A model for diffusional cavity growth in superplasticity

Acfa mefaN. Vol. 35, No. 5, pp. 1089-I 101, 1987 Printed in Great Britain. All rights reserved

A MODEL

OOOI-6160/87 $3.00 + 0.00 Copyright 0 1987 Pergamon Journals Ltd

FOR DIFFUSIONAL CAVITY GROWTH IN SUPERPLASTICITY

ATUL H. CHOKSHIt and TERENCE G. LANGDON Departments of Materials Science and Mechanical Engineering, University of Southern California, Los Angeles, CA 90089-1453, U.S.A. (Received 18 April 1986; in revisedform

6 August 1986)

Abstract-There is an enhancement in the diffusional growth of cavities in superplastic materials if the cavity size exceeds the grain size so that vacancies diffuse into the cavity along a number of grain boundary paths. A model for this process is developed, and it is shown that the rate of change of cavity radius with strain due to superplastic diffusion growth is given by

where r is the cavity radius, c is the total strain, a is the atomic volume, 6 is the grain boundary width, Dgb is the coefficient for grain boundary diffusion, d is the spatial grain size, k is Boltzmann’s constant,

Tis the absolute temperature, u is the applied stress and i is the strain rate. Superplastic diffusion growth is therefore independent of the instantaneous cavity radius and inversely proportional to the square of the grain size; in practice, this process becomes important at the lower testing strain rates and when the specimen grain size is typically less than -5 pm. It is demonstrated that the occurrence of superplastic diffusion growth is capable of providing an explanation for the experimental observations of large rounded cavities in several superplastic materials. Rksmn&Dans les materiaux superplastiques, la croissance des cavites par diffusion s’intensifie lorsque la taille de la cavite d&passe celle du grain de sorte que les lacunes diffusent vers la cavitt le long de plusieurs trajectoires intergranulaires. Nous developpons un modtle de ce mtcanisme, et nous montrons que la vitesse d’ivolution du rayon de la cavite en fonction de la deformation due a la croissance superplastique par diffusion est don&e par

od r est le rayon de la cavite, t la deformation totale, f2 le volume atomique, 6 la largeur du joint de grain, le coefficient de diffusion intergranulaire, d la taille du grain dans l’espace, k la constante de Boltzmann, T la temperature absolue, cr la contrainte appliquee et t‘ la vitesse de deformation. La croissance superplastique par diffusion est done independante du rayon instantant de la cavitt et inversement proportionnelle au carre de la taille de grain; en general, ce mtcanisme devient important pour les vitesses de deformation les plus basses et lorsque la taille de grain de l’echantillon est nettement inferieure a - 5 ym. Nous dtmontrons que la croissance superplastique par diffusion permet d’expliquer les observations experimentales sur les grosses cavids rondes rencontrees dans plusieurs materiaux D,

superplastiques. Zusammenfsssung-Das diffusionsgesteuerte Wachstum von Hohlraumen in superplastischen Werkstoffen ist beschleunigt, wenn die Hohlraumgrijge die KomgrBBe iiberschreitet und Leerstellen iiber eine Anzahl von Komgrenzen in die Hohlraume diffundieren kiinnen. Fiir diesen ProzeD wird ein Model1 entwickelt. Die Rate, mit der sich der Hohlraumdurchmesser mit der Dehnung durch superplastisches Diffusionswachstum lndert ist gegeben durch

dr z= Hierbei ist r der Hohlraumradius, t die gesamte Dehnung, R das Atomvolumen, 6 die Komgrenzdicke, D,, der Koeffizient fiir Komgrenzdiffusion, d die rlumliche KorngriiDe, k die Boltzmann-Konstante, T die absolute Temperatur, u die iiuBere Spannung und Z die Dehngeschwindigkeit. Das superplastische Diffusionswachstum hangt daher nicht vom instantanen Hohlraumradius ab und ist umgekehrt proportional zum Quadrat der KorngrBBe. In der Praxis wird dieser Prozeg wichtig bei kleineren Dehngeschwindigkeiten und wenn die KomgriiBe typischerweise kleiner als N 5 pm ist. Es wird gezeigt, da8 das Auftreten des superplastischen Diffusionswachstums die Beobachtung von groBen runden Hohlraumen in verschiedenen superplastischen Werkstoffen erkllren kann.

SNOW at Division of Materials Davis, CA 95616, U.S.A.

Science and

Department

1089

of Mechanical

Engineering,

University

of California,

1090

CHOKSHI and LANGDON:

DIFFUSIONAL CAVITY GROWTH IN SUPERPLASTICITY

1. INTRODUCTION Superplastic materials give very high tensile elongations over strain rates where the strain rate sensitivity is high. However, many superplastic metals exhibit extensive internal cavitation when tested in tension, although the precise nature of the cavitation depends critically upon the material and the testing conditions [l]. To date, the occurrence of cavitation in superplasticity has received relatively little attention. There are numerous reports of rounded cavities in superplastic materials [2-141 and there are also many reports of cavities elongated along the tensile axis [2,4,5,7-l 1, 14191. Attempts have been made to rationalize the experimental observations of the cavity morphologies by using the cavity growth mechanisms developed for conditions of high temperature creep. There are two basic growth mechanisms depending upon whether the growth is controlled by absorbing vacancies through diffusion or by the plastic flow of the surrounding material: if surface diffusion is reasonably rapid, these two mechanisms are expected to lead either to spherical cavities or to cavities which tend to be elongated along the tensile axis, respectively. It is now clear that it is not entirely satisfactory to apply the growth mechanisms developed for high temperature creep, where the grain size is large, to superplastic materials where the grain size is typically < 10 pm.This work was therefore motivated by the need to develop a new model for cavity growth for conditions where the grain size is very small. The following section outlines briefly the limitations of the present creep models, and the subsequent sections describe the proposed model and the application of this model to published data on cavitation in superplasticity. 2. THE APPLICATION OF CAVITY GROWTH MODELS FOR HIGH TEMPERATURE CREEP There are two basic models for cavity growth in high temperature creep. First, if the cavities grow by absorbing vacancies by diffusion, the growth rate is given by [20,21] dV _= dt

k-7 - c&J/r)1 ' 4ln(a/2r)

