High Temperature Strain of Metals and Alloys

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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 [email protected] Cover: “Blish” turbine University of Applied Sciences Gießen-Friedberg, Department MND, MTU

All books published by Wiley-VCH are carefully produced. Nevertheless, authors, editors and publisher do not warrant the information contained in these books, including this book, to be free of errors. Readers are advised to keep in mind that statements, data, illustrations, procedural details or other items may inadvertently be inaccurate. Library of Congress Card No.: applied for British Library Cataloguing-in-Publication Data: A catalogue record for this book is available from the British Library. Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at http://dnb.ddb.de c 2006 WILEY-VCH Verlag GmbH & Co KGaA, Weinheim All rights reserved (including those of translation into other languages). No part of this book may be reproduced in any form – by photocopying, microfilm, or any other means – nor transmitted or translated into a machine language without written permission from the publishers. Registered names, trademarks, etc. used in this book, even when not specifically marked as such, are not to be considered unprotected by law. Typesetting: Steingraeber Satztechnik GmbH, Ladenburg Printing: Strauss GmbH, Mörlenbach Binding: Litges & Dopf Buchbinderei GmbH, Heppenheim Cover: aktivComm, Weinheim Printed in the Federal Republic of Germany Printed on acid-free paper ISBN-13: 978-3-527-31338-9 ISBN-10: 3-527-31338-9

V

Contents Introduction 1

1

Macroscopic Characteristics of Strain of Metallic Materials at High Temperatures 5

2

In situ X-ray Investigation Technique

2.1 2.2 2.3 2.4 2.5 2.6 2.7

Experimental Installation 13 Measurement Procedure 15 Measurements of Structural Parameters Diffraction Electron Microscopy 20 Amplitude of Atomic Vibrations 21 Materials under Investigation 23 Summary 24

3

Structural Parameters in High-Temperature Deformed Metals Evolution of Structural Parameters 25 Dislocation Structure 30

3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 4

4.1 4.2 4.3 4.4

13

17

25

Distances between Dislocations in Sub-boundaries 34 Sub-boundaries as Dislocation Sources and Obstacles 34 Dislocations inside Subgrains 35 Vacancy Loops and Helicoids 39 Total Combination of Structural Peculiarities of High-temperature Deformation 40 Summary 41 Physical Mechanism of Strain at High Temperatures Physical Model and Theory 43 Velocity of Dislocations 45 Dislocation Density 49 Rate of the Steady-State Creep 51

43

High Temperature Strain of Metals and Alloys, Valim Levitin (Author) c 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim Copyright ISBN: 3-527-313389-9

VI

Contents

4.5 4.6 4.7 4.8 4.9 4.10

Effect of Alloying: Relationship between Creep Rate and Mean-Square Atomic Amplitudes 54 Formation of Jogs 55 Significance of the Stacking Faults Energy 57 Stability of Dislocation Sub-boundaries 58 Scope of the Theory 62 Summary 64

5

Simulation of the Parameters Evolution

5.1 5.2 5.2.1 5.2.2 5.2.3 5.2.4 5.2.5 5.2.6 5.2.7 5.3 5.4 5.5

Parameters of the Physical Model Equations 68 Strain Rate 68 Change in the Dislocation Density 68 The Dislocation Slip Velocity 69 The Dislocation Climb Velocity 69 The Dislocation Spacing in Sub-boundaries 70 Variation of the Subgrain Size 71 System of Differential Equations 71 Results of Simulation 71 Density of Dislocations during Stationary Creep 77 Summary 80

6

High-temperature Deformation of Superalloys γ Phase in Superalloys 83

6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9

67 67

83

Changes in the Matrix of Alloys during Strain 88 Interaction of Dislocations and Particles 89 Creep Rate. Length of Dislocation Segments 95 Mechanism of Strain and the Creep Rate Equation 96 Composition of the γ Phase and Atomic Vibrations 102 Influence of the Particle Size and Concentration 104 The Prediction of Properties 106 Summary 109

7

Single Crystals of Superalloys

7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8

Effect of Orientation on Properties 111 Deformation at Lower Temperatures 116 Deformation at Higher Temperatures 124 On the Composition of Superalloys 129 Rafting 130 Effect of Composition and Temperature on γ/γ Misfit Other Creep Equations 137 Summary 141

111

136

VII

8

8.1 8.2 8.3

Deformation of Some Refractory Metals The Creep Behavior 143 Alloys of Refractory Metals 149 Summary 155 Supplements

143

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 Index

169

168

1

Introduction Whoever controls the materials, controls the science and the technology E. Plummer Modern civilization is based on four foundations: materials, energy, technology, 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 remarkable properties. One – their strength at high temperatures – is of great scientific and practical importance. The durability of gas 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 investigation 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 dependences 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 scattered in numerous articles. Previous investigations improved our knowledge High Temperature Strain of Metals and Alloys, Valim Levitin (Author) c 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim Copyright ISBN: 3-527-313389-9

2

Introduction

of the problem and stimulated further experimental approaches. It is essential, however, to emphasize that the physical nature of the high-temperature strain in metals, especially industrial superalloys, is not yet understood sufficiently. 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 diffusion mobility of atoms and 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 lead to a definite macroscopic behavior of the material, 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 structural and kinetic characteristics which are related to physical fundamentals of the high-temperature strain of metallic materials. A number of specific parameters 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, structural 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 peculiarities 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 investigation 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 subgrains are obtained for various metals. Attention is given to the dislocation

Introduction

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 hightemperature deformation processes. A system of ordinary differential equations models the phenomenon under study. Evolution of structural parameters and the effect of external conditions on the parameters are analyzed. High-temperature deformation of the creep-resistant superalloys is the subject of the sixth chapter. Structure changes in modern materials and the interaction between deforming dislocations and particles of the hardening phase are analyzed. A physical mechanism of deformation and a strain rate equation are considered. Data are presented on the connection between meansquare 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 the crystal lattice parameters of the matrix and of 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].

3

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 under 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 “hightemperature 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 steadystate 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 associated 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 decreases 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 deformation event gets additional energy from local thermal excitation. It is generally agreed that above 0.5 Tm ( Tm is the melting temperature) the activation energy of steady-state deformation is close to the activation energy of selfdiffusion. The correlation between the observed activation energy of creep, Qc , and the energy of self-diffusion in the crystal lattice of metals, Qsd , is illustrated 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 investigation of the problem under consideration. The functions σ(ε, ˙ T ) and the rupture life (durability) τ (σ, T ) have also been studied. For the dependence 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. Q ε˙ = A1 exp − (1.1) σn kT

Q − vσ ε˙ = A2 exp − kT

ασ Q ε˙ = A3 exp − sinh kT kT

(1.2)

(1.3)

where A1 , A2 , A3 , n, v, α are constant values; Q is the activation energy of the process; k is the Boltzmann constant and T is temperature. If we suppose that constants A1 , A2 , Q, n, v do not depend upon temperature then it is easy to obtain ∂ ln ε˙ Q = −k (1.4) ∂ T1 σ

7

Fig. 1.2 Comparison of the activation energy of creep, Qc , and the activation energy of self-diffusion, Qsd , for pure metals. The activation volume, ∆Vc 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 A2 and Q do not depend upon stress ∂ ln ε˙ v = kT (1.5) ∂σ T 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 different metals. Grains in polycrystalline materials as well as in single crystals disintegrate during high-temperature deformation to smaller parts called subgrains 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


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 elongated in the directions of the face diagonals of the cubic face-centered crystal lattice. Several theories of dislocation mechanisms of high-temperature deformation were proposed in early studies on the problem. According to the theories 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 group of theories consider creep as a diffusion controlled motion 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 investigators 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 between 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) becomes 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 basis for various dislocation models of high-temperature deformation. It would be much better to use the real parameters of the structure which could be measured 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 observed in many studies on various metals, none of the previous strain rate equations contains these parameters directly. It appears that very little systematic 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 structure 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 satisfactory fit between experimental and calculated deformation curves. However, one should not draw any conclusion about the correctness of a physical theory from this fit.

10


6. The various directions of research are somewhat separated from each other. 7. The physical nature of the creep deformation behavior of industrial superalloys has not been investigated sufficiently. Quantitative physical theories are still being worked out. I consider that the essence of the problem of the physical fundamentals of high-temperature strain consists in structural evolution under specific external conditions. My approach to the problem is based on the concept that the effect of applied stresses upon the crystal lattice at high temperatures results in distinctive structural changes and these specific changes lead to the definite macroscopic behavior of a material, especially, to the strain rate and to the stress resistance. A key to the problem is the response of the structural elements of a material. In some way the situation is in accordance with the Le Chatelier rule. The changes in a metallic system which take place under the influence of external conditions are directed so as to relax this influence. The formation of an ordered dislocation structure is just an evolution process which tries to act against applied stresses. The point is that the high temperature conditions give the possibility of supplying the dislocation rearrangement with energy and which results in the substructure formation. That is why our aim is first to investigate quantitatively and in detail the interaction of dislocations with each other, the formation of subgrains, the interactions between dislocations and particles in superalloys, and only then to conclude a physical mechanism for the process. The nature of microscopic processes should be revealed as a result of experiments that enable one to observe the events on the atomic, microscopic scale, and not on the basis of the general properties of crystal lattice defects nor on the basis of mechanical tests. This approach enables us to find unequivocal and explicit expressions for the high-temperature steady-state strain rate. These expressions contain substructural characteristics, physical material constants and external conditions. The essence of this approach is defined as the physics of the processes, which are the structural background and the kinetic basis of the macroscopic deformation of metals and solid solutions in the interval (0.40–0.70) Tm , where Tm is the absolute melting temperature. Superalloys operate at higher temperatures. Thus, the planned path can be shown schematically as follows. Systematic investigations of the structure of metals strained at high-temperature. ⇒ The determination of the physical mechanism of strain, which should be based upon experimental data. ⇒ Calculation of the macroscopic strain rate on the basis of this mechanism. ⇒ Comparison with experiment.

11

This plan demands first an efficient structural investigation and detailed proofs of correctness of physical models. An in situ investigation of metals is necessary in order to address the problem of the physics of the high-temperature deformation.

13

2

The Experimental Equipment and the in situ X-ray Investigation Technique 2.1 Experimental Installation

The experimental installation for direct measurements of substructural changes in the massive metallic specimens during deformation at high temperatures must meet the following requirements. 1. The sensors used to measure stress and elongation of the specimen must be compact because of the relatively small distance between the axis and the slits of an X-ray goniometer (for example, a typical distance may be 65 mm). 2. It is advantageous to mount a loading mechanism on the frame of the chamber in order to provide rotation of the specimen around the axis of the goniometer during its exposure to X-rays. The mechanism must be able to create a load of the order of thousands of Newtons while preserving the vacuum. 3. The windows in the chamber should be arranged in such a way that they are transparent to X-rays. Recording of the scattered irradiation must be provided in the interval of the Bragg angles that is important for measurements. Figure 2.1 shows a set-up of an experimental installation designed according to these requirements [13]. A specimen 1 is fastened in holders 2 and 3. A double-shovel shaped specimen with gauge diameter 1.5 to 3.0 mm and gauge length up to 20 mm is used. The lower holder 3 is fastened rigidly to the frame of the chamber. The upper holder 2 can move along the axis direction. Movement of the upper holder is achieved with an electric motor, a reducer and a worm-and-worm gear 5. The gear rotates on the external thread of a hollow rod 6. The speed of the holder 2 may be continuously adjusted with the electric motor. One may also use different reducer gears. The speed may be varied from 5 × 10−7 to 3 × 10−3 s−1 . Silphon 7 enables one to move the rod relative to the chamber while preserving the vacuum. Due to the motion of the rod 6 it is possible to apply a load of up to 2000N to the specimen. A special dynamometer 8–12 is used to measure the stress. It High Temperature Strain of Metals and Alloys, Valim Levitin (Author) c 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim Copyright ISBN: 3-527-313389-9

14

2 In situ X-ray Investigation Technique

consists of a frame 8 and an electronic valve 9 (a so-called diode mechanotron). Deformation of the frame because of stress is transmitted through the screw 10 to the stem of the valve. The elongation of the specimen is measured with a clock indicator 14 with an accuracy of 0.01 mm. The chamber 18 and covers 17 have special windows 19 made of beryllium. The initial beam enters and the scattered irradiation leaves through this window. The location and the size of the window enables one to measure angles of 2θ in the interval from 50◦ to 165◦ . The Wilson packing 24 enables one to rotate the chamber around the goniometer axis during X-ray irradiation. The second cover has a window made of glass in order to measure the temperature of the specimen with an optical pyrometer. The temperature can also be measured with a thermocouple fastened to the specimen. The chamber 18 and covers 17 are cooled with flowing water. Adjustment of the chamber relative to the initial X-ray beam is provided by sledges 22, 23 in two perpendicular directions. The specimen 1 is heated by electrode 16 by passing an alternating current. The electrode is cooled with flowing water. The specified temperature is maintained with an electric circuit with an accuracy of 5K. A mechanical pump and sorption pump ensure, through the hollow rod 6, a vacuum level in the chamber of less than 1.33 × 10−4 Pa (1 × 10−6 torr).

Fig. 2.1 The experimental installation for

the X-ray structural investigations of metallic specimens during high-temperature tests.

2.2 Measurement Procedure

2.2 Measurement Procedure

Polycrystal specimens of metals and alloys are investigated. A method for measuring the irradiation intensity which is diffracted with separate crystals has been worked out. Dependence of the X-ray intensity upon the double Bragg angle, I(2θ), is recorded. The measurement procedure is as follows: the specimen is placed between holders in the chamber. The thermocouple is fastened to the operating point of the specimen. The specimen is then adjusted relative to the initial beam; the vacuum is established in the chamber and the heating is turned on. First it is necessary to choose a number of crystals to be monitored and to determine the exact coordinates of their reflections. There are three angle values that enable one to define a reflecting position of a crystal: the rotation of the specimen with the chamber around the goniometer axis (angle ω) and the rotation of the detector of the scattered X-rays in the meridianal plane (angle ψ). These two rotations make possible the selection of reflecting crystals when the detector is installed in advance in the horizontal plane at the double Bragg angle 2θ relative to the initial X-ray beam. It is possible to obtain the maximum of intensity as a control point by means of thorough adjustment of all three angles. The monochromatic irradiation Kβ of an X-ray tube is used. X-ray irradiation is performed before loading the specimen, then straight after loading and subsequently at the regular intervals. Recording of a diffraction curve usually takes from 5 to 10 min and is repeated three times. In Fig. 2.2 the formation of diffracted radiation is presented. Sections of the Evald sphere are shown. A projection of the reflecting plane is seen as a short line segment at the center of Fig. 2.2(c). A node of a so-called reciprocal lattice is in a reflected position, i.e. on the surface of the Evald sphere. Three angles conform to this position: angle ω of the crystal rotation and the two angles, 2θ and ψ, of the detector motion. Angles ω and 2θ are measured in the equatorial (horizontal) plane of the goniometer and angle ψ in the meridianal (vertical) plane. The diffracted beam is recorded permanently with a fixed scintillation detector with a rectangular split. The dimensions of the node in the reciprocal lattice are known to be dependent upon the misorientation angle δ of subgrains (cells) in the metal under examination, the divergence χ of the initial X-ray beam, the interval between wavelengths and the crystal dimensions. The divergence results in the appearance of a stroke χ, which is formed by the ends of the diffracted beam vectors. This stroke is directed to the reciprocal crystal lattice vector H at an angle θ.

15

16


Fig. 2.2 Formation of the diffracted beam for the method of investigation: (a) the Evald sphere; s0 and s are unit vectors of the initial and of the diffracted is the reciprocal beams, respectively; H lattice vector; EP, MP are equatorial and meridianal planes, respectively. (b) The

reflection strip (shaded) at the intersection of a node of the reciprocal lattice and the Evald sphere. (c) Directions of erosion of the reciprocal lattice node due to the divergence of the initial beam χ and to the finite size, l, of the studied crystal.

The experimental technique that has been worked out by us enables one to study structural changes in the same crystallites of the polycrystalline specimens during high-temperature deformation. For this purpose the angular dependence of the diffracted intensity, I(2θ), is measured. The most typical range of conditions was chosen: temperatures in the interval from 0.40 Tm to 0.80 Tm , stresses between 10−4 µ and 2 × 10−3 µ, where Tm is the melting temperature and µ is the shear modulus.

2.3 Measurements of Structural Parameters


It was revealed that the high-temperature deformation does not result in a broadening of the X-ray reflections. Therefore dynamic effects were used to obtain data about the material structure. The multiple wave reflections from parallel crystalline planes of the same crystal lead to a reduction in the wave energy. This phenomenon is called primary extinction. The X-ray intensity loss depends upon the number of reflecting planes, i.e. upon the subgrain size. Measurements of a relation between irradiated and initial intensities make it possible to determine the dimensions of the reflecting crystal. According to the classical theory of Darvin [14] the decrease in intensity factors of irradiation due to primary extinction is given by f=

tanh(nq) nq

(2.1)

where n is the number of the parallel reflecting planes in the crystal, q is the so-called reflection power of the crystal plane. q=

e2 N |F |λ csc θ mc2

(2.2)

where e is the charge of an electron, m is the mass of an electron, c is the velocity of light, N is the number of elementary cells in the unit of irradiated volume, F is the structural amplitude and θ is the Bragg angle. The size of a subgrain is equal to D = nd

(2.3)

where d is the interplane spacing in the crystal lattice. A screening effect is also observed. Internal subgrains are screened with subgrains which are situated in external layers of the material. This phenomenon is called secondary extinction. Secondary extinction results in an increase in the absorption coefficient, µ. The increment of X-ray absorption is equal to gQ, where 1 g= √ (2.4) 2 πη Q=

e2 mc2

2

2 λ3 2 1 + cos 2θ |F | a6 2 sin 2θ

(2.5)

where η is the mean angle of misorientation of neighboring subgrains, Q is the reflectivity of the crystal, λ is the wavelength, a is the crystal lattice

17

18


parameter. Measurements of subgrain dimensions should be performed in conditions where secondary extinction does not play a considerable part. Thus, we can write the following conditions: f < 1; gQ µ

(2.6)

Assume that gQ = 0.1µ. It follows that nq = 2Dd

Q Lλ3

(2.7)

where L is the angle coefficient, which appears in Eq. (2.5). Inequalities (2.6) are satisfied when nq > 0.59. Therefore the following inequality must also be satisfied: Dd

µη > 0.495 Lλ3

(2.8)

Consequently, interferences with a large interplane spacing d should be chosen for measurements of subgrain sizes. For example, the minimum values of D to be measured are equal to 0.29, 0.34, 0.13µm for Ni, Fe, W, respectively. We have used the following method to calculate the values of substructure parameters. The full power of a diffracted X-ray beam, which is scattered by a crystal, is expressed as I = I0

1 f QV 2(µ + gf Q)

(2.9)

where I0 is the power of the initial beam, V is the crystal volume, the other variables have been described above. Denote the intensity (power) of a beam diffracted by a crystal in the initial strainless state by Iin , after high-temperature deformation by IT , after strong deformation at room-temperature by Id . It follows from the general formula (2.9) that 1 (2.10) Iin = I0 fin QV 2(µ + gin fin Q) Strong “cold” deformation of a specimen results in an increase in the density of dislocations and other crystal lattice defects. Under these conditions both types of extinction are suppressed, and fd = 1; gd fd Qd µ. Thus Id can be expressed as QV (2.11) Id = I0 2µ


We have IT = I0

1 fT QV 2(µ + gT fT Q)

(2.12)

We may neglect the difference between values Qin and QT because the increase in temperature influences Q and the fraction in Eq. (2.9) in opposite directions. Measurements of Iin and Id as well as Iin and IT are performed for the same crystallite. Therefore taking the ratios in pairs we obtain the following equations for calculation: Iin fin µ (2.13) = Id µ + gin fin Q Iin fin (µ + gT fT Q) = IT fT (µ + gin fin Q)

(2.14)

The order of calculation is as follows. First values of gin , gT are calculated. In order to be able to compute gin , gT from Eq. (2.4) one needs the values of the angles η. These have to be found from independent measurements. Then Eq. (2.13) is used to calculate fin . Next one calculates fT from Eq. (2.14) and finally calculates the subgrain sizes D from Eqs. (2.1) and (2.3). This method of measurement gives a relative accuracy of 5–7%. In Fig. 2.3 the distribution curves for misorientation angles δ in the subgrain are presented. These data were obtained by rotating the specimen around the axis of the goniometer while the detector was motionless. It goes without saying that monochromatic irradiation was used.

Fig. 2.3 Distribution of angle misorientations of subgrains in nickel. Symbols correspond to the Gaussian distribution. Solid curves are the experimental dependences. Test temperature 1073K. 1, stress 20MPa; 2, stress 14MPa.

19

20


This distribution was found to be a Gaussian distribution as was verified by means of a the so-called Kolmogorov test. In Fig. 2.3 the theoretical dependence is marked with symbols. From the fact that the distribution of misorientations conforms with the Gaussian law one may calculate the mean angle between adjacent subgrains: η = 0.35δ

(2.15)

The density of dislocations within subgrain walls may be estimated as [15] ρs =

η bD

(2.16)

2.4 Diffraction Electron Microscopy

High-resolution transmission electron microscopy (TEM) enables the direct observation of metal structure and therefore has an advantage over other methods. There are some typical difficulties one faces when using TEM: the field of view is relatively small; the specimen must be thin enough, of the order of 100 nm, so that it is transparent to the electron beam; it is possible to deform thin foils during preparation. It is appropriate to apply both the X-ray method and TEM so that they complement each other and this combination is particularly valuable for studying high-temperature strain. Electron waves are scattered by the thin crystal specimen. The electron intensity distribution in the specimen brings about a variable brightness on the screen of the microscope. The direct beam generates a so-called lightfield image. Deflection of the diffracted beams from the optical axis of the microscope is about 20 mrad. Diffracted beams are usually absorbed with an aperture. Crystal lattice defects cause displacements of atoms from their equilibrium positions. These distorted areas scatter electron waves differently, and a diffraction contrast can be seen on the screen of the instrument. Diffracted beams also form images. To study them one has to decline the illuminating system of the microscope in order to shift the image to the center of the screen, where result dark-field images are formed. Thus a diffraction contrast from defects is observed if the aperture passes either the direct or the diffracted electron beam. Atomic displacements, which are parallel to a reflecting crystal plane do not contribute to the diffraction contrast but perpendicular displacements of atoms lead to a contrast image.

2.5 Amplitude of Atomic Vibrations

Dislocations as line defects of the crystal lattice cause regular, ordered displacements of atoms from their equilibrium positions. The equality (g · b) = 0

(2.17)

is the condition for disappearance of the contrast. Here g is the vector that is perpendicular to the reflecting plane (vector of the reciprocal lattice); b is the Burgers vector (i.e. vector of atomic displacements). The dislocation becomes visible if this scalar product differs from zero: (g · b) = 0. Equation (2.17) is used to determine the Burgers vectors. Modern electron microscopes are supplied with a goniometric holder, which enables one to incline and to rotate the foil in order to change diffraction conditions. The chosen dark-field image can be placed in the center of the screen and studied as well. (g · b) analysis as a method of determination of Burgers vectors is described in detail in, for example, [16]. In our work discs were cut from the tested specimens for transmission electron microscope analysis. A special machine was used to cut off thin plates from the specimens. An artificial filament is applied as a working element. The moving filament carries a chemical agent. Afterwards the plates of 0.1–0.4 mm thick were electropolished; naturally, the composition of the electrolyte depends on the material of the specimen. Thin foils were examined in an electron microscope operating at 200keV. Light-field- and dark-field pictures in 3 or 4 reflections of the diffracted beams were examined. The dislocation structures observed in the micrographs were analyzed with respect to the Burgers vector b. The images were also used to determine the associated slip planes of the dislocations. The dislocation density, N , in the specimens was measured after the interrupted high-temperature tests by means of the intercept method. The number of intersections of dislocations in a foil with a square grid was calculated. From 10 to 15 pictures of total area 200µm2 , of each specimen were used for computation. This method of measurement has an accuracy within 15–25%.

2.5 Amplitude of Atomic Vibrations

Atom and ions, which are bonded with each other by considerable interatomic forces, are not motionless; due to the constant vibrations, they permanently deviate from their equilibrium positions. A typical order of the atomic vibration amplitudes is 10−11 m and that of the frequency 1013 Hz. The amplitude–frequency characteristics of the vibrating spectrum strongly influence, in particular, the heat-resistance of metals and alloys. This fact is

21

22


explained as being due to a relation between atomic amplitudes and diffusion parameters. The smaller the amplitudes of the atomic oscillations the smaller is the diffusion mobility of the atoms and the greater the resistance to applied stresses. Therefore measurements of amplitudes of atomic oscillations are of great interest. The wavelength of electromagnetic radiation in the X-ray range is of the same order as the interatomic distances in solids. When a crystal is irradiated, the X-rays excite the electrons in the atomic shells. They are forced to vibrate with a frequency equal to that of the electric field intensity vector of the initial electromagnetic wave. The reflected beams interfere with each other and the resulting electromagnetic vibration propagates in certain selective directions. The result of the interference depends on the distance between the atoms. The heat vibration motion of atoms has a great influence on the interference pattern. The intensity of the scattering of the X-rays by a group of atoms subjected to independent heat vibrations is weakened by the factor exp(−2M ), where −2M = −

16 2 2 sin2 θ π u 3 λ2

(2.18)

In Eq. (2.18) θ is the Bragg angle, λ is the wavelength, u2 is the mean-square atom amplitude. The vibrating displacements of atoms from equilibrium positions occur in different directions. The arithmetic mean of the atomic displacements is equal to zero, because all directions of displacements of atoms from the equilibrium positions in the crystallattice are equiprobable. By introducing the mean-square atom amplitude u2 one can eliminate negative values of displacements. Displacements are directed along perpendicular to the reflecting crystal plane. The mean-square amplitude is measured as follows. A specimen of a metal or an alloy is studied at a range of high temperatures as well as at room temperature. From formulas for the intensities of scattered X-rays at two temperatures one can obtain an expression for the intensity ratio

I 16 sin2 θ ln = − π 2 (u2 − u2 ) 2 I 3 λ

(2.19)

where the primed quantities refer to a high temperature and those without primes refer to room temperature. Thus, for calculation of the vibration amplitudes in conformity to Eq. (2.19) one should measure the ratio of intensities at two temperatures. The method allows one to determine the difference ∆u2 = u2 − u2 in the mean-square displacements of atoms at two temperatures.

2.6 Materials under Investigation

The amplitude of atomic vibrations increases with increasing temperature. For instance, in iron at 673K u2 = 21.0 pm. At T = 873K the amplitude increases to 28.1pm. The experimental technique of the X-ray measurement of amplitudes is described by the present author elsewhere [17].

2.6 Materials under Investigation

The materials under investigation were pure metals, binary substitutional solid solutions and superalloys. Metals with face-centered and body-centered crystal lattices were examined: nickel, copper, iron, vanadium, niobium and molybdenum. Nickel and iron are two of the most important materials for practical use and are applied as the base for numerous alloys for high-temperature operation. Vacuum-melted materials of 99.99% purity were used. Binary nickel-based alloys contained about 10at.% of the second component: chromium, aluminum, tungsten or cobalt. The compositions of the superalloys studied are presented in Table 2.1. They were melted in industrial vacuum furnaces. The specimens for hightemperature tests were prepared from hot-rolled rods. A standard heat-treatment of every superalloy included the solution treatment and one-step or two-step ageing followed by air cooling. The amount of the hardening phase and the creep strength increase in the sequence EI437B → EI698 → EP199 → EI867. Intermetallic compounds Ni3 Al and Ni3 (Al,W) were vacuum-melted in a laboratory furnace. Refractory metals (niobium and molybdenum) were produced in industrial arc-heating and electron-beam furnaces. Refractory materials were of commercial purity. Tab. 2.1 Chemical composition (wt.%) of superalloys under investigation. Alloy Ni3 Al Ni3 (Al,W) EI437B EI698 EI867 EP199

C

Cr

Al

Ti

W

Mo

Co

B

Nb

Ni

– – 0.06 0.08 0.02 0.05

– – 20.1 14.0 9.5 19.8

12.64 9.29 0.70 1.65 4.47 2.14

– – 2.52 2.70 – 1.42

– 9.90 – – 5.26 9.10

– – – 2.99 9.82 4.54

– – – – 5.12 –

– – 0.006 0.003 0.020 0.008

– – – 2.04 – –

87.36 80.81 rest rest rest rest

23

24


2.7 Summary

An experimental installation has been developed for in situ X-ray investigations of metals and alloys directly during high-temperature deformation. The method of measurement of the irradiation intensities, which are diffracted by separate crystallites of the investigated material, has been worked out. Dynamic effects are used to obtain data concerning the material structure. The sizes of subgrains and the angles of subgrain misorientations have been measured. The structural peculiarities of the high-temperature strained metals have also been studied by transmission electron microscopy. A combination of these two methods sheds light on the physical microscopic processes that are the basis of the macroscopic strain behavior. The values of the mean-square amplitudes of atomic vibrations have been determined by the X-ray method at high temperatures. Pure metals, solid solutions and nickel-based superalloys have been studied.

