Materials Science and Engineering, 74 (1985) 5573
55
The Effect of Cavitation and Microstructural Damage on the Intergranular Creep Fracture of Nickelbase Superalloys G. SUNDARARAJAN Defence Metallurgical Research Laboratory, Hyderabad 500258 (India) (Received July 6, 1984; in revised form January 2, 1985)
ABSTRACT The creep behaviour o f nickelbase superalloys is unusual in the sense that they rarely exhibit a wellestablished steady state creep regime. Rather, the creep rate progressively increases from a m i n i m u m value. The early onset o f the tertiary regime can be explained either by assuming that the timedependent coarsening o f the "y' precipitates reduces the back stress and thus increases the creep rate or through the concept o f straininduced loss o f coherency at the ~[7' interface and the resulting higher rates o f recovery (and hence creep) o f the matrix dislocations. In this paper the combined effect o f precipitate coarsening, straininduced loss o f coherency and cavitation (which is inevitable) on intergranular creep fracture is considered. In particular, the effect o f cavitation and microstructural damage mechanisms on time and strain to fracture and the M o n k m a n  G r a n t constant is analysed in detail. It is s h o w n that both the time to fracture and the M o n k m a n Grant constant are reduced significantly in the presence o f various forms o f damage. In contrast, the strain to fracture is largely unaffected by these damage mechanisms.
1. INTRODUCTION Pure metals and many of the singlephase alloys are characterized by an extensive constantcreeprate regime followed by a much shorter tertiary creep regime [ 1 ]. In such materials, extensive cavitation is responsible for the onset of the tertiary stage. However, for multiphase alloys such as the nickelbase superalloys, the constantcreeprate regime is rarely observed [ 2  4 ] . Rather, the creep rate progressively increases from a minimum 00255416/85/$3.30
value. In such materials the minimum creep rate emo is most appropriately expressed using the concept of friction or back stress Spo as [57] •
emo 
v
oO
kT
(1  K0)"
 ~/
(1)
where ~mo is the minimum creep rate, A a material constant, Dv the lattice diffusion coefficient, b0 the Burgers vector, G the shear modulus, S the applied stress, Spo the initial value of the friction or back stress, n ~ 4 the stress exponent, K0 a constant less than unity which increases in proportion to the extent of solid solution strengthening in the matrix phase and k T the thermal energy. The extensive tertiary creep regime in the superalloys can be rationalized in three different ways. (1) The most obvious way to rationalize the observation is to postulate that extensive cavitation occurs quite early during the creep test, resulting in an early onset of the tertiary creep regime. However, experiments have clearly shown that, even in these materials, extensive cavitation occurs only in the later stages of tertiary creep [3, 8], thereby indicating that some other factor is responsible for the early onset of tertiary stage. Nevertheless, it must be noted that, in these materials, cavitation is inevitable and will contribute to the overall strain of the sample, although mainly during the final stages of creep. (2) A second way to rationalize the observation is to assume that the friction stress Sp0 comes about mainly because of the resistance offered by the precipitates to dislocation motion and that this friction stress decreases with increasing time because of precipitate coarsening [3, 7, 9]. The coarsening of the © Elsevier Sequoia/Printed
in The
Netherlands
56 precipitate results in an increase in interprecipitate spacing for a constant volume fraction of the precipitate. This, in turn, reduces the magnitude of Spo if either the Orowan bowing of the dislocations between the precipitates [10] or the local climb of the dislocations over the precipitates [ 11 ] is responsible for the observed friction stress. (3) Recently, Henderson and McLean [12, 13] have shown that for many of the superalloys, even when precipitate coarsening is largely absent, a progressively increasing creep rate can be obtained right from the start of the creep test. This observation has been rationalized by Henderson and McLean, using the concept of straininduced loss of coherency (SILC) at the precipitatematrix interface. According to these researchers, since incoherent interphase boundaries should be more efficient sources and sinks for vacancies than coherent interphase boundaries are, SILC should make it easier for the matrix dislocations to climb. Consequently, the recovery rate will be enhanced and the parameter A will n o t be a constant b u t will increase continuously with increasing strain or equivalently time. Thus the early onset of tertiary stage can be explained. The loss of coherency at the ~  ~ ' interface can also make the local climb of the dislocation over the precipitate easier, thereby causing a continuous reduction in Sp0 (eqn. (1)) with increasing strain. However, experiments [13] have generally indicated this to be a secondary effect. In a real material, all the degradation mechanisms, namely cavitation, precipitate coarsening and SILC, can occur simultaneously b u t to different levels of severity. As a result, it is important to study the effect of one or more of these degradation mechanisms on intergranular fracture. The present paper serves this purpose since its aim is to analyse the combined effects of cavitation and microstructural damage (coarsening and SILC) on the various parameters characterizing intergranular creep fracture such as the time to fracture, strain to fracture and the M o n k m a n Grant constant (the product of the minimum creep rate and the time to fracture). In the present paper a typical nickelbase superailoy will be considered and the effect of cavitation, precipitate coarsening and SILC on its fracture behaviour under creep conditions will be analysed. Towards this purpose,
the following format will be adhered to. In Section 2 the effect of cavitation on the time and strain to fracture, the overall sample strain rate and the M o n k m a n  G r a n t constant will be investigated in detail and the basic equations characterizing these effects will be formulated. In Section 3 the combined effects of cavitation and microstructural damage on intergranular creep fracture will be explored in detail. The usefulness of these basic equations derived in Sections 2 and 3 will be illustrated in Section 4 b y applying them to a typical nickelbase superalloy with known properties.
2. THE EFFECT OF CAVITATION ON CREEP FRACTURE 2.1. Introduction In this section the effect of intergranular cavitation on the time and strain to fracture will be considered in detail. Intergranular cavitation essentially involves the nucleation and growth of cavities situated along the grain boundaries. A number of theoretical models exist in the literature in which either diffusive [ 1 4  1 7 ] or dislocationcreepcontrolled [18] growth of cavities is considered. The actual local stress Scar experienced by the cavitated boundary facet determines which of these models will apply. At one extreme, either if the extent of cavitation is very high or if the matrix creep at a rapid rate (either because the applied stress is high or because the ma. terial has an inherently poor creep strength), the cavities grow without any constraint and the local stress at the cavitated boundary facet will essentially equal the applied stress S. At the other extreme, if the extent of cavitation is low and if the matrix has a high creep strength, the cavity growth is severely constrained and the actual local stress driving cavity growth will be considerably smaller than the applied stress [19]. In this paper, we are concerned with materials with a high creep strength, such as superalloys. In addition, experiments have shown that under typical creeptesting conditions the extent of cavitation of the grain boundaries is quite low [20]. Hence the effect of matrix constraint on cavity growth should be considered while modelling intergranular fracture of superalloys.
57
2.2. The model Since the cavity growth is expected to be severely constrained on the materials of interest to us, the local stress S¢~ on the cavitated facets will be considerably lower than the applied stress S. At such low stress values the most appropriate cavity growth mechanism is the one first proposed by Hull and Rimmer [14] and later refined by a number of workers [1517]. This model postulates that the cavity growth is controlled by grain boundary diffusion and also assumes that the deformability of the grains adjoining the cavitated facet can be neglected• The resulting expression for cavity growth rate for the cavity configuration shown in Fig. 1 is obtained as [21]
da Yc=
dt
D ( S ~  (1  
d2)So} (2)
b2d~hQ(d)
where V¢ represents the constrained cavity growth rate, a is the cavity radius on the grain boundary plane, t is the time, D = D b ~ b ~ / k T (Db~b is the grain boundary diffusivity and is the atomic volume), S~a~ is the local stress on the cavitated facet, b is the halfcavity spacing, d = a/b, h ( ~ 0.6) is a function dependent on the cavity tip angle, So = 2 R J b d (Rs is the surface free energy), Q(d) = ln(1/d 2)  (3  d2)(1  d2)/2 and k T has the usual meaning. In eqn. (2), Scar is still undetermined. By equating the rate of opening of the cavitated boundary (which equals 4d2hVc) with the average opening rate of the isolated crack sub
jected to a remote stress of S and a restraining stress of Scar, Rice [21] obtained the following expression for Scar: 8  (1  d2)So + (1   d2)S0 I+M
Scav=
where S is the applied stress, M = 4DS/ o~b2LQ~mo, L is the grain size, era0 is the minim u m creep rate given by eqn. (1) and c~ is a parameter dependent on the e x t e n t of cavitation and grain boundary sliding. Equation (3) derived by Rice [21] does not include explicitly, the effect of the extent of cavitation on the severity of the constraint. In fact, eqn. (3) is strictly valid only when the extent of cavitation is sufficiently low to ensure that no interaction exists among the cavitated facets. If such an interaction exists, as is usually the case, the Rice equation (eqn. (3)) has to be modified as indicated below. Let f be the fraction of the boundaries oriented normal to the stress axis, which are cavitated. Alternatively, if we assume that the grains are cubic (an idealization), f will equal L2/B 2 where L is the grain size and B is the average spacing between two cavitated facets along the plane perpendicular to the applied stress axis. Compatability requires that the grains adjoining the cavitated facet should creep at the same rate as the grains adjoining the uncavitated facet. This, in turn, implies a stress redistribution among the cavitated facets (and the adjoining grains) and the uncavitated grains. Thus, neither the local stress Scar in the cavitated region nor the local stress Sun¢ in the uncavitated regions will equal the applied stress S. In addition, equilibrium with the externally applied traction demands that Sun¢(1  f) + Sc~vf = S
Scar
t 2b
(3)
(4)
As a result, when interaction is allowed between the cavitated facets, each of the cavitared facets (modelled as a discshaped crack) is actually subjected to a " r e m o t e " stress of Sun¢ and not S. The corresponding " r e m o t e " strain rate, which also equals the overall sample strain rate, is given as •
.
/Sunc~ n (1
es° = e m ° ~   ~ )
SpO/Suno) ~
(5)
( l _ Spo/S)n
Scar Fig. 1. A schematic view of the cavity configuration on a grain boundary.
Substitution o f eqn. (4) in eqn. (3), with Sunc and eso replacing S and emo respectively,
58
results in the following final expression:
Scav   (1

