A microscopic theory of brittle fracture in deformable solids: A relation between ideal work to fracture and plastic work

0001.6160.'80.1101-1479102.00/0

A MICROSCOPIC THEORY OF BRITTLE FRACTURE IN DEFORMABLE SOLIDS: A RELATION BETWEEN IDEAL WORK TO FRACTURE AND PLASTIC WORK M. L. JOKL, V. VXTEK and C. J. McMAHON

JR

University of Pennsylvania. Department of Materials Science and Engineering, Philadelphia. PA 19104. U.S.A. (Receiued 23 June 1980) Abstract-A criterion is developed for brittle fracture of a crystalline solid which is capable of being plastically deformed. The theory starts with the experimental fact that during extension of a brittle crack, energy is consumed not only by bond stretching and breaking, but also by dislocation emission from the crack tip. The latter is the “plastic work”. TV.which in the present theory depends on the ideal work of fracture. An empirical relationship between stress and dislocation velocity is employed to calculate the work connected with dislocation emission; then an approximation of the dynamics of bond stretching is made. ~mb~nation of the two allows the development of a Grifith-type thermodynamic fracture criterion which is applicable to any deformable solid. This theory contains all the basic features of ductile vs brittle behavior of solids. regardless of whether the brittle mode is transgranular cleavage or intergranular fracture. As a consequence, a relationship between yp and 7 is obtained which enables one to estimate y from measurements of local fracture stress. The use of this is illustrated in the form of estimates of reductions in 7 due to segregation of several impurities. calculated from measured local integranular fracture stresses in experimental steels. pr~~ntons un critbe de rupture fragile d’un sotide cristallin susceptible de se d&former plastiquement. Cette thiorie repose sur le fait exptrimental qu’au tours de la propagation d’une fissure R&WI&-Nous

fragile, I’tnergie est utilisCe non seulement pour ttendre ou briscr da liaisons, mais kgalement pour Cmettre des dislocations P I’extrimitC de la fissure. Cette dernitre Cnergie reprCsente le “travail plastique” yp qui d&end dans notre thiorie du travail de rupture idial. Nous utilisons une relation empirique entre la contrainte et la vitesse des dislocations pour calculer le travail mis en jeu au tours de 1’Cmission de dislocations; puis nous utilisons une approximation pour la dynamique de l’extension des liaisons. La combinaison des deux permet d’ttablir un c&&e de rupture du type de Griffith, applicable d tout solide d&formable. Cette th&orie contient toutes ies caractCristiques principales des solides en ce qui conceme Ies ruptures ductile et fragile, que Ie mode fragile soit un clivage transgranulaire ou une rupture intergranulaire. Nous en dCduisons une relation entre yp et 7 qui permet d’cstimer y B partir des mesures de la contrainte locale de rupture. A titre d’illustration, nous obtenons une estimation de la diminution de p due ii la sCgr&ation d’un certain nombre d’impuret&, B partir des contraintes locales de rupture intergranulaire obtenues dans des aciers exptrimentaux.

Zollmmeaf~Es wird ein Kriterium fir Spr&ibruch eines kristallinm Festkiirpers, der sich plastisch verformen kann, mtwickelt. Die Theorie ruht auf der experimentellen Tatsache, da8 wshrend der &w&rung eina sprtien Risscs nicht nur durch Dehnung und Aufbrechen von Bindungen sondern such durch Vcrsetzungsemission aus der RiDspitze verbraucht wird. Diese letztere ‘plastische Arbeit’ y,, hiiap bei der vorlicgenden Theorie von der idealen Brucharbeit ab. Mit einem empirischen Zusammenhang zwischen Spannung und Versctzungsgeschwindigkeit wird die bci der Verxtzungsemission auftretende Arbeit berechnet; danach wird eine Ngihcrung ftir die Dynamik der Bindungsdehnung durchgcfiihrt. Die Kombiition bcider e~~~icht, ein th~~~arni~h~ B~chkrjt~jum vom Garth-Typ aufzusteflen, welches auf &den vcrformbaren Festkiirper anwendbar ist. Diese Theorie enthiitt siimtiiche Merkmale des duktilen oder spr6den Bruchverhaltens, unabhiingig davon, ob der Sprtibruch transgranularcs Spaltm oder intergranularer Bruch ist. Daraus folgt ein Zusammenhang twischen yc und y, der alaubt, )I BUSMcssungen der lokalm Bruchspannung zu bestimmen. Als Anwendung wird die Verringcrung von y durch die Segregation verschiedener Verunreinigungen aus den an experimentellen Stihlen gemessenen lokalen intergranularen Bruchspannungen berechnet.

