Mechanics and mechanisms of large-scale brittle fracture in structural metals

Review Paper Mechanics and Mechanisms of Large-scale Brittle Fracture in Structural Metals J. F. K N O T T Department of Metallurgy, University of Cambridge, Cambridge ( Gt. Britain) (Received March 30, 1970)

CONTENTS 1. Introduction

1

8. Large-scale brittle fracture in mild steel

21

2. Elastic stress concentrations and triaxiality

9. A mechanistic basis for the fracture toughness of steels

27

3

5. Elastic fracture and Griffith's theory

10

10. Discussion (a) General aims of research (b) Effect of fabrication (c) Application to design

6. (Quasi-)brittle fracture

12

11. Conclusions

34

7. Fracture toughness testing (a) Plane stress (b) Plane strain

15

Acknowledgements and apologia

35

References

35

3. Plastic constraint 4. Stress fields near cracks

6 8

1. INTRODUCTION

An ideally brittle fracture is one preceded by deformation which does not include any non-elastic component, but this situation is seldom encountered in the fracture of crystalline materials, because plastic flow is usually produced at the tip of a crack nucleus before the atomic bonds at the crack tip have been stretched to their failure point. A few notable exceptions to this general behaviour have been reported for single crystals in which yielding was particularly difficult to produce. Experiments, among others, on the cleavage of mica 1 (layersilicate structure), basal-plane cleavage of zinc 2'3 (hexagonal, with large axial ratio) and {001} cubeplane cleavage of tungsten at low temperatures 4 (body-centered cubic, with a high lattice friction stress) have been carried out to determine values of surface energies in what are claimed to be ideally brittle fractures. One of the more glamorous examples of a brittle fracture in a single crystal is met with in the cutting of facets on a diamond gem-stone. Again, some intergranular fractures in polycrystalline materials have been regarded as completely brittle 5.

29

For the purposes of general structural engineering, use would never be made of materials which were liable to fail in such a brittle fashion unless components made from them were subjected to a compressive stress system, as in columns or prestressed members, or to a very low applied tensile stress. In any application where the weight or volume of the piece is important, the choice of materials for members which have to bear high tensile loads is limited mainly to metals which possess a large number of slip systems, or to fibre composities. When cost is taken into account, the choice is virtually limited to steels, titanium and aluminium alloys, and the remainder of this review will be concentrated on the behaviour of these three types of material, which may be grouped together under the heading of "structural metals". The first problem is to define what is meant by "brittle" behaviour when referring to these metals. The interpretation most generally accepted is that "a metal is brittle if it breaks before it yields". However, in practice, this definition means different things to different people and so it must be examined in more detail. The starting-point is to be found in the title of this article in the association of "large-

Materials Science and Engineering American Society for Metals, Metals Park, Ohio, and Elsevier Sequoia S.A., Lausanne - Printed in the Netherlands

2

J . F . KNOTT

scale" and "brittle". For structural metals, apparent "brittleness" is a function of scale and may be said to lie in the eye of the beholder; what is "brittle" to a structural engineer, thinking in terms of feet and inches, may be quite otherwise to the materials scientist, concerned with events on a microscopic scale. The crux of the definition given above is the meaning of the term "yield stress". Consider any piece which contains a stressconcentrator. Structural metals are never perfectly homogeneous and therefore always contain some form of local stress raiser, whether it be a hatch corner in a ship's deck, a fatigue crack in a fracture toughness specimen, or an inclusion or second-phase particle in a macroscopically smooth-sided tensile testpiece. In each case it is of crucial importance to distinguish between two yield stresses: (1) The applied stress at which yielding first occurs in the vicinity of the stress concentrator; this is the local yield stress. (2) The applied stress at which yielding completely crosses the net cross-section of the piece, so that it is no longer possible to trace a path between opposite loading points through elastically deformed material only; this is the general yield stress. Scale enters into the problem because it is the general yield stress that an engineer or materials scientist is thinking of when he talks of "fracture before yield" and the general yield stress has different conventional meanings for different situations. It corresponds to the stress at which macroscopic yielding is readily detected by the techniques commonly associated with the particular application. In a uniaxial tensile test on mild steel, for example, the general yield stress is synonymous with the lower yield stress, which is readily observable because of the large Luder's band extension. Local yielding can be detected only by techniques, such as microstrain extensometry, etch-pitting, or thin-film electron microscopical observation of dislocations, which are much more sensitive than are those usually employed for conventional tensile tests. Similarly, on a large scale, the engineer's general yield stress is the applied stress at which a component, containing a gross stress concentrator, suffers general plastic collapse; it bears a relationship to the material's uniaxial yield stress which has to be calculated, for example by slip-line field theory or by computing techniques. The difference between the local and general yield stresses is increased for a large, thick piece, because a gross stress concentrator can produce substantial amounts of local yielding

at a low applied stress, yet may raise the (netsection) general yield stress by plastic constraint to a maximum of some two-and-a-half times the uniaxial yield stress. The conclusion to be drawn is that what is really meant by a brittle fracture in a structural metal is that it is a fracture which occurs before general yield, although it has necessarily been preceded by local yielding around some form of stress-concentrator. An example of such a fracture is shown in Fig. 1.

Fig. 1. Specimen which has t?actured before general yielding (macroscopically brittle) but after local yielding around the stress concentrator. Notched mild steel bar, broken at -90°C. The plastic deformation has been revealed by etching with Fry's reagent. × 6. (After ref. 6.)

It is important also to include the concept of the unstable, catastrophic propagation of fracture. If fracture is to be regarded as the complete separation of two halves of a piece, then the final unfractured ligament of a structural metal containing a fastrunning crack will always yield some fraction of a microsecond or so before it separates, because a small plastic zone travels just ahead of the crack tip. In practical terms, t h e engineer is interested in the point at which the crack begins to run in an unstable manner, and will regard this point of instability as fracture. The best practical definition of a brittle Mater. Sci. Eng., 7 (1971) 1-36

3

LARGE-SCALE BRITTLE FRACTURE IN STRUCTURAL METALS

fracture is therefore one which relates to the instability behaviour. For the purposes of this paper, a brittle fracture is a "fracture which is produced by an applied stress less than the general yield stress of the remaining ligament when crack growth becomes unstable". As will be seen later, this definition can embrace even fractures produced by local shear strains of the order of unity, which on other definitions would be classed as fully ductile. It is probably better to reserve the term "fracture" for the total macroscopic effect and use a term such as "mode of separation" to describe micromechanisms. However, "fracture" is generally used in both senses. The association of brittle fractures with largescale pieces arises from two main causes: one, due to geometrical effects, the other, due to the mode of fabrication. As examples of the former, it is possible to obtain very high local strains around a stress concentrator before general yield if the piece is large; also, increase of thickness may increase the triaxiality in the region of the concentrator. As will be seen, both these effects may promote the onset of brittle fractures. Typical examples of the effect of mode of fabrication are the introduction of stress concentrations at bolt-holes (although riveted constructions can provide natural barriers for crack arrest) and material embrittlement effects due to welding. The number of brittle fractures that have been reported is distressingly large and it seems that most major fields of structural engineering have experienced catastrophic failures; many have occurred in bridges, ships, heavy electrical machinery, pressure vessels, pipe-lines and aircraft. It is not the intention of this review to describe the details of individual failures of this type; ample sources are given in references 7-10. The main features of such failures are summarized below. In general, brittle fracture in service is associated with macroscopic stress-concentrators in largescale pieces. Brittle fractures in steels are particularly prone to occur at low temperatures. In some cases, an original stress-concentrator has been made more severe by the growth of fatigue or stresscorrosion cracks. Often, fracture initiation occurs in an embrittled region near a stress-concentrator, the embrittlement having been induced, for example, by welding or localized cold-working. This paper will review the research that has been carried out to explain the above effects and will describe the methods by which it is hoped to prevent future catastrophic failures. Some indication will be

given of fields in which it is considered beneficial to carry out further research. The approach will be kept as physically-based as possible, working through from stress-concentrations to crack theory, the basis of linear elastic and general yielding fracture mechanics, materials properties, fracture mechanisms, design considerations and finally a summary of the present situation and future possibilities. The field covered is really "the approach to instability", a theme which unfortunately excludes much discussion on crack-arrest philosophies 11, although these are also very important in practice. The starting point is taken as stress-concentrations, because these are generally the starting points for brittle fractures. The treatment given is fairly full, because it is necessary, particularly for materials scientists who are unfamiliar with the concepts, to establish the basic groundwork thoroughly. More detailed coverage is to be found in several textbooks 12- 14.

2. ELASTIC STRESS CONCENTRATIONS AND TRIAXIALITY

It was not until the latter half of the nineteenth century that experimentalists began to realize that the introduction of a notch into a specimen did something more than uniformly increase the stress across the load-bearing cross-section in proportion to the decrease in area. Despite earlier work by Kitsch 15 on stresses around circular holes, the "notch effect" remained a cause of mystification until 1913, when Inglis 16 produced an analytical solution for the elastic stress distribution in a plate containing a small elliptical hole. He was able to show (Fig. 2) that the longitudinal stress, O-yy,is concentrated at the ends of the major axis of the

STRESS 0~,~, t

~x Fig. 2. Distribution of stresses around a stress concentrator.

Mater. Sci. Eng., 7 (1971) 1 36

4

J.F. KNOTT

ellipse and decreases with distance into the specimen. The maximum value of ayr (at x = 0, y = 0) is given as ayy(max) ---- 0.app 1 + 2 ~ (1)

(a)

w h e r e 0.app is the uniform stress applied to the gross

cross-sectional area and a and b are the major and minor semi-axes of the ellipse. If the radius at the end of the ellipse is p, we may write

O'yy(max)= O'app (1 + 2V~ . .)

(2)

For a circular hole, p = a, 0.yy(max) = 30"app' For a very long, sharp ellipse, approximating to a crack (see later), 2

/~

~> 1;

hence

ayy(max)--~

2aappv~

Probably the best physical picture of a stress concentration is given in terms of lines of force, or stress trajectories, by which the load is transferred, from one end of the specimen to the other. These behave like elastic strings (they m a y be thought of as the atomic bonds joining atoms to each other) and crowd together at the tip of the hole to provide a region of stress concentration (Fig. 3). It will be

-×

×

Fig. 3. Pictorial representation of stress concentration, employing concept of stress trajectories.

noticed that there are areas (shaded) above and below the crack which do not bear any stress. This lines of force analogy invites comparison with magnetic and electric field situations, and in fact the basic phenomena are very similar; the differential equations which have to be solved have the same form and an electrical potential model has been used to calculate elastic stress concentrations for notches 17. Other methods of determining the elastic stress concentration factor (ESCF _= Kt in fatigue terminology - = 0.yy(max)/0.app)have included : analytical calculation (much work of this type is that carried out by Neuber is, who approximated notch profiles to hyperbolae), numerical computation and photoelastic analysis, using materials or coatings which exhibit stress birefringence 12. The present situation is that stress concentration factors have been determined for a large number of profiles (notches, keyways, fillets, etc.) in standard states of stress 13, (plane strain, plane stress, radially symmetric, etc.), but rather few results have been published for stress states intermediate between plane stress and plane strain. Computer techniques, such as finite-element analysis, should be able to deal with these intermediate stress states relatively easily, although it is doubtful whether purely elastic solutions of this type are particularly helpful for the solving of fracture problems in structural metals, where yielding precedes fracture. As a consequence of the stress distribution in a plate containing a stress concentrator (e.g. Fig. 2) there is a decrease of the longitudinal stress Cryyfrom its maximum value at the tip of the ellipse until it attains the value of the average applied stress on the specimen at large distances from the tip. For a circular hole of radius a subjected to a uniform uniaxial stress O'app in the Y direction, for example, 0.yy along the X axis is given by 14 ~ ( aZ a4 ) ¢ryy~y= o) = 2 + x~ + 3 ~ (3) for a plane strain state of stress, i.e. at the circumference of the hole (x = a), O-yy= 3aapp, at large distances from the hole (x >>a, a2/x2~ 0), O-yy= O'app. As an elliptical hole becomes longer and sharper, not only does the maximum value of 6yr increase, but the stress gradient immediately ahead of the notch also steepens. It will be seen from Fig. 2 that O-yyis not the only stress ahead of the notch, but that a tensile stress axx is also produced. Given the distribution of O-yy, the presence of a ~rxx stress may be understood Mater. Sci. Eng., 7 (1971) 1-36

LARGE-SCALE BRITTLE FRACTURE IN STRUCTURAL METALS

qualitatively as follows. Consider Fig. 4, which shows the distribution of tensile stress, Oyy,and a set of small elements : a, b, c ... p, q .... etc. along the X axis. Each element, if deforming freely, would suffer a tensile strain eyy in response to the appropri(,p(a)>,q(b) ate value of tensile stress, O'yy \_yy __=yy etc.). We see that an elastic strain gradient is produced below the notch. However, each tensile strain eyy will also have associated with it a lateral Poisson's ratio contraction, exx, which, if each element were

I

, STRESS

¢ I I I I [

I I

. x

,obe

fl

I w, I I I

I I

5

representation ; the continuity problem is automatically taken care of in a full elastic analysis, and Poisson's ratio does not appear in the expression for axx. For example, for the circular hole in plane strain, axx along the X axis is given by 14 : 3

O'xx(y=0) = ~ O'ap p

(a~

a¢ )

X4

(4)

which is zero at the surface of the hole, when x = a, and exhibits a maximum at x = x/2a of o-xx= ~3 aapp or 10.yy(max)" Calculations19 for other uniaxially stressed notches have shown that the maximum value Ofaxx is usually less than 20 % of the maximum value of O'yy. Further, if the deformation occurs in plane strain, as in the median section of a specimen which has a large total thickness B in the Z direction, we have ~zz=0; hence ~zz = v (Oyy-t-Oxx ). The elastic stress distribution along the X axis in a thick notched bar is then as shown in Fig. 5 ; az~ of course becomes zero at both surfaces z = ___ 13/2. Calculations have been made for the circular hole geometry to decide on the thickness of specimen, B, required to ensure that the median section is deforming in plane st rain 2°. The conclusion is that B must be at least four times the radius of the hole, a. However, in the calculation of this value O'zz(max) is taken as 2VO'app, n o t 3VO'app. The effect of taking 3Vaapp would be to increase the required B/a ratio. For thinner pieces, the effect of thickness on the magnitude and distribution of

Fig. 4. Schematic drawing, showing strains induced by the stress Cryy in elements along the X axis (see text).

~ STRESS

deforming independently of its neighbours, would be -Very. (v= Poisson's ratio.) In this case, since .q(a) .e(c) ~yy >,q(b)> ~yy ~yy, etc., so ~x) > ~x~-(b)>~x), etc., and all the interfaces between a and b, b and c, etc. would pull apart. To retain continuity, a tensile stress, axx, must exist across each interface. At the free surface of the notch (x = a), a~x = 0, because element "a" can undergo lateral contraction without any restraint from one side. The stress a~, then increases with distance into the specimen, but falls again at large values of x, because the difference in longitudinal strain between adjacent elements becomes small as the O'yydistribution flattens out (e.9. elements p and q). The difference between lateral contractions is then almost negligible. It is therefore expected that axx will rise fairly steeply with distance and then fall more slowly, as observed. It must be emphasised that this is only a rough pictorial

~X

I I

~z

-Z

, ,~

f3

, Z ,,i

Fig. 5. Distribution of the three principal stresses O-yy,O'xx and crzz along the X axis and a,~ (x = 0) along the Z axis ; cryyand crx are virtually independent of Z.

