Intergranular creep cavitation with time-discrete stochastic nucleation

Acra metal/. Vol. 34, No. 7, pp. 1433-1441,1986 Printed in Great Britain

INTERGRANULAR TIME-DISCRETE

oool-6160/86$3.00+ 0.00 Pergamon Journals Ltd

CREEP CAVITATION WITH STOCHASTIC NUCLEATION

S. J. FARIBORZ, D. G. HARLOW and T. J. DELPH apartment

of Mechanical Engineering and Mechanics, Lehigh University, ~thlehem, (Received 14 December

PA 18015, U.S.A.

1984; in revised form 25 Juiy 1985)

Abstract-An empirical model for the time-discrete stochastic nucleation of intergranular creep cavities is proposed. Nucleation is assumed to occur randomly in time, with the temporal behavior being governed by an inhomogeneous Poisson process. Based upon experimenta evidence, the mean function of the Poisson process is taken to be a power-law function of time. The nucteation model is then implemented in conjunction with a recent analysis which treats the growth of randomly spaced populations of voids in a simplified bicrystal configuration. Numerical simulations with the resulting model indicate, among other things, that both the times-to-failure and the cavity radii are distributed according to a Weibull cumulative distribution function. Both predictions have qualitative experimental support. Moreover the presence of nucleation is found to reduce the time-to-failure by creep rupture by approximately a factor of two to eight compared to simulations without nucleation.

R&am~Nous proposons in mod&e empirique pour la germination stochastique disc&e des cavitks de fluage intergranulaire. Nous supposons que la germination se produit alkatoirement dans le temps, le comportement temporel &ant gouvernt par une loi de Poisson h&t&rog&e. En nous appuyant sur des resultats exfirimentaux, nous admettons que la moyenne de la loi de Poisson varie en fonction du temps selon une loi de puissance. Nous dkveloppons ensuite le modble de germination, suivant une analyse r&ente qui traite la croissance de populations de cavites dispostes aleatoirement en une configuration de bicristal simplifite. Les simulations numkriques effect&es & l’aide de ce modtle montrent, entr’autres chases, que la d&e de vie g la rupture et le rayons des cavit& pr&entent une fonction de distribution cumulative de Weibull. Ces pr&visions sont confirm&es par des r&.ultats exp&imentaux qualitatifs. De plus, nous trouvons que la germination diminue la durCe de vie B la rupture au fluage d’un facteur de deux $ huit environ, lorsqu’on compare avec des simulations sans germination. Zusammenfassung-Es

wird ein empirisches Model1 filr die zeit-diskrete stochastische Keimbildung von intergranularen Hohltiumen beim Kriechen vorgeschlagen. Es wird an~nommen, daD die Keimbildung zufaig in der Zeit auftritt, wobei das zeitliche Verhalten durch einen inhomogenen Poisson-ProzeB bestimmt wird. Nach experimentellen Hinweisen wird fiir die mittlere Funktion des Poisson-Prozesses eine Potenzfunktion der Zeit angesetzt. Das Keimbildungsmodell wird dann in Verbindung mit einer neueren Analyse aufgestellt, welche das Wachstum zut?illig verteilter Hohlrlume in einer vereinfachten BikristallAnordnung behandelt. Die numerische Durchrechnung dieses Modelles ergibt neben anderem, daD sowohl die Bruchstandzeiten als such die Radien der Hohlrlume der kumulativen Verteilungsfunktion von Weibull gehorchen. Beide Voraussagen sind durch Experimente qualitativ gestiitzt. AuDerdem ergibt sich, daB die Zeit bis zum K~~hbruch durch die beriicksichtigte Keimbildung urn einen Faktor zwei bis acht gegeniiber den Simulationen ohne Keimbildung verkiirzt wird.

