mechanical-properties-and-working-of-metals-and-alloys_compress

Springer Series in Materials Science 264 Amit Bhaduri Mechanical Properties and Working of Metals and Alloys

Springer Series in Materials Science Volume 264 Series editors Robert Hull, Troy, USA Chennupati Jagadish, Canberra, Australia Yoshiyuki Kawazoe, Sendai, Japan Richard M. Osgood, New York, USA Jürgen Parisi, Oldenburg, Germany Udo W. Pohl, Berlin, Germany Tae-Yeon Seong, Seoul, Republic of Korea (South Korea) Shin-ichi Uchida, Tokyo, Japan Zhiming M. Wang, Chengdu, China

The Springer Series in Materials Science covers the complete spectrum of materials physics, including fundamental principles, physical properties, materials theory and design. Recognizing the increasing importance of materials science in future device technologies, the book titles in this series reflect the state-of-the-art in understand- ing and controlling the structure and properties of all important classes of materials. More information about this series at http://www.springer.com/series/856

Amit Bhaduri Mechanical Properties and Working of Metals and Alloys 123

Amit Bhaduri Department of Metallurgical and Materials Engineering Indian Institute of Technology Kharagpur Kharagpur, West Bengal India ISSN 0933-033X ISSN 2196-2812 (electronic) Springer Series in Materials Science ISBN 978-981-10-7208-6 ISBN 978-981-10-7209-3 (eBook) https://doi.org/10.1007/978-981-10-7209-3 Library of Congress Control Number: 2017959907 © Springer Nature Singapore Pte Ltd. 2018 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Printed on acid-free paper This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. part of Springer Nature The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

This book is dedicated to the memory of my father LATE ASIM RANJAN BHADURI

Preface The book has evolved from the author’s lecture notes used for the past several years in the teaching of two upper-level undergraduate (B.Tech.) and one postgraduate (M.Tech.) courses in the Department of Metallurgical and Materials Engineering, Indian Institute of Technology Kharagpur. The book provides a comprehensive account of the basic principles of mechanical testing and metalworking and can be used as a text for two undergraduate courses taught in the disciplines of metallurgical engineering, mechanical engineering and production or manu- facturing engineering. Parts of the book can be used as study material at the undergraduate level in the branches of civil engineering, aerospace engineering and material science. The book also contains advanced study material suitable for postgraduate students of metallurgical engineering. In addition to the above multifaceted academic applications, a unique feature of the book is its practical content of direct use to metallurgists and practising engineers in industry. The book is designed to help them understand and apply the theories of mechanical testing and working to determine, control and improve the mechanical properties of metals and alloys as well as for improving production in the metalworking industry by analysing working problems—both the mechanics of working processes and how the properties of metals interact with the processes. Mechanical properties of metal play an important role in the processes of metalworking. Hence, the first part of this book contains a discussion on evaluation of mechanical properties by testing, relationship among different mechanical properties, factors and variables affecting mechanical properties, engineering aspects of mechanical properties and their applications in design, etc. This is continued in the second part of this book that deals with detailed analysis of various types of metalworking or forming operations to produce useful shapes under the application of stress arising primarily from mechanical source. The analytical methods in the text for the treatment of mechanical working processes include slab method of analysis or free-body equilibrium approach. The slab method is used to estimate the deformation loads for hot working processes, such as forging and extrusion under conditions of Coulomb’s sliding friction, sticking friction and mixed sticking–sliding friction, and for cold working processes, such as drawing of strip, rod or wire, and tube and deep drawing of cup under sliding frictional condition. Using the slab method, this book provides an exhaustive analysis of Bland and Ford theory of cold rolling for sliding frictional condition and Sims’ theory of hot rolling for sticking frictional condition. Elaborate and easy-to-understand mathematical analyses without skipping any intermediate steps have been incorporated in this book. Not only complete derivations of mathematical treatments have been considered, but also a comprehensive and reasonably wide coverage in sufficient depth has also been attempted in this book. High energy rate forming, which is an unconventional metalworking process, and where chemical, mag- netic and electrical sources of energy are used, has also been included as the last chapter in this book. For better understanding of the theory, several solved problems have been included in each chapter of the book. Similarly, many numerical problems along with multiple choice questions are given as exercise in each chapter, and answers to the problems and questions have also been provided. vii

viii Preface In addition to the slab method of analysis, this book also contains slip-line field theory, its application to the static system, such as plane-strain indentation with flat frictionless platens for cases of various heights of work-piece with respect to the breadth of platen, and its application to the steady state motion, such as plane-strain frictionless extrusion and strip drawing. Further, this book includes upper-bound theorem, and upper-bound solutions for indentation of a semi-infinite slab, for compression, for plane-strain frictionless extrusion and strip drawing. The field of mechanical metallurgy may be roughly divided into three modules taught at the undergraduate level of engineering discipline. These modules are deformation behaviour, mechanical properties or mechanical testing and mechanical forming or working. Since the subject matter of this book comprises the last two modules, which are dependent on the first module, knowledge of deformation behaviour, consisting of elastic stress–strain relations, plastic deformation including theory of plasticity, dislocation theory and strengthening mechanisms, is desirable for a better understanding of the contents of this book, though relevant aspects of deformation behaviour have been briefly included in the first chapter of this book. However, calculus and engineering mechanics are essential prerequisite subjects for this book, and thus, the reader must have knowledge of these subjects. Although the text has been carefully scrutinized, a few errors may still exist in the first edition of this book. Hence, the author will be happy to receive comments along with sug- gestions and constructive criticism. The author acknowledges here that he was inspired by a few lectures on metal rolling offered by Prof. (Late) R. Roy, Metallurgical Engineering Department, National Institute of Technology Durgapur. The author is extremely thankful to all of his departmental colleagues, especially Prof. K. Biswas, Prof. R. Mitra, Prof. S. Biswas, Prof. S. Kar, Prof. T. Laha, Prof. R. N. Ghosh, Prof. K. K. Ray, Prof. S. K. Pabi, Prof. S. K. Roy, Prof. D. Chakrabarti, Prof. S. Mandal, Prof. T. K. Bandyopadhyay, Prof. N. Chakraborti, Prof. J. Das, Prof. J. Datta Majumdar, Prof. G. G. Roy, Prof. M. Roy, Prof. S. Ghosh and Prof. T. K. Kundu, for their assistance, cooperation and valuable suggestions. The author is really indebted to Prof. A. Guha, Mechanical Engineering Department, and Prof. S. B. Singh, Metallurgical Engi- neering Department, IIT Kharagpur, for their helpful comments leading to improvement in the manuscript. The author is also indebted to his mother, wife, brother, sister, daughter, son-in-law and grandson for their forbearance, encouragement and constant support during the preparation of the manuscript. Finally, the author is thankful to the publisher ‘Springer’, especially its executive editor (applied science and engineering) Ms. Swati Meherishi, its senior editorial assistant Ms. Aparajita Singh and its editorial assistant Ms. TCA Avni, for editing the manuscript and giving it the shape of a book. The author is also thankful to production department of Springer (Scientific Publishing Services), especially its project manager Sri Nandhini, and its other production administrators (books), such as Ms. Krati Shrivastava, and Vinoth. S. Kharagpur, India Amit Bhaduri

Contents Part I Mechanical Properties of Metals and Alloys 1 Tension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.1.1 Description of Stress at a Point . . . . . . . . . . . . . . . . . . . . . . 4 1.2 Strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2.1 True Versus Engineering Strain . . . . . . . . . . . . . . . . . . . . . . 6 1.2.2 Advantages of True Strain Over Engineering Strain . . . . . . . . 6 1.2.3 Poisson’s Ratio and Volume Strain . . . . . . . . . . . . . . . . . . . 7 1.3 Conventional and True Stresses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.3.1 Relationship Between True and Engineering Stresses During Plastic Deformation. . . . . . . . . . . . . . . . . . . . . . . . . 8 1.4 Elastic Stress–Strain Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.4.1 Three-Dimensional State of Stress . . . . . . . . . . . . . . . . . . . . 9 1.5 Elements of Plastic Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.5.1 Relationship Between Principal Normal and Shear Stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.5.2 Mohr’s Stress Circle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.5.3 Slip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.5.4 Twinning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 1.5.5 Strain Hardening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 1.5.6 Stacking Fault . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 1.5.7 Strengthening Methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 1.5.8 Spherical and Deviator Components of Stress . . . . . . . . . . . . 35 1.5.9 Yielding Criteria for Ductile Metals . . . . . . . . . . . . . . . . . . . 35 1.5.10 Octahedral Shear Stress and Shear Strain . . . . . . . . . . . . . . . 37 1.5.11 Invariants of Stress and Strain . . . . . . . . . . . . . . . . . . . . . . . 38 1.5.12 Levy–Mises Equations for Ideal Plastic Solid . . . . . . . . . . . . 38 1.5.13 Yielding Criteria Under Plane Strain . . . . . . . . . . . . . . . . . . 40 1.6 Types of Tensile Stress–Strain Curve. . . . . . . . . . . . . . . . . . . . . . . . . 41 1.6.1 Type I: Elastic Behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . 41 1.6.2 Type II: Elastic–Homogeneous Plastic Behaviour . . . . . . . . . 42 1.6.3 Type III: Elastic–Heterogeneous Plastic Behaviour. . . . . . . . . 51 1.6.4 Type IV: Elastic–Heterogeneous–Homogeneous Plastic Behaviour. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 1.6.5 Type V: Elastic–Heterogeneous–Homogeneous Plastic Behaviour for Some Crystalline Polymers. . . . . . . . . . . . . . . 54 1.7 Linear Elastic Properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 1.7.1 Modulus of Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 1.7.2 Proportional and Elastic Limit . . . . . . . . . . . . . . . . . . . . . . . 56 1.7.3 Resilience and Modulus of Resilience . . . . . . . . . . . . . . . . . 56 ix

x Contents 1.8 Nonlinear Elastic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 1.8.1 Secant and Tangent Modulus . . . . . . . . . . . . . . . . . . . . . . . 58 1.8.2 Elastomer or Rubber Elasticity . . . . . . . . . . . . . . . . . . . . . . 58 1.8.3 Elastic Resilience or Resilience . . . . . . . . . . . . . . . . . . . . . . 59 60 1.9 Inelastic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 1.9.1 Yield Strength. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 1.9.2 Ultimate and True Tensile Strength . . . . . . . . . . . . . . . . . . . 63 1.9.3 Ductility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 1.9.4 Fracture Strength and True Fracture Strength . . . . . . . . . . . . 69 1.9.5 Toughness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 1.10 Influence of Temperature on Tensile Properties. . . . . . . . . . . . . . . . . . 72 1.10.1 Effect of Temperature on Stress–Strain Curve 73 of Mild Steel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 75 1.11 Strain Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.11.1 Relation Between Flow Stress and Strain Rate . . . . . . . . . . . 77 1.11.2 Superplasticity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 1.11.3 Effect of Strain Rate on Stress–Strain Curve of Mild Steel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 80 1.12 Testing Machine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 1.12.1 Influence of Testing Machine on Strain 85 and Strain Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 1.13 Notch Tensile Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.14 Tensile Fracture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.15 Solved Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Compression. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 2.2 Standard Specimen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 2.3 Elastic Range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 2.4 Plastic Range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 2.4.1 Stress and Strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 2.4.2 Strain Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 2.4.3 Brittle Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 2.4.4 Ductile Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 2.5 Bauschinger Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 2.6 Advantages of Compression Over Tension Test . . . . . . . . . . . . . . . . . 105 2.7 Problems in Compression Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 2.7.1 Buckling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 2.7.2 Barreling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 2.8 Compressive Failure of Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 2.9 Solved Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 3 Hardness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 3.2 Classification of Hardness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 3.3 Precautions to Avoid Erratic Hardness Measurement . . . . . . . . . . . . . . 121 3.4 Mohs’ Scale of Hardness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 3.5 File Hardness Test. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

Contents xi 3.6 Brinell Hardness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 3.6.1 Principle of Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 3.6.2 Derivation for BHN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 3.6.3 Indenters, Loads and Loading Periods . . . . . . . . . . . . . . . . . 126 3.6.4 Method of Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 3.6.5 Anomalous Behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 3.6.6 Advantages and Disadvantages . . . . . . . . . . . . . . . . . . . . . . 129 3.7 Meyer Hardness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 3.7.1 Meyer’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 3.7.2 Load Sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 3.7.3 Influence of P/D2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 3.8 Rockwell Hardness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 3.8.1 Principle of Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 3.8.2 Loads. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 3.8.3 Indenters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 3.8.4 Direct-Reading Hardness Dial . . . . . . . . . . . . . . . . . . . . . . . 133 3.8.5 Hardness Scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 3.8.6 Method of Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 3.8.7 Advantages. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 3.9 Rockwell Superficial Hardness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 3.9.1 Principle of Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 3.9.2 Superficial Hardness Scale . . . . . . . . . . . . . . . . . . . . . . . . . 139 3.9.3 Merits and Demerits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 3.10 Vickers Hardness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 3.10.1 Indenters and Loads. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 3.10.2 Principle of Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 3.10.3 VHN Versus BHN. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 3.10.4 Operational Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 3.10.5 Minimum Thickness of Test Section . . . . . . . . . . . . . . . . . . 144 3.10.6 Anomalous Behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 3.10.7 Advantages and Disadvantages . . . . . . . . . . . . . . . . . . . . . . 144 3.11 Microhardness (Knoop Hardness) . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 3.11.1 Penetrators and Loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 3.11.2 Principle of Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 3.11.3 Advantages and Disadvantages . . . . . . . . . . . . . . . . . . . . . . 147 3.12 Monotron Hardness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 3.12.1 Indenters and Hardness Scales. . . . . . . . . . . . . . . . . . . . . . . 148 3.12.2 Principle of Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 3.12.3 Advantages and Disadvantages . . . . . . . . . . . . . . . . . . . . . . 149 3.13 Shore Scleroscope Hardness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 3.13.1 Principle of Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 3.13.2 Mass Effect of Test Piece . . . . . . . . . . . . . . . . . . . . . . . . . . 150 3.13.3 Advantages. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 3.14 Poldi Impact Hardness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 3.14.1 Principle of Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 3.14.2 Use of Supplied Table to Determine BHN . . . . . . . . . . . . . . 152 3.14.3 Advantages and Disadvantages . . . . . . . . . . . . . . . . . . . . . . 153 3.15 The Herbert Pendulum Hardness . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 3.15.1 Time Test. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 3.15.2 Scale Test. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 3.15.3 Time Work-Hardening Test . . . . . . . . . . . . . . . . . . . . . . . . 154 3.15.4 Scale Work-Hardening Test . . . . . . . . . . . . . . . . . . . . . . . . 154

xii Contents 3.16 Nanohardness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 3.16.1 Indenters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 3.16.2 Derivation for Berkovich Hardness . . . . . . . . . . . . . . . . . . . 156 3.16.3 Determination of Contact Depth of Penetration . . . . . . . . . . . 157 3.16.4 Correction for Machine Compliance. . . . . . . . . . . . . . . . . . . 159 3.16.5 Indenter Shape Function . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 3.16.6 Errors Due to Pile-Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 3.16.7 Martens Hardness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 163 3.17 Relationship to Flow Curve and Prediction of Tensile Properties . . . . . . 164 3.18 Solved Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Bending . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 4.2 Pure Bending . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 4.2.1 Bending Stresses and Flexure Formula . . . . . . . . . . . . . . . . . 174 4.2.2 Experimental Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 4.3 Beam Design in Pure Bending . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 4.4 Linear Elastic Behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 4.4.1 Important Variables Affecting Modulus of Rupture . . . . . . . . 182 4.4.2 Modulus of Elastic Resilience . . . . . . . . . . . . . . . . . . . . . . . 182 4.5 Yielding. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 4.5.1 Discontinuous Yielding and Shape Factor . . . . . . . . . . . . . . . 183 4.6 Nonlinear Stress–Strain Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 4.7 Shear Stresses in Elastically Bent Beam . . . . . . . . . . . . . . . . . . . . . . . 186 4.8 Solved Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 5 Torsion—Pure Shear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 5.2 State of Stress and Strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 5.2.1 Shear Strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 5.2.2 Relation Between Shear Modulus ‘G’ and Young’s Modulus ‘E’ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 5.3 Relation Between Shear Strain and Angle of Twist . . . . . . . . . . . . . . . 202 5.4 Torsional Stresses in Elastic Range . . . . . . . . . . . . . . . . . . . . . . . . . . 203 5.4.1 Relation Between Torque, Shear Modulus and Angle of Twist . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 5.4.2 Computation of Torque in Practical Applications . . . . . . . . . . 204 5.4.3 Polar Moment of Inertia, Shear Stress and Angle of Twist . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 5.4.4 Thin-Walled Tube of Arbitrary Cross-Section . . . . . . . . . . . . 207 5.5 Torsional Stresses for Plastic Strains . . . . . . . . . . . . . . . . . . . . . . . . . 208 5.6 Behaviour of Material in Torsion . . . . . . . . . . . . . . . . . . . . . . . . . . . 210 5.6.1 Testing Equipment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 5.6.2 Specimen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 5.7 Elastic Properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 5.7.1 Shear Modulus and Torsional Proportional Limit . . . . . . . . . . 212 5.7.2 Torsional Modulus of Elastic Resilience . . . . . . . . . . . . . . . . 213 5.8 Inelastic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 5.8.1 Torsional Yield Strength. . . . . . . . . . . . . . . . . . . . . . . . . . . 215