Mater. Sci. Eng., 7 (1971) 1-36

0

J.F. KNOTT

ayy and ax, is small (less than 10%) but the peak value of az~ is affected strongly (in plane stress, a~, is naturally zero). It is important to note that the maximum value of azz depends critically on Poisson's ratio: it is therefore dangerous to extrapolate results from three-dimensional photoelastic stress analyses obtained in resins (v = 0.5) directly to calculate the azz stress in metals (v = 0.3). Figure 5 then shows that the stressing of a thick piece containing a stress concentrator produces three principal tensile stresses; elastic triaxiality is induced. Some theories have considered that an ideally brittle fracture can be initiated at the point of maximum elastic triaxiality 21, but this is unlikely to happen in structural metals. In these, plastic flow will occur relatively easily at the surface of the notch, where the maximum difference in principal stresses exists, and yielding always precedes fracture.

3. PLASTIC CONSTRAINT The development of plastic flow in a thick, notched piece produces its own marked effects on the distribution of stresses. In plane strain, yielding starts at the root of a notch when, on Tresca's criterion, O'yy--O'xx=2Zy-----O'v, where O'y is the material's tensile yield stress. Since Oxx=0 at the surface, yielding starts at an applied stress of ay/3 for a circular hole. This is the onset of local yielding, as described in the introduction. We now consider the spread of yield across the total notched cross-section when deformation is confined to the X-Y plane. It is immediately apparent (Fig. 5) that the applied stress has to be increased substantially to raise the value of (O'yy-- O'xx) sufficiently to enable the second element in from the surface (say, element b in Fig. 4) to yield, because O-yydecreases, and axx increases, with increasing x. However, in reality matters become even more difficult, because once element "a" has yielded, it deforms at constant volume, with v = 0.5, rather than 0.3 for elastic deformation. Consequently, a larger strain exx is generated and, to maintain cohesion at the a/b interface, axx must increase. In spreading a plastic zone from a stressconcentrator in plane-strain, we therefore anticipate a very sharp increase in axx with distance; the maximum value of axx is found at the plastic-elastic interface. Since the zone has yielded, Oyyis given by O'yy~---O'xx-~-O'y ; and azz by azz = 0.5 (O'yy-~-O'xx) throughout the plastic zone; O-zz= v(O-yy+ O-xx) in the elastic

regions. Analytical solutions for the variation of the principal stresses with spread of plasticity and with applied load are rare, but slip-line field theory has been used to provide results for a semi-circular notch, where the slip-lines take the form of logarithmic spirals 22. If rv represents the spread of plasticity along the X axis and is measured from the surface of the semicircle of radius a, we have, for the local stress, tTyy,within the plastic zone :

ayy=6vll+loge

(l+~ff)}.

(5)

If two such semicircular notches form the exterior boundary of a specimen of minimum notched cross-section 2w times unity then the general yield stress is given by

(a) (w)

O-6v=Ov 1 +

log,

i +-a

.

We see that the general yield stress of a notched bar is greater than the uniaxial yield stress because it is more difficult to spread yield in regions containing triaxial stresses. The ratio O'Gy/O"Y is the "constraintfactor" (L), and has a maximum value derived by Orowan 23, using a "plastic punch" analogy, of 2.57 for deep parallel-sided external slots, in plane strain deformation. As the applied stress is raised from that necessary to produce local yielding to the general yield stress, the maximum principal stress beneath the notch, O'yy, increases 24- 26. Details of this increase have not been widely determined, but a schematic diagram of the general behaviour is shown in Fig. 6. The maximum value that tYyy ever attains in thick notched pieces subjected to uniform applied stress and plane strain loading is 2.57 av at general yield. For a 45° "V" notch in pure bending, the maximum is 2.18 O'V27'28. The ratio O'yy(max)/O"Y is the "stress intensification" (R) or "plastic stress concentration factor ''29. The generalised form of eqn. (5) is then O-ry= Ray

(6)

where R is a function of geometry and extent of plasticity. If the notched piece is thin in the Z direction, a z z ~ 0 and the deformation mode changes from shear in the XY plane to shear in the YZ plane, because azz is now the smallest principal stress. If azz remains at zero throughout deformation O-yydoes not increase with spread of plasticity and neither ayy nor a6v exceeds the uniaxial yield stress av. The importance of these stresses will be considMater. Sci. Eng., 7

(1971) 1-36

7

LARGE-SCALE BRITTLE FRACTURE IN STRUCTURAL METALS

II

STRESS

31 ~'

:

GENERAL

YIELD

..,~ I

7/\\\ ,L. . . . .

I

/ PLASTC I/j ,

i i

-

Z

-

r

,

4/3r~'~

O

I

I°

O

, '

0.5

l.O

ELASTIC

~°~//L~y --~

4 / 3 " 3, ~,'

Fig. 6. Variation of m a x i m u m local tensile stress, (Oyy(max)), the plastic/elastic interface, with applied stress, aapp.

i i

at

ered later, in relation to fracture behaviour. The main characteristics are that the presence of the triaxial stress state increases the load required to produce a given plastic strain; strain-controlled fracture at notches in ductile metals therefore often leads to observations of "notch-strengthening'. On the other hand, any cracking process favoured particularly by tensile stress will be likely to occur in a notched bar at lower strains than in a smooth tensile testpiece, because the tensile stress %y is increased by constraint rather than by strain-hardening. Since the state of stress may be affected by the thickness of the specimen, it would be useful to know the thickness required to ensure that azz reached its full plane-strain value of 0.5 (ayy+ axx). Computing techniques are available which can be used to determine the values of azz for specimens of different thicknesses, but detailed results for the plastic/elastic situation have not been published. (The calculations become complex, because the common simplifying assumption that Poisson's ratio, v, is the same in both elastic and plastic regions cannot be made.)As a rough estimate, one could suppose that az~ increased, from zero at the free surfaces on the side faces to its full plane-strain value in the centre, at the same rate as axx increases from zero at the free surface of the notch root to its maximum value at the plastic/elastic interface (see Fig. 7). On this basis, if we took a typical value of, say, 1.5av for the maximum value of axx at the plastic/ elastic interface, the minimum value of the thickness,

Fig. 7. Distribution of the principal stresses for plastic/elastic stressing: ayy decreases to a v on the side surfaces (z= _+B/2) throughout the plastic zone.

B, required to ensure that the median plane was deforming in plane strain would be about 2,7 rv, where r v is the total extent of plastic zone ahead of the notch. Such a value is compatible with results on the effect of thickness on the cleavage fracture at general yield of "V"-notched specimens of mold steel 3°. A relationship between minimum thickness and extent of spread Of plasticity is entirely reasonable for notched specimens, because the maximum required value of azz at the median plane increases with the zone size. The specimen must be made thicker to contain some plane strain deformation, because the higher median plane stresses are more likely to produce "45 ° shear" in the YZ plane. The effect has been beautifully demonstrated by etch-pitting the plastic zones in notched plates 31. This detailed survey of the stress distributions around blunt notches has been given as an introduction to the various considerations that must be taken into account when dealing with naturally cracked specimens. To summarize : for a piece containing a stress concentrator, we may be interested in any of the following. For elastic stress states : The concentration of stress at the til~ of the concentrator. The elastic strain concentration due to the stress concentration. The increase in stress- or strain-rate at the tip, arising from the higher local stresses or strains. The elastic triaxiality. The thickness required to produce a plane-strain stress state in the median plane. Mater. Sci. Eng., 7 (1971) 1-36

8

J . F . KNOTT

For plastic/elastic stressing : The production of local yielding at low applied stresses, hence the local plastic strain concentration and increase in plastic strain-rate at the tip. Local stress "intensification"; i.e. the raising of the local tensile stress O'yy(max) by plastic constraint. The increase in general yield stress produced by constraint. The thickness required to produce a plane-strain stress state in the median plane. The rest of the paper will concentrate mainly on stresses and strains around cracks and on how these affect brittle fracture. As for blunt notches, we consider first the elastic stress system.

4.

0"yyis given by tryy=K1/x/2rcr. In the configuration described,/_~_is given as K1=aappx/~da, hence O'yy= aappx/a/2r, close to the crack tip. The advantage of expressing the crack tip stress field in terms of the single parameter Kt lies in the fact that the tip region knows only the stress field; it is immaterial whether this has been produced by an increase in O'ap p or in crack 1,~ngth, and the K~ factor conveniently embraces both effects. An alternative approach to the calculation of crack tip stress fields has been that of Westergaard 37, by analog), with bearing pressures. He gives Oyy(y= O ) = 0"app/X/1 -- a2/X2 where x is measured from the centre of the crack. Making the substitution r = (x-a) and remembering that r,~ a, we again obtain

STRESS FIELDS NEAR CRACKS tTyy(y= 0) ~---O'app

The stresses near cracks have been calculated analytically by subjecting various elastic stress solutions, known for other stress concentrator geometries, to appropriate limiting processes. For example, the Inglis solution ~6 for an elliptical crack has been taken to the limit, p (radius)~ 0; Neuber's solution for hyperbolic notches a8 to a limit where the asymptotes of the hyperbolae close together; Williams's solution 32 for "V" notches to the limit where the included angle becomes zero, and so on. All these limits naturally lead to the same answer for the stresses around a crack 33- 35, which may be written as follows, for a situation similar to that in Fig. 2, i.e. total crack length =2a, subjected to a uniform applied tensile stress in the Y direction: O'yy -

K, 0 ( l + s i n 0 s i n -3-0 ) + 2N///~ COS.2 \ . .2 . 2

gl 0 (1 '_sin_O sin ~ ) + axx - v / ~ r cos .2 . . 2. .

azz = 0 (plane stress) a~z = v (%y + Crxx) (plane strain)

(7)

where r is the distance measured from the crack tip, and 0 is the angular co-ordinate, measured anticlockwise from the X axis (cf Fig. 2). There are several important points to notice about these expressions. Firstly, the stresses are given by a series of terms, but, close to the crack tip (r small), only the first term is significant, and the local crack tip stresses can then be characterised by a single-valued parameter, K~, termed the stress intensity 36. For example, along the X axis (0=0),

o r O'yy(y= 0) =

KI/

This simple expression for ayy(y=0) in an infinite plate is about 4 % inaccurate when r/a has reached 0.05. The second important point involves the presence of singularities in the stress field. As r ~ 0 , so O'yy---+o(],O'xx---~ooand, in plane strain~ r z z - ~ . This situation is physically unreasonable. It arises because the crack tip radius has been reduced to zero, and the material has been considered as an elastic continuum, deforming in accordance with the relationships of linear elasticity. In fact, the material is discontinuous on an atomic scale and the stressstrain relationships at the crack tip are not linearly elastic. It is only some little way ahead of the tip that the expressions for the stresses have physical significance. We contemplate that a crack under stress will have a very small, but finite, tip radius and that there will not be any infinities in the local stresses and strains, although the atomic bond at the crack tip will be stretched almost to failure. Even on the continuum approach, any stress applied to the specimen will produce some elastic opening at the crack tip. Westergaard 37 gives 6 - 4 ( 1 - v 2) aappx/2a r E in plane strain, 4(1 = nE in plane stress, 4K x/~x r

6= 5

(8)

Mater. Sci. Eng., 7 (1971) 1-36

9

LARGE-SCALE BRITTLE FRACTURE IN STRUCTURAL METALS

for the total elastic opening within the crack. Here, K~ is the crack tip stress intensity and r is the distance into the crack from the tip. At any given value of KI, the crack tip therefore assumes an elliptical shape (6 oc x/r). Similarly, the position of maximum tensile stress ahead of a crack depends on the treatment applied to the problem of root radius. Equations (7), derived on the basis that p =0, of course produce infinite singularities at r = 0. Even so, the maximum tensile s t r e s s , O-yy, can be shown to occur, not across the plane y = 0, but across planes inclined at some 60° to it. Once a finite root radius has been admitted, the position of maximum tensile s t r e s s (O'yy=O'00) appears to be across the plane y = 0. We have so far considered only an internal crack subjected to a uniform tensile stress which tends to produce a simple opening in the X-Y plane, but the form given for the crack tip stress field applies to a variety of possible geometries and loading configurations if KI is replaced by the appropriate stress intensity factor. Table 1 lists a number of common examples of this type; those to which most frequent reference is made are: (i) The opening mode, in plane stress or plane strain (K0: applied tensile stress O'yy, opening disp l a c e m e n t , (~yy, parallel to Y axis. (ii) The shear mode (KH): applied shear stress ryx; sliding displacement, 4)yxparallel to X axis. (iii) The antiplane strain mode (Kin) : applied shear

stress Zyz ; sliding displacement (~yx parallel to X axis. One great advantage of the common form for crack tip stress fields is that it is possible to calculate the stress field for a crack subjected to a combination of applied loads by simple superposition of the components arising from the individual stress fields. (As, for example, for a crack subjected to both uniform applied tension and point wedging forces within the crack.) In these analyses, nothing has been said about the variation of azz with plate thickness. Similar stress fields are to be expected for ayy, axx and rxy in both plane stress and plane strain opening modes (see eqn. (7), and refer back to the calculation of stresses for the circular hole), but azz is bound to vary substantially, from zero at the side surfaces to a higher value in the median plane, reaching eventually the figure V(ayy+axx) if the piece is sufficiently thick to withstand some plane strain deformation. No figures have been given for this critical thickness for a purely elastic stress field, but if one could carry over an analogy from the circular hole calculations, it might be expected that a thickness B = 4a would represent a minimal value, since the required stress in the median plane is high near the crack tip. As was the case with other stress concentrators in structural metals, the elastic stress distributions around cracks are not of final importance to the fracture problem, because yielding is produced at a

TABLE 1 : SOME FORMULAE FOR STRESS INTENSITY FACTORS

Type of crack

Applied stress

Crack tip displacement

Mode

Stress intensity

0"yy

by)

Kl = a,,'~a

ry x ryz

~brx

~byx

Opening (Plane stress or plane strain) Shear Antiplane strain

Central, of length 2a in plate of width W

O-yy

~yy

Opening

Kl = a

Central, penny-shaped in infinite solid

O-yy

6yy

Opening (radially symmetric)

2 K = ~N/7~ a ,."a

O'yy ryz

•yy (~yz

Opening Antiplane strain

K l = 1.12a\.'n~a Kll I = "tN,"~a

O'yy

6yy

Opening

Central, of length 2a ( + a along i X axis) in infinite plate,

Edge, of length a, extending into semiinfinite plate Two coplanar edge cracks, each oflength a, in plate o f t o t a l w i d t h W

Ktl=zV/rC-o Knl = z. ,'~a [IV tan (\ ~w a ]]" Jj

sin

Mater. Sci. Eng., 7 (1971) 1-36

10

J . F . KNOTT

crack tip before fracture and this alters the stress distribution. However, elastic stress distributions have been widely used, in but slightly modified form, to explain the initiation of fracture from cracks and it is as well to see how they were used before proceeding to situations involving yielding.

5. ELASTIC FRACTURE AND GRIFFITH'S THEORY

To provide a theory of elastic fracture, one would initially examine the concentrated stress field around an crack tip and consider that fracture occurs, i.e. that the crack extends, when sufficient stress exists to pull apart the atomic bond at the crack tip. Thus, if we were to take the Inglis solution 16 for the stress at the tip of a crack of length 2a and tip radius bo equal to a lattice spacing, we would obtain 2aap p ~

=

47a

2 2 7ZO'appa

E

in plane stress. This change in energy is shown graphically in Fig. 8. It can be seen that the energy passes through a maximum and then decreases with increase in crack length. Griffith considered the change in energy produced as the crack of length

S..--~UKFACE- - a ~e~ER,G¥ ~'~'

/ ./INSTABILITY ,~,-~=~.j,'TOTA L ENERGY -\ ~ C ~ACK Z ~ LENGTH

o .__..