INTRODUCTION above approximately half the melting point, the nucleation and growth of intergranular voids is the primary phenomenon responsible for creep rupture in metals. Void growth at these temperatures is generally thought to occur as the result of the diffusion of vacancies from the grain boundary to the surface of the void. This process was first quantified by the well known Hull-Rimmer model [I], which has since been refined by other investigators [2-4]. Such void growth models have now achieved a fairly substantial degree of success in predicting some of the salient features of the creep rupture process. The process by which voids form or nucleate, is, however, not as well understood, although it has also At t~m~rature~

A.M. 3417-s

been the subject of extensive study. A good deal of work in this area has been reviewed in survey papers by Perry [5], and by Yoo and Trinkaus [6]. As was pointed out in the latter of these papers, two different mechanisms have commonly been invoked to model void nucleation: microcracks nucleated as a result of grain boundary sliding (athermal nucleation) and ithermal void nucleation via vacancy clustering. Both of these mechanisms depend upon the existence of stress concentrations at sites such as grain boundary irregularities, grain boundary inclusions and triple junctions. Due to the difficulties involved, precise experimental measurements of nucleation rates are quite scarce in the literature. However a number of measurements of the increase in the number of voids of observable size with time have been made. This of

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FARIBORZ ef al.: NUCLEATION OF INTERGRANULAR CREEP CAVITATION

course is not quite the same thing as measuring the nucleation rate, but the two quantities are obviously closely related. A number of these experimental results are discussed by Greenwood [7], who notes that, at constant applied stress, the data have the common form N(t) = cl@

(1)

reiation given by equation (1). An assumption basic to the model is that nucleation proceeds in time as an inhomogeneous Poisson process [lo]. Although there appears to be no experimental evidence which would either support or contradict this assumption, the Poisson process has some attractive features which lead to the sort of behaviour which one would intuitively expect of a nucleation process. Furthermore the Poisson process is frequently used to model the probability of occurrence of other phenomena which appear to have statistical characteristics similar to void nucleation, e.g. the random emission of Alpha particles from a radioactive materiai [lo]. From a mathematical standpoint, the Poisson process is a continuous time process with a discrete state space in the time domain 0 < t < co, where the state space counts the nucleation events. The inhomogenous Poisson process can be completely characterized by the following two properties:

where N is the number of observed voids and t is the elapsed time. The quantities tl and /? are constants at a given stress level. Greenwood argues for the value /i = 1 which agrees with much of the experimental data cited by him. Under steady-state creep conditions, a vaiue of /I = 1 would indicate a linear relationship between N and strain. Lower values of /I, however, have also been reported, e.g. Gittins [8], who obtained /i =OS, and Chen and Argon 191, whose data yield values of 8 between 0.38 and 0.74. Some support for the value @= 1 can also be found in the thermal nucleation model put forward by Yoo (i) the mean function of the process p(t) is an and Trinkaus [6], if it is assumed that variations in the increasing continuous function of time t, with grain boundary normal traction due to grain boundp(O)=% ary sliding or variations in the chemical potential may (ii) the number of events with epochs in disjoint be neglected. Then the grain boundary normal tractime intervals are independent random variables, tion may be computed simply by multiplying the The first property simply states that the mean, or remote stress by the term (1 -A,), where Ar, the average, number of nucleated voids increases concavity area fraction, is the ratio of cavitated to total tinuously with time. This is a quite reasonable asgrain boundary area. As long as A, is sufficiently sumption which has substantial experimental supsmall, as it is over most of the lifetime of the port, as was noted in the Introduction. The second specimen, the grain boundary normal traction during property implies that the number of void nucleations this period will be essentially equal to the constant in disjoint time intervals are independent of each remote normal stress. Because in the model of Yoo and Trinkaus the nucleation rate depends only upon other. In other words, the number of nucleations the normal traction and not upon time, a linear occurring in a given interval of time is completely independent of the number of nucleations outside variation with time in the number of nucleated voids that time interval. This latter assumption may not be would be predicted so long as the grain boundary normal traction remained approximately constant. in complete accord with the physics of the void In the present paper we propose a stochastic model nucleation process, because of effects such as exhausfor cavity nucleation based upon equation (I), which, tion of void nucleation sites and the influence of as noted, is itself based upon available experimental previously nucleated voids upon conditions at potenevidence. In accordance with reasons to be set forth tial nucleation sites. If, however, it is assumed that subsequently, nucleation is assumed to proceed as an these effects are not substantial, then (ii) represents a inhomogeneous Poisson process [lo]. This leads to a reasonable postulate. Before proceeding, it is worth model which envisions nucleation as occurring at noting that the sequence of assumptions given here discrete, random points in time. Because nucleation can be turned around so as to start with (i) and (ii) actually seems to occur in this manner, such a model as the basic postulates. If this is done, then it can be is felt to be in at least qualitative agreement with shown, [IO], that any process satisfying (i) and (ii) is physical reality. necessarily an inhomogeneous Poisson process. The resulting nucleation model is then used in Based upon the experimental results outlined in the conjunction with recent work of the authors [ill, previous section and summarized by equation (1), we dealing with the growth of randomly spaced popuassume that the average number of nucleated voids lations of voids, to make estimates of the time-tohas a power-law dependency upon time. Specifically, failure of a simplified bicrystal configuration. we assume that the mean function for the number of nucleated voids for the process is given by NUCLEATION MODEL In this section, we develop a stochastic, timediscrete model of cavity nucleation which represents a generalization of the deterministic, time-continuous

