PANEL ON MICROMECHANICS AND PHYSICS OF FRACTURE · plastic deformation process itself [ 3 ], (4) the study of atomistic processes of fracture, with special regard to the influences

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 electromag- netic measurements, and neutron and X-ray scattering. These methods have not been em- ployed 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 approxi- mation, 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 charac- terize 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 ex- tending 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

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 propa- gating 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

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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

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though more complex and difficult to imple- ment (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 typical- ly 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 mag- nitude 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

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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 ap- proaches 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.

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(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

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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

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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 possi- bility, 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.

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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 com- parison 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

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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

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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 in- creases. 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 polycrys- talline 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

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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

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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 distin- guish 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 pri- marily 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 con- stants, 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 polycrystal- line 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,

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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.

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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, b

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PANEL ON MICROMECHANICS AND PHYSICS OF FRACTURE · plastic deformation process itself [ 3 ], (4) the study of atomistic processes of fracture, with special regard to the influences