25

3

Structural Parameters in High-Temperature Deformed Metals 3.1 Evolution of Structural Parameters

In situ X-ray studies allow conclusions to be drawn concerning the effect of stresses at high temperatures on the evolution of structure in metals. The diffractometer curves change in shape after a stress is applied to a specimen. As a rule, the integral intensity of reflections grows and the angle width of the curve base increases appreciably, Figs. 3.1 and 3.2. The increase in the diffracted irradiation energy indicates that the reflecting structural elements become smaller. In Fig. 3.3 the lower curve, ε(t), is typical for the creep of nickel specimens. The primary and steady-state stages of deformation are seen. Variations in

Fig. 3.1 Change in the X-ray (111) reflection during high-

temperature strain. Copper tested at 610K under stress 14.7MPa. (a) Before loading, (b) the end of the primary stage, the integral intensity has increased. High Temperature Strain of Metals and Alloys, Valim Levitin (Author) c 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim Copyright ISBN: 3-527-313389-9

26

3 Structural Parameters in High-Temperature Deformed Metals

Fig. 3.2 As in Fig. 3.1 at the end of the

steady-state stage.

subgrain sizes D (upper curves) and in their misorientations η are presented on the same graph. Here and in all following figures each type of symbol corresponds to one crystallite of the same specimen. The initial mean size of the subgrains, D, is equal to 3.0µm, in the primary stage of deformation it decreases to 0.8µm and then is almost unchanged during the steady-state creep.

Fig. 3.3 Structural parameters versus time and creep curve

for nickel. Tests at temperature T = 673K (0.39 Tm ), σ = 130MPa (1.7 × 10−3 µ).

3.1 Evolution of Structural Parameters


for copper. Tests at temperature T = 610K (0.45 Tm ); σ = 19.6 MPa (4.0 × 10−4 µ).

The misorientation angle, η, increases from 2 to 5–7mrad. The change in η is observed during the primary stage. The smaller changes occur in the crystallite with the larger initial value of η (open circles). Estimation of the dislocation density in sub-boundaries in conformity with Eq. (2.16) gives a quantity of the order of 1013 m−2 . Subgrains and sub-boundaries are formed easily in copper, Fig. 3.4 and Fig. 3.5. The same result is observed under σ = (1.2–2.7) × 10−4 µ at all temperatures: the crystallites are reduced to fine cells and sub-boundaries are formed during the primary stage of creep. The value D decreases and the angle of misorientation increases. The steady-state strain occurs at almost constant mean values of both parameters. D and η depend strongly upon stress; the greater the applied stress the greater the misorientation angles and the smaller the sub-boundaries’ dimensions. Thus, the substructure is formed inside crystallites during the primary, transitive stage of creep. The origin of the steady-state strain coincides with the end of the substructure formation. These peculiarities are seen well in Figs. 3.4 and 3.5.

27

28



for copper. Tests at temperature T = 610K (0.45 Tm ); σ = 29.4 MPa (6.1 × 10−4 µ).

Processes naturally occur differently in different crystallites. Equilibrium values of D and η are somewhat distinct. There is a distribution in the size of these values, however, one may consider the mean values. The accuracy of the method is of concern. We have used the t-distribution for evaluation of the relative error of the average values. Accepting a confidence factor of 0.9 we find the minimum number of necessary measurements, n = 12. Under these conditions we obtain a mean relative error of 12% for D and 8% for η. In accordance with this result we usually investigated in situ at least 3 to 5 crystallites of 3 or 4 specimens of each material under each set of external conditions (temperature and stress). The substructure formation during high-temperature strain in vanadium is shown in Figs. 3.6 and 3.7. The data are obtained at the same temperature 0.6 Tm , but under different stresses. The rate of steady strain increases from 9 × 10−7 to 5 × 10−6 s−1 . The change in stress leads to a sharper increase in η and decrease in D. The values of the structural parameters in this metal are also dependent upon stress.

3.1 Evolution of Structural Parameters

Fig. 3.6 Structural parame-

ters and strain as a function of time for vanadium. Tests at temperature T = 1318K (0.60 Tm ); σ = 5.9 MPa (1.3 × 10−4 µ).

Fig. 3.7 Structural parame-

ters and strain as a function of time for vanadium. Tests at temperature T = 1318K (0.60 Tm ); σ = 9.8 MPa (2.1 × 10−4 µ).

The average values of the subgrain size and the subgrain misorientation at the beginning of the steady-state stage for face-centered metals are listed in Table 3.1.

29

30


Tab. 3.1 Average substructure parameters in nickel and copper at steady-state creep. Metal

Ni

Cu

T /Tm

T, K

σ/µ, 10−4

σ, MPa

D,µm

η, ¯ mrad

0.39

673

11.0 17.0 20.0

85 130 152

1.9 1.1 0.7

4.4 6.0 5.3

0.51

873

6.7 8.9 9.7

50 66 72

1.4 0.9 0.7

4.5 4.8 4.5

0.62

1073

1.3 2.0 2.7

10 14 20

2.0 1.7 1.0

5.3 6.3 7.5

0.45

610

3.0 5.1 6.1

14.7 24.5 29.4

1.5 1.0 0.6

3.1 3.8 4.8

0.50

678

1.8 3.4 4.2

8.8 16.7 20.6

1.8 1.2 0.9

3.1 4.2 4.7

0.55

746

1.2 2.0 2.8

5.8 9.8 13.7

1.8 1.7 0.8

3.5 3.5 5.4

In Table 3.2 the average values of the parameters at the steady-state creep are presented for three body-centered metals. D and η have the same order in various metals. D tends to increase with temperature. The value of η increases when the applied stress rises. In Fig. 3.8 one can see the effect of stress on the average subgrain size in nickel. The dependence is almost linear. Investigations of single-phase two-component alloys Ni–9.5Cr (at.%), Ni–9.9Al, Ni–10.1 Co, Ni–9.5W do not show any qualitative differences in the structure evolution from that in the pure metals. The formation of substructure inside crystallites also occurs in the substitutional solid solutions at the primary stage. However, solid solutions differ in having greater initial values of η. In solid solutions one observes, at the stationary deformation, greater values of η than in pure metals.

3.2 Dislocation Structure

Some regularities are revealed as the result of systematic examination of the bright- and dark-field image pictures and diffraction patterns of a large number of specimens. Most of the dislocations in specimens after hightemperature tests are associated in sub-boundaries. The parallel sub-boundary


Tab. 3.2 Average substructure parameters in niobium, vanadium and α-iron at the steady-state creep. Metal

Nb

V

T / Tm

T, K

σ/µ, 10−4

σ, MPa

D,µm

η, ¯ mrad

0.50

1370

7.8 9.4 12.0

29.4 35.3 44.1

1.3 1.3 1.2

2.9 3.3 3.3

0.55

1508

4.6 7.2 7.6

17.2 27.0 28.4

1.5 1.3 1.1

2.4 2.5 3.4

0.60

1645

2.1 2.6 3.1

7.8 9.8 11.8

1.6 1.3 1.1

2.2 2.7 3.2

0.50

1096

5.3 6.3 7.4

24.5 29.4 34.3

1.9 1.5 0.8

3.6 4.1 4.4

0.55

1206

2.6 3.7 4.2

12.8 17.2 19.6

2.2 1.6 1.4

4.3 4.5 4.6

0.60

1318

1.3 1.7 2.1

5.9 7.8 9.8

1.8 1.5 1.2

4.1 5.6 5.1

0.51

923

0.7 1.3 1.7

6.0 11.0 14.0

1.5 1.6 0.8

3.8 4.4 4.5

0.54

973

0.4 0.7 1.2

3.0 6.0 10.0

1.5 1.4 1.2

3.5 3.8 3.9

Fe

dislocations are situated at an equal distance from each other. It follows from the results of the Burgers vector determinations and from the repeating structural configurations that the parallel sub-boundary dislocations have the same sign. Two intersected dislocation systems are often observed inside sub-boundaries. These systems form the small-angle boundary. The electron micrographs of typical subgrains and sub-boundaries in niobium are presented in Fig. 3.9. Creep tests were carried out until the second stage of creep was reached. In Fig. 3.9(a) the Burgers vector of the dislocations is b = a[¯ 100], i.e. it is directed along the rib of the elementary cell of the cubic body-centered crystal lattice. The plane of the foil is the face (100). Figure 3.10 illustrates the dislocation sub-boundary in α-iron. Two systems of dislocations, which intersect each other at right angles, are observed. Dislocation lines are parallel to face diagonals, i.e. they are directed along crystalline directions [110] and [1¯ 10]. One of the systems is inclined noticeably to the foil plane. This is the cause of an oscillating contrast in the dislocation images.

31

32


Fig. 3.8 Subgrain dimensions versus applied stress. Nickel

tested at 673K, steady-state stage. Errors of measurements are shown with vertical bars.

¯ The Burgers vectors were determined to be b1 = a/2[111] and b2 = a/2[111]. Atomic displacements are directed along the body diagonals of the elementary cubic cell. The typical small-angle boundary in α-iron, which consists of pure screw dislocations, is shown in Fig. 3.11. Dislocations form a network with cells

Fig. 3.9 Transmission electron micrographs showing

dislocation sub-boundaries in niobium, which are formed in the steady-state creep. (a) T = 1370K, σ = 44.1MPa; (b) T = 1233K, σ = 39.2MPa. ×39 000.


Fig. 3.10 Transmission electron micrograph showing the dislocation sub-boundary in α-iron, which is formed in the steady-state creep. T = 813K and σ = 49.0MPa. The first dislocation system is directed along [110] with b1 = a/2[111], the second along [1¯ 10] with b2 = a/2[1¯ 11]. The plane of the foil is (2¯ 10). ×84 000.

of hexagonal shape. The dislocation lines are located along directions [1¯11], [¯ 111] and [001]. The Burgers vectors are b1 = a/2[1¯11] and b2 = a/2[¯111]. The third side of the network with b3 = a[001] appears to be formed as a result of reaction b3 = b1 + b2 . Hexagonal cells of the other sub-boundary in iron are seen in pattern (b). The regular dislocation networks as low-angle sub-boundaries are found to be typical for the high-temperature tested metals.

Fig. 3.11 Sub-boundaries in α-iron tested at T = 923K and σ = 12.0MPa. (a) The pure screw sub-boundary formed by dislocations along [1¯ 11], [¯ 111], [001] directions. (b) Network with hexagonal cells. Plane of foil is (110). ×66 000.

33

34


3.3 Distances between Dislocations in Sub-boundaries

The distance λ between parallel dislocations of the same sign in a small-angle boundary can be represented (when η 1, tan η η) by an expression of the form: b λ= (3.1) η where b is the modulus of the Burgers vector. Two methods were used in this work in order to measure the average spacing between sub-boundary dislocations. Note the satisfactory fit between the electron microscopy results and the X-ray data (Table 3.3). Tab. 3.3 Distances between dislocations in sub-boundaries. Metal

T, K

σ, MPa

TEM data λ, nm

X-ray data λ, nm

Ni

1073

14.0 20.0

42 ± 5 34 ± 4

45 ± 3 38 ± 3

Fe

773 813 973

50.0 49.0 11.0

36 ± 2 59 ± 5 41 ± 2

34 ± 3 73 ± 5 69 ± 7

Nb

1233

39.2

67 ± 13

74 ± 6

1370

29.4 44.1

109 ± 14 60 ± 3

107 ± 10 94 ± 6

746

7.8

87 ± 6

83 ± 6

Cu

3.4 Sub-boundaries as Dislocation Sources and Obstacles

The sub-boundaries that have been formed seem to be sources of slipping dislocations. The process of generation of mobile dislocations by sub-boundaries is readily affected by the applied stress. The TEM technique allows one to observe the beginning of a dislocation emission. The creation of dislocations occurs as if the sub-boundary blows the dislocations loops like bubbles. These loops broaden gradually and move further inside subgrains. One can see this effect for nickel in Fig. 3.12. The sub-boundary in α-iron that generates dislocations is shown in Fig. 3.13. The subsequent dislocation semi-loops are blown by the ordered boundary.

3.5 Dislocations inside Subgrains

Fig. 3.12 Transmission electron micrographs showing the

dislocation sub-boundary as a source of mobile dislocations in nickel. T = 1073K; σ = 20MPa. ×48 000.

Fig. 3.13 Emission of dislocation loops from the sub-boundary in αiron. Tests at 813K and σ = 49MPa. ×66 000.

At the same time sub-boundaries act as obstacles for moving dislocations. One can often observe a sequence of dislocation lines which are pressed to the sub-boundary and these can enter the boundary.


Some dislocations, which are observed in specimens after the high-temperature deformation, are not associated in sub-boundaries. They are located inside subgrains and have the Burgers vector a/2 < 111 > in metals with the body-centered crystal lattice, i.e. α-iron, vanadium, and niobium. The slip plane is generally of the {110} type. Screw dislocations are observed, as well

35

36


Fig. 3.14 Dislocations inside subgrains in niobium tested at T = 1370K and σ = 44.1MPa. s, Screw dislocations; j, jogs; h, helicoids; l, vacancy loops. ×26 000.

as edge or mixed ones. Screw dislocations are located at the left-hand side of Fig. 3.14 (marked with the letter s). These dislocations have the Burgers vector a/2 < 111 > and are found to be in the {¯110} plane. The second family of screw dislocations is seen on the right-hand side. Bends and kinks in the dislocations, marked with j, attract one’s attention. They give an impression that certain points of mobile dislocations are pinned up. This can be easily seen in the left lower corner of Fig. 3.14 and in other areas marked with the letter j. These kinks at mobile dislocations turn out to be of great importance for our understanding of the physical mechanism of the steady-state creep. Figure 3.15 illustrates the dislocation structure in nickel. Again screw components with kinks are observed. Another effect is the appearance of small dislocation loops. The dark-field technique allows one to conclude that these are vacancy loops. There are good reasons to assume that kinks and bends that have been described by us are jogs. A jog is known to be a segment of a screw dislocation, which does not lie in its plane of slipping. In fact, the jog is a segment of the

Fig. 3.15 Dislocations inside subgrains in nickel tested at T = 1023K and σ = 49.0MPa. s, Screw dislocations; j, jogs. ×48 000.


edge extra-plane and therefore it can move with the slipping screw dislocation only with emission or absorption of point defects (vacancies or interstitial atoms). During movement the jog slows the dislocation and lags behind. Even the highest resolution of the electron microscope is not sufficient for direct observation of jogs, since their length is of the order of one interatomic distance. However, kinks and loops that have been observed in this work for different metals show convincingly that the formation of jogs takes place during high-temperature deformation. Assuming that the kinks and bends in dislocation lines are produced by jogs we have measured distances z0 between adjacent bends. The histograms of these density distributions are presented in Fig. 3.16. Under the mentioned strain conditions the most probable quantities of z0 in nickel and niobium are 4–5 hn and 9–10 hn, respectively. ¯ between sub-boundary dislocaA comparison of the average distances λ tions, determined by the X-ray method, and the spacings z0 between jogs in mobile dislocations, measured with the aid of electron microscopy, is given in Table 3.4. n is the number of measurements of z0 values. Confidence intervals by probability 0.95 are also shown in the table. The two values are close to each other. In our opinion, the new experimental result that has been obtained ¯ z¯0 ≈ λ

(3.2)

is of great importance for our understanding of the physical mechanism of high-temperature deformation.

Fig. 3.16 Histograms of the distribution of distances between jogs in screw components of dislocations: (a) nickel, 1073K, 14.0MPa, number of measurements n = 129; (b) niobium, 1645K, 11.8MPa, n = 185.

37

38


Tab. 3.4 Comparison of average distances z0 between jogs

¯ between in mobile dislocations and of average distances λ subgrain dislocations. n is the number of measurements.

Metal

Nb

V

Ni

T, K

σ, MPa

¯ nm λ,

z0 , nm

n

1370

29.4 35.3 44.1

107 ± 10 94 ± 6 94 ± 6

120 ± 10 100 ± 8 97 ± 12

75 94 24

1508

17.2 28.4

130 ± 10 92 ± 7

120 ± 10 95 ± 11

120 37

1645

9.8 11.8

120 ± 10 97 ± 8

93 ± 9 95 ± 9

68 185

1096

24.5 34.3

80 ± 8 67 ± 6

90 ± 8 60 ± 7

115 35

673

110.0 140.0

55 ± 3 50 ± 3

56 ± 5 48 ± 3

85 211

873

50.0 58.0 72.0

63 ± 6 58 ± 4 58 ± 4

67 ± 7 54 ± 5 46 ± 5

60 125 75

1073

10.0 14.0 20.0

54 ± 5 45 ± 3 38 ± 3

65 ± 5 48 ± 4 45 ± 7

96 129 85

873

25.0 30.0

64 ± 5 60 ± 4

63 ± 5 53 ± 5

78 88

923

6.0

71 ± 5

76 ± 12

45

746

7.8

87 ± 6

83 ± 7

46

Fe

Cu

The density of dislocations, which are not associated in sub-boundaries, N , has been measured, and the results are presented in Table 3.5. Dislocation densities during the high-temperature deformation for the metals under study are estimated to be from 1011 m−2 to 1012 m−2 . Tab. 3.5 The density of dislocations inside subgrains. Metal

T, K 873

Ni 1023

Cu

678

σ, MPa

N , 1011 m−2

29.4 68.7 98.1 9.8 40.0

2.4 5.7 9.5 1.3 6.3

8.8 20.6

2.2 9.6

Metal

T, K 1370

Nb 1508

V

1096

σ, MPa

N , 1011 m−2

24.9 29.4 35.3 17.2 28.4

1.6 3.1 3.8 2.9 5.3

29.4 34.3

1.6 1.4

3.6 Vacancy Loops and Helicoids

3.6 Vacancy Loops and Helicoids

Closed dislocation loops as well as helicoids are observed very often in the structure of the high-temperature tested metals. Dark-field analysis makes it possible to determine the sign and the type of loops. The loops have been found to be of the vacancy type. Helicoids are known to be formed usually by screw dislocations under conditions of volume supersaturation by point crystalline defects. We can see the loops, marked with the letter l, in Fig. 3.14 and also in 3.15. In Fig. 3.17 the typical structures of helixes and vacancy loops are presented. The helicoid looks like a spiral in electron patterns. The foil in Fig. 3.17(a) and (b) coincides with the crystal plane (111). One third of the loops lie in the plane (1¯ 10), but two thirds are in planes (0¯11) and (¯101). Thus, vacancy loops are generated in the dislocation slip planes. In Fig. 3.17(c) a very interesting effect can be observed. Three chains of loops have been left behind two segments of screw dislocations. These moved in the slip plane (110). The dislocations have the Burgers vector a[¯110]; the loops are of the vacancy type. One can also see helicoids.

Fig. 3.17 Transmission electron micro-

graphs showing vacancy loops and helicoids: (a), (b) Iron tested at T = 973K and σ = 10MPa; ×46 000. (c) Niobium tested at T = 1508K and σ = 17.3MPa; ×39 000.

39

40


3.7 Total Combination of Structural Peculiarities of High-temperature Deformation

The structural peculiarities can be generalized from the observed facts. Our experimental data have led to the conclusion that there are several distinctive structural features of strain (creep) at high temperatures and it turns out that these features are caused by a certain physical mechanism. The first distinctive feature is the simultaneous formation of sub-boundaries within crystallites and a decrease in the strain rate ε˙ to an approximately constant value. Each of the three curves [ε(t), η(t) and D(t)] have a “nearlinear” segment. The abscissae at the start of these segments coincide with each other. The following conclusion look obvious: it is the process of substructure formation that is the cause of the decrease in the plastic strain rate. The subgrain dimensions are of the order of micrometers, one or two orders less than the grain size in polycrystalline materials. Subgrains are separated from each other by small-angle boundaries, which give a rotation angle of the order of several milliradians. ¯ and η¯ during The relative constancy of the average structural parameters D the steady-state period is the second essential feature that is intrinsic for facecentered metals at temperatures of (0.40–0.70) Tm , and for body-centered ¯ and η¯ depend upon external pametals at (0.45–0.65) Tm . The values of D ¯ decreases, and η¯ increases when σ increases. rameters, especially stress. D One should distinguish between the immobile dislocations associated in sub-boundaries and the mobile dislocations inside subgrains. Dislocation sub-boundaries that have been formed are regular networks or ordered junctions. Usually one or two system of parallel equidistant dislocations have been observed inside small-angle boundaries. The results of the Burgers vector, b, determination indicate that the parallel sub-boundary dislocations are of the same sign. Sub-boundaries contain dislocations with a considerable screw component. Mixed dislocations along [110] with b = a/2[111] are characteristic for body-centered metals. Pure screw sub-boundaries have also been found with b1 = a/2[¯ 111] and b2 = a/2[1¯11]. Screw dislocations and 60-degree dislocations have been observed in the structure of face-centered metals. The presence of a screw component in the boundary structure is the third distinctive feature of the structure of the high-temperature deformed metals. The fourth peculiarity is as follows. It is obvious from our investigations that sub-boundaries play a double role. They are both the sources and the obstacles (sinks) for mobile dislocations. Only the mobile dislocations are known to make a contribution to the elementary events leading to the deformation of the specimen. Some dislocations are located inside subgrains. They move in slip planes i.e. in planes of the {111} type for the cubic face-centered and the {110}

3.8 Summary

type for the cubic body-centered crystal lattice. The fifth distinctive structural feature is the presence of jogs in mobile dislocations. Moreover, the distances between jogs are very close to the distances between immobile dislocations in small-angle sub-boundaries. The following conclusion can be drawn: the mobile dislocations arise from the sub-boundary dislocations. It is as if the former bear a “stamp”, an imprint of the latter. The generation of vacancies during the process under consideration is the last feature. Loops and helicoids are formed when vacancies collapse. It is logical conclude that sources of vacancies are activated during the hightemperature deformation process. It should be noted that some features described above, especially, the formation of small-angle boundaries were observed in many studies. However, we should to take into account all the structural peculiarities for an adequate understanding of the phenomenon under consideration. Now we can proceed to describe the physical mechanism of high-temperature deformation of pure metals and single-phase alloys. Our aim is to relate the microstructural observations to the measured strain rates.

3.8 Summary

Typical structural features are observed in pure metals and solid solutions, which are loaded at high temperatures. These features are caused by certain physical mechanisms of deformation. The average subgrain sizes, D and the average subgrain misorientations, η, have been systematically directly measured during high-temperature strain of the metals and alloys under investigation. The values of D are of the order of 0.7–2.0 µm, the values of η are of the order of 2.9–6.0 mrad. The substructure is formed inside crystallites during the primary stage of creep. The value of D decreases and η increases during the primary stage. The origin of the steady-state stage coincides with the end of substructure formation. The steady-state creep occurs at almost constant mean values of both parameters. It is the process of substructure formation in the primary stage that causes the decrease in the strain rate and the beginning of the steady-state stage. The values of D and η are strongly dependent upon stress. The greater the applied stress the greater the misorientation angles and the smaller the subgrains. Investigations of single-phase two-component alloys do not reveal any qualitative differences in the structure evolution from the processes occurring in

41

42


pure metals. Larger values of η are observed in solid solutions as compared with metals. Most of the dislocations in specimens are associated in sub-boundaries. The parallel sub-boundary dislocations are situated at equal distances from each other. It follows from results of the Burgers vector determinations and from the repeating structural patterns that the parallel sub-boundary dislocations are of the same sign. Two intersecting dislocation systems are often found inside sub-boundaries. The sub-boundaries that have been formed are the sources of slipping dislocations. At the same time the sub-boundaries act as obstacles to the movement of deforming dislocations. Kinks and bends are observed in dislocations inside subgrains. They are caused by jogs in the mobile screw components. A great number of vacancy loops and helicoids are present in the structure. The slipping dislocations with equidistant one-sign jogs generate vacancies. It has been found that the average distances between sub-boundary dislocations, λ, and the mean spacings between jogs in mobile dislocations, z0 are close in value. The conclusion is drawn that the sub-boundary dislocations generate mobile dislocations with vacancy-producing jogs in their screw components.

43

4

Physical Mechanism and Structural Model of Strain at High Temperatures 4.1 Physical Model and Theory

It follows from the obtained data that complex processes occur in the crystal lattice at high-temperature strain. These processes are inherent to pure metals and to solid solutions at certain temperatures and under the applied stresses. The quantitative theory, which we are going to develop, will be based on the experimental results presented in Chapter 3. The dislocation density increases at the beginning of the plastic strain. In the primary stage of deformation some of the generated dislocations form discrete distributions. They enter into low-angle sub-boundaries. The interaction of dislocations having the same sign is facilitated by the high-temperature conditions and applied stress. These conditions make it easy for dislocations to move and for the edge components of dislocations to climb. The edge dislocations can change their slip planes. Why do the dislocations form ordered sub-boundaries spontaneously? The immediate cause of the formation of the dislocation walls is the interaction between dislocations of the same sign that results in a decrease in the internal energy of the system. The elastic energy of dislocations that are associated in subgrains, Es , is less, than the energy of dislocations distributed chaotically in the whole volume of the material, Ev . One can compare these energies. The values of Ev and Es are expressed as [18] Ev ≈ Es ≈

ρµb2 4π

ρµb2 8π

L ln b

ln

F ρb2

(4.1)


(4.2)

44

4 Physical Mechanism of Strain at High Temperatures

where ρ is the dislocation density, µ is the shear modulus, b is the Burgers vector, L is the size of a crystal, and F is the fraction of the crystal volume that is occupied by sub-boundaries. Assuming reasonable values of ρ = 1012 m−2 , L = 0.3cm, F = 0.05, one can obtain Ev = 2.4 Es It follows from this ratio that an appreciable decrease in internal energy takes place due to the formation of sub-boundaries. Well-formed sub-boundaries were observed in the experiments described in Chapter 3. In fact the dislocation subgrains are two-dimensional defects. They are sources, emitters of mobile dislocations, which contribute to strain. In Section 4.6 and in Supplement 2 one can find evidence that emission of mobile dislocations from sub-boundaries leads to the formation of jogs in them. It is the dislocation sub-boundary that generate jogs in mobile dislocations. The screw components of emitted dislocations “keep their origin in their memory”. They contain equidistant one-signed jogs, although this is only in the physical model, actually, the distances between jogs, z0 , are distributed values. A jog is a segment of dislocation, which does not lie in the slip plane. The jog cannot move without generation of point defects, i.e. vacancies. The jogged dislocation can slip if there is a steady diffusion of vacancies from it. The nonconservative slipping of jogged dislocations is dependent on the material redistribution. The shorter the distance between jogs, the lower the dislocation velocity. Hence, it is the diffusion process that controls the velocity of the slip of deforming dislocations. Note that we consider vacancy-producing jogs only. The interstitial-producing jogs are practically immobile because the energy of formation of the interstitial atoms is several orders greater than that of the vacancies. That is why we have observed so many vacancy loops and helicoids in the structure of tested metals (Section 3.6). It appears that a subsequent coalescence of vacancies leads to cavity formation and rupture. This seems to be the essence of the tertiary stage of creep. The distance between dislocations in sub-boundaries decreases during the primary stage of deformation and the role of sub-boundaries as obstacles for slipping dislocations increases. At the end of the substructure formation the dislocation arrangement is ordered and a steady-state stage begins. During this stage a dislocation emission from sub-boundaries takes place. The emitted dislocations are replaced in sub-boundaries with new dislocations, which move under the effect of applied stress. Having entered a sub-boundary a new dislocation is absorbed by

4.2 Velocity of Dislocations

it. Jogs of the same sign appear along the screw components of the emitted dislocations. The distance between jogs at mobile dislocations is equal to the distance between the immobile sub-boundary dislocations, see Eq. (3.2). Figure 4.1 demonstrates the described model of steady-state strain (creep). It can be seen that atoms are required in order to complete the extra-planes during the motion of the jogs. Vacancies are generated in the vicinity of jogs since atoms are consumed in completing the extra-planes. Consequently, the relay-like motion of the vacancy-emitted jogged dislocations from one subboundary to another is the distinguishing feature of the high-temperature strain.

