MATERIALS SCIENCE SERIES EDITORS ALLEN M. ALPER JOHN L. MARGRAVE A. S. NOWICK GTE Sylvania Inc. Department of Chemistry Henry Krumb School Precision Materials Group Rice University of Mines Chemical & Metallurgical Houston, Texas Columbia University Division New York, New York Towanda, Pennsylvania Anelastic Relaxation in Crystalline Solids A. S. N O W I CK and B. S. B E R RY Henry Krumb School of Mines Thomas J. Watson Research Center Columbia University IBM Corporation New York, New York Yorktown Heights, New York 1972 A C A D E M IC P R E SS New York and London COPYRIGHT ' 1972, BY ACADEMIC PRESS, INC. ALL RIGHTS RESERVED NO PART OF THIS BOOK MAY BE REPRODUCED IN ANY FORM, BY PHOTOSTAT, MICROFILM, RETRIEVAL SYSTEM, OR ANY OTHER MEANS, WITHOUT WRITTEN PERMISSION FROM THE PUBLISHERS. ACADEMIC PRESS, INC. Ill Fifth Avenue, New York, New York 10003 United Kingdom Edition published by ACADEMIC PRESS, INC. (LONDON) LTD. 24/28 Oval Road, London NW1 7DD LIBRARY OF CONGRESS CATALOG CARD NUMBER: 70-154378 PRINTED IN THE UNITED STATES OF AMERICA To Joan and Maureen Preface Anelasticity may be said to have originated as a distinct discipline in 1948 with the publication of Zener's pioneering monograph "Elasticity and Anelasticity of Metals/' This book defined anelasticity and demon- strated the usefulness of separating it from the more general viscoelastic behavior treated in the earlier literature. In the years since Zener's book, an enormous growth in the literature of the subject has taken place, as is became apparent that the study of anelastic relaxation could contribute greatly to our understanding of almost the whole gamut of physical and chemical phenomena in crystalline solids. Accordingly, there has been a clear need both for a textbook and an up-to-date monograph on anelasticity in crystals. In writing the present book the authors have hoped, perhaps too ambitiously, to fulfill both requirements. The first six chapters comprise the formal theory of the subject, the heart of which is covered in Chapters 3-5. It is this theory that provides the glue which holds together the diverse topics to be covered later. In this sense, anelasticity differs from the related topic of "internal fric- tion," which does not possess such a unifying theoretical base. These first six chapters, together with selected readings from the later chapters, can constitute a text for self study, or for a graduate course such as one of the authors (A.S.N.) has given at Columbia University. These early chapters are by no means confined to crystals, and are therefore equally suitable as an introduction to anelasticity in the noncrystalline state, or, with simple changes in notation, to dielectric relaxation. To further aid the usefulness of the book as a text, we have designated with heavy as- terisks those sections which may be omitted on a first reading. Also, a rather substantial number of problems are included in the early chapters. The remainder of the book deals in some detail with the various phys- ical phenomena in crystalline substances which can give rise to anelastic relaxation. In view of the fact that the book is intended for readers with diverse backgrounds, such as mechanical or metallurgical engineers, materials scientists, and solid-state physicists, we have felt it desirable xiii xiv PREFACE to present the essential background material on each of the phenomena covered. In the case of the topics dealt with most extensively, namely point defects and dislocations, full chapters (7 and 12) of background material are presented; for other topics, a single section of background material seemed to be sufficient. For each of the physical phenomena, we have attempted to describe the current state of understanding of the anelastic effects in such a way that the reader can then go directly to the literature. With the enormous growth of the literature in the past decade, however, we regret that a complete or exhaustive literature survey could not be attempted, and that often work of some significance had to be omitted. Phenomena which involve ultrasonic attenuation or internal friction but which are not strictly describable as anelastic effects have generally been omitted, although (in Chapters 14 and 17-19) some bor- derline topics are discussed briefly because of their special importance. Finally, due to the limitations of the book as well as our own backgrounds, we have omitted consideration of organic crystalline compounds. Acknowledgments The authors owe a debt of gratitude to many colleagues and students who read individual chapters and contributed valuable criticisms and suggestions, particularly to D. N. Beshers, C. Elbaum, M. C. Franzblau, C. W. Garland, R. Gibala, D. F. Gibbons, W. R. Heller, G. M. Leak, W. P. Mason, D. I. Paul, D. S. Richter, J. T. A. Roberts, Ε. T. Ste- phenson, J. C. Swartz, T. J. Turner, and G. P. Williams. We are espe- cially grateful for the support of much of the research leading to this book, as well as partial support of the writing itself, by the U. S. Atom- ic Energy Commission. We thank Miss Maureen Weaver for her skill- ful typing of a substantial fraction of the manuscript. Finally, we wish to express our deepest gratitude to our wives for their encouragement and understanding over the several years during which this book was being written. XV Chapter 1 / Characterization of Anelastic Behavior The first part of this book (Chapters 1-6) deals with the formal theory of anelasticity. We begin with the introduction of a set of postulates which serve to define the subject, and then proceed to develop the con- sequences of these postulates. The formal theory does not include the physical origins or atomistic mechanisms of anelasticity, but does em- brace the interpretation of anelastic behavior as a manifestation of in- ternal relaxation processes. The function of the present chapter is to lay the groundwork for the formal theory, first, by introducing the postulates, and second, by de- ducing the characteristic response of an anelastic material to the imposi- tion of certain simple histories of stress or strain. The response functions which emerge from this treatment are then taken as the basic manifesta- tions of anelastic behavior. These objectives are, in fact, accomplished in Sections 1.1-1.3, so that the reader interested in the purely theoretical development of the subject may omit the remainder of the chapter. In Sections 1.4-1.7 we turn to the various dynamical methods that are widely used in studying anelasticity. These methods are of such im- portance in practice that we have felt it necessary to present them here, even though the material constitutes a digression from the main develop- ment of the theory. 