Micromechanisms of Brittle Fracture ANTHONY W. THOMPSON and JOHN F. KNOTT Mechanical processes operating in materials on the scale of the microstructure have come to be called "micromechanisms." The fundamental science and the micromechanisms of brittle fracture are reviewed here, w i t h particular emphasis on cleavage and intergranular fracture. Extant micromechanisms for these fracture types are evaluated. The role of solutes, particularly in intergranular fracture, is also discussed in terms of the fundamentals of brittle fracture.

I.

INTRODUCTION

I N metals, it is possible to produce brittle fractures which exhibit virtually no plasticity. More commonly, deformation accompanies fracture, and it becomes neeessary t o define the term "brittle." In this article, w e use "brittle" to refer to fractures which typically have the following characteristics: (1) unstable or catastrophic failure occurs at applied stresses less than the general yield strength of the uncracked ligament (in a cracked specimen) at the start of instability; tz] (2) little or no macroscopic plastic strain t o failure is observed; and (3) little or no evidence, e.g., from fractography, of local or microstructural-scale plastic strain accompanies failure. It will be noted that the first characteristic is, in essence, an engineering one, the second a classical metallurgical one, and the third relates t o relatively sophisticated observations on a microscopic or microstructural scale. These characteristics thus reflect the long history of interest in brittle fracture a m o n g engineers, metallurgists, and materials scientists. The most commonly identified brittle fractures in metals are cleavage and intergranular fracture, and w e devote most of our attention here to those two types. Both are of significance in the world of technology. The dire practical consequences of the occurrence of cleavage fracture in engineering structures made in mild steel are well recognized and documented, while temper embrittlement and other manifestations of intergranular fracture are also well known. In recent years, much emphasis has been placed on the engineering aspects of brittle fractures in metals, and the techniques of fracture mechanics have been developed t o relate applied fracture stresses t o the size of any defect present in a structure. Rather more interest has been shown in the reproducible measurement of parameters which can be used in engineering design and rather less in the micromechanics of cracking, whether for cleavage or for other brittle fracture types. The present review is intended partially to redress this balance, by starting with the fundamentals of cleavage crack propagation and showing how the associated mechanisms on a microstructural scale, which ANTHONY W . THOMPSON, Professor, is with the Department of Materials Science and Engineering, Carnegie Mellon University, Pittsburgh, P A 15213. JOHN F. K N O T T , Head, is with the School of Metallurgy and Materials, University of Birmingham, Birmingham B12 2TT, United Kingdom. This article is based on a presentation made in the symposium "QuasiBrittle Fracture" presented during the T M S fall meeting, Cincinnati, OH, October 21-24, 1 9 9 1 , under the auspices of the T M S Mechanical Metallurgy Committee and the A S M / M S D Flow and Fracture Committee. METALLURGICAL TRANSACTIONS A

have come t o be called "micromechanisms," may relate to appropriate engineering design parameters.

II.

FUNDAMENTALS

OF B R I T T L E F R A C T U R E The basic ideas underlying brittle fracture theories are briefly summarized here, emphasizing points which are developed below. The ultimate mechanism of brittle fracture is usually regarded as a simple elastic extension of atomic bonds up t o the point of final separation. In such a situation, an approximate value for the ideal fracture strength may be deduced by considering the force needed to separate two atoms. The familiar schematic curve for the energy of interaction of two atoms as a function of the distance between them is shown in Figure l ( a ) . In the crystalline lattice of a metal such as iron, this curve would represent the resultant of repulsion between bare (positive) iron ions and the attraction between each ion and other (electron) screened ions. The simple curve exhibits a minimum at the characteristic initial spacing of the atoms, b0. The force required to separate the atoms may be derived by taking the differential of the interaction energy at any g i v e n separation. As Figure l(b) shows, the force is zero at the stable rest position of the atoms, where the energy is at a minimum, and rises t o a m a x i m u m at the point of inflection in the energy/distance curve. It is to be noted that there is a negative interaction energy and thus a positive force even at quite l a r g e separations of the atoms ( ~ 2 b 0 , i.e., a displacement of about b0). The calculation of theoretical fracture stress proceeds by transforming the force/displacement curve (Figure l(b)) first into a stress/displacement curve and then into a stress/strain curve. The ordinate is converted t o stress by dividing the force by the square of the interatomic spacing in the fracture surface, and the abscissa is converted t o strain by dividing the displacement by the initial separation, b0. The total amount of energy which must be expended in producing fracture, i.e., the "work of fracture" per unit area, 2 y , is given by the area under the stress/displacement curve, which is 2 3/= f o ' ( b ) db. Then, by approximating the stress/strain curve t o half a sine wave and using the fact that stress is equal to Young's modulus multiplied by strain, at low elastic strains, it is possible to derive for the theoretical fracture stress:

~rth = X/~3,/b0)

[11

where E is Young's modulus. Equation [1] is a very oversimplified expression but does give a reasonable order VOLUME 24A, MARCH 1993--523

displacement required to produce fracture. In uniaxial tension, only a hypothetical machine would be sufficiently stiff to prevent unstable separation o f a specimen into two halves once a stress o f E / 3 had been reached, but as will be seen from the following, uniaxial tension is not the appropriate stress state to consider for atomic fracture, either at c r a c k tips in a laboratory experiment or for engineering applications.

(a)

Distance of separation,b=__ ¢W

L,

, |

I

I I

I I I I I

I I

I I i

(b) I! o

U 0 U,

rlb0

Disptacement,b

Fig. 1--Interatomic force and energy curves: (a) energy U as a function of distance of a t o m i c separation, s h o w i n g magnitude of b o n d i n g energy Uo, and (b) force-displacement curve corresponding to (a), with lattice parameter b0 indicated, t~l

o f magnitude for the i d e a l fracture stress. For a perfect elastic fracture, the work to fracture 2~/may be taken as twice the true surface energy o f a material, %, since two surfaces are produced. The substitution o f typical experimental values for iron gives o'th - E~oo/3 for fracture across the observed {t013} cleavage planes, where E~00 is measured normal to these planes. From E q . [1], the fracture plane should be close-packed, to maximize b0, provided that the variation o f E with crystallographic orientation does not have a greater effect. The bodycentered cubic (bcc) lattice o f iron does not possess a truly close-packed plane, but {110} planes are more closely packed than are {100} planes. It is arguable that iron cleaves on {100} because E has a minimum value f o r (100) directions. A further point to note is that the nature o f metallic bonding is such that the point o f inflection in the energy/ distance curve (Figure l(a)), and hence, the maximum in the force/displacement curve (Figure 1(b)), occurs at a displacement o f the o r d e r o f 0.25 to 0.4b0. It may well be that the simple sine wave approximation to the f o r c e / displacement curve is inappropriate in framing a definition o f "fracture." The maximum stress is achieved at a displacement o f 0.25 to 0.4b0; the assumed displacement required to pull atoms sufficiently f a r apart to reduce the stress to zero is then 0.5 to 0.8b0. However, in reality, there is still an energy o f interaction at these distances, and the simple m o d e l may underestimate the 524--VOLUME