- [1 - (2r/~)~][3 - (2r/a)2]

(1)

where V is the cavity volume, t is the time, n is the atomic volume, 6 is the grain boundary width, Dgb is tMore detailed analyses predict growth rates which exceed equation (4) by a small geometric factor: for typical superplastic conditions, this factor is less than -2. Thus, in view of the general uncertainties in the various models of cavity growth, it is reasonable to use equation (4) in the present anlaysis.

the coefficient for grain boundary diffusion, k is Boltzmann’s constant, T is the absolute temperature, Q is the applied stress, y is the surface energy, r is the cavity radius and a is the cavity spacing. Thus, for a spherical cavity growing by diffusion, the rate of change of the cavity radius with the total strain, 6, may be expressed as (!!!)~+?)[c-pyjr)]

(2)

where the subscript d denotes diffusion growth, L is the strain rate, and c( is a cavity size-spacing parameter defined as 1 a = 4ln(a/2r)

- [1 - (2r/a)2][3 - (2r/a)2]’

(3)

Second, if cavity growth is controlled by plastic flow of the surrounding material, the rate of change of radius with strain is given by [22] 3Y =‘-, where the subscript pl denotes power-law growth due to plasticity in the crystal matrix.t Inspection of equations (2) and (4) shows that, since (dr/dt), is proportional to l/r2 and (dr/dt),, is proportional to r, there is generally a transition from diffusion growth to power-law growth as the cavities increase in size. This trend is illustrated schematically in Fig. 1 by plotting dr/dc vs r for the two growth mechanisms. Since the faster growth process is dominant, the cavity grows by diffusion growth up to a critical radius r, and thereafter it grows by power-law growth. The approximate value of r, may be estimated by making the assumption of c( N 1 and equating equations (2) and (4) [23]. Thus

It follows from equation (5) that diffusion growth dominates to a larger cavity radius when the strain rate is reduced. The prediction from Fig. 1 of a transition from diffusion growth at small radii to power-law growth at large radii is consistent with experimental data on a superplastic microduplex alloy steel [24], Zn-22% Al [24], an Al-Zn-Mg alloy [25] and 7475 Al [26]. However, close inspection shows that, at low strain rates, the values of r, from equation (5) consistently underestimate, by over an order of magnitude, the limiting radius for rounded cavities. For example, in a superplastic Cu-2.8% Al-l .8% Sia.4% Co alloy, Miller and Langdon [24] estimated r, N 1 pm under experimental conditions where there were rounded cavities with radii of up to N 40 pm. Al-

CHOKSHI

and LANGDON: I

’

’

DIFFUSIONAL

I”“‘1

/

POWER-LAW GROWTH

_

DIFFUSION GROWTH

1

1091

CAVITY GROWTH IN SUPERPLASTICITY HIGH TEMPERATURE

CREEP

SuPERPLASTIC

DEFORMATION

v’ /

/

/

Grain

Grain size > SOpm

size - 5pm

Fig. 2. Schematic illustration of the effect of a cavity in high temperature creep with a grain size of > 50 pm (left) and in superplastic deformation with a grain size of _ 5 pm (right): the stress axis is vertical.

and thereafter the model is compared with experimental data.

Fig. 1. Schematic illustration of diffusion growth and power-law growth, showing the critical cavity radius, rc.

3. A MODEL FOR SUPERPLASTIC DIFFUSION GROWTH 3.1. The efict

though it was noted that many of the larger cavities probably arose from cavity coalescence and rapid

spheroidization, there were many smaller individual cavities, having typical radii in the range of -l-15pm, where the appearance was consistent with diffusion growth but the radii were substantially larger than the value estimated for r,. Similarly, Livesey and Ridley [I l] calculated a value of r, N 0.3 pm for an IN836 alloy, although an examination of published photomicrographs for this material under the selected experimental conditions [5, 111 reveals several rounded cavities with radii up to -15pm. A qualitative explanation for this discrepancy was presented by Miller and Langdon [24] by noting that the cavity size in superplastic materials often exceeds the grain size, so that there is an enhancement in the diffusive flux because vacancies enter the cavity along a number of grain boundary paths. The situation is illustrated schematically in Fig. 2. In high temperature creep, where the grain size is large, the cavity is isolated on a single boundary and equation (1) gives the theoretical growth rate (left); but in superplastic deformation, where the grain size is typically of the order of -5 pm, a cavity of similar size intersects a number of grain boundaries and each boundary provides a path for vacancy diffusion (right). Thus, there is an enhancement of diffusion growth in superplastic conditions, and a corresponding increase in the value of the critical radius marking the transition to power-law growth. It is interesting to note in this context that the grain sizes of the Cu alloy examined by Miller and Langdon [24] and the IN836 alloy examined by Livesey and Ridley [1 I] were both very small and of the order of 3 pm. The following section develops a quantitative model to incorporate the enhancement in diffusion growth when the cavity size exceeds the grain size,

of more than one diffusion path

It is assumed that the gradient in chemical poten-

tial provides the driving force for the diffusional growth of cavities. The diffusive flux to the cavity is given by

where VP is the chemical potential gradient. If the cavity size is larger than the grain size, so that there are a total of N diffusion paths, the total flux, J, is given by J = Nj. In practice, it is necessary to incorporate the variation in chemical potential on the different grain boundaries intersected by the cavity. This problem is illustrated schematically in Fig. 3. When the cavity is located on a single grain boundary lying perpendicular to the stress axis, as illustrated on the left in Fig. 3 and incorporated into the standard model for diffusion growth in high temperature creep, the excess chemical potential at the boundary is given by

J-OJFig. 3. The variation in stress normal to the grain boundaries in high temperature creep with a large grain size (left) and in superplasticity with a small grain size (right): the stress axis is vertical.

