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The propagation of cracks by cavitation: A general theory
OVERVIEW THE
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PROPAGATION OF CRACKS BY CAVITATION: A GENERAL THEORY D. S. WILKINSON
Department of Metallurgy and Materials Science. McMaster University, 1380 Main St. W.. Hamilton. Ontario, L8S 4M 1 Canada and V. VITEK
Department of Materials Science and Engineering. University of Pennsylvania, 3231 Walnut St.. Philadelphia. PA 19104. U.S.A.
Abstract-In many materials. both ductile and brittle. cracks propagate by the nucleation and growth of cavities in advance of the crack tip. During stable crack growth. a steady-state can be achieved in which the size of cavities depends only on their position ahead of the crack tip. In the present paper we h.tv: modelled this process in a general way. We assume that the stress field ahead of the crack tip can be determined independently of the cavitation process. We then use general parametric equations for the variation of local stress with position ahead of a crack tip [cr = (4 .6’]. and for the growth veloctty of cavities as a function of local stress (c = &v”), to determine the average rate of crack extension. I: rs shown that these equations are of the right form to be used with established models of plastic relaxa;isn ahead of a crack tip and of cavity growth. We find that the nucleation process influences the rate of crack growth only if the parameter, x/?, is less than I. TWO approaches to cavity nucleation are ccnsidered. and their affect on crack growth assessed. Finally. several examples of the use of the model ore presented for both ductile and brittle materials, in which cavities grow by a variety of mechanisms involving both diffusion and plasticity. A large range of crack growth behaviour is predtcted. and we tnd that many of the published models for creep crack growth. fit into our solution as spectal cases. R&sum&-Dans de nombreux materiaux. ductiles ou fragiles. les fissures se propagent par germination et croissance de cavites devant I’extremite de la fissure. Au tours de la croissance stable dune cavitt. un regime permanent peut s’etablir. dans lequel la taille des cavitis ne depend que de lettr position en al-ant de I’extrimite de la fissure. Dans cet article. nous presentons un modele tris general pour ce phenomtne. Nous supposons que Ton peut determiner le champ de contrainte en avant de I’extrimite de la fissure indipendamment de la cavitation. Nous utilisons alors des equations parametriques getterales pour ia variation de la contrainte locale en fonction de la position en avant de I’extremtte de la hssttre [u = (.+Is)‘] et pour la vitesse de croissance des cavites en fonction de la contrainte locale (L‘= r$a”t_arin de determiner la vitesse moyenne de propag,ation de la fissure. Ces equations ont une forme adequate pour Otre utilisee avec des mod&es bien ttablts pour la relaxation plastique en avant de l’extremiti dune fissure et pour la croissance dune cavite. La germination n’influence la vitesse de croissance dune fissure que si le parametre zfi est inferieur a 1. Nous considerons deux approches pour la germination dune cavite et nous pricisons leur influence sur la croissance dune fissure. Enfin, nous prisentons plusieurs exemples d’application du modtle a des mattriaux ductiles et fragiles dans lesquels les cavites croissant par differents micanismes mettant en jeu a la fois la diffusion et la plasticite. Nous prevoyons une large gamme de croissance des fissures et nous trouvons que la plupart des modeles de croissance des fissures de Huage deja publies apparaisxnt comme des cas particuliers de notre solution. Zusammenfassung-In vielen duktilen und sprijden Materialien breiten sich Risse durch die Keimbildung und das Wachstum von Hohlrlumen vor der RiBspitze aus. Wahrend des stabilen RiBwachstums kann ein stationirer Zustand erhalten werden, bei dem die HohlraumgriiGe nur von Abstand des Hohlraumes von der RiBspitze abhangt. In dieser Arbeit wird dieser Prozess mit einem Model1 sehr allgemein beschriebon. Wir nehmen an. da8 das Spannungsfeld vor der RiBspitze unabhangin von der Hohlraumbildung bestimmt werden kann. Mit allgemeinen parametrischen Gleichungen fur die .%ndcrung der lokalen Spannung als Funktion des Abstandes von der RiDspitze [a = (XX)‘] und fur die Wachstumsgeschwindigkeit der Hohlraume als Funktion der lokalen Spannung fc = 4~“) wird die durchschnittliche Ausbreitungsgeschwindigkeit des Risses bestimmt. Diese Gleichungen haben w-ie gezeigt wird, die richtige Form fur die Anwendung im Rahmen bekannter Modelle der plastischen Relaxation vor der RiDspitze und des Hohlraumwachstums. Wir finden. daB der Nikleationsprozees die RiBwachstumsgeswindigkeit nur beeinfluBt, wenn der Parameter zfl kleiner als 1 ist. Die Keimbildsng der Hohlraume wird mit zwei Nlherungen behandelt und auf ihren EinHuB auf das RiBwachstum bin gepriift. SchlieDlich wird die Anwendung des Modelles demonstriert mit verschiedenen Beispielen fLir duktile und spriide Materialien, in denen Hohlrlume mit unterschiedlichen Mechanismen einschlieDlich
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Diffusion und Plastizitlt wachsen. Es wird ein umfassender Bereich dcs RiDwachstums beschrieben. Viele der publizierten Modelle fir RiDwachstum Ghrend des Kriechens sind in unserer Liisung als SpezialMlle enthalten.
1. IKI’RODUCI-ION Catastrophic failures of structural materials usually occur by the propagation of cracks, which originate either at pre-existing flaws or at other regions of high stress concentration. A crack may propagate either in a fast mode, in which case the crack might move through a sample or a component in a fraction of a second. Or, it may propagate by a slow mode. taking perhaps a number of years before failure finally occurs. The former case. encountered in both brittle and ductile failures, occurs commonly in tests under rising load, and the propensity of the material to this type of fracture is usually characterized by its fracture toughness. Slow crack growth is most commonly observed under cyclic loading. However, it also takes place under static loading, both at ambient and elevated temperatures. Because such cracks can grow at stress intensity factors much lower than those determined by the fracture toughness [I, 21, their existence in practical situations may be particularly dangerous. At ambient temperatures, slow crack growth usually occurs as a precursor to fast fracture [2] and it proceeds in either a transgranular or an intergranular manner. The physical mechanism involved is often the same as in fast ductile fracture. In particular, cavities are formed ahead of the crack (e.g. at manganese sulphide inclusions in steels [3]). These then grow in a ductile manner and link up with the main crack. Creep crack propagation at elevated temperatures, has been observed in several types of steels [ClO], nickel-base super alloys [I, 11,121 as well as in some ceramics [13]. It is also associated with cavity formation and growth. The crack path is usually intergranular and cavities grow either by diffusion, plasticity or both. Creep crack extension driven by the diffusional growth of cavities ahead of the crack has recently been modelled by a number of authors. These studies vary in their assumptions, with regard to both the mechanism of cavity growth, and the way in which the stress is relaxed in the vicinity of the crack tip. In several studies [ 14-163, growth laws for individual cavities which depend linearly on the local stress ahead of the crack have been used. These are based on a model of cavitation [17-193 in which one assumes that material is deposited uniformly over the ligament between the cavities. However, different assumptions have been made in each of these studies regarding stress relaxation, and the number of cavities which grow simultaneously ahead of the crack tip. For example, Raj and Baik [14] consider a crack in an elastic solid, and calculate the stress relaxation contributed by the growth of the cavities. The number
of cavities which grow ahead of the crack tip is determined by assuming that a critical stress needs to be achieved for nucleation of cavities. The crack is supposed to advance when the cavity nearest to the crack tip reaches a critical size. On the other hand, Riedel [lj] assumed a fully relaxed stress due to plasticity in the vicinity of the crack tip, while Bassani [16] considered the time-dependence of the stress relaxation. In both studies. transient effects were taken into account. An alternative description of creep crack growth has been developed by Vitek and Wilkinson [ZO]. Only one cavity. that nearest to the crack tip, is assumed to grow (by diffusion) independently of other cavities. i.e. material deposition varies with distance from the cavity. As a result. cavity growth no longer depends linearly on the stress, but on some higher power of stress. Because of the large stress-sensitivity of the growth laws, the assumption that only the cavity nearest the crack tip grows at a significant rate is reasonable, as will be seen later in this paper. Because of the proximity of the cavitation process to the crack tip, no long-range stress relaxation is assumed and the elastic field of the crack-cavity system [30] is used to determine the cavity growth rate. The ductile growth of cavities ahead of a crack has also been studied [21,22]. and applied to creep crack growth [I$ 16,23,24]. Nix et nl. [23,24] assume that a very large (effectively infinite) number of cavities grow in the elastic stress field of a crack. More realistic models, in which ductile cavity growth is assumed to occur within a plastic creep zone ahead of the crack tip, have been produced by Riedel [lS] and Bassani [16]. They use the same representative cavity model, and the same assumptions with regard to crack advancement, as in their models for diffusional cavitation. In the present paper a comprehensive procedure is developed for modelling stable crack growth by cavitation when, in general, iV cavities are growing simultaneously ahead of the crack tip. This procedure has been developed from the study of a special case. in which the crack is assumed to be embedded in an elastic medium and cavities grow by diffusion [25]. However, that development is not applicable when applied to cavity growth laws which are non-linear in stress. The approach developed here applies generally for any power dependence of the rate of cavity growth on the local stress. In addition, no assumptions need initially be made about the stress distribution, except that it decays as a power of the distance ahead of the crack, and that it is time-independent (i.e. we are treating steady-state crack growth only). The result of our analysis is a
&‘Il_KIYjOX
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PROPAGATION
general equation for crack growth. It contains the effect of cavity nucleation. which determines the number of cavities that grow simultaneously; the dependence on the mechanism of cavity growth: and a dependence on the stress field in the vicinity of the crack tip. We then consider a number of examples invoiving different conditions for cavity nucieation, different mechanisms for cavity growth tinvolving both diffusion and plasticity) and different types of crack tip stress field.