1. INTRODUCI’ION

energy released per unit crack extension in glass agreed fairly well with surface tension measurements which he made on the same glass. Using the later-developed terminology of linear elastic fracture mechanics [23 the Griffith criterion can be written as strain

The fundamental thermodynamic criterion for unstable fracture of a cracked stressed solid is that the energy stored in the specimen and ‘loading system be sufficient to supply the energy needed for the increase in the area of the crack. For the limiting elastic case Grifith El] identified that energy with the surface free energy, and he showed that the predicted critical

f, = 2Y

(11

where 8, is the critical strain energy release rate and y is the ideal work of fracture; i.e., the work needed in

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BRITTLE FRACTURE

the absence of inelastic modes of energy dissipationt. In the purely elastic case the energy release rate is given, for example [73. by i(: + Ki: ‘=

2G

K,:, (1 - v) + 2G

(2)

where G is the shear modulus, v Poisson’s ratio, and the K’s are the stress intensity factors for modes I, II. and III loading. respectively. GritTith had recognized [l] that his criterion would not apply. as written. to deformable solids such as metals. since it does not account for inelastic modes of energy dissipation during crack extension, the most important of which is plastic deformation. Later Orowan 183 suggested a modification of the Griffith criterion by the addition of a plastic work term such that 8, = 2Y + Yp

(3)

The implication. which was accepted by many subsequent investigators. was that ;‘r is a material parameter in the same sense as ;: independent of crack geometry. loading configuration, etc. The analysis embodied in equations (2) and (3) has been widely used in linear elastic fracture mechanics as a macroscopic failure criterion for cracked solids in which the extent of plastic flow prior to crack extension is negligibly small. In this case, the right hand side of equation (3) need not have any we11 defined physical interpretation, and equation (3) can be regarded simply as a statement that a critical energy release rate must be achieved for brittle fracture to occur. On the other hand. when one is studying the micromechanisms of fracture the situation is different. On the microscopic scale crack extension from a notch or pre-crack is not a continuous process. Rather, a microcrack must first be pucleated ahead of the notch or precrack, for example at a carbide [9] or other hard inclusion$ This microcrack is then loaded by the stress field of the notch or precrack and the attendant plastic zone. There may also be a significant loading contribution from the dislocation pile-up or twin which started the fracture of the inclusion [IO-121. Brittle fracture may take place if the microcrack can propagate immediately after nucleation into the metaltic matrix. This can occur if equation (3) is satisfied. the stress intensity factors being those applicable to the tip of the microcrack. For the development of a microscopic theory, however. it is essential to have a proper physical interpretation of the right hand side of equation (3).

+ It should not be expected that :, which is irreversible work. should be equivalent to the surface energy measured riversibly [3-63. However. they are presumably of the same order of magnitude. t If a sharp microcrack were to exist ab inirio, it would become blunted by plastic flow as soon as a load, however small. is applied.

IN DEFORMABLE

SOLIDS

The use of a constant plastic work term ;‘p regarded as a material parameter. is one possible physicaf assumption, In this framework a material behaves in a brittle manner (or is below its ductible brittle transition temperature) if equation (3) is satisfied before plastic rupture or general yielding occurs. otherwise. it is ductile. However, with this postulate it is not possible to rationalize the embrittlement which occurs in steels and other alloys when certain impurity elements segregate to grain boundaries. These embrittlement phenomena are quite u-idely recognized: examples are found in steels with phosphorus. tin, or antimony, in nickel alloys with sulfur. in copper with bismuth, in molybdenum with oxygen. and a number of other systems. In all cases. the segregation is limited to a very narrow region along the grain boundaries, and the plastic properties of the bulk material are not affected. Hence. there is no reason to expect that parameters such as yield strength or ;‘p which are functions of dislocation mobility. would be directly influenced by segregation. and yet very pronounced embrittlement does occur. On the other hand, the cohesive strength. or the ideal work of fracture, 2;. can be strongly affected by segregation. The importance of this is obscured in the Orowan formulation [equation (311, since 2; is much less than ;‘p and is therefore only a small fraction of the tota work of fracture. The very large embrittlement effects due to ~~egation are therefore unexplainable with the Orowan postulate as originally stated. This dilemma can be resolved by eliminating the postulate that yr, is an independent material parameter and by recognizing that it must be a function of ;, [13J an idea apparently first mentioned in the context of macrocrack propagation by Rice [ 143. Previously. it was pointed out 1131 that the sustainable level of stress at the tip of a propagating microcrack in any material depends on the value of ;‘. At the same time it is this stress which causes dislocation emission at the microcrack tip, and the energy thereby expended constitutes yp Hence. a ~gregation-induct reduction in j may produce a large reduction in y,, since dislocation velocity depends strongly on stress. When viewed in this way, it can be seen that the study of impurity-induced embrittlement presents an op~rtunity to investigate the details of the interdependence of the two terms on the right hand side of equation (3). By varying the amount and type of impurity segregation one, in effect, can vary 7 without chan,mg any other material parameter. Implicit in this approach is the idea that in brittle fracture of a deformable crystaIIine solid the processes of bond-breaking and dislocation emission at the microcrack tip are concomitant. rather than mutually exclusive. It must be recognized that this position is opposite to that taken by a number of other investigators who have started with the assumption that these processes are mutually exclusive [ 1S-20]. These analyses have all been based on the postulate that piflier