24A, MARCH 1993

A . Griffith Cracks It is well known that, in practice, samples o f crystalline materials and glasses fracture in an apparently elastic manner at stresses very much less than the ideal fracture strength. The reason for this was first given by Griffith,[2~ who supposed that such samples contained inherent cracklike flaws. He was able to circumvent the awkward features of crack tip stress fields by considering the global changes in potential energy and surface energy (the only "work o f fracture" considered) in a stressed body as the length o f a flaw was increased. F o r an infinite body in plane strain, containing a central, through-thickness crack o f length 2 a , normal to an applied stress, O'app, the total energy o f the system decreases if a o r O'app is greater than given by the relationship o'avp = [2Ey/rra(1

-

1 ' 2 ) ] 1/2

[2]

where v is the Poisson ratio. Substituting typical figures for i r o n , E = 200 GPa and setting 3' = % , the true surface energy, with Ys = 2J/m2, it can be seen that a c r a c k of total length o f roughly 1 /xm can produce a fracture stress o f E / 2 0 0 . (The suitability o f Ys for insertion into E q . [2] for metals is discussed at greater length below.) For elastic bodies with a well-defined work o f fracture, this "Griffith equation" should describe the behavior o f most kinds o f testpieces. On a macroscopic scale, the Griffith energy balance has been used to derive "fracture toughness" values for precracked specimens o f commercial alloys. In such materials, l o c a l plastic deformation occurs at the c r a c k tip before fracture, but it has been found experimentally that unstable propagation is still characterized by a critical value o f the elastic energy release rate, G~nt, provided that the extent o f the plastic yielding is small compared with the dimensions of the testpiece, tl,3~ The fracture stress, for an infinite body in plane strain, is then given by O'F-=

[EGcrit/rra(1

- - 12)]1/2

[3]

and the critical energy release rate Gent plays the role o f 3' and is expended partly in producing two new surfaces and partly in producing plastic deformation. O r o w a n originally suggested a relation between the two components o f G c r i t ,[41 and shortly afterward,* Irwin TM in*Irwin's 1957 articles are often cited in support of this p o i n t , leaving the i m p r e s s i o n that his idea came years after, and i n ignorance of, Orowan's 1945 article; tal but Irwin cited Reference 4 i n his 1948 overview.t5]

dependently

made

essentially

the same

suggestion,

w h i c h i s fl,6] Gcrit =

2 Ys + Yp

[4]

where yp is the measure o f the "plastic work" associated METALLURGICAL TRANSACTIONS A

with fracture. Two contributions to yp can be distinguished: the plasticity associated with loading a c r a c k t o the point where it can propagate, called "precursor" workt7] and written a s ~tp(p . . . . . . .r); and the plasticity associated w i t h the propagation or crack growth process itself, called Yp(growth~- Thus, the plastic work term could be written as yp = yp(p. . . . .r) + ")tp(growth)" Experimental methods to separate these two contributions t o yp have been devised, as discussed further below. It is naturally tempting t o convert G,it t o the conventional plane strain fracture toughness, K i t , as discussed below, through the elastic relation G = K2(1 - v 2 ) / E

[5]

and thus, using estimated values of Ys, to calculate yp from a measured value of KI¢. For any reasonably tough material, such a calculation shows that yp is much greater than 2y~, often by factors of 103 t o 105. We return t o the interpretation of Eq. [4] in later sections. It is clear, however, that the meaning, in microstructural terms, of a given value of fracture toughness, K~c or G,it, can be derived only if it can be explained why a particular amount of plastic work was necessary before the material could fracture. But whenever yp significantly exceeds 7s, it might be appropriate to define the circumstances as "quasibrittle" fracture. Note that the definitions g i v e n for "brittle fracture" in Section I are not necessarily violated in this situation. There are several fundamental problems concerned with application of the foregoing to brittle fracture, e . g . , cleavage, in metals. On the atomic scale, it must be decided whether it is reasonable t o expect a pre-existing crack nucleus to propagate by cleavage in a truly brittle manner, i . e . , without any yielding at the crack tip. On the microstructural scale, those features which give rise to crack nuclei must be identified. Experimental results from mechanical tests made at different temperatures or strain rates and under different stress states must be examined t o decide whether the fracture stress is that necessary to nucleate cracks or to propagate them in the Griffith manner. Finally, it is instructive to use the fracture criteria derived from a study of the micromechanisms of fracture t o predict values of "fracture toughness" which are of use in engineering design. W e begin by examining events on the atomic scale, to assess whether truly brittle fracture occurs. B.

Conditions at a C r a c k T i p

The question of whether an atomically sharp crack can or cannot propagate in a brittle manner has been treated in terms of the ease of fracturing the bond at the crack tip by tensile stress, compared w i t h that of creating and propagating dislocations, which cause the crack tip to blunt. Dislocation production is controlled by shear stress, the m a x i m u m value of which lies at an a n g l e t o the line of the crack. The simplest practical example of brittle fracture occurs in the basal-plane cleavage of mica and other layer silicates. Here, it is easy t o break the weak bonds between the close-packed layers and virtually impossible to propagate dislocations on any plane at an angle to the basal plane. Mica can therefore be cleaved across the close-packed layers in an ideally brittle manner. METALLURGICAL

TRANSACTIONS A

Kelly e t al.t8~ were the first to assess the situation in physically realistic terms, for a number of metals and ionic solids, using linear elastic theory to calculate crack tip stress fields in those materials for which interatomic potentials were not available, and m a k i n g use of experimental observations of slip systems and cleavage planes. The principle of their calculation was t o derive values for the theoretical shear stress and the theoretical brittle fracture stress pertaining to the appropriate planes and t o compare the ratio of these with the m a x i m u m values of local normal stress and local shear stress that could be obtained from the applied stress system. In such a manner, it can be shown, for example, that cracks in diamond (high Peierls-Nabarro force) will always p r o p agate in a brittle manner, while cracks in face-centered cubic (fcc) metals will always generate plasticity and, thus, blunt.18.91 The situation for metals of some importance, such as iron, is less clear, and it appears that a definitive calculation cannot be made on the basis of present knowledge. An alternative approach to this problem is that of fracture mechanics. The crack produces a singular stress field, the dependence of which upon distance from the crack tip, r, is as r -1/2 and the strength of w h i c h is g i v e n by K, the stress intensity factor, which has the value tl'3J K = O-appVe-~

[6]

for the crack of length 2a in an infinite body. In passing, it is instructive to noteE7~ that an elegant calculation of the Griffith energy release rate, G, defined as d W / d a , may be made by considering the work done when a crack is virtually extended from crack length a to a + 6a. Using Eqs. [5[ and [6] gives, for the failure condition, G c r i t = OrzFTra(1