2. THE CRACK GROWTH
MODEL
The model of crack growth used in this work is shown schematically in Fig. 1. The cavities are assumed to nucleate ahead of the crack and grow under the effect of a crack stress field which varies with separation from the crack tip. The centers of the cavities are separated by a fixed distance c which may be identified with the averaging spacing of cavity nucleation sites (e.g. inclusions or large carbides). When the cavity nearest the crack tip grows to a critical half-length. I,. it is assumed to link up with the main crack causing the crack the crack length to increase by c. While in thick specimens the crack can always be regarded as infinite in one direction the cavities are generally three dimensional posessing a spherical or penny-shaped form. Nevertheless, in this model we shall regard both the crack and the cavities as infinite in the direction of the crack front so that the problem studied is two dimensional. This corresponds. approximately to a consideration of rows of cavities parallel to the crack front and as shown in various studies of diffusional cavitation. the two and three dimensional cases differ only by numerical factors [X-29]. The modelling of crack growth requires that we understand: (if the spatial and possibly time dependent variation of the stress field ahead of the crack and Iii) the process of cavity growth in an arbitrary local stress field. In general. of course. the stress ahead of a crack is also affected by the presence of cavities
OF CRACKS BY CAVIfTfON
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and thus the stress analysis and cavity growrh are not strictly separable probiems. There are two aspects to this. First. the deposition of matter by diffusion from the cavity relaxes the local stresses. which may slow the rate of cavitation near the crack tip. This is of concern at high temperatures. Raj and Baik [13] have considered this effect. and their result is compared with ours in Section 6. More generally, the simpte presence of cavities may alter the local stresses. However. it has been shown [30] that in an elastic body, the average stress in the ligament between a crack and a cavity positioned ahead of the crack tip. is essentially constant. until the cavity is nearly linked up with the crack. This means that the cavity is loaded by the same stress field as that of the crack alone until the ligament between the crack and the cavity is considerably smaller than the cavity size. We shall, therefore, assume that the stress ahead of a crack can be determined without considering the presence of the cavities. In the present study. we have also neglected any possible time dependence of the stress field ahead of the crack, dealing instead with stress distributions which depend only on the position ahead of a crack tip. This is a good approximation provided the rates of plastic zone formation and of the cavity growth are appreciably different. When the plastic zone develops more slowly than the growth of cavities. the stress held ahead of the crack is simply the elastic field as determined by K, the stress intensity factor, i.e. o = Ki;(Znsf, where .Yis the distance from the crack tip. If the rate of the growth of the plastic zone is faster than that of the cavities, the corresponding plastic stress field should be applied. For the case of power hardening materiais (6 5 Go, where m is the hardening exponent). this field [3 I.321 depends on the distance from the crack tip as .x-l lrn- ‘I. It has been shown recently [33] that the stress field ahead of a crack in a power-law creeping material depends on .Yin the same way when m is replaced by the creep component m, in the relation i -- P=. in the following. we shall assume that the stress at a distance x from the crack tip is gisen by
where both A and r are positive and il is a function of external loading, crack length and specimen geometry. (This expression encompasses all the possibilities mentioned above.) Thus, the nth cavity removed from the crack (Fig. 2) is loaded by a local stress
-30
0000 k--C-+
Fig. I. A crack (half-length a) loaded in tension by a farfield stress G_ propagates by the growth of cavities ahead of it. Under steady state conditions. the length of each cavity 3, depends only on its distance from the crack tip.
G, = (;l,'C?l)'.
(II
The growth of cavities under the effect of the local applied stress proceeds either by diffusion, plastic flow or some combination of the two. A number of models have been developed for these mechanisms [e.g. 17-19, 34-361. Each predict the dependence of the rate of cavity growth on the applied stress. cavity size and spacing. Since they generally result in a power dependence on the stress, the rate of growth of the nth
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cavity can then be written as dl c, = $ = &I,. c)a$
(2)
where f, is the half-length of the cavity (Fig. 2) and fi is a positive number. 4 is a function of the cavity size and cavity spacing as well as various material parameters (such as diffusion coefficients). Its specific form depends on the mechanism considered. It should be noted that for small cavities and/or low stresses the sintering stress cron,due to surface curvature should be included in equation 2. This would lead to a growth law of the form [ 19,343
While we have not treated this type of law in the following development, it is often possible to deal with limiting cases for which growth laws conforming to equation 2 result (see also Section 5). 3. DERIVATION OF THE RATE OF CRACK GROWTH Consider an array of N cavities. spaced a distance c apart, growing in the stress field of a macroscopic crack of length a (Fig, 1). A steady-state form of crack growth occurs when the number of cavities :V, which grow simultaneously ahead of the crack, remains constant as the crack advances.* In this case the cavity size, the focal stress loading the cavities, and thus the cavity growth rate, are determined solely by the position of the cavity relative to the crack tip. This is implicitly assumed in equation I. Whenever the cavity nearest to the crack tip grows to a length, IO = 21, sufficient for linkage to occur, the crack advances by c. According to the steady-state condition, a new cavity must then be nucleated with a length, 21zi= 21,, a characteristic length for nucleated cavities. At this point in time there are :V cavities ahead of the crack, with half-lengths, I,, I,, I,, . . . , 1,~. After a time interval At, the cavity nearest to the crack tip becomes large enough for linkage, and the whole process is repeated. During this time interval each cavity grows from the length it has at the beginning of the time interval, i.e. 21,, to that of its neighbor nearer the crack at the same time, i.e. 21,_ 1 (Fig. 2). The average rate of crack growth by this process is thus c C’=-$
Qn t
S-1 t
Fig. 2. The n”’ cavity is loaded by a locrtl stress 6,. and arows at velocity c, along the grain boundary. During the bme interval At, this cavity grows from a Iength of 21, to 21,_ r,
in a set of N equations At
+ Strictly this is true only if the loading does not change as the crack advances (e.g. constant-K specimens under K-control). In general, so long as the loading parameter A changes slowly as the crack grows, a steady state is achieved in all specimen geometries.
. .V.
(4
These are N simultaneous equations for 1,. I&..., &_ 1 and AL The initial and final cavity lengths, I0 and Is, are known since rhey are determined separately according to the physics of the linkage and nucleation mechanisms, respectively. They therefore represent the boundary conditions of the problem studied. Using equations 1 and 2, equation 4 can be written as At = A-=~~=%‘~[,f-(f~_it c) - f(l,,, c)]. n=l
.2,...,iv
(5)
where
f(f,C) =
I -5 4(L
4
The function~(~, c) can be evaluated. so long as I$(/,c). which depends on the specific cavity growth model considered, is known. Examples will be presented below. The set of equations 5, does not need to be solved explicitly to determine At. Instead, the equations can be summed, giving
(31
The time interval AL can be determined for each cavity from the growth velocity (equation 2). resulting
n = I,.
=
=
~-‘5C’5[f(lo, c) -
f(1.c c,-J.
(7)
Thus, the solution for Af depends on the initial and final cavity sizes, and the form of the cavity growth equation (which determines the function f). However we still need to evaluate the summation of the Ieft side of equation 7. To do this we use the foliowing approximate formula: k-7
z
&
[(k
+ I)‘-:‘-
(k - I,‘-:‘I.
(8)
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1727
This is valid to a very good approximation when k > 1. Applying this equation to n-‘@, we obtain
x [(.V +
I)‘--‘”
+ .vl-=@ - >*-xtl -
1-j.