1092

CHOKSHI and LANGDON: p =a,R=aR

DIFFUSIONAL CAVITY GROWTH IN SUPERPLASTICITY 60-

(8)

where CT~is the stress normal to the boundary. For an hexagonal grain array in superplastic deformation, shown on the right in Fig. 3, there are differences in gN between the two boundaries perpendicular to the stress axis and the four boundaries inclined at 30” to the stress axis. Thus

50-

40-

anr, = fl and C& = a/2

.t

(9)

1

For a cavity intersecting a regular hexagonal array of grains, there are four inclined boundaries for every two normal boundaries so that the average chemical potential may be expressed as p a”g2 0.7 UQ

i

30-

20-

(10)

and the total flux becomes JEO.7Nj.

(11)

Thus, there is an enhancement in the diffusive flux by a factor of -0.7 N when there are N diffusion paths, and equation (2) becomes dr

0 z

40,7N)(s)(o-jZY’r))

(12)

spd

where the subscript spd denotes superplastic diffusion growth when the cavity size exceeds the grain size. The precise form of equation (12) is complicated because the number of grain boundaries intersected by a cavity depends both upon the cavity size and the grain size. It is therefore necessary to obtain the relationship between N and r/d, where d is the spatial grain size of the material. 3.2. The variation in number of intersected boundaries with cavity radius

The perimeter, P, of a section of a spherical cavity is related to the perimeter, p, of an hexagonal grain by P CCN,P

IO -

c I

0 0

2

I

4 r/d

I

6

Fig. 4. The relationship between the number of grain boundaries intersected by a cavity in two dimensions, N,, and the ratio of cavity radius to grain size, r/d.

regular hexagonal array and counting the number of intersections, N,, for each circle. Using this procedure, Fig. 4 shows the variation of N, with r/d, giving p N 8. This approach may be extended to three dimensions by considering a spherical cavity of radius r contained within a regular three-dimensional hexagonal array of grains, as illustrated in Fig. 5 where OA (= r) is the radius of a diametrical section. From equation (14) with p N 8, the number of intercepts, N,, made by the circular diametrical sec-

(13)

where N, is the number of intercepts between the cavity section and the hexagonal grain array. Since P is proportional to r and p is proportional to d, equation (13) becomes N,=B(r/d) where b For a possible drawing

(14)

is a constant of proportionality. two-dimensional hexagonal grain array, it is to deduce the variation of N, with r/d by a series of circles with different radii on a

Tin specimens exhibiting grain boundary sliding, the normal stresses are slightly different from those shown in equation (9). However, calculations indicate that this difference is less than a factor of 2, and hence the simple expression given in equation (9) is employed in the present analysis.

Fig. 5. Schematic illustration of a large spherical cavity of radius r( = OA) contained within a regular threedimensional hexagonal array of grains.

CHOKSHI

and LANGDON:

DIFFUSIONAL

CAVITY GROWTH IN SUPERPLASTICITY Thus, equation

(17) reduces

1093

to

N = 8(r/d) + 16n(r/d)

(18)

where n is the number of sections, above the diametrical section, that must be included for a spherical cavity of radius r. It is apparent from Fig. 5 that, since the analysis requires taking two sections for each hexagonal grain, n N 2(r/d). Thus, it follows that

(a)

N N 8(r/d) + 32(r/d)‘.

(19)

For conditions where the cavity size is significantly larger than the grain size, the diffusive paths outside of the diametrical section become dominant and equation (19) reduces to N N 32(r/d)2.

(b) Fig. 6. The geometric construction used to calculate the superplastic diffusion growth rate: (a) dimensions of an hexagonal grain, and (b) the relationship between a grain centered at X and the arc of a large spherical cavity ABC.

Although equation some approximations, the same relationship geometric procedure.

(20)

(20) was obtained by making it is shown in Appendix II that follows also from a more rigid

3.3. The relationship for superplastic d@ision growth tion of the cavity perpendicular to Fig. 5 with radius r is -8 (r/d). Now, considering the sections with radii r,, r,, etc., it follows from Fig. 6(a) that the height h is equal to $d/4 so that, using Fig. 6(b) and noting that OB = OC = = r, the various radii of the different sections are given by r, = [r’-

(3d’/l6)]”

r2 = [r2 - (3d2/4)]“’ rj = [r’ - (3d2)]‘j2

(15) I

etc. with the spherical Thus, made by

Substituting equation (20) into equation (12), and simplifying for large values of r so that surface tension effects are negligible, the rate of change of cavity radius with strain due to superplastic diffusion growth is given by

relationships continued to the top of the cavity. from equation (14), the number of intercepts circular sections with radii rl, r,, etc. is given

by N, N (8/d)[r2 - (3d2/16)]‘j2 N2 1: (8/d)[r2 - (3d2/4)]‘/2 NX21 (8/d)[r2 - (3d2)]‘!’

(16) I

etc. Summing N,, N2, N,, etc., multiplying by 2 to include both hemispherical caps, and then adding N,, for the diametrical section, the total number of diffusion paths is given by N = 8(r/d) + 16(r/d) {[1 - (3/16)(d/r)2]‘/2 +[1 -(3/4)(d/r)2]‘i’+...}.

(17)

Considering the second term on the right of the equality in equation (17), there is an expression of the form [l - x(d/r)2]‘/2 with x = 3/16, 3/4, etc. When the cavity size is larger than the grain size, (d/r)’ < 1, and it is reasonable to put [1 - x(d/r)2]“2 1 1: the validity of this approximation is demonstrated in Appendix I.