--

v2)/E

[7]

which is also Eq. [3]. There are obvious problems in using linear elastic stress analysis to treat crack tip fracture processes, due t o the singular nature of the stress and displacement fields. As r approaches zero, the stress and, therefore, strains become infinite; yet as negative r goes t o zero, i . e . , as the crack tip is approached from the crack side rather than from the plastic zone side, the displacement between the crack faces tends to zero. It is clearly necessary t o replace the assumed linear relationship between stress and strain by a more realistic form, such as that in Figure l ( b ) , in the region of the c r a c k tip. One method that has been used is to represent the crack as a distribution of dislocations, for example, of edge character, with Burgers vector b = bl. A simple rectangular approximation t o the force law is m a d e , assuming no failure up t o a limiting stress, o"1, and then separation at constant stress t o a limiting displacement, bl, at which complete fracture is deemed to have occurred. This is termed the "cutoff" fracture displacement. The m o d e l is then identical to that investigated by Bilby et a l . ~°~ for the equilibrium of a crack and plastic zone. Infinities in stress and discontinuities in displacement disappear. At low stresses, the fracture criterion becomes O'app

~-~

[Eorlbl/,lra(1

- v2)] 1/2

[8]

V O L U M E 24A, M A R C H 1 9 9 3 - - 5 2 5

which is identical t o the Griffith equation (Eq. [2]), if the area, o-~bl, under the assumed stress/displacement curve is equated to 2% Choosing values of o'1 = 0.2E and bl --- 0.5b0 then gives y = 0.05Eb0. With E~00 = 140 GPa and b0 = 0.3 nm, which are the values taken by Kelly e t al., f81 y would be about 2 J / m2, in good agreement with % values for iron measured at 1723 K but about half the value obtained by extrapolation of hightemperature data to 0 K. Realistically, fracture surface energies are "effective" energies which should not be expected IH-15J t o be equal to true thermodynamic surface energies, Ys, but instead should be larger than %, as most experimental determinations t~2,~5-18~ have found. The difference is usually described as a combination of irreversible work, y~, which includes the effects of local heating as stretched atomic bonds relax, effects of surface roughness such as cleavage steps and other factorst~31 and of any contribution from Yp~g~owth) in metals. It is then appropriate for metals to define ~/eff as the sum of these contributions, i . e . , "~eff = Ys -4- "~irr -~- Yp(growth)" Gerberich and Kurman, tlSJ for example, have made a recent determination of 3'~f in Fe-4 pct Si and arrived at a value of 23 J / m2. This alloy cleaves with extensive formation of ligaments on the fracture surface, so it is understandable that the value of Y~e is larger than the 14 J / m2 determined for iron. In summary, the work done to determine whether or not a crack can propagate in a brittle manner in iron has led to rather indefinite results. For many materials, either dislocation can be generated easily before a crack can propagate, or the activation energy for dislocation creation at a crack tip is so high that the crack must run in an ideally brittle manner. For iron, the situation is ambiguous. It is, perhaps, possible that a crack can propagate with a number of dislocations e n s u i t e , although these would apparently "blunt" the crack. Kelly et a l . 181 compared shear stresses with a bond "fracture strength," o'th (Eq. [1]). The use of linear elastic equations in the c r a c k tip region leads, however, to infinities in stresses. The removal of these by use of a dislocation model (taking 2 7 = o'Lb~) g i v e s an estimate of the size of the "end region," R, in which nonlinear behavior holds, but the value of R depends critically on that chosen for the fracture "cutoff" displacement, b~. Use of Barenblatt's more realistic force/displacement law ~9~ leads to a smoother crack profile, but the use of a "modulus of cohesion" as a material property, dependent a g a i n on a cutoff displacement, b,, demands an identity with surface energy, the derivation of which produces serious conceptual difficulties in definition of the "work to fracture." (These are compounded if surfaces rearrange or reconstruct when they become free, as happens, for example, in silicon.) Rice and Thomson,t9~ and later extensions to their work, r2°'2~'z2J have treated the general problem in a less doubtful manner, by calculating the ease of producing dislocations at the Griffith stress, which is derived from the macroscopic energy balance, properly using linear elasticity. The resultant equilibrium distances are clearly within a region at the tip where nonlinear theory should be used, but their conclusions seem appropriate to the observed behavior of iron. They still treat the dichotomy 5 2 6 - - V O L U M E 24A, M A R C H 1993

as "propagation with the sharp tip" vs "crack blunting," and it may be worth examining this situation more closely. The type of stress/displacement law shown in Figure 1(b) indicates that the attainment of O'th (of order E / 3 ) is a necessary, but perhaps not sufficient, criterion for separation. The Barenblatt and, t o a smaller extent, the dislocation m o d e l of a c r a c k b o t h incorporate the existence of some cohesion across crack faces at displacemerits greater than that corresponding to crth. The fracture energy is then a function of the cutoff fracture displacement, bt or bl. Estimates of this lie (probably) in the r a n g e of 1 to 2b0. In view of the discrepancies observed between experimental values for iron of work to fracture (see Section III), y~ff = 14 J / m 2, and that deduced from surface energies, % = 2 t o 4 J / m 2, it is of interest to e x a m i n e what effects m i g h t be produced by taking the critical event not as the attainment of o'th but of a critical value of cutoff displacement. In Griffith's Eq. [2], w e substitute a typical fracture stress, crF = 1 GPa, a value of Ej00 = 140 GPa, and three values of 3' = 2, 4, and 14 J i m e, respectively. The equilibrium half crack lengths, a, under this stress are then: 0.19 /xm, 0.38 /zm, and 1.33 /zm. These cracks are sufficient to reduce the fracture stress from --48 to 1 GPa. If the c r a c k could be treated simply as a long, thin ellipse of half length a and root radius p , the stress concentration at its tip would be 2 X / - a / p , using a linear elastic stress analysis. The values of p required to give the necessary stress of 48 GPa for the three equilibrium cracks would then be of the order of b0, 2b0, and 7b0, respectively. The first figure, although derived by an improper method, is clearly in agreement with assumed displacements at a and at the "cutoff": the figure of 2b0 is twice as l a r g e simply because the extrapolated value for y taken by Kelly et a l . r8J is 4 J/m 2 rather than 2 J / m2. The figure of 7b0, which corresponds to the experimental value, Yeff = 14 J / mz, is the bluntest that a 1.33-/zm-long Griffith crack can be and still give the necessary stress, ~rth, at its tip. For any crack sharper than this, the Griffith (macroscopic energy balance) fracture stress will a p p l y . Now s u p p o s e that a further condition is applied that the "cutoff" displacement must be, say, 1 to 2b0, t o give separation. In the crack tip environment, unlike a smooth tensile specimen, displacement of the crack faces can occur only by producing strains elsewhere in the (stiffer, because stressed below ~th) body. In uniaxial tension, failure at o'th = E l 3 can occur at a displacement of 0.25 to 0.4b0 in the tip region. The production of crack tip dislocations must be attained in this 1 t o 2b0. The production of crack tip dislocations in this circumstance may, then, not simply cause blunting but may become an integral part of the process by which the crack faces separate, essentially by allowing displacement which would not be allowed by a dislocation-free, stiff matrix. Calculations of the Rice and Thomson type 0,22,23j involve a crack at the Griffith stress: provided that it is sufficiently sharp t o allow O'th t o be attained, a higher cutoff displacement, together w i t h a few, essential c r a c k tip dislocations, can be contemplated but does not appear t o greatly affect the result,r22j At l o w e r fracture stresses, say, O'F = 500 MPa, figures for a and p (with Yeee = 14 J / m 2) are 0 . 3 3 / ~ m and 1.75b0, respectively. Such a METALLURGICAL T R A N S A C T I O N S A