Contents xiii 5.8.2 Ultimate Torsional Shear Strength or Modulus 216 of Rupture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 217 5.8.3 Ductility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218 5.9 Torsion Test Versus Tension Test . . . . . . . . . . . . . . . . . . . . . . . . . . . 218 5.9.1 Comparison in Terms of State of Stress and Strain . . . . . . . . 219 5.9.2 Comparison of Ductile Behaviour . . . . . . . . . . . . . . . . . . . . 220 5.9.3 Torsional Shear Stress–Strain Diagram from Tensile 221 225 Flow Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.10 Failure Under Torsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.11 Solved Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Impact Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 6.1 Dynamic Loading and Brittle Fracture . . . . . . . . . . . . . . . . . . . . . . . . 227 6.1.1 Factors Responsible for Brittle Behaviour . . . . . . . . . . . . . . . 228 6.2 Notched-Bar Impact Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 6.2.1 Single-Blow Pendulum Impact Test . . . . . . . . . . . . . . . . . . . 230 6.3 Calculation of Energy Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 6.3.1 Correction for Energy Losses . . . . . . . . . . . . . . . . . . . . . . . 234 6.4 Impact Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 6.4.1 Transition Temperature Curves . . . . . . . . . . . . . . . . . . . . . . 238 6.4.2 Various Criteria of Transition Temperature . . . . . . . . . . . . . . 239 6.5 Metallurgical Factors Affecting Impact Properties . . . . . . . . . . . . . . . . 240 6.5.1 Embrittlement During Tempering. . . . . . . . . . . . . . . . . . . . . 243 6.6 Instrumented Charpy Impact Test . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 6.7 Additional Large-Scale Fracture Test Methods . . . . . . . . . . . . . . . . . . 248 6.7.1 Explosion-Crack-Starter Test. . . . . . . . . . . . . . . . . . . . . . . . 248 6.7.2 Drop Weight Test (DWT). . . . . . . . . . . . . . . . . . . . . . . . . . 249 6.7.3 Robertson Crack-Arrest Test . . . . . . . . . . . . . . . . . . . . . . . . 250 6.7.4 Dynamic Tear (DT) Test . . . . . . . . . . . . . . . . . . . . . . . . . . 250 6.8 Fracture Analysis Diagram (FAD) . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 6.8.1 Design Philosophy Using FAD . . . . . . . . . . . . . . . . . . . . . . 252 6.9 Solved Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 7 Creep and Stress Rupture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 7.1 Long-Time Loading at High Temperature. . . . . . . . . . . . . . . . . . . . . . 258 7.2 The Creep Curve. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259 7.3 Strain–Time Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 7.4 Creep Rate–Stress–Temperature Relations . . . . . . . . . . . . . . . . . . . . . 265 7.5 Steady-State Creep . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268 7.5.1 Effect of Grain Size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268 7.5.2 Activation Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269 7.6 Creep Deformation Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271 7.6.1 Dislocation Creep or Climb–Glide Creep . . . . . . . . . . . . . . . 272 7.6.2 Diffusional Creep . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274 7.6.3 Grain-Boundary Sliding . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 7.7 Deformation Mechanism Map. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 7.8 The Stress-Rupture Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280 7.9 Concept of ECT and Elevated-Temperature Fracture . . . . . . . . . . . . . . 281 7.9.1 Wedge-Shaped Cracks and Round or Elliptically Shaped Cavities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283

xiv Contents 7.10 Presentation of Engineering Creep Data . . . . . . . . . . . . . . . . . . . . . . . 287 7.10.1 Prediction of Creep Strength . . . . . . . . . . . . . . . . . . . . . . . . 288 7.10.2 Prediction of Creep-Rupture Strength . . . . . . . . . . . . . . . . . . 290 291 7.11 Parameter Methods to Predict Long-Time Properties . . . . . . . . . . . . . . 293 7.11.1 Larson–Miller Parameter . . . . . . . . . . . . . . . . . . . . . . . . . . 294 7.11.2 Orr–Sherby–Dorn Parameter . . . . . . . . . . . . . . . . . . . . . . . . 295 7.11.3 Manson–Haferd Parameter . . . . . . . . . . . . . . . . . . . . . . . . . 296 7.11.4 Goldhoff–Sherby Parameter . . . . . . . . . . . . . . . . . . . . . . . . 297 7.11.5 Limitations of Parameter Methods . . . . . . . . . . . . . . . . . . . . 297 299 7.12 Stress Relaxation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299 7.12.1 Step-Down Creep Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302 303 7.13 Materials for High-Temperature Use . . . . . . . . . . . . . . . . . . . . . . . . . 304 7.13.1 Rules to Develop Creep Resistance . . . . . . . . . . . . . . . . . . . 306 7.14 Creep Under Multiaxial Stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308 7.15 Indentation Creep . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314 7.15.1 Method to Obtain Creep Curve Using Rockwell Hardness Tester. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.16 Solved Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Fatigue. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317 8.1 Fatigue Failure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318 8.2 Stress Cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319 8.3 Standard Fatigue Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321 8.4 The S–N Diagram and Fatigue Properties . . . . . . . . . . . . . . . . . . . . . . 323 8.4.1 Reason for Existence of Fatigue Limit . . . . . . . . . . . . . . . . . 325 8.5 Statistical Nature of Fatigue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326 8.6 Fatigue Crack Nucleation and Growth . . . . . . . . . . . . . . . . . . . . . . . . 328 8.6.1 Fatigue Crack Growth Rate. . . . . . . . . . . . . . . . . . . . . . . . . 330 8.7 Effect of Mean Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332 8.8 Stress Fluctuation and Cumulative Fatigue Damage . . . . . . . . . . . . . . . 335 8.8.1 Overstressing, Understressing and Coaxing . . . . . . . . . . . . . . 335 8.8.2 Cumulative Fatigue Damage . . . . . . . . . . . . . . . . . . . . . . . . 336 8.9 Stress Concentration Effect. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338 8.10 Size Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341 8.11 Surface Effects and Surface Treatments . . . . . . . . . . . . . . . . . . . . . . . 342 8.11.1 Surface Roughness and Treatment . . . . . . . . . . . . . . . . . . . . 343 8.11.2 Surface Properties and Treatment. . . . . . . . . . . . . . . . . . . . . 343 8.11.3 Surface Residual Stress and Treatment . . . . . . . . . . . . . . . . . 344 8.11.4 Metallurgical Processes Detrimental to Fatigue . . . . . . . . . . . 346 8.12 Effect of Metallurgical Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346 8.13 Frequency of Stress Cycling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348 8.14 Temperature Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349 8.14.1 Low Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349 8.14.2 High Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349 8.14.3 Thermal Fatigue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350 8.15 Chemical Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350 8.16 Cyclic Strain-Controlled Fatigue . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352 8.16.1 Low-Cycle Fatigue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356 8.16.2 Strain–Life Equation and Curve. . . . . . . . . . . . . . . . . . . . . . 356

Contents xv 8.17 Creep–Fatigue Interaction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358 8.18 Increasing Amplitude Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361 362 8.18.1 Step Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362 8.18.2 Prot Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364 8.19 Solved Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 370 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Fracture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374 9.2 Theoretical Cohesive Strength. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375 9.3 Inglis Analysis of Stress Concentration Factor . . . . . . . . . . . . . . . . . . 376 9.4 Effects of Notch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378 9.5 Characteristic Features of Fracture Process . . . . . . . . . . . . . . . . . . . . . 382 9.5.1 Energy to Fracture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382 9.5.2 Macroscopic Mode of Fracture . . . . . . . . . . . . . . . . . . . . . . 383 9.5.3 Microscopic Mode of Fracture or Fractography . . . . . . . . . . . 384 9.6 Griffith Theory of Brittle Fracture . . . . . . . . . . . . . . . . . . . . . . . . . . . 387 9.6.1 Applicability of Griffith Theory. . . . . . . . . . . . . . . . . . . . . . 390 9.6.2 Modification of Griffith Theory. . . . . . . . . . . . . . . . . . . . . . 391 9.7 Elastic Strain Energy Release Rate . . . . . . . . . . . . . . . . . . . . . . . . . . 392 9.8 Stress Intensity Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394 9.8.1 Different Crack Surface Displacements. . . . . . . . . . . . . . . . . 396 9.8.2 Relationship Between Energy Release Rate and Stress Intensity Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397 9.8.3 Fracture Toughness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397 9.9 Plastic Zone at Crack Tip. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397 9.9.1 Effective Stress Intensity Factor. . . . . . . . . . . . . . . . . . . . . . 398 9.10 Fracture Toughness: Plane Stress Versus Plane Strain . . . . . . . . . . . . . 399 9.11 Plane-Strain Fracture Toughness ðKIcÞ Testing . . . . . . . . . . . . . . . . . . 400 9.11.1 Specimen Size, Configurations, and Preparation . . . . . . . . . . 401 9.11.2 Test, Interpretation of Result and Calculation of ðKIcÞ . . . . . . 403 9.11.3 Kc from KIc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404 9.12 Design Philosophy with Fracture Toughness . . . . . . . . . . . . . . . . . . . . 405 9.13 Solved Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 410 Part II Mechanical Working of Metals and Alloys 10 Fundamentals of Mechanical Working. . . . . . . . . . . . . . . . . . . . . . . . . . . . 413 10.1 Classification of Mechanical Forming Processes . . . . . . . . . . . . . . . . . 414 10.1.1 Aims of Mechanical Working . . . . . . . . . . . . . . . . . . . . . . . 414 10.1.2 Different Forming Processes . . . . . . . . . . . . . . . . . . . . . . . . 414 10.2 Temperature and Strain Rate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417 10.2.1 Cold-Work-Anneal Cycle . . . . . . . . . . . . . . . . . . . . . . . . . . 417 10.2.2 Temperature Limits for Hot Working . . . . . . . . . . . . . . . . . . 420 10.2.3 Hot Working Versus Cold Working . . . . . . . . . . . . . . . . . . . 421 10.2.4 Warm Working . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422 10.2.5 Temperature Change During Working . . . . . . . . . . . . . . . . . 423 10.2.6 Strain-Rate Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425 10.2.7 Choice of Allowable Hot Working Temperature Range . . . . . 426

xvi Contents 10.3 Friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427 10.3.1 Coulomb’s Law of Sliding Friction . . . . . . . . . . . . . . . . . . . 428 10.3.2 Shear Friction Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431 10.3.3 Measurement of Friction. . . . . . . . . . . . . . . . . . . . . . . . . . . 432 10.3.4 Adverse Effects of Friction . . . . . . . . . . . . . . . . . . . . . . . . . 434 10.3.5 Beneficial Effects of Friction . . . . . . . . . . . . . . . . . . . . . . . 435 435 10.4 Lubrication. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435 10.4.1 Material Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436 10.4.2 Functions and Characteristics of a Lubricant . . . . . . . . . . . . . 436 10.4.3 Lubrication Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . 439 440 10.5 Mechanics of Working Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 440 10.5.1 Slab Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441 10.5.2 Uniform-Deformation Energy Method . . . . . . . . . . . . . . . . . 451 10.5.3 Slip-Line Field Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456 10.5.4 Upper-Bound Technique. . . . . . . . . . . . . . . . . . . . . . . . . . . 456 10.5.5 Finite Element Method. . . . . . . . . . . . . . . . . . . . . . . . . . . . 457 458 10.6 Deformation-Zone Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462 10.7 Anisotropy of Mechanical Properties . . . . . . . . . . . . . . . . . . . . . . . . . 10.8 Solved Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Forging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465 11.1 Classification of Forging Processes . . . . . . . . . . . . . . . . . . . . . . . . . . 465 11.2 Types of Forging Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 466 11.3 Forging Equipments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 470 11.3.1 Drop Forging Hammer . . . . . . . . . . . . . . . . . . . . . . . . . . . . 470 11.3.2 Forging Press . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474 11.4 Open-Die Forging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476 11.5 Closed-Die or Impression-Die Forging . . . . . . . . . . . . . . . . . . . . . . . . 477 11.5.1 Flash . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 480 11.5.2 Draft . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482 11.5.3 Radii . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483 11.5.4 Parting Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484 11.5.5 Design Steps. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485 11.6 Material Loss During Forging. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487 11.7 Plane Strain Forging of Flat Rectangular Plate . . . . . . . . . . . . . . . . . . 488 11.7.1 Coulomb Sliding Friction . . . . . . . . . . . . . . . . . . . . . . . . . . 490 11.7.2 Sliding with Shear Friction Factor and Sticking Friction . . . . . 491 11.7.3 Mixed Sticking–Sliding Friction . . . . . . . . . . . . . . . . . . . . . 492 11.7.4 Selection of Proper Equation for Forging Load . . . . . . . . . . . 495 11.8 Plane Strain Forging of Strip with Inclined Dies . . . . . . . . . . . . . . . . . 495 11.8.1 Strip Thickness at Neutral Plane and Its Location . . . . . . . . . 498 11.9 Forging of Flat Circular Disk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 498 11.9.1 Coulomb Sliding Friction . . . . . . . . . . . . . . . . . . . . . . . . . . 500 11.9.2 Sliding with Shear Friction Factor and Sticking Friction . . . . . 501 11.9.3 Mixed Sticking–Sliding Friction . . . . . . . . . . . . . . . . . . . . . 503 11.9.4 Selection of Proper Equation for Forging Load . . . . . . . . . . . 505 11.10 Forging of Circular Disk by Conical Pointed Dies. . . . . . . . . . . . . . . . 505 11.11 Forging Defects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 508 11.12 Solved Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 511 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 519

Contents xvii 12 Rolling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 521 12.1 Fundamentals of Rolling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 522 12.1.1 Terminology of Rolled Product . . . . . . . . . . . . . . . . . . . . . . 523 12.1.2 Different Methods of Rolling . . . . . . . . . . . . . . . . . . . . . . . 523 12.1.3 Quantities Characterizing Deformation . . . . . . . . . . . . . . . . . 523 12.2 Classification of Rolling Mills . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525 12.2.1 Cluster Mill . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 527 12.2.2 Sendzimir Cold-Rolling Mill . . . . . . . . . . . . . . . . . . . . . . . . 527 12.2.3 Sendzimir Planetary Hot-Rolling Mill . . . . . . . . . . . . . . . . . 528 12.2.4 Pendulum Mill . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 530 12.2.5 Contact-Bend-Stretch Mill . . . . . . . . . . . . . . . . . . . . . . . . . 531 12.2.6 Universal Mill. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532 12.3 Rolling Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532 12.3.1 Hot Rolling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532 12.3.2 Cold Rolling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533 12.4 Deformation Zone in Rolling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534 12.4.1 Angle of Bite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535 12.4.2 Neutral Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 537 12.5 Ekelund Expression for No-Slip Angle . . . . . . . . . . . . . . . . . . . . . . . 539 12.6 Forward Slip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 540 12.6.1 Relation with No-Slip Angle. . . . . . . . . . . . . . . . . . . . . . . . 540 12.6.2 Measurement of Forward Slip . . . . . . . . . . . . . . . . . . . . . . . 542 12.6.3 Importance of Forward Slip . . . . . . . . . . . . . . . . . . . . . . . . 542 12.7 Elastic Deformation of Rolls. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 542 12.7.1 Roll Flattening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 542 12.7.2 Roll Deflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543 12.8 Simplified Assessment of Rolling Load . . . . . . . . . . . . . . . . . . . . . . . 544 12.8.1 Ekelund Equation for Rolling Load . . . . . . . . . . . . . . . . . . . 546 12.9 Theory of Rolling: Derivation of Differential Equation of Friction Hill . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 546 12.10 Bland and Ford Theory of Cold Rolling. . . . . . . . . . . . . . . . . . . . . . . 548 12.10.1 Cold Rolling with no External Tensions . . . . . . . . . . . . . . . . 550 12.10.2 Cold Rolling with Back and Front Tensions . . . . . . . . . . . . . 551 12.10.3 No-Slip Angle in Cold Rolling . . . . . . . . . . . . . . . . . . . . . . 552 12.10.4 Cold-Rolling Load. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 552 12.10.5 Cold-Rolling Torque . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553 12.10.6 Maximum Allowable Back Tension . . . . . . . . . . . . . . . . . . . 554 12.10.7 Estimation of Friction Coefficient . . . . . . . . . . . . . . . . . . . . 555 12.11 Sims’ Theory of Hot Rolling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555 12.11.1 No-Slip Angle in Hot Rolling . . . . . . . . . . . . . . . . . . . . . . . 557 12.11.2 Hot-Rolling Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 557 12.11.3 Hot-Rolling Torque . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 560 12.11.4 Limitations of Sims’ Theory . . . . . . . . . . . . . . . . . . . . . . . . 561 12.11.5 Mean Strain Rate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 561 12.12 Lever Arm Ratio, Roll Torque and Mill Power . . . . . . . . . . . . . . . . . . 561 12.12.1 Estimation of Lever Arm Ratio from Sims’ Theory . . . . . . . . 562 12.12.2 Mill Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 562 12.13 Minimum Thickness in Rolling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563 12.14 Factors Controlling Rolling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564 12.15 Gauge Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 566 12.16 Defects in Rolled Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 569

xviii Contents 12.17 Roll Pass Design Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 571 12.17.1 Types and Shapes of Passes . . . . . . . . . . . . . . . . . . . . . . . . 571 12.17.2 Gap and Taper of Sides in Pass. . . . . . . . . . . . . . . . . . . . . . 573 12.17.3 Pass Arrangement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573 12.17.4 Pass Sequences Used in Rolling of Billets to Rods . . . . . . . . 575 12.17.5 Pass Sequences Used in Rolling of Billets to Square Bars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 577 578 12.18 Manufacture of Tubes and Pipes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.18.1 Production of Seamless Tube and Pipe 580 by Hot Rolling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584 597 12.19 Solved Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Extrusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 599 13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 599 13.1.1 Comparison with Rolling . . . . . . . . . . . . . . . . . . . . . . . . . . 600 13.2 Two Basic Methods of Extrusion . . . . . . . . . . . . . . . . . . . . . . . . . . . 600 13.2.1 Direct Versus Indirect Extrusion . . . . . . . . . . . . . . . . . . . . . 602 13.3 Extrusion Equipments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 602 13.3.1 Extrusion Dies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603 13.4 Metal Flow During Extrusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605 13.5 Factors Influencing Extrusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 608 13.6 Estimation of Extrusion Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 616 13.6.1 Open-Die, Indirect and Hydrostatic Extrusions . . . . . . . . . . . 616 13.6.2 Direct Extrusion Through Conical Converging Die . . . . . . . . 619 13.6.3 Direct Extrusion Through Square Die. . . . . . . . . . . . . . . . . . 623 13.6.4 Selection of Proper Equation for Ram Load and Stress. . . . . . 625 13.7 Strain Rate in Extrusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625 13.8 Extrusion Defects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 626 13.9 Impact Extrusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 629 13.10 Hydrostatic Extrusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 630 13.10.1 Basic Difference Between Hydrostatic and Conventional Extrusion . . . . . . . . . . . . . . . . . . . . . . . . 631 13.10.2 Conventional Hydrostatic Extrusion . . . . . . . . . . . . . . . . . . . 631 13.10.3 Differential Pressure Hydrostatic Extrusion . . . . . . . . . . . . . . 632 13.10.4 Advantages of Hydrostatic Extrusion . . . . . . . . . . . . . . . . . . 632 13.10.5 Disadvantages of Hydrostatic Extrusion . . . . . . . . . . . . . . . . 633 13.11 Seamless Tube Production by Extrusion. . . . . . . . . . . . . . . . . . . . . . . 633 13.11.1 Extrusion of Cable Sheathing . . . . . . . . . . . . . . . . . . . . . . . 635 13.12 Application of Slip-Line Field to Steady-State Motion . . . . . . . . . . . . . 636 13.12.1 50% Plane-Strain Frictionless Extrusion . . . . . . . . . . . . . . . . 636 13.12.2 2/3 Plane-Strain Frictionless Extrusion . . . . . . . . . . . . . . . . . 638 13.13 Upper-Bound Solution for Plane-Strain Frictionless Extrusion . . . . . . . . 639 13.14 Solved Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 641 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 646 14 Drawing: Flat Strip, Round Bar and Tube. . . . . . . . . . . . . . . . . . . . . . . . . 647 14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 648 14.2 Strip Drawing Through Wedge-Shaped Dies. . . . . . . . . . . . . . . . . . . . 648 14.2.1 Drawing Stress with Friction. . . . . . . . . . . . . . . . . . . . . . . . 649 14.2.2 Frictionless Ideal Drawing Stress . . . . . . . . . . . . . . . . . . . . . 651 14.2.3 Maximum Reduction of Area in a Single Pass With and Without Friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 651 14.2.4 Drawing Stress for Work-Hardening Strip. . . . . . . . . . . . . . . 652