Fig. 4.1 The physical model of the steady-state strain at high

temperature. The sub-boundaries are built of two systems: the pure screw and the 60◦ -dislocation systems. Emission of mobile dislocations from sub-boundaries is shown. The jogs at the mobile screw components have the same sign and generate vacancies.


Consider a screw dislocation that is situated along the Oz axis of the coordinate system. A dislocation with jogs moves in the direction Ox (see Fig. 4.2). The material parameters are different in the volume and inside the dislocation “tube”. Let us denote the coefficients of the diffusion of the vacancies by Dv and Dd , respectively. In Figs.4.2 and 4.3 the energy of vacancy generation is denoted by E. The energy of vacancy diffusion is denoted by U . Subscripts j, d, and v refer to the jog, dislocation and volume, respectively. For example, Ujd is the energy of

45

46


Fig. 4.2 Scheme of energetic barriers for the motion of the screw dislocation with vacancy-producing jogs. The dislocation is slipping along Ox from left to right. (see text for further explanation).

the vacancy displacement from the jog in the dislocation “tube”; Udv is the energy of the vacancy displacement from the dislocation in the crystal volume and so on. The following elementary events determine the process of the dislocation slip: the generation of vacancies near jogs (energy of activation Ed ); the diffusion of vacancies along the dislocations (energy of activation Ud ); transition of vacancies to the volume (Udv ); diffusion of vacancies in the volume (Uv ). The external applied stress performs work, dA, and facilitates the generation

Fig. 4.3 Scheme of energetic barriers for the motion of the

screw dislocation with vacancy-absorbing jogs.


of vacancies: Ujd = Ed + Udj − dA

(4.3)

The vacancy concentration in the vicinity of the jogs, cjp , increases under the effect of stress: cjp = c0 exp

dA + ε0 kT

1 = 3 b

Ev − kT

exp

dA + ε0 kT

(4.4)

where c0 is the equilibrium vacancy concentration in the volume at a given temperature, ε0 is the energy of bonding a dislocation and a vacancy, ε0 = Udv − Uvd . In the case of vacancy-absorbing jogs the sequence of events is the inverse, i.e., the generation of vacancies in the volume (Ev ), diffusion of vacancies in the volume (Uv ), transition of vacancies in the dislocation “tube” Uvd , diffusion of vacancies in the “tube” Ud , joining the vacancy in the jog Udj . The applied external stress facilitates joining of vacancies: Ujd = Ed + Udj + dA

(4.5)

In both cases the work of applied stress is given by dA = σzz b3 + σyz

z0 3 b a

(4.6)

where σzz and σyz are the components of the stress tensor; a is the height of the jogs. As super-jogs are unstable we may assume that a ≈ b and b z0 . Hence dA = σyz b2 z0

(4.7)

Now we will consider the velocity of the dislocations. The velocity of the screw components with vacancy-producing jogs is given by the following expression [19] σyz b2 z0 + ε0 Ev + Uv + ε0 πνr0 z0 Vp = exp − exp −1 bF (α) kT kT

(4.8)

where ν is the Debye frequency, r0 is the radius of the dislocation “tube”, α = Vp r0 /2Dv , F (α) is a weak function. The vacancy-absorbing jogs have velocity Ev + Uv + ε0 πνr0 z0 σyz b2 z0 + ε0 exp − Va = 1 − exp − (4.9) bF (α) kT kT

47

48


In practice we always have σyz b2 z0 kT . Therefore Vp Va . The velocity of the dislocations is exponentially dependent on stress. One can see that the exponent (4.8) contains the sum Ev + Uv . This implies that the effective energy of the jogged dislocation motion is close to the activation energy of diffusion. The activation volume in Eq. (4.8) is equal to b2 z0 . The computed values of Vp vary in the range 10−11 to 10−2 cm s−1 . As is shown in Fig. 4.4 the velocity of dislocations with vacancy-absorbing jogs, Va , is less by many orders. These dislocation components are immobile and do not control the strain rate. In Fig. 4.5 the velocity of jogged dislocations in α-iron is shown. The distance between jogs strongly influences the velocity of the dislocations.

Fig. 4.4 Velocity of screw dislocations in nickel. The distance between jogs is 36nm. 1, 2, 3: jogs generate vacancies; 1 , 2 , 3 : jogs absorb vacancies. 1 and 1 : 673; 2 and 2 : 873; 3 and 3 : 1073K.

4.3 Dislocation Density

Fig. 4.5 The effect of temperature, stress and distance between jogs on the velocity of screw dislocations in α-iron. 1: 773K, z0 = 35nm; 2: 813K, z0 = 57nm; 2 : 813K, z0 = 75nm; 3: 973K, z0 = 52nm; 3 : 973K z0 = 75 nm.

4.3 Dislocation Density

The slip strain rate is given by [20] γ˙ = bN V

(4.10)

where N is the density of deforming dislocations and V is their average velocity. The total mobile dislocation density is assumed to be equal to the sum ρ = N + Na

(4.11)

where Na is the density of annihilating dislocations. Unlike N the density Na does not contribute to the macroscopic strain. Processes of the dislocation multiplication, annihilation, sub-boundary emission and immobilization, occur in metals during the high temperature strain. The balance equation, which characterizes the change in the mobile dislocation density, can be written as m a a ρ˙ = ρ˙ m d + ρ˙ s + ρ˙ d + ρ˙ s ,

(4.12)

49

50


where ρ˙ m d is the rate of the density increase due to the dislocation multiplication, ρ˙ m s is the rate of the density change on account of the sub-boundary emission, ρ˙ ad is the annihilation rate, and ρ˙ as is the rate of the immobilization of dislocation by sub-boundaries. (Subscript d refers to dislocations, subscript s to sub-boundaries, superscript m to multiplication and emission, superscript a to annihilation and immobilization). Therefore, two terms of Eq. (4.12) are determined by the interactions of dislocations and the other two are related to the effect of sub-boundaries. Consider each term of Eq. (4.12) separately. The number of newly generated dislocation loops is directly proportional to the dislocation density and to the dislocation velocity. Hence the rate of multiplication of the mobile slip dislocations is given by ρ˙ m d = δρV

(4.13)

where δ is a coefficient of multiplication of the mobile dislocations. The coefficient has the unit of inverse length. Similarly, the rate of emission of the mobile dislocations out of sub-boundaries is directly proportional to the sub-boundary dislocation density and to the dislocation velocity: ρ˙ m s = δ s ρs V

(4.14)

where ρs = η/bD = 1/λD is the density of dislocations in sub-boundaries; δs is a coefficient of dislocation emission. Let us assume there are n+ positive dislocations and n− negative ones inside a subgrain. The annihilation rate of dislocations of opposite signs is given by n+ n− V (4.15) n˙ ad = −2 l where l = ρ−0.5 is the mean distance between dislocations inside subgrains. Since ρ = n/D2 , ρ+ = ρ− = ρ/2, one can obtain ρ˙ ad = −0.5D2 V ρ2.5

(4.16)

Immobilization of the mobile dislocations occurs when they are captured by sub-boundaries. The rate of immobilization is assumed to be directly proportional to the dislocation density and velocity and inversely proportional to the subgrain size: V ρ˙ as = −ρ (4.17) D Only deforming dislocations of density N come out of sub-boundaries; annihilating dislocations of density Na are eliminated inside subgrains.

4.4 Rate of the Steady-State Creep

The ratio of the densities of annihilating and deforming dislocations is given by Na ρ˙ a = as = 0.5D3 ρ1.5 (4.18) N ρ˙ d Taking into account the obtained results we may rewrite the differential equation for the dislocation density evolution during high-temperature deformation of metals as follows: ρ˙ = δρV + δs ρs V − 0.5D2 V ρ2.5 − ρ

V D

(4.19)

It will be noted that Eq. (4.19) does not contain any arbitrary parameters, only values which have physical meaning determine the rate of the dislocation density evolution. It is obvious from the derived formulas that the structural parameters λ and D both influence the dislocation density. Consequently, the strain rate γ, ˙ Eq. (4.10) is dependent on structural parameters not only due to the dislocation velocity V but also due to the dislocation density N . Equation (4.19) can be solved for the steady-state stage (see Section 5.4). We shall also make use of Eq. (4.19) in Chapter 5 to develop a computer model of processes that take place during the high-temperature deformation of metals.


We should take into account that the creep tests are realized as a one-axis tension. However, Eqs. (4.7) and (4.8) contain the stress component σyz . In accordance with the rules for tensor component conversions we have σy z = γy y γz y σyy

(4.20)

where the primes relate to the new coordinate system; γi i are cosines of angles between the respective axes. Averaging the product of the cosines over all orientations of the new coordinate system and finding the average value of the function of two variables we arrive at 4 σy z = 2 σyy (4.21) π The conversion from γ˙ to ε˙ is similar to Eq. (4.21).

51

52


Now we can calculate the rate of the steady-state high-temperature deformation ε. ˙ Three groups of physical parameters are needed: • •

•

External parameters: temperature T and stress σ. Diffusion parameters: the energy of vacancy generation Ev and the energy of vacancy diffusion Uv . Structural parameters: the subgrain size D and the distance between dislocations in the sub-boundaries λ [the last value depends on the subgrain misorientation angle η, see Eq. (3.1)].

These parameters and constants are the input data for calculations. Values of η and D are listed in Tables 3.1 and 3.2. The diffusion constants can be easily found in the literature, e.g. [21, 22]. For example, Ev = 2.56 × 10−19 Jat.−1 , Uv = 1.92 × 10−19 Jat.−1 for α-iron, and 2.88 × 10−19 Jat.−1 and 1.68 × 10−19 Jat.−1 , respectively, for nickel. The dislocation density has been measured by us, see Table 3.5. Thus, both multipliers in Eq. (4.10) are known. In other words we have gone from a microscopic level of thinking to a macroscopic one. The ability of the physical model to represent the macrocreep behavior of metals correctly and quantitatively is shown in Figs. 4.6–4.9. The theoretical curves are in very close agreement with the experimentally observed values of the steady-state creep rate in α-iron at 813, 873, 923 and 973K. However, the calculated data lie somewhat lower. The difference between computed and experimental curves corresponds to a coefficient of about 1.5, which one should insert into the right-hand part of Eq. (4.10). It can be seen that the theory does not represent the experimental results at 773K. The data obtained for nickel are illustrated in Figs. 4.8 and 4.9. The solid lines that present the calculated data fit satisfactorily the experimental steady-

Fig. 4.6 The steady-state creep rate of α-iron as a function

of stress. Comparison of the experimental results (symbols and dotted lines) and computed data (solid curves); 1, 813K; 2, 923K.


state strain rates at the test temperatures 673, 873, and 1023K. When the temperature is increased to 1173K the theoretical model fails.

Fig. 4.7 The steady-state creep rate of

α-iron as a function of stress. Comparison of the experimental results (symbols and dotted lines) and computed data (solid curves): 1, 773K; 2, 873K; 3, 973K.

Fig. 4.8 The steady-state creep rate of nickel as a function of stress. Comparison of the experimental results (symbols and dotted lines) and computed data (solid curves) for nickel: 1, 673; 2, 873K.

Fig. 4.9 The steady-state creep rate of

nickel as a function of stress. Comparison of the experimental results (symbols and dotted lines) and computed data (solid curves) 1, 1023K; 2, 1173K.

53

54


4.5 Effect of Alloying: Relationship between Creep Rate and Mean-Square Atomic Amplitudes

The alloying of metals within the solubility limit was found not to change the physical mechanism of strain, however, the strain rate is noticeably affected by the diffusion parameters. For example, the activation energy of self-diffusion in nickel is 4.64 × 10−19 Jat.−1 [22]; in alloy Ni+9.5%W this value increases to 4.88 × 10−19 Jat.−1 [23]. In Table 4.1 the data for nickel-based alloys are presented. The calculated quantities of ε˙ are in accordance with the measured ones. Tab. 4.1 Comparison of computed and observed values for the nickel-based solid

solutions. Alloy, at.%

T, K

σ, MPa

Measured λ, nm

Measured ε, ˙ 10−7 s−1

Calculated V, nm s−1

Calculated ε, ˙ 10−7 s−1

Ni+9.9Al

723

125 202

52 34

1.4 10.1

4.3 7.6

2.2 11.5

873

70 136

54 33

0.9 13.9

2.8 8.2

1.0 13.0

873

138 170

41 33

5.6 17.6

12.0 8.6

7.8 26.0

1023

50 90

46 32

1.6 7.6

6.0 17.0

1.8 10.8

Ni+9.5W

Comparison of the temperature dependence of the mean-squared atomic displacements, u2 , and the steady-state creep rates, ε, ˙ for nickel and for its solid solutions clearly shows a connection between the two quantities (Fig. 4.10). The amplitudes of atomic vibrations correlate with one of the main characteristics of the high-temperature strength – the rate of the stationary creep. One can see in Fig. 4.10 that the creep rate ε˙ depends almost exponentially on the mean-squared atomic displacements over the temperature range 850–1050K, where the deformation is controlled by diffusion of vacancies in the solid solution. The effect of alloying on ε˙ increases with increasing temperature. The decrease in the vibration amplitude under the influence of solution atoms of Al, Cr, and W in nickel results in an increase in the values of Ev and Uv . This causes a decrease in the values of ε. ˙ The most noticeable is the decrease of the creep rate under the influence of the tungsten additions (by three or four orders of magnitude at 850–1050K).

4.6 Formation of Jogs

Fig. 4.10 Comparison of the temperature dependence of

the creep rates ε˙ under stress of 80 MPa (a) and of the mean-square atomic amplitudes (b) for nickel-based alloys. The composition of alloys (at.%): 1, nickel; 2, Ni+9.9Al; 3, Ni+9.5Cr; 4, Ni+9.6W.

4.6 Formation of Jogs. Low-Angle Sub-boundaries in f.c.c. and b.c.c. Crystal Lattices

The distance between the sub-boundary dislocations, λ, is related to the angle between adjacent subgrains. If we assume that the low-angle sub-boundary is constructed by two crossing dislocation systems, then a node N belongs to both intersecting dislocations (Fig. 4.11). A dislocation node is simply a point where three or more dislocations meet. At a dislocation node the sum of all Burgers vectors is equal to zero, bi = 0. Vectors b2 and b4 enter the node, vectors b1 and b3 leave the node, Fig. 4.11(a). In Fig. 4.11(b) the same situation is presented after changing the signs of the vectors ξ3 and ξ4 . During parallel slip of dislocations ξ1 and ξ2 the displacement of the crystal part takes place in 1 and ξ2 × V 2 . Thus, after the plane which is determined by the vectors ξ1 × V an emission from the sub-boundary a jog of length b2 must be created in the mobile dislocation ξ1 , Fig. 4.11(c). We prove in Supplement 2 that dislocations of at least one sub-boundary systems contain the screw component. Hence, the slipping jog has to generate lattice vacancies or interstitial atoms.

55

56


Fig. 4.11 The jog formation in a dislocation emitted by a

low-angle sub-boundary. (a) The initial position, bi are the Burgers vectors; ξi are the unit dislocation vectors. (b) The same as (a) after changing the signs of the vectors ξ3 and ξ4 . P1 and P2 are the slip planes; V1 and V2 are the velocity vectors. (c) Dislocations with jogs after emission of the dislocation ξ1 .

The typical Burgers vectors, slip planes and unit dislocation vectors have been selected for examination of sub-boundaries. The results are presented in Table 4.2. The angles < b1 ξ1 and < b2 ξ2 are not equal to 90◦ . This means that the dislocations of both systems contain screw components. Tab. 4.2 The crystallography of low-angle sub-boundaries. Lattice

Slip plane

< ξ1 ξ2

f.c.c.

{111}

90◦ 60◦ 60◦ 60◦

b.c.c.

{110}

90◦ 73.2◦

b1 a [110] 2 a [110] 2 a [011] 2 a [110] 2 a [111] 2 a [111] 2

1 ξ √ 2 [110] 2 √ 2 [110] 2 √ 2 ¯¯ [ 110] 2 √ 2 [110] 2 √ 2 [110] 2 √ 2 ¯¯ [110] 2

< b1 ξ1 0◦ 0◦ 120◦ 0◦ 35.3◦ 144.7◦

b2 a ¯ [110] 2 a [10¯ 1] 2 a ¯ [ 110] 2 a ¯ [110] 2 a ¯ [111] 2 a ¯ [111] 2

2 ξ √

2 ¯ [110] 2 √ 2 [10¯ 1] 2 √ 2 ¯ [0 11] 2 √ 2 [011] 2 √ 2 ¯ [110] 2 √ 6 ¯ [121] 6

< b2 ξ2 0◦ 0◦ 120◦ 60◦ 35.3◦ 19.5◦

4.7 Significance of the Stacking Faults Energy

4.7 Significance of the Stacking Faults Energy

The processes of high-temperature strain are dependent upon the nature of a metal, especially, upon peculiarities of dislocations in its crystal lattice. Metals have different values of the stacking fault energy which results in a different ability to change the slip plain, i.e. to climb into parallel slip plains. This difference leads to various types of macroscopic behavior at high temperature. In Ref. [24] four crept metals with face-centered crystal lattices: aluminum, nickel, copper and silver were investigated. The subgrain misorientations were measured with the X-ray rocking method at discrete time moments. Tests were carried out at the tensile rate of 0.5MPah−1 . The total dislocation density was calculated from the misorientation angles. All four metals reveal linear dependences of the misorientation angle on strain at room temperature. In Fig. 4.12 the data of tests at 0.45 Tm are shown. The linear dependence remains only for silver. At T = 0.68 Tm all dependences η(ε) have a certain curvature (Fig. 4.13). The curves are ordered in the order of the stacking fault energies: Al, Ni, Cu, Ag, 290, 150, 70, and 25mJ m−2 , respectively. Pishchak [24] considers the dependence of deformation ε on stress σ. At room temperature there is a linear dependence, ε ∼ σ. At high temperatures he assumes the empirical equation ε = Aσ m to be the most appropriate. The exponent of the power function, m, turns out not to be a constant value but to increase with temperature from m = 1 to m = 2. The temperature of the m change is equal to 0.30, 0.35, 0.40 and 0.60 Tm for Al, Ni, Cu, Ag, respectively.

Fig. 4.12 The average subgrain misorientation versus strain

in four metals with face-centered crystal lattice. Temperature is equal to 0.45 Tm . B, aluminum; C, nickel; D, copper; E, silver. The results were calculated from the data of Ref. [24].

57

58


Fig. 4.13 The same as in Fig. 4.12, but at temperature

0.68 Tm .

From our point of view, in metals with little stacking fault energy, the climb of the dislocation edge components is hindered and dislocations cannot change their slip plane. A higher temperature is needed in order for regular sub-boundaries to be formed.

4.8 Stability of the Dislocation Sub-boundaries

As has been noted above, the sub-boundaries are both the sources of and the obstacles for deforming dislocations. Let us consider the effect of external stress and temperature on the subboundary dislocation emission. By dislocation emission we mean a thermally activated release of a dislocation from an immobile sub-boundary and its subsequent transformation into a mobile deforming dislocation. Our aim is to determine a threshold stress, above which the sub-boundaries are unstable and can be destroyed without the thermal activation. We shall analyze the effect of applied stress and temperature on the sub-boundary dislocation emission. Consider the boundary built by two perpendicular systems of equidistant parallel screw dislocations (Fig. 4.14). Assume first that there is no dislocation 2 in a slip plane P1 . The components of stress affecting a sub-boundary dislocation 1 in the slip plane (y = 0) are given by [18]

σyz

(µb sinh 2πX)(1 − cos 2πZ) µb = − 2λ(cosh 2πX − 1)(cosh 2πX − cos 2πZ) 2πλX

(4.22)

4.8 Stability of Dislocation Sub-boundaries

Fig. 4.14 A sub-boundary formed in the yOz-plane by two

systems of screw dislocations. λ is the distance between adjacent dislocations, D is the subgrain size, P1 (xOz) is the slip plane; a boundary dislocation under consideration is denoted by 1, another dislocation in the slip plane outside the boundary is denoted by 2.

σxz =

µb µb sin 2πZ ; σxy = 2πλX 2λ(cosh 2πX − cos 2πZ)

(4.23)

where µ is the shear modulus, X = x/λ, Y = y/λ, Z = z/λ. When the dislocation 1 deviates from the boundary the shear stress component σyz acts on it. (For a screw dislocation the stress component is parallel to the dislocation line.) The value of this component depends upon the coordinates. The results of the calculations of the shear stress are shown in Fig. 4.15.

Fig. 4.15 The stress component σyz in units of µb/2λ as a function of distance. On the left: z/λ = const, on the right: x/λ = const.

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60


The curves have singularities at x = 0. Within the sub-boundary (in the initial position) the stress components are therefore equal to zero. Thus, the dislocation inside the boundary is affected by the force F (0) = 0. The force reaches its maximum value near the node at a distance equal to the dislocation core radius r0 . It is reasonable to assume that F (r) is a linear function within the range 0 < r < r0 . Further F (r) = −bσyz if r0 ≤ r < r1 , where r1 is a distance at which the interaction force between the dislocation and the boundary is close to zero. The calculated dependences of force and energy on the distance from the deviated dislocation 1 to sub-boundary are shown in Fig. 4.16. One can see that the maximum returning force is achieved at a distance of the order of the dislocation core. This force acts in the opposite direction.

Fig. 4.16 The force at which the sub-boundary acts on the emitted dislocation 1, and the activation energy versus the distance. r0 = 2b is assumed.

Assume that the applied external stress is σ. The energy to be consumed by the emission is expressed as U =− 0

r0

F (r)dr −

r1

r0

F (r)dr

(4.24)

The stress field of the sub-boundary tends to return the dislocation 1 to the sub-boundary. Thus, the dislocation is pinned with pinning point density 1/λ and is emitted by means of thermal activation. According to the theory of the rates of reactions [25] the dislocation can be regarded as a linear crystal with D/b degrees of freedom. The number of thermal activations per unit of time

4.8 Stability of Dislocation Sub-boundaries

can be represented by an expression of the form ∆U Γ = νeﬀ exp kT

(4.25)

where νeﬀ is a pre-exponential factor; ∆U is the activation energy and kT has its usual meaning. From Eqs. (4.22), (4.23) and (4.24) we obtain for one degree of freedom (z = 0) µb3 αeα r1 U= (4.26) ln b 2π where α = b/r0 . Taking into account the work of the external stress we obtain ∆U = U − σb2 λ

(4.27)

The activation energy is essentially less if there are n ≥ 2 slipping dislocations in the same slip plane. One can show that in this case the factor n appears before the second term on the right-hand side of Eq. (4.27). In Fig. 4.17 the calculated curves of the influence of temperature and stress on the Γ value are shown for two metals. The probability of dislocation emission from the sub-boundary is strongly affected by the temperature and the number of dislocations in the slip plane.

Fig. 4.17 The number of thermal activations per unit time as a function of stress and temperature. Solid lines, one dislocation in slip plane, dashed lines, two dislocations in slip plane. (a) Nickel, (b) vanadium. r0 = 2b and r1 = λ is assumed.

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The condition of the stability of the boundary during strain is 1 > τcreep . Γ Here Γ −1 is the time interval before the emission begins and τcreep is the time interval during which the creep deformation occurs; e.g. for a creep time of 105 s then Γ < 10−5 s. The results in Fig. 4.17 show the temperature and stress intervals where the sub-boundaries are observed. From Eqs. (4.26) and (4.27) we obtain the condition of the inactivated emission of dislocations from the sub-boundaries: σ≥

µb αeα λ ln 2πnλ b

(4.28)

Assuming λ = 50nm, n = 2, α = 0.5 we obtain σ ≥ 2 × 10−3 µ for nickel. When the external stress is higher than this value then the sub-boundaries are unstable and are destroyed.

4.9 Scope of Application of the Theory

A well-read reader may ask: what is the distinction between this theory and the model published by Barrett and Nix [11]? This excellent article was the first to examine deeply the motion of jogged dislocations as a process which controls the strain rate. However, the authors conceived the jogs as being a result of thermal activation. The equations proposed by them take into account only a thermodynamic equilibrium number of jogs in the dislocations. In their opinion, the screw components therefore contain equidistant alternating jogs of opposite signs. They wrote: “The average spacing between jogs, λ, has never been measured directly”, so they assumed a parameter λ which could not be measured. The quantitative evaluation of the strain rate was out of the question at that time, of course. As a matter of fact, the adjacent jogs of opposite signs slip along the dislocation line easily and would simply annihilate each other. The equilibrium values of λ can affect neither the dislocation velocity nor the creep rate. According to our experimental results the sources of jogs of the same sign in mobile dislocations are the immobile sub-boundary dislocations and we believe that the substructure formation plays a key role during high-temperature strain, being the process that affects the strain rate. The present theory is understood to be valid within certain limitations. When the temperature is relatively low, the dislocation climb is depressed

4.9 Scope of the Theory

and hence regular sub-boundaries cannot be formed. The lower limit to give a sufficient climb rate is about 0.40 or 0.45 Tm . The low-temperature deformation is controlled by other processes, e.g. the overcoming of the Peierls stress in the crystal lattice. The stable sub-boundaries are of major significance in the process of hightemperature strain for pure metals and solid solutions. The upper stress limit of the sub-boundary stability depends upon the metal properties and temperature. The lower the shear modulus µ and the higher the temperature, the lower the limit. An estimation, for instance, shows that in nickel at 0.6 Tm sub-boundaries are destroyed by a stress of 2 × 10−4 µ in 30h. The analysis shows that when the applied external stress is higher than about 2 × 10−3 µ inactivated emission of dislocations from sub-boundaries occurs and the sub-boundaries break up. The upper limit of temperature is (0.70– 0.75)Tm . Diffusion creep takes place (the mechanism of Herring-Nabarro) at higher temperatures and relatively lower stresses. It is necessary to emphasize that an adequate understanding of dislocation processes in these ranges of temperature and stress is of great practical importance. Most heat-resistant metals, steels and alloys operate at temperatures between 0.40 and 0.75 Tm . The area of temperature and stress where the proposed mechanism of hightemperature deformation takes place, is shown in normalized coordinates in Fig. 4.18. Construction diagrams (maps) of this type were proposed by Ashby, e.g. in Ref. [26].

Fig. 4.18 The deformation map of nickel. The shaded area

represents the interval of temperature and stress where the physical mechanism under consideration takes place. The numbers on the curves denote strain rates in s−1 .

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Fig. 4.19 The same as in Fig. 4.18

but for iron.

It is known that in iron allotropic transformation occurs at 0.65 Tm (Fig. 4.19). The mutual arrangement of the deformation areas is in other respects similar to the previous one, however, there is a quantitative difference. The strain rate of iron, which has a body-centered crystal lattice is considerably greater. For example, at 0.5 Tm under a stress of 6 × 10−4 µ strain rates for Ni and Fe are equal to 10−7 and 10−3 s−1 , respectively.

4.10 Summary

The dislocation density increases at the beginning of plastic strain. In the primary stage of deformation a part of the generated dislocations form discrete distributions. Dislocations penetrate low-angle sub-boundaries. The interaction of dislocations having the same sign is facilitated by high-temperature and applied stress. These conditions make it easier for edge components of dislocations to climb. The immediate cause of the formation of dislocation walls is the interaction between dislocations of the same sign resulting in a decrease in the internal energy of the system. At the end of the substructure formation the dislocation arrangements are ordered. Then a steady-state stage of strain begins. During this stage a dislocation emission from sub-boundaries takes place. In metals with small stacking fault energy the climb of the dislocation edge components is hindered. The ordered dislocation sub-boundaries require a higher temperature in order to form. The low-angle sub-boundaries are built up of parallel equidistant dislocations that contain screw components. The sub-boundary dislocations are

4.10 Summary

sources, emitters for mobile dislocations, which contribute to the specimen strain. Emission of mobile dislocations from sub-boundaries leads to the formation of the equidistant one-signed jogs. The distance between jogs at mobile dislocations is close to the distance between the immobile sub-boundary dislocations. The jogged dislocations can slip when there is a steady diffusion flux of generated vacancies from jogs. The emitted dislocations are replaced in subboundaries with new dislocations, which move under the effect of applied stress. Having entered a sub-boundary a new dislocation is absorbed by it. The relay-like motion of the vacancy-emitted jogged dislocations from one subboundary to another one is the distinguishing feature of the high-temperature strain of single-phase metals and solid solutions. The velocity of dislocations depends exponentially on the applied stress. The exponent contains the sum of the activation energies of vacancy generation and vacancy migration. Processes of dislocation multiplication, annihilation, sub-boundary emission and immobilization occur in metals during the high-temperature strain. The balance equation, which characterizes the change in the mobile dislocation density, has been derived. Three groups of physical parameters are needed for estimation of the steady-state strain rate ε: ˙ •

External parameters: temperature T and stress σ.

