Scripta METALLURGICA Vol. 26, pp. 1239-1244, 1992 Pergamon Press Ltd. et MATERIALIA Printed in the U.S.A. All rights reserved

OBSERVATIONS ON DIFFUSIONAL CAVITY GROWTH IN SUPERPLASTIC MATERIALS

Yan Ma and Terence G. Langdon Departments of Materials Science and Mechanical Engineering

University of Southern California Los Angeles, CA 90089-1453

(Received February 5, 1992) (Revised February 18, 1992)

i. Introduction

It is well known that internal cavities are often nucleated at the grain boundaries during deformation at elevated temperatures. This phenomenon was first reported by Greenwood et al. [i] in experiments on large-grained copper, alpha- brass and magnesium, and subsequently it was confirmed in creep experiments on a wide range of materials.

The occurrence of cavitation in superplastic alloys, where the grain size is very small (usually <i0 ~m) and the elongations to failure are very high, may be traced to more recent work on various Cu alloys [2-5] and the Zn-22% A1 eutectoid [6,7]. As discussed in a detailed review [8], the nucleation and growth of internal cavities is now an established characteristic of many superplastic materials.

A grain boundary cavity may grow by absorbing vacancies which diffuse into the cavity from an adjacent zone in the grain boundary plane. This process has been modeled for creep conditions [9-11] where a cavity grows on the grain boundary in a large-grained material, and the same models have been used for superplastic alloys with very small grain sizes [12-14]. Very recently, the model for diffusional cavity growth was modified to cover the situation where the cavity size exceeds the grain size and vacancies diffuse into the cavity along a number of grain boundary paths [15].

Close examination of the diffusional cavity growth models [9-11] shows that, although they are satisfactory for large-grained materials under creep conditions, they are not generally satisfactory, at least in their present form, for cavity growth in small-grained superplastic materials. The purpose of this paper is to draw attention to their deficiencies and to suggest an alternative procedure.

2. The Standard Models for Diffusional Cavity Growth

The first model for diffusional cavity growth in high temperature creep was due to Hull and Rimmer [9]. Subsequently, several models were proposed [10,11,16,17] with different modifications and improvements.

In the field of superplasticity, it has become standard practice to use the model of Speight and Beer~ [I0] in which the rate of change of the cavity radius, r, with the total strain, ~, is given by

dr fl6DRb[a - (2F/r) ] (i) _ _ = ~! ds 2r2kT~

1239 0036-9748/92 $5.00 + .00

Copyright (c) 1992 Pergamon Press Ltd.

1240 CAVITY GROWTH Vol. 26, No. 8

where N is the atomic volume, 6 is the grain boundary width, D~b is the coefficient for grain boundary diffusion, k is Boltzmann's constant, T is the absolute temperature, a is the applied stress, F is the surface energy, % is the strain rate, and ~i is a cavity size-spacing term defined as

1 a~ = (2)

in(a/r) - (i - r~/a 2) (3 - r2/a 2)/4

where 2a is the center-to-center spacing between two adjacent cavities on the same grain boundary.

There are two other models leading to relationships for dr/d~ which are very similar to equation (i). A model by Raj and co-workers [17,18] gives an almost identical relationship to equation (i) and a model by Speight and Harris [16] leads to equation (I) but with a cavity size-spacing term ~2 in place of ~l where u 2 is defined as

1 - r2/a 2 (3)

in(a/r) - (i - r2/a 2)/2

It should be noted that the earlier model by Speight and Harris [16] is not as complete as the later model by Speight and Beer~ [i0].

When a large-grained polycrystal is subjected to creep at a high temperature, inspection shows that most of the parameters in equation (i) are constant and only the cavity size-spacing term, ~i, is in a complex form. Since the cavity spacing is relatively large in the early stages of cavity growth, it is usually assumed that this term is constant [12,19,20] and with a value which depends on the precise nature of the model. This leads to a simplification of equation (i) so that the cavity growth rate, dr/ds, becomes essentially inversely proportional to r 2 with a lower cut-off value determined by the magnitude of the surface energy.

The assumption of a constant value for ul is reasonable when the grain size is large. For example, curve A in Fig. 1 shows a logarithmic plot of dr/d8 versus r calculated from equation (i) using a constant value of ~ = 0.4 and with typical values of N = 10"m m ~, 6Dsb = 2.9 x 10 21 m 3 s I, T = 800 K, a = i00 MPa, F = 1 N m "l and

= 104 s "l. This curve is essentially linear above a value of r = 0.i ~m with a slope of -2.

Curve B in Fig. 1 shows the corresponding relationship between dr/d8 and r, also calculated from equation (i), where the instantaneous values of ul were calculated using equation (2) with a cavity spacing of i00 ~m (equivalent to a value of a = 50 ~m). Since ~i in equation (2) increases continuously with increasing r, it is clear that the assumption of a constant value of ul = 0.4 leads to a slight overestimation of the cavity growth rate in the very early stages of growth but, more important, there is a very substantial underestimation in the value of dr/dE in the later stages of growth. As shown in Fig. i, curve B increases rapidly as the cavity radius approaches the half cavity spacing and subsequent interlinkage. Thus, the difference between curves A and B is less than a factor of 3 at r = i0 ~m, but thereafter the difference becomes increasingly significant.