(9) Thus Ar can be readily evaluated and the rate of crack growth determined using equation (3). For S > I. we can write (.V + I)’ -‘s 2 ,V’ -“fl + (1 - rfi):V-‘” and the rate of crack growth is then
Fig. 3. The function iit$. JV).as definrd by equation 11 is sensitive. to .V. the number of trowing cavities only if x/‘I< 1. For :V = 1. II is practically constant. varying between I and I.5 for all vaiues of z&
where
For @3-
1, this reduces to the limit h(l..V)=
1-~lo?cloSi~.
(lfb)
The dependence of h on up and N can be seen in Fig. 3. It is instructive to calculate the crack growth rate for the case of .V = 1. This corresponds to the situation when a single cavity grows ahead of the crack from I, to a critical Iength ic. Equation (4) is then used directly, in conjunction with equations (I)-(3) to obtain an equation for ci with &’= I:
applied to nucleation associated with the brittle fracture of the particle or particle-matrix decohesion. Another example is nucleation associated with a local clustering of vacancies. as proposed by Raj and Ashby [ lg.371 for creeping solids. The latter mechanism can be shown [3SJ to operate only near stress concentrators which. at nucleation. reach a critical local stress. Since the stress applied to the nth cavity is given by equation 1. the maximum number of cavities. N, which can exist if a critical stress. G<.is needed for their nucleation. is
(13)
(12) If instead, we use equation LO. and evaluate the function hfr& .V) directly for X = 1, we find that h ranges from 1.0 for x,5 = 0, to I.5 as rb+ X. Thus, equation 10 is a good approximation for the rate of crack growth even when a small number of cavities (S - 1) grows ahead of the crack. Note that the effect of multiple cavity growth is entirely contained in the function ~l(%~.X). This function is insensitive to the value of N for $ > t and depends on !V logarithmjcaliy for rfi = 1. On the other hand for #3 <: 1, h increases as a power of 5 and thus the rate of crack growth is sensitive to the number of cavities growing simultaneously. This number is not known o priori but depends on the physics of the nucleation process and the specific problem considered. Two different approaches will be considered in the next section.
1. SC’CLEATION
OF C.%\-‘IWES
Since cavities usually nucleate on second phase particles. one of the simplest possible criteria for their nucleation is that a critical stress. G,, needs to be attained at the nucleation site. This criterion can be
This can be substituted into equation (10) to obtain the crack growth rate. In particular. for .V >>I and r/? < 1 (the only case in which the .V-dependence is significant) we can write (14) and A ii=
(1 _
r/j)G:’
-2m
[f(l,. cl -
_fcr,. c,] .
(lj)
Note that the rate of crack propagation depends upon the external loading only through a linear dependence on A. while the mechanism of cavity growth determines the dependence on the separation of nucleation sites. When intergranular creep crack growth occurs at elevated temperatures. cavitation is commonly observed on suitably oriented grain boundary facets ahead of the crack [ll. 39.401. In addition. concomitant grain boundary sliding is often seen. Such observations lead [25] to the suggestion that sliding may
WILKINSON
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PROPAGATION OF CRACKS BY CAVITATION
The average crack growth rate can now be written as b=_,_
d
(d
.A’“d’-ad
rfl 1 -
(20) 2s
f&.c,
-f(l,.c)’
This result differs only by the factor r/3. from equation 16, which was derived using the steady-state model, and X = Iv’*_For this model of cavity nucleation, the functional dependence of the rate of crack growth upon external loading (contained in A) is determined solely by the stress dependence of the mechanism of the growth of cavities (cf. equation 10). Since it was assumed that cavities nucleate simultaneously on whole grain facets. the rate of crack growth now also depends on the grain size and because ~$3< 1, the model indicates that cracks should grow more rapidly in large grain size material,
1
5. IMODELS OF INTERGRAXLLAR CREEP CRACK GROWTH
B
Fig. 4. A model for nucleation. in which cavities are nucleated simultaneously on a grain boundary facet. result in the simultaneous nucleation of all cavities on a single grain boundary facet. fn this case no further nucleation of cavities occurs while the crack grows across this facet, and the maximum number of growing cavities is N, = d/c, where d is the grain size (see Fig. I). Since it is reasonable to assume that d D c, it follows that No x 1. A crude approximation for the crack growth rate can then be found by using N = N, in equation 10. Thus (when r/3 < 1)
However, since no nucleation of cavities takes place on each facet as the crack grows across it, the number of cavities ahead of the crack decreases as the crack extends. To take this into account we evaluate in more-detail the time td, required for the crack to propagate across the facet. The crack always grows in steps of length c, each jump taking a time interval hr..,. as determined by equation 7, when N is the current number of cavities between the crack tip and the far side of the facet. Thus
The general description of crack growth driven by cavitation ahead of the crack tip, which we have just developed, can be applied to specific cases of creep crack growth in which cavities grow by diffusion and by plastic creep. By considering specific physical mechanisms we can evaluate the functionf‘(l, c) which appears in the general expressions for the rate of crack growth and principally determines its dependence on the separation of cavities. Inserting these functions into equations 10, 15 or 16. as physically appropriate, we obtain the corres~nding rates of creep crack growth. (a) Consider first, the case of difihmcd cacitation, for which we assume, as in previous work [14-161, that the rate of material deposition between the cavities is uniform and thus the rate of cavity growth depends linearly on the locat stress. A different situation, in which individual cavities grow independently of each other, and the rate of material deposition varies with the distance from the cavity. will be discussed elsewhere [41]. The rate of cavity growth due to grain boundary diffusion is, in the two-dimensional case, given by
(17) Using equation (14) (which is a good approximation even for N J: 1 unless z/I is very close to unity), we obtain Ar,v = A-‘flc=@(l - d)Cf(fc, cl - jv,, c)3N=g-‘. To perform the summation
(18)
in equation 17 we again make use of equation 8 with No >z I, to get ld
Here, y1 is the surface energy and i. = 8$I&G/kT, where D, is the grain boundary diffusion coefficient, Sb the diffusional width of the boundary, R the atomic volume, G the shear modulus, k Boltzmann’s constant and T the absolute temperature. r, is the radius of curvature at the cavity tip. & is determined by the geometry of the cavity, such that the area occupied by the cavity is a,&. If surface diffusion is rapid and thus the cavity maintains an equilibrium shape (Fig. 5), then a* = MO)
(22a)
WILKINSON
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where g(N) =
‘!% - sin t, cos 0)
(22b)
sin’ tl
@is the equilibrium dihedral angle at the cavity-grain boundary interface. If surface diffusion is slow and the cavities are, therefore. crack-like & = X,. where d, is the cavity width (Fig. 5). The first term in equation 21 represents the effect of the applied load while the second term shows the effect of the sintering stress. kVe consider two cases of diffusional cavitation. First. when cnriries possess eqctilibriam sktrpr. r, is large and the sintering term in equation 21 can be neglected. Second. when cacirirs nre crack-like. d, becomes growth rate dependent [27] and the sintering term significant. (i) For equilibrium cavities equation 21 simplifies to 6.
rn=go l,(c This has the form required fi = 1 and &l.
C)
=
--!L
C
EOUILIBRIUM
CAVITY
CRACK-LIKE
CAVITY
Fig. 5. If surface diffusion is rapid. a cavity c3n maintain an equilibrium shape with constant curvature tlcft). If however. surface diffusion is too slow. the cavity becomes crack-like in shape (right).
limiting forms. When
then
o-”
21,Yz-’ by equation
g(O)G i(c-
(28a)
(2) with and when the opposite inequality is valid
c
(21) rfl
h
Inserting #J into equation (6) yields f(l,C) = g
(+c’l” - ;c1-’ -+ I”).