Yuc
Vc =
= S(lfScav/S)  (I d2)(lf)So (6)
(1f)(l+M) M in the above equation is obtained as
[O~moLb2Q(1
fS~JS)"1] I x

X [(1   Spo/S(1  
Vuc
Vc
For creepresistant materials, So is considerably smaller than S and hence eqn. (6) simplifies to S
lfSc~v/S (1/)(1 + M)

Vc 
S
S
(Scav(ld2)So}
= Vuc(1f/(l+M)) 4198
(8)
(13a)
(1 + M)(1 f) M
(lf) n1
[email protected] (lf/(l + M)} n1
The constrained cavity growth rate Vc can now be estimated as
Vuc
(12c)
Ol~moLb2Q
f ) / ( 1   fS¢~,,/S)}"] 1 (7)
Scav(ld2)So=
4DS
M 
Inspection of eqns. (11) and (12) indicates that a single set of equations which will be valid over the whole range of M (0 to ¢~) or fSca,,/S values (0 to f) should be of the form
M = 4 D S ( 1   f ) "  ~ ( 1   Spo/S)" X X
(12b)
I+M
d2)So
X
(1Spo/S)" X
[1Spo(1f)IS{1f/(1 +M)}]"
(13b)
(Sc~  (l d2)So}
VuA I fSc.,/S) (9) (1f)(l+M) In eqn. (9), Vuc represents the unconstrained cavity growth rate and M is given by eqn. (7). The appropriate expression for 'Cue [21] is DS V.c   (10) d2b2hQ =
Estimation of Vc using eqn. (9) requires the value of Scar. The value of Scar can be obtained by numerically solving eqn. (8), but eqn. (8) can be simplified further as follows. When fScav/S is considerably smaller than unity (this implies that M >> 1), eqns. (7)(9) reduce to the following: Sc~v  (1 d2)S0 = S
1
(11a)
(lf)(l+M)
Figure 2 illustrates the variation in the norrealized constrained cavity growth rate Vc/Vuc with the parameter M, as predicted by eqn. (13b). Each curve in the figure is valid for one value of f. From Fig. 2 it is obvious that either a low f or a high M causes the cavity growth to be severely constrained. Incidentally, when f ,~ 1, eqn. (13a) becomes identical with the equation obtained by Rice [21]. This is so since Rice's model is valid
1 
# '~u
V¢ =
(lib)
0.1
>
f
(1f)(l+M)
M =
4 D S ( 1   f ) "  l ( 1   Spo/S)"
[email protected](1 S p o ( 1   f)/S}"
(11c) ool J 001
In contrast, when fSca,,/S assumes values approaching f ( o r M + 0), eqns. (7)(9) simplify to 8¢,~ (1S
d 2 ) S 0 __
1 I+M
I
1
I 01
I
I
J 1
l
I
I 10
I
I
I 100
PARAMETER M (12a)
Fig. 2. The variation in the normalized constrained cavity growth rate Vc/Vuc with the parameter M at different cavitation levels f: graphical representation of eqn. (13a).
59 only when f ~ 1. Figure 2 also indicates that the cavity growth is unconstrained when M assumes values much lower than unity. Thus, it appears that eqn. (13a) is capable of predicting the expected behaviour with regard to cavity growth rate over the whole range of M and f values. Using eqn. (13) the time to fracture can be calculated as follows. The constrained cavity growth rate Vc can be calculated for given b, d and f values (it should be noted that Q is a function of d) by first numerically solving eqn. (13b) for M. Then, using this M value, V~ can be obtained from eqn. (13a). Now, the time tt to fracture is given as
ao
N
Aa
bAd
(14)
= i~1V~(di)
where Vci is the value of Vc at the ith increment (ai = a0 + i Aa and di = a d b ) , ao and a~ are the initial and final values of a, Aa = a~ ao/N, N is the number of increments, Ad = df  do~N, d~ = a~/b = 1.0 and do = ao/b. Thus, eqn. (14) should be solved numerically to obtain the value of tf. Once t~ is known, the M o n k m a n  G r a n t constant CMG can be easily estimated since it is merely the p r o d u c t of em0 (eqn. (1)) and tf. The overall strain rate e~0 experienced by the sample can be estimated using eqn. (5), once the value of Sun¢ (the local stress in the uncavitated regions) is known. Using the expression for SCar obtained from eqn. (13a) and the relation between Sexy and Sure (eqn. {4)), the following expression is obtained for Sunc: Sunc
eso


OmO (1 U)" X (lf) n (1Spo/S)" 

× (1_
S lU/
where f U
1fl(l+M)
 
