Nanomaterials: Mechanics and mechanisms

Nanomaterials Mechanics and Mechanisms

K.T. Ramesh

Nanomaterials Mechanics and Mechanisms

123

K.T. Ramesh The Johns Hopkins University Baltimore, MD 21218 USA [email protected]

ISBN 978-0-387-09782-4 e-ISBN 978-0-387-09783-1 DOI 10.1007/978-0-387-09783-1 Springer Dordrecht Heidelberg London New York Library of Congress Control Number: 2008942797 c Springer Science+Business Media, LLC 2009  All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

To Arjun, Rohan and Priti, for your love and patience.

Preface This book grew out of my desire to understand the mechanics of nanomaterials, and to be able to rationalize in my own mind the variety of topics on which the people around me were doing research at the time. The field of nanomaterials has been growing rapidly since the early 1990s. Initially, the field was populated mostly by researchers working in the fields of synthesis and processing. These scientists were able to make new materials much faster than the rest of us could develop ways of looking at them (or understanding them). However, a confluence of interests and capabilities in the 1990s led to the explosive growth of papers in the characterization and modeling parts of the field. That confluence came from three primary directions: the rapid growth in our ability to make nanomaterials, a relatively newfound ability to characterize the nanomaterials at the appropriate length and time scales, and the rapid growth in our ability to model nanomaterials at atomistic and molecular scales. Simultaneously, the commercial potential of nanotechnology has become apparent to most high-technology industries, as well as to some industries that are traditionally not viewed as high-technology (such as textiles). Much of the rapid growth came through the inventions of physicists and chemists who were able to develop nanotechnology products (nanomaterials) through a dizzying array of routes, and who began to interface directly with biological entities at the nanometer scale. That growth continues unabated. What has also become apparent is that much of the engineering community continues to view nanomaterials as curiosities rather than as the potentially gamechanging products that they can be. This book seeks to provide an entr´e into the field for mechanical engineers, material scientists, chemical and biomedical engineers and physicists. The objective is to provide the reader with the connections needed to understand the intense activity in the area of the mechanics of nanomaterials, and to develop ways of thinking about these new materials that could be useful to both research and application. Note that the book does not cover the areas associated with soft nanomaterials (polymer-based or biologically-derived), simply because I am not knowledgeable about such systems (and the mechanics can be quite different). This is not intended to minimize the importance of soft nanomaterials or the potential of soft nanotechnology; the reader will simply have to go elsewhere to encounter those areas. This book is intended to be read by senior undergraduates and first-year graduate students who have some background in mathematics, mechanics and materials science. It should also be of interest to scientists from outside the traditional fields of mechanical engineering and materials science who wish to develop core expertise in this area. Although senior undergraduates should be able to read this book

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from cover to cover, they may find it somewhat heavy going as a textbook. First and second year graduate students should find this book challenging but accessible. My intent has not been to provide a review of the field but rather to provide a basic understanding. If the reader puts the book down with an appreciation of the excitement of the field of nanomaterials and the potential applications in mechanical engineering, materials science and physics, this book will have achieved its objectives. My personal objective with this book is to provide a means for integrating the discussions that currently go on in the mechanics and materials communities, and at the same time to provide an accessible path to those from outside these disciplines who wish to get involved in one of the most exciting fields of our time. While this book is intended to be read cover to cover, it can also be used as a reference work. The book should serve effectively as a textbook for a course in nanomaterials or nanomechanics as it relates to materials. It is possible that mechanical engineers using this book will also want to have a materials science reference available as they move through this volume, and conversely, material scientists using this book may want to have a basic mechanics of solids book as a companion volume. An effective scenario would be for graduate students who have taken courses in the mechanics of solids or the mechanical properties of materials to then take a course that uses this book as a text. The suggested reading at the ends of the chapter will be useful for those who wish to pursue a particular topic in greater depth than can be covered in a book such as this. I am aware that the book in its current form does not have enough problems at the end of each chapter to make it ideal as a textbook. However, this is a field at the cutting edge of nanotechnology, rather than an established area with standard problems that can be given to students without considering their specific backgrounds. The problems that are provided at the ends of the chapters are intended to provoke discussion and further study, rather than to provide training in specific solution methods. We are in the midst of a veritable explosion in nanotechnology, and nanomaterials are at the heart of it. I hope you put this book down as excited about the field as I have been in the writing of it. The Johns Hopkins University Baltimore, MD

K.T. Ramesh

Acknowledgements This book developed as a result of conversations with many students and colleagues working in the area of the mechanics of nanomaterials. What began as a simple attempt to rationalize my own thinking in this area became a concept for a book after discussions with Elaine Tham, my editor at Springer. Both she and her assistant, Lauren Denahy, deserve my thanks for their patience with me as I learned how to write a large document. My thanks go also to my assistant, Libby Starnes, who has given me a great deal of help in organizing my materials for this volume, and in rearranging my schedule to make time for this book. Her resourcefulness and ability to handle details have been terrific. I would like to give special thanks to the several anonymous reviewers of this book. Their suggestions have greatly improved the manuscript, and I appreciate all the effort involved. My research group has been bearing the weight of my absentmindedness as I worked on this book for nearly a year now, and I would like to express my gratitude to all of them. They have helped me immeasurably by doing the research that I get to talk about, by being willing to critique ideas, and by helping me make up for the times that I was unavailable. During the writing of this book, the research group included Dr. Shailendra Joshi, Reuben Kraft, Bhasker Paliwal, Dr. Qiuming Wei, Dr. Fenghua Zhou, Jessica Meulbroek, Emily Huskins, Sarthak Misra, Dr. Jamie Kimberley, Dr. Krishna Jonnalagadda, Dr. Buyang Cao, Cindy Byer, Rika Wright, Guangli Hu, Cyril Williams, Brian Schuster, Dr. Bin Li, Dr. Nitin Daphalapurkar and Dr. George Zhang. My thanks to all! Several students have been involved in reviewing chapters of this book, and to them my special thanks: Guangli Hu, Cindy Byer, Justin Jones; Christian Murphy made up some of the illustrations. My colleagues at Hopkins have been a big help, explaining (patiently) to me the physical concepts in nanomaterials and nanomaterial behavior. This is an exploding field, and my knowledge of it comes almost entirely through my interactions with the faculty and students around me at Johns Hopkins. Special thanks to Evan Ma, Kevin Hemker, Bill Sharpe, Jean-Francois Molinari and Todd Hufnagel. Finally, this book would not have been possible without the love and support of my family, who had to put up with my occasional pre-occupation, my prolonged absences, and a lack of social interaction as I wrote. They kept me sane during the difficult times, and cheerfully stepped in to help when they could. Rohan reviewed some of the chapters, and everyone has been involved with the proofreading. Arjun has been a source of good cheer throughout the process, and Priti, I could not do this without you. To my family, my love and my thanks for all that you do!

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Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix Acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv 1

Nanomaterials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Length Scales and Nanotechnology . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 What are Nanomaterials? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Classes of Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Making Nanomaterials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1 Making dn Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.2 Health Risks Associated with Nanoparticles . . . . . . . . . . . . . . 1.4.3 Making Bulk Nanomaterials . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Closing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Suggestions for Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 Problems and Directions for Research . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 3 5 6 6 7 8 17 18 18 19

2

Fundamentals of Mechanics of Materials . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Review of Continuum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Vector and Tensor Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Kinematics of Deformations . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.3 Forces, Tractions and Stresses . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Work and Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Field Equations of Mechanics of Materials . . . . . . . . . . . . . . . . . . . . . 2.4 Constitutive Relations, or Mathematical Descriptions of Material Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Plastic Deformation of Materials . . . . . . . . . . . . . . . . . . . . . . . 2.4.3 Fracture Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Suggestions for Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Problems and Directions for Research . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

21 21 21 25 29 34 35

Nanoscale Mechanics and Materials: Experimental Techniques . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 NanoMechanics Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Characterizing Nanomaterials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61 61 62 64

3

35 36 43 53 57 57 59

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3.3.1 Scanning Electron Microscopy or SEM . . . . . . . . . . . . . . . . . . 3.3.2 Transmission Electron Microscopy or TEM . . . . . . . . . . . . . . 3.3.3 X-Ray Diffraction or XRD . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.4 Scanning Probe Microscopy Techniques . . . . . . . . . . . . . . . . . 3.3.5 Atomic Force Microscopy or AFM . . . . . . . . . . . . . . . . . . . . . 3.3.6 In situ Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Nanoscale Mechanical Characterization . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Sample and Specimen Fabrication . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Nanoindentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.3 Microcompression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.4 Microtensile Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.5 Fracture Toughness Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.6 Measurement of Rate-Dependent Properties . . . . . . . . . . . . . . 3.5 Suggestions for Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Problems and Directions for Research . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

64 65 66 66 68 68 71 71 72 74 82 86 86 91 91 91

4

Mechanical Properties: Density and Elasticity . . . . . . . . . . . . . . . . . . . . . 95 4.1 Density Considered as an Example Property . . . . . . . . . . . . . . . . . . . . 95 4.1.1 The Rule of Mixtures Applied to Density . . . . . . . . . . . . . . . . 96 4.1.2 The Importance of Grain Morphology . . . . . . . . . . . . . . . . . . . 101 4.1.3 Density as a Function of Grain Size . . . . . . . . . . . . . . . . . . . . . 103 4.1.4 Summary: Density as an Example Property . . . . . . . . . . . . . . 105 4.2 The Elasticity of Nanomaterials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 4.2.1 The Physical Basis of Elasticity . . . . . . . . . . . . . . . . . . . . . . . . 106 4.2.2 Elasticity of Discrete Nanomaterials . . . . . . . . . . . . . . . . . . . . 107 4.2.3 Elasticity of NanoDevice Materials . . . . . . . . . . . . . . . . . . . . . 110 4.3 Composites and Homogenization Theory . . . . . . . . . . . . . . . . . . . . . . . 111 4.3.1 Simple Bounds for Composites, Applied to Thin Films . . . . . 113 4.3.2 Summary of Composite Concepts . . . . . . . . . . . . . . . . . . . . . . 116 4.4 Elasticity of Bulk Nanomaterials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 4.5 Suggestions for Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 4.6 Problems and Directions for Research . . . . . . . . . . . . . . . . . . . . . . . . . 118 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

5

Plastic Deformation of Nanomaterials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 5.1 Continuum Descriptions of Plastic Behavior . . . . . . . . . . . . . . . . . . . . 121 5.2 The Physical Basis of Yield Strength . . . . . . . . . . . . . . . . . . . . . . . . . . 122 5.3 Crystals and Crystal Plasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 5.4 Strengthening Mechanisms in Single Crystal Metals . . . . . . . . . . . . . 132 5.4.1 Baseline Strengths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 5.4.2 Solute Strengthening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 5.4.3 Dispersoid Strengthening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 5.4.4 Precipitate Strengthening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 5.4.5 Forest Dislocation Strengthening . . . . . . . . . . . . . . . . . . . . . . . 135

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5.5 From Crystal Plasticity to Polycrystal Plasticity . . . . . . . . . . . . . . . . . 136 5.5.1 Grain Size Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 5.5.2 Models for Hall-Petch Behavior . . . . . . . . . . . . . . . . . . . . . . . . 138 5.5.3 Other Effects of Grain Structure . . . . . . . . . . . . . . . . . . . . . . . . 150 5.6 Summary: The Yield Strength of Nanomaterials . . . . . . . . . . . . . . . . . 154 5.7 Plastic Strain and Dislocation Motion . . . . . . . . . . . . . . . . . . . . . . . . . . 155 5.8 The Physical Basis of Strain Hardening . . . . . . . . . . . . . . . . . . . . . . . . 156 5.8.1 Strain Hardening in Nanomaterials . . . . . . . . . . . . . . . . . . . . . 158 5.9 The Physical Basis of Rate-Dependent Plasticity . . . . . . . . . . . . . . . . 160 5.9.1 Dislocation Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 5.9.2 Thermal Activation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 5.9.3 Dislocation Substructure Evolution . . . . . . . . . . . . . . . . . . . . . 166 5.9.4 The Rate-Dependence of Nanomaterials . . . . . . . . . . . . . . . . . 167 5.10 Case Study: Behavior of Nanocrystalline Iron . . . . . . . . . . . . . . . . . . . 172 5.11 Closing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 5.12 Suggestions for Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 5.13 Problems and Directions for Research . . . . . . . . . . . . . . . . . . . . . . . . . 176 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 6

Mechanical Failure Processes in Nanomaterials . . . . . . . . . . . . . . . . . . . . 179 6.1 Defining the Failure of Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 6.2 Failure in the Tension Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 6.2.1 Effect of Strain Hardening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 6.2.2 Effect of Rate-Sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 6.2.3 Multiaxial Stresses and Microscale Processes Within the Neck . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 6.2.4 Summary: Failure in the Simple Tension Test . . . . . . . . . . . . . 189 6.3 The Ductility of Nanomaterials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 6.4 Failure Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 6.4.1 Nucleation of Failure Processes . . . . . . . . . . . . . . . . . . . . . . . . 194 6.4.2 The Growth of Failures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 6.4.3 The Coalescence of Cracks and Voids . . . . . . . . . . . . . . . . . . . 196 6.4.4 Implications of Failure Processes in Nanomaterials . . . . . . . . 196 6.5 The Fracture of Nanomaterials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 6.6 Shear Bands in Nanomaterials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 6.6.1 Types of Shear Bands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 6.6.2 Shear Bands in Nanocrystalline bcc Metals . . . . . . . . . . . . . . . 203 6.6.3 Microstructure Within Shear Bands . . . . . . . . . . . . . . . . . . . . . 207 6.6.4 Effect of Strain Rate on the Shear Band Mechanism . . . . . . . 210 6.6.5 Effect of Specimen Geometry on the Shear Band Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210 6.6.6 Shear Bands in Other Nanocrystalline Metals . . . . . . . . . . . . . 211 6.7 Suggestions for Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 6.8 Problems and Directions for Research . . . . . . . . . . . . . . . . . . . . . . . . . 211 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212

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7

Scale-Dominant Mechanisms in Nanomaterials . . . . . . . . . . . . . . . . . . . . 215 7.1 Discrete Nanomaterials and Nanodevice Materials . . . . . . . . . . . . . . . 215 7.1.1 Nanoparticles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 7.1.2 Nanotubes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 7.1.3 Nanofibers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 7.1.4 Functionalized Nanotubes, Nanofibers, and Nanowires . . . . . 226 7.1.5 Nanoporous Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226 7.1.6 Thin Films . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 7.1.7 Surfaces and Interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 7.2 Bulk Nanomaterials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228 7.2.1 Dislocation Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228 7.2.2 Deformation Twinning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 7.2.3 Grain Boundary Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 7.2.4 Grain Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236 7.2.5 Stability Maps Based on Grain Rotation . . . . . . . . . . . . . . . . . 251 7.3 Multiaxial Stresses and Constraint Effects . . . . . . . . . . . . . . . . . . . . . . 256 7.4 Closing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256 7.5 Suggestions for Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256 7.6 Problems and Directions for Research . . . . . . . . . . . . . . . . . . . . . . . . . 257 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257

8

Modeling Nanomaterials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 8.1 Modeling and Length Scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 8.2 Scaling and Physics Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . 267 8.3 Scaling Up from Sub-Atomic Scales . . . . . . . . . . . . . . . . . . . . . . . . . . . 268 8.3.1 The Enriched Continuum Approach . . . . . . . . . . . . . . . . . . . . . 269 8.3.2 The Molecular Mechanics Approach . . . . . . . . . . . . . . . . . . . . 269 8.4 Molecular Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274 8.5 Discrete Dislocation Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 8.6 Continuum Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278 8.6.1 Crystal Plasticity Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278 8.6.2 Polycrystalline Fracture Models . . . . . . . . . . . . . . . . . . . . . . . . 279 8.7 Theoretically Based Enriched Continuum Modeling . . . . . . . . . . . . . . 280 8.8 Strain Gradient Plasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287 8.9 Multiscale Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 8.10 Constitutive Functions for Bulk Nanomaterials . . . . . . . . . . . . . . . . . . 292 8.10.1 Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292 8.10.2 Yield Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293 8.11 Closing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294 8.12 Suggestions for Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295 8.13 Problems and Directions for Future Research . . . . . . . . . . . . . . . . . . . 295 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311

Acronyms There are several acronyms that are commonly used in the field of nanomaterials. Some of the primary acronyms are listed here for use throughout the book. AFM cg CNT DDD EAM FEM LEFM MD MFM MWCNT nc ns OIM QC SEM STEM STM SWCNT TEM ufg XRD

Atomic Force Microscopy coarse-grained Carbon NanoTubes Discrete Dislocation Dynamics Embedded Atom Method Finite Element Method Linear Elastic Fracture Mechanics Molecular Dynamics Magnetic Force Microscopy Multi-Walled Carbon NanoTube NanoCrystalline NanoStructured Orientation Imaging Microscopy Quasi Continuum Scanning Electron Microscopy Scanning Transmission Electron Microscopy Scanning Tunneling Microscopy Single-Walled Carbon NanoTube Transmission Electron Microscopy ultra fine grained X-Ray Diffraction

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1.1

1.2

1.3

1.4

1.5

1.6

Length scales in mechanics and materials and in nature. The topics of interest to this book cover a large part of this scale domain, but are controlled by features and phenomena at the nm scale. Note the sophistication of natural materials and systems at very small length scales. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Transmission electron micrographs (see Chapter 3) of nanocrystalline nickel material produced by electrodeposition (Integran). (a) Conventional TEM micrograph showing the grain structure. The average grain size is about 20 nm. (b) High resolution electron microscopy image of the same material. Note that there is no additional or amorphous phase at the grain boundaries, although many models in the literature postulate the existence of such a phase. Micrographs by Qiuming Wei. Reprinted from Applied Physics Letters, 81(7): 1240–1242, 2002. Q. Wei, D. Jia, K.T. Ramesh and E. Ma, Evolution and microstructure of shear bands in nanostructured fe. With permission from American Institute of Physics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Transmission electron micrograph of nanocrystalline iron produced by consolidation of a nanocrystalline precursor made by ball milling. There is a range of grain sizes, and many of the grains show evidence of the prior plastic work produced by the ball-milling process. The width of the photograph represents 850 nm. . . . . . . . . . . Schematic of the die used in Equal Channel Angular Pressing (ECAP). The included angle φ in this case is about 120◦ , but can be as small as 90◦ . The angle Φ subtended by the external radius is also an important parameter. Various obtuse die angles and dimensions may be used. Very large forces are required to move the workpiece through such dies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (a) Microstructure of tantalum produced by Equal Channel Angular Extrusion (ECAE), four passes at room temperature through a 90◦ die. (b) Selected Area Diffraction (SAD) pattern for this sample, showing the presence of many low angle grain boundaries. The fraction of grain boundaries that are low-angle rather than high-angle can be an important feature of the microstructure. . . . . . . . Schematic of High Pressure Torsion (HPT) process. A thin specimen is compressed and then subjected to large twist within a constraining die. The typical sample size is about 1 cm in diameter and about 1 mm thick. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

5

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13

14

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Microstructure of tungsten produced by high-pressure torsion (HPT). (a) A bright field TEM micrograph. (b) A dark field TEM micrograph. (c) A selected area electron diffraction pattern from this field. The grains contain a high density of defects (because of the plastic work associated with the HPT process). The grains are also elongated, with widths on the order of 80 nm and lengths of about 400 nm. Grain orientation appears to be along the shearing direction. The selected area electron diffraction shows nearly continuous rings, with no obvious intensity concentration along the rings, indicating large angle type grain boundaries in the sample. Reprinted from Acta Materialia, Vol. 52, Issue 7, P. 11, Q. Wei, L. Kecskes, T. Jiao, K.T. Hartwig, K.T. Ramesh and E. Ma, Adiabatic shear banding in ultrafine-grained Fe processed by severe plastic deformation. April 2004, with permission from Elsevier. . . . . . 16

2.1

The stress tensor σ maps the normal vector n at any point on a surface to the traction vector t at that point. . . . . . . . . . . . . . . . . . . . . . . Kinematics of deformation: a body initially in the reference configuration B0 , with material particles occupying locations X (used as particle identifiers) is deformed into the current configuration Bt with each particle occupying spatial positions x. . . . Internal forces on surfaces S and S’ within a body, and the traction t at a point on a surface with normal n. The entire body must be in equilibrium, as must be all subparts of the body. (a) The entire body, in equilibrium with the external forces. (b) Free body diagram corresponding to part B1 of the whole body, showing the traction developed at a point on the surface S because of the internal forces generated on B1 by part B2 . . . . . . . . . . . . . . . . . . . . . . . Schematic of a simple tension test. The loading direction is the x1 direction, and the gauge length is L. Note the specimen shape, designed to minimize the influence of the gripping conditions on the ends. Specimen shapes may be specified by testing standards. . . . (a) Schematic of tensile stress-strain curve for ductile metal, showing yield, an ultimate tensile strength, and subsequent failure. (b) Experimentally obtained curve for nanocrystalline nickel, with an average grain size 20 nm. Note the unloading line after initial yield, with the unloading following an elastic slope. Data provided by Brian Schuster. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (a) Illustration of the biaxial tension of a sheet. (b) 2D stress space corresponding to the sheet, showing the stress path and possible yield surface bounding the elastic region. . . . . . . . . . . . . . . . . . . . . . . . .

2.2

2.3

2.4

2.5

2.6

23

26

30

44

45

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List of Figures

2.7

2.8

3.1

3.2

3.3

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A sharp planar crack in a very large block of linear elastic material. The coordinate axes are defined so that the crack front (the line drawn by the crack tip) is the x3 or z axis. Note that all constant z planes look identical, so that this is essentially a plane strain problem in the x1 − x2 plane. The mechanics of this problem is dominated by the behavior as one approaches the crack tip. . . . . . . . . 54 Two views of the crack in Figure 2.7, with (a) being the side view and (b) the top view looking down on the crack plane. The crack front is typically curved, but this can be neglected in the limit as one approaches the crack tip. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 Schematic of a Scanning Probe Microscopy system (not showing the feedback loop, tunneling amplifiers and other electronics). A wide variety of such systems exist, some of which contain only a subset of the features shown here. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Single crystal copper specimen and tension gripper for in situ tensile testing in the SEM, from the work of the group of Dehm, Kiener et al. (this group has also performed compression experiments using a similar apparatus) (a) SEM image showing a single-crystal copper tension sample and the corresponding tungsten sample gripper before the test at a low magnification. (b) Sample and gripper aligned prior to loading. Image taken from (Kiener et al., 2008). Reprinted from Acta Materialia, Vol 56, Issue 3, P. 13, D. Kiener, W. Grosinger, G. Dehm, R. Pippan, A Further step towards an understanding of size-dependent crystal plasticity: In situ tension experiments of miniaturized single-crystal copper samples. Feb. 2008, with permission from Elsevier. . . . . . . . . . . . . . . . Schematic of the NanoIndenter, showing both actuation and force and position sensing. Such a device may fit comfortably on a large table top, although vibration isolation may be desirable. . . . . . . . . . . . Example of force-displacement curve obtained during the nanoindentation of a nanocrystalline nickel material with an average grain size of 20 nm. Data provided by Brian Schuster. . . . . . . (a) Schematic of a microcompression experiment, showing micropillar and flat-bottomed indenter tip. (b) SEM micrograph showing a micropillar of a PdNiP metallic glass produced by focused ion beam machining. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

67

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(a) Finite element mesh used for 2D axisymmetric model of the micropillar in a microcompression experiment. (b) Computed effective (von Mises) stress distribution in a sample after plastic compression. Note the nonuniformity at the root of the pillar, indicating the importance of the root radius (also called the fillet radius). Reprinted from Scripta Materialia, Vol 54, Issue 2, page 6, H. Zhang, B.E. Schuster, Q. Wei and K.T. Ramesh, The design of accurate microcompression experiments, January 2006, with permission from Elsevier. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Input and simulated stress-strain curves for various assumed root radii (detailed differences observable in the inset) as computed from the finite element model presented in the last figure. The “input” curve is the input material behavior used in the finite element simulations, and the “output” stress-strain curves are obtained from the forces and displacements computed from the simulations and then processed in the same way as the force and displacement data is processed in the experiments. If the experimental design were to be perfect, the output curves would be identical to the input curve. Note that the simulated curves are all above the input data, demonstrating the effect of the end condition. The outermost curve has the largest root radius, equal to the radius of the cylinder. As the root radius is decreased, the simulated curves approach the input curve but are always above it. Reprinted from Scripta Materialia, Vol 54, Issue 2, page 6, H. Zhang, B.E. Schuster, Q. Wei and K.T. Ramesh, The design of accurate microcompression experiments, January 2006, with permission from Elsevier. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8 Computed variation of apparent elastic modulus (normalized by true elastic modulus) with fillet radius size at fixed aspect ratio, using the finite element simulations discussed in the text. The effects of the Sneddon and modified Sneddon corrections are also shown. Reprinted from Scripta Materialia, Vol 54, Issue 2, page 6, H. Zhang, B.E. Schuster, Q. Wei and K.T. Ramesh, The design of accurate microcompression experiments, January 2006, with permission from Elsevier. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.9 Schematic of a microtension testing apparatus, showing the actuation and sensing systems. The air bearing is an important component. The strain is computed from displacement fields measured using Digital Image Correlation (DIC) software analysis of images from the camera. Illustration due to Chris Eberl. . . . . . . . . . 3.10 Example of stress strain curve obtained on nanocrystalline nickel (20 nm grain size) in uniaxial tension using a microtension setup. The sample was a thin film. Data due to Shailendra Joshi. . . . . . . . . . .

