Brittle to ductile transition in cleavage fracture

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Acta metall. Vol. 35, No. I, pp. 185-196, 1987 Printed

in Great

Britain.

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BRITTLE TO DUCTILE TRANSITION CLEAVAGE FRACTURE

0

1987 Pergamon

Journals

IN

A. S. ARGON Massachusetts Institute of Technology, Cambridge, MA 02139, U.S.A. (Received 17 February 1986) Abstract-In many cleavable solids, cleavage cracks can propagate at steady state by laying down a trail of dislocations emanating from the crack tip and lying on planes inclined to the crack front. These crack-tip initiated dislocations produce shielding at the crack tip that reduces both the crack tip tensile stresses and the shear stresses on the inclined planes. They also blunt the crack. Cleavage cracks, can nevertheless, still propagate under appropriately increased stress intensity conditions to keep the crack tip tensile stress constant. A condition is reached in the propagation of such slightly blunted cracks where a small increment in temperature or a decrement in the crack velocity permits the nucleation of a new set of dislocations that produce additional shielding and blunting which tip the balance against the crack-tip tensile stresses. This results in a transition from brittle cleavage to ductile behavior. The steady state specific plastic work that can just be tolerated by a propagating cleavage crack before it catastrophically blunts is calculated to be only of the order of 10% of the specific surface energy. Although most geometrical details of the dislocation emission process are adequately modeled, the calculated brittle to ductile transition temperatures are found to be more than an order of magnitude higher than those that have been experimentally measured. This discrepancy is a result of the present inadequate methods of modeling activation configurations by considering the dislocation loop radius as the only activation parameter, while proper modeling of such configurations must consider also the Burgers shear displacement of the loop as an activation parameter. Such two parameter analyses, however, require accurate information on interlayer atomic shear resistance profiles for specific crystals which are presently not available. The analysis furnishes ready explanations of the toughening effects of so-called “ductilization” treatments, and embrittling effect of aging and dislocation locking, as well as the relatively large difference between the lowest levels of toughness between fracture in polycrystals and in single crystals.

R&sum&-Dans de nombreux solides clivables, les fissures de clivage peuvent se propager a l’etat stationnaire en laissant derriere elles un faisceau de dislocations prenant naissance a l’extremite de la fissure, et situ&es dans des plans inclines par rapport au plan du front de fissure. Ces dislocations initiees en t&e de fissure ont un effet de protection qui rbduit a la fois les contraintes de tension a l’extremite de la fissure et les contraintes de cisaillement sur les plans inclines. Elles emoussent egalement la fissure. Les fissures de clivage peuvent cependant se propager encore avec un accroissement de la contrainte destine a maintenir constante la tension a l’extrtmitt de la fissure. Au tours de la propagation de ces fissures legtrement imousstes, on atteint un etat oti un petit accroissement de temperature ou bien une diminution de la vitesse des fissures permet la creation d’une nouvelle strie de dislocations qui provoquent un ecrantage et un Bmoussage supplementaires qui s’opposent aux constraintes de tension a l’extremite de la fissure. I1 en resulte une transition clivage fragile-comportement ductile. Nous avons calcult que la plasticite stationnaire specifique que peut toltrer une fissure de clivage qui se propage, juste avant qu’elle ne &mousse de facon catastrophique, ne vaut qu’environ 10% de l’energie specifique supeficielle. Bien que la plupart des details geometriques du mode d&mission des dislocations soient convenablement mod&lists, on trouve que les temperatures de transition fragile-ductile calculees sont superieures de plus dun ordre de grandeur a celles que Ton a mesurees expbrimentalement. Ce d&accord resulte de l’imperfection des mbthodes actuelles de modelisation des configurations d’activation oi l’on considere le rayon des boucles de dislocation comme le seul parametre d’activation, alors qu’une modtlisation correcte de ces configurations devrait tenir compte egalement du cisaillement de Burgers de la boucle. Nianmoins, ces analyses a deux parametres ntcessitent une connaissance precise des profils de resistance au cisaillement atomique intercouche pour des cristaux specifiques, qui ne sont pas encore disponibles. Cette analyse permet deja d’expliquer les effets de durcissement des traitements dits “de ductilisation”, l’effet de fragilisation du vieillissement et du blocage des dislocations, ainsi que la difference relativement importante entre les resistances a la rupture les plus basses dans les polycristaux et les monocristaux. vielen spaltbaren Festkdrpern konnen sich Spaltrisse stationar ausbreiten, indem sie eine Spur von Versetzungen hinterlassen; diese Versetzungen entspringen der RiBspitze und hegen auf Ebenen, die geneigt zur RiBfront sind. Diese an der RiBspitze erzeugten Versetzungen schirmen die Rigspitze ab, wobei die Spannungen an der Rigspitze und die Scherspannungen in den geneigten Ebenen verringert werden. AuBerdem wird der Rig abgestumpft. Spaltrisse kiinnen sich trotzdem weiter unter entsprechend erhijhten Spannungen, urn die Zugspannungen an der Rigspitze konstant zu halten, ausbreiten. Damit wird in der Ausbreitung eines solchen schwach abgestumpften Risses ein Zustand erreicht, bei dem ein kleiner Anstieg in der Temperatur oder ein kleiner Abfall in der RiOgeschwindigkeit die Bildung einer neuen Schar von Versetzungen erlaubt. Diese neue Schar fiihrt zu weiterer Spannungsabsenkung und Rigabstumpfung und halt damit das Gleichgewicht gegen die Zugspannung an der RiBspitze. Dieser Vorgang fiihrt zu einem ffbergang von der spriiden Spaltung zum duktilen Verhalten. Die spezifische plastische Arbeit im stationiiren Zustand, die gerade noch von einem sich ausbreitenden RiB vor einer katastrophalen Abstumpfung geduldet werden kann, IaBt sich berechnen und betragt nur Zusammenfassung-In

185

Ltd

186

ARGON:

BRITTLE TO DUCTILE

TRANSITION

IN CLEAVAGE

FRACTURE

etwa 10% der speziflschen Oberflachenenergie. Obwohl die me&en geometrischen Einzelheiten der Versetzungsemission im Model1 entsprechend erfaBt sind, ergeben sich aus den R~hnungen spr~d~uktile Ubergangstemperaturen, die urn iiber eine Crdbenordnung iiber den experimentellen liegen. Diese Diskrepanz ist eine Folge der gegenwtirtigen ungeeigneten Methoden, die Aktivierungskonfigurationen dadurch beschreiben, indem der Radius der Versetzungsschleife als der einzige Aktivierungsparameter angesehen wird. Dabei muR eine geeignete Modellbeschreibung such die Burgers-Scherung der Versetzungsschleife als Aktivierungsparameter berficksichtigen. Solche Analysen mit zwei Parametern erfordern jedoch genaue Kenntnisse iiber die Profile des Sche~iderstandes in atomaren Zwischenschichten, die gegenw~~ig nicht verfilgbar sind. Die Analyse liefert fertige Erkl~rungen fur die Z~higkeitswirk~ng der sogenannten “Duktilisierungs’‘-Beh~dlungen und fiir den Verspriidungseffekt durch Alterung und Versetzungsverankerung, wie such fur den vergleichsweise grogen Unterschied zwischen den untersten Niveaus der Ziihigkeit beim Bruch von Polykristallen und Einkristallen.

1. I~RODU~ION The transition of fracture in many crystalline solids between a high energy absorbing ductile mode and a brittle cleavage mode is an important limitation to their use in structures subjected to low temperatures and high rates of loading. Although this subject has received serious attention by many investigators from both phenomenological and mechanistic points of view, there is presently no comprehensive and definitive treatment of this bifurcation of behavior. The brittle response of a stressed and initially crack-free polycrystalline solid depends on may complicating factors such as the need to initiate a cleavage crack in local surroundings capable of supporting a low energy absorbing cleavage mode of propagation in the presence of possible repeated arrests by micros~uctural barriers, the presence or absence of readily mobilizable dislocations in the immediate surroundings, the state of tri-axiality of the local stress, etc. In its most basic form, however, a brittle or ductile form of behavior is governed by the response of the stressed solid in the immediate surroundings of an atomically sharp crack. Starting with Kelly et al. [l], the response of sharp cracks has been considered theoretically by a large number of investigators from the point of view of both continuum elastic-plastic crack tip mechanics and atomistic simulations. Many of these studies have been reviewed recently by de Celis et al. [2] in the context of an atomistic simulation of crack tips in Fe and Cu. While the most satisfactory answers are likely to come from atomistic simulations of crack tips, these have proved to be difficult to implement with the required local detail. In view of the above, two previous continuum studies have been noteworthy. In the first study of Rice and Thomson [3], the crack tip bifurcation behavior is considered to be a competition between an impending state of propagation of a cleavage crack at a constant critical energy release rate and the nucleation of a dislocation loop on a highly stressed inclined slip plane going through the crack front. If the activation energy barrier to the formation of a dislocation loop on the inclined plane is too high at the point of impending propagation of the crack brittle behavior is assumed to result. If on the other hand the critical activation configuration of the dislocation on the inclined plane is of the order of

dislocation core dimensions with a negligible activation barrier then emission of a dislocation will result. It is assumed that this will tip the balance toward plastic blunting of the crack tip. Using the computed levels of the energy barriers to dislocation nucleation around the crack tip, Rice and Thomson have had considerable success in rank ordering materials into intrinsically brittle or intrinsically ductile ones. On the basis of the Rice and Thomson analysis, the activation energies for dislocation nucleation, at the point of im~nding motion of a stationary cleavage crack, in W, Si, LiF, and Zn were calculated to be 329, Ill, 58, and 107 eV, respectively, which should make all of these materials intrinsically brittle with little hope of showing crack-tip initiated plastic dissipation at any finite temperature below the melting tem~rature. Yet, in all four of these materials, detailed experimental observations have shown conclusively that dislocation emission occurs not only from stationary but often even from propagating cleavage cracks (W: Liu and Shen [4], Kobayashi and Ohr [5]; Si: St. John [6], Brede and Haasen [7]; LiF: Gilman et al. [S], Burns and Webb 191,Kitajima [lo]; Zn: Burns [1 11). In the case of E-Fe, the Rice and Thomson analysis gives an activation energy of only 2.2 eV, which indicates that plastic dissipation at tips of cleavage cracks should occur quite readily at finite tem~ratures. This is found to be indeed the case [IO], but not markedly more so than in the cases W, LiF, and Zn. In Si, while stationary cleavage cracks were found to catastrophically blunt above a strain rate dependent sharply defined brittle to ductile transition temperature [6,7], no dislocation emission was found to occur from moving cleavage cracks. Additional TEM observations of disiocation emission from cleavage cracks in a number of other materials have been reported in a comprehensive survey by Ohr [ 121. These observations, which are particularly conclusively established through the experiments of St. John [6] and Brede and Haasen [7] on Si, have established that dislocation emission from sharp cleavage cracks in dislocation-free material does not at all guarantee ductile behavior in a cleavable solid. More precise considerations are necessary, particularIy for the purpose of understanding the level of the brittle to ductile transition temperature. In the second study by Freund and Hutchinson

ARGON:

BRITTLE

TO DUCTILE

TRANSITION

[13], the ductile to brittle transition in fracture is attributed to the ability of fast cracks to outrun their plastic accommodation zones which are assumed to adjust by a combination of thermally activated glide at low strain rate (or stress) level and phonon drag controlled glide at high strain rates (or stress). The analysis which considers also typical inertial distortions of the travelling crack-tip fields pre-supposes that the crack tip remains atomically sharp and demonstrates that ultimately at very high velocities of the crack where the local plastic resistance is dominated by phonon drag, the crack-tip field is characterizable by a Mode-I stress intensity K,. Thus, Freund and Hutchinson find that the plastic accommodation zone around the travelling crack shrinks as the crack moves faster and faster under a condition of a constant crack tip energy release rate evoking a corresponding decrease of the overall energy release rate. For very fast cracks where inertial effects dominate, the overall energy release rate rises sharply. In the Freund and Hutchinson study, even though the crack-tip is assumed to remain sharp, the crack is found to be unable to outrun the plastic accommodation zone and the minimum energy release rates just prior to the sharp relativistic rise are found to be orders to magnitude larger than what would be required for only surface energy production. These very large energy release rates should normally produce very substantial crack-tip blunting and a concomitant steep drop of the crack tip tensile stress below what is necessary for continued cleavage. In what follows, we present an approximate analysis of a travelling cleavage crack in a perfect crystal in which the consideration is entirely directed at the details of dislocation emission from the crack tip where plastic adjustment of the background material is not possible. The analysis is quasi-static and does not consider inertial effects. The goal of the study is to demonstrate that a certain level of steady dislocation emission is possible from propagating cleavage cracks and that a final brittle to ductile transition can occur abruptly when the temperature increases above a certain level for a given crack velocity or the crack velocity drops below a certain level at a given temperature. Although the results are not definitive on an absolute scale, they give correct relative behavior of materials. They also point out dramatically the need for a true atomistic and nonlinear consideration of the problem of dislocation emission from the highly stressed regions of a crack tip.