displacement m i g h t not need any dislocations; again, doubts are difficult t o resolve because the situation is nonlinear. The situation in iron may be so delicately balanced that the necessary compliance in the crack tip region can be obtained only by the creation of a few dislocations, which give rise t o an experimental "fracture work" term, presumably reflecting "Yp(growth), significantly greater than the surface energy. It is possibly fortuitous that the diagram in Figure 1 of Rice and Thomson's article,tgJ showing a "sharp" and a "blunted" crack tip, indicates that the crack tip bonds in the "blunt" crack are stretched relative to those in the "sharp" crack. However, more detailed treatment of cutoff displacementsE22,23] could, perhaps, be undertaken with benefit. A second problem which is becoming resolved is the role of dislocations nucleated near but not at the tip of the crack, whose motion can blunt or stress-relieve, i . e . , shield, the crack, t22,24~ T o summarize, the calculations show t h a t , at observed fracture stresses, crack nuclei of total length 0.4 to 2.67 /zm exist to account for the reduction in fracture strength from ~48 t o ~1 GPa. In Section III, w e turn to specifics of cleavage fracture, and the experiments which have been used to obtain values of fracture strength, and t o identifying the nature of the crack nuclei in the best-studied material, mild steel.

III.

CLEAVAGE FRACTURE

We define cleavage as brittle fracture occurring on a low-index crystallographic plane,t~,25~ which means that intergranular fracture cannot, in general, be termed cleavage. Additional fractographic criteria for cleavage in metals [25~ are also relevant to interpretation of experiments, although they need not be detailed here. The dramatic change in ductility and fracture appearance over a narrow temperature r a n g e that occurs when mild steel is tested in uniaxial tension at low temperature has been a subject of study for many years,t~J Typically, a steel which fractures with some 65 pct reduction in area and a fibrous appearance in tests carried out at all temperatures down to 140 K may break at 100 K with minimal signs of plastic deformation and a fully crystalline appearance, composed of {100} cleavage facets. The first hypotheses proposed to explain this behavior, by Orowan[4] and others, postulated the existence of a brittle fracture stress, O'F, which varied only slightly with temperature (Figure 2). This could he thought of as the Griffith stress (Eq. [2]), deriving from a sensibly temperature-independent surface energy. In contrast, the yield stress of iron, as that of other bcc transition metals, increases markedly with decrease in temperature (Figure 2). In simple terms, ductile behavior is expected at high temperatures, because the specimen yields before it fractures, and brittle behavior at low temperatures, where the reverse situation holds. The nature of this brittle fracture requires examination. In uniaxial tension, the cleavage fracture stress is about 800 MPa, as opposed to the value of 1 GPa used earlier, which derives from notched tests. Using an average value for Young's modulus of 200 GPa and the surface energy % = 2 J/mz for iron gives (Eq. [2]) a critical crack length METALLURGICAL TRANSACTIONS A

I

i

I

',\ ! I

,.I Ta

-Brittle

f r a c t u r es t r e s s

',\ \

', i

I ~A

\ ~

U

n

a imum stress i

a

x

i

o

l yield stress

T e m p e r a ture

i Both

• . J L U n i a x a duct e : l ~ . Drittle . ',' . . ,~otn ducti(e . I c r a c K e u • o rl t t I e ÷ I I

Fig. 2 - - T h e Orowan proposal for brittle fracture, with an a p p r o x i mately temperature-independent brittle fracture stress. S t r e s s elevation due t o a notch or crack increases the m a x i m u m normal s t r e s s , thus assisting brittle fracture in notched or cracked bars relative to t e n s i l e specimens.~J

acrit = 0.4 /xm. That is, the length of crack nucleus which could propagate by a completely brittle fracture is only 2a = 0.8 txm in size, g i v e n two assumptions. The first is that the nucleus is a through-thickness one. This is a rather unrealistic situation: the nucleus is far more likely to be penny-shaped, for which the stress intensity is red u c e d by a factor of about 1.5, which increases the critical nucleus size to approximately 2 / z m . The second is that the "work t o fracture" 3' has been equated to the surface energy %. Instead, y should probably be interpreted as the effective energy, 3'~fe, especially if necessary dislocations are involved in crack propagation, because the displacement of bonds must be accommodated through 3'i~ as well as through contributions from 3"p(growth), SO that the size of the critical nucleus is further increased. If calculations are made for zinc,C26] the critical nucleus size turns out t o be of the order of millimeters, and zinc crystals of diameter smaller than this size can be made which still fracture at low stress in a brittle manner. The situation for iron is not so clear-cut, although pre-existing nuclei of 1- to 2-/zm size are unlikely in a pure ferrite matrix. Even if brittle cementite particles can be regarded as inherently cracked, it is possible to reduce the carbon content t o a level such that particles of 1 /zm in size will not exist. The presence of pre-existing cleavage fracture nuclei in nominally pure iron has not been seriously contemplated for over two decades. In technical materials, however, nuclei for cleavage can exist. Few investigators have been able t o identify fracture nucleating features in their experiments, in part due t o the small size of such nuclei and t o lack of scanning electron microscopy capability in earlier work. However, recent studies on pearlitic s t e e l s [27,28'29] and weld metals t3°l were successful in identifying fracture nuclei and subsequently using these flaws as the basis for fracture calculations. In pearlite, nuclei were found t o be VOLUME 24A, MARCH 1993--527

either small o x i d e inclusions [27,28'291 or microcracks nucleated across a number of pearlitic carbides, t27,28j while a variety of inclusions were identified in weld metals,t3°1 The application of dislocation theory to metallic deformation processes in the 1950s was paralleled by similar applications t o fracture processes. In 1951 and 1953, Hall and Petcht31,3z} obtained the classical relationship between yield stress ~rr and grain diameter, d: t r y = cri + k r d -1/2

/

~

[9]

which could be explained in terms of the production of mobile dislocations in unyielded grains ahead of a Liiders front by the local shear stress at the tip of a dislocation pileup in the yielded grain. A similar form was proposed for the fracture stress:[32] CrF = cri + k e d -1/2

-

[7rE3'/4(1

- T~)

-

~

[11]

1/2)]1/2d-1/2

[12]

The form of Eq. [10] is obtained and a meaning is given t o ke. The derivation, however, shows t h a t , provided that 3' remains constant throughout crack growth, either a crack nucleus forms and propagates under the action of the shear stress or no nucleus forms. This is called "nucleationcontrolled" fracture: the critical event is formation of a crack nucleus. There are two main problems w i t h this as a general conclusion. First, in a g i v e n type of test on mild steel, such as uniaxial tension, there is a clear transition from brittle to ductile behavior as the temperature is raised; yet if fracture occurred at, say, the yield stress at low temperature, there is no reason, from Eq. [12], why it should not occur at the yield stress at room temperature, since the relation Tv - T i = "/'eft (Eq. [12]) is independent of temperature in annealed mild steel (where dislocations are fully pinned), unless deformation twinning is involved in the low-temperature fracture. One possibility m i g h t be that the value of 3' in Eq. [12] is greater than the surface energy and involves necessary dislocations, 5 2 8 - - V O L U M E 24A, M A R C H 1993

a

(a)

Stress l

~[. T T"T .~.

d/2?~

j.