76

77

79

84

85

List of Figures

3.11 Experimental techniques appropriate for various ranges of strain rate. The range of rates identified as “specialized machine” is very difficult to reach. Very few laboratories in the world are able to achieve strain rates higher than 104 s−1 . . . . . . . . . . . . . . . . . . . . . . . . . . 3.12 Schematic of the compression Kolsky bar (also known, incorrectly, as the split-Hopkinson pressure bar). The projectile is usually launched towards the input bar using a gas gun. Specimen surfaces must be carefully prepared for valid experiments. . . . . . . . . . . . . . . . . . 3.13 Stress strain curves obtained on 5083 aluminum using the compression Kolsky bar at high strain rates (2500 per second). The lowest curve represents quasistatic behavior. In general, the strength appears to increase with increasing strain rate, but note the anomalous softening at the highest strain rate (perhaps due to thermal softening). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.14 The Desktop Kolsky Bar – a miniaturized compression Kolsky bar arrangement developed by Jia and Ramesh (2004). This device is capable of achieving strain rates above 104 per second, and can fit on a standard desktop. Specimen sizes can be on the order of a cubic millimeter. The large circular object behind the bar is part of a lighting system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1

4.2

4.3

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87

88

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90

The linear rule of mixtures. Calculated variation of density in a nanocrystalline material with volume fraction of grain boundary, based on Equation (4.6). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 Schematic of (a) a cubical grain of size d with a grain boundary domain of thickness t, and (b) the packing of such grains to form a material. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 Variation of grain volume fraction, grain boundary volume fraction, triple junction volume fraction, and corner junction volume fraction with normalized grain size β = dt for the cubic grain morphology. The junction volume fractions become major contributors when d ≈ t. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 Scanning electron micrograph showing the presence of pores (dark regions) on the surface of a nanocrystalline nickel material produced by an electroplating technique. . . . . . . . . . . . . . . . . . . . . . . . . 101 Another possible space-filling morphology, using hexagonal tiles of side s and height h = α s, where α is the aspect ratio of the tile. The grain boundary domain thickness remains t. . . . . . . . . . . . . . . . . . 102 Variation of grain volume fraction, grain boundary volume fraction, triple junction volume fraction, and corner junction volume fraction with normalized grain size β = dt for the hexagonal prism grain morphology of Figure 4.5 (with an aspect ratio α = 1). The junction volume fractions become major contributors when d ≈ t. . . . 103

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4.8

4.9

4.10

4.11 4.12 4.13 4.14

4.15 4.16

5.1

Variation of grain volume fraction, grain boundary volume fraction, triple junction volume fraction, and corner junction volume fraction with normalized grain size β = dt for the hexagonal prism grain morphology of Figure 4.5, with two aspect ratios (10 and 0.1) corresponding to rods (top) and plates (bottom). . . . . . . . . . . . . . . . . . . 104 Predicted variation of overall density of polycrystalline material (normalized by ρsc ) with grain size for the cube and hexagonal prism morphologies (with an aspect ratio α = 10). The assumed parameters are ρgb = 0.95ρsc , ρt j = 0.9ρsc , and ρc j = 0.81ρsc . . . . . . . 105 Typical interatomic pair potential U(r), showing the equilibrium position r0 of the atoms. The curvature at the bottom of the potential well (at the equilibrium position) corresponds to the effective elastic stiffness of the bond. . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 Surface effects on a spherical nanoparticle (e.g., surface tension). Such effects can have a significant impact on the behavior of the nanoparticle, particularly with respect to its interaction with the environment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 Schematic of a thin film on a substrate. The film thicknesses of interest to industry are typically submicron. . . . . . . . . . . . . . . . . . . . . . . 110 Schematic of a polycrystalline thin film on a substrate. Note the typical columnar microstructure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 Process of homogenization of a composite material. (a) Original heterogeneous material. (b) Equivalent homogenized material. . . . . . . 112 (a) Schematic of a columnar microstructure for a thin film. (b) Schematic of a layered microstructure, which can be viewed as the columnar microstructure loaded in the orthogonal direction to that shown in (a). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 Variation of effective modulus (for fictitious material) with grain boundary volume fraction, based on Equations (4.30) and (4.34). . . . . 115 Variation of effective Young’s modulus with normalized grain size in a bulk nanocrystalline material, where the modulus in the grain boundary domain is defined to be ζ Eg = ζ Esc with ζ = 0.7 in this figure. The grain size is normalized by the effective thickness t of the grain boundary domain, typically assumed to be about 1 nm. If the latter thickness is assumed, the horizontal axis corresponds to grain size in nm. E1 and E2 correspond to Equations 4.30 and 4.34. . . 117 Schematic of slip in a crystalline solid under shear. (a) Original crystal subjected to shear stress. (b) Crystal after deformation (slip) along the shearing plane shown in (a). The final atomic arrangement within the deformed crystal remains that of the perfect crystal, except near the free surface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

List of Figures

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5.3

5.4

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5.6

5.7 5.8

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xxiii

Dislocations are visible in the transmission electron microscope. The picture shows dislocations (the lines marked by the black arrow) in a magnesium alloy (ZK60). Note the length scale in the picture. This micrograph was taken by Bin Li. . . . . . . . . . . . . . . . . . . . 124 Expanded version of Figure 5.1 showing the primary deformation mechanism that leads to plasticity in metals: the motion of line defects called dislocations. Edge dislocations are shown gliding along the slip plane. The inset shows an expanded view of an edge dislocation, amounting to an extra plane of atoms in the lattice (with the trace of the extra plane on the slip plane defined as the line defect). The motion of many dislocations results in the macroscopic slip step shown in (c). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 Schematic of slip with an edge dislocation (top figure) and screw dislocation (bottom figure), showing the Burgers and line vectors in each case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 Examples of basic crystal structures that are common in metals: body centered cubic (one atom in the center of the cube), face centered cubic (atoms at the center of each cube face), and hexagonal close packed structures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 Typical planes in a cubic crystal defined using Miller indices. The (111) plane and similar {111} planes are close-packed planes in a face-centered-cubic (fcc) crystal and therefore are part of typical operating slip systems (together with < 110 > type directions). . . . . . 129 Schematic of gliding dislocation bowing around dispersoids that are periodically spaced a distance L apart along a line. . . . . . . . . . . . . . 134 Schematic of a polycrystalline microstructure with a variety of plastic strains (represented by color or shading) within each individual grain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 Schematic of grain boundary ledges in a polycrystalline material. In the grain boundary ledge model, these ledges are believed to act as dislocation sources and generate a dislocation network. . . . . . . . . . 140

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5.10 A summary of the three basic models for the observed Hall-Petch behavior. (a) A dislocation pileup at a grain boundary. (b) Grain boundary ledges in a polycrystalline material. (c) A schematic of the geometrically-necessary dislocation model. On the left side of this subpart of the figure, the individual crystals that make up this polycrystalline material are assumed to have slipped along their respective slip systems, resulting in misfits between the grains. However, the polycrystalline material must remain compatible at the grain boundaries, assuming that voids do not open up at the boundaries. These misfits can be accommodated by creating a new set of dislocations (the so-called geometrically-necessary dislocations) that then result in a dislocation distribution near the individual grain boundaries. Only one such GND distribution is shown in one grain on the right. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 5.11 The variation of the flow stress (measured at 4% strain) with grain size for consolidated iron, taken from the doctoral dissertation of Dexin Jia at Johns Hopkins. The variation of hardness (defined here as H3v , where Hv is the Vickers hardness) is also shown. . . . . . . . . . . . . 144 5.12 Evidence of reduction of hardness with decreasing grain size in a Ni-P material, from Zhou, Erb et al., Scripta Materialia, 2003 (Zhou et al., 2003). The right-hand axis represents the Vickers Hardness number HV. Reprinted from Scripta Materialia, Vol. 48, Issue 6, page 6, Y. Zhou, U. Erb, K.T. Aust and G. Palumbo, The effects of triple junctions and grain boundaries on hardness and Young’s modulus in nanostructured Ni-P. March 2003, with permission from Elsevier. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 5.13 Compendium of data on copper in a Hall-Petch-style plot by Meyers et al. (2006), showing the scatter in data at small grain sizes but the general deviation from classical Hall-Petch behavior. Reprinted from Progress in Materials Science, Vol. 51, Issue 4, Page 130, M.A. Myers, A. Mishra, D.J. Benson, Mechanical properties of nanocrystalline materials. May 2006, with permission from Elsevier. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 5.14 Hall-Petch plots for copper, iron, nickel and titanium compiled by Meyers et al. (2006), showing the deviation from classical Hall-Petch at small grain sizes. Note the strength appears to plateau but not decrease with decreasing grain size. Reprinted from Progress in Materials Science, Vol. 51, Issue 4, Page 130, M.A. Myers, A. Mishra, D.J. Benson, Mechanical properties of nanocrystalline materials. May 2006, with permission from Elsevier. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

List of Figures

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5.15 Measured grain size distribution in a consolidated iron sample, based on 392 grains measured from TEM images such as that shown above (Jia et al., 2003). While some of the grains are clearly in the nanocrystalline range, others are more than 100 nm in size. Reprinted from Acta Materialia, Vol. 51, Issue 12, page 15, D. Jia, K.T. Ramesh, E. Ma, Effects of nanocrystalline and ultrafine grain sizes on constitutive behavior and shear bands in iron. July 2003, with permission from Elsevier. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 5.16 TEM micrographs showing microstructure of a cryomilled 5083 aluminum alloy in (a) extruded and (b) transverse directions (transverse to the extrusion axis). Note that the two images were taken at slightly different magnifications. However, elongated grains are evident in the direction of extrusion (Cao and Ramesh, 2009). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 5.17 Stress strain curve for a material, showing the elastic unloading response and the increased yield strength upon reloading (this is called strain hardening, since the material has become harder because of the increased plastic strain). . . . . . . . . . . . . . . . . . . . . . . . . . 156 5.18 Variation of normalized strain hardening with the square root of grain size for a variety of nanocrystalline and ultra-fine-grain materials (from Jia’s doctoral dissertation, Jia et al., 2003). The reference numbers correspond to those in Jia’s dissertation. . . . . . . . . 159 5.19 The dependence of dislocation velocity on shear stress for a variety of materials, from Clifton (1983). Note that a limiting velocity is expected, corresponding to the shear wave speed. There is a domain in the figure where the dislocation velocity is linear with the applied shear stress, and this is called the phonon drag regime. Reprinted from Journal of Applied Mechanics, Vol. 50, p. 941–952, R.J. Clifton, Dynamic Plasticity. 1983, with permission from original publisher, ASME. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 5.20 Rate-dependence of the flow stress of A359 aluminum alloy over a wide range of strain rates. All of the flow stresses are plotted at a fixed strain of 4% (because of the nature of high strain rate experiments, it is not generally possible to measure accurately the yield strength of materials at high strain rates). This result is due to the work of Yulong Li, and includes compression, tension and torsion data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

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5.21 Results of strain rate jump tests on severely plastically deformed copper (both cold-worked and ECAPed). Note the jump in the stress associated with the step increase in the strain rate on a specimen. One of the curves has been shifted to the right for ease of discrimination. Step increases in strain rate (jump from a lower strain rate to a higher strain rate) are used because a step decrease will exacerbate the effect from the machine compliance, and the interpretation of experimental data becomes more involved. In this case the strain rate was increased by a factor of 2 between consecutive rates. Reprinted from Materials Science and Engineering, Q. Wei, S. Cheng, K.T. Ramesh, E. Ma, Effect of nanocrystalline and ultrafine grain sizes on the strain rate sensitivity and activation volume: fcc versus bcc metals. Sep. 2004, with permission from Elsevier. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 5.22 The rate-sensitivity of copper as a function of the mean grain size, including the nanocrystalline, ultra-fine-grain and coarse-grain domains. Data is presented for materials made through a variety of processing routes, some of which involve severe plastic deformation. The figure is taken from the work of Wei et al. (2004a). Reprinted from Materials Science and Engineering, Q. Wei, S. Cheng, K.T. Ramesh, E. Ma, Effect of nanocrystalline and ultrafine grain sizes on the strain rate sensitivity and activation volume: fcc versus bcc metals. Sep. 2004, with permission from Elsevier. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 5.23 The rate-sensitivity of bcc metals as a function of grain size from a variety of sources (the figure is a variant of one published by Wei et al. (2004a). The decrease of rate sensitivity with decreasing grain size is the opposite behavior to that observed in fcc metals. Reprinted from Materials Science and Engineering, Q. Wei, S. Cheng, K.T. Ramesh, E. Ma, Effect of nanocrystalline and ultrafine grain sizes on the strain rate sensitivity and activation volume: fcc versus bcc metals. Sep. 2004, with permission from Elsevier. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 5.24 Schematic of motion of screw dislocations by kink pair nucleation and propagation. The dislocation is visualized as a line that needs to go over the energy barrier. Rather than move the entire line over the barrier, the dislocation nucleates a kink, which jumps over the barrier. The sides of the kink pair have an edge orientation, and so fly across the crystal because of their high mobility, resulting in an effective motion of the screw dislocation. . . . . . . . . . . . . . . . . . . . . . . . 173 6.1

Stress strain curve for a material obtained from a standard tensile specimen tested in uniaxial tension, showing the final fracture of the specimen in uniaxial tension. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

List of Figures

6.2

6.3

6.4

6.5

6.6

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The development of a neck during plastic deformation of a specimen within a simple tension experiment. The position and length of the neck are determined by geometric and material imperfections in the specimen. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 The influence of rate-sensitivity on the total elongation to failure of materials, from the work of Woodford (1969). Note the strong effect of the rate-dependence, and the similarity in behavior of a wide variety of materials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 Form of the fracture developed in a simple tension test. (a) Original specimen configuration before loading. (b) Tensile failure of a brittle material, showing that the fracture surface is essentially perpendicular to the loading axis. (c) Tensile failure of a ductile material, showing the cup and cone failure morphology (in section). . 189 The total elongation to failure of copper materials of varying grain sizes, as presented by Ma et al. (Wang et al., 2002). It is apparent that the ductility decreases dramatically as the yield strength is increased. The point labeled E in the figure corresponds to a special copper material produced by Ma and coworkers that included both nanocrystalline and microcrystalline grain sizes, with the larger grains providing an effective strain hardening in the material. Reprinted by permission from Macmillan Publishers Ltd: Nature, Vol. 419, Issue 6910, pages 912–915, High tensile ductility in a nanostructured metal, Yinmin Wang, Mingwei Chen, Fenghua Zhou, En Ma. 2002. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 The total elongation to failure of fcc materials of nanocrystalline and microcrystalline grain sizes, as presented by Dao et al. (2007). Reprinted from Acta Materialia, Vol. 55, Issue 12, page 25, M. Dao, L. Lu, R.J. Asaro, J.T.M. DeHosson, E. Ma, Toward a quantitative understanding of mechanical behavior of nanocrystalline metals. July 2007, with permission from Elsevier. . . . . . . . . . . . . . . . . . . . . . . . 192 A standard compact tension (CT) specimen used for fracture toughness measurements. The specimens are several centimeters in size. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198

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A sequence of in situ TEM micrographs obtained by Kumar et al. (2003) during the loading of nanocrystalline nickel using multiple displacement pulses. Images a–d show the microstructural evolution and progression of damage with an increase in the applied displacement pulses. The presence of grain boundary cracks and triple-junction voids (indicated by white arrows in (a)), their growth, and dislocation emission from crack tip B in (b–d) in an attempt to relax the stress at the crack tip as a consequence of the applied displacement, can all be seen. The magnified inset in (d) highlights the dislocation activity (Kumar et al., 2003). Reprinted from Acta Materialia, Vol. 51, Issue 2, page 19, K.S. Kumar, S. Suresh, M.F. Chisholm, J.A. Horton, P. Wang, Deformation of electrodeposited nanocrystalline nickel. Jan. 2002, with permission from Elsevier. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 6.9 Postmortem microscale fracture morphology observed by Kumar et al. (2003) after the loading of nanocrystalline nickel in tension. There is no direct evidence of the presence of dislocations in this image, although some of the grains appear to have necked before separation. Reprinted from Acta Materialia, Vol. 51, Issue 2, page 19, K.S. Kumar, S. Suresh, M.F. Chisholm, J.A. Horton, P. Wang, Deformation of electrodeposited nanocrystalline nickel. Jan. 2002, with permission from Elsevier. . . . . . . . . . . . . . . . . . . . . . . . 200 6.10 Postmortem fracture surface morphology of nanocrystalline (average grain size of 38 nm) gold thin film tested in tension till failure by Jonnalagadda and Chasiotis (2008). Dimples are seen on the fracture surface, with an average size of more than 100 nm. The void size just before coalescence is therefore much larger than the grain-size, and the growth of the voids must involve plastic deformation in many many grains. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 6.11 Progressive localization of a block (a), with initially uniform shearing deformations (b) developing into a shear band (c). The final band thickness depends on the material behavior. . . . . . . . . . . . . . 202 6.12 Stress-strain curve for a material showing softening after a peak stress. This is a curve corresponding to a material undergoing thermal softening, but similar behaviors can arise from other causes such as grain reorientation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202

List of Figures

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6.13 Gross deformation features of coarse-grain and ultra-fine-grain iron samples after compressive deformations to nearly identical strains (Jia et al., 2003). (a–b) represent quasistatic compression, while (c) represents high-strain-rate compression. (a) Homogeneous deformation of coarse-grain (20 µm grain size) Fe. (b) Shear band pattern development in ultra-fine-grain (270 nm grain size) Fe. (c) Shear band pattern development in ultra-fine-grain Fe after dynamic compression at a strain rate of ≈ 103 s−1 . Reprinted from Acta Materialia, Vol. 51, Issue 12, page 15, D. Jia, K.T. Ramesh, E. Ma, Effects of nanocrystalline and ultrafine grain sizes on constitutive behavior and shear bands in iron. July 2003, with permission from Elsevier. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 6.14 Shear band patterns evolve with strain in compressed ultra-finegrain Fe (Wei et al., 2002). Note the propagation of existing shear bands, the nucleation of new shear bands, and the thickening of existing shear bands. The development of families of conjugate shear bands is also observed. Reprinted from Applied Physics Letters, Vol. 81, Issue 7, pages 1240–1242, Q. Wei, D. Jia, K.T. Ramesh, E. Ma, Evolution and microstructure of shear bands in nanostructured fe. 2002, with permission from American Institute of Physics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 6.15 (a) Shearing deformation across one band and (b) the stable intersection of multiple shear bands in 270-nm Fe after quasistatic compression. The shear offset is clearly visible across the first band. Note that no failure (in terms of void growth) is evident at the intersection of the shear bands. Reprinted from Acta Materialia, Vol. 51, Issue 12, page 15, D. Jia, K.T. Ramesh, E. Ma, Effects of nanocrystalline and ultrafine grain sizes on constitutive behavior and shear bands in iron. July 2003, with permission from Elsevier. . . 206 6.16 TEM micrographs (a) within and (b) outside a shear band in quasistatically compressed Fe with an average grain size of 138 nm (Wei et al., 2002). The shearing direction is shown by the arrow. Note the preferred orientation of the grains within the shear band, while the grains outside the band are essentially equiaxed. Reprinted from Applied Physics Letters, Vol. 81, Issue 7, pages 1240–1242, Q. Wei, D. Jia, K.T. Ramesh, E. Ma, Evolution and microstructure of shear bands in nanostructured fe. 2002, with permission from American Institute of Physics. . . . . . . . . . . . . . . . . . . 208

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6.17 TEM micrograph showing the microstructure near a shear band boundary in nanocrystalline iron (Wei et al., 2002). The boundary between the material within the band and that outside the band is shown by the solid line. Note that the transition occurs over a transition width that is about one grain diameter. Reprinted from Applied Physics Letters, Vol. 81, Issue 7, pages 1240–1242, Q. Wei, D. Jia, K.T. Ramesh, E. Ma, Evolution and microstructure of shear bands in nanostructured fe. 2002, with permission from American Institute of Physics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 7.1

7.2

7.3

7.4

7.5

7.6

Schematic of a cubic nanocluster consisting of 64 atoms, approximately 1 nm on a side. The majority of the atoms are on the surface of this nanocluster, and so will have a different equilibrium spacing than atoms in a bulk sample of the same material. . . . . . . . . . . 218 Volume fraction of atoms on the surface of a cuboidal nanoparticle as a function of nanoparticle size, based on Equation (7.1) and an assumed interatomic spacing of 0.3 nm. Significant surface fractions are present below about 20 nm. Note that particle sizes below 0.6 nm are poorly defined. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 Core-shell model of a nanoparticle viewed as a composite, with a surface layer that has different properties as a result of the surface energy and its effects on binding. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 A spherical nanoparticle carrying several organic molecules on its surface. The molecules are chosen to perform a specific function, such as recognizing a molecule in the environment, and so are called functionalizing molecules. The spacing of the functionalizing molecules defines the functionalization density. Since the molecules modify the surface stress state when they attach to the surface, the functionalization density modifies the surface stresses and can even modify the net conformation of the nanoparticle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 Schematic of the structure of carbon nanotubes, showing the armchair, zigzag and chiral conformations. This beautiful illustration was created by Michael Str´ock on February 1, 2006 and released under the GFDL onto Wikipedia. . . . . . . . . . . . . . . . . . . . . . . . 223 Schematic of two possible modes of deformation and failure in nanotubes. (a) Buckling of a thin column. (b) Telescoping of a multi-walled nanotube. The fracture of nanotubes is a third mode, but is not shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224

List of Figures

7.7

7.8

7.9

7.10

7.11

7.12

7.13

7.14

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The development of a twinned region in a material. The figures show the results of a molecular dynamics calculation of the simple shear of a single crystal of pure aluminum. (a) Atomic arrangement in undeformed crystal before shearing. The straight lines are drawn to guide the eye to the atomic arrangement. (b) After shearing, a region of the crystal has been reoriented (this region is called the twin). The twin boundaries can be viewed as mirror planes, and the lines show the new arrangement of the atoms. . . . . . . . . . . . . . . . . . . . . 231 Example of deformation twins in a metal (the hcp metal hafnium). (a) Initial microstructure before deformation. (b) Twinned microstructure after compressive deformation at low strain rates and at 298K. The twinned regions are the lenticular shapes within the original grain structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 Evolution of twin number density in titanium with applied stress, over a variety of strain rates and temperatures. The twin density is not correlated with strain rate or temperature (or strain), but only with applied stress (Chichili et al., 1998). Reprinted from Acta Materialia, Vol. 46, Issue 3, page 19, D.R. Chichili, K.T. Ramesh, K.J. Hemker, The high-strain-rate response of alpha-titanium: experiments, deformation mechanisms and modeling. Jan. 1998 with permission from Elsevier. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 High resolution electron micrograph of deformation twins developed in nanocrystalline aluminum subjected to large shearing deformations (Cao et al., 2008). The diffraction pattern on the right demonstrates the twinned character. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 The possible grain boundary motions: displacements δn in the direction normal to the GB (grain growth or shrinkage), and δt in a direction tangential to the plane of the boundary (grain boundary sliding). The unit normal vector is n and the unit tangent vector is t, as in Equation (7.11). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236 (a) Schematic of a grain showing its soft and hard orientations (with respect to plastic deformation, not elastic stiffness). (b) The grains are initially randomly oriented in the material, but begin to orient themselves so that the soft direction is in the direction of shearing, and the process of grain rotation into the soft orientation results in the localization of the deformation into a shear band. . . . . . 238 Schematic of simple shearing of an infinite slab, showing the terms used in examining the shear localization process in simple shear (Joshi and Ramesh, 2008b). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 Schematic of the rotation of ensemble of nano-grains occupying region ℜ embedded in a visco-plastic sea S subjected to shear. The background image shows the undeformed configuration. . . . . . . . . . . . 240

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7.15 A hierarchical approach to homogenization of grain rotation due to interface traction. (a) Material continuum. (b) Collection of bins in sample space. (c) Grains within a RVE. The colored shading represents the average grain orientation in that bin. (d) Interaction at the grain level. Grains with individual orientations are described by different colors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240 7.16 Enlarged view of ℜ (Figure 7.14) showing intergranular interaction in the region. Rotation of the central grain is accomodated by rotation of the surrounding grains over a length L. . . . . . . . . . . . . . . . . 243 7.17 Evolution of grain orientation fraction (φ ) around the band center. Note the rapid early growth, the spatial localization, and the saturation of φ . Also note the spreading of the band, i.e., the increasing band thickness - this is also observed in experiments. . . . . 248 7.18 Evolution of plastic shear strain rate (γ˙p ) around the band center. The greatest activity in this variable is at the band boundaries, where the grains are reorienting into the soft orientation for shear. . . . 248 7.19 Evolution of the plastic shear strain γ p around the band center. After localization, γ p evolves slowly indicating that the plastic flow inside the band develops at the rate of strain hardening at higher nominal strains. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248 7.20 Overall stress-strain response for a defect-free sample (curve A) and a sample with an initial defect in φ (curve B). Curve B’ indicates the development of the shear band thickness corresponding to curve B. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 7.21 Evolution of shear band thickness for different grain sizes, assuming grain rotation mechanism. The material hardening parameters are held constant for all the grain sizes. The applied strain rate is 10−3 s−1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 7.22 The critical wavelength for instability as a function of grain size for three different metals (note that this is a log-log plot), with the critical wavelength computed using Equation (7.60). This figure has been obtained assuming that j = 10 below d = 100 nm, while for 100 nm < d ≤ 1 µm, we have L = 1 µm and j = Ld . The dashed straight line represents the condition that λcritical = d, which we call the inherent instability line. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254 7.23 Stability map, showing the domains of inherent instability in materials as a consequence of the rotational accommodation mechanism. The map is constructed in terms of the strength index and grain size, so that every material of a given grain size represents one point on the map, and a horizontal line represents all grain sizes of a given material. Reprinted figure with permission from S.P. Joshi and K.T. Ramesh, Physical Review Letters, Stability map for nanocrystalline and amorphous materials, 101(2), 025501. Copyright 2008 by the American Physical Society. . . . . . . . . . . . . . . . . 255

List of Figures

8.1

8.2

8.3

8.4 8.5 8.6

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Typical sequence of steps involved in modeling a mechanics of nanomaterials problem. Note the many layers of approximations involved, pointing out the danger of taking the results of simulations at face value. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262 The typical modeling approaches of interest to the mechanics of nanomaterials, and the approximate length scales over which each approach is reasonable. Note the significant overlap in length scales for several of the modeling approaches, leading to the possibility of consistency checks and true multiscale modeling. An example of the observations that can be made at each length scale is also presented, from a materials characterization perspective. . . . . . . . . . . . 263 The length scales associated with various microstructural features in metallic materials. For most crystalline materials, the behavior at larger length scales is the convolution of the collective behavior of features at smaller length scales. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 Schematic of the Lennard-Jones interatomic pair potential V (d). . . . . 272 The pair potential corresponding to the EAM potential for aluminum, using the parameters provided by Mishin et al. (1999). . . . 273 Nanocrystalline material constructed using molecular dynamics. The material is nickel, and the atoms are interacting using an EAM potential due to Jacobsen. About 30 grains are shown. The color or shade of each atom represents its coordination number (number of nearest neighbors), with the atoms in the crystals having the standard face-centered-cubic coordination number. The change in coordination number at the grain boundaries is clearly visible. This collection of atoms can now be subjected to mechanical loading (deformations) and the motions of the individual atoms can be tracked to understand deformation mechanisms. . . . . . . . . . . . . . . . . . . 275 High resolution micrograph showing the boundary between two tungsten grains in a nanocrystalline tungsten material produced by high-pressure torsion. There is no evidence of any other phase at the grain boundary, contrary to pervasive assumptions about the existence of an amorphous phase at the grain boundary in nanomaterials in crystal plasticity and composite simulations at the continuum level. Note the edge dislocations inside the grain on the right. Such internal dislocations are rarely accounted for in molecular dynamics simulations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279

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Finite element model of a polycrystalline mass of alumina, with the crystals modeled as elastic solids (Kraft et al., 2008). The model seeks to examine the failure of the polycrystalline mass by examining cohesive failure along grain boundaries. Reprinted from Journal of the Mechanics and Physics of Solids, Vol. 56, Issue 8, page 24, R.H. Kraft, J.F. Molinari, K.T. Ramesh, D.H. Warner, Computational micromechanics of dynamic compressive loading of a brittle polycrystalline material using a distribution of grain boundary properties. Aug. 2008 with permission from Elsevier. . . . . . 280 8.9 Schematic of the structure of a carbon nanotube, viewed as a sheet of carbon atoms wrapped around a cylinder. The sheet can be arranged with various helix angles around the tube axis, which are most easily defined in terms of the number of steps in two directions along the hexagonal array required to repeat a position on the helix. Illustration by Volokh and Ramesh (2006). Reprinted from International Journal of Solids and Structures, Vol. 43, Issue 25–26, Page 19, K.Y. Volokh, K.T. Ramesh, An approach to multi-body interactions in a continuum-atomistic context: Application to analysis of tension instability in carbon nanotubes. Dec. 2006, with permission from Elsevier. . . . . . . . . . . . . . . . . . . . . . . . 281 8.10 Connection between continuum deformations and atomic positions as defined by Equation (8.21). We associate atomic positions in the two configurations with material and spatial vectors. . . . . . . . . . . . . . . 284 8.11 Range of experimental measurements of the elastic modulus of carbon nanotubes, as summarized by Zhang et al. (2004). Note the theoretical predictions discussed here were 705 GPa (Zhang et al., 2002) and 1385 GPa (Volokh and Ramesh, 2006). Reprinted from Journal of the Mechanics and Physics of Solids, P. Zhang, H. Jiang, Y. Huang, P.H. Geubelle, K.C. Hwang, An atomistic-based continuum theory for carbon nanotubes: analysis of fracture nucleation. May 2004, with permission from Elsevier. . . . . . . . . . . . . . 285 8.12 Range of modeling estimates of the elastic modulus of carbon nanotubes, as summarized by Zhang et al. (2004), including a wide variety of first principles and MD simulations. Note the theoretical predictions discussed here were 705 GPa (Zhang et al., 2002) and 1385 GPa (Volokh and Ramesh, 2006). Reprinted from Journal of the Mechanics and Physics of Solids, P. Zhang, H. Jiang, Y. Huang, P.H. Geubelle, K.C. Hwang, An atomistic-based continuum theory for carbon nanotubes: analysis of fracture nucleation. May 2004, with permission from Elsevier. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286

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8.13 Simulation of a copper grain boundary by Warner et al. (2006) using the quasicontinuum method. The arrows correspond to the displacement of each atom between two loading steps. Reprinted from International Journal of Plasticity, Vol. 22, Issue 4, Page 21, D.H. Warner, F. Sansoz, J.F. Molinari, Atomistic based continuum investigation of plastic deformation in nanocrystalline copper. April 2006, with permission from Elsevier. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290 8.14 Computational approach for simulations of nanocrystalline copper by Warner et al. (2006). The grayscale represents different orientations of the crystals. Reprinted from International Journal of Plasticity, Vol. 22, Issue 4, Page 21, D.H. Warner, F. Sansoz, J.F. Molinari, Atomistic based continuum investigation of plastic deformation in nanocrystalline copper. April 2006, with permission from Elsevier. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 8.15 Comparison of the predictions of the multiscale simulations of Warner et al. (2006) (identified with the FEM symbol) with molecular dynamics (MD) simulation results and experimental data on nanocrystalline copper. Note that the experimental results are derived from hardness measurements, while the FEM simulation results correspond to a 0.2% proof strength. Both MD and FEM simulations predict much higher strengths than are observed in the experiments. Reprinted from International Journal of Plasticity, Vol. 22, Issue 4, Page 21, D.H. Warner, F. Sansoz, J.F. Molinari, Atomistic based continuum investigation of plastic deformation in nanocrystalline copper. April 2006, with permission from Elsevier. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291

List of Tables

1.1

A broad classification of nanomaterials on the basis of dimensionality and morphology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

2.1

Young’s modulus and Poisson’s ratio for some conventional grain sized materials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.1

Some measured moduli of nanotubes, nanofibers, and nanowires, viewed as 1D nanomaterials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

5.1 5.2

Slip systems with typical numbering scheme in typical fcc metals. . . 130 Known slip systems in a variety of metals, from a table put together by Argon (Argon, 2008). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 Hall-Petch coefficients for a variety of materials. . . . . . . . . . . . . . . . . . 143 Hall-Petch coefficients for pure aluminum and select aluminum alloys. Data taken from the work of Witkin and Lavernia (2006). . . . . 145

5.3 5.4 7.1 7.2 7.3

Scale-dominant mechanisms in nanomaterials, categorized in terms of materials classification, morphology, and length scale. . . . . . . . . . . 216 Typical intrinsic length scales that arise from dislocation mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 Basic parameters for grain rotation model in polycrystalline iron, as developed by Joshi and Ramesh. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247

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There’s plenty of room at the bottom. Richard Feynman

Nanomaterials 1.1 Length Scales and Nanotechnology Nanotechnology demands the ability to control features at the nanoscale (10−9 m), and a variety of techniques have been developed recently that give humanity this ability. From a fundamental science perspective, issues of physics and chemistry must be addressed at these scales. Surface and boundary effects can dominate the response. Many of the classical distinctions between mechanics, materials and physics disappear in this range of length scales, and a new kind of thinking emerges that is commonly called nanoscience (sometimes humorously interpreted as “very little science”). The recent rapid development of nanoscience is the result of a new-found ability to observe and control structure at small length and time scales, coupled with the development of computational capabilities that are most effective at small scales. It is useful to develop a physical idea of length scale, and Figure 1.1 shows the range of length scales of common interest in mechanics and biology (the latter is included because it allows one to develop a human sense of scale). Beginning at small ˚ (one angstrom, scales, features associated with atomic radii are on the order of 1 A ˚ 10−10 m) in size. The atomic lattice spacing in most crystals is of the order of 3 A. ˚ and this correlates well The diameter of a carbon nanotube is about 2 nm or 20 A, with the diameter of a double helix of DNA (which indicates, incidentally, that the nanotube is a good approach to handling DNA). A tobacco mosaic virus is about 50 nm across (this corresponds approximately in scale with the typical radii of curvature of the tips of nanomanipulators such as AFM probes). Grains in most polycrystalline metals have sizes that range from about 1 µm to about 20 µm (grain boundary thicknesses, to the extent that they can be defined, are typically <1 nm). A number of bacteria (living organisms) are also about 1 µm in size, a reminder of the remarkable sophistication of nature. Small-scale failure processes, such as the voids developed in spallation, are typically of the order of 10 µm in separation. Many cells in eukaryotic organisms are of this size-scale. Some of the most sophisticated small devices in engineering, integrated circuit chips, are of the order of 1 cm in size (the corresponding natural “device” might be a beetle). The author arrives on the scale at about 1.8 m, while an M1A1 Abrams tank is about four times bigger. Some of

K.T. Ramesh, Nanomaterials, DOI 10.1007/978-0-387-09783-1 1, c Springer Science+Business Media, LLC 2009 

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Nanomaterials

Fig. 1.1 Length scales in mechanics and materials and in nature. The topics of interest to this book cover a large part of this scale domain, but are controlled by features and phenomena at the nm scale. Note the sophistication of natural materials and systems at very small length scales.