2. THEORETICAL

MODEL

2.1. Basic outline At absolute zero temperature, the analysis of Rice and Thomson, [3], or a suitable modification of it, will establish whether or not a dislocation loop can nucleate on the most highly stressed plane at the

IN CLEAVAGE

FRACTURE

187

crack tip before the crack can begin to propagate on its plane. We term a material in which the crack will propagate before any dislocation can nucleate at absolute zero temperature a cleavable material, and consider cleavability a necessary but insufficient condition for a ductile to brittle transition in fracture. In distinction to this, is the possibility of spontaneously nucleating a dislocation loop, without any energy barrier, before propagation of the crack can occur at absolute zero. Materials that exhibit such behavior are not cleavable. It should be impossible to propagate sharp cracks in such materials in the characteristically low energy absorbing mode of cleavage under any condition short of altering the crack tip chemistry. Face-centered cubic metals, with the exception of iridium, fall into this category. We now consider a cleavage crack as shown in Fig. 1, travelling on a cleavage plane in a cleavable cubic crystal in a cube direction, with the crack tip being parallel to another cube direction. The details of our model are guided by the observations of Burns and Webb [9] on LiF and by those of a similar earlier study of Gilman Knudsen and Walsh [8]. Some of the analysis also parallels that of Burns and Webb [14]. These authors have established that dislocation emission does not occur on inclined planes containing the crack front in a plane-strain geometry as assumed by Rice and Thomson [3] but on planes making a 45” angle with the crack front, as pictured in Fig. 1. These planes have a resolved shear stress of nearly twice the magnitude of that on the inclined plane at 45” containing the crack front. They have the further advantage, where the laying-down of continuous curved dislocation lines emanating from the crack tip, as pictured in Fig. 1, is possible without the need of repeatedly nucleating loops, as would be the case in the plane strain alternative. Although at absolute zero temperature dislocation emission is assumed not to occur before or during the propagation of the cleavage crack, at higher temperatures, such emission can occur if the thermal energy is sufficient to aid in the nucleation process. If this were to occur, the crack could, however, still propagate in a cleavage mode, provided that the blunting that results from the laying-down of dislocation trails, as illustrated in Fig. 1, and the associated crack tip shielding due to the dislocations does not lower the crack-tip tensile stress below the required critical value. We will show that if the kinetics of dislocation glide permits dislocation velocities equal to the crack velocity under stresses less than the largest crack-tip stresses, then a certain amount of such steady state laying-down of dislocation trails is possible without stopping the cleavage crack. For a given crack velocity above a certain temperature or at a given temperature below a certain crack velocity, the increasing ease of layingdown additional dislocation trails will result in too large a level of crack tip blunting and shielding to maintain the required critical tensile stress to propagate the cleavage crack even under increasing stress

188

ARGON:

BRITTLE TO DUCTILE TRANSITION IN CLEAVAGE FRACTURE

/

crack

shape in trail

Kclined plane containing dislocation trail Fig. 1. Geometrical details of dislocation initiation from crack tips in a cubic crystal such as LiF.

intensities. At this point, a brittle to ductile transition should occur in the behavior of the material. 2.2. Steady state crack tip shielding and blunting of a propagating cleavage crack Burns and Webb [ 141have given a detailed analysis of the shapes of dislocations in the dislocation trails emitted by a travelling cleavage crack. Their analysis does not consider inertial distortions of the crack tip field, nor modifications of the stress field resulting from the presence of the dislocation trails. These limitations, however, are not serious. As Freund and Hutchinson [ 131 demonstrate, inertial effects become important only at near relativistic crack velocities. Little modification of the crack tip field beyond the normal crack tip shielding considerations should be expected for the relatively low densities of dislocation trails and wide dislocation free zones that are compatible with a cleavage mode of crack propagation. Thus, we consider the dislocation trails of Burns and Webb [9] as representative, and on this basis, consider a steady state consisting of n dislocations, each of Burgers vector b, in both sets of y’ and z’ planes, at a spacing h apart. As Burns and Webb discuss, the trajectory of individual dislocations that emanate from the crack tip, and lie on the y’z’ planes reach a distance q (see Fig. 1) from the crack tip at the border of the dislocation free zone that can be given as

K: ? =u-p where u has been evaluated by Burns and Webb to be = 0.018, and r is the level of the resolved shear stress on the y’z’ plane that drives the screw components of the loops at points P and P’ in Fig. 1, at the same velocity i of the cleavage crack. We assume here that the principal resistance to dislocation motion comes from a lattice friction stress, that this is high, and that the shapes of loops calculated by

Burns and Webb will not change significantly even when the dislocations being laid down are not individuals, but are in the form of small groups of n parallel dislocations. Thus, if trails of n parallel dislocations are laid down at the border of the dislocation free zone in both sets of y’ and z’ planes, at a plane spacing of h, the uniform plastic strain L,, in the dislocation free zone will be

where 6 ( = nb) is the total slip displacement in any given plane. Since no plastic strain L, should result from such motion of dislocations, there will be a corresponding uniform plastic strain cyYof the opposite sign from that given in equation (2). This will result in an average half plastic crack opening displacement (relative to the symmetry plane of the crack)

This will be taken as the radius of the now slightly blunted tip of the advancing cleavage crack. The consequence of laying-down these dislocation trails is two fold. First, the dislocations of total Burgers vector 6 in these inclined slip planes will produce a certain amount of crack-tip shielding, lowering both the tensile stress across the cleavage plane, and the maximum resolved shear stress in the inclined slip planes where additional dislocations can be nucleated. Second, since the dislocations emanate from the crack tips, there will be an increase in the crack tip radius from b/2 to 6,, which decreases all crack-tip stresses for a constant applied stress intensity factor. The crack tip shielding in shear in any one inclined slip plane y ’ or z’, due to the set of parallel edge dislocations with total Burgers vector 6, in the same