1f
I+M
Finally, the strain ef to fracture can be estimated using N b Ad ef = ~ e s o ( d i ) i=l Vc(di)
af da
N
Substitution of eqn. (15) in eqn. (5) results in the following expression for eso:
S 
1f
X
f(1d,)Sol
x{1: f 1f
S [ 1f
~]1
l+m
f 1f/(l+M)) ~ 1f iTM J
)
] (15)
(17)
where di is the value of d at the ith interval, A d = d~  d o / N and N is the number of increments. Thus, all the parameters characterizing intergranular fracture (tf, eso, ef and CMG) can be estimated using numerical methods. In derivmg the various equations given above, we have made a number of assumptions, none of which has been stated explicitly. So let us consider in detail the various assumptions made and their validity. (a) The model described above assumes that the incubation period for cavity nucleation can be neglected in comparison with time to fracture. A number of experiments [22, 23] have shown that the incubation period is either zero or at the most a few hours, thereby supporting our assumption. (b) The model also assumes that all the cavity nucleation occurs quite early during the creep test, b u t experiments [ 2 4  2 6 ] do n o t support this assumption since cavities are found to nucleate continuously during the creep test until fracture intervenes. It should be noted that the effect of continuous nucleation can be easily incorporated in the present model by prescribing a timedependent value for b. However, such a procedure leads to an additional set of variables, thereby complicating the model further. As a result, the effect of continuous nucleation will be accounted for implicitly in the model, by assuming that b represents an average value. Thus the average value of b will be smaller than its initial value b u t greater than its final value (prior to fracture).
60 (c) The model makes an implicit assumption, namely that the fraction f of boundaries cavitated is a constant independent of time. Experiments have shown f to be time independent except during the late stages of the creep test when f increases exponentially with time until fracture intervenes [20]. Once again, this time dependence of f can be accounted for by the proper choice of an "average value" for f. (d) According to the model, macroscopic fracture of the sample is assumed to occur, once the isolated cavities on the cavitated boundary facet coalesce to form a macroscopic intergranular crack of one facet length. This assumption implies that the interlinking of these cracks, once they form, occurs quite easily. Usually, the final fracture occurs when f attains a critical value fc characteristic of the material, at which time some form of plastic instability develops between the isolated intergranular cracks. If the initial value o f f in the creeping material is reasonably close to fc, the time to fracture calculated using the above model will be reasonably accurate. However, if the initial value of f is considerably smaller than fc, the time to fracture predicted by the present model will be lower than the actual value and hence will represent the lower bound value. In the literature, there are almost no data for either fc or the initial value of f. Thus, it is not possible to ~ m m e n t o n how reasonable fl~is last assumption is.
2.3. L o w e r b o u n d and upper b o u n d values As noted earlier, the various parameters characterizing intergranular creep fracture (tt, eso, e~ and CMG) Can be estimated only by numerical methods. Nevertheless, as shown below, the upper and lower bound values of these parameters can be expressed in terms of analytical equations.
2.3.1. Upper b o u n d The upper bound values for t~, e~o, ef and CMG (denoted by (ub) after the appropriate parameter) can be obtained if we assume that the cavity growth is fully constrained, i.e. M >~ 1. Under such conditions, eqn. (13) can be simplified as follows:
da Vcm dt
Y~o M(1f) = °~em0Lb2( 1  ( S p o / S ) ( 1   f ) ) n 4a2h(1f)"(1Spo/S) n
(18)
Integration of eqn. (18) with the assumption that d = d~ at fracture results in an expression for the upper bound value of the time t~ to fracture: t~(ub) =
4hb(1f)n(1Sp°/S)n 3a~moL { 1  Sp o (1  f ) / S } " (dr3  d°a)
(19) where do is the initial value of d and under most conditions d~3  d o s = 1 to a good approximation. Thus, tf exhibits a stress and temperature dependence which is an exact inverse of the stress and temperature dependence of minimum creep rate. Now, CMG is obtained as C M G ( U b ) ~mot~(ub)
_ 4hb(1 f)n(1 S ~ / S ) n (d~a  do s) 3aL{1Spo(1f)/S} n
(20) Hence CMG is really a constant nearly independent of stress and temperature. From eqn. (16), the upper bound value of the sample strain rate ~so is found to be dmo { 1   8 p o ( 1   f ) / S } " ~so(ub) =  (lf)" (1Spo/S)"
(21)
Finally, the upper bound value for the strain e~ to fracture is obtained by multiplying eqns. (21) and (19): 4hb s e~(ub) = ~ (dr  dos)
(22)
2.3.2. L o w e r b o u n d The lower bound values for t~, ef and CMO (denoted by (lb) after the appropriate parameter) are obtained when the cavity growth is unconstrained, i.e. M ~ O. Integration of eqn. (2) with S replacing Scar results in the following expression for tf [21]: hb3dt 3 tf(lb) =  19.7DS
(do < 0.12)
(23)
61 3. C A V I T A T I O N A N D M I C R O S T R U C T U R A L
The M o n k m a n  G r a n t constant is then obtained as
DAMAGE
CMG(Ib) = em0Q(lb)
~mohb3df 3
(24)
19.7DS
Thus, tf is proportional to S1 and CMGis a strong function of S and T and hence is actually not a constant. The overall sample strain rate eso can be estimated only for special situations. For example, if the absence of constraint is due to extensive cavitation (f ~ 1), then e~o can be obtained as 4DS ~so(lb) = en~o +  
(25)
b2QL
In eqn. (25) the first term represents the strain rate due to matrix creep while the second term is the strain rate due to cavitation only. The corresponding value for ef is then obtained as ef(lb)
hbdf a = C M G ( I b ) h 

