the grain size [235]. In the case of nickel, ∆ ~ d –1g in a wide range of d g. Similar dependences were also recorded in the case of iron and copper of commercial purity [236]. If ∆ depends on grain size, this is observed in a specific range of d g and has the form ∆ ~ d –1g . The particles distributed at the grain boundaries form internal stresses around them and complicate grain boundary sliding. The phenomenon is independent of the elasticity coefficient of the particle. The change of the relaxation time of the peak associated with grain boundary relaxation and its activation enthalpy are not in agreement with these values determined for the Kê model. For example, for pure metals τ ~ dgn, where 1< n < 2 (n = 2 for Fe, n = 1.86 for Al). The activation enthalpy of the peak associated with grain boundary relaxation for the pure metals is lower than the activation energy of self diffusion. The results show [237] that the value of H with increasing purity of the metal tends to the activa215

Internal Friction of Materials

F ig .5.13. Dependence of the degree of relaxation (a) and relative activation enthalpy ig.5.13. (b) of different metals on the extent of splitting r ( H 0 is the enthalpy of self diffusion of pure metal).

tion enthalpy of grain boundary diffusion. Since the relaxation time of the peak can be described by the Arrhenius equation, τ 0 is similar to the value of this factor for the relaxation of point defects (~10 –14 s). On the other hand, it was observed [236] that in the case of pure iron the value of τ 0 decreases with decreasing d g which means that it depends on the microscopic parameters of the structure. This phenomenon has not been completely explained. The peak on the Q –1 (T) dependence, associated with grain boundary relaxation, is described by a set of relaxation times. The parameter of the logarithmico-normal distribution β changes over a wide range, and in some cases β = 7. The value of this parameter depends on temperature which means that there is a set of the values of τ 0 and H. The peak has several components forming a wide profile which also indicates that the distribution of the relaxation parameters is not in agreement with the logarithmico- normal distribution. 5.3.2 Solid solutions The interaction of the grain boundaries with the atoms of the solutes in the solid solution results in the redistribution of the relaxation spectrum, and the behaviour of the interstitial and substitutional solid solutions differs. Pearson and Rotherham [238] examined copper and silver with an addition of ~1 wt.% of the solute and concluded that in comparison with the pure metals, the peak on the Q –1 (T) dependence (RM peak) is lower and a ss peak is recorded at high temperature and is caused by the interaction of solute atoms with the grain boundaries. The activation energy of the peak resulting from the presence of the solutes is close to the activation en216

Structural Instability of Alloys

ergy of diffusion of the solutes [239] and the activation enthalpy of self diffusion of the solvent. The value of τ 0 was 10 –14 to 10 –16 s, as in the case of the RM peak. The relaxation spectrum of the ss peak is wider than the RM peak. When the concentration of the substitutional additions in copper is increased, the original peak for relaxation at the grain boundaries (RM peak) and the peak due to the presence of the solutes (ss peak) increases. With the decrease of the height of the RM peak its activation enthalpy increases. In the binary and ternary alloys based on chromium and containing La, Ni, Fe and V, the spectrum of the Q –1 (T) dependence at high temperatures contained complicated boundary peaks [240]. At present, there are insufficient experimental data on grain boundary relaxation for the case in which the solid solution is interstitial. The Fe–C and Fe–N alloys have been studied in detail [224]. The main difference of the relaxation spectra for the substitutional and interstitial solid solutions is that the interstitial additions already have an effect at low concentrations, i.e. they change the temperature at which the peak is formed and also the activation enthalpy of its formation. The interstitial solid solutions did not contain the isolated peak corresponding to the presence of the solutes. The concentration dependences of the parameters of the peaks in the Fe–C and Fe–N alloys are shown in Fig. 5.14. The saturation of the grain boundaries with the solute atoms is found at c ≅ 10 –2 at% in the Fe–C alloys. A large decrease of the height of the peak with a further increase of the concentration is associated with the precipitation of the particles of the corresponding phase at the grain boundaries. In the iron alloys, the internal friction, associated with grain boundary relaxation, decreases with increasing grain size. 5.3.3 Relaxation models Roberts and Barrand [242] were the first authors to propose the quantitative interpretation of the peak on the Q –1 (T) dependence, associated with grain boundary relaxation. Their interpretation was based on the empirically determined relationship between the degree of relaxation of internal friction and the width of splitting of the dislocations. They assumed that, depending on the stacking fault energy of the metal, lattice dislocations in the vicinity of the grain boundaries move under the effect of stress either by climb or slip. In the metals such as aluminium or nickel with a high value of γ, the internal friction, caused by the relaxation at the grain boundaries, is the result of dislocation climb and the activation enthalpy 217

Internal Friction of Materials

wt.%

wt.%

F ig .5.14. Dependence of the degree of relaxation (a) and activation enthalpy (b) ig.5.14. on the content of interstitial elements in alpha iron.

of the process is close to the activation enthalpy of self diffusion. In the metals with the low value of γ (for example, Ag), the extent of dislocations splitting is very high and the relaxation process takes place by dislocation climb at the grain boundaries because nonconservative displacement of the split dislocations is not advantageous from the energy viewpoint. The activation enthalpy of the peak on the Q –1 (T) dependence as a result of grain boundary relaxation: in the metals with average values of γ (Au, Cd), it is assumed that the dislocations move by a combined mechanism consisting of slip and climb at the grain boundaries. This is also in agreement with the activation of the enthalpy of the process whose value is between the values for diffusion at the grain boundaries and in the volume of the metal. In single crystal aluminum, the Q –1 (T) dependence shows an internal friction peak at a temperature of 365°C and a frequency of ~ 1 Hz. This peak is the result of the process of climb of the split dislocations under the effect of stress [243]. Other relaxation phenomena, associated with the polygonisation process in aluminium and diluted Al–Cu solid solutions, have been published in Ref. 244 and 245. Good correlation has been found between the formation of the peak on the Q –1 (T) dependence at 265°C and the changes of the microhardness of aluminium. This can be used as an additional method for the indirect evaluation of the hardness of materials. The spectrum of the peaks on the Q –1 (T) dependence for complicated systems, such as fibre-reinforced composite materials, is not a single superimposition of the phenomena taking place in the components of the composite at the fibre–matrix interface. The phenomena associated with relaxation at the grain boundaries or sub-boundaries can 218

Structural Instability of Alloys

also be detected when the size of grains all subgrains is smaller than the spacing between the fibres [246].

5.4 ANALYTICAL PROCESSING OF THE RESULTS OF MEASUREMENTS The discussed mechanisms of anelasticity in metallic materials and the analysis of the mechanisms indicate that the internal friction measurements can be used efficiently in quantification of a large number of specific relationships in metals and alloys. Some examples are given in the following part of the book. 5.4.1 Solubility boundaries The solid interstitials solutions of metals with the bcc lattice are characterised by the linear relationship between the height of the Snoek peak Q –1max (if the background of internal friction is subtracted) and the concentration of interstitial atoms n (wt.%) in the form −1 n = p Qmax ,

(5.24)

Here p is the coefficient of proportionality whose values is determined by the type of atoms forming the solid solutions and depends on the grain size, texture and other factors [66]. The value of p for different interstitial atoms is not a quantity changing in a simple manner (for example, for C in polycrystalline α-iron p ≈ 1.28 – 1.3, and for N p ≈ 1.26 – 1.3). In fact, it is not necessary to know the value of p when determining of the solubility boundaries. A suitable example is the analysis of the Fe–C system. Specimens with different carbon content (to 0.02 wt.%) should be annealed at different temperatures, followed by rapid cooling and measurements of the Q –1 (T) dependence. After evaluating the height of the peak and subtracting the internal friction background, with determination of the activation enthalpy of the peak, it is possible to plot the dependence shown in Fig. 5.15. The graph summarises the results of measurements of the solubility of carbon in ferrite obtained by the method of depleting the solid solution in carbon [247] with the values obtained from the height of the Snoek peak of the specimens containing 0.04 wt.% C, grain size > 1 mm [248]. At p = 0.9, the solubility values are in agreement. If the temperature of the specimens with the C content lower than the solubility limit is increased, 219

Internal Friction of Materials

the value of Q –1max rapidly decreases (Fig. 5.15b). The authors of Ref. 247 and 248 assume that the concentration of C in the areas of grain boundary segregation is close to the eutectoid concentration and austenitic layering forms at subcritical temperature. This the decreases the C content in the lattice of residual α-iron. In the specimens with 0.4 wt.% C, cooling from temperatures higher than A c1 decreases the value of Q –1max as a result of dissolution of carbon in austenite. Consequently, it is possible to determine the limiting concentration of the solubility of carbon in α-iron [249], the solubility limits in ternary and more complicated systems (for example, Fe–Cr– C) and also evaluate the vacancy–interstitial atom complexes. Similar measurements by the internal friction method can also be taken at low temperatures (to 150°C) when the solubility of carbon in αiron is very low. The possibility of evaluating the content and solubility limit of interstitial atoms in the interstitial solutions in the metals and alloys with fcc and hcp lattices is based on the Finkel’stein and Rozin principles. The values of n and Q –1max are linked by the equation −1 n − n0 = K Qmax ,

(5.25)

here n 0 is the critical concentration of the interstitial atoms (at

wt.%

F ig .5.15. Dependence of the height of Snoek peak ( Q S–1= Q 40–1 , f 1 Hz) and carbon ig.5.15. concentration in ferrite (a), dependence of the height of the Snoek peak on temperature (b) for the carbon contents in wt.%. 1) 0.003, 2) 0.010, 3) 0.040, 4) 0.400. 220

Structural Instability of Alloys

n < n 0 the dependence of Q –1 on n is quadratic), K is a coefficient which takes into account the structure of the alloy. The deviation from the linear dependence (equation 5.25) is utilised when determining the solubility limit. The substitutional solid solutions are characterised by Zeener relaxation. The following equation is valid for many specific systems −1 n 2 = m Qmax ,

(5.26)

where m is the structure-sensitive coefficient of the alloy. This method has been used successfully for evaluating the solubility curves of Ag–Al, In–Tl, etc. alloys [26]. 5.4.2 Activation energy and diffusion coefficient The condition for the formation of a relaxation peak in the form ωτ 0 exp (H/kT) = 1 enables several methods to be selected for determination of the activation enthalpy. By changing the loading frequency, it is possible to determine H from equation (5.1). The accuracy in determination of H can be increased by constructing the dependence Q –1x = Q –1 /Q –1max on T –1 , in which the internal friction background is subtracted from the internal friction peak so that it is possible to decide whether the spectrum has activation energy or not. If the change of the frequency of external loading causes no change of the form of the Q –1x – T –1 curve, the process is characterised by a single relaxation time. Otherwise, the spectrum has activation enthalpy H. In most cases, the value of H is determined using the method based on the determination of the shift of the temperature at which the peak is recorded on the Q –1 (T) dependence, utilising the equation derived by Wert and Marx (equation 5.3), taking into account the fact that the contribution T ∆S changes only slightly the value of H. Consequently, we can use the equation in the following form

H = RTmax ln

kTmax , hωmax

(5.27)

where ω max is the frequency of external loading at the temperature at which the peak appears. The approach proposed by Wert and Marx is limited to the relaxation processes induced by the thermal activation of atom migration under the effect of external stress so 221

Internal Friction of Materials

that the relaxation mechanism must be known in advance. The value of H is determined from the angle of inclination of the low- and high-temperature part of the peak of the Q –1 (T) dependence in the coordinates ln Q –1 vs 1/T. This method is suitable only for very narrow peaks of the Q –1 (T) dependence. The value of H can also be determined from the width of the peak on the Q –1 (T) dependence measured at the height 1/2 Q –1max. The results are satisfactory when the relaxation process is characterised by a single relaxation time. Since the peak is formed by a spectrum of relaxations, the value of H will be lower than the physically substantiated value of H. The determination of the bulk diffusion coefficient by internal friction measurements is based on linking the relaxation time τ with the diffusion coefficient D (equation 5.4). The method was described in details when discussing the interpretation of the Snoek peak. It should be noted that the actual displacement of the point defect (the frequency of jumps) is f times smaller than the corresponding values obtained within the framework of the model of random experiments. The value of the correlation factor depends on the type of lattice and the diffusion mechanism [2,26]. For example, for self diffusion with the vacancy mechanism, the value of f changes from 0.78 (for fcc lattice) to 0.5 (for rhombic lattice). If the value of factor f is available, it is possible to calculate more accurately the actual diffusion coefficients from the change of the relaxation time. Internal friction measurements also make it possible to determine the pipe diffusion coefficient, from the change of the temperature of formation of the peak on the Q –1 (T) dependence and also from the time dependence of internal friction. The utilisation of the thermalfluctuation peaks (TF relaxation) for the evaluation of the activation parameters of diffusion of solute atoms along the dislocations is discussed in section 5.2.4. The coefficient of diffusion of the solute atoms along the core of the edge dislocation can be determined by measuring the time dependence of internal friction in accordance with Ref. 249. The phenomenon of time dependence of Q –1 (T) is determined by the migration of solute atoms along the dislocations under the effect of external stress in the areas of strong pinning which changes continuously the distribution of dislocations segments and results in the formation of large adsorbed atoms of the solutes, free from dislocations segments. When the stress σ k is higher than some critical value σ c , the value of Q –1 becomes time-dependent. If σ is reduced in such a manner that σ < σ c , the Q –1 (T) dependence is character222

Structural Instability of Alloys

ised by a single relaxation time. Consequently, we obtain the dependence of τ on the pipe diffusion coefficient D d in the form

2 lc2 , Dd = τ

(5.28)

Relaxation time is determined by transformation of the experimental data to suitable coordinates, for example

ln =

Q(−T1) − Q∞−1 Q0−1 − Q −1

νs t ,

where Q –10 and Q –1∞ are the values obtained under the effect of σ and in its absence, t is time, and the values of l c are determined by an independent method (for example, from the curves of the Q –1 (T) dependence). The method is described in detailed in Ref. 2. 5.4.3 Breakdown of the solid solution The kinetics of precipitation of a phase in alloys is characterised by the exponent n in the equation proposed by Wert and Zeener [250] according to which the fraction of the particles precipitated from the solid solution q is determined by the equation n

 t q = 1 − exp  −  ,  τ

(5.29)

where t is time, τ is the characteristic process time. The value of n depends on the mechanism of the precipitation process. The efficiency of this approach can be seen in, for example, Ref. 251. When evaluating the precipitation of carbon or nitrogen from the solid solution during the thermal and deformation breakdown of the solid solution of α-iron, the Snoek maximum can be described by the following equation:

q =1−

−1 −1 Qmax 0 − Qmax t −1 −1 Qmax 0 − Qmax ∞

,

(5.30)

223

Internal Friction of Materials

where Q –1max t and Q –1 are the heights of the Snoek peak at time t max 0 and t = 0, with Q –1max ∞ being the same, but at t → ∞. The transformation of the results to the coordinates log ln (1/1 – q) vs ln t can be utilised when determining the precipitation factor n which characterises the breakdown mechanism. At the same time, it is possible to calculate the activation enthalpy and the rate of the process of breakdown of the solid solution [252]. The carbon content of structural steels is higher than the concentration associated with the saturation of the solid solution. In principle, 'reversible' dissolution of the phases, containing carbon and nitrogen, in ageing can be evaluated from ‘reversible’ dissolution during time-limited heating of the alloy to high-temperature. Another example is from the area of the evaluation of the development of low-temperature brittleness observed at 475°C. From the kinetic parameters of the change of the height of the Snoek maximum and from the splitting of the peak in the breakdown stage, it is possible to plot the diagram of breakdown of ferrite with a high chromium content (Fig. 5.16), as described in Ref. 254. In the case of short-term ageing, the height of the Snoek peak rapidly decreased and carbon and nitrogen precipitated at the dislocations (n = 0.5 – 0.7). In this stage, 50–70% of interstitial atoms in ferrite precipitates from the solid solution (region A, Fig. 5.16). In continuing ageing of quenched Kh25 steel (the concentration of C and N is 0.01–0.2 wt.%) in a narrow temperature range (450–525°C) the Snoek peak is split as a result of the formation of microregions (s–i complexes) with different content of chromium and interstitial atoms into regions in later stages of the process; this is verified by another, objective method (region C). It is also possible to examine the processes of distribution of the interstitial elements during isothermal annealing of supercooled austenite in the steel on the basis of the Finkel’stein and Rozin peaks (FR) which follow the redistribution caused by quenching to the formation of the Snoek maximum and Snoek and Köster maximum [255]. 5.4.4 Intercrystalline adsorption The peak on the Q –1 (T) dependence, associated with grain boundary relaxation, is sensitive to intercrystalline absorption also at a low solute content. Piguzov and Glikman [256] observed a case of intercrystalline adsorption of phosphorus in steel. The peak on the Q –1 (T) dependence at 300°C (f ≈ 1 Hz) is caused by the presence of hundredths 224

Structural Instability of Alloys

F ig .5.16. Regions of breakdown of ferrite with high Cr content: A) breakdown ig.5.16. of the interstitial solid solution (complete – 1), to ∆HV max (2), to 50% breakdown (4); B) formation of s-i complexes at the total C + N content of 0.2 wt.% (1), 0.04 wt.% (2) and 0.02 wt.% (3); C) formation of Cr-enriched zones.

of a percent of P in alloyed iron and low-alloy steels. Temper brittleness is the result of adsorption enrichment of the grain boundaries with phosphorus. Investigations were also carried out on an iron alloy alloyed with Ni and Cr where the enrichment of the regions in the vicinity of the grain boundaries cannot develop by the nonadsorption mechanism [257]. The development of temper brittleness was evaluated by the authors by measuring the notch toughness of specimens fractured at – 196°C and also on the basis of the etchability of the grain boundaries in the saturated aqueous solution of zirconic acid. The kinetics of the changes of plasticity at low temperature (Fig. 5.17a) and of the size of the area below the peak belonging to phosphorus (Fig. 5.17b) in relation to the embrittling time of alloyed iron, containing 0.032 and 0.005 wt.% P, is shown in Fig. 5.17. For the alloy with 0.95 wt.% Si, 0.7 wt.% Ni and 0.032 wt.% P, the (the peak initially increases and reaches the maximum value Q –1 gb max concentration at the grain boundary c gb = 0.5) and then decreases, and after tempering for 2 hours completely disappears (c gb = 1). The changes of plasticity at low temperature and of the etchability of the grain boundaries indicate that the phosphorus concentration at the grain the boundaries increases during the first two hours at 500°C and then shows no significant changes. In the alloy with 0.95 wt.% Si, 0.7 wt.% Ni and 0.005 wt.% P, examination showed only the increasing section of the dependence on tempering time; this is as225

Internal Friction of Materials

sociated with slight embrittlement and decrease of the etchability of the grain boundaries [257]. The increase of the carbon content in iron with Ni and Cr results in the phenomenon in which the concentration of phosphorus at the grain boundaries and the susceptibility to temper brittleness decrease. At a constant content of carbon and phosphorus, the phosphorus concentration at the grain boundaries and the susceptibility to temper brittleness in iron, containing manganese, is higher than in iron with Ni and Si in cases in which the alloying elements not forming carbides result in the enrichment of the zones in the vicinity of the grain boundaries in carbon thus impairing intercrystalline adsorption of phosphorus and reducing the rate of embrittlement. The change of the c gb vs t ratio during intercrystalline adsorption of P for five alloys with different phosphorus and carbon content can be described by the equation cgbt − cgb 0 cgb ∞ − cgb 0

 4 Dt = − exp  2 2  α dz

 2 Dt ,  erfc αd z 

(5.31)

where c gb 0 is the initial concentration at the grain boundaries, c gb t is the concentration after tempering for time t, c gb ∞ is the equilibrium concentration at a specific tempering temperature, α is the c gb ∞ /c gb 0 ratio, D is the diffusion coefficient. The experimental points for different alloys are distributed on the same curve. Analysis of the results indicates that of the possible mechanisms, the mechanism with the highest rate is the controlling

F ig .5.17. Change of the reduction in area determined at –196°C (a), the height ig.5.17. of the P addition peak of internal friction (b) in relation to tempering time at 500 °C of alloyed iron with 0.032 wt.% P (solid circles) and the internal friction peak (broken lines). 226

Structural Instability of Alloys

one. According to the authors of [258], the mechanism of inter-crystalline adsorption is associated with bulk diffusion to the nearest sub-boundary, i.e. dislocations, and is then transferred at a high rate along the sub-boundaries to the grain boundaries. Table 5.7 gives, for several alloys, the experimentally determined data of the concentration at saturation and the temperature at which the peak caused by the atoms forms, and also the data on the binding energy of the solute atoms with the grain boundaries. 5.4.5 Transition of the material from ductile to brittle state In a specific temperature range T t , the strain rate or hydrostatic pressure P t, the material may show a change from the nature of failure from ductile to brittle. The aim of several investigations was to determine whether it is possible to find a correlation between the changes of internal friction and the transition of the material from the ductile to brittle state (DB transition), for example, [259, 260]. Measurements of the Q –1 (T) dependence were taken, for example, at strain amplitudes of ε ≅ 10 –7 , frequencies of f = 10 7 , 10 5 , 10 3 , and 1 Hz, and also at ε = 10 –7 to 10 –5 at frequencies of f = 10 5 , 10 3 and 1 Hz, and also at ε = 10 –5 to 10 –4 at a frequency of 1 Hz. The investigations were carried out using brittle polycrystalline metals with different transition temperatures T t (–50 to 200°C) and pressures P t (1 × 10 8 – 7 × 10 8 N m –2 ): Zn, Bi, Mo, Cr, steel 09G2, steel 35KhGSA and, for comparison, also single crystals of Zn, Bi, Mo, fine-grained Zn, polycrystalline aluminium and iron which are ductile in the examined temperature range. It was expected that the selected non-cubic brittle polycrystalline materials Zn and Bi would show, in the range T t and P t, two phenomena: the DB transition and plastic deformation caused by the anisotropy of expansion and compression. Only the DB transition was expected in the case of the cubic metals. The change of the T t by ±(10 – 40)°C was obtained by annealing, deformation and alloying of the material. The brittle materials in the range of T t and P p show a peak on the Q –1 (T) curve. No peak is recorded in the ductile materials. In order to be able to explain the nature of the peak, it is necessary to know in detail the measurement conditions characterised by the frequency of oscillations and strain amplitude. For example, the peak at ε ≅ 10 –7 is recorded at a frequency of 10 7 Hz in the range T t (Zn, Mo) and P t (Zn, Bi, Mo, Cr). At a frequency of 10 5 Hz, the peak is recorded only in individual cases in the expected range of T t (Zn). At a strain amplitude of ε ≥ 3 × 10 –7 the peak on the 227