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bond-breaking. leading to microcrack propagation. or dislocation emission. leading to blunting of the microcrack, would occur. An early microscopic criterion for brittle fracture was based on a comparison of the ideal fracture stress with the yield stress on appropriate slip planes [ 15.16]. A later, more complex model of Rice and Thomson [17,18] suggested that brittle crack propagation takes place provided there is a barrier for dislocation nucleation at the crack tip when the energy release rate connected with the microcrack propagation is equal to the ideal work of fracture. The idea of ‘screening’ of the microcrack by the plastic zone of the precrack was subsequently introduced [ 19,20-J in order to reconcile the large critical stress intensity of the pre-crack with the very small fracture stress required for microcrack propagation in this case. The main difficulty with this kind of approach is that the initial assumption is inconsistent with an increasing body of experimental evidence which shows clearly that the processes of bond breaking and dislocation emission at the microcrack tip are concomitant. As discussed previously [ 131. it has been found by X-ray diffraction [21,22], SEM channeling patterns [23] and transmission electron microscopy [24] that the dislocation density adjacent to inter~anular fractures was less than that found adjacent to cleavage fractures and that it decreased with increasing concentrations of embrittling impurities. In addition it has been found that measured fracture stresses can be correlated with measured sizes of microcrack nuclei only if a value of 211+ yP > l@; is used [ 12,25,26]; this implies that dislocation emission must occur along with brittle crack extension and must account for most of the total fracture work. A realistic theory of brittle fracture of a deformable crystalline solid must deal with the interrelation between the ideal work to fracture and plastic work. That is, it must recognize that the plastic work is controlled by the attainable stress ahead of a microcrack, and, therefore, that yr, is a function of 7. Hence, the proper form of equation (3) should be 8, = 2Y + YIxy)

IN DEFORMABLE

how a reduction transition.

SOLIDS

i481

in 7 can lead to a ductile-brittle

2. PHYSICAL MODEL

AND THEORY

2.1 Node1 The basic assumption of the model is that the bond breaking and dislocation emission at the microcrack tip are concomitant processes. We consider a loaded propagating microcrack of length c (originally nucleated, for example. at a carbide). as shown schematically in Fig. 1. The loading of the microcrack is due principally to the local applied tensile stress and may also be partly due to the stress of a dislocation pile-up, e.g.. as analyzed by Smith [I2]. This involves a crack-tip stress field which, in the elastic case. can be described by a microcrack stress intensity factor. k. This may correspond to mode I. II, or III loading or to a mixture of the three modes. We assume further that the extent of the plastic deformation near the microcrack tip is always small in comparison with c: thus, the full description of the crack tip stress field by k is valid even when dislocation emission takes pJace. The time ta needed for breaking a bond at the tip of the microcrack, can be found from the solution of equations of motion of the atoms near the microcrack tip whose bonds are stretched under the effect of a stress r(r). This stress is the sum of the applied stress of the tip of the microcrack and of the back stress of previously emitted dislocations. It decreases with time as the dislocations are being emitted and depends on the parameters governing the dislocation mobility. Another parameter entering the equations of motion u

*

I I I I

(4)

Because the cohesive strength of grain boundaries can be altered by segregation of various impurities. it is possible to carry out experimental studies of the variation of the total work with 7. However. in order to be able to understand the physical nature of cohesion and the dependence of ;’ on composition, it is necessary to develop a theory capable of separating the cont~butions of 7 and pP to the total work of fracture. The present paper gives the first develop ment of such a theory. In the framework of this approach, which has been described in a preliminary form in Ref. [27], a microscopic Griffith-type fracture criterion for deformable solids is established and the dependence of yp on ;’ is obtained. It is then shown

I Iu* I I Fig. I. Schematic representation of microcrack nucleation under a local tensile stress u*. showing the initiating dislocation pile-up and the dislocation emission at the microcrack tip which results in the expenditure of plastic work. ;‘.