I*(t) = at*+ ‘/(b + 1); - 1 < b < 0.

(2)

The power-law form for the mean p(t) given in equation (2) is one of the simplest possible choices for an inhomogeneous Poisson process, and hence has often been used to model various physical processes.

FARIBORZ et al.: NUCLEATION OF INTERGRANULAR

It should be noted that when b = 0, p(t) becomes a linear function of 1. In this case, the Poisson process becomes homogeneous. The mean function p(t) is related to the intensity function of the process, n(t), by ’ l(s) ds.

p(t) = Equations

L/2

L/2 X x,

(3)

s0 (2) and (3) then give

I_ :,ilF

/

n(t) = at*.

(4)

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CREEP CAVITATION

R 2r,

Consequently, the Poisson probability that the number of events, or nucleations, in the interval (t,, t2) equals k is Pr{N(t,) - N(t,) = k} = exp{ - W)

Fig. 1. Distribution of voids on grain boundary of bicrystal.

- ~c(~)lH~(fd -

Ml)lk/k! (5)

The conditional probability density function (p.d.f.) of the inhomogeneous Poisson process is defined as

0 < t, < f,+,

i = 1,2,3,.

.

with the p.d.f. in the initial time interval being given by fr,(tl) = A(t,)exp ( - [;’ M)di).

(7)

The integration of the p.d.f. yields the conditional cumulative distribution function (c.d.f.) which is F r,+,ir,(tz+,Iti)=

I,+I s I,

(8)

.h,+,I~,(tlti)dt

i = 0, 1,2, 3,

Equations

(6)<8) and (4) yield

FT,(f,) = 1 -exp[ -at:+

‘/(b + l)]

(9)

x &,+,rr,(‘~+~l’J = 1 -exp[ -a(tl++, -t;+‘)/(b

+ l)]

(10)

The random times at which void nucleation occurs may now be obtained from equations (9) and (10). These are t, = {(b + l)[ -ln(l 4, I = {(b + I)[ -Ml

-FT,(f,))]/n}“‘*+‘)

(11)

-FT, +,ir,

x (ri + 11t,))]/a + t;+ I}‘/(*+ ‘)

(12)

IMPLEMENTATION OF MODEL

In earlier work [l 11, the authors considered the growth of a one-dimensional configuration of randomly spaced voids by a quasi-equilibrium diffusive growth mechanism similar to the standard HullRimmer model. In this section, we extend this work to take account of the effects of nucleation by making

use of the stochastic nucleation model developed in the previous section. Due to the numerical difficulties involved in treating a two-dimensional distribution of voids, we consider the simplified, one-dimensional configuration shown in Fig. 1, consisting of a perfect bicrystal of length L containing N voids placed randomly along the grain boundary. The number of voids N will in general vary with time because of sintering, coalescence, and, in the present case, nucleation. The bicrystal is assumed to be loaded at t = 0 by a remote uniform tensile stress (T applied transversely to the grain boundary. At the instant of load application, a given number, No, of randomly located voids are assumed to form spontaneously. The initial void locations are assumed to follow a uniform distribution. With respect to a coordinate system located at the center of the bicrystal, the center of the i-th void is placed at x = x, and spans a distance of 2ri along the grain boundary. The voids are further assumed to be initially equisized, cylindrical through the thickness of the bicrystal, and sufficiently long so that the variation of any quantity on the thickness direction may be neglected. Thus the problem becomes onedimensional. A detailed analysis of the kinetics of void growth in such a system by a quasi-equilibrium diffusive growth mechanism was presented in [ll]. This analysis led to a system of N coupled, first order, nonlinear differential equations whose numerical solution gave the growth of the N void radii as functions. Void nucleation was, however, explicitly excluded. In this section, we modify this analysis by assuming that voids are nucleated discretely in time according to the stochastic process discussed in the previous section, with the random nucleation times given by equations (11) and (12). The nucleation sites are moreover assumed to be randomly distributed according to a uniform distribution, subject to the restriction that nucleation must not occur inside a pre-existing void.

