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PANEL ON MICROMECHANICS AND PHYSICS OF FRACTURE

PANEL MEMBERS

Professor Ali S. Argon Department of Mechanical Engineering Massachusetts Inst i tute of Technology Cambridge, MA 02139 U.S.A.

Professor Robert Asaro Division of Engineering Brown University Providence, RI 02912 U.S.A.

Dr. Michael I. Baskes Theoretical Division Sandia National Laboratories Livermore, CA 94550 U.S.A.

Professor John L. Bassani Department of Mechanical Engineering and

Applied Mechanics l l l A Towne Building D3 University of Pennsylvania Philadelphia, PA 19104 U.S.A.

Professor Howard K. Birnbaum Department of Metallurgy and Mineral Engineering University of Illinois 1304 West Green Street Urbana, IL 61801 U.S.A.

Professor Anders E. Carlson Department of Physics Washington University St. Louis, MO 63130 U.S.A.

Dr. James R. Chelikowsky Materials and Chemical Theory Exxon Research and Engineering Corporation Clinton Township Route 22 East Annandale, NJ 08801 U.S.A.

Professor William A. T. Clark Department of Metallurgical Engineering The Ohio State University Columbus, OH 43210 U.S.A.

Dr. Ronald Gronsky Department of Materials Science and Mineral

Engineering University of California Berkeley, CA 94720 U.S.A.

Professor John W. Hutchinson Division of Applied Sciences Harvard University Cambridge, MA 02138 U.S.A.

Dr. Masao Kuriyama Metallurgy Division National Bureau of Standards Materials Building 223 Gaithersburg, MD 20899 U.S.A.

Professor Che-Yu Li Department of Materials Science and Engineering Bard Hall Cornell University Ithaca, NY 14853 U.S.A.

Professor Charles J. McMahon, Jr. Department of Materials Science and Engineering University of Pennsylvania 3231 Walnut Street Philadelphia, PA 19104 U.S.A.

Dr. David Pett ifor Department of Mathematics Imperial College of Science and Technology Huxley Building Queen's Gate London SW7 2BZ U.K.

Professor James R. Rice Division of Applied Sciences Harvard University Cambridge, MA 02138 U.S.A.

Dr. Manfred Riihle Max-Planck-Institut fiir Metallforschung Inst i tut fiir Werkstoffwissenschaften Seestrasse 92 7000 Stuttgart 1 F.R.G.

Present address: Materials Engineering Program University of California Santa Barbara, CA 93107 U.S.A.

Dr. R. Bruce Thompson (U.S. Department of Energy) Ames Laboratory Ames, IA 50011 U.S.A.

Professor Vaclav Vitek (Chairman) Department of Materials Science and Engineering University of Pennsylvania 3231 Walnut Street Philadelphia, PA 19104 U.S.A.

Dr. Haydn Wadley Metallurgy Division, Room A163 Materials Building 223 National Bureau of Standards Gaithersburg, MD 20899 U.S.A.

Professor Julia R. Weertman Department of Materials Science and Engineering Northwestern University Evanston, IL 60201 U.S.A.

Dr. Calvin White Metals and Ceramics Division Oak Ridge National Laboratory Post Office Box X Oak Ridge, TN 37381 U.S.A.

Present address: Department of Metallurgical Engineering Michigan Technological University Houghton, MI 49931 U.S.A.

PANEL ADVISOR

Dr. Joseph Darby Division of Materials Science Office of Basic Energy Sciences Department of Energy Washington, DC 20545 U.S.A.

Materials Science and Engineering, 94 (1987) 7-8 7

EXECUTIVE SUMMARY

V. VITEK (PANEL CHAIRMAN)

Department o f Materials Science and Engineering, University o f Pennsylvania, 3231 Walnut Street, Philadelphia, PA 19104 (U.S.A.)

In this report a summary is given of the discussions and recommendations of a panel convened under the auspices of the Council of Materials Sciences for the Division of Materials Sciences, U.S. Department of Energy, to review the present status and future trends of basic research on the fracture of crystalline materials with an emphasis on metallic materials. The at tent ion was focused on the microscopic processes controlling the fracture behavior of materials. The more traditional fracture mechanics problems dealing with macroscopic aspects of fracture were not considered by the panel. However, at tempts were made to assess the impact of understanding of the micromechanisms of fracture on the predictions of materials reliability.

While the result of fracture is always a catastrophic failure of the material, the mechanisms involved in the fracture process can vary greatly for different materials and may be different under different circumstances. A variety of fracture phenomena, such as brittle cleavage and interfacial cracking, creep fracture and stress corrosion cracking, have been identified. Each of these is a very complex phenomenon often involving simultaneous operation of several physical processes, such as decohesion, localized dislocation motion, diffusion and local chemical changes. Hence, a comprehensive study of these processes requires an interdisciplinary approach involving materials science, solid state physics and chemistry, and solid mechanics. Accordingly, the panel was chosen to represent all these fields and an at tempt was made to define the required contributions from these different fields to achieve the common goal of understanding the microscopic phenomena and basic material properties controlling the propensity to fracture.

The report is divided into four chapters, each dealing with a different aspect of the research discussed here. However, these sec-

tions are not exclusive, and the same phenomena are frequently discussed more than once, although from different points of view. The conclusions and recommendations of this panel are presented in detail in the individual parts of the report and only the most important general conclusions are summarized here. In Chapter 1 the micromechanics of fracture are dealt with and the mechanics of microcracking, cavitation and strain localization, including consideration of constitutive behavior, are concentrated on. The important point made throughout this chapter is that the primary role of micromechanics studies is to connect the understanding of the atomic level processes with microscopic fracture phenomena. In Chapter 2 microstructural aspects of fracture are dealt with and various embrit t lement phenomena in both disordered alloys and ordered intermetallic compounds are concentrated on. It is emphasized here that an integral part of basic fracture research should be studies of interfacial properties, of dislocation behavior, and, in general, of those crystal defects which play an important role in the cracking process. It was concluded in both these chapters that more experiments using model materials are needed in order to elucidate in a systematic and well~lefined way both the physical and the micromechanical phenomena of fracture.

A substantial part (Chapter 3) of the report is devoted to recently developed experimental techniques which can be used to study the micromechanisms of fracture. These include acoustic emission, ultrasonic and electromagnetic measurements, and neutron and X-ray scattering. These methods have not been employed traditionally in fundamental research of micromechanisms of fracture but it was concluded that they have been developed now to such a level of sophistication that they can be applied successfully. An experimental technique not reviewed here is transmission

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electron microscopy, which has played a very significant role in studies of interfaces and lattice defects and is, therefore, also of primary importance in studies related to fracture phenomena. However, the possibilities as well as limitations of this technique have been discussed in a number of publications and it is fully recognized to be one of the most important research tools in materials science in general.

In the final chapter of the report, Chapter 4, the theory of interatomic forces which is a

basic precursor for atomistic studies of processes and phenomena relevant to fracture is discussed. It is concluded here that recent advances based on the local density approximation, together with the availability of very fast computers, have made it possible to advance qualitatively the atomistic studies of fracture phenomena and related lattice defects, such as interfaces. Hence, significant theoretical developments in understanding the propensity to fracture from the atomistic point of view are envisaged in the near future.


CHAPTER 1

MECHANICS AND MICROMECHANICS OF FRACTURE

1. INTRODUCTION

From its beginnings as a means of understanding the tensile failure of brittle materials, fracture mechanics has blossomed into a field of considerable breadth and importance. The subject is concerned with failure by cracking of materials (structural, geophysical, biological etc.) under a wide variety of loadings and environments. Applications range from the microscale where crack sizes may be a small fraction of a millimeter to earthquake rupture where faults are many kilometers in extent. Fracture mechanics is now being actively developed and applied around the world. Hardly a week goes by wi thout some new cracking problem reaching the front pages of the nation's newspapers. The present panel reflected the growing interest in fracture from a broad communi ty of engineers and scientists. This chapter of the report on promising research areas in this field will focus on possibilities which are fairly closely tied to the development of engineering fracture mechanics, i.e. the aspects of fracture used to characterize material fracture resistance and to predict lifetimes and load-carrying capacities of structural components . Examples will be given which demonstra te the successes and potential in coupling a detailed characterization of the processes of material separation with rigorous (often numerical) micromechanics and macromechanics models.

The scientific field concerned with the mechanics of materials, especially relating to the inelastic deformat ion and fracture of solids, has developed rapidly over the past decade. During this period, new perceptions and major advances have been made in important areas such as (1) non-linear fracture mechanics, especially in relating effects of microstructure and material properties to crack tip mechanics and fracture toughness [1], (2) the collapse and buckling of structures, especially in under-

standing how material constitutive behavior influences the development of non-uniform and localized deformat ion patterns which serve as failure modes and thus limit toughness and ducti l i ty [2], (3) the s tudy of localized inelastic deformation, both as a result of material damage by microvoid and microcrack initiation and as an inherent feature of the plastic deformat ion process itself [ 3 ], (4) the study of atomistic processes of fracture, with special regard to the influences of crystal symmetry and alloy chemistry (including environments) on the basic physics of material separation [4] and (5) the incorporation of micromechanics and microstructural effects into continuum models.

These research areas have progressed through coordinated programs involving both experiment and analysis, an approach which should be strongly encouraged in future research. The successful theoretical work has been done with careful at tention to mathematical, and more recently computational , rigor. This, in particular, has led to several notable examples where the phenomenology of complex deformation and fracture processes has been described theoretically in sufficient detail to serve as a guide for suggesting and interpreting critical experiments. The suggested experiments thereby become more definitive and in turn have a more direct influence on the formulation of new models.

2. CURRENT RESEARCH TRENDS

As a background, we note that the stage was set for the modern development of engineering fracture mechanics in all its aspects by Irwin, who in the 1950s and early 1960s reinterpreted and extended Griffith's classical work of the 1920s on brittle materials. Irwin's new way of approaching fracture permit ted engineers and metallurgists to apply fracture

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mechanics to relatively brittle structural materials such as high strength metal alloys. A precise measure of fracture toughness was one of the early successes of the new approach. Irwin, together with the growing fracture mechanics community, quickly extended the new approach to deal with fatigue cracking, a problem which had then begun to plague the aircraft industry and which still today presents many challenges.

The early methods and concepts were based on the theory of linear elasticity and were limited to applications in which plastic (non- linear} deformations were confined to the immediate neighborhood of the crack tip itself. The theory, known as linear elastic fracture mechanics, could not be applied to a number of important cracking problems involving some of the tougher, more ductile structural materials, such as many steels for example, which often experience fracture only after extensive non-linear deformation. Indeed, for many of these materials the linear approach did not even permit a practical means of assess- ing fracture toughness. Crude ways for extending the linear theory were devised, but these have now been supplanted by non-linear fracture mechanics which was developed by theoretical and experimental mechanics researchers over the last 15 years [5]. The new methods were largely developed in this coun- try, with assistance from researchers from abroad. The tough, more ductile materials tend to give rise to stable crack growth. Tear- ing resistance can now be characterized and small amounts of stable crack growth can be analyzed with confidence. It is not yet possible to predict the stability of a crack undergoing large amounts of crack advance (as in a thick wall section of a pressure vessel} bu t this is one area in which progress should be possible. The more basic problem of predicting the tearing resistance, as well as initiation toughness, of a material from micromechanical considerations represents an exciting major challenge, as discussed further below.

Advances in non-linear fracture mechanics have also contr ibuted to major progress on high temperature cracking problems where non-linear creep deformat ion must be taken into account. In addition, the past decade has seen significant advances in the understanding of the microscopic mechanisms of high temperature fracture and in at tempts to establish

macroscopic rupture and cracking behavior in terms of these microscopic mechanisms [6]. U.S. Department of Energy support of research in this area has been instrumental. Coincident with the evolution of non-linear fracture mechanics has been the development of effective numerical methods for analyzing flaws in structures under linear and non-linear situations. Only recently has computing power been sufficient to permit the analysis of some of the most practical three<limensional flaw problems. Further work on such problems is needed to extend the range of applications.

Today, fracture mechanics is progressing on many fronts with activity in many countries. Dynamic fracture and the fracture mechanics of composite materials are two particularly exciting areas with important potential applications. So too are compressive fracture and earthquake faulting analysis, bo th of which are now being tackled from the vantage point of fracture mechanics. Recent work on the analysis and observations of dynamic crack propagation has substantially advanced the understanding of unstable crack growth and how it is influenced by material behavior. Criteria for dynamic crack propagation are starting to emerge, and companion methods for analyzing dynamic crack initiation, growth and arrest should appear with adequate support. The aim of Such research is to guide the design of structures against dynamic crack initiation or for crack arrest should a running crack get started. Promising work is already under way in this area. Material strain rate effects play a critical role in dynamic fracture which is just now starting to be unraveled. This recent work on dynamic fracture will shed light on the ductile-to-brittle transition in metals that can cleave. Indeed, the basic question of ductile v s . cleavage mechanisms of fracture in structural metal alloys such as steels is increasingly recognized as an issue of the conditions for initiating a cleavage crack which can outrun most crack tip plasticity.

Successful composite material development, particularly for discontinuous reinforcements, is inexorably tied to an improved fracture mechanics of composites since the performance of these materials tends to be limited by a number of competing fracture modes. From a relatively slow start, the fracture mechanics of composites now seems to be poised for progress in the next decade.

Cooperat ion be tween researchers in mechanics and materials science has long been a hall- mark of fracture research. Such collaboration is now more apparent, and even more essential, in several areas on the forefront of research. The design of materials with improved fracture properties, while still maintaining desired levels of ducti l i ty and yield strength, requires a more fundamental understanding of the mechanics of crack initiation and growth as it is influenced by material failure mechanisms at the microscopic level. The methods of both mechanics and materials science are needed to make progress on the challenging problem of characterizing macroscopic fracture properties in terms of material microstructure. Research which cuts across the boundaries of mechanics and materials science has already made some noticeable progress in two areas: understanding the fracture of metal alloys designed to carry stress at high temperatures, and the design of ceramics with enhanced fracture toughness by exploiting certain inelastic microstructural mechanisms such as phase transformation and stable microcracking. Much remains to be achieved in these fundamental areas of research. Even for metal systems the connections between macroscopic features, such as fracture toughness and the ductile-to-brittle transition, and microstructural features are only qualitatively established at present. For example, even some of the qualitative dependences of fracture toughness on microstructural parameters, such as second- phase particle size and spacing and grain size, as predicted by existing models, are less than adequate. There has been much recent progress in dealing with the mechanics of void growth as a basic failure mechanism [7]. Many of us working in the micromechanics of materials feel that the time is ripe for developing quantitative models for fracture toughness and tearing resistance in the ductile hole growth regime.

Selected examples of critical issues that enter into a quantitative description of fracture of a particular material under a particular set of loading and environmental conditions are presented below. Although a comprehensive discussion is not the intent, we hope to demonstrate through these examples that a complete macromechanical description of any fracture phenomenon naturally builds up from a micromechanical description of mechanisms

causing material separation. References are occasionally, but not uniformly, cited.

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3. CONNECTION BETWEEN MICROMECHANICS AND MACROMECHANICS OF FRACTURE

Fracture is a local process of material separation at the tip of a crack, capable of propagating by a large family of mechanisms, ranging from pure bond breakage to pure microrupture by "blowing apart". The driving forces of this separation are externally applied as tractions or displacements at a distance from the crack where they arise from known service conditions of the structure. The local driving forces are not known a pr ior i , but their effect on the development of concentrated or dispersed damage is in principle determinable by separate experiments or modeling. Nevertheless, the development of the local damage field is related to the distant field "driving forces" by the t ime-dependent inelastic response of the solid surrounding the crack tip region. There- fore, accurate knowledge of the inelastic behavior of the surrounding solid is necessary to determine how action is transmitted from the distant field to the local crack tip volume elements. This is done by elastic and inelastic continuum mechanics.

When information on the micromechanisms of the local separation process is insufficient, correlations of fracture can be based only on distant-field characterizing parameters which, hopefully, will gauge the criticality of the local process. However useful that might be as a temporary engineering expedient, it rarely provides satisfactory answers for the nature of the local separation process or its control by deliberate microstructural alternations.

In a few instances, engineering solids are homogeneous, albeit anisotropic, substances. In the majority of cases, however, they are microcomposites composed of grains, and separate phases of a variety of forms, each with a different deformat ion resistance. For the purpose of accounting for the deformations in the vicinity of the crack tip that control the local damage processes, it is usually adequate to consider these materials as heterogeneous continua. Within this framework, the following ingredients are usually necessary to obtain accurate deformat ion field solutions around the crack tip: (1) three-dimensional incremen-

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tal inelastic constitutive relations for deforma- t ion resistance of the heterogeneous continua that are based on physical mechanisms for all individual phase components; (2) incremental evolution laws for deformation resistance for individual component phases based on physically sound mechanisms of hardening and recovery, precipitation, coarsening, aging etc.; (3) the establishment of bet ter constitutive relations and evolution laws based on defect structure studies by microscopy and model experiments; (4) since service conditions involve cyclic deformations and aggressive environments, experimental studies and modeling considerations must include some of these effects.

4. MECHANISMS OF DAMAGE AND THEIR MODELING

Although most ingredients of material separation or progressive damage processes in structural alloys are now qualitatively under- s tood, quantitative understanding is of ten very inadequate. We view the following as requiring attention.

(1) Intrinsic bifurcation of crack tip response between cleavage and ideal crack-tip- initiated plasticity by product ion of dislocations needs to be bet ter defined. Modeling of such bifurcations at the crack tip should be pursued by atomistic studies utilizing pair potentials or more appropriate schemes that respond better to the non-central nature of interatomic interactions. As a parallel to model calculations, there is a great need for discriminating experimental studies on border- line materials such as cleavable transition metals and covalent solids. Since bifurcation in crack tip behavior is sensitively dependent on the rate of crack propagation, some additional modeling and experimentation of such rate<lependent behavior is necessary.

(2) Extrinsic effects of segregated species at grain boundaries and aggressive environments are known to affect the bifurcation. Addition- al modeling and experimental studies of either a mechanistic nature or utilizing thermodynamic arguments of surface energy should be fruitful in many such cases.

(3) Identification and better characterization of potential damage sites are necessary. These should include grain boundary structure

and segregation, grain boundary and inter- phase boundary strengths, precipitation of particles at grain boundaries, possibilities for controlling the nature of grain boundary particles and their distribution to advantage.

(4) The effect of environments on damage sites must be determined.

(5) Many damage processes involve concur- rent evolution of microstructure in parallel with the actual evolution of damage on existing sites. Such phenomena are often quite interactive and require careful experimentation to separate different component phenomena.

The connection between fracture mechanisms and macromechanics as suggested in this and the preceding sections has favored the development of basic component phenomena that enter into most complex fracture processes, such as stress corrosion cracking, stress relief cracking, environmentally assisted creep cracking, and interaction between cyclic and monotonic crack growth. However, many of these phenomena have not been adequately explored to decompose them into basic component phenomena, nor is such decomposi t ion always advisable. It is therefore essential to explore these phenomena in more simple surroundings of model materials from which some of the inessential complexities are removed. These may include experiments on bicrystals, relatively clean alloys with only known segregants, and alloys with well-defined particle or microcomposi te structures that are more readily amenable to definitive modeling.

5. LOCALIZATION OF PLASTIC DEFORMATION

Localization (e.g. the formation of shear bands or necks) plays an important role in determining the strength of solids loaded far into the plastic range. The intensely localized plastic strains within shear bands or other localized deformation modes lead to rapid material damage and failure. Localization thus not only sets limits to achievable strains and load-bearing capacity bu t also plays an important role in determining the patterns of failure and fracture. In many cases, localized plastic flow appears to involve (and is evidently caused by) softening. At high strain rates, softening may result from local heating and thermal softening; at low or high strain rates, the formation of microvoids or microcracks may

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cause an overall softening. In contrast, there is now a large body of observation which also shows that localized plastic deformat ion of ten occurs wi thout softening and thus should be viewed as an inherent feature of the plastic flow process. These observations suggest that the development of localized modes of plastic deformations of ten require the solution of boundary value problems, provided that the constitutive relations that are used accurately reflect the stress-strain response of the material.

Analyses of localized plastic deformat ion have been performed using various constitutive models [3]. This work has generally indicated that constitutive models which account for strain-path<lependent material stiffness lead to descriptions of localized deformat ion that are in good accord with experiment. This is particularly true of analyses of the so-called plane strain tension test [8]. In contrast, material models that account for "fractures in progress" due to void growth have also been recently used to analyze failure modes in common laboratory test specimens [7] and have demonstrated that localization is promo- ted by the softening caused by void growth.

The results of these analyses are encouraging to the point where it seems important to extend the implementat ion of such constitutive models to a wider range of laboratory and test configurations. In particular, it seems necessary to extend the models and calculations to situations involving high strain rates and temperature-dependent flow. A possible program outline is given below.

(1) Develop constitutive relations that account for both material rate and temperature dependences; these models should specifically account for strain-softening behavior.

(2) Develop analytical and numerical procedures to perform computat ions of deformation processes, in particular those that involve flow localization.

(3) Develop numerical algorithms for efficiently integrating the full tensorial form of strain-rate-dependent constitutive relations.

(4) The material modeling and the computational procedures should specifically account for strain-softening phenomena. The strain softening could be modeled as resulting from non-uniform heating and thermal softening due to strain rates or as the result of ~fractures in progress" due to void growth or microcrack formation.

(5} The specific analyses should also account for the role of imperfections in localization phenomena. Imperfections can be modeled as non-uniformities in material properties, specimen geometry, or loading or temperature conditions to predict the onset o f strain localization phenomena and limits to ductility. In recent years the models have been refined to represent bet ter the loss in material stiffness which accompanies damage on the microscale.

Modeling of this type combined with experimental studies of void initiation and growth processes together with studies of the mechanisms and patterns of ductile failure represent an effective quantitative means of understanding the role of microstructure and chemistry on macroscopic failure. Some recommended areas for future work include (1) experimental studies aimed at document- ing the micromechanical and macromechanical phenomenology of fracture and the critical c.onditions of stress and strain rate, strain hardening and strain-rate-sensitive material properties leading to failure, (2) the extension of theoretical material modeling to incorporate various forms of material damage and, in particular, to account for the interactions among voids and microcracks which reduce stiffness, (3) experimental studies, including those on "model systems" as well as on structural alloys, of the influence of chemistry and microstructure on void initiation and early growth mechanisms and (4) the implementation of the material models developed in (2) into full-scale (computer) analyses of deformation and fracture processes.

6. CONTINUUM MODELS FOR LARGE-STRAIN

METAL DEFORMATION

Another kind of constitutive modeling that is necessary is concerned with the development of deformation-induced anisotropy and of path-dependent strain-hardening behavior. These models are concerned with large-strain deformat ion of polycrystals. Existing polycrystal models can be grouped into two categories, namely (1) upper-bound isostrain models which are typically derived from the original model proposed by G. I. Taylor and (2) self- consistent models. Taylor-like models have been successful in predicting deformat ion textures [9, 10]. Self-consistent models, al-

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though more complex and difficult to implement (particularly for finite strains), provide an alternative averaging scheme to the isostrain models and seem to yield a bet ter description of the behavior especially of two-phase materials. Approximate versions of these models have been recently introduced which account for non-linear behavior [11]. It is not clear, however, at this point how these two classes of models differ in all their predictions, but both types do provide a useful framework for studying large-strain constitutive behavior, including the development of texture and its influence on the stress-strain response.

It is important that these micromechanical- ly based models be developed with a clear view toward numerical implementation. For example, it is important to understand how the new models and their possible implemen- tations fit within the structure of existing models and numerical algorithms already in use. This is important so that their implications can be more fully assessed and so that they may have appropriate impact on problem solving. Particular strain-path-dependent characteris- tics, such as strain hardening, should be studied and compared with experiment; where useful experimental results do not yet exist, the models should make clear suggestions for future experiments.