f28b) (2s)
We will assume that the cavities, as nucleated are much smaller than when they achieve the critical length for linking with the crack (i.e. 1, >>I,). Thus, ,f(L c) >>jf(r,.c). Furthermore, the maximum possible cavity length corresponds to actual contact with the crack tip (i.e. I, = c 2) and it is reasonable to assume that !_will be close to that value. Using these assumptions 126)
Both of these limits have the form required by the present analysis with p = 3 2 in the first case and /? = 3 in second. Using the same conditions on I, and I, as previously we find that f(/,, c) _ /(I,. c) = [20 - cos @13” 24, 7;. X
(29a)
in the first case, and [2(1 - cos 8)]“’ &D,
(ii) In the case of crack-like cnciiies. the shape of the cavity is controlled by the rate of surface diffusion in the region near the cavity tip. Chuang and Rice [27] have show-n that both n, and r, then depend on the rate of cavity growth and [ 19.321
CG3
f29b) in the second. fb) We next consider the case where cavitation is controlled by power-law creep as described by the constitutive equation (30) iO. ~~ and m, are material constants. The rate of cavity growth is given approximately as [34]
where D, is the surface diffusion coefficient and 6, the surface diffusion path width. This equation has two
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and cavity growth develop concomitantly, special considerations are required [33, 16-J The microscopic metallurgical parameters pertinent to the growrh of cavities. including the separation of cavity nucleation sites, are all contained in the function f# of equation 2. Once we arrive at an equation for the rate of crack growth. these parameters are 6. DISCLSSIOS then contained in f(/,, c) - f(r.. c), derived from 4 using equation 6. In general. the crack growth rate In this paper the problem of crack growth due to cavitation ahead of a crack tip has been analysed. A also depends upon the number of cavities N, growing simuhaneously ahead of the crack. This number is general expression for the rate of crack propagation controlled by the ~~~~e~riotzmechanism. In general, A’ has been derived (equation 10) assuming that (it an may be a function of both the loading parameter A unspecified but constant number of cavities, X, grow and metallurgical parameters such as grain size. simultaneously ahead of the crack tip, (ii) the stress cavity separation. etc. Two examples of nucleation field ahead of the crack decreases with the distance criteria have been discussed in Section 4. However, from the crack tip, X, as .Y-* where x > 0. (iii) the rate of cavity growth depends on a power fi > 0 of the the significance of the nucleation process to the overlocal stress loading the cavity. (iv) the cavities are all crack growth rate can vary widely. depending on spaced uniformly, and (v) all the cavities grow by the the parameter r/I. It can be seen from equations 10 same mechanism. Relaxation of the stress ahead of and 11 that for rb > 1 the rate of crack growth is practically independent of .V and cavity nucleation the crack resulting from cavity growth has not been does not si~ni~cantly influence the rate of crack considered. This enables us to treat the development of the stress field ahead of the crack tip and the growth. On the other hand, for rJ < 1, the nucleation of growth of cavities in the stress field, as separable cavities has to be considered when studying crack problems. propagation. The stress field which is appropriate to a given situLet us turn now to some specific cases. for both the ation depends primarily on the macroscopic response of the material to crack loading. This determines the stress field ahead of the crack. and the mechanism of parameters .1 and r in equation 1. Engineering par- cavity growth in a creeping solid. (a) If the effect of plastic creep near the crack tip is ameters such as load, crack length and specimen geometry are al1 contained in the parameter A, while x is negligible, and the material is essentially hear elastic only affects the rate (2 = 1'2). then cavity nucieation primarily influenced by the degree of plasticity near of crack growth for /3 c 2. In this type of material. the crack tip. When the extent of the plastic deformacavities are expected to grow by diffusion. For tion near the crack tip is negligible in comparison cavities (5 = 1) this with the size of the zone of cavitation, then the field rorrnderi (equilibrium-shaped) 10 and 26) that d _ ahead of the crack can be taken as that of a crack in means (using equations KNi :ce3. while for crack-like cavities in the limiting an elastic body case given by equation 28a (p = 1.51, then A = K’12n, x = I;2 (33) ci _ K1.5N1~4~-q”, The actual dependence of ci on K and c depends on the nucleation mechanism which where K is the crack stress intensity factor. If. on the influences N. If we use a critical stress criterion. as other hand, cavitation is largely confined to the plasdiscussed earlier, then ri c K”cb3 for rounded cavitic zone of a material for which the plastic behaviour is described by the constitutive law. & = (~//a~)“. ties. and d c K2cv5 ’ for the cracklike cavities. This result for rounded cavities agrees with both the Kwhere eO, ITS and m are constants, then the approand c-dependence found by Raj and Baik [14]. in priate stress field ahead of the crack is of the HRR their model for steady-state crack growth. In that type [31,37] study, equilibrium shaped cavities only are considered, and the same assumptions regarding a critical (34) stress for cavity nucleation are made. In addition. m-+1 ORI stress relaxation due to cavity growth is considered. It where J is the J-integral and I, a constant given in would thus appear that relaxation. while affecting the Ref. [31]. Finally, if the plastic zone is developed size of the cavitation zone. does not alter the depenthrough power-law creep deformation, then a stress dence of crack growth on K. field analogous to the HRR fields develops [333 and The second nucleation criterion u’e may use assumes that N is related directly to the grain size of cT$c* 1 material, 8 = die, in which case we get 2-p A=(35) the . m, + I co 1% a _ Kdi”cm3 and ci _ K3jzd1 ‘cm5 ’ for the two cases of rounded and crack-like cavities, respectively. In this case, crack growth rate is also a function of grain where C* is the path independent integral, analogous to J [43,44]. In this case however, if the plastic zone size.
In this case /3 = m, and
A=$,
x7-i
WILKINSON
,~ND VITEK:
PROP.AGXTION
(b) If the cavitation occurs primarily inside 3 phscreep zonr. then x = 1 (m, + I). and the mechanism of cavity nucleation will affect the rate of crack growth. whenever 0 < m, f 1. Since m, is usu3))y much larger than 2. this means that nucleation cannot be neglected for any mode of diffusional cavitation we have considered, When cavities nucleate at a critical stress ti - C+. If. on the other hand, the number of growing cavities is related to the grain size, then ri _ (C*):. where ; = 1 (m, T 1) in the case of equilibrium cavities. ; = 3 2(m, + 1) for the crack-like cavities in the case of equation 283. and ; = 3 (m, + 1) in the case of equation Bb. Hence the present work show-s that substantially different dependencies on C* may be obtained if different modes of cavity nucleation and growth are considered. If cavity growth is controlled not by diffusion. but by creep deformation. then the mechanism of cavity nucleation affects the rate of crack growth for WI,< 1 1. Consider tvvo cases. First. when the effect of plastic zone formation can be neglected (r = 1’2) nucleation does not affect the rate of crack growth so long 3s tt~, > 2, which is usually the case. This situation has been studied by Nix et (I/. [?I]. They were able to calculate 3 finite rate of crack growth assuming an unlimited number of nucleated cavities. only if no, > 2, consistent with our analysis. We find in this case J . c(K’, c)“* In(c 21,). which agrees with the result of Nix er al. [24]. On the other hand for 171,< 2. a nucleation criterion must be included. When cavities nucleate at 3 critical stress. ir _ K’ In (c 21,). while ic _ ti(K , d)“~,In(c’2/,) if the number of growing cavities is related to the grain size. Ductile cavity growth is unlikely to occur in an elastic solid. Instead. we should consider cavities growing in a plastic creep zone. In this case. rfl = nt, (m, + 1). which is always smaller than 1 and the nucleation of cavities must be considered. If nucleation occurs at 3 critical stress il _ C* In (c 21,). while if the number of cavities is related the grain size. ci _ d(C*,‘ti)‘“~mc- “‘In (c 21,). For m, >> 1, which is common in the creep situation, the difference between the two types of C* dependence on creep crack growth is practically indistinguishable. Furthermore. the dependence of crack growth rate on grain size is extremely weak. We have considered a number of examples using the assumption that cavities nucleate simultaneously on grain boundary facets ahead of the crack tip. This leads. when r/3 < 1. to a dependence of crack growth rate on grain size, of the form ri - d’ -I!‘. Experimentally. Gooch and King [45] find that crack growth rates in 1 Z-CrMoV steel change as 3 crack grows through successive coarse- and fine-grained regions of a weld heat-affected zone. This type of grain size dependence has been generally observed in low alloy steels and fits well with the model we have proposed.
OF
CR.ACKS
BY CAVIT;\TION
1731
REFERESCES
ric
.-lr~,~~\~Iniyrments--This research was supported Arm) Research Office under grant No. 29-81-K-002.5 (V.V.) and by a grant from Imperial
(D.S.CV.).
by the DAAG Gil Ltd.
1. K. Sadananda and P. Shahirnan. J. Enyny .Lfarrr. 7-echnol. loo, 381 [ 197s). 2. I. Milne and G. G. Chell. Elosric-Pfasric Fr~turr (edited by J. D. Landes. J. A. BegIcy and G. .A. Clarke). o. 3%.
1732
WILKINSON
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VITEK:
PROPAGATION
by R. M. Pelloux and N. S. Stoioff). p. 46 TMS-AIME 11980). 39. D. S. Wilkinson, K. Abiko. N. Thyagarajan and D. P. Pope, Merall. Trans. 1 IA. 1827 f1980). 40. C. L. Jones and R. Pilkington, Metall. Trans. 9A, 865 f 1978). 11. T. Takasugi. D. S. Wilkinson and V. Vitek. to be published.
OF CRACKS BY CAVITATION
42. T. Takasugi and V. Vitek. &ferall. Trans. 12A, 659 (1981). 43. J. D. Landes and J. A. Begley. .Llrchanics of Crack Growth. p. 178. ASTM STP 590 (1976). 44. K. M. Nikbin, G. A. Webster and C. E. Turner, Cracks and Fracture, p. 47. ASTM STP 601 (1976). 45. D. J. Gooch and B. L. King, Metals. Technol., p. 405 (1979).