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the largest animals are blue whales, coming in at about 100 m, comparable in size to some aircraft. Phenomena and features at the nanoscale and microscale can dominate the behavior and performance of devices and structures in the centimeter to 100 m range. Most of the behaviors of interest to this book arise from length scales of 0.1–100 nm, and this will be our definition of “the nanoscale.” One way to distinguish between nanotechnology and nanoscience is by distinguishing between what we can control and what we can understand. Most of what we interact with as humans has structure at the nanoscale, that is, there is a nanoscale substructure to most materials. Understanding what the nanoscale structure does (in terms of behavior or phenomena) is the core of nanoscience. Controlling nanoscale structure so as to achieve a desired end is the essence of nanotechnology. Nanotechnology cannot succeed without nanoscience, and the most efficient growth of nanotechnology (and growth with the smallest risk) occurs when the necessary nanoscience is already available. The enabling science in much of nanotechnology today is the science of nanomaterials (indeed in the broadest sense, nanotechnology would not be possible without nanomaterials). From a disciplinary viewpoint, most nanoscale phenomena are either controlled or modulated by the mechanics at the nanoscale, and so mechanics plays a critical role in nanoscience. Nanomechanics controls phenomena as immediately obvious as the interaction of interfaces between nanosize crystals and as subtle as the folding of proteins (controlling and organizing the living cell). The mechanics of nanomaterials is therefore the focus of this book.

1.2 What are Nanomaterials? A nanomaterial is a material where some controllable relevant dimension is of the order of 100 nm or less. The simple presence of nanoscale structure alone is not sufficient to define a nanomaterial, since most if not all materials have structure in this range. The ability to control the structure at this scale is essential. One could argue, in this sense, that many of the classical alloys and structural materials that contained nanoscale components by design (e.g., Oxide-Dispersion-Strengthened or ODS alloys) could be called nanomaterials. Conventionally, however, the modern usage of the term does not include the classical structural materials. In modern usage, nanomaterials are newly developed materials where the nanoscale structure that is being controlled has a dominant effect on the desired behavior of the material or device. There are three different classes of nanomaterials: discrete nanomaterials, nanoscale device materials, and bulk nanomaterials. Discrete nanomaterials or dn materials are material elements that are freestanding and 1–10 nm in scale in at least one dimension (examples include nanoparticles and nanofibers such as carbon nanotubes). Nanoscale device materials or nd materials are nanoscale material elements that are contained within devices, usually as thin films (an example of an nd material would be the thin film of metal oxide used within some semiconductor fabrication).

4

Table 1.1 A broad classification of nanomaterials on the basis of dimensionality and morphology. Type of nanomaterial

Dimensionality

Morphology

Characteristics

Remarks

Discrete nano (dn) materials

0D or 1D

Particles, fibers

Large surface functionalization

Potential health hazard

Nanoscale device (nd) materials

Usually 2D, occasionally 1D

Thin films, sometimes wires

Functionalization, electrical/thermal characteristics

Semiconductor fabrication

Bulk (nc or ns) nanomaterials

3D

Minimum mm3

Mechanical and structural applications

May be built up from dn and nd materials

Nanomaterials

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5

Fig. 1.2 Transmission electron micrographs (see Chapter 3) of nanocrystalline nickel material produced by electrodeposition (Integran). (a) Conventional TEM micrograph showing the grain structure. The average grain size is about 20 nm. (b) High resolution electron microscopy image of the same material. Note that there is no additional or amorphous phase at the grain boundaries, although many models in the literature postulate the existence of such a phase. Micrographs by Qiuming Wei. Reprinted from Applied Physics Letters, 81(7): 1240–1242, 2002. Q. Wei, D. Jia, K.T. Ramesh and E. Ma, Evolution and microstructure of shear bands in nanostructured fe. With permission from American Institute of Physics.

Bulk nanomaterials are materials that are available in bulk quantities (defined here as at least mm3 volumes) and yet have structure controlled at the nanoscale. Bulk nanomaterials may be built up of discrete nanomaterials or nanoscale device materials (for example one may construct a bulk nanomaterial that contains a large number of nanofibers). It is possible to view this classification approach (Table 1.1) in a mathematical light: discrete nanomaterials are either zero-dimensional (particles) or one-dimensional (fibers), nanoscale device materials are typically two-dimensional (thin films), and bulk nanomaterials are three-dimensional. The vast majority of materials are polycrystalline, that is, they are made up of many crystals (which are also called grains within the materials community). Most conventional engineering materials have grain sizes of 10–100 µm. In this regard, we distinguish between two subclasses of bulk nanomaterials: nanocrystalline materials (nc materials) with crystal or grain sizes that are <100 nm, and nanostructured materials (ns materials) with mixtures of nanoscale and conventional crystal sizes. An example of a nanocrystalline material is shown in Figure 1.2. Polycrystalline materials with grain sizes between 100 nm and 1 µm are conventionally called ultrafine-grained or ufg materials. Note that some of the literature includes the range between 100 nm and 500 nm in the “nanocrystalline” domain.

1.3 Classes of Materials There are three basic classes of materials categorized on the basis of atomic bond type and molecular structure: metals, ceramics and polymers. The character of polymers is typically determined by the interactions of large numbers of long-chain

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Nanomaterials

molecules (typically more than 100 nm long). There are several discrete nanomaterials and nanostructured materials based on polymers, including microspheres and microballoons (hollow spheres), and some two-dimensional structures (such as lamellae) that can be used to build three-dimensional bulk nanostructured materials. Some composites consisting of a polymer matrix with clay nanoparticles or carbon nanotubes as reinforcements are also considered to be bulk nanomaterials. This book does not examine soft nanomaterials in any detail, simply because the author is not knowledgeable about the field. The book focuses on nanomaterials that are either metals or ceramics or composites of these two.

1.4 Making Nanomaterials The ways in which nano materials are made vary widely, and there is not enough room in this book to discuss all of them. On the other hand, it is important to understand some features of the processes used in the synthesis of nanomaterials, because the processing route often dominates the behavior of any given material. The focus here is on processing issues in nanomaterials, rather than on the broader issues associated with processing nanotechnology – in particular, issues related to nanobiotechnology are ignored in this discussion. The fundamental issue associated with the making of nanomaterials is that one is trying to control the structure of a material at a very fine scale. Stated differently, one is trying to introduce order into the material at a very fine scale (or equivalently, trying to reduce the entropy density all over this material). This attempt to reduce the local entropy density has two direct consequences. First, such processes will require a significant amount of energy and therefore may incur significant expense. Second, the material that is generated is often thermodynamically unstable, and may attempt to revert to a higher entropy state (for example through growth of features that were initially nanoscale). Some of the more creative processing techniques use this fact to their advantage: they begin by generating extremely disordered states, which on equilibration generate the desired nanostructured state. In a broad sense, the approaches used to make materials can be put into two categories: top-down approaches, in which one begins with a bulk material that is then processed to make a nanomaterial, and bottom-up approaches, in which the nanomaterial is built up from finer scales (in the limit, building the nanomaterial up one atom at a time). It is evident that bottom-up approaches require control of processes at very fine scales, but this is not as difficult as it sounds, since chemical reactions essentially occur molecule by molecule (indeed, the nanomaterials made by nature are grown through bottom-up approaches).

1.4.1 Making dn Materials The processes required to generate discrete nanomaterials are usually quite distinct from the processes required to generate bulk nanomaterials. Many bulk nanoma-

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terials are created from dn materials using a second processing step (such bulk nanomaterials are said to have been created using a two-step process). Discrete nanomaterials themselves are often made using bottom-up approaches, sometimes relying on self-assembly to generate the desired nanoparticles or nanofibers. Under some conditions, a properly tuned bottom-up process (such as condensation from a vapor) can be used to generate significant quantities of discrete nanomaterials such as nanopowders. The majority of processing routes for dn materials rely on controlling the nucleation and growth process, since nucleation dominated processes tend to generate small sizes. Thus, one might nucleate nanoparticles by condensation from a vapor or precipitation from solution, and then intervene to control the growth of the particles, resulting in nanoparticles. Some of the typical approaches to creating discrete nanomaterials include: • • • • • • •

Condensation from a vapor phase (Birringer et al., 1984) Precipitation from solution (Meulenkamp, 1998) Chemical vapor deposition (Ren et al., 1998) Chemical reactions, particularly reduction or oxidation (Brust et al., 1994) Processes used to generate colloidal phases (Ahmadi et al., 1996) Self-assembly using surfaces (Li et al., 1999) Mechanical attrition (Nicoara et al., 1997)

A very large number of processes are used to generate discrete nanomaterials, using technologies borrowed from a variety of fields. Sometimes nanoparticles and nanotubes are byproducts of reactions that are already industrially relevant, and the usefulness of these byproducts has only recently been recognized. The characterization and transport of these dn materials are major issues in the development of nanotechnology. The definitive characterization of nanoparticle size and size distribution may require expensive high-resolution electron microscopy, and optical and X-ray techniques that obtain a reasonable measurement of the average size of nanoparticles over reasonably large volumes are of industrial interest.

1.4.2 Health Risks Associated with Nanoparticles There is very little data on the interaction of nanoparticles with biological systems and the environment. Particles with nanoscale dimensions can easily pass through biological systems, and may accumulate in undesirable locations within cells, and so there are potential health risks associated with nanoparticles. Scientific study of these effects is only just beginning in the early twenty-first century, on two fronts: the potential health risks, and the potential use of nanoparticles for clinical treatments. Given the lack of data and the ease with which nanoparticles can be redistributed in the environment given their small sizes, it is wise to take precautions in handling nanoparticles. Industrial operations with nanoparticles should consider such particles to be a potential hazard, and develop handling protocols that are consistent with potential hazards. Bulk nanomaterials (fully dense nanomaterials) are

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Nanomaterials

not likely to be a danger in themselves, but note that some nanoparticles may be generated during the fracture or failure of such materials.

1.4.3 Making Bulk Nanomaterials There are two basic approaches to the production of bulk nanomaterials, commonly referred to as the bottom-up and top-down approaches. In bottom-up approaches, the material is built up one atom, molecule, particle or layer at a time, with the focus on precise control at this scale. While materials made in this way can have very clean microstructures, the approach is inherently expensive and is generally not suitable for making large quantities of material. In top-down approaches, on the other hand, the nanomaterial is made by the application of macroscale processes, with the nanoscale structure caused to develop afterwards. Such approaches generally do not generate a high-quality material, but are relatively inexpensive. There are also some approaches that defy such broad categorization.

1.4.3.1 Deposition Techniques Deposition techniques used to generate nanocrystalline materials include electrodeposition (Elsherik and Erb, 1995), chemical (Bhattacharyya et al., 2001) and physical (Musil and Vlcek, 1998) vapor deposition (CVD and PVD), and a variety of techniques associated with semiconductor fabrication. These are typically slow processes used for the development of relatively thin films of material, with the best control of microstructure usually obtained if the film thickness is less than 1 µm (although some commercial electrodeposited nanocrystalline materials are now available with thicknesses up to 1 mm). These techniques are well adapted to the development of coatings, e.g., with mechanical applications in wear resistance and friction control. The major advantage of these techniques is that very precise control of the environment allows one to develop very precisely controlled microstructures, and so bulk materials produced in this way can be made in relatively reproducible fashion. Further, it is possible to generate microstructures with specific contaminants and specific doping (approaches that are particularly useful in sensor applications). Figure 1.2 shows the microstructure of a commercially developed electrodeposited nanocrystalline nickel film from Integran. The substantial process control that is available does not, however, imply that the materials that are generated will be free of defects. The development of controlled fully dense nanocrystalline structures is not an easy task. Process optimization is expensive, and the classic business problem remains of defining a sufficiently large market to make process optimization economically viable. For most nanocrystalline materials, therefore, it remains difficult to obtain reproducible films through deposition techniques. Indeed, much of the early confusion about the properties of

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nanocrystalline materials arose from the variability in material microstructures that are generated through these various processes (Agnew et al., 2000). Specific issues of concern include porosity, inhomogeneous microstructures and particularly columnar grains, and variable grain size distributions. This makes it difficult to compare the results obtained by one laboratory on a material that they have made with the results obtained by another laboratory on another nominally identical material that was made differently. As a consequence, both intensive and extensive characterization of a nanomaterial is essential if one is to understand the behavior. Intensive characterization procedures attempt to determine structure at a very fine scale, e.g., using TEM to determine local grain size. Such intensive procedures are incapable of providing information over large volumes. Extensive characterization procedures seek to determine the degree of variation of a structural feature (such as the grain size) with location over a larger length scale. Such procedures are usually incapable of resolving detail at the same fine scales as intensive procedures. We discuss some possible characterization procedures in chapter 3. Deposition techniques such as those discussed in this section are typically only capable of making films of material. Thin films are the most common products, although the use of multiple layers may allow one to build up thicker twodimensional structures, and occasionally up to the “bulk” mm thickness. These techniques, particularly those that have been optimized for semiconductor fabrication, have tremendous potential for making nanomaterials for applications within devices, i.e., nd materials. Since such devices represent a large fraction of the current commercial value of nanotechnology, these techniques have much to offer the nanotechnologist. The optimization of semiconductor fabrication processes for volume and cost allows one to develop relatively large films in terms of surface area, and thus the potential for volume production should not be minimized.

1.4.3.2 Consolidation of dn Materials The second major approach to making bulk nanomaterials relies on the already well developed capacity for generating large volumes of nanoparticles or nanopowders. The idea is to take a large volume of nanoparticles and then consolidate this to generate a bulk nanomaterial. Many of the techniques used in the powder metallurgy and ceramics industries can be applied to these nanopowders – these industries have generated bulk materials from powders for decades. The primary difficulty faced in the direct application of these classical powder techniques results from the difference in powder size. The typical steps in the consolidation of bulk materials from powders are: 1. Mixing of the powders, if necessary (it is sometimes advantageous to start with a variety of powder sizes, and different material powders might be used if a composite material is desired). 2. An initial consolidation step, typically cold isostatic pressing (CIPing) to generate a solid that can be handled, called a green body.

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Nanomaterials

3. Further consolidation of the green body, either through sintering or hot isostatic pressing (HIPing) to obtain the desired density. This step usually involves a high temperature because it is usually controlled by diffusion mechanisms. 4. A finish machining process to remove regions of the material that have not been completely consolidated. A variety of other steps are possible, including plastic deformation through rolling and forging, and the precise recipes used are often jealously guarded as proprietary information by laboratories and companies. The specific temperature histories and pressure histories used in a process have a major influence on the final microstructure. The relative instability of nanomaterials implies that grain growth is a major issue at high temperatures, and a compromise that must often be considered is between the desired final density and the desired final grain size. Every group making materials by powder consolidation also appears to have a different definition of “full density,” with some industries defining full density as 98% of the theoretical density (which is calculated using the known densities of the elements, compounds or phases involved and assuming no porosity). This is a major problem because the mechanical properties of the nanomaterial can be very strongly dependent on the presence of even small amounts of porosity. The properties of a material with 98% density can be very different from that of the same material with 99.5% density (particularly with regard to fracture and failure). Part of the difficulty with density arises because the more traditional powder metals produced by the powder metallurgy industry are designed for applications where small amounts of porosity would be removed by subsequent processes such as forging, or the small porosity might even be beneficial (as in lubrication systems). Such finishing processes are often not used in the development of the nanomaterial, resulting in residual porosity. A related difficulty is that it can be very difficult to see the porosity within a nanomaterial, because unlike traditional powder processed materials the porosity might be very finely distributed as very small voids. The consequence is that lower final densities may be considered acceptable by a materials processing vendor in nanomaterials because the quality control step is not able to find significant pores. An example of a bulk nanomaterial (nanocrystalline iron) made by the consolidation of powders is presented in Figure 1.3. First, nanostructured iron particles were generated by a mechanical attrition process: commercial iron powder with a purity of 99.9% and a range of powder sizes averaging around 5 µm was used as a starting material for ball milling. Ball milling (Malow and Koch, 1998) was conducted using a SPEX 8000 mill with a vial and balls made of stainless steel (SS440). The weight ratio of balls/powder was 4:1 and the milling time was 18 h. The vial was sealed in an argon atmosphere and cooled using a fan during milling. The powder particles after milling had internal grain sizes on the order of 10 nm (Malow and Koch, 1998) as revealed by X-ray diffraction line broadening. A two-step consolidation procedure was used to form bulk nanocrystalline iron with the desired grain size. In the first step, the powder was compacted at a pressure of 1.4 GPa at room temperature for about 10 h using a tungsten carbide (WC) die set. The resulting compact had a

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Fig. 1.3 Transmission electron micrograph of nanocrystalline iron produced by consolidation of a nanocrystalline precursor made by ball milling. There is a range of grain sizes, and many of the grains show evidence of the prior plastic work produced by the ball-milling process. The width of the photograph represents 850 nm.

green density of 72–75% of the theoretical density. In the second step, the compact was transferred into a larger WC die set for hot consolidation in a hot press. A temperature of 753 K and a consolidation time of 30 min at a pressure of 850 MPa was used to obtain an average grain size of just less than 100 nm and a final density of 99.2% of the theoretical density. One interesting feature of the nanocrystalline iron shown in this figure is that there are no visible voids in the material observable within the transmission electron microscope (TEM observations represent very small volumes), even though there must be 0.8% of porosity somewhere within the material. It is possible that the voids are very small and so not easily seen, or perhaps the porosity is eliminated by the TEM specimen preparation procedure. Another interesting feature of the nanocrystalline material shown in this figure is one that is to be expected from a consolidated material: many of the grains show evidence of the substantial prior plastic deformation involved in the consolidation process (there is extensive dislocation activity within the grains). This can make it difficult to decouple the effects of plastic deformation and the effects of the small grain size on the mechanical properties of the nanomaterial, and from a science viewpoint this can be viewed as one of the disadvantages of the consolidation process. From an engineering viewpoint, however, the consolidation process allows one to generate substantial quantities of bulk nanomaterials, and to take advantage of existing technologies in the powder metallurgy and ceramics industry. At the date of publication of this book, reproducibility of the quality of bulk nanomaterials made by the consolidation process remains a contentious issue.

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1.4.3.3 Severe Plastic Deformation Processes A variety of processing techniques for bulk nanomaterials fall under the rubric of severe plastic deformation (SPD) processes (Valiev et al., 2000; Valiev and Langdon, 2006). The primary idea in all of these is to develop very large plastic deformations within metals, with associated dislocation substructures, and then to use the processes of recovery, recrystallization and rearrangement of microstructure to develop the crystalline microstructures of interest. Recovery involves the thermallyenhanced motion of dislocations to reduce internal stresses through rearrangement and annihilation, while recrystallization involves the growth of new nearly strainfree crystals out of a collection of heavily-worked prior crystals. Large plastic deformations of metals have been known to develop patterned dislocation substructures for many years (Mughrabi, 1983), and the metalworking industry has used this extensively to their advantage. Large plastic deformations can result in dislocation structures that reorganize themselves to form cells, subgrains (regions with small angular misalignment, usually separated by dislocation walls), and eventually large angle grain boundaries (Hughes and Hansen, 1997), resulting in progressive grain refinement. The final grain size that is obtained in these processes depends strongly on the details of the thermomechanical processes involved – in general, larger strains develop finer grains, but the minimum grain size that can be obtained varies from material to material. Truly nanocrystalline grains (≤ 100 nm) have been observed within the centers of adiabatic shear bands developed within some materials, and have also been developed during high-pressure torsion processes. However, most SPD processes result in ultra-fine-grain sizes in most materials. 1.4.3.4 Equal Channel Angular Processing Equal channel angular processing or Equal Channel Angular Pressing, commonly known as ECAP, is a technique in which very large plastic deformations are developed by forcing a metal rod (the sample) through a die, with a die axis that changes direction suddenly (resulting in substantial shear deformation). Some variants of this technique are known as equal channel angular extrusion (ECAE) processes. A schematic diagram representing such a process is shown in Figure 1.4, with a hard plunger forcing a metal sample between two channels of equal size connected by an angular section (thus the name Equal Channel Angular). The magnitude of the shear strain that is developed in a single pass of the specimen through the die is determined by the die geometry, and particularly the included angle φ shown in the figure. The angle Φ subtended by the external radius is also an important parameter. Various obtuse die angles and dimensions may be used, and die design is a critical part of an effective ECAP process. Angles closer to 90◦ give larger shear strains, but require greater workability of the material. The metal specimen that is forced through the die is typically about 1 inch in diameter and several inches long and is made of conventional grain size materials. Substantial forces are required to develop these deformations, and so large presses must be used and particular care must be taken with die design for high-strength metals.

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φ Φ

Fig. 1.4 Schematic of the die used in Equal Channel Angular Pressing (ECAP). The included angle φ in this case is about 120◦ , but can be as small as 90◦ . The angle Φ subtended by the external radius is also an important parameter. Various obtuse die angles and dimensions may be used. Very large forces are required to move the workpiece through such dies.

Since the angle change renders the process inherently anisotropic, it is common practice to run the specimen through the die multiple times (these are called multiple passes), rotating the specimen between consecutive passes. Multiple passes may also be used to increase the fraction of the sample that has been subjected to the large deformations (portions of the specimen ends are typically much less deformed). The various protocols used for these multiple passes and rotations are called processing routes (commonly referred to as A, B, C . . . routes). Each route has a different yield in terms of grain size and the likely development of texture (relative orientation of grains), and also in the fraction of the specimen that achieves significant grain refinement. The use of multiple passes typically yields ultra-fine-grain (ufg) size materials relatively easily, but nanocrystalline grain sizes are very difficult to obtain. These techniques have been used on a variety of metals successfully. In particular, metals with body-centered cubic (bcc), face-centered cubic (fcc) and hexagonalclose packed (hcp) crystal structures have been processed in this way. An example of the microstructure of a material made by equal channel angular pressing is shown in Figure 1.5. The average grain size of this material is 200 nm. A selected area diffraction (SAD) pattern is also shown, and indicates the presence

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Nanomaterials

a

b

Fig. 1.5 (a) Microstructure of tantalum produced by Equal Channel Angular Extrusion (ECAE), four passes at room temperature through a 90◦ die. (b) Selected Area Diffraction (SAD) pattern for this sample, showing the presence of many low angle grain boundaries. The fraction of grain boundaries that are low-angle rather than high-angle can be an important feature of the microstructure.

of low-angle grain boundaries (typical of this SPD approach). The high dislocation density in many of the grains is a result of the large plastic work that must be introduced in order to generate this microstructural grain refinement. As in the consolidated case, materials produced by this SPD process contain large internal dislocation densities because the processes of grain refinement result in significant numbers of retained dislocations. There is often a significant texture produced by this process, although the degree of texture does depend on the specific route used and varies strongly from one material to another. Elongated grains are another characteristic feature of such processes, and as a result these materials may have anisotropy in the yield strength (that is, the yield strength – defined in Chapter 2 – may be different in different directions). The nature of the grain boundaries produced by equal channel angular pressing is a matter of some controversy. It appears that the structure of the grain boundary may be strongly dependent on the temperatures at which the processing is conducted for a given material. In a related issue, the degree to which rotational recrystallization plays a part in the development of the fine grains also remains uncertain.

1.4.3.5 High-Pressure Torsion Another kind of SPD process is shown in Figure 1.6, and is known as the highpressure torsion or HPT process (Liao et al., 2006). Relatively small samples are subjected to combined high-pressure and severe torsional deformations using large

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15 Applied Pressure

Press

Sample

Die

Applied Torque

Fig. 1.6 Schematic of High Pressure Torsion (HPT) process. A thin specimen is compressed and then subjected to large twist within a constraining die. The typical sample size is about 1 cm in diameter and about 1 mm thick.

multi-axial presses. Pressures of several GPa are used, and the torsional strains can be very large (depending on the total angle of twist). Since thin specimens are being considered, the strains generated by the compression are relatively small. For any given increment in the angle of twist θ , the local increment in shear strain is given by

δγ =

rδ θ , h

(1.1)

where r is the radial distance from the center of the disc and h is the disc thickness. Very large strains can thus be generated by using small thickness samples. For example, a full rotation of 360◦ will lead to a shear strain of approximately 30 in a sample that is 10 mm in diameter and 1 mm thick. These very large strains can lead to much more refined microstructures than can be obtained through ECAP processes. The superimposed pressure is necessary to keep the sample from breaking apart, and issues of workability typically still require performing this HPT operation at elevated temperature. The major difficulty with this technique, however, is the small size of the specimen that is subjected to the very large strain. Further, the strain is quite inhomogeneous, with the center of the specimen seeing no strain at all. At this point, therefore, the technique is a good way to understand what the effects of severe plastic deformation will do to materials, but this is not a good approach for generating significant quantities of nanomaterials.

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(a)

(b)

(c)

Fig. 1.7 Microstructure of tungsten produced by high-pressure torsion (HPT). (a) A bright field TEM micrograph. (b) A dark field TEM micrograph. (c) A selected area electron diffraction pattern from this field. The grains contain a high density of defects (because of the plastic work associated with the HPT process). The grains are also elongated, with widths on the order of 80 nm and lengths of about 400 nm. Grain orientation appears to be along the shearing direction. The selected area electron diffraction shows nearly continuous rings, with no obvious intensity concentration along the rings, indicating large angle type grain boundaries in the sample. Reprinted from Acta Materialia, Vol. 52, Issue 7, P. 11, Q. Wei, L. Kecskes, T. Jiao, K.T. Hartwig, K.T. Ramesh and E. Ma, Adiabatic shear banding in ultrafine-grained Fe processed by severe plastic deformation. April 2004, with permission from Elsevier.

An example of a microstructure generated by HPT is shown in Figure 1.7. The material shown is pure tungsten, which is a notoriously difficult material to deform plastically. The final grain size has a minimum dimension of about 80 nm, and some elongation of the grains is observed. An SAD pattern is also shown, demonstrating the nanocrystalline nature of the structure. Note again the large number of dislocations within the grains, a characteristic of severe plastic deformation processes. Elongation along the shearing direction is evident, and a range of grain sizes is present. The combination of finer grains but greater heterogeneity of the microstructure is characteristic of HPT processes as compared to ECAP processes. An excellent discussion of the controllable process parameters is provided by Zhilyaev (Zhilyaev et al., 2003). There appears to be a limit to the smallest achievable grain size via HPT, with some indications that the limitation is due to dynamic recrystallization.

1.4.3.6 Surface Mechanical Attrition Treatment Large plastic deformations can also be obtained on the surface by impacting the surface repeatedly with hardened materials (as in shot peening). The typical approach involves seating the sample within a chamber that is loaded with a large number of hardened steel balls, and then shaking the entire chamber using a vibration generator (Tao et al., 2002). By proper tuning of the frequencies, the steel balls can be caused to impact the specimen at relatively high velocities and generate fairly high strains

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17

in a surface layer on the sample. Continued vibration can lead to microstructural refinement of the surface layer. This process is very similar to the mechanical attrition process used in ball milling to generate nanoparticles; since it is constrained to a surface, the process is called surface mechanical attrition treatment or SMAT. The distinction is the presence of a surface layer and related strain gradients and microstructural gradients. Control of the atmosphere in the chamber, the frequency of vibration, and temperature allows one to control the microstructures that are generated.

1.4.3.7 Rolling and Accumulative Roll Bonding Very large plastic deformations can also be generated by repeated rolling. Indeed, the processes of grain refinement during severe plastic deformation were first examined in repeated rolling processes, because of the applications in the metalworking industry. While repeated rolling does requires substantial workability, the development of protocols involving rolling followed by annealing and then subsequent secondary rolling allows for the development of very fine microstructures. Rolling can also be performed at cryogenic temperatures, which has been shown to allow the development of finer grains. A version of rolling called Accumulative Roll Bonding (ARB) involves rolling a material to a thin-sheet, then cutting the sheet in two, stacking the sheets and rolling again to further reduce the thickness, and then repeating this process (Saito et al., 1999). Accumulative roll bonding has been used to develop bulk ultra-fine-grained and nanostructured metals, both from alloys and pure metals, and such an ARB process can be adapted for continuous production of substantial quantities of material (this is a key difficulty with a number of the other SPD techniques).

1.4.3.8 Crystallization from the Amorphous State Another approach to making bulk nanomaterials involves the crystallization of the material from an amorphous state or from the molten state. Control of the nucleation and growth process to emphasize nucleation can lead to very fine crystal sizes (Hono et al., 1995). Such control is, however, quite difficult to maintain, and so very few bulk nanomaterials are made in this way at this time.

1.5 Closing Now that we have some sense of how to make nanomaterials, we develop the mechanics of materials in the next chapter. We can then use that understanding to focus on the mechanics of nanomaterials in subsequent chapters.