4.92QL
(26)
If the absence of constraint arises because the matrix creep rate ~mO is much higher then the local strain rate (4DS/bgQL) at the cavitated facet, the situation is more complex. Under such circumstances, interaction between the creeping grain and cavity growth will be inevitable, resulting in an acceleration of cavity growth rate [17, 27]. Therefore the HullRimmer type of model will not be valid under these conditions. The actual value of the various parameters such as tf, es0, ef and CMG will lie within the bounds given above, b u t their actual values for a given material can be obtained only by resorting to numerical solutions. Nevertheless, it should be obvious that the present set of equations will be accurate only when some a m o u n t of constraint is present (say, M > 10) since only then will the stress Scar in the cavitated region be sufficiently low for the H u l l  R i m m e r t y p e of mechanism to operate. Fortunately, for materials of interest to us (superalloys), such a situation is easily obtained.
In Section 2 the effect of grain boundary cavitation on the various parameters characterizing intergranular creep fracture was considered. The aim of the present section is to analyse in detail the combined effects of cavitation and microstructural damage on intergranular creep fracture. For superalloys, precipitate coarsening or Ostwald ripening and SILC at the 7  7' precipitatematrix interface are the two c o m m o n forms of microstructural damage. We shall investigate the effect of these two forms of microstructural damage on intergranular fracture in Sections 3.13.3.
3.1. Cavitation and precipitate coarsening In superalloys the uniformly distributed precipitates resist or slow down dislocation motion in a number of ways. The extent of this resistance to dislocation motion is characterized by the magnitude of the back stress Spo in eqn. (1). The higher the value of Sp0, the greater is the resistance to dislocation motion and hence the higher is the creep strength. At this juncture it is worth noting that the strengthening due to the solid solution is taken care of by the Ko term and hence Sp0 represents the back stress due to precipitates alone.
3.1.1. Precipitatestrengthening mechanisms The dislocation can overcome the obstructing precipitate by a variety of competing processes, each of which is characterized by a threshold stress which must be exceeded before the particular mechanism can become operative. The three c o m m o n mechanisms by which dislocations usually overcome the precipitate and the relevant equations are as follows. (a) Orowan bowing. In this case the dislocations remain in their glide plane and overcome the precipitate by bowing to a critical radius between the precipitate. The threshold stress or the back stress Spo that must be exceeded is given as [10] o0(
ow) =
rrl0
Gb
lo
\re / (27)
62
where lo is the particle spacing (initial value), G is the shear modulus, b is the Burgers vector and re is the dislocation core radius. (b) Cutting o f precipitates. When the 7  ? ' interface is coherent, the matrix dislocation can shear the particles. During such a cutting process the antiphase boundaries are generated in the precipitate by the leading dislocation while the trailing dislocation destroys the antiphase boundary. So the dislocations move in pairs and the m i n u m u m stress Spo required for cutting the precipitate under such circumstances is given by [10]
Sp°(cut)  R ApB { (4f~O1/2
(28)
In eqn. (28), RApB is the antiphase boundary energy and f~ is the volume fraction of precipitates (?'). Thus, unlike the back stress due to Orowan bowing, the threshold stress for cutting is independent of lo and depends only on fv. (c) Localized climb around precipitates. According to this mechanism, the dislocation moves past the precipitate by climbing out of the glide plane in the vicinity of the precipitate alone. Shewfelt and Brown [11] have estimated the threshold or back stress for localized climb to be about 40% of the Orowan bowing stress (eqn. (27)). Hence the threshold stress for localized climb can be obtained as
0.4Gb Spo(loc climb) ~  lo
(29)
Of these three parallel mechanisms, the one which operates with the lowest threshold or back stress will be rate controlling since that mechanism allows the highest creep rate (creep rate ~ (S Spo) 4) to be obtained. At creep temperatures, localized climb around the particles is inevitable. Hence the localized climb mechanism always predicts a lower threshold stress than the Orowan bowing mechanism does. Therefore, cutting and localized climb are the two competing mechanisms. Since f~ = roa/los (where ro is the initial Drecinitate size), I, in eqn. (29) can be substit u t e d by ro/f~ 113. The critical size roc of the precipitate, above which the localized climb mechanism operates and below which cutting of precipitates occurs, is then obtained by
equating eqns. (28) and (29) as
0.4Gb2f~ 113 roe
RAPB((4fv[~) 1/2 fv}
(30)
For typical values of the various parameters in eqn. (30) (G(0.6Tm) = 5.74 × 10 l° Pa, b = 2.5 × 10 l° m, fv = 0.4 and RApB = 0.17 J m2), roe is around 200 • (20 nm). If Orowan bowing is the operating mechanism, r0c will roughly equal 500 A (50 nm).
3.1.2. Ostwald ripening o f precipitates and back stress A number of experiments with different superalloys [2, 3, 28] have clearly shown that the timedependent coarsening of the 7' precipitates is volume diffusion controlled. In addition, the rate of coarsening has been found to be independent of applied stress [ 2]. Therefore, we obtain r 3 = ro8 + K't
(31)
where K' = 8Ceq~t2DsRp/9kT, r is the precipitate radius at a time t, ro is the initial radius of the precipitate, Ceq is the concentration of the precipitateforming c o m p o n e n t in the matrix in equilibrium with the precipitate, ~t is the atomic volume, Ds is the volume diffusivity of the slowest diffusing species, Rp is the precipit a t e  m a t r i x interface energy and k T has the usual meaning. The effect of precipitate coarsening on the back stress Spo depends on the actual mechanism by which the dislocation overcomes the precipitate. In general, for superalloys the initial volume fraction fv of precipitates is quite close to the equilibrium value. As a result, except for a short initial period, fv remains reasonably constant with time during the creep test. Consequently, if the cutting of precipitates (eqn. (28)) is the ratecontrolling mechanism, Spo will be a constant independent of time and coarsening will have no effect on the minimum creep rate (eqn. (1)). However, if either Orowan bowing (eqn. (27)) or local climb (eqn. (29)) is the operating mechanism, Ostwald ripening will cause the magnitude of the back stress to decrease continuously with increasing time, as shown below. For either bowing or local climb, the back stress Sp at any time t is given by
63 care of by allowing the parameter A in eqn. (1) to be strain dependent as follows:
1o Sp = Sp0 l ro = Spor
(32)
where I is the value of interprecipitate spacing at time t, l0 is the initial value of I and Sp0 is the initial value of Sp. In the derivation of eqn. (32), fv has been assumed to be constant. Hence, loft can be replaced by ro/r. If we n o w substitute the value of ro/r from eqn. (31) into eqn. (32), the final expression for Sp is obtained as
Sp0 Sp
( l + P t ) 1/3
(33)
where P = K'/ro a is the precipitatecoarsening rate. The equations describing the combined effects of cavitation and precipitate coarsening can n o w be simply obtained by substituting Sp0/(1 + Pt) 1/3 for Spo in the various equations derived in Section 2.
3.2. Cavitation and straininduced loss o f coherency For an Inconel superalloy, Henderson and McLean [12, 13] observed that the creep rate progressively increased with time, even in the absence of cavitation and particle coarsening. On the basis of a detailed analysis, Henderson and McLean showed that this acceleration of creep rate is related to a continuously increasing (with time) recovery rate. On the basis of transmission electron microscopy (TEM) examination of the crept samples, Henderson and McLean also came to the conclusion that the increasing rate of recovery can be correlated with the increasing loss of coherency at the 7  7 ' interface with increasing strain. The SILC at the interface can affect the creep rate of the sample in three different ways [13]. (1) A semicoherent interface can act as a better source and sink for vacancies than a coherent boundary can. As a result, the climb rate of the matrix dislocation can be increased progressively with strain as the interface becomes increasingly incoherent. Therefore, as long as the matrix creep is recovery controlled, SILC will result in an acceleration of the creep rate. This effect can be taken
Ao em= ernoA = ~m0(1 + Ce)
(34)
where era0 is given by eqn. ( 1 ) , A o = A ( I ÷ Ce), e is the sample strain and C is a constant for a given material with constant microstructure. According to Henderson and McLean [ 13], the parameter C in eqn. (34) increases with increasing volume fraction f, of 7', increasing temperature, increasing deviation from zero 7  7 ' mismatch and decreasing precipitate size. The applied stress, however, does n o t have a significant effect on the value of C. (2) SILC can affect the threshold or back stress Spo directly. For example, the semicoherent 7  7 ' interface can permit the dislocations to climb over the precipitate more easily since it is characterized by a lower strain field and further acts as a more efficient source and sink for vacancies when compared with the coherent 7  7 ' interface. Consequently, SILC should cause the back stress Sp0 to decrease continuously with strain. (3) SILC also causes the 7  7 ' interface energy Rp to increase progressively. An increase in the value of Rp, as in eqn. {31), increases the coarsening rate P of the precipitate. Thus the value of P will increase continuously with increasing strain, once again resulting in a continuous reduction in Spo. The experimental results of Henderson and McLean indicate that, of these three effects, the first is the most important. As a result, only this effect will be considered in the subsequent parts of this paper. The combined effects of cavitation and SILC can n o w be simply obtained by substituting Ao, which equals A(1 + Ce), for A in the various equations derived in Section 2.
3.3. Cavitation, coarsening and straininduced loss o f coherency The combined effect of cavitation, coarsening and SILC can be accounted for by substituting Spo/(1 +Pt) 1Is for Spo and A(1 + Ce) for A in the equations derived in Section 2. The resulting equations are as follows:
64
vo= y.o
1f/(l+M)
(35)
(l+M)(1f)
where
M =
41)8(1 f)n l( l__ 8po/S) n ~moLb2Q(1 + Ce){1f/(1 + M ) } n [ 1   S p o ( 1   f ) / S { 1   f / ( 1 + M)}(1 + Pt)l/3] n
em°(1U)n(l+Ce) {1S p ° ( 1   f) ~n es  (l__f)n(l__Spo/S)n S ( 1   U)(1 + Pt)l/3J
(36)
Using eqns. (35) and (36), t~, CMG and e~ can be e s t i m a t e d in the m a n n e r indicated in S e c t i o n 2.
4. NUMERICAL RESULTS In this section the e f f e c t o f t h e p a r a m e t e r s P and C o n t h e intergranular f r a c t u r e parameters (t~, CMG and e~) will be illustrated in detail b y considering a t y p i c a l superalloy, n a m e l y U d i m e t 700. T h e values o f t h e various material c o n s t a n t s f o r U d i m e t 700 are given in Table 1. T h e values o f A, Ko and Sp0 given in Table 1 are t h e values r e p o r t e d b y Ajaja et al. [30] f o r U d i m e t 700. T h e c o m p u t a t i o n s were d o n e f o r o n e t e m p e r a t u r e , T = 0.6Tin, and t h e value o f Spo given in Table 1 is valid f o r this p a r t i c u l a r t e m p e r a t u r e only. It is usually f o u n d t h a t Spo decreases with increas
ing t e m p e r a t u r e while C and P increase with increasing t e m p e r a t u r e [13, 28]. T h u s t h e e f f e c t o f P a n d C o n t~ will be stronger at higher t e m p e r a t u r e s , especially in view o f t h e f a c t t h a t t h e higher t h e o p e r a t i n g t e m p e r a ture, the l o w e r will be t h e applied stress. Nevertheless, t h e general f e a t u r e s s h o u l d r e m a i n t h e same, i n d e p e n d e n t o f t e m p e r a ture. T h e actual values o f t~, e~ and CMG are also strongly d e p e n d e n t o n the values o f the halfcavity spacing b, t h e initial cavity size do, t h e final cavity size d~ a n d the material grain size L. In o r d e r t o r e d u c e the n u m b e r o f vari
TABLE 1 The material constants for Udimet 700
Parameter
Value
Units
Reference
point Tm Atomic volume ~ Burgers' vector b o Shear modulus Go at 0 K Temperature coefficient C' of shear modulus Surface free energy R s Dimensionless c o n s t a n t A Creep component n Volume diffusivity D v Dov Qv Boundary diffusivity Db D0b~ b Qb Solidsolutionstrengthening c o n s t a n t K 0
1726 1.09 × 1029 2 . 4 9 × 10l° 7.89 X 1 0 1 0 3.70 X 1 0  4 1.725
K m3 m Pa J m2
29 29 29 29 29 16
Particle back stress Spo Cavity tip angle
2.36 X l 0 s
Pa
78 0.7
deg 
Melting
Cavity volume parameter h
K 1
2 . 6 5 × 103