'

\ Fig. 8. Change in total energy of a cracked plate with variation in crack length.

=- aF ~-- (E/EO)v~o/a.

(9)

We see that aF is proportional to a -~. Such an approach provides both a sufficient and a necessary condition for fracture. To obviate the difficulties produced by the infinite singularities in the stress field near a crack tip, Griffith 38 considered only the energy balance in a cracked body to decide whether or not unstable propagation could occur. His method may be understood by reference to Fig. 3. The introduction of a crack into a body strained between fixed grips alters the total energy of the body in two ways. The energy is increased owing to the energy of the two crack surfaces ; if the surface tension is ?, this increase is given by S = 4 7 a per unit thickness in the Z direction. On the other hand, the distribution of the stress trajectories means that there are areas above and below the crack which are stress free (the shaded areas in Fig. 3). There is therefore a decrease in stored elastic strain energy, U, per unit thickness released, given by 2 2 7~O'appa

S(increase) -I- U(decrease) :

fracture strength, say E / I O .

Thus, aapp at fracture

U --

ence of the crack is given by

in

2a was extended by a small amount da, and established a critical energetic condition - - above which the crack runs catastrophically to failure and below which it heals up completely. Thus if we differentiate (S + U) with respect to crack length, we obtain ~aa0(4?a

rctrZppa2~-) = 4 7

2trZPPZm-~-

(11)

At the critical point, we have 47 - 2O-a2ppna E ' i.e. the critical applied stress required to cause an internal crack of length 2a to run in an unstable manner is given by 2~/~/na

in plane stress ;

a v = x/2ET/7c(1 - v 2 ) a

in plane strain.

aF =

(12)

plane stress

E

(10)

U = - ~a2pp(1- v2)a2 in plane strain. E Thus, the total change in energy due to the pres-

These are Griffith's expressions, and represent a necessary, but not sufficient, condition for brittle fracture, in the sense of our definition, to occur. For example, if the crack were blunt at its tip, there might not be sufficient stress concentrated to pull Mater. Sci. Eng., 7 (1971) 1-36

11

LARGE-SCALE BRITTLE FRACTURE IN STRUCTURAL METALS

apart the atomic bonds. The Griffith criterion is probably best regarded simply as the thermodynamic condition for unstable propagation; in this way, the conceptual difficulties associated with the way in which the critical condition is approached (elastic Griffith-type cracks of less than critical length, or any crack of given length subjected to less than the critical applied stress, would be expected to heal up) are avoided. In any real situation concerning structural metals, yielding at the crack tip precedes unstable propagation, and the Griffith condition must be modified to take account of plastic/ elastic stress conditions. However, it is instructive to examine a second method of deriving the Griffith energetic condition, since this introduces further commonly-used relationships. In considering the change of total energy-of'h body with crack length (eqn. (11)), we derived a term ?~U/Oa, which represents the rate of release of elastic strain energy with respect to crack length. This parameter is given a symbol, 2G, where G is called the elastic strain-energy release rate G ~- O'a2ppz~a/E GI = aZpp(1-v2)rca/E

in plane stress in plane strain.

(13)

If we make the standard substitution for crack opening modes of K=o-appx/ga, we obtain G in terms of stress intensity : K 2 = EG K2 _

EGI

( 1 - v 2)

in plane stress

(8)), remembering that in one case r is measured from the tip of a crack of length a in the + X direction, whereas in the other case it is measured along the - X direction from a tip situated at (a+da)), and the work done is equated to the new surface, over length da, produced. Again, eqns. (12) are obtained for plane stress and plane strain. The approach can be generalised to calculate values of strain energy release rate for combinations of modes I, II and III stressing +°. It must be emphasised that these relationships are formally equivalent to the Griffith approach and therefore establish a thermodynamic condition for unstable propagation. Only if the surface tension ? is expressed in terms of bonding forces and atomic displacements can the energetic approaches produce a sufficient condition for the onset of fracture. To make the critical values of K mechanistically based, further assumptions have to be made; for example, that the O-yystress is equal to E/IO when r is one lattice spacing (Kcrt~-,,/2~bo'(E/10)), or that the elastic displacement attains a critical value, say bo/2, one lattice spacing behind the crack tip (Kcrit ~ x/(Ttbo/2).(El8) . An example has been given by Rice 41 of the way in which the energetics and mechanistic approaches may be reconciled (see Fig. 9). He considers the

T

~

(14)

in plane strain.

The criterion for unstable crack propagation becomes, in terms of G G = 27 G~ = 2?

t

in plane stress in plane strain

o - -

(is)

~(S)= A c s v ~ m ~

,s EV*LUAV.O

]

and in terms of stress intensity K 2 = 27E K 2 = 27E/(1 __y2)

in plane stress in plane strain.

It is possible to derive the Griffith relationship between fracture stress and crack length purely by considering the stress-field at the crack tip. Instead of determining the change in total energy in the body as a whole, the amount of work done by the stresses around the crack tip when the crack extends by an amount da is calculated; Colonetti's theorem shows that the two approaches are formally identical 39. Both the stress field and the crack tip displacement are expressed in terms of K (eqns. (7) and

,Wgt,~,~]

/--. \

/R~A-2;

SF SEPARATION DISTANCE

~i

Fig. 9. Model used to relate the Griffith and Barrenblatt approaches. (After ref. 41.)

Mater. Sci. Eng., 7 (1971) 1-36

12

J.F. KNOTT

crack model due to Barrenblatt 42, who removed the singularities from the stresses and strains predicted by linear elasticity theory by postulating that the interatomic attractions in the "cohesive zone" immediately in front of the crack tip could be represented by a restraining force a(6) which is a function of separation distance of the atoms along the X axis from their rest positions. Rice evaluates a line integral J, given by

fracture. It is important to remember, however, that K 2 and G as originally conceived express the rate of release of elastic strain energy and that they are therefore treated in the literature more in terms of the necessary thermodynamic condition, rather than the mechanistic condition for brittle fracture. This interpretation may sometimes be changed when yielding at the crack tip precedes crack extension.

OU ds J = fr T "O~x 6. (QUASI-) BRITTLE FRACTURE

where T is the traction vector O'ij nj, defined according to the outward-facing normal of the curve F, U is the displacement vector and ds is an element of the curve. By shrinking his F contour down to the boundaries of the cohesive zone, he shows that J is given by

J = f 'o-(6)d6 where 6t is the separation distance of the bond at the crack tip. For a mechanistic criterion for fracture, 6t may be given the value 6F, the distance at which atoms may be considered to have been just pulled out of the range of attraction of their neighbours (o-(6)=0). Thus, on the cohesive mechanistic approach, the critical value of J is given by

J =flFo-(6)d6.

(16)

The Griffith formula as it stands is not directly applicable to the fracture of structural metals because this does not occur as a result of a simple strain energy : surface energy balance. For fractures in which the extent of yielding at the point of instability is substantially smaller than the width of the specimen (i.e. brittle fractures), the fracture stress is proportional to a -~, but the constant of proportionality is much greater than that predicted by the Griffith formula. The first explanation of this higher proportionality factor was provided by Orowan 2a and Irwin 36, who suggested that the surface tension term in eqn. (12) should be modified to include the amount of plastic work done as the crack extended. Thus, O'F=/E(27-+- ~p) V

On the other hand, Rice shows that, ifJ is evaluated around a contour which is situated at a distance which is large with respect to the size of cohesive zone, in, e.g., the plane strain opening mode, then J-(1-v

(17)

ga

in plane stress where 7p is a plastic work term. From the experimental values it was deduced that 7p was very much greater than the surface energy 27, hence, approximately,

2) K z = G I , E

the elastic strain energy release rate. On the energetic criterion for instability, J = G, = 27, where 7 is the surface tension. But the area under the forceseparation curve, given by the integral in eqn. (16) above, is by definition equal to 27. For small perturbations in the linear elastic stress field, the energetic and mechanistic approaches therefore lead formally to identical results. The result appears to be general; J is always given by the rate of decrease of potential energy with respect to increase in crack length. In practice, complications arise because we use results for elastic stressing situations and try to apply them to materials where plastic flow precedes

as a condition for the crack to spread. In terms of strain energy release rate, the critical value of G is given by Go=Tp (cfi eqn. (15)) and this substitution has often been made in the expression above, to give

for the applied fracture stress, or

Ko =

(18)

for the critical value of stress intensity. The critical Mater. Sei. Eng., 7 (1971) 1-36

13

LARGE-SCALE BRITTLE FRACTURE IN STRUCTURAL METALS

value of G is commonly referred to as a metal's fracture toughness 36. Similarly, in plane strain, the critical value of strain energy release rate is given by Gm= 7~, of applied stress by

EGlc v2)a

09 )

av =Vrc(l -

and of stress intensity by:

EG]c K'c = ~ / (1 - ve) Similar forms of expression hold for the other modes of stressing indicated in Table 1. These expressions, as developed by Irwin 36, provide the basis for the whole field of fracture toughness testing of materials, which will be described in detail in a later section. Here, we shall consider in a little more detail the ways in which

T~

brittle fracture from cracks in structural metals may be treated theoretically. There are primarily two approaches: one, which uses modifications of formulae derived from elastic stress analyses; the other, which relies on full plastic/elastic analyses. The former has the virtue of being applicable to a much wider range of situations; the latter, that of being more rigorous when it can be used. We consider first the approach from elastic stress distributions, or linear elastic fracture mechanics (LEFM) 1°'43-47. Here, we rely mainly on results obtained in the previous section, which must, however, be modified to take account of local yielding around the crack tip. The Orowan/Irwin expressions represent a first attempt at this, but can obviously be bettered on two counts. Firstly, the way in which plastic flow modifies the stress distribution around the crack tip must be explored ; secondly, a physical meaning must be given to the critical value G~, the "plastic work" term.

*Y I

Stress (y =0)

--+X

Stress

,

Cry ~Y

~- a

X

ry

a

ry

X

(B) Stress '~

Stress

~

l

I

I

I

J - - J = .,'~, . . . . a

ry

ry

Co)

a+ry

~

ry

~

..~ X

(o)

Fig. 10. Basis for modification of elastic stress analysis to take account of small plastic zone. (After ref. 44.)

Mater. Sci. Eng., 7 (1971) 1 36

14

J . F . KNOTT

A first approximation to the extent by which the plastic zone modifies the stress distribution is given by an argument 4s based on Fig. 10. Here, we assume plane stress, so that we are interested only in the magnitude of O'yy (as the stress which produces yielding). We envisage that our final stress distribution should be as in Fig. 10A, i.e. that, ahead of the crack tip, O-yyshould be equal to the yield stress av throughout the plastic zone (up to rv) and vary as K / x / / 2 ~ at larger distances. We might try to estimate rv by finding the value of r at which O'yy= O'y, i.e. ry = K2/2na 2 (Fig. 10B), but this would be very much an underestimate, because it does not allow for the yielding produced by all the stress depicted in Fig. 10A by the area between the lines ayy(r< rv), a = a v ( u p to rv) and the ordinate. It so happens that this area is exactly the same as that shown shaded in Figs, 10A and 10B ; thus, the total spread of plasticity* may be taken approximately as K 2

2

2rv = no---~- a,ppa a2

(20)

The stress distribution ahead of a real crack of half-length a with extent of plastic zone 2rv may be regarded as identical to the elastic stress field ahead of a "notional" crack of half-length (a + rv). Thus, the failure stress is given by

O'v=

~/ Eac --.-:.~/EGc/7~a(1--~ O'ap2P ) n(a+ry)

2a 2

(21)

and the tip of the notional crack is located at the centre of the real plastic zone (Fig. 11). Wells 49 has used this concept to provide a physical interpretation for G~. The opening 3 at the boundary of the plastic zone (a distance rv in the - X direction from the notional crack tip) is given by 4K 2 6nEay (eqn. (8), substituting r = r v = K2/2na 2 or, since K 2 = e 6 (eqn. (14)), 7~

G = ~ av 6 -~ o"v 6.

(22)

The interpretation given to this expression is that, for a plastically deforming region at a crack tip, the energy release rate in the body as a whole corresponds to stretching an element of width da ahead

* N.B.: Change in definition of r v with respect to Section 3.

I I I I I I

I

~X

I I I I l_

I

I ~ a

~ Iql.-------------,,l~I

I *~,

I

~'~, I

I

I

I

I I Fig. 11. Diagram of crack + plastic zone, showing location of "notional" crack tip and opening displacement, 6. (After ref. 49.)

of the tip by a displacement 6 in a region of yield stress av. Separation will occur when a critical opening displacement, 6orit, has been attained. At first sight, we then have G ¢ - O'v6erit for substitution into eqn. (21), for example. It should be emphasised that the fracture toughness Go is still expressed as a measure of the critical energy release rate, i.e. it still represents the extra work done when the crack is extended by an amount da. It is often supposed that the attainment of this critical value of G automatically implies that instability has set in. Strictly, this can no longer be justified for plastic/elastic stress states, because we are no longer dealing with the Griffith energy balance. Equation (22) merely recharacterises G; to use the Griffith approach, for example, we equate av6 to 27. This would then be virtually equivalent to the Barrenblatt-Griffith identification (eqn. (16) et seq.) for elastic stress states. Much the same interpretation of G~ is provided by analyses which treat the stress system as plastic/ elastic throughoutS°'51; these have the advantage that they are able to calculate displacements even when the extent of spread of plasticity is large, but, unfortunately, there is no full analytical solution of this type corresponding to the plane-strain opening Mater. Sci. Eng., 7 (1971) 1-36

LARGE-SCALE BRITTLE FRACTURE IN STRUCTURAL METALS

mode. The situation here, of course, is complicated enormously by the development of plastic constraint as the yielded zone grows (cf. Section 3). In plane stress, Dugdale 5~ gives for the displacement at the crack tip 3

80"y

/ ~O-app\

--- -nE - - a IOge sec~ 2~-av )"

(23)

Fracture is assumed to occur when 3 reaches a critical value 6~r~tand the fracture stress is then the value, or, of the applied stress, aapp, corresponding to 3erit in the above expression. For very low applied stresses, we may expand the "log~ sec" term and obtain

15

of large structures has been covered. The approach adopted from here on in some ways traces the history of the subject as tests have been made, and the results obtained from them have entailed modification of the original concepts. Much of the quantitative background has been developed on a semiempirical, rather than on a purely theoretical, basis. The main emphasis is placed on the prevention of crack initiation or crack instability in a statically loaded piece containing a sharp stress-concentrator. Wherever possible, some mechanistic explanation of fracture results will be attempted; often, such explanations are no better than a wild extrapolation from a simple model to real behaviour, but it is felt that the problems, at least, will be highlighted.