111~I

-

Suppose that a Griffith crack nucleus forms at an a n g l e O, e . g . , by the Zener mechanism. Then, if certain improvements are made to the original Stroh calculations, [36'371 it can be shown that the nucleus will propagate if Tef f "m-- Tape - - T i >

l

[10]

although the experimental data used t o justify this form had been manipulated, by others, in such a manner (fracture stresses were "corrected for the reduction of area preceding fracture") that it does not stand up to rigorous examination, t331 Zenert34j had proposed an appealing mechanism for the formation of cleavage crack nuclei by the coalescence of edge dislocations (Figure 3(a)), and Stroht3sl made use of this to explain the significance of Eq. [10]. Suppose, as in Figure 3(b), that there is a yielded grain containing a dislocation pileup. This can be regarded as a mode II (shear) crack under a stress Teff = ('Tap p Ti), where T~pp derives from the applied tensile stress and T~ is the lattice friction stress. Then the distribution of tensile stress ahead of the crack is as o'~pr, = K u / V ' - 2 ~ r f ( O ) = (T~pp

Lattice ,~resistance

(b) Fig. 3 - - T h e role of slip in cleavage fracture. (a) Pileups of dislocations at grain boundaries under a shear stress ~" resulting from an a p p l i e d s t r e s s a t . ( b ) T h e Z e n e r c r a c k n u c l e u s idea d e v e l o p e d b y Stroh,C3s~ i n w h i c h an e f f e c t i v e s t r e s s , z - r~, f o r c e s a p i l e u p o f l e n g t h d/2 t o open a c r a c k . T h e m a x i m u m s t r e s s ~r00 o c c u r s at an a n g l e t o t h e p l a n e o f the p i l e u p .

i . e . , that 3" should be replaced with %ff because of contributions from 3'p
The ideas o f Smith expressed in E q . [12] were extended, in light o f McMahon and C o h e n ' s results,t38] to include a role f o r nucleation at carbides, t391 as sketched in Figure 4. It is sometimes assumed that these would have to be grain boundary carbides, as in m u c h original work, [38,4°] but in f a c t , spheroidal, intragranular carbides give rise to similar effects.[17] This model for propagationcontrolled fracture from carbides has been quite successful f o r mild steel. Yet this conclusion o f propagation control in steel is also not general. Equation [12], with its implication o f nucleation control, does appear to describe the behavior o f zinc ~261 and perhaps o f the titanium aluminide alloy Ti-24 AI-1 1 N b ,t41j while for mild steel, p~bpagation control appears to dominate behavior. It seems clear, in both brittle ~12,42-441 and quasi-brittle materials, that additional experimental work would be valuable to examine the extant theoretical ideas in more detail. When a distinct microstructural constituent, such as a carbide in mild steel, operates as the nucleus for fracture, the fracture behavior o f that constituent must be understood. A statistical approach, pioneered by Curry and KnotttlTJ and l a t e r extended in some detail, t45J shows promise and may warrant extension to other microstructurally controlled brittle fractures. More complexmicrostructures, such as pearlitic steels t46] o r multiconstituent titanium alloys like Ti-24 AI-1 1 Nb, t4~1 are a challenge to these concepts w h i c h should repay further work. The role o f microstructure in cleavage fracture may involve variables other than those emphasized above, namely, the size o f grains and grain boundary carbides. This is particularly true when, for example, grain boundary particles are sparse and their relative distances from a c r a c k tip may be more important than their size alone. Models based on the original Ritchie and co-workers "critical distance" idea [47'48'49] are a logical extension o f effort to incorporate microstructural factors into brittle fracture, as Figure 5 shows. In this figure, a critical fracture stress, 0-*, is supposed to be maintained over a sufficient distance, 1", that an effective c r a c k nucleus can fail and thus support advancement o f the crack. Several experimental results have been found to be adequately described by such an interpretation, t4t,491 IV.

INTERGRANULAR FRACTURE

In intergranular fracture, fracture is characterized primarily by its path, namely along the grain boundaries. Since at relatively l o w temperatures, metals normally exhibit greater grain boundary strength than grain interior strength, that is, fracture is transgranular, the evident grain boundary weakness requires explanation. It is often found that brittle intergranular fracture at ambient temperatures is associated with segregation o f solute or impurity elements to grain boundaries, altering their cohesion, although there are counterexamples in w h i c h the intergranular weakness appears to be intrinsic and not a function o f segregated elements, tS°,51j Further work has examined the toughness o f these "intrinsically" brittle materials in some detail, t52,s31 There have been numerous efforts to develop the fundamental basis for fracture at boundaries, flL13,14,~2,54-57] One aspect has been to apply the Rice-Thomson approacht9] to grain boundaries, as first outlined by Rice.t1~1 METALLURGICAL TRANSACTIONS A

Ferrite matrix: surface energy Ys

(7

=

~Grrain boundary carblde s u face energy, y¢

I ~L

d

.J

ICo I

Grain diameter Carbide thick ness Fig. 4 - - T h e c o m p l e t e S m i t h modelisgj for cleavage in mild steel, nucleated at a grain boundary carbide of thickness Co. T h e effective stress, ~" - ~'i, is from Fig. 3.

5

~

" 3- $ ~

=l"

l. ~ .,

x.~2~ 0

i I

I I O.OI

1 I 0.02 I

i 0.03

x / ¢ K l a ' , Iz

Fig. 5 - - T h e "critical distance" idea, as originally formulated b y Ritchie e t al. 1471 T h e critical distance 1" is expected to be of the order of the graindiameterdg for the microstructure s h o w n , with y i e l d strength o0 normalizing stresses. [49]

Another aspect has been directed at the phenomena o f segregation-induced embrittlement, f o r example, the extensive, experimentally based articles o f McMahon and co-workers, ts4,sS,s61 These articles p o i n t out that E q . [4] is readily misinterpreted: although yp may o f t e n be f a r larger than %, that should not be a basis for concluding that the contribution o f Ys is unimportant to fracture. Indeed, if Ys along the grain boundary fracture path were to fall to zero, so that no cohesion between grains remained, then presumably Gcrit (or any other measure o f fracture resistance) would also fall to zero. Thus, an interrelation must exist a m o n g %, yp, and y (or Gcrit), and it must be recognized that a large value o f yp in one circumstance cannot imply the maintenance o f that v a l u e if % is reduced. The complication here is that the net VOLUME 24A, MARCH 1993--529

surface energy of the fracture is given by 2 y~ - Ygb, not by 2 y~ alone, so the interrelation among boundary cohesion, as expressed by y , along the (intergranular) fracture path, and the grain boundary energy itself, Ygb, must be understood. McMahon and co-workers, [54'55,56] however, have stated the dependence among these variables entirely in terms o f the energies, aiming to make explicit the relation between the plastic energy yp and the surface energy Ys. This is an awkward approach experimentally, because the energies themselves are difficult to measure independently (as opposed to inferring them from fracture experiments). The equations o f McMahon and co-workers, [54,55,56] Rice and Wang[~7] and others [2°,2q can describe the interrelation which occurs when large decreases occur in y,, for example, by the segregation o f metalloid elements to grain boundaries in steels. The subject o f solute effects is treated in Section V. In dealing with fracture experiments, the concept o f a fixed fracture stress[4] is helpful, since it clarifies how the amount o f plastic deformation accompanying fracture, i . e . , yp, varies naturally with y,. The interrelation o f these two variables can be envisioned as in Figure 6, where the effect o f changes in the brittle fracture stress o-F on the magnitude o f yp is evident in the (conceptual) stress-strain curve at the c r a c k tip. The total plastic work Wp is then given by an equation such as the following, for parabolic hardening according to tr = O-y + ke", [13a]