Contents xix 14.3 Drawing Stress of Strip Through Cylindrical Dies . . . . . . . . . . . . . . . . 654 14.4 Treatments of Work Metal Prior to Drawing. . . . . . . . . . . . . . . . . . . . 656 14.5 Drawing Equipments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 658 659 14.5.1 Conical Converging Die . . . . . . . . . . . . . . . . . . . . . . . . . . . 661 14.6 Drawing of Rod and Wire . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 662 14.6.1 Drawing Load and Power with Friction 666 and Back Tension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 666 667 14.6.2 Frictionless Ideal Drawing Stress . . . . . . . . . . . . . . . . . . . . . 14.6.3 Maximum Reduction of Area in a Single Pass . . . . . . . . . . . 668 14.6.4 Redundant Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . 671 14.6.5 Drawing Stress Versus Die-cone Angle: Optimum Cone 673 676 Angle, Dead Zone and Shaving . . . . . . . . . . . . . . . . . . . . . . 677 14.7 Tube Drawing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 678 14.7.1 Close-Pass Plug Drawing Stress and Load . . . . . . . . . . . . . . 678 14.7.2 Close-Pass Mandrel Drawing Stress . . . . . . . . . . . . . . . . . . . 680 14.7.3 Maximum Reduction of Area in a Single Pass . . . . . . . . . . . 681 14.7.4 Tube Sinking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 685 14.7.5 Equilibrium Condition of Forces Acting 691 on a Floating Plug . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.8 Application of Slip-Line Field to Strip Drawing . . . . . . . . . . . . . . . . . 14.9 Upper-Bound Solution for Strip Drawing . . . . . . . . . . . . . . . . . . . . . . 14.10 Solved Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 Deep Drawing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 693 15.1 Fundamentals of Deep Drawing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 693 15.1.1 Stresses and Deformation in a Deep-Drawn Cup . . . . . . . . . . 695 15.2 Deep-Drawing Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 697 15.2.1 Derivation of Mathematical Expression . . . . . . . . . . . . . . . . 698 15.3 Formability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 702 15.3.1 Strain Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 702 15.3.2 Maximum Strain Levels: The Forming Limit Diagram . . . . . . 703 15.4 Deep Drawability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 703 15.4.1 Plastic Strain Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 704 15.4.2 Drawing Ratio. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 705 15.5 Effects of Process and Material Variables. . . . . . . . . . . . . . . . . . . . . . 706 15.5.1 Effect of Drawing Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . 706 15.5.2 Radii of Die and Punch . . . . . . . . . . . . . . . . . . . . . . . . . . . 708 15.5.3 Punch-to-Die Clearance . . . . . . . . . . . . . . . . . . . . . . . . . . . 708 15.5.4 Drawing Speed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 709 15.5.5 Friction and Lubrication . . . . . . . . . . . . . . . . . . . . . . . . . . . 709 15.5.6 Restraint of Metal Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . 710 15.5.7 Material Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 711 15.6 Evaluation of Formability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 711 15.6.1 Marciniak Biaxial Stretching Test . . . . . . . . . . . . . . . . . . . . 711 15.6.2 Swift Cup Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 712 15.6.3 Ericksen and Olsen Cup Tests. . . . . . . . . . . . . . . . . . . . . . . 712 15.6.4 Fukui Conical Cup Test . . . . . . . . . . . . . . . . . . . . . . . . . . . 712 15.6.5 Hole Expansion Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 713 15.6.6 Forming Limit Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . 713

xx Contents 15.7 Deep Drawing Defects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 714 15.8 Solved Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 716 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 719 16 High-Energy Rate Forming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 721 16.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 721 16.2 Fundamentals of HERF Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . 722 16.2.1 Advantages and Limitations . . . . . . . . . . . . . . . . . . . . . . . . 722 16.3 Explosive Forming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 724 16.3.1 Standoff or Unconfined Technique. . . . . . . . . . . . . . . . . . . . 725 16.3.2 Contact or Confined Technique . . . . . . . . . . . . . . . . . . . . . . 727 16.4 Electromagnetic Forming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 728 16.5 Electrohydraulic Forming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 731 16.6 High-Energy Rate Forging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 732 16.7 Other HERF Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 733 16.8 Solved Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 734 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 735 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 737

About the Author Amit Bhaduri is a faculty member in the Department of Metallurgical and Materials Engineering, IIT Kharagpur, where he has taught since 1984, prior to which he was a lecturer at Regional Engineering College Durgapur (presently, National Institute of Technology Durgapur). He has earned his B.E. and M.Tech. in Metallurgical Engineering from University of Calcutta and IIT Kanpur, respectively. He was awarded the ‘Indranil Award’ from Mining, Geological and Metallurgical Institute of India as well as other awards and prizes for his subject knowledge. He has over 34 years of teaching experi- ence, over the course of which he has taught several subjects. His special focus has been on teaching ‘Mechanical Testing of Materials’ and ‘Mechanical Working of Materials’, which he has taught for about 20 years and 10 years, respectively, at IIT Kharagpur. xxi

Part I Mechanical Properties of Metals and Alloys

Tension 1 Chapter Objectives • Engineering and true strain and stress, and their relationship, Poisson’s ratio and volume strain. • Generalized Hooke’s law for a triaxial state of stress. • Relationship between principal normal and shear stresses and explanation with Mohr’s stress circle. • Mechanisms of plastic deformation of crystalline solids: slip and twinning. A brief introduction to dislocations. • Reasons of strain hardening, relationship between yield strength and grain size, and effect of stacking fault energy on the extent of strain hardening and twinning. • Various strengthening methods. • Yielding criteria, invariant functions of stress and strain, and flow rules. • Discussion of various types of stress–strain curve in tension. • Empirical relationship for flow curve, strain-hardening coefficient, tensile instability, stress field at the neck. • Different elastic (linear and nonlinear) and inelastic tensile properties, geometry of tensile specimen and ductility measurement. • Influence of temperature, strain rate and testing machine on tensile properties. • Notch tensile test and tensile fracture. • Problems and solutions. 1.1 Introduction tensile loading) or repulsive force (as in case of compressive loading) that opposes the applied force. If the applied force The deformation of a solid may be constituted of change in is not too large, the interatomic force may be sufficient to size or volume, called dilation, or change in shape, known as completely resist the applied force, allowing the object to distortion. Hence, mechanical deformation can be defined as return immediately to its original state on complete release a change in the size or shape of an object due to an applied of the applied load. This type of instantaneous self-reversing force, which may be tensile or pulling, compressive (push- deformation is called recoverable or elastic deformation, ing), shear, bending or torsion (twisting). When deformation which involves stretching or squeezing of the bonds, but the occurs, the atoms are displaced from their equilibrium atoms do not slip past each other. However, recoverable interatomic spacing. This atomic displacement causes to deformation may also be time dependent and is known as develop internal interatomic attractive force (as in case of anelastic deformation. A larger applied force may lead to a permanent set or plastic deformation of the material or even © Springer Nature Singapore Pte Ltd. 2018 3 A. Bhaduri, Mechanical Properties and Working of Metals and Alloys, Springer Series in Materials Science 264, https://doi.org/10.1007/978-981-10-7209-3_1

4 1 Tension to its structural failure. Plastic deformation involves the r ¼ P cos h ð1:1Þ breaking of a limited number of atomic bonds by some of the A atoms or molecules in the material and the movement of these atoms or molecules from their original positions to new And the shear stress lying on this plane will be equilibrium sites. This atomic movement in the material results in an extension under tension or reduction under s ¼ P sin h ð1:2Þ compression, which will be more than that produced by A displacement of atoms due to elastic stretching or squeezing of the bonds. These atoms or molecules in their new posi- Since the shear stresses are generally inclined at arbitrary tions will form new bonds, which will not allow the atoms or angles to the coordinate axes lying on the plane of action, molecules to return to their original states when the applied each shear stress may be further resolved into two compo- load is completely removed. Thus on complete release of nents parallel to the directions of those coordinate axes. In applied load, the deformation becomes inelastic or irrecov- the above example if the shear stress is inclined at an angle erable or irreversible, which is called permanent or plastic of / with the y-axis, then it will make an angle of ðp=2Þ À / deformation. Permanent plastic deformation discussed in this with the x-axis, where both axes are lying on the plane of the chapter is a function of load and not a function of time. area A: Hence, the shear stress in the x direction is Time-dependent permanent deformation called creep will be discussed in Chap. 7. In general, if a material exhibits the s ¼ P sin h sin / ð1:3aÞ ability to undergo plastic deformation under load it is duc- A tile, otherwise it is brittle. In a ductile material, atomic or molecular bond is reformed easily, whereas it cannot And in the y direction is be reformed easily in a brittle material. A completely brit- tle material like glass will fracture almost at the end of s ¼ P sin h cos / ð1:3bÞ elastic range, whereas usually all brittle metals, e.g. white A cast iron, show some slight amount of plasticity prior to their fracture. Therefore, in general one normal stress and two shear stresses may act on a given plane. The static tensile test performed under uniaxially applied load is the most extensively used experimental test method To describe the stress at a point, it is convenient to which characterizes many important mechanical properties (i) construct an elemental cube around it and (ii) describe the of materials. From this test, one can know the material’s stresses on its faces, as shown in Fig. 1.1a. It appears from elastic properties such as elastic limit, proportional limit, Fig. 1.1a that nine quantities rx; ry; rz; sxy; syx; syz; szy; szx; modulus of elasticity and modulus of resilience in tension. sxz; are required to define the state of stress at a point. The For engineering purposes, most commonly used inelastic subscript of each normal stress indicates the direction in properties such as yield strength, tensile strength, breaking which it acts. Obviously, the normal stress must act on the or fracture strength, ductility and toughness can also be plane normal to this direction. For example, rx is the normal measured from this test record. Just like chemical compo- stress acting in the direction of the x-axis, i.e. acting on the sition of a metal or an alloy, the tensile strength of a material plane normal to the x-axis. By convention, if the value of is used to identify it. As this test provides so much infor- normal stress is positive it indicates tension (pulling), while mation, the importance of this test in research as well as in negative value of normal stress denotes compression industry may easily be understood. (pushing). For each shear stress, the first and second sub- scripts identify, respectively, the plane and the direction in 1.1.1 Description of Stress at a Point which the shear stress acts. As any plane is designated by its normal, so the first subscript of each shear stress also refers Stress is defined as the applied load or force per unit area. to this normal. For example, sxy is the shear stress on the Often, these stresses are inclined at some arbitrary angles to plane normal to the x-axis acting in the direction of the y-axis the areas over which they act. In such cases, to describe and syx is the shear stress lying on the plane normal to the y- stresses conveniently, each of those stresses is resolved into axis acting in the direction of the x-axis. If a shear stress two components, a normal stress r, perpendicular to the area directs in the positive or negative direction, respectively, on of action, and a shear stress or shearing stress s lying on the the positive or negative face of a unit cube, it is positive, but plane of action. For example, if a force P applied on a plane if a shear stress directs in the positive direction of a negative of area A; makes an angle h with the normal to this plane, face of a unit cube and vice versa, it is negative. Disre- then the normal stress is given by garding the type of normal stresses that are present, the shear stresses shown in Fig. 1.1b(i) are all positive and those shown in Fig. 1.1b(ii) are all negative shear stresses. For simplification, the areas of the faces of the unit cube are assumed to be small enough so that there are negligible

1.1 Introduction 5 (a) z 1.2 Strain σz τzx τzy Change in dimension of the body is observed in all solid τyz ∆z materials when they are deformation with externally applied τyx τxz axial loads. The ratio of change in length to the original σy τxy τyx σy length is called linear or normal strain or simply, strain, and ∆x y this ratio, i.e., strain, is a dimensionless quantity. Strain τyz σx measured during elastic deformation is known as elastic or O +y recoverable strain and that during plastic deformation is permanent or plastic strain. ∆y x To achieve uniform tensile deformation of a given material, original length of the specimen is fixed by putting (b) + y two gage marks on its surface in its undeformed condition. The distance between these marks is the original gage –x +x –x +x length, say L0. The fixation of gage length depends on the original cross-sectional area of the specimen, which will be –y –y discussed later. If the specimen is loaded uniaxially in ten- (b)(i) (b)(ii) sion, the original gage length will increase by a certain amount, say DL; called the extension or deformation, which Fig. 1.1 a Description of stresses acting on faces of an elemental unit will be accompanied by a decrease in cross-sectional area of cube. b Figure showing sign convention for shear stresses: b(i) Positive the specimen to maintain the constancy in volume during shear stresses. b(ii) Negative shear stresses plastic deformation, to be discussed subsequently. The dis- tance between the two gage marks after tensile deformation is measured, and let this gage length at the moment of measurement be L: Conventional strain or engineering strain, denoted by e; is the linear strain referred to the original gage length; i.e., it is the ratio of change in the gage length to the original gage length. Engineering strain or conventional strain, changes in the stresses over the faces. By taking the sum- e ¼ DL ¼ L À L0 L0 L0 mation of the moments of the forces about the z–axis, it has ð1:5Þ been shown below that sxy ¼ syx: À ÁÀ Á This definition of strain does not give a true picture about sxyDyDz Dx ¼ syxDxDz Dy the deformation behaviour of the material because it is based on original constant dimension, L0; which changes contin- ) sxy ¼ syx ð1:4aÞ uously during tensile deformation. To remove this difficulty, Ludwik (1909) first proposed the definition of true strain or Similarly, it can be shown that natural strain, which will be denoted by e: It is defined as the summation of change in length at any instant divided by syz ¼ szy ð1:4bÞ the instantaneous gage length. This means true strain or szx ¼ sxz ð1:4cÞ natural strain, Thus, the state of stress at a point is completely described e ¼ X L1 À L0 þ L2 À L1 þ L3 À L2 þ L4 À L3 þ Á Á Á by six components—three normal stresses, rx; ry; rz, and L0 L1 L2 L3 three shear stresses, sxy; syz; szx: ð1:6Þ Similarly, the state of strain at a point is completely If the increment of true strain, de; is caused by an increase described by six components—three linear or normal strains, ex; ey; ez, and three shear strains, cxy; cyz; czx: Linear or in length, dL; based on the instantaneous gage length L; i.e., normal strain or, simply, strain is discussed in this chapter, and shear strain has been defined and described in Chap. 5. if de ¼ dL ; then L

6 1 Tension True strain or natural strain; ð1:7Þ Tension +2 Engineering strain +1 True strain Ze ZL e ¼ de ¼ dL ¼ ln L L L0 0 L0 Therefore, true strain is the natural logarithm of the ratio Strain of the gage length at the moment of observation to the 0 1 2 3 4L original gage length. L0 Similar to true strain, the total engineering strain e can be –1Compression obtained from an infinitesimal increment by integration, as –2 follows: Fig. 1.2 Comparison of engineering and true strain as a function of Ze ZL 1 ZL dL ¼ L À L0 ð1:8Þ ratio of instantaneous to original gage length e ¼ de ¼ dL ¼ L0 L0 L0 for very small values of L=L0; which is generally in the range of elastic deformation, the values of e and e are nearly 0 L0 L0 equal, as shown in Table 1.1. Strain is a dimensionless quantity, since both numerator and denominator of this parameter are expressed in units of length. 1.2.1 True Versus Engineering Strain Assuming a homogeneous distribution of strain along the 1.2.2 Advantages of True Strain Over gage length of the tensile specimen, the relationship between Engineering Strain true strain, e; and conventional linear strain, e; is shown as follows: The use of true strain has some advantages over engineering strain. One advantage is that at any value of L=L0 in tension From (1.5), and reciprocal value of that in compression, the absolute value of true strain is always same, whereas the absolute e ¼ L À L0 ¼ L À 1 or, L ¼ 1 þ e ð1:9Þ value of engineering strain is always more in tension than in L0 L0 L0 compression. Therefore, equivalence is obtained for tensile and compressive deformation if true strain is used. This is From (1.7) and (1.9), illustrated in Table 1.2. e ¼ ln L ¼ lnð1 þ eÞ ð1:10Þ Another advantage of true strain is that the total true L0 strain calculated from initial and final length of a specimen is equal to the sum of true strains for every small length Strain is plotted against different values of instantaneous increment from initial to final value. This means the total to original length ratio, L=L0; as shown in Fig. 1.2, which strain is: shows a comparison of true and engineering strain. From eL0ÀLn ¼ eL0ÀL1 þ eL1ÀL2 þ eL2ÀL3 þ eL3ÀL4 þ Á Á Á ð1:11Þ Fig. 1.2, it can be seen that as the values of L=L0 increase the engineering strain, e; becomes always greater than the true But, this is not the case with the engineering strain; i.e., strain, e; for uniform tensile deformation and the difference the total engineering strain is: between e and e increases with L=L0: Similarly, for uniform compressive deformation, when the value of L=L0 decreases eL0ÀLn 6¼ eL0ÀL1 þ eL1ÀL2 þ eL2ÀL3 þ eL3ÀL4 þ Á Á Á ð1:12Þ below 1, e is always less than e: If the length of a specimen could be compressed to zero, i.e., at L=L0 ¼ 0; e ¼ À1; whereas e is infinite which is much more logical. However, Table 1.1 Comparison of true Nature of deformation L=L0 e ¼ lnðL=L0Þ e ¼ ðL=L0Þ À 1 strain, e; and engineering strain, e; 0.01 for values of L=L0 close to 1 Tension 1.01 0.0099 −0.01 Compressiona 0.99 −0.01005 aStrain value with negative sign indicates compressive deformation