ARGON:

BRITTLE TO DUCTILE

TRANSITION

IN CLEAVAGE

FRACTURE

= -E+)(;)K,

189

(8)

where, in order to obtain the final result, we have again made use of equation (1). 2.3. Propagation

Fig. 2. Effective dislocation packets producing the same crack-tip blunting as the dislocation loop trails formed on planes inclined at 45” to the crack front.

of a partially

blunted cleavage crack

In order to maintain the continued propagation of a cleavage crack, the crack tip tensile stress must be maintained at a critical level ec, the cleavage strength of the solid. This strength is considered to be temperature independent, or at most, have the same temperature dependence of the Young’s modulus. Thus, it is required that

at the other end of the dislocation free zone, on one side of the cleavage crack [12] is

slip plane,

K;,_,,=

E6

-

where

2(1 -v’>JG/’ Since there are dislocations also on the other side of the cleavage crack and since for each slip plane y’ there is a conjugate plane z’, the shielding will be 4 times as high at the line of intersection of any pair of planes y’ and z’, at the crack tip. Because of the presence of similar neighboring loops on other parallel inclined planes separated by a distance h, the total average shielding at any one point along the crack tip is amplified further by a superposition factor of q/h, giving a total shielding contribution K;,.,Y on any inclined plane, K;,_,=;K;,_,,=

4/V&

-

(1 -v)J%r& = -fi&

(;)(;)K,

(5)

where, in order to obtain the final form, we made use of equation (1). The crack-tip shielding on the cleavage plane due to the dislocations that are being laid down at the ends of the dislocation free zone, on the inclined planes, is obtained through the effective half crack opening displacement 6, that results from this operation. Considering this crack opening to result from an effective edge dislocation packet with total Burgers displacement 6,, as shown in Fig. 2, where 6, =

J5 6, =

2

K ,L,,,=K,+K,-s.

(10)

Substitution of equations (3), (8) and (10) into equation (9) gives the following characteristic equation for (6/h), for the limiting condition where 6, % (b/2)

where again T is the shear stress level at distance VI, on the inclined slip plane that is able to drive screw dislocations at the same velocity as that of the crack, i. In equation (11) above, the left hand term is the net crack tip stress intensity necessary for propagation of the crack. The first term on the RHS of the equation is the applied stress intensity and the second term the Mode I shielding stress intensity. For a given crack velocity, hence, a given shear resistance t, there will be a certain characteristic plastic shear strain (6/h) on the inclined planes when the equation is just satisfied. At levels of (S/h) greater than this value, there will be too much shielding, and the crack tip stress intensity will be too low, for cleavage propagation. At levels of (6/h) smaller than the characteristic value for the given crack velocity, the crack tip stress intensity will be in excess of what is required, and the crack will accelerate to seek a new equilibrium. The solution for 6/h of interest is

we have for the Mode I crack tip shielding [12] K I-s=

-2

a n sin - cos -. 4 8

3E6, 4(1-v2)Jnrl

(7)

In equation (7), the first factor 2 accounts for the dislocation packets both above and below the crack plane. Substitution of equation (6) into (7) and where the final result is obtained by evaluation of some of the constants gives for K, _-5 r IP G WP)‘. The condition given by equation 1.959 una for cleavage crack propagation, but K -_-I-‘(I -v) Jj&h It is also necessary that the largest a

I

recognizing that (9) is necessary is not sufficient. crack tip shear

190

ARGON:

BRITTLE

TO DUCTILE

TRANSITION

stresses have insufficient time to nucleate additional dislocations from the crack tip as the travelling crack with its dislocation trails subjects crack tip elements of volume to high stress. We state this condition as

where z, is the largest crack-tip shear stress on the inclined y’ or z’ planes at the crack tip radius 6,, tC(k, i”) is a critical value of this stress to initiate a dislocation loop, K;,, K;,_,, and K;,_,ip are the K,, type stress intensities produced by the applied stress (through the applied K,), the shear shielding resulting from the dislocation trails, and the net crack tip stress intensity, all on the inclined y’ or z’ planes. 2.4. Brittle to ductile transition in cleavage crack

IN CLEAVAGE

in the intensified crack tip shear stress field that exists at the partially blunted crack surface on the inclined plane 1 t,=-exp v,

AG:(B, x) kT

(14)

’

Here, v, is a frequency factor of the order of atomic frequencies, and AG*(/?, x) is the activation free energy of the critical configuration, given as a function of the local peak shear stress represented by a dimensionless parameter /?, and the surface free energy x of the atomic size ledge that must result from the formation of the loop. The result of the analysis gives AG*=@$exp(

propagation

On the basis of the above discussion of the necessary conditions for cleavage crack propagation, a condition for transition from brittle cleavage to a form of ductile fracture can now be stated. Thus, during the steady state propagation of a partially blunted cleavage crack, whenever the inequality of equation (13) becomes an equality or reverses sign, under a constant applied driving force of K,, a brittle to ductile transition will occur. At this stage, a further increase in blunting and crack-tip shielding will reduce the crack tip tensile stresses below the required level for cleavage propagation. Since in both the equality of equation (11) prescribing the condition of steady state cleavage propagation and in the equality form of equation (13) presenting the onset of the brittle to ductile transition, the level of the applied stress intensity K, does not appear explicitly, when all terms are evaluated, it is clear that increasing the level of the applied K, will not affect the condition of the transition. For operational implementation, the condition for the brittle to ductile transition must be stated somewhat differently. The new dislocation loop that must achieve the critical incremental blunting must be nucleated on the inclined plane at the crack tip under conditions of near homogeneous nucleation from the perfect crystal-albeit having the benefit of a highly intensified stress field. The nucleation of dislocation loops under these conditions at crack tips has been studied in some detail by Rice and Thomson [3] under certain restrictions. We will adopt their development in form here, but modify it to rectify some of its operational shortcomings. Further and more substantial needed improvements to this development will not be attempted here. The clear need for such improvements will come out of our results. The details of a homogeneous nucleation analysis from a crack-up field, based on a linear elastic approach using dislocation line properties is given in the Appendix. The principal result of the analysis is an expression for the mean waiting time t, for the nucleation of a critical dislocation loop configuration