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1.6 Suggestions for Further Reading 1. L. Foster, Nanotechnology: Science, Innovation, and Opportunity. Prentice Hall, Upper Saddle River, NY, 2005. 2. G. Cao, Nanostructures & Nanomaterials: Synthesis, Properties & Applications. Imperial College Press, London, 2004. 3. Y. Gogotsi, Nanomaterials Handbook. New York, CRC Press, 2006. 4. B. Bhushan, Springer Handbook of Nanotechnology. Springer, New York, 2007. 5. P.C. Hiemenz and R. Rajagopalan, Principles of Colloidal and Surface Chemistry. CRC Press, New York 3rd ed., 1997. 6. D.D. Brandon and W.D. Kaplan, Microstructural Characterization of Materials. Wiley, New York, 2nd ed. 2008.

1.7 Problems and Directions for Research 1. Identify a source of silver nanoparticles, and find out what particle size distribution is available. How does the company determine this distribution? What is the cost of the nanoparticles, and what are the sample masses (or volumes) that are available? 2. Identify the risks associated with handling silver nanoparticles. What literature is available on their toxicity, if any? How should they be stored? Is there a danger of fire if the particles are stored in air? 3. Repeat the problem above, but this time consider aluminum nanoparticles. As always with respect to safety and risk management, the key is to (a) know and document the risks, (b) know and document safe handling procedures and (c) know and document emergency response procedures in case of an accident or mishandling. Write a three page report that includes (a), (b) and (c) above (one page each) for aluminum nanoparticles. 4. Consider the ECAP process shown in Figure 1.4. The included angle is φ ≥ 90◦ , and often the outside corner is not sharp but rounded, subtending an angle Φ with the sharp inner corner. Then the total shear strain developed by the ECAP process is given by γ = √N3 [2 cot ( φ2 + Φ2 ) + Φ csc ( φ2 + Φ2 )], where N is the number of passes (the number of times the workpiece is reinserted). Using the literature, identify an ECAP route and the number of passes needed to achieve a shear strain of 4. 5. Consider the problem of the industrial production of bulk nanomaterials. In your opinion, which of the processes described in this chapter is best suited to scaleup? Why? What are the potential limitations of this process? What do you think would be the most expensive part of the facility required for scale-up of this process? You will need to define an expected output capacity.

1 Nanomaterials

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References Agnew, S. R., B. R. Elliott, C. J. Youngdahl, K. J. Hemker, and J. R. Weertman (2000). Microstructure and mechanical behavior of nanocrystalline metals. Materials Science and Engineering a-Structural Materials Properties Microstructure and Processing 285(1–2), 391–396. Ahmadi, T. S., Z. L. Wang, T. C. Green, A. Henglein, and M. A. ElSayed (1996). Shape-controlled synthesis of colloidal platinum nanoparticles. Science 272(5270), 1924–1926. Bhattacharyya, S., O. Auciello, J. Birrell, J. A. Carlisle, L. A. Curtiss, A. N. Goyette, D. M. Gruen, A. R. Krauss, J. Schlueter, A. Sumant, and P. Zapol (2001). Synthesis and characterization of highly-conducting nitrogen-doped ultrananocrystalline diamond films. Applied Physics Letters 79(10), 1441–1443. Birringer, R., H. Gleiter, H. P. Klein, and P. Marquardt (1984). Nanocrystalline materials an approach to a novel solid structure with gas-like disorder. Physics Letters A 102(8), 365–369. Brust, M., M. Walker, D. Bethell, D. J. Schiffrin, and R. Whyman (1994). Synthesis of thiolderivatized gold nanoparticles in a 2-phase liquid-liquid system. Journal of the Chemical Society-Chemical Communications 7, 801–802. Elsherik, A. M. and U. Erb (1995). Synthesis of bulk nanocrystalline nickel by pulsed electrodeposition. Journal of Materials Science 30(22), 5743–5749. Hono, K., Y. Zhang, A. Inoue, and T. Sakurai (1995). Atom-probe studies of nanocrystalline microstructural evolution in some amorphous alloys. Materials Transactions of JIM 36(7), 909–917. Hughes, D. and N. Hansen (1997). High angle boundaries formed by grain subdivision mechanisms. Acta Mater 45(9), 3871–3886. Li, M., H. Schnablegger, and S. Mann (1999). Coupled synthesis and self-assembly of nanoparticles to give structures with controlled organization. Nature 402(6760), 393–395. Liao, X. Z., A. R. Kil’mametov, R. Z. Valiev, H. S. Gao, X. D. Li, A. Mukherjee, J. F. Bingert, and Y. T. Zhu (2006). High-pressure torsion-induced grain growth in electrodeposited nanocrystalline ni. Applied Physics Letters 88, 021909. Malow, T. R. and C. C. Koch (1998). Mechanical properties, ductility, and grain size of nanocrystalline iron produced by mechanical attrition. Metallurgical and Materials Transactions aPhysical Metallurgy and Materials Science 29(9), 2285–2295. Meulenkamp, E. A. (1998). Synthesis and growth of zno nanoparticles. Journal of Physical Chemistry B 102(29), 5566–5572. Mughrabi, H. (1983). Dislocation wall and cell structures and long-range internal stresses in deformed metal crystals. Acta Metallurgica 31(9), 1367–1379. Musil, J. and J. Vlcek (1998). Magnetron sputtering of films with controlled texture and grain size. Materials Chemistry and Physics 54(1–3), 116–122. Nicoara, G., D. Fratiloiu, M. Nogues, J. L. Dormann, and F. Vasiliu (1997). Ni-zn ferrite nanoparticles prepared by ball milling. In Synthesis and Properties of Mechanically Alloyed and Nanocrystalline Materials, Pts 1 and 2 - Ismanam-96, Volume 235- of Materials Science Forum, pp. 145–150. Transtec. Ren, Z. F., Z. P. Huang, J. W. Xu, J. H. Wang, P. Bush, M. P. Siegal, and P. N. Provencio (1998). Synthesis of large arrays of well-aligned carbon nanotubes on glass. Science 282(5391), 1105– 1107. Saito, Y., H. Utsunomiya, N. Tsuji, and T. Sakai (1999). Novel ultra-high straining process for bulk materials - development of the accumulative roll-bonding (arb) process. Acta Materialia 47(2), 579–583. Tao, N. R., Z. B. Wang, W. P. Tong, M. L. Sui, J. Lu, and K. Lu (2002). An investigation of surface nanocrystallization mechanism in fe induced by surface mechanical attrition treatment. Acta Materialia 50, 4603–4616. Valiev, R. and T. Langdon (2006). Principles of equal-channel angular pressing as a processing tool for grain refinement. Progress in Materials Science 51, 881–981.

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Valiev, R. Z., R. K. Islamgaliev, and I. V. Alexandrov (2000). Bulk nanostructured materials from severe plastic deformation. Progress in Materials Science 45, 103–189. Zhilyaev, A. P., G. V. Nurislamova, B.-K. Kim, M. D. Baro, J. A. Szpunar, and T. G. Langdon (2003). Experimental parameters influencing grain refinement and microstructural evolution during high-pressure torsion. Acta Materialia 51, 753–765.

...there is no philosophy which is not founded upon a knowledge of the phenomena, but to get any profit from this knowledge it is absolutely necessary to be a mathematician. Daniel Bernoulli

2

Fundamentals of Mechanics of Materials We begin with a brief introduction to the mechanics of materials for the nonspecialist, and then develop a description of elastic and inelastic constitutive behavior. The objective is to develop a common mathematical language that can be used to describe the mechanics of nanomaterials. A basic familiarity with the concepts of undergraduate strength of materials is assumed, particularly the concepts of stress and strain.

2.1 Review of Continuum Mechanics 2.1.1 Vector and Tensor Algebra The equations of interest are written in terms of the independent variables (x,t) representing spatial position and time, respectively. In our notation, boldface letters will represent vectorial or tensorial quantities. A quick description of basic vector and tensor analysis is presented below. An excellent exposition of vector and tensor algebra as used here is presented by Asaro and Lubarda (2006). In three dimensions, any vector x can be represented by a linear combination of three unit vectors {e1 , e2 , e3 } (this collection of “base vectors” is sometimes called a basis). The most commonly used basis is the Cartesian basis, in which the unit base vectors are perpendicular to each other and are typically written in the x, y and z directions in order, so that {e1 , e2 , e3 } = {ex , ey , ez }. In a Cartesian basis the base vectors do not change from point to point, whereas in a basis such as that used with a cylindrical coordinate system the base vectors actually change direction from point to point (consider for example the unit vectors at two points with different polar angles θ in an r, θ , z coordinate system). Thus, x is represented by x = x1 e1 + x2 e2 + x3 e3 = xi ei

K.T. Ramesh, Nanomaterials, DOI 10.1007/978-0-387-09783-1 2, c Springer Science+Business Media, LLC 2009 

(2.1)

21

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Nanomaterials

where the scalars xi are called the components of the vector x in the basis {ei }. The choice of a different basis (e.g., a basis representing a rotated set of base vectors {˜ex , e˜ y , e˜ z }) will result in a different set of components representing the same vector: x = x˜1 e˜ 1 + x˜2 e˜ 2 + x˜3 e˜ 3 = x˜i e˜ i .

(2.2)

Therefore, the components of a vector alone do not represent the vector unless the base vectors are defined. The second half of either Equation (2.1) or (2.2) demonstrates a condensed way of writing this linear combination through the use of the summation convention: the use of a repeated index (in this case i) implies summation of similar terms over all values of that index, i.e., xi ei = Σ xi ei = x1 e1 + x2 e2 + x3 e3 .

(2.3)

If one chooses to retain the same base vectors throughout the discussion, as we do in this book, then each vector can be represented simply as an ordered set of the components, for example, as a row or column matrix. Thus a vector may be represented by v = v1 e1 + v2 e2 + v3 e3

(2.4)

  v1 v2 v3 .

(2.5)

    u(x,t) = u1 (x1 , x2 , x3 ,t) u2 (x1 , x2 , x3 ,t) u3 (x1 , x2 , x3 ,t)

(2.6)

or as a row matrix by

Examples of vectorial dependent variables that will be used in this book include the displacement vector u(x,t), the velocity vector v(x,t), and the force vector f(x,t), all of which are “field” variables in that they are generally functions of position x. Thus the components of the displacement vector u(x,t) can be represented by a column matrix:

where the square brackets around the vector in Equation (2.6) indicate that the matrix of components of the vector is specifically being considered. Note that the matrix represents the vector only when the choice of base vectors is explicitly agreed upon and does not change throughout the course of the discussion. Unless otherwise stated, throughout this book a Cartesian basis is used with unit vectors defined along the x, y and z directions in order, so that {e1 , e2 , e3 } = {ex , ey , ez }. The scalar product of two vectors a and b is defined by a.b = (ai ei ).(b j e j ) = ai b j (ei .e j ) = ai b j δi j = ai bi = a1 b1 + a2 b2 + a3 b3 ,

(2.7)

2 Fundamentals of Mechanics of Materials

23

where the fact that the base vectors are orthonormal (i.e., unit base vectors that are perpendicular to each other) has been used, and the Kronecker delta δi j is defined by

δi j = 1 if i = j = 0 if i = j.

(2.8)

The definition of the scalar product can be used to define the magnitude (or length) of a vector through √ u = |u| = u.u  (2.9) √ = ui ui = u1 2 + u2 2 + u3 2 .

It is assumed that the reader is familiar with the “vector product” commonly used to define areas. For our purposes, a tensor is defined as a linear operator that maps one vector into another vector. The classical example of a tensor in continuum mechanics is the stress tensor σ , which maps the normal vector n normal to a surface at any point to the corresponding force density (or traction) vector t at that point (Figure 2.1). This operation is written as (2.10) t = σ n which is typically read as t equals σ operating on n, where the dot between the σ and the n represents the operation. The stress tensor can be written as a 3×3 matrix, and has the property of being symmetric (because of the balance of angular momentum), so that the matrix corresponding to the tensor σ can be written as ⎡ ⎤ σ11 σ12 σ13   σ = ⎣σ12 σ22 σ23 ⎦ . (2.11) σ13 σ23 σ33

As with vectors, the matrix can be used to represent the tensor only when the choice of base vectors is explicitly agreed upon and does not change throughout the course of the discussion (Equation 2.2). Similarly, the components of a tensor will change when the base vectors used to describe the vector are changed; that is, the matrix of components of a tensor will change when the basis changes, even though the tensor itself (defined as always by its operation) does not change. Thus, in terms

n

t = σ.n

Fig. 2.1 The stress tensor σ maps the normal vector n at any point on a surface to the traction vector t at that point.

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Nanomaterials

of a new set of base vectors {˜ex , e˜ y , e˜ z }), the identical tensor σ results in a different matrix of components: ⎤ ⎡ σ˜ 11 σ˜ 12 σ˜ 13   σ |e˜ i = ⎣σ˜ 12 σ˜ 22 σ˜ 23 ⎦ . (2.12) σ˜ 13 σ˜ 23 σ˜ 33

The matrix notation is particularly useful to show the effect of operating with the tensor: e.g., Equation (2.10) can be written (in matrix form) as ⎡ ⎤ ⎡ ⎤⎡ ⎤ t1 σ11 σ12 σ13 n1 ⎣t2 ⎦ = ⎣σ12 σ22 σ23 ⎦ ⎣n2 ⎦ t3 σ13 σ23 σ33 n3

(2.13)

where the ti and the ni are the components of the vectors t and n respectively and the operation of matrix multiplication is used. This equation provides us with the components of the resulting vector t (after the mapping) in terms of the components of the original vector n, since the first component of t in this basis is now t1 = σ11 n1 + σ12 n2 + σ13 n3

(2.14)

ti = σi j n j .

(2.15)

and more generally Here the summation convention has been used. Note that the repeated index is j, so that we are summing over the possible values of j, while the index i is the free index and takes on each of its three values separately, i.e., Equation (2.15) represents three scalar equations of the type of Equation (2.14). Equation (2.15) is essentially the component form of Equation (2.10). Several simple tensors are now defined in terms of their operation. The identity tensor is that tensor that operates on every vector a to return the same vector: a = I  a for all possible arguments a,

(2.16)

and the Zero tensor operates on all vectors to give the zero vector: 0 = 0  a for all possible arguments a.

(2.17)

Tensors are always defined in terms of what they do, i.e., in terms of their operation. Two tensors are defined to be equal to each other if they always operate in the same way. i.e., L ≡ G if and only if L  v = G  v for all possible arguments v.

(2.18)

For reasons of brevity, from this point on the operation of a tensor on the vector immediately following it is no longer explicitly written in terms of a centerdot (), but is understood from the relative placement of the vector following the tensor.

2 Fundamentals of Mechanics of Materials

25

Thus Equation (2.18) will be written as Lv = Gv, and so forth. It is understood that the result of a tensor operating on a vector is always a vector, so that the quantity Lv is known to be a vector, generally different from the operand v. The transpose LT of a tensor L is defined as a specific operator that provides the following identity: a.(Lv) = v.(LT a) for all possible arguments a and v.

(2.19)

The matrix of the transpose LT of a tensor can be shown to be the transpose of the matrix of the original tensor L: [LT ] = [L]T , or LiTj = L ji .

(2.20)

The product of two tensors amounts to a sequence of operations, so that we can write ABv and understand that this means that the tensor A operates on the vector that results from the operation of the tensor B on the operand v: ABv = A(Bv) = Cv where by definition C = AB.

(2.21)

One product that is of particular interest is that of the tensor with itself, e.g., S2 = SS. This quantity and the transpose of a tensor from Equation (2.19) will be used extensively in this book. From a physical point of view, note that all a tensor is doing is mapping one vector into another vector. Since each vector is defined by a magnitude and a direction, all that a tensor does is change the magnitude and the direction, i.e., the tensor stretches and rotates the vector on which it operates. The action of the tensor can thus be decomposed into a sequence of two operations: the first a pure stretch, where magnitude changes but not direction, and the second a pure rotation, where direction changes but not magnitude.

2.1.2 Kinematics of Deformations This section develops an approach to describe the deformation of a body, i.e., the kinematics of deformation. Consider a body B0 consisting of a collection of material particles (the subscript 0 indicates that this is the initial configuration or reference configuration of the body). It is conceptually useful to imagine that each material particle can be identified by a distinct color (so that their motions through space can be tracked by watching the motion of the red particle, the green particle, etc.), but for mathematical purposes it is assumed that each particle can be identified individually through its initial location X. As a result of deformation, the various material particles that comprise the body will occupy new locations x at any given time t, resulting in a new or current configuration of the body that is called Bt (see Figure 2.2). Since each material particle X will occupy different spatial positions x as a function of time, we have

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Nanomaterials χ

Fig. 2.2 Kinematics of deformation: a body initially in the reference configuration B0 , with material particles occupying locations X (used as particle identifiers) is deformed into the current configuration Bt with each particle occupying spatial positions x.

x = χ (X,t),

(2.22)

where χ (X,t) is called the motion of the material particle X. The function χ (X,t) is assumed to be invertible, i.e., two material particles cannot occupy the same spatial position at any given time. Thus it is possible in principle, to find the inverse map (the inverse of the motion) X = χ −1 (x,t). (2.23) Equation (2.23) can be interpreted as allowing us to compute which material particle will be at a given spatial position at any given time, whereas Equation (2.22) tells us which spatial positions a given material particle will occupy at any time. The latter two equations allow us to take any quantity that is described in terms of material particles (this is called a material description) and convert it to a quantity described in terms of spatial positions (this is called a spatial description), and vice versa. Such conversions are very useful in mechanics problems, particularly those that involve large deformations (such as fluid flow). The displacement u(X, t) of the material particle is defined as the difference between the current spatial position and the initial position of the particle: u(X, t) = x − X = χ (X, t) − X.

(2.24)

Note that possible displacements include rigid translations (e.g., where u is independent of X) and rigid rotations, but neither of these is of interest to us in this book. Much of this book considers only situations involving small displacements. Given that the displacement field is a vector field in three dimensions, there are three independent components of the displacement vector, written in matrix form as     u(X,t) = u1 (X1 , X2 , X3 ,t) u2 (X1 , X2 , X3 ,t) u3 (X1 , X2 , X3 ,t) . (2.25)

Note that displacements in general can include rigid translations, rigid rotations, and deformations. This book considers primarily problems involving small displacements, and rigid motions (rigid translations and rigid rotations) are not of interest to

2 Fundamentals of Mechanics of Materials

27

us. The only displacement fields that are of interest in this book are those representing deformations, with nonzero gradients of the displacement field. The displacement gradient Grad u(X,t) is a tensor, with the matrix of components given by ⎡ ∂u ∂u ∂u ⎤ 1

∂ X1   ⎢ ⎢ ∂ u2 Grad u(X,t) = ⎢ ∂ X ⎣ 1

1

1

∂ X2 ∂ X3

⎥

∂ u2 ∂ u2 ⎥ . ∂ X2 ∂ X3 ⎥ ⎦

∂ u3 ∂ u3 ∂ u3 ∂ X1 ∂ X2 ∂ X3

(2.26)

This displacement gradient tensor is denoted by ∇u (read as Grad u) for short. The Grad notation, with a capital G, emphasizes that this is a derivative with respect to capital X (the material particle descriptor). This tensor is the same as the zero tensor for rigid-body motions, with all the components of the corresponding matrix being identically zero. Now, every tensor can be decomposed into symmetric and antisymmetric parts (like matrices). For example, the displacement gradient tensor ∇u can be decomposed as follows: ∇u = ε + ω ,

(2.27)

where the symmetric part of the displacement gradient tensor is called the infinitesimal strain tensor ε , and is given by

ε=

∇u + (∇u)T 2

(2.28)

while the anti-symmetric part of the displacement gradient tensor is called the infinitesimal rotation tensor ω and is given by

ω=

∇u − (∇u)T . 2

(2.29)

In component form, we can write Equation (2.27) as

∂ ui = ui, j = εi j + ωi j , ∂xj

(2.30)

where the notation ui, j has been used to show the differentiation of ui with respect to x j . The infinitesimal rotation tensor ω will not be used in this book. The infinitesimal strain tensor ε in Equation (2.28) has the components  1 ∂ ui ∂ u j εi j = + (2.31) 2 ∂ X j ∂ Xi where we have used the fact that [(∇u)T ]i j = u j,i . Thus, for example, we have   1 ∂ u1 ∂ u1 ∂ u1 ε11 = + = (2.32) 2 ∂ X1 ∂ X1 ∂ X1

28

while

Nanomaterials

1 ε12 = 2



∂ u1 ∂ u2 + ∂ X2 ∂ X1



.

(2.33)

The first of these quantities (Equation 2.32) represents what is called a “normal strain,” while the second of these (Equation 2.33) represents a “shear strain.” Note that ε is the infinitesimal strain tensor, rather than a measure of finite strain (i.e., this ε is a reasonable measure only for small strains). Indeed, for most of this book, the displacement gradient itself (Equation 2.25) is assumed to be small. Finite deformations are not considered here, but excellent discussions of such deformations can be found in books on continuum mechanics and mechanics of solids, such as the books by Lai, Rubin and Krempl (Lai et al., 1999) or Asaro and Lubarda (2006). In Equation (2.26), the displacement vector was differentiated with respect to one of its independent variables, X, to obtain the displacement gradient tensor. Differentiation of the displacement vector with respect to the other independent variable (time) gives us the velocity of each material particle: v(X,t) =

∂ u(X,t) ∂t

(2.34)

where it is understood that the differentiation occurs with the value of X (identifying the material particle) held fixed. The resulting velocity vector (like the displacement vector) is a vector in the material description (i.e., it is a function of X). However, it can be easily converted into a function of spatial position instead, i.e., into the spatial description, using Equation (2.23), resulting in a vector field that we call v˜ (x,t): v˜ (x,t) = v(χ −1 (x, t),t)

(2.35)

The “tilde” symbol above the v is used to remind us that this function has a different form than v(X,t). The “spatial velocity field” obtained in Equation (2.35) is the “flow field” that one typically sees in fluid flow applications, where arrows at each spatial position represent the magnitude and direction of the velocity vector at each position (with different material particles occupying a given spatial position at different times). The gradient of the spatial velocity vector field is a tensor called the velocity gradient or grad v˜ (x,t) and is typically denoted by the symbol L. That is, the velocity gradient tensor L is given by L(x,t) = grad v˜ (x,t) =

∂ v˜ . ∂t

(2.36)

The symmetric part of the velocity gradient tensor is called the rate of deformation tensor D: L + LT D= . (2.37) 2 The rate of deformation tensor D corresponds to what is traditionally called the strain rate, and has the following components:

2 Fundamentals of Mechanics of Materials

Di j =

1 2

29



∂ v˜i ∂ v˜ j + ∂ x j ∂ xi



.

(2.38)

For the particular case of small deformations and small velocity gradients of interest to this book, this corresponds identically to the time derivative of the infinitesimal strain tensor, i.e., Di j =

∂ εi j = ε˙i j . ∂t

(2.39)

These latter strain rate components provided in Equation (2.39) will be used in the rest of this text. It is useful to remember that the strain rate arises from the symmetric part of the velocity gradient; the antisymmetric part of the velocity gradient leads to the concept of spin, and is related to the rate of rotation in the kinematics. The key quantities that we will take forward to our discussion of the mechanics of nanomaterials are the infinitesimal strain ε and the strain rate ε˙ .

2.1.3 Forces, Tractions and Stresses The reader is assumed to be familiar with the concepts of statics and strength of materials, including the concept of force equilibrium and the calculation of resultant forces. The physical quantity that we know as a force is mathematically described as a vector F, and the statement of force equilibrium (which is for the static problem, with no accelerations) is Σ F = 0 (which can of course be written as three scalar force equations along the three axes of a Cartesian coordinate system, Σ F1 = Σ Fx = 0, Σ F2 = Σ Fy = 0, Σ F3 = Σ Fz = 0). The two major types of forces of interest are surface forces (such as contact forces) and body forces (such as the force due to gravity). Our interest here is in relating the relatively intuitive concept of a force applied on a body to the somewhat more complex concept of internal forces exerted by one part of a body upon another, and this will lead us to the related concept of the stress at a point in the body. The key physical concept is presented in Figure 2.3. Consider the current configuration of a body deforming under the action of external forces. The body shown is subjected to a number of external forces acting on its boundary. Consider an imaginary surface S that separates the body into two parts, denoted in the figure by B1 and B2 . Clearly part B2 of the body must exert a force on part B1 of the body (and vice versa), across surface S, in order to keep the entire body together. Now imagine that one makes a cut along the surface S, everywhere except over a small element of area around the point A on the surface. Since parts B1 and B2 remain connected through the small area around the point A along the surface S, there must be forces acting on this small area. Since both subparts of the body remain in equilibrium, part B2 must exert a resultant force on part B1 at the point A. The point A on the surface S was chosen arbitrarily, and the choice of a different element of area on the surface would generally produce a different resultant force

30

Nanomaterials

t n

A

B2

B1 S

(a)

t

B1

(b)

Fig. 2.3 Internal forces on surfaces S and S’ within a body, and the traction t at a point on a surface with normal n. The entire body must be in equilibrium, as must be all subparts of the body. (a) The entire body, in equilibrium with the external forces. (b) Free body diagram corresponding to part B1 of the whole body, showing the traction developed at a point on the surface S because of the internal forces generated on B1 by part B2 .

on the part B1 acting on the new elemental area. It is clear, therefore, that there is a distribution of internal forces on the surface S. Consider again the point A shown in Figure 2.3. The normal to the surface S at the point A is the vector n. The resultant force acting on the point A will in general have a direction and magnitude that is different from that of the normal vector (the direction of this normal vector has been chosen so that it is the outward normal from the part B1 ). Remembering that there is a distribution of internal forces along the surface S in the uncut body, it is useful to consider the force density on the surface S at the point A, given by the resultant force acting on part B1 at point A divided by the area of the element of the surface that has remained uncut. Since the force is a vector, and the area is a scalar, the force density remains a vector; this vector is denoted by the symbol t, and is called the traction vector. This is the traction vector first discussed in Equation (2.10). The direction of the traction vector is identical to the direction of the resultant force acting at point A, and the magnitude of the traction vector is equal to the magnitude of the force divided by the elemental area at A. Note that the resultant force on the point A in the body is not by itself a welldefined quantity, although the resultant force on the point A on the surface S is a

2 Fundamentals of Mechanics of Materials

31

well-defined quantity. To see this, consider a different surface S’ (shown with the long-dashed line in Figure 2.3) which also passes through the point A. Clearly the element of area at the point A defined on the surface S’ is different from the original element of area on the surface S that was considered in this discussion (although the elements of area may have the same size, they have different normal vectors). The body is divided into two slightly different parts by this new imaginary surface, and the resultant force acting at the point A on the surface S’ will in general be a different resultant force than that acting at the point A on the surface S. Therefore, the traction vector acting at the point A on the new element of area (belonging to the surface S’) will be a different vector than that acting on the point A on the old element of area (belonging to S). Since the point A is defined by the spatial position x and the element of area is defined through the normal vector n, the traction vector depends not only on x but also on the normal n to the element of area passing through x: t = t(x, n). (2.40) If it is assumed that the dependence of the traction vector on the normal vector is linear, then the traction t is a linear (vector-valued) function of the vector variable n, and we can define a linear operator (called a tensor) that maps n into t, as in Equation (2.10): (2.41) t = σ (x,t)  n. The tensor σ (x,t) defined by Equation (2.41) is called the Cauchy stress tensor, and is the most commonly used stress tensor in the mechanics of solids. There are other stress tensors, defined for example with reference to the initial or reference configuration rather than the current configuration (This is like the engineering stress versus the true stress in strength of materials). In this book we will confine ourselves to the Cauchy stress, and refer to this simply as the stress tensor. The components of the stress tensor (which is defined at a point, note) were first presented in Equation (2.11) (as always, assuming a Cartesian basis): ⎤ ⎡ σ11 (x,t) σ12 (x,t) σ13 (x,t)   σ (x,t) = ⎣σ12 (x,t) σ22 (x,t) σ23 (x,t)⎦ . (2.42) σ13 (x,t) σ23 (x,t) σ33 (x,t) Thus, for example, the matrix of components of the stress tensor corresponding to the problem of the biaxial extension of a thin sheet in the x and y directions in the plane of the sheet, with no applied stresses in the z direction normal to the sheet, would be ⎤ ⎡ σxx 0 0   σ = ⎣ 0 σyy 0⎦ . (2.43) 0 0 0

In the discussion following Equation (2.2) it was pointed out that the components of a vector change when the base vectors used to describe the vector are changed. Similarly, the components of a tensor will change when the base vectors used to describe the vector are changed; that is, the matrix of components of a tensor will

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change when the basis changes, even though the tensor itself (defined as always by its operation) does not change. This process of finding a different set of components of the matrix representing the stress tensor at a point by changing the basis or coordinate system is called the transformation of the stress components. One particular set of transformations is especially useful. Suppose that we can find a direction e (that is, a normal vector e to an internal surface) such that when the stress tensor acts on e the resulting traction only has components along e, that is, the traction vector only has a normal component:

σ  e = σev e,

(2.44)

where σev is the only component of the traction. Such a statement (where the tensor operates on a vector to give a resultant vector that is in the same direction as the original vector but scaled by some magnitude) is called an eigenvalue statement. The corresponding vector (here e) is called an eigenvector, and the scaling factor (here σev ) is called the eigenvalue corresponding to that eigenvector. Second-order tensors such as the stress tensor have three eigenvalues and at least three eigenvectors (it is possible for all vectors to be eigenvectors, as when the tensor in question is the identity tensor). With this in mind, consider again the general matrix of components of the stress presented in Equation (2.42). The eigenvalues σ1 , σ2 , σ3 of this matrix are called the principal stresses, and the corresponding eigenvectors are called the principal directions. It can be shown that for a symmetric tensor like σ , the eigenvectors can be chosen to be orthonormal, and can be used to provide another set of base vectors. The stress tensor σ can now be written as a matrix of components in the basis of the eigenvectors, and in this basis the matrix of the (components of the same stress tensor) will now be diagonal: ⎤ ⎡ σ1 (x,t) 0 0   σ (x,t) |eigenvector basis = ⎣ 0 σ2 (x,t) 0 ⎦. (2.45) σ3 (x,t) 0 0

Transformation of the stress components so that the matrix has only diagonal components (Equation 2.45) is the result of solving the eigenvalue problem for the matrix and then representing the tensor in terms of a basis made up of orthonormal eigenvectors. The principal stresses σ1 , σ2 , σ3 can be very useful in descriptions of the response of materials, but it is important to remember that they are associated with the directions of the eigenvectors, which can also vary from point to point. Since one tensor can be represented by many different matrices, these matrices must be connected in some way (i.e., there must be some way to identify the different matrices as representing the same tensor). For example, it can be shown that all of the matrices representing a single tensor must have the same determinant, and all of the matrices representing a tensor must have the sum of the diagonal components be the same. Such properties are said to be invariant, in that they do not change even when the base vectors used to describe the tensor change (and therefore when the matrix representing the tensor changes). The invariants are in fact properties of the

2 Fundamentals of Mechanics of Materials

33

underlying tensor, independent of the basis in which that tensor is described. The two specific (scalar) invariants that were just discussed are known as the determinant (det T) of the tensor T and the trace (trace T) of the tensor: det T = det Ti j = det T˜i j ,

(2.46)

trace T = Tii = T˜ j j = T11 + T22 + T33 = T˜11 + T˜22 + T˜33 .