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FARIBORZ et al.: NUCLEATION OF INTERGRANULAR CREEP CAVITATION

Because of the sparsity of experimental nucleation data, the estimation of the constants a and b required in the Poisson intensity function, equation (4), is somewhat problematic. In the absence of reliable data, two values of b were selected: case (A) with b = -0.59 and case (B) with b = 0. If we consider N(t) as given by equation (1) and the mean function p(t) given by equation (2) to represent the same quantity, then these two values of b correspond respectively to /I = 1 + b = 0.41 and 1. As noted in the Introduction, these values appear to lie near the lower and upper limits of observed values. Values of the constant a were then selected by a trial-and-error process so as to lead to calculated values of void number density at failure which are in approximate agreement with experimentally observed values of around 1 x lO’m_’ [9,12]. These values were a = 2.5 x 10m4for case (A) and a = 2.505 x 10-i’ for case (B). It is worth noting that the values of a and b may in general be expected to vary with temperature and applied load. In this paper no attempt has been made to clarify these dependencies. Two further cases were also considered, case (1) with a large number of initial voids and case (2) with a small number. Specifically, in case (1) we assume that 50 voids form spontaneously at the instant of load application, while in case (2), only three voids form initially. The general void growth equations developed in reference [l l] remain valid in the present case. The numerical techniques used for their solution were applied here as well, using the same material and geometric constants. Briefly, these were the free surface energy y, = 1 J/m3, the atomic volume Tr = 1 x 1O-29m3, the tip angle x = 75”, the length of the bicrystal L = 5 x 10m4m, the initial radius of nucleated voids r = 5 x 10e9 m, and the remote applied stress e = 250 MPa. The criterion for failure was taken to be A, = 0.5 where A, is the fraction of grain boundary covered by cavities. The same criteria for void sintering and coalescence used in reference [l l] were applied here also. Using the Monte-Carlo method, 25 different simulations for each of the four cases obtained by combining cases (A) and (B) with cases (1) and (2) were carried out [the combinations will henceforth be referred to as case (Al), etc.]. The results of the simulations are expressed in terms of a nondimensional time-to-failure defined as 7, = t&,S,a/(KT)

(13)

where D,, is the grain boundary diffusivity, & the grain boundary thickness, K is Boltzman’s constant Table

1. Lowest,

Case

No

b

Al A2 Bl B2

50 3 50 3

-0.59 -0.59 0.00 0.00

-3

A Case (811

20

0.745 1.176 1.096 2.838

x x x x

10’2 IO” IO’* IO”

2.2

2.4

2.6

2.6

3.0

3.2

In ( i, x lo-“)

Fig. 2. Weibull plots of failure times for Cases (Al) and (RI). and T is the temperature. As was the case with the simulations reported in reference [l 11, the computed times-to-failure exhibited substantial amounts of scatter. The table below gives the mean nondimensional time-to-failure for each of the four cases, as well as the lowest and highest failure times. The observed variations for time-to-failure in each case were analyzed by using the Weibull cumulative distribution function (c.d.f.) to model the distribution of the times-to-failure. The analysis here parallels that in Ref. [ll]. The two-parameter Weibull c.d.f. used here was F(1) = 1 - exp{ -(t/COP}

(14)

where p is the shape parameter and a is the scale parameter. We order the failure times, T,, . . . ,T, in magnitude from smallest to largest, Tr,j < Tr21 < * 9 . < T,, . Then the cumulative probability plotting point Pi associated with the ith ordered failure time Tt,l is assumed to be Pi = i/(n + 1)

