High Temperature Strain of Metals and Alloys Part 1 ppt

Valim Levitin
High Temperature Strain
of Metals and Alloys
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Valim Levitin
High Temperature Strain
of Metals and Alloys
Physical Fundamentals
The Author
Prof. Valim Levitin
National Technical University
Zaporozhye, Ukraine

Cover:
“Blish” turbine
University of Applied Sciences
Gießen-Friedberg,
Department MND, MTU
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 2006 WILEY-VCH Verlag GmbH & Co KGaA,
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ISBN-13: 978-3-527-31338-9
ISBN-10: 3-527-31338-9
High Temperature Strain of Metals and Alloys, Valim Levitin (Author)
Copyright
c
 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
ISBN: 3-527-313389-9
V

Contents
Introduction 1
1 Macroscopic Characteristics of Strain of Metallic Materials
at High Temperatures 5
2 In situ X-ray Investigation Technique 13
2.1 Experimental Installation 13
2.2 Measurement Procedure 15
2.3 Measurements of Structural Parameters 17
2.4 Diffraction Electron Microscopy 20
2.5 Amplitude of Atomic Vibrations 21
2.6 Materials under Investigation 23
2.7 Summary 24
3 Structural Parameters in High-Temperature Deformed Metals 25
3.1 Evolution of Structural Parameters 25
3.2 Dislocation Structure 30
3.3 Distances between Dislocations in Sub-boundaries 34
3.4 Sub-boundaries as Dislocation Sources and Obstacles 34
3.5 Dislocations inside Subgrains 35
3.6 Vacancy Loops and Helicoids 39
3.7 Total Combination of Structural Peculiarities
of High-temperature Deformation
40
3.8 Summary 41
4 Physical Mechanism of Strain at High Temperatures 43
4.1 Physical Model and Theory 43
4.2 Velocity of Dislocations 45
4.3 Dislocation Density 49
4.4 Rate of the Steady-State Creep 51
VI Contents
4.5 Effect of Alloying: Relationship between Creep Rate

and Mean-Square Atomic Amplitudes
54
4.6 Formation of Jogs 55
4.7 Significance of the Stacking Faults Energy 57
4.8 Stability of Dislocation Sub-boundaries 58
4.9 Scope of the Theory 62
4.10 Summary 64
5 Simulation of the Parameters Evolution 67
5.1 Parameters of the Physical Model 67
5.2 Equations 68
5.2.1 Strain Rate 68
5.2.2 Change in the Dislocation Density 68
5.2.3 The Dislocation Slip Velocity 69
5.2.4 The Dislocation Climb Velocity 69
5.2.5 The Dislocation Spacing in Sub-boundaries 70
5.2.6 Variation of the Subgrain Size 71
5.2.7 System of Differential Equations 71
5.3 Results of Simulation 71
5.4 Density of Dislocations during Stationary Creep 77
5.5 Summary 80
6 High-temperature Deformation of Superalloys 83
6.1 γ

Phase in Superalloys 83
6.2 Changes in the Matrix of Alloys during Strain 88
6.3 Interaction of Dislocations and Particles 89
6.4 Creep Rate. Length of Dislocation Segments 95
6.5 Mechanism of Strain and the Creep Rate Equation 96
6.6 Composition of the γ


Phase and Atomic Vibrations 102
6.7 Influence of the Particle Size and Concentration 104
6.8 The Prediction of Properties 106
6.9 Summary 109
7 Single Crystals of Superalloys 111
7.1 Effect of Orientation on Properties 111
7.2 Deformation at Lower Temperatures 116
7.3 Deformation at Higher Temperatures 124
7.4 On the Composition of Superalloys 129
7.5 Rafting 130
7.6 Effect of Composition and Temperature on γ/γ

Misfit 136
7.7 Other Creep Equations 137
7.8 Summary 141
VII
8 Deformation of Some Refractory Metals 143
8.1 The Creep Behavior 143
8.2 Alloys of Refractory Metals 149
8.3 Summary 155
Supplements 157
Supplement 1: On Dislocations in the Crystal Lattice 157
Supplement 2: On Screw Components in Sub-boundary
Dislocation Networks
161
Supplement 3: Composition of Superalloys 163
References 164
Acknowledgements 168
Index 169
High Temperature Strain of Metals and Alloys, Valim Levitin (Author)