Wp = f t r d e = o'r" ee + f k e " d e

and, for integration between zero and 6g, W e ~- [(OrF -- O'y)/k]l/n[(OrF + n t r v ) / n + 1]

[13b]

Implementation o f Eq. [13] requires normalization by the plastic v o l u m e , w h i c h is complicated by the complex shape o f the plastic zone, as well as by the strain gradients within it; but the concept o f o-F as a limiting stress in the plastic behavior at the c r a c k tip is clear. One then needs to formulate a relation between/3r r and % (or the net energy from 2 y~ - Ygb), a complex and well-studied problem~l.20,2~,54-57] which, however, remains difficult to test experimentally.

(~F1

(~F2 /Nl I

d,,.I] \f

\11

\ 1 Nil I Nl Nil EF2

EEl

Strain E

Fig. 6 - - A sketch of a crack tip stress-strain curve, with termination at a brittle fracture stress ~rr which has two different values due to changes in 7~- The area under the curve is proportional t o the plastic work of fracture. 530--VOLUME

24A, MARCH 1993

The summary equations derived by McMahon and co-workers for intergranular fracture I54,55] are o f the Griffith form (Eq. [3]) and employ concepts quite similar to those o f Smith;t37,39] indeed, Smith's equations were the starting point for their derivation. F o r brittle intergranular fracture, as for cleavage, the role o f grain boundary particles w h i c h may act as fracture nuclei is a relevant conc e r n , and similar analytical approaches have been used. ]54,55,'~81 A sketch o f the m o d e l basis is given in Figure 7. These ideas are now being extended to brittle fracture o f interfaces between dissimilar materials. ~59-63] Although both mechanics and experimental techniques are complicated, the topic is o f v i t a l importance to c o m p r s ite materials for a range o f applications from engineering structures to electronic devices.

V.

R O L E OF S O L U T E S

A n u m b e r o f solutes are known experimentally to affect brittle fracture at boundaries. Examples include metalloid elements such as S, P, As, and Sb in steels, ts4'64~ lead in aluminum alloys,[651 and hydrogen in a n u m b e r o f structural alloys,C66,67,681 all o f w h i c h are damaging to fracture resistance, and such elements as boron in Ni3A1, which increase fracture resistance.I69,7°] A n u m b e r o f approaches to understanding these effects have been suggested, most recently in a series o f articles by Cottrell. t7~,72'731 The simplest is to begin with the atomic scale effects o f changes in cohesion, as summarized by Hirth.i~3] Figure 8 shows some o f the possible cohesion changes which have been suggested. The salient points in these diagrams are these: the location o f the curve at zero stress identifies the lattice parameter; the low-stress slope o f the curve at that point is a measure o f the elastic modulus; the maximum stress, although not the appropriate criterion for fracture m o d e l s ,t13] is a measure o f the cohesive force; and the area under the curve is a measure o f the work of fracture Y. Each o f these may or may not change with solute additions. For c r a c k tip analyses, it would appear that the area under the curve is the most important parameter, so that all the sketched curves in Figure 8 would imply lowered fracture stress, although some o f the curves imply further, testable effects o f the solute, such as changed lattice parameter. Note that changes in lattice parameter o r modulus, if they occur, would take place within distances very near the boundary and thus would be difficult to measure experimentally. N o w that atomic calculations by techniques like the embedded atom method ( E A M ) are possible at surfaces and grain boundaries, ]7°,74] greater understanding o f cohesion changes can be expected. The E A M technique has already been applied, for example, to the p r o b l e m o f B in Ni3A1,]7°J and the expected change in cohesion energy quantified as about 15 pct. Given that changes in boundary cohesion occur when segregants are present, it is important for the brittle fracture case to evaluate the concentration o f segregant needed to "embrittle" the boundary, i . e . , to cause at least some areas o f intergranular fracture to occur a m o n g the prevailing, nonsegregated fracture m o d e . Some published METALLURGICAL TRANSACTIONS A

O"

a"

// /'4

// /

/,¢

"•?," ~k " ~\\

~~7777777~ X

"'"~'"

~"

Xp

1111

J

\\

\

1111

o"

17"

(b)

(a)

Fig. 7--Concepts for grain boundary fracture nucleation, analogous t o Figs. 3 and 4. (a) Stroh-like crack located at a grain boundary. (b) Strohlike crack which nucleates at a particle-matrix interface rather than a grain boundary.[ 54j

U

O-

I I 0

~ I I I

; t

.._.b

0

"

b

/.,~ ,,5,"

(a)

(b)

(c)

(d)

O"

0

Fig. 8--Cohesion changes due t o solutesY 31 Here, the lattice parameter i s designated b0 and the m a x i m u m force era- (a) An energy-distance curve with solute-induced changes (cf. Fig. l(a)). (b) The corresponding force curve (cf. Fig. l(b)). (c) Changes i n the force curve which leave lattice parameter and modulus unchanged. (d) A force curve change which keeps crm constant. METALLURGICAL TRANSACTIONS A

VOLUME 24A, MARCH 1993--531

AVERAGE PHR Sb454

(PCT

Fe---~-~03 )

I0

20

!

I

80

so o ~

60

w z -r •o

40

6O

--

4 0

-

2 0

"~

0 kW '~ F-

I

u_

Cc

r

CH

(a)

20

I

o.,

I

I

0.2 _ 0.3

ESTIMATED Xsb ( FRACTION OF A LAYER)

(b)

I

OH (c)

Ccr

OH

(d)

Fig. 9--Dependence of fracture stress O'Fon soluteconcentration, here expressed for hydrogen, C.. (a) Threshold model, in which fracture stress is lowered at a solute concentration of C,. (b) Measured data for antimony at grain boundaries,164}where .~ is the estimated fraction of an atomic layer at the boundary. (c) Concentration dependence sketched from (b) results. (d) Linear version of (c), with modified critical value C cr'.[67]

models have assumed a critical or threshold concentration, which is a simple approach; but experimental evidence suggests that it is probably oversimplified.[66] Figure 9 illustrates the p o i n t . Figure 9(a) shows a threshold-type model, compared in Figure 9(b) to an actual determination of effect of segregation on fracture.[64] A smoothly curving dependence of fracture on segregation (Figure 9(c)) or at least a linear transition from segregation-free to fully segregated effects (Figure 9(d)) would be more realistic than a sharp threshold. Because local, not global, concentrations of solute are relevant to problems of this kind, [671 determination of the value of Ccr or Ccr' is not simple. Hydrogen is often associated with intergranular fracture.[66'68[ Numerous practical instances of hydrogen embrittlement have been observed in which hydrogen damage occurs in the form of a brittle intergranular fracture mode not present without hydrogen. Yet careful experiments have shown that in nearly all such cases, the grain