•

Diffusion parameters: the energy of the vacancy generation Ev and the energy of the vacancy diffusion Uv .

•

¯ and the mean distance Structural parameters: the average subgrain size D ¯ between dislocations in sub-boundaries λ.

The computed values of ε˙ fit the experimental data satisfactorily at certain temperature and stress conditions. The rate of the stationary creep correlates with the amplitude of atomic vibrations at high temperatures. The developed theory is valid within certain limits of the temperature and applied stress. When the temperature is relatively low, the dislocation climb is depressed and hence the regular sub-boundaries cannot be formed. The lower limit for sufficient climb rate is about 0.40 Tm or 0.45 Tm , the upper limit is 0.70 Tm or 0.75 Tm . The upper stress limit of the sub-boundary stability depends upon the metal properties and temperature. The lower the shear modulus µ and the higher the temperature, the lower the limit. An inactivated emission of dislocations from sub-boundaries occurs when the applied external stress is higher than about 2 × 10−3 µ, the sub-boundaries then break up.

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5

Simulation of the Evolution of Parameters during Deformation

5.1 Parameters of the Physical Model

In the previous chapter the physical model of the high-temperature dislocation deformation in metals was worked out. Recall that the model and the equations deal exclusively with “natural” parameters, which have well-defined physical meaning. Our next step is to study this model in detail. The processes progress in time. The approach is to make a system of ordinary differential equations and to solve the system numerically. The results of the simulation are used to validate the correctness of the model as well as to study further the processes under consideration. The forming of subgrains occurs during the high-temperature strain. This phenomenon was described in Chapter 4. The model under study is as follows. Let us consider two intersecting crystallographic systems of parallel slip planes. The Schmid factors are generally different on the planes of these two systems, therefore the values of the applied shear stresses are also different. Jogs are generated as a result of the intersections of the moving dislocations in both systems. The nonconservative slipping of jogs is controlled by vacancy diffusion and determines the velocity of the dislocations. The velocity of the mobile dislocations in the first system depends on the average spacing between jogs, z0 . In turn this spacing depends on the distances between dislocations in the second system, λ. The velocity of dislocations in the second system is affected by the average distance between the sub-boundary dislocations in the first system. Six values are of interest for a complete description of the physical mechanism under consideration:


68

5 Simulation of the Parameters Evolution •

the relative shear strain γ

•

the total dislocation density ρ

•

the slip velocity of dislocations in their slip planes V

•

the climb velocity of dislocations to the parallel slip planes Q

•

the mean spacing between parallel dislocations in sub-boundaries λ

•

the mean subgrain size D.

All enumerated values depend on time t. To make a system of ordinary differential equations we should derive formulas that describe the changes in each parameter as a function of time and of the other parameters. We use subscript 1 to denote parameters in the first system of parallel planes and subscript 2 for those in the the second one. The following equalities hold true: (z0 )1 = λ2 ; (z0 )2 = λ1

(5.1)

This is one of our main experimental results [see Table 3.4 and Eq. (3.2)]. Let us consider equations, which relate to each of the enumerated parameters.

5.2 Equations 5.2.1 Strain Rate

Combining Eqs. (4.10), (4.11) and (4.18) we arrive at bρ1 V1 dγ1 = dt 0.5D13 ρ1.5 1 +1

(5.2)

for the first system of planes. One can see from this equation that the strain rate depends on all structural parameters via the dislocation density and the dislocation velocity. 5.2.2 Change in the Dislocation Density

We have obtained Eq. (4.19). Hence dρ1 V1 = δρ1 V1 + δs ρs1 V1 − 0.5D12 V1 ρ2.5 1 − ρ1 dt D1

(5.3)

5.2 Equations

It has been noted that the first term of the right-hand side describes the multiplication rate of mobile dislocations. The second term is related to the emission of mobile dislocations from sub-boundaries. The third term corresponds to annihilation of the mobile dislocations of opposite sign. Finally, the fourth term is dependent on the capture, i.e. on the immobilization of slipping dislocations by sub-boundaries. 5.2.3 The Dislocation Slip Velocity

One ought to use Eq. (4.8) to derive a formula for the time derivative of V but the equation is rather cumbersome to work with. We therefore use a simplified version of the equation obtained in Ref. [11]: σb2 z0 −1 V = 4πDv b2 c0 exp kT

(5.4)

where V is the slip velocity of dislocations with vacancy-producing jogs, Dv is the coefficient of vacancy diffusion, c0 is the equilibrium concentration of vacancies, σ is stress, and the values k and T have their usual meaning. After differentiating Eq. (5.4) and substituting the value of λ2 for (z0 )1 we arrive at 4π(Dv )b4 c0 σ1 dV1 dλ2 σb2 λ2 = exp (5.5) dt kT kT dt One can see from Eq. (5.5) that the distance between immobile dislocations in the second system influences the dislocation slip velocity in the first one. The velocity of slip decreases after loading since λ2 decreases, dλ2 /dt < 0. 5.2.4 The Dislocation Climb Velocity

The velocity of climb of the edge dislocation components is given by [21] Q=

Ev + Uv 11νb2 σb2 λ exp − exp λ kT kT

(5.6)

Taking the derivative of Eq. (5.6) one obtains dQ1 11νb2 =− 2 dt λ2

σ 1 b2 λ 2 Ev + Uv σ1 b2 λ2 dλ2 − 1 exp − (5.7) exp kT kT kT dt

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5 Simulation of the Parameters Evolution

The velocity of climb increases with time. 5.2.5 The Dislocation Spacing in Sub-boundaries

In Fig. 5.1 a scheme is shown for an edge dislocation slipping near the subboundary. The dislocation wall repulses the approaching mobile dislocation. There is a “corridor” (shaded), inside which the dislocation can enter the wall under a given stress. In order to get into the “corridor” the dislocation must climb into another parallel plane. We may describe the permeability of the boundary by the ratio χ = p/λ, where p is the width of the “corridor” and λ is the sub-boundary dislocation spacing. χ increases when σ and p increase. The mean path of climb is (λ − p)/4. The time needed for climbing is equal to t = (λ − p)/4Q. The spacing λ is changed to (λ − p)/2 during the time interval t. Let us assume that the probability of getting into the wall (independent event) is directly proportional to the permeability of the wall. If χ equals to zero the dislocation cannot enter the wall. The number of onesigned dislocations in the band of width λ is equal to ρλD/2, hence the rate of change of λ inside the first system of planes is dλ1 1 + χ1 = −ρ1 λ1 D1 Q1 χ1 dt 1 − χ1

(5.8)

The author [27] has obtained a semi-empirical formula: χ1 = 0.45λ1 σ1 − 0.23

(5.9)

where λ is measured in meters and σ in megapascals.

Fig. 5.1 Motion of mobile dislocations to a sub-boundary. λ is the distance between sub-boundary dislocations, p is the width of the “corridor” (shaded). 1, A slipping dislocation has to change its slip plane to be able to enter into a “corridor”. 2, A dislocation slipping to the sub-boundary under a given stress.

5.3 Results of Simulation

5.2.6 Variation of the Subgrain Size

The equation for the dependence of the mean subgrain size on time has been derived [28]: dD1 3.17 dρ1 = − 1.81 (5.10) dt ρ dt

5.2.7 System of Differential Equations

Equations (5.2), (5.3), (5.5), (5.7), (5.8), (5.10) constitute half the equations of the required system. The second half of the system is obtained by replacing subscript 1 with subscript 2 in these equations. We have obtained a system of 12 ordinary differential equations. This is the system to be used for computer simulation. The general form of a set of N first-order differential equations for the unknown functions yi , i = 1, 2, ..., yN is dyi = fi (t, y1 , y2 , ..., yN ) dt

(5.11)

where the functions fi on the right-hand side are known. In our case N = 12. The initial conditions are feasible variables that have certain numerical values. One should specify the initial values of the parameters. We use a Runge-Kutta method [29] for integration of differential equations. As is known, Runge-Kutta methods propagate a numerical solution over an interval by combining the information from several Euler-style steps (each involving one evaluation of the right-hand side of equations) and then using the information obtained to match a Taylor series expansion up to some higher order. Program MATLAB enables one to solve the system and achieve a specified precision of the fourth order. We use the so-called ODE45 RungeKutta method with a variable step size. The step size is continually adjusted to achieve a specified precision.

5.3 Results of Simulation: Changes in the Structural Parameters

In Fig. 5.2 the data from the model are presented for nickel tested at 1073K. The following initial values of parameters were chosen.

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Fig. 5.2 Dependence of strain and structural parameters on time for nickel. The computer simulation uses the set of 12 ordinary differential equations. Two curves in each of six graphs correspond to two intersecting systems of parallel slip planes. T = 1073K, σ1 = 17.5MPa, σ2 = 16.5MPa.

time t=0 deformation γ1 = γ2 = 0 dislocation density ρ1 = ρ2 = 2 × 108 m−2


dislocation spacing in sub-boundaries λ1 = λ2 = 50 nm subgrain size D1 = D2 = 3 µm coefficients of the dislocation multiplication and emission, respectively, δ = 2 × 104 m−1 ; δs = 4 × 104 m−1 The test time is 5h = 1.8 × 104 s. Comparison of the obtained results (Fig. 5.2) with the experimental data shows remarkable overall agreement. Further analysis of data from the model leads to some interesting conclusions: There are some differences in how the processes in both plane sets proceed. One can see an increase in strain in Fig. 5.2. The steady-state stage of creep occurs earlier under the lower stress. The strain value of 2% is observed in 1.5 × 104 s after the load has been applied. For comparison with the model data the experimental results are presented in Fig. 5.3 and Fig. 5.4. The inter-dislocation spacings in Fig. 5.4 were determined from X-ray measurement data, as described in Chapter 2. There is an obvious fit of model and experimental data which is evidence that the physical model is adequate.

Fig. 5.3 Strain versus time for nickel tested at 1073K. To be

compared with the first graph in Fig. 5.2. Two specimens. B, σ1 = 10MPa; C, σ2 = 14MPa.

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Fig. 5.4 The sub-boundary dislocation spacing versus time

for nickel tested at 1073K. To be compared with the third graph in Fig. 5.2. Experimental data for the same two specimens as in Fig. 5.3.

The density of dislocations increases during the high-temperature strain from 2 × 108 m−2 to (4.0–4.5) × 1011 m−2 . The dislocation density increases very quickly after loading, within 75–100 s. At the steady-state stage the values ρ1 and ρ2 are almost constant, hence the rates of the mobile dislocation generation and of the dislocation annihilation (immobilization) are equal. The sum of the positive terms on the right-hand side of Eq. (5.3) is equal to the sum of the absolute values of the negative terms. The velocity, V , of the mobile dislocations decreases gradually. At the steadystate it is of the order of 20 nm s−1 , i.e. 80 interatomic spacings per second. V is less in the planes where the applied stress is less. In contrast, the climb velocity Q increases with time. Q is two orders less than V . The spacing between jogs in mobile dislocations1) , λ, decreases from 50 to 35 nm in exactly the same way as in reality, see Fig. 5.4. The decrease in λ = l correlate with the decrease in the V value. The change in the subgrain size D (the last graph) differs somewhat from the observed one. The calculated values drop too quickly. The proposed model enables one to examine the influence of structural parameters on strain and on the strain rate. It is convenient to study the evolution of the investigated values. The strain decreases when the initial value λ is decreased or the initial value D is increased by means of suitable treatment. For example, if the initial average subgrain size D is 12µm instead of 3µm then the strain drops from 0.02 to 0.005. As regards λ this value is in the exponents in Eqs. (4.8), 1) In Fig. 5.2 and 5.5 the sub-boundary dislocation spacing is denoted as l.


(4.10), (5.4). Hence, the strain rate is strongly affected by the sub-boundary dislocations spacing. A seeming paradoxical result is of interest. When the initial value of the mobile dislocation density, ρ, increases sharply then the annihilation of dislocations progresses. As a result the strain of the specimen decreases. In contrast, when the initial value ρ is decreased, e.g. from 2 × 108 to 5 × 107 m−2 , the strain increases from 0.02 to 0.10. It might seem that the coefficients of the dislocation multiplications are chosen somewhat arbitrarily. Undoubtedly, the real values δ and δs are unknown. We are forced to consider them as fitting coefficients. However, it

Fig. 5.5 Dependence of strain and structural parameters on

time for niobium. The computer simulation uses the set of 12 ordinary differential equations. Two curves in each of the six graphs correspond to two intersecting systems of parallel slip planes. T = 1370K, σ1 = 17.0MPa, σ2 = 16.5MPa.

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Fig. 5.6 Strain versus time for niobium tested at 1370K,

σ = 44.1MPa.

Fig. 5.7 The sub-boundary dislocations spacing versus time for niobium tested at 1370K. B and C are two crystallites of the same specimen. To be compared with the third graph in Fig. 5.5.

turned out that changes in these values, even if by orders of magnitude, have only a small effect on the results of the calculations. For example, varying δ from 2 × 104 to 3.2 × 105 m−1 does not affect the deformation curve or the dislocation density. The increase in δs from 4 × 104 to 1.6 × 105 leads to an increase in ρ at the steady-state stage up to 7 × 1011 m−2 . In Fig. 5.5 the model data are presented for niobium. The typical experimental curves for niobium tested at 1370K are shown in Figs. 5.6–5.8. These curves show the parameters ε, λ = l, and D, respectively, during steady-state strain. One can see that the physical model fits the experimental data well.

5.4 Density of Dislocations during Stationary Creep

Fig. 5.8 The average subgrain size versus time for niobium

tested at 1370K. B and C are two crystallites of the same specimen.


At the steady-state deformation the rates of the dislocation generation and annihilation seem to be equal and the parameters λ and D are constant. Thus, we should solve the system of equations to calculate the density of deforming dislocations during the constant strain rate (see Section 4.3): N + Na = ρ

(5.12)

Na /N = 0.5D3 ρ1.5

(5.13)

where ρ is the real root of the following equation: δρV + δs ρs V − 0.5D2 V ρ2.5 − ρ

V =0 D

(5.14)

These equations were solved numerically (the Newton method was used). The dislocation densities were computed for different values of the structural parameters D and λ. The results are presented in Fig. 5.9. The obtained results seem to be quite reasonable. The density of deforming dislocations is of the order of 1011 m−2 . This density is strongly affected by the subgrain size as well as by the distance between sub-boundary dislocations. The larger the subgrains the smaller the density of dislocations that contribute to the deformation process. It is obvious that sub-boundaries of relatively large misorientation are sources for moving dislocations.

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Fig. 5.9 The computed density of deforming dislocations N

versus the sub-boundary dislocation spacing. B, the subgrain size D = 60 µm; C, D = 30 µm; D, D = 10 µm.

The experimentally measured data (Table 3.5) are of the same order, 1011 m−2 . For example the measured density is (1.3–9.5) × 1011 m−2 in nickel, (1.6–5.3) × 1011 m−2 in niobium and so on. From Figs. 5.9 and 5.10 it can be seen that the total dislocation density, ρ, is one order greater than the deforming dislocation density, N . Since the dislocation density is affected by structural parameters the minimal strain rate depends on their values, too. The different values of D and λ can be obtained by a preliminary treatment of the metal.

Fig. 5.10 The computed total dislocation density ρ versus the sub-boundary dislocation spacing. B, the subgrain size D = 60 µm; C, D = 30 µm; D, D = 10 µm.


Tab. 5.1 Structural parameters in nickel after preliminary deformation and annealing. Deformation 0 0.03 0.07

D, µm

η, mrad

λ, nm

1.83 1.18 1.00

3.40 5.40 7.07

73.2 46.1 35.2

In order to test the influence of the D and λ parameters, we deformed specimens of nickel by 3% and 7% at room temperature and then annealed them at 873K. As a result we obtained the various average subgrain sizes and misorientations (Table 5.1). The specimens revealed after treatment an improved creep resistance (Fig. 5.11). The difference in the creep rate for specimens with 7% deformation is one order less than for specimens without preliminary deformation. However, at relatively high stress the values of the steady-state strain rates become equal. The sub-boundary dislocation distance decreases if the stress increases (Fig. 5.12).

Fig. 5.11 The steady-state strain rate of the preliminary de-

formed nickel specimens versus applied stress. Temperature 873K. B, without deformation; C, deformation 3%; D, deformation 7%.

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Fig. 5.12 The distance between sub-boundary dislocations

in the preliminary deformed nickel specimens versus applied stress. Temperature 873K. B, without deformation; C, deformation 3%; D, deformation 7%.

5.5 Summary

A system of differential equations has been proposed to simulate the processes of high-temperature deformation in metals. Two intersecting crystalline systems of parallel slip planes are considered. The dislocation slip velocity in each system is controlled by vacancy-producing jogs and depends on the distances between the sub-boundary dislocations in the parallel planes of another system. The evolution of six values in each system was studied: the shear strain; the total dislocation density; the slip velocity of the dislocations; the climb velocity of the dislocations to the parallel slip planes; the mean spacing between parallel dislocations in sub-boundaries; the mean subgrain size. The formulas that describe the changes in each parameter as a function of time and of the other parameters have been derived; a system of 12 ordinary differential equations was obtained. The Runge-Kutta methods were used for integration of the system. The quantitative model results show a satisfactory fit with experiments. The processes in each plane set happen somewhat differently. The density of mobile dislocations increases during the high-temperature strain from 2 × 108 m−2 to (4.0–4.5) × 1011 m−2 . Experimental data have the same order of 1011 m−2 . The total dislocation density, ρ, is one order greater than the deforming dislocation density, N . The coefficient of multiplication of mobile dislocations is found to be of the order of 2 × 104 m−1 .

5.5 Summary

The velocity of the mobile dislocations, V , decreases gradually. At the steady-state stage it is of the order of 20 nm s−1 . The value of V is less in the plane set where less applied stress operates. In contrast, the climb velocity, Q, increases with time. The value of Q is two orders less than that of V . The jog spacing in mobile dislocations decreases when the strain increases, as in reality. A decrease in λ correlates with a decrease in the V value. The strain rate of the specimen is strongly affected by the sub-boundary dislocation spacing. A preliminary decreased value of λ and increased value of D lead to the strain rate decreasing when the applied stress is relatively low.

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6

High-temperature Deformation of Superalloys 6.1 γ Phase in Superalloys

The high-temperature strength requirements of materials have increased with new developments in engine design. The continual need for better fuel efficiency has resulted in faster-spinning, hotter-running gas turbine engines. One of the most important requirements is resistance to high-temperature deformation. This has created a need for alloys that can withstand higher stresses and temperatures for the hot zones of modern gas turbines. The development has led, during past decades, to a steady increase in the turbine entry temperatures (5K per year averaged over the past 20 years) and this trend is expected to continue. Other crucial material properties are crack resistance, stiffness, resistance to oxidation and an acceptable density. Such alloys – superalloys – have been developed. The largest applications of superalloys are in aircraft and industrial gas turbines, rocket engines, space vehicles, submarines, nuclear reactors and landing apparatus. The structure of the majority of nickel-based superalloys consists of a matrix i.e. of the γ phase and particles of the hardening γ phase. The γ phase is a solid solution with a face-centered crystal lattice and randomly distributed different species of atoms. By contrast, the γ phase has an ordered crystalline lattice of type L12 (Fig. 6.1). In pure Ni3 Al phase atoms of aluminum are placed at the vertices of the cubic cell and form the sublattice A. Atoms of nickel are located at the centers of the faces and form the sublattice B. In fact the phase is not strictly stoichiometric, there may exist an excess of vacancies in one of the sublattices, which leads to deviations from stoichiometry. Sublattices A and B of the γ phase can dissolve a considerable amount of other elements. Many of the industrial nickel-based superalloys contain, in addition to chromium, aluminum, and titanium, also molybdenum, tungsten, niobium, tantalum and cobalt. These elements are dissolved in the γ phase.


84

6 High-temperature Deformation of Superalloys

Fig. 6.1 Crystal structure of the γ phase. The face-centered

cubic cell contains 3 atoms of B-type (6 atoms/2 adjacent cells) and 1 atom of A-type (8 atoms/8 cells). The chemical formula is B3 A.

The crystal lattice parameter of the γ phase is close to the parameter of the solid solution, so the misfit between two lattices is relatively small. The misfit, δ, between precipitates and matrix is defined as δ = (aγ − aγ )/

aγ + aγ 2

(6.1)

The value of δ is negative for current commercial superalloys. The magnitude and sign of the misfit also influence the development of microstructure under the operating conditions of stress and high temperature. Furthermore, because their lattice parameters are similar, the γ phase is coherent with the γ phase with a simple cube–cube relationship ([001]γ [001]γ ). Dislocations in the γ phase nevertheless find it difficult to enter γ , because of stresses on the boundary and partly because the γ is the atomically ordered phase. The particles reduce the velocity of the deforming dislocations and thus act as obstacles. The order interferes with the dislocation motion and hence strengthens the alloy. The small misfit between the γ and γ lattices is important. When combined with the cube–cube orientation relationship, it ensures a low γ/γ interfacial energy. The ordinary mechanism of precipitate coarsening is driven entirely by the minimization of the total interfacial energy. A coherent or semi-coherent interface therefore makes the microstructure stable, a property which is useful for high-temperature applications. The solubility of the γ phase is dependent on temperature. This dependence has been studied in [31] using a high resolution triple crystal diffractometer and high energy synchrotron radiation (150keV, λ = 0.08). The re-

6.1 γ Phase in Superalloys

Fig. 6.2 Dependence of the solubility of the γ phase on

temperature.

sults are shown in Fig. 6.2. Solution of the γ phase in the matrix, i.e. in the γ solid solution, is observed above 1173K (900 ◦ C). The usual heat treatment of superalloys consists of heating above the temperature of the γ -solution and subsequent hardening. The γ phase has remarkable properties, in particular, an anomalous dependence of strength on temperature. The γ phase first hardens, up to about 1073K, and then softens. This peculiarity is reflected in a similar dependence of the yield strength upon temperature in superalloys. This is shown in Fig. 6.3. The yield stress increases as the temperature increases from 573 to 1073K.

Fig. 6.3 Dependence of the yield strength of a superalloy on temperature. Experimental data (symbols) and predicted quantities (dashed lines, see below). Reprinted from [30] with permission from Maney Publishing.

85

86


Fig. 6.4 Creep curves for EI437B superalloy. The steady-state

and the tertiary, accelerating stages of creep are observed. T = 973K. Stress: 1, 410; 2, 450; 3, 490; 4, 530MPa. The symbols × show when specimens were taken for transmission electron microscopy study.

The creep curves of polycrystalline superalloys differ from the curves of pure metals. After the incubation stage the material reaches a minimum value of strain rate. The steady-state stage creep is sometimes short. This minimum creep rate stage is followed by a relatively long tertiary deformation, which results in rupture. The representative curves of high-temperature tests for two superalloys are shown in Figs. 6.4 and 6.5. For superalloys it is often preferred to measure the time tr until a rupture occurs instead of determining the minimum creep rate ε˙ (T and σ are chosen as independent parameters in this case). Thus, one determines the rupture life (durability), tr , at the given temperature and stress. Another method consists in determination of the stress that leads to the rupture at a given temperature in a given time (T and t are chosen as independent parameters). In other words one measures the long-time strength 1173 (creep rupture strength). The notation σ100 = 150MPa means that the minimal rupture life of a specimen at temperature 1173K under stress 150MPa is equal to 100h. It is generally agreed that the lower the minimum creep rate of a specimen the longer its rupture life. The rupture life seems to be inversely proportional to the steady-state strain rate: ε˙ · tr ≈ const.

(6.2)

6.1 γ Phase in Superalloys

Fig. 6.5 Creep curves of a superalloy at 1023K. Reprinted

from [32] with permission from Maney Publishing.

The creep properties of some superalloys are presented in Table 6.1. One can see that Eq. (6.2) is held to an accuracy of 18 to 50%. In other words, the equation is suitable as a rough approximation only. Tab. 6.1 Correlation between the minimum creep rate and the rupture life. Superalloy

T, K

σ, MPa

ε, ˙ s−1

tr , h

ε˙ · tr

Ref.

10−9

20.0Cr + 2.2Al+ +2.2Ti + 3.3W + 5.0Fe

1023

230 260 290 320

2.3 × 8.4 × 10−9 2.8 × 10−8 8.3 × 10−8

2290 756 418 118

0.019 0.029 0.042 0.035

32

A286

913

490 513 553 588 623 692

3.7 × 10−8 6.3 × 10−8 1.7 × 10−7 5.3 × 10−7 1.5 × 10−6 2.2 × 10−6

190 153 66 38 18 9

0.025 0.035 0.040 0.073 0.097 0.071

33

MC544

1033

840

5.2 × 10−7 2.2 × 10−7 1.0 × 10−7

42 143 250

0.079 0.113 0.090

34

AM1

1033

840

1.4 × ×10−7 0.7 × ×10−7 1.4 × ×10−7

124 165 83

0.062 0.042 0.042

34

87

88


6.2 Changes in the Matrix of Alloys during Strain

The specimens of industrial superalloys (see Table 2.1) were investigated in situ by means of the X-ray method described above; transmission electron microscopy studies were also carried out. For this purpose the high-temperature tests were interrupted and thin films were prepared from the specimens (Fig. 6.4). EI437B is a nickel-based superalloy widely used in gas turbine jet engines and also in various applications up to 1023K (750 ◦ C) such as turbine blades, wheels and afterburner parts. The specimens were solution treated for 1h at 1273K, air cooled and aged for 16h at 973K. Figure 6.6 presents the results of X-ray investigations on this superalloy. The loading of specimens results in an increase in the misorientation angle, η, and a decrease in the average subgrain size, D. The mean values of the parameters under investigation are almost unchanged at the steady-state stage and are equal to 3mrad and 0.6µm, respectively. Consequently, fragmentation of the γ crystallites is also intrinsic to superalloys. This is due to the formation

Fig. 6.6 Dependence of the elongation, average subgrain

size and subgrain misorientation angle on time for the EI437B superalloy. T = 973K. ◦, •: σ = 570MPa; : σ = 700MPa.

6.3 Interaction of Dislocations and Particles

Fig. 6.7 Dislocation sub-boundaries in the matrix of the tested EI437 superalloy. The steady-state stage of creep. T = 973K; σ = 450MPa. ×150000.

of dislocation sub-boundaries in the γ matrix. Decrease in D and increase in η is observed when the applied stress increases. Transmission electron microscopy is difficult to apply to the alloy because it contains many small coherent γ particles. The contrast at the matrix–particle boundary is known to have a deformation origin and hence the borders of the particles seem to be fuzzy. The average particle size in EI437B superalloy was found to be 14nm after the initial heat treatment. The dimensions of the particles increase to 22nm after creep tests and the borders of the particles become more distinct. One can see the dislocation sub-boundaries in Fig. 6.7. The dimensions of the subgrains are about 0.3–0.5µm. This is close to the values estimated with the X-ray method.

6.3 Interaction of Dislocations and Particles of the Hardening Phase

Typical pairs of deforming dislocations are seen in Fig. 6.8. The dislocation lines, which slip under the effect of the applied stress, are parallel and intersect the particles. Transmission electron microscopy evidence supports the cutting of γ particles by slipping dislocations. The dislocations cut the coherent particles of the γ phase without changing the slip plane which is mainly of the type {111}. However, during the tertiary stage of creep the particles coarsen and their coherent bond with the matrix is broken. Orowan bowing

89

90


Fig. 6.8 Electron microphotographs of the EI437 superalloy during the steady-state stage of creep. T = 973K; σ = 450MPa. ×150000 (a), ×200000 (b).

occurs as the rate-controlling strain mechanism. It is the Orowan mechanism that dominates in tertiary creep deformation. In Fig. 6.9 one can see that dislocations cut small particles and bow the big ones. The dislocation loops around particles remain when the dislocation lines have passed. The bowing of particles takes place till cavitation occurs and the specimen ruptures. EI 867 is a superalloy strengthened by chromium, aluminum, molybdenum, tungsten and cobalt. The standard heat treatment consists of solution treatment at 1493K for 2h, quenching in air and ageing at 1223K for 8h. This heat treatment produces cuboidal γ particles, which are on average 130 nm in size along the cube edge. The edges of the cubes are oriented along the < 100 > direction (Fig.6.10). The electron micrographs taken during the steady-state stage of creep are presented in Figs. 6.11–6.13. Parallel deforming dislocations are seen. They move inside the ordered zones one after the other. It is at once apparent from Fig. 6.11 that the particles are obstacles for the moving dislocations. A plane sequence of dislocations is pressed to the edge of the γ particle. The spacing between successive dislocations decreases as the distance to the particle is reduced, as if the dislocations “are waiting” to enter the particle. After entering the particle the dislocations continue to move. The dislocation loops that expand from the interface of the phases are seen in Fig. 6.13.