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is the strength of interatomic bonds which is related to the ideal work to fracture ;‘. Clear]). two cases may arise: (i) te is finite and the microcrack extends, or (ii) tg is infinite, i.e. the bond never breaks. In the latter case, the dislocation activity relaxes rhe stress so quickly that atomic bonds cannot break and brittle fracture does not occur. In the former case brittle fracture occurs accompanied by dislocation emission. If the problem of bond breaking were solved exactly, i.e. if equations of motion of all the atoms whose bonds are being stretched during the process of bond breaking and dislocation emission were solved simultaneously, a critical value of k, k,i,, for which fracture can occur would be found from the condition that for k > kmi,.,fg is finite. Clearly. at the same time the total energy of the system, E. would be decreasing with microcrack extension for k > kmin and increasing for k < km,. However, in the present study only the bond between the atoms nearest to the microcrack tip will be considered explicitly. To include the effect of all the other atoms which are not considered explicitly a thermodynamic criterion has to be invoked which requires that at fracture E d 0. This is analogous to the situation encountered in the case of the Griffith crack where no dislocation emission occurs and the bond between the atoms nearest to the crack tip can be assumed to be loaded to its breaking stress. if the atom pair is considered as isolated from the rest of the medium. However. the effect of the other bonds must be considered, and then the crack propagation may only take place if the above-mentioned thermodynamic criterion is satisfied. Hence, the present calculation does not determine directly the critical value of k, but it allows us to estimate for a given value of k the amount of dislocation emission during the time r8 and thus evaluate the resulting work, Y\*~as a function of lB and k. Since ta depends on k and 7, wP is at this stage obtained as a function of k and ;‘. The thermodynamic criterion for the unstable microcrack extension can then be written as E = -$I

- v) + 27 + H’&;‘,k) < 0

(5)

This inequality is superficially similar to the Orowan modification [83 of the Griffith condition, but wP is now a function of both k and y. Thus, equation (5) is an implicit equation from which the dependence of k at fracture, kG, on y can be calculated. Once this dependence is known, HI,,at fracture yr can also be determined as a function of y, and thus a relationship between the plastic work produced at the tip of the m&raack and the ideal work to fracture is established.

proportional to the velocity c of the dislocations as they move away from the tip. Designating the microcrack opening displacement (which is equal to the total Burgers vector of emitted dislo~tions), as 4, its rate of increase is

-d4 = gbv

dr where b is the Burgers vetor of a unit dislocation and 0 -’ is the linear spacing of the emitted dislocations, or the distance over which any emitted dislocation has to move away from the microcrack tip before the next dislocation can be emitted. Equation (6) is. therefore, a one-dimensional analog of the plastic strain rate equation. The dependence of c on the local stress, 7, can be expressed as a power law: c = I;~(s/G)”

2.2. Dislocation emission When studying the time-dependent emission of dislocations from the microcrack tip we assume that it is always possible and that the rate of the emission is

tit) =

$-&) 1 - v ka(t)

#(t, = -t Values > 30 have been reported.

(7)

where r,, and II are empirical parameters, and G is the shear modulus. This form has been frequently used to represent experimentally measured dislocation velocities and plastic strain rates [28-301. The value of PI may be large for relatively low dislocation vefocitiest; however, II must tend to zero asymptotically as the terminal velocity is approached. In the present work the calculations have been carried out in detail for values of II between 1 and 2. which should be apprqpriate for the high dislocation velocities which we expect near the microcrack tip due to the high stress in this region. A limited study has also been made for several values of II > 2. An analogous problem has been solved previously in the investigation of time-dependent plastic flow near a crack tip in connection with creep deformation. Here also the stress dependence of strain rate was approximated by a power law [31-333. As a result of these calculations the time dependences of the stress T ahead of the crack, of the total Burgers vector of dislocations emitted, 4, and of the extent of the local plastic_zone, s, have been obtained. in general, these calculations have been made nume~~lly, but since only very short times are considered here, we can assume that s 4 c and use the small scale yielding approximation. In these calculations, it is assumed that the emitted dislocations lie in the plane of the crack, i.e., a time-dependent Dugdale-BilbyCottrell-Swinden (DBCS) model is employed. In general, the emitted dislocations will aIways move along some well-defined crystallographic planes which are inclined to the crack plane; however, in the smallscale-yielding limit the quantities r,#, and s, indicated in Fig. 2. are calculated at the crack tip, and they can be expected to be well approximated by the in-plane model. Therefore, folIowing Ewing [333, we write:

GJ;;

(8)

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Thus, at any time wP is a function of material parameters and of the local stress intensity k. 2.3. Bond breaking In order to investigate the process of bond breaking we have to study the dynamics of bond stretching in the stress field ahead of the microcrack. Since we assume that the dislocation emission and the bond stretching are simultaneous processes, we have to take into account the fact that the stress ahead of the miFig. 2. Schematic representation of the stress relaxation crocrack decreases with time as dislocations are being ahead of the microcrack and of the physical quantities emitted. In our calculations we have identified this defined by equations (S)-(lO). stress with I given by equation (8). As mentioned before, stretching of only the bond s(t) = fa’(t) (10) between the first atom pair (nearest to the microcrack tip) was considered implicitly. Furthermore, it was where k is the microcrack stress intensity factor, as assumed that the restoring force is harmonic. If J is before, and a(t) is a function defined by equations (8) the coordinate perpendicular to the plane of the miand (9). The rate of dislocation emission from the crocrack. i.e., in the direction of the bond stretching, microcrack tip, d4/dt, can be obtained both by differthe equation of motion of the atom pair is entiating equation (9) and by substituting equations mj; = -KY + b2r(r) (8) and (7) into equation (6). Thus, (16)

where #,, = gbo,,. The constant g can be expected to be of the order of I/b, but it is not known (I priori; thus & is an adjustable parameter. The present calculation was carried out in detail for such values of & which produce yP values about an order of magnitude larger than y, as suggested by experiments [25,26]. Integration of equation (11) yields