The overall goal of this research is to con- struct an accurate, physically based constitutive theory that accounts for stress-strain path dependence of strain hardening. Useful starting points would be rate-independent "corner theories" which model path dependence, together with phenomenological models which account for material rotations and anisotropic hardening and the rate-dependent polycrystal models that predict the development of aniso- t ropy and its effects on path dependence. The physically based models provide, on the one hand, a useful framework for constructing simpler and analytically tractable theories while, on the other hand, they provide a necessary tool for interpreting and guiding experiments. The following are possible suggestions.

(1) The stress and strain path dependences of metal strain hardening predicted by existing constitutive theories, including Ja flow theory and Ju corner theory, should be characterized. This characterization is t imely in view of available experimental evidence.

(2) The existing theories which account for path dependence should be extended to more general stress states and to arbitrary stress paths.

(3) A convincing physical basis (preferably based on experiments) should be provided for the type of phenomenological model described above. In doing this, appeal should be made to the predictions of physical models, such as polycrystal models, since experimental information will almost certainly be incomplete.

(4) The models should be tested and they should be implemented in computat ional solutions of a number of carefully selected boundary value problems. These problems should involve the development of non-uniform deformations which in turn involve abrupt and large departures from proportional loading.

7. STRESS-TO-RUPTURE PROPERTIES OF

HIGH TEMPERATURE ALLOYS

The temperature range (0.4-0.6)Tin, where Tm is the absolute melting temperature, is of technological importance. In this temperature range the stress-to-rupture properties of high temperature engineering alloys vary with stress. As many as three different stress ranges can be found. At high stresses, the failure mode is transgranular rupture produced by plastic instability, which is characterized by a strong stress dependence of the rupture life and a large strain to rupture. In the intermediate stress range, the failure mode becomes intergranular creep fracture accompanied by a reduced strain to rupture and a reduced stress dependence of the rupture life. The failure mechanism involved has been identified to be grain boundary cavitation which is strongly affected by grain boundary sliding and particle spacing. As the stress is further reduced with failure times beyond 104 h, the failure mode in the lowest stress range is still intergranular fracture, bu t the stress dependence of the rupture life becomes strong again, suggesting an increased stress-to-rupture resistance which is beneficial and needs to be quantified for engineering applications.

High temperature engineering alloys typically are used in service for long periods, up to 30 years for example, in a fossil power plant. The stress-to-rupture properties of the lower two stress ranges have been given much atten-

t ion in the literature because of their technological importance. Experimental data for a long time to rupture are not readily available. There has been a continued effort to develop methodologies for data extrapolation based on short-term test data although much of the past work has not taken full advantage of the fundamental knowledge of grain boundary cavitation. This task becomes more difficult if at a given temperature more than one failure mechanism is operating, albeit possibly at different stages of service. Current interest in the assessment of the remaining life of an engineering component which has been exposed to elevated temperature service has made the need in this area more apparent.

Mechanistically, intergranular creep fracture involves the nucleation and growth of grain boundary cavities and their dependence on metallurgical structures. The latter will evolve with time at elevated temperatures. The nucleation of grain boundary cavities often is a direct consequence of grain boundary sliding, which produces stress concentrations at grain boundary inhomogeneities [12]. In the intermediate stress range, cavity nucleation is an easy process and is directly related to creep strain. The stress-to-rupture time is determined essentially by the rate of cavity growth. In the lowest stress range, the rate of grain boundary sliding is reduced so that the stress concentra- t ion at a grain boundary inhomogeneity can be relaxed and the rate of cavity nucleation is lowered. The low cavity nucleation rate can contribute significantly to the increased stress- to-rupture resistance in this lowest stress range discussed previously. Experimental data on cavity nucleation in this latter stress range are too scarce, however, to allow the quantifica- t ion of this contribution. More controlled experiments are needed on relatively simple but non-trivial alloys such as type 304 stainless steel with controlled grain boundary carbides.

The growth mechanism of grain boundary cavities has been the subject of extensive study. The concept of constrained cavity growth first suggested by Dyson is believed to be most applicable at low stresses. In this process, matrix deformation constrains cavity growth so that the cavity growth rate becomes proportional to the creep rate of the material. At elevated temperatures, the microstructural evolution in a good engineering alloy should strengthen its grain matrix and produce in-

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creased creep resistance. Since the rate of cavity growth is proportional to creep rate, increased creep resistance will lead to increased rupture life. This beneficial effect has also not been well quantified either experimentally or theoretically.

The evolution of the microstructure at elevated temperatures will vary depending on the particular class of engineering alloys of interest. The characterization of the kinetics of microstructure evolution during creep and its influence on the creep strength of a material are expected to be a more difficult task compared with that of predicting the yield strength of a material based on its microstructure.

Rice [13] has examined the consequences of constrained cavity growth based on Dyson's concept and has been able to show that the Monkman-Grant relation follows naturally. This relation suggests that the stress-to-rupture life of a material is related to its creep rate through a constant. If this type of concept can also be shown to be applicable in a material containing prior damage, it may be possible to measure the creep rate of a material to esti- mate its remaining life. The demonstrat ion and application of these theoretical concepts can potentially yield significant technological benefits and provide ample opportunities for future research.

8. HIGH TEMPERATURE CRACK GROWTH

At elevated temperatures the growth of a macroscopic crack in a polycrystal usually is intergranular since the crack follows a path along damaged grain boundaries where cavities develop or where environmental effects are most pronounced. The mechanism of damage under the high crack tip stresses depends on various material factors such as creep properties, relative rates of surface and grain boundary diffusion, and microstructure. Under higher stresses where plastic cavity growth dominates over mechanical factors, the magnitude and triaxiality of the local stress, in particular, are important. The history of crack growth depends, in a complex and coupled manner, on the spatial and temporal variations of both the crack tip stresses and damage. The aim of micromechanical and macromechanical studies of creep crack growth is both to

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improve high t e m p e r a t u r e mate r ia l s and to iden t i fy f rac tu re p a r a m e t e r s t h a t can be used fo r design and remain ing life predic t ions .

Models fo r c rack g r o w t h u n d e r extens ive creep cond i t ions have d e m o n s t r a t e d t h a t C* is the re levant f r ac tu re p a r a m e t e r and have expl ic i t ly p red ic ted the d e p e n d e n c e o f crack veloci t ies on, fo r exam p l e , creep proper t i es , grain b o u n d a r y d i f fus iv i ty and inclusion spacing. E x p e r i m e n t s on stainless steels have con f i rmed these pred ic t ions . Howeve r , since s t ruc tu ra l design unde r c reep cond i t ions general ly is conservat ive , m o s t o f the life o f a c o m p o n e n t wi th a relat ively small c rack is l ikely to be in the small-scale-yielding regime. F u r t h e r m o r e , m o s t e x p e r i m e n t s on supera l loys t end to be in this regime.

Unde r small-scale-yielding cond i t ions , even when c o n t i n u u m analyses neglect damage , the c rack t ip stress and s t rain fields d e p e n d s t rongly on the crack ve loc i ty as well as on the mac roscop i c loading. The compe t i t i ve e f fec t s o f c rack t ip stress r e l axa t ion due to cons t ra ined creep and the elastic r e sponse to crack t ip , coup led wi th the h i s to ry of mate r ia l damage , leads to very c o m p l e x t i m e - d e p e n d e n t behav io r for which a single m a c r o s c o p i c load p a r a m e t e r {such as KI ) c a n n o t cor re la te the ra te o f c rack g rowt h [14] . Never theless , extens ions o f the mode l s which ef fec t ive ly descr ibe creep crack g rowt h unde r ex tens ive creep cond i t ions are beginning to he lp to clarify the small-scale-yielding regime. The re is a s t rong need for long- te rm crack g r o w t h e x p e r i m e n t s in this regime.

The re is a long h i s to ry wi th an extens ive l i te ra ture o f the analyses and e x p e r i m e n t s o f grain b o u n d a r y cav i t a t ion f r o m the nuc lea t ion o f cavit ies up to rup tu re . A l though the re are m a n y m e c h a n i s m s and cor re la t ions for cav i ty g rowth , unde r a wide range o f cond i t ions the cav i ta t ion t ha t leads to the f o r m a t i o n o f grain b o u n d a r y mic roc racks is, t h r o u g h o u t m o s t o f life, cons t ra ined b y c reep o f the su r round ing grains. Ve ry recent ly , this n o t i o n has led to the d e v e l o p m e n t o f phys ica l ly based cons t i tu - t ive equa t ions which inc lude the e f f ec t o f m ic roc rack d a m a g e [15] , and p re l imina ry a t t e m p t s to i n c o r p o r a t e these in c rack g rowth analyses look very promis ing . Once again, however , t he re is a t r e m e n d o u s need fo r ex- p e r i m e n t s and, m o s t p robab ly , new e x p e r i m e n - tal t echn iques t ha t are capab le o f quan t i fy ing the gradient o f d a m a g e a round a crack. With

respec t to the la t ter , new t echn iques are being deve loped t h a t can sample a large vo lume o f d a m a g e d mate r ia l using s y n c h r o t r o n radia t ion.

REFERENCES FOR CHAPTER 1

1 P. Bowen, S. G. Druce and J. F. Knott, Effects of microstructure on cleavage fracture in pressure vessel steel, Aeta MetaU., 34 (1986) 1121-1131.

2 D. Peirce, R. J. Asaro and A. Needleman, An analysis of nonuniform and localized deformation in ductile single crystals, Acta Metall., 30 (1982) 1087-1119.

3 J.R. Rice, The localization of plastic deformation, in W. T. Koiter (ed.), Proc. 14th Int. Contr. on Theoretical and Applied Mechanics, North- Holland, Amsterdam, 1976, pp. 207-220.

4 M. S. Daw, M. I. Baskes, C. L. Bisson and W. G. Wolfer, Proc. Syrup. on Modelling o f Environmen- tal Effects on Crack Initiation and Propagation, 1986, Metallurgical Society Inc., Warrendale, PA, 1986, p. 99.

5 M. F. Kanninen and C. H. Poplar, Advanced Fracture Mechanics, Oxford University Press, New York, 1985.

6 J. C. Earthman, J. C. Gibeling and W. D. Nix, High-temperature intergranular crack growth processes in copper and copper with I wt.% antimony, Acta Metall., 33 (1985) 805.

7 A. Needleman and V. Tvergaard, An analysis of ductile rupture in notched bars, J. Mech. Phys. Solids, 32 (1984) 461-490.

8 L. Anand and W. A. Spitzig, Initiation of shear bands in plane strain, J. Mech. Phys. Solids, 28 (1980) 113.

9 C. Tome, G. R. Canova, U. F. Kocks, N. Christo- doulou and J. J. Jonas, The relation between macroscopic and microscopic strain hardening in f.c.c, polycrystals, Acta MetaU., 32 (1984) 1637.

10 R. J. Asaro and A. Needleman, Texture development and strain hardening in rate dependent polycrystals, Acta Metall., 33 (1985) 923-953.

11 M. Berveiller and A. Zaoui, An extension of the self-consistent scheme to plastically flowing polycrystals, J. Mech. Phys. Solids, 26 (1979) 235.

12 A. S. Argon, I. W. Chen and C. W. Lau, Inter- granular cavitation in.creep: theory and experiments, in R. M. N. Pelloux and N. Stoloff (eds.), Creep-Fatigue-Environment Interactions, TMS- AIME, Warrendale, PA, 1980, pp. 46-85.

13 J. R. Rice, Constraints on the diffusive cavitation of isolated grain boundary facets in creeping polycrystals, Acta Metall., 29 (1981) 675-681.

14 F.-H. Wu, J. L. Bassani and V. Vitek, Transient crack growth under creep conditions due to grain boundary cavitation, J. Mech. Phys. Solids, 34 (1986) 455-475.

15 J. W. Hutchinson, Constitutive behavior and crack tip fields for materials undergoing creep-constrained grain boundary cavitation, Acta Metall., 31 (1983) 1079-1088.


CHAPTER 2

MICROSTRUCTURAL AND MICROSCOPIC ASPECTS OF FRACTURE

1. INTRODUCTION

The development of an understanding of the physics and micromechanics of fracture requires an interplay between a number of disciplines. In the past the experimental inputs were based on relatively macroscopic measurements and observations, while the theoretical developments addressed both the macroscopic and the microscopic aspects of fracture and phenomena affecting the fracture processes. As there is a lack of suitable microscopic experimental input, the theoretical developments have been limited with respect to their predictions of macroscopic behavior. This situation largely accounts for the significant success of fracture-mechanics-based theories and the less effective nature of the microscopic theories of fracture. However, even in the fracture mechanics treatments, difficulties have arisen in at tempts to model the detailed role of plasticity in fracture, as again there has been a lack of detailed experimental observations with which the theories could make contact.

This situation has been changing in the past few years as the available experimental techniques have greatly improved [1-3]. Improve- ments to the spatial resolution of transmission electron microscopy (TEM) and X-ray topography methods, new ways of utilizing signals in scanning electron microscopy (SEM), improvements in the sensitivity and spatial resolution of the various microchemical methods, and development of techniques for carrying out in situ experiments in these instruments have all contributed to closing the gap between the abilities of theorists and experimentalists to address questions of interest to fracture. These techniques allow significant data to be obtained at a level at which a direct interaction between calculations and measurements is possible. The field of fracture lies now at a point analogous to that of crystal deformation

at the time when methods of imaging lattice defects were developed and basic dislocation theory was available. A marked improvement in our understanding of the fracture processes will result from the synergism of high resolu- t ion microchemical and microstructural experiments and the new theoretical methods for the study of fracture at the atomic level which are being developed at present.

Experimental input needed for understanding fracture processes {from the basic microscopic point of view) are the following.

(a) The nature of the plastic deformation processes at the crack tip and the role of dislocations in the detailed fracture process should be studied. Experimental input is required on the scale of 1-10 nm to determine the interactions of dislocations at the crack tip and on the scale of 10-106 nm to determine the distribution of dislocations in the plastic zone around the crack tip. Techniques for obtaining this information have been developed based on TEM [4], SEM [5] and X-ray topography methods [6]. Interactions of dislocations with crack tips have been studied in a variety of materials using TEM methods, with a recent series of significant advances being made by in situ deformation and fracture studies [7-10]. Studies of phase transitions at crack tips and of their role in the fracture process have also been carried out at high spatial resolution using TEM [ 11-13 ]. The longer-range deformation processes have been studied using SEM methods, most notably the use of channel patterns to measure distribution of strains at the crack tip [3, 5, 14]. Imaging of the defects near the crack tip can also be carried out in a limited number of cases with the use of X-ray topography methods [15- 18]. Recent developments in X-ray techniques allow determination of the stress tensor at the crack tip with a spatial resolution of the order of 10 pm [16].

These observations have interacted with the


18

development of the theory of fracture by providing detailed experimental support of the Bilby-Cottrel l-Swinden model of fracture, by suggesting that a "dislocation-free zone" exists at crack tips [19-22] (a matter which is in some dispute at the present time [10]) and by contributing to the understanding of the concept of dislocation shielding of the stress field at the crack tip [23]. Furthermore, the concept of transformation toughening in ceramics has received direct confirmation [ 11, 12] as has the suggestion that crack tip phase transformations are involved in the hydrogen embrit t lement of a number of materials [13].

(b) The microchemistry at the crack tip and along the path traversed by the crack should be investigated. This information is required at a variety of levels and is becoming increasingly available. Techniques such as scanning Auger electron spectroscopy (SAES), electron energy loss spectroscopy (EELS), energy-dispersive X-ray analysis (EDXA) and secondary ion mass spectrometry (SIMS) are being exploited to provide the corresponding chemical information to the microstructural information discussed above. The concentration sensitivity of some of these techniques is already quite good and developments are in progress to improve the sensitivity of some of the others. Similarly, the spatial sensitivity of these techniques is good but can be expected to improve with further instrumental developments. Thus, SIMS measurements [24, 25] can be carried out with concentration sensitivities in the parts per million range and with a lateral spat ia l resolution of about 1 pm and a depth resolution of about 10 nm. Recent developments suggest that the lateral resolution can be decreased to about 100 nm. A significant drawback of the method is that absolute concentration determinations are fraught with difficulties, but this is partially remedied by the development of the secondary neutral mass spectrometer [26]. EELS measurements can be carried out for a limited but important range of elements on a spatial scale ranging down to 10 nm, as can EDXA measurements for a somewhat complementary set of elements [27]. SAES measurements are being continual- ly improved and measurements at a spatial resolution of 50-100 nm have been reported [28]. While some of the composit ional sensitivities and accuracies of the EELS, EDXA and SAES methods are adequate, improve-

ments in these aspects of the techniques will be required for continued application of the techniques to fracture problems. At the present time the analysis of grain boundaries generally (but not always) requires that a crack be propagated along the boundary so that the energetic probe can illuminate the boundary area. This approach requires the assumption that the fracture occurred along the grain boundary, an assumption which may not always be justified [29]. This limitation will be removed as the spatial resolution of the methods improves, allowing measurements to be made transverse to the grain boundaries.

The qualitative and quantitative analysis of segregated elements is usually performed by Auger electron spectroscopy (AES). However, this requires that the specimen must be broken intergranularly within an ultrahigh vacuum apparatus, and consequently the segregation can only be studied at those grain boundaries at which the fracture occurs. No data can be obtained concerning the "stronger" boundaries; Auger studies of segregation are selective for the weakest grain boundaries. It is hoped that future development of analytical TEM will allow the determination of segregated layers at grain boundaries using scanning transmission electron microscopy and EELS and/or EDXA. However, detailed information is required concerning inelastic scattering mechanisms within a thin specimen before these methods can be optimized.

The available analytical techniques can be used for many of the elements of interest. For hydrogen, however, the choice of technique is very limited. At present, only SIMS methods are applicable to the microanalysis of hydrogen distributions [30, 31].

In addition to compositional information, it would be highly desirable to be able to obtain measurements which could be interpreted in terms of effects of solutes on the atomic bonding at the fracture regions. Unfortunately, most of the available methods, principally X-ray photoelectron spectroscopy and UV photon spectroscopy are large-area methods and cannot now be applied with the required spatial resolution. Methods which give configurational information about the lattice and grain boundary structures as affected by solutes, such as extended X-ray absorption fine structure and its variants, and electron-stimulated (or photon-stimulated) desorption are

also limited by their spatial resolution. Infor- mation of this type is becoming increasingly available for solutes at surfaces, and it may be expected that methods will be developed to obtain such bonding and configurational information about grain boundaries and interfaces. Improvements in the spatial resolution of these methods will require increased bril- liance of the primary beam, e.g. the possible use of synchrotron sources. However, the cross- sections for signal generation are generally so small that it is unlikely that the resolutions will improve to the level where measurements could be made at crack tips. It is probable that other methods of obtaining this information will need to be developed. Direct imaging of the atomic positions at grain boundaries and in the lattice using high resolution TEM and field ion microscopy (with the complementary use of the atom probe) holds great promise for many aspects related to fracture. It does not appear possible to apply these methods to deformed material, such as crack tips, however.

A major need in this area is the interaction between the theoretical and experimental practitioners, which is necessary to prevent the theory from degenerating into speculation and the experiments from becoming stale and repetitious.

(c) While it is often sufficient to obtain the type of information discussed above ex p o s t fac to , by interrupting the fracture process and studying the behavior of the crack tip region, many situations require that observations be carried out during the fracture process. A typical example where this is necessary is environmental fracture. In situ deformation and fracture experiments can be carried out in the transmission electron microscope and in many of the surface analysis instruments. These experiments are limited principally by the availability of the instruments and the imagination of the investigators. Since the in si tu experiments often require modification of the basic instruments, dedicated facilities may be required.

Theoretical studies must be carried out in parallel with the above experimental approaches in order to provide a general basis for understanding fracture processes and to develop models on the basis of which microscopic experiments can be interpreted. These studies can be divided into two major groups.

19

(a) The first group consists of studies of individual crystal defects and phenomena which play a role in either crack nucleation or crack propagation. The most important here are interfaces such as grain boundaries and particle-matrix interfaces. In the past, extensive thermodynamic treatments of interfacial phenomena have been developed, but it is now necessary to advance physical microscopic theories of these processes which take into account local atomic and electronic structures at interfaces. This has been done to a certain extent in crystallographic treatments and modeling studies using pair potentials (for reviews and recent papers, see refs. 32-35), but problems of cohesion, interfacial chemical effects etc. can only be treated when developing the theoretical methods discussed in Chapter 4 on theory of interatomic forces and cohesion.

(b) The second group is studies of crack tip processes that lead to energy dissipation during crack propagation. These include the mechanisms of bond breaking and the effects of chemical and physical parameters on this process, as well as mechanisms of irreversible deformation occurring at or near the crack tip. Ideally, a complete investigation of all these processes should be made by modeling the crack tip and its environment atomistical- ly. However, such an approach is probably too ambitious at present, and the task needs to be divided into stages, some of which are carried out in model atomistic studies while in others the developments of dislocation theory and of the theory of plastic flow in the vicinity of cracks are utilized.

2. FRACTURED INTERFACES VS. RELAXED CRYSTAL SURFACES

One of the components of an understanding of interfacial fracture is the ability to describe the structure and bonding of the newly fractured interface on the atomic scale, taking account of adsorbed solutes or impurities. The obvious reference state for this is the relaxed crystal surface of the appropriate orientation. In current thermodynamic theories of fracture, for example, these two types of surface are taken to be identical, but this is an expedient based on a lack of information to the contrary. Investigation of the relationships between the

20

s t ruc tu re o f the f rac ture surface and the corresponding in ter face (grain boundary , interface be tween mat r ix and particles, fibers e tc . ) is one of the main goals o f b o t h the experi- menta l and the theore t ica l s tudies in the physics o f f racture .

At present , we have at ou r disposal a large and growing n u m b e r of ways to character ize crystal surfaces. This requires appl ica t ion of a var ie ty o f surface techniques , including those shown in Table 1. These t echn iques are at present being applied to low index crystal surfaces, b u t t h e y could be applied to newly f rac tu red surfaces, e.g. f r om bicrystals wi th flat grain boundar ies . In addi t ion , a tomic reso lu t ion TEM can be used to character ize surfaces in some special cases.

The exper imen ta l diff icul t ies in applying these me thods are fo rmidab le for grain bound-

TABLE 1

Surface techniques for characterizing crystal surfaces

Technique Information

AES

Rutherford back- scattering, e.g. 1-2 meV He +

Low energy ion scattering, e.g. 10 keV Ne +

Medium energy ion scattering, e.g. 400 keV He +

Low energy electron diffraction, e.g. 30 V electrons

Surface extended X-ray absorption fine structure

Reflection electron microscopy

Scanning tunneling microscopy

UV photoelectron spectroscopy

X-ray photoelectron spectroscopy (also calle d electro n spectroscopy for chemical analysis)

Chemical composition of top one to three atom layers

Chemical composition of the top 50-100 atom layers; can be used to calibrate AES for a segregated surface

Identity and position of atoms on the top one to three atom layers; very low damage rate

Position of top-layer atoms with an accuracy of better than 0.01 nm

Symmetry of the top-layer atoms; diffraction patterns easy to obtain and difficult to analyze

Identity of first neighbors of a given atom species

Crystal surface topography, e.g. steps in atom layers

Crystal surface topography

Shifts in electronic energy levels of surface atoms due to chemical interactions

Same, but for core levels only

aries. In addi t ion to those c o n n e c t e d wi th specimen prepara t ion , the specimen mus t be f rac tured in an ul t rahigh vacuum and t h e n it mus t be manipu la ted for analysis. Fur ther , the necessi ty o f f rac tur ing the grain boundar ies allows on ly a selective sampling o f the grain boundar ies to be studied. The boundar ies which can be s tudied o f t en can n o t be specified as to the i r crys ta l lographic characterist ics. Since it is unl ikely tha t m o r e t h an two or th ree o f the above techn iques would be available in any given vacuum system, a mul t ip l ic i ty o f samples would be required. Also, at least at present , these studies would be best d o n e as col laborat ive e f for t s be tween laborator ies , because of the range o f facilities and exper t i se involved.