Copyright
c
 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
ISBN: 3-527-313389-9
1
Introduction
Whoever controls the materials,
controls the science and the technology
E. Plummer
Modern civilization is based on four foundations: materials, energy, tech-
nology, and information.
Metals and alloys are materials, which have been widely used by mankind
for thousands of years, and this is no mere chance: metals have many re-
markable properties. One – their strength at high temperatures – is of great
scientific and practical importance.
The durability ofgas turbine engines, steam pipelines, reactors, aeroplanes,
and aerospace vehicles depends directly on the ability of their parts and units
to withstand changes in shape. On the other hand, a significant mobility of
crystal lattice defects and of atoms plays an important role in the behavior
of materials under applied stresses at high temperatures and is also of great
interest for materials science research and practical applications.
Mechanical tests were historically the first method of investigating the
high-temperature deformation phenomenon. The technique originated from
practical needs to use metallic materials for various machines. A deep inves-
tigation of material structure was impossible in early studies because of the
lack of suitable equipment and appropriate techniques. Even now mechanical
tests are a source of indirect information about physical processes that take
place in the atomic crystal lattice of metals and alloys. However, if we want
to understand the nature of these processes and to be able to use them in
practice we should try to investigate them directly.

The phenomena of high-temperature strain and creep have been studied
for many years. Numerous theories have been developed, based on the de-
pendences of the strain rate upon stress and temperature. The structure of
tested metals was also studied. The obtained results are of great value and
have been described in books and reviews and important data are also scat-
tered in numerous articles. Previous investigations improved our knowledge
2 Introduction
of the problem and stimulated further experimental approaches. It is essen-
tial, however, to emphasize that the physical nature of the high-temperature
strain in metals, especially industrial superalloys, is not yet understood suffi-
ciently. By this we mean the physical background of the deformation on the
atomic microscopic scale.
The problem of the high-temperature properties of metallic materials has
a number of experimental, theoretical and applied aspects. Naturally, it is
necessary to identify the scope of the problem considered in this book.
My idea is as follows. The high-temperature diffusionmobility of atomsand
the effect of applied forces are the conditions under which special processes
occur in the crystal lattice of metallic materials. Thus, external conditions
result in a distinctive structural response of the material. In their turn these
specific structural changes leadto adefinite macroscopicbehavior ofthe mate-
rial, especially, to a definite strain rate and to a stress resistance. Consequently,
structure evolution is the primary stage of response; mechanical behavior is
the secondary result. The response in the crystal lattice is a cause, while the
plastic strain of a metal or an alloy is a consequence. The structural evolution
is therefore a key factor, which determines the mechanical properties of the
metallic materials at high temperatures.
This book treats data from experimental measurements of important struc-
tural and kinetic characteristics which are related to physical fundamentals
of the high-temperature strain of metallic materials. A number of specific pa-
rameters of substructure, which have been directly measured, are presented.

Theories that have been worked out on the basis of these experiments are
quantitative and contain values which have a definite physical meaning. A
method of calculation of the steady-state strain rate from the material, struc-
tural and external parameters is developed for the first time.
The book consists of eight chapters.
A summary of the problem is presented in the first chapter. The peculiar-
ities of the strain of metallic materials at high temperatures are described.
The reader’s attention is drawn to the shortcomings of existing views and
the author’s approach to the problem is substantiated. It is advisable for the
reader to remind himself of the main principles of dislocation theory by first
reading Supplement 1.
The second chapter is devoted to experimental techniques. The unique
equipment developed by the author is intended for the in situ X-ray investi-
gation of various metals, i.e. for direct structural measurements during the
high-temperature tests. The method of transmission diffraction microscopy
is briefly considered. The studied metals and alloys are described.
Data on measurements of structural parameters are presented in the next
chapter. Dependences on time of the size and misorientations of the sub-
grains are obtained for various metals. Attention is given to the dislocation
Introduction 3
structure of sub-boundaries that are formed during strain. The experimental
data concerning dislocations within subgrains are presented and discussed in
more detail. The totalities of the structural peculiarities of the metals, which
have been deformed at high-temperatures, are formulated.
In the fourth chapter the physical mechanisms of the high-temperature
deformation of pure metals and solid solutions are worked out on the basis of
the obtained data. The quantitative model of creep is considered and validated.
Equations are presented for the dislocation velocity and for the dislocation
density. The physically based forecast of the minimum strain rate is given.
The subject of the fifth chapter is a computer simulation of the high-