FRACTURE

-3){[(2-8)

+J(C@-=12-$1

-B)

+ JImm131 where the various defined as

(2-v) 4 = 8(14 =

p

dimensionless

(19 parameters

(0

are

Wd

0

J2 50 5

=,.,dj!+[, -zg] 6h

(16~)

(164 Of these terms, r, is the well-known dislocation core cut-off radius; (6/h) is the steady state ratio of dislocation strength to plane spacing in the dislocation trails laid down by the cleavage crack, as defined by equation (12); t is the shear stress on the inclined planes at the border of the dislocation free zone where screw components of the loops must move at the same speed as that of the crack; x is the surface free energy; and v the Poisson’s ratio. The loop nucleation analysis that follows the lines of thought of Rice and Thomson has been modified as follows: (a) the total work done by the local stress is considered over the entire loop area and not only over the increment between the critical configuration and the core size of a dislocation; (b) the activation energy was made to go to zero for the largest stress r,,,,,,,, for which only one solution for the critical configuration exists; (c) the free energy was then expanded by a Taylor series for lower stress levels Br,Y,,X (/I < 1) around this critical configuration; (d) the radius of the new critical configuration under such a lower stress was calculated to obtain its radius c,( = RJr,); (e) this value was substituted into the now Taylor expanded form, to obtain the explicit form of the activation free energy AG* given in equation (15). The critical configuration radius of the dislocation, nucleated under a stress /37y.r8max that results from this

ARGON:

BRITTLE TO DUCTILE TRANSITION

dislocation

analysis is R

cc=A=exp

-“I

r,

(

[(2-B)

(17)

C

where i is the crack velocity. The brittle to ductile transition temperature TB-n for any given crack speed i is now obtained by substituting equations (14), (I 5) and (17) into equation (18) and solving for the temperature in equation (14). This gives

pb 3u. exp(- duo)

0 k

velocity expression as [ 151

s=~exp{-~[l-(;~~3~.

2Rc t, = y--

-

191

(20)

>

We note that when fi -+ 1, the critical configuration radius [, goes to the limiting form [,, = exp (- u,/u,), which is the smallest configuration achievable under a stress of z,.~.,,,~~( = 0.278~ in alpha iron). The activation free energy expression given in equation (15) has been “adjusted” to vanish for this limiting configuration. At higher levels of stress, no critical activation configurations exist for the form of the free energy expression in equation (15), developed to model the nucleation process. We now re-state the brittle to ductile transition condition of equation (13) as a rate condition in which the nucleation time of equation (14) must equal the time of residence of the peak crack tip shear stress field over a volume element having a size equal to twice the critical radius given by equation (17). Thus, a brittle to ductile transition in fracture is expected to occur when

=

FRACTURE

110

+J_].

T B-D

IN CLEAVAGE

21n (Zv,R,/i)

x {K2-B)+&=BQ2

- $(l - B) + J-1’).

(19)

To evaluate this expression for specific materials, with the help of equations (16ad), it is necessary to have an explicit expression that gives the slip plane shear resistance z as a function of dislocation velocity where the dislocation velocity is set equal to the crack speed d to obtain the resistance to the motion of the screw segments of the dislocation trails laid down by the moving crack. Considering the shear resistance to arise from a lattice resistance in a nearly perfect crystal with no strong extrinsic obstacles, where a dynamic balance must exist between double kink nucleation rate and neighboring double kink anihilation governed by the phonon-drag controlled drift of the expending kinks, we state the required

Table

I.

Material

properties

Material

G-,/b)

y

a-Fe W LiF Si Reference

0.67 0.67 0.25 0.25 (3)

0.291 0.278 0.187 0.215 (3)

influencing

(xl@) 0.115 0.039 0.038 0.052 (3)

In this expression, B = kT/Rv, is the generally accepted form of the phonon drag where vi, [ =0(1013 s-‘)] is the atomic frequency, R the atomic volume, To an abreviation of the double-kink energy in units of the Boltzmann’s constant (given below), and z^the threshold lattice resistance occurring under athermal conditions [15]. We take these terms to be

@Ia) zb 00 =

-

B (taken as 105cm/s, a terminal limit)

-r^=

(21b)

10-2

(appropriate

for many b.c.c. metals)

(21~)

for our evaluation of the brittle to ductile transition. The brittle to ductile transition temperature Ta_n, has been evaluated for four prominent materials, Fe, W, LiF, and Si that are known to exhibit such a transition in a prominent way, and for which considerable experimental information is available. The required materials data are given in Table 1. The first three columns have been taken from the data given by Rice and Thomson [3], while the values for z^/p and o,/p are best assumptions considered reasonable for these materials on the basis of known plastic response and bonding. The evaluation of TB_n from equation (19) starts with assignment of a crack velocity which is then considered to be the same as the velocity of screw dislocations travelling with the crack. This leads to a plastic shear resistance from equation (20) for a given temperature T. This plastic shear resistance, normalized with the shear modulus, together with a,/~ prescribes 6/h, from equation (12) which in turn gives /I, from equation (16~) which in turn permits the evaluation of R, from equation (17) and together with the other material parameters, the T B_D. As might be expected, the choice of a temperature in the evaluation of the plastic resistance t affects the RHS of equation (19) only in a very weak way in the temperature range between 200-600K, primarily because of the rather high temperatures To. Thus, the brittle to ductile transition temperature TB_nis principally dependent on the crack velocity C,

the brittle

to ductile transition

To = (yb”/4k)(il~) 1940K 5960 K 1830K 4230 K

in fracture

(i/P)

@Jr )

0.01 0.01 0.01 0.02

0.26 0.26 0.26 0.52

(rb’/K) 7.74 2.39 7.37 1.69

x x x x

IO4 K 10’ K IO* K IO5 K

192

ARGON:

BRITTLE TO DUCTILE TRANSITION IN CLEAVAGE FRACTURE

LIF

al =

i

c

IO3

IO4 6, Cleavage

Crack

Velocity,

IO3 cm/set

Fig. 3. Calculated dependence of the brittle to ductile transition temperature on crack velocity for LiF, Fe, Si, and W, compared with experimental measurements made on LiF crystals by Gilman et al. [8].