228

α - iro n- C C u– B

C u– N i N i– C u F e–C r Ag– C d C u– S b Al– Mg N i– C r C u– Zn Allo ye d α – iro n- P

S o lute a d d itio n syste m

– –

– 730–770

580 660

920 1070 900 700 800 650 1040 770

With sa tura tio n " " " " Witho ut sa tura tio n " "

–

Te mp e ra ture a t whic h p e a k is a t ma ximum p o int o r sa tura tio n o f c urve Qp r – 1 ( c ) (K )

De p e nd e nc e o f he ight o f p rima ry peak on c o nc e ntra tio n Q–1 (c ) 0.12 0.2 0.15 0.17 1 0 –3 7 · 1 0 –3 0 . 11 0.2 0.2 – – – – – –

C o nc e ntra tio n c a t p o int o f p e a k c gb= 0 . 5 o r sa tura tio n c gb= 1 o f d e p e nd e nc e Qpr–1 (c ) – – – – – – – – 923 923 923 823 11 2 3 673–873 –

Anne a ling te mp e ra ture T0 (K )

920 1070 900 700 800 650 1040 770 923 923 923 823 11 2 3 673–873 –

T p use d in c a lc ula tio n (K )

0.25 0.24 0.22 0.17 0.77 0.38 0.21 0.1 0.8 0.96 0.88 1.1 0.7 0.96 0.76

Bind ing e ne rgy o f so lute a to m with gra in b o und a rie s (J · 1 0 19)

Ta b le 5.7 Binding energy of solute atoms with grain boundaries, obtained by internal friction measurements

Internal Friction of Materials

Structural Instability of Alloys

Q –1 (T) dependence is recorded at a frequency of 10 5 – 10 3 Hz in the expected range of T t and P t (Zn). At a frequency of approximately 1 Hz in the expected range of T t the peak is not recorded (Zn) and is recorded at T t (Zn, Mo, W, steels 09G2 and 35KhGSA) only in a specific range of strain amplitude, ε = 1 × 10 –5 to 1 × 10 –4 for Zn. No peak is recorded at lower or higher strain amplitude. With the change of temperature or at high hydrostatic pressure, the peaks of Q –1 are recorded in different ranges. At low strain amplitudes in materials with a cubic lattice, there is no hysteresis and in the materials with a non-cubic lattice hysteresis is evident also at ε ≥ 10 –7 . The peaks on the Q –1 (T) dependence, recorded at different values of f and ε, can be found in the same temperature and pressure range corresponding to T t and P t . On the other hand, after a specific increase of ε, resulting in significant changes in the substructural the material, changes are recorded in the position of the peak on the Q –1 (T) dependence, denoted by 1 and 2, and the height of the peak also changes (in Zn). The change of the initial structure of the material results in the shift of the peak after annealing to higher temperatures (Zn, Mo, W) and pressures, after prior deformation results in the shift to lower temperatures and pressures (Zn), and after alloying to high or low temperatures, or no effect is recorded. The height of the peak does not change. Internal friction peaks can be recorded at lower temperatures in comparison with the range of changes of microhardness and toughness, by approximately 10–40°C (Fig. 5.18). The application of internal friction measurements in quantification of the temperature of the DB transition has certain general characteristics. In brittle polycrystalline materials with cubic and non–cubic lattices, the range of the values of T t and P t maybe characterised by the formation of an internal friction peak under certain measurement conditions. As the oscillation frequency increases, the strain amplitude decreases. This indicates changes of the dislocation substructure at T t and P t . Plastic deformation affects the height and position of the internal friction peak. Consequently, an internal friction peak can also be recorded after plastic deformation, after prior deformation at low strains. The height of the internal friction peak is determined by various reasons which depend on the type of material, the frequency of oscillations, amplitude of the material, etc. The internal friction peak, recorded at high loading frequency in the materials with a cubic lat229

Internal Friction of Materials

F ig .5.18. Effect of temperature on the properties of zinc at the DB transition ig.5.18. ( Q –1 measured at ~1 Hz).

tice, is associated with the DB transition. The phenomenological model proposed in Ref. 261 explains this conclusion by the relaxation of the density of mobile dislocations. The position of the peak is determined by T t and P t and its height depends on the frequency of oscillations. The internal friction peak, determined at a high loading frequency of materials with a non-cubic lattice, may be associated with the DB transition and the peak determined at a low loading frequency and high values of ε may be associated with microplastic deformation taking place during the DB transition. The temperature or pressure position of the peaks in the range of the values of T t and P t is associated, according to the authors of Ref. 261, with the start of the development of the DB transition (T tx , P tx ). The value T max tx depends on the structure of the material, like T t. To conclude this section, it should the added that the value of T t can be obtained from the results of the notch toughness test, the dynamic tensile test, the torsion test, etc. [262]. Every method gives different values of T t . It is evident that it is necessary to carry out extensive experiments to determine the correlation of mechanical measurements and internal friction measurements, especially for the correct physical interpretation of the observed dependence with special attention given to the fact that the DB transition is characteristic of the bcc materials with additional interstitial atoms [262]. 5.4.6 Relaxation movement of microcracks Armco iron, subjected to thermomechanical and thermomagnetic 230

Structural Instability of Alloys

treatment, shows another internal friction peak at –50°C and a frequency of ~1 Hz [263]. Measurements taken on 20GS2 high-strength steel at ~1 Hz, subjected to long-term strength tests, after different holding periods in corrosive media, also showed a similar peak (Fig. 5.19) [264] in the range 20–50°C under the effect of tensile stresses (0.5–0.8) R p 0.2. The activation energy of the relaxation peak, measured at a loading frequency change, was ~57 kJ⋅mol –1 . The peak on the Q –1 (T) dependence was recorded at a specific test time. The time to the appearance of the peak and the rate of its development (increase of the height of the peak) depend on the acting stress, for example, when σ = 0.5R p0.2 the peak is detected at the time t > 0.5 h, whereas at higher stresses this time is significantly shorter. Specially organised experiments showed that the formation of the peak is not typical of strengthened steel in the condition in which microcracks can form [264]. For example, after tensile loading the specimens of 20GS2 strengthened steel up to the formation of a neck in the central part where the microcracks appear, the height of the peak is almost doubled. The relaxation peak, associated with the formation of microcracks, was described in Ref. 275. Measurements, taken on grey cast iron, show that T max ≈ 50°C at 0.83 kHz and the activation enthalpy is 69 + 5 kJ⋅mol –1 . Differences in the mobility of the front of the areas containing graphite are the result of dif-

F ig .5.19. Temperature dependence of internal friction (a) of 20GS2 steel after ig.5.19. tensile loading without necking (1), after necking (2), after long-term strength test at σ = 0.6 R m after 0.25 h (3), 1 h (4) and 2 h (5) loading and the time dependence of internal friction (b) under the effect of σ: 6) 0.5 R m , 7) 0.6 R m , 8) 0.7 R m , 9) 0.8 R m for 20GS2 steel. 231

Internal Friction of Materials

ferent curvature of the front of these ‘microcracks’. This results in the broadening of the peak on the side of the internal friction which depends on higher temperature. Tests of Cr18Ni9 steel in a solution 160 g CuSO 4·5H 2 O, 100 ml H 2 SO 4 and 1 l of water showed always an anomalous increase of Q –1 at temperatures of approximately 80°C (f ≈ 0.9 kHz), which corresponds to the formation of submicroscopic cracks in the material. The formation of the peak on the Q –1 (T) dependence corresponds to the time to the formation of mobile microcracks and also the build-up of other structural defects, for example, dislocation clusters, cavities, etc. After recording the maximum peak of internal friction, it can be seen that the surface of the specimens contains networks of visible cracks, and the modulus of elasticity decreases by up to 30% of its initial value. The interaction of external stress at the front of the moving microcracks and around clusters of dislocations at the front of the propagation results in the formation of suitable conditions for the acts of opening and closure of microcracks and scattering of mechanical energy by relaxation. The process of migration of microcracks under the effect of stress can be divided on the basis of the ductility of the material. In the case of brittle material and at a high stress cracking takes place by the disruption of cohesion bonds at the front of the crack up to complete failure, or until the crack encounters a barrier (grain boundary, including, particles of other phases, etc.). In ductile materials, crack migration is a gradual process of merger of many microcracks into a main crack. The plastic relaxation at the tip of the cracks results in intersection of planar dislocation clusters (Fig. 5.20). When individual dis-

F ig .5.20. Interaction of the crack with planar dislocation clusters (a) and thermal ig.5.20. fluctuation formation of the microcrack (b). 232

Structural Instability of Alloys

locations approach each other, they merge and form a microcrack (Cottrell mechanism for the fcc metals). The microcracks can form by the thermal fluctuation mechanism also in the absence of external stress, for example, in planar clusters at the crack tip [266]. The equilibrium depth of thermal fluctuation microcracks, even in the case of a large number of dislocations in a planar cluster, is the value equal to several multiples of the Burgers vector, so that the merger of the microcrack with the main crack may or may not take place. In the case of cyclic loading, the process of formation of microcracks is interpreted as the emission of a bow of the dislocations to the side and subsequent joining of the parallel parts. The process of crack closure can be interpreted as the movement of the bow back to the original dislocation. The energy for opening of the crack under the effect of the stress, generated at the tip of the main crack, is several times lower than the energy required for emission of the bow.

233

Internal Friction of Materials

6 CYCLIC MICROPLASTICITY Taking into account the promising nature and prospects of the practical application of the results, and taking into account chapter 3.3, it is useful to pay special attention to the effect of the strain amplitude on the internal friction of the materials and the defect of the elasticity (Young) modulus in the range characterised by the extensive generation, movement and interaction of the dislocations, i.e. in the region of cyclic microplastic response of the materials. Mott [267] assumed that in the presence of randomly distributed dislocations situated at certain distances from each other, the random distribution of the stress exerts an effect

σi =

Gbρ1/ 2 . π

The formation of new dislocations requires the critical stress σ kr= σ i + Gb/l. If a new dislocation, generated by a source, moves by the distance d and the number of these dislocations is n, the strain increases by the value nbd. It is assumed that the sources generate dislocations in an avalanche like manner and that only some sources operate. The avalanche–like nature of generation of the dislocations is determined by the fact that the activity of the sources depends on the sum Gb/l and the local value of σ i and not on their means value. Local stress fluctuations generate the avalanche–like formation of dislocations whose intensity increases with the increase of the degree of nonuniformity of distribution of stress in the volume. Under the effect of the stress with the reverse orientation, new sources operate and generate the dislocations of the opposite sign. Golovin [268] determined the internal friction in the microplastic region in the form 234

Cyclic Microplasticity

F ig ig.. 6.1. Diagram of displacement of dislocations in the stress field.

( nb) =

2

Qp−1

ρa d 2

(6.1)

q εtm

where ρ a is the density of active sources of dislocations, q is the distance between the slip planes, m, B are constants depending on the dispersion of the stress distribution curve, ε t is the strain at which microplasticity appears. The actual conditions of dispersion of the energy in the material are complicated by the fact that the dislocation overcomes obstacles during its movement. It is assumed that ρ 0 is the number of dislocations with the length L n . During displacement, the dislocation changes its shape, for example, in accordance with Fig. 6.1. The probability of the jump from the position M to position N is described by the equation

 U − (σ − σi ) bdl  exp − 0  kT  

(6.2)

Consequently, internal friction is

Q1p =

rρ0 Ln b4G U D  U  1 (σ01 − σi ) b dl exp  − 0  , 2 kT ν πl C0  kT  σ0

(6.3)

Here r ≈ 0.5–0.75, σ i is the internal stress caused by the presence of point defects, U D = W M for non–concentrated solutions, and U 0 ~ 0.5W M for concentrated solutions. The value of l/b changes from

235

Internal Friction of Materials

1/3

  2G    (σ0 − σi ) C0 

for the non–concentrated solutions to 1/3

 Gb3   2  2WM C0 

for concentrated solutions. Davidenkov [269] used the structural heterogeneity of the material as the basis for scattering of energy in the material. In repeated loading, some subgrains and grains are subjected to microplastic deformation. From many equations, some of the equations are used for the numerical description of the hysteresis loop in the form

ε σ = Eε ± af   ε0n ,  ε0 

(6.4)

where f(ε/ε 0 ) is the function describing the shape of the hysteresis loop. This approach was developed further by Pisarenko [270] who introduced the stress distribution function in the grains and assumed that the specimens, loaded to a level lower than the yield limit in the grains, will differ and some of the specimens may show stresses higher than the yield limit. The amount of energy scattered in the unit volume of the material during a single loading cycle is ∞

∆W = 2 NK s

∫ P (σ ) d σ,

(6.5)

σx

whereN is the number of grains in the unit volume of the material, K s is the mean value of the capacity for absorption of energy in the microvolume. The integral in equation (6.5) is the quantity that determines the number of plastically deformed grains in the unit vol236

Cyclic Microplasticity

ume of the material at a specific mean which depends on the yield limit of the grain, the mean stress distribution in the grains and on the shape of the size distribution curve of the grains. The stress distribution in the grains has the form of a Gaussian dependence. Consequently m   σs   % ∆W = 2 NK s  A0 + B    ,   σk  

(6.6)

where m, B, A are constants whose value depends on the dispersion of the stress distribution in the grains. After further transformations, the following equation is obtained for the determination of internal friction

Qp−1 = −

E N% Ks B m−2 σ , πσmk

(6.7)

where σ k is the yield limit. The equation describes satisfactorily the distribution of internal friction in relation to stress amplitude σ. The problem of the effect of the magnitude of the stress amplitude on internal friction and the defect of the modulus of elasticity at higher values of the repeated loading is directly linked with the problems of cyclic microplastic deformation and quantification of the cyclic plastic response of materials. Taking into account the attempts for the quantification of the relationship between the values of the strain amplitude of the intensity of changes of Q –1 , in relation to strain amplitude (α = d log Q –1 /d log ε ac ) and the change of the response of the material to loading, it is useful to introduce a convention in the designation of the critical amplitudes (ε kr ) and the intensity of changes of characteristics (α).

6.1 CRITICAL STRAIN AMPLITUDES AND INTENSITY OF CHANGES OF CHARACTERISTICS According to Puškár [24] and the latest experimental results, the dependence Q –1 – ε or δ – γ can be conventionally divided into four sections (Fig. 6.2). 237

Internal Friction of Materials

F ig ig.. 6.2. Changes of internal friction with the change of strain amplitudes indicating critical strain amplitude and intensity of changes of the characteristics.

Under the effect of strain amplitudes with the value to ε kr1 (the first critical strain amplitudes), the increase of ε ac is accompanied by internal friction independent of the strain amplitudes. This is the background of internal friction Q –10 (section I, where the intensity of changes of Q –1 is α 0 = 0). Under the effect of strain amplitude higher than ε kr1 but lower than another critical value ε ac , there is a slight increase of the magnitude of internal friction, and with the increase of ε ac recording shows a large increase of internal friction, which is the manifestation of cyclic microplasticity. This region is (seccharacterised by the occurrence of plastic internal friction Q –1 p tion III, where α 2 > α 1 ). With a further increase of the strain amplitude it is possible to reach ε kr3 (third critical strain amplitude), and at values above this amplitude internal friction depends strongly not only on the strain amplitude but also the loading time Q t –1 (section IV, where α 3 > α 2 ). This convention and the occurrence of all or several strain critical amplitudes and the intensity of the changes of the characteristics are typical of many materials and different test and measurement conditions, and in most cases, some manifestations of the changes of the dependence Q –1 – ε are not recorded. This may be caused by the low sensitivity of measurements but in most cases by the fact that the acting physical mechanisms in heterogeneous ma238

Cyclic Microplasticity

terials operate do not operate in different subgrains, grains and phases under the effect of different strain amplitudes. Therefore, in detailed investigations in this area, in addition to determining the dependences Q –1 – ε or ∆E/E – ε, it is useful to determine reproducible values of Q –1 , ε kr1 , ε kr2, ε kr3 , but also α 1, α 2 and α 3 . The interpretation of the intensity of the changes of the characteristics (α) is only in the initial state of its development. Usually, it is necessary to use two groups of methods, based on analysis of the shape or temperature at which the relaxation maxima occur. 6.1.1 Physical nature of the critical strain amplitude The main problem in the interpretation of internal friction that depends on the strain amplitude is the behaviour of dislocations. It was assumed for a long period of time that the dislocations move in a reversible way in accordance with external alternating loading. The difference in the nature of the interaction of dislocations with the lattice and lattice defects is restricted by the second critical strain ε kr2 . Consequently, in the models used for explaining the effect of the strain amplitude on internal friction it was assumed that in the range ε ac < ε cr2 the dislocations structure does not change and that no displacement of the dislocations will be recorded after unstressing. The mechanical hysteresis loop will be closed. This type of the dependence of internal friction on strain amplitude is referred to as hysteresis internal friction Q –1 (ε). r Under the effect of strain amplitude higher than ε kr2 the increase of internal friction is associated with the irreversible movement of the dislocations, accompanied by the change of the characteristics of the dislocation structure. Since residual deformation is recorded in the material after the effect of strain amplitudes higher than ε kr2 , attention was given to the value of ε kr2 which was regarded as the boundary indicating the start of microplasticity, and the internal friction in this range of the strain amplitude was referred to as plastic internal friction Q –1 . p The division of the internal friction curve in relation to strain amplitude into the hysteresis and microplastic region is conventional. Examination by transmission electron microscopy and internal friction measurements, simultaneous evaluation of the dependences Q –1 (ε) and the curves of microdeformation in single crystals and polycrystalline materials such as Cu, Al, Ni [271] and also Cu– Al, Cu–Si, Cu–Sn and Fe–Co alloys [272] showed that the multiplication and irreversible movement of the dislocations already take place at amplitudes lower than ε kr2 . 239

Internal Friction of Materials

Initially, the processes of microplastic deformation and internal friction were studied theoretically and experimentally in separate investigations and the physical mechanisms proposed for these processes were differentiated. The dependence Q –1 (ε) was interpreted as the result of reversible movement of the dislocations around the initial positions which do not change during measurements. The process of microplasticity is associated with the generation and subsequent displacement of the dislocations. After finding that in a certain region of the dependence Q –1 (ε) some materials showed the generation and irreversible movement of the dislocations, studies appeared in which both mechanisms were combined into a single mechanism [273]. The linear section of the curves Q –1 (ε) was denoted as the first stage of microplasticity caused by the slip of dislocations in the individual grains. The second, parabolic stage of microplasticity was associated with the occurrence of plastic internal friction. The boundary, separating the linear and parabolic stages of microplasticity, is close to the value of the second critical strain amplitude. The mechanisms of release of the initial dislocations, blocked by the atoms of the solute elements, and of the generation of dislocations have different physical nature. In the former case, the process is associated with overcoming the segments of dislocations of the short–range stress fields formed by the atoms of the solute elements. As a result of the rapid decrease of the forces of dislocation– solute interaction, the distance between the equilibrium positions, corresponding to the blocked and unblocked states, is equal to several lattice spacings. The time required by the moving dislocation to travel this distance is comparable with the duration of existence of the thermal fluctuation increase of the energy of the ‘dislocation– saolute atom’ complex so that the release of the dislocation from the atmosphere of the solutes is a thermally activated process. The enthalpy of activation of the processes close to the value of H B and the activation volume is V *a ≈ L pb 2 . The activity of the dislocation sources is accompanied by the displacement of dislocations to large distances L n . The external stress overcomes the interaction forces of the dislocations with the atoms of the solute elements, and the reversible force of the linear stretching of the dislocations and of the load–range field acting over a large distance where the wavelength is comparable with the length of the dislocation segments. In this case, the process of thermal activation is of lesser importance and the activation volume increases several tens of times in comparison with value Va*. These differences 240

Cyclic Microplasticity

are even larger when the dislocations generate grain boundaries where fresh dislocations form as a result of movement of special objects — dislocations in the active part of the grain boundaries. In a real material subjected to repeated loading, the mechanisms of hysteresis and plastic internal friction operate simultaneously. The magnitude of internal friction, independent of strain amplitude Q –1 , can be interpreted using one or several of the already men0 tioned mechanisms (section 2.2.3). The strain (stress) amplitude is too low to enable dislocations segments to vibrate (Fig. 2.30a) and scatter the mechanical energy. Under the effect of stress amplitude with the value from ε kr1 to ε kr2 where Q m– 1 is recorded, the latter can be expressed by the equation proposed by Granato and Lücke [28] B

Qm−1 =

A −ε e , ε

(6.8)

here ε is the strain amplitude

L3n , L2p

(6.9)

ΩF , 4 π s aE

(6.10)

a . Lp

(6.11)