(15)

which is the mean value of the fraction of the sampled values that fails prior to Tril. Figures 2 and 3 are the plots of (T,,], Pi) on Weibull probability paper or of on linear paper for cases {M~iJ,

14

mean, and highest Lowest 7f

o Goss IA11

A

-Ml

pi)l}

time-to-failure

Mean tf 1.077 3.306 1.571 3.834

-

x x x x

10’2 10” 10’2 lOI

Highest 1.422 4.709 2.451 5.240

x x x x

7, IO” 10’2 10” IO’*

FARIBORZ

o

Case

et al.: NUCLEATION

OF INTERGRANULAR

2.0

2.4

(A2)

3.2

2.8

In

4.4

40

3.6

( 7I x 16”)

Fig. 3 Weibull plots of failure times for Cases (A2) and (B2).

(Al), (’Bl) and (A2), (B2), respectively. It is apparent that in all four cases the Weibull c.d.f. fits the data quite well. The estimated Weibull c.d.f. was obtained by finding the linear least squares fit to the data. The estimate p for p is the slope of the linear least squares fit, while the estimate of a can be obtained from the Y-intercept of the fit. Given in Table 2 below are the resulting Weibull parameters and the measures of central tendency obtained from the estimated c.d.f. for the four cases. The average of the time-to-failure for the four cases obtained from the 25 simulations were given previously, and they are quite consistent with the estimated values of the true mean p. In order to check the appropriateness of the Weibull c.d.f., 90% confidence bands for the data were constructed using the technique outlined in Ref. [13]. The times-tofailure for all simulations were found to lie inside the confidence bands, which indicates that the Weibull c.d.f. can be used with confidence to represent the failure data. Some further insight on the results of the simulations may be obtained by an inspection of Fig. 4, which shows the progression from initial conditions Table 2. Weibull

Darameters

DISCUSSION In the present work, we have developed a stochastic, time-discrete model of void nucleation based on the assumption that voids are nucleated in time as

and measures ,x

Al A2 Bl 82

1.163 3.596 1.706 4.090

x x x x

IO’* IO’* lo’* 10’2

5.578 5.044 5.111 7.010

1437

of a typical simulation for the case (Al). Here the location of the voids is shown on the vertical axis, while the diameter of each void is shown in the horizontal axis. Figure 4(a) shows the initial conditions, while Fig. (4b) shows the situation just prior to failure, with the nucleated voids being shown as dashed lines. Despite the fact that the simulation began with 50 randomly spaced voids (some of which sintered as time progressed), the nucleated voids appear to be dominant. This was found to be the case in most, if not all, of the 50 Case (1) simulations conducted. It may be noted from Fig. 4(b) that the distribution of void sizes just prior to failure shows a considerable amount of variation, due both to the effects of nonperiodic spacing and random nucleation in time. Figure 5 shows this distribution plotted on Weibull paper, along with a least-squares linear fit to the data. It may be seen that the data closely follow a Weibull distribution. Figure 6 shows how this distribution evolves from initial conditions for a typical Case (Al) simulation. At f = 0, all the voids are assumed to be equisized, which implies no scatter and p = co, and hence the resulting distribution is represented by a vertical line on the Weibull plot. As time progresses, the distribution inclines to the right as the distribution becomes more scattered, and also shifts to the right as the mean void size increases. Figure 7 shows the variation in number of voids with time for typical Case (Al) and (Bl) simulations. The sintering process dicussed in Ref. [1 1] results in a very rapid decrease in the number of initial, nonnucleated voids. However as time progresses, the number of voids begins to continuously increase on the average, despite a number of fluctuations due to sintering and coalescence of voids. Figure 8 shows a similar plot for the Case (2) simulations with three initial voids. Also plotted here are the mean functions p(t) for the Poisson processes for Cases (A) and (B) as given by equation (2). In the absence of any tendency for sintering to occur, the N vs t data for these two cases would tend to fluctuate randomly about the mean function. However the sintering and nucleation processes compete with each other, the one removing and the other adding voids. Hence the N vs t data from the simulations tend to fall considerably below the mean functions.