[““,“L”

a_ f

(3-l

y”+‘)

(12)

The work produced due to dislocation emission during the time r can then be approximated as , wp0) =

I0

M(r) + 4(t) t@! 70) s(r) dr dr [

1 dt

wdr) =

lacidr I0

8G

(13)

(14)

and substituting equation (12) we obtain

1

- vr-‘(n 23n+

h’=-

+ I)2

(15)

4yb2 62

(17)

At the same time K = mw2, where w is the eigenfrequency of the corresponding atoms. If we take this frequency as the Debye frequency, w,,,

d

where q(t) is the position of the “center of mass” of the dislocation density ahead of the microcrack, which can be taken as s/2. The first term represents the work done when moving the dislocations newly nucleated during the time dr to the ‘center of mass’, while the second term is the work done when moving the dislocations existing before dt through the distance the center of mass moves during the time dt. Inserting equations (8-10) into equation (13) gives 3(1 - v)k2

where m is the mass of one atom, K is the ‘spring constant’ of the bond, and b2 the area per atom. We then assume that the restoring force vanishes when a bond is stretched through a displacement 6. A relation between K and 6 follows from the condition that the work produced when the atoms are separated by S is equal to the ideal work of fracture, i.e., j?cS2 = 2yb’ and thus

,2b y UD J m

(18)

However, whilst equation (17) represents a definition of K in the framework of the model, equation (18) cannot be regarded in the same way. Rather it should be taken as giving a reasonable order-of-magnitude estimate of b for a given 7 within our model. In the calculation we then consider S to be an independent parameter, but the values used were always such as to be of the same order as those given by equation (18). 3. RESULTS In general, the solution of equation (16) cannot be obtained analytically, but the following two limiting cases can be investigated: When 7 is large or k small the second term in equation (16) can be neglected and the equation simply becomes that of an harmonic oscillator. In this limit the atoms oscillate with an amplitude less than 6 and the bonds never break. Hence, in this case, the time rB can be taken as infinitely long. On the other hand when ;’ is small or k

JOKL,

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McMAHON JR: BRITTLE FRACTURE IN DEFORMABLE SOLIDS

large the second term dominates and

where

In this case the time to break the bond, defined by fits) = 6 is

The plastic work per unit length of the virtual microcrack extension produced during the time t, is 3x -(lm+lMZn+1)(1 _

ii2

,,(& X

+

Fig. 4. Dependence of E/G&[equation (S)] on k/G, x for n = 1.6, &, = 8.37 x lo5 cm s-l. 6 = 4 x 10-gem and three different values of y/Gb; arrows indicates the values of kc/G,!b

y)(2n-3MZn+l)

1

]),,#&t”-

1’2 2!(2n+ 1)

2b2 lM2n+f)

(21)

For intermediate values of ;’ and k the solution of equation (16) can only be found numerically. This has been done for a wide range of values of y and k and for various values of I < n < 2, & and b. The time tB was evaluated in each case and was found to have identical characteristics for all cases studied. A typical dependence of re on k is shown in Fig. 3 for n = 1.6, &, = 8.37 x 10s ems-‘, 6 = 4 x 10-9cm and for several values of 7: 17= l/3 was used in this and all other calculations. As discussed above, for large values of k the time ts is given by equations (20) whilst for small values of k it becomes infinite; it is seen from Fig. 3 that the transition between these two extremes is always very sharp. Hence, to a very good approximation we can define a kmin (cf. Fig. 3) such that for k < kin bond breaking cannot occur and for k > krnin tB is given by equation (20) and the corresponding plastic work by equation (21). However, as explained in Section 2.1 kin is a critical value for

t

f IO2

I

f

10'

to6

I

to@

f 10’0

t lOf2

k fNm-s’2)

Fig. 3. Dependence of ra on k for n = 1.6, & = 8.37 x 10scm s-‘ and d = 4 x lo-$cm and 3 values of Y/Gb.