A = k1 ρ d

k1 =

B = k2 η

In these equations, ρ d is the dislocation density, Ω is the orientation factor, s is the factor of shear stress in the slip plane, E is the Young modulus, k 2 is the constant of proportionality, and η is the difference of the atom sizes of the main metal and the solutes. The value of F can be determined from the stress corresponding to the amplitude causing bending of the dislocation segment, i.e. ε kr1 . The increase of internal friction in this part of the Q –1 – ε dependence is interpreted by the non–elastic interaction of the dislocation segments with energy barriers. The dislocation seg241

Internal Friction of Materials

ments vibrate in a quasiviscous environment formed by the point defects (in steels, these are the atoms of C and N). The critical strain ε kr2 corresponds to the stress causing the generation of dislocations, i.e. ε kr2 = Gb/L n, where b is the Burgers vector of the dislocations. Therefore,

ε kr 2 =

b . Ln

(6.12)

Processing of the results of measurements of internal friction that depends slightly on the stress amplitude, using these equations, givesthe quantitative data on the dislocation structure of the material. The binding energy between the dislocation and the solute atoms (r) is expressed by the equation [274]

ε kr1 =

rc b3 E

(6.13)

where c is the concentration of solute atoms around the dislocation. For temperatures other than T = 0 K, for which the equation (6.13) was derived, it is necessary to take into account the temperature dependence of the solute concentration in the region of the dislocation [275]

c = c0 e

−

∆G kT

(6.14)

,

where ∆G is the difference of free enthalpies, k is the Boltzmann constant, T is absolute temperature. Therefore, ∆S

ε kr 1 =

r

− r − T ∆S c0e k e kT , 3 b E

(6.15)

where ∆S is the change of entropy, and c 0 is the average composition of the alloy, in at.%. The entropy change can be determined using the following equations [69] 242

Cyclic Microplasticity

∆H , T1

(6.16)

E ∂  E ∆S = n∆H  0  , ∂T

(6.17)

∆S = β

where

E ∂  E β=  0 , T  ∂   T0  E and E 0 are the elasticity moduli at experimental temperature and at absolute zero, ∆H is the activation energy of self diffusion. The quantity ∂(E/E 0) and coefficient β are determined from the temperature dependence of the square of frequency (f 2 ) of the vibrations. The activation entropy can be determined by processing the experimental results and using the above equations. Coefficient n must be determined by evaluating the entropy of self diffusion and sublimation temperature (n = 0.5–0.6). From experimental measurements, taken on Armco iron (0.03 wt.%C + N) after different heat treatments and deformation (ε) and on Mo (99.98 wt.%), using a torsion pendulum with determination of γ kr1 and γ kr2 and other characteristics, the authors of Ref. 276 obtained the experimental data presented in Table 6.1. The increase of strain is accompanied by increase of the dislocation density and the length of the dislocation segments decreases. The interpretation of the results of the Granato–Lücke theory, taking into account the atomic structure of the crystal and thermally activated unpinning of the dislocations from the pinning points, makes it possible to take into account the data on bows on dislocation lines [277]. Under the effect of strain amplitude higher than ε kr2 (section III in Fig. 6.2) we determined the internal friction Q –1 which depends p strongly on the strain amplitude. For the case in which dislocations 243

Internal Friction of Materials Ta b le 6.1 Change of critical amplitudes, parameters of dislocation structure and binding energy of the dislocations with solute atoms for materials subjected to different treatment

Ma te ria l a nd tre a tme nt Armc o iro n a nne a le d a t 9 3 0 º C /1 h a fte r e = 2 . 8 % a fte r e = 7 . 0 % a fte r e = 11 . 2 % q ue nc he d fro m 9 5 0 º C Mo lyb d e num a fte r e = 6 0 % a fte r re c rysta llisa tio n a t 1 2 0 0 º C /4 h

T [º C ]

γ k r1

γ k r2

Lp [mm]

Lp [mm]

β

r [J· 1 0 19]

ρ [m–2]

100 100 100 100 100

8 · 1 0 –5 – 1 0 –4 2 · 1 0 –4 2 . 4 · 1 0 –4 – 1 0 –4

3 . 2 · 1 0 –4 6 . 1 · 1 0 –4 8 . 0 · 1 0 –4 – 1 0 –3 5 . 9 · 1 0 –4

– 1 0 –3 4 . 5 · 1 0 –4 3 . 5 · 1 0 –4 2 . 8 · 1 0 –4 4 . 7 · 1 0 –4

9 . 8 · 1 0 –5 6 . 7 · 1 0 –5 6 . 0 · 1 0 –5 5 . 4 · 1 0 –5 8 . 0 · 1 0 –5

0.4014 0.4263 0 . 4 5 11 0.4712 0.4360

0.288 0.294 0.336 0.362 0.320

2 · 1 0 11 1 · 1 0 12 7 · 1 0 12 3 · 1 0 13 4 · 1 0 12

350

1 . 1 · 1 0 –4

2 · 1 0 –3

1 . 8 · 1 0 –4

1 . 7 · 1 0 –4

0.55

–0.416

1 · 1 0 15

350

7 . 8 · 1 0 –5

6 · 1 0 –4

5 . 0 · 1 0 –4

2 . 8 · 1 0 –4

0.555

–0.416

6 · 1 0 13

overcome obstacles by a thermally activated process [278], plastic internal friction is characterised by the equation

Q −p1 =

C D1 (ε−ε1 ) e , εh

(6.18)

where Q

C=

2ρ bν − kT e πf

and

D1 = α

VG . kT

In the equations, h is a constant with the value 0.5–1 [278], ε is the strain amplitude, ε i is the strain amplitude required for overcoming barriers [278], υ is the frequency of oscillations of the dislocations, v is the area occupied by the dislocation during a oscillation, f is the frequency of changes of loading, Q is the activation energy, α is the orientation factor (~0.5), V is the activation volume, G is the shear modulus of elasticity. Another interpretation [279] assumes that at ε ac > ε kr2 dislocation sources start to generate dislocations. In the range from ε kr2 to 2ε kr2 unstable dislocation loops interact together and with a dislocation forrest, but they do not manage to move to the barriers of the type of grain boundaries. The movement of dislocations to the grain boundaries is possible only at ε ac ≥ 2ε kr2 and dislocation clusters form in the vicinity of the obstacles. When this boundary is ex244

Cyclic Microplasticity

ceeded, plastic internal friction, determined by the equation

Q p−1 = X ε m ,

(6.19)

takes place, where X and m are the material and experimental constants. Therefore, when ε kr2 or 2ε kr2 is exceeded, microplastic deformation takes place. This microplastic deformation is of saturation nature, i.e. after a specific number of repeated loading cycles, Q –1 stap bilises at a certain value and it is assumed that the density and distribution of dislocations no longer changes significantly. The repetition of the strain amplitudes higher than ε kr3 (region IV in Fig. 6.2) results in the process of fatigue damage cumulation, i.e. internal friction Q –1 is a function of the number of load cycles. Dependp ing on the value of ε, fatigue failure of the component is recorded after a certain number of load cycles. The experimental data and interpretation of these data for describing region IV have been published only in a small number of cases, also owing o the fact that the method of measuring internal friction in evaluation of the fatigue process is being supplemented by other procedures. The form of the Q –1 –ε dependence, Fig. 6.2, is a function of many substructural and structural factors, the type and composition of the material, and the method of processing the material. Depending on the sensitivity of measurement of Q –1 and overlapping of different partial mechanisms of scattering of mechanical energy, all three critical strain amplitude may or may not be recorded. Therefore, the dependence of internal friction on strain amplitude can be characterised by the equation

Q −1 (ε ) = Q0−1 + Qm−1 ( ε, ρb ) + Qp−1 (ε, ρ p ) + Qt−1 (ε, ρ p , N ) ,

(6.20)

where ρ b is the density of pinned (stationary) dislocations with ρ b ≥ ρ 0 , where ρ 0 is the density of dislocations in the annealed material, and ρ p is the density of mobile dislocations, N is the number of load cycles. 6.1.2 Methods of evaluating critical amplitudes In the evaluation and classification of the processes, taking place in the material during its loading in the region in which the internal friction depends strongly on strain amplitude, it is necessary to use 245

Internal Friction of Materials

reproducible and substantiated (from the viewpoint of physical metallurgy) procedures of determination of critical strain amplitudes. Despite the significance of ε kr1 and ε kr3 , in determination of these amplitude there are a number of disputes and unexplained features. Several methods and procedures are still used for the determination of the second critical strain amplitude. The value ε kr2 is determined as the strain amplitude which causes rapid rising of the Q –1 curve [280] − procedure I, and as the strain amplitude which causes the Q –1 /ε – ε dependence change from a curve to a straight line [2] — procedure II. Other studies indicate that ε kr2 can also be determined using other methods in which the effect of ε kr2 and higher strain amplitude results in a sharp increase of the effect of the Young modulus (in the ∆E/E – ε dependence), i.e. procedure III [281]. The large increase of ∆E/E above ε kr2 is caused by the increase of the density of mobile dislocations. This is a typical sign of plastic deformation. The experimental studies, concerned with the evaluation of the effect of the prior strain amplitude ε d on the extent of the change of background Q –10 and the magnitude of the second critical strain amplitude, Fig. 6.3, show [281] that the evaluation of results of the measurements taken using the procedures I, II and III, gives different values of ε kr2 . In a specific range of ε d , the procedures II and

F ig ig.. 6.3. Dependence of the first and second critical strain amplitude (solid lines) and internal friction background (broken line). 246

Cyclic Microplasticity

III give similar values when determining ε kr2 , whereas procedure I proved to be unreliable, because in comparison with the data published in Ref. 2 and 282, it gives physically unjustified low values of ε kr2 . In addition to these methods, there are also methods which determine ε kr2 as the strain amplitude whose effect results in the formation of the first slip lines on the surface of the specimen [273], or as the strain amplitude resulting in irreversible changes of the internal friction background [2, 283]. The principle of the second method is based on determining the background Q –1 at ε 0 < ε kr1 . This 0 is followed by loading with a strain amplitude higher than ε kr1 and by repeated measurements of the background Q –1 at ε 0 . With gradual 0 increase of ε and measurement of Q –1 , after exceeding a certain value of ε, the recorded internal friction is already higher than Q –1 . This value then belongs to the second critical strain amplitude. 0 Experiments with the automated measurements of internal friction carried out in equipment VTP–A (VSDS) showed the possibility of determination of ε kr2 by another procedure [284]. Automatic recording of the dependence of internal friction on the loading time and of the defect of the Young modulus on loading time at 7×10 –7 ≥ ε ≥ 5×10 –4 at every experimental point, i.e. under the effect of the selected values of ε, showed that the form of the Q –1 – t dependence is complicated and is at present very difficult to interpret. However, the ∆E/E – t dependences are of two shapes. The first shape, at low values of ε, shows a decrease of ∆E/E with increasing loading time, and the decrease of ∆E/E with increasing loading time becomes smaller with increasing strain amplitude up to the state in which it no longer changes with time under the effect of a specific value of ε. The second shape, at the values close to ε kr2 and higher than ε kr2 , ∆E/E increases with increasing loading time. This shape no longer changes at high values of ε. Since at values higher than ε kr2 the ∆E/E – t curves are characterised by the saturation nature of the changes ∆E/E, which is a manifestation of the generation of dislocations, it may be assumed that ε ac at which the ∆E/E – t dependence changes from decreasing to increasing has the value ε kr2 . It is useful to note that different materials are characterised by different form of the Q –1 – ε dependences and, consequently, in experimental investigations, it is efficient to select the methods for determining ε kr2 as the important characteristic of microplasticity of the material. Some other physically substantiated method for determination of ε kr2 are presented in section 6.2. 247

Internal Friction of Materials

6.2 CYCLIC MICROPLASTIC RESPONSE OF MATERIALS The above results show that the Q –1 – ε, ∆/E – ε dependences and also Q –1 – t or ∆E/E – t dependences at certain strain amplitudes, provide, in addition to the specific values of the critical strain amplitude and their sensitivity to the history of manufacture and loading, a large number of valuable physical–metallurgical and engineering data. A number of specific examples of the measurement and application of these dependences will be discussed later. 6.2.1 Dislocation density and the activation volume of microplasticity The plastic strain rate is a function of the speed of dislocations v and the density of mobile dislocations ρ p, with the Burgers vector b, in accordance with the equation ρ p = ρ pvb; it is well known that the total dislocation density ρ = ρ 0 + ρ p , where zero is the initial density of the dislocations, and ρ p = Aε np , where A, n are material characteristics. In experiments with the materials having the structure of fcc substitutional solid solutions at the microplastic strain (ε p < 0.1), where the flow stress and plastic strain ε p are linked by a parabolic relationship, n = 1–2. The value of n for the initial stage of microplastic deformation (ε p << 0.1) is not available. It is difficult to determine ρ p in the process of continuing deformation because all experimental methods for evaluating the dislocation density are basically static, or the dislocation density is determined by indirect methods on the basis of certain assumptions, or it is assumed that ρ p in this range of microplasticity is a variable quantity. The experiments show that dislocations in, for example, annealed material (ρ 0 ) do not contribute to microplastic deformation and also that the forest of these dislocations is not a source of new dislocations. Fresh dislocations generate on the surface of the specimens, at the boundaries of grains and twins, at the second phase particles and in areas with heterogenities in the structure of the material. Measurements and analysis of the internal friction mechanisms Q –1 with the change of the strain amplitude ε ac provide a large amount of information on the dynamics of changes of the dislocations structure of the material. No quantitative relationships have been determined for describing the changes of ρ p in the strain range 10 –6 – 10 –4 . Measurements and analysis of the internal friction mechanisms, such as the dynamic method, can provide valuable information assuming we use a suitable model which would take into account the type of dislo248

Cyclic Microplasticity

cations sources, the nature and dynamics of movement of the fresh dislocations and also the nature of dislocations structures, examined by transmission electron microscopy. Cu–Al alloys are highly suitable from this viewpoint; the generation of dislocations in these alloys takes place in sources at the grain boundaries and the dislocations structure after microdeformation has the nature of planar rows of the dislocations with the same sign [285]. For Cu–Al alloy with an aluminium content of 9.2 and 13.8 at% after annealing at 600°C for 1 hour measurements were taken of the Q –1 (ε) curve on each specimens (diameter 1 mm, length 100 mm, pressure 0.5 Pa, frequency 2–2.5 Hz in equipment RKM–TPI) initially with the value of ε gradually increasing from ε = 0 to ε = ε m, where ε m > ε kr2 . The curve was obtained from 25 measurements taken on new specimens and denoted by Q –1 (↑ε) (Fig.6.4). After obtaining the selected values of ε m, we immediately measured the internal friction for the gradually decreasing value of ε ac , i.e. from ε ac = ε m to ε ac = ε 0 , and the curve was denoted Q –1 (↓ε). The experiments show that at a specific value of ε ac Q –1 (↓ε) > Q –1 (↑ε) [286]. The difference Q –1 (↓ε) – Q –1 (↑ε) was denoted by ∆Q –1 . This is the contribution of the scattering power of the material by mobile dislocations with the density ρ p which formed as a result of loading the material with the selected value of ε m . The phenomenon in which Q –1 (↓ε) ≠ Q –1 (↑ε) at a specific value of ε is universal. It does not depend on the loading method (tension–compression, torsion, bend-

ε kr1

ε kr2

F ig ig.. 6.4. Diagram for deriving the increment of internal friction as a result of increasing dislocation density. 249

Internal Friction of Materials

ing) nor on loading frequency (for example, 1 Hz or 23 kHz) as shown in other investigations [282, 287]. Analysis of the measurement results shows [282] that with increasing value ε, when ε > ε kr2 , the density of mobile dislocations ρ p in the material increases. With decreasing value ε from the selected value ε m the density of mobile dislocations remain unchanged but with increase of ε it may increase in the microplasticity range by several orders of magnitude but the measurable change of the total dislocation density is recorded only after exceeding the stress corresponding to the cyclic yield limit of the material. For example, in Cu +13.8 at.% Al alloy, the relative increase of dislocation density

ρ (ε −ρ0 ) , ρ0 is 0.1 only when ε m ≈ 0.2% [289]. Therefore, when evaluating the changes of the dislocation density during microplastic deformation, i.e. when ε m < 7×10 –4 , it can be assumed that the total dislocation density ρ and also ρ n (the density of stationary dislocations) do not change. This shows that in the examined range ε (to 7×10 –4 ) the value of Q –1 is an inversed function of strain. Consequently,

∆Q−1 (ε, εm ) = Q−p1 ε, ρ p (εm ) − Qp−1 ε, ρ p (ε ) .

(6.21)

Generally, the magnitude of ∆Q –1 depends on the time required to reach ε m, with the gradually increasing amplitude ε ac . The time dependence of Q –1 is important for pure metals, but in the Cu–Al alloys with an aluminium content of 1–17.3 at.% it is not evident, because the curves Q –1 (↓ε) and the subsequently plotted curves Q –1 (↑ε), when ε ac < ε kr2 , are more or less identical. Golovin and Levin [290] quantified microplastic internal friction for the case in which the following conditions are fulfilled: a) only dislocation sources at the grain boundaries operate during microplastic deformation and deformation is accompanied by the formation of planar rows of dislocations with the same sign; b) the dynamics of movement of the dislocations is controlled by the forces of Newtonian viscous friction. The first condition is fulfilled in the case of copper–aluminium alloys. The second condition requires a 250

Cyclic Microplasticity

comment. The materials examined were homogeneous solid solutions with short-range ordering. The movement of the first dislocation of the planar row of the dislocations is associated with overcoming the resistance of the lattice whose value in the slip plane is influenced by the short-range ordering of the solid solution. However, after movement of several dislocations in the slip plane, the solid solution is no longer ordered and the movement of further dislocations of the planar row is controlled by the viscous friction mechanisms. After the arrest of the first moving dislocation in front of an obstacle of the type of sub–boundaries, dislocation clusters, etc., a planar row of dislocations with length 2L starts to form behind the dislocation. Alternating loading does not change this arrangement of the dislocations, because the latter do not return to the sources. Local vibrations of the segments of the moving dislocations in the planar row of the dislocations then contribute to microplastic internal friction. This model shows that

Q−1 (↑ ε ) = Qp−1 ε, ρ p ( ε ) =

1.24BωL2ρ p ( L ) Gε2

Q−1 (↓ ε) = Qp−1 ε, ρ p (εm ) = 1.178

,

BωL2ρ p ( L)  ρ p (εm )  − ... , 1 + 24ε2m Gε2m  

(6.22)

(6.23)

where G is the shear modulus of elasticity, ω is the frequency of external loading, B is the viscous factor of movement of the dislocations. The low values of ε made it possible to simplify the form of equations, because at ε/ε m << 1; this is fulfilled under the given conditions. The series on the right–hand side of equation (6.23) can be replaced with sufficient accuracy by its first term. Consequently, at low strain amplitudes

∆Q−1 =

3πBωL2 ρ p ( εm ) . 8G ε2m

(6.24)

Equation (6.24) makes it possible to characterised the function ρ p (ε m). Taking into account the given model, it can be seen that the experimental points are distributed on straight lines whose tangent 251

Internal Friction of Materials

is approximately 3. This shows that

ρ p = A ε3m ,

(6.25)

which indicates that the density of moving dislocations in the initial stage of microplasticity of the Cu–Al alloys increases with the cube of the plastic strain. The intensity of generation of dislocations during the microplasticity phenomena has two values n in the equation ρ p = Aε np . In the first part of the deformation curve, where the strain is a linear function of stress, the plastic strain ε p is proportional to the maximum strain ε m so that the density of moving dislocations is expressed by the equation ρ p = Cε np , where n = 3. In the second part of the deformation curve, where strain ε is a parabolic function of stress, ε p ~ ε 0.5 . Consequently, ρ p = Dε 1.5 , where m p n = 3/2. These data should be included in the information on the behaviour of an ensemble of the dislocations, on the annihilation of dislocations of the same time with the reverse sign and on the establishment of the saturated value of dislocation density [24, 291]. Analysis of the dependences Q –1 (ε) makes it possible to quantify certain characteristics of the dislocation network, the stress conditions of the start of microplasticity, the dynamics of changes of the microstructure, differentiate the mechanisms of scattering of mechanical energy in the material or also determine, for example, the size of the activation volume of microplastic deformation. In the range ε kr1 ≤ ε < ε kr2 , the scattering of the mechanical energy in the material can be interpreted by the spring model in accordance with Granato and Lücke [28], using equation (6.8). Interpretation of the factors A, B makes it possible to quantify certain parameters of the dislocations structure and also the characteristics of the interaction of dislocations with point defects, especially interstitial atoms. Therefore, it can be expected that in this range of ε we obtain straight lines when the experimental data are plotted in the coordinates ln Q –1 vs ε –1 with the straight lines havm ing the slope B. Several models have been proposed for the range ε kr2 ≤ ε ≤ ε kr3 . Peguin, Perez and Gobin [278] assumed that the dislocations overcome obstacles in their movement by a thermally activated process. The appropriate activation energy depends on the level of acting mechanical stress. Consequently, plastic internal friction 252

Cyclic Microplasticity

Q –1p (= Q –1 – Q 2–1 , where Q –1 or Q –12 is internal friction at ε > ε kr2 , or ε ε1 at ε = ε kr2) is determined by equation (6.18). Burdett [280] assumed that the activation volume depends on the level of acting stress in accordance with the equation