Fig. 6.9 Electron micrographs of the tested EI437 superalloy during the tertiary stage of creep. T = 973K; σ = 450MPa. ×200000.

Fig. 6.10 Electron micrographs of the EI867 superalloy in

the initial state. Particles of γ phase. (a) Replica, ×20000; (b) thin film, ×100000.

At the stage of the tertiary, accelerating creep the shape of the particles becomes irregular. A rafting process of the γ structure occurs because of development of diffusion coarsening. Now the incoherent irregular rafted

91

92


Fig. 6.11 Electron micrographs of the EI867 superalloy during the steady-state stage of creep. T = 1173K; σ = 215MPa. Interaction of deforming dislocations with γ particles. ×130000.

Fig. 6.12 Interaction of deforming dislocations with γ

precipitates in the EI867 superalloy. ×90000 (a); ×40000 (b).

Fig. 6.13 Interaction of deforming dislocations with particles in the EI867 alloy. ×60000.


particles cannot be obstacles for deforming dislocations. As a result dislocation networks are formed (Fig. 6.14). The networks fill the volume between the particles and spread inside the particles. At the same time the strain rate of the specimen increases.

Fig. 6.14 Dislocation networks during the tertiary stage of

creep in the EI867 superalloy. ×90 000.

Electron micrographs of the EP199 superalloy (Table 2.1) are shown in Fig. 6.15. Parallel dislocations can be observed. The dislocations move one after the other and intersect particles of the γ phase ((a), (b)). The dislocation sets are formed at the tertiary stage of creep ((c), (d)). Electron microstructural examination of the crept test specimens of superalloys has indicated that a rate-controlling process is the precipitate cutting, or shearing. During high-temperature exposure the precipitates coarsen, and the rate-controlling mechanism becomes dislocation bowing. It follows from the obtained data that deforming dislocations are slowed by the coherent particles and then cut them. Hence, the thermally activated overcoming of particles is the process that controls the constant strain rate. However, under the effect of applied stress and high temperature the rafting of particles occurs and the deforming dislocations can bow between the obstacles. This results in accelerating tertiary creep and rupture.

93

94


Fig. 6.15 Electron micrographs of the EP199 superalloy. T = 1173K; σ = 110MPa. (a), (b) At the end of the steady-state stage of creep; ×65000. (c), (d) The stage of the tertiary creep; ×48000.

6.4 Creep Rate. Length of Dislocation Segments

6.4 Dependence of Creep Rate on Stress. The Average Length of the Activated Dislocation Segments

The experimental dependences of the minimum creep rate, ε, ˙ on the applied stress for five superalloys are presented in Fig. 6.16, where ln ε˙ is plotted against σ. Results for three alloys are shown in Fig. 6.17. A linear dependence is observed for all superalloys. Hence, the minimum creep rate is dependent exponentially on stress. The activation volume v of an elementary deformation event can be calculated from these data according to Eq. (1.5). Further, we may compute the average length of a dislocation segment ¯l that must be activated in order that the dislocation can move ahead: ¯l = v (6.3) b2 r are listed. In Table 6.2 the values of l and the average particle dimensions 2¯ The lengths of the activated dislocation segments are one order less than the average particle sizes. Values of the ratio of ¯l/2¯ r lie within the range 0.07 to 0.14, more precisely 0.12 ± 0.04.

Fig. 6.16 Logarithm of strain rate versus stress for superalloys: B, Ni+18Cr+2.6Al. T = 1023K. Data from Ref. [35]. C, Ni + 19Cr + 0.8Al + 2.1Ti. T = 1023K. Data from Ref. [35]. D, Ni + 9Cr + 4.5Al + 5W + 14Co (EI867).

T = 1173K. Data of the present author. E, Ni + 19Cr + 0.8Al + 2.5Ti. T = 973K. Data from Ref. [35]. F, Ni + 20Cr + 2.2Al + 2.0Ti + 3.3W + 5Fe. T = 1023K. Data from Ref. [32].

95

96


Fig. 6.17 Logarithm of strain rate versus

stress for superalloys: B, Ni + 9Cr + 5.0Al + 2.0Ti + 1Nb + 12W + 10Co. T = 1144K. Data from Ref. [36]. C,

Ni + 21Cr + 0.8Al + 2.5Ti (EI437B). T = 973K. Data of the present author. F, superalloy C263. T = 973K. Data from Ref. [37].

Tab. 6.2 The length of activated dislocation segments. Alloying elements in Ni-based alloy

T, K

l, nm

2¯ r , nm

Particle shape

Ref.

19Cr + 0.8Al + 2.1Ti 18Cr + 2.6Al 19Cr + 0.8Al + 2.1Ti

973 1023 1023

4.8 8.3 7.8

42 65 65

cub. spher. spher.

35

21Cr + 0.8Al + 2.5Ti 9Cr + 4.5Al + 5W + 4Co

973 1173

4.7 10.7

22 133

spher. cub.

author author

20Cr + 2.2Al + 2.0Ti + 3.3W + 5Fe 9Cr + 5Al + 2.5Ti + 12W + 10Co Superalloy C263

1023 1144 973

8.9 5.7 7.6

100 83 54

spher. cub. spher.

32 36 37

6.5 Mechanism of Strain and the Creep Rate Equation

The applied stress is insufficient to let a dislocation cut a particle under normal creep conditions. The ordered structure of the γ phase requires that two dislocations in the γ phase must combine in order to enter the γ phase as a superdislocation. The associated anti-phase energy in γ posseses a large barrier to the dislocation entry. A mechanism involving diffusion-controlled movement of dislocations in the ordered γ phase seems to be the most probable one. There is good reason to believe that, in these conditions, the slip of the deforming dislocations is


controlled by diffusion processes, indeed, with ordering, the activation energy of diffusion increases as well as the creep strength. A mechanism of diffusion-controlled dislocation displacement through the ordered γ phase is presented in Fig. 6.18. An arrangement of atoms in a superdislocation is shown. The superdislocation is dissociated into two partial dislocations that are separated by the band of the anti-phase boundary. A vacancy approaches the first partial dislocation as a result of thermal activation. The atomic row shears under the effect of the applied stress, and a relaxation in the vacancy area occurs, thus, a double bend is formed in the dislocation line and the adjacent rows displace. This is equivalent to the expansion of both

Fig. 6.18 The atomic mechanism of the dislocation diffusion displacement in γ phase. Arrangement of atoms in two parallel slip planes of [111] is shown. (a) Ideal crystal lattice. Twelve rows are shown. Along the face diagonal [10¯ 1] atoms of aluminum and nickel are altered. Atoms of Ni that are denoted as 1 and of Al that are denoted as 3 are located in the first slip plane. Atoms of Ni 2 and of Al

4 are located in the second parallel slip plane. (b) Partial dislocations and the anti-phase-boundary (APhB). The Burgers vector (arrows) is [10¯ 1]. At the row 11 a vacancy 5 () is formed. (c) The shear of the atomic row. The vacancy has migrated to the next atom. A double bend has been formed at the moving superdislocation.

97

98


branches of the double bend in opposite directions parallel to the dislocation line. When the bend passes the particle the leading dislocation displaces. The shift is equal to the Burgers vector length b. In such a way the elementary event of plastic strain occurs. One vacancy is enough to displace all atoms in the particle section. It is obvious that increasing the activation energy of diffusion in the γ phase is the necessary condition to raise the creep strength. This goal is achieved by means of an optimal composition design of superalloys. The anti-phase boundary energy also plays an important role. This value has been determined to be approximately 180 mJ m−2 for industrial single crystal superalloys. For example, the energy of the anti-phase boundary of Ni3 Al increases from 180 to 250 mJ m−2 when substituting 6 at.% Ti for 6 at.% Al. Thus, thermal activation is necessary in order for the segment l to advance. The work which is performed is equal to the increment of the thermodynamic potential of the system dislocation-obstacle. The average slip velocity, V , of the dislocation segment ¯l is given by A V =Γ ¯ l

(6.4)

where Γ is the frequency of attempts of the dislocation to overcome the potential barrier, A is the average area swept out by the segment released during the thermal activation, ¯l is the average length of segment, which is activated inside the particle. The displacement of the segment is equal to the Burgers vector length |b|. On the basis of the theory of the activated reactions rate [25] the value of Γ can be represented by an expression of the form n ∆Φ j=1 νj Γ = n−1 exp − (6.5) kT j=1 νj In Eq. (6.5) νj are the normal frequencies of oscillations of the segment in a crystal lattice in the initial state, the total number of these oscillations is n (all the atoms that take part in overcoming the potential barrier are considered). The values of νj are (n − 1) frequencies of oscillations in the activated state at the peak of the potential barrier. The increment of the thermodynamic potential, ∆Φ, (of the Gibbs free energy) is given by ∆Φ = ∆U − T ∆S − τ v

(6.6)

where ∆S is the increment of entropy, ∆U the increment of internal energy as the segment overcomes the barrier, τ the shear stress, v the activation volume.


Applied stress affecting the crystal does some work τ v in order for the segment to move forward. It is possible to express ∆S for vibrations with a small amplitude as n−1 νj ∆S = k ln (6.7) νj j=1 The first frequency ν1 = νb/2¯l is an efficient frequency of attempts to overcome the barrier, ν is of the order of the Debye frequency. Combining Eqs. (6.4)–(6.7) we find that the dislocation segment velocity can be expressed as 2 ¯ νbA ∆U τb l V = ¯2 exp − exp (6.8) kT kT 2l The value of ∆U in Eq. (6.8) is close to the activation energy of generation and migration of vacancies in the ordered phase. The sum of these values is known to be the energy activation of diffusion in the γ phase. The value of area A is assumed to be expressed as A = b2 ¯l

(6.9)

Substituting Eq. (6.9) in Eq. (6.8) we obtain V =

2 τb l νb2 ∆U exp − exp kT kT 2l

(6.10)

where τ is the shear stress in the slip plane. We have obtained the theoretical equation for the velocity of movement of dislocations through particles. Recall that it describes the diffusion-controlled mechanism of the cutting of particles by dislocations. It is important to connect the velocity of dislocations with the average particle size. We can use the correlation which has been obtained experimentally: l = 0.12 ± 0.04 2r

(6.11)

Substituting ratio (6.11) into Eq. (6.10) we find V =

∆U 0.12τ b2 · 2r νb2 exp − exp 0.24 · 2r kT kT

(6.12)

The last equation is a semi-empirical one, because we have used the results of measurements of the particle sizes and the data of the strain rate tests.

99

100


Finally, the minimum strain rate of a superalloy is supposed to be directly proportional to the dislocation velocity, V , and to the dislocation density, N : ε˙ = bf (c)N

∆U 0.12τ b2 · 2r νb2 exp − exp 0.24 · 2r kT kT

(6.13)

where f (c) is a decreasing function of concentration of the γ phase. 0 < f (c) < 1. First, we can conclude from Eq. (6.13) that the dependence of the strain rate upon the average particle size 2r is a function with a minimum. There is a particle size, under which the deforming dislocations move at the least velocity. Indeed, if the value of 2r is small the pre-exponential factor is large and the exponent factor is close to unity. In this case, if the particle size 2r increases, the exponent increases more rapidly than the pre-exponential factor. Hence there is an optimal size of particles 2r0 , which is dependent on T and σ. Taking the derivative and solving ∂V /∂(2r) = 0 we find that for EI437B superalloy at 973K and σ = 400MPa the value of 2r0 = 12.2 nm. The result fits the measured value 14 nm satisfactorily. It is of importance to estimate the energy ∆U . Taking the logarithm of Eq. (6.13) we obtain ∆U = −kT ln ε˙ + 0.12τ b · 2¯ r + kT ln 2

νb3 0.24 · 2¯ r

+ kT ln[f (c)N ] (6.14)

It is essential to know the last term on the right-hand side of Eq. (6.14) in order to calculate a mean activation energy ∆U . Rae et al. [38] have measured the dislocation density in the crept CMSX-4 superalloy. The density of dislocations was estimated to be 2 × 1012 m−2 by measuring the total length of dislocations in an area. One may reasonably assume that the product of the concentration and the dislocation density will be in the range 1010 –1012 m−2 , most likely 1012 m−2 . Applying Eq. (6.14) for tests of the same superalloy under two stresses σ1 and σ2 and denoting the sums of the first three terms as A1 and A2 one can write ∆U = A1 + kT ln[f (c)N1 ]

(6.15)

∆U = A2 + kT ln[f (c)N2 ]

(6.16)

Thus, the logarithm of the ratio of the dislocation densities is given by ln

A1 − A2 N2 = N1 kT

(6.17)


Tab. 6.3 The calculated activation energies of the elementary deformation event in the γ phase of superalloys. The values of f (c)N that have been assumed (m−2 ) are shown above columns 4 to 6. ∆U , 10−19 Jat.−1

σ2 /σ1 , T, K

MPa/MPa

923 1023 973 1023 1173 1144

600/400 280/80 360/280 320/200 280/120 520/280

Ref.

N2 /N1

1010

1011

1012

30.3 115.1 6.0 10.3 32.0 7.2

3.56 3.68 3.97 4.11 4.45 4.50

3.87 4.01 4.28 4.44 4.82 4.86

4.18 4.33 4.59 4.76 5.19 5.22

author 35 37 32 author 36

One can see from Table 6.3 that the dislocation density increases by one or two orders when the applied stress is increased twofold or less. In Table 6.3 the calculated values of the activation energy are presented. The superalloys are listed in the order of increasing activation energy. The experimental data from Figs. 6.5, 6.6 and Table 6.1 are used to calculate f (c)N . If the dislocation density is assumed to be 1012 m−2 the activation energy ranges from 4.18 × 10−19 to 5.22 × 10−19 Jat.−1 . The apparent activation energy of creep, ∆Qapp was measured by Picasso and Marzocca [39] for the IN-X750 superalloy1) . They used the technique of differential temperature steps. The method is based on Eq. (1.4). A specimen is subjected to a deformation under constant stress and temperature T1 until a given strain is reached. At this point the temperature is changed abruptly to T2 which may be above or below T1 . The value of ∆Qapp is calculated using the relationship ln(ε˙2 /ε˙1 ) ∆Qapp = −k (6.18) (1/T2 − 1/T1 ) where ε˙1 is the initial strain rate at the temperature T1 and ε˙2 is the final strain rate at temperature T2 . This method has an essential drawback: an instability of structure during the temperature change results in too high measured values of Qapp . If the differential change of temperature did not modify the substructure the authors [39] obtained Qapp = 5.08 × 10−19 Jat.−1 . The energy of self-diffusion in Ni is (4.64 ± 0.21) × 10−19 Jat.−1 [22]. The data are averaged over results presented for temperatures from 953K to 1673K in seven publications. Thus, the value of the activation energy ∆U for superalloys is somewhat greater than the energy of self-diffusion in nickel. 1) The activation energy of an elementary event of the high-temperature strain (creep) is denoted differently by different authors, as Q, Qc , ∆Qapp , ∆U .

101

102


Manonukul et al. [37] have proposed an expression for the dependence of the strain rate on the volume fraction of the γ phase of the form f (c) =

(4π/3c)1/3 − 2 (4π/3c)1/3

(6.19)

However, the expression ceases to be valid for a γ fraction in a superalloy of more than 0.45 (Table 6.4). Tab. 6.4 Dependence of the factor f (c) on the fraction of hardening phase. c f (c)

0.10 0.42

0.15 0.34

0.20 0.28

0.25 0.22

0.30 0.17

0.35 0.13

0.40 0.09

0.45 0.05

6.6 Composition of the γ Phase and Mean-square Amplitudes of Atomic Vibrations

We have studied the composition and have measured the mean-square amplitudes of the atomic vibrations in solid solutions based on Ni3 Al. The alloys Ni3 Al, Ni3 (Al, Ti) and strengthening phases representing solid solutions of different elements in the intermetallide of the type B3 A were investigated (Table 2.1). The γ phases were extracted electrolytically from aged superalloys. An X-ray technique for the measurement of mean-square amplitudes separately for each of the two sublattices of a γ phase has been developed [17]. We determined the chemical compositions of the γ phases by chemical analysis and X-ray diffractometer studies of the specimens were also carried out. We measured the intensities, I, of the (100) and (200) reflections and compared the ratios I(100) /I(200) with calculated values. The ratios have been calculated for the distribution of all kinds of elements between the γ sublattices, B and A. The ratio under study is dependent on the average factors of the X-ray scattering in the sublattices, f¯B and f¯A , respectively. The final formulas of the phases have been established by us under the condition of a coincidence of the experimental and calculated values of this ratio. The obtained data are summarized in Table 6.5. According to the obtained results Ti atoms occupy places in the sublattice A (Fig. 6.1). Cr, Fe and Co atoms are located preferentially in sublattice B. Mo, W and Nb partition between the two sublattices, but are mainly located in A. There is a connection between the amplitudes of heat-induced atomic vibrations and the forces of interatomic bonds in the crystal lattice of the hardening γ phase. It seems natural that the greater the vibration amplitude the greater

6.6 Composition of the γ Phase and Atomic Vibrations

Tab. 6.5 Compositions of B3 A (γ ) phases and the mean-square

atomic displacements. 2 u2 B , pm

Phase 723 K

873 K

1023 K

2 u2 A , pm

723 K

873 K

1023 K

Ni2.96 Fe0.02 Al1.00

330

470

660

380

580

810

Ni2.95 Fe0.05 (Al0.85 Ti0.15 )

380

440

600

290

370

460

Ni2.83 Fe0.04 Cr0.07 Mo0.06 (Al0.51 Ti0.31 Nb0.07 Mo0.08 Cr0.03 )

250

380

550

180

230

330

Ni2.83 Fe0.04 Cr0.08 W0.03 Mo0.02 (Al0.43 Ti0.28 W0.08 Mo0.03 Cr0.10 )

230

340

450

30

60

80

Ni2.76 Co0.09 Fe0.01 Cr0.06 Mo0.02 (Al0.79 W0.08 Mo0.11 Cr0.03 )

260

340

580

20

40

50

the diffusion mobility in the crystal lattice. The probability of an atom transfer to an adjacent node of the crystal lattice also increases. The smaller the amplitudes of the atomic vibrations, the greater the high-temperature strength of the γ phase. A decrease in amplitudes results in an increase in the value of the activation energy ∆U in Eq. (6.13) and the creep rate decreases. In the first phase, practically pure Ni3 Al (Table 6.5), the atomic displacements of the lighter atoms of aluminum (nodes A) appreciably exceed the displacement of the nickel atoms (nodes B) at all tested temperatures, u2A > u2B . The replacement of approximately one sixth of the aluminum atoms with titanium atoms decreases the amplitudes of the heat vibrations in the aluminum sublattice, so that now u2A < u2B . The effect of titanium dissolution in the sublattice A on decreasing amplitudes becomes more evident if the temperature rises. The effect of adding molybdenum and tungsten to the γ phase is even greater. One can see from Table 6.5 that the values u2A are 10–16 times less at 1023K in phases containing both these elements. The vibration amplitudes decrease less in the sublattice B. The decrease of u2A is not proportional to the growth of the mean atomic mass because of Mo and W dissolution, but exceeds it essentially. Actually, the effective mass of atoms A in the investigated strengthening phases increases only by 64–78% in comparison with the effective mass of aluminum atoms. The mass of atoms dissolved at the B nodes is close to that of nickel. This fact is evidence of the growth of interatomic bonds in the crystal lattice of the alloyed γ phase. Comparing the values of the mean-square atomic displacements in the sublattice A for phases of different composition we can conclude that they are related, mainly, to the sum of the atomic parts of molybdenum and tungsten. In Table 6.5 the phases are ordered by increasing this sum while decreasing u2A .

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An energetically efficient mechanism of an elementary diffusion event in the γ phase consists of a certain cycle of a vacancy shift. The nodes of both sublattices must be included in this cycle which is why the diffusion mobility of elements in the crystal lattice of the γ phase can be decreased essentially although the mean-square amplitudes are decreased only significantly in the aluminum sublattice. We have compared the mean-square atomic amplitudes in strengthening phases of superalloys with the rate of the steady-state creep ε˙ of the same alloys. The superalloy EI437B with u2A = 460 pm2 at 1023K has the least high-temperature strength of the three alloys listed in Table 2.1 whereas the superalloy EI867 with u2A = 50 pm2 , has the greatest strength.

6.7 Influence of the Particle Size and Concentration

It is the cut of the precipitates that is the rate-controlling process when the average distance between particles is smaller than a critical value. However, the interaction of dislocations with particles is dependent on the size of the particles and on the distance between them. The Orowan mechanism of obstacles bowing can also be rate-controlling. The mean particle size and the average distance between particles are dependent on one another. Manonukul et al. [37], by assuming that the precipitates of the γ phase are arranged in a three-dimensional array, have proposed to a simple, first approximation, that λp =

4π 3c

1/3

− 2 2r

(6.20)

where λp is the average distance between particles, c is the volume fraction of the γ phase, 2r is the mean particle size, as before. The results of calculations according to Eq. (6.20) are presented in Table 6.6. Particle sizes change during ageing at a constant temperature. Both the particle size and the volume fraction increase. This process is called coarsenTab. 6.6 The mean particles distance λp , nm as a function of the average particle size and of the γ phase concentration. c

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

Ratio λp /2r λp when 2r = 20nm λp when 2r = 60nm

1.47 29.4 88.2

1.03 20.6 61.8

0.76 15.2 45.6

0.56 11.2 33.6

0.41 8.2 24.6

0.29 5.8 17.4

0.19 3.8 11.4

0.10 2.0 6.0

6.7 Influence of the Particle Size and Concentration

ing and has an appreciable effect on creep rate. This is reflected in Eq. (6.13), in which the average particle size, 2¯ r, appears within the exp term. The dependence of precipitate size on time can be represented by the empirical equation given in [37] 2r = 2r0 + 2.96 × 105 t1/3 exp(0.012T )

(6.21)

where the time t is in hours. In Fig. 6.19 the corresponding dependence is shown for two temperatures. There is a critical value λpc as well as a critical particle size. If λp ≤ λpc the rate-controlling mechanism is the cutting of γ -particles by slipping dislocations. If λp > λpc the rate-controlling mechanism becomes dislocation bowing. The C263 superalloy is strengthened by titanium, aluminum, chromium, molybdenum and cobalt. The volume fraction of γ phase is equal to 0.095. In Fig. 6.20 the creep curves for this superalloy are shown. At 1073K and stress 160MPa the critical particle size was found to be 85 nm. Therefore the dislocations overcome obstacles with thermally activated cutting. Coarsening of particles takes place, however, and after approximately 1.7×106 s (472 h) the critical particle size, 85 nm is achieved, and the rate-controlling mechanism becomes dislocation bowing. This is indicated in Fig. 6.20 where the transfer from steady-state to tertiary creep is clearly seen. It is reasonable, for completeness, to consider the case in which the precipitate dissolves out such that c = 0. The slip of dislocation is controlled by dislocation networks. The length of the activated dislocation segment l equals the distance between pinning points.

Fig. 6.19 The increase in the average particle size with time. B, 973K; C, 1073K.

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Fig. 6.20 Creep curves in C236 superalloy at 1073K. B, stress 160MPa; C, stress 200MPa. Data from Ref. [37].

6.8 The Prediction of Properties on the Basis of Integrated Databases

Comprehensive tests and investigations are needed in order to design new superalloys. Variation of the chemical composition of superalloys is known to be the main method for optimization of alloy properties, however, there are an astronomical number of combinations. Therefore another approach has been developed, which consists in the prediction of properties of new alloys. The idea is as follows [30]. The known data (from the literature and industrial sources) on dependences of certain characteristics (such as the rupture life, the yield stress, the lattices parameters) on the element concentration have to be gathered and put into a computer database. A special program is used to predict the properties of a new superalloy. This approach does not take into account the physical processes. The structure of alloys, solution and precipitation of phases, the influence of elements on interatomic bonds, the mechanism of deformation are not considered. Moreover, the additive action of alloying elements on properties is also ignored. Nevertheless, such an approach is substantiated, because our knowledge about the extremely complex processes in superalloys is too limited to be used to predict the properties. The investigators try to evaluate the errors of prediction and to compare the results with experimental data. Tancret et al. [30] write: “Because the influence of the composition and processing parameters on the material properties is extremely complex and multivariate, designing an alloy “to measure” is not feasible using experience alone. Modern alloys contain many chemical elements added to achieve particular properties. The influence of individual alloying elements on me-

6.8 The Prediction of Properties

chanical properties can be measured and understood in isolated cases; simple interactions between two or three elements can be formulated, but describing all the interactions as a whole is generally impossible”. The Gaussian processes of the modelling of mechanical properties have been described. A stochastic process is called a Gaussian process if all the distributions are subjected to the corresponding distribution. Suppose that N alloys have been studied. Each tested ith alloy is associated with an input vector xi . The obtained data D are considered as N input vectors {x1 , x2 , ...xN } = [XN ]. There are N corresponding outputs or targets {t1 , t2 , ..., tN } = tN , each target being a measurement. The joint probability distribution, in an N -dimensional space, of the target vector tN is denoted by an expression P (tN |[XN ]). The known data are denoted by D = {tN , [XN ]}. One may consider the data D as an input matrix, where every column represents one superalloy and every row contains the values of the same characteristic throughout all alloys. The aim is to predict the output value, tN +1 , corresponding to a new input vector, xN +1 (i.e. a new alloy or test conditions). This means to calculate the one-dimensional probability distribution over the predicted point P (tN +1 |xN +1 , D), given a knowledge of the corresponding input vector, xN +1 , and the data D. The new point is given by the following relationship P (tN +1 |xN +1 , D) =

P (tN +1 , tN |xN +1 , [XN ]) P (tN |[XN ])

(6.22)

The model assumes that the joint probability distribution of any N output values is a multivariate Gaussian. Equation (6.22) can be reduced to a univariate Gaussian of the form 1 (tN +1 − tm )2 P (tN +1 |xN +1 , D) = √ exp − (6.23) 2σt2 2πσt where tm is the mean value, σt is its standard deviation. We refer the reader for details to Ref. [30] and to references therein. The authors have collected the data concerning mechanical properties and, especially, creep rupture stress. In Fig. 6.21 the predicted and actual effects of stress on rupture life for IN939 superalloy are presented. This figure shows that the error of the method is about ±15–20%. Using the described technique the authors try to estimate the influence of alloying elements upon the properties of superalloys. Titanium and aluminum, when used as the γ formers, increase creep rupture stress. However, the respective influence of Al and Ti atoms seems to be inverted. Since titanium atoms are bigger than aluminum atoms (by 4%) they induce an increase

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Fig. 6.21 Predicted and actual relation between creep rupture stress for IN939 superalloy and its rupture life, tr . Temperature of tests 1143K. Reprinted from Ref. [30] with permission from Maney Publishing.

in the γ parameter and γ /γ mismatch and thus influence the strain fields. Ti also increases the anti-phase boundary energy of the γ phase, which makes the cutting of particles by dislocations more difficult. Titanium is thus expected to increase the yield strength more than aluminum and this has been recognized by the Gaussian process. On the other hand, it has been shown that an excessive increase in the γ /γ misfit reduces the creep strength. Thus, adding titanium, although it increases creep strength by promoting γ precipitation, is less effective than adding aluminum because of a higher lattice misfit, which is also predicted by the model.

Fig. 6.22 Predicted influence of Co, Mo and W content on the creep rupture stress at 1023K of the Ni–20Cr–1Al–1Ti–10Co superalloy. Reprinted from [30] with permission from Maney Publishing.

6.9 Summary

The authors’ predicted influence of Co, Mo and W atoms is presented in Fig. 6.22. All these elements increase the creep resistance of nickel-based superalloys, in agreement with known results. It is obvious from this figure that the error of prediction of about 50% and more shown by the bars is too great. The value of the method seems to lie in the possibility to cast out “bad” solutions even though it fails to find “good” solutions.

6.9 Summary

Superalloys have been developed for aircraft and industrial gas turbines, rocket engines and space vehicles. The structure of nickel-based superalloys consists of the γ matrix and the γ phase precipitations. This phase is a solid solution of various elements in the intermetallic compound Ni3 Al. The γ phase has remarkable properties; in particular it hardens with increasing temperature. Dislocation sub-boundaries form in the matrix of superalloys under the influence of the applied stress. The loading results in an increase in the subgrain misorientations and a decrease in subgrain size in the matrix. The moving dislocations cut the particles during the steady-state stage of creep. The most probable mechanism of the cutting is a diffusion-controlled motion of partial dislocations with the anti-phase boundary in the ordered phase. At the stage of the tertiary creep the shape of the particles becomes irregular as the coarsening of precipitates takes place. The experimental dependences between the logarithm of the minimum strain rate and the applied stress for superalloys are linear. Hence the minimum creep rate of superalloys depends exponentially on stress. The computed lengths of the thermally activated dislocation segments are one order less than the average particle sizes. The minimum strain rate of superalloys is expressed as ε˙ = bf (c)N

νb2 0.12τ b2 · 2r ∆U exp − exp 0.24 · 2r kT kT

(6.24)

where f (c) is a decreasing function of the concentration of the γ phase, N is the dislocation density, 2¯ r is the average particle size, ∆U is an activation energy, τ is the shear stress in the slip plane. The activation energies, ∆U , for the high-temperature steady-state creep of superalloys are estimated to vary from 4.18 × 10−19 to 5.22 × 10−19 Jat.−1 when the dislocation density is assumed to be of the order of 1012 m−2 .