1.2 Strain 7 Table 1.2 Comparison of true Nature of deformation L=L0 e ¼ lnðL=L0Þ e ¼ ðL=L0Þ À 1 strain, e; and engineering strain, e; Tension 2 ln 2 1 for equivalent tensile and 3 ln 3 2 compressive deformation 4 ln 4 3 1=2 À ln 2 À 1=2 Compression 1=3 À ln 3 À 1=3 1=4 À ln 4 À 1=4 The following example will help to understand the above: denoted by the symbol, m: If ex is longitudinal strain along the Let; L0 ¼ 10 cm, L1 ¼ 11 cm, L2¼ 12:1 cm, L3 ¼ loading axis x and the lateral strains in other two transverse 13:31 cm and L4 ¼ 14:641 cm; directions y and z are, respectively, ey and ez; then Therefore, ey ¼ ez ¼ Àm ex ð1:13Þ eL0ÀL1 þ eL1ÀL2 þ eL2ÀL3 þ eL3ÀL4 Only the absolute value of v is used in calculations. Poisson’s ratio ðmÞ is a dimensionless quantity, which usu- ¼ ln 11 þ 12:1 þ 13:31 þ ln 14:641 ally varies from 1=4 to less than 1=2 during elastic defor- 10 ln ln 12:1 13:31 mation of nonporous solids. For a perfectly isotropic elastic material, Poisson’s ratio m is 1=4; but for most materials the 11 values of m are close to 1=3 during elastic deformation. For an ideal plastic material or during plastic deformation of any which proves (1.11). nonporous solid, the value of m is considered to be 1=2: Again, The volume strain is the change in volume per unit vol- eL0ÀL1 þ eL1ÀL2 þ eL2ÀL3 þ eL3ÀL4 ume. Consider an element of initial volume, V0; which is subjected to tensile loading along x-axis. Assume that, initial ¼ 1 þ 1:1 þ 1:21 þ 1:331 ¼ 0:4 volume, V0 ¼ L0w0h0; where L0 is the initial length of the 10 11 12:1 13:31 specimen along x-axis, i.e. along longitudinal axis, and the Whereas total conventional strain is original lateral dimensions along y and z-axes, i.e. along transverse axes are, respectively, w0 and h0: After loading at eL0ÀL4 ¼ 4:641 ¼ 0:4641 any instant, the length, L; in longitudinal direction and the 10 respective lateral dimensions, w and h; are obtained with the help of (1.5), as shown below: and so, (1.12) is justified. The use of true strain offers another additional advantage ÀÁ L ¼ L0ð1 þ exÞ; w ¼ w0 1 þ ey and h ¼ h0ð1 þ ezÞ: when considering the constant volume deformation process in which ex þ ey þ ez ¼ 0; [see (À1.21)] ÁIn contrast, a less The new volume, V; is then given by V ¼ L w h: convenient relationship ð1 þ exÞ 1 þ ey ð1 þ ezÞ ¼ 1; [see (1.19)] is found for the case of engineering strain. These ÀÁ above relationships have been developed in the next ) V ¼ L0 w0 h0ð1Àþ exÞ 1Áþ ey ð1 þ ezÞ Sect. 1.2.3. ¼ V0ð1 þ exÞ 1 þ ey ð1 þ ezÞ 1.2.3 Poisson’s Ratio and Volume Strain ð1:14Þ If a body is subjected to a tensile or compressive load in a Hence, the volume strain, Δ, is given by particular direction, then not only strain takes place in the direction of loading, known as longitudinal strain, but also D ¼ DV ¼ V À V0 ð1:15Þ strains, known as lateral strains, are observed in directions V0 V0 perpendicular to the direction of loading. Extension along the axis of loading causes lateral contraction, whereas compres- Putting the value of V from (1.14), in (1.15), we get sion along the axis of loading will cause lateral expansion. The strains in the lateral directions have been found by  ÀÁ à experience to be a constant fraction of the longitudinal strain. D ¼ V0 ð1 þ exÞ 1 þ ey ð1 þ ezÞ À 1 The ratio of lateral strain to longitudinal strain under condi- À ÁV0 tions of uniaxial loading is called ‘Poisson’s ratio’, usually ¼ ð1 þ exÞ 1 þ ey ð1 þ ezÞ À 1 ð1:16Þ From (1.13), as 1 þ ey ¼ 1 þ ez ¼ 1 À mex; so (1.16) can be written as

8 1 Tension hi 1.3 Conventional and True Stresses À mexÞ2 D ¼ Âð1 þ exÞð1 2mÞ þ e2x À1 Á m2ex3Ã ¼ 1 þ exð1 À Àm2 À 2m þ À 1 ð1:17Þ The ratio of applied load at any instant to average original cross-sectional area along the gage length of a specimen is If ex is small, say 0.01 or less as in elastic deformation, called conventional stress or engineering stress, which will e2x and ex3 can be neglected in comparison with ex; then be denoted by the symbol S: If the applied load is P and the (1.17) takes the following form: original cross-sectional area is A0; then D ¼ DV ¼ exð1 À 2mÞ ð1:18Þ Conventional stress or engineering stress; S ¼ P ð1:24Þ V0 A0 Hence, volume strain is always positive as long as Natural stress or true stress is defined as the ratio of applied load at any instant to the minimum cross-sectional m \\ 0:5: During elastic deformation, the value of m is always area of a specimen along the gage length at that same instant. The symbol r; will be used for the true or natural stress. If less than 1=2 and therefore, (1.18) indicates that an increase A is the instantaneous cross-sectional area over which the load P acts, then in volume or dilation accompanies elastic extension. During plastic deformation, considering m ¼ 0:5 in (1.18), volume strain comes out to be zero; i.e., no volume change will occur during plastic deformation. It has been observed that any nonporous solid after large Natural stress or true stress; r ¼ P ð1:25Þ A plastic strain shows the density change less than 0.1%. As the mass of solid remains same, for engineering purpose it During elastic deformation where the difference between the original and the instantaneous cross-sectional area is can be considered that the volume of a nonporous solid insignificant, it is not necessary to make the above distinc- tion regarding the stress. But during plastic deformation, the remains constant during plastic deformation. With this instantaneous cross-sectional area, A, differs remarkably from the original cross-sectional area, A0: With the progress engineering approximation, the value of m during plastic of the tension or compression test in the plastic region, the value of A will, respectively, show gradual decrease or deformation can be derived from (1.16) as follows: increase from the constant value of A0: So, it is important to distinguish between the above two definitions of stress Substituting D ¼ DV=V0 ¼ 0 into (1.16), we get during plastic deformation. ÀÁ ð1:19Þ ð1 þ exÞ 1 þ ey ð1 þ ezÞ ¼ 1 Taking natural logarithm to both the sides of equality sign 1.3.1 Relationship Between True in (1.19), and Engineering Stresses During Plastic Deformation ÀÁ lnð1 þ exÞ þ ln 1 þ ey þ lnð1 þ ezÞ ¼ ln 1 ¼ 0 ð1:20Þ ÀÁ From (1.10), ex ¼ lnð1 þ exÞ; ey ¼ ln 1 þ ey ; ez ¼ lnð1 þ ezÞ; and substituting them into (1.20), we obtain ex þ ey þ ez ¼ 0 ð1:21Þ Equation (1.13) can be written in terms of true strains as Assuming both constancy in volume and a homogeneous follows: distribution of strain along the gage length of a tensile specimen, the following relationship between engineering ey ¼ ez ¼ Àm ex ð1:22Þ and true stress has been derived, which can only be used From (1.21) and (1.22), we get during uniform plastic deformation. ex À 2mex ¼ exð1 À 2 mÞ ¼ 0 ð1:23Þ From the constancy of volume, V; during plastic defor- mation as mentioned in Sect. 1.2.3, one can write, As ex 6¼ 0; rather ex is a large finite quantity for plastic V ¼ A0L0 ¼ A L; or, A0 ¼ L ð1:26Þ deformation, so in (1.23), ð1 À 2 mÞ ¼ 0 : Therefore, A L0 m ¼ 0:5; for plastic deformation.

1.3 Conventional and True Stresses 9 where A0 and A are, respectively, initial and instantaneous which no shear stress act; i.e., the shear stress is zero and cross-sectional areas and L0 and L represent, respectively, only normal stress act. The direction perpendicular to the initial and instantaneous gage length. From (1.25) and (1.26), principal plane is called the principal direction along which the principal stress acts. The three principal directions are True stress; r ¼ P ¼ P A0 ¼ P L ð1:27Þ often designated by the orthogonal axes ‘1’, ‘2’ and ‘3’. As A A0 A A0 L0 any plane is designated by its normal, so the principal planes are also represented by their corresponding principal axes. In From (1.9) since L=L0 ¼ 1 þ e; and from (1.24) since three-dimensional state of stress, all of the three principal P=A0 ¼ S; hence we get from (1.27): stresses acting at a point are generally unequal. This is called a triaxial state of stress. The state of stress is known as r ¼ Sð1 þ eÞ ð1:28Þ cylindrical, if two of the three principal stresses are equal, and is said to be hydrostatic, or spherical, if all three prin- From (1.10), since 1 þ e ¼ expðeÞ; cipal stresses are equal. ) r ¼ S expðeÞ ð1:29Þ The generalized three-dimensional Hooke’s law in terms of true stress, r; and true strain, e; is narrated below, but 1.4 Elastic Stress–Strain Relations similar equations will also hold good for the engineering stress, S; and engineering strain, e;—the only requirement is In general, the applied load P in the elastic range is found to to put ‘S’ in place of ‘r’ and ‘e’ in place of ‘e’, in the be proportional to the elastic deformation DLe; which means following equations. Let r1; r2 and r3 are the respective principal true elastic stresses acting along the three mutually P ¼ MDLe ð1:30Þ perpendicular principal axes 1, 2 and 3. Similarly, the principal true elastic strain along the three orthogonal prin- where M ¼ proportionality factor expressing the body cipal strain axes 1, 2 and 3 are, respectively, represented by stiffness, often referred to as the spring constant. Since the e1; e2 and e3: Principal strain axes are those coordinate axes conventional or engineering stress S / P and the conven- along which there are no shear strains. An element oriented tional or engineering strain e / DL; (1.30) can be written as along one of the principal strain axes will undergo pure extension or contraction without any rotation or shear strain. S ¼ Ee ð1:31aÞ For an isotropic body, the directions of principal strain coincide with those of principal stress. Under the action of where E ¼ proportionality constant, known as the ‘Young’s the triaxial state of stresses on a body, the total true elastic modulus’ or the ‘modulus of elasticity’, whose dimension is strain along any principal axis will be the summation of the the same as that of the stress, i.e. N m2 or Pa; because the normal strain occurring due to the principal stress acting strain is a dimensionless quantity. The magnitude of ‘E’ is along that axis and the transverse strains resulting from the considered to be the same in both tension and compression. Poisson’s effect of the principal stresses acting along the other two axes. With the help of (1.31b) and (1.22), the During elastic deformation, the values of engineering elastic stress–strain relations for a triaxial state of stress can strain, e, and true strain, e, are nearly equal as shown in be developed as shown in Table 1.3. Table 1.1, i.e. the engineering strain, e % the true strain, e: Since the difference between the original and the instanta- Now, due to application of principal elastic stresses along neous cross-sectional area is insignificant, i.e. A0 % A in the the three orthogonal principal directions, the total principal range of elastic deformation, so the engineering stress, S % strain produced in each of the principal directions of ‘1’, ‘2’ the true stress, r: Hence, (1.31a) may also be expressed as and ‘3’ is obtained by adding the respective strain compo- nents in each column of Table 1.3 and expressed by the r ¼ Ee ð1:31bÞ following equations: Equations (1.31a) and (1.31b) are called the ‘Hooke’s e1 ¼ 1 ½r1 À mðr2 þ r3ފ ð1:32aÞ law’, which describes that stress is proportional to strain in E the linear elastic region of the stress–strain curve. e2 ¼ 1 ½r2 À mðr3 þ r1ފ ð1:32bÞ E 1.4.1 Three-Dimensional State of Stress e3 ¼ 1 ½r3 À mðr1 þ r2ފ ð1:32cÞ The normal maximum stress acting on a principal plane is E called the principal stress. A principal plane is a plane on

10 1 Tension Table 1.3 Elastic stress–strain relations Principal Principal strain in principal direction Principal strain in principal direction Principal strain in principal direction stress ‘1’ ‘2’ ‘3’ r1 Normal strain, Transverse strain, Transverse strain, e1 ¼ r1 e2 ¼ Àm r1 e3 ¼ Àm r1 E E E r2 Transverse strain, Normal strain, Transverse strain, e1 ¼ Àm r2 e2 ¼ r2 e3 ¼ Àm r2 E E E r3 Transverse strain, Transverse strain, Normal strain, e1 ¼ Àm r3 e2 ¼ Àm r3 e3 ¼ r3 E E E The principal elastic stress can be expressed in terms of comparison with the other two dimensions. For example, if a principal elastic strains by solving the above (1.32) as follows: thin plate is loaded in the plane of the plate, there is no stress normal to a free surface. If we assume r3 ¼ 0; then (1.32a) By adding the (1.32a), (1.32b) and (1.32c), we get reduces to as follows: e1 þ e2 þ e3 ¼ 1 ðr1 þ r2 þ r3Þ À 2m ðr1 þ r2 þ r3Þ 1  E  E E e1 ¼ ½r1 À m r2Š ð1:35aÞ 1 À 2m ¼ E ðr1 þ r2 þ r3Þ  E e2 ¼ 1 ½r2 À m r1Š ð1:35bÞ Or, r1 þ r2 þ r3 ¼ 1 À 2m ðe1 þ e2 þ e3Þ ð1:33aÞ E From (1.32a), e3 ¼ À mðr1 þ r2Þ ð1:35cÞ E e1 ¼ 1 ½r1 þ mr1 À m r1 À mðr2 þ r3ފ It is to be noted that if r1 ¼6 Àr2; the strain e3 along the E  principal strain axes ‘3’ is not zero even when the stress 1þm m ð1:33bÞ along that axis, r3 ¼ 0: E E ¼ r1 À ðr1 þ r2 þ r3Þ By adding (1.35a) and (1.35b) and subtracting (1.35b) With (1.33a) and (1.33b), from (1.35a), we get, respectively, the following (1.36a) and (1.36b).     e1 ¼ 1 þ m À 1 m m ðe1 þ e2 þ e3Þ  E À2 r1 e1 þ e2 ¼ 1 À m ðr1 þ r2Þ; E    & ' ð1:36aÞ E mE E ) r1 ¼ 1þm e1 þ ð1 þ mÞð1 À 2 mÞ ðe1 þ e2 þ e3Þ Or, r1 þ r2 ¼ 1Àm ðe1 þ e2Þ ð1:34aÞ  þ m e1 À e2 ¼ 1 E  ðr1 À r2Þ;  Similarly for other two principal axes,  & ' E ð1:36bÞ 1þm r2 ¼ E e2 þ mE ðe1 þ e2 þ e3Þ Or, r1 À r2 ¼ ðe1 À e2Þ 1þm ð1 þ mÞð1 À 2mÞ ð1:34bÞ Solving (1.36a) and (1.36b), the principal stresses r1 and r2 for a biaxial state of plane stress is obtained, as And shown in the following (1.37a) and (1.37b).  & mE ' &'  E ð1 þ mÞð1 À 2 mÞ r3 ¼ 1þm e3 þ ðe1 þ e2 þ e3Þ 2r1 ¼ E 2ðe1 þ me2Þ ; or, r1 ¼ E ðe1 þ m e2Þ 1 À m2 1 À m2 ð1:34cÞ ð1:37aÞ Two-dimensional state of stress is called plane stress &'  condition, in which the stress is zero in one of the primary 2ðe2 þ m e1Þ E directions. This condition is often approached in practice 2r2 ¼ E 1 À m2 ; or, r2 ¼ 1 À m2 ðe2 þ m e1Þ when one of the dimensions of the object is much smaller in ð1:37bÞ

1.4 Elastic Stress–Strain Relations 11 Very often in experimental stress analysis, the principal P1=A1; r2 ¼ P2=A2; and r3 ¼ P3=A3; although for a cube strains e1 and e2 are measured directly by strain gages and A1 ¼ A2 ¼ A3: An arbitrary plane ‘aa1b1b’ that is normal to then the corresponding principal stresses r1 and r2 can be the principal plane ‘3’ denoted by ‘OABC’ or ‘GDEF’, but calculated from (1.37). inclined to the principal axes ‘1’ and ‘2’ is shown by dotted If the strain is zero in one of the primary directions, the lines in Fig. 1.3 for subsequent analysis. Figure 1.3 shows situation is called plane strain condition. It is found typically when one of the dimensions of the object is much larger than that the plane ‘aa1b1b’ is inclined to the principal plane ‘1’, the other two dimensions as in a long rod or if the object is i.e. plane ‘ABED’ or ‘OCFG’ at an angle of h: physically restrained to deform in one direction. If we assume principal strain in direction ‘3’ is zero, i.e. e3 ¼ 0; it Let us first consider a plan view of the cube of Fig. 1.3, can be written from (1.32c) as follows: which is shown in Fig. 1.4. In Fig. 1.4, the angle of incli- Stress in the principal direction ‘3’; r3 ¼ mðr1 þ r2Þ ð1:38Þ nation of ‘ab’ or ‘a1b1’, which is the trace of the plane ‘aa1b1b’, is indicated by an angle h between the normal ‘N’ Therefore, although the strain in a direction is zero, a to ‘ab’ or ‘a1b1’ and the principal axis ‘1’. restraining stress is present in that direction. Substituting the value of stress from (1.38) into (1.32), we get the following On the oblique plane ‘aa1b1b’, i.e. on ‘ab’ or ‘a1b1’ in equations for strain in terms of stresses under plane strain Fig. 1.4, normal stress r and shear stress s will act as a result condition. of the applied principal stresses r1 and r2: Since the area e1 ¼ 1 ½r1 À mfr2 þ mðr1 þ r2ÞgŠ ‘A1’ of plane ‘ABED’ or ‘OCFG’ that is normal to the E principal axes ‘1’, is inclined to the area of the plane ‘aa1b1b’ by the angle h; the stress, say r1ab; acting on this oblique plane ‘aa1b1b’, i.e. on ‘ab’ or ‘a1b1’ along the principal axis ‘1’, corresponding to the principal stress r1; is given by 1 ÂÀ Á à ð1:39aÞ ra1b ¼ P1 ¼ r1 cos h ð1:40aÞ E 1 r1 mÞr2 A1=cos ¼ À m2 À mð1 þ h e2 ¼ 1 ½r2 À mfmðr1 þ r2Þ þ r1gŠ Similarly, as the area ‘A2’ of plane ‘BCFE’ or ‘AOGD’ E ÂÀ Á à ð1:39bÞ that is normal to the principal axes ‘2’, is inclined to the area 1 1 r2 mÞr1 of the plane ‘aa1b1b’ by the angle ð90 À hÞ; so the stress, ¼ E À m2 À mð1 þ say ra2b; acting on this oblique plane ‘aa1b1b’, i.e. on ‘ab’ or e3 ¼ 0 ð1:39cÞ ‘a1b1’ along the principal axis ‘2’, corresponding to the principal stress r2; is given by r2ab ¼ P2 À hÞ ¼ r2 sin h ð1:40bÞ A2=cosð90 1.5 Elements of Plastic Deformation 1.5.1 Relationship Between Principal Normal 3 σ3 and Shear Stresses Let us assume that all six faces of a cube shown in Fig. 1.3 G a1 D b1 F are principal planes, which are subjected to a triaxial state of O E σ2 true tensile principal stress. Suppose these stresses acting on 2 aC the principal planes, i.e. faces of the cube along the three A b θ principal directions or axes designated by ‘1’, ‘2’ and ‘3’ are, 1 σ1 respectively, r1; r2 and r3: Let us further assume that r1 is B the algebraically greatest principal normal stress, r2 is the algebraically smallest principal normal stress, and r3 has an Fig. 1.3 A cubical element subjected to a triaxial state of tensile intermediate value, i.e. r1 [ r3 [ r2: As any plane is des- ignated by its normal, so the principal planes are also rep- principal stress, showing an arbitrary plane ‘aa1b1b’ inclined to the resented by their corresponding principal axes, i.e. by ‘1’, ‘2’ principal plane ‘1’, i.e., plane ‘ABED’ or ‘OCFG’ at an angle of h; but and ‘3’. Let the cross-sectional areas of the principal planes normal to the principal plane ‘3’, i.e., plane ‘OABC’ or ‘GDEF’ lying normal to the principal axes ‘1’, ‘2’ and ‘3’ are denoted, respectively, by A1; A2 and A3 ; on which the respective applied loads are P1; P2 and P3; so that r1 ¼