V=

F

( σ − σi )x

,

(6.26)

where F and x are constants. According to Spitzig [292], x = 0.5. After substituting and transforming equation (6.18), where the given model is realistic, it may be expected that in the range of cyclic microplasticity in the coordinates ln (Q –1 ε) vs. (ε – ε i ) 1/2, the experimentally determined points will fit straight lines with the slope corresponding to D. Jon, Mason and Beshers [279] derived equation (6.19). If this modelling assumption is valid, we can expect linearisation of the experimental measurements, when the results are plotted in the coordinates log Q –1 – log ε. These hypotheses have been verified in Ref. 293. In the first part of the experiments, the authors used pure copper (99.994 wt.%), denoted Cu, iron with 0.03 wt.% C, denoted Fe, and an alloy of iron with titanium (0.043 wt.% Fe, 0.03 wt.% Ti), denoted Fe–Ti. The internal friction of Cu, Fe and Fe–Ti in relation to the strain amplitude was measured in low–frequency (approximately 1 Hz) vacuum equipment using RKM–TPI torsion pendulum, on specimens of Fe and Fe–Ti in the presence of a magnetic field with an intensity of 1.9×10 4 A⋅m –1 , at a temperature of 23°C. In the second part, the experiments are carried out on iron with 0.03 wt.% C (Fe) and on CSN 412013 steel (0.07 wt.% C, 0.27 wt.% Mn, 0.03 wt.% Si, 0.013 wt.% P, 0.08 wt.% S, 0.07 wt.% Cr, and 0.006 wt.% N) after heat treatment; the ferrite grain size of Fe was 0.032 mm, that of the steel 0.022 mm. Experiments were carried out in RKM–TPI equipment in the presence of a magnetic field and at the same temperature as in the first part of the experiments. Measurements were taken at a frequency of 1 Hz in the laboratory of the Tula Polytechnic Institute, Russia. The ferrite grain size of CSN 412013 steel after heat treatment was d 1 = 0.022 + 0.004 mm, d 2 = 0.29 + 0.045 mm, d 3 = 0.620 + 0.085 mm. The specimens of CSN 412013 steel, diameter 3 mm, were subjected to stabilisation annealing at 220°C for 0.5 hour and 253

Internal Friction of Materials

chemically polished. The dependence of internal friction on the strain amplitude in tension–compression loading was measured in a modified Mason system at a loading frequency of 23 kHz in the presence of a magnetic field with an intensity of 1.9×10 A⋅m –1 , at a temperature of 23°C. The experiments were carried out at the laboratories of VSDS Technical University in Zilina, Slovak Republic. The first modelling assumption, characterised by equation (6.18), brings linearisation only if we use Burdett’s interpretation. For other materials (Cu, Fe – Ti), the ln Q –1p ε vs. (ε – ε i ) 1/2 dependence is characterised by curves, which may indicate that this model does not characterise efficiently the interaction of materials with repeated mechanical loading. Appropriate atmospheres of C and N formed in iron with 0.03 wt.% C t temperatures lower than the condensation temperature of the atmospheres of solute elements. In the Fe–Ti alloy, the dislocations are freed from C and N atoms and only very weak interaction of the solute substitutional atoms with the dislocations is observed in copper. The second modelling assumption, characterised by equation (6.19) results in linearisation in the case of copper when ε > 6.3×10 –5 , or Fe – Ti, when ε > 1.6×10 –4 . The authors of the modelling assumptions [279] prepared this model for materials such as bronze, copper, etc., i.e. the materials characterised by a very weak effect of the substitutional solute atoms of the dislocations. The first part of the experiments showed that in the case of the materials characterised by high–intensity interaction of the solutes with the dislocations, it is convenient to use Burdett’s approximation [280] also for the quantification of other parameters of the activation of cyclic microplasticity. The results of measurements of the internal friction of materials with high–intensity interaction of the solutes with the dislocations in relation to the strain amplitude at a frequency of 1 Hz and 23 kHz are shown in Fig. 6.5, where each curve is the average of 5 measurements. The magnitude of the first critical strain amplitude, determined by the method presented in Ref. 281 for the examined materials, is shown in Table 6.2, and if we use equation (6.8), the results can be plotted in Fig. 6.6 and the graph can be used to determine the values of ε i , presented in Table 6.2. The processing of the results of measurements in accordance with the first model [278], supplemented by Burdett [280], again confirms the linearisation of ln Q –1p ε vs (ε – ε i ) 1/2 for the region of cyclic microplasticity (Fig. 6.7). This enables the equations for cal254

Cyclic Microplasticity

F ig ig.. 6.5. Dependence of internal friction on strain amplitude for different materials loaded with a frequency of 1 Hz and 23 kHz.

F ig .6.6. Results of measurements for the case of validity of equation (6.8) and ig.6.6. different materials loaded with a frequency of 1 Hz and 23 kHz

255

Internal Friction of Materials Ta b le 6.2 Activation parameters of cyclic microplasticity ( b is Burgers vector) Ma te ria l

Gra in size [mm]

Lo a d ing fre q ue nc y

ε 1· 1 0 4

ε i· 1 0 4

F e+0.03 wt . % C 12 013

0.032

1 Hz

1.4

4.0

0.022

1 Hz

2.0

4.4

12 013 12 013 12 013

0.022 0.290 0.620

2 3 k Hz 2 3 k Hz 2 3 k Hz

1.3 1.0 0.7

2.2 1.9 1.3

V [mm3]

V/b 3

–18

53

9 . 1 5 · 1 0 –19

39

1.25·10

F ig .6.7. Results of measurements in Fig.6.7 for the case of validity of modified ig.6.7. equation (6.18) and different materials loaded with a frequency of 1 Hz and 23 kHz

culating D 1 (6.18) and (6.26) to be used for the determination of the activation volume of the microplasticity process. For the selected value (σ – σ i ) = 14.7 N⋅mm –2 , the activation volume at a frequency of 1 Hz is 1.25×10 –18 mm 3 , and at a frequency of 23 kHz it is 9.15×10 –19 mm 3 . In Ref. 292, the activation volume under the effect of the same effective stress (σ – σ i ) = 15 N⋅mm –2 ) and at a frequency of approximately 1 Hz was 1.2×10 –18 mm 3 . For iron with a grain size of 0.048; 0.057 and 0.076 mm, the activation volume according to Burdett [280] is 1.3×10 –18 mm 3 . The difference in the activation volume of the valid materials at 256

Cyclic Microplasticity

a loading frequency of 1 Hz and 23 kHz is relatively small, Table 6.2. The experiments showed that the activation volume of the microplasticity of the examined steels at a temperature of 23°C and an effective stress of 14.7 N⋅mm –2 is approximately 10 –18 mm 3 . The results of the experiments indicate that in the case of the materials with significant interaction of the solute atoms with the dislocations in the range of cyclic microplasticity it is possible to use the interpretation proposed by Peguin, Perez and Gobin [278], after supplementing the approximation according to Burdett [280]. The mechanism of cyclic microplasticity is determined by the intensity of the interaction of the atoms of the solutes with the dislocations and is not influenced by the frequency of loading changes. The importance of measurements of internal friction and the Young modulus defect with the change of strain amplitude will be shown on other examples.

6.2.2 Condensation temperature of the atmospheres of solute elements The transition from low–density (Maxwell) to dense (Cottrell) atmospheres of the solute interstitial elements on the dislocations controls the microplastic behaviour of the material also under repeated loading. Reproducible values of condensation temperature T c can be determined by internal friction measurements under different thermal and amplitude conditions [294]. Low–carbon unalloyed steel with composition: 0.07% C, 0.27% mn, 0.03% Si, 0.013% P, 0.018% Cr, 0.07% Cr and 0.006% N (CSN 412013) was annealed during the measurements of internal pressure (dependent on temperature and strain amplitude), using the method of the torsional pendulum [92] with a vibrational frequency of ~1 Hz in the strain amplitude range γ from 5×10 –6 to 7×10 –4 . Some of the measurements were taken under the effect of a constant magnetic field with the intensity H = 1.7×10 –7 A⋅m –1 , and other measurements were taken at H = 0. The experiments were carried out at the laboratories of the Tula Polytechnic Institute in Tula, Russia. Bars with a diameter of 20 mm were also used to produce testpieces similar to tensile bars with a diameter in the central part of 3 mm [281]. Internal friction Q –1 and ∆E/E at a frequency of 23 kHz were measured by the method described by Mason [282]. The specimens welded with symmetric tension–compression at a tempera257

Internal Friction of Materials

ture of 23°C in the strain rate range ε from 1.5×10 –6 to 6×10 –4 . Some of the measurements were taken under the effect of a constant magnetic field with the intensity H = 1.9×10 –4 A⋅m –1 , or at H = 0. The experiments were conducted at the laboratories of the VSDS Technical University in Zilina. Prior to the measurements, the specimens in both experiments were initially annealed at 720°C for 0.5 hr and cooled at a rate of 100°C/h in a furnace; this was followed by chemical etching of the surface of the specimens. The ferrite grain size of the specimens was 0.022 ± 0.004 mm in both cases. The measurements of internal friction at a loading frequency of ~1 Hz in the temperature range 20 – 440°C on specimens of the low–carbon unalloyed steel after annealing (Fig. 6.8, curve 2) and also after rapid cooling from 725°C (Fig. 6.8, curve 1) show that in both conditions of the material, the maximum on the Q –1 – T curve is recorded at 40°C (Snoek maximum). The cabin content of the solid solution of α-iron, calculated from the Snoek maximum, was 0.003 wt.%% after annealing and 0.014 wt.%% after rapid cooling from 725°C. The activation energy of the maximum on the Q –1 – T curve, determined from the Wert–Marx equation, was ~80.8 kJ ⋅mol –1, which corresponds to the activation energy of carbon diffusion in the solid solution of iron.

F ig ig.. 6.8. Temperature dependence of internal friction for mild steel after rapid cooling from 725°C (curve 1) and after annealing (curve 2) at a loading frequency of ~ 1 Hz. 258

Cyclic Microplasticity

Fig ig.. 6.9. Dependence of internal friction on strain amplitude for different temperatures at a loading frequency of ~1 Hz (arrows indicate the magnitude of τ kr 1 ).

Figure 6.8 shows that the low value of the internal friction of the annual material does not change when temperature is increased from 100 to 400°C. When 400°C is exceeded, the characteristic increases. Measurements were taken at γ ≅ 5×10 –6 and under the effect of a magnetic field with intensity H = 1.7×10 4 Am –1 . The data on the unstable pinning of the dislocations by the atmospheres of the interstitial elements at higher temperatures confirm the results of internal friction measurements (internal friction depends on strain amplitude), at different temperatures in the range from 20 to 550°C, as indicated in Fig. 6.9. Internal friction measurements at different strain amplitude changing with a frequency of ~1 Hz, under the effect of a magnetic field with an intensity of 1.7×10 4 A⋅m –1 , show that the form of the Q –1 – dependence in the temperature range 100 – 400°C does not change greatly. The strain amplitude indicating the start of internal friction, dependent on the strain amplitude, is denoted γ kr1 and its value is ~7×10 –5 . At temperatures of 425°C and higher temperatures, the authors detected a significant increase of the magnitude of internal friction, and a large decrease of the strain amplitude amplitude the start of the region of internal friction, dependent on strain amplitude (γ kr1 ), was recorded. At γ > γ kr1 , the magnitude of 259

Internal Friction of Materials

F ig ig.. 6.10. Temperature dependence of the first critical strain amplitude during loading with a frequency of ~1 Hz.

increase of internal friction per unit increase of the strain amplitude also changes. Figure 6.10 shows that up to 400°C γ kr1 ≅ 7×10 –5 , and with increase of temperature this value decreases exponentially so that at 550°C γ kr1 = 10 –5 . The experimental dependences of the internal friction on the strain amplitude, determined on specimens of mild steel with the same structure at a frequency of ~1 Hz and 23 kHz under the effect of a constant magnetic field or in its absence (Fig. 6.11 and 6.12) indicate that regardless of the different nature of loading and the accuracy of the methods used, the curves Q –1 – γ and Q –1 – ε are similar. The Q –1 – ε curves, plotted at a loading frequency of 23 kHz (Fig. 6.12) and ε < ε kr1 shows the strain amplitude ε. Consequently, in the range from ε to ε kr1 we can see a slightly higher value of internal friction in comparison with that obtained at a loading frequency of ~1 Hz and the corresponding method of determination of internal friction in this region. The value was reproducible and exceeded the extent of scatter of internal friction values. At γ > γ kr1 or ε > ε kr1 (γ kr1 = 2×10 –5 or ε kr1 = 1.2×10 –5 for a loading frequency of ~1 Hz or 23 kHz at H = 0 and γ kr1 = 7×10 –5 , or ε kr1 = 6.4×10 –5 at a loading frequency of ~1 Hz or 23 kHz at H = 1.7×10 –4 or 1.9×10 –4 A⋅m –1 ) the curves show a strong dependence of internal friction on strain amplitude. The position of the 260

Cyclic Microplasticity

F ig ig.. 6.11. Q –1 – τ dependence at a loading frequency of ~1 Hz without the effect of the magnetic field (solid lines) and with the effect of the field (broken fields) indicating the Q–1 – τ dependence and also changes of the internal friction background at the second critical amplitude.

F ig ig.. 6.12. The Q –1 – ε and ∆ E / E – dependences at a loading frequency of 23 kHz without (solid lines) and with the effect of the magnetic field (broken lines).

critical strain amplitude ε kr1 can be recorded quite efficiently by the measurements of ∆E/E at a loading frequency of 23 kHz, because the first values of ∆E/E with a gradually increasing strain amplitude are obtained at ε kr1 . 261

Internal Friction of Materials

The modified methods of internal friction measurements in the range of the dependence of internal friction on strain amplitude (after determining the value of internal friction at we determined internal friction at ε = 5×10 –6 , and this was followed by internal friction measurements at ε x1 ≤ ε kr1 and again by internal friction measurements at ε = 5×10 –6 , etc.), we determined the strain amplitude at which the internal friction background (Q –10 ) increases irreversibly. These strain amplitudes were denoted γ kr2 or ε kr2 ; they represent the strain amplitudes in the material. At a loading frequency of ~1 Hz or 23 kHz, ε kr2 = 3.4×10 –4 at H = 1.7×10 –4 , or 1.9×10 –4 A⋅m –1 . The critical strain amplitude γ kr2 was inspected at a loading frequency of up of 1 Hz also by the method of ‘zero shift’ on the scale of the measuring device. Another method was applied at a loading frequency of ~1 Hz as well as at 23 Hz Q –1 /ε – ε using Golovin’s procedure. The results are in the top righthand corner of Fig.6.11. At a loading frequency of 23 kHz the investigations were supplemented by microscopic examination of the specimens. The results show that at ε > ε kr2 the first slip lines appear on the surface of the specimens [282]. These methods, characterising the second critical strain amplitude, give, at both frequencies, the values in the range from 2×10 –4 to 3×10 –4 . The application of a sufficiently strong magnetic field to suppress the magnetomechanical component of internal friction in the ferromagnetic material for both loading frequencies decreases the internal friction background, increases the critical strain amplitude γ kr1 or ε kr1 , decreases the width of the range between γ kr1 and γ kr2, or ε kr1 or ε kr2 , and increases the value of γ kr2 or ε kr2 . The position of the critical amplitude characterising the microplastic strain γ kr2 or ε kr2 can be efficiently recorded by all the given methods with the application of the magnetic field to the loaded material. In the independent experiments with the measurements of internal friction in mild steel, the authors of Ref. 249 determined the cyclic strain amplitude range characterised by the operation of different energy scattering mechanisms at a frequency of ~1 Hz and 23 kHz, and showed that the evaluation criteria for the results are comparable. Several methods were used to determine the cyclic strain amplitudes at which microplastic deformation starts. On the basis of the agreement of results it can be concluded that in cyclic loading of mild steel in the annealed condition, without the magnetic field, the cyclic strain amplitude is (3.5–6)×10 –4 . 262

Cyclic Microplasticity

The condensation temperature of the atmospheres of the solute elements in the evaluated steel is ~425°C. The activation energy of occurrence of γ kr1 or γ kr2 at a loading frequency of ~1 Hz is 24.1 or 16.7 kJ⋅mol –1 , and the activation energy of occurrence of ε kr1 or ε kr2 at a loading frequency of 23 kHz is 16.2 or 13.6 kJ⋅mol –1 . The differences in the activation energies of the occurrence of the critical amplitudes were interpreted by the authors from the viewpoint of the differences in the shape and mobility of the dislocation segments at the given loading frequencies. The cyclic plastic response of the materials is significantly influenced by many factors of the history of the specimens, especially by prior cyclic plastic deformation. Its effects can be efficiently evaluated by measuring the internal friction and the defect of the Young modulus [295]. 6.2.3 Deformation history In order to investigate the effect of prior cyclic microplastic deformation on internal friction and changes of the dislocation density, the author of this book carried out [281] experiments on mild unalloyed steel (0.07 wt.% C, 0.006 wt.% N, 0.27 wt.% Mn, 0.03 wt.% Si, 0.013 wt.% P, 0.018 wt.% S, 0.07 wt.% Cr), the ferrite grain size was 0.022 ± 0.004 mm. In the preparation of the specimens with a diameter of 3 mm in the central part, the specimens were annealed for 30 minutes at 200°C in a shielding atmosphere. This was followed by chemical polishing of the surface of the specimens. Symmetric cyclic loading with a frequency of 23 kHz of the pull–push type with the strain amplitude in the central part of specimens of 1 × 10 –6 – 7 × 10 –4 was carried out for 45 seconds, i.e. 1×10 6 load cycles. This was accompanied by the measurements of internal friction Q –1 and relative changes of the Young modulus, the so–called defect of the Young modulus ∆E/E. The experiments were carried out at a temperature of 21°C, using a constant magnetic field with a strength of 1.9×10 A⋅m –1 to eliminate the magnetomechanical component of the internal friction in the ferromagnetic material. At a gradually increasing strain amplitude in a series of specimens, the main dependences Q –1 – ε or ∆E/E – ε (Fig. 6.13, curves 1) were initially determined. After evaluating the critical strain amplitudes, other batches of the specimens were initially loaded with strain amplitudes of ε d = 2.6×10, ε d = 3.5×10 –4 , ε d ≥ 5.1×10 –4 , ε d = 6.5×10 –4 for 120 seconds, representing 2.8×10 6 cycles; this was followed immediately by gradual loading of the specimens from a strain amplitude of 263

Internal Friction of Materials

F ig ig.. 6.13. Dependence of Q –1 for prestrained steel: 1) ε d = 0; 2) 2.6 × 10 –4 , 3) 3.5 × 10 –4 ; 4) 5.1 × 10 –4 ; 5) 6.5 × 10 –4 . The arrows pointing upwards indicate ε kr2 and those downwards ε kr 2, where ε kr 1 and ε kr2 were determined from the occurrence and changes of ∆ E / E in relation to ε.

1×10 –6 to 7×10 –4, with simultaneous measurements of Q –1 , ∆E/E, always for 45 seconds. This gave the corresponding dependences Q –1 – ε or ∆E/E (Fig. 6.13, curves 2, 3, 4, 5). The dependence of internal friction on strain amplitude for the specimen series subjected to different cyclic prior deformation (Fig. 6.13) indicates that the form of the curves 1–4 is identical in a wide strain amplitude range and the magnitude of prior cyclic deformation changes the rate of increase of Q –1 only at ε ≥ 1 – 5 × 10 –4 . Significantly different Q –1 –ε dependences were obtained for the specimens subjected to prior deformation at ε d = 6.5 × 10 –4 (curve 5). The main value of internal friction slightly increases with increasing ε d , but after the application of ε d = 6.5 × 10 –4 the increase of Q –10 is significant. Valuable information on the activation of microplastic processes is provided by the measurements of the defect of the Young modulus ∆E/E. The first reproducibly determined values of ∆E/E were obtained at relatively low values of ε kr1. With the increase of ε the defect of the Young modulus slowly increases, and after the application of certain strain amplitudes and the increase of these amplitudes there was a significant increase of ∆E/E. Recording of the first values of ∆E/E detects the strain amplitude at which the dislocation segments start to oscillate in the 264

Cyclic Microplasticity

stress field around the equilibrium positions (the first critical strain amplitude ε kr1 ). The absolute values of internal friction and the defect of the Young modulus are higher for the specimens subjected to preliminary cyclic deformation at a higher strain amplitude. In the case of mild steel in accordance with the experimental results [283] we can use the modelling assumptions of the vibration of a spring for the vibration of dislocation segments around their equilibrium positions in the quasiviscous environment. In this modelling assumption, the ratio of the square of the defect of the Young modulus and internal friction determines the instantaneous dislocation density in the material using the equation [296] 2

 ∆E     E  = k ρ, Q −1

(6.27)

where k is a proportionality factor. In accordance with a large number of experimental results, it is assumed that in low–carbon unalloyed steel after annealing ρ = 10 12 m –2 . Cyclic prior deformation changes the dislocation density only when ε d > ε kr2. The increment of the strain amplitude ∆ε = ε – ε kr2 in the microplastic region of cyclic loading results in the increase of the dislocation density by the value ∆ρ = ρ – ρ 2, where ε are the values of the strain amplitude and the dislocation density in the range ε > ε kr2 , or ρ > ρ 2 , and the values ε kr2 , ρ 2 are the second critical strain amplitude and the initial dislocation density in the material. In the coordinates log ∆ρ – log ∆ε the results were plotted in the form of straight lines which can be described by the equation

∆ρ = a ∆ε m ,

(6.28)

and the values of the exponent m are also a function of the magnitude of prior cyclic deformation of the material. The validity of equation (6.28) indicates the process of irreversible changes in the structure of the steel by the increase of dislocation density with increasing strain amplitude. In the range of the start of microplasticity (ε > ε kr2) the increase of ε is accompanied by an increase of the dislocation density and of the magnitude of ‘plastic’ internal friction Q –1p , indicating the 265

Internal Friction of Materials

elastic–plastic interaction of mechanical loading with structural changes of the material, i.e. the existence and effect of the plastic strain amplitude ε p. To a first approximation, it will be assumed that

ε p = ∆ε = ε − ε kr 2 .