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In a phase of type B3 A (Ni3 Al) Ti atoms occupy places in sublattice A. Cr, Fe and Co atoms are located mostly in sublattice B. Atoms of Mo, Nb and W partition between the two sublattices but are mainly located in A. Under the effect of solution of W, Mo and Ti atoms in the Ni3 Al phase the meansquare vibration amplitudes of A-atoms decrease from 810 to 50–80 pm2 at 1023K. The smaller the mean-square amplitudes of the atom vibrations in the hardening phase the higher the creep strength of a superalloy. Methods of prediction of the properties of new superalloys are being developed on the basis of integrated databases of superalloy composition and test results. The error of prediction is from 15 to 50%.

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7

Single Crystals of Superalloys

7.1 Effect of Orientation on Properties

Single crystals of the nickel-based superalloys were developed for gas turbine applications in the 1970s and 1980s. Single-grained castings replaced polycrystalline moldings for some engines with a view to avoiding slip and fracture along grain boundaries. The single-crystal superalloys have superior creep, fatigue and thermal properties compared to conventional cast alloys because the grain boundaries have been eliminated. Their components are able to operate at temperatures 200–300K higher, up to 1373K. The rupture life is considered to be at least 20–40% greater than that of polycrystalline superalloys. Cast nickel-based single-crystal superalloys are utilized for blades in aircraft and in stationary gas turbines. The blades of very complex geometry are often constructed with channels so that cooler air can be forced to flow within the blades during operation. Single crystals are manufactured by means of the directional crystallization technique. The preferred crystal growth direction is the < 001 >, i.e. along the cube side, as compared with < 011 > and < 111 >. The Young’s modulus has a minimal value in the < 001 > direction, so that the thermal stresses in this direction are minimal. The anisotropy coefficient of the Young’s modulus is equal to 2.2, whereas the mechanical properties have the anisotropy coefficient of 1.4 at high temperatures. Many manufacturers recommend to produce the single crystal blades with < 001 > orientation, however, in practice, there is a misalignment from exact < 001 > orientation. The creep behavior of single crystal superalloys is highly anisotropic. Under otherwise equal conditions the orientation of a superalloy single crystal is the factor which contributes to the overall creep strength. High Temperature Strain of Metals and Alloys, Valim Levitin (Author) c 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim Copyright ISBN: 3-527-313389-9

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7 Single Crystals of Superalloys

The shear stress in a crystalline plane is known to be determined by the Schmid law τ = σ · cos ϕ · cos ψ (7.1) where τ is the shear stress in the slip plane, σ is the applied stress, ϕ is the angle between the applied stress and the perpendicular to the slip plane, ψ is the angle between the applied stress and the slip direction. The values of the Schmid factors are shown in Table 7.1. The < 001 > orientation is a multiple slip orientation. All the {111} slip planes are equivalent. When an orientation moves from < 001 > the Schmid factor increases since the angles ϕ and ψ decrease. One can see that the Schmid factors are less for the < 111 > tensile direction than for < 100 > (0.27 in comparison with 0.41). A < 111 > oriented single crystal has to have a higher creep strength. Some dislocation splittings are also shown in Table 7.1. For crystals oriented near the < 001 > direction (the tensile axis is also directed along the < 001 >) the dominant crystalline shear system is {111} < 110 >. This is the socalled octahedral slip. The system {111} < 110 > operates most frequently at temperatures in the vicinity of 1023K. The dislocations with the Burgers Tab. 7.1 The Schmid factors in the cubic face-centered γ, γ

crystal lattices and examples of the dislocation splitting. (hkl), indexes of the slip plane; [uvw], indexes of the slip direction. APB, anti-phase sub-boundary; SISF, the superlattice intrinsic stacking fault. Axis of tension

(hkl)

[uvw]

[001]

(111)

[¯ 101] [0¯ 11] [110] [0¯ 11] ¯ [101] [¯ 110]

0.41 0.41 0

a/2[¯ 101] → a/6[¯ 112] + a/6[¯ 2¯ 11]+APB

0.41 0.41 0

a/2[0¯ 11] → a/3[1¯ 12] + a/6[¯ 2¯ 1¯ 1]+SISF

(¯ 1¯ 11)

[011] [101] [¯ 110]

0.41 0.41 0

(1¯ 11)

[011] [¯ 101] [110] ¯ [110] [¯ 101] [0¯ 11]

0.41 0.41 0 0.27 0.27 0

a/2[¯ 110] → a/6[¯ 121] + a/6[¯ 21¯ 1]+APB

[110] [101] [0¯ 11]

0.27 0.27 0

a/2[110] → a/3[211] + a/6[¯ 11¯ 2]+SISF

(¯ 111)

[¯ 111]

(111)

[111]

(¯ 111)

Schmid factor

Eventual dislocation splitting


Fig. 7.1 Bar graphs of the orientation

distributions of single-crystal blades fabricated from ZhS6UVI superalloy. Along the x-axis the angle of a misalignment from the < 100 > direction is plotted.

The statistical population consists of 2000 blades. Data are shown for cooled blades (a) and for uncooled blades (b). Unallowed orientations are shaded.

vector a/2 < 110 > tend to dissociate, as shown in Table 7.1. Therefore slip occurs in the < 001 > oriented crystals on the system {111} < 112 > at 1023–1123K. A high primary creep strain and a short rupture life are observed for the < 011 > oriented single crystals. This is caused by the predominance of a single {111} < 112 > system. The specimens crept under relatively high stresses in the < 111 > orientation reveal very low resolved stress in all {111} < 112 > equivalent slip systems and so creep strength in this orientation is maximal. The cube slip {001}< 110 > appears when the material is loaded uniaxially along a direction far from < 001 >. The higher the temperature the more propensity for cube slip. Rubel [40] presented the distributions of orientations in a large group of cast single-crystal gas-turbine blades. These data are shown in Fig. 7.1. The relative frequency of misalignment from the < 001 > direction ranged from 20 to 55 ◦ . He investigated a series of single blades. In Table 7.2 the initial orientations of the specimens and the results of the creep tests are presented. Tab. 7.2 The crystal orientations of the ZhS26VI superalloy single crystals and results of creep tests at T = 1248K and σ = 226 MPa. Number 1 2 3 4 5 6 7 8 9 10

∆θ[001]

∆θ[011]

∆θ[111]

Rupture life, h

42.6 23.6 24.4 39.9 39.5 14.5 42.0 45.8 41.5 33.5

8.8 20.4 19.8 20.0 20.6 30.9 23.8 25.1 25.7 27.6

26.5 38.1 38.0 18.3 18.6 42.4 14.8 10.2 14.4 21.4

58.0 64.0 88.3 103.0 106.0 118.0 143.3 162.8 163.3 228.0

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Fig. 7.2 The crystallographic orientation of the stress

axes and the data of creep tests for ZhS26VI superalloy. Temperature 1248K, stress 226MPa. The numbers denote the rupture life in hours.

One may imagine the effect of orientations on properties more obviously using the standard stereographic triangle. The durabilities of tested specimens are shown in such a way in Fig. 7.2. It is obvious that the properties are distinguished on account of the high anisotropy of single crystals. The deformation slip along the system {111} < 110 > is the most probable. The system of slip {111} < 211 > is also active, though to a lesser degree. The following empirical equation was proposed on the basis of analysis of the experimental data. τ = −0.0261 · T · (lg tf + 20) + 823.7

Fig. 7.3 Comparison of dependence (7.2) (line) with experimental data of tests (symbols). Rupture life of the ZhS26VI superalloy is 100h in this temperature range.

(7.2)


where τ is the shear stress in the active slip system in MPa, tf is the rupture life in hours. The correlation coefficient was found to be 0.88. The comparison of the linear dependence (7.2) with results of tests shows that the equation can be used to calculate the long-time strength for durability of 100h (Fig. 7.3). The main results of integrated studies of the properties of the ZHS26VI superalloy specimens are shown in Fig. 7.4. The data are taken from [40]. One can see that if the orientation is moved from < 001 > and the angle ∆θ001 ranges from 0 to 25 ◦ the properties at first decrease. However, they then begin to increase (when ∆θ001 varies from 25 to 54 ◦ ). Single crystals whose

Fig. 7.4 Effect of orientations of the

ZhS26VI superalloy single crystals on properties. Isolines of mechanical properties (MPa) are shown. (a) Yield strength σ0.2 at 293K; (b) ultimate stress limit

1248 at 293K; (c) long-time strength τ100 : under the indicated stress the rupture life at 1248K is 100h; (d) fatigue strength at 293K, number of cycles till fracture is 2 × 107 . Data from Ref. [40].

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axes are close to the < 111 > orientation have the greatest high-temperature creep strength. In this case very low resolved shear stress operates on all three principal {111} < 110 > slip systems, so the creep strength is maximal. At the same time the plasticity decreases.

7.2 Deformation of Single-crystal Superalloys at Lower Temperatures and Higher Stress

Single-crystal superalloys have salient features in response to the applied stress conditions. The deformation behavior of a single crystal depends on the superalloy composition, the crystal orientation, the temperature and the applied stress. The structure of the cast single-crystal superalloy consists of a high volume fraction of the coherent γ phase (up to 75%) separated by thin channels of γ-matrix. One may conclude that such a crystal is in a sense a composite material. The directed strengthened γ areas are alternated with more plastic γ areas. The sizes of the particles and the channel widths are distributed quantities, even in the initial cast state. The typical size of the γ precipitates is 300–600 nm, while the γ channels are much narrower, approximately 50–100 nm. The macroscopic strain rate of single-crystal superalloys is strongly dependent on the temperature and the stress loading conditions. It is appropriate to distinguish between two temperature areas, where the superalloys are really used and tested. A difference in the physical processes proceeding in superalloys is a consequence of their operations within the different temperature intervals. The first interval is from approximately 1023 to 1123K (750–850 ◦ C). We term this area the lower temperature interval. From the practical standpoint this temperature range is known to be of crucial importance for cooled gas turbine blades. The roots of blades and the air cooling channels operate in turbines at relatively low temperatures, such as 1023K. The turbine entry temperatures continue to rise. “Such developments have, over the past 20 years or so, contributed significantly to a steady increase in the turbine entry temperatures of modern aero-engines. . . ; the rate of approximately 5 ◦ C per year average over the past two decades shows no sign of slowing” [42]. The other temperature area ranges from about 1223 to 1373K (950–1100 ◦ C). The tests are understood to be conducted under lower stresses at such high temperatures. Single-crystal superalloys reveal the usual creep curves within the lower temperature interval. All four stages of creep are observed. A significant

7.2 Deformation at Lower Temperatures

Fig. 7.5 Creep curves for CMSX-4 superalloy. Reprinted from

Ref. [41] with permission from Elsevier Science Ltd.

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amount of primary creep, 10 to 20%, is exhibited. The extent of the primary creep is found to vary with temperature and deviation of the orientation from the < 001 > direction. A long steady-state stage is typical. This constitutes a considerable proportion of the available rupture life of the material. The research group from the University of Cambridge and Rolls-Royce University Technology Center have been carried out investigations on the problem of deformation of the single-crystal superalloys in detail [37, 38, 41, 42] using the CMSX-4 superalloy (see the chemical composition in Table S1, Supplement 3). The obtained creep data for CMSX-4 superalloy are shown in Fig. 7.5 [41]. The orientations of the tested single crystals were found to lie within 10 ◦ of the < 001 > direction. The test results are plotted in two forms: strain versus time (A) and strain rate versus strain (B). At 1023K (750 ◦ C) the usual creep curve (with primary, steady-state and tertiary stages)is observed [Fig. 7.5(a)]. At 1223K (950 ◦ C) the tertiary deformation begins almost at once [Fig. 7.5(b)]. One can see in the inset of Fig. 7.5(b) that, to a good approximation, the creep strain rate can be taken to be proportional to the accumulated creep strain. At higher temperature 1423K (1150 ◦ C) (and under a lower stress) a small amount of strain is accumulated but there exists a plateau where the creep strain is almost constant in time. Later the creep strain dramatically increases with rupture eventually occurring [the typical curve is Fig. 7.5(c)]. The dependence of the creep on the orientation of the CMSX-4 single crystals is shown in Fig. 7.6 [42]. The orientations of the specimens after the creep tests are plotted in the standard stereographic triangle in Fig. 7.7.

Fig. 7.6 Creep curves from CMSX-4 superalloy single crystals tested at temperature 1023K and stress 750MPa. The test temperature belongs to the lower temperature interval. Orientations of specimens are shown in Fig. 7.7. Reprinted from Ref. [42] with permission from Elsevier Science Ltd.


Fig. 7.7 The orientation of the CMSX-4 superalloy specimens

within the stereographic triangle.

The effect of misorientation on the behavior of specimens at lower temperatures is very strong. The authors noted that the magnitude of the steady-state creep rate correlates with the maximum rate in the primary creep stage. The further from the < 001 > orientation the less the minimum creep rate (specimens M, N, Fig. 7.6, 7.7). Primary creep at 1023K is due to a deformation on the {111} < 112 > slip system. After some deformation and a lattice rotation the strain is associated with at least two slip systems of type {111} < 112 >. The influence of stress on strain is of interest. When the stress is large enough to promote primary creep, then the creep strain evolution is relatively insensitive to its value (Fig. 7.8). The “threshold” stress for primary creep at 1023K appears to lie between 600 and 750 MPa. A physical mechanism of strain at a given temperature is supposed to be the cause of the observed creep behavior. This is confirmed by transmission electron microscopy. The γ/γ structure of the cast single crystal may be considered as two interdependent systems. Therefore the properties of phases influence the generation and motion of dislocations as do the phase boundaries. In the conditions under consideration the γ phase, being a soft constituent, is the source that generates deforming dislocations. The precipitates are permitted to deform only elastically, while the matrix can be deformed both elastically and plastically. The slip in the γ phase occurs on octahedral planes, {111} < 110 >. The matrix dislocations have Burgers vectors of the type a/2 < 110 > in a plane of this type. The a/2 < 110 > dislocations spread through the γ channels between the dendrites of the cast structure.

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Fig. 7.8 The creep curves from CMSX-4 superalloy single crystals tested at 1023K at various stress levels for specimens oriented within 10 ◦ of < 100 >. Reprinted from [42] with the permission from Elsevier Science Ltd.

During the incubation period of creep the dislocations propagate from each dislocation grow-in source. The local density of grow-in dislocations was estimated to be about 1011 m−2 in CMSX-3 superalloy tested at 1098K [43]. The deforming dislocations have most probably the same Burgers vectors for an orientation with the highest resolved shear stress. During creep deformation the ordered structure of the γ phase requires that two dislocations in the γ phase must combine to be able to enter the γ phase as a superdislocation (Fig. 6.18). The associated energy of the antiphase boundary is a high barrier to the entry of dislocations, so the γ phase remains largely undeformed during the majority of the specimen rupture life under low and moderate stresses. In CMSX-4 superalloy tested at 1023K the ribbons of net Burgers vector a/6 < 112 > are the most striking feature of the structure of crept specimens. The term “ribbon” is used to describe the observed configuration of overall Burgers vector a < 112 > separated by superlattice stacking faults and anti-phase boundary faults [38]. The authors note that large numbers of a/2 < 110 > dislocations are present in the structure of tested specimens. The dislocations expand in the γ channels when creep deformation proceeds. The two dislocations a/6 < 112 > are separated as the partials pass through the γ phase. Where the ribbon of two partial dislocations passes the γ phase they are constricted by the anti-phase boundary energy. As the primary creep deformation proceeds, the creep rate drops since secondary slip systems become activated. The lattice rotation which occurs is considered to be a superposition of the deformation associated with two or more {111} < 112 > slip systems.


The authors [38] describe the physical mechanism of the primary creep of the < 001 > oriented single crystals as follows. “The first step”: two dislocations, both slipping in the γ phase on the (111) plane, combine to produce the leading half of the a < 112 > ribbon: a/2[10¯1] + a/2[01¯1] → a/2[11¯2] “The second step” consists in the dissociation: a/2[11¯2] → a/3[11¯2] + a/6[11¯2] The leading a/3[11¯ 2] dislocation is able to penetrate into the γ -particle, shearing it to leave a superlattice intrinsic stacking fault bounded by the a/6[11¯ 2] partial which remains at the γ/γ boundary. The trailing dislocation cannot enter the γ phase without forming an anti-phase boundary. Entry would increase the energy on the deforming plane. In order to form the full a[11¯ 2] dislocation, which is able to pass through the γ phase and to restore the unfaulted structure, presumably two further dislocations of the same Burgers vectors are necessary: a/2[10¯ 1] + a/2[01¯1] + a/2[10¯1] + a/2[01¯1] → → a/3[11¯2] + a/6[11¯2] + a/3[11¯2] + a/6[11¯2]

In Fig. 7.9 one can see the early stages of the ribbon nucleation that have been proposed by Rae et al. [38]. Two dislocation loops of Burgers vectors a/2[10¯ 1] and a/2[01¯ 1] have moved in the γ phase on the same (111) plane [see the right lower corner of scheme (b)]. They have combined to form a leading a/3[11¯ 2] dislocation. This slips through γ particles leaving a small circle of the superlattice intrinsic stacking fault surrounded by the trailing a/6[11¯2] dislocation lying at the γ/γ boundary. To complete the ribbons two further dislocations of the same Burgers vector would be needed to thread their way into the narrow channel of the γ phase. Once nucleated the dislocation ribbons move until they are halted by interaction with other dislocations. The ribbon velocity has been estimated to be of the order of 1.2 nm s−1 . This corresponds to a ribbon crossing one γ particle every 340 s. The primary creep is related to a rise in the population of a/2 < 110 > dislocations. A primary creep strain of the order of 5% can be achieved by the movement of ribbons. Dislocations accumulating at the γ/γ interfaces provided effective pinning for the mobile dislocation ribbons and play a role

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Fig. 7.9 The first stage of the nucleation

of a dislocation ribbon. The CMSX-4 single-crystal superalloy tested at 1023K. (a) Electron micrograph. (b) Diagram identifying the dislocations in (a). The

area of the superlattice intrinsic stacking fault (SISF) is shaded. Reprinted from [38] with permission from Elsevier Science Ltd.

in terminating the primary creep. Thus, the primary creep at lower temperatures starts with the propagation of a/2 < 110 > dislocations through the γ channels. The sources of these dislocations supposedly are grow-in dislocations that are located between the dendrites. A primary creep can also occur when a population of a/2 < 112 > dislocations has become established and if the shear stress is sufficiently high. The addition of rhenium and ruthenium and the ageing heat treatment have an important effect on the properties of a new generation of singlecrystal superalloys. Experiments have revealed that as the alloys have become more creep resistant, the propensity for primary creep has increased. Two different types of creep behavior have been observed at 1033K under a stress of 840MPa with the superalloys denoted as MC544, CMSX-10M and MC544 (Fig. 7.10) [34]. The first is characterized by a high amount of primary creep (3–10%). As a result the rupture life is reduced. The primary creep stage is preceded by an incubation period and is followed by a distinct secondary creep stage. The minimum creep rate increases with the primary creep strain. The tertiary stage is short. If the primary stage is limited (0.25 %, MC534 superalloy) then a continued steady state creep leads to relatively long rupture life of the specimen. The increase in the size of the γ particles promotes glide of a/2 < 110 > dislocations within the γ matrix and leads to a decrease in the primary creep amplitude. However, the excessive increase in the size of the γ particles


Fig. 7.10 The results of tests on various single-crystal

superalloys at 1033K and stress 840MPa. Nominal chemical compositions are listed in Table S1. (a) Entire creep curves, (b) enlarged view of the primary creep stage. Reprinted from Ref. [34] with permission from Elsevier Science Ltd.

facilitates the deformation by Orowan bypassing and reduction of the rupture life. The various structure factors are likely to affect the physical mechanism of strain of the investigated superalloys. There exists a competition between the Orowan mechanism of bowing of the dislocations between the γ particles and the cutting of particles by partial dislocations. The difficulty for matrix dislocations in moving between γ precipitates by {111} < 110 > slip results in the short incubation period. It is energetically more favorable for γ dislocations to dissociate to produce a/3 < 112 > partial dislocations, which will cut the γ particles than to by-pass them by the Orowan bowing. The a/2 < 110 > matrix dislocation dissociates creating two partial dislocations (Table 7.1). An a/3 < 112 > partial shears the precipitate by leaving a superlattice stacking fault which stays at the γ/γ interface.

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7.3 Deformation of Single-crystal Superalloys at Higher Temperatures and Lower Stress

Increase in temperature up to 1223K results in a change in the shape of the curves. The tertiary stage follows the short steady-state stage (Fig. 7.11, specimens I G, H and J). The tertiary creep rate depends more upon the applied stress than on the misorientation (Fig. 7.12). It is of interest to compare the shape of curves in Figs. 7.12 and 7.8. The effect of misorientation from the < 001 > direction on creep strain is relatively weak at this temperature. According to the data obtained in [42] the mode of lattice deformation of CMSX-4 superalloy changes from {111} < 112 > at 1023K to {111} < 110 > at 1223K. The authors suggest that the occurrence of {111} < 112 > and {111} < 110 > strain should be modelled as two separate curves which are associated with the hardening and the softening, respectively. All dislocations were found to have Burgers vectors consistent with the type a/2 < 110 >. The authors observed neither any dislocations in the γ phase nor partials separated by the anti-phase boundary. The activity of the {111} < 110 > slip dominates at higher temperatures where the γ matrix is relatively weak for the tension tests close to the < 100 > orientation. The strain occurs on planes with greater Schmid factors. Moreover, slip is confined witin the matrix channels. Mayr et al. [44] studied the dislocation structure of CMSX-6 single crystals at 1298K. The strain rate was studied under a pure shear stress. It is the shear testing that allows a direct loading of a specific macroscopic crys-

Fig. 7.11 Creep curves from the CMSX-4 superalloy single crystals tested at 1223K under stress 185MPa. Orientation of specimens is shown in Fig. 7.7. Reprinted from [42] with permission from Elsevier Science Ltd.

7.3 Deformation at Higher Temperatures

Fig. 7.12 Creep curves from CMSX-4 superalloy single crystals tested at temperature 1223K under various stress for specimens oriented within 10 ◦ of the < 100 > direction. Reprinted from Ref. [42] with permission from Elsevier Science Ltd.

talline slip system. Two macroscopic shear systems, namely the octahedral {111} < 110 > and the cubic {100} < 010 >, were analyzed. The effect of the crystal orientation on the shear strain rate is presented in Fig. 7.13. The obtained curves exhibit initially a decreasing primary creep rate, followed by a creep rate minimum and then a slow increase in the shear rate after the minimum. An incubation period for creep is not observed. The primary creep begins with a relatively high strain rate in both systems. Initially the macroscopic shear system {111} < 110 > deforms by a factor

Fig. 7.13 The shear strain rate as a function of shear strain

in the CMSX6 superalloy. B, The slip system {111} < 110 >; C, the slip system {100} < 010 >. Temperature 1298K, stress 85MPa. Experimental data from Ref. [44].

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of five faster than the system {100} < 010 >. This difference between creep rates increases during deformation. At the minimum creep rate the system {111} < 110 > is deformed by a factor of ten faster than the crystalline system {100} < 010 >. Recall that the repetition factor is 48 for the former slip system and 18 for the other. Furthermore, the {111} type slip plane is the most close-packed plane in the face-centered crystalline lattice. The distance between the nearest parallel {111} planes is maximal in the fcc lattice. Thus, this is an expected result. During the primary creep a considerable part of the dislocations on the {111} < 110 > slip system is associated with long dislocation segments. Their Burgers vector was found to be of the type a/2[0¯11]. After a minimum shear rate the dislocation segments enter the γ interface. A common microstructural feature of both macroscopic crystalline shear systems in the above quoted study is that cutting of γ particles is not observed during primary creep but does occur after the shear rate minimum. This is also typical for ther superalloys in the higher temperature, lower stress regime. Macroscopic shear loading of the CMSX-6 superalloy results in slip in several microscopic systems of type {111} < 110 > and {100} < 100 >. Slip on {111} < 110 > systems is clearly favored in the case of the (111)[01¯ 1] macroscopic system compared with the (001)[100]. In Table 7.3 are listed ratios of the resolved to the applied shear stresses calculated for all microscopic slip systems. The shear creep deformation is revealed to be always associated with the multiple slip. The single slip (which sometimes is intuitively associated with the pure shear deformation) is never observed. Thus, matrix channel deformation always precedes the cutting of the γ particles. The γ/γ -interface represents an obstacle to the motion of channel Tab. 7.3 Ratios of resolved to applied shear stresses in the 12 crystallographic {111} < 110 > slip systems for the two macroscopic crystallographic shear systems (111)[01¯ 1] and (001)[100]. Reprinted from Ref. [44] with permission from Elsevier Science Ltd. Macroscopic system

Number of microscopic slip systems

Ratio of resolved to applied stress

(111)[01¯ 1]

1 2 4 1 4

1.00 0.67 0.50 0.33 0.17

(001)[100]

8 4

0.41 0.00

7.3 Deformation at Higher Temperatures

dislocations. In order to start pairwise cutting a second channel dislocation segment must approach the first segment against the repulsive forces, which the two segments of equal sign exert on each other (see Fig. 6.18). This results in a critical interface stress for cutting. It is important that cutting of γ particles can only start when the resolved shear stress, which acts on two interface dislocation segments, exceeds the critical interface stress. The dominant mechanism of strain is likely to be a function of the applied loading condition. Crystal twin formation occurs in superalloy CMSX-4 under unaxial load [45]. The twinning was observed during the compression, demonstrating a mode distinction from results of the tensile tests. The dislocation-free γ particles are resistant to shearing by dislocations in the temperature range 1073 to 1123K under compressing stresses of 550MPa or lower. This was observed by Pollock and Argon [43] for the CMSX-3 superalloy single crystals. The incubation period lasts for 4h at 1073K, while at 1123K it is reduced to about 10 min. During the incubation stage the grown-in dislocations serve as sources, and deforming dislocations spread from these areas throughout areas that were previously dislocation free. During the primary creep the material continues to be filled by dislocations. All creep deformation is accomplished by movement of the dislocations on {111} planes through matrix channels. The movement of dislocations through matrix channels seems to be a general tendency. It is related to high levels of misfit stresses present in the matrix due to coherent precipitates. When the steady-state creep is reached the matrix is completely filled with dislocations. The multiple slip systems are activated during the steady-state stage. There are very few dislocations left which are not associated with the three-dimensional network. We have discussed the formation of the dislocation sub-boundaries in the matrix in Section 6.1, see Figs. 6.4 and 6.7. It turns out that this phenomenon is intrinsic to single crystals of superalloys as well as to polycrystals. However, in the former case, dislocations fill the narrow channels between dendrites. The structure of cast single crystals is more heterogeneous than that of polycrystals. The spacing of dislocations in the network of stressed specimens was found to be in the range 50–120nm. Dislocations of the network lie in the {111} planes. The networks that consist of a/2 < 110 > slipping dislocations are a common characteristic of the steady-state stage creep. Dislocations of this type have been observed in many electron microscopy studies. During the later stages of steady-state creep the γ particles are sometimes sheared. In the tertiary stage creep strain is accumulated in the matrix and shearing increase steadily. The particles become coarse at the same time. Srinivasan et al. [46] investigated the dislocation cutting of the γ particles in the single-crystal superalloy CMSX-4. Shear creep curves were obtained at

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1293K under shear stress 80MPa for the macroscopic crystal shear system (111)[01¯ 1]. The dislocation density of the produced single crystals is low. The γ channels are filled with dislocations in the early stage of high-temperature deformation of the GMSX-4 superalloy. γ particles are free from dislocations. The increase in dislocation density in the γ channels (the hardening process) is accompanied by the annihilation (the recovery process). Dislocations from two intersecting γ channels move towards the γ corner where they annihilate each other (Fig. 7.14). It is obvious that the climb of dislocation edge components is necessary in order for this process to occur. The cutting of γ particles decreases the dislocation density in the γ channels. The minimum creep rate corresponds to this mechanism. Later the morphological instability of the

Fig. 7.14 Annihilation of dislocations in a cast superalloy

single crystal. (a) Motion of dislocation of opposite signs in the γ channel; (b) motion due to cutting of γ particles. Reprinted from Ref. [46] with permission from Elsevier Science Ltd.