temperature deformation processes. A system of ordinary differential equa-
tions models the phenomenon under study. Evolution of structural parame-
ters and the effect of external conditions on the parameters are analyzed.
High-temperature deformation of the creep-resistant superalloys is thesub-
ject of the sixth chapter. Structure changes in modern materials and the inter-
action between deforming dislocations and particles of the hardening phase
are analyzed. A physical mechanism of deformation and a strain rate equa-
tion are considered. Data are presented on the connection between mean-
square amplitudes of atomic vibrations in the hardening phase and the creep
strength.
The seventh chapter is devoted to the single-crystal superalloys. The effect
of orientation, temperature and stress on the properties of single crystals
is considered. The physical mechanisms of the dislocation deformation are
described. Attention is given to the phenomenon of rafting and to the role of
misfit between thecrystal latticeparameters of thematrix andof the hardening
phase.
The subject of the last chapter is the peculiarities of the strain behavior of
refractory metals.
A detailed review of all aspects of the problem under consideration for pure
metals goes beyond the scope of this book. Therefore known principles and
established facts are mentioned only briefly.
The reader can find reviews concerning the creep of metals in different
books and articles, for example [1–8].
High Temperature Strain of Metals and Alloys, Valim Levitin (Author)
Copyright
c
 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
ISBN: 3-527-313389-9
5
1

Macroscopic Characteristics of Strain of Metallic Materials at
High Temperatures
The deformation of a metal specimen begins with the application of a load.
There are two kinds of high-temperature strain, namely, deformation un-
der constant stress σ (i.e. creep) and deformation under constant strain rate
˙ε. Physical distinctions between these two processes are not essential. In
this book we shall use the definitions “high-temperature strain” and “high-
temperature creep” almost as synonyms.
In Fig. 1.1 one can see the dependence of strain upon time, ε(t), when the
applied stress remains constant. In the general case the curve contains four
stages: an incubation, primary, steady-state and tertiary stages. The steady-
state stage is the most important characteristic for metals, because it takes
up the greater part of the durability of the specimen. Correspondingly, the
minimum strain rate during the steady-state stage, ˙ε, is an important value
because it determines the lifetime of the specimen. The tertiary stage is associ-
ated with a proportionality of the creep strain rate and the accumulated strain.
It is observed to a certain extent in creep resistant materials. The tertiary stage
is followed by a rupture.
Fig. 1.1 The typical curve of creep.
6 1 Macroscopic Characteristics of Strain of Metallic Materials at High Temperatures
Thus, the following stages are observed:
1. The incubation deformation. For this stage the strain rate ˙ε =const;
¨ε>0.
2. The primary stage, during which ˙ε = const;¨ε<0. The creep rate de-
creases when the strain increases.
3. The steady-state strain. The plastic strain rate is a constant value.
˙ε =const.
4. The tertiary stage. ˙ε =const; ¨ε>0. The tertiary creep leads to a rupture.
High-temperature strain is a heat-activated process. An elementary defor-
mation event gets additional energy from local thermal excitation. It is gener-

ally agreed that above 0.5 T
m
( T
m
is the melting temperature) the activation
energy of steady-state deformation is close to the activation energy of self-
diffusion. The correlation between the observed activation energy of creep,
Q
c
, and the energy of self-diffusion in the crystal lattice of metals, Q
sd
,isil-
lustrated in Fig. 1.2. More than 20 metals show excellent correlation between
both values.
The measurement of the dependences ˙ε(σ, T ) was the first step in the in-
vestigation of the problem under consideration. The functions σ(˙ε,T ) and
the rupture life (durability) τ (σ, T ) have also been studied. For the depen-
dence of the minimum strain rate ˙ε upon applied stress σ several functions
have been proposed by different authors. The explicit function ˙ε(σ, T ) is still
the subject of some controversy. The power function, the exponent and the
hyperbolic sine have been proposed.
The following largely phenomenological relationships between ˙ε, σ and T
are presented in various publications.
˙ε = A
1
exp

−
Q
kT


σ
n
(1.1)
˙ε = A
2
exp

−
Q −vσ
kT

(1.2)
˙ε = A
3
exp

−
Q
kT

sinh

ασ
kT

(1.3)
where A
1
,A

2
,A
3
,n,v,αare constant values; Q is the activation energy of the
process; k is the Boltzmann constant and T is temperature.
If we suppose that constants A
1
,A
2
,Q,n,vdo not depend upon tempera-
ture then it is easy to obtain
Q = −k

∂ ln ˙ε
∂

1
T


σ
(1.4)
7
Fig. 1.2 Comparison of the activation energy of creep, Q
c
,
and the activation energy of self-diffusion, Q
sd
, for pure
metals. The activation volume, ∆V

c
is also shown. Data of
Nix and Ilshner [7].
Thus, the activation energy can be found from experimental curves of ln ˙ε
vs. 1/T .
If A
2
and Q do not depend upon stress
v = kT