(2.47)

and Here Ti j and T˜i j are the components of the matrices representing the tensor T using two different sets of base vectors ei and e˜ i respectively (consider, for example, a standard Cartesian coordinate system and a rotated coordinate system). Note that we have made use of the summation convention in Equation (2.47), so that the repeated index i in the equation implies summation: Tii = T11 +T22 +T33 , and similarly with T˜ j j . The two scalar invariants of the tensor T defined in Equations (2.46) and (2.47) are often referred to as TIII = det T and TI = trace T respectively. There is a third scalar invariant that one can also define for every tensor T, given by TII =

 1 (trace T)2 − trace T2 . 2

(2.48)

Together, the set of scalars TI , TII , TIII are called the set of scalar invariants of the tensor T, and these identical invariants must be obtained from every matrix that represents T. It turns out that if the set (TI , TII , TIII ) is known, then the set (T1 , T2 , T3 ) of eigenvalues of the tensor T can be completely determined, because (considering the corresponding diagonal matrix obtained after solving the eigenvalue problem and using the eigenvectors as base vectors) TI = T1 + T2 + T3

(2.49)

TII = T1 T2 + T2 T3 + T3 T1 and TIII = T1 T2 T3 .

(2.50) (2.51)

It is possible to decompose any tensor T into a hydrostatic part, − 13 (trace T)I, ˆ with the latter defined by and a deviatoric part T, ˆ = T − 1 (trace T)I. T 3

(2.52)

The deviatoric part of a tensor is particularly useful when we are considering shearing deformations and shearing stresses. The trace of a tensor has a special meaning for the Cauchy stress tensor σ . The mean stress at a point, σm , is defined as the scalar measure: 1 1 σm = trace σ = (σ11 + σ22 + σ33 ), 3 3

(2.53)

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and physically represents the average hydrostatic tensile stress at the point. The negative of this quantity is called the hydrostatic pressure p: 1 1 p = −σm = − (σ11 + σ22 + σ33 ) = − (σ˜ 11 + σ˜ 22 + σ˜ 33 ) 3 3

(2.54)

which has the conventional physical meaning of the pressure. The deviatoric part of the Cauchy stress tensor is a particularly useful quantity, since it represents the part of the Cauchy stress tensor that is left when the hydrostatic (or volume changing) part has been removed, and thus is a measure of the degree of shear. It is normally given a special name, S, and is known as the deviatoric stress: 1 (2.55) S = σˆ = σ − (trace σ )I = σ + pI, 3 and is particularly useful in discussions of plasticity. Now that both the kinematics of deformations and the relationships between stresses and forces can be described mathematically, it is possible to construct rigorous ways of approaching problems in the mechanics of materials by using the conservation laws of physics. The next section presents the governing equations of mechanics of materials using the terminology developed above.

2.2 Work and Energy From a thermodynamics perspective, the deformation of a body caused by external forces acting on the body must involve some work (that is, the external forces do work on the body in causing it to deform). In basic physics terms, a work or energy can be described through terms of the type Fdx, where F is the force and dx is the displacement that occurs through the applied force. A similar concept can be developed for deformable solids, considering elements of the solid, by defining a strain energy density W defined by W = σi j d εi j ,

(2.56)

and the total energy within the body due to the deformation is then the integration of  the strain energy density over the volume of the body, W dV . This total energy must of course be related to the work done on the body by the external forces. Similarly, one can define a kinetic energy density in the body ( 12 ρ v  v). Other thermodynamic quantities can also be defined for application in continuum mechanics through such “density” approaches. Although we do not pursue the thermodynamics of continua further in this book, there is a great deal that can be learned through thermodynamic approaches, and an excellent theoretical description of the thermodynamics of solids is provided by Ericksen (1998).

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2.3 Field Equations of Mechanics of Materials There is a core set of field equations, derived from continuum mechanics, which represent most problems of interest in the mechanics of materials. These equations are: (i) The conservation of linear momentum

∂ σi j ∂ vi =ρ ∂xj ∂t

(2.57)

(ii) The conservation of angular momentum

σi j = σ ji (iii) The conservation of energy, which in one form is  2 ∂T ∂ T ∂ 2T ∂ 2T ∂W p =κ ρ cp + + +β 2 2 2 ∂t ∂t ∂ x1 ∂ x2 ∂ x3

(2.58)

(2.59)

Here ρ is the density of the material, T is the temperature field, κ is the thermal conductivity, β is the ratio of plastic work converted to heat, and W p is the plastic work (this book deals principally with elastic-plastic solids). Note that quantities such as the stress σ and the strain ε are defined at every point, and so the list of conservation laws of physics presented above are in a form that must be satisfied at every point in the continuum. The second law of thermodynamics is not explicitly considered here, although it does place some restrictions on the kinds of material behaviors that are possible. When a mathematical description of the behavior of the material (known as a constitutive relation or constitutive law, and described in the next section) is added to this list of the conservation laws of physics, a complete set of governing equations is obtained for describing the mechanics of materials. Since the governing equations presented above are partial differential equations in space and time, the solution of a problem in the mechanics of materials also involves the definition of boundary conditions and initial conditions. The solution procedures relevant to many such problems can be found in standard textbooks in continuum mechanics or the mechanics of solids. Examples of useful references of this type include the books by Lai, Rubin and Krempl (Lai et al., 1999) or Asaro and Lubarda (2006).

2.4 Constitutive Relations, or Mathematical Descriptions of Material Behavior An equation or equations representing the behavior of the materials of interest (e.g., stress-strain relationships) must be added to this list (Equations 2.57–2.59) of the conservation laws of physics. Such equations are commonly referred to as constitu-

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tive laws or constitutive relations. Note that constitutive laws are generally expected to be satisfied pointwise, that is, they are valid at every point in the solid (in a composite material, for example, different constitutive laws may apply at different material points). Several examples of constitutive laws that may be of interest to the mechanics of nanomaterials are discussed in subsequent sections.

2.4.1 Elasticity A material is defined as being elastic if the stress in the material depends only on the current deformation (rather than the history of the deformation) and if all of the work done by the stresses on the deformations is fully recoverable upon unloading. During the deformation, this work is stored in the material in a form that is called the strain energy. Since all of the information of interest about the deformation is contained in the displacement gradient tensor ∇u, an elastic material is one in which the stress tensor is σ = σ (∇u). (2.60) i.e. the stress is a function of ∇u The focus here is on linear elastic materials, in which the stress depends linearly on the displacement gradient. Now the simplest way to evaluate a function is in terms of a Taylor-series expansion: consider a function y = f (x), and let us suppose that you know the value of the function (and all of its derivatives) when x = x0 , but you do not know how the function behaves anywhere else. You want to evaluate the value of this function at a different value of the argument, i.e., you want to evaluate the function at x = x0 , but where x is close to x0 (so that the distance x − x0 is small). The easiest way to get an estimate of the value of the function at x is to use the information that you have at x = x0 , through a series expansion called a Taylor series:    2  ∂f ∂ f f (x) = f (x0 )+ (x−x0 )+ (x − x0 )2 +. . . higher order terms.   ∂ x x=x0 ∂ x2 x=x0 (2.61)

If you keep only the second term in the expansion (the term that is linear in x − x0 ), you have a linear approximation of the function (if all of the higher order terms are actually zero, then the function is in fact a linear function, so that a curve y = f (x) will be a straight line). Similarly, considering a Taylor-series expansion of the function σ (∇u) about the initial value ∇u = 0 corresponding to zero strain, and retaining only the first order terms, we have  ∂ σ  σ = σ |∇u=0 + : ∇u + . . . higher order terms. (2.62)  ∂ ∇u ∇u=0 Here the first term on the right hand side corresponds to the stress in the material under zero displacement gradient (i.e., no deformation, so that σ |∇u=0 is like a residual stress). The second term represents the linear term in the Taylor-series σ expansion. The second term on the right hand side involves the derivative ∂∂∇u ,

2 Fundamentals of Mechanics of Materials

37

which is the derivative of the second-order tensor σ with respect to another secondorder tensor u, and is thus a fourth-order tensor (i.e., it will map a second-order tensor into another second-order tensor, just like a second-order tensor like σ maps a σ when ∇u = 0 vector into another vector). The result of evaluating the derivative ∂∂∇u is usually denoted C and referred to as the elastic modulus tensor. The colon in the second term on the right hand side of equation (2.62) is used to show that the fourthorder tensor is operating on the second-order tensor ∇u, the result of the operation being another second-order tensor which is the stress tensor (just as a second-order tensor maps a vector into another vector, a fourth-order tensor maps a second-order tensor into another second-order tensor). If we assume the material has zero stress at zero displacement gradient (so that the initial state is stress free), we can write the elastic constitutive relationship in component form as

σi j = Cijkl

∂ uk ∂ xl

(2.63)

where Cijkl are the components of the elastic modulus tensor and ∂∂ uxk are the compol nents of the displacement gradient tensor (note the use of the summation convention on the dummy indices k and l in the equation (2.63)). The components of the elastic modulus tensor are constants that represent material properties (known as the moduli), and must be measured independently. Using Equation (2.30), we can rewrite the linear elastic constitutive relationship in terms of the infinitesimal strain and infinitesimal rotation tensors:

σi j = Cijkl {εkl + ωkl }.

(2.64)

Assuming that pure rotations give zero stress, the linear elastic constitutive relationship reduces to σi j = Cijkl εkl . (2.65) Since the stress and strain tensors each have nine components (remember the 3x3 matrices), the elastic modulus tensor will in general have 81 components. The constitutive law of Equation (2.65) is appropriate to anisotropic linear elastic solids (i.e., linear elastic solids in which the properties vary with direction in the material, such as a fiber-reinforced material or a crystal of quartz). However, the symmetry of the stress and strain tensors results in some simplifications, since it then requires that (2.66) Cijkl = Cjikl and Cijkl = Cijlk

(2.67)

respectively. This reduces the number of independent components of the elastic modulus tensor to 36. There is a further expectation that every linear elastic material is associated with the existence of a strain energy function W (εi j ) from which the stress can be derived: ∂W σi j = . (2.68) ∂ εi j

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It follows that the linear elastic moduli can be derived as (using Equation 2.65) Cijkl =

∂ σi j ∂ 2W ∂ 2W ∂ σkl = = = = Cklij . ∂ εkl ∂ εkl ∂ εi j ∂ εi j ∂ εkl ∂ εi j

(2.69)

Thus the existence of the strain energy function results in the requirement that Cijkl = Cklij

(2.70)

and this reduces the number of independent components of the elastic modulus tensor to 21. The restriction represented by Equation (2.70) is observed to be a reasonable restriction for all metals and ceramics deformed to small strains. Thus, the most general anisotropic linear elastic material would have 21 independent constants that need to be experimentally determined for complete specification of the material behavior in an elastic domain. Note that so far no material symmetry has been invoked in our description of the elastic constants. Invoking such material symmetries (for example, crystal symmetries) further reduces the number of elastic constants that are needed. Thus, while a general anisotropic linear elastic material requires 21 constants, a monoclinic linear elastic material requires only 13 independent elastic constants, an orthotropic material requires 9, and a transversely isotropic material requires 5. The limiting condition of a completely isotropic linear elastic material requires only two independent elastic constants (e.g., the Young’s modulus and the Poisson’s ratio). Excellent derivations of the symmetries are presented from the mechanics perspective by Lai, Rubin and Krempl (Lai et al., 1999) and from a materials perspective by Nye (1950). The fact that the stress and infinitesimal strain tensors are symmetric implies that each of them has six independent components, and this leads to another way of describing the anisotropic linear elastic relationship of Equation (2.65), using Voigt notation, which is useful when the base vectors are Cartesian and unchanging. In this approach, we first construct column matrices representing the stress and strain. Now, we use the row index in this column matrix to define a new index called the Voigt index that takes values 1, 2, . . . 6. For example, in the Voigt notation the matrix of stress components would be ⎡ ⎤ ⎡ ⎤ σ11 σ1 ⎢σ22 ⎥ ⎢σ2 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢σ33 ⎥ ⎢σ3 ⎥   ⎥ ⎢ ⎥ σi j |Voigt = ⎢ (2.71) ⎢σ23 ⎥ ≡ ⎢σ4 ⎥ . ⎢ ⎥ ⎢ ⎥ ⎣σ31 ⎦ ⎣σ5 ⎦ σ12 σ6 Similarly, a matrix of strain components can be constructed:

2 Fundamentals of Mechanics of Materials

⎡ ⎤ ⎡ ⎤ ε11 ε1 ⎢ε22 ⎥ ⎢ε2 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ε33 ⎥ ⎢ε3 ⎥   ⎥ ⎢ ⎥ εi j |Voigt = ⎢ ⎢ε23 ⎥ ≡ ⎢ε4 ⎥ . ⎢ ⎥ ⎢ ⎥ ⎣ε31 ⎦ ⎣ε5 ⎦ ε12 ε6

39

(2.72)

These two matrices for the stress and the strain are then related by (for general anisotropic linear elasticity, following Equation 2.65) a new matrix of Voigt constants CIK , each of which can be explicitly derived from Cijkl :

σI = CIK εK , with I, K taking values 1, 2, . . . , 6

(2.73)

where the components σI and εK are the components of the column matrices in Equations (2.71) and (2.72) respectively, and the Voigt constants CIK are obtained from the moduli Cijkl by replacing i j with a value for I following the pattern in Equation (2.71). Thus, if i j = 11, then I = 1; if i j = 22, then I = 2; if i j = 23, then I = 4, and so forth (similarly with the kl being matched to the K index). Thus the matrix of Voigt constants is a 6x6 matrix, and there are 36 constants. Note, however, that CIK are NOT the components of a second-order tensor, just as σI are not the components of a vector—these are simply convenient matrices to use to describe linear elasticity. The existence of the strain energy function makes (following Equation 2.70) the CIK = CKI , and this symmetry provides the 21 independent constants in the most general anisotropic linear elastic material. The corresponding matrix of Voigt constants is ⎤ ⎡ C11 C12 C13 C14 C15 C16 ⎢C12 C22 C23 C24 C25 C26 ⎥ ⎥ ⎢   ⎢C13 C23 C33 C34 C35 C36 ⎥ ⎥ CIK = ⎢ (2.74) ⎢C14 C24 C34 C44 C45 C46 ⎥ . ⎥ ⎢ ⎣C15 C25 C35 C45 C55 C56 ⎦ C16 C26 C36 C46 C56 C66

The various material symmetries can now be easily seen in the Voigt matrix (sometimes called the stiffness matrix in materials science). For example, consider a transversely isotropic material (such as might be produced by a severe extrusion process). In such a material, every plane perpendicular to a single specific plane is a plane of symmetry for the material. Consider, for example, a material that has been extruded along the e3 direction, so that every plane that contains the e3 axis (and therefore is perpendicular to the plane of isotropy that has normal e3 ) is a plane of symmetry. For such a transversely isotropic material the Voigt constants would be ⎤ ⎡ C11 C12 C13 0 0 0 ⎥ ⎢C12 C11 C13 0 0 0 ⎥ ⎢ ⎥ ⎢C13 C13 C33 0 0   0 ⎥, ⎢ CIK |transversely isotropic = ⎢ (2.75) ⎥ 0 0 0 C 0 0 44 ⎥ ⎢ ⎦ ⎣ 0 0 0 0 C44 0 0 0 0 0 0 12 (C11 −C12 )

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and there are only five independent constants. Another case that is of common interest in materials science is that of cubic materials (as in the face-centered-cubic and body-centered-cubic systems), for which case we have ⎡ ⎤ C11 C12 C12 0 0 0 ⎢C12 C11 C12 0 0 0 ⎥ ⎥ ⎢ ⎢C12 C12 C11 0 0 0 ⎥   ⎥, ⎢ CIK |cubic = ⎢ (2.76) ⎥ ⎢ 0 0 0 C44 0 0 ⎥ ⎣ 0 0 0 0 C44 0 ⎦ 0 0 0 0 0 C44 so that there are three independent constants in this case. In the limiting case of the fully isotropic material, the Voigt matrix reduces to ⎡ ⎤ C11 C12 C12 0 0 0 ⎢C12 C11 C12 ⎥ 0 0 0 ⎥ ⎢ ⎥ ⎢C12 C12 C11   0 0 0 ⎥ , (2.77) CIK |isotropic = ⎢ ⎥ ⎢ 0 0 0 1 (C11 −C12 ) 0 0 2 ⎥ ⎢ 1 ⎣ 0 0 0 ⎦ 0 0 2 (C11 −C12 ) 1 0 0 0 0 0 2 (C11 −C12 ) and now there are only two independent constants. In the more conventional notation for the elastic moduli used in Equation (2.65), this fully isotropic material has the moduli Ci jkl = λ δi j δkl + µ {δik δ jl + δil δ jk }, (2.78)

and the two independent constants, λ and µ , are called the Lame´ moduli. These are related to the Voigt constants C11 and C12 in Equation (2.77) by C11 = λ + 2µ and 12 C12 = λ , so that C11 −C = µ. 2 The Lame´ moduli λ and µ are related to the conventional Young’s modulus and Poisson’s ratio by Eν λ= (2.79) (1 + ν )(1 − 2ν ) and

µ=

E . 2(1 + ν )

(2.80)

Since only two constants are required to define an isotropic linear elastic material, we can use either the pair λ , µ or the pair E, ν . Returning to Equation (2.65), we now see that for an isotropic linear elastic material the stress is related to the strain by σi j = λ εkk δi j + 2µεi j . (2.81) Here εkk uses the summation convention and therefore is ε11 + ε22 + ε33 ; this quantity is the total volumetric strain at a point in the material. Thus, for example, we have

σ11 = λ (ε11 + ε22 + ε33 ) + 2µε11

(2.82)

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41

whereas

σ12 = σ21 = 2µε12

(2.83)

and so forth. In terms of the Young’s modulus and Poisson’s ratio, Equation (2.81) can be rewritten as  ν E εkk δi j . σi j = εi j + (2.84) 1+ν 1 − 2ν Although many mechanical engineers and most materials scientists are most comfortable with linear elastic materials described through relations of the form of Equation (2.84), many problems in mechanics of materials are best thought of in terms of changes in volume (corresponding to the volumetric or hydrostatic response) and changes in shape (corresponding to the distortional or shearing response). Recalling Equation (2.55), the stress tensor can also be broken up into hydrostatic and deviatoric parts, so that σ = S − pI, and this decomposition allows one to break down the constitutive behavior into volumetric and shearing behaviors: 1 p = − σkk = −K εkk , 3

(2.85)

and Si j = 2µ εˆi j ,

(2.86)

where εˆi j are the components of the deviatoric strain tensor given, as in Equation (2.52), by 1 εˆi j = εi j − εkk δi j . (2.87) 3 The coefficient K in Equation (2.85) is called the bulk modulus, and µ is called the shear modulus. Thus three different pairs of material properties are commonly used to describe isotropic linear elasticity: (λ , µ ), (E, ν ), and (K, µ ). The first of these is often convenient while writing governing equations, the second of these is convenient when the primary data available is obtained in uniaxial tension experiments, and the third is useful when separating out volumetric and distortional effects. Any of these pairs can be used, or other combinations involving just two of these properties. Since there are only two independent properties for an isotropic linear elastic material, there are relationships between all of these quantities, as in Equations (2.79) and (2.80). Some additional relationships are particularly useful: 2µ 3 9K µ E= 3K + µ 1 2µ − 3K and ν = − . 2 3K + µ K=λ+

(2.88) (2.89) (2.90)

The linear elastic relationships given above are all presented in terms of calculating the stress components given the strain components. It is possible to invert

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all of these relationships to calculate the strain components when given the stress components, e.g., inverting Equation (2.84) to solve for the strains gives

εi j =

ν 1+ν σi j − σkk δi j . E E

(2.91)

Similar equations can be written in terms of the other moduli. It is possible to show, on physical grounds, that E > 0, µ > 0, K > 0 and −1 < ν ≤ 0.5 (although in most engineering materials, 0 ≤ ν ≤ 0.5). From Equation (2.90) above, we see that ν = 0 only when µ = 3k 2 , but the bulk modulus is usually greater than the shear modulus, i.e., usually K > µ , so that it is difficult to find materials with zero Poisson’s ratio (the closest approximation is cork, which is why this material is used to stop up wine bottles – it does not expand as you push it into the bottle). Materials with negative Poisson’s ratio are unknown in nature, but efforts are underway to construct metamaterials that have this property. For a single crystal, the linear elastic response represents the stiffness of the interatomic bonds within the material near the equilibrium positions of the atoms in the lattice, and the elastic moduli for a single crystal are not generally isotropic. Thus one must use the full set of linear elastic constants (making use of available symmetries) to describe the mechanical behavior of a single crystal in the linear elastic range, and one must find values for the coefficients Ci jkl corresponding to Equation (2.74) for each material. For example, for single crystal α -iron, the cubic symmetry allows us to use Equation (2.76) and one only needs to find values for C1111 , C1122 , and C2323 corresponding to the Voigt constants C11 , C12 , and C44 . Experimental determination of these constants provides C11 = 230.37 GPa, C12 = 134.07 GPa, and C44 = 115.87 GPa at a temperature of 300 K (the elastic moduli are functions of temperature). A scalar measure of the elastic anisotropy of cubic crystals that is often used is 2C44 A= . (2.92) (C11 −C12 ) This measure is usually much greater than 1 for most metals (e.g., A = 2.41 for α -iron at 300K), with the exception being tungsten, where A ≈ 1. However, most engineering materials consist of large polycrystalline assemblies, with the crystals oriented randomly, and the effective behavior of this polycrystalline mass in the linear elastic range might be nearly isotropic (assuming that there is no significant texture). For polycrystalline pure iron, therefore, an isotropic linear elastic behavior is often a reasonable approximation, and the experimental values are E ≈ 210 GPa and ν ≈ 0.29. Note that the averaging performed over a large number of randomly oriented crystals has effectively reduced the number of independent constants. A table of typical values of the moduli for a number of materials (for conventional grain sizes, in the randomly oriented polycrystalline case) is presented in Table 2.1. Note the high variability in the elastic data on some polycrystalline ceramics such as B4 C, which arises because of differences in processing approaches and the resulting flaw distributions, as well as variations with measurement processes. Up to this point, the discussion has assumed that the material behavior is linear elastic. However, most materials demonstrate linear elastic behavior only in a finite

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Table 2.1 Young’s modulus and Poisson’s ratio for some conventional grain sized materials. Material Young’s modulus E, GPa Poisson’s ratio ␯ Iron 210 0.29 Copper 130 0.34 Aluminum 70 0.33 Vanadium 128 0.37 Tantalum 186 0.34 Nickel 200 0.31 Titanium 116 0.32 Alumina, Al2 O3 396 0.25 Silicon carbide, SiC 410 0.14 Boron carbide, B4 C 362–460 0.18–0.21

range of stresses (or strains), beyond which the deformations are not reversible and some of the work done by the stresses is associated with permanent changes in the material. This non-elastic behavior is discussed in the next section.

2.4.2 Plastic Deformation of Materials Most materials undergo permanent, non-reversible deformations if they are stressed beyond some critical point, so that the basic assumptions made in the section on elasticity above are no longer valid. In particular, the behavior of the material may become such that the stress in the material depends not just on the current state of deformation, but on the history of the state of deformation. For example, the material behavior can be different during a second loading cycle than it was during the first loading cycle, a behavior sometimes called hardening. This history-dependent constitutive behavior is called plasticity – a material that obeys such a constitutive law is said to be plastic. Most metals in our everyday experience show plastic behavior at some point, and such a response is used heavily in industry in the forming of materials, components and structures. Understanding plasticity is important both from a design viewpoint (if permanent deformations are to be avoided) and from a manufacturing viewpoint (if the material is to be formed into a desired shape). The classical example of plastic behavior is observed when a rod or strip of a ductile metal is subjected to uniaxial tensile loading (Figure 2.4) . This is called the simple tension test, although—as an experiment—there is little that is simple about it. The shape of the sample is typically the dogbone shown in the figure, and it is loaded along the x1 direction with a force P. The central section of the sample has a gauge length L and a cross-section area A0 , and sees a nearly uniform stress state with only one nonzero stress component, σ11 . Note that σ11 is a component of the Cauchy stress, also sometimes called the true stress, which is defined in the deformed configuration as σ11 = σtrue = PA and so requires a knowledge of the current cross-section area A (usually very difficult to measure). The corresponding engineering stress σeng is easier to measure and is defined as the ratio of the applied

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P

x

P

Fig. 2.4 Schematic of a simple tension test. The loading direction is the x1 direction, and the gauge length is L. Note the specimen shape, designed to minimize the influence of the gripping conditions on the ends. Specimen shapes may be specified by testing standards.

force P to the initial cross-section area A0 :

σeng =

P . A0

(2.93)

As the sample is loaded in tension, it elongates, and the change in the length Δ L can be measured using a variety of devices. The ratio of this change in length to the original length is a strain measure called the engineering strain:

εeng =

ΔL . L

(2.94)

This is a strain measure that uses the initial configuration of the specimen as a reference. The strain measure that corresponds to the current configuration of the specimen is called the true strain, and considers the change δ l in the current length l of the specimen through ε = δll . The true strain should be used whenever the true stress is used, since they both represent variables in the current configuration of the specimen. The engineering stress is plotted against the engineering strain, providing a stress-strain curve that is characteristic of the material rather than of the specimen (at least until necking of the specimen begins, when the deformation localizes to a small region that thins rapidly and results in sample failure). A schematic of such a tensile stress-strain curve is presented in Figure 2.5a. The early part of the stressstrain curve is linear, and represents the linear elastic response of the material (the slope of this line corresponds to the Young’s modulus). In this linear domain, unloading the sample by decreasing the force P will result in retracing the stress-strain curve along the straight line, as elastic behavior demands. As the engineering stress continues to increase, the stress-strain curve deviates from linearity. The point at which the deviation from linearity begins is called the proportional limit and the corresponding stress is called the proportional limit stress σ pl . With a further increase in the stress, the point is reached at which permanent deformation of the sample begins, and the corresponding stress is called the yield strength (σy ). In the regime between the proportional limit σ pl and yield σy , unloading and reloading of the sample results in retracing the stress-strain curve along this nonlinear portion, which is a nonlinear elastic response. For most metallic materials this nonlinear elastic regime is negligible, and so one often identifies the proportional limit with yield. Note that by convention, a stress level that represents material behavior is referred to as a strength, as in the yield strength.

Engineering Stress

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σUT σY σ PL E

Engineering Strain True Strain 2000

0

0.01 0.02 0.03 0.04 0.05 0.06 0.07

Stress, MPa

Nanocrystalline Nickel

1500 1000 500

Engineering Stress True Stress

0

0

0.01 0.02 0.03 0.04 0.05 0.06 0.07

Engineering Strain Fig. 2.5 (a) Schematic of tensile stress-strain curve for ductile metal, showing yield, an ultimate tensile strength, and subsequent failure. (b) Experimentally obtained curve for nanocrystalline nickel, with an average grain size 20 nm. Note the unloading line after initial yield, with the unloading following an elastic slope. Data provided by Brian Schuster.

Once the applied stress has been taken beyond σy , permanent (plastic) deformation occurs, and unloading of the sample results in an unloading stress-strain response which does not follow the original stress-strain curve, but instead typically unloads along a straight line parallel to the linear elastic domain (this is called linear elastic unloading, as shown by the unloading line in Figure 2.5). The plastic deformation process after yield is called plastic flow. In practice, measuring the true onset of permanent deformation is not possible, and one instead identifies the yield strength by determining the point at which a permanent offset strain of 0.2% would be achieved. If loading is continued beyond the yield point, the stress-strain curve is nonlinear in two senses: first, the loading behavior does not follow a straight line response, and second, the unloading path is different from the loading path. If the stress in the sample increases after yield, the material is said to show hardening, while if the stress remains constant after yield the material is said to be perfectly plastic. Since the unloading is always elastic with a slope corresponding to the original Young’s modulus E, the permanent plastic strain left over after elastic unloading evolves

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with increasing load. Even in a hardening material, continued loading results in a maximum in the engineering stress called the ultimate strength σut , where the subscript ut signifies that the strength referred to is the ultimate strength in tension. While there is a maximum in the engineering stress, the true stress (if it could be calculated) generally shows no maximum – what is happening is that the sample begins to develop a neck, with a region of the specimen becoming rapidly thinner and consequently with a rapidly decreasing current cross-section area A. Even though the load carried by the specimen decreases once necking begins, the current crosssection area A carrying the load is typically decreasing faster, so that the true stress continues to increase. The tensile stress-strain curve for the material thus provides two new material properties that describe the plastic response: the yield strength σy and some measure of the hardening of the material with increasing plastic strain. For example, a very common description of the tensile stress-strain curve has the power law form  n ε σ = σy (2.95) εy σ

where εy = Ey is the yield strain and the parameter n is called the strain hardening index and is a new material property. For the purposes of constitutive modeling the hardening index of the material should be defined in terms of the true stress vs true strain response, rather than the engineering stress vs engineering strain response. An example of real experimental data is presented in Figure 2.5b for a nanocrystalline nickel sample with an average grain size of 20 nm. The sample was tested using a microtension apparatus at a low strain rate of 10−4 s−1 . Both the engineering stress vs engineering strain curve and the true stress vs true strain curve are shown in the figure. Several points are worthy of note. First, the onset of yield is not immediately obvious – what should be chosen as the yield strength? The departure from linear elastic response is so smooth that yield is not clearly defined (the 0.2% onset condition would suggest that σY ≈ 1000 MPa). Second, the elastic modulus can be obtained directly by allowing a small amount of elastic unloading, as shown in the figure, and measuring the slope of the unloading line. In this case, a Young’s modulus of 160 GPa is obtained, quite close to the known bulk response for coarse-grained nickel. Third, as expected, the true stress is always greater than the engineering stress (in this tension experiment). Fourth, the material is hardening for substantial strains, up to about 6%. The necking regime and the ultimate tensile strength are not shown in this dataset.