7.4 On the Composition of Superalloys

γ/γ system makes it easy for dislocations to enter the γ channel, and as a result the tertiary stage of creep begins. The transmission microscopy investigations show that two partial γ channel dislocations with different Burgers vectors combine and form an a[010] superdislocation in the γ phase. Two possible scenarios of this process have been proposed: a/2(111)[01¯1] + a/2(1¯11)[011] → a[010]

(7.3)

a/2(111)[01¯1] + a/2(1¯11)[0¯1¯1] → a[00¯1]

(7.4)

or

The separation distance between the individual a/2 < 110 > dislocations is determined from the electron micrograph images to be 2.5nm. The authors [46] say that the a[010] superdislocation has a noncompact core and can only move by combined slip and climb processes. The second noncompact superdislocation is of the type [00¯1]. The movement of both types can be either “self-fed" or externally coupled. In the former case the vacancies produced by one superpartial are adsorbed by the other partner. A coupled movement of these dislocations allows vacancy equilibrium to be maintained everywhere. The vacancy equilibrium can also be maintained by a coupling between a rafting and a climb-controlled γ cutting. The authors believe that the dislocation pairs can only move when a dislocation climb contributes to the process significantly. They estimate the a/2[01¯1] climb velocity to be approximately 4 × 10−8 m s−1 . The dislocation density was assumed to be 1011 m−2 . The secondary creep rate is calculated to be 4.9 × 10−7 s−1 . This agrees satisfactorily with the experimentally observed value of the secondary creep rate. This result implies that the cutting of the γ phase may indeed be a possible rate-controlling step in the high-temperature and low-stress creep of single crystals.

7.4 On the Composition of Superalloys

Some data about the influence of alloying elements on the strength of nickelbased superalloys are available in the literature for a limited number of elements. The rate of solution hardening per at.% of solute, dσ/dc, was measured at 77K. The coefficient dσ/dc in MPa (at.%)−1 is equal to 6.5 for titanium, 15.0 for molybdenum, 43.0 for tantalum and 24.8 for tungsten [34]. It is appropriate to mention briefly the current trends in the development of superalloy compositions (see Table S1).

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7 Single Crystals of Superalloys •

The concentration of titanium has decreased. Some contemporary superalloys do not contain any titanium (EI867, Rene N5, Rene N6, MC534, TMS75, TMS-138). Titanium increases the energy of the anti-phase boundary of the γ phase, which makes the cutting of particles by dislocations more difficult. However, titanium atoms are bigger than aluminum atoms. Therefore they induce an increase in the γ parameter and cause an increase in the γ/γ misfit stresses.

•

The concentration of chromium has decreased over the last decades. Superalloys of the first and second generation contained up to 10–20 wt.% chromium. Superalloys of the third and fourth generations have a lower chromium content 2–3% (CMSX-10, CMSX-10M, TMS-75, TMS-138, Rene N6).

•

Addition of rhenium and ruthenium are under investigation for superalloys which must withstand high stresses at temperatures from 1273 to 1373K (GMSX-4, GMSX-10, TMS-138, Rene N6 and others). The reason for the positive influence of the Re and Ru additions has not been studied sufficiently. It is possible that these elements make the lattice γ/γ misfit more negative (see below).

•

The volume fraction of the γ phase in contemporary superalloys is greater than 50%. The maximum rupture life of single crystals is found with a γ fraction as high as 60 to 75%.

7.5 Rafting

At temperatures above 1273K the γ morphology becomes unstable. Under the combined influence of high temperature and centrifugal stresses the cubic particles are transformed into flat shapes, which are called rafts. At 1423K, for example, the evolution of the γ/γ structure from cuboidal to a plate-like structure takes place at a very early stage of the creep strain. The driving force for rafting has been shown to be proportional to the applied stress, to the γ/γ -misfit and to the difference in their elastic constants [47]. It has also been found that the direction of rafting depends upon the direction of loading and on the sign of the lattice misfit. The rafted structure seems to resist creep strain under low stresses only. It appears that rafting accelerates the creep more often than not. A scheme illustrating the rafting process is shown in Fig. 7.15 [48]. Deformation is different in the horizontal and vertical channels. The misfit strain is usually negative at high temperature. The negative misfit implies that the

7.5 Rafting

Fig. 7.15 Rafting process in a single-

crystal superalloy. (a) Sketch illustrating stresses in γ channels. The squares represent the cubic γ particles. The tensile

stress is applied in a negative misfit alloy. (b) Morphology of the γ phase as a result of rafting. Reprinted from Ref. [48] with permission from Elsevier Science Ltd.

γ channels parallel to the interface must be loaded in compression. At the same time the stress due to misfit can be higher than that due to the applied stress. The total stress results in the change of the cuboid shape. γ precipitates take on a pancake shape with the flat direction is perpendicular to the tension loading direction.

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Fig. 7.16 Electron micrograph showing dislocation networks

in the CMSX-4 superalloy. T = 1150K, σ = 100MPa, rupture after 10.8% strain. Reprinted from Ref. [41] with permission from Elsevier Science Ltd.

Reed et al. [41] studied the deformation of the CMSX-4 superalloy at temperatures above 1273K; the < 001 > ±10◦ oriented crystals were investigated. Examination of specimens by transmission electron microscopy revealed that a well-developed network of dislocations has already been formed after 10 h. With increasing deformation the networks become more regular (Fig. 7.16). The average cell size of the equilibrium network is about 50 nm. Families of six different dislocations form the cells of the network as shown in Fig. 7.17. Dislocation activity decreases and the networks become more sta-

Fig. 7.17 Scheme of the dislocation network seen in Fig. 7.16.

Reprinted from Ref. [41] with permission from Elsevier Science Ltd.

7.5 Rafting

ble with increasing strain. Thus, the morphological instability retards the evolution of creep strain, especially at lower stress levels. This effect is attributed by the authors to the reduction in the number of vertical γ-channels, which is caused by rafting. Stable networks of dislocations are formed and the passage of dislocations from the γ phase in γ particles is hindered. Intersections of the dislocation networks become obstacles for mobile dislocations. Zhang and coworkers [49, 50] have reported new data concerning the creep strength of the fourth generation superalloys. The base superalloy TMS-75 has been developed by increasing the concentration of molybdenum or ruthenium or both. The latter alloy was denoted TMS-138. The alloys were tested at 1373K under a stress of 137MPa. The creep curves obtained by these authors are of the type shown in Fig. 7.5 (c) and have a relative long steady-state stage. The steady-stage creep continues up to 3–4% strain and is followed by the tertiary creep of a very high trajectory. The TMS-138 superalloy which contains 3% Mo, 5% Re and 2% Ru reveals the largest rupture life. This superalloy has significantly improved properties compared with the third generation superalloy (TMS-75). The low creep rate turned out to be due to a fine interfacial dislocation network and a solid solution strengthening. Transmission electron microscopy after creep testing reveals the typical regular dislocation networks (Fig. 7.18). These networks are similar to those observed for the CMSX-4 superalloy and are shown in Fig. 7.16. The CMSX-4 material was crept at 1423K under stress 100MPa, while the TMS-138 superalloy was tested at 1373K under stress 137MPa. Thus, the formation of the fine interfacial networks is not an accidental finding, but is also found for the rafted crept superalloys at temperatures above 1323K. Zhang et al. [49] have measured the mean dislocation spacing in the dislocation networks. The relationship between the minimum creep rate and

Fig. 7.18 Dislocation network after creep rupture in the TMS138 superalloy. Reprinted from Ref. [50] with permission from Elsevier Science Ltd.

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Fig. 7.19 The minimal creep rate versus the interfacial dislocation spacing in networks. The symbols correspond (from left to right) to the superalloys TMS-138, TMS75+3.0Mo, TMS-75, TMS-75+1.6Ru. Experimental data from Ref. [49].

the dislocation spacing is illustrated in Fig. 7.19. The creep strain rate increases almost linearly with the increase in spacing between the interfacial dislocations. Moreover, it is of interest to compare these data with results obtained by ¯ us for pure metals. One can see (Chapter 3, Table 3.4) that the distance λ ranges from 38 to 80 nm for Cu, Fe, Ni, V. Almost the same interval from 40 to 70 nm was found by Zhang et al. [49] for dislocation networks in rafted superalloys. Thus, quite different materials have similar values of spacings between interacting dislocations in networks. This allows us to conclude that the network formation is related to general dislocation properties such as defects of the crystal lattice. We prove in Supplement 2 that when a low-angle sub-boundary consists of two crossed systems of parallel and equidistant dislocations then dislocations of at least one system have a screw component. Zhang et al. [50] have revealed that almost all dislocation segments have screw character or a screw component in creep tested superalloys. They believe that the dense and regular dislocation networks play a key role in creep resistance. The key role of dislocation networks as effective obstacles for the dislocation motion has been discussed already in Section 4.1. The formation of vacancy-emitted jogs is the result of the dislocation intersections and the cause of the slowing of the dislocation motion. The formation of the regular dislocation network on the phase interface is shown in Fig. 7.20 [50]. The tensile stress is applied along [001]. For every {111} slip plane there are three slip directions. However, the directions that

7.5 Rafting

Fig. 7.20 Formation of 60◦ dislocations on the γ/γ

interface. Reprinted from Ref. [50] with permission from Elsevier Science Ltd.

are shown by hollow arrows will not operate due to their unfavorable Schmid factors. The two systems indicated by the bold arrows operate. These screw dislocations move and interact with the γ/γ interface network. These network dislocations may then change their line directions from [101] or [011] to [110] or [1¯ 10] in the interface plane (001). Thus, they become deposited as segments on the interfacial dislocation network. The character of the cell dislocations changes from screw in the γ channels to 60◦ in the γ/γ interface. Thus, a paradoxical result is observed. At higher temperatures the cubic γ particles disintegrate. This results in destruction obstacles to the slip of the deforming dislocations. At the same time dislocation networks in γ/γ surface are created. These networks are able to hinder dislocation motion and to increase the creep strength.

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7.6 Effect of Composition and Temperature on γ/γ Misfit

The γ/γ misfit has a great influence on the creep of superalloys containing a high volume fraction of the γ phase. The misfit can be positive or negative depending on the particular composition of the superalloy. It can be controlled by altering the chemical composition, particularly the aluminum to titanium ratio. Moreover, the misfit is dependent on the heat treatment the alloy is subjected to and it also varies with temperature. The sign of the misfit plays an important role in the evolution of the microstructure of the superalloy. The misfit stresses rise with increasing misfit. This promotes resistance to entering dislocations in γ and improves creep properties. The negative misfit stimulates the formation of rafts of γ , essentially layers of the phase in a direction normal to the applied stress. This can reduce the creep rate if the mechanism involves the penetration of dislocations across the precipitate rafts. The misfit, δ, between the γ and γ phases was found to be 0.02% at room temperature [48]. An in situ neutron diffraction technique was used to measure the lattice parameters of the phases and elastic microscopic strains in a polycrystalline superalloy CM247 LC. The linear expansion factor of the γ phase increases faster than that of the γ phase as the temperature rises. As a result the misfit is negative and its module increases. The dependence of the lattice parameters on temperature is shown in Table 7.4. One can see that the misfit increases with temperature and at 1173K is approximately −0.17%. Other researchers have measured misfits varying from +0.08% to −0.17% at 1173K. Evidently the difference in values is related to the composition of the superalloys under study. The intergranular elastic strains were found to be dependent on the grain orientation. The [001] and [111] grains oriented along tension accumulate the applied load, whereas grains close to [110] orientation shed load during strain. Tab. 7.4 Lattice parameters of the γ and γ phases and value of misfit. Reprinted from Ref. [48] with permission from Elsevier Science Ltd. Temperature, K

aγ , nm

aγ , nm

Misfit, %

293 473 673 873 1173

35.734 35.803 35.908 36.023 36.254

35.740 35.795 35.891 36.005 36.193

0.019 −0.020 −0.047 −0.048 −0.167

7.7 Other Creep Equations

The [110] oriented grains seem to be off-loading part of their load during the creep process. These results are similar to those described in Section 7.1. Ma et al. [48] have estimated the critical resolved shear stress for the γ phase. At the end of the rupture life of a specimen this value was found to be equal to 210 MPa at 1173K. The specimens were tested under stresses from 110 to 425 MPa. The increase in γ/γ mismatch promotes the activity of {111} < 110 > dislocation slip in the matrix between the precipitates. This limits the extent of the primary stage and therefore greatly improves the rupture life of the specimens [34].


Manonukol et al. [37] proposed that the steady-state strain rate is given by τ bld ρνb2 A(P 1/3 − 2) ∆F γ˙ = exp − sinh kT kT P 1/3 l2

(7.5)

where P = 4π/3c, l is the length of the pinned dislocation segment, d the length of the thermal activated event, A the area swept out by a dislocation segment released during thermal activation, ∆F the Helmholz free energy. The increment of free energy ∆G = ∆F − τ v. The activation volume is v = bld. This equation is similar to Eq. (6.13). Following Gibbs [51] the authors consider both forward and backward activation events. Therefore the equation contains the sinh-function instead of the exponent-function. Many authors assume that the creep strain rate is directly proportional to the accumulated creep strain. This assumption is used for quantitative interpretation of the creep data. In fact formulas of this kind are empirical and one needs to choose parameters to fit calculated curves and experimental results. Reed et al. [41] consider a quantitative interpretation of the creep data like those shown in Fig. 7.5 (b). The strain rate is assumed to be proportional to the accumulated creep strain. A suitable empirical relationship appears to be ε˙ = (Γ˙ + Ω ε)

(7.6)

where Γ˙ is a constant that is a measure of the initial creep rate, Ω is assumed to be related to the dislocation multiplication rate and is considered as a softening coefficient. Both of these parameters in Eq. (7.6) are dependent on temperature T and applied stress σ and have to be derived from experimental

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curves:

Q ˙ Γ , Ω = a exp bσ − RT

(7.7)

where a, b, Q are constants. It is clear that they are different for Γ˙ and for Ω . R is the gas constant. The constants were calculated by the authors. The values of the minimum creep strain rate are plotted against applied stress in Fig. 7.21. At temperatures 1173 and 1273K the predicted curves correspond satisfactorily to the experimental creep data. However, above 1273K the model does not represent the data. They modified Eq. (7.6) by introducing an attrition coefficient φ which accounts for the blocking of a/2{111} < 110 > dislocations by the rafted structure and the gradual formation of the equilibrium network: ε˙ = (Γ˙ + Ω ε) exp(−φε)

(7.8)

For the sake of simplicity they assumed that φ is a constant value. Equation (7.8) was used to estimate the attrition coefficient φ and to fit calculated curves to creep data.

Fig. 7.21 The minimum strain rate of creep in the CMSX-4 superalloy plotted against the applied stress. The calculated constants are shown. The solid lines represent a fit to the experimental data

between 1123 and 1273K. The data above 1273K are not satisfactorily represented by extrapolation. Reprinted from Ref. [41] with permission from Elsevier Science Ltd.


From fundamental considerations Pollock and Argon [43] take the form of the solid-solution-resistant-governed velocity of dislocations to be Q − (τ − τOr )v V = V0 exp − (7.9) kT where V0 is a pre-exponential factor, Q is the asymptotic activation free energy for dislocation motion at vanishing stress, v is an activation volume, τ is the resolved shear stress on the {111} plane in the active < 110 > direction for the specific slip system, τOr is the Orowan resistance of the narrow γ channels. The resolved shear stress is assumed to be equal to √ τ = (σ3a + σ3t − σ1t )(1/ 6)

(7.10)

where σ3a is the applied tensile stress, σ3t is the tensile misfit stress acting across the horizontal γ channel, −σ1t is the biaxial compressive misfit stress √ acting in the plane of the γ channel, 1/ 6 is the appropriate Schmid factor for the active {111} < 110 > slip system. Their article [43] is of interest because they have evaluated physical values in equations starting from experimental data. The stereo pair of dislocation structure on electron micrographs have been analyzed in detail. They consider the incubation time, ti , as a period of percolation of dislocation loops between regions of ingrown dislocations at an average spacing of Λ in the volume. They introduce a multiplying factor, β, accounting for the need for dislocation motion along low stressed vertical γ channels and arrive at Λ Q − (τ − τOr )v = βV0 exp − (7.11) ti kT Incubation times for initiation of strain are given in Table 7.5. In this table σ is the applied tensile stress, τm is total resolved misfit stress, τ is the total resolved stress, ti is incubation time. The distance Λ was estimated from electron micrographs to be about 50µm. From the incubation data given in Table 7.5 the authors have determined that Q = 10.9 × 10−19 J at.−1 Tab. 7.5 Incubation times of creep and related driving stress [43]. T, K

σ, MPa

τm , MPa

τ , MPa

ti , s

1073 1123 1123

552 450 552

178 189 189

403 373 415

1.39 × 104 4.50 × 103 6.00 × 102

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140


v = 7.44 × 10−28 m3 Other constants were found to be V0 = 1.64×1019 m s−1 and β = 5.24×10−2 . The authors recognize fairly that the calculated value of the activation energy, Q, is too high. They believe that while the dependence given by Eq. (7.9) is “mechanistically preferable", it is also possible to state it in power law form as m (τ − τ0 )v Q V = V0 exp − (7.12) kT kT The power law exponent m is found to be 11.97. Nevertheless, from the present author’s point of view the exponential dependence of the strain rate on stress is physically well founded. The shear stress which acts on a dislocation segment in a given slip system depends on time t and on the position and orientation of this segment in the γ/γ microstructure. The shear stress can be represented as [44] τ (t) = KSchm τappl + τcoh (t) + τint (t) + τint ,

(7.13)

where KSchm is the Schmidt factor, τappl is the applied stress, τcoh is a coherency stress term, τint represents an interface stress term. The velocity of a dislocation segment depends on τ : V (t) = ϕ(τ )

(7.14)

The dependence of the dislocation velocity on stress Eq. (7.14) written in an explicit form is exponential, see Eqs. (6.10), (6.12). Mayr et al. [44] express the overall shear rate in the general form: m

γ(t) ˙ =

n

bl gj Vi (t) Q j=1 i=1

(7.15)

where Vi is the velocity of the ith dislocation segment, n is the common number of segments, gj the number of activated slip systems, which contribute to shear displacement, m the common number of slip planes, l the mean segment length, Q the crystal volume under consideration. Some authors assume that the function ϕ(τ ) in Eq. (7.14) can be represented by a simple power law, such as V (t) = V0 [τ (t)]n

(7.16)

7.8 Summary

However, it follows from experimental data that the power exponent n is not a constant. The physical meaning of Eq. (7.16) seems difficult to substantiate.

7.8 Summary

Single-crystal nickel-based superalloys have been developed for gas turbine applications. The cast single crystals consist of a high volume fraction of the coherent γ phase separated by thin channels of the γ matrix. The hightemperature strain behavior of the single-crystal superalloys is strongly dependent on their composition, their crystal orientation, as well as on temperature and applied stress. Single crystals of < 001 > and close directions are utilized for gas-turbine blades. The crystal shear system {111} < 110 > operates most frequently at temperatures in the vicinity of 1023K. The other systems {111} < 11¯2 > and {001} < 110 > appear after a lattice rotation. If the initial orientation is moved from < 001 > the creep strength decreases but afterwards begins to grow towards < 111 > orientation. However the plasticity decreases. It is appropriate to consider two temperature regions of single-crystal superalloy application, 1023–1123K and 1223–1373K. At lower temperatures the single-crystal superalloys reveal the usual creep curves. The properties are strongly dependent on orientation. In a typical CMSX-4 superalloy ribbons of dislocations with Burgers vectors a/6 < 112 > are observed. Two dislocations, both slipping in the γ phase on the {111} plane, combine to produce the leading half of the ribbon. The dislocation reactions take place before the dislocations are able to penetrate into the γ phase. The trailing dislocation cannot enter the γ phase without forming an anti-phase boundary. The activation of {111} < 112 > systems plays a major role in the overall creep response of the material, particularly in regard to the regime of primary creep. A very rapid tertiary stage of creep occurs at higher temperatures. The effect of misorientation from the < 001 > direction is relatively weak. The lattice deformation changes from {111} < 112 > at 1023K to {111} < 110 > at 1223K. The matrix channel deformation always precedes the cutting of the γ particles. Dislocations fill narrow channels between dendrites. Cutting of γ particles decreases the dislocation density in the γ channels during the steady-state creep. Two partial γ channel dislocations with different Burgers vectors may combine and form a superdislocation in the γ phase. Shear creep deformation of single-crystal superalloys is always associated with multiple slip. A common microstructural feature of the macroscopic

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crystal shear systems is that cutting of γ -particles is not observed during primary creep but does occur after the shear rate minimum has been reached. At temperatures above 1273K, under the influence of high temperature and centrifugal stress, the cubic particles are transformed into flat shapes called rafts. It appears that the rafting accelerates the creep in most operational conditions. However, the opposite data have been reported for superalloys of the fourth generation containing Re and Ru. The fine interfacial dislocation networks that are formed at this temperature are able to improve the creep strength. The γ/γ misfit influences on the creep strain. Their value was measured to be about −0.17% at 1173K. The current trends in the development of superalloy compositions are as follows. The volume fraction of γ is increased up to 75%. The concentration of titanium is decreased with some contemporary superalloys containing no titanium. The concentration of chromium is decreased to 2–3%. Additions of rhenium and ruthenium are promising for increasing the operation temperatures.

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8

High-temperature Deformation of Some Refractory Metals

8.1 The Creep Behavior

Refractory metals and alloys attract the attention of investigators because of their remarkable properties and on account of promising practical prospects. They are characterized by their extremely high melting points, which range well above those of iron and nickel. If refractory metals are considered to be those metals melting at temperatures above 2123 K then twelve metals constitute this group: tungsten (melting point 3683 K), rhenium, osmium, tantalum, molybdenum, iridium, niobium, ruthenium, hafnium, zirconium, vanadium, and chromium. The physical properties of refractory metals, such as molybdenum, tantalum and tungsten, i.e. their strength and high-temperature stability, make them suitable materials for hot metalworking applications and for vacuum furnace technology. Many special applications exploit these properties: for example, tungsten lamp filaments operate at temperatures up to 3073 K, and molybdenum furnace windings withstand temperatures up to 2273 K. However, a poor low-temperature fabricability and an extreme oxidizability at high temperatures are shortcomings of most refractory metals. Interactions with the environment can significantly influence their high-temperature creep strength; application of these metals requires a protective atmosphere or a coating. The refractory metal alloys of molybdenum, niobium, tantalum, and tungsten have been applied for the space nuclear power systems. These systems were designed to operate at temperatures from 1350 K to approximately 1900 K. The environment must not interact with the material in question. Liquid alkali metals are used as heat transfer fluids as well as ultrahigh vacuum. The high-temperature creep strain of alloys should not exceed 1–2%. An additional complication in studying the creep behavior of the refractory metals High Temperature Strain of Metals and Alloys, Valim Levitin (Author) c 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim Copyright ISBN: 3-527-313389-9

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8 Deformation of Some Refractory Metals

is interactions with the environment, which can significantly influence the creep behavior. Figures 8.1 and 8.2 show the results of our mass tests of niobium at temperatures from 0.50 Tm to 0.65 Tm . We use testing machines with constant strain rate tensions, ε. ˙ The vacuum is of the order of 4 × 10−4 Pa (3 × 10−6 torr). At every strain rate three specimens are tested. The average data are presented. The accuracy of the measurements is from 5 to 10%. As one can see the obtained curves log ε˙ − σ are almost straight lines. Thus, the exponential dependence (1.2) holds in the investigated temperature interval. The slope of the curves increases when the test temperature increases from 1373 to 1573 K. In Fig. 8.3 the deformation map for niobium is presented [26]. The map shows strain mechanisms in niobium with a grain size of about 100µm. The data obtained by us lie in the area denoted by the authors as the “powerlaw creep”. The values of creep rates and stresses that have been measured (Fig. 8.2) correspond satisfactorily to the curves of Fig. 8.3. The activation energy of strain was calculated according to Eq. (1.4). The data are presented in Table 8.1. The mean value of the activation energy

Fig. 8.1 The logarithm of strain rate versus stress in niobium. Test temperatures are from 0.50 to 0.65 Tm : B, 1273; C, 1373; D, 1573; E, 1673; F, 1773 K.


Fig. 8.2 The logarithm of strain rate versus the inverse absolute temperature for niobium. The applied stress,MPa, is equal to: B, 1.96; C, 4.92; D, 9.81; E, 19.42.

Fig. 8.3 The strain rate map for niobium with a grain size of 100µm. Reprinted from Ref. [26].

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Tab. 8.1 The measured activation energy of high-temperature strain for niobium. σ/µ, 10−4 Q, 10−19 Jat.−1

5.2 8.1

7.9 7.2

10.5 6.7

13.1 7.9

of high-temperature strain for niobium is found to be Q = (7.5 ± 0.6) × 10−19 Jat.−1 . The activation energy of self-diffusion in niobium is reported to be from 6.66 × 10−19 to 7.99 × 10−19 Jat.−1 at temperatures from 1151 to 2668 K [22]. Frost and Ashby [26] cite a value of 6.66 × 10−19 Jat.−1 for self-diffusion in niobium. The activation volume was calculated from the data shown in Fig. 8.1. The data are presented in Table 8.2: v is the activation volume, ¯l the average length of an activated dislocation segment calculated as l = v/b2 , λ the mean spacing between sub-boundary dislocations, z0 the mean spacing between jogs in mobile dislocations (see Section 3.5). The activation volume is of the order of 10−27 m3 . There is a boundary between 0.50 Tm and 0.54 Tm , where the values of l and z0 increase half as much. It is likely the increase in temperature causes instability of subboundaries. We have shown some typical structures of the tested niobium in Figs. 3.9, 3.14, 3.17(c). Tab. 8.2 The activation parameters of high-temperature strain for niobium. T, K

1273

1373

1473

1573

1673

v, 10−27 m3 l, nm λ, nm z0 , nm

4.39 53.4 68.1 74.0

4.60 56.2 90.2 100.5

7.68 93.8 – –

7.30 89.2 102.1 109.7

7.34 89.7 105.9 119.5

The evolution of structural parameters is presented in Fig. 8.4. The subgrain size and subgrain misorientations change during the primary stage of deformation. It follows from the experimental data that the rate-controlling mechanism in niobium is the slip of deforming dislocations with one-signed jogs (see Section 4.1). We have also tested specimens of molybdenum in the temperature range 1973 to 2773 K (the relation T / Tm varies from 0.68 to 0.96). The applied stress ranged from 7.3 × 10−5 µ to 3.7 × 10−4 µ. The dependences of strain rate on stress and temperature for molybdenum are presented in Fig. 8.5. The exponential dependence (1.2) describes the strain rate satisfactorily.


Fig. 8.4 Dependence of strain and structural parameters in niobium on time at T = 1645 K (0.60 Tm ) and σ = 9.8MPa (2.6 × 10−4 µ).