(6.29)

The cyclic strain curve, stress amplitude σ a vs. the plastic strain amplitude ε p , characterising the plastic response of the material to cyclic loading, has the form

σ a = χ ε np ,

(6.30)

where χ is a constant of the material that is sensitive to the structure and experimental conditions, and n is the cyclic strain hardening exponent. Using the equations (6.28), (6.29) and (6.30), at ε > ε kr2 , we obtain the relationship between the stress amplitude and the increase of the dislocation density in the form n

(6.31)

σ a = λ ∆σ m ,

where λ = χ a –(n/m) . Substituting equation (6.27) into equation (6.31), using the given symbols, we obtain the relationship between stress amplitude σ a , the value of the effect of the Young modulus ∆E/E and the magnitude of internal friction Q –1 . These equations and the values of exponent m, determined from the experiments at the same value of the exponent n for mild steel [297], with the change of the amplitude of prior strain ε d , yield some qualitative and also quantitative relationships characterising the cyclic plastic response of the material. At the selected value of ∆ε, the increase of the dislocation density ε d is more marked (equation 6.31, values m). These qualitative relationships indicate the softening effect of cyclic microplastic deformation with strain amplitudes from 2.6 × 10 –4 to 6.5 × 10 –4 in the case of mild steel in the region of the start of microplasticity; this has also been confirmed indirectly by other studies [279, 296]. The experimentally determined values of Q –1 (and also Q –10, Q –1p ), ∆E/E, ε kr1 and ε kr2 with the increase of the strain amplitude, after prior cyclic loading of the steel, are the results of more or less ex266

Cyclic Microplasticity

tensive release of the dislocation segments from the area of weak pinning by the atmospheres of C and N at the given prior strain amplitude. The processes taking place under the effect of ε ≥ ε kr2 can be interpreted in the sense of extensive microplastic deformation on the basis of statistical considerations regarding the structural heterogeneity of the material. The cyclic strain curve and its parameters are important for the quantification of the cyclic plastic response of materials [298]. Measurements of internal friction and the defect of Young modulus with the change of the strain amplitude also provide important information in this area. 6.2.4 Cyclic strain curve It has been shown in many studies that the plastic strain amplitude is the controlling factor of fatigue damage and the formation and propagation of static cracks. This amplitude can be characterised as half width of the hysteresis loop of the material. In loading with frequencies of up to 300 Hz the hysteresis loops are recorded by the testing machine, but at ultrasound frequencies (for example, approximately 20 kHz) direct measurements and recording is not possible at the current state of measuring devices. Since the measurement of internal friction and of the defect of the Young modulus provides information on the integral representation of the microplasticity process, these measurements can also be utilised in the quantification of the plastic strain amplitude at ultrasound loading frequencies. Low–carbon, unalloyed steel CSN 412013 (0.07 wt.% C, 0.27 wt.% Mn, 0.03 wt.% Si, 0.013 wt.% P, 0.08 wt.% S, 0.07 wt.% Cr and 0.006 wt.% N) after annealing at 720°C ± 20 hours for 4.25 and 108 hours with slow cooling in the furnace was characterised by the ferrite grain size of d z1 = 0.022 ± 0.044 mm or d z2 = 0.29 ± 0.045 mm, or d z3 = 0.620 ± 0.085 mm. After completion of heat treatment, in order to carry out measurements in VTP equipment, the specimens were annealed in Ar at 200°C for 330 min. and cooled in the furnace at a rate of 100°C/h and, in the final stage, subjected to chemical polishing. Symmetric cyclic pull–push loading with a frequency of 23 kHz in the central part of the specimens was applied by a resonant system described in a previous study [282]. Equipment makes it possible to generate the total strain amplitude ε ac in the evaluated cross section of the specimen in the range from 6×10 –6 to 4×10 –4 , with a reproducibility better than 1.5×10 –7 . The total strain amplitude ε ac 267

Internal Friction of Materials

was measured with strain gauges fixed to the essential part of the specimens and evaluated using a Wheatstone bridge with a selective nanovoltmeter. The calibration of the proportionality factor of the strain gauges was measured by the deviation of the free end of the system with an accuracy of ±1 µm, using TW5/2A sensors and Vibrometer AG equipment with a frequency range from 0 to 100 kHz. The proportionality factor of the strain gauges at a loading frequency of 23 kHz was 1.8 2%, whereas at the usual frequencies the proportionality factor is 1.95 + 2%. The stress amplitude in the central part of the specimen was evaluated by approximation in accordance with Fig. 6.14. In the elastic loading range σ = E ε a, and the value E (= tan α) is related with the main resonance frequency fr. In the elastic–plastic region of loading (for example, point x in Fig. 6.14) σ ax = E x (ε aex + ε apx ) = E x ε acx , and the value E x (= tan α x) corresponds to the resonance frequency of the system fr x. Consequently, E x = E – ∆E x , where ∆E x is the change of the Young modulus associated with the microplastic deformation of the specimen. These considerations show that the stress amplitude, for example, at point x

F ig ig.. 6.14. Determination of relationship between stress and strain. 268

Cyclic Microplasticity

 ∆E x  σ ax = ε acx ( E − ∆E x ) = ε acx E 1 −  E  

(6.32)

and the amplitude of the plastic component of strain

ε apx = ε acx − ε acx = ε acx −

σ ax ∆E x = ε acx . E E

(6.33)

The advantage of experimental equipment is that it makes it possible, at the selected value ε ac , to measure not only the magnitude of internal friction Q –1 but also evaluate fr x and, in accordance with Mason [299], determine directly

∆E x 2 M fr − frx , = E Mν fr

(6.34)

where M and M v the effective mass of the entire system and of the specimen. The measuring equipment and corrections made it possible to evaluate ε ac with the accuracy of ±1×10 –6 and the changes of ε ac with a scatter of ±1.5 × 10 –7 , the relative change of the Young modulus with the accuracy of 6 × 10 –4 and the changes are also made by connecting strain gauges to the reduced section of the titanium attachment of the system at the distance of λ/4 from the end of the attachment, in which no microplastic deformation took place up to ε ac = 4 × 10 –3 [299]. After evaluating the changes of the value of the amplification factor in the central part of the specimens, the author found that the agreement of the results determined by this approximation and by the direct measurement of the stress amplitude is better than 98%. The experiments were carried out at a temperature of 21°C under the effect of a constant magnetic field with an intensity of 1.9 × 10 4 A m –1 , parallel with the axis of the specimen. Series of the specimens with the grain sizes of d z1 , d z2 , and d z3 , were loaded gradually with increasing strain amplitudes ε ac . At every selected value of ε ac, the author determined internal friction Q –1 , the relative change of the Young modulus ∆E/E and, using equations (6.32) and (6.33), also the stress amplitude σ a and plastic strain amplitude ε ap . In calibration, it was observed that the loading of the specimens for 1 hour with ε ac = 6 × 10 –4 does not cause any measurable changes of the temperature of the specimen so that there cannot be 269

Internal Friction of Materials

any changes of the Young modulus of the evaluated material as a result of the change of the temperature of the specimen. Measurement of every experimental point at ε ac = const was carried out over a period of 100 s, which represents 2.30 × 10 6 of loading, sufficient for the stabilisation of the changes of the properties of the material in the evaluated strain amplitude range. The characteristic course of the changes of internal friction in relation to the total strain amplitude (Fig. 6.15) showed the effect of the ferrite grain size on the internal friction background, determined at a low value ε ac , and the differences in the form of the Q –1 – ε ac dependence in the region of the dependence of Q –1 on strain amplitude. The relative change of the Young modulus ∆E/E is recorded after obtaining certain values of ε ac , and rapidly increases with increase of the total strain amplitude of the specimen. Conventionally, the value ε ac resulting in a sharp increase of ∆E/E, can be denoted as the second critical strain amplitude ε kr2 [294]. The values of ε kr2 are the function of the size of the ferrite grain, i.e. the critical strain amplitude decreases with increasing grain size (Table 6.3). The intensity of the changes of ∆E/E in relation to ε ac in the range ε ac > ε kr2 is higher in the case of steels with larger ferrite grains. In the coordinates log ∆E/E – log ε ac , Fig. 6.15, the experimental points fit straight lines so that the experimentally obtained points can be expressed by the equation

F ig .6.15. Q –1 – ε (solid lines) and ∆ E / E – ε dependences (broken lines) for different ig.6.15. grain sizes: 1,1') 0.022 mm; 2,2') 0.290 mm; 3,3') 0.620 mm. 270

Cyclic Microplasticity

∆E = A ε aac , E

(6.35)

where A is the constant that depends on the structure and experimental conditions, a is the exponent with the values given in Table 6.3, with both factors dependent on the ferrite grain size. The experimentally determined dependence of stress amplitude σ a on the total strain amplitude ε ac (Fig. 6.16) shows that the deviation of the dependence from the straight line, determined by the equation σ = E ε ac , starts in the case of specimens with different ferrite grain sizes at different values of ε ac . The ‘stress amplitude vs. plastic strain amplitude’ cyclic curves, shown in Fig. 6.16, indicate that the ferrite grain size has a significant effect on the form of these dependences. The cyclic strain curves can be characterised by equation (6.30). Figure 6.17 shows that the equation of the cyclic strain curve is also fulfilled for a loading frequency of 23 kHz, with the values of the factor χ, given in Table 6.3. With the increase of the ferrite grain size, the value of χ decreases from 0.41 to 0.32, with a simultaneous decrease of the value of the factor χ from 9700 to 2700 MPa. The relationship between the stress amplitude and the total strain amplitude has the form

F ig .6.16. σ a – ε ac (solid lines) and σ a – ε ap (broken lines) dependences for steels ig.6.16. with different ferrite grain size: 1,1') 0.022 mm; 2,2') 0.290 mm; 3,3') 0.620 mm. 271

Internal Friction of Materials Ta b le 6.3 Experimentally determined exponents and factors of microplastic response of steel with different ferrite grain size Ma te ria l S te e l 12013 Ti Mo Nb

σ a = χ apn n χ [MP a ]

Gra in size [mm]

ε c r2

0.02 0.29 0.62

1 . 3 · 1 0 –4 1 . 0 · 1 0 –4 7 . 3 · 1 0 –5

0.410 9700 0.357 4400 0.320 2700

1.313 2.30 1.092 3.260 1.661 0.95 2.60 0.790 3.610 1.967 2.97 0.732 3.917

1 . 1 · 1 0 –3 5 . 3 · 1 0 –4 7 . 8 · 1 0 –4

0.257 5700 0.308 25600 0.249 1800

3.106 2.063 1.903

a

b

c

3.02 3.06 3.68

d

1.243 0.771 0.748

h

8.19 4.69 5.74

Fig ig.. 6.17. Experimentally determined dependences of the microplasticity characteristics, determined at a frequency of 23 kHZ for steel with different ferrite grain size: solid lines 0.022 mm, broken lines 0.290 mm, dot-and-dash lines 0.620 mm.

σ a = B ε bac ,

(6.36)

where B (= –χA n), b [= (a +1)n] are constants that depend on structure and experimental conditions, the value of n is the same for all ferrite grain sizes examined (Table 6.3). The validity of equation (6.36) is shown in Fig. 6.17. The plastic strain amplitude in the range ε ac > ε kr2 can be determined from the equation

ε ap = C εcac ,

(6.37)

where C = (B/χ) 1/n, c = b/n, or C = A, c = a + 1. The values C, c 272

Cyclic Microplasticity

are constants depend on the structure and experimental conditions, and the values of exponent c are presented in Table 6.3. To a first approximation, the reciprocal value of exponent c is comparable with the value of the coefficient of cyclic strain hardening for the evaluated ferrites grain sizes. The validity of equation (6.37) is confirmed in Fig. 6.17, which also shows that the application the selected value of ε ac results in a higher plastic strain amplitude in a material with larger ferrite grains. Internal friction in the strain amplitude range resulting in microplastic deformation (ε ac > ε kr2 ), referred to as plastic internal friction Q –1 = Q –1 –Q –1εk , where Q –1εc is the internal friction at the p εc selected value of ε ac , Q is the internal friction under the effect of the critical strain amplitude, depends on ε ap (Fig. 6.17) in accordance with the equation

Q p−1 = D ε dap ,

(6.38)

where the values of the factors D, d depend on the ferrite grain size, and the values of the exponent n are presented in Table 6.3. The resultant dependences indicate the strong sensitivity of Q –1 on the ferp rite grain size, because the same value of ε ap results in more extensive scattering of mechanical energy in the material with a larger ferrite grain. This scattering is associated with the movement and generation of the dislocations. Internal friction Q –1 can be expressed as the ratio of the energy, p scattered in a single load cycle ∆W = Fσε ap [300] (where F is the characteristic of the shape of the hysteresis loop), to the total supply of energy W = 1/2 E ε 2ac . Consequently

Q p−1 =

δp π

=

F σ a ε ap ∆W , = 2 2πW πEε ac

(6.39)

where δ p is the friction decrement. If the equation for the cyclic strain curve is substituted into equation (6.39], the plastic component of the total strain can also be expressed by the equation in the form 1

ε ap

1

 πQ p−1E 2  n+1  δ p E 2  n+1 = ε ac  =  ε ac  .  Fχ   F χ  273

(6.40)

Internal Friction of Materials

In equation (6.40), the values Q –1 , δ p , F, σ, or χ depend on the p magnitude of ε ac and, consequently, on ε ap . If the equations (6.32), (6.33) are substituted into equation (6.39), by measurements of plastic internal friction and the relative change of the Young modulus it is possible to determine, for the selected values of ε ac , the factor characterising the shape of the hysteresis loop by the equation

F=

π Q p−1  ∆E ∆E 2  − 2   E   E

. (6.41)

The area of the hysteresis loop in cyclic loading increases with increasing total strain amplitude ε ac , i.e., with increasing ε ap . If we use equation ∆W = F σε p and equation (6.35), (6.37), we can obtain the equation

∆W = H ε hac ,

(6.42)

where the values of the factors H, h increase with increase of the ferrite grain size, as indicated in Table 6.3 using the value of the exponent h as an example. The validity of equation (6.42) is illustrated in Fig. 6.17. The equation show that H = FχA n+1 , h = (a + 1) (n + 1). If in derivation of equation (6.42) we use the equation (6.36), then H = Fχ –1/n B (n+1)/n , h = b(1 + 1/n). Detailed evaluation of the experimental results show that none of the factors and exponents in these equations changes its magnitude in accordance with the Hall–Petch equation, i.e. in the region of cyclic microplasticity of mild steel it is not possible to obtain the linear dependence of any of these quantities on d z1/2 . The increase of the main value of internal friction with increasing grain size of the material in a wide range of strain amplitude was also detected and interpreted by several authors [279, 280], like the decrease of the values of the critical strain amplitude with increase of the grain size. The activation of oscillations of the dislocation segments, release of the dislocations from pinning areas, and the generation of new dislocations and their movement after exceeding the value ε ac > ε kr2 require lower values of ε ac for the specimens with larger ferrite grains. In the evaluated range of the 274

Cyclic Microplasticity

cyclic strain amplitude there is no large–scale movement of the dislocations to the grain boundaries, and this is reflected in the fact that none of the factors, characterising the microplastic response of the material to cyclic loading with a frequency of 23 kHz, fulfils the Hall–Petch equation. The validity of this equation is confirmed in the majority of processes characterised by higher plastic strains, for example, in the static tensile test or in fatigue test where the specimens are loaded with stress amplitudes close to or higher than the fatigue limit. For identical steel loaded at a frequency of 70 Hz, χ = 835 MPa, n = 0.156 [297]; these values are lower than those obtained at a loading frequency of 23 kHz. The values of the microplastic strain differ at the high loading frequency. The experimentally determined and derived analytical dependences are exponential which increases the importance of the amplitude of plastic strain ε ap in the fatigue damage cumulation process. When loading a material with a high frequency (for example, 23 kHz), the maximum value of ε ap is 3.5% ε ac in the evaluated strain amplitude range. This is also consistent with the results of experiments carried out on copper at a loading frequency of 21 kHz [301]. At usual loading frequencies, for example, 70 Hz [297], the fraction of ε ap in the value ε ac is considerably higher, which means that the same total strain amplitude ε ac results in far more intensive microplastic processes in the material that in loading at a high loading frequency. The difference in the fraction of ε ap in the value of ε ac may be associated with the greatly restricted time available for the movement of dislocations over larger distances under the effect of the given stress amplitude, with the fact that the stress field cannot be relaxed by movement of dislocations over large distances, and with the significant localisation of microplastic strains in loading the material with a high frequency [281]. Equation (6.40) forms a link between the highly sensitive microplastic characteristic Q –1p or δ p, the shape of the hysteresis loop F and the value of the ratio ε ap /ε 2ac , or the significant characteristic of the material in cycling loading (n), which is included in the general equation for the determination of the fatigue life of materials. The results obtained in Ref. 302 indicate that the given approximation (equations 6.32, 6.33) provides the appropriate factors of cyclic microplasticity also for titanium (0.39 wt.% Al, 0.04 wt.% Fe, 0.04 wt.% Si and 0.0 wt.% Cr), molybdenum (0.09 wt.% Fe, 0.03 275

Internal Friction of Materials

wt.% Si and trace amounts of Cu, Nb, Ni, Ca, Mg) and also for niobium (0.25 wt.% Mg, 0.18 wt.% Ca, 0.07 wt.% Fe, 0.04 wt.% Si and trace amounts of Ba, Mn), as presented in Table 6.3. The cyclic strain curve and hardening of materials under repeated loading are significantly influenced by temperature. 6.2.5 Temperature and cyclic microplasticity Measurements of internal friction and the effect of the Young modulus with the increase of strain amplitude provide important information on the change of the response of the material to increasing temperature, as shown in Ref. 303 by Puškár and Letko using a titanium alloy. Titanium alloy VT 3–1 (according to Russian standard GOST) is a two–phase creep–resisting martensitic alloy with the following chemical composition (in wt.%): 6.49 Al, 2.36 Mo, 1.47 Cr, 0.42 Fe, 0.24 Si, 0.03 C, balance titanium. Artificial ageing was carried out at a temperature of 550°C for 5 hours. After removing from the furnace, the specimens were cooled in air. The microstructure consisted of the grains of the α–phase and β–phase at a surface area ratio of 1:1 in the form of equiaxed of globular formations, without any distinctive boundaries of the initial β–phase. After this treatment, the properties of the material at 20°C were: R m = 1200 MPa, R p 0.2 = 12 000 MPa, A 5 = 14.2%, Z = 52.3%. Internal friction Q –1 and effect of the Young modulus ∆E/E of the VT 3–1 alloy were determined in equipment VTP, described in detail in Ref. 304. For the experiments, equipment was fitted with a furnace and facilities for supplying power to the furnace, and for measuring and recording temperature. The VTP equipment operates on the resonance principle; therefore, cylindrical specimens with two heads and the cylindrical central part must also fulfil the resonance condition. The measurements were taken at temperatures of 20, 200, 300, 400 and 550°C, with a scatter of ±1%. A change of the temperature of the VT3–1 alloy results in the change of the velocity of propagation of sound (v) and also the dynamic Young modulus (E d ) at a loading frequency of 23 kHz (Table 6.4). Therefore, the length of the specimens was changed, whilst maintaining the given shape of the specimens. Experimental equipment VTP makes it possible to measure internal friction and the defects of the Young modulus in the total strain amplitude range ε ac from 5 × 10 –7 to 3 × 10 –3 , also at elevated temperatures, where the dimensions of the specimens are adapted in relation to the resonance condition. 276

Cyclic Microplasticity Ta b le 6.4 Experimental values of VT3-1 alloy at 23 kHz and different test temperatures

Te mp e ra ture , º C Q ua ntity v [m s –1] E d, MP a D2 ' d' D2 A a χ [MP a ] R

20

200

300

400

550

5068 1.14·105 1 . 0 9 · 1 0 –3 0.137 3 5 3 · 1 0 –3 243.47 1.777 1.56·104 0.360

4905 1.07·105 1 . 6 4 · 1 0 –3 0.180 5 . 2 · 1 0 –3 430.05 1.762 1.17·104 0.361

4750 1.01·105 2 . 3 9 · 1 0 –3 0.196 4 2 6 · 1 0 –3 739.66 1.779 9.4·103 0.360

4640 0.95·105 1 . 0 5 · 1 0 –3 0.063 3 · 1 0 –3 852.00 1.793 8.2·103 0.356

4440 0.87·105 0 . 9 4 8 · 1 0 –3 0.077 4 · 1 0 –3 32.91 1.214 1.7·103 0.448

F ig .6.18. Dependence of internal friction on the strain amplitude of VT3-1 alloy ig.6.18. at 23 kHz and different temperatures.