30
4.0

30
1.9 × 104 280000
m 2 s1 J tool 1
31 31
3.5 × 1015 115000 0.93
m 3 s1 J tool 1
31 31 30 30
16 16
The shear modulus G at temperature T is given by G = G o p ( 1  C ' ( T  300)}; Dv = Dov exp( Qv/RT); D b = D0b exp( Qb/RT); R = 8 . 3 1 4 J tool 1 K 1 ; h is a f u n c t i o n o f cavity tip angle; Spo was measured at T = 0.6Tin.
65 TABLE 2 T h e range o f t h e variables e m p l o y e d in t h e n u m e r i c a l model 7
Variable
Value
Unit
Halfcavity spacing b
5 × 10 6
m
Initial n o r m a l i z e d cavity size do
0.1
F i n a l n o r m a l i z e d cavity size df
0.95
Temperature T
1 0 3 6 (0.6Tin)
K
G r a i n size L
10 4
m
SPo ~ 236 MPa
UNCONSTRAINED
6
Variables held constant
S
4.
3 2 CONSTRAINED 1
Range o f other variables N o r m a l i z e d applied stress
1.053.0
S/Spo
I
I
025
Cavitated boundary fraction f
0.050.75

Precipitatecoarsening rate P
10 15_ 10 2
s1
C
0500

ables, we assumed constant values for these parameters as indicated in Table 2. The effect of varying these parameters on tf, CMO and e~ will be specifically considered at the end of this section. The other parameters, namely the applied stress S, the fraction f of boundary facets cavitated, the precipitatecoarsening rate P and C, were varied over a wide, b u t plausible, range. The actual range for each of these variables is also indicated in Table 2. In eqn. (1), Sp0 represents the back stress due to particles alone since the solidsolutionstrengthening effect is taken care of by Ko. For the numerical computation, we shall assume that all the back stress (Sp0 = 236 MPa) arises because the dislocations overcome the precipitates by either bowing between or locally climbing over them. We shall n o t unduly worry a b o u t whether this assumption is reasonable since our primary objective is to illustrate the use of the proposed numerical model. Nevertheless, it should be noted that our assumption is such that the maximum possible effect of particle coarsening on Q, ef and CMG will be obtained.
4.1. Extent of constraint A necessary condition for the existence of an interaction between cavitation and micro
0.50
075
f Fig. 3. T h e b o u n d a r y b e t w e e n t h e c o n s t r a i n e d a n d t h e u n c o n s t r a i n e d cavity g r o w t h regimes in t h e norm a l i z e d s t r e s s  c a v i t a t i o n level (S/Spof) p l a n e (M = 1.0; T = 0.6Tin; b = 5 ~ m ; L  100 pro). T h e t h r e e lines c o r r e s p o n d t o t h r e e d i f f e r e n t a s s u m e d conditions.
structural damage experienced by the creeping material is that the cavity growth should be significantly constrained. A value of unity for M in eqn. (13b) should roughly correspond to the boundary between constrained and unconstrained cavity growth. At low values of f, a value of unity for M reduces the actual cavity growth rate b y a factor of 2 when compared with the unconstrained cavity growth rate (eqn. (13a)). In Fig. 3 the boundary lines between the constrained and the unconstrained regimes are indicated for Udimet 700 (T = 0.6Tm) in an S/Spof plane. The three lines correspond to three different assumed conditions. The t o p line (Sp0 = 236 MPa) corresponds to a situation in which cavitation is the only damage process. The middle line (Sp ~ Spo) represents a situation in which the precipitate coarsening rate is so high that the back stress reduces to negligible values compared with its initial value Spo soon after the start of the creep test. Finally, the b o t t o m line characterizes the boundary between the constrained and unconstrained regimes when SILC of 7' precipitates occurs in addition to cavitation and causes the constant A0 (in eqn. (34)) to increase from an initial value of A to 20A b y the end of creep test (thus the average
66
value of A0 is 10A). It is obvious from Fig. 3 that SILC is more effective in reducing the extent of the constrained regime than precipitate coarsening is. It should be remembered that A0 = 10A represents an extreme case and usually the effect of SILC will be less than that depicted in Fig. 3. The same situation applies in particle coarsening also. As a result, in the range of applied stresses of interest to us (S/Spo = 1.053.0), significant constraint on cavity growth will exist even in the presence of particle coarsening and SILC for reasonable values of f ( ~ 0.25). Nevertheless, the extent of constraint will be reduced (but still M ~ 1) b y these microstructural damage mechanisms. It should be noted that, in Fig. 3, as we move away and downwards from the boundary lines, the value of M increases and so does the constraint on cavity growth. So we can conclude that, for Udimet 700, t~, e~ and CMG will be significantly affected by coarsening and SILC, if the assumptions made by us are valid.
4.2. The effect o f cavitation In Fig. 4 the variation in the time t~ to fracture is plotted as a function of the normalized
10 t
~\ \\
~.u.
,o,
lO s
1
2
S/Spo
3
Fig. 4. T h e variation in t h e t i m e tf t o f r a c t u r e w i t h the normalized stress S / S p O at d i f f e r e n t c a v i t a t i o n levels f (b = 5 p r o ; do = 0.1; P = C = 0; T = 0.6Tin ; Sp0 = 236 MPa):    , u p p e r b o u n d a n d l o w e r b o u n d value o f tf.
applied stress S/Spo for different cavitation levels f. It has been assumed that neither precipitate coarsening nor SILC occurs to any significant degree, i.e. P = C = O. In the same figure, the upper bound as well as the lower b o u n d values of tf (eqn. (19) and eqn. (23) respectively) are also indicated. The figure indicates that tf decreases with increasing f at any stress level, the effect being most dramatic at the lowest stress values. The time to fracture also decreases with increasing applied stress, once again the effect being most significant at the lower stress levels. Both these observations can be explained with the help of eqn. (13). The parameter M decreases as either f or the applied stress increases. A decrease in M value, equivalent to reducing the constraint on cavity growth, causes the cavity growth rate to increase (eqn. (13a)) and hence reduces tf. The dramatic effect at low applied stress levels is largely due to the parameter [ 1 Sp0(1 f)/ S ( 1   f / ( I + M ) } ] " in eqn. (13b). At low applied stress S values, when Spo/S is close to unity, even a small change in the value of either f or S causes a dramatic change in the value of the above parameter. As a result, tf is also affected in a similar fashion. Figure 5 shows the variation in the Monkm a n  G r a n t constant CMG with applied stress, for different values of f. The upper