A Case (821

-55

CREEP CAVITATION

1.075 3.293 1.568 3.830

I

x x x x

10’2 IO’* IO’* lOI

of central ,

0.214 0.698 0.338 0.613

tendencies I

x x x x

IO” lOI lOI IO’*

19.9% 21.2% 21.6% 16.0%

1438

FARIBORZ et al.: NUCLEATION OF INTERGRANULAR CREEP CAVITATION -

--

Void

--

Ortginal void Nucleated void

(bl

diameter

Fig. 4. (a) Typical initial configuration (dia. x 20) for Case (Al). (b) Configuration just prior to failure.

a Poisson process. As was noted earlier, this assumption depends upon two underlying assumptions, first that the number of nucleated voids increases continuously with time, and second that void nucleation in a given time interval is unaffected by nucleation events occurring outside the given time interval. Although the latter of these assumptions is not completely correct from a physical standpoint, it seems likely to represent a fairly reasonable approximation to the actual situation, and hence the assumption of a Poisson process is not without justification. There is, moreover, the practical consideration that if the assumption of a Poisson process is abandoned, then the determination of a more appropriate

temporal distribution function becomes highly problematical. Based upon the experimental evidence discussed earlier, the mean function for the Poisson process was taken to be a power-law function of time. The parameters characte~zing the process, namely the constants a and b, were estimated from available experimental data. The nucleation model was then combined with an earlier model developed by the authors which describes the growth of randomly spaced voids in a simplified bicrystal configuration. Numerical simulations of the resulting model were conducted to obtain estimates for the time-to-failure of the bicrystal.

FARIBORZ

et al.: NUCLEATION

OF INTERGRANULAR

A

40

0

CREEP

I 2

1439

CAVITATION

I 6

I 4

i

I 6

I 10

I 12

x lo-"

Fig. 7. Variation in number of voids with time for typical Case (Al) and Case (Bl) simulations.

-51 -7.0

I -6.6

I -6.2

I - 5.6

I -5.4

1

I - 5.0

- 4.6

In Lp)

Fig. 5. Weibull plot of distribution of cavity radii just prior to failure. Among the principal results of the present study are the fact that both the time-to-failure data and the void radii appear to be distributed according to a Weibull cumulative function. Moreover the results are found to be strongly affected by the existence of a void sintering process active throughout most of the specimen lifetime, as well as being somewhat less affected by a void coalescence process active late in the specimen lifetime. Several of these results parallel 3.6

those obtained in the authors’ earlier study, Ref. [ 111, in which nucleation was not taken into account. The times-to-failure, for example, reported in [l 11,were also found to obey a Weibull c.d.f. As noted there, the Weibull c.d.f. appears to fit to a reasonable degree the results of some carefully conducted creep rupture tests carried out by Garofalo et al. [14]. Hence this finding has at least qualitative experimental support. Interestingly enough, the scatter in the predicted times-to-failure was reduced from 45% in the nonnucleating case considered in [l 11, with N,, = 50, to a value of approx. 20% in the present Case (1) simulations (Table 2). This result is somewhat surprising, because the inclusion of random nucleation would be expected to add an extra degree of randomness to the results. The Weibull c.d.f. was also found to describe the variations on void radii resulting from nonperiodic spacing and random nucleation. This result is in excellent qualitative agreement with some recent experimental data of Liu et al. [IS]. Using automated image analysis techniques, large numbers of creep

80

0 Case

(A2)

A Case

(82)

60 zz 40

20

0

In(p)

Fig. 6. Evolution

with time of cavity

radii distribution.

6

16

24

32

40

46

Fig. 8. Variation in number of voids with time for typical Case (A2) and Case (B2) simulations, along with mean functions for Poisson processes.

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FARIBORZ et al.: NUCLEATION OF INTERGRANULAR CREEP CAVITATION