7. FRACTURE TOUGHNESS TESTING

for the "fracture" stress. This bears very close resemblance to eqn. (18), if Gc is written as av 3crit, but we have said nothing about any instability criterion for spreading the crack; we have merely said that some crack extension will occur when a critical displacement is achieved at the crack tip. When considering various fracture mechanisms in later sections, we shall see that it may become necessary to distinguish between two critical values of 3: that at which the crack tip begins to move forward and that at which crack propagation becomes unstable. The majority of plastic/elastic crack problems investigated have been those in antiplane strain shear (Kin mode, Table 1). Here, the yielded zone may be represented by an inverse pile:up of screw dislocations to which methods derived for continuous distributions of dislocations are applied 52'53. The results for a single crack in an infinite body are very similar to those for the plane stress opening mode, the crack tip (shear) displacement, for example, being given by q5 = 4Zvp~a log~ see/

~Tapp~2zY /

(24)

where "~app is the shear stress Zy~; Zv is the shear yield stress and p is the shear modulus. The antiplane strain model has been applied to many configurations to examine effects due to finite boundaries 54, to the interaction of two or more cracks 55'56, and to blunt notches 5v. In turn these further analyses have been used to predict the fracture behaviour of, e.9., inclusion stringers 5s. We have now reached the stage where sufficient formal theory for assessing the fracture behaviour

(a) Plane stress The early application of the Orowan/Irwin formula was to the assessment of the resistance to fracture of structural metals in the form of thin, centrally notched sheets 59'6°. The practical importance of testing in this form was that the materials studied were those liable to be used in similar plane-stress situations in service (aircraft and spacecraft fuselage). The appropriate failure stress for a central crack of length 2a in an infinite body is given (eqn. (18)) by

7za

If the stress-free boundaries are not infinitely removed from the crack tip, this must be modified 59 to give

[ t n( tl where the total width of the specimen is If. In general, good agreement with the a ~ formula was obtained, but, in some cases, it was found that the critical value of strain energy release rate at instability, which we shall now denote by G~ritrather than Go, increased with initial crack length, a, for otherwise geometrically similar specimens of constant thickness. The effect was first discovered in ~-in. thick panels of an age-hardened aluminium alloy (7075-T6) 6° and was explained in terms of an increasing resistanMater. Sci. Eng., 7 (1971) 1-36

16

J.F. I,:YOTT

ce to crack growth with length. It was observed that the first stages of cracking from the starter crack were confined progressively to the midthickness of the piece (PQR in Fig. 12): this cracking was of the "square" type (i.e. normal t o O-yy).Associated with this square fracture were two shear lips PRS and QRT (Fig. 12) which occupied progressively more and more of the cross-section until at point R the entire fracture was of the slant type. It was found that the distance, d, that a crack had to grow before a fully slant fracture was obtained was sensibly independent of the original length of the starter crack. If we now accept that the critical value of fractut~e toughness in such a test is calculated from the applied load at which instability sets in 45, we must identify this instability with the point at which the fully slant fracture runs catastrophically. The initial square fracture produced at low load does not necessarily give rise to instability in the specimen as a whole, because the shear lips require increasing strain and load before they separate. The general form of loading curve for this situation is shown in Fig. 13 ; the square fracture may produce a step in the curve if it occurs quickly (a "pop-in") or it may simply decrease the slope if it occurs more gradually. The failure of the Irwin relationship for the cracked Y

I

~'j.,,JJ~

Q

Z

¢..~

[~

~-.~.L;~.f~.~

i

I

ER NOTCH÷ FATIGUECRACK SQUARE FRACTURE T

SLANT

FRACTURE

S

s

×

TRANS-THICKNE$$ PROFILES OF CRACK

Fig. 12. Development of fracture in ~-in. thick aluminium alloy panels. Note the change from square to slant fracture as it grows.

FINAL INSTABILITY

"PoI

FRACTURE I

,

CRACK EXTENSION Fig. 13. Load-extension curves for thin cracked panels. The speed with which the square fracture grows in the centre of the panel determines whether or not a "pop-in" is observed. (After refs. 45 and 60.)

aluminium alloy sheets was attributed to the fact that the amount of(square + shear lip) crack growth, d, to the point of instability was independent of initial crack length, a. Since, for short cracks and small specimens, a given increment d produces a larger change in strain energy release rate than it would for a long crack in a wide specimen (the stress distribution in the uncracked ligaments becoming more strongly perturbed), the scaling with a- + that the Irwin formula demands does not follow. For such stressing situations, a curve must be obtained of resistance to crack growth (an R curve) versus initial crack and specimen dimensions before critical values of G or K relevant to specific service applications rr~y be determined. A material's behaviour, when tested in sheet form, depends on the mode of fracture. If it is brittle (predominantly square fracture) then Gerit will be independent of crack length (instability coincident with "pop-in"). If the modes are mixed, then the phenomena described above occur. If the fracture is 100~ slant throughout, then one would again expect instability to be coincident with the first stages of crack extension. This third possibility has been modelled in terms of a Km (antiplane strain) mode of separation at the crack tip as fracture occurs by sliding-off on a through-thickness plane inclined at 45 ° to the tensile a x i s 61'62. The process is completely ductile (shear strains of the order of unity are involved) but the total fracture would be regarded as brittle by our definition. It has been described as a "cumulative" mechanism of fracture. If, in Fig. 14, the main block Mater. Sci. Eng., 7 (1971) 1-36

17

LARGE-SCALE BRITTLE FRACTURE IN STRUCTURAL METALS

associated tensile displacement (LN or MO) as B, we have

E .y B O'app

7~a

This also gives the condition for instability, because the tensile displacement B remains constant as the screw dislocations advance into the material and increase the crack length "a". Any increase in "a" subsequent to initial crack extension both decreases the value of aap p required to cause further extension and at the same time raises the applied stress available to extend the crack by decreasing the loadbearing cross-section. Hence a maximum in applied load is obtained and Gcrit may be written as:

I~CEMWNT AT CRACX TIP

45°= ~-B

Gcrit -~ (7v B

,- ~

~v~"

PROOUCED llY SlNGLi[ SCREW DIIK.OCATION XY

(~app

Fig. 14. Model to illustrate fracture by "sliding off'. In practice, some crack opening will be associated with this K m mode and the dislocations will be part screw, part edge. (After refs. 53 and 61.)

drawing represents the thin sheet material, we visualise the 45° plane emerging at LM as that along which separation occurs. (The centre points of LM and NO would represent a 45° section through the original crack tip.) Complete sliding along this plane separates LM and N O ; at this stage, initial crack extension has been achieved by a pure K.~ mode, with a relative shear displacement, q5= xf2B. As in the dislocation models of plastic/elastic crack deformation, the displacement can be regarded as having been produced by an inverse pile-up of screw dislocations S, each of which contributes a shear displacement b (see inset diagram in Fig. 14) to the total. This is precisely the situation leading to eqn. (24) which therefore gives, as the criterion for first crack extension, "gapp

=~J

2x/2/~ rv B

(27)

(26)

~a

provided that the extent of pile-up is small compared with the width dimension of the sheet. Taking the

(28)

Taking a value of approx. 104 lb./in. 2 for the yield stress of pure aluminium, we see that G~rit- 104 B, i.e. that the toughness of thin sheet increases very rapidly with thickness. For example, in pure aluminium, a value of 700 in. lb./in. 2 would be obtained in 0.07 in. thick sheet. In commercial microstructures, perfectly plastic rupture, if it occurs, will be initiated from second-phase particles. One might then suspect that the appropriate value of displacement would be related to the through-thickness particle spacing rather than to sheet thickness. On the other hand, the yield stress will be increased by the presence of particles, so the resultant effect o n Gcrit is not easy to calculate, particularly since the displacement in practice is a combination of opening and sliding. However, in a 7075 T-6 aluminium alloy 6°, the slope, Gcrit= 104 B, would be quite consistent with the experimental results shown in Fig. 15. (The solid line drawn there is derived from a slightly differem model, but again shows G oc thickness for very thin sheets.) The main feature of the experimental results on the aluminium alloy (Fig. 15), however, is the rapid decrease in toughness for thicknesses greater than about 0.1 in. to a constant level of about 115 in. lb./in. 2 at large thickness (say those greater than 0.6 in.). At the same time, as the toughness drops, the percentage of square fracture (cf Fig. 15) increases, becoming about 85 ~o of the total fracture surface in the 0.6 in. thick specimen. The low limiting toughness level and associated high percentage of square fracture are closely related to the "pop-in" type of behaviour described earlier. If the specimen is very thick, the proportion of total cross-sectional area Mater. Sci. Eng., 7 (1971) 1 36

18

J . F . KNOTT -

I

1200

I

I

I

[

[

I

I

I

1

I(300 --901 I

-

7O

-

£-

~r,~

400 -30

~

-

200

Gk

- 10

~'~

0.1

1

0.2

I

0.3

I

I

0.4 0.5 B, inches

I

0.6

I

0.7

I

0.8

I

0.9

1.0

Fig. 15. Variation of fracture toughness with thickness in aluminium alloy 7075-T6. The steep rate of increase in very thin sheet is drawn schematically. (See refs. 45 and 60.)

occupied by the square fracture at the initial "pop-in" may be so great that instability occurs immediately, giving rise to a low toughness. As B is decreased, the "pop-in" may be stabilised at low loads and instability occur only at higher loads, as described above. The concept is straightforward, but some confusion may arise because of terminology. Sometimes the maximum in the curve is called-the "plane stress" fracture toughness, "G¢"; sometimes the whole curve (apart from the limiting level) is called this, and usually the limiting value is called the "plane strain" fracture toughness, Gic. These definitions, in particular "plane stress", are not synonymous with the mathematical definitions of plane stress (az~= 0) or plane strain (ezz= 0). As we have seen, the initial portion of the curve, representing the toughness of thin foil, rises rapidly (Gerit ~ B). A maximum is reached only when a small amount of square fracture occurs. In the aluminium alloys, this square fracture corresponds to a stable "pop-in" produced by approximately the same applied stress level as that required to cause complete "plane-strain" fracture in thicker sections. The reason for a "pop-in" occurring at a particular applied stress depends entirely on the mechanism by which the square fracture is produced, but, in these alloys, the process is obviously made easier by the presence of a hydrostatic component of tensile stress. In these circumstances, it is tempting to attribute the square fracture to a "cracking process" (as, for example, in the cleavage of mild steel) rather than to a "flow process" (plastic flow is made more difficult by the presence of the hydrostatic component). Two points arise immediately. Firstly, at no stage have we encountered truly plane stress deformation. We therefore

see that "Go" (better written as Gerit ) is the critical toughness level (derived from the applied stress level sufficient to produce instability) pertaining to a given thickness; it also depends on crack length, as described earlier. Secondly, we are interested in the conditions leading to the onset of square fracture and the common identification of "pop-in" with "plane-strain" toughness. In mild steel, there is much evidence to show that the onset of low strain cleavage cracking is greatly enhanced by the presence oftriaxial stresses, because the value of O'yy(max) is raised by plastic constraint (see Section 3) and is hence more easily able to propagate crack nuclei 63. If a similar cause is responsible for the aluminium alloy behaviour, it is instructive to compare Fig. 15 with Fig. 16, which shows the effect of thickness on the fracture stresses of mild steel specimens which failed by cleavage 3°. On a macroscopic scale, the mild-steel fractures for thicknesses greater than ~-in. were all completely square. Now consider the "plane-strain"/"pop-in" relationship. In very thick specimens of aluminium alloys, firstly the proportion of square fracture at the limiting level of toughness (Fig. 15) w.ill be so great that the contribution of the shear lips to the total toughness will not be detectable. Secondly, we assume that azz is at its maximum (plane-strain) value throughout most of the thickness. This is, then, fairly, plane-strain fracture and the openingmode plane strain toughness, Gic, may be determined. It is of interest to compare the figure for the "critical" proportion of square fracture (85~) in these alloys with a figure for the critical proportion of cleavage fracture (70~) supposed to lead to brittle fracture in mild steel ship plates 64. The problem that arises is whether in, say, the 0.1 in. thick section (Fig. 15) the square "pop-in"

8°i 7c

x\

\

'O

fracture

stress

.o general yield etres~ e-

specimen

thickness

(in.)

Fig. 16. Variation of general yield and fracture stresses with thickness in mild steel at -95°C. (After ref. 30.)

Mater. Sci. Eng., 7 (1971) 1-36

19

LARGE-SCALE BRITTLE FRACTURE IN STRUCTURAL METALS

occurs at the applied stress corresponding to G~c or at some higher level. It is quite possible that the median value of azz may not have reached its plane strain magnitude, although it has induced sufficient triaxiality to change the fracture mode from slant to square. In mild steel, for example, (cf. Fig. 16), square cleavage fracture is indistinguishable, on a macroscopic scale, in the fully plane-strain stress-state from that occurring under less severe triaxiality. It is difficult to estimate the critical full thickness at which azz in the median plane attains the full plane strain value, but the estimate in Section 3 gives B > 2.7 x total extent of plastic zone ; i.e. B > 5ry. Giving r v the value

K2c rv

--

EGIc

2ha2 - 2n(1-v2)o -2

(eqn. (20))

and substituting appropriate figures for 7075-T6 aluminium alloy - say, G i c = 115 lb./in. (Fig. 15), E = 107 lb./in. 2, a v = 7 2 × 103 lb./in. 2 - we obtain rv=0.04 in. and hence B_~0.2 in. Boyle et al. 4s'6° observed experimentally that "pop-in" first occurred when B = 4 r v = 0 . 1 6 in. On another basis, from results obtained by Hahn and Rosenfield 31 on the nature of the spreading plastic zone, a value of B-~ 0.1 in. is obtained. Since the calculations are extremely sensitive to the precise criterion used to decide when azz attains its full plane strain magnitude, but place the critical thickness in the range where toughness is varying rapidly, the identification of "pop-in" toughness with "plane strain" toughness must be established experimentally for any given material. It is unwise to assume that "pop-in" toughness automatically gives the value of G~c for any new alloy system, for which the experimental evidence is lacking. The situation depends entirely on the local fracture criterion for square fracture and on how it depends on triaxiality. We therefore conclude, from this extensive discussion on the results obtained from the aluminium alloy, that : (1) In very thin foil, the critical value of fracture toughness, G~it, increases with specimen thickness. A dislocation model predicts that the relationship is linear. (2) In very thick specimens, it is possible to measure meaningful values of plane strain fracture toughness, G1c. (3) The maximum in the G~it curve occurs when sufficient triaxiality to produce square fracture is first developed in the centre of the sheet. This does not necessarily correspond to the first achievement

of plane strain in the centre; hence "pop-in" determinations of G,c may not always be valid. (4) The decrease in Gcrit to G~c at large thicknesses is caused mainly by an increase in the amount of square fracture, but also possibly by an increase in triaxiality in the centre of the specimen. (5) The peak value of Gcr, and the corresponding value of B depend on the mechanism by which square fracture is produced in an alloy, and on how this is influenced by the triaxial stress state. Both values may be expected to be quite different for different alloy systems. (6) Since the value of Gcrit depends on plate thickness and width, it is recommended that G~rit or "G~" toughnesses be determined only in full plate thickness for specific service applications where geometry is known (e.g. aircraft or spacecraft fuselages). The width effect may be allowed for by means of an "R" curve (see above).

(b) Plane strain Recently the emphasis on toughness testing has been turned to the determination of plane strain fracture toughness (expressed as K~c) in specimens sufficiently thick to ensure that instability occurs as soon as the first plane strain crack extension begins47,65. Various specimen geometries have been considered, which all basically have to meet the following requirements : (1) If the linear elastic stress analysis is to be applied, the size of the plastic zone at fracture must be small compared with the uncracked ligament width of the specimen (total width - crack length = (W-a) for an edge-cracked testpiece). This consideration is reinforced by the need to simplify corrections to the elastic stress field occasioned by the near presence of a free surface to the crack tip. (2) The plastic zone must also be small compared with the crack length, a, if the crack tip stress field is to be characterized by a single-valued stress intensity factor, K (see page 8 and ref. 66). (3) The thickness, B, must be sufficient to give a plane-strain toughness. The compact tension specimen design is shown in Figs. 17 and 1865,67. Here, the three specimen dimensions (W-a), a and B are all equal t'o one another, and the general rule 31,66 is that the specimen must be sufficiently large for any of these to be approximately sixteen times the radius, rv, of the plane stress plastic zone at fracture, which is given by: Mater. Sci. Eng., 7 (1971) 1-36

20

J . F . KNOTT

ry = ~

\ Cry/

at the outer surfaces. The expression has K~c and not Kc because the critical value of K on the side surfaces at fracture is still, of course, K~c in magnitude. Thus, one sees as a criterion laid down for plane strain testpieces that

(W-a)=a=B>25(@) 2 \ •v/

To allow for the effects of triaxiality on yielding, it is, however, supposed that the corresponding size of the plane strain plastic zone riy is only about onethird of that in plane stress: Fig. 17. Compact tension fracture toughness specimens: 1, 2, 4, 6 and 12 in. thick. (After ref. 67.)

f 2H

I> W1 W =2.OB

D = O.5B

a =LOB H = 1.2B

Wl= 2.5B Hz= 0.65B

Estimated Tentative Measurement Capacity* Overall Dimensions Type

K1c/oys

ITCT 2T-CT 31"%3" 41"-CT 6T-CT 8T-CT IO]'-CT 12T-CT

0. 63 0.90 I. I0 1.30 1. (30 1.80 2.00 2.20

(K1c/Oys32 Thickness lin.) 0. 40 0.80 1.20 LeO 2. 40 3.20 4.00 4.80

1 2 3 4 6 8 10 12

Height (in.)