b o u n d (eqn. (20)) and the lower b o u n d (eqn. (24)) values of CMG are also depicted in the figure. At low values of f ( ~ 0.25), CMG is fairly independent of the applied stress, except at S/Spo values close to unity. The dramatic decrease in the value of CMG at low S/S~o values comes a b o u t because the term (1Spo/S)'/ {1Spo(1f)/S} ~ in eqn. (20) decreases dramatically even with small changes in the value of S (or f). When the extent of cavitation is high ( f ~ 0.5), the dependence of CMG on S is quite similar to that of its lower b o u n d value (eqn. (24)), indicating the presence of minimal constraint on cavity growth. The dependence of the strain ef to fracture on the applied stress as well as on the level of cavitation is presented in Fig. 6. For f ~ 0.25, e~ is quite close to the upper b o u n d value predicted by eqn. (22) and further is independent of applied stress as is the upper b o u n d value. At higher values of f, e~ becomes increasingly stress dependent and further approaches the lower b o u n d value (eqn. (26)).
67 10I
I
This b e h a v i o u r is indicative o f a d e c r e a s e in c o n s t r a i n t w i t h increasing stress as well as increasing level o f c a v i t a t i o n .
UPPERLIMIT (CONSTRAINED)
4.3. Cavitation and coarsening
.~
• •
I 1
n
I 1.5
l 2
i 2,5
S / S pO
Fig. 5. The variation in the MonkmanGrant constant CMGwith normalized stress S/Spo and cavitation level f (b = 5 pm; d o = 0.1 ; P = C = 0; T = 0.6 Tm; Sp0 = 236 MPa). The upper and the lower bound values of CMG are also indicated in the figure.
I n this s e c t i o n t h e e f f e c t o f c a v i t a t i o n a n d coarsening on intergranularfracturerelated p a r a m e t e r s will be e x p l o r e d . In Fig. 7 t h e v a r i a t i o n in tf w i t h t h e p r e c i p i t a t e  c o a r s e n i n g rate P at t h r e e d i f f e r e n t n o r m a l i z e d stress levels (S/Spo = 1.05, 1 . 5 0 a n d 3.0) a n d t w o levels o f c a v i t a t i o n (f = 0 . 0 5 a n d 0.25) is p r e s e n t e d . T h e figure s h o w s t h a t , at all S a n d f values, t~ decreases w i t h increasing P value, o n c e a critical value o f P, d e n o t e d Pc, is exceeded. T h e m a g n i t u d e o f Pc increases w i t h increasing S/Spo as well as w i t h increasing f. F o r e x a m p l e , w h e n S/Spo = 1 . 0 5 a n d f = 0.05, Pc ~ 10z3/S. H o w e v e r , P ~ 10 e f o r S/Spo = 3.0 a n d f : 0 . 2 5 b u t , even at S/Spo = 3.0 a n d f = 0.25, t h e value o f t~ is still a b o u t a f a c t o r o f 15 higher t h a n its l o w e r b o u n d value, indicating t h e p r e s e n c e o f sizable c o n s t r a i n t . T h e o b s e r v a t i o n s d e t a i l e d a b o v e can be easily e x p l a i n e d in t e r m s o f eqn. (35). As in this e q u a t i o n , t h e p a r a m e t e r M decreases in magnit u d e as t h e p r o d u c t Pt increases. T h u s , at
0" [
UPPERBIXJNOISf/Si~,l.0S) I
UPPERBOUND f~ 0.25 ~ ~  . . . . ~ = 0
s0
I0~ IC~ uC~
I
~
,~ \
s/spo
2
z
f
Q~ t~
UPPERN
(T~/S~=3.0)
~
1~ .00
1 LOWERBOUND I 2
LO~[R Imum (sr/sl, o,l.OS) ~o3.
10"vs
S/Spo Fig. 6. The effect o f n o r m a l i z e d stress S/Spo and c a v i t a t i o n level f o n the strain ef t o fracture (b = 5 pro; d o = 0.1 ; P = C = 0, T = 0 . 6 T m ; Spo = 236 MPa). T h e upper and lower bound values of ef are also indicated in the figure.
Lo~m m~u.o (s../s.o.j.~,)
10"
.
l07
a 10.3
i
P (s 1} F i g . 7. T h e e f f e c t o f p r e c i p i t a t e  c o a r s e n i n g
rate P on
to fracture ( b = 5 p r o ; d o = 0 . 1 ; C = 0; T = 0.6Tin ; Spo  236 MPa):   , f = 0.05; . . . . , f = 0.25. Each line in the figure represents one combination of f and S/Spo values. the time
68 lower stress levels characterized by high t~ values, even when the precipitatecoarsening rate is low (i.e. P is low), the coarsening still affects tf considerably since the p r o d u c t Ptf is high. Alternatively, at high stress levels the same effect is obtained only when the value of P is high since t~ is low at these stress levels. In addition, since the extent of constraint is lower at higher stress levels, the effect of P on t~ becomes diluted still further. The fact that at a given stress level the effect of P on t~ is more dramatic at lower f values can once again be understood in terms of eqn.
(35). The effect of the precipitatecoarsening rate on the M o n k m a n  G r a n t constant CMG at different stress levels and at t w o levels of cavitation is indicated in Fig. 8. In general, an increase in the value of P causes the value of CMG to decrease. This behaviour can be traced to the reduction in constraint with increasing values of P. For the same reasons cited earlier, the reduction in CMG with increasing P is maximum at the lowest stress levels. Finally, the effect of P on the strain ef to fracture is indicated for a range of f values and at two stress levels in Fig. 9. The figure indicates that ef is largely independent of the
precipitatecoarsening rate P. This behaviour can be explained as follows. In the constrained limit the decrease in t~ due to coarsening is balanced exactly by the corresponding increase in the sample strain rate es0 (eqn. (21)) and hence ef is independent of P. At the other extreme, when the constraint is negligible, there is no interaction between the creep fracture process and the microstructural damage mechanism. Consequently, e~ is once again independent of P. Thus, irrespective of whether constraint exists or not, ef should be independent of P. The figure also shows that the absolute magnitude of ef decreases with increasing f and S/Spo values. This behaviour is entirely due to the reduction in the extent of constraint and its effect on cavity growth.
4.4. Cavitation and straininduced loss o f coherency Figure 10 shows the effect of parameter C on time to fracture at three stress levels (S/Spo = 1.05, 1.50 and 3.0) and at two f levels of cavitation (f = 0.05 and 0.25). It is obvious that the effect of C on tf is not as dramatic as the effect of P on t~ (Fig. 7). The magnitude of t~ decreases at the most by a
10.t tIPPER ~
IK~ONSTRAIN~J
]
S/SpO
i
102
UPPER BOUNO
/
~  " ~
o   ' ° ' ' ' 
"
3.00 . ~ 1.05 ~ " 1.05
/
3.00
I.=.
Z ~ 106.
%. % 1.05
ev" g.
C~ 0 E .QJ
z ,,v"
. . . . 10"w i I
,
I 15
I 2
I 215
S/Spo Fig. 8. The variation in the M o n k m a n  G r a n t c o n s t a n t CMG with t h e n o r m a l i z e d stress S/Spo for different values of P (0, 106 and 10 4) and f (0.05 and 0.25) (b ~ 5 p m ; d 0 = 0.1; C  0; T = 0 . 6 T m ; S p o = 236MPa): , f = 0 . 0 5 ;    , f = 0.25.
1015
LOWER" BOUNO I 1015
i
I 10 .7
• 'r