In Fig. 8.6 the dependences of log ε˙ on the inverse temperature, 1/T , are shown. From this data the values of the activation energy of strain were calculated according to Eq. (1.4). The energy was found to be dependent on the applied stress (Fig. 8.7). At stresses from 30 to 50MPa we obtain Q = (5.59 ± 0.35) × 10−19 Jat.−1 . This value is less than that for crept molybdenum in Fig. 1.2, where Qc = 6.81 × 10−19 Jat.−1 On the other hand, the activation energy of self-diffusion in molybdenum is reported to be from 6.41 × 10−19 to 7.64 × 10−19 Jat.−1 at temperatures from 1873 to 2813 K [22]. Frost and Ashby [26] reported the activation energy of self-diffusion in molybdenum as 6.73 × 10−19 Jat.−1 . In Fig. 8.8 the deformation map for molybdenum is shown [26]. The map presents areas of the different strain mechanisms in molybdenum. According to Frost and Ashby the power exponent n is equal to 4.85 in the power-law creep area. The values of the creep rates that have been obtained by us (Fig. 8.5) correspond satisfactorily to the curves of Fig. 8.8. Moiseeva and Pishchak [52] described investigations of polycrystalline molybdenum crept at lower temperatures, namely from 1303 to 1973 K (from 0.45 Tm to 0.68 Tm ). The steady-state creep rate as a function of applied stress

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Fig. 8.5 The logarithm of strain rate versus stress in molybdenum. The testing temperatures are B, 1973; C, 2173; D, 2373; E, 2573; F, 2773 K (from 0.68 to 0.96 Tm ).

is presented in Fig. 8.9. The authors plot experimental data in log ε˙ − log σ coordinates. Two segments of straight lines are observed at every test temperature. At low stresses the creep rate is directly proportional to stress, so that the factor in the power-law equation (1.1), n = 1. At higher stresses factor n increases abruptly to 8–9. The slope of the curves changes at critical stresses, σcr , 10, 20, and 70MPa at 1973, 1633 and 1033 K, respectively. No creep strain with n = 1 was observed earlier. In Ref. [26] the authors call it a high-temperature power-like creep. The primary stage covers nearly 80% of the strain to rupture. The polished surface of specimens exhibits slip bands, thus, the slip of dislocations occurs. The creep mechanism of molybdenum, which has a body-centered crystal lattice, differs from the creep in metals that have a face-centered crystal lattice. Unlike face-centered metals the minimum creep rate of molybdenum depends weakly on temperature. The critical creep rates in the studied temperature range (changes of flection coordinates) are from 2.2 × 10−8 to 1.3×10−8 s−1 . The only visible effect is a shift of the function log ε˙ = f (log σ) to higher stresses. Molybdenum specimens have a random dislocation distribution after creep in the range n = 1. The ordered dislocation sub-boundaries cannot be formed

8.2 Alloys of Refractory Metals

Fig. 8.6 The logarithm of strain rate versus the inverse abso-

lute temperature for molybdenum. The applied stress,MPa, is equal to: B, 9.81; C, 19.62; D, 29.43; E, 39.24; F, 49.05.

under n = 1 conditions. Only some grains close to the break stress between segments n = 1 and n = 8 have sub-boundaries, obviously due to a local overstress. At n = 8 one can observe well formed sub-boundaries. With increasing stress the dislocation density in both sub-boundaries and subgrains increases, and the structure eventually becomes cellular.


A review of the creep behavior of refractory metal alloys has been published [53]. A so called Larsom-Miller parameter, P , is widely used to estimate the creep strength of alloys: P = T [15 + log t1% ]103

(8.1)

where T is temperature of the tests, t1% is the time of 1% deformation of a specimen, 15 is an empirically determined value.

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150


Fig. 8.7 The measured activation energy of high-temperature strain

for molybdenum.

Fig. 8.8 The strain rate map for molybdenum with a grain

size of 100µm. Reprinted from Ref. [26].


Fig. 8.9 The effect of stress on the steady-state creep in molybdenum at temperatures: 1, 1973; 2, 1633; 3, 1303 K. Experimental data from Ref. [52].

One can see in Fig. 8.10 the typical creep behavior of the refractory metal alloys. The corresponding nominal compositions of alloys are presented in Table 8.3. Molybdenum-, niobium- and tantalum-based alloys have been developed, studied and utilized. The creep properties of the refractory alloys are very sensitive to composition, structural features, and test environment. Small quantities of interstitial atoms such as C, O and N may also have an important effect on the properties. Moreover, additional factors are possible, such as even the geographic location from which the metal ore was obtained and technological features during the production process. Other factors affecting creep behavior include grain size, which can be attributed to the annealing temperature (Fig. 8.11). Many studies have been devoted to the search for potential strengtheners of refractory metals. Incoherent or semi-coherent particles have been the most commonly investigated. These precipitates are based on carbides. Hafnium Tab. 8.3 Nominal composition of some refractory alloys. Data from [53]. Curve

Alloy

Mo

Ti

Zr

Nb

Ta

W

Hf

Re

C

1 2 3 4 5

Mo-TZM Nb-1Zr PWC-11 T-111 ASTAR-811C

bal. – – – –

1.0 – – – –

0.75 1.0 1.0 – –

– bal. bal – –

– – – bal bal.

– – – 8.0 8.0

– – – 2.0 0.7

– – – – 1.0

– – 0.10 – 0.025

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152


Fig. 8.10 Applied stress to produce 1% creep strain in some refractory alloys. Composition of alloys given in Table 8.3. Reprinted from Ref. [53] with permission from Elsevier Science Ltd.

carbide possesses the highest melting point. Tungsten–rhenium–hafnium carbide alloys seem to be promising for operation at high temperatures.

Fig. 8.11 Effect of annealing temperature on applied stress

to produce 1% creep in ASTAR-81C alloy. 1, annealed at 1923 K; 2, annealed at 2273 K. Reprinted from Ref. [53].


Fig. 8.12 Logarithm steady-stage creep rate versus the logarithm stress for W–4Re–0.32HfC alloy. The testing temperatures are B, 2200; C, 2300; D, 2400 K. Experimental data from Ref. [54].

Park [54] compares some creep models with the experimental data on the creep behavior of W–4Re–0.32HfC alloy. He obtained strain–time creep curves of the tested alloy at 2200 K. Three regions of a creep curve are normally observed: primary, secondary and tertiary strain. The secondary creep rate is assumed by the author to be expressed as ε˙ ∼ σ n [see Eq. (1.1)]. Three straight parallel lines were obtained from this log ε˙ − log σ plot, Fig. 8.12, implying that the secondary creep rate and the applied stress have a power-law relationship. The value of n was obtained from the slope of each straight line, and a least-squares analysis yielded n = 5.2. Three creep models for secondphase particle-strengthened alloys were applied to the creep behavior of the alloy in this research. Park [54] studied the Ansell-Weertman, the Langeborg, and the Roesler-Arzt models (the reader can find references in the quoted article). The conclusion was as follows: “The results showed that none of these models predicted the creep behavior of the alloy”. Some models predicted the secondary creep rate approximately five orders of magnitude different from the value obtained experimentally. However, the same experimental data satisfy another dependence, for example, an exponential one. In Fig. 8.13 the same strain rates are plotted as log ε−σ. ˙ We also obtain straight lines which imply the dependence ε˙ ∼ exp σ. We have noted (Chapter 1) that a functional dependence only makes it not possible to conclude unequivocally about a physical mechanism of strain. The orientation relationship between a matrix structure and a precipitate structure have a dramatic effect on the creep deformation. The preferred

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Fig. 8.13 Logarithm steady-stage creep rate versus the stress for W–4Re–0.32HfC alloy. The same experimental data as in Fig. 8.12 from Ref. [54] are used.

orientation relationships between coherent and semi-coherent precipitates and matrix may result in an improved resistance against slip of deforming dislocations. A niobium–titanium-based alloy has been investigated by Allamen et al. [55]. The alloy under study contains 44Nb–35Ti–6Al–5Cr–8V–1W–0.5Mo– 0.3Hf. The microstructure of extruded and recrystallized material consists of a solid solution and of particles of titanium carbide, TiC. The particle sizes are between 200 and 500 nm. Creep curves were obtained at 977 K. At relatively low stress, 103MPa, the slipping dislocations were attracted to TiC particles. The attraction is energetically favored when the modulus mismatch between the phases is decreased by diffusion. In contrast, a higher density of dislocations is observed at the higher stress 172MPa, along with bowed dislocations that are pinned by carbide particles. The lattice periodicity in the [200]-type direction of the cubic body centered matrix is about 0.33 nm. On the other hand, for the [220]-type direction of the cubic face-centered precipitate, the lattice periodicity is about 0.32 nm. The misfit is about 3%. This may explain why these two directions are nearly parallel at the precipitate/matrix interface. A specific orientation relationship, namely: [100](110) matrix parallel to [220](111) precipitates, was observed in the specimens subjected to the highest stress level. The development of superalloys for operation at temperatures up to 2073 K continues. New classes of alloys attract investigators and engineers. Refractory superalloys based on the platinum group metals have a cubic face centered crystal lattice, high melting temperature, and a coherent two-phase structure.

8.3 Summary

A two-phase iridium-based refractory superalloy has been proposed recently [56]. The alloy is strengthened by a coherent phase of L12 type. This structure is similar to that of nickel-based superalloys. The authors investigated the strength behavior and the structure of some binary iridium-based alloys. The systems Ir–Nb and Ir–Zr are found to be the most promising alloys for study at temperatures up to 1473 K. The rupture life of Ir–Nb alloys was found to be increased dramatically by the addition of nickel. The strengthening phase was determined to be (Ni, Ir)3 Nb. The steady-state creep rate at 1923 K for the Ir–15Nb–1Ni alloy was 1.2 × 10−8 s−1 , about three orders of magnitude lower than that of the binary Ir–17Nb alloy (10−5 s−1 ). This shows that the iridium-based alloys may possibly be regarded as ultrahigh temperature materials. However there is a lot of work ahead before new alloys of this type can be used practically.

8.3 Summary

The physical properties of refractory metals are related to their high melting points. They look very promising from the practical point of view. The most refractory metals have, however, drawbacks such as poor low-temperature fabricability and an extreme high-temperature oxidizability. When used they need a protective atmosphere or a coating. The minimum strain rate of niobium and molybdenum is dependent exponentially on the applied stress at high temperatures. The mean value of the activation energy of the high-temperature strain for niobium is found to be Q = (7.5 ± 0.6) × 10−19 Jat.−1 , for molybdenum Q = (5.59 ± 0.35) × 10−19 Jat.−1 . It follows from the experimental data that the rate-controlling mechanism of strain for niobium is the slip of deforming dislocations with one-signed jogs. Molybdenum-, niobium- and tantalum-based alloys have been developed. These alloys are able to operate at temperatures up to 1900 K. The creep properties of the refractory alloys are very sensitive to composition, structural features, and test environment. Other factors have yet to be studied in any detail. The alloys of the systems Ir–Nb and Ir–Zr are found to be the promising for future study.

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Supplements Supplement 1: On Dislocations in the Crystal Lattice

The concept of dislocations is known to be important in the theory of strength and plasticity [18, 20, 21]. Let us recall the main theses of the theory of dislocations. A crystal lattice is not ideal. The arrangement of atoms differs from a regular order. This is the immediate cause of the great discrepancy between the theoretical strength of materials and the measured values. The practical strength is about three orders less than the strength that would follow from the concept of a regular atomic lattice. Any crystal lattice contains defects, i.e. there are areas where the structure is irregular. The point is that atoms on a slip plane do not displace simultaneously under the effect of the applied stress. The atomic bonds do not break all at the same time. The dislocation lines move along slip planes. A dislocation is a one-dimensional defect. This means that the dislocation extent is compared with the crystal size in only one dimension. In the two other dimensions the dislocation has the extents of the interatomic order. The crystal lattice is disturbed along the dislocation line. So the dislocation is the line defect in the crystal lattice. It is like a stretched string. There are two vectors, which determine the dislocation line at any point. The Burgers vector is denoted by b. The dislocation line vector is denoted by ξ. The unit vector ξ is directed along the tangent to the dislocation line at every point. It may be directed in a different way at different points of the same dislocation line. The Burgers vector b is related to the atomic displacements, which the dislocation causes in the crystal lattice. The Burgers vector is the same along a given dislocation, i.e. it does not change with the coordinates. The magnitude of the Burgers vector is the interatomic distance b. It is a measure of deformation associated with the dislocation. The Burgers vector is always directed along a close-packed crystallographic direction. This provides


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Supplements

Fig. S1 Motion of the edge dislocation (⊥) in a crystal lattice

under the effect of shear stress.

the smallest value of b and, therefore, the lowest energy per unit length of dislocation. Dislocations move under the influence of external forces, which cause an internal stress in a crystal slip plane. The force per unit length of dislocation, F , exerted on the dislocation by the shear stress τ is F = bτ. The area swept by the dislocation movement defines a slip plane, which always (by definition) contains the vector ξ. In Fig. S1 the edge dislocation formation and its movement is shown. Figure S1(a) demonstrates the generation of an edge dislocation by a shear stress, dislocation is denoted as ⊥. In Fig.S1(b) movement of the dislocation through the crystal occurs and an extra-plane appears above the slip plane. The shift of the upper half of the crystal takes place after the dislocation emerges from the crystal (Fig.S1(c)). The relative displacement of the two crystal halves is normal to the dislocation. The Burgers vector of the edge dislocation is perpendicular to the line vector, so the scalar product =0 (b · ξ) The edge dislocation can change its slip plane by means of a climb process. In this connection completion of the extra-plane occurs. A diffusion flow of vacancies or interstitial atoms is needed for the climb of the edge dislocation. The climb is a slower process than the slip. In Fig. S2 screw dislocation is shown, for screw dislocation vector b is parallel to vector ξ: =b (b · ξ) All dislocations have a character that is either pure edge, pure screw or a combination of the two. In fact a dislocation is a boundary of a slip area. It separates the area where the slip has occurred from the area where the slip has not yet occurred. Dislocation lines may be arbitrarily curved. In Fig. S3 the arrangement of atoms in a mixed dislocation is shown. Atoms denoted

Supplement 1: On Dislocations in the Crystal Lattice

Fig. S2 A screw dislocation in the crystal lattice.

by large circles are situated over the diagram plane; those denoted by small circles are situated under the diagram plane. We observe a transfer from the pure screw to the pure edge dislocation. In the general case one may consider the edge and screw components of the mixed dislocation. In reality dislocation lines can have any shape, they can form loops and networks and they can contain jogs, nodes, junctions, kinks. The dislocation possesses an energy. The total energy per unit length is the sum of the energy contained in the elastic field and the energy in the dislocation core. The self-energy per unit length of dislocation, Eel , depends upon the magnitude of the Burgers vector and the shear modulus of the material, µ, as Eel ≈ µb2 . The atoms nearest to the dislocation core are displaced most from their equilibrium positions and therefore they have the highest energy. In order to minimize this dislocation self-energy, the dislocation tries to be as short as possible. That is, a dislocation prefers to minimize its length rather than

Fig. S3 A mixed dislocation

in a crystal lattice.

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160

Supplements

meander through the crystal. This tendency to shorten itself, gives rise to the concept of a dislocation line tension. Inherent properties of dislocations are mobility and multiplication. Dislocations move easily in their slip plane. The stress that a dislocation needs to begin to move is of the order of 10−4 µ, where µ is the shear modulus. The velocity of dislocations is related to the applied stress and temperature. Dislocations can multiply under the effect of external stress. The quantity of dislocations in a crystal is measured by the dislocation density: ρ = N/S, where N is the number of dislocations which intersect the area S. The strain of a crystal is given by equation ε = bρL, where L is the length of the crystal. A dislocation may dissociate into two so-called partial dislocations. A reason for this dissociation is a gain in energy. Instead of a pure one dimensional defect, the perfect dislocation, some kind of ribbon stretching through the crystal is formed. This stacking fault ribbon may be constricted at some knots or jogs. It is clear that a dislocation dissociated into two partials is able to slip on the same plane as the perfect dislocation; the stacking fault just moves along. It can also change its length. In a face-centered crystal lattice the deformation occurs usually as the dislocation slip in the crystal plane of type {111} in the < 110 > direction. The Burgers vector is e.g. b = a/2[110]. The dissociation happens according to the reaction a a a [110] = [121] + [21¯1] (S.1) 2 6 6 Two Shockley dislocations are formed that can slip in the same plane {1¯11}. The sum of Burgers vectors of two partial dislocations must be equal to the Burgers vector of the complete dislocation: b1 + b2 = b. One must expect that there is an equilibrium distance d, which gives a minimum energy for the split dislocation and the stacking fault. This equilibrium distance depends mostly on the stacking fault energy γ. The smaller γ the larger distance d between the partial dislocations; d is equal to four interatomic distances in nickel and ten interatomic distances in copper. The dissociation into partial dislocations hinders the climb of the dislocation into parallel slip planes. When a dislocation travels past two precipitates that are sufficiently far apart, there is resistance set up to hinder the movement. More energy has to be provided to move the dislocation past the barriers. Hence, the precipitates strengthen the material via this mechanism. The evidence for the Orowan mechanism lies in the residual dislocations that are often deposited around the precipitates.

Supplement 2: On Screw Components in Sub-boundary Dislocation Networks

If a precipitate is sufficiently hard, it cannot be readily sheared by a dislocation. In these cases, the dislocation will sometimes bow around the particle. The applied stress exerts a force on the dislocation causing it to move. Points along the dislocation are pinned by strong precipitates that are resistant to dislocation penetration and shearing. The dislocation is able to bow out between the precipitates, but the bowing process is resisted by the dislocation tension. (Remember, the bowing of the dislocation creates more dislocation line and increases the energy of the system). The dislocation is able to continue slipping and a dislocation loop is left behind, circling the precipitate. This process of circumventing a particle is called Orowan looping or dislocation bypass.

Supplement 2: On Screw Components in Sub-boundary Dislocation Networks

Prove the following theorem. Let us suppose that a low-angle dislocation subboundary consists of two crossed but not perpendicular systems; within each of the networks the dislocations are parallel and equidistant. The theorem states that in these conditions dislocations at least of one system have a screw component. The notations are illustrated in Fig. S4. 1 and 2 are planes, in which are located the systems under consideration. ξ1 and ξ2 are the unit vectors of dislocation lines. b1 and b2 are Burgers vectors. e1 = (b1 × ξ1 )/|b1 |; e2 = (b2 × ξ2 )/|b2 | are unit vectors of perpendiculars 1 and N 1 are inverse vectors, they lie in the boundary plane to slip planes. N and are perpendicular to dislocation lines. By definition i = Ni (n × ξi ) N

(S.2)

where i = 1, 2; Ni = 1/ηλi . n = (ξ1 × ξ2 ) is the unit vector of the perpendicular to the sub-boundary plane. It is known in the theory of low-angle sub-boundaries [18] that 1 = b2 − n(n · b2 ) N |b1 × b2 |

(S.3)

2 = −b1 − n(n · b1 ) N |b1 × b2 |

(S.4)

and

1 and ξ2 as scalars and taking into account Eq. (S.2) Multiplying vectors N we obtain (b2 · ξ2 ) − (n · ξ2 )(n · b2 ) N1 · [(n × ξ1 ) · ξ2 ] = (S.5) |b1 × b2 |

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Supplements

Fig. S4 The dislocation sub-boundary that consists of two crossed systems of parallel equidistant dislocations.

2 and ξ1 as scalars we obtain Similarly multiplying vectors N (−b1 · ξ1 ) + (n · ξ1 )(n · b1 ) N2 · [(n × ξ2 ) · ξ1 ] = |b1 × b2 |

(S.6)

However (n × ξ1 ) · ξ2 = (ξ1 × ξ2 ) · n = n · n = 1 ; n · ξ2 = 0 and (n × ξ2 ) · ξ1 = (ξ2 × ξ1 ) · n = −n · n = −1 ; n · ξ1 = 0 Consequently we have N1 =

(b2 · ξ2 ) (b1 · ξ1 ) ; N2 = |b1 × b2 | |b1 × b2 |

(S.7)

vary from 0 (for an edge dislocation) to ±b (for a screw The values of (b · ξ) dislocation). According to the theorem condition it is impossible for N1 = 0 and N2 = 0 at the same time. One can see from Eq. (S.7) that at least one scalar multiplication is not equal to zero. This means that at least one system contains a screw component.

Supplement 3: Composition of Superalloys

Supplement 3: Composition of Superalloys

In Table S1 the nominal contents of the main alloying elements are presented. Typical third generation alloys include CMSX-4, EI867, ZhS26VI, TMS-75, Rene N6, and the fourth generation include CMSX-10M, TMS-138. Tab. S1 Nominal chemical composition (wt.%) of some nickel base superalloys. Alloy

Cr

Al

Ti

Mo

W

Co

Ta

Nb

Re

Others

Ref.

AM1 AM3 RWA 1480 Rene N4 SRR99 AF56 RR2000

8.0 8.0 10.0 9.0 8.0 12.0 10.0

5.2 6.0 5.0 3.7 5.5 3.4 5.5

1.2 2.0 1.5 4.2 2.2 4.2 4.0

2.0 2.0 4.0 2.0 – 2.0 3.0

6.0 5.0 – 6.0 10.0 4.0 –

6.0 6.0 5.0 8.0 5.0 8.0 –

9.0 4.0 12.0 4.0 3.0 5.0 –

– – – 0.5 – – –

– – – – – – –

– – – – – – 1.0V

57

C263

20.0

0.5

2.2

5.8

–

20.0

–

–

–

0.7Fe

58

CMSX-2

8.0

5.6

1.0

0.6

8.0

5.0

6.0

–

–

–

57

CMSX-3

7.9

5.6

1.0

0.5

8.0

4.6

6.0

–

–

0.1Hf

43

CMSX-4

6.2

5.6

1.0

0.6

6.5

9.4

6.4

–

2.8

0.1Hf

46

CMSX-6

9.8

4.8

4.7

3.0

–

5.0

2.1

–

–

0.1Hf

44

CMSX-10

2.0

4.8

0.2

0.4

5.0

3.0

8.0

–

6.0

0.03Hf

57

IN-X750

14.9

0.7

2.5

–

–

–

0.9

–

6,5Fe

39

EI437B EI698 EI867 EP199 ZhS6UVI ZhS26VI

20.1 14.0 9.5 19.8 9.4 5.0

0.7 1.7 4.5 2.1 5.6 5.8

2.5 2.7 – 1.4 2.5 0.9

– 3.0 9.8 4.5 1.5 1.1

– – 5.3 9.1 10.1 11.5

– – 5.1 – 9.6 8.9

– – – – – –

– 2.0 – – 1.1 1.4

– – – – – –

– – – – 0.5Hf 0.9V

Auth.

Rene N5 Rene N6

7.0 4.2

6.2 5.8

– –

2.0 4.0

5.0 6.0

8.0 12.5

7.0 7.2

– 0.1

3.0 5.4

0.2Hf 0.2Hf

57

CMSX-10M MC544 MC534

2.0 4.0 4.0

5.8 6.0 5.8

0.2 0.5 –

0.4 1.0 4.0

5.4 5.0 5.0

1.8 – –

8.2 5.0 6.0

0.1 – –

6.5 4.0 3.0

– 0.1Hf 0.1Hf

34

Rene 80

14.5

3.8

3.8

–

–

10.0

–

–

–

0.20C

59

MC2 SC180 IN-100

8.0 5.0 12.4

5.0 5.2 5.0

1.5 1.0 4.3

2.0 2.0 3.2

8.0 5.0 –

5.0 10.0 18.5

6.0 9.0 –

– – –

– 3.0 –

– 0.1Hf –

57

CM247LC

9.2

13.3

0.8

0.3

2.6

10.1

0.9

–

–

0.5Hf

48

TMS-75 TMS-138

3.0 3.0

6.0 6.0

– –

2.0 3.0

6.0 6.0

12.0 6.0

6.0 6.0

– –

5.0 5.0

0.1Hf 0.1Hf 2.0Ru

49 50

LCAstroloy MAR-M200 NASAIIB-7 Waspaloy

15.1 9.0 8.9 13.3

4.0 5.0 3.4 1.3

3.5 2.0 0.7 3.6

5.2 – 2.0 4.2

– 12.5 7.6 –

17.0 10.0 9.1 13.6

– – 10.1 –

– 1.0 – –

– – – –

– 0.05Zr 1.0Hf –

59

163

164

References

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167

168

Acknowledgements

My sincere gratitude to my wife Lydia for her support and patience. I would like to express my deep gratitude to Dr. Vik. V. Levitin for valuable assistance with discussions. Special thanks to Dr. O.V. Rubel for help concerning the computer simulation. I gratefully acknowledge Dr. L.K. Orzhitskaya for many years of her participation in numerous experiments. I am grateful to Dr. V.I. Babenko for his participation in the development of equipment for in situ X-ray studies.


169

Index a activated dislocation segments – length 95, 96 activation energy of creep – apparent 101 – in pure metals 6, 7 – in refractory metals 146, 147, 150 – in superalloys 101 activation volume – equation 7 alloys – Ir–Nb, Ir–Zr 155 – Ni–Cr, Ni–Al, Ni–W 55 – of refractory metals 143, 149, 151, 152, 153 – W–Re, W–Hf 153 amplitudes of atomic vibrations – in γ phases of superalloys 102, 103 – in nickel base solid solutions 54, 55 – measurements 21–23, 102 c creep – curve 5, 6 – dislocation theories 8, 9 – in refractory alloys 151, 152 – in refractory metals 143–145, 147–150, 152

– in solid solutions 54 – in superalloys 86, 87, 95, 96, 116–120, 124, 125 – at higher temperatures 124 – at lower temperatures 116 – dislocation splitting 112, 120–122, 129 – equations 99, 100 – influence of orientation, temperature and stress 111–120 – primary stage 118, 119 – tertiary stage 118 – physical mechanism 43–45, 67, 68 – steady-state stage 51, 77 – calculation for pure metals 51–53 – equations 49, 51–53, 95, 96, 100, 137–140 – structural peculiarities 40 d deformation map – iron 64 – molybdenum 150 – nickel 63 – niobium 145 density of dislocations – differential equation 49–51, 77, 78


170

Index

– in metals 38 – in superalloys 100, 101 diffraction electron microscopy 20 dislocation networks 30–33, 89, 132–135 dislocations – annihilation 49–51 – coefficients of multiplication 50, 73, 75 – in γ phase 90, 92, 94, 97 – in crept metals 35–38 – interactions with particles 89–94 – jogged 35, 36 – mobile 35, 36 – partial 112, 160 – ribbons 120–122 – screw components 36, 161 – splitting 121, 129 – subgrains 35 – theory 157

– for metals 43–45, 67, 68 – for superalloys 95–97 – shear deformation 124, 125

e evolution of structural parameters – in matrix of superalloys 88, 89 – in metals 25–33

r rafting 130, 131 refractory metals – molybdenum 146–151 – niobium 144–147 – refractory alloys 149, 151, 152 rupture life 86, 87, 114, 115

g γ/γ misfit – influence of temperature 136 γ phase – amplitude of atomic vibrations 102, 103 – coarsening 104, 105 – composition 83, 103 – crystal lattice 84 – lattice parameter 136 – rafting 130, 131 – solubility 85 h high-temperature strain rate – physical model

i interaction of dislocations with particles 89–94 j jogs in dislocations – formation 55, 56 – in crept metals 36–38 m metals – copper 27, 28, 30 – iron 31–35 – molybdenum 146–151 – nickel 26, 30, 32, 34–37 – niobium 144–147 – vanadium 29, 31 misfit 136

s Schmid factor 112 simulation – by the system of differential equations 67–71 – data for metals 71–77 – of structural parameters evolution 67 single crystal superalloys – blades 113 – creep curves 117–120, 123–125 – influence of orientation on 114–119

Index

– influence of stress on 120 – influence of temperature on 116–118, 120 – dislocation mechanisms of strain 119–127, 129 – properties 115 – shear strain 125, 126 solid solutions – Ni-based 55 stacking faults – energy 57 structural parameters – average values 30 – evolution 25–30 – measurements 17–20 structural peculiarities – of crept metals 40 – of superalloys 83, 88 sub-boundaries – as sources and obstacles for mobile dislocations 34, 35 – crystallography 55, 56 – distances between dislocations 31–35, 37, 38 – stability 58–62

superalloys – composition 129, 163 – equations of strain rate 95–100, 137–140 – physical mechanism of strain 96–98 – prediction of properties 106–108 – trends of development 129 v vacancies – energies of formation 46, 52 – energy of diffusion 46, 47, 52 – loops and helicoids 39 velocity of dislocations – with vacancy-absorbing jogs 46, 47 – with vacancy-producing jogs 46–49, 72, 75 x X-ray in situ studies – data 26–31 – equipment 13, 14 – technique 15 – measurement of structure parameters 17–20

171

High Temperature Strain of Metals and Alloys

High Temperature Strain Of Metals And Alloys

Plasticity of Metals and Alloys

Plasticity of Metals and Alloys

Processing of Metals and Alloys

Melting and Crystallization of Metals and Alloys

Parametric Analyses of High-Temperature Data for Aluminum Alloys

Fundamentals of Creep in Metals and Alloys

Fundamentals of Creep in Metals and Alloys

Mechanisms of high temperature of high temperature superconductivity. Vol. 2

Introduction to the High-Temperature Oxidation of Metals

Fatigue Crack Propagation in Metals and Alloys

Phase transformations in metals and alloys

Theory of High Temperature Superconductivity

High Temperature Corrosion of Ceramics

Theory of High Temperature Superconductivity

High-Temperature Measurements of Materials

High-Temperature Superconductors

High Temperature Coatings

High Temperature Superconductors

Theory of High Temperature Superconductivity

High-Temperature Levitated Materials

High Temperature Superconductors

Theory of Weldability of Metals and Alloys (Materials Science Monographs)

The Theory of the Properties of Metals and Alloys

Microstructure and Properties of High-Temperature SuperConductors

High-temperature Structural Materials

High Temperature Superconductors

Modern Theory of Magnetism in Metals and Alloys

High Temperature Strain of Metals and Alloys