As an al ternat ive to bicrystals, and m u ch more easily ob ta ined , studies can be made on materials in which grain boundar ies develop facets , such as b i smuth -doped coppe r [36] and t e l lu r ium-doped i ron [37 ]. In these cases, we have the o p p o r t u n i t y to s tudy grain boundaries which are planar on the scale of a tomic dimensions and which are those selected by the material , p resumably on the basis o f mini- m u m energy. The compos i t ions o f the facets can be measured b y high reso lu t ion AES, and the a tomic ar rangements and defec ts of the unf rac tu red face ted boundar ies might be observed by high reso lu t ion TEM. The a tomis t ic s t ruc tu re o f these grain boundar ies and corresponding f rac ture surfaces can be mode led by c o m p u t e r s imulat ion, as descr ibed below. The results o f such studies can be co m p a red wi th those of crystal surfaces having the relevant or ien ta t ions , using the surface-analyt ical t echn iques ou t l ined above. In this way, it should be possible to get a m u ch clearer under- s tanding of the fu n d am en ta l basis for the ef fec ts o f at least some impuri t ies on interfacial cohesion. Expe r imen t s on po lycrys ta l s would also be a useful pre lude to the prepara- t ion of bicrystals, since t h e y would indicate the relevant grain b o u n d a r y or ien ta t ions for the bicrysta l exper iments .

3. MODEL EXPERIMENTS

I t is to be expec t ed tha t full charac te r iza t ion is on ly possible for special grain boundar ies which possess a high s y m m e t r y and a shor t

periodicity. For example, Bourret [38], Pen- nison and Bourret [39] and Pennison et al. [40] have performed high resolution electron microscopy studies for grain boundaries in both semiconductors and metals, and Sass and coworkers [41, 42] and Ichinose and Ishida [43] have carried out studies of the structure of high coincidence twist boundaries in gold by X-ray diffraction. It should be emphasized that the possibilities and limitations are not yet determined for the different experimental methods (including the analytical techniques). Well-defined experiments have to be performed, and it is essential that quantitative analytical results, as well as quantitative contrast experiments, can be done either by structure imaging (high resolution electron microscopy) or by quantitative diffraction studies with X-rays, electrons or neutrons. Mechanical tests have to be performed on specimens which include the same boundaries for which the structure was analyzed. The dependence of mechanical properties on the structure of the boundary can then be studied. Furthermore, the interaction of dislocations with grain boundaries, pile-ups of dislocations at grain boundaries and finally microcracking could be analyzed.

Such experiments require that bicrystals of metals containing a well-defined grain boundary be available. This has already been achieved for silicon and germanium, where bicrystals are available with a very accurate knowledge of the misorientation of the two adjacent grains. Bicrystals of other model materials such as Cu-Bi, Ni-S, NiaA1 and molybdenum must become available. The area of the grain boundary and the volume of the adjacent crystals must be large enough that the atomic structure of the boundary can be determined and the mechanical behavior can be studied. Knowledge of the misorientation and the amount of impurity at the boundary should allow a deeper understanding of the mechanisms of intercrystalline fracture.

The fabrication of bicrystals of any desired orientation requires that single crystals of a very high purity be available for the different model materials. The single crystal should not possess subboundaries, and the crystals must be cut parallel to well-defined orientations with an accuracy of bet ter than 0.01". The crystals forming the grain boundary must be diffusion bonded under ultrahigh vacuum

21

conditions and the surfaces to be joined must have been cleaned by ion sputtering prior to the bonding procedure. The preparation of such bicrystals requires special experimental facilities, especially since the boundaries must be flat. Such an ultrahigh vacuum bonding apparatus is under construction at the Max- Planck-Institut fiir MetaUforschung in Stutt- gart. The facility should be in operation in 1986 and some well-defined bicrystals of model materials may become available at that time.

4. IN SITU EXPERIMENTS

The mechanisms leading to intergranular failure (dislocation movement , dislocation pile-ups and microcrack formation) can be studied by in situ SEM or TEM experiments. The two methods are complementary as SEM allows only surface studies, while TEM is limited to the plane stress state. In situ TEM studies may give insights into the mechanisms occurring at interfaces during deformation. Large foil thicknesses of the investigated material may reveal important features which are essential for an understanding of the mechanisms; this requires high voltage electron microscopy.

Recently, in situ experiments were performed within a high voltage microscope on zirconia-containing ceramic materials [44]. It was possible to understand the essential mechanisms of the stress-induced martensitic transformation of small zirconia inclusions confined in an alumina matrix. Quantitative results concerning the transformation zone could be extracted, since the "thin foil e f fec t" could be excluded. Microcracking in the ceramic could also be studied by in situ experiments. For metals, equivalent in situ experiments are easier to perform than for ceramics; however, the interpretation may be much more difficult, since the essential processes in a thin foil may be very different from those in a thick specimen, because of the plasticity of metals. There is the possibility, however, that essential mechanisms can be extracted by in situ experiments on metallic bicrystals in metals, if experiments are performed for different orientations of the boundaries within the foil in a high voltage microscope.

22

5. THERMODYNAMICS OF GRAIN BOUNDARY

COHESION

If we consider the uniform quasi-static separation of two grains, as depicted schemat- ically in Fig. 1, the cohesive strength Ogb and the work w of separation are the maximum and the shaded areas respectively in Fig. 2. If the two grains were to be pulled apart uniformly, stretching all bonds equally, then the fracture criterion would be that o = Og b . However, a real fracture process, such as the process of plastic flow by slip, is consecutive rather than simultaneous and thus occurs by the spread of a crack. In this case, the fracture criterion is that the applied stress o supplies energy equal to w.

Rice [45] has considered the effects of solute segregation on such a separation process and has shown that, when it occurs at a

GRA,N ~-- r~ BOU N DARY

PLANE

.~- O"

Fig. 1. Schematic drawing of the simplest atomic force model used to discuss fracture along crystal planes or grain boundaries.

Z T o

o /

Fig. 2. Force us. separation curve for the model shown in Fig. 1.

sufficiently low temperature and/or sufficiently rapidly, to prevent exchange of solute atoms between the fracture surface and the bulk lattice, then the change in the chemical potential of the solute as it is transformed from the grain boundary environment to the free-surface environment is indicative of its effect on the cohesive energy of the boundary. This effect is expressed as

dw = ~ --P0 (1)

dF

where F is the solute excess at the boundary and P0 is the chemical potential of the solute in the unseparated boundary. The quanti ty #~ in eqn. (1) is the chemical potential of the solute as it exists on the fracture surface.

The chemical potentials in eqn. (1) can be estimated from the relative tendencies of the solute to segregate to grain boundaries and free surfaces. If the solute in question tends to segregate much more strongly to free surfaces than to grain boundaries, then/~® will be less than/~0, and the derivative in eqn. (1) will be negative, leading to a decrease in grain boundary cohesion on segregation. Conversely, if the solute segregates more strongly to the grain boundaries, an increase in cohesion is predicted. We should note, however, that estimates of these chemical potentials based on a comparison of the free-surface and grain boundary segregation behavior neglects the specific crystallography of the separating grains, as well as any differences in surface structure between the reversibly separated grains and those observed in a segregation experiment. It is well known that the segregation tendency can vary markedly with surface orientation and with grain boundary structure; the same is also presumably true of w(F). These factors would need to be accounted for in a fully developed theory.

Most solutes that segregate to grain boundaries appear to segregate even more strongly to free surfaces and therefore would be expected to lower w and to embrittle grain boundaries. NiaA1 appears to have intrinsically weak grain boundaries, and small boron additions are known to segregate and enhance grain boundary cohesion. If present at levels above a few parts per million, sulfur is also known to segregate strongly to grain boundaries in NiaA1 and to embrittle them further. The AES studies of grain boundary and surface segregation

in NiaA1 indicate that both the embrittling segregant (sulfur) and the toughening segregant (boron) exhibit the relationship between free- surface and grain boundary segregation indicated by eqn. (1). It would clearly be desirable to study other such systems where beneficial segregation is known to occur, to test the applicability of eqn. {1) and its relevance to intergranular failure processes. If the relative tendencies for grain boundary and free-surface segregation are found to be generally relevant to segregation effects on grain boundary failure, then this knowledge could be valuable both as a tool for alloy design and for guiding fundamental investigations of intergranular failure.

6. ATOMIC STRUCTURE OF INTERFACES

The structure of interfaces was originally analyzed using crystallographic geometrical concepts (see for example ref. 46). More recently the atomic structure of grain boundaries has been studied by computer modeling using pair potentials to describe interatomic forces (for reviews see refs. 32, 33 and 47). The results of these calculations, combined with the crystallographic considerations, have led to the establishment of a number of basic concepts common to the structure of boundaries in materials crystallizing in cubic lattices. These are, for example, the recognition of the fundamental atomic formations found in various boundaries [48, 49] and the structural unit model relating short-period boundaries to more general boundaries and establishing a relationship between atomic structures of boundaries and their physically significant dislocation contents [33, 47, 50]. At the same time, these calculations showed a number of new features, again general in nature, which could not be recognized purely crystallograph- ically. An example is the extensive multiplicity of boundary structures which may have important consequences for interpretation of structural observations and when developing theoretical models of boundary phenomena [ 51, 52]. Similarly, atomistic studies have suggested basic structural features of point defects in boundaries [32, 52], possible ocur- rences of phase transitions and/or melting at boundaries [53, 54] and some general rules governing segregation [55, 56]. However, the

23

atomistic studies using pair potentials are not able to address the problems of intergranular cohesion and the relation between the boundary structure and the structure of surfaces formed by splitting the material along a boundary, which are both problems of primary importance in fracture. Similarly, these calculations cannot deal with chemical and electronic phenomena at interfaces. The latter problem has been studied using cluster calculations [57 ], but only for relatively small clusters chosen to represent possible typical atomic configurations in boundaries, and without relaxation. In contrast, the recently emerging descriptions of interatomic forces, discussed in more detail in Chapter 4 on interatomic forces and cohesion, will allow these problems to be studied more fully by large-scale relaxation calculations. In particular, the embedded- atom method [58] and its modifications [59], as well as new classes of pair potentials that are different from those used until now [60], are well suited to studies of boundary structures and newly formed surfaces. The approximate tight-binding methods [61] and the ab initio calculations allow us to include chemical and electronic effects and to s tudy problems of cohesion in detail. Hence, exploration and utilization of these new more fundamental descriptions of atomic interactions should be the principal goal in the future studies of the atomistics of fracture of interfaces.

7. CRACK TIP PROCESSES

As noted above, the bond-breaking process at a crack tip generally does not involve the uniform quasi-static separation process depicted in Fig. 1. Even in the simplest case involving purely elastic atomic separation, the atoms adjacent to the fracture plane experience interatomic separations (and therefore tensile stresses) that depend on their distance from the crack tip. In general, the stress at the crack tip will be considerably in excess of the applied stress and will decrease as a function of distance from the crack. If the crack tip stress is sufficient to cause separation, then the crack advances. The purpose here is not to discuss the details of fracture mechanics that lead to mathematical descriptions of crack tip stress intensification but instead to cite briefly

24

the microstructural properties and processes that should be considered in such descriptions. The crack length and crack tip radius are the two principal geometrical parameters influen- cing continuum mechanics descriptions of crack tip stress fields. In intergranular failure, the crack is assumed to have a length that corresponds to some microstructural feature, such as precipitate size or grain size.

The crack tip stress decreases and spreads over larger distances in front of the crack as the radius of curvature at the crack tip increases. The crack tip radius in the core of a perfectly brittle crack is likely to be of the order of atomic dimensions and this exhibits an intense, but short-range, stress field. Any process that blunts such a crack, such as plastic deformation, lowers the stress and spreads it out over larger distances in front of the crack. Such processes are also expected to involve irreversible conversion of elastic strain energy into work, thus increasing the apparent energy required to propagate the crack.

Both direct observation of crack tip plasticity and measurements of apparent fracture energies indicate that the cohesive strength of all but the most brittle grain boundaries is greater than the stress required to initiate plastic flow near the crack tip. This inelastic deformation can take several forms, including emission of dislocations from the crack tip and activation of nearby dislocation sources in the adjacent grains. It is also likely that the fracture surface undergoes some relaxation on a time scale that is long compared with the separation time for an individual pair of atoms but is short compared with the time required for the entire fracture event. Such processes represent additional energy that must be supplied by the crack tip stress field over that required for reversible breaking of bonds.

The most obvious way in which segregated solutes can influence crack tip plasticity is through the cohesive strength of the boundary. As the cohesive strength of a boundary is increased or decreased by a segregating solute, so will the extent of the stress field and the plastic zone adjacent to the fracture path increase or decrease respectively. It is no doubt possible for segregants to influence crack tip plasticity in more subtle ways, however; we might expect segregation to influence the ability of a grain boundary to transmit or emit dislocations. Such an effect could result

from either a change in boundary structure or a change in the stiffness and directionality of atomic interactions at grain boundaries.

8. DIFFUSION-CONTROLLED BRITTLE FRACTURE

The preceding considerations of the effects of solutes on interface cohesion were confined to the low temperature regime where diffusion during the fracture process can be ignored. There exists an entirely different set of phenomena in which atoms adsorbed on the surfaces of a crack can diffuse into the grain boundary at the crack tip under the influence of an applied stress normal to the grain boundary. The process is identical with the classical Hull-Rimmer mechanism of diffusive cavity growth during creep except that, instead of surface atoms of the matrix phase, the surface atoms are one or more foreign elements which, when concentrated in a grain boundary, act to reduce grain boundary cohesion. As the tensile stress becomes sufficiently large, decohesion can occur at a rate controlled by a complex combination of surface and grain boundary diffusion and the matrix diffusive processes which govern power law creep in the crack tip region.

This process was first revealed in a study of brittle intergranular cracking in steels at high temperatures [62]. Here the most important foreign element was sulfur, and the crack rate at around 600 °C ranged from 10 -9 to 10 -5 m s -1, depending on the stress intensity raised to the power of 3.5.

It was realized that this type of fracture should not be limited to the special case of stress relief cracking in steels and that the same mechanism should occur in any polycrystalline material whose surfaces are exposed to an embrittling species, regardless of whether the source is a liquid, a vapor or a solid surface coating.

Since this discovery is still quite new, only few experiments to explore this area have been carried out. Conceptually, it is a new area of fracture research, and it is of great significance from both the scientific and the technological points of view. It is doubtless an area which will attract much interest in the next few years.

9. ENVIRONMENTAL ASPECT OF FRACTURE

A very extensive literature exists in the field of environmental effects on fracture, and no a t tempt will be made to review the subject here. However, a number of critical problems remain to be understood and the discussion will focus on these. The subject is commonly understood to encompass hydrogen embrit t lement (HE), stress corrosion cracking (SCC), and liquid metal embrit t lement (LME) or solid metal embrit t lement. The common feature in all these phenomena is the presence of a mobile species (hydrogen, the liquid environment etc.) which is aggressive toward the system being considered. These systems include both metallic and non-metallic systems (including polymeric systems). The phenomena are caused by an aggressive species sufficiently mobile to be able to follow the crack tip during the fracture process (for external environments) or to be able to under- go rearrangements in the period of stressing prior to the actual fracture process. Within the scope of these comments, many species which are not normally considered to cause environmental failures must be included. Thus the presence of chlorine in the environment may cause more than an external corrosive attack of the metal, if chlorine is soluble and mobile in the metal. Failure of systems by stress-induced oxide precipitation in front of a crack may have to be considered at high temperatures where the oxygen solubility and mobility are high, whereas external oxidation alone may be a problem in other temperature ranges. Other examples of possible environmental effects on fractures can be cited, many of which are at present unrecognized by the general materials community .

In addition to the conditions outlined above, the mobile species has to be aggressive toward the host solid in the sense that it decreases the stress and/or strain for fracture. In some cases the mode of fracture may remain the same, e.g. pearlitic steels fail by microvoid coalescence, and the presence of hydrogen may serve to enhance the formation of microvoids and their coalescence. In others, a macroscopically ductile fracture mode may be altered to be a fracture which appears macroscopically brittle but which may, in fact, be microscopi- caily ductile, e.g. LME of nickel or HE of nickel. In still others, the ductile failure may

25

be altered to a truly brittle fracture process, e.g. the stress-induced formation of a brittle hydride in systems such as Nb-H or Ti-H, or the HE of iron and steel in which the grain boundaries have already been partially weak- ened by impurities.

Another complicating feature of environmental failures is that the aggressive species may be created by reactions at the solid- environment interface. This may be the case in the SCC of certain systems where the corrosion reaction serves to create high fugacity hydrogen, which then enters at the crack tip and causes crack propagation by HE. The details of this phenomenological sequence are sufficiently complex that they have not been understood despite many serious efforts, and the relationship between SCC and HE remains a controversial subject to this day. The same is true of many of the interactions of solids with their environments, as these depend on the interactions of gases and liquids with the solid surface (adsorption, dissociation, chemical reactions etc. }, on the transport of the aggressive species in the material at the crack tip {diffusion, permeation and stress effects on the thermodynamics and kinetics of solid solutions) and on the effects of the aggressive species on the fracture process itself. All these, and many other pertinent factors, are poorly understood and are deserving of study.

Progress has been made in understanding many of these problems in recent years through the use of microstructurai and microchemical methods. These, which are discussed elsewhere in this report, allow the determination of the effects of the environment on the dislocation structure and the phases present at the crack tip and on the crack tip chemistry and of the changes caused by the environment and the solid. Most significantly, these experiments can now often be carried out during the interactions with the environment and during the fracture process.

While many environments can have deleteri- ous effects on the mechanical response of solids, the two phenomena which have been extensively studied are HE and SCC. The relation between these has been extensively discussed with no complete resolution of the matter. In some systems it does appear that SCC is caused by the evolution of hydrogen in the corrosion reaction; hence, there is a close relation between the fracture processes

26

in the two phenomena. In the following discussion, we shall avoid this controversy and discuss HE with the understanding that SCC, and indeed other environmental phenomena, may differ in at least some respects. Despite these differences, it appears that the critical questions are common to most of the forms of environmental degradation.

In discussing HE, it is important to distinguish between the thermodynamics and kinetics of the process and the micromechanisms of fracture. Common to all systems is the observation that HE occurs when the concentration of hydrogen at the crack tip exceeds a critical composition, the value of which depends on the temperature and stress intensity. This concentrat ion can result from the diffusion of hydrogen to the crack tip in the solid solution, in which case the kinetics are controlled by the stress-affected diffusion or by the entry of hydrogen from the environment, in which case the kinetics are functions of the temperature, the stress intensity, the hydrogen fugacity in the environment and the surface reactions which control the entry of the hydrogen. Another important distinction which should be made is based on whether the system is a hydride former or not. In the former case the mechanism of embri t t lement seems to be well established as the stress- induced formation and cleavage of a brittle hydride at the crack tip. In non-hydride- forming systems, the fracture mechanism is not well established and will be discussed below. While there remain many interesting problems to be treated in this area, much of the basic understanding of the thermodynamics and kinetics is in hand.

The mechanism of fracture in a hydride- forming system depends on the formation of a brittle hydride phase and is characterized by repeated hydride formation and cleavage. Since this process requires significant diffusion of the hydrogen and a large increase in the hydrogen concentration at the crack tip, it is possible to prevent the hydride fron~ forming by decreasing the temperature or increasing the loading rate. The result is either ductile failure or embri t t lement by an alternative mechanism. In the Ti-4wt.%A1-H system, this is what is observed. At intermediate loading rates the crack can propagate faster than the hydride can form, and the fracture pro- ceeds by the hydrogen-enhanced localized-

plasticity mechanism described below. This system is only one of many in which there is evidence for the occurrence of several different micromechanisms of hydrogen-related fracture.

In non-hydride-forming systems such as nickel- and iron-base solid solution alloys, one mechanism of hydrogen-related fracture which has been observed is a localized-plasticity-based fracture in which the plastic deformation is enhanced at the crack tip by the high local concentration of hydrogen. This enhanced plasticity presumably results from the increased dislocation velocity in the presence of hydrogen and the resulting decrease in the flow stress. While these results are based primarily on in situ TEM environmental cell observations, they are supported by in situ SEM observations and by careful fractography on macroscopic specimens. The detailed mechanism by which this enhanced dislocation mot ion leads to fracture has not been developed.

In many of the non-hydride-forming systems, the hydrogen-related fracture occurs along grain boundaries or interfaces. In the Ni-H system, intergranular fracture occurs in hydrogen gas when impurities such as sulfur are allowed to segregate at the grain boundaries, while in the absence of these segregated impurities the fracture is transgranular. Nickel with hydrogen in solid solution exhibi t s hydrogen segregation at the grain boundaries, and the resulting fracture is transgranular. A detailed observation of the intergranular fracture suggests that it also is the result of hydrogen-enhanced local plasticity which takes place in the vicinity of the grain boundary and that the fracture does not actually occur along the boundary itself. In contrast with these observations, most of the reports of HE of high strength steels suggest that the fracture occurs along prior austenitic boundaries and that the fracture occurs by a brittle mode. The HE of the steels has been shown to be sensitive to the presence of other species segregated at the boundaries.

The question of whether the fracture mode in any given system is ductile or brittle is of ten unresolved. Both mechanisms may be possible, depending on the actual stressing conditions and on impurity effects in grain boundaries. Interpretations of the fracture in terms of a brittle mode have focused on a

postulated decrease in the cohesive energy of the system by species such as hydrogen. Direct evidence for such a decrease in atomic bonding has not been obtained, however. Some of the available evidence bearing on this subject has been obtained from measurements which probe the lattice potential curve in the vicinity of the equilibrium lattice spacing; phonon dispersion measurements, atomic force constants, elastic moduli etc. These generally are consistent with an increase in the atomic bonding due to hydrogen, rather than a decrease. However, the large atomic volume of hydrogen in a metal such as iron would suggest a reduct ion in cohesive energy due to hydrogen, and the above physical measurements have not been made at large atomic displacements, nor are the techniques for these determinations available. The issue remains one of the most significant for the understanding of environment-induced fracture. It is a matter in which close interplay between the experimental determinations and theory may be expected. The same issues which are addressed in the effects of hydrogen on atomic bonding also apply to the influence of other species on the atomic bonding, particularly at interfaces.

10. INTERMETALLIC COMPOUNDS

The nickel aluminide, NisA1, exhibits several attractive properties for high temperature structural applications. Unlike conventional alloys, Ni3A1 gets stronger with increasing temperature and, because of its high aluminum content, it forms an adherent A1203 scale and exhibits good oxidat ion resistance. In spite of its good ducti l i ty when tested as a single crystal, technological interest in NisA1 has been limited because of its propensi ty for brittle intergranular failure in the polycrystalline form at room temperature. Intergranular brittleness in NisA1 was originally at tr ibuted to segregation of trace impurities (e.g. sulfur or oxygen) at grain boundaries. Recent studies indicate, however, that the alloy remains brittle even if the impurity levels are reduced below the detect ion limit of AES (approximately 0.1 at.% S in the two to three atom layers constituting the grain boundary) . It appears that grain boundaries in NisA1 are intrinsically weak and, although segregation of impurities can further embrit t le the alloy,

27

this is not the primary cause for the grain boundary brittleness. Such intrinsic grain boundary brittleness is relatively rare, but not unprecedented. Iridium, some platinum-base alloys, molybdenum and even iron have been reported to have intrinsically brittle grain boundaries, although some of these observations have been contested.