2.4.2.1 Plastic Constitutive Relations Although Equation (2.95) is commonly used by materials scientists to describe the plastic behavior of metals, it does not provide a true constitutive relation in that it does not allow one to relate the general stress in the material (and the stress is a tensor) to the strain in the material at that point (the strain is also a tensor). The classical theory of plasticity provides a mechanism for translating the concepts presented

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in Equation (2.95) into a generalized constitutive equation that can be used to describe the behavior of materials under arbitrary loading conditions. The first step is to generalize the concept of a limiting stress (beyond which the material develops permanent or plastic deformations) to arbitrary stress states rather than the uniaxial stress state discussed above. In other words, one asks the question: when will a material deform permanently if it is loaded under a stress state that involves multiple stress components? The second step is to describe how the stresses and strains are related once plasticity begins. Here one immediately sees the need to distinguish between two parts of the total strain tensor: the elastic strain εiej (t) and the plastic strain εipj (t), since the former is recoverable on unloading while the latter accumulates with the deformation. The elastic strain components are always related to the stress components through the elastic moduli (e.g., through Equation 2.91). The generic problem in plasticity therefore reduces to the following: given σi j (t), determine the plastic strain εipj (t) that is developed, i.e., given the stress history, find the strain history. The simplest way to think about the first step is to visualize the problem in stress space. Recall that there are six independent stress components in the general stress tensor, but that we can solve the eigenvalue problem to find the three principal stresses and the three principal directions. If we assume that the material is isotropic, so that the behavior of the material is the same in all directions, then all we need to consider are the three principal stresses in order to decide whether or not the material has reached a limiting condition where permanent deformations begin. Imagine, therefore, a three-dimensional space in which the three principal stresses are plotted along the three axes of the space. This three-dimensional space is called the stress space (for the anisotropic material we would have to use a sixdimensional space corresponding to the six independent components of the stress tensor). Note that the stress tensor is defined at every point in the body, so that the three principal stresses are defined at every point in the current configuration of the body, and so there is a three-dimensional stress space associated with every point in physical space currently occupied by the body. The stress state at a point in the material is represented by the local principal stresses (σ1 , σ2 , σ3 ) which represent a point in the stress space. The stress history is the locus of stress states in stress-space (followed by a material point in the physical body). Now consider the origin in the stress space, corresponding to the case where σ1 = 0, σ2 = 0, σ3 = 0. This corresponds to an unstressed material. As we load the material, the various stress components in the stress tensor begin to increase, and so do the principal stresses. For example, the uniaxial tension experiment that we described earlier corresponds to the case where the only nonzero (and increasing) principal stress is (say) σ1 . For small stresses, our expectation is that the material will remain linear elastic, even though we have a multiaxial stress state, and Equation (2.65) will hold. Thus, there must be a region around the origin in stress-space where the material behavior is always linear elastic. What is the boundary of this region? This question is identical to the question of when permanent deformations will begin under multiaxial loading, i.e., when will yield begin, and we see that the answer to this question must represent a surface in this three-dimensional stress

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space. This surface, bounding the elastic region in stress-space, is called the yield surface, and is analogous to the definition of σy in the preceding subsection. Indeed, by definition the intersection of this yield surface with the σ1 axis will be σy . Now consider the stress state at a point. If a single principal stress is increased sufficiently in magnitude, plastic deformation will occur. If more than one principal stress is changing, the onset of plasticity will depend on the relative magnitude of the principal stresses. To see this a little more physically, consider a thin sheet of material under biaxial tension (see Figure 2.6a), so that both σ1 = 0 and σ2 = 0 in the plane of the sheet, but σ3 = 0. The corresponding two-dimensional stress space is shown in Figure 2.6b. Consider first the case where σ2 = 0, which amounts to the simple tension problem; this case corresponds to moving out along the horizontal (σ1 ) axis in Figure 2.6b, and clearly yield will occur when σ1 = σy . If instead we consider the case where σ1 = 0, corresponding to uniaxial tension along the vertical direction, the material behavior should remain similar to that in the first case since the material is isotropic, and so yield will occur when σ2 = σy along the vertical (σ2 ) axis in Figure 2.6b. This much follows just from the simple tension test. σ σ σ

σ

σ

σ σ

σ

Fig. 2.6 (a) Illustration of the biaxial tension of a sheet. (b) 2D stress space corresponding to the sheet, showing the stress path and possible yield surface bounding the elastic region.

What happens if both σ1 = 0 and σ2 = 0? In this case the stress state will follow some arbitrary path in the σ1 − σ2 space of Figure 2.6b, for example the curved path shown. When will the onset of permanent deformations occur along this path? In general, the answer to this depends on the material and can only be determined by experiment; one possible solution is the arc shown in Figure 2.6b and defined as the yield surface (this is in fact the intersection of the full yield surface with the plane corresponding to σ3 = 0). The area within this arc and containing the origin is the elastic region in this stress space; the yield surface is the boundary of this elastic region. There are many possible shapes for this yield surface, and some of the common shapes are discussed later in this book. It is already clear that plasticity is a far more complex constitutive behavior than linear elasticity. However, some of the basic properties of yield surfaces can be determined simply by considering the physics of plasticity, considered next.

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Mathematically, the yield surface is defined by those stress states that lie on it, that is, the yield surface is defined by f (σ ,Yi ) = 0

(2.96)

where the Yi represents a set of material parameters (such as the yield strength in uniaxial tension). It is expected that a finite stress magnitude is required to produce plastic deformation, so that (2.97) f (0,Yi ) = 0, that is, the yield criterion is not satisfied when the stress tensor is zero. Further, for isotropic materials, the yield surface can only depend on the principal stresses, so that (2.98) f (σ ,Yi ) = 0 =⇒ f (σ1 , σ2 , σ3 ,Yi ) = 0, and this relationship describes the yield surface in our 3D stress space. So far we have used the concepts of the existence of an elastic region, and the isotropy of the material. Further determination of the form of this function f (σ ,Yi ) requires the use of the physics of plastic deformation in metals. Plastic deformation in crystalline metals occurs through the motion of line defects known as dislocations, and dislocation slip occurs largely due to shear stresses. Indeed, it is generally observed that crystalline metals do not yield under purely hydrostatic stress states. We thus conclude that we can replace the function f (σ ,Yi ) by a function f (S, Yi ), remembering (Equation 2.55) that the Cauchy stress tensor σ can be broken down into deviatoric and hydrostatic parts: σ = S − pI. Since p cannot cause yielding, the yield surface must be represented by f (S,Yi ) = 0,

(2.99)

and the concept of material isotropy then leads to f (S,Yi ) = 0 =⇒ f (S1 , S2 , S3 ,Yi ) = 0 =⇒ f (SI , SII , SIII ,Yi ) = 0,

(2.100)

where we have used the fact that the set of eigenvalues of S can be directly obtained from the three invariants SI , SII , SIII of the deviatoric stress tensor S. The three invariants SI , SII , SIII of the deviatoric stress tensor are conventionally called J1 , −J2 , and J3 respectively (note that J2 = −SII ). Since S is deviatoric, J1 = trace S = 0, and the yield surface can only be described by the yield function f (J2 , J3 , Yi ) = 0.

(2.101)

Thus, the assumption of dislocation driven plasticity and isotropy leads to Equation (2.101) describing the yield surface (appropriate for crystalline metals). However, what is the form of this function? Further reduction of the form of the yield surface requires experimental measurements, since a specific functional form depends on the specific deformation mechanism, and even crystalline metals show a wide variety of such mechanisms. The most common additional assumption is that

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the dependence of the function on J3 = det S is negligible, and that there is only one material strength parameter Y that is needed to describe the behavior so that f (J2 , Y ) = 0,

(2.102)

where Y would be related, for example, to the yield strength observed in the simple uniaxial tension test. The simplest possible form of this function is f (J2 , Y ) ≡ J2 −Y = 0 =⇒ yield will occur when J2 = Y.

(2.103)

An equation such as Equation (2.103) describing the condition that must be satisfied for yield to occur (that is, for the onset of plastic deformation for an arbitrary stress path) is called a yield criterion. All yield criteria have a similar form, in that on the left-hand side of the criterion will be a function of the applied stresses, while on the right-hand side of the yield condition is a material property such as the yield strength (there are more complex forms of the yield criterion, but they are not discussed in this book). Yield occurs when the function of the applied stresses is equal to the material property. In this case, for example, we know that (using Equation 2.48) J2 = −SII = −

 1 (trace S)2 − trace S2 2

1 = trace S2 , because trace S = 0 2 1 = Si j Si j 2 1 2 2 2 2 2 2 = {S11 + S22 + S33 + 2(S12 + S23 + S31 )} 2 1 = (S12 + S22 + S32 ) 2

(2.104) (2.105) (2.106) (2.107) (2.108)

where we have used the summation convention. In terms of the principal stresses (the eigenvalues of the Cauchy stress tensor), we have 1 1 J2 = {σ12 + σ22 + σ32 − (σ1 + σ2 + σ3 )2 } 2 3 1 2 = {(σ1 − σ2 ) + (σ2 − σ3 )2 + (σ3 − σ1 )2 }. 6

(2.109) (2.110)

The yield condition at Equation (2.103) can thus be written as 1 Si j Si j = Y, 2

(2.111)

1 {(σ1 − σ2 )2 + (σ2 − σ3 )2 + (σ3 − σ1 )2 } = Y. 6

(2.112)

or in terms of the principal stresses,

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What is Y ? Consider the onset of yield in the simple tension test, where only σ1 = 0. It follows from Equation (2.112) that at yield we have Y = 13 σy2 . It follows that for general loading as in Equation (2.112) we must still have 1 Y = σy2 , 3

(2.113)

where we remember that σy represents the yield strength in tension. Thus the full yield criterion for multiaxial loading of a crystalline metal would be 1 {(σ1 − σ2 )2 + (σ2 − σ3 )2 + (σ3 − σ1 )2 } = σy2 , 2

(2.114)

which essentially says that yield will occur when the combination of applied stress components on the left side reaches the value defined by the material property on the right side. This represents a surface called the yield surface in our 3D space of the principal stresses. This yield criterion is sometimes called the von Mises yield criterion, and appears to work very well for a large variety of metals. The quantity on the left-hand side of Equation (2.114) is often used to define a scalar measure of the applied stress, usually called the von Mises or effective stress σ¯ :  1 σ¯ = √ {(σ1 − σ2 )2 + (σ2 − σ3 )2 + (σ3 − σ1 )2 }, (2.115) 2 and in terms of this von Mises stress, the yield criterion can be very simply written as yield occurs when σ¯ = σy . (2.116) Note that in the two-dimensional case corresponding to the biaxial tension problem of Figure 2.6a, Equation (2.114) reduces to

σ12 − σ1 σ2 + σ22 = σy2 ,

(2.117)

which is the equation for the curve in Figure 2.6b. As the stress increases along any arbitrary stress path in Figure 2.6a, yield will occur when the path reaches the yield curve in Figure 2.6b. Equation (2.116) represents a surface called the yield surface in our 3D space of the principal stresses. It predicts, for example, that yield will occur in a torsion test when the shear stress in torsion τ satisfies 1 τ 2 = σy2 , 3

(2.118)

so that the yield strength τy observed in a simple shear or torsion experiment should be related to the yield strength in simple tension σy by 1 τy = √ σy . 3

(2.119)

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This is experimentally observed in a very large number of engineering metals. Let us consider an example of plastic behavior in nanocrystalline metals. Suppose the yield strength in uniaxial tension of a nanocrystalline nickel material with a grain size of 20 nm (made by electrodeposition) has been measured to be 1.1 GPa (Figure 2.5). Then it follows that σy = 1.1 GPa. Let us assume that von Mises yield (Equation 2.116) describes the onset of plastic deformation in nanocrystalline nickel. This yield criterion would then predict that if we were to take a tube of nanocrystalline nickel and subject it to torsion, it would yield plastically when the applied shear stress reaches √13 σy which is 635 MPa. Unfortunately the yield stress of nanocrystalline nickel has not been measured in torsion as of this writing, so we do not know if von Mises yield is a good description of the behavior (even though it is known that conventional grain sized nickel follows von Mises yield behavior). We now have a relationship (Equation 2.114) that predicts the onset of plastic deformation, but we still do not have what we set out to obtain: a relationship between the total strain history and the stress history for a plastically deforming material. Remember that the yield surface is the boundary of the elastic region. Within the elastic region, we know how the stresses and strains are related (note that in the elastic region the history of the strain is irrelevant – the current strain determines the current stress). How are these quantities related once we reach the yield surface? This is a far more complicated problem, and we merely summarize the results here. The reader should refer to textbooks on plasticity for a detailed development of the theory. Since plasticity is a history dependent behavior, it is useful to think about changes in the stress and the strain rather than the current stress and strain. Consider the situation when the stress path has just reached the yield surface and we now have an additional increment δ σi j of the stress along the path. Our goal is to find the increment of the strain δ εi j that results from this increment in the stress. Assume that the stress increment is a loading increment, that is, the stress path is trying to move out of the elastic region. By definition, this stress increment will result in a permanent deformation of the material, that is, in a permanent plastic strain increment δ εipj . On the other hand, if the additional stress increment were to be removed, the material would recover a certain amount of elastic strain, so that there must be an elastic strain increment εiej also associated with the loading stress increment. It is therefore appropriate to view the total strain increment as having an elastic part and a plastic part, so that δ εi j = δ εiej + δ εipj , (2.120) with the additional proviso that the plastic strain increment δ εipj would only be developed if the increment in the stress is the loading increment rather than an unloading increment (all unloading is assumed to be linear elastic). The elastic strain increment can be related to the stress increment using the equations of linear elasticity (e.g., Equation 2.91):

δ εiej =

 ν 1+ν δ σi j − δ σkk δi j . E E

(2.121)

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If we can now find a relationship between the plastic strain increments and the stresses, our objective of relating the stresses and strains will have been achieved, at least in an incremental sense. First, it is important to recognize that the material behavior may or may not show hardening (usually called strain hardening). If the plastic behavior of the material is perfectly plastic, without any strain hardening, then it is not possible to apply a loading stress increment, and the plastic strain increment is indeterminate (in terms of material response – any number of deformed states can be attained after yield has occurred, limited only by the boundary conditions and the available energy). If the material behavior shows hardening after initial yield, then it is possible to determine the plastic strain increment uniquely for a loading increment in the stress. A textbook on plasticity will demonstrate that the plastic strain increment is related not so much to the stress increment as it is to the current deviatoric stress:

δ εipj =

1 δ σ¯ Si j , if and only if J2 = Y and δ σi j is a loading increment (2.122) 2 h(ε p ) σ¯

where σ¯ is the effective stress measure defined in Equation (2.115) and h(ε p ) is a function that describes how the material hardens with increasing plastic strain (this function is itself a material property, relating for example to the strain hardening index n). Note that Equation (2.122) also contains δ σ¯ , which is the scalar increment in σ¯ and which depends on δ σi j , but that individual components of the plastic strain increment depend only on the deviatoric stress components and not on the Cauchy stress components. Such a model is referred to as a J2 f low model, since it is derived from a J2 -based plasticity model and is written in an incremental strain form (the flow). It is important to recognize that δ εipj = 0 if the stress state is in the elastic region, or if δ σi j is an unloading increment. Putting Equations (2.120) and (2.122) together, we have

δ εi j =

 ν 1+ν 1 δ σ¯ Si j δ σi j − δ σkk δi j + . E E 2 h(ε p ) σ¯

(2.123)

This is the relation between the strain increments and the stress increments (and the stresses themselves, because of the history dependence). Together with the yield criterion (Equation 2.114), this incremental relationship defines the plastic constitutive response. This can be used to describe material behavior under multiaxial stress states along arbitrary stress paths, but must be solved incrementally, with a process of continually updating the stresses and strains as the nonlinear deformation develops.

2.4.3 Fracture Mechanics The concept of a stress concentration is a nearly intuitive concept in elasticity – it is apparent to most people, based on daily experience, that sudden changes in cross-section, notches and holes make it easier to cause the failure of materials

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(imagine, for example, how easy it is to tear a sheet of paper that has already had a hole made in the middle). The concepts of fracture mechanics have similar roots, and are key concepts in structural design and failure analysis. The fundamental question of fracture mechanics is this: if a structure has a crack within it, and the structure is placed under load, will the crack grow? A secondary question is: if the crack does grow, how much will it grow, and how fast? We will address these questions in this section. The simplest form of fracture mechanics involves a crack in a linear elastic material (this is called Linear Elastic Fracture Mechanics or LEFM). Consider a sharp planar crack within a large block of material (Figure 2.7), loaded in tension in the x2 -direction at some distance. Define the coordinate system as in the figure. In real plates, the crack front (the locus of crack tips) is a curve rather than a straight line, and is sometimes called a thumbnail crack (Figure 2.8). This is because the surface of the block is essentially in a different stress state (close to plane stress) while the center of the block is closer to plane strain for a sufficiently large block, and cracks appear to grow more easily in plane strain. The crack front is the curve separating the cracked plane from the rest of the material (Figure 2.8).

Fig. 2.7 A sharp planar crack in a very large block of linear elastic material. The coordinate axes are defined so that the crack front (the line drawn by the crack tip) is the x3 or z axis. Note that all constant z planes look identical, so that this is essentially a plane strain problem in the x1 − x2 plane. The mechanics of this problem is dominated by the behavior as one approaches the crack tip.

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x

x

x x

Fig. 2.8 Two views of the crack in Figure 2.7, with (a) being the side view and (b) the top view looking down on the crack plane. The crack front is typically curved, but this can be neglected in the limit as one approaches the crack tip.

From a mechanics viewpoint, we would like to know all of the stresses, strains and displacements everywhere in the problem. However, this is more information than we perhaps need to answer the primary question in fracture mechanics. Far away from the crack tip, the material and structure do not feel the influence of the crack, and the stresses and so forth are identical to what one would have from a simple uniaxial tension problem. In fact, therefore, we only need to investigate what happens as we get close to the crack tip. This is a recurring theme in linear elastic fracture mechanics – the stress, strain and displacement fields as one approaches the crack tip are those of interest, and are indeed expected to dominate the problem (because the crack tip can be viewed as a stress concentration, and so the stresses should be highest there). Most of the results in LEFM are only valid in the limit as one approaches the crack tip (such solutions are called asymptotic solutions). Thinking of the crack tip as a stress concentration, note that the radius of the crack tip in the x1 − x2 plane is much smaller (essentially zero) than the radius of the crack front in the x1 − x3 plane (Figure 2.8b). Thus the dominant behavior as one approaches the crack tip is the behavior in the x1 − x2 plane. Another way to think of this is as follows: as one approaches the crack tip, all variations in the x3 direction are small in comparison to variations in the x1 and x2 directions, that is, none of the quantities of interest depend on x3 . Note that this implies that (as one approaches the crack tip) ε33 = 0, that is, the problem looks like a plane strain problem as one approaches the crack tip. For this reason, most measurements of the toughness of a material (the resistance to fracture) are performed in plane strain. As we approach the crack tip, therefore, we have (in terms of functional dependence) σi j = σi j (x1 , x2 ,t), εi j = εi j (x1 , x2 ,t) and ui = ui (x1 , x2 ,t). In quasistatic problems the dependence on time can also be dropped. Note that u3 = 0 in general. The problems in LEFM are typically separated into problems with u1 , u2 = 0, u3 = 0 and u1 , u2 = 0, u3 = 0, using the principle that for linear problems we can add the solutions for the two problems to obtain solutions to more general problems. The

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case where u1 = 0, u2 = 0, u3 = 0 is called the plane problem, and the case where u1 = 0, u2 = 0, u3 = 0 is called the antiplane problem (this corresponds to the tearing mode in fracture problems). Let us define the crack front line as the x3 = z-axis. Note that symmetry requires that every x3 = constant plane will look the same in terms of stress and strain fields. Consider the plane problem (Figure 2.8a), so that the problem reduces to considering the stresses in the x1 − x2 or x − y plane. Thus the only stresses of interest are σxx , σxy , or σyy , because the σ33 = σzz can be computed directly from the linear elastic constitutive law, given that ε33 = 0. The crack tip can be defined as the origin in this 2-D space. A transformation to polar coordinates (r, θ ) from the Cartesian coordinates (x, y) has the crack tip at r = 0, and the crack faces at θ = ±π , and in this coordinate system the stresses of interest are σrr , σrθ , or σθ θ . In either case, distance from the crack tip can be defined as r = x2 + y2 . Far away from the crack tip, the influence of the crack is negligible, and the stress state will just be σyy = σ∞ , assuming that the far field tension σ∞ is in the x2 = y direction. As we approach the crack tip, the influence of the crack is increasingly felt, and the stresses at the crack tip itself are infinite (for a sharp crack – like a stress concentration of infinity). How rapidly do the stresses grow as one approaches the crack tip? A mathematical analysis of this problem, found in any number of books on mechanics (e.g., Asaro and Lubarda, 2006), shows that all of the stress components increase in the same way: KI σi j = √ fi j (θ ), r

(2.124)

where the angle-dependent function fi j (θ ) is the only difference between the three stresses in the plane. Thus all of the stresses increase as √1r , going to ∞ as r → 0. The derivation of Equation (2.124) uses a specific physical concept: that although the stresses at the crack tip can be infinite, the displacements and energies at the crack tip must be finite. The fact that the stresses are infinite at the crack tip is called a singularity, and the crack tip in the elastic material is said to have a √1r singularity (other singularities are possible, for example in viscous materials). The coefficient KI in Equation (2.124) is called the stress intensity factor, and scales the magnitude of the stress: it is a function of the applied loading and the geometry of the particular problem. Since the stress intensity factor scales all of the stresses, it is a measure of the effect of the applied load in the crack problem. The stress intensity factor KI is tabulated for a very large number of problems, so often one only needs to look it up for a specific geometry and loading condition (e.g., three-pt bending of a cracked beam). As the applied load increases, KI increases. The answer to the first question that was raised at the beginning of this section is that the crack will grow when the applied stress intensity factor KI reaches a critical value that is a material property called the plane strain fracture toughness KIc , i.e., the fracture criterion is KI = KIc .

(2.125)

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As with all criteria of this type, the left-hand side defines something related to the applied load and geometry, and the right-hand side defines a material property (recall the yield criterion σ = σY ). The plane strain fracture toughness is a material property that must be measured in specific experiments called fracture toughness experiments, in which specimens containing a precrack are loaded until the initi√ ation of crack growth is perceived. The units of fracture toughness are MPa m. The fracture toughness is a fundamental material property in terms of the application of materials to problems involving mechanical loading, since fracture is one of the most common failure modes for materials and structures, and tables of fracture toughness can be found in most materials and mechanical √ design textbooks. 15 and 60 MPa m, while for most In general, for most metals KIc lies between √ ceramics it lies between 1 and 6 MPa m. Materials with low values of fracture toughness are said to be brittle; most ceramics are brittle. Many nanocrystalline metals are believed to be more brittle than the conventional grain size metal, but there is little real experimental data on the fracture toughness of nanomaterials (largely because of the difficulty of doing valid fracture experiments on very small scale samples). The second question in fracture mechanics, that of how the crack will grow once it begins to grow, must be answered on an energetic basis. The primary concept here is that it takes energy to grow a crack, for instance in terms of the addition of energy required for the free surface corresponding to the crack faces. The amount of crack growth that would be observed depends on the amount of energy available to grow the crack. The energetic approach to fracture mechanics is a very powerful approach, resulting in the ability to solve fairly complex problems and also suggesting ways in which we can increase the toughness of materials.

2.5 Suggestions for Further Reading 1. A. Bedford and K.M. Liechti, Mechanics of Materials. Good if you have little or no background in mechanics. Prentice Hall, Upper Saddle River, NJ, 2000. 2. Y.C. Fung and P. Tong, Classical and Computational Solid Mechanics. World Scientific Publishing, Singapore, 2001. 3. J.R. Barber, Elasticity (Solid Mechanics and its Applications). Springer, New York, 2003. 4. R. Hill, The Mathematical Theory of Plasticity. Clarendon Press, Oxford, 1986. 5. M.F. Kanninen and C. Popelaar, Fundamentals of Fracture Mechanics. Butterworths, London, 1973.

2.6 Problems and Directions for Research 1. Which of the following expressions are meaningful in our notation? Write out each of the meaningful ones completely for a 3D space: aii , ai j b j , aii bi , ai j bi .

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⎤ 1 0 −1 Given the matrix A = ⎣−1 2 0 ⎦, find the quantities Akk , Ai j Ai j , Ai j A ji . 3 0 −1 Consider the simple tension of a circular cylindrical bar of elastic/plastic material (6061-T6 aluminum alloy). Assume that the tension test is performed very rapidly at a nearly constant strain rate ε˙0 = 10−3 s−1 . Assume further that the material flows with no strain hardening at a stress σy = 500 MPa. Solve the 1D energy equation (Equation 2.59) to obtain the temperature distribution in the bar 500 s after the deformation commences, assuming no heat conduction through the grips, and ignoring convective and radiative losses through the circumfer ential surface. Note that the plastic work W p is given by W p = σy ε˙ dt. Use handbook data as necessary. Provide the temperature distribution in the form of a plot. Consider the plastic deformations of a thin sheet of metal deformed in biaxial extension. The problem is one in which the forces in the two orthogonal directions in the plane are varied independently. Apply the von Mises yield criterion, and sketch the yield surface in the 2D stress space. If the metal were to be made of a sheet of nanocrystalline aluminum 1 cm square and 100 µm thick with a yield strength in tension of 1 GPa, at what load would it yield under equibiaxial stretching? (Simple torsion) Equilibrium in torsion of a circular bar gives τ = Tr J , where τ is the shear stress in the bar, T is the torque, r is the radius, and J is the polar moment of inertia of the cross-section. The constitutive law then gives γ = rLθ  µτ , where γ is the shear strain, θ is the angle of twist in radians, and L is the length of the bar. Now consider a specific problem. A 1 m long steel tube with a diameter of 3 cm and a wall thickness of 1 mm is subjected to torsional loading until a total angle of twist of 5◦ is achieved. Compute the torque required to accomplish this, and then calculate the shear stress in the tube. Suppose the tube is simultaneously subjected to a tensile deformation, so that an axial elongation of 1 mm is obtained. Compute the axial force required to accomplish this elongation, and compute the resulting tensile stress in the tube along the axial direction. Sketch the 2D stress state on the surface of the tube after the combined tensile and torsional deformations. Now compute the principal stresses and principal directions (the directions in which the principal stresses act) for this stress state. The yield strength of a nanocrystalline aluminum sample is 700 MPa in tension. If the material’s behavior were to obey the Mises yield surface, at what shear stress would it yield in torsion? How about in compression? The surface of a 6061-T6 aluminum component of a machine tool is subjected to principal stresses of 200 MPa and 100 MPa. The third (out-of-plane) principal stress is zero. What tensile yield strength is required to provide a safety factor of 2 with respect to initial yielding? ⎡

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References Asaro, R. and V. Lubarda (2006). Mechanics of Solids and Materials. New York: Cambridge University Press. Ericksen, J. (1998). Introduction to the Thermodynamics of Solids. New York: Springer. Lai, W., D. Rubin, and E. Krempl (1999). Introduction to Continuum Mechanics. Burlington, MA: Butterworth Heinemann. Nye (1950). Physical Properties of Crystals. Oxford: Oxford Science.

Experimental science does not receive truth from superior science: she is the mistress and the other sciences are her servants. Roger Bacon

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Nanoscale Mechanics and Materials: Experimental Techniques 3.1 Introduction One of the fundamental difficulties associated with the measurement of the mechanical properties of nanomaterials is that nanomaterial samples generally cannot be obtained in sufficiently large sizes to do traditional mechanical testing. From another perspective, one must use experimental techniques that can operate at scales that are relevant to the nanomaterial problem. For both of these reasons, there has been a drive towards the development of small-scale experimental techniques that can provide mechanical information on materials or structures. This chapter discusses a number of these techniques. Note that this is not a comprehensive discussion because these techniques are in rapid evolution, and the reader would be well advised to review the current literature in this area. We classify these techniques into two types, based on the objectives: 1. Techniques that attempt to identify or quantify a phenomenon at the nanoscale, which we refer to as nano mechanics techniques (NMT). Examples of such phenomena include grain boundary failure, dislocation motion, and adhesion of interfaces. 2. Techniques that attempt to measure the mechanical properties of materials using small sample sizes, which we refer to as nanoscale mechanical characterization techniques (NMCT). Examples of mechanical properties of interest include elastic modulus, yield strength, and fracture toughness. Both NMT and NMCT approaches have been developing rapidly since the 1990s. In practice, some of these techniques involve specimens that are microscale (10−7 to 10−4 m) in dimension, rather than nanoscale, because the observational resolution may well be nanoscale even when the overall specimen size is in the microscale regime.