In the measurements, the experiments were carried out with the lowest suitable value of ε ac to higher values in such a manner that measurements at every experimental point was carried out within 50 seconds, i.e. 1.15 × 10 6 load cycles. The results of the measurements show that the characteristics of the functional dependence Q –1 vs. ε ac do not change with increasing temperature, Fig. 6.18. The curves in the range of ε ac from 3 × 277

Internal Friction of Materials

10 –5 to 2.5 × 10 –3 show that the value Q –1 is independent of ε ac up to ε kr1 . This is the internal friction background Q –10. The values of ε kr1 can be determined as the magnitude of ε ac at which the defect of the Young modulus ∆E/E is recorded for the first time [304]. With the increase of ε ac to ε kr1 and in further measurements of Q –1 at ε ac << ε kr1 , for example at ε ac = 3 × 10 –5 , the internal friction background does not change up to the value ε ac = ε kr2, and then the value of Q –10 starts to change [281]. This method was used to determine the values of ε kr2 for all temperatures in the tests. Between the values of ε kr1 and ε kr2 there is a slight dependence of Q –1 on ε ac and at values above ε kr2 the internal friction depends in a significant manner on the value ε ac . In the temperature range 20–400°C, the internal friction background of VT3–1 alloy gradually increases (Fig. 6.19), but at a temperature of 550°C there is an anomaly in the dependence of Q –1 on T. When test temperature is increased, the magnitude of the first critical strain amplitude ε kr1 decreases (Fig. 6.19) in accordance with the equation

ε kr 1 = ε x1 − J T ,

(6.43)

F ig ig.. 6.19 6.19. Change of internal friction background and the first and second critical strain amplitude in relation to temperature for VT3-1 alloy at 23 kHz. 278

Cyclic Microplasticity

where ε x 1 = 4.1 ⋅ 10 − 4 ,

J = 4.23 ⋅ 10 − 7 ° C − 1 .

The increase of test temperature results in a faster decrease of the second critical strain amplitude ε kr2 than in the case of ε kr1 (Fig. 6.19). The decrease of ε kr2 with increasing temperature can be expressed by the equation ε kr 2 = ε x 2 − KT ,

(6.44)

where ε x 2 = 1.14 ⋅ 10 − 3 ,

K = 1.13 ⋅ 10 − 6 ° C − 1 .

Figures 6.18 and 6.19 show that increasing temperature decreases the width of the range between ε kr1 and ε kr2 . At a test temperature of 550°C, the authors of Ref. 303 recorded anomalies in the temperature dependence of ε kr1 and ε kr2 . In the range of ε ac from ε kr1 to ε kr2 but also above ε kr2 (to 2.5 × –3 10 ), internal friction can be expressed analytically by the equation

Q −1 = D2 ε dac ,

(6.45)

here D′2 , d', or D′2 , d (for the range of ε ac from ε kr1 to ε kr2 , or ε kr2 and higher) are the experimentally determined coefficients or exponents, presented in Table 6.4. With the increase of ε ac at a specific value of this amplitude, the response of the material changes from elastic to elastic–plastic, reversible (ε kr1 ) up to the region of microplastic deformation (ε kr2 ). This is reflected in the change of the resonance frequency of the VTP system. After evaluating this change, it is possible to determine the defect of the Young modulus of the experimental material ∆E/ E [304] which reflects the integral cyclic microplasticity in the examined volume of the material. The results of measurements of ∆E/E with increasing ε ac at the selected temperatures are presented in Fig. 6.20. Measurements made it possible to determine more accurately the value of ε kr1, because ∆E/E is recorded only when this value is reached or exceeded. The curves can be expressed analytically by equation (6.35). The 279

Internal Friction of Materials

F ig ig.. 6.20. Change of the defect of the Young modulus on the total strain amplitude of VT3-1 alloy at 23 kHz and different temperatures (arrows indicate the values of ε kr 1 ).

values of the factors of this equation are presented in Table 6.4. Whilst in the temperature range from 20 to 400°C, the value of a is approximately 1.7, at a temperature of 550°C it is already 1.2, which reflects the anomaly in the behaviour of the initial material at this temperature. For every experimental point it is possible to determine ε ac and ∆E/E. Using equations (6.32) and (6.33), we can determine the values of stress amplitude σ a and the corresponding values of ε ap . The results are shown graphically in Fig. 6.21. The cyclic strain curves can be expressed in the form σ a = χε nap ε nap , where χ or n is the coefficient of proportionality of the exponent of the cyclic strain curve. The values of these quantities for different temperatures are presented in Table 6.4. When the temperature is increased from 20 to 400°C, the value of χ decreases whereas n does not change, and at 550°C the values of χ and n greatly differ from their temperature dependence, obtained in the given temperature range. The results of the measurements show that increasing tempera280

Cyclic Microplasticity

F ig ig.. 6.21 6.21. Cyclic strain curves of VT3-1 alloy at 23 kHz and different temperatures.

ture results in a gradual increase of the internal friction background of VT3–1 alloy in loading with a frequency of 23 kHz. The increase of temperature during mechanical loading activates the process of microplastic deformation also in the case of VT3–1 titanium alloy, as indicated by the change of the values of ε kr1 , ε kr2, which decrease when the temperature is increased in the range 20–400°C. The anomalous behaviour of the VT 3–1 alloy during tests at 550°C is the result of changes of the microstructure of the alloy which is stable up to 450°C. After exposure of the alloy at 550°C, the structure shows a significant heterogeneity in the shape of the α–and also β–phase. This is also associated with the fact that the temperature of 550°C is the processing temperature for the process of ageing of this alloy and high–intensity ultrasound greatly shortens the ageing nd coarsening time of the phases [305]. If we consider the accuracy of the measurements and interpretation of the results, Table 6.4 shows that the value of n does not change when the temperature is increased in the range 20–400°C. 281

Internal Friction of Materials

This is in agreement with the results which shows that at frequencies of up to 300 Hz the increase of temperature in the range of structural stability of the alloy results in no significant changes in the shape and nature of the cyclic strain curve, but with the increase of temperature the curve is displaced along the stress amplitude axis [298]. The measurements of internal friction and the defect of the Young modulus make it possible to evaluate the important microplastic characteristics of the materials under different service conditions. It is important to stress the fact that the results of measurements of the microplasticity factors may be significantly influenced or even overlapped by, for example, the magnetomechanical component of internal friction when testing ferromagnetic materials. 6.2.6 Magnetic field and microplasticity parameters Taking into account section 3.6, it may be noted that the dislocational internal friction which depends on the strain amplitude, being a frequency–independent component, is the basis of a useful indirect method used in many investigations for evaluating the cyclic plastic response of the materials, whereas the second, frequency-independent component, i.e. magnetomechanical damping, in ferromagnetic materials may influence these results from both the qualitative and quantitative viewpoint. Taking into account the link between the extent of internal friction and the defect of the Young modulus, it may be expected that magnetomechanical friction will also influence the response of the material, characterised by the defect of the Young modulus. The magnetomechanical component of internal friction in ferromagnetic materials can be suppressed by placing the component or specimen in a unidirectional magnetic field with the intensity causing the magnetically saturated state in the ferromagnetic material. Consequently, the magnetomechanical component of internal friction and the defect of the Young modulus can be separated from the dislocational friction that depends on the strain amplitude. Taking into account different response of different alloys with different fractions of the ferromagnetic phases, the minimum strength of the external magnetic field required for obtaining magnetic saturation differs. Puškár [306] carried out a number of experiments to determine experimentally the conditions for magnetic saturation and evaluate the effect of the magnetic field on the extent of internal friction and the defect of the modulus of elasticity of the CSN 412032.1 steel 282

Cyclic Microplasticity

in loading with a frequency of 22.9 kHz at different values of the total strain amplitude. Specimens of the shape and dimensions shown in Fig. 6.22, were produced from 41 2032.1 steel with the following chemical composition, wt.%: 0.3 C, 1.2 Mn, 0.8 Cr, 0.15 V. The steel was in the condition after normalizing annealing. After completing preparation, the specimens were annealed at 500°C, for 1 hour, in vacuum. Internal friction Q –1 and the defect of the Young modulus ∆E/ E in relation to the total strain amplitude ε ac , at ε ac = const, during measurements at a specific point, were determined in automatic equipment for internal friction measurements VTP–A (VSDS Technical University), described in Ref. 155. The experiments were carried out at a temperature of 22°C. The specimens were loaded in symmetric pull–push loading at a frequency of approximately 22.9 kHz. In contrast to the measurements taken on non-ferromagnetic materials, specimens of 12032.1 steel were placed in a coil with the shape and dimensions shown in Fig. 6.22. The coil contained 1000 turns of copper wire, diameter 0.5 mm. When measuring the dependence of the intensity of the magnetic field H of the coil in relation to the intensity of direct current I j , the author used the Hall probe, placed in the centre of a central circular hole in the coil, when the specimen was not in the coil. The dependence of H on I s is shown in Fig. 6.23. To determine the conditions of saturation of specimens of 12032.1 steel, after placing the specimen in the hole in the coil, the author used the circuit shown in Fig. 6.22. The stabilised source of direct current Z with the possibility of regulating the intensity of direct current (transformer 220 V/0.7 V) was used. The ohmic resistance of 1 Ω is characterised by a voltage loss, and the measur-

F ig ig.. 6.22. Circuit of connection of the coil, shape and dimensions of the specimens and the shape and dimensions of the coil used in the experiments. 283

US/IS

H×10–3, A m–1

Internal Friction of Materials

IJ , A F ig ig.. 6.23. Saturation characteristic of the coil, determined by the dependence U s / I s (solid line) and the dependence of the intensity of the magnetic field H on the intensity of direct current supplied to the coil I j (broken line).

ing equipment gives the intensity of alternating current I s . Another measuring equipment controls the intensity of alternating voltage U s. The circuit makes it possible to determine the dependence of direct current I j on the value of the ratio of alternating voltage U s and alternating current I s . With the gradual increase of I j , the ratio U s /I s does not initially change. In a specific range of I j (from 0.2 A to 0.5 A), the ratio decreases. At values higher than I j ≥ 0.5 A, increase of I j to 2 A no longer causes any changes in this ratio. Measurements of the magnetic field show that I j > 0.5 A in the given coil with the inserted ferromagnetic core of the given shape and dimensions causes complete magnetic saturation (Fig. 6.23). Therefore, for the given arrangement, I j = 0.5 A and H = 8 000 A⋅m –1 is sufficient for the magnetic saturation of the specimens of the given shape, dimensions and material. The measurement procedure used in VTP–A equipment is based on the measurement of Q –1 and ∆E/E at 30 different values of ε ac from the total strain amplitude range from 1.1 × 10 –6 to 7 × 10 –4 , by applying, at each measurement point, the selected value of ε ac during 300 s, followed by selection of a higher value of ε ac . The specimen was placed in the coil (Fig.6.22) through which the regulated direct current I j passed; the intensity of the current was such that the intensity of the magnetic field in the coil without the specimen was H = 0, 800, 1600, 2400, 3200, 4000, 4800, 6400, 9600, 12800, 16000 and 19200 A⋅m –1 , at a temperature of 22°C. 284

Cyclic Microplasticity

F ig ig.. 6.24. Dependence Q –1 – ε of 12 032.1 steel under the effect of magnetic field of different intensity.

The results of measurements show that the internal friction background Q –10 changes only slightly with the change of the intensity of the magnetic field H: increasing H decreases Q –10 (Fig. 6.24). At ε ac > 10 –5 , the value of Q –1 at H = 0 rapidly increases. At ε ac = 2 × 10 –4 and it reaches the maximum and then slowly decreases with increasing ε ac . This qualitative description is similar for the values of H of up to 6400 A⋅m –1 . For higher intensities of the magnetic field H (from 9600 to 19200 A⋅m –1 ), the form of the dependence Q –1 vs ε ac is identical, i.e. the value of Q –1 continuously increases with increasing ε ac . The slow decrease of Q –1 after reaching the maximum value, in the log–log coordinates, is probably the indication that the tested ferromagnetic material is not examined in the magnetically saturated condition. The dependence of Q –1 on H (Fig. 6.25) shows that with increase of H the value of Q –1 , determined at different values of ε ac , initially decreases, and the magnitude of the decrease is a function of the value of ε ac . At H ≥ 9600 A⋅m –1, the value of Q –1 no longer changes and, consequently, this phenomenon is independent of the value of ε ac . The results show that the contribution of magnetomechanical friction to the total value of internal friction depends on the total strain amplitude and increases with increasing ε ac . The experimen285

Internal Friction of Materials

F ig ig.. 6.25. Dependence of internal friction on the intensity of the magnetic field of 12 032.1 steel at different total strain amplitudes.

tal results also show that the component of magnetomechanical friction can be suppressed in cases in which the intensity of the magnetic field in the coil with the given characteristics is H = 9600 A⋅m –1 , for the specimens of the given shape and dimensions and made of 12032.1 steel. For practical purposes, the value almost 100% higher is used, i.e. the value of H of approximately 20 000 A⋅m –1 . Figure 6.26 shows that the defect of the Young modulus is the function of not only the total strain amplitude but also of the intensity of the magnetic field. With increasing H ≥ 9600 A⋅m –1 , the increase of the intensity of the magnetic field has no longer any effect on this dependence. In a number of investigations, the criterion for the determination of the second critical strain amplitude ε kr2 is represented by the value ε ac at which the dependence ∆E/E vs. ε ac rapidly increases [302]. For the purposes of this chapter of the book, this characteristic will be denoted by ‘ε 2 ’,because in measurements on ferromagnetic materials this characteristic is not exclusively associated with cyclic microplastic deformation but also with the magnetomechanical response of the material. This conclusion results from the comparison which shows that the value of "ε 2 " increases 286

Cyclic Microplasticity

F ig ig.. 6.26 6.26. Dependence of the defect of the Young modulus on the intensity of the magnetic field for 12 032.1 steel under the effect of the magnetic field of different intensity H × 10 –3 (A m –1 ).

with the increase of the intensity of the magnetic field H up to H = 9600 A⋅m –1 , and from the values of H ≥ 9600 A⋅m –1 the increase of H has no longer any effect of the value of "ε 2", Table 6.5. The ∆E/E – H dependence, Fig. 6.26, indicates that the value of the defect of the Young modulus at a specific value of H is higher at higher values of ε ac . With increasing H the value of ∆E/E initially decreases, until H reaches approximately 9600 A⋅m –1 . With a further increase of H the value of ∆E/E no longer changes, in the entire range of 0 current of the defect of the Young modulus. In the log–log representation, using the intensity of the magnetic field lower than H = 9600 A⋅m –1 , the experimental dependences ∆E/E – ε ac have the form of curves, but at H ≥ 9600 A⋅m –1 , the straight lines overlap (Fig. 6.26). If it is assumed that the curves ∆E/E – σ ac at H < 9600 A⋅m –1 are replaced by the straight lines, the experimental dependences can be expressed by the equation (6.35). The determined characteristics a, presented in Table 6.5, show that with increase of the intensity of the magnetic field the value of a increases up to H ≥ 9600 A⋅m –1 , and it then remains constant. The comment on the conventional notation "ε 2 " also relates to the values of A, a, with the exception of the case in which the steel is already in the magnetically saturated condition, i.e. H ≥ 9600 A⋅m –1 . 287

Internal Friction of Materials Ta b le 6.5 Change of characteristics of 12 032.1 steel with change of the intensity of the magnetic field

H· 1 0 – 3 (A. m–1)

"ε 2"

A . 1 0 –3

a

0 0.8 1.6 2.4 3.2 4.0 4.8 6.4 9.6; 12.8 16; 19.2

3 . 7 · 1 0 –6 5 . 3 · 1 0 –6 1 . 3 · 1 0 –5 2 . 1 · 1 0 –5 3 . 0 5 · 1 0 –5 4 . 0 · 1 0 –5 5 . 8 · 1 0 –5 1 · 1 0 –4 2 . 4 · 1 0 –4

513 2094 230 209 442 210 224 210 212

0.7364 0.7581 0.8186 0.8551 0.8857 0.9095 0.9441 2.0000 2.1053

χ (M P a )

383 321 207 142 107 80 57 46 10

873 361 855 688 050 521 822 667 679

n

0.6039 0.5889 0.5496 0.5200 0.4962 0.4744 0.4466 0.4166 0.4186

In accordance with the approximation for the determination of the plastic strain amplitude ε ac and stress amplitude σ a in high– frequency loading [302], these quantities can be quantified by the equations (6.32) and (6.33). The cyclic strain curve is defined by equation (6.30). Comparison of the experimental data gives the values of χ, n (Table 6.5). It can be seen that the increase of the intensity of the magnetic field H results in a decrease of the value of n from 0.60 for H = 0 to 0.3 at H ≥ 9600 A⋅m –1 , and the value of χ also decreases. Figure 6.23 and also 6.24 and 6.26 show that magnetic saturation of the given specimen of 12032.1 steel in the coil, used in the investigations, takes place up to the intensity of passing direct current I j = 0.5 A, which corresponds to H = 8 000 A⋅m –1 in the coil without the ferromagnetic core. The form of the curves in Fig. 6.25 and 6.27 shows that at H > 9600 A⋅m –1 the value of Q –1 or ∆E/E shows no longer any measurable changes with the increase of H at different values of ε ac . The conventionally quoted intensity of the magnetic field for suppressing the magnetomechanical component of friction of 20000 A⋅m –1 [2] is consequently a safe value of H for obtaining guaranteed saturation in coils of different shapes, with different technically required air gaps, for the specimens with different cross sections and dimensions, and also for steels with different fractions of the ferromagnetic phases. When evaluating the engineering properties of the materials with the ferromagnetic phase, the intensity of internal friction is significantly higher than in the case in which the magnetomechanical component is suppressed by 288

Cyclic Microplasticity

F ig .6.27. Dependence of the defect of the Young modulus on the intensity of ig.6.27. the magnetic field for 12 032.1 steel at different total strain amplitudes.

the external direct magnetic field. Consequently, it is then possible to evaluate the cyclic plastic response of the material on the basis of the component of dislocational friction which depends on the strain amplitude. On the other side, heating of the material is proportional to the area of the hysteresis loop, and its formation is determined not only by the dislocational friction component, which depends on the strain amplitude Q –1ε , but also the component of magnetomechanical friction Q –1 . The total internal friction is Q –1 = m c –1 –1 –1 Q ε + Q m . Measurements show [306] that Q m also depends on the magnitude of ε ac and, consequently, it can significantly overlap the values of Q –1ε in a wide range of ε ac . The magnetomechanical component of friction, associated with the direction of the orientation of magnetisation in domains and with the movement of Bloch walls under repeated loading of ferromagnetic materials, represents by its contribution a significant part of the total internal friction of steel Q –1c . For example, at ε ac = 3 × 10 –4 , Q c–1 = 1.6 × 10 –3 , but Q –1ε = 4 × 10 –4 , which means that the contribution from the magnetomechanical component of friction is Q m–1 = 1.2 × 10 –3 . The measurements also show that the defect of the Young modulus in ferromagnetic materials has two components, i.e. (∆E/E) c = (∆E/E) ε + (∆E/E) m . One of these components is associated with the magnetomechanical processes in the material (∆E/E) m and the other 289

Internal Friction of Materials

one with the generation and interaction of the dislocations in the material, i.e., with the cyclic microplastic deformation (∆E/E). Below ε ac = 2.4 × 10 –4 , the material shows mainly the component (∆E/ E) m and its value is approximately an exponential function of the value of ε ac . The conventionally denoted characteristic "ε 2 " can be regarded as the threshold strain amplitude at which the magnetomechanical mechanism of interaction of repeated loading with the ferromagnetic material at different intensities of the external magnetic field is activated (6.5). Also, at ε ac > 2.4 × 10 –4 , the (∆E/E) m component is significantly higher than the component (∆E/E) ε. For example, at ε ac > 3 × 10 –4 the value of (∆E/E) m at H = 0 is 2.04 × 10 –3 , and the value of (∆E/ E) ε at H = 19200 A⋅m –1 is 1.4 × 10 –4 . These considerations show that the determination of the actual second critical strain amplitude ε kr2 at which the process of cyclic microplastic deformation starts, is determined by fulfilling the condition of complete suppression of the magnetomechanical mechanism of scattering of energy in the ferromagnetic material. Using this comment in characterisation of the cyclic strain curve (equations 6.32, 6.33 and 6.30) shows that after complete magnetic saturation, i.e. when H ≥ 9600 A⋅m –1 , n = 0.30 at a loading frequency of 22.9 kHz. For steels with a similar content of carbon and other elements, at a loading frequency of approximately 100 Hz, n = 0.10–0.20 [307]. The results of our measurements and processing of results should be critically reviewed especially from the viewpoint of the time required to measure every experimental point, because it is likely that changes of the properties in relation to the loading time are of the saturation nature and loading for 300 s is too short to stabilise the changes of the properties at a loading frequency of approximately 2.9 kHz. 6.2.7 Saturation of cyclic microplasticity Changes of the dislocation structure and mechanical properties of the materials under repeated mechanical loading with stress amplitude lower than the yield limit are of the saturation nature, which means that after a certain number load cycles they no longer change with their increase at a specific stress amplitude [298]. The saturation characteristics, expressing a certain part of the cyclic plastic response of the material in relation to the number load cycles, at conventional loading frequencies are characterised in many studies [307], and it has been shown that they differ for different groups of the materials [298, 307]. Measurements and evaluation of the 290

Cyclic Microplasticity

coefficient of cyclic strain hardening, present in the equation of the cyclic strain curve, also depend on the assumption that the response of the material will be evaluated only after stabilising the changes of the properties in relation to the number of cycles. These results can be verified by measurements of internal friction and the defect of the Young modulus in relation to the loading time, as carried out in a study by Puškár [284]. Bars made of electrically conducting copper, purity 99.98%, R p0.2 = 37 MPa, R m = 220 MPa, were annealed at 600°C/h, 850°C/ h or 1050°C/h in vacuum. The resultant grain size of the specimens was z 1 = 0.062 mm, the dynamic modulus of elasticity E = 1.387 × 10 5 MPa, the grain size z 2 = 0.250 mm, E = 1.274 × 10 5 MPa, or the grain size z 3 = 0.707 mm, E = 1.263 × 10 5 MPa. Experiments were carried out using equipment for measuring internal friction and the defect of the Young modulus VTP–A (VSDS Technical University, Zilina) with completely automatic control, measurements and processing of the measurement results [155]. Equipment loads the specimens with symmetric pull–push loading with a frequency of approximately 22.9 kHz (R = –1), with the controlled total strain amplitude ε ac which was 2 × 10 –7 to 3 × 10 –4 in the given case. The advantage of completely automatic equipment is that the measurement of a single point can be carried out in a very short time of 6 s and, consequently, it is also possible to evaluate the time dependence of the values of internal friction Q –1 and the defect of the Young modulus ∆E/E at the selected value ε ac . The accuracy of measurement in this equipment at Q –1 = 10 –3 is 10 –2 percent, and at ∆E/E = 10 –3 is 10 –4 %. The first critical strain amplitude ε kr1 was determined as ε ac at which the defect of the Young modulus of approximately 10 –4 is detected for the first time [302], since the high accuracy of measurements gives non–systematic changes of the quantity at ∆E/E < 10 –4 . The value of the second critical stress amplitude ε kr2, which can be determined by the currently available methods (see section 6.1.2), was the determined by a new method described later, since the currently available methods are suitable in most cases for materials in which the dislocations are strongly pinned by the solute atoms; this is not typical of copper [293]. All the measurements were taken during loading of the specimens in air at a temperature of 22 ± 1°C. The experimental dependences, presented in Fig. 6.28, show that the form of the curves Q –1 – ε ac and ∆E/E – ε ac for the given grain size of copper is very similar. The internal friction background, i.e. 291

Internal Friction of Materials

F ig .6.28. Q –1 – ε (solid lines) and ∆ E / E – ε (broken lines) dependences for copper ig.6.28. with different grain sizes, where triangles refer to ε kr 1 and squares to ε kr 2 .