In order to estimate a value for a, it is first noted that the term a in equation (3) represents the spacing between two adjacent cavities on a single grain boundary [20]. In diffusion growth during high temperature creep, it is customary to divide the intercavity spacing into two contiguous zones [21]: an inner zone surrounding the cavity where vacancies diffuse along the grain boundary and an outer zone where the cavity growth is accommodated by plastic deformation of the matrix. In practice, there is a coupling between the diffusion and power-law growth mechanisms, and it is possible, as noted by Chen and Argon [27], to incorporate this coupling by replacing the term a in equation (3) with a diffusion length parameter, A. In superplastic diffusion growth, it is anticipated that vacancy diffusion along the grain boundaries will be limited to a width of approximately one grain diameter beyond the periphery of the cavity, with the remaining material between the cavities undergoing normal superplastic deformation. Thus, it is appropriate for superplastic materials to replace the term a in equation (3) with the term (r + d). This leads to a value of CI of the order of unity, so that the relationship for superplastic diffusion growth becomes 45R6D,, yqy-

0

Q 2 .

(22)

1094

CHOKSHI and LANGDON:

DIFFUSIONAL CAVITY GROWTH IN SUPERPLASTICITY

Two important implications arise from equation (22). It follows that the rate of change of the cavity radius with strain for superplastic diffusion growth is (i) independent of the instantaneous cavity radius and (ii) inversely proportional to the square of the grain size ofthe material. Both of these trends arise because of the presence of the term (r/d)2 in the relationship for N given by equation (20).

The physical examined.

significance

of the model is now

4. PHYSICAL SIGNIFICANCE OF THE MODEL 4.1. The variation of dr/dc with r Figure 1 presented a schematic illustration of the variation of dr/dc with r using the mechanisms for diffusion growth [equation (2)] and power-law growth [equation (4)]. It is clear from the preceding analysis that, when the grain size is very small, it is necessary also to include superplastic diffusion growth [equation (22)]. The three mechanisms of cavity growth are depicted schematically in Fig. 7. As indicated in equation (22), superplastic diffusion growth is independent of the cavity radius and it therefore appears as a horizontal line in Fig. 7. However, since this growth process exists only when the cavity intersects more than one grain boundary, the line is shown only for cavity radii larger than -d/2. Cavity growth occurs by the fastest process, as indicated by the upper broken line in Fig. 7. Thus, under the conditions depicted in Fig. 7, the cavity initially grows by diffusion growth until the radius

reaches a critical value, roSP,where the cavity intersects more than one grain boundary. There is then an increase in drldc, and this rate of growth remains constant until a second critical radius, r,,, where there is a transition to rapid growth by the powerlaw mechanism. In practice, there is a distribution of grain sizes in any superplastic alloy and, consequently, it is anticipated that there will be a smooth, rather than an abrupt, transition between the diffusion and the superplastic diffusion growth mechanisms. It is important to note also that, since superplastic diffusion growth and power-law growth act independently, the experimental cavity growth rate in the transition region is obtained from a summation of the contributions of the two individual mechanisms. Since the model for superplastic diffusion growth is essentially a modification of the standard diffusion growth process, it is reasonable to assume that surface diffusion is significantly more rapid than grain boundary diffusion so that the cavities will appear rounded throughout the regime of superplastic diffusion growth. Thus, in the schematic illustration in Fig. 7, the rounded cavities persist until a radius of rcsp,and this radius may be as much as one order of magnitude larger than the critical radius r, given by equation (5). The occurrence of superplastic diffusion growth is therefore capable of providing an explanation for the large rounded cavities observed experimentally. The superplastic diffusion growth process is given by equation (22). It is appropriate, therefore, to examine the various parameters contributing to this relationship in order to delineate the precise experimental conditions where this process becomes dominant. 4.2. The dependence on grain size Equation (22) shows that drlde varies with l/d2 for superplastic diffusion growth, so that this process becomes of less importance when the grain size is increased. It is possible to estimate the range of grain size where superplastic diffusion growth is dominant. Equating equations (4) and (22) for large cavity radii, the critical radius marking the transition from superplastic diffusion growth to power-law growth is given by

DIFFUSION GROWTH

For experimental conditions of constant T and i, the value of rFsp [equation (23)] decreases towards the value of r, [equation (5)] when the grain size is increased. The maximum grain size for superplastic diffusion growth, a&,,, is obtained by equating equations (5) and (23), so that Fig. 7. Schematic illustration of diffusion growth, superplastic diffusion growth and power-law growth, showing the critical radii rasp,r, and resp.

CHOKSHI and LANGDON: Using equation

(5) equation

DIFFUSIONAL CAVITY GROWTH IN SUPERPLASTICITY

(24) is equivalent

d,,,,, N 5r,.

to (25)

Typically, the value of r, is estimated as - 1 pm in superplastic materials deformed at low strain rates. Thus, the superplastic diffusion growth process becomes important at low strain rates only when the grain size is less than - 5 pm. At high strain rates, the value of r, is typically estimated as -0.1 pm, so that superplastic diffusion growth is then only important in materials having extremely small grain sizes below about 0.5 pm. This upper limit is smaller than the grain size of most superplastic materials in current use, but it would become important in, for example, the future utilization of materials having ultra-fine microcrystalline structures [28]. As indicated in Fig. 7, superplastic diffusion growth occurs only when the cavity is sufficiently large that it intersects more than one grain boundary. Typically, this requires that the cavity size is larger than rOspN d/2. The grain size range for superplastic diffusion growth is therefore defined uniquely by two parameters: (i) a lower bound of d 1~2r and (ii) an upper bound of d N 5r,. It is possible to increase the range of dominance of superplastic diffusion growth in three independent ways. First, by decreasing the grain size, since this decreases rOspand increases rCsp. Second, by performing experiments at lower strain rates to thereby increase the value of a/i in equation (23). Third, by performing experiments at higher temperatures to increase the value of GD,,/kT in equation (23) provided the temperature is not sufficiently high that lattice diffusion dominates. 4.3. The SigniJicance of the term a/i As noted in equation (23), the upper limiting cavity radius, rCsp,is proportional to a/i. Since the strain rate sensitivity, m (= 8 In a/a In i), is of the order of 0.5 for many superplastic materials, a decrease in strain rate leads to an increase in the value of rCsp. Most laboratory tests on superplastic materials use machines having a constant rate of cross-head displacement, so that there is a continuous decrease in the effective strain rate with increasing elongation of the sample. This effect may be incorporated into equation (23) by replacing i with an average strain rate, iavg. defined as 1 11 1 -=iavg 2 [ ; + 6