as it should. The calculated dependence of T,_, on crack speed in the range of lo3 to 5 x lo4 cm/s for the four materials that have been considered is given in Fig. 3. Crack speeds in excess of these, produce important relativistic modifications around them, for which no allowance was made in the approximate analysis presented above. 3. DISCUSSION There are few available measurements of the brittle to ductile transition in the propagation of a cleavage crack in any material where the transition occurs as a result of increasing temperature at a given crack speed or decreasing crack speed at a given temperature. The experiment that comes closest to what is required is that of Gilman et al. [8], who have made observations at room temperature on the critical velocities for dislocation emission from propagating cleavage cracks in LiF. In their experiments, they found that at room temperature, when the cleavage crack velocity drops below 6 x lo3 cm/s, it has time to emit a large density of dislocations. At wedge insertion rates less than what is required to drive the cleavage crack at this critical velocity, the propagation of the crack was found to be unstable and going through cycles of blunting, and reinitiation. At liquid nitrogen temperature, at 77 K, Gilman et nl. found that brittle cleavage cracks in LiF could be propagated without such instabilities at speeds as low as

1 cm/s, although etching indicated that some dislocations were still being emitted from the crack tip, but were apparently insufficient in density to affect the overall response. These two points for LiF, at 300 and 77 K, were used to construct the experimental line in Fig. 3 within the range of velocities for the calculated T,, for the other materials. Clearly, the brittle to ductile transition temperatures calculated by our model for LiF are somewhat more than one order of magnitude too high. The transition temperatures for the other materials, are also far too high-well above their respective melting temperatures. Although the mode1 presented above lacks certain refinements, such as no inertia1 effects, approximations in computing the shielding effects, and some uncertainties on the values chosen for the material parameters, the gross overestimate just referred to is for a very clear and different reason. The analysis used by Rice and Thomson, which was also retained in the model here, is restricted to linear behavior and line properties of dislocations that are to be nucleated. In this model, the only concession made to the atomic and discrete nature of the material is to introduce a dislocation core cut-off parameter while no allowance is made for the nonlinear response of the highly stressed material in which the dislocation is to nucleate and perturb its surroundings. The activation analysis searches for a critical loop radius R of a fully formed dislocation in which the Burgers displacement is that of a total lattice dislocation. The modifications on the results due to nucleating a partial dislocation was briefly discussed by Rice and Thomson, but was not pursued. Thus, in their analysis, there is only one activation coordinate, which is taken to be the radius of the critical loop configuration. Clearly, a proper analysis of the nucleation of a dislocation loop in the very highly stressed material of the crack tip must consider the process in the context of an interlayer shear resistance between atom planes, characteristic of the material. In a sinusoidal interaction model used in the initial Peierls-Nabarro analysis, the peak interlayer shear resistance occurs at one quarter of the full Burgers displacement 6. In most metallic materials as approximated first by Foreman et al. [16] by a skewed resistance profile, the peak of the interlayer shear resistance occurs at even smaller relative displacements. Thus, in a proper nucleation analysis, to improve on that of Rice and Thomson, two activation coordinates, the Burgers shear displacement of the loop, as well as the loop radius must be considered in the context of an interlayer shear resistance. This would also quite naturally account for the much reduced effective shear stiffness of the highly stressed solid at a state of nucleating a dislocation loop. The crack-tip stress field has a long range that falls off as r-II2 while the critical unstable dislocation loop requires a stress for equilibrium that falls off with increasing size somewhat less rapidly that r-l (to

ARGON:

BRITTLE

TO DUCTILE

TRANSITION

account for the logarithmic dependence of the line energy on the loop radius). Thus, it can be expected that the critical activation configuration is not governed by the loop size but by the Burgers displacement which reaches its critical condition with a shear displacement of less than one quarter of the interatomic spacing b. If this were the case, the main scaling factor, pb 3, of the activation free energy could be reduced by a factor of as much as l/64. The reduced shear stiffness of the solid due to the nonlinear response of the highly stressed solid near the crack tip would provide a further reduction. An analysis following some of these considerations where the loop size was considered fixed but the interlayer shear displacement was taken as the principal activation coordinate has been carried out by Argon [17] for the nucleation of shear patches in metallic glasses under stress in the context of a skewed interlayer shear resistance of the type used by Foreman et al. [16]. The critical configuration Burgers displacement, and consequently, the energy was indeed found to be only a fraction of that where the Burgers displacement of the loop was the full interatomic displacement. Although such an activation analysis is tractable for the crack tip problem by numerical procedures using line notions, it requires accurate interlayer shear resistance profiles for the specific materials to be considered. These are presently not available. Additional experimental evidence suggests that the above explanation is indeed the source of the discrepancy between the calculated and the expected brittle to ductile transition temperature. Thus, the Rice and Thomson analysis in which the activation energies of the critical loop configurations are scaled by p’b3 establishes for the activation energies for such loops the values of 329, 111, and 58 eV in W, Si, and LiF, respectively, at the stress level of impending cleavage propagation. These energies are in the ratio of 5.67/l .91/l, and compare very well with the ratios 5.64/1.91/l of the TB~,,of these materials at a constant fraction (3.5 x 10m2by arbitrary choice) of the respective velocities of sound in these materials. Since the basic assumptions and computation techniques used here are nearly identical with those of Rice and Thomson, this agreement is not surprising. Nevertheless, these levels of activation energy are all well beyond the realm of what is thermally activable at any finite temperature below the respective melting points of these materials. Yet, in all three of these materials, the experiments of Liu and Shen [4] for W, of St. John [6] and Brede and Haasen [7] for Si, and of Gilman et al. [8], and Burns and Webb [9] for LiF have indicated directly that dislocations are indeed nucleated from crack tips at energy levels far below those calculated by Rice and Thomson. In the experiments of St. John and those of Brede and Haasen, the brittle to ductile transition of the response of stationary cracks in initially dislocation free Si crystals has been found to be governed by the rate of spreading