∂ ln ˙ε
∂σ

T
(1.5)
where v is an activation volume. The latter value can be calculated from the
dependence of ln ˙ε on σ.
Transmission electron microscopy is used, in particular, for the study of
crept metals. Investigators have observed the formation of subgrains in dif-
ferent metals. Grains in polycrystalline materials as well as in single crystals
disintegrate during high-temperature deformation to smaller parts called sub-
grains or cells.
First, we show an electron micrograph of subgrains and sub-boundaries
in crept nickel, Fig. 1.3. One can see a clean area in the center of (a), i.e.
8 1 Macroscopic Characteristics of Strain of Metallic Materials at High Temperatures
a subgrain or cell, surrounded by dislocation aggregations. The cell walls
separate relatively dislocation-free regions from each other. Subgrains are
also seen at the borders of the picture.
Aggregations of dislocations in sub-boundaries seem to be more or less
ordered. We observe regular dislocation lines elongated in the same direction.

The dislocation lines form low-angle sub-boundaries unlike the large-angle
boundaries between crystallites (grains). Thus, the subgrains are misoriented
to each other. The misorientation is of the order of tens of angle minutes i.e.
of milliradians.
Fig. 1.3 Subgrain in nickel tested at 1073K, stress 20MPa.
(a) Bright-field image. Screw dislocations along [
¯
101] are
denoted as B. (b) Electron diffraction pattern. (c) Scheme
of the arrangement of dislocations inside the boundary.
In Fig. 1.3(a) the so-called diffraction contrast is observed. It is created
by separate dislocations in sub-boundaries. Strictly speaking, the electronic
beam generates an interference contrast due to stresses near the dislocation
line. In Fig. 1.3(c) the screw sub-boundary dislocations are shown to be elon-
gated in the directions of the face diagonals of the cubic face-centered crystal
lattice.
Several theories of dislocation mechanisms of high-temperature deforma-
tion were proposed in early studies on the problem. According to the the-
ories of one group a glide of dislocations along slip planes occurs during
the creep process and this is followed by a climb of edge dislocations at the
rate-controlling distances [9, 10]. The climb velocity depends upon the flux of
vacancies in the crystal lattice.
Another groupof theoriesconsider creepas adiffusion controlledmotion of
screw dislocations with jogs [11]. The jog is known to be a bend, a double kink
at the dislocation line. The jog cannot move further without diffusion of the
lattice vacancies or interstitial atoms. Only thermal equilibrium generation
of jogs was considered. The probabilities of the heat generation of alternating
9
jogs that have opposite signs (vacancy-emitting and interstitial-emitting) are
equal to each other. Thus, from Barrett and Nix’s [11] point of view a screw

dislocation contains both types of thermally generated jogs, equally spaced
and alternate along the dislocation line. They emphasize that the average
spacing between jogs was never measured directly.
Attention has been devoted in the literature to other theories. Some inves-
tigators developed a model for creep based on the Frank dislocation network
[12]. Concepts of internal stresses were discussed in subsequent publications
as well as steady-state substructures and possible values of n in the power
law (1.1). The dislocation theories of creep have been considered in detail in
a review [7].
I would like to emphasize certain shortcomings in these studies and in the
state of the problem under consideration.
1. The researchers pay special attention to the functional connections be-
tween the external parameters of deformation: i.e. between the strain rate
and stress. For example, principal concern is paid to the numerical value of
the steady-state stress exponent, n, in the power law (1.1). However, the same
experimental data can satisfy both Eq. (1.2) and Eq. (1.1). The more so when
graphs are plotted usually in logarithmic coordinates. Moreover, Eq. (1.3) be-
comes Eq. (1.2) if the stresses are not small enough. According to my point
of view, an analysis of the dependences ˙ε(σ, T ) or σ(˙ε,T ) cannot allow one to
conclude unequivocally about the physical mechanism of the phenomenon
under consideration.
2. Some properties of dislocations as defects of the crystal lattice are the ba-
sis for various dislocation models of high-temperature deformation. It would
be much better to use the real parameters of the structure which could be mea-
sured experimentally. On the contrary, some parameters of theories, which
have been proposed, cannot be measured.
3. It is surprising that though the substructural elements have been ob-
served in many studies on various metals, none of the previous strain rate
equations contains these parameters directly. It appears that very little system-
atic data for correlation between the structure and the creep behavior have

been reported. Dimensions and misorientations of substructural elements
have not been measured sufficiently.
4. No attempts have been made to calculate or even to estimate the strain
rate of metals and solid solutions based on the test conditions, observed struc-
ture and material constants.
5. Some authors introduce equations, which contain 3–5 or more so-called
fitting parameters. Varying these parameters enables one to obtain a satisfac-
tory fit between experimental and calculated deformation curves. However,
one should not draw any conclusion about the correctness of a physical theory
from this fit.