Width (in.)

2. 4 4.8 L2 9.6 14. 4 19.2 24.0 28.8

2. 5 5.0 1.5 10.0 15.O 20.0 25.0 30.0

• Basedon currently su99estedASTME-24 minimum size criterion, a

B zs(Ki 2

Fig. 18. Recommended design for compact tension toughness specimen.

1 {K,v] z In terms of the considerations above, the criteria imply: (1) that the linear elastic approach is applicable to fractures at stresses below about one-third of the general yield stress ; (2) that the error in using a single value of K to characterize crack tip stresses could be as much as 8 ~ (at rlv/a ~-0.02) 66. The critical value of B is deduced more from experimental results than from examination of mathematical formulae, but all three criteria require a judgement as to acceptability. The aluminium alloy results (Fig. 15) may be used to support the critical value of B. Taking K~c=EG~c/(1-vE); E=107 lb./in. 2 Gzc/(1-v 2) approx. 125 in. lb./in. 2 (from Fig. 15) and a v = 7 2 × 10 a lb./in, z, we obtain BCrit= 0.6 in. Figure 15 in fact shows that to all intents and purposes we have attained the plane strain fracture toughness level at 0.6 in. It is of interest to compare the proportion of square fracture observed in fractured specimens of this thickness (approx. 85 ~) with estimates that we might make of the proportion of the specimen expected to be in true plane strain conditions (ezz= 0). On either a rate of increase of az, with thickness (B=5ry) or a plastic zone geometry 3~ ' (B = 4rv) basis, we would expect that the non-planestrain deformation would occupy rather more (32 ~ or 26 ~ rather than 15 ~) and hence the plane strain deformation rather less (68 ~ or 74 ~ not 85 ~) of the cross-section than is observed. This may imply that in the aluminium alloys, as in mild steel (Fig. 16)it is possible to obtain square fractures even though the stress state is not fully plane strain. This casts further doubt on the validity of the "pop-in" technique. Mater. Sci. Eng., 7 (1971) 1-36

LARGE-SCALE BRITTLE FRACTURE IN STRUCTURAL METALS

The best practical determinations of K~c involve specimens of the type shown in Figs. 17 and 18 (compact tension) and 3-point bend 4v. Crack opening is monitored by a clip gauge (Fig. 19) and results are accepted as valid only if the deviation from linearity on the load: clip-gauge trace is negligible. (For the "KQ" testing procedure see reference 47.) Other forms of K~c testing have been extensively reviewed by Srawley and Brown ¢s'¢~, but the general trend has been to settle for compact tension and threepoint bend, although the presence of an additional shear stress component in the crack tip stress field of the latter 2s might be expected to affect the single K] opening-mode assumption (this problem would not arise in pure four-point bending). In all cases, starter notches are sharpened by fatigue cracks prior to testing, since fracture toughness has been shown to vary with root radius (Fig. 20); care is taken to grow these cracks at low cyclic stresses, so that the effect of fatigue damage on the material properties at the crack tip is minimized.

500~ poperwi boroeked I

I__~_i--{----____I~

strain gouge ( r e f . KX 8 / 5 0 0 ]

GAUGE MOUNTEO

ON SPECIMEN

Fig. 19. Design of clip gauge for toughness testing.

600

500

EFFECTOFNOTCHROOT RADU ISONAPPARENTKC STEEL {1.5mmTHC I K} o C Cr MO V

0.4 % 5.0 % 1.3 % 0.5%

/

- - 150

o o

400 8

o

--

IOO

_z _z

°o @

2o0

~x

50

~

TEST SPEICIME N /

I 0'.1

0.2

plt~ IN mml/2

iI

0.3

o12

Fig. 20. Variation of fracture toughness with notch root radius.

21

Consistent values of Klc are regularly being obtained by such tests, but size limitations on specimens restrict the field mainly to fairly brittle, high-strength alloys. The next section will describe the approach followed for lower-strength, more ductile material. In particular, discussion will concentrate on the large-scale brittle fracture of mild and low-alloy steels.

8. LARGE-SCALE BRITTLE FRACTURE IN MILD STEEL

Up to now, we have been considering fractures that are brittle in an engineering sense, i.e. those associated with the onset of instability at stresses less than the general yield stress. Behaviour of individual materials has been restricted to the way in which particular mechanisms for producing slant and square fracture might vary with thickness. The main endpoints have been the establishment of a test to determine plane strain fracture toughness and the considerations to be borne in mind when attempting to determine critical values in thinner sections. The behaviour of low-strength steels introduces another major variable into any assessment of resistance to brittle fracture, because these steels undergo a transition in mode of fracture from tough fibrous fracture at high temperatures to brittle cleavage fracture at low temperatures. The transition and arises from the basic change in micromechanisms by which the fracture is produced at different temperatures. Until recently, there was controversy as to whether the observed magnitude of the transition was not due simply to the impossibility of maintaining sufficient triaxiality in small specimens (we shall consider this in more detail later). However, Wesse167 has now carried out experiments in which a large transition in K]c with temperature is observed in specimens which satisfy all the linear elastic fracture mechanics plane strain criteria. Typical results are given in Fig. 21, which shows the very marked increase in K~c with temperature. It must be emphasised that the specimens needed at the higher temperatures are extremely large. At 50°F, for example, the valid figure for Kic is about 140 k.s.i. ,,/in. and the yield stress is 70 k.s.i. Using the criteria (W-a), a, B > 2.5 (K,c/av)=, we see that all dimensions must be at least 10 inches; the "compact tension specimen" turns out to have dimensions 24 in.×25 in.× 10 in. On the other hand, at low temperatures (say -250°F), plane strain fracture Mater. Sci. Eng., 7 (1971) 1 36

22

J.F. KNOTT I

I

I

I

I

i

I

i

I

ASTYB Class I Steel Plale

tee

"'" -

9 ;

12" Thick IHSST) Opan Points: Valid ASTM

I

~n

17C -- Closed Points: DONM Satisfy Size Requirements

I II

IKQI

o 1; wol.{ ewR

I

1~- ~ 2TWOU q

I.~

I~

t

oa .

4T CT)

I

0 6T CT~ lIES

-- o ] m c u

+,,ok 5o I -

,..,

-300

1'o+

j

~o

l- ...... 30/ I

bOO:

Jr

+

L

--

~

~

_

"

1

T -2OO

100

~.

2,,~_..9,~ "_,=

o I

I

1

-I00

~. I 0

i

I

i

I00

Temperature,

Fig. 21. Dependence of Kic on temperature in alloy steel. (After ref. 67.)

can be obtained in pieces less than h~f-an-inch thick. This drastic reduction in toughness occurs simply because it is easier to produce lowZstrain cleavage fracture at low temperatures. Most of the research carried out on the brittleness of mild steel has been concentrated on the factors which control the ductile/brittle transition. As mentioned above, the transition ~as seldom, if ever, been observed, prior to Wesse]'s experiments, in conditions which could be regarded as "plane strain" by fracture mechanics criteria. Usually in tests on smaller specimens, the increasing fracture strain due to increasing temperature required such large plastic zones relative to the specimen size that the axx and/or azz stresses were relaxed at relatively low temperatures. Nevertheless, transitions have been studied in wide, fairly thin precracked plates 68, in smooth tensile specimens 69'7°, in slow notch-bend 63'71 and in notched impact v2'73. The last mode of testing has been in great favour for many years as a means of assessing the toughness of low-strength steels 7'74, presumably because the

completely arbitrary combination 7s of hammer speed, notch geometry and specimen size gives a transition temperature around room temperature for many steels. In fact, the stress state in service (fairly thin ships' plates, for example) is often not plane strain, unless fracture initiates by low-strain cleavage, and much useful information of a practical nature has been gained by the use of tests, such as the Wells wideplate test 68, in which plates of full service thickness are fractured. An important situation observed in service failures is one in which fracture has started by fibrous mechanisms in a large plate at a stress less than the general yield stress, has grown slowly and has then changed to a catastrophic cleavage crack76, induced by the increasing strain-rate ahead of the growing fibrous rupture ~7 (the effect is shown in Fig. 22). The total fracture is brittle, because it runs catastrophically at a stress less than the general yield stress; also, it may show a large proportion (70 ~ or more) of bright cleavage facets. Nevertheless, a small specimen cut from the large Mater. Sci. Eng., 7 (19711 1-36

23

LARGE-SCALE BRITTLE FRACTURE IN STRUCTURAL METALS

differences in large-scale and small-scale behaviour. The method of approach adopted provides further insight into the general assessment of a material's toughness. In the previous section, we have considered toughness in engineering terms, as the critical value of strain energy release rate or stress intensity at fracture. We have not yet tried to produce much physical e.xplanation for the critical values in terms of criteria for fracture in the material, but it is now necessary to do this to explain the large-scale/small-scale discrepancies. We suppose that fibrous fracture is initiated at the tip of a stress concentrator when a critical displacement is produced there 49'52'53,7s. (This is in fact similar to the criterion for Km sliding in "plane stress" fracture (see page 16 last para.) In an opening mode the displacement is the critical opening displacement (C.O.D.) and is perhaps best visualised if we think of the region just ahead of the crack tip as being a miniature tensile specimen which has to be strained by a critical amount before it breaks 78. An alternative formulation for C.O.D., consistent with this view, in fact derives it in terms of a critical strain attained over a gauge length determined by the microstructure of the material 5°. To recapitulate from Section 6, the displacement at a crack tip subjected to the plane stress opening mode may be written (eqn. (23)) as 51 80-y

6 = ~-

~ T/70"app} a log esec~

(23)

for all stresses less than the general yield stress. On the C.O.D. criterion for the onset of fracture, the situation is simply that when the critical value 6crit has been achieved, the applied stress O'app has reached the fracture stress, o-v. At very low values of o-app/aY we may write Fig. 22. Fracture mechanism in mild steel at room temperature. Note the sharp change from fibrous to cleavage as the fracture propagates, x 60. (After ref. 6.)

plate and tested at a comparable strain rate exhibits substantial general plastic flow before failure and a fracture appearance which is completely fibrous TM. The problem of brittle fracture in mild steel is therefore concerned mainly with (1) the prevention of cleavage initiation in service and (2) the assessment of large-scale behaviour, using small specimens; this is particularly important if meaningful quality control tests are to be developed. It is probably best to start with the issues raised by the apparent

E/E~Y~crit and hence obtain a relationship between toughness and critical displacement, Gerit = O'y0crit. We therefore interpret the critical value of strain energy release rate per unit thickness as the work done by the yield stress in stretching a crack tip mini-specimen of cross-sectional area (da x unity) by the critical opening displacement 6erit" The important point with regard to fracture behaviour is that the plastic displacement, refit , c a n be achieved at the crack tip only by spreading yield Mater. Sci. En~].. 7 (1971) l 36

24

j.F. KNOTT

into the specimen. The amount of yielding (size of plastic zone) required to produce this displacement is given at low stresses by Dc =

K2 2 r y = ~ztr2 -

E6cri' 7~O'y

(29)

(In other words, for a circular zone of radius rv, 6 may be taken as the strain at the circumference, i.e. av/E, multiplied by the length of the circumference, i.e. 2nry.) At higher stresses, the (total) spread of plasticity required to accommodate a critical displacement, CSerit, is D~ = a (sec/~aavF}-- 1) • We now consider the width of the uncracked ligament (W-a). If De is less than 0.1(W-a), we have brittle fracture with linear elastic behaviour. If De is less than ( W - a ) we will still have a brittle fracture, but now the "loge sec" formula (eqn. (23)) must be used to give values of fracture stress. If De is greater then ( W - a ) , general yielding precedes fracture. We therefore see that, for a given stress-concentrator and a given value of 3¢rit , the apparent brittleness of a specimen depends on the position of the far boundary. The effect is illustrated in Fig. 23 which shows central sections from three thick specimens of identical notch geometry which have all been broken under the same conditions of temperature and applied strain rate. The plastic

deformation preceding fracture has been etched, using Fry's reagent6, 28, and it can be seen that the specimen with the free surface nearest the notch tip (i.e. the narrowest specimen) has broken just after general yield whereas the two larger specimens fractured before general yield, but after similar amounts of plastic deformation had spread into the cross-section. At the point of fracture, the C.O.D. at the bottom of the notch root (measured with a "paddle" gauge 79) was similar for all three specimens, even though the smallest exhibited general plasticity whereas the other two did not. Macroscopically, the small specimen exhibits a lower ductile/brittle transition temperature than do the larger pieces. We can therefore relate the apparent difference in fracture ductility of large and small cracked pieces directly to the ease of accommodating a given C.O.D., 6trOt, before the spreading plastic zones produce general yield. In the practical example given, it was also stated that the fracture appearance could vary. This is most probably attributable to a rate effect77 ; in a small specimen, tested at the same temperature as the plate, the initial fibrous crack does not build up sufficient speed to change to the cleavage mode. The crack opening displacement is a most valuable measure of a material's toughness, because, ideally, it is simply the ductility of a unit element of microstructure at the crack tip. In principle, it may, therefore, be measured on a small specimen which

Fig. 23. Sections of mild steel specimens fractured at - 70° C, the notch geometry being the same in all specimens. The plastic deformation preceding fracture has been etched, using Fry's reagent, x 0.6.

Mater. Sei. Eng., 7 (1971) 1-36

LARGE-SCALE BRITTLE FRACTURE IN STRUCTURAL METALS

breaks after general yield, and employed to calculate critical values of strain energy release rate in large structures which break well before general yield. The feasibility of developing practical C.O.D. tests to measure material toughness is currently being investigatedS°; the problems are likely to come, however, not so much in measurement technique as in the application of the small-scale critical values to large pieces. Difficulties arise as a result of the differences in fracture criteria for the cleavage and fibrous fracture modes, the ways in which these modes are affected by the hydrostatic stress state, and the factors which affect the hydrostatic stress state in specimens of different geometries. We begin by considering the way in which fracture is produced at different temperatures in narrow, but thick, notched specimens of annealed mild steel, tested in slow bend 6'63'72. Typical behaviour is indicated in Fig. 24, which shows both general yield stress and fracture stress as a function of temperature. Fracture occurs before general yield at temperatures lower than rGy ; fracture at T~y corresponds to the amount of deformation shown by the smallest specimen in Fig. 23. In region A (Fig. 24), specimens are very brittle and cleavage fracture is initiated by deformation twins. The fracture criterion is simply that a "burst" of twins shall be produced81 -83. This is not a mode of initiation that is of practical importance with regard to the slow strain-rate loading of cracked specimens at higher temperatures, but there are two interesting features worthy of further mention. First, there is an effect of the stress gradient ahead of the notch on the production of the twins T M . This leads to a decrease in the effective elastic

\l

I

I

l\

I

,~AX,WUMLO,,O~

; \, '

;

A

I

8

c

,,/ ~

I

/ ,,~..~

m Y I

TGY

~

I

-2~o

ACTORE ST.ESS

/ ',

/i[ ' ' ~- ~ O

1/,

TEMPERATURE C°

INITIATION

GENERAL Y ELD STRESS

Tw I I

-,6o

FRACTURE

6

Fig. 24. Schematic diagram, illustrating the general fracture behaviour of notched bars of mild steel as a function of temperature.