breaking an isolated bond, but it is not a critical value of k for microcrack extension. This can only occur if the overall energy of the system decreases. which will not generally be the case for k > k,i, in the framework of the present model. The condition for microcrack extension is expressed mathematically in the Griffith-type equation (5). A typical dependence of the energy E on k is shown in Fig. 4 for n = 1.6. #0 = 8.37 x IO5cm s-‘, 6 = 4 x 10e9 cm and for several values of ;‘. The arrows in Fig. 4 denote the values of k = kc for which the microcrack becomes unstable. The uppermost curve in Fig. 4 corresponds to a case where such an instability cannot occur; physically this means that ta becomes so large that the energy wP dissipated during this time cannot be compensated by release of elastic strain energy. The former two cases correspond to brittle materials and the latter case to a ductile material. Obviously, whether a kG exists and the material is brittle or not depends on the value of y; a decrease in p can convert a ductile material into one in which brittle fracture can occur. When the critical stress intensity kG exists, it is always found to be much larger than the corresponding value of k,i,+ showing again that it is kc and not krninwhich has to be attained for microcrack propagation to start. Both kG and the corresponding plastic work yP = w,,(kJ depend on the ideal work of fracture, 7. and also other material parameters, namely. I&, n, 6. and G. The dependence of ki on y is shown in Figs Sa,b for n = 1.6 and 2.0, respectively, 6 = 4 x 10m9cm and for several values of &. (The dependence of kc on 7 which follows from the purely elastic Griffith criterion, i.e., kG = t’4C;/(l - V) is drawn dashed). The corresponding plastic work ;)P ($ is shown in Figs 6a,b. The curves shown in Figs 5 and 6 terminate at points corresponding to ‘r’= ym,

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SOLIDS

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075

Le

Gb 0.5

0.25

/ I

0.05 Y

(a) 0.375

I

I

0.025

0.05

0.075

ijI

I

L Gb

(a)

I

0.25 kt GJrj

0.20 VP Gb

0.125

Y (W

!

0 15

75

s

Fig. 5. Dependence of k&G, ‘t; on p/Gb for d = 4 x 10-qcm and several values of & (a) n = 1.6. (b) n = 2.0 which is the maximum value that y may take and still

permit quation (5) to be satisfied. The parameters tpO and 6 are not known u priori but must be regarded as adjustable parameters, although reasonable values of 6 have to be of the order given by equation (18). As mentioned earlier, when calculating the dependences shown in Figs 5 and 6 these parameters were chosen so that yP is about an order of magnitude larger than y. The calculations were also carried out for 17 z 2 and although the dependences of kc on y are similar to those shown in Fig 5 the corresponding plastic work yr is very close to. or smaller than, y for any choice of CpOor b. Hence, only for n < 2 can the ratio of ;& suggested by exoeriment (25.26) be obtained. On the other hand. cal-

0.05

0

0025 (b)

0.050 Y =

01:

Fig. 6. Dependence of y&b on y/Gb for 6 = 4 x lo-’ cm and several values of (b. (a) n = 1.6, (b) n = 2.0.

culations for n < 1.5 lead to dependences of kc vs 7 which do not terminate for any large value of p. This means that brittle fracture can occur whatever is the ideal work to fracture. Hence, the range of values of n which leads to physically interesting situations is 1.5 < n < 2.0. 4 DISCUSSION In this paper a microscopic theory of brittle fracture in deformable solids is developed. It is important

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to re-emphasize that brittle fracture must start with the nucleation of a fresh microcrack which then immediately continues to propagate into the metallic matrix. Any microcracks formed prior to the onset of fast fracture will be blunted by plastic flow during the loading process. In steels the nucleation takes place at carbides or other hard inclusions, and this has been studied by a number of authors assuming that stress intensification due to dislocation pile ups or twins aids the fracturing process [e.g. 9, 11, 12,251. There is ample experimental evidence. cited earlier in this paper, that when the microcrack enters the deformable metallic matrix the dislocation emission and breaking of the bonds occur simultaneously at the microcrack tip. The plastic work, yp is produced in connection with this dislocation emission, and it represents the principal mode of energy dissipation which occurs during the microcrack extension. In the earlier theories of microcrack propagation this plastic work was simply regarded as a constant, i.e., as a material parameter. The purpose of the present paper is to show that yp is a function of the ideal work to fracture y and to establish a possible relationship between yp and y. The heart of the present theory lies in equation (5) which is a Griffith-type criterion for microcrack extension in a deformable solid. However, unlike the original Orowan modification of the Griffith criterion [8] the plastic work, wp in equation (5) is a function of both y and the microcrack stress intensity factor, k. If this function can be determined, equation (5) represents a condition which determines the critical value of k at fracture, kc. At the same time the plastic work at fracture yp = wp(kG) is then determined as a function of y. The first aim of the analysis presented here was, therefore, to find the function y&y). The principal approximation made is that we have used a static analysis to describe a dynamic process in which a brittle microcrack moving through the nucleating elastic particle attempts to penetrate the deformable matrix. A partial justification for this is that the dislocations of interest move and do work in the frame of reference of the moving crack tip. Thus, it may not be unreasonable to use a static approach in which the microcrack tip defines the origin of a moving frame of reference. Certainly, a dynamic analysis is preferable, ideally an atomistic (rather than continuum) one, and we will have to wait for this to be able to judge how closely the present attempt approaches reality. Furthermore, in the present analysis we have not attempted to solve the equations of motion of all the atoms involved in bond stretching and dislocation emission, but have carried out the calculation only for the two atoms adjacent to the microcrack tip where the bond breaks when the microcrack extends. The effect of all the other atoms is then implicitly included in the thermodynamical Griffith-type criterion of equation (5). Another approximation is the use of equation (7) to describe dislocation motion under stress. This is