Recent research in Japan and the U.S.A. has led to remarkable improvements in the ductil i ty of NisA1, largely as the result of small (0.1 wt.%) boron additions. The boron appears to be effective only when the NisAl is slightly substoichiometric, having aluminum contents of less than 24.5 at.%. Alloys containing 24 at.% A1 and 0.02-0.1 wt.% B, however, exhibit room temperature tensile elongations of approximately 50%, in contrast with the negligible ducti l i ty of the same alloy wi thout boron. The boron is in solid solution and segregates strongly to the NisA1 grain boundaries, being present at levels between 10 and 20 at.% in the two to three atom layers that make up the boundary.

It appears that boron, unlike most other grain boundary segregants, strengthens the grain boundaries in NisA1, making them more resistant to separation than in the pure alloy. Again, this behavior is unusual bu t not unprecedented. It has been suggested that boron improves the grain boundary strength in iron, and there is evidence for improvement of grain boundary strength resulting from boron segregation in platinum-base alloys [63]. In addition to boron, carbon has been shown to strengthen grain boundaries in iron and steel [64], and thorium has been observed [65] to segregate to and to strengthen grain boundaries in iridium alloys*.

In addition to exhibiting an unusual (strengthening) effect on grain boundaries in NisA1, boron also exhibits unusual segregation behavior in the alloy. Grain boundary segregants normally segregate even more strongly to free surfaces. Although the reasons for this are not known, it is commonly held that strain energy associated with a misfitting solute a tom in the bulk crystal is more completely

* The above examples are res t r i c ted to cases where t h e t e m p e r a t u r e and crack p r o p a g a t i o n ra te effectively preclude diffusional mechanisms for crack propaga- t i on and d i f fus iona l r e d i s t r i b u t i o n of t he segregat ing so lu te at t he crack t ip.

28

relieved at the relatively unconstrained environment of a free surface site than at a grain boundary site, thus yielding a higher binding energy of the solute to the surface. Observa- tions of surface segregation in boron-doped NisA1 seem to contradict this conventional wisdom, however. Surface segregation studies [66] on alloys containing 24 at.% A1, 0.05 wt.% B and less than 1 wt .ppm S have revealed strong segregation of sulfur to the free surface, wi thout any detectable segregation of boron. This indicates that, like many other segregants, sulfur segregates much more strongly to free surfaces than to grain boundaries, while boron appears to behave in the opposite manner. In spite of being present in sufficient concentration to segregate extensively to the grain boundaries, boron does not segregate to the free surface [67].

It is worth noting that, at elevated temperatures in vacuum, boron-toughened NisA1 retains its intergranular strength but that this is lost in the presence of oxygen [68, 69]. This could well be another example of diffusion- controlled brittle fracture, as in the sulfur- induced stress-relief cracking of steels [62].

11. SIGNIFICANT ISSUES

(a) Identification of the aggressive species in the sense discussed earlier remains a key issue in many situations. Too often, hydrogen is identified as the culprit in the absence of any direct evidence. The ability to apply microchemical methods to determine the composi t ion of the regions in the vicinity of the crack tip should alleviate this problem.

(b) Understanding the effect of chemical species such as hydrogen, sulfur and boron on the atomic bonding in the lattice and at the grain boundaries is a significant problem. Theoretical treatments of this problem are not yet very satisfactory, bu t they are improving faster than are the experimental measurements. In a number of b.c.c, systems, principally the group Vb metals, extensive measurement of the effect of hydrogen on the phonon dispersion curves, atomic force constants, elastic moduli etc. have been carried out. These all indicate a strengthening of the atomic bonding bu t suffer from the fact that they are all measurements taken near the equilibrium

atomic positions. What is required is a technique to determine the effect of solute species on the atomic bonding at large displacements, i.e. on the force-displacement relationships. Alternatively, measurements of the true fracture (i.e. cohesive) energy would be useful. Improvement in the theoretical treatments of this problem is also required. This problem should be treated both for the lattice and for interfaces.

(c) The detailed structural changes at the crack tip must be understood in greater detail. It is now possible to do so using a variety of techniques, as discussed above. The resolution of the modern methods is becoming sufficiently high as to allow resolution of many of the problems about which speculation was only possible in the past. Dislocation generation at the crack tip and the formation of the plastic zone have been shown to be affected by the presence of aggressive species. In a number of non-hydride-forming systems, one of the environmental failure modes has been shown to be hydrogen-induced local plasticity at the crack tip. This appears to result from enhanced dislocation mobil i ty in the presence of hydrogen. The mechanisms by which these effects occur are not well understood and require further study. Additionally, the possibility that similar effects occur with other solutes should be explored.

(d) Significant advances are being made in the understanding of grain boundary structure, bo th experimentally and theoretically. The understanding of the role of solutes in determining grain boundary structures and the properties of these structures is much less advanced. Since many of the fractures which occur in the presence of aggressive species are associated with grain boundaries, this subject is of great importance. The availability of the new microanalytical and microstructural techniques for studying grain boundaries, including the possibilities of t h e synchrotron sources for diffraction from the boundaries and the use of high-resolution TEM for directly imaging the boundary structure, promises great advances in our understanding of this subject. Similarly, the newly developed descriptions of interatomic forces, together with increasing power of computers, will allow more fundamental theoretical studies of intergranular cohesion, in both pure materials and alloys.


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CHAPTER 3

NOVEL EXPERIMENTAL TECHNIQUES FOR THE STUDY OF FRACTURE MICROMECHANISMS

1. INTRODUCTION

Our understanding of the fundamentals of fracture has always progressed hand in hand with the emergence of new experimental techniques. Early experimental studies of fracture relied on the monitoring of load and strain (extension) and, more recently, potential drop to follow the fracture process. These measurements were the integrated responses of a complete sample and, with the realization of the microscopic nature of fracture, it was recognized that more sophisticated measurement methods were called for. Numerous studies coupling destructive post-test metallography and interrupted loading have subse- quently been directed at identifying the critical microstructure consti tuents controlling fracture. These have recently been augmented by in situ transmission electron microscopy (TEM) and scanning electron microscopy (SEM) straining studies, providing an in situ measurement capability although not wi thout serious problems related to thin foil and surface effects.

As we look to the future and a t tempt to address the need for a science-based lifetime predictive capability, we recognize the severe limitations of the traditional fracture measurement techniques. Crack initiation measurement is too insensitive, crack length determinations are too inaccurate and accumulated damage processes are not readily characteriza- ble. Furthermore, post-test metallography, while indispensable, is seriously limited in its ability both to locate damage sites (which are of ten sparsely distributed) and to expose truly the chronology of fracture.

A number of experimental techniques are emerging which show promise of addressing the measurement needs of fundamental fracture studies. They include acoustic emission, ultrasonic scattering, electromagnetic nondestructive evaluation, the use of synchrotron

radiation and small-angle (neutron and X-ray) scattering. Here, we review the current status of each technique and identify possible opportunities for their use in the fundamental s tudy of fracture. It will become apparent that, while they all share a non-invasive capability, the information which they each provide is complementary and points to opportunit ies for collaborative research.

2. ACOUSTIC EMISSION

2.1. Introduct ion Acoustic emission refers to the transient

elastic waves sometimes spontaneously generated by a body when it is subjected to stress. The frequency of the elastic waves extends into the megahertz region and they can thus be detected with sensitive ultrasonic transducers attached to the surface of the body. Dislocation mot ion and crack growth are thought to be the origin of this naturally created ultrasound. We can intuitively understand how this arises with the following argument. Suppose a stressed body is at mechanical (static) equilibrium, when an abrupt deforma- t ion or fracture event occurs. An exchange of elastic information occurs be tween the site of the event (the source) and the boundaries of the body. This information exchange is achieved through the propagation of elastic waves, the same elastic waves monitored as acoustic emission [ 1 ]. These waves enable the body to change its shape and eventually to reach a new state of mechanical equilibrium. Thus the measurement of acoustic emission is the dynamic analog of a quasi-static strain measurement.

The monitoring of acoustic emission then offers a potential to observe the dynamics of microscopic deformat ion and fracture events and also to monitor the location of each event. Location is possible since the velocity at which


32

elastic waves propagate is usually well known. The simultaneous recording of signals at different positions on a body enables a set of time-of-flight differences to be deduced and, from these, source location by triangulation is achieved. The accuracy of this location depends on the shortest-wavelength components present in the detected signals. This is usually controlled by the bandwidth of the recording system. For recording systems available today (of bandwidth 10-100 MHz) this could be of the order of 30-300 pm, compa- rable with the dimensions of the source event itself [2].

The dynamic source information contained within signals is much more difficult to unfold because of the tensor nature of the problem. Conceptually, the signal can be thought of as a time-varying vector displacement caused by an abrupt localized change in stress (a second- rank tensor). The magnitudes and time dependences of all the components of this stress change are the quantities that we wish to determine. It is not possible to determine the six components of stress (for an isotropic linear elastic medium) from a single displacement waveform (although, if all the components have the same temporal character, the temporal character alone may be approximately determined from a single signal and used as a "signature" of the event). Rather, it seems necessary to make six (and preferably more) displacement measurements at different positions around the source and to perform a simultaneous deconvolution to determine the magnitudes of all six stress components. A further micromechanical modeling step would then enable the micromechanism responsible for the stress change to be quantitatively deduced.

Acoustic emission in principle has a great deal to offer the materials scientist attempting to probe the mechanisms by which materials respond to stress. This has been recognized for at least 10 years. However, there is a relative absence of widespread application of this tool, indicating that difficulties have been encountered in practice.

One problem is that not all the micromechanisms of interest are detectable [3]. This stems in part from the fact that the transducers used to measure surface displacements associated with elastic waves are not infinitely sensitive and in part because of background

noise levels present in all experiments. Noise in the low kilohertz range is particularly large and of ten obscures total emission in this spec- tral region. Thus, it is usual to high pass filter signals to improve the signal-to-noise ratio. Typically the modern generation of piezoelec- tric transducers can measure displacements greater than 10 -la m in the approximate frequency range of 20 kHz-2 MHz. More nearly ideal broader-bandwidth devices are available but their sensitivities are reduced (10 -12 V). Narrow-band (resonant) transducers with bet ter sensitivities {down to 10 -14 m) also exist, but they so distort the displacement signal that it is at present not considered feasible to use them for quantitative work. While advances in transducer sensitivity and background noise reduction continue, it turns out that a wealth of micromechanisms (from the propagation of individual dislocations in perfect single crystals to cleavage microcracking of ferrite grains in steels) are in fact detectable (a signal-to-noise ratio greater than unity over the bandwidth) and the more energetic of these may be measured with broad-band transducers and thus be charac- terizable with today 's technology.

The more serious problem is that of the characterization process itself, a problem akin to that in seismology of characterizing the faulting mechanism of an earthquake from remotely recorded seismograms. Current research at laboratories in the U.S.A., U.K., F.R.G., France, Japan and elsewhere is attempting to apply the formalisms initially developed for earthquake characterization to the characterization of acoustic emission sources. The first successes of this are now emerging. It seems that properly performed acoustic emission measurements during conventional mechanical property testing indeed reveal the dynamics of fracture micromechanisms and consti tute a new materials probe.

2.2. Phenomenological observations Since acoustic emission is, after all, a natu-

rally occurring phenomenon, it is reasonable to begin with a series of phenomenological observations utilizing systematic microstructure control and intuitive reasoning to deduce the origin of the emission signals in terms of deformat ion and fracture micromechanisms. Perhaps the most controlled experiment to consider first would be that of the constant-

strain-rate deformat ion of an f.c.c, crystal oriented initially for single slip. Numerous research groups have begun their studies at this point [4-7] and a typical result showing the acoustic emission power variation with strain is shown in Fig. 1 [7]. In the experiment the high purity aluminum sample was deformed at a slow strain rate (10 -4 s -1) and acoustic emission in the 0.1-1.0 MHz frequency range detected with a transducer whose signal-to-noise ratio exceeds uni ty for displacements 10 -z4 m.

Beginning during nominal elastic loading, it was observed that many individual signals were emitted and their summed power (intensity) quickly reached a maximum shortly after general yield. Less emission is detected as the strain increases beyond this point. We see that the amount of acoustic emission per unit plastic strain varies with strain, the ex- planation for which lies in the microscopic nature of slip.

The imposed strain rate ~ is satisfied by the formation and mot ion of dislocations, i.e.

= p b v where p is the mobile dislocation density, b the Burgers vector and v the dislocation velocity. Early during deformation, dislocations on slip planes at 45 ° to the tensile axis (maximum resolved shear stress plane) glide considerable distances at a potentially large velocity through an almost perfect lattice. In exceptionally pure material containing few grown-in defects, this distance can be as great

0.5 60

0.4

~- , 0 03

0~ 0.2 o 20 m

O.t

0 " ,£i 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Strain

Fig. 1. The acoust ic emission (AE) intensi ty (power) de tec ted during the tensile de format ion of single crystal o f a luminum decreases wi th increasing plastic strain during parabolic work hardening. The measurements were made over a f r equency band f rom 0.1 to 1.0 MHz wi th an ampl i f ica t ion o f 98 dB. A smooth relat ionship is observed because many individually small signals are emi t t ed wi th in the response t ime of recording ins t rumenta t ion , result ing in the observa- t ion of a mean emission intensi ty [7].

33

as the slip plane diameter (a few millimeters). However, toward the end of a mechanical test, a dislocation cell structure develops, and individual dislocations are only able to propagate distances of the order of the cell wall spacing, typically a few microns. Thus the intense emission early in the test arises from the unimpeded coordinated mot ion of a few (small p) dislocations over large distances at a high velocity. This is much more likely to be the origin of a detectable signal than the motion of numerous dislocations moving in an uncoordinated manner over short distances (e.g. during the latter part of stage III deformation).

This hypothesis may be rather straightfor- wardly tested by systematically varying the spacing of obstacles in the microstructure that are impenetrable to dislocations and then observing a standardized acoustic emission response. Grain boundaries serve this purpose. The grain size can be varied over a wide range in, say, aluminum by appropriate cold work or recrystallization treatments. The distribution of slip distances will now be a function of the grain size which, initially at least, controls the grain interior dislocation-dislocation interactions. When samples of aluminum polycrystals are tested under conditions such as those of the single crystal, a similar dependence of the emission on strain to that of the single crystals is observed, reflecting again the development of a cell structure. However, the total emission detected for the test depends sensitively on grain size as shown in Fig. 2 [7]. It can be seen that, as the grain size decreases,

400 ¸

300

200

w

o

ALUMINUM - I 3 wt % ~ . ~ G N E S I U M ~ @

I I I I L 31o 315 05 I0 I ~ 20 25

GRAIN SIZE/ ram

Fig. 2. A compar ison of the grain size dependence of acoust ic emission (AE) f rom pure a luminum and an A1-Mg solid solut ion al loy shows that , for small grain sizes, solute a toms enhance the emission f rom dislocations while, for large grain sizes, the emission f rom dislocations is suppressed [ 7 ].

34

so does the detectable acoustic emission, consistent with the view that the dislocation propagation distance has an important effect on signal detectability.

Interestingly, this observation is profoundly changed if substitutional impurities are introduced. In alloys of aluminum, substitutional impurity segregation to dislocations occurs even when the bulk impurity concentration is below the solubility limit. During dislocation motion, either this solute atmosphere diffuses with the dislocation as it propagates or, if the stress is sufficiently high, the dislocation may escape, leaving behind the solute atmosphere. The breakaway phenomenon manifests itself as a load drop during constant-strain-rate testing and is the origin of yield point phenomena (upper and lower yield points, and Lfiders band propagation) and dynamic strain aging in aluminum alloys.

Figure 2 shows the effect of grain size on the total acoustic emission of a (substitutional) Al-l .3wt.%Mg alloy [7 ]. When the grain size was large (several millimeters), almost no detectable acoustic emission was observed even though the total plasticity was compara- ble with that of the pure material (a copious emitter at this grain size). This material also did not exhibit significant yield point or serrated yielding (dynamic strain aging) behavior. In contrast, higher yield strength samples with a smaller grain size (about 80/~m) exhibited a maximum acoustic emission (much greater than that of pure aluminum of equivalent grain size) and a maximum yield point phenomenon and serrated yielding activity. Apparently, the origin of the emission is related to the breakaway of dislocations which depends on grain size (probably because of a flow stress dependence on grain size). In these alloys the cooperative high velocity mot ion of groups of dislocations is the origin of the emission and serves to emphasize that the dislocation velocity and the cooperative slip of numerous dislocations are possibly as important as the propagation distance in determining the detectability of signals emitted by a deformation event.

The micromechanisms of dislocation motion in polycrystals are known to be strongly influenced by the presence of a second-phase particle distribution within the grains. There are two extreme classes of dislocation-precipitate interaction.

(1) When the precipitates are large, strong and widely spaced and have an incoherent interface with the surrounding matrix, dislocations bow between particles, leaving a dislocation loop around the particle.

(2) When the precipitates are small, weak, closely spaced and coherent with the matrix, dislocations may penetrate the precipitates and shear them.

Process (1) encourages uniformly distributed slip because it is harder to force a second dislocation past a strong obstacle that is already surrounded by a dislocation loop. Process (2) can lead to slip localization because, if the precipitates on a particular slip plane have been sheared once, subsequent shear on that plane is often easier, resulting in a local strain instability. Studies in which the precipitate strength, size distribution and coherence have been systematically varied in coarse-grained aluminum alloys (where dis- locat ion-impuri ty interactions are not active acoustic sources) indicate that process (1) leads to weak emission while process (2) leads to intense signals {Fig. 3) [9, 10].

From theoretical considerations of the surface displacement amplitude of acoustic emission signals, it has been possible to deduce the detectabil i ty criteria that a source must meet if the signal is to exceed background noise [11]. For the slip case, the criterion is of the form nav >t k where a is the dislocation slip distance, v is the dislocation velocity, n is the number of cooperatively moving disloca-

7 0

~ Z 6 0

' . no 50 ~Eu~

u J z 4 0

~ 30

I0 --

0 "'"~ QUENCHED

. DISLOCATION - ~ I N T E N S E SHEAR B A N O S ~ CELL STRUCTURE

J /

IO IOO IOoo IO,00o IOO,0O0 AGING TIME AT 120°C (rain)

Fig. 3. Aging supersaturated aluminum alloys (in this case, A1-5.5wt.%Zn-2.5wt.%Mg) results in "shear- able" precipitates which "focus" dislocation motion onto closely spaced slip planes (intense shear bands). The acoustic emission from this mechanism of dislocation motion is much more detectable than that leading to a cellular network in solid-solution- or dispersion-strengthened microstructures.

tions and k is a constant that depends on the material elastic properties and is proportional to the transducer sensitivity and source-to- receiver distance. For the aluminum single crystal tests, k ~ 0.035 m 2 s -1 for a transducer sensitivity of 10 -14 m. For v ~ 350 m s -1, na >1 100 pm, implying that individual dislocations, if they move fast enough, can give rise to detectable events.

Following a similar procedure, detectabili ty criteria for mode I microfracture events have also been deduced [11]. For steel, the criterion is oa2v >1 5 × lOZ4rT for a signal of detectable strength to be emitted. Here, a is the local stress, a the crack radius, v the radial crack velocity, r the source-to-receiver distance and T the transducer sensitivity. Substituting realistic values gives

o a 2 ~ 0 . 1 W (1 )

For cleavage microfracture of a ferritic steel, a is half a grain size (say, 10/Jm), v ~ 113 m s -1, o ~ 500 MPa, and thus aa2v = 50 W. Cleavage in steels (and other brittle materials) is clearly a source of readily detectable signals.

Other mechanisms of fracture have different a and v distributions and thus some will be detectable whilst others will not. A con- venient way to summarize this information is to plot the anticipated ranges of a and v for each micromechanism on a map on which is superimposed a detectabili ty criterion [ 3]. In Fig. 4 such a map for steel is shown assuming that T ~ 10 -12 m. Only those events to the right of the detectabili ty criterion in Fig. 4 are detectable.

These studies serve to show that , in contrast with stress-strain measurements or microhard-

Detectability t,

. ' If I 1 I, Shear Velocity Limit Detections l i I ~ ' ~ r . . . . . . .

- ntergranular looo I Th.sho0 irl(l - ; i n c l u s i o n s 100 ~:i:i :i:i ::i ~ " - . a ~ i ~ ~ 1 Alternating ~ - - - ' ~ ~. ~ ,uues . / , ~ HI ~ Shear . . . . . . / "

, - - - - . , . ~ - - l l l r l h

D t i n el / ~ " ~ Microvoid I I I I , ~ e ect'o c.~, -Coalescenc~ ~ I Threshold

t 1 I I i J J I ~m 2 1 ~m 2 10~m 2 100pro 2 1000~m 2 .01 mm 2 .1 mm 2 1 mm 2

Incremental Crack Area

Fig. 4. Detec tab i l i ty o f f racture events in steels. The de t ec t ion th resho ld assumes a t ransducer sensi t ivi ty of 10 -13 m, a dis tance be tween the source and the t ransducer of 0.1 m and a stress o f 500 MN m -2 [3].

T

35

ness testing where an integrated static strain response to load is measured, acoustic emission signals emanate from individual microdefor- mation events, and these are extraordinarily sensitive to microstructure {compared with a stress-strain curve). Furthermore, the passive nature of the technique (it does not perturb the source process) and its 100% volume coverage make it an ideal complement to traditional materials-testing tools. However, its full potential will only be realized if the dynamics of the source can be observed and for this a predictive theoretical formulation and a robust inverse modeling approach are required.

2. 3. Theoret ical f o r m u l a t i o n The sequence of events giving rise to an

acoustic emission signal are summarized in Fig. 5. A casual sequence of processes occurs following the occurrence of the source event, which can be likened to the creation of a dynamic force field at the source position. This is propagated as a mechanical disturbance through the body, causing a surface displacement u ( t ) a t the receiver location. The receiver detects the surface disturbance and produces an output voltage waveform that is subse- quently electronically processed and digitally recorded. One goal of a theoretical formulation is the precise prediction of acoustic emission signals for prescribed sources; a second is the formulation of a robust approach to the inverse problem.

If the system in Fig. 5 is linear elastic, then, when viewed in the frequency domain, information is transmitted independently, frequency by frequency, from the source to the observed signal. Thus the observed signal

i ui(Lt) •, , / i i , i i / , , i i 1 1 1 / , 1 . , , , , i i

P r o p a g a t e

a v ~ D e f e c t P r o d u c e s St ress C h a n g e at

_r ~, t

Fig. 5. Schemat ic i l lustrat ion o f the acoust ic emission process.

14.00

can be represented as a convolution between the source function and the impulse responses of the body, transducer and electronic system (this latter assumed here a perfect 5 function of unit strength for simplicity).

Then, for buried sources that can be modeled as dislocation combinations, the general formulation of Simmons and Clough [12] can be simplified and the displacement at r as a function of time t is given by

u,(r, t )=fdrfG,j .~,(r ,r ' ; t - - t ' ) Aaik(r' , t') dt' (2)

where Gij(r, r', t) is the dynamic elastic Green tensor representing displacement in the xi direction at r as a function of time due to a unit force impulse applied at r ' and t = 0 in the xj direction. Thus the Green tensor is the solution of the wave equation for a unit force source. The corresponding solution for a dipole force of unit strength is obtained by spatial differentiation of G~j (denoted ~, above). Aaj~ corresponds to the stress change (or dipole density) at r ' and its volume integral for a point source Aojk corresponds to the dipole (or moment ) tensor of the source. Thus, for a point source, eqn. (2) is a convolution between the dipole (or moment) tensor of the source and the dynamic elastic Green function of the body.