1.1 The Meanin g of Anelasticit y In order to describe anelasticity in a formal way, it is convenient to consider first an ideal elastic material, for which Hooke's law defines the relation between stress a and strain (cid:229) as (cid:243) = (cid:204)(cid:229) (1.1-1) or (cid:229) =Ja (1.1-2) with M=\\J (1.1-3) 1 2 1 CHARACTERIZATION OF ANELASTIC BEHAVIOR The constant (cid:204) is called the modulus of elasticity (or often just the modulus) while its reciprocal / is called the modulus of compliance (or simply the compliance). For an arbitrary deformation, the stress and strain must be expressed as second-order tensors, and Hooke's law then becomes a set of linear equations expressing each component of the stress tensor in terms of all the components of the strain tensor (or vice versa). For present purposes, this generalization would merely add unnecessary complications, so we shall proceed from the statement of Hooke's law as given by Eqs. (1.1-1) and (1.1-2), leaving the more elaborate treatment to Chapter 6. We note however that the present approach implies a simple mode of deformation, such as pure shear, uniaxial deformation, or hydrostatic deformation. For such cases, the appropriate modulus (cid:204) will be the shear modulus, Young's modulus, or the bulk modulus, respectively. There are three conditions defining ideal elastic behavior which are implicit in Eqs. (1.1-1) and (1.1-2). These are: (1) that the strain response to each level of applied stress (or vice versa) has a unique equilibrium value; (2) that the equilibrium response is achieved instantaneously*; (3) that the response is linear (e.g., doubling the stress doubles the strain). It should be noted that a characteristic feature of elasticity, namely the complete recover ability of the response upon release of the applied stress or strain, is a corollary of condition (1). In order to generalize upon ideal elastic behavior, the three conditions listed above may be lifted in various combinations. The possibilities are shown in Table 1-1 together with the name given to the discipline of study in each case. As already mentioned, if all three conditions apply, we have the case of ideal elasticity. If only the restriction of linearity is lifted, "nonlinear elasticity" is the obvious result. If both linearity and complete recoverability are dropped, the material becomes capable of exhibiting "instantaneous plasticity," which is often called "crystal plas- ticity" because this type of plasticity is best exemplified by the yielding of ductile crystals under high stress. In contrast to the first three entries in Table 1-1, the remaining two entries introduce time dependence into the response, by lifting the condition of instantaneity. If this is the only condition lifted, we produce the type of behavior known as anelasticity, the subject of this book. The more general behavior obtained by addition- + Because of the finite velocity of sound, the response is actually instantaneous only in an infinitesimally small sample. The essential point is that Eqs. (1.1-1) and (1.1-2) do not contain time as a variable. 1.1 THE MEANING OF ANELASTICITY 3 TABLE 1-1 DIFFERENT TYPES OF MECHANICAL BEHAVIOR, CLASSIFIED ACCORDING TO THE CONDITIONS OBEYED BY THE STRESS-STRAIN RELATIONSHIP Unique equilibrium relationship Instantaneou s Linear (complete recoverability ) Ideal elasticity Yes Yes Yes Nonlinear elasticity Yes Yes No Instantaneou s plasticity No Yes No Anelasticity Yes No Yes Linear viscoelasticity No No Yes ally lifting the condition of complete recoverability is known as linear viscoelasticity, which thus includes anelasticity as a special case. To summarize the definition of anelasticity, we may employ the fol- lowing three postulates: 1. For every stress there is a unique equilibrium value of strain, and vice versa. 2. The equilibrium response is achieved only after the passage of sufficient time.+ 3. The stress-strain relationship is linear. The first and third postulates are, of course, merely a repetition of those for ideal elasticity. It should again be recalled that complete re- coverability is a corollary of postulate (1), only now the recovery will, in general, be time dependent. It should be noted that lifting the condition of instantaneity does not imply that all of the response of an anelastic material must develop in a time dependent manner, since to do so would produce the special (and unrealistic) case of a material without any component of elastic behavior. Rather, anelasticity implies that, in addition to an instantaneous (elastic) response, there also exists a time dependent nonelastic response.* Another point of clarification is concerned with the meaning of the term "linear" + There is no restriction on the time scale in these considerations , i.e., the achievement of equilibrium may require anything from microseconds (or less) to extremely long periods of time. t Thus, while the term "anelasticity " carries the connotation of being "without elasticity," this meaning applies in a literal sense only to a fraction (and often, a very small fraction) of the total response.