5 3 2 - - V O L U M E 24A, M A R C H

1993

boundaries which are fractured have contained various concentrations of damaging impurities, such as metalloids, although not in sufficient quantities to cause intergranular fracture by themselves. One of the best-studied materials is nickel, in which it appears that sulfur plus hydrogen can cause brittle intergranular embrittlement,[75-781 but that hydrogen alone merely causes a highly localized plastic fracture within a few tens of nanometers of the boundary, thus appearing brittle. [77J In other words, it would appear that hydrogen by itself ordinarily cannot weaken metal g r a i n boundaries sufficiently for intergranular fracture but can only cause such fracture when "assisting" other segregants which partially weaken the boundary. It would be of interest, and of some fundamental worth, to have determinations of 7elf values for such cases of fracture, so that relative magnitudes of effects of various solutes could be quantified; but t o date, it appears that none have been determined. Such measurements could well be implemented for

METALLURGICAL

TRANSACTIONS A

intergranular fracture with tests using the established methodology of notched bar fracture mechanics. [1'7'17"26-30'40'46'67'79-82] AS in the case of cleavage, there is a balance between yield and fracture behavior which can be exploited to minimize contributions from plasticity, O'7,:6,2s'46,sll

i.e.,

f r o m yp(p. . . . . .), a n d t h u s a c h i e v e a

good estimate o f yCff. It would also be useful to attempt systematic variation in microstructural parameters of interest, such as grain boundary particle size, type, and distribution, to provide a robust test of extant models. Much underlying fracture sciencetu-13,21-23~ and experimental knowledge I1'28'29'44'46'62"66'831 exists, but additional experimental work is still needed.

ACKNOWLEDGMENTS We appreciate helpful discussions with J.J. Lewandowski, R.O. Ritchie, J.P. Hirth, and J.C. Williams o n this topic. Our original collaboration o n this review was supported b y the SERC in the Department of Metallurgy, University of Cambridge. One of us (AWT) acknowledges the hospitality of the Materials Science Division, Lawrence Berkeley Laboratory, during the preparation of this manuscript, and partial support from the David Jackson Fund for Scholarly Endeavors.

REFERENCES 1 . J.F. Knott: Fundamentals o f Fracture Mechanics (Revised), Butterworth's, London, 1979. 2 . A.A. Griffith:Phil. Trans. R. Soc. A, 1921, vol. 221A, p p . 163-98. 3 . D. Broek: Elementary Engineering Fracture Mechanics, 4th ed., Martinus Nijhoff, Dordrecht, T h e Netherlands, 1987. 4 . E. Orowan: Trans. Inst. E n g . Shipbuilders Scot., 1 9 4 5 , vol. 8 9 , p p . 162-215. 5 . G. Irwin: in Fracturing o f Metals, ASM, Cleveland, OH, 1 9 4 8 , p p . 147-66. 6 . E. Orowan: Rep. Prog. Phys., 1 9 4 8 , vol. 1 2 , p p . 185-212. 7 . J.F. Knott: in Physics and Chemistry o f Fracture, Proc. N A T O Adv. Study Inst., R.M. Latanision and R.H. Jones, eds., Martinus Nijhoff, Dordrecht, T h e Netherlands, 1 9 8 7 , p p . 44-59. 8 . A. Kelly, W . Tyson, and A.H. Cottrell: Phil. Mag., 1 9 6 7 , vol. 15, p p . 567-86. 9 . J.R. Rice and R. Thomson: Phil. Mag., 1 9 7 4 , vol. 2 9 , p p . 73-97. 1 0 . B.A. Bilby, A.H. Cottrell, and K.H. Swinden: Proc. R. Soc. A , 1963, vol. 272A, p p . 304-14. 11. J.R. Rice: in Effects o f Hydrogen on the Behavior o f Materials, A.W. Thompson and I.M. Bemstein, e d s . , TMS-AIME, New York, NY, 1 9 7 6 , p p . 455-66. 12. A.G. Evans and T.G. Langdon: Prog. Mater. Sci., 1976, vol. 2 1 , p p . 171-441. 13. J.P. Hirth: Phil. Trans. R. Soc. A , 1 9 8 0 , vol. 295A, p p . 139-49. 14. R. Asaro: Phil. Trans. R. Soc. A, 1 9 8 0 , vol. 295A, p p . 151-63. 15. W . W . Gerberich and E. Kurman: Scripta Metall., 1 9 8 5 , vol. 1 9 , p p . 295-98. 16. J.J. Gilman: J. Appl. Phys., 1 9 6 0 , vol. 3 1 , p p . 2208-18. 17. D.A. Curry and J.F. Knott: Met. Sci., 1 9 7 8 , vol. 1 2 , p p . 511-14. 18. J.M. Liu and B.-W. Shen: Metall. Trans. A, 1 9 8 4 , vol. 15A, p p . 1247-51. 19. G.I. Barenblatt: Adv. Appl. Mech., 1 9 6 2 , vol. 7 , p p . 55-129. 2 0 . D.D. Mason: Phil. Mag., 1 9 7 9 , vol. 3 9 , p p . 455-68. 2 1 . P . M . Anderson and J.R. Rice: Scripta Metall., 1 9 8 6 , vol. 2 0 , p p . 1467-72. 2 2 . I.-H. Lin and R. Thomson: Acta Metall., 1 9 8 6 , vol. 3 4 , p p . 187-206. 2 3 . R. Thomson: in Solid State Physics, H. Ehrenreich and D. Tumbull, e d s . , Academic Press, New York, NY, 1 9 8 6 , vol. 3 9 , p p . 1-129.