12 1 Tension G, O a, a Thus, r is shown to be bounded by the magnitudes of r1 1 C, F and r2: When h ¼ 45; the shear stress on such a plane attains the maximum value because sin 2h ¼ 1: This maxi- θ σ τ1 22 mum shear stress is known as principal shear stress, which b, b σN has been denoted by s3; as this plane of maximum shear 1 B, E stress is parallel to the principal axis ‘3’. Hence, substituting sin 2h ¼ 1 in (1.41b) we get D, A r1 À r2 2 Principal shear stress; s3 ¼ ð1:42Þ σ From (1.41a) and (1.41b), the magnitudes of normal and 1 shear stresses can be calculated on any plane normal to the Fig. 1.4 Plan view of Fig. 1.3, showing element subjected to biaxial tensile principal stresses principal plane ‘3’ denoted by ‘OABC’ or ‘GDEF’ in Fig. 1.3. If we desire to get the normal stress r0 and shear Each of the above respective stresses ra1b and r2ab; cor- stress s0 on an oblique plane whose normal makes an angle responding to the principal stresses r1 and r2; can be h0 ¼ 90 þ h with the principal axes ‘1’, i.e. on an oblique resolved into components perpendicular and parallel to the plane normal to the plane ‘aa1b1b’, (1.41a) and (1.41b) become: oblique plane ‘aa1b1b’. The component perpendicular to the oblique plane ‘aa1b1b’, i.e. to ‘ab’ or ‘a1b1’ in the direction r0 ¼ r1 cos2ð90 þ hÞ þ r2 sin2ð90 þ hÞ of the normal ‘N’, is the normal stress, r; which is given ¼ r1ðÀ sin hÞ2 þ r2ðcos hÞ2 below: From ra1b : normal stress, r ¼ r1ab cos h ¼ r1 1 À cos 2h þ r2 cos 2h þ 1 ð1:43aÞ ¼ r1 cos2 h; ½from ð1.40aފ: 2 2 From r2ab : normal stress, r ¼ r2ab cosð90 À hÞ ¼ r1 þ r2 À r1 À r2 cos 2h ¼ r2 sin2 h; ½from ð1:40bފ: 22 s0 ¼ r1 sinð90 þ hÞ cosð90 þ hÞ Hence, from both r1ab and ra2b : total normal stress, À r2 sinð90 þ hÞ cosð90 þ hÞ ð1:43bÞ r ¼ r1 cos2 h þ r2 sin2 h ¼ ðr1 À r2Þ cos hðÀ sin hÞ ¼ À r1 À r2 sin 2h 2 ¼ r1 cos 2h þ 1 þ r2 1 À cos 2h The stresses given by (1.41a) and (1.43a) are called 2 2 complementary stresses. It is evident from (1.41a) and ¼ r1 þ r2 þ r1 À r2 cos 2h ð1:41aÞ (1.43a) that r þ r0 ¼ r1 þ r2; i.e. the sum of the comple- 22 mentary normal stresses r and r0 is equal to the sum of the two given principal stresses r1 and r2: Thus, the sum of the The component parallel to the oblique plane ‘aa1b1b’, i.e. normal stresses on two perpendicular planes is constant, i.e. to ‘ab’ or ‘a1b1’, is the shear stress, s; which is given below: an invariant quantity, which does not depend on the angle h; i.e. the orientation of the plane. Equations (1.41b) and From ra1b : shear stress; s ¼ r1ab cosð90 À hÞ (1.43b) show that the complementary shear stresses s and s0 ¼ r1 cos h sin h; ½from ð1:40aފ: are equal in magnitude but opposite in sign, i.e. opposite in From ra2b : shear stress; s ¼ ra2b cosð180 À hÞ ¼ r2 sin hðÀ cos hÞ; ½from ð1:40bފ: the sense of rotation about any point inside the body. Thus, Hence, from both r1ab and r2ab: total shear stress; the absolute values of the shear stresses are the same for all s ¼ r1 sin h cos h À r2 sin h cos h angles h differing by 90 for which the values of sin 2h are ¼ r1 sin 2h À r2 sin 2h ¼ r1 À r2 sin 2h ð1:41bÞ the same, i.e. the absolute values of the shear stresses on 222 planes at right angles are always equal, but they are of It can be concluded from (1.41a) and (1.41b) that opposite sense. • When the angle h ¼ 0; r ¼ r1; and s ¼ 0: • When h ¼ p=2; r ¼ r2; and s ¼ 0: If we consider the elevation view of the cube of Fig. 1.3, • For 0\\h\\p=2; r1 [ r [ r2: an analysis similar to previous one will result in the fol- lowing equations for the normal and shear stresses acting on any arbitrary oblique plane, which is perpendicular to the principal plane ‘1’ denoted by ‘ABED’ or ‘OCFG’ in Fig. 1.3, and has its normal inclined at an angle of say ‘u’ to

1.5 Elements of Plastic Deformation 13 the principal axis ‘3’. Hence, each of the principal stresses ) Maximum shear stress; smax ¼ s3 ¼ r1 À r2 ð1:48Þ r2 and r3 will contribute to the normal stress and shear 2 stress on this oblique plane. These are: Since shear stresses are involved in the process of plastic Total normal stress; r ¼ r3 cos2 u þ r2 sin2 u; ð1:44aÞ deformation like in theories of yielding and mechanical Or, r ¼ r3 þ r2 þ r3 À r2 cos 2u working operations, it is important to know the planes on which the maximum or principal shear stresses occur. The 22 planes of the principal shear stress are shown in Fig. 1.5 for a cube whose faces are the principal planes. It is to be noted Total shear stress; s ¼ r3 sin h cos h À r2 sin h cos h; that for each pair of principal stresses, there are two planes of Or, s ¼ r3 À r2 sin 2u principal shear stress, which bisect the directions of the principal stresses. 2 ð1:44bÞ When the angle u ¼ 45; the principal shear stress is 1.5.2 Mohr’s Stress Circle denoted by s1; since the plane of the maximum shear stress is parallel to the principal axis ‘1’. Hence, substituting Equation (1.41) in the preceding Sect. 1.5.1 give the normal sin 2u ¼ 1 in (1.44b) we get stress r and the shear stress s on any plane which is per- pendicular to the principal plane ‘3’ denoted by ‘OABC’ or Principal shear stress; s1 ¼ r3 À r2 ð1:45Þ ‘GDEF’ in Fig. 1.3 and has its normal inclined at an angle of 2 h to the principal axis ‘1’ as shown in Fig. 1.4, and these equations are repeated below for convenience of reference: Similar to previous one, it can be obtained from (1.44a) and (1.44b) that when the angle u ¼ 0; r ¼ r3; and s ¼ 0; Normal stress; r ¼ r1 þ r2 þ r1 À r2 cos 2h ð1:41aÞ when u ¼ p=2; r ¼ r2; and s ¼ 0; and for 0\\u\\p=2; 22 ð1:41bÞ r3 [ r [ r2: Shear stress; s ¼ r1 À r2 sin 2h From the consideration of the side view of the cube of 2 Fig. 1.3, the following equations can similarly be obtained for It has been shown below that these are the equations of a circle in a r À s plane, with the angle h as a parameter. Let the normal and shear stresses acting on any arbitrary oblique us denote ðr1 þ r2Þ=2 ¼ rav, and recall (1.42) where plane which is perpendicular to the principal plane ‘2’ denoted by ‘OGDA’ or ‘CFEB’ in Fig. 1.3 and has its normal inclined at an angle of say ‘w’ to principal axis ‘1’. Hence, each of the principal stresses r1 and r3 will contribute to the normal stress and shear stress on this oblique plane. These are: Total normal stress; r ¼ r1 þ r3 þ r1 À r3 cos 2w σ3 22 ð1:46aÞ Total shear stress; s ¼ r1 À r3 sin 2w ð1:46bÞ σ2 τ3 2 σ1 σ2 When the angle w ¼ 45; the principal shear stress is σ1 denoted by s2; since the plane of the maximum shear stress is parallel to the principal axis ‘2’. Hence, substituting τ2 σ3 sin 2w ¼ 1 in (1.46b) we get τ3 = σ1 – σ2 τ2 = σ1 – σ3 2 2 r1 À r3 σ3 2 Principal shear stress; s2 ¼ ð1:47Þ It can similarly be obtained from (1.46a) and (1.46b) that σ2 σ2 when the angle w ¼ 0; r ¼ r1; and s ¼ 0; when w ¼ 45° τ1 p=2; r ¼ r3; and s ¼ 0; and for 0\\w\\p=2; r1 [ r [ r3: τ1 = σ3 – σ2 σ3 Since it is assumed that r1 [ r3 [ r2; the value of any 2 normal stress r on any plane lies between the limits of r1 and r2; i.e. r1 ! r ! r2: Accordingly, s3 has the highest Fig. 1.5 Shaded planes are planes of principal shear stress value of shear stress and is called the maximum shear stress, denoted by smax:

14 1 Tension ðr1 À r2Þ=2 ¼ s3 (principal shear stress). Now, (1.41) can τ be written in the following compact form as Normal stress; r ¼ rav þ s3 cos 2h ð1:49aÞ σ1 Shear stress; s ¼ s3 sin 2h ð1:49bÞ σ2 F B D From (1.49b), we can get A pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C 2θ E cos 2h ¼ 1 À sin2 2h ¼ 1 À ðs=s3Þ2: Substitution of cos 2h in (1.49a) will eliminate the O σ parameter h: pffiffiffiffiffiffiffiffiffiffiffiffiffiffi r ¼ rav þ s3 s32 À s2 ; or; ðr À ravÞ2¼ s23 À s2; s3 ) ðr À ravÞ2 þ s2 ¼ s32 ð1:50Þ D' Equation (1.50) represents the standard form for the σ1 + σ2 equation of a circle which is centred at rav ¼ ðr1 þ r2Þ=2; and has a radius given by s3 ¼ ðr1 À r2Þ=2: This circle is 2 called Mohr’s stress circle whose abscissa and ordinate represent, respectively, the normal stress r and the shear Fig. 1.6 Mohr’s stress circle for biaxial tensile stresses stress s; as shown in Fig. 1.6. It is convention that if the sense of rotation of the shear stress about any point inside Thus, the coordinates ðOE; DEÞ of the point D in Fig. 1.6 the body is clockwise, the shear stress is positive and if the represent the values of the normal stress r and the shear normal stress is tensile in nature, i.e. acts away from the stress s for the plane whose orientation is given by h with body, it is taken as positive. Whereas the shear stress causing respect to the principal axes ‘1’ in Fig. 1.4. an anticlockwise rotation is taken as negative, the com- pressive normal stress acting towards the body is negative. For each different orientation of plane in Fig. 1.6, as In Mohr’s circle, the positive shear stress is plotted above defined by h ; there is a corresponding point on this circle and the negative one below the horizontal axis and the whose coordinates give the normal stress r and the shear positive normal stress is plotted in the right of the vertical stress s on that plane. For example, when h ¼ 0 in Fig. 1.4, axis and the negative one in the left. Mohr’s circle is very the normal ‘N’ of the oblique plane ‘aa1b1b’, i.e. of ‘ab’ or helpful for graphically solving of (1.41), (1.44) or (1.46). To ‘a1b1’, coincides with the principal axes ‘1’ and, in Fig. 1.6, work with Mohr’s circle, it is to be remembered that the the corresponding position of point D is at the point A on the angular orientation h of the plane in the body subjected to circle, the coordinates of which give the normal stress r ¼ stress is represented by 2h on the Mohr’s circle and the sense r1 (principal stress) and the shear stress s ¼ 0: When h ¼ of rotation in each case must be the same, either clockwise or 45; 2h ¼ 90; and the point D is positioned at the point F anticlockwise. Hence, the stresses on any plane with orien- on the circle in Fig. 1.6, resulting r ¼ ðr1 þ r2Þ=2; and s ¼ tation h are given by the coordinates of the corresponding ðr1 À r2Þ=2 ¼ s3 ¼ smax (maximum principal shear stress). point on this circle making an angle of 2h at the centre. When h ¼ 90 in Fig. 1.4, the normal ‘N’ of the oblique plane ‘aa1b1b’, i.e. of ‘ab’ or ‘a1b1’, coincides with the The following example will help to understand the principal axes ‘2’ and, in Fig. 1.6, the point D coincides with function of Mohr’s stress circle. Let us consider any arbi- the point B on the circle, indicating r ¼ r2 (principal stress) trary point D on the Mohr’s circle, where the angle \\ACD ¼ and s ¼ 0: The complementary stresses r0 and s0 as indicated 2h; (say), as shown in Fig. 1.6. The coordinates of the point by (1.43) can also be obtained from the coordinates of point D can be obtained from geometry as follows: D0 that is located at the diametrically opposite side of the point D on the circle. Thus, the Mohr’s circle provides us all OE ¼ OC þ CD cos 2h ¼ r1 þ r2 þ r1 À r2 cos 2h necessary information about the stresses on various planes in 22 the body. ¼ rðnormal stressÞ ½From ð1:41aފ: The Mohr’s stress circle for planes which are perpen- DE ¼ CD sin 2h ¼ r1 À r2 sin 2h ¼ sðshear stressÞ dicular to the principal plane ‘3’ denoted by ‘OABC’ or ‘GDEF’ in Fig. 1.3 is redrawn in Fig. 1.7a for convenience. 2 ½From ð1: 41bފ:

1.5 Elements of Plastic Deformation τ 15 σ (a) τ3 = τmax O σ2 σ3 σ1 τ σ σ1 O σ2 (b) τ2 σ Fig. 1.8 Mohr’s stress circle for triaxial tensile stresses, formed by O σ1 superimposition of all three Mohr’s circles shown in Fig. 1.7a–c σ3 decreases the stress required for slip and that is why slip is preferred on closely packed planes. The slip plane jointly with (c) σ2 τ1 σ the slip direction creates the slip system, and the number of slip σ3 systems is obtained by multiplying the number of slip planes O with the number of slip directions lying on the slip planes. When more than one slip system is operative during the Fig. 1.7 Mohr’s stress circle for biaxial tensile stresses for planes deformation, it is often described as duplex or multiple slip. normal to a principal plane ‘3’, b principal plane ‘2’, and c principal When a polished specimen is deformed by slip, a step is plane ‘1’ with reference to Fig. 1.3 produced on the polished surface. When viewed under the microscope, the step appears as a line, called as a slip line, which is due to the change in the surface elevation. Similarly, the Mohr’s stress circles for planes which are 1.5.3.1 CRSS for Slip perpendicular to the principal plane ‘2’ denoted by ‘OGDA’ In uniaxially loaded tension test, the applied tensile stress or ‘CFEB’ in Fig. 1.3 and perpendicular to the principal plane ‘1’ denoted by ‘ABED’ or ‘OCFG’ in Fig. 1.3 are will result a stress on any plane whose normal is inclined to shown, respectively, in Fig. 1.7b, c. All three Mohr’s stress circles shown in Fig. 1.7a–c are superimposed to construct the loading axis, but the resulting stress acts along the the composite Mohr’s stress circle diagram for the complete loading axis and can be resolved into components perpen- triaxial state of stress, as shown in Fig. 1.8. Now, the stresses on any plane perpendicular to one of the principal dicular and parallel to the oblique plane. The component of planes can be determined by applying (1.41), (1.44) or (1.46) or using the corresponding Mohr’s circle. the stress perpendicular to the inclined plane is its normal stress while that parallel to it is the shear stress on it. This 1.5.3 Slip shear stress component is responsible for slip. Schmid The mechanisms of plastic deformation of crystalline solids (1931) first recognized that slip begins in a single crystal are slip and twinning. Slip is the most important mechanism of when the shear stress resolved on the slip plane in the slip plastic deformation. It refers to the sliding of the blocks of direction reaches a critical value called the ‘critical resolved crystal over one another under the application of shear stress. It shear stress’ or ‘CRSS’ and this law of critical resolved shear occurs most readily on definite crystallographic planes of stress is known as Schmid’s law. greatest atomic density called slip planes, in the closest- packed directions on the slip planes known as slip directions. Let us consider a cubic single crystal as tensile specimen, Since the distance between like planes varies inversely with which is subjected to uniaxial tensile load ‘P1’ acting along their atomic density, the slip planes are widely spaced that the axis ‘1’. Assume that the tensile load ‘P1’ is applied on the principal plane which is normal to the plane of paper and represented by AB or CD in Fig. 1.9. If this plane, which is normal to the loading axis ‘1’, has a cross-sectional area of ‘A1’, then the stress acting along the loading axis ‘1’ is r1 ¼ P1=A1: The cross-section ‘ABCD’ of the tensile spec- imen parallel to the loading axis ‘1’ has been shown in Fig. 1.9, where ‘ab’ is a slip plane. The normal to this slip