25

stress concentration factor for the notch in a small specimen and predicts that the fracture stress for twin-initiated fracture will drop as specimens of constant grain size are scaled geometrically. Secondly, it is found that there is an effect of thickness on the fracture stress for this type of fracture. In thin specimens, fracture is no longer coincident with the initial "burst" of twinning 82. To return to Fig. 24, fracture in regions B and C is still produced by cleavage mechanisms, but twinning is no longer involved in the initiation stage, because the temperatures are too high. Instead, cleavage cracks are initiated underneath the notch by slip-bands - an example is given in Fig. 25, which shows a crack produced at a fairly high temperature 84. The basic reason for the increase of fracture stress with temperature lies in the attainment of the critical local condition foT fracture instability beneath the notch. In the cleavage of annealed mild steel, we have an example of an instability due to cracking, which is worthy of study, because it may serve as a model for other forms of instability due to cracking in other materials. The original Orowan criterion 23 for cleavage supposed that fracture was produced at a critical tensile fracture stress, ~rF,which was independent of temperature. The critical value of o-F was interpreted first in terms of Cottrell's theory of brittle fracture 6s, and subsequently by Smith 85, who considered the growth of a cleavage crack nucleus, initiated by slip or twinning, under the combined action of a shear stress and a superimposed hydrostatic tensile stress. He shows that if the surface energy of the lattice through which the crack grows remains constant, then nucleation is the controlling event and the superimposed hydrostatic component cannot affect fracture. If, however, the crack nucleus is produced by cracking a brittle secondphase particle and has to spread into a tougher matrix, growth becomes the critical stage and the value of the tensile stress on the specimen is of crucial importance. For annealed mild steel with coarse carbides (and ky constant), the value, aF, is virtually independent of temperature for slipinitiated cleavage; this is not the case, however, for twin-initiated fracture s6, The application of the constant critical value, av, to the behaviour of thick notched bars involves the way in which the local tensile stress below a notch depends on the extent of the plastic zone and on the yield stress. Mater. Sci. Eng., 7 (1971i 1 36

26

J.F. KNOTT a grain diameter into the specimen, we can assume that a large number of crack nuclei have been produced by the fracture of brittle carbides, because these have been subjected to high concentrated stresses at the ends of slip bands. Smith's theory s5 shows that the criterion for growing the nuclei to form a cleavage crack which can spread catastrophically may be restated, for this material, in the form that the local tensile stress must attain a critical value, crv. The magnitude of the maximum tensile stress, ayy(max), is given by O'yy(max ) =

Fig. 25. Initiation of a cleavage crack in the region of high triaxiality ahead of a notch. (a) × 120. (b) Detail of crack × 600. Note absence of twins and possible association of crack with carbide particle. (After ref. 84.)

At low temperatures in region B (Fig. 24), the uniaxial yield stress of the mild steel is high, because the lattice friction stress increases markedly with decrease in temperature. We visualize the cleavage fracture process as follows: Yielding is produced at the notch root when the local value of ayy (max) at the notch surface reaches the uniaxial yield stress (cf Section 3). Once yield has spread by about

Rav

(eqn. (6))

where R is a function of the extent of the plastic zone (and so of aapp/aV, Fig. 7) and a v depends on temperature. Hence the criterion becomes O'yy(max) = Ray= aF for the cracks to grow. At low temperatures, av is high; therefore, R need not be very large, and thus (Fig. 7) the specimen breaks well before general yield, as observed (Fig. 24). As the testing temperature is raised, so av decreases and the value of R required to satisfy the growth criterion becomes larger. We therefore see, from Fig. 7, that the value of aap~/Lav at fracture is expected to increase with temperature until, at Toy, the ratio becomes unity, and fracture is coincident with general yield. Although the notch root strains increase with temperature, the local strain just behind the plastic/elastic interface, where growth of the critical crack nucleus occurs, remains constant, as the position of the interface varies with temperature. The critical local stress is raised by plastic constraint rather than by strainhardening, and, for fully annealed mild steel, ~rF is virtually independent of temperature in region B. It is important to note that no effect of size on the fracture behaviour of 9eometrically similar specimens (i.e. with root radius also scaled) is anticipated, and there does seem to be experimental evidence 8v which supports this viewpoint. In region C, specimens still fracture by cleavage, but only after the specimen has yielded generally. Initially, post-general yield deformation is of the "plastic hinge" type observed at general yield (cf ref. 6 and, e.9., the smallest specimen in Fig. 23) and fracture occurs soon after general yield. For this situation, it is reasonable to assume that the local conditions are still meeting the previous fracture criterion, but that the local value of the yield stress is increased by work-hardening 63. Thus, we write Ray (o-v + As) = o-F

(30)

Mater. Sei. Eng., 7 (1971) 1-36

27

LARGE-SCALE BRITTLE FRACTURE IN STRUCTURAL METALS

where Roy is the maximum (general yield) value of stress intensification due to plastic constraint, and Aa represents the increase in yield stress due to strain-hardening. Although the critical value of o-F increases with strain, experiments on uniformly pre-strained notched bars 88 have shown that the effect of strain on strain-hardening is greater than that on av. In any case, the critical fracture event still occurs close to the plastic/elastic interface. Since the notch root strain increases rapidly with applied stress after general yield, the divergence between fracture stress and general yield stress is not large. At the temperature, Tw (Fig. 24), the strains required for strain-hardening become so large that overall yielding of the gross cross-section by plastic flow on 45° straight slip-lines (plastic "wings") becomes possible 6a. Immediately, much of the plastic constraint which produces the high local tensile stresses is lost, and the effective value of R in eqn. (30) drops sharply. Additionally, throughthickness yielding may relieve the hydrostatic component of stress 89. To try to meet a cleavage fracture criterion, the strains in the notch root region would now have to be increased enormously. In practice, what happens is that the strains at the notch root become so large ,that fibrous fracture is produced at the surface of the notch before any cleavage crack nuclei can grow below the notch, viz. the formation of holes around inclusions near the notch surface in Fig. 22. We recognise that the criterion for the initiation of fibrous fracture is fundamentally different from that for cleavage. Fibrous rupture is aflow process (strain-controlled); cleavage is a crackin9 process (stress-controlled). The reason why a growing fibrous fracture gives rise to a catastrophic cleavage crack (Fig. 22) as it accelerates is due simply to the fact that the uniaxial yield stress increases with increase in strain-rate as well as with decrease in temperature 73'7~'9°. Thus, the high strain-rate conditions ahead of a growing fibrous rupture can lead to cleavage. The ductile/brittle transition in small specimens of mild steel has been described in detail; firstly, to explain the importance of local fracture mechanisms and criteria in governing macroscopic ductility and, secondly, to try to make use of the results in the interpretation of large-scale behaviour. The next section will attempt to relate the large-scale toughness of low-strength steel to probably local fracture events; to some extent, the basic concepts may also be applicable to the behaviour of other materials.

9.

A MECHANISTIC BASIS FOR THE FRACTURE TOUGHNESS AND STEELS

The argument presented in the last section to explain the ductile/brittle transition in slow notchbend tests (which applies equally w611 to the energy transition in notched impact, once allowance has been made for the higher applied strain-rate ~'73) needs to be modified slightly to provide a basis for explaining the K~c transition in large sharply-cracked pieces (Fig. 21). Again, we suppose that cleavage crack nuclei are produced in the yielded zone ahead of the stress-concentrator by the action of slip-bands on second-phase particles, and that the local shear stresses must be reinforced by a critical level of tensile stress before the nuclei can grow catastrophically through the matrix. In sharply-cracked specimens, however, the local elevation of tensile stress by plastic constraint should remain c o n s t a n t (O'yy(max)/Oy~---R= 2.6) right from the onset of yielding at the crack tip 22'23'41. Any increase in local tensile stress with applied stress must therefore be attributed to the strainhardening produced by the increased crack tip str,ains. If, as in notched bars, the cleavage fracture criterion can be expressed in terms of a critical tensile stress, OF, which is independent of temperature, we have, for the cracked specimen 2.6(o v + Aa) = OF or

(31) Ov + ke" = ov/2.6

(assuming that the flow stress is given by of = Ov + ken) to express the relative contributions of (temperature dependent) yield stress and strain-hardening in the achievement of the local fracture conditions. At any particular temperature, therefore, the crack tip strain, eF, required to produce the critical amount of hardening can, in principle, be obtained from eqn. (31) by solving for e. This strain is then assumed to be distributed uniformly over a microstructure-dependent gaugelength (say, a grain diameter) to obtain the critical crack-tip displacement at fracture: 3,it=ev'd. From this, the size of plastic zone, applied stress and macroscopic fracture toughness may be derived. Even without detailed calculations it is clear that, as temperature increases and the yield stress decreases, more strain is required for hardening and the observed toughness increases. The steep rise of the K~c transition most probably occurs at temperatures where the crack-tip displacements start to become Mater. Sci. Eng., 7 (1971) l 36

28

J.F. KNOTT

so large that fibrous rupture is initiated at the crack tip before cleavage nuclei can be propagated ahead of the tip. In the alloy steel (Fig. 21) the voids or dimples are presumably on a much finer scale than those observed in mild steel (Fig. 22). Alternatively, fibrous fracture may occur by a localized shearing mechanism rather than by internal necking (see e.g. ref. 91 and Fig. 26). Even so, calculating from

Fig. 21, the C.O.D. at fracture at 40° F is at least 0.003 in. - a value which is more than sufficient to initiate fibrous rupture in mild steel 92. A detailed study of the fracture surfaces of the higher temperature fractures reported in Fig. 21 has, in fact, revealed "stretch zones" and fine dimples immediately ahead of the fatigue crack tip 6v. The fracture process at high temperatures may therefore be very similar to that in mild steel (Fig. 22), i.e. a small amount of ductile fracture which accelerates and then changes to a fast-running cleavage crack. Krafft 9a has proposed an alternative model for the variation of K~c with temperature based on a fibrous necking mechanism. Again, a critical strain must be attained over a microstructure-dependent gauge-length, dT, which lies along the X axis. If the tensile stress-strain curve can be represented by a=ke n, it may be shown that instability in the tensile test sets in when e = n. Krafft argues that the strain required to produce fracture at a distance dT ahead of the crack tip is directly proportional to n. Since, for low stress, the strain at a distance dT is given by e= K / E ~ taking a fracture strain, e ~ n, gives KIC -----n E ~ .

Fig. 26. Formation of shear cracks at the notch root in a 3 ~ CrMoV steel. (a) x 500, tempered to 80 ton/in. 2 proof stress; showing initiation from notch surface. (b) × 100, fractured specimen, tempered to 60 ton/in. 2 proof stress; the natural banding in the material shows how the shear fracture develops. (See also ref. 91.)

(32)

Krafft explains the variation of Kic with temperature and strain-rate in terms of the ways in which these affect n. The model may apply to high-strength materials in which fast plane-strain fractures are produced by hole-joining fibrous mechanisms, but it seems to be fundamentally inapplicable to the fracture mode transition in steels. We therefore explain the K~c transition in terms of a local tensile stress-controlled cleavage fracture process (a craekin9 mode) at low temperatures and, at high temperatures, a strain-controlled (or. shearstress-controlled) fibrous rupture initiatiot~'(a flow mode), which reverts to cleavage as it accelerates. These processes are quite consistent with the features seen on the fractured surfaces of specimens. The stress criterion helps to explain both the effect of tensile yield strength on K~c (Kic decreases as yield stress increases, because the tensile stress in any yielded zone increases; hence the amount of strain required to harden the region drops, and o'app/ay at fracture decreases) and the initial effect of strain-rate o n K i c 94'95. Full results are shown in Fig. 27, where it can be seen that the curve of K~c versus strain-rate passes through a minimum. Initially, an increased rate of strain increases the Mater. Sci. Eno., 7 (1971) 1-36

29

LARGE-SCALE BRITTLE FRACTURE IN STRUCTURAL METALS

3/4tnSemi-kliled steel plol'e NDT-15"20~ 5OO

500

400 300 200 150

c cK

20

|

10o 10l 102 10,1 10~

L~ING

RATE(~), s"

Fig. 27. Dependence of fracture toughness, K,c, on applied strain rate. (After ref. 95.)

level of tensile stress in the yielded region at the crack tip; hence, the crack tip fracture strains and K~c decrease. When the strain-rate is very high, as for example ahead of a fast-running crack, it is possible for the deformation to occur so quickly that the heat generated by plastic flow cannot be dissipated into the bulk of the specimen, and it raises the temperature locally, ahead of the propagating crack. Under these adiabatic conditions, the toughness of the region just ahead of the crack tip may be raised substantially. There is, therefore, a case to be made for measurement of the minimum dynamic K,c at operating temperature if "safe" values are required. An important point concerning the fibrous initiation model for K~c at high temperatures (Fig. 21) is connected with the fracture behaviour of specimens which were too small to satisfy the planestrain requirements. At 75°F, for example, no genuine K~c values could be determined, but the inferred KQ values increase with specimen thickness and/or width. There are two possible effects to be considered, both of which are compatible with the concept of a fibrous initiation changing to cleavage as it accelerates. Firstly, as thickness increases, plastic restraint increases, and it becomes more difficult to obtain a given plastic displacement at the crack tip. The results, therefore, give a strong indication that fracture intitiation is strain-controlled at these high temperatures. For cleavage initiation, the more common decrease in toughness with increase in thickness would be observed. Secondly there is the possibility that, somewhat as in the aluminium alloys (page 15/16) the fibrous rupture grows to a length that is independent of the initial crack length

before producing cleavage and hence instability. This constant fibrous growth would perturb the stress field more strongly in smaller specimens and so would lead to an apparently lower value for K~c. The marked increase in Klc with temperature when the first stages of fracture proceed by fibrous rupture must be attributed to the need to arrive at a condition for instability. The initial fracture is probably "non-cumulative" in type and will not become unstable at stresses less than the general yield stress. To give unstable propagation, a fastrunning cleavage crack must be produced ahead of the advancing fibrous rupture. The temperature variation of K~c then arises because increase in temperature decreases Gy in the yielded region. Hence a faster strain-rate is required there to raise the local tensile stress to its critical value, so the fibrous rupture has to grow further before the mode change occurs. (This is also the reason for the increase in depth of fibrous "thumbnail" with temperature throughout the transition range in a Charpy test.) This further growth produces an increase in the original crack tip displacement, hence in the applied stress at instability and in fracture toughness. There is some difficulty in quantifying the effect, firstly because the acceleration characteristics of a fibrous rupture are not known; secondly, because the increase in C.O.D. with fibrous growth is largely uninvestigated. Some recent results have shown that this increase is quite small in mild steel 92, which lends support to earlier predictions 96 that the correlations between critical values of C.O.D. and fracture toughness in mild steel might be difficult to establish.