1

where i, and i, are the strain rates calculated from the initial and final gauge lengths, respectively. This trend is not important in materials exhibiting relatively low elongations to failure (up to - 500%), since the minor effect on rCsp(to within a factor of -2) is within the experimental scatter; but the trend becomes important at high elongations as the value of rCspis increased by a factor of - 15 if i is replaced

1095

in equation (23) for a total elongation by iavg

of - 3000%. A secondary effect arises because the presence of cavitation leads to an increase in the net flow stress. However, the total extent of cavitation in superplastic materials is generally relatively low, ranging from -2% in Zn-22% Al[lO] to -20% in some Cu-Ni-Zn alloys [5], and this increases the average net flow stress by up to only - 10%. This change in flow stress is negligible by comparison with the decrease in the effective strain rate in tests conducted at constant cross-head displacement. 4.4. The efSect of concurrent grain growth Superplastic materials have a very small grain size and there is often some grain growth during the deformation process. It is possible to incorporate concurrent grain growth into the superplastic diffusion growth model by expressing the strain rate in the standard form

where A is a dimensionless constant, D is the appropriate diffusion coefficient, G is the shear modulus, b is the Burgers vector, and p and n are constants. In a test at a constant strain rate, it follows from equation (27) that an increase in grain size leads to a corresponding increase in flow stress through the relationship

where the subscripts 1 and 2 refer to condition and some later condition growth, respectively. It follows from equation (23) that radius marking the upper limit of diffusion growth, rCsp,is proportional to equation (28) may be expressed as

the original after grain the critical superplastic a/d’. Thus,

whererc’cspcl) and rcsti2) are the critical radii for the original condition 1 and the later condition 2, respectively, and q is a constant [ = 2 - (p/n)]. Generally, the behavior of superplastic materials divides into three distinct regions depending upon the imposed strain rate. At low strain rates in region I, the material exhibits limited superplasticity and the values of p and n are - 2 and -4, respectively; at intermediate strain rates in region II, the material exhibits maximum superplasticity and, typically, p rr n N 2; and at high strain rates in region III, there is only limited superplasticity and the values of p and n are essentially zero and N 5, respectively [29]. Using these values of p and n, the values of q in equation

1096

CHOKSHI and LANGDON:

DIFFUSIONAL

CAVITY GROWTH IN SUPERPLASTICITY 5.1. A commercial Cu alloy

Fig. 8. Schematic illustration of the cavity growth processes shown in Fig. 7 for the condition where rorD> rc.

(29) are estimated as N 3/2, - 1 and - 2 for regions I, II and III, respectively.

4.5. The implication of rosP> r, It was tacitly assumed in the construction of Fig. 7 that rasp< r,. Although this is often true, situations may arise where rosP> r,. the experimental condition for Since rosP-d/2, since superplastic rasp > r, is d > 2r,. Further, diffusion growth occurs within a limited range of grain sizes with an upper bound of d N jr,, the experimental condition for rasp> rc is a grain size lying within the range from -2r, to -5r,. Within this grain size range, Fig. 7 is replaced by the schematic illustration in Fig. 8. Under these conditions, there is diffusion growth until r,, powerlaw growth for radii from rc to rosP, superplastic diffusion growth for radii from rosPto rcsprand then power-law growth again at even higher radii. Since power-law growth leads to elongated cavities, Fig. 8 suggests the possibility of a subsequent rounding of the cavities within the radius range from rosPto rcsp.

As noted in section 2, detailed experiments on a Cu-2.8% Al-1.8% Si-O.4% Co alloy revealed large rounded cavities with radii up to -40 pm [4]. Subsequent calculations by Miller and Langdon [24] gave r, N 1 pm, so that it was not possible to explain the rounded cavities in a satisfactory manner. To check the importance of superplastic diffusion growth in this material, the cavity growth rates were calculated using equations (2), (4) and (22) for diffusion growth, power-law growth and superplastic diffusion growth, respectively, and for the conditions of T=823K, d=3pm and i =1.67x lo-‘s-l. Taking the experimental value of c = 14 MPa so that a/i = 8.4 x 10’ MPa s, and with R = 1.2 x 1O-29m3, 6 = 1 x 10m9m, Dgb = lo5 exp( - 103,00O/RT) m* s-’ where R is the gas constant (8.31 J mol-’ K-‘) [30] and y = 1.1 J m-* [31], the cavity growth rates were plotted as dr/de vs r as shown in Fig. 9. Inspection of Fig. 9 shows that the critical radius is r, N 2 /~rnt in the absence of superplastic diffusion growth, but the occurrence of diffusion along many boundary paths increases this critical radius to rcsPN 12 pm. This is in excellent agreement with the experimental observations when it is noted that Miller and Langdon [24] suggested that the larger rounded cavities were due to cavity coalescence but that the smaller individual cavities, in the range of N 1-15 pm, were evidence for growth by a diffusion process. Thus, the mode1 for superplastic diffusion

I

I

C~-2.SXAI-l.BKSi-0.4%Co

’

’ ’ ’ ““1

I

Shei and

l_angdon(A

T=823K

102-

IO-

f

za 3 <

c

l.Or -

-

IO’-

I I

5. A COMPARISON WITH PUBLISHED EXPERIMENTAL DATA

lci2The predictions of the model were compared with several sets of experimental data for a commercial copper alloy and the Zn-22% Al eutectoid alloy.

tThe earlier estimate of r, = 1 pm [24] was obtained by using a slightly different, and less accurate, relationship for diffusion growth.

