Internal Friction of Materials

INTERNAL FRICTION OF MATERIALS i ii INTERNAL FRICTION OF MATERIALS Anton Puškár Transport and Communications Techni...

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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

iv

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

v

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


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


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)


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


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


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 ( 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 ( 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


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


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


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


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 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


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


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


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


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


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 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


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


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


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


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


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


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


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


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


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


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


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 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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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)


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


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


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


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


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


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


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


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


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


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

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