This formulation provides a basis, in principle, for predicting surface displacement waveforms caused by homogeneous stress changes within a body. In practice, the formulation is computat ionally cumbersome for anything other than point sources in isotropic linear elastic bodies with either one or two parallel surfaces (infinite half-space or infinite plate). The approach to sources that are spatially extended is to apply a Taylor expansion to the Green tensor and to represent the source as a linear combination of moments (zeroth, first, second etc.) [12].

The final step of the formulation is to introduce the effect of transduction process itself on the signal. For "non-disturbing" transducers with a surface area ST the voltage as a function of time t due to a point dipole source Ao~j at position ro' is

v(t) = f f Tf(r, t --t ') Glj, k'(r, to'; t - t")

ST × Aa~(t") dr dt" (3)

where Tl(r ) represents the transducer sensitivity at a point r on the transducer face due to a displacement in the xt direction.

For a point receiver, eqn. (3) simplifies to the form of a convolution between a source function, the impulse response of the body and the impulse response of the transducer. In the frequency domain the convolution integrals are replaced by products and so the (scalar) voltage spectrum is given by

V(co) = rjk(c~) AOjk(CO) (4)

where Fik(co ) is now the combined transfer function of both the body and the transducer.

The role of theory then is to determine the combinations of forces (dipoles, quadrupoles etc.) that are equivalent to those that occur when various types of defect source occur and to combine these with appropriate Green functions and transducer responses to predict acoustic emission signals. An example of this for a simulated acoustic emission, a thermo- elastic expansion source (due to pulsed laser absorption at the surface of an aluminum plate), is shown in Fig. 6 [13]. Also shown for comparison is the experimentally measured waveform. The agreement is good for this isotropic material. Discrepancies associated with grain scattering are observed, however, in anisotropic materials such as stainless steel

12.00

/ /

10.00

0.oo

"~ 6.00

> 4.00

F, 2,00

0.00

-2.00

- 4.00 3.00

36

6.00 9.00 12100 15100 18100 21.00 24100 27100 30.00 Time in Microseconds

Fig. 6. Comparison of experiment ( ) and theory (- - - ) for laser-generated acoustic emission in aluminum alloy 2024-T6 [13l.

[13] and points to the need for research to deduce the Green tensors for realistic engineering materials.

2.4. Inverse problem From eqn. (4) it is clear that a deterministic

approach to characterizing the acoustic emission is to evaluate Fjk(¢o) and to divide V(co) by Fj~ on a frequency-by-frequency basis. This will reveal Aojk(¢o) and the temporal behavior can be deduced by inverse transformation. While simple in concept, the approach must recognize several key issues.

Even for a point source in an isotropic linear elastic medium the dipole (or moment) tensor contains up to six independent components. However, a voltage is only a single {scalar) parameter that depends on an unknown combination of the source components. Ideally, six (or more) voltage waveforms should be recorded (with multiple transducers located around the source) and a simultaneous deconvolution performed to reveal independently the six unknowns.

Characterization schemes based on the analysis of only a single transducer voltage are commonly incorporated in commercially available instruments. However, unless a priori information exists to link the stress components to each other (constraining relations) so that there is only a single unknown quantity, it is difficult to understand the validity of such a characterization.

Schemes are evolving for both the full vector calibration of acoustic emission transducers [4-16] and the experimental evaluation of Green's tensors for realistic bodies [17 ]. As a simple first step in applying the inverse modeling approach, Hsu and Hardy [18] have determined the source funct ion for a pseudo- force step (the force unloaded when a glass capillary is broken by compression against a plate). Using a capacitance transducer responding only to out-of-plane displacement u3, the one-dimension problem to be solved was

t

u3(t) = f GH3(t -- T) Fs(r) d r

o

(5)

where F3(t) is the force- t ime funct ion to be determined. Using matrix inversion techniques, Hsu and Hardy deduced the result shown in Fig. 7 from waveforms measured either at the

37

epicenter or on the same plate surface as the source. It should be noted that, if Fs(t) is known, this equation can be solved to deduce Gas(t) experimentally.

The extension of this approach to naturally occurring sources in a half-space has received extensive recent interest. Wadley et al. [ 19] set up isolated mode I loading situations, in modified tensile samples. Since the source orientation was known, and elastic loading was assumed, constraining relations between source stress components could be derived so that only a single unknown (the crack diameter) was to be determined. Epicenter waveforms were then used to characterize the source in terms of the crack diameter and its rate of change (the radial crack speed). By using a very wide bandwidth (80 kHz-25 MHz), short-duration events could be measured, and cracks propagating at significant fractions of the sound wave velocity resolved (Fig. 8).

% , , , , , , , , , , , , , , , , , ,

Fig. 7. Source force-time function of breaking glass capillary obtained by time domain deconvolution of recorded epicenter displacement [ 18] (full scale, 20ps).

2000 I ~ ° - 0.6V~

,500

OAV~ i" ~ o 8 °o o° I

tO00~ , ~°o ,e ° • o ° ," ~ °o o~: . . . . oO o

/ , . ° o{"4,.i ~o °o ° / °°% ~ ~ o o

~o 5°°1:"1,' '"~J:'".~--'~'~"~ ° ' " 2:. o _ . . . . . . . o2v~

O 50 IOO 1.50 2 0 0 2 5 0 3OO 5 5 0 CRACK D I A M E T E R / / z m

Fig. 8. The crack diameter and growth rate for every emission recording during two fracture processes: o, cleavage fracture of mild steel; e, intergranular fracture of electrolytic iron [19].

38

Ohira and Pao [20] have recently reported experiments on compact tension samples using a full six-channel recording system. They have been able to detect signals from inclusion interface failure in A533B (several millimeters) ahead of a pre-existing fatigue crack as they loaded the samples. They deduced the moment tensor for each microcrack and from this determined the mode (i.e. type I, II or III) and plane of fracture.

This progress in the development of acoustic emission for such fracture characterization is encouraging. However, much still needs to be resolved before the technique can be considered routine. Perhaps of most immediate importance is the effect of a pre-existing crack on the emitted acoustic emission signal.

A large pre-crack can be viewed either as perturbing the Green tensor of the bodies or the mechanism of the source. If the Green tensor of an uncracked body is used for deconvolution purposes, the pre-crack effect will be present in the deduced source function. In experiments on the geometries of compact tension samples, Wadley and Scruby [21] indeed found that, if the pre-crack presence was ignored during deconvolution, the deduced cleavage crack lengths were too large (by as much as a factor of 10). Physically, it was believed that the microscopic advance of the pre-crack (by a cleavage microfracture) caused a large pre-crack volume change (compared with that of the microcrack if isolated) and this in turn resulted in an overestimate of the microcrack crack length deduced assuming no pre-crack presence.

Achenbach et al. [22], using a two<limensional model, have since theoretically investigated the effect in some detail and have shown that significant signal amplification indeed does occur in the presence of the pre-crack. The disconcerting feature at this point is that the amplification factor varies with the particular microcrack-macrocrack configuration and, unless this is independently known, the possibility of introducing errors into the analysis exists. Very precise three~limensional location of the microcrack position might overcome the problem, however.

2.5. Fu ture research needs Research in basic measurement method-

ologies for acoustic emission has resulted in development of quantitative measurement and

analysis methods that potentially can reveal the dynamics of deformation and fracture micromechanisms. With the technology available today, the possibility exists of passively monitoring fracture tests on compact-tension- type geometries over a wide range of temperatures and environmental conditions. Coupled with traditional testing methods the potential exists to advance radically our understanding of the basic micromechanisms of fracture.

It is now possible to locate the site of a microscope fracture event to a precision com- parable with the dimensions of the source. The data to do this can be recorded sufficiently rapidly that this could be done conceivably at realistic microfracture rates (the limit is as high as a thousand per second; higher rates cause overlapping signals because of sample reverberations). Thus the critical sites where fractures first occur could be identified and correlated with microstructure features such as second-phase or inclusion particles or ab- normal impurity element concentrations. Additionally, it is feasible to observe, in three dimensions, the evolution of a distributed damage state and its coalescence to form a macroscopic fracture, a competence that could be critically significant as we a t tempt to develop fracture criteria for composite materials.

With existing technology the local adhesion at, for instance, grain boundaries, inclusion- matrix interfaces or particle-matrix interfaces in composite materials can be measured. By locating the site of a microfracture and either deducing the stress change of the source or numerically evaluating the local stress state at fracture, the strength of an interface can be deduced. The coupling of such measurements with systematic variations in thermomechani- cal t reatments offers the possibility of unique- ly determining the physical quantities that control adhesion at the microscopic level. The pay-off would be materials with scientifically tailored interface properties for opt imum performance.

Present fracture measurement methods are unable to resolve truly the dynamics of fracture when cracks are propagating at speeds close to the sound wave velocity. In this velocity regime, acoustic emission (if suitably performed) can observe, wi thout perturbation, dynamic fracture, even if the crack extends no more than one grain diam-

eter. These experiments, while at the fringes of today 's expertise, do offer the possibility of providing experimental insights to guide the development of micromechanical modeling of dynamic fracture and even the critical evaluation of modeling predictions.

These opportunit ies exist for traditional engineering materials. They promise to make valuable contr ibutions to our understanding of material-fracture relations. However, there exists an extra incentive to apply them to the emerging generation of materials (advanced ceramics and composi te materials) because these materials rely on the use of hard brittle consti tuents to bestow superior properties. The micromechanisms of failure in these materials promise to be strong sources of acoustic emission and, in turn, this acoustic emission could prove invaluable in guiding the development of fracture-resistant properties and attaining the so far intangible goal of lifetime prediction.

Just around the comer await numerous improvements to the basic measurement methodology of acoustic emission, developments that would extend the range of sample geometries and material types for which quantitative work can be performed. New analysis methods, based on frequency domain measurement, are in the offing and could solve some of the problems that may arise in the application of existing methods to experimental fracture studies. The incentive for this would stem from the successful fulfillment of some of the opportunit ies discussed above.

3. ULTRASONIC AND ELECTROMAGNETIC MEASUREMENTS OF FRACTURE-RELATED MICROSTRUCTURAL FEATURES

3.1. Presen t s ta tus Motivated by a pressing need to detect and

characterize cracks in engineering structures, a variety of ultrasonic and electromagnetic non-destructive evaluation techniques have been developed over the past 15 years. Some have a good deal to offer in fundamental studies of fracture. These measurement techniques can be categorized according to the material condit ion being examined. Their most traditional use is in the detect ion and sizing of macrocracks [23, 24]. Here a variety

39

of techniques are employed involving the scattering of ultrasound from cracks, the change in the impedance of an electrical coil as it is passed over a crack, the leakage of magnetic flux out of a part surface in the vicinity of a crack in a ferromagnetic material etc. In the first-order implementation of these techniques, the crack is viewed as a smooth, simply shaped reflector (e.g. a circular pair of stress-free surfaces in the ultrasonics case) and the information sought is its size, shape and orientation.

At the next level of sophistication, the crack is recognized as having structure. For example, asperities on opposite faces of the crack may be in contact, providing a path for the transmission of mechanical or electromagnetic signals across the crack faces. Proper interpretation of these changes can provide important information on plastic deformat ion and the resulting stresses which have developed in the vicinity of the crack during its growth and which would influence its future propagation. For ceramics, Kino and coworkers [25], by extending the theoretical predictions of Budiansky and Rice [26], showed and experimentally confirmed that the reduced stress intensity factor of an open crack could be deduced directly from the scattering of long-wavelength elastic Rayleigh waves. Evans and coworkers [27] used this technique to s tudy indentation-induced cracks in Si3N and observed the influences of plastic deformat ion at the crack mouth. These concepts are at present being investigated further by Khuri- Yakub and coworkers [28, 29] in the context of the inspection of machining-induced cracks in ceramic bearings.

For cracks grown in metals under fatigue loading, similar developments have occurred. Buck e t al. [30] made one of the first direct observations of fatigue crack closure, as pre- dicated by Elber [31], by monitoring variations in the transmission of Rayleigh waves past the crack during a load cycle. Yuce e t al. [32] have applied the aforementioned long-wavelength Rayleigh-wave-scattering ideas to the s tudy of closure effects on microcrack growth, and Golan and/krone [33], Thompson e t al. [34] and Whapham e t al. [35] have used various features of the scattering of ultrasound from macrocracks to investigate closure effects. Based on these ideas, a scenario for mapping the contact stress along the faces of

40

re) E ,r,J,

t -

ro

- o m

2

OF STAGE If GROWTH

END OF STAGE I GROWTH

0 ~ 400 500 600 700 800

POSITION, MILS

Fig. 9. Local interfacial stiffness along a fatigue crack grown in two stages.

the crack has been proposed by Buck e t al. [36].

Figure 9 shows the result of a partial implementat ion of this approach to a fatigue crack grown in two stages in aluminum alloy 7075- T651. There the local stiffness associated with the stress-induced contact of asperities (having values of zero and infinity for the cases of no contact and perfect contact respectively) is plotted as a function of distance from the crack tip. The rapid change near the position of 400 X 10-3 in corresponded to the partially contacting region near the current tip position while that at 700 X 10 -s in corresponded to residual contacts at the end position of the first stage of growth. This partial information on the history of the crack growth promises to be of importance to the prediction of future growth, and hence the deduct ion of remaining life.

In addition to crack face contact influences on the linear scattering of ultrasound by cracks, important non-linear effects also occur which to date have only received limited attention. As an example, Buck e t al. [37] have shown that the generation of harmonics by a Rayleigh wave during the microcrack initiation regime of fatigue can be used as a

o...o

o ~ c

E

O_

2 0 -

4 0 -

6O

8O

/

( I00

! !

/

/ /

O/~ 0 f

I /

/

I I I 80 60 40

Acfual remaining life (%)

/ 0

I 2O 0

Fig. 10. Fatigue crack lifetime prediction based on Rayleigh wave harmonic generation.

predictor of remaining life. Figure 10 illustrates this relationship for aluminum alloy 7075-T6 fatigued at 90% of the yield stress.

A number of other relationships have been observed which will not be discussed in detail here. Included are the relationships between (microcracks) distributed damage developed during the early stages of fatigue and ultrasonic at tenuation and a variety of electromagnetic interactions. Although quite useful and often easy to implement, in metals the electromagnetic techniques tend to give less detailed information about the structure near the tip of a crack because of electromagnetic shielding by the intervening conducting material.

At the microstructural level, a variety of features can be detected by both ultrasonic and electromagnetic techniques. Inference of grain size from ultrasonic at tenuation was pioneered by Papadakis [38] and field-imple- mentable techniques based on backscattered signals which could be observed from single- sided access were developed by Goebels [39]. For single phase materials consisting of random [40-43] or a preferred [44] orientation distribution of equiaxed grains, a solid theoretical foundation exists. However, empirical techniques must be used to interpret data obtained on more complex alloys [45, 46].

At the subgrain level, important progress has been made in the ultrasonic measurement

of porosity, a proper ty which is of interest because of its role in creep, fracture etc. A number of theories have been recently reported which describe the influence of porosi ty on the ultrasonic velocity and at tenuat ion [47- 52]. At low porosi ty volume fractions, these all show that, for spherical pores, the decrease in velocity is linearly proport ional to volume fraction of pores and independent of pore radius, whereas the at tenuat ion depends on both the volume fraction and the pore radius. The theories differ in detail at higher volume fractions where multiple scattering events must be considered.

Several experimental techniques have evolved and been applied to a variety of materials problems. Included are inferences of porosi ty in powder metallurgy nickel-based superalloys [53] and in cast aluminum [54] from the frequency dependence of the backscattered signals and the at tenuat ion informa- t ion that could be derived therefrom. Use of this information requires that the pores are either large with respect to grains or that the materials have a small elastic anisotropy, e.g. as in aluminum [55]. If this condit ion is not satisfied, porosi ty can still be estimated from the shift in velocity [56], and this effect has recently been used to s tudy the initial porosi ty in iron powder compacts [57 ] and the porosi ty nucleated during the high temperature deformation of copper [58]. These application studies have indicated that the aspect ratio as well as the volume fraction of porosi ty enter into the determination of velocity shifts.

These techniques for measuring grain size and porosi ty all make use of ultrasonic energy with wavelengths large with respect to the size of an individual microstructural feature. Con- sequently, a statistical average of the material properties is obtained. Recent developments in high frequency acoustic microscopy have made it possible to image directly microstructure on a scale of 1 pm or less. Features which have been observed include grain structure (including twins), microcracks, subsurface delaminations and various other forms of mechanical damage [59-62] . As an example, Fig. 11 consists of an optical micrograph (Fig. 11(a)) and an acoustic micrograph {Fig. l l ( b ) ) of a plain bearing A1-Si alloy on a steel substrate [59]. The high contrast image of a surface crack observed in the acoustic case results from the strong interaction of Rayleigh

41

Fig. 11. High contrast surface crack detection in a bearing alloy with acoustic microscopy: (a) optical micrograph ; (b) acoustic micrograph.

waves with the face of the crack. This is distinct from the much weaker interaction of optical waves with the thin (submicron) mouth of the crack, which can be total ly obscured by any metal smearing.

In addition to these intrinsic microstructural features, extrinsic properties such as temperature [63] and stress [64] can be sensed through their effects on the ultrasonic velocity. At their present state of development, these techniques do not have adequate resolution to measure local values, e.g. as would occur in the plastic zone near the top bf a crack. However, they are suitable for measuring more slowly varying fields in parts of simple geometry.

In each of the above ultrasonic cases, a well- defined physical mechanism exists which is believed to be responsible for the microstructure-measurement relationship. A number of

42

other correlations with physical properties have been reported, e.g. fracture toughness and hardness, for which such an understanding is lacking. Their omission in this discussion should not be taken to imply that they may not become of future importance after further study.

In electromagnetic interactions, a number of correlations have also been observed whose mechanistic interpretation is not complete. In non-magnetic materials, the list is fairly short, a good example being the relationship between electrical resistivity and hardness in the aluminum alloys [65, 66]. In magnetic materials, the phenomena are much richer. A number of important effects have been observed and reduced to practice to varying degrees, including even the development of commercial instrumentation [67-70]. Parameters sensed such as coercive field, remanent magnetiza- tion, and magnetic and acoustic Barkhausen noise, can be correlated with stress, hardness etc. The basic foundations on which these techniques are based, e.g. the interaction of Bloch walls with dislocations and point de-

fects, were studied extensively over the last 20 years [71, 72]. However, a quantitative understanding is not available in commercially used alloys. Recent work addressed at these issues include studies of the effect of carbon content on the hysteresis of low carbon steels [ 73 ], and the variatio ns in magnetoacoustic and Barkhausen emission with dislocation structures in iron polycrystals [74], precipitates in Incoloy 904 [75], and dislocation structure and grain boundaries in nickel polycrystals [76].

3.2. Future directions The ultimate goal of predicting the lifetime

of materials and hence achieving materials reliability can only be realized through a two- step process as illustrated in Fig. 12. The first involves the development of basic materials knowledge. How to synthesize the material must be learnt, its microstructure must be characterized, and the resulting physical and mechanical properties must be determined. An important element is the development of an understanding of the failure mechanisms of

MATERIALS

I MATERIALS R E L I A B I L I T Y

KNOWLEDGE MATERIALS UTIL IZATION

Synthesis I Characterization ] I

I Properties J I I s'Iv'c'l

MICROMECHANICS OF FRACTURE

• FAILURE MODELS

• MEASUREMENTS

-- MECHANICAL PROPERTIES

-- DAMAGE DETAILS

Fig. 12. Elements of materials reliability.

FUNDAMENTAL

KNOWLEDGE

FITNESS FOR SERVICE

• LIFE PREDICTION MODELS

• NON-DESTRUCTIVE EVALUATION FOR

m PROCESS CONTROL

MANUFACTURING QUALITY CONTROL

MONITORING SERVICE DEGRADATION

the material with the ultimate objective being the formulat ion of failure models which can be used in a predictive rather than a post-mortem interpretative manner. As in any scientific endeavor, this requires close coupling between theoretical and experimental activities. The traditional experiments are various forms of mechanical testing under static and/or dynamic loads with strain measurement, followed by analysis of the degraded material by microscopic techniques such as SEM, TEM, Auger analysis etc. The role of the ultrasonic and electromagnetic techniques discussed herein is to moni tor non<lestructively the progress of the damage in situ.

The second step is the utilization of the material, including the design of a part, its manufacture and service. The above nondestructive evaluation techniques can be employed (i) to control the quality of incoming material, (ii) to control the process, {iii) to check the quality of the fully manufactured part or (iv) to detect and characterize service- induced degradation. The last is of primary interest in the prediction of lifetime, although all have a role in ensuring that the material has the desired initial condition. It is important to note that the measurement techniques and interpretative procedures emerging in conjunction with the development and testing of failure models can often provide the basic foundations for these non-destructive evaluation techniques.

To illustrate these ideas, let us consider the example of the influence of fatigue crack asperity contact on ultrasonic scattering as discussed in Section 3.1. This work is at present in the stage of development of materials knowledge, and the ultrasonic measurements are being used to obtain a detailed characterization of the closure state of fatigue cracks

g r o w n under various conditions of variable R ratio loading and hostile environments. This information will be used to differentiate between different theoretical descriptions of the closure phenomenon. After experiment and theory have reached an appropriate agreement, the second stage of materials utilization will be entered. The previously developed understanding of the ultrasound closure interaction could then be used as the basis for the design of a fieldable non<lestructive evaluation technique and interpretative procedures for more improved predictions of lifetime.

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Several other specific examples can be cited. During monotonical ly increasing loads, an important precursor of failure is void nucleation, often at grain boundaries at high temperatures. Ultrasonic velocity (and in some cases attenuation) measurements can be used to detect this condition. In ferritic steels, various ferromagnetic properties are influenced by microstructural changes such as dislocation structure, and precipitate number and size. These interactions should be bet ter under- s tood and more extensively utilized. Higher resolution techniques such as acoustic and thermal wave imaging should be investigated. Although still in their infancy, they show great promise of providing information not readily obtained by other non-destructive means.

New physical phenomena need also to be explored further. Ultrasonic harmonic generation was briefly discussed above in the context of the detect ion of microcracks in the early stages of fatigue. This is just one example of the more general class of non-linear energy- material interactions. Other recently reported exploratory results include relationships between non-linear elastic constants and second- phase components in metals [ 77 ], harmonic generation by electromagnetic fields and hardness in steels [78], non-linear thermal properties [79] and fatigue crack closure effects [80]. These, as well as others, merit more careful s tudy so that they can be employed in material studies alongside the more traditional techniques discussed above.

3.3. New instrumentation For these non-destructive evaluation tech-

niques there are two sets of instrumentation issues: those associated with laboratory measurements and those associated with field or manufacturing measurements. In the former case, the major needs are the rapid acquisition and interpretation of large amounts of data. The factors defining this need are twofold. It is desirable to make as detailed a test of theoretical models as possible and clearly, the more the data, the more critical the test. For example, for spherical pores (as might be nucleated during fracture) it is well known that the ultrasonic velocity shifts linearly with porosi ty volume fraction. However, when the pores have oblate or prolate shapes, the proportion-

44

ality constant is a function of both the ellipticity of the pores and their orientation with respect to the wave propagation direction. Since the ellipticity and orientation change during deformation, it would be desirable to make measurements of several angles of illumination.

This specific example suggests that, in general, scanned measurements, including as a special case imaging systems, and multi- frequency measurements will be particularly important. Two examples which appear to have particular promise in the next few years are recently developed acoustic and thermal wave microscopes. These devices have been under development for the last decade and are just reaching the market-place in a form con- venient for general materials science studies. In a similar spirit, instruments which incorporate more than one measurement modali ty should be highly encouraged.