usually applied to low velocity dislocations, and I’ is taken to be an average velocity over some distance B b, in contrast with the present case of high speed dislocations moving over much shorter distances. Although in reality the dislocations actually accelerate and decelerate, the average velocity should provide a good approximation for the purpose of calculating the plastic work. The dislocations in this model must be moving close to their limiting velocity and thus small values of n should be used. Physically reasonable values of yp have, indeed, been obtained only for n < 2 which indicates consistency of the present theory. However, the calculated values of yr, depend on the choice of do (or t.e) and the values of & for which reasonable values of yp are obtained are different for different values of n. This is so because the ordinate intercept of the slope of the log r vs log I plot changes rapidly in the high c regime. One of the difficulties which should be dealt with in future developments of this approach is this great sensitivity of ;‘p to the choices of do (or a,) and II. This will presumably require casting the V-T relationship in another form or taking some other approach to the problem of characterizing dislocation mobility. In this study we have also assumed that the plastic work arises solely due to the lattice resistance to dislocation motion but that the nucleation of dislocations at the microcrack tip is an easy process and its contribution to yp may be neglected. The nucleation of dislocations at the tip of a sharp crack has been studied in Ref. [!7], where an expression for the activation energy of the nucleation of a dislocation loop has been derived. Using this expression and values of y and kc encountered in our calculation, this activation energy is always found to be negative for the corresponding saddle point configuration and thus there is no barrier for dislocation nucleation. The results of the present study, summarized in Figs. 5 and 6, exhibit all the principal features of the ductile vs brittle behaviors of materials. First, there is obviously a maximum value of the ideal work to fracture, y’“, beyond which brittle fracture cannot occur and only plastic blunting of a crack will take place. The magnitude of this critical value y”’ is, of course, also related to the plastic properties of the material as described by the relationship between dislocation mobility and the applied stress. Thus, materials with cohesive strength higher than ;* are inherently ductile. On the other hand in materials with cohesive strength smaller than y’” brittle fracture will occur. However, in this case the principal mode of energy dissipation is due to the crack tip dislocation emission, and the corresponding plastic work may be more than an order of magnitude higher than the cohesive energy. It is also seen from the present results why drastic embrittlement may occur due to segregation which lowers the cohesion at grain boundaries. When the cohesive energy of a normally ductile material is decreased below y’“, the material will become brittle.

JOKL, VITEK AND McMAHON

ESTIMATED

(a)

(FRACTION

JR:

BRITTLE FRACTURE

Xyx

OF A LAYER 1

IN DEFORMABLE

SOLIDS

1487

02i2_-

(b)

’

0.2

COVERAGE

0.4

( fraction

06

0.8

of o layer

IO

)

Fig. 7. (a) Dependence of the critical local tensile stress for brittle intergranular fractures. c*, on estimated grain boundary concentration of three impurity elements in a Ni-Cr steel (Ref. [35]) (b) Dependence of ;’ on the grain boundary concentration of three impurity elements. calculated using the present theory and measured u*; 17= 1.6, #,, = 8.37 x IO’ cm SC-I, d = 4 x 10-9cm. initial crack length c = I pm. Examples of embrittlement

this situation are the pronounced of copper by bismuth [34] and the

often-observed drastic changes in the ductile-brittle transition temperature in steels embrittled by metalloid elements. At the same time, a decrease in the toughness of a relatively brittle material (e.g. molybdenum or tungsten) with decrease of cohesion due to grain boundary segregation can also be rationalized, since a relatively small decrease in the cohesive strength y may lead to a large decrease in yp (Fig. 6). The present theory applies to both transgranular cleavage and intergranular brittle fracture. However, in the former case the value of y is fixed by the bulk composition and thus the dependence of yp on y cannot be studied directly. On the other hand, in the latter case 7 varies with the concentration of various metalloid impurities in grain boundaries and the dependence of y,, on y plays a major role in the relationship between the impurity concentration and material toughness. Kameda [35] has measured the dependence of the critical local stress for brittle intergranular fracture, a*, as a function of the intergranular concentration of three impurity elements: P, Sn, and Sb, as shown in Fig. 7a. If we assume a size c of 1 pm for the initiating brittle particle (corresponding to an average-size nonmetallic inclusion in these vacuum-melted steels), and if we approximate k 2 B*(~Ic)~/~(ignoring the dislocation pile-up term, which is negligible in a tempered steel, due to the fineness of the microstructure) then the values of y derived from Fig. Sa for n = 1.6 are as shown in Fig. 7b. We cannot take these absolute values of y too literally, since they depend on somewhat arbitrary choices of n and &,. However, the > values are of the proper order of magnitude, which supports the choices of n and 40. More importantly, however, Fig. 7 indicates that it is possible in principle using the present theory to relate measured