10. DISCUSSION

(a) General aims of research The main theme of this review has been the extent to which it is possible to explain fracture phenomena on the macroscopic scale in terms of a material's microscopic mechanisms of fracture. Attention has been concentrated on the onset of unstable crack propagation at stresses less than the general yield stress in pieces containing sharp stress-concentrators. Crack-arrest philosophies (prevention of unstable propagation by choice of material with a sufficiently low crack-arrest temperature 11) have not been described in detail, but the mechanistic basis for the transition observed is identical to that Mater. Sei. Eng., 7 (1971) 1-36

30

J . F . KNOTT

for the Kic transition (Fig. 21) once allowance has been made for stress-state and applied strain-rate. In this context, it is important to distinguish carefully between situations which arise with conventional testing machines or hydraulically pressurized containers on the one hand and with pneumatically pressurized vessels on the other 97, because the elastic strain energy stored in the testing system may relax as the crack extends. The fracture mechanisms which produce a cleavage/fibrous transition can, to some extent, be related to a material's microstructure. The local cleavage fractures stress is, for example, a function of both grain size and carbide distribution, and, through the latter, of at least one alloying element 63's5'86 (manganese). Currently, research is proceeding which characterizes the critical C.O.D. for fibrous initiation in terms of microstructure 92. The experimental difficulties increase as the scale of microstructure is refined, but the long-term aim must be to be able to provide a quantitative basis for the toughness of commercial microstructures, e.g. for a quenched and tempered martensite, in terms of its alloy content, dislocation density, cell size and distribution of inclusions and second-phase particles. The metallurgical challenge is formidable. Additionally, much work has to be done to attempt to establish local fracture criteria in other structural alloys. The division into cracking or flow processes is extremely general and there is no direct evidence, for example, that the square fracture in aluminium alloys is attributable to any form of cracking process. Hole-joining by viscoelastic flow may give rise to brittle behaviour if the hydrostatic component of tensile stress is sufficiently large98; this might be evident in the shape of dimples on the fracture surface. Criteria also need to be established for intergranular fractures, and for multiaxial stress systems (mixtures of K l, KII and KIIl modes). A fundamental point is that, in general, the criterion for any type of fracture will be affected by both local stress and strain 99. Only in cases, such as those which we have tacitly assumed, where the stress concentration/strain concentration relationship is of the same order in different pieces, can we hope to be able to use either stress or strain singly as a criterion.

(b) Effect of fabrication Even when fracture criteria have been established for various materials and a metallurgical descrip-

tion of toughness has been obtained for laboratory specimens, there is still the problem that, in service, the stresses or the microstructure of the material may be so affected by fabrication procedure that the whole situation is altered. One such problem arises from effects of welding on the toughness of lowstrength steels. Here there are two aspects to consider: firstly, the production of residual stresses; secondly, the formation of brittle microstructures in heat-affected zones (HAZ). Residual stresses are created as a result of the strains set up when a constrained weld pool solidifies and contracts. The relative strengths of weld metal and parent plate are important in determining the position and magnitude of the stresses. It is generally argued that the level of residual stress will not exceed the yield stress of the softest part of the welded joint, because higher stresses would be relieved by plastic flow. However, it is not inconceivable that, in a thick pipe joint, for example, triaxial tensile stresses might be produced and give rise to a high hydrostatic component which could not be removed by thermal stress-relief treatments. Such a high hydrostatic component could promote low-strain cracking processes and the toughness of material subjected to this stress-state might be substantially less than its plane strain fracture toughness. Wells 1°° argues that the maximum hydrostatic component obtainable is no greater than the material's yield stress because yielding will occur near free surfaces, but he does not consider the possibilities of large stress gradients. It is observed that thermally induced residual stresses produce plastic strains around defects as the weld cools. Material embrittlement produced in the heataffected zones ~HAZ) of welded steels is of two types. In "non-transformable" low-carbon steels of lean alloy content, the main embrittlement is produced by strain-ageing dislocations in the regions of plastic strain induced by residual stresses 1°1 ; such embrittlement occurs in "subcritical" (temperatures less than the At1) regions of the HAZ. In "transformable" alloy steels, however, the embrittlement is produced by the formation of untempered martensite in "super-critical" regions 1°1. For either situation, it is advisable to employ a stress-relief heat-treatment after welding. This also improves toughness by removing the strain-aged dislocation density in the former case and tempering the martensite in the latter. Weldable aluminium alloys are confined to those containing zinc and magnesium, because the ageing precipitate at peak hardness is predomiMater. Sei. Eng., 7 (1971) 1-36

31

LARGE-SCALE BRITTLE FRACTURE IN STRUCTURAL METALS

nantly coherent and reverts rather than overages in the HAZ during welding; subsequent natural or artificial ageing appears to restore joint properties adequately 1°3. The fracture of a welded joint containing an embrittled HAZ may be regarded as a two-stage process. Consider the situation shown in Fig. 28. Here, a defect of length 2c is embedded in a region

T

O"

(A) 2d

A

\

(B)

I O" Fig. 28. Diagram to illustrate the two-stage propagation of fracture from a defect located in a brittle weld. (After ref. 96.)

B of low toughness. The applied stress at which this defect will run catastrophically throughout region B is given by :

% = K,clm/x//-~ where KIC(B1 is the static fracture toughness of the embrittled material in region B. At this stress level, region A is therefore subjected to a propagating crack of length 2d. This crack will spread through A at a stress

(7"A = K I D ( A ) / N / ~

where KID(A ) is the dynamic fracture toughness of region A at the strain-rate appropriate to the crack acceleration from 2c to 2d (see Fig. 27). If O"g is greater than aa, the crack will arrest at the boundaries between A and B. The effect is shown by an experiment o11 an artificially strain-aged plate (Fig. 29). Here, a stress concentrator was cut into the edge of a high nitrogen steel plate and a welding torch was run along the edge to induce thermal strains and strain-ageing (faint smudges near the notch tip indicate these strains in Fig. 29). On the testing of the specimen at -50°C, first extension occurred by cleavage at a low applied stress level (equivalent to %), but final fracture took place only at a much higher stress (equivalent to ag). The point of crack arrest and re-initiation is clearly shown in Fig. 29 both by the Fry's etch pattern and by the appearance of the fracture surface. The effect might be expected to be even more pronounced for "transformable" steels where the untempered martensitic HAZ is very much more brittle than the normalised or quenched and tempered parent plate. Additionally, the strain-rate dependence of yield stress in these materials is small, so the value of KID would not be expected to be much less than that of Kic; there is some evidence to support this viewpoint t°4. A crack arrest approach, based on values of minimum KID , may therefore be particularly viable for initially short defects in weld/HAZ/parent plate configurations.

(c) Application to design The main purpose of research on brittle fracture is to gain information which can assist the engineer to design against such fracture in service. There are two main philosophies on which design tends to be based. One of these involves reliance on the concept of a transition temperature, above which fractures grow only by non-cumulative mechanisms and unstable propagation does not occur. The engineer then has to use materials which have transition temperatures lower than the operating temperature of his structure. In determining the effective transition temperature, it is of the utmost importance precisely to reproduce the characteristics of the loading system (e.g. pneumatic pressurizing of pressure vessels 9v) and to test realistically large pieces, to allow for the Mater. Sci. Eng., 7 (1971) 1-36

32

J . F . KNOTT

Fig. 29. Example of two-stage fracture propagation. Fracture starts in a brittle strain-aged region and then arrests. The reinitiation point is located both by Fry's etching of the midsection and by markings on the fracture surface, x 1.25. The stress concentrator and initial strain-aged region are at right-band side of the photograph.

possibility of the acceleration of non-cumulative growth leading to a change in fracture mode and thence to unstable propagation. The ~im must be to simulate as closely as possible the transition curve that would be exhibited by the structure in service. The position of the transition curve with respect to temperature depends on initial flaw length. The most common reference point is the nil-ductility

temperature (NDT), where by "ductility" is really meant general ductility. The relationship between the NDT and the operating temperature at which unstable propagation is judged to be no longer a problem depends on tbe applied stress level and on the degree of reliability required. For example, to avoid unstable propagation at a design stress equal to the yield stress, experience shows that the operating temperature should be at least N D T + 6 0 ° F . For severe demands, it may prove to be very expensive to use material with a sufficiently low NDT. Additionally, for high strength metals, the shape of the transition curve is quite flat. The guarantee against unstable propagation (perhaps by localized shear mechanisms) at a given temperature above the N D T may now not be as great as for mild steel, which shows a very sharp transition. A further problem arises in tbe design of quality control specimens. These have to be small pieces, which will be fractured under conditions of stressing pertaining to a particular type of laboratory testing machine. Transitions determined using quality control specimens will not, in general, be the same as those found in larger pieces or in the structure. It is necessary to produce a correlation between the results on small and large pieces and this correlation must be established separately for each material used, because the way in which fracture mechanism depends on stress system, strain-rate, etc. is very much an individual material property. The alternative philosophy of design against brittle fracture in service is based, not on a transition temperature approach, but on preventing initiation of unstable fracture from a pre-existing flaw. Structures can then be operated at temperatures even below the "nil-ductility" temperature, provided that the applied stress and flaw size are sufficiently small. Below the NDT, the fracture stress may be related to flaw size by the appropriate fracture mechanics expression, e..q. at very low temperatures (fracture well before general yield) we have CrFX/~ = Kcrlt for tbe opening mode. To make use of this expression in structural design, three parameters: stress (a,pp), fracture toughness (Kcrit) and defect size (a), must each be investigated thoroughly. The value of applied stress in service will be known with precision only as a result of detailed stress analysis, taking into account stress concentrations, the presence of free surfaces, multiaxiality of the stress state, residual stresses and extent of yielding. The operative value of Kerit depends on stress system, temperature, strain-rate and on a material's microMater. Sci. Eng., 7 (1971) 1-36

33

L A R G E - S C A L E BRITTLE F R A C T U R E IN S T R U C T U R A L METALS

structure and yielding characteristics. Inherent flaws arise from the casting, working and fabrication history of a material and from its non-metallic inclusion content; detection and measurement of flaw size rely on non-destructive testing techniques. Accumulation of all the information needed to provide a completely a priori basis for design against brittle fracture using fracture mechanics concepts becomes impracticable for any situation that is at all complex. It is usually necessary to begin by testing models of the snucture which are sufficiently large to reproduce service conditions, and then to follow by testing specimens which conform to the linear elastic fracture mechanics criteria. By measuring fracture toughness on such specimens, it is possible to establish inspection limits for flaw sizes in the structure, at the known applied stress. In highstrength applications, the reduction of non-metallic inclusion content in materials by processes such as vacuum degassing means that the procedure can be reversed; knowing that he has very clean material, an engineer can raise his design stresses. Finally, quality control specimens are needed. Because these must be small, it is no longer possible to specify material quality directly in terms of a fracture toughness value, and an alternative measure, which can be correlated with fracture toughness, is required. The most promising measurement of this type would appear to be the crack opening displacement, but work remains to be carried out to determine how the critical value of C.O.D. varies with specimen size in the structural metals of interest. In some cases it may prove possible to relate C.O.D. directly to the behaviour of the model structure, bypassing the need to determine linear elastic fracture toughness. This may be of particular benefit in situations where the stress state in service cannot be classed as either plane stress or plane strain. It is most important to realize the implications of

fracture mechanics concepts with regard to the metallurgical design of materials in conjunction with the engineering design of structures. Traditionally, the engineering design stress has been expressed as a fraction of a material's yield stress (assuming a design philosophy based oil plastic collapse) and this has tended to lead to the u~e of materials designed to have high yield stresses. Unfortunately, as a material's yield stress is raised, so its notch ductility and fracture toughness decrease, and the risk of brittle fracture increases. In some circumstances, it would be safer to alter the design code and raise the working stress to a higher fraction of the yield stress, rather than use higher yield strength material. This conclusion is supported by figures in Table 2, where permissible defect sizes (calculated from C.O.D. measurements) l°s in 3°,/, CrMoV steel ale shown for various applied stresses and proof stresses. Suppose that the designer wishes to load his structure to a stress of 30.75 ton/in. 2. On a traditional approach, using a "safety factor" of 2 on the uniaxial yield stress, he would have to use steel with a proof stress of 61.5 ton/in, z, which Table 2 shows can tolerate a defect size of only 0.35 in. at this applied stress level. However, if the design code is altered, he can obtain the same working stress of 30.75 ton/in, z in material of lower proof stress (e.g. 54, 50 ton/in. 2) which permits a greater tolerable defect size (e.g. 0.5 in. and 0.82 in. respectively). This increase in local ductility with decrease in proof sttess holds only for this particular tempered martensite microstructure : at the 40 ton/in. 2 proof stress level, a ferrite/pearlite structure is coarser and less tough than the martensite at the 50 ton/in. 2 level. The general concept of design should logically be to prevent failure of the weakest link of a structure. If plastic collapse is the mode of failure, then the yield stress must be increased by conventional heat treatment and choice of material. If fracture is the mode of failure, the optimum condition for fracture resis-

" F A B L E 2: PERMISSIBLE DEFECT SIZES 1N 3 % C r M o V STEEL AT OPERATING TEMPERATURE ( + 80 ° C)

P r o o f sTress

c~,wp/'C~y=0.5

c; vv/c;~ =0.57

~,,vp, tTy 0.615

50 ton/in. 2

1.37 in.

0.99 in.

0.82 in.

54 ton/in. 2

0.68 in. f J ~

0 . ~ ~ 0 . 4 2

61.5 t o n J, i n ~

~

0.35 in. f _ / ~ /

0.26 in. ~'-~

~

j

in. 0.22 in.

W o r k i n g = 30.75 stress ton/in. 2

Mater. Sci. Eng., 7 (1971) 1 36

34

J.F. KNOTT

tance needs to be developed in an alloy. At present, there is rather little basis for the scientific development of such an optimum condition; it is hoped that this review has drawn attention to some of the problems and to possible methods of approach. Eventually, in an efficient structure, the chance of failure from plastic collapse ought to be as great as that from fracture, the design stress being slightly less than that needed to produce failure by either process.

l 1. CONCLUSIONS

The main objectives of fracture research have been described in the previous section. Starting with a detailed examination of the ways in which fracture is produced at stress concentrations, whether these are initially sharp cracks, inclusions or abrupt changes in geometrical cross-section, it is hoped that a two-fold basis of design against brittle fracture in structural metals can be established: designing both the macroscopic features of the structure and the microscopic features of the metal. The present paper has reviewed the extent to which such design principles can be pursued at the moment. The elastic stress analysis of notches and cracks, and associated compliance calculations, have reached a high degree of precision for both plane stress and plane strain stress states, and can adequately take account of the presence of free surfaces. Intermediate thicknesses barely affect the O-yy and axx stresses, but az~ varies strongly with thickness (Fig. 5). Given the distribution of ayy, the failure stress of a brittle cracked component may be readily calculated, using values of fracture toughness measured from test specimens. The linear elastic assumptions have to be substantially modified for tougher materials, where the plastic zone size at instability is large. The stress analysis for plastic/elastic stressing is well established for plane stress and may be used to calculate toughness figures in thin sheet material. Little work has been published on the stresses and strains associated with the spread of plasticity in plane strain. The maximum value of Oyyis raised, by plastic constraint and/or work-hardening, and this strongly influences the mode of fracture in thick sections. The extent of yielding is often taken to be about one-third of that in plane stress, but etching studies have shown that the situation is not as simple as this. More effort in numerical computation is

needed to calculate how O'yy(max) varies with thickness and the size of the plastic zone. The variation of a~z with thickness is not well established, although indications are given by the etching results. If az~ attains the plane strain value of ½(ayy+axx) in the median plane only when the plate thickness is 4--5 times greater than the radius of the plastic zone, the nature of the "pop-in" toughness needs to be reexamined; a square fracture does not guarantee a plane-strain stress state. The safest procedure for thin sheet applications is to test in full sheet thickness and to determine "R-curves" as necessary. The determination of plane strain fracture toughness values is now a well-established practice. Specimens may sometimes have to be unrealistically large to satisfy the linear elastic plane strain criteria in tough materials; in some cases, fracture behaviour in thinner section may also have to be studied to ensure that the Kic value measured on the "approved" specimen really is the lowest toughness that the material exhibits. In many ways, the crack opening displacement is more satisfactory as a measure of toughness; it is easier to picture physically and may, in principle, be measured on both small and large specimens. More research is needed to understand how critical values of C.O.D. depend on stress system and to try to verify the relationship between C.O.D. and fracture toughness: aerit = O'y6crit. The toughness that a metal exhibits in practice depends on the mechanism by which it fractures when subjected to the combination of stress system, temperature, strain-rate, etc., appropriate to its service application. This mechanism is also a property of the metal's microstructure and yielding characteristics. An explanation for the toughness of mild steel in terms of microscopic fracture processes has been presented, but, again, the type of work described needs to be extended and applied to other materials if a general quantitative relationship between large-scale fracture behaviour and microstructural phenomena is to be established. In general structural engineering practice, it seems that improvements will be made, not by the introduction of new, fundamentally different, alloys, but by the more efficient usage of materials which are already available, taking advantage of improvements in quality and cleanliness. The design code should lose its arbitrary "safety factor" and replace it by a design stress based logically on the maximizing of resistance to failure, whether elastic buckling (resistance provided by general structural configuration), plastic collapse (high yield stress) or fast Mater. Sci. Eng., 7 (1971) 1-36