K.T. Ramesh, Nanomaterials, DOI 10.1007/978-0-387-09783-1 3, c Springer Science+Business Media, LLC 2009 

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3.2 NanoMechanics Techniques Understanding the behavior of nanomaterials involves understanding the phenomena that are active at the nanoscale, and this requires two steps: (i) identifying the phenomena and (ii) being able to model them. The identification of phenomena is the core objective of nanomechanics techniques. The phenomena themselves may be mechanics (physics)-dominated or chemistry-dominated, although often it is difficult to determine which parameters are the most relevant ones to measure. Examples of such phenomena include the mechanical behavior of grain boundaries in terms of the forces required to cause separation or sliding, the forces across interfaces (for example, the interfaces between two elements of a nanoelectromechanical systems or NEMS device), and the development of instabilities at the nanoscale such as the buckling of a carbon nanotube. Many of the advances in this area are fueled by the rapid growth in techniques that can examine atomic scale phenomena, particularly using scanning probe microscopy and transmission electron microscopy. What are the typical nanoscale phenomena that are relevant to the mechanical behavior of nanomaterials and the functioning of nanomaterials or nanostructures? Since the nature of nanomaterials and nanostructures implies the presence of boundaries at the nanoscale, the dominant question is the influence of such boundaries, whether these are interfaces between entities or simply surface features. For example, the behavior of grain boundaries is critical to understanding the mechanical behavior of nanomaterials, while the behavior of the surface of a cantilever beam can dominate the sensitivity of a cantilever sensor. Thus most nanomechanics techniques focus on the observation of boundaries and interfaces. Measuring such boundary phenomena requires that one be able to observe at the appropriate scale as well as be able to quantify at that scale. The classic questions of precision, accuracy and uncertainty are important for such measurements. It is often the case that we can observe phenomena at smaller scales than the scales at which we can accurately quantify those phenomena (the nanoscience version of “one’s reach is often beyond one’s grasp”). A more fundamental question is whether one can be certain that one is measuring the phenomenon under the same conditions each time the measurement is made. For example, we cannot obtain two identical grain boundaries for examination, because it is difficult to prepare two identical grain boundaries. Because of the difficulty of preparing samples, we observe what we can get as specimens, even though there is a very large variety of boundaries that should be characterized. However, the number of phenomena at the nanoscale may be very large, and our technology is just beginning to grasp the variety of features present here, even as foretold by Feynman, when he said that “there’s plenty of room at the bottom” (Feynman, 1959). In this regard, our difficulties in identifying and quantifying nanoscale phenomena have strong similarities to the travails of astronomers who are trying to identify the behavior of stars and understand the stellar lifecycle. Since we cannot make stars, we have to build our understanding on the basis of those that we can observe.

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Just so, in the nanomaterials world, we have to build understanding on the basis of the boundaries that we can observe. Note that the richness of phenomena, and the difficulty of control, arise for exactly the same reasons in the two cases of astronomy and nanotechnology: a vast difference of scale from the human condition. We must turn the difficulty that we are presented with (the variety of structures and phenomena available) into an opportunity, even as the astronomers do, and use these riches to our advantage. An analogy might be useful. Imagine an alien spacecraft arrives at the Earth, and the spaceship commander wishes to obtain an understanding of the human lifecycle. She could indeed choose to observe a human evolve from an infant to a full-grown adult and then to old age, but this would take more than 70 years. On the other hand, if she chose to examine a very large cross-section of the population, she would observe every stage of the human lifecycle in a very short time and would be able to obtain a fairly good understanding of the process (although certainly she would see variations based on race, economic status, and environment). This is what astronomers do with stars, and what nanotechnologists need to do with phenomena in nanostructures. We cannot control all of the nanostructures, but we can observe a large variety of them and try to extract an understanding of the phenomenon of interest. The difficulty would be identifying the persistent features across a spectrum of nanostructures (comparable to identifying the common aging processes across a spectrum of races and environmental conditions). As an example of this approach, consider understanding the mechanical behavior of grain boundaries in a nanocrystalline material. Since the grain boundary is the boundary between two crystals, there is a very large variety of boundaries that are possible (depending upon the relative orientations of the two crystals). It is not physically feasible to prepare a sample of every boundary with a given relative orientation of the two crystals. However, in any nanocrystalline material, it is likely that we will find representative examples of many of these possible grain boundaries. Thus by examining the population of grain boundaries in a given nanocrystalline material, and learning what their mechanical behaviors have in common, we may obtain an understanding of the likely mechanical behavior of all grain boundaries. This approach inherently requires us to use both experimental and modeling techniques simultaneously to study these phenomena, so that we can extract the persistent behavior across a variety of examples (as in grain boundaries). With that insight, let us consider NanoMechanics Techniques. The two major experimental techniques that comprise NMT are scanning probe microscopy and in situ deformation. The latter may be performed within an AFM (atomic force microscope), a TEM (transmission electron microscope) or an SEM (scanning electron microscope). Each of these techniques is described in the following sections. Note that these are complex techniques that are comprehensively discussed in the literature, so the descriptions here are brief. Other techniques such as optical tweezers and magnetic tweezer arrangements are more appropriate for biological problems, and are not discussed here.

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3.3 Characterizing Nanomaterials The characterization of nanomaterials is no easy task. By definition, the feature sizes that must be observed are very small, being less than half the wavelength of visible light. This makes it impossible to see these features using traditional optical microscopy. The primary characterization techniques that are used are therefore scanning electron microscopy, X-ray diffraction, and transmission electron microscopy. Of these three, the highest spatial resolution is obtained with transmission electron microscopy or TEM, and this is the final arbiter of whether a specific dimension has been achieved and is necessary for the determination of specific deformation mechanisms. However, transmission electron microscopy is a difficult thing to do, and specimen preparation in particular is a laborious task requiring substantial skill. As a consequence, many researchers in the field use X-ray diffraction to obtain a quick estimate of grain size. Unfortunately X-ray diffraction is unable to distinguish between low angle grain boundaries and high angle grain boundaries, leading to some confusion in the literature. The scanning electron microscope is sufficient for the observation of many device nanomaterials, and to examine failure mechanisms (often a question when reliability of devices is an issue). When surface examination is sufficient, various forms of scanning probe microscopy may be appropriate. The atomic force microscope is often used to provide images of surfaces with sufficient resolution. In any case, the use of nanomaterials implies the ability to characterize at the nanoscale, and a substantial investment has to be made in the infrastructure required to observe at these scales. Each of these techniques is described very briefly below; the reader should examine the suggested reading to obtain more detail.

3.3.1 Scanning Electron Microscopy or SEM A scanning electron microscope generates an image by scanning a surface with an electron beam. The electron beam is usually generated from a tungsten filament, and the energy associated with the electrons can be controlled. The beam itself can be manipulated using electron optics, analogous to traditional optics (lenses and so forth). The interaction of the electron beam with the surface results in the interrogation of a certain depth of material below the surface. Electrons are scattered off the material being investigated through both elastic and inelastic scattering, and by collecting the scattered electrons one can obtain images of the surface (in practice, the scattered electrons themselves interact with some kind of display, generating intensity variations that provide the image shown in typical scanning electron micrographs). In the most common mode of operation, the scattered electrons that are collected are those that have been inelastically scattered, and this interrogates a very small depth below the surface (several nanometers). Most SEM images thus provide surface topography information, and resolutions down to a nanometer are possible. In comparison to optical microscopy, scanning electron microscopy is capable of much greater magnification and resolution, and at the same time retains

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great depth of field (in light microscopy, high magnification usually results in low depth of field). Scanning electron microscopy is therefore a great way of looking at surface information on nanomaterials (the depth of field is particularly useful when looking at fracture surfaces). If the scattered electrons that are collected are the elastically scattered electrons (so-called back-scattered electrons), then the image that results contains a significant amount of information on the chemical composition of the surface, because different materials provide different degrees of scattering in this mode (higher atomic numbers – that is, heavier atoms – scatter more). When used in combination with some form of spectral analysis, a great deal of information on chemical composition can be obtained.

3.3.2 Transmission Electron Microscopy or TEM Unlike the SEM, the transmission electron microscope uses the transmission of an electron beam through a sample to characterize the sample. The sample must be thin enough for transmission to occur, and so a good deal of specimen preparation is typically necessary. This technique is capable of much higher resolution, and in some versions is capable of examining individual atomic layers. Since the beam goes through the sample, the interaction of the beam with the sample is much more complex, and this can be used to advantage in generating various imaging modes. Some of these modes include the ability to examine the diffraction of the beam through the crystal lattice, and to select specific diffraction spots for imaging purposes. This combined with the ability to tilt the sample in the beam makes it possible to identify individual crystal structures, to see and identify the character of dislocations in the crystals, and to examine twin structures. TEM micrographs are presented throughout this book, together with selected area diffraction (SAD) patterns that provide information on the crystal structures interacting with the beam. SAD patterns display the diffraction pattern that results from the interaction of the electron beam with some volume of the sample. In the specific case of the beam interacting with only one crystal, the SAD pattern is a pattern of dots that represents the crystal structure (actually the Fourier transform of the crystal structure). When the beam interacts with a nanocrystalline material, very many grains are averaged over, resulting in patterns that are more ring-like than spot-like. The technique of electron energy loss spectroscopy or EELS can be used to provide composition information, and the combination of scanning with transmission modes in a scanning transmission electron microscope or STEM can provide very useful information on the location of specific elements at specific regions in the material. In high resolution electron microscopy or HREM, individual atomic layers can be examined, the locations of atoms can be rigorously interrogated, and grain boundaries can be examined individually. The TEM is without question the definitive instrument for use with nanomaterials, since it can examine the interior of materials with the appropriate resolution.

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The difficulty with the use of the TEM is twofold: first, specimen preparation is tedious and exacting and can change the structure being observed if not done properly, and second, the use of the TEM and interpretation of the TEM images requires a very high degree of expertise. As instruments go, TEMs are expensive to maintain and operate (much more so than SEMs).

3.3.3 X-Ray Diffraction or XRD X-rays interact with crystalline structures through diffraction, and X-rays are relatively easily generated, so X-ray diffraction is the easiest way to examine crystal structures. A large variety of specialized techniques have been developed based on X-ray diffraction, and careful analysis can reveal information even at the level of the locations of nearest neighbor atoms in a material. Resolution can thus be very high, particularly using high-energy beamlines such as those obtained at synchrotrons. Imaging is not usually the objective, but information can be obtained on grain size (although low-angle and high-angle grain boundaries are hard to separate), elastic strain in the crystal, and composition.

3.3.4 Scanning Probe Microscopy Techniques Scanning Probe Microscopy (SPM) is a generic description for a class of techniques where a needle tip (called the probe) is scanned across the surface of the sample, and the interactions between the probe tip and the sample surface are used to generate an image that represents some characteristic of the surface. In many SPM modes, the probe (Figure 3.1) never actually touches the surface of the sample, instead being held a fixed distance (∼1 nm) away from the sample through a feedback circuit. The interactions between the tip and the surface are extremely small in scale, including atomic level interactions. Indeed, it is possible to scan across the surface of the sample and image every atom on the surface. It is also possible to image at the micrometer scale. These techniques are thus extremely powerful, and have revolutionized the study of surface physics and surface chemistry; Gert Binnig and Heinrich Rohrer won the Nobel Prize in Physics in 1986 for inventing the technique in 1981 (Binnig et al., 1983). Such systems can be used both to image and to manipulate atoms on the surface. A very large variety of interactions are possible between the probe and the surface. These include electron tunneling interactions–leading to scanning tunneling microscopy or STM (Binnig and Rohrer, 1985); contact forces–leading to atomic force microscopy or AFM (Binnig et al., 1986); and magnetic interactions–leading to magnetic force microscopy or MFM (Martin and Wickramasinghe, 1987). A variety of technologies have developed that utilize the extreme sensitivity of the measurements to these interactions. The particular interaction that is measured using

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Laser Diode Source

Piezoelectric Scanning Actuators (x,y,z) To Position Sensitive Photodetector Cantilever

Sample Surface Fig. 3.1 Schematic of a Scanning Probe Microscopy system (not showing the feedback loop, tunneling amplifiers and other electronics). A wide variety of such systems exist, some of which contain only a subset of the features shown here.

the scanning probe system determines the characteristic of the surface that will be imaged and the name of the kind of microscopy that is being used. There is a large variety of commercial versions of scanning probe microscopy instruments on the market, and many online resources that provide detailed descriptions of these instruments. There are four critical components to scanning probe microscopy techniques: • First, the nature of the probe tip determines the resolution of interaction. • Second, the ability to scan in a controllable way over very small distances with high-resolution determines the spatial resolution of the technique (this is usually done with a piezoelectric system). • Third, a feedback system is usually used to maintain the probe a fixed distance away from the surface being scanned, and there is a very important relationship (which can be designed to some degree) between the feedback control and the specific quantity being measured. • Fourth, the ability to develop images that visualize the distribution of these interactions across the surface of the sample through appropriate software is critical to the practical use of these techniques. The interpretation of these images often requires substantial expertise, and may require a real understanding of the physics of the interactions. These are very powerful techniques, but come with a strong caveat. The ease with which an image is obtained using a commercial SPM instrument may hide very

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complex physics that might be very important to the issue of interest, because the software that provides the image may make assumptions that are hidden from the viewer. It is critical to understand (a) what the hardware is capable of doing and (b) what the software assumes in generating the image (for example, what are the defaults in the parameters used by the software). Examples of applications of scanning probe microscopy that are of interest in this book include the examination of grain boundary character at the atomic scale, the determination of the development and motion of ledges on crystal surfaces, the examination of compositional variation close to boundaries, and the characterization and fabrication of nanomaterial surfaces.

3.3.5 Atomic Force Microscopy or AFM Atomic force microscopy (AFM) is a very high-resolution SPM technique for examining the surfaces of materials. The image is formed by scanning a probe over the surface and using the interaction of the probe with the atoms on the surface to generate an image. Many different kinds of interactions can be examined, and images can be generated of surface topography, surface magnetic character, and so on. This technique is often used with discrete nanomaterials and thin films, and less so with bulk nanocrystalline materials.

3.3.6 In situ Deformation In situ deformations are defined as deformations that are developed while the sample is within an imaging environment. The ability to image during the deformation allows us to determine the mechanisms that are active and to examine the deformation processes at the microscale. The development of an in situ deformation technique requires two components: • First, an arrangement that allows one to develop controlled deformations within the microscope • Second, an imaging capability through microscopy (in order to be able to examine the length scales of interest) From a mechanics viewpoint, it is useful to be able to measure the forces within the system at the same time as the deformations are being imaged through the microscope. This can be very hard to do at nanometer scales, because the forces that are developed can be in the nano-Newton range. The primary in situ deformation techniques that are available are accomplished within AFM, TEM, and SEM systems. The major difficulty in in situ deformation techniques is that it is difficult if not impossible to obtain samples of identical structure; one has to look at what one has

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available, rather than being able to examine model structures for a given material (such as identical grain boundaries).

3.3.6.1 In Situ Deformation Within an Atomic Force Microscope Atomic force microscopy involves the measurement of the forces involved in the scanning of a probe along the surface of a material (often including contact). The probe tip is often made from single crystal silicon. Probe tip radii of curvature can be of the order of 1 nm. Typically, the probe is carried on a cantilever beam called a microcantilever. The force of interaction between the probe tip and the surface results in the bending of the microcantilever, and the forces are measured by recording the bending of the beam using an optical arrangement. The forces that are measured might be contact forces, van der Waals forces, or electrostatic forces. In terms of spatial resolution, this technique often does not have the spatial resolution of STM but is much more robust in operation. A common approach to in situ deformation within the atomic force microscope involves the use of thin films of material that are deformed using a mechanical arrangement (Nishino et al., 2000). The large majority of such experiments do not record the forces within the sample during the deformation. However, recent approaches to mechanical loading using a MEMS type device show significant promise (Chasiotis et al., 2004). Images of the deformed sample can then be correlated to measures of the current load to estimate the relationship between the stress state and the deformation. Note that the observation of deformation processes using imaging may be worthwhile even without load measures. For example, the evolution of surface features in a nanocomposite can be imaged effectively using this technique, as can grain boundary migration.

3.3.6.2 In Situ Deformation Within a Transmission Electron Microscope The Transmission Electron Microscope (TEM), as the name implies, is an instrument that uses the transmission of electrons through a material to determine the structure of the material. An electron beam is accelerated using a high voltage and is passed through a sample; an image is constructed on the basis of the electrons that have been transmitted through the sample and that strike a detector placed on the opposite side of the sample. The most common application of the TEM is to provide microstructural characterization of crystalline materials. The interaction of the electron beam with the crystal can be studied by using the principles of diffraction, and highly sophisticated techniques have been developed to infer the structure of the material on the basis of the distributions of the transmitted electrons. Among the characteristics of the material that can be inferred are the local grain size, the crystal structure, and the defect density, in addition to the characterization of boundaries and identification of composition. At nanometer length scales, the TEM is often believed to provide “ground truth” in terms of materials characterization. The primary difficulty with the use of the TEM is that the samples must be made electron

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transparent, and the production of good TEM specimens is essentially an art that requires a great deal of experience. Most TEM specimens are disks about 3 mm in diameter with a central dimple containing a region of material that has been thinned to about 100 nm in thickness. The process of generating a good TEM specimen may itself (if done incorrectly) result in substantial modification of the structure that is being examined. With these caveats, there is no substitute for high-quality transmission electron microscopy work in the nanocrystalline materials area for determining the material characteristics. In situ deformation experiments within a TEM are also complicated by the need to develop an electron transparent region within the sample. This essentially limits the in situ samples to thin films. There are a number of commercially available testing stages that can be mounted within TEM vacuum chambers for such experiments, but the development of an effective technique requires substantial effort to ensure that the technique is optimized to the specific microscopy facility. A common way to develop local strains in the sample is to use a thermal misfit rather than to apply controlled mechanical loads. Again, recent developments in MEMS based mechanical testing techniques show strong promise in this area (Haque and Saif, 2004). Some instruments have been developed that accomplish compression and tension within a TEM using a combination of focused ion beam machining to generate the specimens and in situ nanoindentor based loading techniques (Minor et al., 2001; Stach et al., 2001), and these are just beginning to have a significant impact on our understanding of deformation mechanisms at the nanoscale.

3.3.6.3 In Situ Deformation Within a Scanning Electron Microscope The scanning electron microscope (SEM) is an imaging instrument that also uses an electron beam, but generates an image by looking at the electrons that are scattered off a surface. The electron beam is scanned across the surface, generating images of the surface over a very wide range of magnifications. This is the premier technique for examining material surfaces in materials science and engineering. One of the major advantages of this technique is that the depth of field is considerable, so that it is possible to look at very rough surfaces such as fracture surfaces. Unlike the TEM, the SEM can be used by a relatively unskilled operator and specimen preparation is not a major problem except for some special cases. The spatial resolution of SEM systems has improved dramatically over the last two decades, and nanometer resolution is now possible. Further, it is possible to identify crystal orientations using SEM systems in Orientation Imaging Microscopy mode (this is very useful for examining polycrystalline systems). In situ deformation experiments within an SEM have a particular attraction because of the ease of use of the instrument, the wide range of magnifications that are possible, and the substantial depth of field. The coupling of these capabilities with the recently developed digital image correlation (DIC) technique for determining displacements and strains from multiple images has made this a very useful technique for in situ deformation studies. Commercial in situ testing stages for the SEM

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Fig. 3.2 Single crystal copper specimen and tension gripper for in situ tensile testing in the SEM, from the work of the group of Dehm, Kiener et al. (this group has also performed compression experiments using a similar apparatus) (a) SEM image showing a single-crystal copper tension sample and the corresponding tungsten sample gripper before the test at a low magnification. (b) Sample and gripper aligned prior to loading. Image taken from (Kiener et al., 2008). Reprinted from Acta Materialia, Vol 56, Issue 3, P. 13, D. Kiener, W. Grosinger, G. Dehm, R. Pippan, A Further step towards an understanding of size-dependent crystal plasticity: In situ tension experiments of miniaturized single-crystal copper samples. Feb. 2008, with permission from Elsevier.

are now available, and are easily fitted to specific facilities. These techniques have been used, for example, to understand the size-dependence of plastic behavior in metal single crystals (Kiener et al., 2008). An example of the specimen size and shape is presented in Figure 3.2 from the work of Kiener et al. (2008).

3.4 Nanoscale Mechanical Characterization 3.4.1 Sample and Specimen Fabrication The objective of a mechanical characterization technique is to measure the behavior of the material. However, what we in fact measure is the behavior of a sample of the material, and we then infer material behavior from sample behavior. This works very well if we (a) understand the relation between sample behavior and material behavior and (b) have not changed the material response while trying to make the samples that we will test. An example might be useful. Suppose one wished to measure the tensile strength of a relatively brittle material like pure tungsten, and so one performs a standard tensile test on dogbone-shaped samples (Figure 2.4) of the tungsten. Suppose the dogbone-shaped samples are made by machining from a larger tungsten block using a milling machine, but one set of samples is tested immediately after milling, while the other set of samples is subject to a finish grinding operation (after milling) before the tensile tests are performed. A very interesting behavior is observed. The strength of the material measured from the finish-ground samples is found to be much higher than the strength of the material measured from the as-milled samples (a similar behavior has been shown for compression; Lennon and Ramesh, 2000). This is because the as-milled samples have small surface cracks generated during the milling process, and these cause premature failure

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of the sample. The finish-ground samples do not have these surface cracks, because the top layer of damaged material has been ground away, and these samples sustain higher strengths (and higher strains to failure). Thus sample fabrication processes can be critical. Note that this is an issue that goes above and beyond the issue of the generation of high-quality material, with few defects and reproducible microstructures. Sample preparation can be a particular problem for nanocrystalline materials, which are typically high-strength and less tolerant of defects introduced by fabrication processes. Sample fabrication processes may include various kinds of machining, extrusion, and thermal cycling, and the potential effects of all of these processes must be considered.

3.4.2 Nanoindentation The mechanics of the indentation of one solid into another have been known for many years. The first technical discussions of such matters examined the scratch testing of materials, asking the question whether one material would scratch another. This led to the famous Moh’s scale of hardness (diamond was ranked as the hardest material). The mathematical solution of the problem of the elastic indentation of one material by another was provided by Hertz, who calculated the stress state beneath an indentor. This mathematical solution is one of the fundamental solutions in contact mechanics (Johnson, 1987), and describes the relationship between the applied force on the indentor P and the depth δ of the indentation: P=

16 9

1 (

1−νi2 Ei

+

2 1−νs2 Es )

Ri Rs 3/2 δ Ri + Rs

(3.1)

where the subscripts i and s refer to the indenter and the sample respectively (e.g., Ri and Rs are the radii of curvature of the indenter and sample). Note that the applied force depends on the displacement to the 3/2 power, and so that relationship would be expected for elastic behavior. If one performs an indentation experiment using a relatively sharp indentor against a flat sample surface (so that Rs is very large), measuring both the force and the displacement of the indentor tip, and then plots the force against δ 3/2 , a straight line should be obtained, and the slope of that line can be used to compute Es using Equation (3.1) if one knows the indentor radius and indentor modulus Ei . If the indented material can develop permanent deformations (i.e., has elasticplastic behavior), then a permanent indentation will remain after the removal of the load from the indentor tip. Extending the Hertz contact analysis to the elastic plastic case, the development of these permanent indentations within materials can be described mathematically (although the description can be quite complicated if the plastic behavior of the material deviates from simple classical plasticity such as described in the chapter on mechanics). For the simple classical plasticity case, the size of the remaining indent depends on the strength of the material. This has led

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to a very common testing technique in materials science called Vickers indentation where the indentor has a specific pyramidal geometry, and Vickers hardness is a common way of estimating the strength of the material. Other indentor shapes can also be used, resulting in different hardness measures such as the Knoop hardness and Brinell hardness. Oliver and Pharr (1992) extended this approach to the problem of very smallscale indentation, calling this technique nanoindentation. The primary characteristic of nanoindentation is that both the force on the indentor and the displacement of the indentor tip are measured simultaneously with very great accuracy, and very small forces and very small displacements can be measured. A force-displacement curve is obtained for each indent. If the indentor is made out of an extremely stiff material such as diamond, the indentation force-displacement curve is dominated by the response of the material being indented. An example of such a curve is presented in Figure 3.4, which shows the result of a nanoindentation experiment on nanocrystalline nickel. Commercial nanoindentation instruments are now widely available. The coupling of this technique to software for automated indentation makes it possible to develop hardness maps that contain substantial information about the mechanical characteristics of materials and structures at the small scales of interest to nanomechanics and nanomaterials. The device that provides a controlled indentation while simultaneously measuring both indentor force and indentor displacement for very small indents (submicron) is called a nanoindenter. A schematic of such a device is shown in Figure 3.3, indicating the critical components. Current commercial nanoindenters have a displacement resolution better than 0.1 A, and a load resolution of the order of 1 nano-Newton. This high precision gives the technique popular application in

Load Frame Actuator Force Sensor

Position Sensor Indenter Tip Sample Fig. 3.3 Schematic of the NanoIndenter, showing both actuation and force and position sensing. Such a device may fit comfortably on a large table top, although vibration isolation may be desirable.

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many areas of materials science and engineering, such as the measurement of mechanical properties of very thin films (thickness < 100 nm), and tribological measurements of coatings. In a conventional nanoindentation experiment, a Berkovich (tetrahedral) tip is usually used to probe the nano-scale mechanical behavior (such as hardness and elastic modulus) as a function of indentation depth. For example, the indentation size has a strong effect on the hardness of single crystal silver (Pharr and Oliver, 1989). The typical output is a force-displacement curve that may deviate from the form presented in Equation (3.1), as shown in Figure 3.4 for nanocrystalline nickel. The primary mechanical properties extracted from a nanoindentation test are the hardness and the elastic modulus. This is extremely useful information, particularly for nanocrystalline materials. However, the nanoindentation test cannot provide a full description of the mechanical behavior of the material in terms of the stress-strain curve, and such a description is needed in order to be able to use a material within a specific application. The microcompression and microtension tests described next can provide such a constitutive description.

3.4.3 Microcompression Microcompression is an experimental technique where a modified indenter tip geometry is utilized within a nanoindenter to probe the nano-scale mechanical behavior of micro-scale or sub-micron pillars (Uchic et al., 2004). In this modification,

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Nanocrystalline Ni, 20 nm 250

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Displacement Into Surface, nm Fig. 3.4 Example of force-displacement curve obtained during the nanoindentation of a nanocrystalline nickel material with an average grain size of 20 nm. Data provided by Brian Schuster.

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the sharp Berkovich tip is truncated, resulting in a flat end tip which enables one to convert the indentation system into a compression system. Since both the force on the indentor tip and the displacement of the indentor tip can be measured, using this flat tip to compress the top of a cylindrical sample allows one to estimate the stress strain behavior of the material of the sample. This technique is commonly called “micro-compression” or µ -compression. This technique is of great use for probing the mechanical behavior of nanomaterials which are difficult to obtain in bulk form, and whose tensile ductility is limited (which makes micro-tension tests difficult). It is currently difficult to produce bulk forms of many metals and ceramics with nanocrystalline microstructure, but they can be readily produced in thin foils and small cubes or cylinders with dimensions less than 1 mm. Conventional mechanical testing techniques are not useful due to the small size of specimens. Micro-compression makes use of a conventional nano-indenter and a flat-end tip to measure the stress-strain curves of materials using posts with a dimension as small as 250 nm in diameter (Schuster et al., 2007). A schematic of a cylindrical post sitting on a base of the same material is shown in Figure 3.5a. Technologies such as focused ion beam (FIB) enable researchers to fabricate such microposts (Figure 3.5b) with ease. These posts usually have one end (the bottom) fixed on the matrix (or the base material), while the top end is pressed by the flat-end tip. In a conventional uniaxial compression test, both ends of the specimen are free to deform in a nearly frictionless manner. When can the data from microcompression measurements be used to represent the bulk materials behavior? How sensitive is the test to variations in material behavior? Finite element simulations can be used to determine the errors resulting from the compliant base condition, and to provide guidelines for the proper design of microcompression experiments (Zhang et al., 2006). Consider first an axisymmetric finite element model (Figure 3.6a) of the micropost. A velocity boundary condition

Fig. 3.5 (a) Schematic of a microcompression experiment, showing micropillar and flat-bottomed indenter tip. (b) SEM micrograph showing a micropillar of a PdNiP metallic glass produced by focused ion beam machining.