METALLURGICAL TRANSACTIONS A

2 4 . B.S. Majumdar and S.J. B u m s : Acta Metall., 1 9 8 1 , vol. 2 9 , p p . 579-88. 2 5 . C.D. Beachem and R.M.N. Pelloux: in Fracture ToughnessTesting and Its Applications, STP 3 8 1 , ASTM, Philadelphia, PA, 1965, p p . 210-44. 2 6 . D.A. Curry, J.E. King, and J.F. Knott: Met. S c i . , 1 9 7 8 , vol. 1 2 , p p . 247-50. 2 7 . J.J. Lewandowski and A.W. Thompson: in Fracture 1 9 8 4 , Proc. ICF 6 , S.R. Valluri, D.M.R. Taplin, P . Rama Rao, J.F. K n o t t , and R. Dubey, e d s . , Pergamon, Oxford, United Kingdom, 1984, vol. 2 , p p . 1515-24. 2 8 . J.J. Lewandowski and A.W. Thompson: Metall. Trans. A, 1986, vol. 17A, p p . 1769-86. 2 9 . D.J. Alexander and I.M. Bernstein: Metall. Trans. A , 1 9 8 9 , vol. 20A, p p . 2321-35. 3 0 . D.E. McRobie and J.F. Knott: Mater. Sci. Technol., 1 9 8 5 , vol. 1 , p p . 357-65. 3 1 . E.O. Hall: Proc. Phys. Soc. (London), 1951, vol. B64, p p . 747-53. 3 2 . N.J. Petch: J. Iron Steel Inst., 1 9 5 3 , vol. 174, p p . 25-32. 3 3 . J.F. Knott: in Yield, Flow and Fracture o f Polycrystals (Perch Festschrift), T.N. Baker, ed., Applied Science Publishers, London, 1 9 8 3 , p p . 81-99. 3 4 . C. Zener: in Fracturing o f Metals, ASM, Cleveland, OH, 1948, p p . 3-31. 3 5 . A.N. Stroh: Adv. Phys., 1 9 5 7 , vol. 6 , p p . 418-65. 3 6 . A.H. Cottrell: Trans. AIME, 1 9 5 8 , vol. 2 1 2 , p p . 192-203. 3 7 . E. Smith and J.T. Barnby: Met. Sci. J., 1 9 6 7 , vol. 1 , p p . 56-64. 3 8 . C.J. McMahon and M . Cohen: Acta Metall., 1 9 6 5 , vol. 1 3 , p p . 591-604. 3 9 . E. Smith: Acta Metall., 1 9 6 6 , vol. 1 4 , p p . 985-89 and 991-96. 4 0 . J.F. Knott: J. lron Steel Inst., 1 9 6 7 , vol. 2 0 5 , p p . 288-91. 4 1 . Wu-Yang Chu and Anthony W . Thompson: Metall. Trans. A, 1992, vol. 23A, p p . 1299-1312. 4 2 . D. Marsh: Proc. R. Soc. A, 1 9 6 4 , vol. 282A, p p . 33-43. 4 3 . A.G. Evans: Phil. M a g . , 1 9 7 0 , vol. 2 2 , p p . 841-49. 4 4 . B.R. Lawn and T.R. Wilshaw: Fracture o f Brittle Solids, Cambridge University Press, Cambridge, United Kingdom, 1975. 4 5 . T . Lin, A.G. Evans, and R.O. Ritchie: Acta Metall., 1 9 8 6 , vol. 3 4 , p p . 2205-16. 4 6 . J.J. Lewandowksi and A.W. Thompson: Acta Metall., 1 9 8 7 , vol. 3 5 , p p . 1453-62. 4 7 . R.O. Ritchie, J.F. K n o t t , and J.R. Rice: J. Mech. Phys. Solids, 1 9 7 3 , vol. 2 1 , p p . 395-410. 4 8 . R.O. Ritchie, W.L. Server, and R.A. Wullaert: Metall. Trans. A , 1979, vol. 10A, p p . 1557-70. 4 9 . R.O. Ritchie and A.W. Thompson: Metall. Trans. A , 1 9 8 5 , vol. 16A, p p . 233-48. 5 0 . C.L. White, R.E. Clausing, and L. Heatherly: Metall. Trans. A , 1 9 7 9 , vol. 10A, p p . 683-91. 5 1 . T . Ogura, S. Hanada, T . Masumoto, and O. Izumi: Metall. Trans. A , 1 9 8 5 , vol. 16A, p p . 441-43. 5 2 . P.S. Khadkikar, J.J. Lewandowski, and K. Vedula: Metall. Trans. A, 1 9 8 9 , vol. 20A, p p . 1247-55. 5 3 . J.D. Rigney and J.J. Lewandowski: Mater. Sci. Eng., 1 9 9 2 , vol. A149, p p . 143-51. 5 4 . C.J. McMahon and V. Vitek: Acta Metall., 1 9 7 9 , vol. 2 7 , p p . 507-13. 5 5 . M.L. Jokl, V. Vitek, and C.J. McMahon: Acta Metall., 1 9 8 0 , vol. 2 8 , p p . 1479-88. 5 6 . M.L. Jokl, J. Kameda, C.J. McMahon, and V. Vitek: Met. Sci., 1980, vol. 1 4 , p p . 375-84. 5 7 . J.R. Rice and J.-S. Wang: Mater. Sci. Eng., 1 9 8 9 , vol. A107, p p . 23-40. 5 8 . G. Irwin: Trans. ASME, Series E (J. Appl. Mech.), 1 9 5 7 , vol. 7 9 , p p . 361-64. 5 9 . T.S. O h , J. Rodel, R.M. Cannon, and R.O. Ritchie: Actu Metall., 1988, vol. 3 6 , p p . 2083-90. 6 0 . B.J. Dalgleish, M.C. Lu, and A.G. Evans: Acta Metall., 1 9 8 8 , vol. 3 6 , p p . 2029-35. 6 1 . M.F. Ashby, F.J. Blunt, and M . Bannister: Acta Metall., 1 9 8 9 , vol. 3 7 , p p . 1847-57. 6 2 . A.G. Evans, M . Riihle, B.J. Dalgleish, and P.G. Charalambides: Metall. Trans. A , 1 9 9 0 , vol. 21A, p p . 2419-29. 6 3 . M.A. Bannister and M.F. Ashby: Acta Metall. Mater., 1 9 9 1 , vol. 3 9 , p p . 2575-82.

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64. J. Kameda: MetaU. Trans. A, 1 9 8 1 , vol. 12A, pp. 2039-48. 65. J.J. Lewandowski, V. Kohler, and N . J . H . Holroyd: Mater. Sci. Eng., 1987, vol. 96, pp. 185-95. 66. A.W. Thompson and I.M. Bernstein: in Advances in Corrosion Science and Technology, M.G. Fontana and R.W. Staehle, e d s . , P l e n u m P r e s s , New Y o r k , NY, 1980, vol. 7, pp. 5 3 - 1 7 5 . 67. A.W. Thompson: Mater. Sci. Technol., 1985, vol. 1, pp. 711-18. 68. I.M. Bernstein and A.W. Thompson: Int. Metall. Rev., 1976, vol. 21, pp. 2 6 9 - 8 7 . 69. K. Aoki and O. Izumi: Nippon Kinzaku Gakkaishi, 1979, vol. 43, pp. 1190-96. 70. M.J. Mills, S.H. G o o d s , S.M. Foiles, and J.R. Whetstone: Scripta Metall. Mater., 1991, vol. 25, pp. 1283-88. 71. A.H. Cottrell: Mater. Sci. Technol., 1989, vol. 5 , pp. 1165-67. 72. A.H. Cottrell: Mater. Sci. Technol., 1990, vol. 6, pp. 121-23. 73. A.H. CottreU: Mater. Sci. Technol., 1990, vol. 6, pp. 325-29. 74. S.M. Foiles: Phys. Rev. B, 1985, vol. 32, pp. 7 6 8 5 - 9 3 . 75. A.W. Thompson: in Grain Boundaries in Engineering Materials,

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76. 77. 78. 79. 80. 81. 82. 83.

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METALLURGICAL TRANSACTIONS A