Fig. 9. Cavity growth rate vs cavity radius for a superplastic Cu alloy, using the experimental conditions of Shei and Langdon [4].

and LANGDON:

CHOKSHI

DIFFUSIONAL

CAVITY GROWTH

IN SUPERPLASTICITY

1097

Table I. Experimental conditions for Zn-22% Al d

a/i

(rm)

UL

W’a

9

rc (pm)

2.5 2.5 2.5 1.5 1.5

503 503 473 473 473

8~10~ 3x IO’ 3x104 1x10s 2x 10’

-2 -0.6 -I -I -0.4

Reference Ahmed et al. [9] Ahmed et al. [9] Ishikawa et al. [l8] Miller and Langdon [8] Miller and Langdon [8]

growth provides a satisfactory explanation experimental observations in this material. 5.2. The Zn-22%

for the

Al eutectoid alloy

The experimental observations on Zn-22% Al fall into two categories depending upon whether the spatial grain size is < 3 pm or > 5 pm. (a) d < 3pm. Three sets of experimental data are available for the Zn-22% Al eutectoid alloy with grain sizes from 1.5 to 2.5 pm [8,9, 181. The experimental conditions are given in Table 1, including the values calculated for r, from equation (5): these values were estimated using the initial strain rates with R= 1.5 x 10m’9m3, 6 = 5.7 x 10-‘Om and D,, = 10m4exp( -64,70O/RT) m2 SC’,t and with the flow stress estimated in the study of Miller and Langdon [8] using equation (28) with u2 corresponding to the flow stress for the experimental grain size of d, = 1.5 pm and u, corresponding to the known flow stress for a grain size of dI. Inspection of the published photomicrographs for the experimental conditions given in Table 1 showed rounded cavities having radii much larger than the values estimated for r,. To check for the occurrence of superplastic diffusion growth, the photomicrographs were examined carefully and the radii recorded for both the largest rounded cavities and the smallest cavities clearly elongated along the tensile axis. It is believed that these two radii provide a reasonable indication of the range of transition in growth mechanisms, and the datum points are plotted in Fig. 10 where the lower open symbols represent the large rounded cavities and the upper closed symbols represent the elongated cavities. The data are plotted on a pictorial representation of equation (23), and the line at 45” marks the predicted transition from superplastic diffusion growth at small radii to power-law growth at large radii. It should be noted that, since Zn-22% Al exhibits very large elongations to failure, the experimental datum points were placed on Fig. IO using the average strain rate, iavgrdefined by equation (26): this has the effect of moving the datum points to the right with respect to the lower axis. Inspection of Fig. IO shows that the datum points exhibit the correct trend but the experimental transition radii are lower than the predicted radii by a factor of, typically, about 3. This discrepancy arises tThis value of D,, was estimated from a detailed appraisal of experimental data for Zn-22% Al [32]. AM.35/1--G

Fig. 10. A cavity growth diagram showing the predicted transition from superplastic diffusion growth at small cavity radii to power-law growth at large cavity radii: the experimental datum points for Zn-22% Al are taken from Miller and Langdon [8], Ahmed et al. [9] and Ishikawa et nl. [ 181.

because of the occurrence of concurrent grain growth during testing. An increase in the grain size around the cavity leads to a decrease in the rate of accumulation of diffusion paths and a corresponding decrease in the growth rate through equation (22). This effect is equivalent to moving the datum points in Fig. 10 to the left with respect to the lower axis. (b) d > 5pn. Livesey and Ridley [IO] performed experiments on samples of Zn-22% Al having initial grain sizes of 5pm or larger. Inspection of a published photomicrograph of a specimen pulled to fracture at 473 K with an initial grain size of 6.5 pm suggests transitions from very small rounded cavities to elongated cavities with radii up to - 5 pm, rounded cavities in the range from N 5 to N 10 pm, and elongated cavities with radii greater than N 10 pm. To check the occurrence of superplastic diffusion growth, it is first’noted that, using the initial value of i and a value for CJestimated from equation (28) the critical transition radius in the absence of superplastic diffusion growth is given by r, N 1.3 pm. It was demonstrated in equation (25) that the maximum grain size for superplastic diffusion growth is -5r,, corresponding to _ 6.5 pm, so that the material used by Livesey and Ridley [lo] appears to have a grain size which is sufficiently large that is precludes the operation of this mechanism. However, since the specimen exhibited a total elongation of = 1200%, it is necessary to use the average strain rate in place of the initial strain rate in the calculations. By this procedure, the critical radii are calculated as rctavsjN 2.4 pm and rcspCavgj N 7 pm, respectively. Figure 11 shows the variation of dr /dc with r under the experimental conditions of d = 6.5 pm, T = 473 K and using iavgin place of the initial strain rate. This is similar to Fig. 8 presented earlier, and it

1098

CHOKSHI and LANGDON:

‘11 Zn-22%

._

10-z

AI

10-I

Livesey

DIFFUSIONAL CAVITY GROWTH IN SUPERPLASTICITY

ond Rldley (19821

1.0

IO

102

rlpm)

Fig. Il. Cavity growth rate vs cavity radius for Zn-22% Al, using the experimental conditions of Livesey and Ridley

UOI. suggests the presence of rounded cavities up to -2pm, elongated cavities growing by power-law growth in the range - 2 to -4 pm, rounded cavities growing by superplastic diffusion growth from -4 to -7 pm, and elongated cavities with radii > 7 pm. These transitions are in reasonable agreement with the experimental observations noted earlier.? 6. DISCUSION

The superplastic diffusion growth model is applicable when the cavity intersects a number of grain boundaries, and it provides a satisfactory explanation for the large rounded cavities, up to an order of magnitude larger than r,, reported in some superplastic materials. However, it has been argued that diffusion growth may be constrained in high temperature deformation, and it is necessary also to consider whether it is possible to explain the observations of rounded cavities in terms only of cavity coalescence. These potential difficulties are now examined. Dyson [33,34] first noted that diffusion growth may be constrained in a polycrystal undergoing high temperature creep if the creep cavities are inhomogeneously distributed amongst the various grain boundaries, and this concept was subsequently further developed by BeerC [35], Rice [36,37] and Stephens and Nix [38]. Caceres and Wilkinson [14] used a similar approach for a superplastic Cu alloy and concluded that the diffusion growth of cavities was always controlled by plasticity of the matrix. tLivesey and Ridley [lo] reported some concurrent grain growth during the test. This effect was not included in the analysis, but it would tend to decrease the value of rcsp in Fig. 11 through equation (29).