These same considerations also apply to manufacturing or field measurements. In addition, the problems associated with measurements in hostile environments and geometrically complex parts must be addressed. Non-contact sensors, which can be operated remotely from the part, are a good example of a solution to the former problem. In the ultrasonic case, this implies the development of optical and electromagnetic probes for the generation and detect ion of ultrasound. Probes with well- controlled and localized beam patterns address the second issue, as do advanced signal-processing techniques, including those falling under the title of "artificial intelligence", which could be used to distinguish various contributions to the total observed signal.

3.4. Impact A discussion of impact must again be sepa-

rated into the categories of laboratory and field measurements. The former is difficult to quantify economically, since the measurements are only one of an entire set of tools, used jointly in the development of a fundamental understanding of the fracture processes. One major role of the measurement is in speeding up the rate of development of this understanding by providing (a) new information about the material state and (b) both new and old information more quickly than would otherwise be possible. For example, if

information can be obtained continuously during a mechanical test, the need for stop- test metallography and the associated sub- jective interpretations may be avoided. In addition, it is possible to monitor the condition of a single sample through an entire sequence of steps, culminating in failure rather than in having to compare the conditions of a number of individual samples after different degrees of loading. An equally important consequence of the laboratory measurements is that they provide a knowledge base for field measurements, whose impact is discussed below.

The impact of field non-destructive evaluation is more readily quantified. In a heavily studied example from the military aircraft engine industry, it has been established that the reliable detect ion of low cycle fatigue cracks could allow a major extension of component lives. Life cycle cost savings of U.S. $250 million have been estimated for the F-100 engine alone [81]. It should also be noted, however, that new failure modes may be encountered during such an extended life, and the need for the study of fracture is increased by such a scenario. A second example involves the nuclear power generation industry. It has been estimated that replacement power costs for 1 day of unscheduled outage of a typical reactor are U.S. 8500 thousand [82]. Unanticipated failure modes, such as intergranular stress corrosion cracking, have come very close, particularly in the summer of 1983, to forcing extensive shut-down of a number of reactors. This is an excellent example of the need for coupled fracture measurement studies. The failure mode had not been anticipated when the reactors were designed. By the early 1980s it had been extensively studied and a significant level of understanding had been reached, particularly from an engineering point of view. However, despite the assumption that field inspection techniques were adequate, the magnitude of the problem was only revealed by the visual detect ion of leaks in a particular reactor a short period of time after in-service inspection crews had pronounced the pipes fit. The improved inspection techniques, which are emerging in response to this crisis, could have been developed more efficiently and economically during the-study of the cracking phenomenon itself.

4. APPLICATION OF SYNCHROTRON RADIATION TO ATOMISTIC AND MICROSTRUCTURE CHARACTERIZATION OF MATERIALS

4.1. Introduct ion Electrons, when constrained to move in

a curved path, experience a centripetal acceleration and thus emit electromagnetic radiation over a wide solid angle, as predicted by Maxwell's equations. As the velocity of electrons approaches the speed of light, the pattern of radiation (the Larmor pattern) is distorted by relativistic effects and exhibits a sharp directionality with extremely high fluxes. Normally, the speed of an electron is maintained constant in a storage ring (thus a constant centripetal acceleration is achieved) after being first accelerated to nearly the speed of light in a synchronous fashion. The radiation is then confined to within an ex- t remely thin and flat horizontal plane which contains the electron orbit in the storage ring. The vertical divergence of the radiation (orthogonal to the storage ring plane) is given by a relation 1/~ = 105/E (GeV)", where E is the electron energy.

The high energy tail of the continuous radiation spectrum moves into the X-ray energy region as the electron energy increases into the gigaelectronvolt range, thus providing applications in materials science. The unique features of this synchrotron radiation include a continuous spectrum with a high flux, brightness and intensity and extreme collimation in the vertical direction. In addition, synchrotron radiation has a pulsed time structure (pulses of less than a nanosecond width with a microsecond interval) and well<iefined polari- zation states. At most synchrotron facilities operated at the gigaelectronvolt range, the spectrum ranging from 3 keV (wavelength, about 4 A) to 30 keV (wavelength, about 0.04 A) can readily be made available for X-ray experiments. The flux is of the order of 10 la photons s -1 rnrad -1 mA -1 in a 1% energy bandpass. The vertical divergence 1/7 of the beam is 21"-52" for electron energies of 2-5 GeV. The divergence in the orbital plane is limited by apertures in the beam transport and is typically several to tens of millirads, which can be made narrower by additional slit systems. The radiation is linearly polarized in the orbital plane of electrons.

45

Three facilities, SPEAR at the Stanford Linear Accelerator Center, CHESS at CESR (Cornell) and NSLS at Brookhaven National Laboratory, can be used as a hard X-ray source in the U.S.A. The first two are electron- positron storage rings. The last, an electron storage ring, serves exclusively as an X-ray (light) source. In all these facilities, similar individual beam lines have been developed for various experiments in solid state physics, physical chemistry and materials science, by utilizing the unique features of synchrotron radiation, particularly high flux and wavelength tunability.

Typically the beam lines using hard radiation whose wavelength is shorter than 3 A include facilities for real-time microradiography and topography, small-angle scattering, inelastic scattering spectroscopy and crystallography. Scientific problems are aimed at advancing materials science by utilizing techniques previously thought to be impractical. For example, the high flux and extreme brightness (which are of the order of 104 higher than those of X-rays generated in conventional X-ray tubes) greatly improve the observation of fine-scale (about 1-10 ~tm) phenomena in materials in real time. The energy and momentum resolution can be made extremely good so as to increase the precision in the determination of the atomic coordination and arrangement. Furthermore, the wavelength tunabil i ty of the radiation with the desired energy bandpass enable focus- ing on a particular atomic species (elemental tunabili ty) for the precise determination of behavior under the influence of neighboring atoms in different local volumes.

Some of the results using synchrotron radiation have already indicated great promise for various material problems. These results are discussed as typical examples to show how synchrotron radiation can be used in a variety of problems, and to propose new experiments specifically for problems related to fracture. Topics will be divided into two areas: research concerning phenomena statistically averaged in space (traditional solid state physics), and research concerning phenomena occurring differently in spatially separated regions (microstructural effects). When necessary, both types of measurement can be made as a function of time.

46

4.2. Statist ically averaged e f fec ts The first example is in the area of crystal-

lography. The high flux and continuous energy spectrum of synchrotron radiation enable the rapid acquisition of data, such as lattice parameters, using energy-dispersive diffractometry, from samples, in situ, under various environmental conditions. For example, Skelton et al. [83] have exploited the energy-dispersive diffractometry technique under extreme high pressures of tens of gigapascals (using a dia- mond anvil pressure cell) and a wide range of temperatures (0 .03-3273 K). Since such a high pressure can be achieved only by reducing the size of the pressure chamber to microscopic dimensions (10 -s mma), it is impossible to obtain data within a realistic time frame using conventional X-ray sources. Use of synchrotron radiation makes it possible not only to obtain such data bu t to obtain it so rapidly that it also permits the s tudy of phenomena hitherto immeasurable, e.g. structural measurements of phase transformation reaction rates in the time frame of seconds to minutes. For example, Skelton et al. observed the first- order phase transition in potassium iodide from the NaC1 structure to the CsC1 structure, as a function of pressure and temperature. At room temperature (298 K), this transition is induced at a pressure of 1.8 GPa, while at a pressure of 1.5 GPa the transition can be thermally induced at about 373 K. For these observations, each diffraction energy spectrum was recorded in 100 s.

The capability of obtaining data in such a small volume fraction leads to another technique that is effective for the measurement of residual strains caused by inhomogeneities in bulk materials [84, 85]. Essentially the entire three-dimensional volume of the sample can be mapped with adequate spatial resolution (0.5 mm X 0.5 mm X 0.5 mm) to obtain the residual strain distribution and, if it is wished, the elastic strain tensor as a function of position. The analysis of the results obtained by this technique does not require any as- sumptions on elasticity and the concept of a continuum medium as an elastic body. A completely fixed geometrical arrangement of the source, sample and detector only is required; unlike the ordinary diffraction techniques, the entire diffraction patterns are obtained in this fixed arrangement. A successful demonstrat ion of the application of this

technique was the detect ion of an unusual concentrat ion of residual stresses in a welded zone [85].

In small-angle scattering, the high flux and extreme collimation of synchrotron radiation permit the further improvement of X-ray angular resolution within a limited space. For ordinary X-ray sources, a distance of the order of 10 m is required to obtain adequate resolution. By contrast, with synchrotron radiation a double-crystal system using asymmetric diffraction can be employed by placing a sample between the two crystals to prepare a nearly parallel beam (divergence, 1") and to detect the scattered beams to within a second of arc of the direct beam [86]. An intensity profile of small-angle scattering from a block copolymer, polyethyl ine-polystyrene, showed a well-resolved isolated peak corresponding to a chain length of 750 A. Normally, the existing "pinhole" small-angle scattering instruments barely detect this peak. From the observed profile, this technique demonstrated the capability of detecting 4000 A or larger particle sizes or chain lengths.

An absorption spectrum near the X-ray absorption edge of an element has a fine structure variation as the incident X-ray energy changes. This is called extended X-ray absorption fine structure (EXAFS). Such a fine structure is closely related to the chemical state and the nearest-neighbor interaction of an a tom of this element in a condensed matter. These fine structures normally appear within a range of several hundred electronvolts above the edge. A high energy resolution is therefore required to obtain these fine structures. The elemental tunabili ty and high flux of synchrotron radiation provide a significant advantage in this area of research. Here we describe two examples of EXAFS experiments.

When amorphous alloys are annealed far below the crystallization temperature, structural changes (i.e. "relaxation") and a con- comitant reduction in density are often seen. Partial radial distribution functions around the nickel atoms in glassy NiP, obtained from EXAFS, have indicated that, in the NiP case, at least two metastable states exist for a single composition. EXAFS spectra have been obtained for glassy Ni75P25 samples d.c. plated and pulse plated as well as for polycrystalline NiaP powder and pure Ni foils. The results indicate that the measured spectra are domi-

nated by the distances between the nickel and the phosphorus atoms, and these distances in the pulse plated sample are stretched by about 0.1 A, compared with the crystalline NiaP and the d.c. plated samples [87]. These results are in agreement with recent MSssbauer measurements on similar samples and with published density results. However, none of the earlier methods could probe the radial distribution directly.

To obtain structural information on thin (less than 30 A) or interfacial layers on materials, a more advanced method is required. This is called surface EXAFS, which permits measurements from samples in situ under environmental conditions. As an example, a structural s tudy of passive films on iron is described here. The samples were vapor- deposited iron films 50 A thick on glass, which were passivated by either a chromate or a nitrite passivating solution. Near-edge and EXAFS spectra were measured on these films, as well as on bulk germanium and on an air-exposed film. The energy position of the K edge discontinuity of a material is a function of the effective charge and the electronic configuration of the absorbing atom. The edge shift of the chromate-formed film was 5.0 + 0.2 eV lower than either the nitrite- formed film or the air-formed film (which were within 2 eV of each other). This indicates a significantly lower coordination charge for the chromate-formed film and a probable greater covalency in the bonding of the iron in this film [88].

The EXAFS data were Fourier t ransformed to obtain the radial distribution function about the iron site. A comparison of the first- neighbor and the second-neighbor coordination shells in the radial distribution funct ion can be used as a measure of " the relative disorder in a material". It was found that the air-formed film exhibited the most order (crystallinity) and the chromate-formed film the most disorder, while the nitrite-formed film was intermediate between the two. This Suggests that the passive layers in the chromate-formed film are arranged in a glass-like structure about the iron atoms [89].

4.3. Microstructural effects The high flux and extreme brightness of a

synchrotron X-ray source provide the unique oppor tuni ty to tackle challenging materials

47

science problems by utilizing techniques hitherto considered impractical. In particular, microstructural effects can be observed and measured in situ on a real-time basis even under simulated environmental conditions. The technique used for microstructural studies is, in general, called topography [90]. When a white beam of X-rays is incident on a crystal, an array of diffracted beams (a Laue pattern) is created, where each spot of the Laue pattern is of roughly the same size as the beam on the sample. Within each spot can be observed a fine structure related to the microstructure in the sample. This fine structure is the basis of the method of X-ray topography. Depending on the wavelength of the radiation, the diffraction geometry and/or the perfection of the crystalline samples, the depth (or thickness) of material observed can range between 1 #m (Bragg geometry) and 1 mm (Laue geometry). Topographs are obtained for the detailed interpretation of images produced by X-ray diffraction by real crystals.

Synchrotron topography can be categorized into two basic classes: white radiation topography and monochromat ic radiation topography. Both utilize the spatial variation in intensity in the direct beam or diffracted beams, due to absorption and strains (caused by imperfections), to image the microstructural details in materials.

With the use of a suitable monochromator system, synchrotron radiation can be used for topography to permit the highest levels of sensitivity to lattice defects in crystals. How- ever, for crystalline materials of ordinary quality, especially metals and alloys, less refined X-ray optical conditions may be used to advantage. Such a technique is white radiation topography [91]. When the sample is placed in the white synchrotron beam, the direct beam contains the microradiographic image of the sample, and the diffracted beams form a Laue pattern. These images are recorded on an imaging detector serving as a video monitor and, if the images are of interest, they are recorded on high resolution film. The simplicity of this technique lies in the fact that grains of arbitrary orientation are imaged because the sample selects the appropriate wavelength to satisfy the Bragg condit ion locally. Particular- ly exciting is the ability to apply X-ray topography techniques to polycrystalline materials and "single crystals" of poor perfection.

48

An example of this application is seen in the area of rapid solidification to understand the role of interfaces during solidification. A sample of A1-5wt.%Sn was observed during heating at 620 °C in real time. In another example, half of the transmission Laue pattern was monitored on video together with the micrographic images of grains and grain boundaries in the direct beam. (By inserting a pinhole, selective area diffraction can be observed.) The recrystallization process of a deformed aluminum sample maintained at 470 °C was observed, and the areas of grains were plotted against t ime to be compared with the current theory of general coarsening [91]. Similar experiments were carried out with CdSn for liquid phase sintering. The melting of a tin crystal observed in real time was interesting. This observation yielded quantitative data on the interface mot ion and interface shape, including the observation of recalescence [92].

The application of white radiation topography alone is not sufficient to determine the details of microstructures and faults in material. For example, the subgrain structure {dendrites) of a monocrystal turbine blade superalloy was observed clearly by monochromatic radiation topography. The size, orientation and population of dendrites are related directly to the creep performance of the turbine blades. An Fe-A1 alloy sample shows more detailed fine structures, such as magnetic domains and subgrain structures, in this method than in white radiation topography. For improvement in the quality of single-crystal materials, such as electro-optical materials and II-VI compound materials, and for research aimed at a fundamental understanding of the nature of interfaces, monochromatic radiation topography is needed [93l.

The assessment of the crystal quality can be done effectively in real t ime by monochromatic radiation topography as well. The growth conditions of materials are connected to their resultant quality, and the effects due to device preparation processes can be analyzed and seen in CdTe, GaAs, lithium niobates and bismuth silicates [94]. The imperfections in these materials have not been seen by the traditional methods. Also, radiation can im- pinge on the sample at an extreme glancing angle. This grazing-angle diffraction enables

us to study the surface of the material as well as the interface of the film on substrate materials, as seen in some of the examples, such as CdTe on InSb, and other electro-optical materials. Also, this technique is useful for identifying the role of microhardness testing for coating evaluation, as seen for silver on copper, and palladium on copper. Another class of high technology materials, such as multilayered thin films, can be studied in a similar fashion.

For quantitative measurements of strain fields around imperfections and near interfaces, a series of topographs, using a monochromatic incident beam, is required as a function of the incident grazing angle and the observation {detecting crystal) angle [95-97] . These images can be stored on video tape for quantitative analysis. Such an experiment was performed with a copper crystal with Knoop indentations to obtain quantitative information on strains associated with indentation impressions [98], in addition to information on the active slip systems and slip direction [99]. Without synchrotron radiation, it is impossible to obtain such a series within a realistic time frame. This approach, in a general line-broadening context, opens up a new avenue of characterizing imperfections locally and quantitatively in terms of a "structure factor" which contains local atomic displacements.

Finally, the most important aspect of synchrotron radiation topography must be mentioned. It is the real-time observation of changes in local phenomena (microstructural changes) within materials, such as phase transitions and tensile testing. To overcome the present limit of spatial resolution {about 10 #m) of imaging detectors, the X-ray image magnification technique has been developed [100]. As shown in Fig. 13, a beam strikes a sample after the monochromator crystal and the condenser crystal prepare the extremely collimated intense beam, and the diffracted (or transmitted) beam that contains microstructural details is magnified by the X-ray magnifier. This utilizes asymmetric diffraction from the surfaces of two perfect {say, silicon) crystals. The first diffraction magnifies the images in the vertical direction, and the second in the horizontal direction. Pairs of silicon crystals provide a different magnification factor, up to 100 times. This

SYNCHROTRON B E A M ~ ~ j J f

O j

MAGNIFIER

Fig. 13. The geometrical arrangement of crystals (optical elements) used for monochromatic synchrotron topography. A pair of asymmetric crystals after the sample provides the magnification of topographic images before detection. The sample can be set in the transmission (shown) as well as surface reflection geometries.

49

Fig. 14. Silicon solar device: combination of microradiography and topography seen on a video monitor as the sample is rotated from the beam head-on position ((a) microradiograph) to (b)-(d) the (220) Bragg condition in transmission.

t e c h n i q u e m a k e s it poss ible to observe micro- s t ruc tu ra l deta i ls ( a b o u t 1 # m ) in the b e a m images in real t i m e using an image d e t e c t o r w i th the conven t iona l spat ia l r e so lu t ion on a video m o n i t o r .

F o r rea l - t ime obse rva t ion , the c o m b i n a t i o n o f m i c r o r a d i o g r a p h y and t o p o g r a p h y is o f t e n f o u n d to be useful . An e x a m p l e is s h o w n in Fig. 14 fo r a si l icon device mate r ia l . When the sample is inser ted in t he beam, the n o r m a l m ic ro r ad iog raph i c image is seen on a video m o n i t o r , showing the device cons t ruc t ion . As

the sample is being b r o u g h t to a Bragg condi- t ion fo r d i f f r ac t ion , the image reveals defec t s and s t rain fields wi th in the sample ; the images n o w show t o p o g r a p h i c details . In this example , the X-ray magni f ie r was n o t used. With the use o f X-ray magni f ie rs , deta i ls d o w n to a lmos t 1 p m can be observed .

4.4. Future research needs As descr ibed above, the avai labi l i ty o f

s y n c h r o t r o n rad ia t ion w i th the r ecen t devel- o p m e n t of new t echn iques opens up a new

50

avenue for obtaining information on microstructural details quantitatively as well as visually on a real-time basis. The areas discussed above represent only the tip of the iceberg and can be applied with little modification to specific problems in many areas of materials science. For the study of fracture mechanisms, some possible experiments using the unique nature of synchrotron radiation will be discussed in the following.

Energy<lispersive diffractometry described in Section 4.2 is directly applicable to the determination of strain fields around existing macrocracks and can be used to identify microcracks by comparing the observed strain field with the calculated strain fields around macrocrack models. For failure analysis, cavitation along or near grain boundaries (intergranular creep fracture) and cavities ahead of cracks are often evaluated by small- angle neutron scattering. This analysis method is valid when there are many cavities in materials. When cracks are isolated and, in turn, cavitation is locally limited and dispersed in materials, small-angle scattering using the technique described in Section 4.2 may be useful. Here a problem may arise concerning the analysis method, because of the lack of small-angle scattering theories for a small number of dispersed cavities.

For problems related to interfacial embrit- t lement, currently Auger electron spectroscopy (or electron spectroscopy for chemical analysis) is used to identify the types of atom in grain boundaries. The chemical and electronic states of these atoms after segregation to the grain boundaries influence the nature of embri t t lement significantly. It is desirable to determine these states by the measurements of near-absorption-edge structures and extended absorption spectra, in order to assess the precise role that these atoms are playing in cohesion at and near grain boundaries. The measurements of fluorescence near the edge and EXAFS spectra from samples having cracked grain boundaries are very useful for this purpose; if necessary, a focused incident beam (microprobe beam) can be used to pinpoint the location of measurements. Syn- chrotron radiation is ideal for this type of measurement.

There are many questions concerning the formation and propagation of cracks. Are there dislocation emissions before cracks initi-

ate, and are crack tips shielded by dislocations? Are there stress (strain) concentrations ahead of cracks before the cracks propagate? Do observations performed by electron microscopy truly represent the behavior of cracks and crack propagation in bulk materials (not thin foil effects)? Can the propagation of cracks be observed in the environmental conditions, such as stress corrosion or hydrogen cracking conditions?

Figure 15 shows a sequence of synchrotron radiation (monochromatic) topographs during the development of dislocation loops around a crack tip in a silicon crystal [101]. The topographs were taken as the sample with a notch and an initial crack was heated at 750 °C under a load. With the use of the X-ray image magnification technique [100], crack propagations can be observed in real time by topography as well as by microradiography, as in the application of the X-ray magnifier to dental microradiography [102, 103].

Particularly useful is the combination of topography and radiography in the real-time observation of crack propagation. This combined technique, as demonstrated in Fig. 14, is capable of showing not only the motion and shape of cracks in the microradiographic mode but also the strain field around cracks, even in front of the crack, when the sample is rotated to a Bragg condition. The sample thickness can be made large to distinguish the foil effect from the bulk effect. Currently, fractography has been put forward with the availability of synchrotron radiation [ 104, 105]. Clearly the increased use of synchrotron radiation in the area of fracture science will stimulate the further understanding of fracture mechanism at the atomistic level.

5. USE OF SMALL-ANGLE SCATTERING IN THE STUDY OF DAMAGE ACCUMULATION

5.1. Introduction The technique of small-angle scattering has

many advantages as a method for microstructural characterization and for detect ion and quantification of damage. The formal theory of small-angle scattering by X-rays and by neutrons is similar. However, a comparison of the two techniques shows that neutron scattering has many attractive features which stem from the behavior of neutrons. Neutrons have

51

1 m m

I I

Fig. 15. A sequence of synchrotron radiation topographs showing the development of dislocation loops around a crack tip in a silicon crystal (the time indicates the total elapsed time measured from the initiation of heating) (T =750 °C;K I =0.66MPam): (a) 380s; (h) 620s;(c) 920s; (d) 1200s; (e) 1800s; (f) 2700s; (g) 3600s. A monochromatic (0.8 A) beam was prepared from synchrotron radiation.

great pene t r a t i ng p o w e r , so t h a t sample volumes o f the o rder o f a few t en t h s to 1 cm s can be i r radia ted. Thus , bu lk p rope r t i e s can be e x a m i n e d . Small-angle n e u t r o n sca t te r ing

is well sui ted to the s t u d y o f such p h e n o m e n a as grain b o u n d a r y cavi ta t ion , in te rgranular mic roc rack ing , duc t i le void f o r m a t i o n or o t h e r f o r m s o f bu lk damage . Di s to r t ion o f the

52

scattering measurements by surface effects, e.g. as introduced by sample preparation, is minimized. Since large volumes of sample are interrogated by the beam, the statistics of the results on the scattering population are excellent. Each measurement may include 10 s- 1012 scatterers or more, whereas many TEM foils must be examined to gather data on a few hundred. The use of large specimens means that large features can be examined. Recently, workers at the National Bureau of Standards [106, 107] have shown that extensive information can be extracted on defects or microstructural features even though these features are in excess of a micrometer and interpretation of the scattering data is complicated by the occurrence of multiple scattering. In contrast, features as small as 1-2 nm may be detected. Thus, small-angle neutron scattering offers the possibility (which cannot always be realized in practice) of characterizing scattering features over the interesting size range from 10 -9 to I0 -s m.