cavities were measured on polished sections of crept specimens of type 304 stainless steel. Statistical analysis of the results indicated that variations in measured cavity cross-sectional area were well approximated by a Weibull distribution. No quantitative comparison can be attempted here because the experimental data were taken on a complicated polycrystalline material, and the present work concerns a simplified bicrystal configuration. However the qualitative agreement between experiment and one of the major features of the results is felt to be quite encouraging, both with regard to the basic validity of the assumptions underlying the model and also with regard to its predictive ability. In both the present study and that reported in Ref. [ll], the simulations indicated the presence of a marked tendency towards void sintering throughout most of the lifetime of the specimen, as well as some tendency towards void coalescence late in the specimen lifetime. Hence it was found that a substantial fraction of newIy-nucIeated voids did not survive until the time of specimen failure, and hence had no direct effect upon the failure time. Despite this fact, it was found in most cases that the largest voids existing just prior to failure were voids which had nucleated subsequent to t = 0, not those formed initially at t = 0. By far the most evident effect of the sintering and coalescence processes, however, was their effect upon the mean failure times as compared to the case in which the voids are taken to be periodically spaced and nucleation is ignored. As reported in [ll], the time-to-failure for the periodically spaced case is r,= 0.542 x 10”. This value is some two to seven times less than the mean times-to-failure for any of the present cases, which are in turn [for case (1) for 50 initial voids] two to eight times lower than the mean time-to-failure for the comparable (No = 50) nonnucleating case studied in [1 I]. The difference between the results reported in [1 l] and the present results is, of course, attributable of the inclusion of nucleation in the present work. However the fact that the failure times in both cases exceed that for the periodically spaced case appears to be due the sintering and coalescence processes. As was noted in [I 11,sintering retards the growth of the area fraction, upon which the failure criterion has been taken to depend. Moreover the sintering process increases the effective dispersion length of those voids which do experience growth, slowing their growth. Finally, because the void growth rates are inversely proportionaf to the void radii, the increases in void radius due to coalescence also act to slow the growth. None of these effects, of course, are present in the periodically spaced case, so that void growth here proceeds under essentially optimum conditions. By way of comparison with experimental results, we cite the work of Raj [16], dealing with the creep rupture of copper bicrystals. Here it was found that the measured creep rupture times were about an

order of magnitude less than that predicted by the equispaced model, which was in turn noted to be substantially less than the mean failure times calculated in the present study. Thus the agreement between analysis and experiment in this area is not good, although significantly better than that reported in [1 11. Some of the possible reasons for this discrepancy are discussed in [l 11. We now briefly discuss the differences between the individual cases considered in the present work. Simulations with two different numbers of initial voids were conducted, Case (1) with 50 initial voids and Case (2) with 3. In physical terms, Case (1) might be thought to correspond to a situation in which large amounts of initial grain boundary sliding lead to the instantaneous, or nearly instantaneous, nucleation of a large number of voids. Case (2) would correspond to a situation in which this initial burst of nucleation does not occur. Not unexpectedly, Case (2) simulations showed a mean time-to-failure which was approximately three times that for Case (I) simulations. The difference may be attributed to the greater number of preexisting voids in Case (1). Somewhat more surprising were the differences observed between Cases (A) and (B), where the exponent in the mean function for the Poisson process was varied between limits suggested by experimental evidence. Here the mean times-to-failure were found to be relatively insensitive to the value of the exponent. In particular, Case (A) simulations, with b = -0.59, were found to exhibit times-to-failure only about 15--30% lower than the corresponding Case (B) simulations with b = 0. The reason for the lower times-to-failure in Case (A) is that, from Fig. 8, the mean function for this case results in more voids entering the system at an earlier point in time, hence leading to earlier failure. Finally, we draw attention to work by several previous investigators who have taken account of nucleation in making creep rupture calculations, e.g. Raj and Ashby (31, who considered a thermal nucleation model, and Leckie [17], who made use of an empirical nucleation model similar to equation (1). These studies were all of a deterministic nature, and considered the growth of some averaged or mean quantity connected with the cavity growth, such as the average volume occupied by the cavities. In contrast, the present study takes a probabilistic point of view and tracks the birth, growth, and occasionally the death, of each cavity in a fairly large population. The drawback of this approach is that it relies heavily upon computational methods because of the detailed nature of the modelling. Its advantage is that it allows the predictions of phenomena which deterministic models cannot describe. Chief among these are the statistical distributions of times-to-failure and the void radii. Moreover it makes allowance for the sintering and coalescence processes which appear to have an important effect upon the calculated timesto-failure.

FARIBORZ

et al.: NUCLEATION

OF INTERGRANULAR

~c~o~~e~geme~ls-Partial support for this work was provided by the U.S. Department of Energy under Grant No. DE-FG02-84ER45088 and by the Energy Research Center at Lehigh University.

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