16 1 Tension P1 If the slip direction ‘S.D.’ lying on the slip plane, as a shown in Fig. 1.9, makes an angle ‘k’ (not shown in figure) with the shear stress component s acting on the slip plane DC given by (1.52), then the shear stress on the slip plane resolved in the slip direction, sR; is given by sR ¼ r1 sin 2h cos k ð1:54Þ 2 θS.D. N The shear stress resolved on the slip plane in the slip B direction, sR; given by (1.54), at which slip begins is the 1 critical resolved shear stress or CRSS, scr: The resolved Ab shear stress sR; is a maximum when h ¼ 45; and k ¼ 0; so that sR ¼ r1=2. For any other orientations of the slip P1 planes, the resolved shear stress will always be less than its maximum value, i.e. r1=2, and a larger tensile stress is Fig. 1.9 Two dimensional view of a cubic single crystal tensile required to bring the resolved shear stress up to the critical specimen subjected to uniaxial tensile load for derivation of critical value. So the plane on which slip will first take place must resolved shear stress have the maximum value of resolved shear stress. If the slip plane is normal to the tensile axis, i.e. h ¼ 0; or if it is plane is ‘N’, which is inclined to the loading axis ‘1’ by an parallel to the tensile axis, i.e. h ¼ 90; the resolved shear angle ‘h’. As the cross-sectional area of this oblique plane stress sR; is zero, i.e. no shear stress acts on the slip plane. ‘ab’ is larger than that of the principal plane AB or CD; the Therefore, for such orientations of the slip planes, slip can- stress acting on this oblique plane ‘ab’ will be smaller than not occur and the material tends to fracture. the applied principal stress r1: Let the cross-sectional area of this oblique slip plane ‘ab’ is A0; which will be given by The value of CRSS is actually equivalent to the yield A0 ¼ A1=cos h: Hence, the stress, say r1ab; acting on slip stress of a usual stress–strain diagram for the single crystal. plane of cross-sectional area A0; corresponding to the applied The values of CRSS and yield strength (since it is related to tensile stress r1; is given by CRSS) depend chiefly on the composition of the material, r1ab ¼ P1 ¼ P1 cos h ¼ r1 cos h ð1:51Þ A0 A1 strain rate and deformation-temperature, on the interactions of dislocations with each other and with other lattice defects This stress r1ab being parallel to the loading axis ‘1’ is inclined to the normal of the slip plane ‘ab’ by an angle ‘h’ such as vacancy, interstitials and impurity atoms. and can be resolved into components perpendicular and During uniaxial tension test of a single crystal, the movement of cross-head of the testing machine imposes parallel to the slip plane. The component parallel to the slip plane with cross-sectional area A0 is the shear stress constraint at the grips of the specimen in order to maintain responsible for slip and given by the grips in line and does not allow the specimen to deform freely and uniformly on every slip plane along the length of s ¼ P1 sin h ¼ P1 sin h cos h ¼ r1 sin 2h ð1:52Þ A0 A1 2 the specimen. As the deformation progresses and the tensile load remains axial, the slip planes near the centre of the specimen rotate, so as to line up themselves parallel with the loading axis and the extent of rotation increases with the extension of the crystal. The slip planes near the grip is not only rotated but also bent. The shearing stress resolved on the slip plane reaches the 1.5.3.2 Dislocations: A Brief Introduction maximum value when sin 2h ¼ 1 or; h ¼ 45; i.e. when the Slip, the usual method of plastic deformation, occurs by the normal to the slip plane makes an angle of 45° with the movement of linear lattice defect called dislocation, which is defined as the localized region of disturbed lattice making direction of applied tensile stress. Hence, the maximum the boundary on a slip plane between the sheared and the resolved shear stress on the slip plane is given by unsheared regions of a crystal. It is the presence of dislo- cation in the real crystal that makes the shear stress required ðsmaxÞuniaxial tension¼ r1 ¼ applied tensile stress ð1:53Þ to cause plastic deformation at least 100 times lower than the 2 2 theoretical shear strength of a defect-free ideal crystal. When a dislocation moves on the slip plane, the interatomic bonds Thus, in a uniaxial tension, a maximum shear stress are broken and reformed at any given moment only between equal to one-half the applied tensile stress always act on any plane whose normal makes an angle of 45° with the axis of applied stress.

1.5 Elements of Plastic Deformation 17 the atoms located near the axis of the dislocation, but in a dislocation of this type is also called a full dislocation or perfect crystal the bodily shearing of planes of atoms would a perfect dislocation. The strength of a dislocation with require very high force to break the bonds between all the Burgers vector a0½uvwŠ is given by atoms in a crystal plane all at once. Thus, the mechanism responsible for slip is in reality the movement of dislocations pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi that allow atoms in crystal planes to slip past one another at a jbj ¼ a0 ½u2 þ v2 þ w2Š; much lower stress levels than in a perfect crystal. The shear strengths of filament crystals (whiskers) without dislocations where a0 is the lattice constant and ½uvwŠ is the Miller may approach the theoretical value. Dislocations are not indices of the slip direction. For example, if b ¼ only connected with slip but are also important to explain ða0=2Þ½110Š for slip in a cubic crystal from a cube several mechanical behaviours such as strain hardening, corner to the centre of one face,ptffiffihffiffieffiffiffiffiffimffiffiffiaffiffignitudepoffiffif yield point, creep, fatigue and brittle fracture. Relevant Burgers vector will be jbj ¼ ða0=2Þ 12 þ 12 ¼ a0 2: aspects of dislocations and their characteristics are briefly described below: 3. Types of dislocation: There are two basic types of dislocation: edge dislocation and screw dislocation. 1. Origin: Dislocations needed for plastic deformation Edge dislocations are those, which lie perpendicular to originate mainly in the following ways. During crys- the Burgers vector and move on their slip plane in the tallization, many dislocations are found as a natural direction of the Burgers vector (slip direction). If an consequence of crystal growth. The most important incomplete extra vertical plane of atoms is introduced source of dislocations is the dislocation-multiplication within a perfect crystal lattice either above or below the mechanism explained by Frank and Reed (1950), where slip plane, it will create an edge dislocation. The the originally present dislocations in the crystal are intersection of this incomplete extra vertical plane of multiplied many fold during the process of plastic atoms with the slip plane is the dislocation line. The deformation. Moreover, the grain boundaries can act as incomplete extra vertical plane of atoms lying above the source as well as sink of dislocations. Further, dislo- slip plane is conventionally called a positive edge dis- cations can also be generated as a result of phase location, indicated by the symbol ┴, and that below the transformation, thermal stress or high local stresses at slip plane is called a negative edge dislocation, indi- second-phase particles. cated by the symbol ┬. In these above symbols, the slip plane is represented by the horizontal line and the 2. Burgers vector: The magnitude and direction of slip incomplete extra vertical plane by the vertical line. If a are defined by a vector, known as slip vector or Burgers crystal is rotated by 180°, a positive edge dislocation vector, usually represented by ‘b’. The slip vector or will be converted into a negative edge dislocation. Burgers vector is a characteristic property of disloca- Figure 1.10a shows a positive edge dislocation pro- tion. The Burgers vector may be one or more atomic duced by slip for an element of crystal with a simple distances. Burgers vector of a dislocation can be con- cubic lattice. Slip has taken place over area ABCD in veniently defined with the help of the Burgers circuit. the direction of Burgers vector. Edge dislocation line is The circuit starts at a lattice point and moves from atom indicated by AD because it is the boundary between the to atom an equal number of steps in each direction, right-hand sheared portion and the left-hand un-sheared always in the direction of one of the unit cell vectors. portion of the crystal. The shaded area adjacent to BC in The circuit will close in a perfect crystal having no Fig. 1.10a indicates the amount of displacement of the dislocation, but when it encloses a dislocation in an portions of the crystal above the slip plane in the imperfect crystal it will not close; i.e., the end point of direction of slip with respect to the portions below the the circuit will fail to meet with the starting point. The slip plane. This amount of displacement is equal to the vector joining the end point of the Burgers circuit with Burgers vector b of the dislocation. Dislocation line AD the starting point is the Burgers vector, ‘b’ of a single lies perpendicular to Burgers vector b and moves in a dislocation. Burgers vector must always connect one direction parallel to the Burgers vector. The exact equilibrium lattice point with another. The connector atomic arrangement along AD is unknown. A close from the end to the start point of a Burgers circuit representation of atomic arrangement in a plane normal enclosing multiple dislocations is the summation of to the edge dislocation AD is shown in Fig. 1.10b, their individual Burgers vectors. If the Bergers vector of which is generally considered. The plane of the paper in a dislocation is equal to one lattice spacing, it is known this figure is equivalent to any plane parallel to the front as dislocation of unit strength or unit dislocation. A unit face of Fig. 1.10a. Figure 1.10b shows that there is distortion of the lattice in the region of dislocation. The

18 1 Tension (a) (b) Slip vector Incomplete extra Sheared area plane of atoms of slip plane τ b DC Direction AB of motion τ of dislocation Dislocation line Fig. 1.10 a Edge dislocation produced by slip in a simple cubic b Arrangement of atoms in a plane normal to the edge dislocation line lattice. Edge dislocation line indicated by AD lies perpendicular to AD. The plane of the paper in this figure (b) is equivalent to any plane Burgers vector b and moves in a direction parallel to the Burgers vector. parallel to the front face of figure (a) Slip has taken place over area ABCD in the direction of Burgers vector. atomic arrangement in a positive dislocation creates a which the edge or screw component of dislocation compressive stress above the slip plane and a tensile makes an arbitrary angle with its slip vector. stress below the slip plane while that in a negative 4. Dislocation movement: Figure 1.12 illustrates how an dislocation results in a tensile stress above and a com- edge dislocation moves through a crystal subjected to a pressive stress below the slip plane. shear stress which is indicated by the vector s: The On the other hand, screw dislocations lie parallel to the plane x; at the top of Fig. 1.12a is a positive edge dis- Burgers vector and move on the slip plane in a direction location and the plane y lying to the left of the plane x is perpendicular to the Burgers vector (slip direction). a complete plane of atoms running continuously from When the lattice planes spiral around a screw disloca- top to bottom of the crystal. Due to the applied shear tion line like a left hand screw, i.e. in anticlockwise stress s; atom o from the plane y may move to the direction, it is conventionally called a negative or position marked o0 in the plane x; as indicated in left-hand screw dislocation indicated by the symbol, Fig. 1.12b. If this movement occurs, the dislocation moves one atomic distance to the left. It now makes the , and when the lattice spirals around a screw dis- plane y to terminate abruptly at the slip plane and the location line in a right-hand fashion, i.e. in clockwise plane x to run continuously from top to bottom of the direction, it is called a positive or right hand screw crystal. On continuous application of the stress, there dislocation indicated by the symbol, . A screw will be movement of the dislocation by repeated steps dislocation (left-hand screw dislocation) produced by along the slip plane of the crystal until the dislocation slip in a simple cubic lattice is shown in Fig. 1.11a. In reaches the edge of the crystal. The final outcome is this figure, the upper portion of the crystal to the front shown in Fig. 1.12c, in which there is shearing of of AD has been displaced in the direction of Burgers crystal by one atomic distance across the slip plane. The vector (to the left) relative to the lower portion, but no amount of slip, which is possible to measure ordinarily, slip has taken place to the rear of AD. Hence, slip has necessitates the migration of many thousands of such taken place over area ABCD in a direction normal to the dislocations. The slip on any one active slip plane is Burgers vector. Screw dislocation line indicated by AD usually of the order of 1000 atomic distances when a lies parallel to Burgers vector b and moves in a direc- crystal yields. To bring about significant yielding, slip tion normal to the Burgers vector. Figure 1.11b shows a on many such active planes is needed. two-dimensional arrangement of atoms around the 5. Peierls–Nabarro or Peierls force: When the disloca- screw dislocation AD in a simple cubic lattice, showing tion lies in a symmetrical position with respect to the ABCD is the sheared area of slip plane. atoms on the slip plane surrounding it, the force nec- Actually, pure edge and screw dislocations are the essary to move a dislocation through the crystal lattice extreme forms of the possible dislocation structures. In would be zero. Cottrell (1953) pointed out that when most cases, dislocations exist in the form of partly edge the dislocation passes through non-symmetrical and partly screw types, called mixed dislocation, in

1.5 Elements of Plastic Deformation 19 (a) (b) τ D Dislocation line A D A C B b τ Sheared area Direction C B of slip plane of motion of dislocation Slip vector Fig. 1.11 a Screw dislocation produced by slip in a simple cubic dislocation AD, showing that ABCD is the sheared area of slip plane. lattice. Screw dislocation line indicated by AD lies parallel to Burgers Atoms in the atomic plane just above the slip plane are indicated by the vector b and moves in a direction normal to the Burgers vector. Slip has open circles and those just below the slip plane are represented by the taken place over area ABCD. b Arrangement of atoms around the screw solid circles (a) y x τ (b) y x τ (c) τ O O' τ ττ Fig. 1.12 Movement of an edge dislocation through a crystal. a The move one atomic distance to the left. c Movement of the dislocation by plane x is a positive edge dislocation. b Movement of atom o from the repeated steps causes shearing of crystal across the slip plane by one plane y to the position marked o0 in the plane x causes the dislocation to atomic distance positions, the above situation would not hold true and the dislocation ‘W’, the more is the distance between the lattice offers some resistance to its movement. like planes, ‘d’, which is related to ‘W’ as: Consequently, a small shear force, called the Peierls– Nabarro or Peierls force, is necessary to move a dis- W ¼ d=ð1 À mÞ: location through the lattice in a particular direction. Although the magnitude of the Peierls force varies The dependence of Peierls stress spÀn; on ‘W’ and periodically with the movement of dislocation through Burgers vector ‘b’ is given by the lattice, it is found to depend largely on the width of the dislocation ‘W’, which represents a measure of the Peierls stress;  h i distance over which the lattice is distorted due to the presence of the dislocation. Again higher the width of  2pd ð1:55Þ 2G 2G eÀ ð1ÀmÞb spÀn ¼ 1Àm eÀð2pbWÞ ¼ 1Àm

20 1 Tension where G ¼ shear modulus of the crystal; m ¼ Poisson’s the fact that the closest packed directions are the slip ratio. directions in crystals. Equation (1.55) shows that the Peierls stress for a given 8. Partial dislocation: If the Bergers vector of a dislo- plane decreases with increase of distance between cation is a fraction of the lattice spacing, then the dis- similar planes. Since the spacing between the slip location is called a partial dislocation. From the energy planes is the maximum due to their highest atomic point of view, dislocations of strengths greater than density, the Peierls stress on the slip planes will be the unity are usually unstable and dissociate into two or minimum and that is why slip is preferred on these more dislocations of lower strength. In real crystals, closely packed slip planes. both edge and screw dislocations may exist as full or 6. Cross slip of screw dislocations: The plane passing perfect dislocations and as partial dislocations. The through the slip vector and the line of dislocation is tendency of a dislocation is to have the smallest pos- called the glide or slip plane for the edge dislocation. sible Burgers vector, i.e. minimum b, in order to min- So, the edge dislocation has a preferred slip plane imize its elastic strain energy, since the energy is because the dislocation line is perpendicular to its proportional to b2. Thus, a dislocation of Burgers vector Burgers vector. Since the screw dislocation line is ‘b1’ will have a tendency to dissociate into two (or parallel to its slip vector, the glide plane for a screw more) partial dislocations with Burgers vector ‘b2’ and dislocation may be any crystallographic plane passing ‘b3’, if b12 [ b22 þ b23: So, it is important to note that through the dislocation line unlike an edge dislocation. dislocations can decompose into partial dislocations if So, a screw dislocation having no fixed glide plane may there is a decrease in strain energy and this energetically overcome obstacles by gliding from one slip plane to favourable situation will facilitate their movements another having a common slip direction and this through a crystal lattice. movement of a screw dislocation is called cross slip. 9. Interaction forces between dislocations: Depending 7. Elastic stress field around dislocations: The regions on the sign of dislocations, the force between disloca- of the crystal in the vicinity of dislocations are in an tions may be either attractive or repulsive. Unlike dis- elastically stressed state. The stresses decrease in locations lying on the same, slip plane will attract each inverse proportion to distance from the dislocation. In other, run together and cancel or annihilate each other case of a positive edge dislocation, the atoms above the and thus the dislocations will disappear, but a single slip plane in the region immediately surrounding it are dislocation cannot vanish on its own. On the contrary, compressed and so they experience an elastic com- similar dislocations lying on the same slip plane will pressive stress, whereas the atoms below the slip plane repel each other. For two screw dislocations lying on are stretched and so they experience an elastic tensile two different parallel slip planes, the force between stress. In case of a negative edge dislocation, there will dislocations depends only on the separation distance be a state of elastic compression below the slip plane between the dislocations. The force is attractive for and a state of elastic tension above the slip plane. The screw dislocations of opposite sign, i.e. for antiparallel atoms in the region surrounding a screw dislocation line screws, and repulsive for screw dislocations of same in its immediate vicinity undergo a shear displacement, sign, i.e. for parallel screws. Now, in case of two par- and thus, an elastic shear strain exists around a screw allel edge dislocations with the same Burgers vectors dislocation. Due to the presence of elastic strain in the lying on two different slip planes, if ‘x’ is the separation form of compressive or tensile around an edge dislo- distance between those two parallel edge dislocations, cation and shear around a screw dislocation, the dislo- ‘h’ is the angle between their Burgers vectors, and ‘y’ is cation is associated with distortional energy or elastic the separation distance between those two slip planes strain energy. The magnitude of the elastic strain having dislocations, then similar dislocations repel each energy ‘UDislocation’, per unit length of a dislocation of other when x [ y; i.e. h\\45, and attract each other Burgers vector ‘b’, is approximately given by when x\\y; i.e. h [ 45: The opposite kind of forces between edge dislocations is found when they are of UDislocation ’ Gb2 ð1:56Þ opposite sign. The force between edge dislocations is 2 zero at x ¼ 0; and x ¼ y: When x ¼ 0; it is a condition of equilibrium where edge dislocations of same sign lie where G ¼ shear modulus of the crystal. vertically above one another and this vertical array of The energy of a unit dislocation is the minimum when similar dislocations exists in a low-angle tilt boundary. its Bergers vector is parallel to the direction of the In case of two parallel edge dislocations with different closest atomic packing in the lattice. This agrees with Burgers vectors lying on two different slip planes,