In practice, however, it is unrealistic, for two reasons, to invoke constrained diffusion growth under the conditions typically used in the deformation of superplastic materials with fine grain sizes. First, the model of Dyson [33] for a constraint on the cavity growth rate was based on a rigid hexagonal array of grains where the formation of cavities would lead, in the absence of grain deformation, to the overlap of material at adjacent noncavitated boundaries. The concept of a rigid structure does not apply in superplastic deformation because of the presence of very extensive grain rotation of up to f30” [3941], the measured high values for the grain boundary sliding contributions [42], and the direct experimental evidence for extensive grain switching and grain rearrangement [39,41]. Second, a necessary condition for constrained diffusion growth is that the strain rate resulting from unconstrained cavity growth is faster than the creep rate of the surrounding grains. In superplastic materials, however, the grain rotation and grain rearrangement occur under conditions where the creep rates are substantially faster than in similar alloys with a large grain size. For example, Shei and Langdon [43] reported experimental creep data on the Cu-2.8% Al-1.8% SiX4% Co alloy for grain sizes of both 3.0 and 100 pm. At a temperature of 823 K, corresponding to Fig. 9, the creep rate at cr = 40 MPa was three orders of magnitude faster for the superplastic material (-2 x 10m3s-’ for d = 3.0 pm compared with -2 x 10m6s-l ford = 100 pm). Thus, the combination of very rapid creep rates and considerable grain rotation and re-arrangement indicates that superplastic diffusion growth will proceed without the constraints inherent in diffusion growth under normal high temperature creep conditions. The possibility of cavity coalescence has been invoked in several sets of experiments [14,4446] but the evidence is generally indirect. There is some limited evidence for cavity coalescence at high strains as, for example, in the work of Clegg et al. [121on the Cu-2.8% Al-1.8% Si-O.4% Co alloy where there was an increase in the cavity growth rate but a corresponding decrease in the total number of voids. There is no doubt that many of the larger rounded cavities in the Cu alloy are due to coalescence, and this may be inferred from the published photomicrographs where there are clear examples of adjacent cavities which are in the process of linking. However, it was concluded earlier for this alloy by Miller and Langdon [24], and it is reiterated here, that the smaller individual rounded cavities, having radii up to - 15 pm, appear to have grown by a diffusion process rather than being formed by coalescence and rapid spheroidization. As noted earlier, these radii are consistent with the predictions of superplastic diffusion growth in Fig. 9. Finally, four additional points should be noted. First, the possibility of mass transport along several grain boundaries for cavity sintering was

CHOKSHI

and LANGDON:

DIFFUSIONAL CAVITY GROWTH IN SUPERPLASTICITY

suggested recently to explain solid state bonding in the superplastic Ti-6% Al4% V alloy where the grain size was smaller than either the cavity,size or the original surface roughness of the two bonded surfaces [47]. This is equivalent, for the situation of cavity sintering, to the cavity growth process developed in this report. Second, it is important to note, from the expression for the superplastic diffusional cavity growth rate, that there are three limiting conditions for the use of the model: (i) low strain rates, (ii) intermediate testing temperatures where vacancy diffusion into the cavities occurs predominantly along grain boundaries rather than through the lattice, and (iii) fine grain sizes of the order of 6 5 pm. Thus, it was demonstrated recently that this mechanism is not important during cavity growth in Al-Li and Al-Zn-Mg alloys deformed at high strain rates [48], nor is it important in commerical Al-based alloys deformed at high temperatures where vacancy diffusion takes place primarily through the lattice [49]. Furthermore, the requirement of a very fine grain size excludes some of the more recent superplastic alloys, such as the Al-7475E alloy investigated by Pilling and Ridley [50] with a mean linear intercept grain size, L, of 11 pm (equivalent to d = 1.74 L N 19 pm). It is interesting to note that the Al-7475E alloy exhibited intergranular cavity networks and this is markedly different from the discrete cavities observed in conventional superplastic alloys with very fine grain sizes. Third, there may be some enhancement in the rate of growth due to an interaction between adjacent cavities. This effect is not included in the present analysis, but there is some evidence for interactive effects from model experiments on superplastic specimens with pm-machined holes [51]. Fourth, it is possible to determine the cavity radius at any stage of the deformation process by integrating the expressions for the diffusion, superplastic diffusion and power-law cavity growth rates. It is demonstrated elsewhere [52] that this procedure yields results which are consistent with the experimental observations in a superplastic copper alloy under conditions where there is only limited cavity interlinkage. 7. SUMMARY AND CONCLUSIONS 1. The diffusional growth of cavities is enhanced in superplastic materials having very fine grain sizes if the cavity size exceeds the grain size so that vacancies diffuse into the cavity along a number of grain boundary paths. 2. A model for this process is presented, and it is shown that superplastic diffusion growth leads to a rate of change of cavity radius with strain which is (i) independent of the instantaneous cavity radius and (ii) inversely proportional to the square of the grain size.

1099

3. In practice, superplastic diffusion growth becomes important at lower strain rates when the specimen grain size is typically less than N 5 pm. 4. Modifications are presented to incorporate within the model both the average strain rate in a test conducted at a constant cross-head displacement and the occurrence of concurrent grain growth. 5. The process of superplastic diffusion growth is capable of providing an explanation for the experimental observations of large rounded cavities in several superplastic materials. It is shown that the predictions of the model are in good agreement with experimental results on a superplastic Cu alloy and the Zn-22% Al eutectoid. Acknowledgements-This work was supported in part by the International Copper Research Association under Project No. 324 and in part by the National Science Foundation under Grant No. DMR-8503224.

REFERENCES 1. T. G. Langdon, Metal Sci. 16, 175 (1982).