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(a)

(b)

Fig. 3.6 (a) Finite element mesh used for 2D axisymmetric model of the micropillar in a microcompression experiment. (b) Computed effective (von Mises) stress distribution in a sample after plastic compression. Note the nonuniformity at the root of the pillar, indicating the importance of the root radius (also called the fillet radius). Reprinted from Scripta Materialia, Vol 54, Issue 2, page 6, H. Zhang, B.E. Schuster, Q. Wei and K.T. Ramesh, The design of accurate microcompression experiments, January 2006, with permission from Elsevier.

corresponding to the imposed strain rate is applied to the post top, and the bottom of the base is not allowed to move in the x2 -direction as shown in the figure. The true stress-strain behavior of the post can be obtained from the reaction force P generated on the post top and the displacement u of the post top using the following equations:

σ¯ 22 =

1 π r2

 r 0

ε¯22 = ln(1 +

P2 |x2 =h dx1

(3.2)

u2 ) h

(3.3)

where the subscripts 2 indicate the x2 components of the corresponding vector, r is the radius of the post, h is the height of the post with respect to the base, and σ22 and ε22 are the stress and strain components that would be appropriate for a uniaxial compression experiment (note that in the microcompression experiment σ22 will not be the only non-zero stress component). These equations are essentially those used in reducing experimental data. In the finite element simulations, the material of the post is assumed to have the behavior described by the Ramberg-Osgood equation

ε¯ =

σ σ0 σ + β ( )( )N E E σ0

(3.4)

where E is the Young’s modulus, N the inverse of the traditional strain hardening exponent (see Equation 2.95), β is a material constant and σ0 is a reference stress (the Poisson’s ratio ν is also needed to complete the description of the linear elastic behavior). For the calculations shown here, the following parameter set is used,

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corresponding to an aluminum (Al) alloy: E = 70 GPa, ν = 0.33, σ0 = 65 MPa, β = 2.69, and N = 4.18. The question of interest is: under what conditions would the calculated σ22 and ε22 have the same relationship as the input material behavior given by Equation (3.4)? Those conditions would represent the design of an appropriate microcompression experiment, since the computed effective behavior would be identical to the assumed material behavior. 3.4.3.1 Post Geometry Effects h ) on the output The effects of the fillet radius rc and the post aspect ratio (α = 2r stress-strain curves are considered first. We begin by fixing α = 2 and varying rc (expressed in terms of the post radius r). Figure 3.7 displays the input stress-strain curve from Equation (3.5), along with the output stress-strain curves from the simulations with different fillet radii rc = 0.1r, 0.2r, 0.5r, and r. For clarity, a portion of the curves has been enlarged in the inset of Figure 3.7. Due to the fixed end effect, all the simulated output curves are above the input curve, indicating an overestimate

Fig. 3.7 Input and simulated stress-strain curves for various assumed root radii (detailed differences observable in the inset) as computed from the finite element model presented in the last figure. The “input” curve is the input material behavior used in the finite element simulations, and the “output” stress-strain curves are obtained from the forces and displacements computed from the simulations and then processed in the same way as the force and displacement data is processed in the experiments. If the experimental design were to be perfect, the output curves would be identical to the input curve. Note that the simulated curves are all above the input data, demonstrating the effect of the end condition. The outermost curve has the largest root radius, equal to the radius of the cylinder. As the root radius is decreased, the simulated curves approach the input curve but are always above it. Reprinted from Scripta Materialia, Vol 54, Issue 2, page 6, H. Zhang, B.E. Schuster, Q. Wei and K.T. Ramesh, The design of accurate microcompression experiments, January 2006, with permission from Elsevier.

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of the flow stress of the material in the plastic range, and the strength of this effect depends on the fillet radius/post radius ratio. The larger the fillet size (with respect to the post radius), the greater the error. When the fillet radius ratio rrc is less than 0.5, the output curves are within 3% of the input data. This suggests that if the fillet size is well controlled the fixed-end compression measurements can still be used to represent the bulk material’s behavior in terms of flow stress. For the aspect ratio α = 2, even with the largest fillet radius/post radius ratio ( rrc = 1) the error in flow stress from the microcompression test is less than 6%. The distribution of the von Mises stresses in the post and the base (Figure 3.6) for various fillet radius/post radius ratios shows that with increased fillet size, the stress concentration at the fillet is alleviated. For rrc ≥ 0.5 there is no significant stress concentration at the fillet, the maximum von Mises stress being along the post axis. Excessive stress concentrations may result in localized specimen failure prior to general yielding. Thus, the choice of fillet radius for an experiment represents a compromise between the need to accurately recover material behavior in the plastic range (requiring small rrc ) and the desire to avoid localized failure at the post root (which requires large rrc ). The simulations suggest that a choice of rrc in the range of 0.2 ∼ 0.5 represents a good compromise. The fillet size can be controlled to some degree by the FIB or micro-machining parameters. The second important geometric factor that may affect the accuracy of the meah ). The aspect ratio of the posts has a relasurement is the post aspect ratio (α = 2r tively small effect on the output flow stress curves if the aspect ratio is larger than 2. Aspect ratios of less than 2 are not appropriate, since the output strain hardening of the post material then deviates from the input value due to the constraints from the post base. However, aspect ratios much larger than 2 can cause buckling (discussed later in this chapter and in Chapter 7).

3.4.3.2 Accuracy of Measurement of Elastic Response In a standardized mechanical experiment the machine compliance is eliminated by using extensometers or strain gages and the elastic modulus of the specimen material can then be calculated. For micro-compression, this type of compliance correction is not possible. Therefore, it is important to examine the factors that will affect the accuracy of elastic modulus measurement. Figure 3.8 displays the error of elastic modulus measurements as a function of the fillet size (the apparent elastic modulus is normalized by the known input modulus). The post aspect ratio is fixed at 2. It is apparent that increased fillet size improves the accuracy with which the elastic modulus is measured, but the apparent modulus still underestimates the input modulus. The reason is that the base will always deform along with the post, and this deformation contributes to the system compliance, which is not accounted for when collecting the displacement data. However, this error may be alleviated by accounting for the base compliance using a result from Sneddon. Sneddon computed the depth of penetration D of an elastic half-space by a flat-ended rigid cylinder of radius a under a load P as

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Normalized Modulus, E/Einput

1.4 Effect of fillet size ( = 2) Sneddon Correction Modified Sneddon Correction

1.3 1.2 1.1 1 0.9 0.8 0.7

(b) 0

0.3 0.6 0.9 1.2 Normalized Fillet Radius, rc /r

Fig. 3.8 Computed variation of apparent elastic modulus (normalized by true elastic modulus) with fillet radius size at fixed aspect ratio, using the finite element simulations discussed in the text. The effects of the Sneddon and modified Sneddon corrections are also shown. Reprinted from Scripta Materialia, Vol 54, Issue 2, page 6, H. Zhang, B.E. Schuster, Q. Wei and K.T. Ramesh, The design of accurate microcompression experiments, January 2006, with permission from Elsevier.

D=

P(1 − ν ) 4µ a

(3.5)

where µ and ν are respectively the shear modulus and Poisson’s ratio of the halfspace. We then obtain the base compliance (Zhang et al., 2006) as Cb =

1−ν 1 − ν2 = 4µ a 2Eac

(3.6)

where ac = η (r + rc ) is the effective contact radius, with η as a proportionality constant to be determined. The choice of a in Equation (3.6) is difficult because of the geometry of the post-base connection and we have set a = ac . By subtracting this base compliance from the total compliance, a corrected post stiffness (and hence modulus) is obtained. If η = 1 (the Sneddon correction), the modulus is overestimated. Setting η = 1.42 provides a significantly improved correction (the modified Sneddon correction). Figure 3.8 also includes both the Sneddon and modified Sneddon corrections. Similar results on the error in the modulus measurement are found with other aspect ratios. 3.4.3.3 Accuracy of Measurement of Plastic Response The accuracy of micro-compression measurements may be affected by the plastic flow properties (strain hardening and strain rate hardening) of the post material. Finite element simulations have shown that the microcompression technique is most accurate for materials that exhibit low strain hardening. Such materials include

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nano-structured metals, metals and alloys with ultrafine microstructure, metallic glasses, and ceramics. Strain rate effects are also important in many applications, especially when plastic instability is of serious concern. This is because the susceptibility to plastic instability of metals and alloys is strongly dependent on the strain rate sensitivity of the material. There are various ways to define the strain rate sensitivity. An engineering definition assumes a power-law strain rate hardenσ ˙ ing such that the strain rate sensitivity is written as m = ∂∂ln ln˙ε , where ε is the imposed strain rate. Finite element simulations assuming different values of m indicate that microcompression can capture the strain rate sensitivity of the material. Note that the practically useful strain rate achievable for microcompression with a nano-indenter is probably below 1 s−1 , i.e., in the quasistatic regime. 3.4.3.4 Buckling, Friction and Taper Effects Buckling, either elastic or plastic, is a common concern for a compression test. Elastic Euler buckling of a column occurs when the applied compressive load reaches a critical value given by Pcr =

π 2 EI Le 2

(3.7)

where I is the moment of inertia of the column (here the micropost) and Le is the effective length of the column (Le = 2h for the end conditions developed in microcompression). The applied load in the microcompression experiment must be at least large enough to cause yielding of the sample, so P > σY A where A is the crosssection area of the post. The minimum aspect ratio at which Euler buckling is likely to occur in microcompression can then be computed, and is found to be greater than 10 given the typical ratio of E/σY for most materials. Thus Euler buckling is not an issue in microcompression for aspect ratios less than 10. The plastic buckling problem must be addressed with 3D finite element analysis. Effects of friction between the indenter (usually made of diamond) and the post of top must also be considered, as must the effect of “taper” of the post and the system misalignment on the simulation output. In these 3D analyses the smallest fillet radius ratio used is rrc = 0.1 for α = 2. Both 3D cylindrical and cuboidal posts are considered. The compression is exerted by a rigid surface representing the flat-end nano-indenter tip. Misalignment of the experimental setup is considered by tilting the rigid surface with respect to the axis of the post. 3D simulations show that when there is no friction, aspect ratios of approximately 2 provide good measures of behavior and avoid buckling, while aspect ratios of 5 cause buckling after small overall strains. However, micro-compression experiments are usually performed without applying lubrication to the tip-post interface. A range of values for Coulomb friction coefficients exist in the open literature for a number of diamond-metal surface pairs: static coefficients of friction in the range of 0.1–0.15, and kinetic values of 0.03 ∼ 0.06. Friction between the indenter tip and the post top delays the plastic buckling

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of cylindrical posts; however, the friction does increase the estimated flow stress. The ease of development of plastic buckling is strongly dependent on the strain hardening of the material. Thus large aspect ratio posts should also be avoided due to the possible occurrence of plastic buckling when one tests low strain hardening materials: aspect ratios of 2 ∼ 3 are recommended. Another important geometric factor that may have a significant effect on the accuracy of microcompression experiments is the “taper” of the post. Taper is typically a result of fabrication inaccuracies. Taper can cause errors in the estimates of the modulus and yield stress and result in a spurious strain hardening response (due to propagation of the plastically deforming zone down the post axis). Taper should therefore be avoided by careful manipulation of the micro-post fabrication process. 3.4.3.5 Effect of Misalignment Misalignment of the system is manifested primarily in the deviation of the elastic part of the stress-strain curve, giving rise to an underestimate of the elastic modulus of the material. A decrease of the measured elastic modulus is observed with increase of misalignment (note that a lower elastic modulus will be observed even for perfect alignment is due to the compliance of the post base). In addition, excessive misalignment will result in buckling of the post. 3.4.3.6 Summary Recommendations for Microcompression Experiments 1. The post fillet radius affects both the accuracy of the microcompression test (in terms of the flow stress) and the stress concentration at the post root. Fillet radius/post radius ratios of 0.2 ∼ 0.5 are recommended to provide sufficient test accuracy. The fillet radius also has a profound effect on the accuracy of the elastic modulus due to the compliance of the base. This effect can be alleviated using a modified Sneddon correction to account for the base compliance. 2. Strain hardening of the material significantly affects the microcompression test accuracy, particularly when the fillet radius is large (close to the post radius). The results suggest that microcompression experiments are better suited for materials with low strain hardening. 3. For aspect ratios ≥ 5, first-mode plastic buckling is suppressed by the friction between the indenter tip and the post top, but higher-mode buckling may still occur, particularly for high aspect ratios and low strain hardening materials. Post aspect ratios of 2 ∼ 3 are therefore recommended. 4. Taper of the post results in overestimated elastic modulus, spurious strain hardening and increased apparent yield and thus should be minimized during micro-post fabrication. 5. Misalignment of the microcompression system significantly affects the test accuracy by underestimating the elastic modulus, and may result in buckling of the post. This should be minimized through careful post sample preparation, tip machining, and experimental alignment.

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3.4.4 Microtensile Testing Conventional tension testing is the standard approach to obtaining the mechanical response of materials (indeed, there are very well-developed ASTM standards for such testing). These standards involve the use of specially shaped specimens, designed to develop a uniaxial tensile stress along a specific gauge length and to obtain accurate measures of strain. The result is a stress strain curve in uniaxial tension (Figure 3.10) that is the basis of most constitutive modeling in the plastic range. However, conventional tension tests require specimens that are at least several centimeters long. As of 2008, there are very few nanomaterials which can be obtained in sizes sufficiently large to permit conventional tension tests. A number of research groups have therefore developed the ability to perform tension tests on very small samples (typically obtained from thin films). Sample sizes of interest are typically lengths of 10–1000 µm, widths of 10–100 µm, and thicknesses of 10 nm to 100 µm. There are three critical requirements for a good tension test. The first is the development of a significant region of the specimen (called the gauge length) within which a uniaxial tensile stress state exists; the second is the ability to measure that tensile stress accurately; and the third is the ability to measure the tensile strain along the direction of the stress within the gauge section. Finally, there is the question of actuation: how is the tension generated?

3.4.4.1 Stress State and Specimen Design The first requirement determines the design of the specimen. The primary complexity in tension tests arises from the need to apply the tensile force to the specimen, which requires the use of some type of grip. Many, many grip types are used in conventional tension testing, ranging from hydraulic and pneumatic grips to threaded ends and adhesively bonded ends. The stress state at the grip is very complicated, and so careful specimen design is necessary to obtain a gauge length that is primarily under uniaxial tension. An important issue is the degree of stress concentration near the grips, which is particularly important for high-strength and relatively brittle materials like nanomaterials. For the case of materials with a significant degree of plastic deformation, a very common specimen design that provides a gauge length containing nearly uniaxial tensile stresses is obtained with the “dogbone” shape of conventional tension testing. Very good computational design of tensile specimens can be performed using the finite element method, accounting for the expected material behavior, and so it is no longer necessary to rely on a standard to ensure that reasonable results are being obtained. When it comes to testing very small nanomaterials samples, the total forces that are developed are small and so unique grip designs are often used that are mated to the specimen shapes that can be easily fabricated. There is also a strong correlation between the fabrication or machining techniques used to make the nanomaterial and the tension test sample design. Many micro-tension specimens end up being essentially thin films, and so many of the techniques associated with thin-film applications are relevant to such experiments.

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3.4.4.2 Accurate Measurement of Stress The second requirement (that of determining the tensile stress within the gauge length of the specimen) is relatively easy to satisfy, because the conditions of force equilibrium demand that the force along the force chain in the testing technique be constant. Thus the simple insertion of a load cell into the force chain provides a measure of the force in the specimen, and then knowing the cross-section area of the specimen allows one to calculate the tensile stress in the sample gauge length. The only difficulty involves ensuring that the load cell is sufficiently accurate and has the resolution to measure the small forces typically developed in small samples. A rough estimate of the force that would be generated in the specimen can be obtained as σy A where σy is the yield stress of the material and A is the cross-section area of the specimen, which scales as the square of the specimen width (A ∝ b2 ). Thus for a nanocrystalline material with a yield strength of 1 GPa and a specimen width of 10 µm, the force that must be measured is of the order of 0.1N. In the elastic range, the force scales as E ε A, and so to have a strain resolution of ε = 10−4 and a Young’s modulus of 100 GPa one needs a force resolution of 1 mN.

3.4.4.3 Accurate Measurement of Strain The biggest difficulty in microscale tension experiments is accurate measurement of the strain in the specimen. In traditional tension testing, strain is measured in one of two ways. The simplest and least accurate is to measure the displacement δ l of the crosshead (or actuator) and divide this by the specimen gauge length l0 :

ε=

δl . l0

(3.8)

A much more accurate approach involves measuring the strain directly on the specimen using some kind of in situ strain measurement technique such as a strain gauge. However, it is difficult to find strain gauges that are of the appropriate size for most micro sample tests. Approaches that are currently used in microsample testing include digital image correlation or DIC (Chu et al., 1985) together with in situ imaging using an optical microscope, SEM, or TEM; the measurement of the displacement of fiducial markers placed on the specimen, and specialized techniques such as the interferometric strain/displacement gauge (ISDG). A simple measure of the difficulty involved can be obtained by recognizing that the displacements that must be measured are of the order of ε l0 , and so for a specimen length l0 of the order of 100 µm and a strain resolution of ε = 10−4 one needs to measure a displacement of δ u = 10 nm. Such displacement resolutions are difficult to achieve.

3.4.4.4 Actuation Systems for Micro-Tensile Testing Uniaxial tension experiments are typically performed either under load control or under displacement control (that is, one controls the load while the displacement is

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measured as the output of the system, or vice versa). A simple (and idealized) example of a load controlled test is a tension test performed using dead weights to provide the load. The actuator system used to generate tension in conventional tension tests is typically either a servohydraulic system or a screw driven system. These systems are capable of carrying very high loads, and control systems for such devices are highly developed. However, such systems are not appropriate for the small loads and small displacements that must be used to study microscale specimens: the excursions in force or displacement that are developed in these conventional systems before full control is established are typically larger than the forces or displacements σ required to cause failure of the microscale specimens. Initial displacements of Ey l0 σ are sufficient to cause yield of the specimen, and since for most materials Ey ∼ 10−3 , it takes only 100 nm of displacement to cause yield in a specimen that is 100 µm long. An actuator with extreme control of displacement is therefore necessary for micro-tensile testing. Examples of actuators with such control include piezoelectric actuators and MEMS-based actuator systems (Figure 3.9). The advantage of piezoelectric actuator systems is that they can be purchased rather than having to be fabricated.

Fig. 3.9 Schematic of a microtension testing apparatus, showing the actuation and sensing systems. The air bearing is an important component. The strain is computed from displacement fields measured using Digital Image Correlation (DIC) software analysis of images from the camera. Illustration due to Chris Eberl.

In most tensile tests, the compliance of the testing machine is a significant factor in the design of the experiment. Under the load, some of the displacement that is observed in the system is a result of the deformation of the machine because of machine compliance. There are standard approaches to correcting for machine compliance so that accurate measurements of specimen strain can be obtained, at least for conventional tension testing. This can be particularly important for micro-tensile testing.

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The result of a microtensile experiment on a nanocrystalline nickel film is shown in Figure 3.10. The specimen was a thin film made by electrodeposition (an Integran material), with a film thickness of 100 nm and a mean grain size of 20 nm. Note the significant flow stress that is obtained, and the unload-reload cycle used to estimate the modulus (to minimize machine compliance issues). The apparent strain hardening that is observed early in the stress-strain curve will be discussed later in the mechanical properties chapter.

True Stress, MPa

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0.07

True Strain Fig. 3.10 Example of stress strain curve obtained on nanocrystalline nickel (20 nm grain size) in uniaxial tension using a microtension setup. The sample was a thin film. Data due to Shailendra Joshi.

A remarkable variety of nanomechanical and micromechanical tensile testing systems have been developed over the last decade (Sharpe et al., 1997; Espinosa et al., 2003; Legros et al., 2000; Yi et al., 2000; Haque and Saif, 2001; Srikar and Spearing, 2003; Jonnalagadda and Chasiotis, 2008). Several of them have been developed with an eye towards thin-film applications rather than testing nanomaterials, but some of them can be used as nanoscale mechanics characterization techniques with application towards nanomaterials. Since these techniques involve multiple innovations and some uncertainty about accuracy and repeatability, it is important to compare the results of these experiments with the results of other experimental techniques on similar materials. Such comparisons are difficult because the research groups that operate such techniques typically do not work on identical materials, and so it is unclear whether the differences in behavior that are observed are the result of experimental technique variations or the material variations. Note, in particular, that comparisons of nanomaterial behavior measured using these systems with the result of macroscale experiments on large-sample coarse-grain materials may not be meaningful, because of the potential differences in materials processing routes and specimen fabrication.

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3.4.4.5 Summary Recommendations for Micro-Tensile Experiments 1. Finite element methods should be used to accomplish specimen design, with the objective being to optimize the specimen design with respect to the mechanical property that must be measured (for example, optimal design for elastic modulus measurement would be distinct from optimal design for strength measurement). 2. For minimum specimen dimensions larger than several microns, piezoelectric actuators systems provide a good compromise between accuracy and controllability. 3. In situ strain measurement techniques such as digital image correlation are recommended, and are essentially required if computational design of the specimen has not been performed. 4. Ideally, experimental results with any new technique should be compared with the results of other experimental techniques on a standard material in order to obtain confidence in the particular experimental technique that is being used. Unfortunately, at this point (2008) it is not possible to recommend a specific standard material.

3.4.5 Fracture Toughness Testing The fracture toughness of nanomaterials is extremely difficult to measure. Standard fracture toughness measurement techniques require standard specimen sizes, all of which are too large to be used with most nanomaterials (as of 2008). Indeed, it is not clear that the concept of fracture toughness as it is normally defined is relevant to most small-scale systems. Thus, most nanomaterials are most easily obtained in thin-film form, and so plain strain fracture toughness is not a sensible concept. Further, it is not clear that there is a significant K-dominant (see Chapter 2) domain within a nanomaterial containing a crack. That said, estimates of fracture toughness are needed in order for materials to be used within the design of complex components, and it is important to have at least a theoretical understanding of how fracture toughness should change with grain size. It is often stated that nanomaterials are brittle, but what is typically meant is that the tensile ductility of a specific nanomaterial specimen is low. The two existing approaches to fracture toughness testing with nanomaterials are (a) examining the failure of a thin film containing a pre-crack and (b) bending techniques developed for understanding thin film behavior. We explore these techniques further in the chapter on the failure of nanomaterials, because these techniques show material behavior that is intimately coupled to test design.

3.4.6 Measurement of Rate-Dependent Properties The primary experimental techniques associated with the measurement of the rate dependent properties of materials are described in Figure 3.11 (note that the stress

10–6

10–4

10–2

100

102

Strain Rate,

104

87

Pressure-shear plate impact

Miniaturized Kolsky bar

Conventional Kolsky bars

Specialized machines

Servohydraulic machines

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106

108

s–1

Fig. 3.11 Experimental techniques appropriate for various ranges of strain rate. The range of rates identified as “specialized machine” is very difficult to reach. Very few laboratories in the world are able to achieve strain rates higher than 104 s−1 .

states developed within the various techniques are not necessarily identical). An excellent and relatively recent review of these methods is presented by Field et al. (2004), and a summary of the key experimental issues is presented by Ramesh in a chapter in the Handbook of Experimental Mechanics (see suggested reading). For the purposes of this discussion, strain rates above 102 s−1 are classified as high strain rates, strain rates above 104 s−1 are called very high strain rates, and strain rates above 106 s−1 are called ultra-high-strain-rates. Conventionally, strain rates at or below 10−3 s−1 are considered to represent “quasistatic” deformations, and strain rates below 10−6 s−1 are considered to be in the “creep” domain. This is an important range for some nanomaterials. Creep experiments are typically performed at relatively high temperatures, and a variety of specialized machines exist for these kinds of loading; dead loads are often of particular interest. Quasi static experiments are typically accomplished through a variety of servohydraulic machines, and ASTM standards exist for most of these experiments. Most servohydraulic machines are unable to develop strain rates larger than 100 s−1 repeatably, but some specialized servohydraulic machines can achieve strain rates of 101 s−1 . We focus here on the higher strain rates (greater than 102 s−1 ).

3.4.6.1 Split-Hopkinson or Kolsky Bars The now-classical experimental technique in the high-strain-rate domain is the Kolsky bar or split-Hopkinson pressure bar (SHPB) experiment for determining the mechanical properties of various materials (e.g., Nicholas and Rajendran, 1990;

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Ravichandran and Subhash, 1994; Walley and Field, 1994 in metals, ceramics, and polymers respectively) in the strain rate range of 102 s−1 to 8 × 103 s−1 . The terms “split-Hopkinson pressure bar” and Kolsky bar are often used interchangeably, although the latter is more appropriate (the technique was invented by Kolsky, and neither John nor Bertram Hopkinson used a split-bar to study material behavior). One should note also that the term “split-Hopkinson pressure bar” implies the performance of compression experiments, whereas the term “Kolsky bar” is more general and may include compression, tension, torsion or combinations of all of these. Since the fundamental concept involved in this technique was developed by Kolsky (1949), we will use the term Kolsky bar. A schematic of the typical compression Kolsky bar experimental apparatus is shown in Figure 3.12. The device consists of two long bars (called the input and output bars) that are designed to remain elastic throughout the test. These bars sandwich a small specimen (usually cylindrical), which is expected to develop inelastic deformations. The bars are typically made of high-strength steels such as maraging steel, with a very high yield strength and substantial toughness. One end of the input bar is impacted by a projectile made of a material identical to that of the bars; the resulting compressive pulse propagates down the input bar to the specimen. Several reverberations of the loading wave occur within the specimen; a transmitted pulse is sent into the output bar and a reflected pulse is sent back into the input bar. Typically, resistance strain gages are placed on the input and output bars for measuring (i) the incident pulse generated by the impacting projectile, (ii) the reflected pulse from the input bar/specimen interface, and (iii) the transmitted pulse through the specimen to the output bar. The strain gage signals are typically measured using high-speed digital oscilloscopes with at least 10-bit accuracy and preferably with differential inputs to reduce noise.

Fig. 3.12 Schematic of the compression Kolsky bar (also known, incorrectly, as the splitHopkinson pressure bar). The projectile is usually launched towards the input bar using a gas gun. Specimen surfaces must be carefully prepared for valid experiments.

Let the strain in the incident pulse be denoted by εI , that in the reflected pulse by εR , and that in the transmitted pulse by εT (these are bar strains as measured by the strain gages). By assuming stress equilibrium, uniaxial stress conditions in the specimen and 1D elastic stress wave propagation without dispersion in the bars, the nominal strain rate e˙s , nominal strain es , and nominal stress ss (all in the specimen) can be estimated using the following equations:

3 Nanoscale Mechanics and Materials: Experimental Techniques

2cb εR (t) l0

(3.9)

e˙s (τ )d τ

(3.10)

Eb Ab εT (t) As

(3.11)

e˙s (t) = − es (t) =

89

t 0

and ss (t) =

where cb = Eρb is the wavespeed in the bar, l0 is the initial length of the cylinb drical specimen, Eb is the Young’s modulus of the bar material, ρb is the density of the bar material, Ab is the bar cross-section area, and As is the specimen initial cross-section area. Note the negative sign in Equation (3.9). This arises because the strain in the reflected pulse has the opposite sign of the strain in the incident pulse; the latter is compressive and is conventionally considered positive in these experiments. Thus the specimen strain rate expressed by Equation (3.9) is a positive strain-rate, i.e., in this convention, a compressive strain-rate. The true strain and true strain rate in the specimen are then given by Ramesh and Narasimhan (1996):

εs (t) = − log (1 − es (t)) and

ε˙s (t) =

e˙s (t) . 1 − es (t)

(3.12)

(3.13)

Note that compression is defined as positive. Assuming plastic incompressibility, the true stress σs in the specimen is obtained as

σs (t) = ss (t)(1 − es (t)).

(3.14)

From Equations (3.12) and (3.14), a stress-strain curve is obtained at a strain rate defined by an average taken over the strain rate history obtained in Equation (3.13); an example of such a stress-strain curve is presented in Figure 3.13. Several fully miniaturized versions of the compression Kolsky bar have been developed (e.g., see Jia and Ramesh 2004). In the Jia and Ramesh version (see Figure 3.14), the bars are 3 mm in diameter and 300 mm long, and may be made of maraging steel or tungsten carbide. Sample sizes are cubes or cylinders on the order of 600 µm to 1 mm on a side; cube specimens are used when the failure mode must be imaged using a high-speed camera, or when the amount of material is so small that only cuboidal specimens can be cut. The technique is simple, and the entire system can be designed to fit on a desktop, and so the technique is sometimes referred to as the “desktop Kolsky bar.” The miniaturized Kolsky bar or “desktop Kolsky bar” experimental technique is particularly effective when it comes to understanding the properties of nanostructured materials, or of materials with very fine grain sizes.

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Fig. 3.13 Stress strain curves obtained on 5083 aluminum using the compression Kolsky bar at high strain rates (2500 per second). The lowest curve represents quasistatic behavior. In general, the strength appears to increase with increasing strain rate, but note the anomalous softening at the highest strain rate (perhaps due to thermal softening).

Fig. 3.14 The Desktop Kolsky Bar – a miniaturized compression Kolsky bar arrangement developed by Jia and Ramesh (2004). This device is capable of achieving strain rates above 104 per second, and can fit on a standard desktop. Specimen sizes can be on the order of a cubic millimeter. The large circular object behind the bar is part of a lighting system.

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3.5 Suggestions for Further Reading 1. K.L. Johnson, Contact Mechanics, Cambridge University Press, Cambridge, 1987. 2. W.N. Sharpe, Jr., Handbook of Experimental Mechanics. Springer, New York, 2008. See, in particular, the chapter on High Strain Rate and Impact Testing by K.T. Ramesh. 3. B. Bhushan, Springer Handbook of Nanotechnology. Springer, New York, 2007.

3.6 Problems and Directions for Research 1. Determine the resolution limits of the transmission electron microscope and scanning electron microscopes in your institution. Will they be able to resolve a 10-nm grain of nanocrystalline aluminum? Can you see a carbon nanotube in your SEM? 2. Look up the ASTM standard for testing hard materials (determining the yield strength). Determine the size of specimen that would be required to meet the standards if the specimen were to be made of a nanocrystalline aluminum that has a yield strength of 1 GPa. 3. What is the smallest force that can be measured in one of the atomic force microscopes in your institution (or the nearest AFM, if your institution does not possess one)? Can you measure the force required to develop a strain of 1% in a single-walled carbon nanotube of Young’s modulus 1 TPa, assuming elasticity? 4. Perform a tensile stress-strain test on any aluminum alloy in a servohydraulic testing machine, and then determine how to correct for the machine compliance. Provide a corrected tensile stress-strain curve.

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From the many, one. – A liberal translation of E Pluribus Unum

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Mechanical Properties: Density and Elasticity Should nanomaterials have properties that are any different from conventional materials? Since we have defined nanomaterials as materials where some controllable relevant dimension is of the order of 100 nm or less, there should not be a difference in properties unless the property of interest is affected by some size scale. A big part of understanding the properties of nanomaterials will therefore be developing a