values of c* to corresponding values of 7 and that their variations with impurity concentration are similar. A further quantitative development of the theory relating the plastic work and the cohesive strength, combined with the measurements of the local critical stress for intergranular fracture, may provide a means for a direct measurement of the dependence of the cohesive energy on the concentration of segregating impurities. This is the ultimate step in the develop ment of a fundamental understanding of embrittling processes. Ackno*ledyemrnts-This research was supported by the National Science Foundation under Grant no. DMR 78-07535 and in part by the Department of Energy, Contract no. DE-A502-79ERI 0429. The experimental results of Dr Jun Kameda have provided valuable insights into the nature of the embrittlement problem.

REFERENCES A. A. Griffith, Phil. Trans. R. Sot. A 221 (1921). :. G. R. Irwin, Handb. Phys 6, 551 (1958). 3: J. R. Rice in Eflecrs of Hydrogen on rhe Behavior oj Moreriols (edited by A. W. Thompson and I. M. Bernstein) p. 455. AIME N.Y. 4. 1. P. Hirth, Phil. Trans. R. Sot. A 295, 139 (1980). 5. R. Asaro, ibid p. 151. 6. C. J. McMahon Jr, V. Vitek and G. R. Belton, Scripta metall. 12. 785 (1978). 7. J. F. Knott, Fundamentals of Fracture Mechanics, Butterworths: London (1973). 8. E. Orowan, Trans. Iastrt Engrs Shipbldrs Scot. 89, 165 (1945). 9. C. J. McMahon, Jr., and M. Cohen, Acre metal/. 13. 591 (1965). 10 C. Zaner, in Frocruring ojMet& p. 3 New York: ASM (1948). 11. A. M. Stroh, Proc. R. Sot. A 223. 548 (1955). 12 E. Smith, Proc. Con& Phys. Basis of Yield and Fracrure, Insr. Phys. and Phys. Sot. Land.. Oxford, p. 36 (1966). 13. C. J. McMahon Jr, and V. Vitek. Acra mercrll. 27, 507 (1979).

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14. J. R. Rice, Proc. Isr Inr. Con/. Fracfure, Sendai. (edited by T. Yokoboti. T. Kawasaki and J. L. Swedlow) p. 309 (1966). 15. A. Kelly, Strong Solids. Oxford University Press: London ( 1966). 16. A. Kelly, W. Tyson and A. Cottrell. Phil. Mug. IS, 567 ( 1967). 17. J. R. Rice and R. Thomson. Phil. Mug. 29, 73 (1974). 18. D. D. Mason Phil. Mag. 39, 455 (1979). 19. R. Thomson. J. Mater. Sci. 13, 128 (1978). 20. J. Weertman, Acra. metoll. 26, 1731 (1978). 21. D. K. Felbeck and E. Orowan, Weld. J. Res. Suppl. 34, 5705 (1955). 22. T. Ogura and C. J. McMahon Jr, Unpublished Research, Univ. of Penn, (1978). 23. D. E. Newbury, B. W. Christ and D. C. Joy, Metull. Trans. AllS, 1505 (1974). 24. A. Kumar and B. L. Eyre, Proc. R. SOC. A 370, 431 (1980).

IN DEFORMABLE

SOLIDS

(edited by 25. J. F. Knott. Pror. 4fh Inf. Con/: Fracture. D. M. R. Taplin) Vol. 1, p. 61 University of Waterloo Press (1977). 26. D. A. Curry and J. F. Knott. Meral Sci. 12, 341 (1979). 27. M. L. Jokl. C. J. McMahon Jr. Jun Kameda and V. Vitek, Inr. Conj Micromechonisms Fracrure. Cambridge. Metal Sci., to be published (1980). 28. W. G. Johnston and J. J. Gilman, J. uppl. Ph~s. 30. 129 ( 1959). 29. D. F. Stein and J. R. Low, .I. appl. Phys. 31. 362 (1960). 30. D. S. Tomalin and C. J. McMahon Jr Acta mrrull. 21. II89 (1975). V. Vitek. Irtr. .I. Frucr. 13. 39 (1977). H. Riedel, Marer. Sci. E~tgng. 30. I87 (1977). 33. D. J. F. Ewing. Int. J. Frucr. 14. IO1 (1978). 34. A. Joshi and D. F. Stein. J. Insr. Mrruls 99. I78 (1971). 35. J. Kameda and C. J. McMahon Jr. Mrrull. Trurls. A. in press.

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