24
PROPAGATION OF CRACKS BY CAVITATION: A GENERAL THEORY D. S. WILKINSON
Department of Metallurgy and Materials Science. McMaster University, 1380 Main St. W.. Hamilton. Ontario, L8S 4M 1 Canada and V. VITEK
Department of Materials Science and Engineering. University of Pennsylvania, 3231 Walnut St.. Philadelphia. PA 19104. U.S.A.
Abstract-In many materials. both ductile and brittle. cracks propagate by the nucleation and growth of cavities in advance of the crack tip. During stable crack growth. a steady-state can be achieved in which the size of cavities depends only on their position ahead of the crack tip. In the present paper we h.tv: modelled this process in a general way. We assume that the stress field ahead of the crack tip can be determined independently of the cavitation process. We then use general parametric equations for the variation of local stress with position ahead of a crack tip [cr = (4 .6’]. and for the growth veloctty of cavities as a function of local stress (c = &v”), to determine the average rate of crack extension. I: rs shown that these equations are of the right form to be used with established models of plastic relaxa;isn ahead of a crack tip and of cavity growth. We find that the nucleation process influences the rate of crack growth only if the parameter, x/?, is less than I. TWO approaches to cavity nucleation are ccnsidered. and their affect on crack growth assessed. Finally. several examples of the use of the model ore presented for both ductile and brittle materials, in which cavities grow by a variety of mechanisms involving both diffusion and plasticity. A large range of crack growth behaviour is predtcted. and we tnd that many of the published models for creep crack growth. fit into our solution as spectal cases. R&sum&-Dans de nombreux materiaux. ductiles ou fragiles. les fissures se propagent par germination et croissance de cavites devant I’extremite de la fissure. Au tours de la croissance stable dune cavitt. un regime permanent peut s’etablir. dans lequel la taille des cavitis ne depend que de lettr position en al-ant de I’extrimite de la fissure. Dans cet article. nous presentons un modele tris general pour ce phenomtne. Nous supposons que Ton peut determiner le champ de contrainte en avant de I’extrimite de la fissure indipendamment de la cavitation. Nous utilisons alors des equations parametriques getterales pour ia variation de la contrainte locale en fonction de la position en avant de I’extremtte de la hssttre [u = (.+Is)‘] et pour la vitesse de croissance des cavites en fonction de la contrainte locale (L‘= r$a”t_arin de determiner la vitesse moyenne de propag,ation de la fissure. Ces equations ont une forme adequate pour Otre utilisee avec des mod&es bien ttablts pour la relaxation plastique en avant de l’extremiti dune fissure et pour la croissance dune cavite. La germination n’influence la vitesse de croissance dune fissure que si le parametre zfi est inferieur a 1. Nous considerons deux approches pour la germination dune cavite et nous pricisons leur influence sur la croissance dune fissure. Enfin, nous prisentons plusieurs exemples d’application du modtle a des mattriaux ductiles et fragiles dans lesquels les cavites croissant par differents micanismes mettant en jeu a la fois la diffusion et la plasticite. Nous prevoyons une large gamme de croissance des fissures et nous trouvons que la plupart des modeles de croissance des fissures de Huage deja publies apparaisxnt comme des cas particuliers de notre solution. Zusammenfassung-In vielen duktilen und sprijden Materialien breiten sich Risse durch die Keimbildung und das Wachstum von Hohlrlumen vor der RiBspitze aus. Wahrend des stabilen RiBwachstums kann ein stationirer Zustand erhalten werden, bei dem die HohlraumgriiGe nur von Abstand des Hohlraumes von der RiBspitze abhangt. In dieser Arbeit wird dieser Prozess mit einem Model1 sehr allgemein beschriebon. Wir nehmen an. da8 das Spannungsfeld vor der RiBspitze unabhangin von der Hohlraumbildung bestimmt werden kann. Mit allgemeinen parametrischen Gleichungen fur die .%ndcrung der lokalen Spannung als Funktion des Abstandes von der RiDspitze [a = (XX)‘] und fur die Wachstumsgeschwindigkeit der Hohlraume als Funktion der lokalen Spannung fc = 4~“) wird die durchschnittliche Ausbreitungsgeschwindigkeit des Risses bestimmt. Diese Gleichungen haben w-ie gezeigt wird, die richtige Form fur die Anwendung im Rahmen bekannter Modelle der plastischen Relaxation vor der RiDspitze und des Hohlraumwachstums. Wir finden. daB der Nikleationsprozees die RiBwachstumsgeswindigkeit nur beeinfluBt, wenn der Parameter zfl kleiner als 1 ist. Die Keimbildsng der Hohlraume wird mit zwei Nlherungen behandelt und auf ihren EinHuB auf das RiBwachstum bin gepriift. SchlieDlich wird die Anwendung des Modelles demonstriert mit verschiedenen Beispielen fLir duktile und spriide Materialien, in denen Hohlrlume mit unterschiedlichen Mechanismen einschlieDlich
1714
WILKINSON
,440
VITEK:
PROPAGATION
OF CRACKS BY CAVITATION
Diffusion und Plastizitlt wachsen. Es wird ein umfassender Bereich dcs RiDwachstums beschrieben. Viele der publizierten Modelle fir RiDwachstum Ghrend des Kriechens sind in unserer Liisung als SpezialMlle enthalten.
1. IKI’RODUCI-ION Catastrophic failures of structural materials usually occur by the propagation of cracks, which originate either at pre-existing flaws or at other regions of high stress concentration. A crack may propagate either in a fast mode, in which case the crack might move through a sample or a component in a fraction of a second. Or, it may propagate by a slow mode. taking perhaps a number of years before failure finally occurs. The former case. encountered in both brittle and ductile failures, occurs commonly in tests under rising load, and the propensity of the material to this type of fracture is usually characterized by its fracture toughness. Slow crack growth is most commonly observed under cyclic loading. However, it also takes place under static loading, both at ambient and elevated temperatures. Because such cracks can grow at stress intensity factors much lower than those determined by the fracture toughness [I, 21, their existence in practical situations may be particularly dangerous. At ambient temperatures, slow crack growth usually occurs as a precursor to fast fracture [2] and it proceeds in either a transgranular or an intergranular manner. The physical mechanism involved is often the same as in fast ductile fracture. In particular, cavities are formed ahead of the crack (e.g. at manganese sulphide inclusions in steels [3]). These then grow in a ductile manner and link up with the main crack. Creep crack propagation at elevated temperatures, has been observed in several types of steels [ClO], nickel-base super alloys [I, 11,121 as well as in some ceramics [13]. It is also associated with cavity formation and growth. The crack path is usually intergranular and cavities grow either by diffusion, plasticity or both. Creep crack extension driven by the diffusional growth of cavities ahead of the crack has recently been modelled by a number of authors. These studies vary in their assumptions, with regard to both the mechanism of cavity growth, and the way in which the stress is relaxed in the vicinity of the crack tip. In several studies [ 14-163, growth laws for individual cavities which depend linearly on the local stress ahead of the crack have been used. These are based on a model of cavitation [17-193 in which one assumes that material is deposited uniformly over the ligament between the cavities. However, different assumptions have been made in each of these studies regarding stress relaxation, and the number of cavities which grow simultaneously ahead of the crack tip. For example, Raj and Baik [14] consider a crack in an elastic solid, and calculate the stress relaxation contributed by the growth of the cavities. The number
of cavities which grow ahead of the crack tip is determined by assuming that a critical stress needs to be achieved for nucleation of cavities. The crack is supposed to advance when the cavity nearest to the crack tip reaches a critical size. On the other hand, Riedel [lj] assumed a fully relaxed stress due to plasticity in the vicinity of the crack tip, while Bassani [16] considered the time-dependence of the stress relaxation. In both studies. transient effects were taken into account. An alternative description of creep crack growth has been developed by Vitek and Wilkinson [ZO]. Only one cavity. that nearest to the crack tip, is assumed to grow (by diffusion) independently of other cavities. i.e. material deposition varies with distance from the cavity. As a result. cavity growth no longer depends linearly on the stress, but on some higher power of stress. Because of the large stress-sensitivity of the growth laws, the assumption that only the cavity nearest the crack tip grows at a significant rate is reasonable, as will be seen later in this paper. Because of the proximity of the cavitation process to the crack tip, no long-range stress relaxation is assumed and the elastic field of the crack-cavity system [30] is used to determine the cavity growth rate. The ductile growth of cavities ahead of a crack has also been studied [21,22]. and applied to creep crack growth [I$ 16,23,24]. Nix et nl. [23,24] assume that a very large (effectively infinite) number of cavities grow in the elastic stress field of a crack. More realistic models, in which ductile cavity growth is assumed to occur within a plastic creep zone ahead of the crack tip, have been produced by Riedel [lS] and Bassani [16]. They use the same representative cavity model, and the same assumptions with regard to crack advancement, as in their models for diffusional cavitation. In the present paper a comprehensive procedure is developed for modelling stable crack growth by cavitation when, in general, iV cavities are growing simultaneously ahead of the crack tip. This procedure has been developed from the study of a special case. in which the crack is assumed to be embedded in an elastic medium and cavities grow by diffusion [25]. However, that development is not applicable when applied to cavity growth laws which are non-linear in stress. The approach developed here applies generally for any power dependence of the rate of cavity growth on the local stress. In addition, no assumptions need initially be made about the stress distribution, except that it decays as a power of the distance ahead of the crack, and that it is time-independent (i.e. we are treating steady-state crack growth only). The result of our analysis is a
&‘Il_KIYjOX
AXJ VITEK:
PROPAGATION
general equation for crack growth. It contains the effect of cavity nucleation. which determines the number of cavities that grow simultaneously; the dependence on the mechanism of cavity growth: and a dependence on the stress field in the vicinity of the crack tip. We then consider a number of examples invoiving different conditions for cavity nucieation, different mechanisms for cavity growth tinvolving both diffusion and plasticity) and different types of crack tip stress field.