1.5 Elements of Plastic Deformation 21 Fig. 1.13 Climb of edge (a) (b) dislocation. a Diffusion of vacancy through a crystal lattice to a positive edge dislocation. b A positive edge dislocation moves up by one atomic layer resulting in positive climb representing the situation of dislocations on two inter- the process of climb can occur if there is an appreciable rate of diffusion of vacancies or interstitial atoms secting slip planes, they will either attract or repel through a crystal lattice to or away from the edge dis- location. The edge dislocation may move in an upward depending on the following conditions. Let us consider or in a downward direction, and it is a convention to denote the former as ‘positive direction of climb’ and the two parallel edge dislocations with Burgers vectors of latter as ‘negative direction of climb’. For positive climb, atoms are removed from or vacancies are added to the b1 and b2; which may or may not attract and combine extra incomplete plane of atoms at a positive edge dis- location so that this incomplete plane moves up by one into a dislocation with Burgers vectors of b3: The two atomic layer. Usually, this occurs by diffusion of a lattice dislocations will attract if b23\\b12 þ b22; or; h [ 90; and vacancy to the edge dislocation and the movement of will repel if b23 [ b12 þ b22; or; h\\90: extra atoms from the incomplete plane in the opposite direction to occupy the vacant lattice site, which is 10. Dislocation loop: In fact, dislocations within a real shown in Fig. 1.13. Instead of atom–vacancy exchange by vacancy diffusion process, extra atoms from the crystal are hardly present as a straight line and seldom incomplete plane may also move to the interstitial site, but this process is not energetically favourable. For remain on a single plane. Dislocations within the crystal negative climb, a row of atoms must be added below the extra incomplete plane of atoms at the site of edge dis- are generally present in the shape of closed loops or location so that this incomplete plane moves down by one atomic layer. For occurrence of this, surrounding curves, which take shape of an interlocking atoms of the lattice usually join the incomplete plane of atoms that will cause the diffusion of vacancies away three-dimensional network of dislocation. Generally, a from the edge dislocation. Instead of vacancy diffusion, the diffusion of interstitial atoms to the edge dislocation dislocation line cannot terminate within the crystal, is also possible, but such probability is very less. The addition or removal of complete rows of atoms at the except at a node, where several dislocation lines inter- incomplete plane rarely occurs. Rather, vacancies dif- fuse in a small cluster or individually to or away from sect. It must either exist in the form of a closed loop the site of the dislocation causing climb to occur over a short section of the dislocation line, and thus, short steps within the crystal or escapes from the crystal at the free or jogs1 are created along the dislocation line. surface because the surface applies a force of attraction 1When two dislocations intersect, a step with a length of a few atom spacing is produced in the dislocation line. If the step in the dislocation on a dislocation and the escape of dislocation from the moves it out of the current slip plane, it is called a jog. If the step lies in the slip plane, it is known as a kink. crystal at surface reduces the strain energy of crystal. 11. Dislocation density: The dislocation density in a crystal is defined as the total dislocationlength present per unit volume, having unit of mm mm3; or the average number of dislocation lines intersecting per unit area having unit of mmÀ2: Dislocation density in a material can be increased by pplqffiaffiffi;stwichedreefosrm¼atthioenshfeoalr- lowing the relationship of s / stress required for the movement of dislocations to cause slip and q ¼ the dislocation density. The dislo- cation density usually varies from 103 to 104 per mm2 in the fully annealed polycrystalline metals and reaches about 1010 per mm2 in heavily cold worked metals. 12. Climb of edge dislocation: The edge dislocation can move directly above or below its slip plane in a per- pendicular direction onto a parallel plane by a mecha- nism known as climb that is fundamentally different from slip. The vertical movement of edge dislocation by

22 1 Tension During positive climb, the removal of atoms causes Twinned region shrinkage of the crystal in the direction normal to the incomplete plane of atoms, while the addition of atoms Twin boundaries during negative climb expands the crystal in the direc- tion normal to the incomplete plane. Therefore, com- Fig. 1.14 Schematic diagram of twinning. The atomic arrangement on pressive stress in the direction normal to the incomplete one side of a twin boundary is a mirror image of that on the other side plane causes positive climb, whereas negative climb is and the portion of crystal between a pair of twin boundaries is a caused by tensile stress. This makes climb different from twinned region. Open circles represent the original positions of atoms slip because only shear stress is responsible for slip. in the lattice before twinning, which change position. Arrow marks Since the rate of diffusion increases exponentially, i.e. show the direction and distance of their movements, and solid circles very sharply with increasing temperature and vice versa, indicate the final position of atoms after twinning in twinned and the movement of edge dislocation by climb being dif- untwinned regions fusion controlled is much slower than in glide and is less likely to occur except at high temperatures. Climb can 1.5.4.1 Types of Twin occur more readily at high temperatures than at low Twin which forms only as a result of plastic deformation is temperatures due to an increase in diffusion of vacancy. called deformation twin or mechanical twin and that which Further, application of stress at elevated temperature forms during annealing of a mechanically deformed material increases the rate of climb. Dislocation climb plays an is called recrystallization twin or annealing twin. Mechani- important role during creep deformation. The formation cal twins are found in body-centred-cubic (BCC) and of low-angle tilt boundary during polygonization also hexagonal close-packed (HCP) metals under conditions of requires dislocation climb for vertical alignment of edge low deformation temperatures or high strain rates (shock dislocation. loading), whereas most face-centred-cubic (FCC) metals, Screw dislocation cannot climb because it does not have especially those having a low stacking fault energy (see an extra incomplete plane of atoms and can slip on any Sect. 1.5.6), usually shows annealing twins. In exceptional crystallographic plane passing through the dislocation cases, mechanical twins are produced in FCC metals, for line. Therefore, for its movement onto another slip example, gold silver alloy at low deformation temperature plane, no diffusion of vacancies or interstitial atoms is and copper under tension at 4 K. Under optical microscope, required. the appearance of a mechanical twin is acicular or The details on dislocation theory and deformation needle-shaped with thickness smaller than an annealing twin mechanism are beyond the scope of the text, and the and an annealing twin appears like a broad band with readers are referred to any book on plastic deformation straighter sides than a mechanical twin. for detailed study of deformation behaviour. 1.5.4.2 Role of Twinning 1.5.4 Twinning Twinning plays an important role in the plastic deformation of materials possessing a low number of slip systems, such The second important mechanism by which plastic deforma- as in HCP metals with only three slip systems. The tion occurs is twinning (Cahn 1954; Hall 1954; Mahajan and Williams 1973). It is shown schematically in Fig. 1.14. Twinning results when a portion of the crystal takes up such an orientation that it becomes a mirror image of the untwinned portion of the crystal, i.e., the parent crystal. The boundary between the twinned and untwinned portion of the crystal is called the twinning plane, which is a surface defect. Twinning planes always occur in pairs so that the change of orientation of parent crystal across one twin boundary is restored by another boundary of the pairs. The applied shear stress displaces homogeneously every atom in the twinned portion a distance proportional to the distance of each atom from the twin plane. Like slip, twinning also occurs on a specific crystallographic plane along a definite crystallographic direction.

1.5 Elements of Plastic Deformation 23 orientation changes caused by twinning may bring new slip in stress with progress of plastic deformation, i.e. with system in a favourable orientation with respect to the axis of increase in plastic strain because of previous plastic defor- applied stress so that additional slip can occur. Twinning is mation, is known as strain hardening or work hardening. not a dominant deformation mechanism in metals of cubic Due to the strain hardening, the material is strengthened. It is crystal system having many possible slip systems, such as in to be noted that till the point of fracture of the material, the BCC or FCC metals with forty-eight or twelve slip systems, stress increases continuously with increase of strain, i.e. respectively. But twinning occurs in metals of cubic crystal strain hardening continues, but the rate of increase of stress system when the process of slip is restricted due to with strain, i.e. the rate of strain hardening, gradually requirement of higher stress for slip than for twinning. decreases with the progress of plastic deformation. In single crystals of ductile metals, an increase of flow stress (stress 1.5.4.3 Twinning Versus Slip required for plastic flow) of over 100% because of strain Twinning differs from slip in several respects as described hardening is not unusual. below: The defects acting as obstacles to the movement of dis- • Twinning causes change of orientation across the twin locations include the point defects such as vacancies, inter- plane within the crystal, whereas the orientation of stitials and impurity atoms, and surface defects such as twin crystal across the slip plane does not change during slip. boundaries, stacking faults, grain boundaries and subgrain boundaries (low-angle grain boundaries), volume defects • When polished specimens are deformed separately by such as microscopic precipitate particles and foreign-particle slip as well as by twinning, both slip lines and twins are inclusions and also line defects such as other dislocations. visible on polished surfaces, but on re-polishing the Hence, interactions of dislocations with each other can serve surfaces both will disappear due to removal of surface as barriers to the motion of dislocations. Whenever dislo- elevations. Now on etching the re-polished surfaces, the cation movement is made more difficult, the material is twins will be again visible due to the orientation differ- strengthened and, thus, the defects contribute to the ence across the twin plane, but the slip lines will not mechanical properties of metals. reappear. When the dislocations pile up on the slip plane at barriers, • Twins cannot extend beyond a grain boundary, but slip a back stress is produced due to the repulsive force created lines can easily cross a grain boundary although they by the similar dislocations lying on the same slip plane when change their directions depending on the orientations of they are brought together during their pile-ups. This back the neighbouring grains. stress opposes the applied stress on the slip plane and increases the stress required for slip. The other obstacles • Atoms move usually in an integral number of the inter- arise when dislocations gliding on two intersecting slip atomic spacing during slip, while the atom movements planes combine to form of a new dislocation that is not in the during twinning is a fraction of the interatomic spacing slip direction and do not lie on the slip plane of low shear depending on their distance from the twin plane. stress. This dislocation of low mobility produced by the dislocation reaction is called a sessile dislocation, which act • The planes on which slip occurs are comparatively as a barrier to dislocation motion until the barrier is broken widely spaced, but each atomic plane of the twinned down by the increase of stress to an appreciable high level. portion of the crystal takes part in twinning. In addition to above, another reason of strain hardening is the intersection of a forest of dislocation. When dislocations • To form a slip band, a time period of several milliseconds thread through the active slip plane, they are often called a is required, while a twin can be formed within a very dislocation forest. When a screw dislocation moving in the short time in the order of few micro-seconds. slip plane intersects another screw dislocation threading through the active slip plane, a step is formed in each of the • The amount of plastic deformation produced by slip is screw dislocation lines, known as jog. The jogs formed are large, whereas that by twinning is small. edge dislocations because they lie perpendicular to the Burgers vector of the original screw dislocations. Any fur- • Microscopically slip appears as thin lines, while twinned ther movement of the screw dislocations in a direction per- region appears as broad lines or bands. pendicular to their Burgers vector would require the newly formed edge dislocations to move only along the screw axis, 1.5.5 Strain Hardening i.e. to move out of their slip planes because an edge dislo- cation can move only on plane having the dislocation line When any defects in the regular lattice structure act as bar- and its slip vector. Since the jogs formed by the intersection riers to the motion of dislocation, resistance to further slip is of two screw dislocations cannot move in a direction normal developed in the material subjected to stress. So the shear stress required to cause further slip increases compared to the stress needed for prior plastic deformation. This increase

24 1 Tension to the screw axis except by the climbing mechanism, they lower the hardening effect in a polycrystalline material. will cause obstruction to the movement of dislocation. On Therefore, the hardening caused by dislocation pile-up at the other hand, the jog formed by the intersection of two grain boundaries is vital during the preliminary period of edge dislocations has an edge component, which can easily plastic deformation but not at the later phase. Its effect will glide with the rest of the dislocation. Hence, it does not be more in an HCP metal having only one basal plane for impede the motion of dislocation, but energy is needed to cut slip than in BCC or FCC metals with many possible slip a dislocation because its length is increased by the creation systems. Since the orientations of the slip planes in the of a jog. All these processes mentioned above require neighbouring grains of HCP metals may be unfavourable for additional consumption of energy, and therefore, they pro- slip, so a substantially higher stress is needed for initiation of duce strain hardening of single crystals as well as of slip in the adjacent grain. Conversely, as the orientations of polycrystals. slip planes in any grain of BCC or FCC metals cannot be highly unfavourable for slip so the stress needed for initia- 1.5.5.1 Effect of Grain Boundaries tion of slip on a slip plane of the adjacent grains can be Grain boundaries play an important role to impart strain slightly more than that required to initiate slip on the most hardening in a polycrystalline material. As the grains on favourably oriented slip plane. Therefore, in BCC or FCC either side of a grain boundary are oriented differently and metals, the rate of strain hardening in polycrystals is a little randomly, dislocations moving on common slip planes in higher than that in single crystals and the flow curves of the one grain can seldom move onto similar slip planes in former do not differ too much from those of the latter. On the adjoining grains. Further, as grain boundaries of a poly- contrary, in HCP metals, the rate of strain hardening in crystalline material are a region of disturbed lattice, they act polycrystals is much higher than that in single crystals, as barriers to dislocation motion. Dislocations move along which causes the flow curves of the former to raise much the slip planes, and when their movements are blocked by above those of the latter and so a great difference in their grain boundaries, they pile up at the grain boundaries. It is flow curves is observed. one of the reasons which will cause a higher rate of strain hardening in polycrystalline material than in single crystal Most of the mechanical properties are affected remark- during the plastic deformation at room temperature. The ably by the change in grain size. Hardness, yield and tensile other reasons for higher strain hardening in polycrystals are strength, fatigue strength and impact toughness—all these easy occurrence of slip on multiple slip systems and the properties at room temperature increase with reduction in complex mode of deformation introduced within the indi- grain size and vice versa. The properties that are associated vidual crystals to maintain continuity between grains during with the early period of plastic deformation show the max- plastic deformation. Here, it is important to mention that in imum dependency on the grain size, because the grain order to maintain continuity between grains, five indepen- boundaries are the most active barriers during that period. dent slip systems must operate in each grain (Taylor 1938). So, the effect of grain size is more on the yield strength than However, the enhanced strain hardening usually increases on the tensile strength. For most polycrystalline materials, the yield and tensile strength of polycrystals compared to the dependency of yield strength on the grain size at room single crystals. temperature is given by the following relationship [see (1.57a)], proposed by Hall (1951) and extended by Petch Pile-up of dislocations along the slip planes at the grain (1953), showing higher strength for finer grain-size materi- boundary produces stress concentration at the head of the als. The increase of yield stress with decrease in grain size pile-up due to the interaction forces among the piled-up can be explained by assuming that dislocations arise from dislocations and also creates back stresses, which act against sources within the grains and pile up at the grain boundaries. the generation of dislocations within the grain by Obviously, the number of dislocations within a pile up will Frank-Read multiplication mechanism and oppose the have a direct relation with the glide path for dislocations additional dislocations to move along the slip plane in the within a grain, which is decided by the size of grain. For slip direction. In an array of pile-up stationary dislocations, finer grain-size materials, pile-ups at the grain boundaries the stress concentration at the leading dislocation (which is contain fewer dislocations, which will have lower stress nearest to the grain boundary) is equal to the product of the concentrations at their tips, and thus, a greater applied stress number of other dislocations in the pile-up and the resolved is required to generate dislocations in the adjacent grain, shear stress on slip planes. The applied shear stress com- resulting in the increase of yield stress. On the other hand in bined with this stress concentration will cause dislocation large grained materials, the magnitude of stress concentra- movement in the adjacent grain across the boundary. This tion at the head of the pile-up of dislocations will be quite will decrease the pile-up of dislocation, which in turn will high and more than sufficient to initiate plastic deformation

1.5 Elements of Plastic Deformation 25 in adjacent grains. Note that finer grain-size materials are atom interactions and range from 100 to 1000 Å. The term weaker above the equicohesive temperature (the temperature at which the strengths of grain boundary2 and grain body are KydÀ1=2 represents very long-range structural size effect that equal). The following Hall–Petch equation [(1.57a)] is not is effective at a spacing greater than 104 Å. Thus, the yield or only applicable to grain boundaries but also to other types of boundaries such as ferrite–cementite in pearlite, mechanical flow stress of a material depends on the stress field of both twins and martensite plates. short- and long-range order that interacts with moving dislocations. The Hall–Petch (1.57a) is based on the r0 ¼ ri þ KydÀ21 ð1:57aÞ grain-boundary-induced dislocation pile-up model and holds well for large pile-ups containing more than 50 dislocations where but not applicable to extremely small grain size with small r0 yield or flow stress of polycrystalline material. pile-ups of dislocations. For example, if (1.57a) is extrapo- ri “lattice friction stress” representing overall resistance lated to extremely fine grain size, in the order of 4 nm, the of lattice to dislocation movement and dependent on composition, strain rate and temperature; ri can be yield strength will approach the theoretical value, which is considered as the yield strength at ‘infinite’ grain size, i.e. single crystal. unrealistic. For small pile-ups of dislocations, the observed Ky “locking parameter,” which measures relative harden- ing contribution of grain boundaries that depends on strength–microstructural size relation can be better described composition and in some cases strain rate and temperature. (Armstrong et al. 1966) by other equations. d diameter of grain Experiments on many nanocrystalline materials with grain sizes of several tens of nanometres have established that their yield strengths would either remain constant or decrease with reduction in grain size. It has been experi- mentally observed for several systems that the value of Ky in (1.57a) (the slope of Hall–Petch plot) is reduced or becomes The reader is referred to the work of Eshelby et al. (1951) negative below a certain grain size (Chokski et al. 1989; for derivation of (1.57a). The above Hall–Petch relation predicts that the finer the grain size, the higher the yield Fougere et al. 1992; El-Sherik et al. 1992; Gertsman et al. strength is. From the tests on a material with different grain sizes, r0 versus dÀ1=2 can be plotted as a straight line and the 1994). This type of relationship between yield stress and slope and intercept with the ordinate of this linear plot will respectively be the values of Ky and ri in (1.57a). The term grain size has been designated as reverse or inverse Hall– Ky is basically independent of temperature. Conrad (1961) has divided ri into two components: rT ; which is not Petch effect (IHPE). Hence, the Hall–Petch relation is only sensitive to structure and applied stress but strongly depen- dent on temperature, and rST ; which is independent of valid for grain sizes over a critical value up to which dis- temperature and applied stress but sensitive to structure. The term rT is related to the Peierls–Nabarro stress, and the term location pile-ups are still possible. When the grain size is rST is a measure of the stress required to move the dislo- cation against the resistance offered by other dislocations, below the critical value, where grains are unable to support precipitate particles, impurities, subgrain boundaries. dislocation pile-ups, there will be a decrease in hardness According to Conrad, the flow stress of a material may then with decreasing grain size (Nieh and Wadsworth 1991). be given by At present, there is a growing doubt on the dislocation pile-up model for the Hall–Petch equation. Instead, presently the role of the grain boundary as a source for dislocations is considered to be important and responsible for the relation of yield strength with the size of the microstructure. Li (1963) proposed a model based on generation of dislocations at grain-boundary ledges and observed that the dislocation density ‘q’ was inversely proportional to the grain size d; i.e; q ¼ 1=d: Considering the influence of grain size on r0 ¼ rT þ rST þ KydÀ12 ð1:57bÞ the dislocation density, the yield strength is expressed in terms of dislocation density as follows: In (1.57b), the term rT gives the short-range-order Peierls r0 ¼ ri þ aGbpffiqffiffi ð1:58aÞ stress field effects on moving dislocations which extend less than 10 Å. The term rST represents the long-range-order where r0 and ri have the same meaning as in (1.57a); dislocation stress field effects which take care of dislocation– dislocation, dislocation–precipitate and dislocation–impurity a A numerical constant that normally range between 0.3 and 0.6; 2Strength of grain boundary means the resistance offered by the grain boundary to the process of deformation. G Shear modulus; b Burgers vector; q Dislocation density.