LARGE-SCALE BRITTLE FRACTURE IN STRUCTURAL METALS

fracture (fracture toughness). Since rather little can be done to alter the elastic moduli of structural metals, design of microstructure should be aimed at optimizing yield strength and toughness. The ideals outlined above may not always prove to be economic if the extra costs of quality control and cleanliness outweigh the advantages to be gained by the ability of the material to withstand higher stresses. These advantages are greatest for mobile and rotating equipment and for any situation where a high degree of reliability is required. In general it is to be expected that higher strength alloys will be assessed by means of fracture mechanics, to obtain precise values of permissible stress level. Low-strength steels will probably continue to be assessed by variations of the transition temperature approach, modified to simulate the situation in service. It is to be hoped that, eventually, all specifications for materials to be used in structural applications will contain, as a matter of course, some suitable measure of the material's toughness. A suitable measure can seldom, if ever, be obtained from uniaxial tensile tests; it cannot often be obtained from notched impact tests. Fracture starts in a large-scale structure when critical conditions are achieved in local regions. These conditions must be understood and reproduced when testing a material in the laboratory if its ability to resists fracture in service is ever to be determined properly. ACKNOWLEDGEMENTS AND APOLOGIA

This review article has been very much a personal interpretation of what are the important factors in the brittle fracture of structural metals. Many of the points made have originated in discussion with colleagues, particularly Dr. G. Oates, Prof. E. Smith, Dr. M. J. May, Dr. D. Elliott and members of the N.D.A.C.S.S. "C.O.D.A." panel and the B.I.S.R.A. Fracture Toughness (High Strength Steels) Committee. None of these must be held responsible for the personal opinions put forward in the article. A book to which a large amount of reference has been made is "Fracture of Structural Materials", by A. S. Tetelman and A. J. McEvily. References to this have not been given in the text, but many of the topics treated here are also covered in the book, often in more detail and usually by a similar approach. It is also an excellent sourcebook for references pre- 1966. The present reference list is abbreviated, giving mainly the "classical" reference and reference

35

to the most recent work. Other sources for reference, apart from the book by Tetelman and McEvily. are: H. Liebowitz (ed.), ~Fracture - An Advanced Treatise" Vols. I-VII, Academic Press 1968. ASTM Spec. Tech. Publ. No. 381, 1965 ASTM Spec. Tech. Publ. No. 410, 1967. Proc. 1st Intern. Conf. on Fracture, Sendai, 1965. Proc. 2nd Intern. Conf. on Fracture, Brighton, 1969. "Practical Fracture Mechanics for Structural Steel" (UKAEA/Chapman and Hall) 1969. REFERENCES 1 I. B. OBREIMOV,Proc. Roy. Soc. (London), A127 (1930) 290. 2 J.J. GILMAN, J. Appl. Phys., 31 (1960) 2208. 3 A. MAITLAND AND G. A. CHADWICK, Phil. May., 19 (1969) 645. 4 D. HULL, P. BEARDMOREAND A. P. VALINTINE, Phil. May., 12 (1965) 1021. 5 N. P. ALLEN, C. C. EARLEYAND J. H. RENDALL, Proc. Roy. Soc. (London), A285 (1965) 120. 6 J. F. KNOTT AND A. H. COTTRELL, J. Iron Steel Inst. {London), 201 (1963) 249. 7 W. D. BIGGS, Brittle Fracture of Steel, Macdonald and Evans, London, 1960. 8 C. F. TIPPER, The Brittle Fracture Story, Cambridge Univ. Press, 1962. 9 E. PARKER, Brittle Behaviour of Enyineering Structures, Wiley, New York, 1957. l0 Fracture toughness testing, A S T M Spec. Tech. Publ. No. 381, Philadelphia, 1965. 11 W. S. PELLINI AND P. P. PUZAK, NRL Rept. 5920, 1963; Weldinq Res. Council Bull. 88, 1963. 12 M. M. FROCHT, Photoelasticity, Wiley, New York, 1941. 13 R. E. PETERSON,Stress Concentration Desifn Factors, Wiley, New York, 1953. 14 S. TIMOSHENKO AND J. N. GOODIER, Theory of Elasticity, (McGraw~Hill) 2nd ed., New York, 1951. 15 G. KIRSCH, Z. Verein Deutsch Inst., 42 (1898). 16 C. INGLIS, Trans. Inst. Naval Architects, London, 55 (1913) 219. 17 P. S. THEOCARIS, Exptl. Mech., 3 (1963) 159. 18 H. NEUBER, Theory of Notch Stresses, Edwards, Ann Arbor, Michigan 1946. 19 J. COOK AND J. E. GORDON, Proc. Roy. Sac. (London), A282 (1964) 508. 20 E. STERNBERG AND M. A. SADOWSKY, J. Appl. Mech., March 1949, 27. 21 H. M. SCHNADT, Brittle fracture in steel, A.A.C.S.S. ConJl, Cambridge, 1959, Rept. P. 3 (HMSO) p. 127. 22 R. HILL, The Mathematical Theory of Plasticity, Oxford, 1960. 23 E. OROWAN, Trans. Inst. Engrs. Shipbuilders Scot., 89 (1945) 165. 24 J. A. HENDRICKSON,D. S. WOOD AND D. S. CLARK,Trans. Am. Soc. Metals, 50 (1958) 656. 25 T. R. WILSHAW AND P. L. PRATT, J. Mech. Phys. Solids, 14 (1966) 7. 26 J. F. KNOTT, J. Mech. Phys. Solids, 15 (1967) 97. 27 A. P. GREEN, Quart. J. Mech. Appl. Math., 6 (1953) 223.

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28 A. P. GREEN AND B. B. HUNDY, J. Mech. Phys. Solids, 4 (1956) 128. 29 T. R. W1LSHAW, C. A. RAU AND A. S. TETELMAN, Eng. Fracture Mech., 1 (1968) 191. 30 J. F. KNOTT, Proc. Roy. Soc. (London), A285 (1965) 150. 31 G. T. HAHN AND A. R. ROSENFIELD,FinalRept. o f Project SR-164 to Ship Structure Committee, NAS-NRC, May 1968. 32 M. L. WILLIAMS,J. Appl. Mech., Dec. 1952 and 20 (1953) 590. 33 M. L. WILLIAMS,Proc. Syrup. on Crack Propagation, Cranfield, 1961, p. 8. 34 M. L. WILLIAMS,J. Appl. Mech., 24 (1957) 109. 35 P. C. PARISANDG. C. SIH, A S T M Spec. Tech. Publ. No. 381, 1965, p. 30. 36 G. R. IRWIN, 9th Intern. Congr. of Appl. Mech., Vol VIII, Paper 101(11), Univ. Brussels, 1957, p. 245. 37 H. M. WESTERGAARD,J. Appl. Mech., June 1939, p. A49. 38 A. A. GRIEEITH, Phil. Trans. Roy. Soc. (London), A221 (1921) 163. 39 G. R. IRWIN, Proc. First Syrup. on Naval Structural Mechanics, Pergamon, New York, 1960. 40 E. SMITH, Intern. J. Fracture Mech., 1 (1965) 204. 41 J. R. RICE, J. Appl. Mech., (1968) 379. 42 G. I. BARRENBLATT,Advan. Appl. Mech., 7 (1962) Academic Press. 43 Fracture Toughness, Iron Steel Inst., 1969, p. 121. 44 G. IRWIN AND A. A. WELLS, Met. Rev., 10 (1965) No. 38. 45 J. E. SRAWLEYAND W. F. BROWN,A S T M Spec. Tech. Publ. No. 381, 1965, p. 133. 46 P. F. LANGSTONE, Metallurgia, Feb. 1967, p. 67; March 1967, p. 117. 47 A S T M Spec. Tech. Publ. No. 410, Jan. 1967. 48 G. R. IRWIN, Appl. Mater. Res., 3 (1964) 65. 49 A. A. WELLS, Brit. Welding J., 10 (1963) 563. 50 J . A . H . HULTANDF. A. McCLINTOCK,Ninth Intern. Congr. Appl. Mech., Vol. 8, 1957, p. 51. 51 D. S. DUGDALE,J. Mech. Phys. Solids, 8 (1960) 100. 52 B. A. BILBY,A. H. COTTRELL AND K. H. SWlNDEN,Proe. Roy. Soc. (London), A272 (1963) 304. 53 A. H. COTTRELL, Proc. Roy. Soc. (London), A285 (1965) 10. 54 B. A. BILBY,A. H. COTTRELL,E. SMITHANDK. H. SWINDEN, Proc. Roy. Soc. (London), A279 (1964) 1. 55 E. SMITH, Proc. Roy. Soc. (London), A285 (1965) 46. 56 E. SMITH,Proc. 1st Intern. Fracture Congr., Sendai, 1965, Vol. 1, p. 133. 57 E- SMITH, Proc. Roy. Soc. (London), A299 (1967) 455. 58 E. SMITH, Proc. Roy. Soc. (London), A282 (1964) 422. 59 G. R. IRWIN, Fracturing o f Metals, Am. Soc. Metals, 1948, p. 147. 60 J. M. KRAFFT, A. M. SULLIVANAND R. W. BOYLE, Proc. Syrup. on Crack Propagation, Cranfield, 1961, p. 8. 61 A. H. COTTRELL, Properties o f Reactor Materials and the Effects o f Radiation Damage, Butterworths, 1962, p. 5. 62 A. H. COTTRELL,Proc. Roy. Soc. (London), A276 (1963) 1. 63 J. F. KNOTT, J. Iron Steel Inst. (London), 204 (1966) 104. 64 J. HODGSONAND G. M. BOYD, Trans. Inst. Naval Architects, London, 100 (1958) 141. 65 A S T M Committee E24 report for publication in A S T M Standards, Part 31, 1969. 66 J. E. SRAWLEY,Practical Fracture Mechanics for Structural Steel, UKAEA/Chapman and Hall, 1969, p. A1. 67 E. T. WESSEL,Practical Fracture Mechanics for Structural Steel, UKAEA/Chapman and Hall, 1969, p. HI.

68 A. A. WELLS,Brit. Welding J., 8 (1961) 259. 69 G. T. HAHN, B. L. AVERBACH,W. S. OWENAND M. COHEN, in Fracture, Swanpscott Conf., Wiley, New York, 1959, p. 91. 70 N. PETCH, in H. LIEBOWlTZ (ed.), Fracture--An Advanced Treatise, Vol. l, Academic Press, 1969, p. 351. 71 T. R. WILSHAWAND P. t . PRATT, Intern. Conf. on Fracture, ,'Sendal, 1965, BIII, p. 3. 72 T. R. WILSHAW,J. Iron SteelInst. (London), 204 (1966) 936. 73 G. D. FEARNEHOUGHAND C. J. HOY, J. Iron Steel Inst. (London), 202 (1964) 912. 74 A.A.C.S.S. Rept. No. P. 2, HMSO, London, 1960. 75 D. E. W. STONE AND C. E. TURNER, Proc. Roy. Soc. (London), A285 (1965) 83. 76 C. CRUSSARD,R. BORIONE,J. PLATEAU,Y. MORILLONAND F. MARATRAY,J. Iron Steel Inst. (London), 183 (1956) 146. 77 D. K. FELBECKAND E. OROWAN, WeMing J. Res. Suppl., 1955, p. 570s. 78 A. H. COTTRELL, Iron and Steel Inst. Spec. Rept. 69, 1961, p. 281. 79 A. A. WELLS,Brit. Welding J., 12 (1965) 2. 80 R. W. NICHOLS, Practical Fracture Mechanics for Structural Steel, UKAEA/Chapman and Hall 1969, p. F1. 81 J. R. GRIFFITHSANDA. H. COTTRELL,J. Mech. Phys. Solids, 13 (1965) 135. 82 J. F. KNOTT, J. Iron Steel Inst. (London), 205 (1967) 285. 83 G. OATES,J. Iron Steel Inst. (London), 205 (1967) 41. 84 J. F. KNOTT, J. Iron Steel Inst. (London), 205 (1967) 288. 85 E. SMITH,Physical basis of yield and fracture, Inst. o f Physics Conf., Oxford, 1967, p. 36. 86 G. OATES,J. Iron Steel Inst. (London), 207 (1969) 353. 87 B. M. WUNDT, A S M E Paper 59-Met-9, 1959. 88 J. F. KNOTT, J. Iron Steel Inst. (London), 205 (1967) 966. 89 F. M. BURDEKIN,Ph.D. Thesis, Univ. of Cambridge, 1967 and B W R A Rept. C. 140/13a, May 1967. 90 D. S. WOOD AND D. S. CLARK, Trans. Am. Soc. Metals, 43 (1951) 571. 91 C. A. GRIFFISAND J. W. SPRETNAK, Trans. Iron Steel Inst. (Japan), 9 (1969) 372. 92 R. F. SMITHANDJ. F. KNOTT, to be published. 93 J. M. KRAFFT,Appl. Mater. Res., 3 (1964) 88. 94 J. M. KRAFFTAND G. R. IRWIN, A S T M Spec. Tech. Publ. No. 381, 1965, p. 114. 95 J.C. RADONANDC. E. TURNER,J. Iron Steellnst. (London), 204 (1966) 842. 96 J. F. KNOTT, Metalluroia, 77 (1968) 93. 97 R. W. NICHOLS,Proc. Roy. Soc. (London), A285 (1965) 104. 98 F.A. MCCLINTOCKANDA. ARGON,in MechanicalBehaviour o f Materials, Addison-Wesley, 1966, Chap. 16. 99 F.A. McCLINTOCK, Second lntern. Fracture Conf., Brighton 1969, Chapman and Hall, p. 904. 100 A. A. WELLS, Fracture--an Advanced Treatise, Vol. IV, Academic Press, 1969, p. 356. 101 A. A. WELLS AND F. M. BURDEKIN,Brit. Welding J., 10 (1963) 270. 102 F. M. BURDEKIN,R. E. DOLBYAND G. R. EGAN, Proc. 2rid Intern. Fracture Conf., Brighton, 1969. 103 P. M. BARTLE, Weldable Aluminium-Zinc-Magnesium Alloys, The Welding Inst., Sept. 1969. 104 A. H. PRIESTAND M. J. MAY,BISRA Rept. MG/E/338/67. 105 J. F. KNOTT, J. Iron Steel Inst. (London), 204 (1966) 1014.

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