From this comparison, it is concluded that the assumption of a constant value of s t is reasonable only for the early stages of growth in large-grained polycrystalline materials with well separated cavities.

2. The APplication of Diffusional Cavity Growth to Superplastic Materials

~o difficulties arise in applying equation (i) to superplastic materials with very small grain sizes.

Vol. 26, No. 8 CAVITY GROWTH 1241

10 4

,o s

10 2

1o

1.0

I0"

i0 -I

f0"~62

I ' ....... I I

o - 5 0 F . m o'- I 0 0 MPO

-- I~- iO'" s" T - 8 0 0 K

I0 "I 1.0 I0 r ( / *m)

,0'

FIG. 1

Cavity growth rate vs. cavity radius for growth by diffusion: curve A is estimated from equation (i) by assuming a constant value for el; curve B is estimated from equation (i) using ul defined by equation (2) with a = 50 ~m.

First, the cavity growth rate is often calculated by incorporating an assumption in which the half cavity spacing is much larger than the grain size [21-23]. This assumption is questionable because the derivation of equation (1) is based on two cavities, separated by a distance of 2a, growing towards each other by absorbing grain boundary vacancies and with the driving force for vacancy flow given by the difference in chemical potential along their common boundary. If the cavity spacing is larger than the grain size, the cavities are no longer separated by a single flat boundary and the vacancy flow between them, if any, is necessarily different from that considered in the derivation.

Second, the cavity growth rate is often calculated by assuming a constant value of u I [13,24-27] but, as noted earlier, this is reasonable only in the early stages of growth when the separation between cavities is relatively large. The assumption breaks down in superplastic materials because the range of cavity radii having a reasonably constant value of e I becomes very small. This is illustrated in Fig. 2, where curves A and B were calculated as in Fig. 1 with u] = 0.4 for curve A and with a half cavity spacing, a, of 5 ~m and with the stress level taken as 70 MPa.

A comparison between Figs 1 and 2 shows little difference in the curves labelled A because, by taking ~, as 0.4 in both calculations, the curves are independent of the cavity spacings and the small change is due only to the difference in the stress level. Curve B is also similar in shape in Figs 1 and 2 but the sharp increase is displaced to much smaller radii in Fig. 2. From these calculations, curves A and B in Fig. 2 now differ by a factor of about 3 at a cavity radius of only 1 ~m.

1242 CAVITY GROWTH Vol. 26, No. 8

A E :L

w

"o

,o'

10 3

lO

LC

lo-Z

10":

O • 5~m

c,= 7 0 MPo = 10"4 ~,

T- 8OOK

\ \ \ \

I , , , ,.,.I I k

I0" 1.0 I0 D z r(/~m)

FIG. 2

Cavity growth rate vs. cavity radius for growth by diffusion: curve A is estimated from equation (i) by assuming a constant value for ~i; curve B is estimated from equation (I) using ~l defined by equation (2) with a = 5 Mm; curve C is estimated from equation (5) using ~2 defined by equation (7).

It is clear from Fig. 2 that it is not appropriate to use equation (i) for superplastic materials. Since the grain size is very small, it is more likely that each grain boundary contains not more than one cavity. This suggests it is preferable to calculate the rate of growth of a single isolated cavity rather than two adjacent cavities on the same grain boundary.

A theory for the growth of a single cavity was developed by Schneibel and Martinez [28]. The model assumes an isolated cavity located in an infinite grain boundary lying perpendicular to the applied stress and with diffusion dominant in a region with a radius of (r + A) where A is the characteristic length for grain boundary diffusion defined by Needleman and Rice [29,30] as

[,6D,ba] '/3 A = [---~j (4)

Outside of this region, it is assumed that plastic flow is dominant.

Using a procedure similar to Speight and Beer~ [I0], Schneibel and Martinez [28] developed a quasi-equilibrium diffusion growth model in which the cavity growth rate is given by

dr n6Dg b (a - 2Fk~) (5) - - = ~2 de r2kT~h(~)

Vol. 26, No. 8 CAVITY GROWTH 1243

where ~ is the dihedral angle at the cavity tip, h(~) is given by

h(,) = [i/(i * cos¢) - (cos¢)/2] sin~

(6)

k~p is the curvature at the crack tip [= (2 sin ~)/r] and ~2 is given by

1 - r2/(r + A) 2 ~2 = 21n(l + A/r) + r2/(r 2 + A 2) - 1

(7)

It should be noted that equation (5) was corrected by including a factor of 2 for the two crack surfaces.

Curve C in Fig. 2 shows the prediction of equation (5) with ~ = 90 ° . This curve is fairly similar in appearance to curve A but the model avoids the problem of incorrectly assuming that the two cavities grow together along a single grain boundary.

4. Summary and Conclusions

1. The standard theory for diffusional cavity growth was developed for two cavities growing together on a single grain boundary. This theory is reasonable for large-grained polycrystalline materials but not for superplastic materials where the grain size is very small and each boundary generally contains not more than one cavity.

2. An alternative model for diffusional cavity growth, based on the growth of single isolated cavities, provides a more realistic description of the growth process in small-grained superplastic materials.

Acknowledqement

This work was supported by the National Science Foundation under Grant No. DMR-9020828.

References

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1244 CAVITY GROWTH Vol. 26, No. 8

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