2. THE CRACK GROWTH
MODEL
The model of crack growth used in this work is shown schematically in Fig. 1. The cavities are assumed to nucleate ahead of the crack and grow under the effect of a crack stress field which varies with separation from the crack tip. The centers of the cavities are separated by a fixed distance c which may be identified with the averaging spacing of cavity nucleation sites (e.g. inclusions or large carbides). When the cavity nearest the crack tip grows to a critical half-length. I,. it is assumed to link up with the main crack causing the crack the crack length to increase by c. While in thick specimens the crack can always be regarded as infinite in one direction the cavities are generally three dimensional posessing a spherical or penny-shaped form. Nevertheless, in this model we shall regard both the crack and the cavities as infinite in the direction of the crack front so that the problem studied is two dimensional. This corresponds. approximately to a consideration of rows of cavities parallel to the crack front and as shown in various studies of diffusional cavitation. the two and three dimensional cases differ only by numerical factors [X-29]. The modelling of crack growth requires that we understand: (if the spatial and possibly time dependent variation of the stress field ahead of the crack and Iii) the process of cavity growth in an arbitrary local stress field. In general. of course. the stress ahead of a crack is also affected by the presence of cavities
OF CRACKS BY CAVIfTfON
1723
and thus the stress analysis and cavity growrh are not strictly separable probiems. There are two aspects to this. First. the deposition of matter by diffusion from the cavity relaxes the local stresses. which may slow the rate of cavitation near the crack tip. This is of concern at high temperatures. Raj and Baik [13] have considered this effect. and their result is compared with ours in Section 6. More generally, the simpte presence of cavities may alter the local stresses. However. it has been shown [30] that in an elastic body, the average stress in the ligament between a crack and a cavity positioned ahead of the crack tip. is essentially constant. until the cavity is nearly linked up with the crack. This means that the cavity is loaded by the same stress field as that of the crack alone until the ligament between the crack and the cavity is considerably smaller than the cavity size. We shall, therefore, assume that the stress ahead of a crack can be determined without considering the presence of the cavities. In the present study. we have also neglected any possible time dependence of the stress field ahead of the crack, dealing instead with stress distributions which depend only on the position ahead of a crack tip. This is a good approximation provided the rates of plastic zone formation and of the cavity growth are appreciably different. When the plastic zone develops more slowly than the growth of cavities. the stress held ahead of the crack is simply the elastic field as determined by K, the stress intensity factor, i.e. o = Ki;(Znsf, where .Yis the distance from the crack tip. If the rate of the growth of the plastic zone is faster than that of the cavities, the corresponding plastic stress field should be applied. For the case of power hardening materiais (6 5 Go, where m is the hardening exponent). this field [3 I.321 depends on the distance from the crack tip as .x-l lrn- ‘I. It has been shown recently [33] that the stress field ahead of a crack in a power-law creeping material depends on .Yin the same way when m is replaced by the creep component m, in the relation i -- P=. in the following. we shall assume that the stress at a distance x from the crack tip is gisen by
where both A and r are positive and il is a function of external loading, crack length and specimen geometry. (This expression encompasses all the possibilities mentioned above.) Thus, the nth cavity removed from the crack (Fig. 2) is loaded by a local stress
-30
0000 k--C-+
Fig. I. A crack (half-length a) loaded in tension by a farfield stress G_ propagates by the growth of cavities ahead of it. Under steady state conditions. the length of each cavity 3, depends only on its distance from the crack tip.
G, = (;l,'C?l)'.
(II
The growth of cavities under the effect of the local applied stress proceeds either by diffusion, plastic flow or some combination of the two. A number of models have been developed for these mechanisms [e.g. 17-19, 34-361. Each predict the dependence of the rate of cavity growth on the applied stress. cavity size and spacing. Since they generally result in a power dependence on the stress, the rate of growth of the nth
1726
WILKINSON
A?;D VITEK:
PROPA~.~TIO~
OF CRACKS BY CAVITATION
cavity can then be written as dl c, = $ = &I,. c)a$
(2)
where f, is the half-length of the cavity (Fig. 2) and fi is a positive number. 4 is a function of the cavity size and cavity spacing as well as various material parameters (such as diffusion coefficients). Its specific form depends on the mechanism considered. It should be noted that for small cavities and/or low stresses the sintering stress cron,due to surface curvature should be included in equation 2. This would lead to a growth law of the form [ 19,343
While we have not treated this type of law in the following development, it is often possible to deal with limiting cases for which growth laws conforming to equation 2 result (see also Section 5). 3. DERIVATION OF THE RATE OF CRACK GROWTH Consider an array of N cavities. spaced a distance c apart, growing in the stress field of a macroscopic crack of length a (Fig, 1). A steady-state form of crack growth occurs when the number of cavities :V, which grow simultaneously ahead of the crack, remains constant as the crack advances.* In this case the cavity size, the focal stress loading the cavities, and thus the cavity growth rate, are determined solely by the position of the cavity relative to the crack tip. This is implicitly assumed in equation I. Whenever the cavity nearest to the crack tip grows to a length, IO = 21, sufficient for linkage to occur, the crack advances by c. According to the steady-state condition, a new cavity must then be nucleated with a length, 21zi= 21,, a characteristic length for nucleated cavities. At this point in time there are :V cavities ahead of the crack, with half-lengths, I,, I,, I,, . . . , 1,~. After a time interval At, the cavity nearest to the crack tip becomes large enough for linkage, and the whole process is repeated. During this time interval each cavity grows from the length it has at the beginning of the time interval, i.e. 21,, to that of its neighbor nearer the crack at the same time, i.e. 21,_ 1 (Fig. 2). The average rate of crack growth by this process is thus c C’=-$
Qn t
S-1 t
Fig. 2. The n”’ cavity is loaded by a locrtl stress 6,. and arows at velocity c, along the grain boundary. During the bme interval At, this cavity grows from a Iength of 21, to 21,_ r,
in a set of N equations At
+ Strictly this is true only if the loading does not change as the crack advances (e.g. constant-K specimens under K-control). In general, so long as the loading parameter A changes slowly as the crack grows, a steady state is achieved in all specimen geometries.
. .V.
(4
These are N simultaneous equations for 1,. I&..., &_ 1 and AL The initial and final cavity lengths, I0 and Is, are known since rhey are determined separately according to the physics of the linkage and nucleation mechanisms, respectively. They therefore represent the boundary conditions of the problem studied. Using equations 1 and 2, equation 4 can be written as At = A-=~~=%‘~[,f-(f~_it c) - f(l,,, c)]. n=l
.2,...,iv
(5)
where
f(f,C) =
I -5 4(L
4
The function~(~, c) can be evaluated. so long as I$(/,c). which depends on the specific cavity growth model considered, is known. Examples will be presented below. The set of equations 5, does not need to be solved explicitly to determine At. Instead, the equations can be summed, giving
(31
The time interval AL can be determined for each cavity from the growth velocity (equation 2). resulting
n = I,.
=
=
~-‘5C’5[f(lo, c) -
f(1.c c,-J.
(7)
Thus, the solution for Af depends on the initial and final cavity sizes, and the form of the cavity growth equation (which determines the function f). However we still need to evaluate the summation of the Ieft side of equation 7. To do this we use the foliowing approximate formula: k-7
z
&
[(k
+ I)‘-:‘-
(k - I,‘-:‘I.
(8)
.ASD
WILKINSON
VITEK:
PROPAGATION
OF CRACKS BY CAvIT.ATION
1727
This is valid to a very good approximation when k > 1. Applying this equation to n-‘@, we obtain
x [(.V +
I)‘--‘”
+ .vl-=@ - >*-xtl -
1-j.