The scattering length density p for neutrons does not vary monotonically with atomic number Z, as for X-rays. Rather the relationship between p and Z is more random, and a number of very light elements are comparatively good scatterers. Microstructural features and phenomena (e.g. the nucleation and growth of precipitates, and spinodal decom- position) in alloys made up of elements close together in the periodic table can be followed by neutron scattering whereas the X-ray scattering contrast between the constituents is too small to make this possible. (Many of the commercially important alloys are made up of elements which lie close together in the transition series.) Both hydrogen and deuterium are comparatively good neutron scatterers, and the scattering length of hydrogen differs considerably from that of deuterium. Neutron scattering has been an exceptionally useful tool in polymer characterization. Certain hydrogen atoms or groups of atoms can be replaced by deuterium, so that the neutron beam sees only a dilute concentration of scatterers whereas actually the polymer molecules are essentially undisturbed from their normal environment by the substitution. Long-standing controversies over polymer chain configurations are definitively answered only through the use of small-angle neutron scattering. The strong scattering behavior of

hydrogen enables neutron scattering to be used to study the behavior of hydrogen in metals and thus to help to establish the underlying mechanisms of hydrogen embrittlement.

Neutrons possess a magnetic moment and therefore are sensitive to the magnetic behavior of the sample material. The use of neutron scattering in establishing the spin lattice of ferromagnetic and antiferromagnetic materials

• is well known. Because two scattering length densities are involved in neutron scattering from ferromagnetic materials, one nuclear and one magnetic, extra information is available from the measurements which may be used to identify the nature of the inhomogeneity or defect causing the scattering.

It is of interest to examine the type of information which can be extracted from small-angle scattering data. If measurements can be taken down to sufficiently small values of the scattering vector q (q = (4 Ir sin 0)/~, where 0 is the Bragg angle and ~ is the wavelength of the neutrons), the slope of a plot of the log of the differential scattering cross- section d Z / ~ vs. q2 should directly give one characteristic measure of the radius of the scattering population, the Guinier radius RG. RG 2 = (RS) / (R6) . Here R is the radius of the scatterers (assumed spherical) and () denotes an average value. It can be seen that RG is heavily weighted to the large end of the size spectrum. The total surface area of the scatterers can be found from the limiting behavior of dZ/d~2 at high values of q. The volume fraction of the scatterers can be calculated from the integration of dZ/d~2 over all the reciprocal space. Combining the values of surface area and volume fraction for the scat2erers yields another characteristic radius Rp, the Porod radius, which is equal to (RS) / (R2) . The Porod radius is also slanted to overrepresent the large scatterers, but not so strongly as R e . Anisotropy of the two<limensional scattering pattern contains information about the shape or orientation of the scatterers. For closely spaced scatterers, the value of the position of a maximum in the curve of dZ/d~2 vs. q can be used to find an average spacing between scatterers. Finally, under favorable circumstances as regards the scattering population (e.g. dilute, sufficient scattering contrast with the matrix, and only one type of scatterer) and the instrumentation (sufficiently large q range to get well into the Guinier range at the

low q end and into the Porod region at high q values, good resolution, and sufficiently high flux of neutrons), it may be possible to invert the scattering data to obtain size distributions for the scatterers.

5.2. Current s tatus It has been demonstrated that small-angle

scattering is an excellent technique for the s tudy of damage accumulation in materials subjected to deformation. In many cases, scattering measurements are the only means currently available whereby the kinetics of a damage process may be followed. Typically, a series of specimens is prepared in which all the test parameters except time of testing are held constant. Examination of the evolution of the size distributions (or other measure of the state of the scattering population) yields information on the kinetics of the phenomenon observed. The nucleation rates, growth rates etc. can be obtained. Dependence on the test parameters can be determined by varying one parameter from series to series. The results obtained concerning damage mechanisms are in themselves of scientific interest. In addition the quantitative information about the dependence of damage accumulation on deformat ion parameters which comes from small-angle scattering experiments is valuable in the derivation of accurate life prediction relationships.

Small single neutron scattering has been used with considerable success to s tudy the phenomenon of grain boundary cavitation [108-113] . Small-angle neutron scattering is very sensitive to the presence of voids. A void volume fraction of 1 0 -6 c a n be picked up; under cyclic stressing, cavitation can be detected after as little as 15 s o f stressing [109] (Fig. 16). Both fatigue- and creep-induced cavitation, as measured by small-angle neutron scattering, have been modeled with some success using existing theories of cavitation [109, 1 1 0 , 1 1 4 ] (Fig. 17). The technique of small-angle neutron scattering also has proved of value in the s tudy of cavitation in ceramic materials. Page e t al. [112, 113] have been able to demonstrate with small-angle neutron- scattering measurements that a fundamental difference exists in cavitation behavior between ceramics with a glassy phase at the grain boundary and those with clean grain boundaries. In one case it appears that void

53

~ IO -'~ ' I

l ...o -''j / ......-

/

I0 -~ : / / /

/ z2"

I x io 2 I x Io 3 I × Io4

FATIGUE TIME , SECONDS

Fig. 16. Void volume fraction AV/Vas a function of fatigue time (normalized to 60pm grain size) in copper fatigued at a stress amplitude Aa of +34.0 MPa and p = 17 Hz at various temperatures: o, 405 °C; A, 486 °C; o, 567 °C. (From ref. 109.)

T E - 20 xlOn ?

E uf £b

5 > u.. IOx IOn 0

z

I00 200 500 VOID RADIUS, nm

#v D ~ c

\1 i

400 500

Fig. 17. Size distribution of voids calculated from small-angle neutron-scattering data (line A), modeled from corrected Hull-Rimmer theory (line B) and from theories of growth by coupled diffusion/plasticity (lines C and D) (creep; T = 405 °C; o = 27.6 MPa; 4.3 X 104s). (From ref. 110.)

nucleation occurs quickly on the onset o f creep deformat ion and then ceases bu t the voids continue to grow throughout creep. In the other case, nucleation is continuous throughout creep bu t the growth of the voids is highly transient and quickly ceases. Observa- tions of the anisotropy of isointensity curves on the two-dimensional position-sensitive detector indicated that changes were taking place in the shape of the voids as they grew.

54

Small-angle neutron scattering has been used to study microcracking in brittle material [115], carbide coarsening in a ferritic stainless steel under simple aging [ 116 ], accelerated coarsening under deformation [117], and radiation damage in the form of voids [118, 119] or of small copper-rich clusters in a precipitation-hardened steel [ 120 ] which cause embrittlement.

5.3. Current problems in the use o f small-angle scattering for damage studies

It is evident that small-angle scattering is a valuable tool in the study of microstructures. However, some difficulties remain in its use.

5.3.1. Adequacy of scattering data and reliability o f analysis Scattering is an indirect measurement tech-

nique, and therefore great care must be taken that the final results are indeed correct. In- complete or slightly inaccurate data as well as poor methods of data analysis may lead to conclusions which are grossly in error. For example, one of the most commonly used parameters obtained from small-angle scattering measurements is the Guinier radius RG. The Guinier radius can be found directly from the slope of a plot of log(dZ/d~2) vs. q2 at the low q end of the scattering data. This plot should be a straight line. The Guinier approximation is valid out to a q value such that qR ~ 1.2, where R is the radius of the largest scatterer which occurs with appreciable frequency in the ensemble of scatterers. It is stated that the Guinier approximation is valid for spherical scatterers even out to qR ~ 2.0. To check whether or not the "Guinier radius" calculated from a particular experimental scattering curve is valid, the product "RG "q is found and compared with 1.2 (or 2). Here q is the largest of the small q values used in the determination of the Guinier slope and therefore it is close to qmin, the smallest q for which a scattering intensity can be measured.

Figure 18 illustrates the grave errors that are likely to be encountered in finding R6 from small-angle scattering measurements if the scattering population contains some large scatterers, as is frequently the case in damage studies. In Fig. 18 we see the apparent RG which is calculated from a small-angle scattering curve obtained from an ensemble of scat-

500- 2.5

4 0 0 - - ± -*- 2.0

"~ 300- k5

o -

~- 2 0 0 - t.o o_ <

iO0 0.5

, , 0.0 0 4.0 x iO -3 8.0'x iO -3 1.2 xlO -2

qmi. (~,-'.)

Fig. 18. Var i a t ion in " R G " wi th qmin for an ensemble of spher ica l sca t te re rs whose size varies f r o m 0 to 70 n m radius, w i th a peak in t he size d i s t r ibu t ion at 25 nm. The s lope of the Guin ie r plot is t a k e n f r o m the least- squares f i t t o po in t s at qmin, ~qmin and 3 ~-q mi n" Also s h o w n is a curve o f " R G " q m i n vs. qmin .

terers whose size distribution ranges from 0 to 70 nm radius and peaks at about 25 nm. Here the apparent RG, denoted "R~" , is plotted against q ~ . It is assumed that the slope of the Guinier plot is obtained from points at qmm, ~ qmm and ~qmm. AS expected, for extremely low values of qm~, "RG" is a constant and equal to the true value. As qmh~ increases, "RG" decreases. The product " R G " q ~ first increases linearly with increasing q~m and then becomes remarkably stable at around 20. Similar behavior is found when other size distributions are analyzed. After some minor oscillations, the product "RG"qm~ reaches a value close to 2.0. It can be concluded that values of R G calculated from Guinier plots can be seriously in error unless "RG"qmin is well below 2.0. It has been observed by a number of investigators that " R e " remains constant in a series of samples in which the time of creep (or fatiguing or coarsening etc.) varies while all other test parameters are held constant. This constancy of "RG" is taken as evidence that the number of scatterers may be increasing with time but their average size does not. Since the product " R G " q ~ approaches a constant value (--2) for a wide variety of size distributions and, since the same q~m is used in the measurements of a series of samples, the conclusion t h a t R ¢ is constant is likely to be quite wrong.

Failure to achieve really the region of Guinier behavior leads to significant errors not only in RG but also in such quantities as the calculated values of volume fraction of scatterers and of size distributions [121]. Instrumental smearing effects and inappropri- ate methods of data analysis also contr ibute to the problem of achieving reliable results.

5.3.2. Difficulties in analyzing small-angle scattering measurements from complex systems In the more successful example described

in Section 5.2 the materials studied consisted of an ensemble of scatterers (voids, cracks etc.) in a matrix whose structure remained constant during deformation. Thus the small- angle scattering from the feature under investigation could be obtained by a subtraction process using a control specimen. However, in most cases of interest, several classes of scatterers are present, all of which may change during deformation. Thus, it is impossible to determine what port ion of the total scattering is coming from a given consti tuent ensemble of scatterers. A good example of ambiguity in the scattering data when more than one ensemble of scatterers is present is furnished by a small-angle neutron-scattering study of carbide coarsening in a ferritic stainless steel which results from aging or high temperature deformat ion [116 ,117 ] . A principal use of this steel is in power generation applications, where it must maintain its mechanical strength during long periods of high temperature service. The carbides MC in the steel contr ibute to its strength; the carbides M23C6 probably contr ibute little. As MC coarsens (or perhaps disappears), the material softens noticeably [122]. If the scattering from the various carbides could be sorted out, small-angle scattering could serve as a useful non-destructive evaluation method for following the microstructural changes in a component made from this type of steel. Another example of ambiguity in small-angle scattering data concerns cavitation at the interface of hard particles. If such particles are present (as is almost always the case in engineering alloys), they provide the favored sites for void nucleation. Since both the particles and the extent of cavitation change in the course of deformation, it is difficult to sort out the contributions from each class of scatterer to the total. The fact

55

that the particles and the voids are correlated also complicates interpretation of the data.

5.4. Future research needs It should be possible to use the phenome-

non of anomalous scattering to separate the scattering components from the various types of scatterer in a complex alloy. The procedure would involve small-angle measurements using X-rays from synchrotron radiation. Such X-rays not only are intense but their wavelength can be changed over a wide range. By measuring the scattering both close to and some electronvolts away from the anomalous scattering edge of an element contained in a particular class of scatterers, it is possible to determine the scattering coming just from those scatterers. For example, it would be possible to determine the scattering coming from each of the carbide species in the ferritic steel mentioned above or to s tudy cavitation in alloys containing hard particles. The field of microstructural studies by small-angle scattering would be expanded from the consideration of model alloys to real complex engineering alloys.

As regards reliability of the small-angle neutron-scattering data, improved neutron- scattering facilities and instrumentation are needed to increase the q range over which measurements can be made and thus permit reliable extrapolations of the scattering data using asymptotic behavior. A higher flux at neutron sources will increase the accuracy of the data and extend the number of systems which can be studied by neutron scattering. Additional theoretical work is needed to develop better methods of correcting and inverting the data to obtain size distributions and other useful information about the scattering population.

In summary, the improvement of scattering facilities and development of the use of new measuring techniques should make it possible to follow in detail the accumulation of damage in complex alloys when it occurs in the form of coarsening, bulk cavitation or microcracking. The information gained on the dependence of the degradation process on the deformat ion parameters will permit the verification of existing damage models as well as assist in the development of new theories and will be a valuable input into life prediction models.

56

5.5. Summary A host of novel techniques are emerging for

the fundamental study of fracture and its control in the field. In contrast with traditional methods based on strain gauging and load monitoring, coupled perhaps with destructive metallography, the new techniques offer non- invasive capabilities. Thus, it is possible to make measurements under conditions suitable for valid comparisons with theoretical work and in such a way as to guide the theorist to new issues, in particular the elusive goal of lifetime prediction.

The techniques assessed here, acoustic emission, ultrasonics, electromagnetic nondestructive evaluation, synchrotron radiation and small-angle scattering each offer complementary information about the fracture process. Their combined use in a fracture study, either sequentially or simultaneously, promises to provide a rich new insight into the physical phenomena at play during fracture.


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74 D . J . Buttle, C. B. Scruby, J. P. Jakubovics and G. A. D. Briggs, Philos. Mag. A (1987), in the press.

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75 D. J. Buttle, C. B. Scruby, J. P. Jakubovics and G. A D. Briggs, Philos. Mag. A (1987), in the press.

76 R. Ranjan, O. Buck and R. B. Thompson, in D. O. Thompson and D. E. Chimenti, (eds.), Review o f Progress in Quantitative Nondestruc- tive Evaluation, Vol. 5, Plenum, New York, 1986, p. 1335-1344.

77 J. S. Heyman, S. G. Allison and K. Salama, 1983 Ultrasonics Syrup. Proc., IEEE, New York, 1983, p. 991.

78 K. J. Law, K. J. Law Inc., Detroit , MI, personal communication, 1986.

79 H. K. Wickramasinghe, IBM Watson Research Laboratories, Yorktown Heights, NY, personal communication, 1986.

80 J. D. Achenbach and A. N. Norris, J. Nondestr. Eval., 3 (1982) 229.

81 C.J . Annis, J. S. Cargill, J. A. Harris and M. C. Van Wanderham, in O. Buck and S. M. Wolf (eds.), Nondestructive Evaluation: Mierostrue- tural Characterization and Reliability Strategies, AIME, Warrendale, PA, 1981, p. 53.

82 G. Dau, EPRI J., 5 (1980) 16. 83 E. F. Skelton, J. Kirkland and S. G. Qadri, J.

Appl. Crystallogr., 15 (1982) 82. 84 M. Kuriyama, W. J. Boettinger and H. E. Burdette,

Proc. Syrup. on Accuracy in Powder Diffraction, in NBS Publ. 567, 1980, p. 479 (National Bureau of Standards).

85 C. J. Bechtoldt, R. C. Placious, W. J. Boettinger and M. Kuriyama, Adv. X-ray Anal., 25 (1982) 329.

86 H. Chen and M. Kuriyama, J. Appl. Crystallogr., 14 (1981) 280.

87 L. H. Bennett, G. G. Long, M. Kuriyama and A. I. Goldman, in G. E. Walrafen and A. K. Revez (eds.), Structure and Bonding in Noncrystalline Solids, Plenum, New York, 1986, p. 385-409.

88 G. G. Long, J. Kruger, D. R. Black and M. Kuri- yama, J. Electroanal. Chem., 150 (1983) 603.

89 J. Kruger, G. G. Long, M. Kuriyama and A. I. Goldman, in M. Froment (ed.), Passivity o f Metals and Semiconductors, Elsevier, Amsterdam, 1983, p. 163-168.

90 M. Kuriyama, W. J. Boettinger and G. G. Cohen, Annu. Rev. Mater. Sci., 12 (1982) 23.

91 W.J. Boettinger, H. E. Burdette and M. Kuriyama, in S. Weissmann, F. Balibar and J. F. Petroff (eds.), Applications o f X-ray Topographic Methods to Materials Science, Plenum, New York, 1984, p. 283-293.

92 M. Kuriyama and G. G. Long, in S. Weissmann, F. Balibar and J. F. Petroff (eds.), Applications o f X-ray Topographic Methods to Materials Science, Plenum, New York, 1984, p. 97-109.

93 W. J. Boettinger, H. E. Burdette, M. Kuriyama and R. E. Green, Jr., Rev. Sci. Instrum., 47 (1976) 906.

94 M. Kuriyama, Argonne National Laboratory 12th Annu. Materials Science Syrup. on In situ Observation o f Phase Chan~es, Kinetics and Microstructures, 1984 and TMS-AIME, Detroit, September 1984, Metallurgical Society of AIME, Warrendale, PA, 1984.

95 M. Kuriyama and G. G. Long, Z. Naturforsch. a, 37 (1982) 465.

96 .M. Kuriyama and W. J. Boettinger, in S. Weiss- mann, F. Balibar and J. F. Petroff (eds.), Appli- cations o f X-ray Topographic Methods to Materials Science, Plenum, New York,.1984, p. 23-31.

97 S. R. Stock, H. Chen and H. K. Birnbaum, in S. Weissmann, F. Balibar and J. F. Petroff (eds.), Applications o f X-ray Topographic Methods to Materials Science, Plenum, New York, 1984, p. 135.

98 R. C. Dobbyn and K. C. Yoo, in S. Weissmann, F. Balibar and J. F. Petroff (eds.), Applications o f X-ray Topographic Methods to Materials Science, Plenum, New York, 1984, p. 241-249.

99 K. C. Yoo, B. Roessler, R. W. Armstrong and M. Kuriyama, Scr. MetalL, 15 (1981) 1245.

100 W. J. Boettinger, R. C. Dobbyn, H. E. Burdette and M. Kuriyama, Nucl. Instrum. Methods, 195 (1982) 355.

101 G. Michot, K. Badawi, Abd el Halim and A. George, Philos. Mag., 42 (1980) 195.

102 S. Takagi, L. C. Chow, W. E. Brown, R. C. Dobbyn and M. Kuriyama, Nucl. Instrum. Methods, 222 (1984) 256.

103 S. Takagi, L. C. Chow, W. E. Brown, R. C. Dobbyn and M. Kuriyama, J. Dent. Res., 64 (1985) 866.

104 J. Billelo, in S. Weissmann, F. Balibar and J. F. Petroff (eds.), Applications o f X-ray Topo- graphic Methods to Materials Science, Plenum, New York, 1984, p. 333-342.

105 S. Weissmann, in S. Weissmann, F. Balibar and J. F. Petroff (eds.), Applications o f X-ray Topographic Methods to Materials Science, Plenum, New York, 1984, p. 1-22.

106 N. F. Berk and K. A. Hardman-Rhyne, Physica B, to be published.

107 K. A. Hardman-Rhyne and N. F. Berk, Proc. 31st Sagamore Army Materials and Mechanics Research Conf., August 1984, Plenum, New York, to be published.

108 T. Saegusa, J. R. Weertman, J. B. Cohen and M. Roth, J. Appl. Crystallogr., 11 (1978) 602.

109 R. A. Page, J. R. Weertman and M. Roth, Acta Metall., 30 (1982) 1357.

110 M. S. Yang, J. R. Weertman and M. Roth, Scr. Metall., 18 (1984) 543.

111 M. H. Yoo, J. C. Ogle, B. S. Borie, E. H. Lee and R. W. Hendrieks, Acta Metall., 30 (1982) 1733.

112 R. A. Page, J. Lankford and S. Spooner, J. Mater. Sci., 19 (1984) 3360.

113 R. A. Page, J. Lankford and S. Spooner, Acta MetaU., 32 (1984) 1275.

114 J. R. Weertman, Can. Metall. Q., 18 (1979) 73. 115 E. D. Case and C. J. Glinka, J. Mater. Sci., 19

(1984) 2962. 116 S. Kim, J. R. Weertman, S. Spooner, C. Glinka,

V. Sikka and W. Jones, in O. Buck and S. M. Wolf (eds.), Nondestructive Evaluation: Appli- cation to Materials Processing, American Society for Metals, Metals Park, OH, 1984, p. 169.

117 S. Kim, Ph.D. Thesis, Northwestern University, Evanston, IL, June 1985.

118 R. W. Hendricks, J. Schelten and W. Schmatz, Philos. Mag., 30 (1974) 819.

119 H. A. Mook, J. Appl. Phys., 45 (1974) 43. 120 F. Frisius, R. Kampmann, P. A. Beaven and R.

Wagner, in Dimensional Stability and Mechanical Behavior of Irradiated Metals and Alloys, British Nuclear Energy Society, London, 1983, p. 171.

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121 M. Yang, Ph.D. Thesis, Northwestern University, Evanston, IL, 1984.

122 S. Matsuoka, S. Kirn and J. R. Weertman, in J. W. Davis and D. J. Michel (eds.), Proc. Top. Conf. on Ferritic Alloys for Use in Nuclear Energy Technologies, Metallurgical Society of AIME, Warrendale, PA, 1984, p. 507.


CHAPTER 4

INTERATOMIC FORCES AND COHESION IN SOLIDS

1. INTRODUCTION

It has been stressed repeatedly throughout this report that the present status of experimental techniques is such that microscopic observations of atomic structures and atomic level processes that are relevant to fracture can now be made. In order to interpret such experiments and to establish predictive theoretical descriptions of fracture phenomena, atomic models of crystal defects involved in fracture as well as models of crack tip processes need to be developed. Typical examples are studies of the structure and properties of interfaces, in particular grain boundaries, and modeling of the bond breaking and dislocation emission at the crack tip.

Almost all such calculations have been made using pair potentials to describe the interatomic forces. These studies which have been performed with bo th experimentally and theoretically constructed interatomic potentials have often yielded basic physical insight but not quantitative results for specific systems. In particular, a feature of the calculated structure found in such studies is most likely to be correct if it is independent of the type of potential used. Results are thus obtained which are characteristic, for example, for all metals having a particular crystal structure. A typical example are studies of screw dislocations in b.c.c, metals that revealed the general features of their cores which then explain the low temperature plastic behavior of these metals [1, 2]. Similarly, a number of general features of grain boundary structures have been found, as discussed in more detail in Chapter 2 on microstructural and microscopic aspects of fracture. However, calculations using pair potentials have not been able to address satisfactorily, for example, problems of cohesion and chemical and electronic effects at interfaces and crack tips. It is demonstrated in this chapter that other approaches are now emerging, some empirical such as the embedded-atom method, and others more funda-

mental, such as the approximate tight-binding model, which are likely to advance qualitatively the atomistic studies of fracture phenomena. Furthermore, ab initio calculations can be carried out for some cases and serve as solid guidelines for other calculations. In fact, ab initio methods can play a role of standard for empirical and semiempirical methods. However, it is also argued here that pair potentials will continue to play an important role in the atomistic studies provided that they are used to describe appropriately restricted classes of problems.