IN CLEAVAGE

FRACTURE

193

of a crack-tip-initiated-plastic-zone, with an activation energy level of 1.8-l .9 eV. This is the activation energy of dislocation glide, indicating that the dislocation nucleation step was early and had an energy barrier less than this amount. Brede and Haasen [7] have estimated the activation energy for the nucleation of a loop from the crack tip in dislocation free Si crystals to require as little as 0.5 eV, if the loop is composed of a partial dislocation and has a prismatic double-kink configuration-without even any further considerations based on an actual interlayer shear resistance. It is interesting to examine some other features of the model at the point of a brittle to ductile transition. The computations that lead to the specific transition temperatures give that the typical levels of /I, for the ratio of the local peak shear stress to the shear stress ry~z,max that lowers the activation energy to zero is about 5-6 x 10M2.This gives typical critical normalized loop configurations <,, that range from about 2.1 in Fe to about 3 in LiF. For these conditions in Fe, the average level of plastic strain t,, = 6/h in the dislocation free zone was found to range, from 5 x 10m4 to about 5 x lo-’ as the temperature decreases from 600 to 200 K, paralleling an increase in the ratio of r/p from about 1.8 x lo-’ to 6 x 10m3. This gives rise to a ratio of crack-tip shielding stress intensity K, _ 5to overall applied stress intensity K,, ranging from 5.9 x lo--* to 1.75 x lo-‘, and dislocation free zone diameters rl ranging from 9 x lo-‘cm down to 1O-5 cm, as the temperature decreases from 600 to 400 K, accompanying an increase in the critical overall stress intensity K, for propagating the cleavage cracks, from 0.88 to 0.98 MPam”*. The plastic work per unit area of fracture x,,, due to the dislocation trails left behind can be readily calculated as

where

are the effective tensile plastic resistance and strain in the dislocation free zone, respectively. Evaluation of equation (22) gives a range of x,,, from 8.2 x 10m2to 0.293 J/m* for Fe as the temperature decreases from 600 to 200 K. These levels of increase in specific work of fracture are small in comparison with the true surface energies x for Fe, of about 2 J/m*. The above estimates are useful and indicate that the capacity of a cleavage crack to propagate in the presence of crack tip initiated plastic dissipation in a perfect crystal is very limited. Clearly, the wellknown effects of substantial increases of fracture work accompanying cleavage fractures in so-called “ductilization” processes of prior plastic deformation, etc. must come from plastic flow around the

194

ARGON:

BRITTLE TO DUCTILE TRANSITION IN CLEAVAGE FRACTURE

crack that is not crack-tip initiated. Such plastic flow is primarily of a crack-tip shielding nature that produces little or no blunting and can be compensated by an appropriate elevation of the applied stress intensity without interfering with the cleavage mode of crack propagation. In the problem considered by Freund and Hutchinson [13], it is assumed implicitly that the propagating crack does not blunt, that the inelastic deformations in the plastic zone are all of a crack-tip shielding nature, and that the crack propagation condition is a critical crack-tip energy release rate. The above estimates show that since the cracktip-initiated specific plastic dissipation, xp is only a small fraction of the true surface energy x, the assumption of Freund and Hutchinson is largely correct. Nevertheless, the brittle to ductile transition in fracture is still controlled by the limited crack-tip initiated plasticity as this is not only of a shielding but also of a blunting nature. Our development presented above clarifies a number of additional common observations on the brittle to ductile transition in polycrystalline materials such as steel. We already pointed out above the connection to the “ductilization” treatments. Cleavage cracks travelling in a velocity range in which the crack-tip initiated blunting is less than the critical value to catastrophically stop them, i.e. below the TB-o that corresponds to this velocity, can nevertheless dissipate very large amounts of energy, often orders of magnitude above the true surface energy x by dissipation in the plastic zone resulting from motion of mobilizable dislocations. When the background dislocations are immobilized by prior work hardening and/or pinning by impurity atmospheres, the plastic dissipation can be radically reduced-all in a cleavage mode of crack propagation. This mode of background response of a cleavage crack is appropriately modelled by Freund and Hutchinson [13]. Finally, the large difference in fracture work or energy release rate that has often been reported between a polycrystalline material with pinned or otherwise immobilized dislocations, and a corresponding single crystal in which a cleavage crack can run in an unencumbered manner (compare a K,, of 22 MPa rn’i2 quoted by F&H [13] for steel at 0 K with 0.9 MPa m-1’2 calculated from specific surface energy and elastic properties) finds a ready explanation. The reason that has often been cited that the cleavage planes on individual grains don’t match and that local plastic flow is necessary to produce tearing is inadequate and unconvincing. The geometrical roughening can provide only factors in the range of 2-3, while ductile tearing would not appear necessary when a multiplicity of cleavage planes are available to produce the bridging. Clearly, however, the ductile tearing is the answer and it comes about when the cleavage cracks are momentarily but repeatedly arrested at the grain boundaries and their velocities are repeatedly brought below that corresponding to the brittle to ductile transition temperature. This

permits substantial crack-tip initiated plastic flow and super-critical crack-tip blunting. In these instances, the continued cleavage cracking from grain-to-grain requires repeated re-initiation of cleavage of the type that was found by Gilman et al. [8] in their nonsteady propagation of cleavage cracks in LiF above the corresponding brittle to ductile transition temperature. 4. CONCLUSIONS In many intrinsically brittle solids that undergo cleavage fracture, it is possible to nucleate dislocations from the tips of propagating cleavage cracks at finite temperatures even though this is not possible at absolute zero tempeature. This is because the decohesion process at the crack tip is primarily temperature independent, while the initiation of dislocation loops from the crack tip can be significantly assisted by thermal activation. Thus, in a certain range of crack velocities at any finite temperature, it is possible for propagating cleavage cracks to surround themselves with steady state dislocation trails that still permit the propagation of the crack in a partially blunted shape and in the presence of a steady state level of crack-tip shielding. Hence, while at a given temperature at crack velocities above this range, the crack can rid itself of the dislocation trails by pulling away from them. Below this range, the increasing density of dislocations in the trails blunt the crack super-critically and lower the crack-tip stress below the level necessary for continued cleavage. At such a velocity, a brittle to ductile transition will occur in the fracture behavior. At higher temperatures, the brittle to ductile transition occurs at higher crack velocities. The analysis presented here shows that the bifurcation in behavior between continued propagation of a cleavage crack and a quasi-static ductile mode of separation is governed by the nucleation of a dislocation loop from the tip of a partially blunted cleavage crack. This occurs when the cleavage crack is travelling under a neutral condition where the crack tip tensile stress is maintained at the decohesion stress level while at the same time the highest cracktip shear stress is just under the required critical shear stress necessary to nucleate another set of dislocation loops along the crack length. Because the brittle to ductile transition is governed by a nucleation condition, the temperature, T,,, depends sensitively on the detail of the thermally activated nucleation process. The analysis presented here demonstrates convincingly that conventional activation analyses that do not include the discrete atomic nature of the crack tip material and its nonlinear response over estimate the energetics of the phenomenon by an order of magnitude or more, even though the relative scaling of the brittle to ductile transition in different cleavable materials is predicted correctly. The analysis indicates that while the total specific plastic work due