(9) Thus Ar can be readily evaluated and the rate of crack growth determined using equation (3). For S > I. we can write (.V + I)’ -‘s 2 ,V’ -“fl + (1 - rfi):V-‘” and the rate of crack growth is then
Fig. 3. The function iit$. JV).as definrd by equation 11 is sensitive. to .V. the number of trowing cavities only if x/‘I< 1. For :V = 1. II is practically constant. varying between I and I.5 for all vaiues of z&
where
For @3-
1, this reduces to the limit h(l..V)=
1-~lo?cloSi~.
(lfb)
The dependence of h on up and N can be seen in Fig. 3. It is instructive to calculate the crack growth rate for the case of .V = 1. This corresponds to the situation when a single cavity grows ahead of the crack from I, to a critical Iength ic. Equation (4) is then used directly, in conjunction with equations (I)-(3) to obtain an equation for ci with &’= I:
applied to nucleation associated with the brittle fracture of the particle or particle-matrix decohesion. Another example is nucleation associated with a local clustering of vacancies. as proposed by Raj and Ashby [ lg.371 for creeping solids. The latter mechanism can be shown [3SJ to operate only near stress concentrators which. at nucleation. reach a critical local stress. Since the stress applied to the nth cavity is given by equation 1. the maximum number of cavities. N, which can exist if a critical stress. G<.is needed for their nucleation. is
(13)
(12) If instead, we use equation LO. and evaluate the function hfr& .V) directly for X = 1, we find that h ranges from 1.0 for x,5 = 0, to I.5 as rb+ X. Thus, equation 10 is a good approximation for the rate of crack growth even when a small number of cavities (S - 1) grows ahead of the crack. Note that the effect of multiple cavity growth is entirely contained in the function ~l(%~.X). This function is insensitive to the value of N for $ > t and depends on !V logarithmjcaliy for rfi = 1. On the other hand for #3 <: 1, h increases as a power of 5 and thus the rate of crack growth is sensitive to the number of cavities growing simultaneously. This number is not known o priori but depends on the physics of the nucleation process and the specific problem considered. Two different approaches will be considered in the next section.
1. SC’CLEATION
OF C.%\-‘IWES
Since cavities usually nucleate on second phase particles. one of the simplest possible criteria for their nucleation is that a critical stress. G,, needs to be attained at the nucleation site. This criterion can be
This can be substituted into equation (10) to obtain the crack growth rate. In particular. for .V >>I and r/? < 1 (the only case in which the .V-dependence is significant) we can write (14) and A ii=
(1 _
r/j)G:’
-2m
[f(l,. cl -
_fcr,. c,] .
(lj)
Note that the rate of crack propagation depends upon the external loading only through a linear dependence on A. while the mechanism of cavity growth determines the dependence on the separation of nucleation sites. When intergranular creep crack growth occurs at elevated temperatures. cavitation is commonly observed on suitably oriented grain boundary facets ahead of the crack [ll. 39.401. In addition. concomitant grain boundary sliding is often seen. Such observations lead [25] to the suggestion that sliding may
WILKINSON
1725
AND
VITEK:
PROPAGATION OF CRACKS BY CAVITATION
The average crack growth rate can now be written as b=_,_
d
(d
.A’“d’-ad
rfl 1 -
(20) 2s
f&.c,
-f(l,.c)’
This result differs only by the factor r/3. from equation 16, which was derived using the steady-state model, and X = Iv’*_For this model of cavity nucleation, the functional dependence of the rate of crack growth upon external loading (contained in A) is determined solely by the stress dependence of the mechanism of the growth of cavities (cf. equation 10). Since it was assumed that cavities nucleate simultaneously on whole grain facets. the rate of crack growth now also depends on the grain size and because ~$3< 1, the model indicates that cracks should grow more rapidly in large grain size material,
1
5. IMODELS OF INTERGRAXLLAR CREEP CRACK GROWTH
B
Fig. 4. A model for nucleation. in which cavities are nucleated simultaneously on a grain boundary facet. result in the simultaneous nucleation of all cavities on a single grain boundary facet. fn this case no further nucleation of cavities occurs while the crack grows across this facet, and the maximum number of growing cavities is N, = d/c, where d is the grain size (see Fig. I). Since it is reasonable to assume that d D c, it follows that No x 1. A crude approximation for the crack growth rate can then be found by using N = N, in equation 10. Thus (when r/3 < 1)
However, since no nucleation of cavities takes place on each facet as the crack grows across it, the number of cavities ahead of the crack decreases as the crack extends. To take this into account we evaluate in more-detail the time td, required for the crack to propagate across the facet. The crack always grows in steps of length c, each jump taking a time interval hr..,. as determined by equation 7, when N is the current number of cavities between the crack tip and the far side of the facet. Thus
The general description of crack growth driven by cavitation ahead of the crack tip, which we have just developed, can be applied to specific cases of creep crack growth in which cavities grow by diffusion and by plastic creep. By considering specific physical mechanisms we can evaluate the functionf‘(l, c) which appears in the general expressions for the rate of crack growth and principally determines its dependence on the separation of cavities. Inserting these functions into equations 10, 15 or 16. as physically appropriate, we obtain the corres~nding rates of creep crack growth. (a) Consider first, the case of difihmcd cacitation, for which we assume, as in previous work [14-161, that the rate of material deposition between the cavities is uniform and thus the rate of cavity growth depends linearly on the locat stress. A different situation, in which individual cavities grow independently of each other, and the rate of material deposition varies with the distance from the cavity. will be discussed elsewhere [41]. The rate of cavity growth due to grain boundary diffusion is, in the two-dimensional case, given by
(17) Using equation (14) (which is a good approximation even for N J: 1 unless z/I is very close to unity), we obtain Ar,v = A-‘flc=@(l - d)Cf(fc, cl - jv,, c)3N=g-‘. To perform the summation
(18)
in equation 17 we again make use of equation 8 with No >z I, to get ld
Here, y1 is the surface energy and i. = 8$I&G/kT, where D, is the grain boundary diffusion coefficient, Sb the diffusional width of the boundary, R the atomic volume, G the shear modulus, k Boltzmann’s constant and T the absolute temperature. r, is the radius of curvature at the cavity tip. & is determined by the geometry of the cavity, such that the area occupied by the cavity is a,&. If surface diffusion is rapid and thus the cavity maintains an equilibrium shape (Fig. 5), then a* = MO)
(22a)
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where g(N) =
‘!% - sin t, cos 0)
(22b)
sin’ tl
@is the equilibrium dihedral angle at the cavity-grain boundary interface. If surface diffusion is slow and the cavities are, therefore. crack-like & = X,. where d, is the cavity width (Fig. 5). The first term in equation 21 represents the effect of the applied load while the second term shows the effect of the sintering stress. kVe consider two cases of diffusional cavitation. First. when cnriries possess eqctilibriam sktrpr. r, is large and the sintering term in equation 21 can be neglected. Second. when cacirirs nre crack-like. d, becomes growth rate dependent [27] and the sintering term significant. (i) For equilibrium cavities equation 21 simplifies to 6.
rn=go l,(c This has the form required fi = 1 and &l.
C)
=
--!L
C
EOUILIBRIUM
CAVITY
CRACK-LIKE
CAVITY
Fig. 5. If surface diffusion is rapid. a cavity c3n maintain an equilibrium shape with constant curvature tlcft). If however. surface diffusion is too slow. the cavity becomes crack-like in shape (right).
limiting forms. When
then
o-”
21,Yz-’ by equation
g(O)G i(c-
(28a)
(2) with and when the opposite inequality is valid
c
(21) rfl
h
Inserting #J into equation (6) yields f(l,C) = g
(+c’l” - ;c1-’ -+ I”).