Q –1 at low values of ε ac , for example 2 × 10 –7 , decreases with increasing grain size of copper. The values of the first critical strain amplitude ε kr1 increase with increasing grain size of copper (Fig. 6.28 and Table 6.6). The second critical strain amplitude ε kr2 (Fig. 6.28 and Table 6.6) also increases with increasing grain size of copper. The form of the ∆E/E – ε ac curve indicates that the dynamics of increase of ∆E/E in relation to ε ac is steeper in the case of the copper specimens with the larger grain size. The curves shown in Fig. 6.28 were obtained after loading the specimens for 500 s, which represents approximately 1.2 × 10 7 cycles, i.e. after the saturation of the changes of the properties. At every experimental point (1 – 11 in Fig. 6.29) we recorded the dependence of Q –1 or ∆E/E on loading time τ. The schematic representation of these dependences in the graph, Fig. 6.29, shows that the dependence of Q –1 on loading time (solid lines) is not systematic and, consequently, is difficult to interpret in this stage of investigations. The dependence of ∆E/E on loading time (broken lines), which is more important from the viewpoint of cyclic microplasticity, is, up to a specific value of ε ac negative or decreases with loading time. However, from a specific value of ε ac the form of the ∆E/E – τ 292

Cyclic Microplasticity

Fig .6.29. Q–1– ε (solid lines), ∆E/ E – ε (broken lines), Q–1– τ and ∆ E/ E – τ dependences ig.6.29. (inserts 1–11) dependences for copper with a grain size of 0.250 mm in loading with a frequency of 23 kHz.

curves is identical (from point 4 and higher) with the course of the changes at the saturation of the characteristics. This amplitude of the total strain is denoted as the second critical strain amplitude ε kr2. The physical meaning of this characteristic is such that at ε ac ≤ ε kr2 the cyclic microplastic deformation of copper starts and the changes of the mechanical characteristics (∆E/E) are saturated in the examined range. The tangent to the origin of the saturation curve (dependence 5 in Fig. 6.29) intersects with the horizontal to the saturation value ∆E/E at the point which determines the time τ 1. The triple value of τ 1 determines the saturation time τ s (analogy with magnetic saturation). Consequently, for each value of ε ac > ε kr2 we obtained a set of data on the saturation time τ s , and, consequently, on the number of cycles of loading in saturation N s (= τ s f, were f is the resonance frequency in measurement of the selected point) determined at the selected values of ε ac , and also the values (∆E/E ) z at the beginning of saturation (τ = 8 s) and at the end of saturation (∆E/E) k . This 293

Internal Friction of Materials

shows that it is possible to determine δ(∆E/E) s = (∆E/E) k – (∆E/E) z . Evaluation of the relationships between the change of the defect of the Young modulus in the relationship between the change of the defect of the Young modulus in relation to time τ of the number of load cycles N at the selected values of ε ac can be expressed analytically by equation (6.5). Evaluation of the value of the parameter a showed that its magnitude depends on the value of ε ac . The dependence can described by the equation in the form

a = B εbac ,

(6.46)

where B, b are the experimentally determined parameters. The values of parameter b are presented in Table 6.6. It can be seen that the increase of the grain size increases the rate of increase of the defect of the Young modulus in relation to the number of load cycles. It can also be seen that the increase of the value of ε ac increases the rate of increase of ∆E/E in relation to the number of load cycles. The magnitude of the increase of the defect of the Young modulus δ(∆E/E) s in relation to the number of load cycles at saturation N s at ε ac > ε kr2 can be expressed in the form

 ∆E  C δ  = C Ns , E  s

(6.47)

where the values of C, c for different grain size of copper are presented in Table 6.6. The increase of the defect of the Young modulus increases with increasing number of load cycles at saturation. The number of load cycles resulting in the saturation of the changes of the properties of copper with different grain size N s depends on the total strain amplitude. The analytical form of the dependence is

N s = D3 ε dac ,

(6.48)

where D 3 , d are the characteristics presented in Table 6.6. The experimental results show that the increase of the total strain amplitude increases the number of load cycles resulting in the saturation of the changes of the properties of copper, and this increase be294

Cyclic Microplasticity Ta b le 6.6 Characteristics of the cyclic microplasticity of copper with different grain sizes Gra in size [mm]

ε k r1

ε k r2

b

C

c

D3

d

z1= 0 . 0 6 2 z2= 0 . 2 5 0 z3= 0 . 7 0 7

8 . 0 7 · 1 0 –7 1 . 7 6 · 1 0 –6 2 . 9 0 · 1 0 –6

5 . 1 · 1 0 –6 5 . 9 · 1 0 –6 9 . 8 · 1 0 –6

0.453 0.599 0.832

2 . 5 · 1 0 15 8 . 5 · 1 0 14 9 . 7 · 1 0 27

1.67 2.17 3.16

1 . 0 · 1 0 10 6.9·109 1.41·109

0.750 0.662 0.482

Gra in size [mm]

Η

h

χz [MP a ]

nz

χ s [MP a ]

ns

z1= 0 . 0 6 2 z2= 0 . 2 5 0 z3= 0 . 7 0 7

4 . 5 5 · 1 0 –4 2.2·108 323.5

1.252 1.436 1.523

53450 36750 24890

0.594 0.567 0.554

453620 29325 22620

0.586 0.556 0.549

comes smaller with increasing grain size of copper. From equations (6.47) and (6.48) we obtain a quantitative correlation between the increase of the defect of the modulus of elasticity during saturation and the total strain amplitude used in loading. The equation has the following form

 ∆E  h   = H ε ac ,  E s

(6.49)

where H = CD c and h = cd, with the values presented in Table 6.6. From the approximation for the calculation of the stress amplitude σ a and the plastic strain amplitude ε ap , which uses the results of measurements of the defect of the Young modulus at specific values of ε ac indicates that the equations (6.32) and (6.33) are valid in this case. The experimental measurements show that the values of ∆E/E at a certain value of ε ac differ at the start of the loading process (denoted by z) and after saturation of the changes of the properties (denoted by s). Processing of the results of measurements using equations (6.32) and (6.33) shows that as a result of the processes taking place during saturation, the cyclic strain curves is displaced to the right, i.e. to higher values of ε ap at a specific value of σ a. The dependence can be expressed by the equation (6.30). The results show that the val295

Internal Friction of Materials

ues of χ, n for the start of loading and after saturation differ (Table 6.6). Increasing loading time decreases the cyclic strain hardening coefficient. Copper with the larger grain size is characterised by a lower value of the cyclic hardening coefficient, both at the start of loading or at saturation of the changes of the properties. In order to obtain saturation in the examined strain amplitude range, the number of load cycles at a loading frequency of 22.9 kHz is approximately 1 × 10 7 . The process of cyclic microplastic deformation is not concentrated only at the grain boundaries but is also associated with the generation and interaction of the dislocations inside the grain. This explains why the values of ε kr1 , ε kr2 increase with increasing grain size. The behaviour of the internal friction background is reversed; this may be associated with the relaxation processes taking place at the grain boundaries which contribute to the extent of internal friction. The dependence of the changes of the properties of the material on the number of load cycles is the same as under conventional loading frequencies [307], although the detailed examination and analytical description provide new information. During saturation, with increasing grain size of copper, i.e. with increasing space in which dislocation interaction takes place, the change of the defect of the Young modulus becomes larger with increasing number of load cycles; this associated with the fact that the magnitude of ∆E/E is the manifestation of the integral cyclic microplasticity in the elementary volumes of the material. The increase of the number of load cycles with the saturation of the changes of the properties results in increase of the difference between the value of the defect of the Young modulus at the beginning and end of loading; this relationship is associated with the result which shows that increasing total strain amplitude requires a larger number of load cycles for obtaining saturation. The coefficient of cyclic strain hardening of copper is, according to Ref. 308, the same at frequencies of 100 Hz and 20 kHz, i.e. n = 0.205, and according to Ref. 309 it is n = 0.209, with the values of n determined at the stress amplitude higher than the fatigue limit of copper. Our experiments and approximation indicate that n z or n s decreases with increasing grain size, and the values of n decrease with increase of the number of load cycles; it should also be added that these characteristics were obtained at stress amplitudes lower than the fatigue limit of copper. The number of load cycles for obtaining the saturated state of the changes of the properties in the examined strain amplitude range is 296

Cyclic Microplasticity

approximately 1 × 10 7 cycles which, at the usual loading frequencies, is approximately 1 × 10 6 cycles [307]. This difference is caused by the fact that the ratio ε ap /ε ac at the high loading frequency is very low in comparison with loading at the conventional loading frequencies and the damaging defect is exerted mainly by the plastic strain amplitude. 6.3 FATIGUE DAMAGE CUMULATION When processing the results obtained in this chapter, it can be assumed that the measurement of internal friction and the defect of the Young modulus with increasing total strain amplitude in the range above ε kr2 provide new possibilities for finding the relationship between the critical strain amplitude and the rate of changes of the characteristics of the materials and their macroplastic behaviour during fatigue loading. 6.3.1 Hypothesis on the relationship of Q –1 – ε and σ a – N f dependences When examining the limiting state of the materials and components [1], it is also useful to include a new concept. The degradation process in the material or a component is a time–dependent process resulting in a change (often for the worse) of the applied properties of the material as a result of the occurrence of internal changes and due to the effect of external factors or their synergic effect [310]. A suitable example of the limiting state of the material is fatigue of the material under mechanical loading, and a good example of the degradation process is the internal response of the material to external loading in the region of inelasticity and also microplasticity in this region. The inelastic behaviour of the material is associated with many processes causing that Hooke’s law is not fulfilled in the submicroscopic dimensions or in the entire volume of the solid, i.e. the magnitude of deformation is not directly proportional to the magnitude of acting stress. The quantification of the representation of inelasticity under static and quasistatic loading, taking into account the fact that the relaxation time is short in comparison with the loading time, is demanding from the experimental viewpoint but can be accomplished by direct methods. In cyclic or repeated loading, depending on the ratio of relaxation time to the loading time in the same direction, a situation may arise in which direct examination is not yet possible. Consequently, it is necessary to use indirect methods and also appropriate models 297

Internal Friction of Materials

for the interpretation of the behaviour of the material. For mild steel, Ivanova [311] proposed to divide the process of fatigue damage cumulation up to fatigue failure in the high–cycle fatigue region to several stages. Puškár [24] included this proposal in the general model for explaining the fatigue curve. Range a (6.30b) characterises the incubation period of the fatigue process, range b is the region of nucleation of submicroscopic cracks and their propagation, range c is the range of propagation of the fatigue crack, and range d the region of increase of the extent of the final fracture of the specimen. The fatigue diagram also shows the lines corresponding to the fatigue limit σ C , the limit of cyclic sensitivity σ Cc and the limit of cyclic elasticity σ Ce [311]. If the fatigue curve (Fig. 6.30b) is shown together with the dependence of internal friction and the defect of the Young modulus on the total strain amplitude (Fig. 6.30a), it is possible to determine a certain phenomenological relationship between the values of the critical strain amplitudes and the given fatigue characteristics. The physical metallurgical similarity of the interpretation of the ranges up to and above ε kr1 , ε kr2, ε kr3 and the ranges of σ, after conversion of the values of ε kr2 and ε kr3 to ε apkr2 and ε apkr3 using equation (6.33), enabled the author of this book to formulate the hypothesis on the mutual relationship of the characteristics using the fol-

F ig .6.30. Dependence of change of Q –1 , ∆ E / E on total strain amplitude (a) and ig.6.30. part of the Wöhler curve. 298

Cyclic Microplasticity

lowing equations: σ Ce = E d ε ke1 ,

(6.50)

σ Cc = χ ε napkr 2 ,

(6.51)

σ C = χ ε napkr 3 ,

(6.52)

where χ is the coefficient of proportionality in the equation for the cyclic strain curve χ and n is the exponent of cyclic strain hardening of the material (equation (6.30)). If it is assumed that the characteristics of the cyclic deformation curve in loading below and at the fatigue limit are the same, the values of χ and n in equations (6.51) and (6.52) are the same. According to some approaches, in loading below and at the fatigue limit the cyclic strain curves differ. Therefore, in equations (6.51) we have χ 1 , n 1, and in equation (6.52) χ 2 , n 2 , and χ 1 ≠ χ 2 , n 1 ≠ n 2 . In equation (6.50) E d is the dynamic modulus of elasticity of the material. The application of the experimental data, obtained on electrically conducting copper (section 6.2.6) for different grain sizes, shows that for the grain sizes of 0.962 mm, 0.250 mm and 0.707 mm the limit of cyclic sensitivity σ Cc is 35.8; 33.5 and 30.0 MPa, whereas the limit of cyclic elasticity σ Ce is 0.11; 0.22 and 0.35 MPa. Further information can be obtained from the experiments carried out on steel CSN 412032.1 (section 6.2.5), with the application of a magnetic field with the intensity H = 19.2 × 10 3 A⋅m –1 in measurement of internal friction and the defect of the modulus of the elasticity in relation to the total stress amplitude. The limit of cyclic sensitivity σ Cc is 198 MPa, and the limit of cyclic elasticity is σ Ce 18.4 MPa. The values of the third critical strain amplitude were not determined in the measurement of the dependences Q –1 vs. ε or ∆E/E vs. ε because of the experimental difficulties determined by the small range of the applicable strain amplitudes in equipment VTP–A (VSDS). The fatigue limit of electrically conducting copper is, however, 80 MPa and the fatigue limit of steel 12032 is 225 MPa [307]. It can be assumed that for certain types of materials, differing mainly in the type of structural lattice and also other morphological features, the ratio of ε kr2 /ε kr3 will change in accordance with a specific dependence. Whilst maintaining the internal and external 299

Internal Friction of Materials

factors, affecting the fatigue limit, it is possible to determine the approximate value of the fatigue limit by measurements of internal friction and the defect of the Young modulus in relation to the total strain amplitude. This procedure can then represent a type of the shortened fatigue test. At present, the author of the book is carrying out extensive experiment to verify this hypothesis. 6.3.2 Deformation and energy criterion of fatigue life The evaluation of the conditions in which the material fails by fatigue fracture is still the subject of discussions. Its solution, in addition to the considerable importance for deeper understanding of the fatigue process, will also provide a basis for the development of shortened fatigue tests. The main question is: what causes fatigue failure at a specific number of load cycles: is it the the limiting amplitude of plastic deformation, at the deformation criterion, or is it the limiting value of the energy scattered irreversibly by the material when using the energy criterion? [300,307,335]. For some materials some of these questions have already been partially answered in tests carried out with the frequency of changes of mechanical loading from 1 to 100 Hz, especially in the low– cycle fatigue range [300,307]. This was carried out using the verified methods of evaluating the plastic strain amplitude ε ap from the total strain amplitude ε ac for the deformation criterion or the verified methods of determining the area of the hysteresis loop ∆W for the application of the energy criterion. Despite the gradual increase of the number of investigations carried out using high-frequency loading (approximately 20 kHz), no investigations have as yet been carried out in which the applicability of the deformation and energy criterion in the quantification of the conditions of formation of fatigue fracture would have been evaluated, as also indicated by the results of international conferences in the USA in 1981 [312] and in the former USSR [313]. Problems are caused mainly by the determination of ε ap or ∆W at high frequencies. This problem was solved in Ref. 314 by Puškár and Durmis. The investigated unalloyed steels differed in the carbon content: steel CSN 412013 0.07 wt.% C, CSN 412040 0.37 wt.% C, CSN 412060 0.56 wt.% C. The internal friction and the defect of the modulus of elasticity of the evaluated materials were determined on three specimens for every steel in the equipment described previously at a frequency of 300

Cyclic Microplasticity

7 × 10 –6 – 2 × 10 –3 in loading with symmetric tension and compression and at a frequency of 22 kHz at a temperature of 22°C. The measurement procedure was described in Ref. 315. Fatigue tests were carried out in resonance equipment described in Ref. 316 and 317 in loading with symmetric tension and compression at a frequency of 22 kHz, always on 25 specimens for every steel. The temperature of the specimens during loading was maintained at 25°C by spraying temperature-controlled water on the specimens. On each of the three stress level the specimens were loaded by a stress 10 MPa lower than the stress level at which the specimens failed at a number of cycles to fracture (N f ) of approximately 10 7 cycles. Figure 6.31 shows the dependence of internal friction Q –1 and the defect of the Young modulus ∆E/E in relation to the total strain amplitude ε ac . The first critical strain amplitude ε kr2 is determined as the value ε ac at which the measurable value of ∆E/E is recorded for the first time. The second critical strain amplitude ε kr2 is determined as the value of ε ac at which there are irreversible changes of the internal friction background, i.e. the value Q 0–1 is determined at ε ac ≤ 10 –5 . The specific values of ε kr1 and ε kr2 are presented in Table 6.7. As a result of processing the data obtained in measurements using equations (6.32), (6.33) and (6.35), the authors obtained the data presented in Table 6.7. The fatigue curves of the examined

F ig .6.31. Internal friction (solid lines) and defect of the Young modulus (broken ig.6.31. lines) in relation to strain amplitude of the steel at a loading frequency of 22 kHz. 301

Internal Friction of Materials

steels, Fig. 6.32, were plotted on the basis of the results of detailed fatigue tests. The fatigue life curves can be described by the equation in the form

σ a = σ′f N bf ,

(6.53)

where σ f is the fatigue strength coefficient, b is the fatigue life exponent. Equation (6.53) is the stress criterion of fatigue life. The specific values of σ′f , a, b for the evaluated steels are presented in Table 6.7, together with the fatigue limit values σ c determined at σ a conventional number of cycles 2 × 10 8 . The ratio σ a /σ f (Table 6.7) shows that this ratio differs from the value 1 for N f = 1, and σ f = F f /S f , where F f and S f are the values of the force and the smallest cross section at the moment of fracture in the static tensile test. The transformation of the fatigue curves from the coordinates σ a – N f to the coordinates ε ap – N f was carried out using the cyclic deformation curves of the evaluated materials (equation 6.30). The values of χ, n are the material and experimental characteristics of the materials presented in Table 6.7. The values of ε ap were calculated using equation (6.33). The fatigue curves in the Manson–Coffin representation (the strain criterion of fatigue life) are presented in Fig. 6.33. The curves can be described by the equation in the form Ta b le 6.7 Characteristics of carbon steels and factors in equations (6.30), (6.35) and (6.54)

C ha ra c te ristic

S te e l 1 2 0 1 3

S te e l 1 2 0 4 0

S te e l 1 2 0 6 0

ε c r1 ε c r2 A a E · 1 0 - 5 (MP a ) σ' f (MP a ) b σ C (MP a ) (σ a/ σ ' f ) N = 1 χ · 1 0 - 4 (MP a ) n ε' f c ε ap/ε' f

4 . 4 · 1 0 –5 2 . 8 · 1 0 –4 628.0 1.41 2.0732 749 –0.079 185 0.75 1.13 0.395 1 . 0 4 · 1 0 –3 –0.200 8 · 1 0 –4

6 . 3 · 1 0 –5 5 . 3 · 1 0 –4 59.2 1.30 2.0720 597 –0.057 215 0.48 3 . 11 0.425 9 . 1 3 · 1 0 –4 –0.134 1 . 3 · 1 0 –4

9 . 1 · 1 0 –5 6 . 5 · 1 0 –4 8.1 1.04 2.0772 461 –0.038 230 0.39 7.42 0.490 3 . 1 3 · 1 0 –5 –0.077 5 . 2 · 1 0 –5

302

Cyclic Microplasticity

F ig .6.32. Fatigue curves of examined steels at a loading frequency of 22 kHz. ig.6.32.

ε ap = ε′f N cf ,

(6.54)

where ε f is the fatigue ductility coefficient and c is the exponent of fatigue life with the values presented in Table 6.7. The values presented in Table 6.7 indicate that the ratio ε ap /ε f , where ε f is the true strain in the area of fracture in the static tensile test, greatly differs for different materials. In the strain amplitude range above ε kr2 plastic internal friction is recorded Q p–1 = Q –1 – Q 2–1 where Q –1 and Q 2–1 are the values of internal friction at ε ac > ε kr2 or at ε ac = ε kr2 (Fig. 6.31, Table 6.7). Using the equations (6.39) and (6.41), gives the equation in the following form

∆W =

πQ p−1σ′f ε′f ∆E  ∆E  − E  E 

2

N bf + c . (6.55)

The total energy consumed by the material up to the formation of fracture (the energy criterion of fatigue life) is

303

Internal Friction of Materials

F ig .6.33. Fatigue curves in Manson and Coffin representation for the examined ig.6.33. steels at a loading frequency of 22 kHz.