2. AB INITIO CALCULATIONS

Within the past few years, it has become possible to describe in a quantitative fashion the ground state properties of condensed matter systems (see for example refs. 3-7). These properties include structural energies, lattice constants, bulk moduli, shear moduli, cohesive energies, phonon spectra, solid- solid transformations and other static and dynamic properties of solids. For example, the values of heat of formation of binary compounds [8] and the energy differences between equilibrium and metastable structures of elemental solids have been evaluated [9] (for reviews see refs. 3 and 10). The only inputs into these calculations are the atomic number and mass of the const i tuent atoms. This development has great significance for many areas of solid state physics and related fields. In principle, it is now possible to consider an ensemble of atoms and to ask what the lowest energy configuration is. For example, a specific arrangement of atomic species could be modeled and consideration given to whether this arrangement is structur- ally stable against perturbations which might replicate the forces present in a fracture process. Such theoretical calculations could even serve as a probe in areas where experimental data are currently inaccessible. Indeed, both


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real and hypothetical geometries of arbitrarily specified atom species could be modeled.

The development of ab initio methods capable of yielding such accuracy has been the result of several different approaches with the common thread being the implementation of the local density approximation [11, 12] to the many-body problem. This formalism is the essential feature of virtually all contemporary methods for evaluation of total energies. The theory leads to an effective SchrSdinger equation (the Kohn-Sham [12] equation):

( h2V2 c / - - - - + Vion + VH + Vx ¢1(r) = El¢l(r)

2m (1)

where Vion is the unscreened potential (e.g. for pseudopotentials, this would be the ion core potential), VH is the Hartree part of the potential and Vxc is the exchange-correlation potential. This equation has the same form as Hartree's approximation except that a given electron now feels an additional attractive potential Vxc arising from the correlations between the electrons. The scheme is made computationally tractable in real systems by adopting the local density function approximation in which the pair correlation function of electrons is that for a uniform gas of electrons. Originally, it was thought that the local density functional approximation would work only for problems with nearly uniform density but it has been found to give very accurate results for inhomogeneous systems such as atoms [13], molecules [14] and solids [3-10] . There are a variety of approaches for solving eqn. (1). V~o n can be taken as an effective core potential, or pseudopotential , or it can be taken to be the bare nuclear charge. The basis used to evaluate the secular equation can be taken to be plane waves, gaussians, Slater orbitals or combinations of the above. Once eqn. (1) is solved, the total energy of the system can be evaluated from a knowledge of the eigenvalues and self-consistent charge density. Several techniques have been designed to evaluate the energy for solids, given this knowledge. Per- haps the most efficient means available is the momentum space formalism of Ihm et al. [15].

As mentioned above, the ab initio calculations have been used very successfully to investigate the structural energy differences of ordered solids. Another area of application for such methods has been the examination of

surfaces in solids. At present, no experimental techniques are able to probe easily the atomic structure of the surface. Low energy electron diffraction and other methods are indirect and require a number of operational assump- tions for analysis. Total energy calculations can be used to determine surface structures and the most recent results are in good accord with experiment [3]. Moreover, total energy methods have recently been applied to compute the surface energy of transition metals such as tungsten [7] with good success. To extract a surface energy, or cleavage energy, for a solid from an experiment is very difficult. The best results are still highly temperature dependent. They also depend crucially on impurities, local deformation etc., i.e. are sample dependent , and from a fundamental point of view are only defined operationally. However, from a theoretical point of view the quanti ty is well defined and can be accurately evaluated by ab initio calculations. A b initio methods have also been used to study defect systems such as dislocations [16] and stacking faults [17]. To handle non-periodic systems, we can use simple techniques such as supercells [3]. Green function methods can be used to handle non-periodic systems in a more rigorous but more complex manner [18]. Although still in a preliminary state, the very important role of temperature is currently being explored with respect to solid-solid state transformations [3-19] . The incorporation of temperature opens up a wide variety of applications. In principle, the inclusion of temperature means that free energies can be computed and thermodynamic quantities can be evaluated from a microscopic point of view.

However, there are limitations in the use of ab initio methods which restrict their applications at present. These limitations can be categorized by whether they are technical in nature, or whether they are inherent in existing formalism. As examples of the former, the above approaches are limited by the number of atoms for which the calculation can be carried out accurately. This is a technical problem in the sense that larger and more powerful computers will allow systems which involve larger number of atoms to be examined. Currently, systems of 20 transition metal atoms are the approximate limit for most ab initio methods. It does not seem unreasonable to expect that, within the next

5 to 10 years, systems of over 100 atoms could be handled. However, it is probably beyond the realm of the foreseeable innova- tions for a system involving more than 10 s transition metal atoms to be handled. Another area which is in the general domain of being a technical limitation is the rigorous inclusion of temperature. For example, it may be possible to compute a phonon contribution to the entropy and to use this as input to determine the temperature dependence of the electronic structure of a solid in a self-consistent fashion, but this would be an extremely expensive calculation.

With respect to inherent limitations of current ab initio methods, these are centered on the local density approximation. This approximation is known to yield certain properties incorrectly. For example, the electronic potential observed by an electron at large distances from an atom should be approximately proportional to l / r ; however, the potential within the local density approximation is exponential in its fall-off, i.e. the local density formalism does not behave correctly in repro- ducing the image charge. Local density formalism also does not lead to correct excited states and magnetic properties may not be correctly reproduced. These problems, while important in general, are probably not crucial for problems related to fracture. However, one problem inherent in the local density theory is its accuracy for ground state properties such as cohesion and this may be important for certain fracture related issues. It is known that a large cancellation of errors must occur for the local density theory to work (see for example ref. 20). For example, errors origin- ating in the local density approximation for the energy of the free atom and similar errors in the energy of an atom in a solid must systematically cancel to yield accurate cohesive energies. In general, it is believed that local density calculations yield cohesive energies which are too large, typically 10% as a result of " incorrect" cancellations. How to remedy this problem is at present an open question.

3. APPROXIMATE TIGHT-BINDING MODEL

3.1. Physical principles As shown in Section 2, ab initio calculations

are now possible in practice but they are very

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expensive and time consuming, even for relatively simple systems. It is difficult to envisage that atomic studies of complex lattice defects such as long-period grain boundaries or of the crack tip phenomena can be carried out by those techniques within the next few years. Hence, simplifications of the theory must be made if quantum mechanics are to be applicable to such problems. Two possible treatments have been developed in the solid state theory. The first, based on pseudopotential theory, writes the structural energy of nearly-free- electron systems as the sum of interatomic pair potentials. Their well-known oscillatory behavior in real space is a direct consequence of the quantum-mechanical nature of the electron bands. The second, based on the linear combination of atomic orbitals approximation, writes the binding energy in terms of the individual bonds between pairs of atoms. The former approach is not directly applicable to transition metals and does not adequately describe the creation of free surfaces and defects with large density variations. However, other types of pair potential can possibly be used for such studies as described in Section 4. The latter approach does not suffer from these problems and its potential is discussed in this section.

The tight-binding approximation has been reviewed recently by numerous researchers (see for example refs. 21-27). It is based on the molecular orbital concepts developed by quantum chemists in their t reatment of bonding in molecules. The bonding molecular orbital or wavefunction CAB linking two atoms A and B can be written as a linear combination of the atomic orbitals ~A and ~B centered on the two sites. The strength of the resulting bond is determined by the matrix element of the atomic potential on one site with respect to the two atomic orbitals on different sites (the hopping integral). In a similar fashion the energy of the electronic states in a solid (consisting of N atoms) can also be found by writing the wavefunctions as linear combinations of the atomic orbitals on all the N different atomic sites. The resultant tight-binding eigenequation is specified by the atomic energy levels E i of the orbital on site i (where i = 1 . . . . , N) and by the relevant tight-binding bond (hopping) integrals between the orbitals on neighboring sites. The analytical form of these integrals is known

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[10] from first-principles theory so that the eigenvalues may be evaluated directly wi thout explicit knowledge of the atomic orbitals themselves.

The total binding energy of a solid is written within the tight-binding model as [22]

U = Urep + Ubo~d (2)

The repulsive energy Urep is expressed as the sum of empirical pair potentials which are fit- ted to obtain the correct equilibrium volume and, for example, compressibility of the elemental solid. The attractive quantum- mechanical bond energy Ubond is given by the sum of the eigenvalues of all the occupied electron states, the energies being measured with respect to the atomic energy level E~ on each site i. It can be writ ten as

EF Nf Ubond = ~ ( E - - E I ) n t ( E ) d E

t----1 (3)

where E F is the Fermi energy and n~(E ) is the local tight-binding density of states associated with site i. This can be evaluated by calculating the band structure of the system of N atoms [23, 24]. A more intuitive expression for the bond energy can be obtained in terms of individual bonds between pairs of atoms. Let us write the local density of states as

1 ni(E ) = - - - Im {Git(E)} (4)

7r

where Gii is the site diagonal element of the Green function G which satisfies the equation (E - - H ) G = 5, where H is the tight-binding hamiltonian and 5 the Dirac delta function. It follows from the definition of G [25] that Im(EG) = Im(HG), so that the bond energy may be written as sum of contributions from individual pairs of bonds, namely

EF

z [ - - 2 I m t f GjI(E) d E } ] (5) Vbond = ~ ~ HI] 1,1 k ?r ~ J

where the expression inside the square brack- ets is called the bond order. Thus the energy associated wi th the quantum-mechanical bond between atoms i and j may be expressed as

( Ubond )i ] = (tight-binding bond integral)u

× (bond order)u (6)

The bond order depends on the intersite Green function G u. This may be evaluated directly from the knowledge of the local atomic environment about the i j th bond, e.g. by using the method of Haydock et al. [26].

3.2. Impor tan t achievements The tight-binding model has been extreme-

ly successful in providing physical insight into the nature of bonding in solids [10]. It helps to close the gap between the fully first-principles local density function calculations and our chemical intuition regarding the importance of the angular character of the s, p and d valence electrons in determining bonding behavior. The tight-binding model also provides a framework for interpreting the parameters which enter semiempirical or phenomenological descriptions of cohesion and structure. For example, it has been shown [27] that the heats of formation of transition metal compounds are controlled by the quantum-mechanical d bond contribution and not by classical ionic terms as the successful semiempirical scheme of Miedema et al. [28] had assumed. This conclusion was later supported by local density functional calculations [8].

The tight-binding bond energy is responsible for the observed structural trends amongst the transition and rare earth metals [10], Laves phases [29] and transition metal coherent phases with respect to f.c.c. [30] and b.c.c. [31] lattices. It accounts for the anom- alies in the phonon spectra of transition metals [32, 33] and semiconductors [24]. The tight- binding bond model has also been used to s tudy possible surface reconstructions in transition metal [34] and semiconductor systems [23]. Recently the model has been shown to predict [35] the observed structural trends amongst the transition metal-metalloid binary AB compounds where NaC1, CsC1, NiAs, MnP and boride structure types compete with one another.

3.3. Future directions The tight-binding model is well suited to

the computer simulation of defect structures. By first-order perturbation theory the force due to the quantum-mechanical bond between any pair of atoms is obtained from eqn. (5) by replacing the bond integral with its derivative with respect to internuclear separation [23 - 33]. Standard atomistic relaxation codes may

then be modified to include quantum-mechanical fo rcesbe tween the atoms. Moreover, if the bond order is evaluated using the real-space recursion [26] rather than conventional band structure approaches which rely on transla- tional symmetry, large blocks of atoms can be treated wi thout the use of periodic boundary conditions. Problems such as grain boundaries, crack tips, dislocation cores etc. can then be tackled.

During the next 5 years, therefore, calculations using the tight-binding bond method should start to make reliable predictions regarding the structure and energetics of defects which are important for a microscopic understanding of fracture. For example, since the model has already accounted for the crystal structures of the transition metal borides and sulphides [15], it should provide for the first t ime quantum-mechanical insight into the embri t t lement of steel. However, it must be recognized that the tight-binding model is semiempirical in that the repulsive contribution is fi t ted with empirical pair potentials [22]. Hence, it is necessary to understand fully the underlying reasons for the success of the tight-binding model by making suitable approximation to the first-principles local density functional theory. This will hopefully allow a consistent set of parameters to be assigned to each element from which the repulsive pair potentials and the tight-binding bond integrals could be evaluated for any given combinations or arrangements of atoms. Finally, much more research is needed to make the tight-binding packages fast and computationally efficient. An order-of-magnitude increase in speed would make the scheme competit ive with pair potentials.

4. P A I R P O T E N T I A L S A N D E M B E D D E D - A T O M

M E T H O D

4.1. General features and justification of pair potentials

It has been pointed out in Section 1 that the majori ty of atomistic studies of structures relevant to fracture problems have been made using pair potentials. These studies often provided basic physical insight, in particular if the structural features considered were insensitive to the type of potential used. At the same

65

time, simulations using pair potentials are generally much faster than even simplified quantum-mechanical electronic structure calculations and thus problems involving much larger numbers of atoms can be treated than in either ab initio or tight-binding schemes. Hence we examine in this section future developments and applications of pair potentials.

It is well established, both empirically and theoretically, that bonding in a metal cannot be accurately described by a universal pair potential. However, there is considerable theoretical justification for the notion of effective pair potentials which describe restricted classes of problems [36]. For each such class, appropriate pair potentials can be derived by systematic approximate solutions of the electronic SchrSdinger equation, or simplifications thereof. Some representative classes of problems relevant to fracture are as follows.

(i) "Bond-breaking"problems. These problems include, for example, the creation of free surfaces during crack propagation, and void and vacancy formation. The loss in bonding energy is primarily associated with the reduced local electron bandwidth. A pair potential description can be generated by analysis of the low order moments of the electronic density of states [37]. The resulting potentials have a deep minimum and short range. They depend on the local environment through the local electron bandwidth. Their range of validity includes problems in which (a) nearest-neighbor bonding energy effects dominate and (b) the fractional loss in coordination number is fairly small. Several empirical and semiempirical schemes are closely related to the "bond- breaking" potential. For example, the electron bandwidth can be obtained from the measured cohesive energy and a phenomenological repulsive term added to obtain the correct elastic constants [38]. Alternatively, the pair potential can be taken to have a spline form with parameters obtained from measured quantities including the vacancy formation energy and the elastic constants [39]. Finally, the "embedded-a tom" method [40] discussed below obtains a local density-dependent energy functional from the measured cohesive and vacancy formation energies, the elastic constants and other quantities. Schemes such as these typically produce roughly 20% accuracy for vacancy formation and surface energies.

66

The "bond strength" in the solid thus appears to be a well<lefined quantity. However, crystal structure energy differences are obtained poorly: the overall magnitudes of these differences are an order of magnitude smaller than band-theoretical estimates, and the systematic chemical trends in preferred crystal structures are not obtained {unless the observed structural energies are used as input).

{ii) Constant-volume problems. This class includes, for example, the studies of the structures and of modes of motion of dislocations and grain boundary studies when little free volume is created on the local scale. The corresponding atomic rearrangements lead to a change in band shape rather than in bandwidth. For simple metals a pair potential description is obtained by expansion of the total energy in powers of the electron-ion pseudopotential, which is assumed to be weak. The "constant-volume" potentials are generally much shallower than the "bond-breaking" potentials and display long-range oscillations. They have in common with the "bond-breaking" potentials a strong dependence on the local environment through the electron gas response function. "Constant-volume" potentials successfully describe phonon spectra, liquid structure factors and crystal structure preferences in a variety of simple metals, together with dislocation and grain boundary properties. However, the vacancy formation energy often is grossly underestimated by pair potential calculations, as expected from the constant-volume restriction, while the migra- tion and formation volume are typically obtained accurately [41]. To obtain accurate estimates for energies involving substantial volume changes, it is necessary to include three-atom and higher order terms.

For metals having significant d band contributions to binding, which are those having the greatest technological importance, constant- volume effective potentials can be obtained by a perturbative treatment [42] of the s-d interaction. The application of these potentials is more difficult than in simple metals because of a strong sensitivity to the electron parameters entering the potential. Nevertheless, such potentials have provided reasonable descriptions of phonon spectra in noble metals. Very recently, potentials for transition metals with partly filled d bands have become available [43]. These obtain reasonably accurate structural

energy differences and phonon frequencies provided that three-atom terms are included.

(iii) "Chemical" rearrangements in alloys. These problems involve the energy change associated with rearranging A and B atoms on an underlying lattice that may be considered fixed, as in surface or grain boundary segregation and ordering. Even for simple metals, the relevant effective pair interactions are hard to calculate because the electronic charge density contains large long-wavelength contributions, reducing the rate of convergence of the pseudopotential expansion. In transition metal alloys, effective pair and cluster interactions have been obtained by several techniques, including perturbative expansions in powers of the short-range order parameters [44] and band-theoretical supercell total energy calculations [45]. These interactions have been quite successful in explaining observed chemical trends in bulk alloy phase diagrams through, for example, the use of chemical maps for binary compounds. However, they are available only for small classes of compounds; there are, for example, no satisfactory pair interactions describing the metal-metalloid bond, which are crucial for studying grain boundary segregation and embrittlement effects. Fur- thermore, lattice distortion effects have not been included. Finally, the applicability of the bulk pair interactions to defect geometries is very uncertain.

4.2. Future uses o f interatomic potentials Despite the advent of faster and faster

computers, expanding the range of problems which can be handled with purely quantum- mechanical techniques, interatomic potentials will continue to be a very important tool in the study of defects and fracture processes. We consider empirically and theoretically obtained potentials separately.

4. 2.1. Empirical potentials Since interatomic potentials are by far

the fastest method for calculating structural properties, they will be needed for problems involving very complicated atomic configurations. In these problems the atoms often see highly variable local environments, which prevents the use of theoretically _obtained "constant-volume" potentials, whose range of validity is narrower than that of empirical "bond-breaking" potentials.

Studies of defects such as grain boundaries and dislocations as well as of crack tip processes with empirical interatomic potentials should continue. These calculations are likely to suggest new modes of defect behavior and to identify general properties of large classes of materials. Future work should treat increasingly complex problems, involving, for example, defec t -defec t interactions and the thermal effects on defect motion. Further- more, future calculations with empirical potentials should take into account chemical distinctions between materials to a greater degree than past work. This can be accom- plished through the addition of physically motivated angle<lependent cluster terms, features such as oscillations in the effective pair potential, or pair terms favoring either like or unlike neighbors.

4.2. 2. Theoretically obtained potentials As mentioned above, theoretical potentials

for structural properties of the technologically important transition metals have been developed only recently. These should be applied to the appropriate class of problems, which is restricted to those involving small density changes. This includes studies of some types of dislocation and grain boundaries, but not direct studies of the fracture process. Although the accuracy of these types of potential in defect studies has not yet been established, a great deal will be learned from the inclusion of the angle-dependent forces which have explicitly been shown to be large in the transition metals [43]. More studies with theoretically obtained potentials should be performed for simple metals as well. In many problems, such as the structure of grain boundaries with small volume changes, these potentials may provide results more accurate than those of all but the most sophisticated quantum-mechanical methods. Furthermore, these calculations will become increasingly meaningful as more precise experimental information on the structure of grain boundaries becomes available.

4.3. Future work in the methodology o f interatomic potentials

Because of the vast technological importance of transition metals, improving existing interatomic potentials of these metals is crucial. Although interatomic potential descriptions of transition metals will inevitably

67

be more complicated than those for simple metals, work currently in progress is aimed at constructing potentials for restricted classes of problems. Typical techniques include nearly- free-electron perturbation theory [ 43], moment analyses [46] and derivation of interatomic potentials by distilling the results of fully quantum-mechanical calculations [47]. A particularly vexing problem is the description of the bond between transition metals and metalloids or simple metals, which can be described accurately neither by nearly-free- electron perturbation theory nor (in many cases) by tight-binding models. Understanding this type of bond is essential because of the crucial effects of impurities on fracture. A large number of elaborate band-theoretical calculations for metal-metalloid compounds have been performed; the insight gained from these calculations should now be translated into a form that can be used for defect studies. Since fracture involves the creation of free surfaces, it is very important to understand the effects of coordination number changes on interatomic potentials. Such effects must significantly influence surface structure and segregation properties. Simple estimates indicate that they are large, particularly for potentials describing the "constant-volume" and "chemical rearrangement" types of problem. Thus potentials for bulk alloys cannot safely be used for surface calculations. However, only a few explicit calculations of coordination number effects have been performed so far.

Work toward obtaining improved interatomic potentials should not focus only on theoretical first-principles developments but should also include semiempirical potential schemes. These combine the physical understanding gained from a variety of calculations with observed values of materials properties that are important in fracture behavior. They have the advantage over the theoretical schemes that a large part of the enormously complicated task of calculating bonding energies is done by nature rather than by the researcher; furthermore, the semiempirical potentials will become available on a much shorter t ime scale than the first-principles potentials. Several semiempirical approaches have been discussed above. In the future, such approaches should further incorporate the increasingly sophisticated understanding of bonding that has emerged from both very

68

simple and very elaborate quantum-mechanical calculations. One such scheme is further discussed in Section 5.

5. EMBEDDED-ATOM METHOD

The embedded-atom method is an empirical approach to the description of interatomic forces based on certain well-established physical principles. As formulated in ref. 40, the basic conclusions of the local density theory [11, 12], described in Section 2, are used to approximate the total energy as

1 Eto t : ~ Fi(Ph, l) + ~ ~ ¢(rll)

l t,J tray

(7)

where Ft is the energy of embedding an atom i into the host electron density p~ ~ at the site i, ¢~ is a short-range pair potential and r~i is the distance between atoms i and j. The elec- t ron density at any point in space is approx- imated by a linear superposition of atomic densities Pl, i.e.

Ph, i = ~ pj(ri~) j¢ i

and the first term in eqn. (7) is then a function of atomic positions. The pair potential has been taken as a Coulomb interaction between screened charges of atoms i and j. Both the screened charge and the embedding function have been determined by fitting a number of bulk properties of the material such as the lattice parameter, elastic constants, sublima- tion energy, vacancy formation energy and, for alloys, the heat o f solution.

A number of calculations have been performed to test the validity and applicability of the embedded-atom method. These include calculations of phonon dispersion curves [48], liquid structure factor [49, 50 ], excess energies in alloys [49], lattice constants in alloys [49], hydrogen solubility and diffusivity [40], hydrogen ordering on surfaces [ 51], alloy surface segregation [52], dislocation mobili ty [53] and fracture [40-47, 49-53] . Results from these calculations show the applicability of this method over a broad range of problems. Molecular dynamics simulations of a fracture process involving more than 1000 atoms [53] clearly show the potential of this

method in adding to our understanding of the atomistic processes at a crack tip.

Recently, a simple bu t well-founded functional form of the embedding energy has been proposed in ref. 38. According to this suggestion, the first term in eqn. (7) is writ ten as

(8) i 1

where/3 (rij) is a positive, rapidly decreasing function of the separation of atoms i and j which is again determined, together with the pair potential ¢, by fitting various bulk properties of materials. The justification for this choice is provided in the framework of the tight-binding approximation when considering the expansion of the energy up to the second moment of the density of states; fl then has the meaning of the hopping integral between sites i and j. This form has been shown to be particularly suitable for transition metals and it has been tested by calculating vacancy formation energies and surface energies and tensions [38, 54] as well as by the study of the surface relaxation [54].

The embedded-atom method overcomes some of the limitations of pair potentials and provides a description of interatomic forces applicable to problems in all three classes: bond breaking, constant volume and chemical rearrangements. Hence, this method, either in the formulation in ref. 40 or in ref. 38, is a very suitable empirical description of interatomic forces in the studies of lattice defects and crack tip phenomena.


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