i
I 103
3.00I 101
P (s') Fig. 9. T h e effect of precipitatecoarsening rate P on the strain ef to fracture (b = 5 pro; d O = 0.1; T = 0.6Tm;C=O;SpO=236MPa): , f ~ 0.25;, f = 0.5; , f = 0.75. Each line in the figure corresponds to one combination of f and S/Spo values.
69 I0I UPPER LIMIT ICONSTRAINIED) 1011
IOZ
1.0S 10 t
10~

.
.
.
.
.
:Se" :/
UJ 1.05
o
, ~ 10 7
104
"OJ
iJ_ ~
~
~
~
~
~ t ~ _

laJ ~E 10 s
I0I
10 3 0
I 100
I 200
I 300
I /.00
10"
SO0
C Fig. 10. The dependence of the time to fracture on the parameter C for different assumed values of S/Spo and f(b = 5pro; do= 0.1;P= 0; T= 0.6Tin; Spo = 236 MPa):   , f= 0 . 0 5 ;    , f= 0.25.
f a c t o r o f 7 or 8 (at C = 500). H o w e v e r , m o r e i m p o r t a n t l y , t h e e f f e c t i v e n e s s o f C in reducing tf is t h e s a m e a t all stress levels. This is so b e c a u s e ef is i n d e p e n d e n t o f a p p l i e d stress (eqn. (22)) a n d h e n c e t h e p r o d u c t Cef w h i c h d e t e r m i n e s t h e e x t e n t o f r e d u c t i o n o f t~ is also stress i n d e p e n d e n t . T h e b e h a v i o u r described a b o v e is in s h a r p c o n t r a s t w i t h t h e d i f f e r e n t i a l e f f e c t i v e n e s s o f P w i t h regard to t h e a p p l i e d stress in r e d u c i n g tf (see Fig. 7). Figure 11 illustrates t h e e f f e c t o f C o n CMG a t d i f f e r e n t S/Spo a n d f values. A n increase in C value ( f r o m 0 to 500) causes t h e value o f CMG t o decrease. T h e d r a m a t i c decrease in CM6 a t l o w S/Spo values is a c o n s e q u e n c e o f t h e c a v i t a t i o n a n d its e f f e c t o n t h e p a r a m e t e r 1   Spo(1   f)/S a n d is n o t c a u s e d b y SILC. T h e e f f e c t o f C o n e~ is p r e s e n t e d in Fig. 12. T h e strain t o f r a c t u r e is i n d e p e n d e n t o f t h e value o f C since t h e decrease in t~ caused b y increasing C is e x a c t l y c o m p e n s a t e d f o r b y t h e c o r r e s p o n d i n g increase in t h e s a m p l e strain rate. A t l o w f a n d S levels, ef is indep e n d e n t o f b o t h f a n d S b e c a u s e u n d e r such c o n d i t i o n s t h e c o n s t r a i n t is severe e n o u g h f o r eqn. (22) t o a p p l y . A t higher f a n d S values, e~ is stress as well as f d e p e n d e n t b e c a u s e o f increasing lack o f c o n s t r a i n t o n c a v i t y g r o w t h .
I
I
I
ts
20
2s
30
S/Spo
Fig. 11. The effect of parameter C on the MonkmanGrant constant CMG at different stress levels (b = 5 pro); d o = 0.1 ;P = 0; T= 0.6Tin ; Sp0 = 236 MPa): ~, f = 0 . 0 5 ;    , f = 0.25.
f ~ 0.25
f ~ 025 ,v
.
.
.
.
.
.
I ~