ARGON:

BRITTLE TO DUCTILE

TRANSITION

to the laying-down of steady state dislocation trails is quite small in comparison with the true surface energy, it nevertheless governs the transition through its capability controlling crack-tip blunting. The very much larger levels of plastic dissipation that are known to accompany cleavage fracture in ductilized crystals and in polycrystals result from plastic flow in the background plastic zone where inelastic strains produce only crack-tip shielding but no blunting. Acknowledgement-The

considerations that resulted in this communization were stimulated by interactions in a Summer Research Group activity at the Center of Materials Science of the Los Alamos National Laboratory in August 1985. REFERENCES 1. A. Kelley, W. R. Tyson and A. H. Cottrell, Phii. Mug. 15, 567 (1967). 2. B. de Celis, A. S. Argon and S. Yip, J. appl. Phys. 54,

IN CLEAVAGE

195

FRACTURE

0R 2 b

tA3) where R is the dislocation loop radius on the inclined y’ or t’ planes and all other terms are as defined in the text. In equation (A3), the first term in brackets is the peak shear stress on the inclined plane at the blunted crack tip of radius 6, induced by the applied opening mode of stress intensity, and the second term is the effective shielding produced by the trail of dislocations laid-down at the end of the dislocation free zone by the crack tip, all as described in detail in the text. Of these three terms, the third one differs from that defined by Rice and Thomson. In ob~ining equation (A3), the work of integration was carried out over the entire slip patch and not only beyond the core region. This eliminates the undesirable indeterminacy in the Rice and Thomson model whose activation energies go to zero for all stress levels when the loop radius becomes equal to the core radius. Introducing the following abreviations

4864 (1983).

(A4)

3. J. R. Rice and R. Thomson, Phil. Mag. 29, 78 (1974). 4. J. M. Liu and B. W. Shen, Metall. Trans. lSA, 1247 (1983): ibid 15A, 1253 (1983).

(A5)

5. S. Kobayashi

and S. M. Ohr, Phil. Mug. A42, 163 (1980). 6. C. St. John, Phil. Mug. 32, 1193 (1975). I. M. Brede and P. Haasen, private communication, to be published. 8. J. J. Gilman, C. Knudsen and W. P. Walsh, J. appl. Phys. 29, 600 (1958). 9. S. J. Burns and W. W. Webb, J. appl. Phys. 41, 2086 (1970).

IO. K. Kitajima, in Second intern. Co@. Fund. Fracture (short abstracts), ORNL/TM-9783 (Oak Ridge, Tennessee), p. 65 (1985); also private communization, to be published. 11. S. J. Bums, Acta metall. 18, 969 (1970). 12. S. M. Ohr, Maier Sci. Engng 72, 1 (1985). 13. L. B. Freund and J. W. Hutchinson, J. Mech. Phys. Solids 33, 169 (1985). 14. S. J. Burns and W. W. Webb, J. appl. Phys. 41, 2078 (1970).

15. U. F. Kocks, A. S. Argon and M. F. Ashby, in Prog. Mater. Sci. (edited by B. Chalmers, J. W. Christian and T. B. Massalski), Vol. 19. Pergamon Press, Oxford (1975). 16. A. J. Foreman, M. A. Jaswon and J. K. Wood, Proc. Ph_vs. SW. (A)64, 1% (1951). 17. A. S. Argon, Acta metall. 27, 47 (1979).

APPENDIX

(A7) the free energy change AC due to the formation of the loop can be written as AG = $rsg([) = ~~3(u~~ln~ + u,(< - 1) - u,[‘)

Depending upon the magnitude of I(,, this equation has either two, one, or no roots, as u, increases. The level of a, where only one root exists gives the critical configuration under the highest stress level. This configuration has a radius

Nucleation of a dislocation loop from a blunted crack tip

Following the analysis of Rice and Thomson 131, we identify three principal contributions to the overall free energy change in the highly stressed crack tip system, due to the nucleation of a dislocation half loop. These are: the dislocation line energy, Us.,r, incorporating an image term by considering a whole loop rather than a half loop; a ledge energy, Utedger due to the surface step: and finally, U,,,,, due to the work done by the local stress field during the activation process. These terms are U&

=

fib’

R

0 ..“I

b

(2-v) ___

S(1 -v)

ln 8(Rlb) ez(rc.b) 1_1

641)

(A8)

where the term 8/e2 in the argument of a logarithmic term of equation (Al) was set equal to unity for convenience. As a first step, we determine the critical configuration under constant stress (n, = constant) by finding the condition for a maximum of the expression in equation (A8), i.e.

5,, = %l2%, at a stress level corresponding

(Al@

to

%I = ? exp (u,/u&

(All)

We now adjust AG to make it vanish at this stress level. This leads to a new form of the free energy expression that is AC =~b’(u,[ln{

+u,[ +3xp

- (u,i’,.

(A12)

In this new form, we consider lower stress levels by n, = Bu,, and expand the free energy expression by a Taylor series around

the critical

configuration

given by equation

(AlO) and (Al 1) where AG = 0. This gives to third order terms a new and more convenient form for the free energy

196

ARGON:

BRITTLE TO DUCTILE

TRANSITION

expression that can be probed more readily for new critical configurations under lower stress levels. This expression is AG = pb3g([) = ~~3[u,,~2(1 - /I) - i F ([ - [,)3]. (A13) cm The new critical configuration straight-forward differentiation,

can now be obtained by i.e.

IN CLEAVAGE

FRACTURE

stress as r = i,, K2 -B) +

JFiF=l.

(A151

Finally, sustitution of this result into the free energy expression of equation (A13) gives the activation free energy that is desired {[(2 -/I) + d-l2

(A14) -f[(l _~ -/I)+J_]3}. This upon substitution of equation (AlO), gives the new (larger of the two roots) critical condition under the lower

_

(A16)

This is what we have quoted in the text as equation (IS).