26 1 Tension Substituting q = 1/d in (1.58a), There is probably a relation between the stacking fault energy and the energy of a coherent boundary of an r0 ¼ ri þ a GbdÀ21 ¼ ri þ Ky0 dÀ21 ð1:58bÞ annealing twin in FCC metals; the former is assumed to be the double of the latter. Hence, the lower the stacking fault Consequently, (1.58b) has the same form as (1.57a). energies, the lower the twin-boundary energy is and the greater the tendency for the formation of annealing twins in 1.5.6 Stacking Fault FCC metals is. This agrees well with metallographic observations of the frequency of occurrence of annealing Stacking fault is a surface defect and created by the error or twins in FCC metals. For example, the stacking fault ener- fault in the stacking sequence of close-packed planes of gies are $ 90 mJ=m2 for copper and $ 250 mJ=m2 for atoms produced mostly by plastic deformation (Warren and aluminium (Hertzberg 1989), and aluminium rarely shows Warekois 1955). The presence of stacking faults can be annealing twins, whereas annealing twins are observed in detected by very precise X-ray diffraction measurements. copper. Further, since addition of zinc to copper decreases Stacking faults are usually observed in close-packed the stacking fault energy in a-brass, annealing twins are face-centred-cubic (FCC) structures having the stacking observed to a greater extent in a-brass than in copper. Thus, sequence of ABCABC and in hexagonal close-packed metals with lower stacking fault energies that have wider (HCP) structures having the stacking sequence of ABAB, stacking faults not only strain-harden more rapidly, but also because it is much difficult to form stacking fault in an twin easily on annealing. open-packed body-centred-cubic (BCC) lattice. A stacking fault is created by the dissociation of a perfect dislocation 1.5.7 Strengthening Methods into two partial dislocations. To make a stacking fault stable, the decrease in energy due to dissociation of dislo- Real crystals due to the presence of dislocations in them cation must be more than the increase in interfacial energy of require very low stresses to cause plastic deformation com- the faulted region. From the point of dislocation theory, a pared to the theoretical shear strength of a defect-free ideal stacking fault can be defined as an extended dislocation crystal. Although the shear strengths of perfect crystals consisting of a faulted region bounded by two partial dis- (whiskers) which are free of dislocations may approach the locations. The faulted region in an FCC or HCP structure is theoretical value, they are so tiny that they cannot be used as respectively a thin HCP region with the stacking sequence of components in service. Material can be made hard and ABAB, or a thin FCC region with the stacking sequence of strong by making the dislocation movement difficult, while ABCABC. The two partial dislocations forming a stacking dislocations can move easily in those materials which tend to fault are nearly parallel and, hence, tend to repel each other, be soft. Hence, to increase the strength of crystalline mate- whereas the surface tension of the stacking fault, which is rials against plastic deformation, the stress required for related to the stacking fault energy, tends to pull the partial movement of dislocations must be increased. The important dislocations closer. Hence, the lower the stacking-fault methods of strengthening crystalline materials against plastic energy, the greater the equilibrium distance of separation yielding are briefly summarized below: between the partial dislocations is, and the wider the stack- ing fault is (Murr 1975). 1. Strain Hardening For the movement on a plane other than the plane of the Strain hardening increases the stress required for slip and fault, the partial dislocations have to recombine together to thereby increases the strength of metals. In addition to the form a perfect dislocation. For materials with high stacking interaction of dislocations with each other, twin boundaries, fault energy, a little stress is required for recombination of the grain boundaries, etc., acting as barriers to dislocation partial dislocations, separated on the order of 1b or less, and if motion cause strain hardening (see Sect. 1.5.5). Stacking this recombined dislocation is of the screw type, it may faults play an important role in strain hardening of metal. cross-slip. On the other hand, for materials of low stacking Metals with a lower stacking fault energy having a wider fault energy, the stress required to recombine the partial dis- stacking fault will strain harden more rapidly. locations, separated on the order of 10–20b, is large and the cross-slip of an extended screw dislocation around obstacles When metals are cold worked, strain hardening causes to and barriers will be restricted and the rate of strain hardening increase their hardness and strength. The dislocation density would be very high. So, it is evident that the extent of strain of metals increases with increasing the amount of cold hardening is greater in a material with lower stacking-fault- working. The dislocation density of an annealed metal is energy than with higher stacking fault energy.

1.5 Elements of Plastic Deformation 27 about 104 mmÀ2; whereas that of a heavily cold-worked Z metal is 1010 mmÀ2: With increasing dislocation density, the stress required for movement of any one dislocation Y increases due to interfering effect of the stress fields of X surrounding dislocations. The dependency of yield strength on the dislocation density has been described by means of Stress (1.58a). 2. Strain Ageing AB C Strain ageing behaviour is usually associated with the Strain yield-point phenomenon, which has been explained in Sect. 1.6.4. When a metal after cold working is heated to a Fig. 1.15 Schematic stress–strain diagram of low-carbon steel illus- relatively low temperature, its strength is increased (although trating the phenomenon of strain ageing. Region A : original steel is its ductility is decreased)—this behaviour is called strain strained plastically through yield-point elongation to a strain corre- ageing. For example, the occurrence of strain ageing phe- sponding to point X: Region B : load is released from point X and nomenon has been explained with respect to the schematic reapplied immediately. Region C : reappearance and increase in yield stress–strain curve (flow curve) of a low-carbon or mild point from Y to Z on reloading after ageing at 400 K ð%130 CÞ steel, which is shown by region A in Fig. 1.15. Suppose a mild steel specimen is strained plastically through the Grain refinement not only increases the strength proper- yield-point elongation to a strain corresponding to point X in ties but also improves the impact toughness and decreases Fig. 1.15. At this point if the load is released and reapplied the ductile–brittle transition temperature. An important immediately, no yield point appears as shown by the region example of this is the development of high-strength B in Fig. 1.15, since the dislocations have been torn away low-alloy (HSLA) steels, or low-carbon microalloyed from atmosphere of carbon and nitrogen atoms dissolved in steels, usually containing less than 0.1% carbon (Union steel and time has not been allowed for diffusion of carbon Carbide Corporation 1977) (see Sect. 6.5 in Chap. 6), in and nitrogen atoms to the dislocations to form new atmo- which ultrafine ferrite grains have been produced by con- spheres of interstitial solutes anchoring the dislocations. trolled rolling. Now, consider that the specimen is strained to point Y and unloaded. If it is aged for several days at room temperature The dependency of yield strength on the grain size has been or several hours at a little higher temperature like 400 K described by means of (1.57a) that shows the finer the grain ð%130 CÞ; the dissolved carbon and nitrogen atoms diffuse size, the higher the yield strength. Note that the strengthening to the dislocations during the ageing period and anchor the effect for a given grain size depends on the magnitude of the dislocations by forming new atmospheres of interstitials. So, constant Ky in (1.57a). For example, the values of Ky are when the specimen is reloaded after ageing, the yield point 0:71 MN mÀ3=2 for BCC iron, 0:11 MN mÀ3=2 for FCC will not only reappear but will also be increased from Y to Z copper and 0:07 MN mÀ3=2 for FCC aluminium. So, for a (region C in Fig. 1.15) due to the ageing treatment. given amount of grain refinement, the strengthening effect in BCC iron is greater than that in FCC copper, which in turn is 3. Grain Refinement greater than that in FCC aluminium. The effect of grain size on the strength properties of a 4. Subgrain Strengthening polycrystalline material has been described in Sect. 1.5.5.1, which reflects that we can obtain stronger materials by Subgrains can exist within the grains surrounded by refinement of grain size. The refinement of ferrite grain size high-energy grain boundaries across which the orientation in ferrite–pearlite steels, or of pearlite colony in eutectoid difference between grains is large. On the other hand, the carbon steels, is obtained by controlling the austenite grain subgrains are bounded by low-angle boundaries having size, which can be performed in at least four ways: energy lower than high-angle grain boundaries. The orien- tation difference across the subgrain boundary may be only a • Normalizing. few minutes of arc or, at most, a few degrees. Low-angle • Repetitive rapid austenitizing. boundaries or subgrain boundaries can be produced in sev- • Controlled rolling. eral ways. They may be produced during crystal growth, as a • Additions of aluminium to steel.

28 1 Tension result of phase transformation, or during creep deformation 5. Solid Solution Strengthening at high temperature. The most general method of producing subgrain network is to apply a small amount of deformation Solid solution formed by introducing solute atoms in the (from about 1 to 10% prestrain) followed by an annealing lattice of a pure metal acting as a solvent is stronger than the treatment to reorganize the dislocation structure into pure metal since the resistance to dislocation movement in low-angle boundaries. This process is called polygonization the solid solution is generally greater than that in the pure or recrystallization in situ. Deformation (say, bending) of a metal. The increase in stress required for plastic deformation single crystal results in the introduction of excess disloca- in a solid solution is due to the interaction of the stress fields tions of one sign which are distributed along the deformed around solute atoms with the stress field of a gliding dislo- (bent) glide planes. When the grain is heated, the disloca- cation. The yield stress and the level of the entire stress– tions rearrange themselves into the lower energy configu- strain curve will usually be raised by solute additions, as ration of a low-angle grain boundary by dislocation climb shown in Fig. 1.17. The solid solution strengthening effect process, resulting in a polygonlike network structure of in a substitutional solid solution increases with increasing low-angle boundaries. Figure 1.16 shows a low-angle tilt mainly boundary, which can be considered to be an array of edge dislocations. • the size difference between the solute and the solvent atoms, because the stress field around the solute becomes It has been found (Parker and Washburn 1952) that a more intense that correspondingly leads to stronger low-angle boundary moves as a unit on application of a interaction with the moving dislocation. shear stress and as the shear distance increases, the boundary angle (the orientation difference across the subgrain bound- • the solute atom concentration, because the gliding dis- ary), h; decreases. Decreasing h means the spacing, L; location interacts with the solute stress fields at many between dislocations lying along the boundary increases, points along its length. since h ¼ b=L; where b is the magnitude of the Burgers vector of the lattice. Again, increasing L implies that the Hence, it appears that the best result from solid solution boundary loses dislocations with its movement. This is an strengthening would be achieved if the size difference between expected fact if dislocations are held up at imperfections the solute and the solvent atoms would be the maximum cou- such as precipitate particles, foreign atoms and other dislo- pled with the maximum solute atom concentration. But dis- cations. Thus, the formation of subgrains makes the metal solution of a large concentration of a solute, which differs stronger. It has been shown from the stress–strain curve of significantly in size from the solvent, is not possible according 1020 steel (Parker and Washburn 1955) that the substructure to Hume–Rothery’s rule, which states the larger the size dif- of low-angle grain boundaries in steel, which has been ference between the solute and the solvent is, the smaller the produced by cold-reduction and annealing, exhibits a higher equilibrium solubility will be. In interstitial solid solution, the yield point and tensile strength than the steel either in solute atoms are usually larger than the interstitial voids they annealed condition only or in cold reduced condition only. occupy. Here, with increasing solute concentration the Moreover the ductility of the steel containing a substructure strengthening effect can be very strong, but the equilibrium is almost comparable to that of the annealed steel. solubility tends to be less. However, this problem of lower (a) (b) Fig. 1.16 a Distribution of positive edge dislocations along the bent glide planes. b Formation of polygonlike network structure of low-angle boundaries by dislocation climb process on heating the crystal

1.5 Elements of Plastic Deformation 29 Stress Solid solution with c2% solute When solute atoms preferentially segregate to the stacking faults contained in extended dislocations and their Solid solution with c1% solute concentration within the stacking fault increases, the stacking-fault energy is reduced increasing the separation Pure polycrystal between the pair of parallel partial dislocations in the stacking fault. The stacking fault energy is found to c2 > c1 decrease more by additions of solutes possessing higher valency to a pure metal than by additions of solutes of lower Strain valency, because a higher-valency solute increases the electron to atom ratio to a greater extent than a Fig. 1.17 Effect of solute additions in alloys on stress–strain curve lower-valency solute. Since the pair of parallel partial dis- (schematic) locations separated from each other in a stacking fault has to recombine together to form a perfect dislocation for the solute solubility can be partly overcome by dissolving higher movement on a plane other than the plane of the fault, the amount of solute in solvent at elevated temperature and then stress required to recombine them together will be higher as quenching the solid solution at lower temperature in order to the separation between them becomes larger. Hence, the obtain a supersaturated metastable solid solution by minimiz- movement of the extended dislocations will be difficult ing the diffusion of solute atoms, which consists of solute leading to strengthening effect, which will be obviously concentration higher than the equilibrium one. For example, greater for solutes of higher valency than of lower valency. the austenite phase in plain carbon steels at high temperature Fortunately, the rise of flow stress due to additions of can dissolve carbon atoms (interstitial solute) more than the higher-valency solutes also tends to be more pronounced equilibrium solubility of carbon in low-temperature ferrite than that due to additions of lower-valency solutes. phase. When this austenite is quenched from the high austen- itizing temperature to room temperature, all the carbon is A large valency difference between solute and solvent retained in the product phase martensite (a super saturated solid atoms can cause strengthening, because the charge associ- solution of C in a-Fe), producing very hard steels. ated with solute atoms of dissimilar valency can interact with dislocations which have electrical dipoles (Cottrell et al. For the same size difference between the solute and the 1953). Hence, higher hardening effect could be achieved if solvent atoms in a given substitutional solid solution, the valency of solutes would differ considerably from the parent solute atom smaller than the solvent atom will produce a metal, but unfortunately, these factors are not favourable for greater strengthening effect than the solute atom larger than extensive solid solubility and, thus, limit the formation of the solvent atom. In addition to the size difference, the higher solid solution. the elastic modulus of the solute, the greater the intensity of the stress field around a solute atom is. For example, the solid Short-range ordering, in which atoms have more dis- solution of Cu–Ni system will be stronger than that of Cu–Zn similar neighbours than expected for a truly random solu- system, where atomic radius is 1.28 Å for solvent Cu, 1.25 Å tion, and clustering (grouping together of like solute atoms) for solute Ni (smaller than the solvent atom) and 1.31 Å for in a solid solution can produce strengthening, because the solute Zn (larger than the solvent atom), and the absolute degree of order is reduced locally when a dislocation moves atomic size difference between the solvent and both the through a region of short-range order or clustering. There solutes is 0.03 Å. Further, since the elastic modulus of Ni is will be an increase in the energy of the alloy due to this higher than that of Zn, the stress field intensity around Ni will process of disordering, and extra work must be provided to be stronger resulting in a greater strengthening effect. In maintain the energetically unfavourable movement of addition to the above factors, one or more of the following dislocations. factors may contribute to solid-solution strengthening by offering resistance to dislocation motion. Long-range-ordered solid solution or superlattice, in which dissimilar atoms are arranged in some regular alter- nating pattern over a large volume of the crystal, also con- tributes to the solid solution strengthening. The movement of an ordinary dislocation through a superlattice creates a region of disorder across the slip plane which is called an antiphase domain boundary (APDB). In order to preserve order across the slip plane, the dislocations are paired with the APDB between them to form an “extended dislocation” in the superlattice. The disorder created by the first dislo- cation of the pair is eliminated by the passage of the second dislocation. Such a pair of dislocations separated by an

30 1 Tension Fig. 1.18 Structure of a superlattice dislocation in an ordered cubic temperature. The solution-treated alloy is then rapidly lattice, in which the dashed line is an antiphase domain boundary lying cooled by quenching into water or some other cooling in between the pair of dislocations medium. This rapid cooling suppresses the immediate sep- aration of the second phase as precipitate so that the alloy is antiphase domain boundary (indicated by the dashed line) in retained at the low temperature as a metastable supersatu- an ordered cubic lattice is shown in Fig. 1.18. The width of rated single-phase solid solution. If, however, after APDB is the result of a balance between the energy of quenching, the alloy is allowed to ‘age’ for a sufficient APDB and the elastic repulsion of the two dislocations of the length of time, the decomposition of the supersaturated solid same sign. The stress required for the movement of a dis- solution causes the second phase to precipitate out. Since location through a long-range-ordered structure is equal to the rate of atomic migration controls the rate at which the the ratio of the energy of an APDB to the spacing of the nuclei grow, so with increasing temperature of ageing the APDBs. The rate of strain hardening in the ordered state is atomic migration increases, which makes the precipitation higher than that in the disordered condition, because more of the second phase rapid. However, the size of the pre- APDBs are produced with the progress of slip. Ordered solid cipitate becomes finer with lowering of the temperature at solution having a fine domain size of about 5 nm is stronger which precipitation takes place. Extensive hardening of the than the disordered solid solution, but the yield stress of the alloy occurs if finely dispersed small-sized particles are former with a large domain size is generally lower than that precipitated during ageing, since fine dispersions present of the latter. strong obstruction to the motion of dislocations. A finer dispersion is produced when particles are nucleated on the 6. Precipitation Strengthening and Dispersion dislocations in the matrix. This can be achieved if the ageing Strengthening treatment is preceded by plastic deformation. Thus, a com- bination of the effects of strain hardening and precipitation The strengthening from distribution of second-phase fine can produce a stronger alloy. particles in a ductile matrix can be achieved by either pre- cipitation hardening, also known as age hardening, (Martin Examples of some common age hardening systems are 1968) or dispersion hardening (Decker 1973). The degree of aluminium alloys, such as Al–Cu, Al–Ag, Al–Mg–Si, Al– strengthening resulting from second-phase particles depends Mg–Cu, Al–Mg–Zn, and copper–beryllium alloys. For on the size (average size), shape, mean interparticle spacing occurrence of precipitation hardening, the second phase and volume fraction of the particles. Of course, factors such must be soluble at a high temperature and the solid solubility as size, spacing and volume fraction are related with each of the solute must decrease with decreasing temperature. other. For example, for a given particle size, the mean Usually, the lattice of the precipitates is crystallographically interparticle spacing increases if the volume fraction of closely matched, or coherent with that of matrix. Coherent second-phase particles is decreased and vice versa. Simi- precipitate particles are particularly powerful barriers to the larly, for a given volume fraction of second-phase particles, motion of dislocation because the large elastic distortion of reduction in particle size decreases the mean interparticle the matrix around the coherent precipitates interacts strongly spacing and vice versa. with the stress field of the dislocations. This effect con- tributes to strengthening and is known as coherency hard- In precipitation or age hardening treatment carried out in ening. Coherency hardening occurs usually in the early the solid state, the alloy is first solution heat-treated at an stages of precipitation strengthening, when there is a for- elevated temperature above the solvus line to produce a mation of zones (cluster of solute atoms) which are normally single-phase homogeneous solid solution by dissolving the coherent with the matrix. For coherent precipitates, there is a second phase, which was present in the alloy at low reduction in the surface energy term, but at the same time coherency leads to an increase in the strain energy. Since fine coherent precipitate causes a large reduction in surface energy but a small increase in strain energy, so coherency is favoured when the size of precipitate is small. However, the problem of precipitation-hardened alloys is that if, at any given temperature, ageing is allowed to proceed too far, coarsening of the precipitate particles might take place; i.e., the smaller ones tend to re-dissolve and the larger ones grow still larger. The driving force for this coarsening process is the reduction in the surface energy per unit volume of the fine particles. While coarsening, the numerous finely dis- tributed, small-sized precipitate particles are replaced by a