"f
ii ~
p
"o.~

f : 0.75 2
IX p,
f ,
100
,
200
J
J
300
~,00
075 500
C
Fig. 12. The variation in the strain ef to fracture with increasing value of C (b = 5 pro; d o = 0.1; e f(u b) = 3.45; T = 0.6Tin ; P = 0; Spo = 236 MPa):   , S/Spo = . 1 . 0 5 ;    , S/Spo = 3.00. Each line represents one S/Spo and f value.
4.5. The effect o f b, L, do and d~ So far in o u r discussion, we have a s s u m e d o n e c o n s t a n t value f o r i n t e r c a v i t y spacing b, grain size L, initial c a v i t y size do a n d final
70
cavity size dr. The effect of varying these parameters, without relaxing the assumption that these parameters are time, stress and temperature independent, on intergranularfracturerelated parameters will be considered in this section. First let us consider the parameters b and L. As in eqns. (13) and (35) the parameter M which determines the extent of constraint is inversely related to b2L. Thus, the smaller the value of b and L, the larger is the value of M and hence the more severe is the constraint. Under such conditions, tf, CMG and e~ are all directly proportional to b/L. However, when b2L is large, there is minimal constraint on cavity growth. The time to fracture and CM6 are proportional to b 3 and further are independent of grain size L when such conditions are realized during the creep test, b u t ef is still proportional to b/L (eqn. (26)). To conclude, a variation in the assumed value of b causes a corresponding change in the values of t~, CMG and el, irrespective of whether constraint exists or not. In contrast, t~ and CMO will depend on the grain size only when the cavity growth is constrained. The effects of d~ and do on tf, e~ and CM6 are straightforward in the sense that either increasing d~ (dr(max) = 1.0) or decreasing do (d0(min) = 2Rs(1 d2)/bS) causes the value of t~, CM~ and e~ to increase.
5. D I S C U S S I O N
In Section 4 the effect of precipitate coarsening and SILC on intergranularfracturerelated parameters was considered separately. A more realistic situation is one in which both these microstructural damage mechanisms act
TABLE
in unison with the cavitation damage mechanism. Before such a combined effect can be analysed usefully, the reasonable ranges of values for f, P and C have to be determined on the basis of data in the literature. The experiments of Lindborg [ 20] with an austenitic steel indicated that, just prior to fracture, the fraction f of cavitated boundaries never exceeded 0.08. Other investigators have also reported [32, 33] low f values in the range 5%20%. Thus, any value in the range 0  0 . 2 5 appears to be reasonable for f. Table 3 indicates the value of P obtained from the literature [2, 3, 28] for three different nickelbase superalloys and also the temperature at which the value of P was obtained experimentally. The corresponding value of P at T = 0.6Tin (the temperature of interest to us) was estimated by assuming that the temperature dependence of P is characterized by an activation energy Q of 268 kJ mo11. This value is an average of the various values reported in the literature [2, 3, 28] for Q. From Table 3 a value in the range 10610 5 appears to be reasonable for P. Since the superalloy of interest to us (Udimet 700) has a P value close to the upper b o u n d of the range assumed, a broader range of P values (10610 4) will be utilized for the subsequent discussion. Finally, on the basis of the data presented by Dyson and McLean [4] for IN738LC, MARM 246 and Nimonic 80A, a value in the range 2 0 250 appears to be reasonable for C if it is assumed that SILC is responsible for all the observed creep acceleration with increasing strain. N o w that the reasonable range of values for f, P and C are know.n, their combined effect can be analysed. Figure 13 shows the variation in tf with the normalized stress S/Spo for various combinations of P and C values. It
3
Values from the literaturefor the precipitatecoarsening rate P
Material
N i m o n i c 105 IN738 U d i m e t 700
P
Temperature
(s1)
(K)
8.9 × 10 5 5.6 × 10 7 1.8 × 10 4
1123 1023 1144
a O n the assumption that Q(coarsening) = 268 kJ tool1 and T m = 1726 K.
Reference
Estimated a value of P at T = O . 6 T m
3 2 28
7.3 x 10 6 7.5 × 10 7 8.8 x 10 6
71
I
t0 l~J
10 v
10'
uJ
¢Y ~.
::::) I
10 1
re" u..
c
~'~
10 ~
u
O F,w
105
10s
I 10,__, ~o5
,
~ 0
,
~so
'
~o0
I
2so
'
300
S/S?o
Fig. 13. T h e v a r i a t i o n in t h e t i m e t o f r a c t u r e w i t h n o r m a l i z e d s t r e s s S / S p o (b = 5 p m ; d o = 0 . 1 ; T = 0 . 6 T i n ; Spo = 2 3 6 M P a ; f = 0 . 0 5 ) : c u r v e a, P = C = 0; c u r v e b, P = O, C = 2 5 0 ; c u r v e c, P = 1 0  4 s1, C = 0; c u r v e d, P = 1 0  4 s1, C  2 5 0 .
was assumed that f  0.05. Curve a represents the situation when coarsening and SILC are absent. Curve b is valid when SILC is the only microstructural damage mechanism (P  0; C = 250). Thus, SILC reduces the t~ value at all stresses b u t does not alter the stress dependence of tf. Curve c illustrates the variation in t~ with S/S,o when coarsening is the only microstructural damage mechanism (P ~ 104/8; C = 0). It is obvious that coarsening is most effective in reducing tf only at low stress values. As a result, the stress dependence of tf is lower in the presence of coarsening, especially at low stresses close to Sp0. In the presence of coarsening as well as SILC, curve e defines the maximum probable effect on t~. This curve is quite similar to curve c, thereby indicating that, for the assumed values of C and P, coarsening is the dominant damage mechanism. Figure 14 is quite similar to Fig. 13 except for the fact that f has now been assumed to be equal to 0.25. The higher f value should cause the constraint on the cavity to decrease with a corresponding decrease in the influence of C and P on t~. Curves a  d in Fig. 14, which correspond to various combinations of C and P, are relativety closer to each other than the curves in
)0 !
1,05
1.20
L50
2.00
2.50
3.00
S/Spo Fig. 14. The variation in the time to fracture w i t h normalized stress S / S p o (b = 15p r o ; d o = 0 . 1 ; T = 0 . 6 T i n ; Spo = 2 3 6 M P a ; / ' = 0.215): c u r v e a, P = C = 0; c u r v e b, P = 0, C = 2 5 0 ; c u r v e c, P = 1 0 . 4 s 1, C = 0; c u r v e d, P = 1 0  4 s 1, C = 2 5 0 .
Fig. 13. Therefore, the influence of C and P on tf is much less, in line with our expectations. From Figs. 13 and 14 it is obvious that precipitate coarsening significantly reduces t~ only when S ~ 1.5 Sp0 (for f ~ 0.25). Thus, experiments have to be carried o u t at stress levels close to Sp0 if it is required to study the interaction between precipitate coarsening and intergranular creep fracture. Unfortunately, most of the experiments reported in the literature pertain to stress levels considerably above Sp0. This is so mainly because the lower fracture times (less than 107 s) at such stress levels are easier to handle experimentally. Nevertheless, it must be noted that the actual operating stress in many of the applications in which the superalloys are preferred is likely to be quite close to Spo. Thus, under actual operating conditions, coarseninginduced reduction in t~ might become quite significant. If that is the case, extrapolation of high stress creep data to lower stresses should be definitely avoided, since such a procedure will result in an overestimation of the creep life. At this juncture, it is worth noting that our assumptions are
72 such that the maximum possible effect of coarsening on t~ is obtained in the numerical computations. In reality, the effect of P on t~ can be considerably less than predicted, especially if only part of the back stress is due to particles or if the Orowantype mechanism is favoured over the cutting mechanism only after a certain "incubation" period. Finally, in certain alloys, even when TEM observations have clearly shown that either Orowan bowing or local climb is the operating mechanism, the experimentally measured back stress Sp0 has been f o u n d to be less sensitive t o particle size than the models for either bowing or local climb would predict. This discrepancy can be explained by modifying the original B r o w n  H a m model [10]. One such model due to McLean [12] predicts that the threshold stress for climb should be relatively insensitive to particle size. If such models are applicable to some superalloy systems, in those alloy systems the particle coarsening will have little or no effect on intergranular fracturerelated parameters. Needless to say, further work needs to be done regarding the modelling of the local climb of dislocations over the precipitate. It is obvious from Figs. 13 and 14 that the effect of SILC is to reduce t~ by more or less the same factor at all stress levels. Consequently, the stress dependence of t~ is hardly affected by SILC, b u t we assumed that the creep value of C will remain constant all through the creep test. This assumption will be valid only in the absence of particle coarsening because coarsening and the associated increase in the precipitate size reduces the value of C [13]. Thus the value of C should decrease progressively with increasing time during the creep test. It follows then that the reduction in tf due to SILC will be the least at the lowest stress and maximum at the highest stress. However, this effect is expected to be of secondary importance.
6. CONCLUSIONS A new model incorporating the effect of cavitation, precipitate coarsening and SILC on the time to fracture, the strain to fracture and the M o n k m a n  G r a n t constant has been formulated. The usefulness of the model has been illustrated b y considering a typical
nickelbase superalloy (Udimet 700) with known properties and by quantifying the effects of cavitation, coarsening and SILC in this particular alloy. It has been shown that both the time to fracture and the M o n k m a n Grant constant are reduced significantly in the presence of various forms of damage. In contrast, the strain to fracture is largely unaffected by these damage mechanisms.
ACKNOWLEDGMENT The author wishes to thank Dr. P. Rama Rao, Defence Metallurgical Research Laboratory, for his constant encouragement and support and for granting the permission to publish this paper.
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