Valim Levitin High Temperature Strain of Metals and Alloys Related Titles Herlach, D.M (ed.) Solidification and Crystallization 322 pages with 204 figures and 20 tables 2004 Hardcover ISBN 3-527-31011-8 Leyens, C., Peters, M (eds.) Titanium and Titanium Alloys Fundamentals and Applications 532 pages with 349 figures and 56 tables 2003 Hardcover ISBN 3-527-30534-3 Westbrook, J.H (ed.) Intermetallic Compounds 4V Set 1310 pages 2000 Softcover ISBN 0-471-60814-9 Mughrabi, H (ed.) Materials Science and Technology A Comprehensive Treatment – Volume Plastic Deformation and Fracture of Materials 710 pages with 436 figures and 19 tables 1992 Hardcover ISBN 3-527-26819-7 Valim Levitin High Temperature Strain of Metals and Alloys Physical Fundamentals The Author Prof Valim Levitin National Technical University Zaporozhye, Ukraine valim.levitin@t-online.de 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 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 Macroscopic Characteristics of Strain of Metallic Materials at High Temperatures 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 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 4.1 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) Copyright c 2006 WILEY-VCH Verlag GmbH & Co KGaA, Weinheim 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 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 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 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 8.1 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 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) Copyright c 2006 WILEY-VCH Verlag GmbH & Co KGaA, Weinheim ISBN: 3-527-313389-9 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 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] 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 High Temperature Strain of Metals and Alloys, Valim Levitin (Author) Copyright c 2006 WILEY-VCH Verlag GmbH & Co KGaA, Weinheim ISBN: 3-527-313389-9 Macroscopic Characteristics of Strain of Metallic Materials at High Temperatures Thus, the following stages are observed: The incubation deformation For this stage the strain rate = const; > ă The primary stage, during which ε = const; ε < The creep rate de ă creases when the strain increases The steady-state strain The plastic strain rate is a constant value ε = const ˙ The tertiary stage ε = const; ε > The tertiary creep leads to a rupture ă 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 in˙ vestigation of the problem under consideration The functions σ(ε, T ) and ˙ the rupture life (durability) τ (σ, T ) have also been studied For the 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 ε = A1 exp − ˙ ε = A2 exp − ˙ ε = A3 exp − ˙ Q kT Q kT σn Q − vσ kT sinh (1.1) (1.2) ασ kT (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 not depend upon temperature then it is easy to obtain Q = −k ∂ ln ε ˙ ∂ T (1.4) σ 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 not depend upon stress v = kT ∂ ln ε ˙ ∂σ (1.5) 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 Macroscopic Characteristics of Strain of Metallic Materials at High Temperatures a subgrain or cell, surrounded by dislocation aggregations The cell walls separate relatively dislocation-free regions from each other Subgrains are also seen at the borders of the picture Aggregations of dislocations in sub-boundaries seem to be more or less ordered We observe regular dislocation lines elongated in the same direction The dislocation lines form low-angle sub-boundaries unlike the large-angle boundaries between crystallites (grains) Thus, the subgrains are misoriented to each other The misorientation is of the order of tens of angle minutes i.e of milliradians Fig 1.3 Subgrain in nickel tested at 1073K, stress 20MPa ¯ (a) Bright-field image Screw dislocations along [101] are denoted as B (b) Electron diffraction pattern (c) Scheme of the arrangement of dislocations inside the boundary In Fig 1.3(a) the so-called diffraction contrast is observed It is created by separate dislocations in sub-boundaries Strictly speaking, the electronic beam generates an interference contrast due to stresses near the dislocation line In Fig 1.3(c) the screw sub-boundary dislocations are shown to be 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 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 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 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 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 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 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  ... Composition of Superalloys 12 9 Rafting 13 0 Effect of Composition and Temperature on γ/γ Misfit Other Creep Equations 13 7 Summary 14 1 11 1 13 6 VII 8 .1 8.2 8.3 Deformation of Some Refractory Metals The Creep... 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, ... Crystals of Superalloys 7 .1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 Effect of Orientation on Properties 11 1 Deformation at Lower Temperatures 11 6 Deformation at Higher Temperatures 12 4 On the Composition of Superalloys