f28b) (2s)
We will assume that the cavities, as nucleated are much smaller than when they achieve the critical length for linking with the crack (i.e. 1, >>I,). Thus, ,f(L c) >>jf(r,.c). Furthermore, the maximum possible cavity length corresponds to actual contact with the crack tip (i.e. I, = c 2) and it is reasonable to assume that !_will be close to that value. Using these assumptions 126)
Both of these limits have the form required by the present analysis with p = 3 2 in the first case and /? = 3 in second. Using the same conditions on I, and I, as previously we find that f(/,, c) _ /(I,. c) = [20 - cos @13” 24, 7;. X
(29a)
in the first case, and [2(1 - cos 8)]“’ &D,
(ii) In the case of crack-like cnciiies. the shape of the cavity is controlled by the rate of surface diffusion in the region near the cavity tip. Chuang and Rice [27] have show-n that both n, and r, then depend on the rate of cavity growth and [ 19.321
CG3
f29b) in the second. fb) We next consider the case where cavitation is controlled by power-law creep as described by the constitutive equation (30) iO. ~~ and m, are material constants. The rate of cavity growth is given approximately as [34]
where D, is the surface diffusion coefficient and 6, the surface diffusion path width. This equation has two
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and cavity growth develop concomitantly, special considerations are required [33, 16-J The microscopic metallurgical parameters pertinent to the growrh of cavities. including the separation of cavity nucleation sites, are all contained in the function f# of equation 2. Once we arrive at an equation for the rate of crack growth. these parameters are 6. DISCLSSIOS then contained in f(/,, c) - f(r.. c), derived from 4 using equation 6. In general. the crack growth rate In this paper the problem of crack growth due to cavitation ahead of a crack tip has been analysed. A also depends upon the number of cavities N, growing simuhaneously ahead of the crack. This number is general expression for the rate of crack propagation controlled by the ~~~~e~riotzmechanism. In general, A’ has been derived (equation 10) assuming that (it an may be a function of both the loading parameter A unspecified but constant number of cavities, X, grow and metallurgical parameters such as grain size. simultaneously ahead of the crack tip, (ii) the stress cavity separation. etc. Two examples of nucleation field ahead of the crack decreases with the distance criteria have been discussed in Section 4. However, from the crack tip, X, as .Y-* where x > 0. (iii) the rate of cavity growth depends on a power fi > 0 of the the significance of the nucleation process to the overlocal stress loading the cavity. (iv) the cavities are all crack growth rate can vary widely. depending on spaced uniformly, and (v) all the cavities grow by the the parameter r/I. It can be seen from equations 10 same mechanism. Relaxation of the stress ahead of and 11 that for rb > 1 the rate of crack growth is practically independent of .V and cavity nucleation the crack resulting from cavity growth has not been does not si~ni~cantly influence the rate of crack considered. This enables us to treat the development of the stress field ahead of the crack tip and the growth. On the other hand, for rJ < 1, the nucleation of growth of cavities in the stress field, as separable cavities has to be considered when studying crack problems. propagation. The stress field which is appropriate to a given situLet us turn now to some specific cases. for both the ation depends primarily on the macroscopic response of the material to crack loading. This determines the stress field ahead of the crack. and the mechanism of parameters .1 and r in equation 1. Engineering par- cavity growth in a creeping solid. (a) If the effect of plastic creep near the crack tip is ameters such as load, crack length and specimen geometry are al1 contained in the parameter A, while x is negligible, and the material is essentially hear elastic only affects the rate (2 = 1'2). then cavity nucieation primarily influenced by the degree of plasticity near of crack growth for /3 c 2. In this type of material. the crack tip. When the extent of the plastic deformacavities are expected to grow by diffusion. For tion near the crack tip is negligible in comparison cavities (5 = 1) this with the size of the zone of cavitation, then the field rorrnderi (equilibrium-shaped) 10 and 26) that d _ ahead of the crack can be taken as that of a crack in means (using equations KNi :ce3. while for crack-like cavities in the limiting an elastic body case given by equation 28a (p = 1.51, then A = K’12n, x = I;2 (33) ci _ K1.5N1~4~-q”, The actual dependence of ci on K and c depends on the nucleation mechanism which where K is the crack stress intensity factor. If. on the influences N. If we use a critical stress criterion. as other hand, cavitation is largely confined to the plasdiscussed earlier, then ri c K”cb3 for rounded cavitic zone of a material for which the plastic behaviour is described by the constitutive law. & = (~//a~)“. ties. and d c K2cv5 ’ for the cracklike cavities. This result for rounded cavities agrees with both the Kwhere eO, ITS and m are constants, then the approand c-dependence found by Raj and Baik [14]. in priate stress field ahead of the crack is of the HRR their model for steady-state crack growth. In that type [31,37] study, equilibrium shaped cavities only are considered, and the same assumptions regarding a critical (34) stress for cavity nucleation are made. In addition. m-+1 ORI stress relaxation due to cavity growth is considered. It where J is the J-integral and I, a constant given in would thus appear that relaxation. while affecting the Ref. [31]. Finally, if the plastic zone is developed size of the cavitation zone. does not alter the depenthrough power-law creep deformation, then a stress dence of crack growth on K. field analogous to the HRR fields develops [333 and The second nucleation criterion u’e may use assumes that N is related directly to the grain size of cT$c* 1 material, 8 = die, in which case we get 2-p A=(35) the . m, + I co 1% a _ Kdi”cm3 and ci _ K3jzd1 ‘cm5 ’ for the two cases of rounded and crack-like cavities, respectively. In this case, crack growth rate is also a function of grain where C* is the path independent integral, analogous to J [43,44]. In this case however, if the plastic zone size.
In this case /3 = m, and
A=$,
x7-i
WILKINSON
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PROP.AGXTION
(b) If the cavitation occurs primarily inside 3 phscreep zonr. then x = 1 (m, + I). and the mechanism of cavity nucleation will affect the rate of crack growth. whenever 0 < m, f 1. Since m, is usu3))y much larger than 2. this means that nucleation cannot be neglected for any mode of diffusional cavitation we have considered, When cavities nucleate at a critical stress ti - C+. If. on the other hand, the number of growing cavities is related to the grain size, then ri _ (C*):. where ; = 1 (m, T 1) in the case of equilibrium cavities. ; = 3 2(m, + 1) for the crack-like cavities in the case of equation 283. and ; = 3 (m, + 1) in the case of equation Bb. Hence the present work show-s that substantially different dependencies on C* may be obtained if different modes of cavity nucleation and growth are considered. If cavity growth is controlled not by diffusion. but by creep deformation. then the mechanism of cavity nucleation affects the rate of crack growth for WI,< 1 1. Consider tvvo cases. First. when the effect of plastic zone formation can be neglected (r = 1’2) nucleation does not affect the rate of crack growth so long 3s tt~, > 2, which is usually the case. This situation has been studied by Nix et (I/. [?I]. They were able to calculate 3 finite rate of crack growth assuming an unlimited number of nucleated cavities. only if no, > 2, consistent with our analysis. We find in this case J . c(K’, c)“* In(c 21,). which agrees with the result of Nix er al. [24]. On the other hand for 171,< 2. a nucleation criterion must be included. When cavities nucleate at 3 critical stress. ir _ K’ In (c 21,). while ic _ ti(K , d)“~,In(c’2/,) if the number of growing cavities is related to the grain size. Ductile cavity growth is unlikely to occur in an elastic solid. Instead. we should consider cavities growing in a plastic creep zone. In this case. rfl = nt, (m, + 1). which is always smaller than 1 and the nucleation of cavities must be considered. If nucleation occurs at 3 critical stress il _ C* In (c 21,). while if the number of cavities is related the grain size. ci _ d(C*,‘ti)‘“~mc- “‘In (c 21,). For m, >> 1, which is common in the creep situation, the difference between the two types of C* dependence on creep crack growth is practically indistinguishable. Furthermore. the dependence of crack growth rate on grain size is extremely weak. We have considered a number of examples using the assumption that cavities nucleate simultaneously on grain boundary facets ahead of the crack tip. This leads. when r/3 < 1. to a dependence of crack growth rate on grain size, of the form ri - d’ -I!‘. Experimentally. Gooch and King [45] find that crack growth rates in 1 Z-CrMoV steel change as 3 crack grows through successive coarse- and fine-grained regions of a weld heat-affected zone. This type of grain size dependence has been generally observed in low alloy steels and fits well with the model we have proposed.
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REFERESCES
ric
.-lr~,~~\~Iniyrments--This research was supported Arm) Research Office under grant No. 29-81-K-002.5 (V.V.) and by a grant from Imperial
(D.S.CV.).
by the DAAG Gil Ltd.
1. K. Sadananda and P. Shahirnan. J. Enyny .Lfarrr. 7-echnol. loo, 381 [ 197s). 2. I. Milne and G. G. Chell. Elosric-Pfasric Fr~turr (edited by J. D. Landes. J. A. BegIcy and G. .A. Clarke). o. 3%.
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by R. M. Pelloux and N. S. Stoioff). p. 46 TMS-AIME 11980). 39. D. S. Wilkinson, K. Abiko. N. Thyagarajan and D. P. Pope, Merall. Trans. 1 IA. 1827 f1980). 40. C. L. Jones and R. Pilkington, Metall. Trans. 9A, 865 f 1978). 11. T. Takasugi. D. S. Wilkinson and V. Vitek. to be published.
OF CRACKS BY CAVITATION
42. T. Takasugi and V. Vitek. &ferall. Trans. 12A, 659 (1981). 43. J. D. Landes and J. A. Begley. .Llrchanics of Crack Growth. p. 178. ASTM STP 590 (1976). 44. K. M. Nikbin, G. A. Webster and C. E. Turner, Cracks and Fracture, p. 47. ASTM STP 601 (1976). 45. D. J. Gooch and B. L. King, Metals. Technol., p. 405 (1979).