W f = ∆WN f =

πQ p−1σ′f ε′f ∆E  ∆E  − E  E 

2

N 1f+b+ c . (6.56)

Using equation (6.53) and substituting into equation (6.32), we obtain the equation 1

 ε E  ∆E   b N f =  ac  1 −  . E    σ′f 

(6.57)

From the equations (6.56) and (6.57) we obtain the functional dependence for the energy consumed by the material up to fracture in the form

πQ p−1σ′f ε′f

 ε ac E  ∆E   Wf = 1 − E   2    ∆E  ∆E   σ′f  −  E  E 

1+ 2+ 3 b

304

(6.58)

Cyclic Microplasticity

F ig .6.34. Dependence of the energy absorbed to fracture on stress amplitude for ig.6.34. the examined steels at a frequency of 22 kHz.

The experimental data processed using equation (6.58) are shown in Fig.6.34. It appears possible to describe the resultant dependences by the equation in the form

σ a = G W fg ,

(6.59)

here G, g are the factors with the values given in Table 6.8. If the energy consumed by the material up to the formation of fracture is expressed in relation to the number of load cycles to fracture, using equations (6.53) and (6.59) we obtain the dependence of the energy absorbed to fracture, Fig. 6.35. The curves can be expressed by the equation in the form

W = H N hf ,

(6.60)

for the material listed in Table 6.8. The experiments carried out in Ref. 314 indicate that the values of the first and, in particular, second critical strain amplitude (Table 6.7) increased with increasing carbon content of the steel. This phenomenon is be determined mainly by the braking and blocking defect of the interphase boundaries which controlled the activity of the mechanisms of cyclic microplasticity above ε kr1 and ε kr2 [294]. The fatigue life exponent b at a loading frequency of 22 kHz is lower for the examined steels (Table 6.7) than for the steels of the 305

Internal Friction of Materials Ta b le 6.8 6.8. Characteristics in equations (6.59) and (6.60) for the examined steels

S te e l

G

g

H [MP a ]

h

b /(1 + b+c)

C SN 41 2013 C SN 41 2040 C SN 41 2060

671 461 391

–0.160 –0.068 –0.045

0.357 0.222 0.025

0.745 0.849 0.848

–0.109 –0.070 –0.043

average slope ≈0.81

F ig .6.35. Dependence of the energy absorbed to fracture on the number of cycles ig.6.35. to fracture for the examined steels at a loading frequency of 22 kHz.

grade 11, 13 and 15 (according to the former Czechoslovak standards) at a loading frequency of 7–100 Hz [318]. The cyclic strain curves for the examined steels at a loading frequency of 7100 Hz are characterised by the exponent of cyclic strain hardening n with the values in the range from 0.06 to 0.15 [297], whereas for a loading frequency of 22 kHz the values are in the range 0.395–0.490 (Table 6.7), and the value of n at both loading frequencies increases with increasing carbon content of the steel. The fatigue curves in the Manson–Coffin representation (equation (6.54)) are characterised mainly by exponent c whose value at a frequency of 22 kHz is significantly lower (Table 6.7) that the value at a loading frequency of 7–100 kHz, where c = –0.75 (for the steels CSN 412013 and CSN 412060), and at a loading frequency of 22 kHz, the values of c decrease with increasing carbon content in the steel. When converting the fatigue stress limit σ c (Table 6.7) to the fatigue strain limit ε apC using the cycling deformation curves, we obtain ε apC = 3.02 × 10 –5 , ε apC = 8.25 × 10 –6 or ε apC = 7.56 × 10 –6 for the steels CSN 412013, CSN 412040, or CSN 412060. At a load306

Cyclic Microplasticity

ing frequency of 7–100 Hz, ε apC = 4 × 10 –5 for the examined steels [318]. The total energy dissipated by the material up to the fatigue fracture increases with increasing number of load cycles in the same manner at a loading frequency of 7–100 Hz as at 22 kHz. For low-frequency loading with a frequency of 70–100 Hz, 1 + b + c = 0.35 [307], whereas for loading with a frequency of 22 kHz 1 + b + c = 0.81. Comparison of the results obtained for the evaluated steels shows that to induce fatigue fracture in CSN 412013 steel, the material dissipates energy 3–4 times more than the steels 412040 and 41262. Discussion of the pseudoelastic behaviour of the CSN 412013 steel was published in Ref. 307. For low–cycle fatigue at a frequency of 70–100 Hz b/(1 + b + c) = – 0.25, whereas at a loading frequency of 22 kHz the average value of the fraction is –0.0 72. Consequently, b = 0.971 and not b = nc, as at a loading frequency of 7 – 100 Hz. Approximation of the relationship between n, b, c at a loading frequency of 22 kHz is not as simple as in the case of loading with a frequency of 70–100 Hz, as shown by our experiments. The mutual relationship between the results obtained in a loading with a frequency of 22 kHz, using the deformation and energy criteria, is therefore very complicated. The evaluation of the fatigue process at a frequency of 22 kHz can be discussed more efficiently on the basis of the deformation criterion, as implicitly concluded in the studies carried out at a loading frequency of 70–100 Hz [300,307]. The applicability of the deformation and energy criteria of fatigue life at elevated temperatures has been described by Puškár and Letko [319]. They based their conclusions on the results which show that mechanical loading at elevated temperatures, acting on the material, are not in a simple addition correlation, which means that it is not possible to carry out fatigue tests at, for example, 20°C, and take into account analytically the changes of the properties with increasing temperature. The experiments were carried out with VT3– 1 two–phase titanium alloy (Russian GOST standards), with the following chemical composition, wt.%: 6.39 Al, 2.36 Mo, 1.47 Cr, 0.42 Fe, 0.24 Si, 0.03C, balance–titanium. The experiments carried out in Ref. 319 where similar to those conducted in Ref. 303 which means that the heat treatment of the material and the parameters of its cyclic microplasticity were published in section 6.2.4. The experimental results were processed using the procedure de307

Internal Friction of Materials

scribed in Ref. 314 and 320. The Wöhler curves of quenched and tempered VT3-1 alloy, obtained at different temperatures, are presented in Fig. 6.36. At 20°C, σ C = 650 MPa, at 400°C σ C = 392 MPa, and at 550°C σ C = 225 MPa, with the scatter of the experimental results of ±7 MPa, with the reference number of load cycles being 2 × 10 8 . Examination of the microstructure of the specimens, loaded with a stress amplitude of σ a = 1.2 σ c , shown that at 20°C, there are no significant changes in comparison with the initial condition. Similar agreement was also found in the case of the microstructure after exposure of the specimens to an appropriate stress amplitude at a temperature of 400°C. The microstructure of the specimens, loaded at a temperature of 550°C, is different. It is heterogeneous, with signs of spheroidisation of α–phase. Figure 6.36 shows that the experimental dependence σ a – N f can be described by equation (6.53), where σ′f and b for different temperatures are presented in Table 6.9. In the Manson–Coffin representation, Fig.6.37, the curves can be

cycles F ig .6.36. Fatigue curves of VT3-1 alloy at different temperatures. ig.6.36. 308

Cyclic Microplasticity Ta b le 6.9 Effect of temperature on the fatigue characteristics of quenched and tempered VT3-1 alloy T, ºC

σ' f [MP a ]

b

σ'f · 1 0 4

c

χ·103 [MP a ]

n

H [MP a ]

h

20 400 550

1052.45 702.32 523.94

–0.027 –0.031 –0.050

5.31 15.5 3.94

–0.072 – 0 . 11 4 –0.109

17.60 4.17 18.80

0.374 0.27 0.457

0.514 0.003 0.025

0.88 0.87 0.86

cycles F ig .6.37. Manson–Coffin curves of quenched and tempered VT3-1 alloy at different ig.6.37. temperatures.

described by equation (6.54), where the values of ε′f and c are presented in Table 6.9. The cyclic strain curves can be described by equation (6.30), where n = b/c and χ = f/f n . The derivation in [302] and measurements of plastic internal friction Q p–1 for VT3-1 alloy at selected temperatures [321] indicate that the amount of energy consumed by the specimen to fatigue fracture can be described by equation (6.60). The evaluation of the tests from the viewpoint of the energy criterion using equation (6.56) gives the graphical dependence shown in Fig.6.38. The dependences can be described by equation (6.60), where H and h are the experimentally determined characteristics with the values presented in Table 6.9. The evaluation of the values of ε ap and ε ac at N f = 10 8 cycles for 309

Internal Friction of Materials

the experimental results show that at 20°C ε ap = 0.022 ε ac, at 400°C ε ap = 0.046 ε ac and at 550°C ε ap = 0.024 ε ac . The values of the fatigue limit, obtained by the authors of Ref.319 at different temperatures and the reference number of cycles of 2 × 10 8 , can be compared with the fatigue limit values obtained in loading at a frequency of 100 Hz and a reference number of cycles of 1 × 10 7 , where the microstructure of VT3–1 alloy was similar to that used in this work. Kalachev et al. [322] carried out tests at a temperature of 20°C and the fatigue limit was in the range 550–620 MPa, whereas in Ref. 323 it was 620 MPa. In fatigue loading at a temperature 400°C, the fatigue limit was 480–500 MPa [323], and in Ref. 324 it was 330 MPa. No values of the fatigue limit have been published in the technical literature for a temperature of 550°C. With increase of temperature from 20°C to 400– 550°C, the strain fatigue limit decreases from 1.85 × 10 –4 to 1.43 × 10 –5 or even 6.1 × 10 –5 and is therefore close to the value of 10 –5 , as generalised for different types of steel [297]. Evaluation of the microstructure prior to and after fatigue loading at different temperatures indicate that in the temperature range 20–400°C the structure of the VT3–1 alloy is stable and, consequently, the changes of the properties with increasing temperature are controlled by the conventional mechanism. However, loading at

cycles F ig .6.38. Application of the energy criterion of fatigue life of VT3-1 alloy at ig.6.38. different temperatures. 310

Cyclic Microplasticity

550°C resulted in significant changes in the microstructure, reflected in the changes of the exponent c and n in relation to temperature (Table 6.9). The values of χ and n, determined at a testing temperature of 20°C and a loading frequency of 22.5 kHz for the VT3–1 alloy are in correlation with the similar characteristics and test conditions for the low-alloyed titanium alloy [302]. Increasing temperature decreases the plastic deformation resistance of the material. This general conclusion is also reflected in fatigue loading at a high frequency where the exponent of the cyclic strain curve n decreases from 0.374 to 0.275, when the temperature is increased from 20 to 400°C. The change of the response of the material to alternating loading with increasing temperature is also evident from the ε ap / ε ac ratio which is more than doubled with the temperature increasing from 20 to 400°C, with the number of load cycles to fracture being 10 8 . However, if there are significant changes in the microstructure during loading at 550°C, the response of the material differs. The values of χ and n of the material are higher than those expected in the case of multiple changes of the properties with increasing temperature. The experimental results and evaluation show that in the analytical evaluation of the results it is possible to use the deformation and energy criteria of fatigue life also at high loading frequencies and elevated temperatures. The total amount of energy required for the formation of fatigue fracture is indirectly proportional to temperature. 6.3.3 Effect of loading frequency on fatigue limit Taking into account section 3.4, it is useful to note that the loading frequency at high total stress amplitudes may have a significant effect on the fatigue characteristics of materials [143]. The results of the experiments carried out by different authors have resulted in a conclusion [325] according to which the fatigue limit at high loading frequency (for example, 20 kHz) is 1.3–1.4 times higher than the fatigue limit determined at a loading frequency of approximately 70 Hz, especially in the case of bcc metals. However, the results are loaded with large errors under the experimental conditions at the compared frequencies. Of special importance is temperature because at a high loading frequency, depending on the extent of internal friction and the rate of its increase with increasing strain amplitude of different materials, the rate of heating of the materials differs. 311

Internal Friction of Materials

The strictly organised experiments carried out on CSN 415313 steel with strictly controlled identical characteristics of the material in the fatigue tests at 25–45 Hz and 20 kHz showed [326] that when the fatigue limit at 20 kHz is determined for 2 × 10 8 cycles, its value is 270 MPa, and when the fatigue limit is determined at 25–45 Hz, at 10 7 cycles, it is 260 MPa. The increase of the loading frequency from 25–45 Hz to 20 kHz results in a significant shift of the fatigue curve to the higher number of load cycles to fracture N f , even though the time required to cause fracture at the high loading frequency is significantly shorter [326]. Evaluation of the rate of fatigue crack propagation at a loading frequency of 120 Hz and 20 kHz indicates that the rate (µm⋅ cycle –1 ) at 20 kHz is up to 100 times lower than in loading at a frequency of 120 Hz, whereas the rate of propagation of fatigue cracks (µm⋅s –1 ) at a loading frequency of 20 kHz is 20 times higher than at 120 Hz. The basic threshold amplitude of the stress intensity factor K ath at a loading frequency of 20 kHz is higher than at a loading frequency of 120 Hz (4.56 MPa⋅m 1/2 , 3.8 MPa⋅m 1/2). These results and further experiments [327] show that the fatigue process at high loading frequency is characterised by the same main characteristics, stages and relationships in comparison with loading at low frequency (for example, at 100 Hz). However, there are certain modifications affecting the physical–metallurgical and engineering characteristics of the materials at the higher loading frequencies. These modifications are the result of the effect of various factors whose influence on the applied characteristics has not as yet been quantified. To understand the problem, it is possible to introduce conventional symbols. The phenomenon increasing the cyclic deformation resistance of the material will be denoted as the (+) phenomenon, the phenomenon increasing this resistance will be denoted as the (–) phenomenon and the phenomenon having a mixed effect in different stages of the process will be denoted by (±). Loading at high frequency is characterised by the preferential absorption of oscillations at lattice defects resulting in a local increase of temperature around these defects (–). The amplitude of the deviation of the dislocation segment at a high loading frequency and a specific shear stress amplitude, used at an ‘arc’ loading frequency is lower because its faster bending is inhibited in a viscous manner by the solute atoms, especially interstitial elements (+). When loading at high frequency, the time for the relaxation of stress concentration is insufficient (+) and, at the same time, there are less suitable condi312

Cyclic Microplasticity

tions for the removal of heat from the area in which heat is generated in a single cycle (–). The temperature difference may result in a gradient of internal stresses (±). The bcc metals with interstitial elements significantly increase the deformation resistance with increasing strain rate (+). At high loading frequency, the time available is insufficient for general corrosion processes (+) to take place. The fraction of the plastic strain amplitude in the total strain amplitude at the selected value of the total strain amplitude and at the high loading frequency is smaller than at the conventional loading frequency (+). At high loading frequency, the slip lines are narrower and are found in a smaller number of grains, with a smaller surface relief in comparison with the conventional loading frequency (+). The width of the plastic zone around the fatigue crack at the high loading frequency is smaller than at the conventional loading frequency (–). The shift of the front of the fatigue crack into the material at high frequency requires a significantly larger number of load cycles than at the conventional loading frequency (+). The size of the activation volume at high and conventional loading frequencies is approximately the same. Each of these reasons is characterised by different intensity of the effect on the appropriate fatigue characteristic. Some of them are mutually connected and other combinations mutually exclude each other. For these reasons, all attempts for the analytical expression of the effect of frequency on the fatigue limit or K ath are of the empirical nature with limited validity. To conclude this chapter, it should be noted that important and valuable information and interpretation can be found in the previously mentioned monographs and also in a compilation edited by Gorczyca and Magalas [328].

313

Internal Friction of Materials

314

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324

References

Index A activation energy of diffusion 86 activation enthalpy 78 activation volume 78 aerodynamic losses 171 allotropic transformations 22 amplitude–frequency spectrum 3 anelastic strain 84 anelasticity 1, 28 atomic density 12 atomic spacing 13

compensation phase method 167 composite materials 37 condensation temperature 257 Cottrell atmospheres 97, 106, 197 Cottrell parameter 106 Coulomb 7 Coulomb forces 153 critical shear stress 86 crystallographic system 7 Curie temperature 19, 36, 68 cyclic microplasticity 79, 276 cyclic strain curve 267 cyclic strain hardening 296

B background of internal friction 79 Barkhausen jumps 129 BHR theory 210 Blair, Hutchison and Rogers model 209 Bloch walls 46, 59, 128, 289 Boltzmann constant 56 Boltzmann distribution 86 Bordoni maxima 98 Bordoni peak 116 Bordoni phenomenon 58 Bordoni relaxation 105, 197 Bordoni’s maximum 100 Bordoni’s relaxation 98, 99 bowout of the dislocation 76 Bridgman equation 67 bulk self-diffusion 85 Burgers vector 74, 115

D d-electrons 22 damping capacity 133 damping decrement 156 damping factor 140 DB transition 230 Debye frequency 186 Debye maximum 90 Debye oscillations 93 Debye peak 208, 212 Debye shape 90 Debye temperature 11, 14 defect of the elasticity modulus 1 degree of dynamic relaxation 49 diffusibility of atoms 188 dilation coefficient 54 dimensional factor 22 dislocation anelasticity 69, 70 dislocation clusters 80 dislocation configuration 71 dislocation kernel 201 dislocation multiplication process 71 dislocation segment 99 dislocational strain 71 dispersion strengthening 38

C characteristic process time 49 circular resonance frequency 51 clustering 126 coefficient diffusion of point defects 86 coefficient of absorption of sound 82 coefficient of anisotropy 10 325

Internal Friction of Materials

dry friction mechanism 69 dynamic methods 142

I infrasound methods 144 inhomogeneous stress 53 instantaneous elastic strain 44 intercrystalline adsorption 226 internal friction background 84 interstitial atoms 193 Invar alloy 15 isotropic pressure 8

E ε-carbide 32, 34 effective energy of bonding 84 Einstein temperature 15 elastic–viscous bond 69 elasticity 1 elasticity characteristics 1 elasticity constant 5 elasticity modulus 1 electron factor 19, 22, 27 electrostatic excitation 153 Elinvar 68 excitation force 138

K Köhler distribution 105 Kurnakov temperature 29

L lattice spacing 20 linear stretching of the dislocation 75 loading frequency 74 logarithmic decrement of vibrations 52

F fatigue stress limit 306 Fermi level 54 Finkel’shtein and Rozin phenomenon 95 Finkel’shtein–Rozin relaxation 102 Finkelstein–Rozin peak 97, 191, 192 Finkel’stein–Rozin peak 192 Frenkel 16 frequency-independent processes 59 friction factor 43

M

G gas constant 16 Granato–Lücke spring model 74, 105 Granato–Lücke theory 60, 114

H magnetostriction 68 Hall–Petch equation 273 Hertz frequency range 86 Hooke law 43, 44 hypersonic methods 167 hysteresis 1 hysteresis anelasticity 71 hysteresis loop 72

326

M–ε dependence 77 magnetic hysteresis 76 magnetic moment 43 magnetomechanical phenomenon 76 magnetomechanical bond 68 magnetomechanical component 281 magnetomechanical phenomenon 59 magnetostriction 46 magnetostriction vibrator 162 Manson–Coffin representation 305 Mason model 86 maximum of cold deformation 98 mechanical hysteresis 69 mechanical hysteresis loop 239 mechanical relaxation 88 microplastic anelasticity 77 microplasticity 76 microstrain 18 molar heat capacity 11

Index References

σ–ε curve 71, 76 saturation of cyclic microplasticity 290 scattering of mechanical energy 1, 73 Schmidt trigger 148 Schoeck characteristic 84 Schoeck model 83 self-diffusion 16 self-diffusion coefficient 54 Shockley 115 Shoeck model 201 Snoek 55 Snoek and Köster maximum 224 Snoek and Köster phenomenon 205 Snoek and Köster relaxation 198 Snoek maximum 88, 90, 91, 103, 181, 186 Snoek peak 219 Snoek relaxation 184 Snoek’s mechanism 95 solute atom 124 splitting 215 stacking fault energy 215 steady-state creep 16 strain amplitude 104 strain hardening coefficient 16 stress sensor 5 sublimation energy 84 sublimation temperature 11 substitutional solid solution 94 superlattice 36

N Nabarro barriers 86 natural frequency of vibrations 73 Newtonian viscous friction 250 non-complanar slip plane 76

O orientation factor 71

P paramagnetic state 68 Peierls barriers 86 Peierls potential energy 99 Peierls stress 115 Peierls–Nabarro barrier 86 phase shift 49 pinning points 60 pipe diffusion coefficient 223 Planck’s constant 92 Poisson number 7, 12, 43 pulse-phase method 166 pulsed methods 12

Q quasi-inelastic strain 44

R Rayleigh waves 135 relaxation maxima 88 relaxation mechanism 87 relaxation time 45, 48, 181 relaxed Young modulus 47 relaxons 87 resonance mechanism of anelasticity 75 resonance methods 12, 142 resonance peak 51 Reynolds number 171 rigidity 1 RM peak 217

T tensors of the second order 6 tetragonality 58 Teutonico model 108, 109 thermal activation 77 thermal–fluctuation relaxation peak 206 torsional pendulum 143 transmission method 165 triclinic 7

S S–K maximum 100, 101, 102 S–K peak 200 S–K relaxation 200

U ultrasound methods 151 327

Internal Friction of Materials

unpinned dislocation 76

Werner’s model 96 whisker crystals 81 Wöhler curve 297

V velocity of propagation 8 viscous friction 74 viscous friction coefficient 74

Z Zener 54 Zeener relaxation 95, 182, 195, 197

W Weert’s equation 10

328