2. S. Saaat. P. Blenkinsoo and D. M. R. Taolin, J. Inst. Metals iO0, 268 (1972)‘.

3. S. Sagat and D. M. R. Taplin, Acta metall. 24, 307 (1976).

4. S-A. Shei and T. G. Langdon, J. Muter. Sci. 13, 1084 (1978).

5. D. W. Livesey and N. Ridley, MetaN. Trans. A 9, 519 (1978).

6. T. Chandra, J. J. Jonas and D. M. R. Taplin, J. Mu/er. Sci. 13, 2380 (1978).

7. C. W. Humphries and N. Ridley, J. Mater. Sci. 13.2477 (1978).

8. D. A. Miller and T. G. Langdon, MetalI. Trans. A 9, 1688 (1978).

9. M. M. I. Ahmed, F. A. Mohamed and T. G. Langdon, J. Mater. Sci. 14, 2913 (1979).

10. D. W. Livesev and N. Ridlev. J. Mater. Sci. 17. 2257 (1982). 11. D. W. Livesey and N. Ridley, MetaN. Trans. A 13, 1619 (1982). 12. W. J. Clegg, J. A. Rooum and A. K. Mukherjee, in Strength of Metals and Alloys (ICSMA 6) (edited by R. C. Gifkins), Vol. 2, p. 689. Pergamon Press, Oxford (1983). 13. N. Ridley, D. W. Livesey and A. K. Mukherjee, J. Mater. Sci. 19, 1321 (1984). 14. C. H. Cdceres and D. S. Wilkinson, Acta metall. 32,423 (1984). 15. C. W. Humphries and N. Ridley, J. Muter. Sci. 9, 1429 (1974). 16. C. I. Smith and N. Ridley, Metals Tech. 1, 191 (1974). 17. K. Matsuki, Y. Ueno, M. Yamada and Y. Murakami, J. Japan Inst. Metals 41, 1136 (1977). 18. H. Ishikawa, D. G. Bhat, F. A. Mohamed and T. G. Langdon, Metall. Trans. A 8, 523 (1977). 19. J. Belzunce and M. Suery, Acta metall. 31, 1497 (1983). 20. M. V. Soeight and W. Beer&, Metal Sci. 9, 190 (1975). 21. W. Beeri and M. V. Speight, Metal Sci. 12, 172 (1978). 22. J. W. Hancock. Metal Sci. 10. 319 (1976). 23. D. A. Miller and T. G. Langdon, Scriptn metall. 14, 179 (1980). 24. D. A. Miller and T. G. Langdon, Metal/. Trans. A 10, 1869 (1979). 25. D. A. Miller and T. G. Langdon, Trans. Japan Inst. Met& 21, 123 (1980).

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DIFFUSIONAL

26. M. J. Stowell, in Superplastic Forming of Structural Alloys (edited by N. E. Paton and C. H. Hamilton), p. 321. Metall. Sot. A.I.M.E., Warrendale, Pa (1982). 21. I.-W. Chen and A. S. Argon, Acta metall. 29, 1759 (1981). 28. H. Gleiter. in Deformation of Polvcrvstals: Mechanisms * and Microstructures (edited by N. Hansen, A. Horsewell, T. Leffers and H. Lilholt), p. 15. Rise National Laboratory, Roskilde, Denmark (1981). 29 T. G. Langdon, in Superplastic Forming of Structural Alloys (edited by N. E. Paton and C. H. Hamilton), p. 321. Metall. Sot. A.I.M.E., Warrendale, Pa (1982). 30. M. F. Ashby, Acta metall. 20, 887 (1972). 31. R. Raj and M. F. Ashby, Acta metall. 23, 653 (1975). 32. P. Yavari and T. G. Langdon, Mater. Sci. Engng 57,55 (1983). 33. B. F. Dyson, Metal Sci. 10, 349 (1976). 34. B. F. Dyson, Can. Metall. Q. 18, 31 (1979). 35. W. Beer&, Acta metall. 28, 143 (1980). 36. J. R. Rice, Acta metall. 29, 675 (1981). 31. P. M. Anderson and J. R. Rice, Acta metall. 33, 409 (1985). 38. J. J. Stephens and W. D. Nix, Metall. Trans. A 17, 281 (1986). 39. A. E. Geckinli and C. R. Barrett, J. Mater. Sci. 11, 510 (1976). 40. K. Matsuki, Y. Ueno and M. Yamada, J. Japan Inst. Metals 38, 219 (1974). 41. K. Matsuki, H. Morita, M. Yamada and Y. Murakami, Mefal Sci. 11, 156 (1977). 42. T. G. Langdon, Metall. Trans. A 13, 689 (1982). 43. S.-A. Shei and T. G. Langdon, Acta metall. 26, 639 (1978). 44. J. Belzunce and M. Suery, Acta metall. 31, 1497 (1983). 45. M. J. Stowell, D. W. Livesey and N. Ridley, Acta metall. 32, 35 (1984). 46. J. Pilling, Mater. Sci. Tech. 1, 461 (1985). 47. J. Pilling, D. W. Livesey, J. B. Hawkyard and N. Ridley, Metal Sci. 18, 117 (1984). 48. A. H. Chokshi, J. Mater. Sci. Lett. 5, 144 (1986). 49. A. H. Chokshi, J. Mater. Sci. 21, 2073 (1986). 50. J. Pilling and N. Ridley, Acta metall. 34, 669 (1986). 51. A. H. Chokshi and T. G. Langdon, in Superplasticity (edited bv B. Baudelet and M. Sutrv). p. 2.1. Centre National-de la Recherche Scientifiquk;.Paris (1985). 52. A. H. Chokshi and T. G. Langdon. To be published. ”

APPENDIX

,,

CAVITY GROWTH IN SUPERPLASTICITY sections have intermediate radii so that (r, + r, +.

. + r”).

(A3)

APPENDIX II A Geometric Procedure to Estimate the Variation in the Number of Grain Boundaries Intersected as a Function of Increasing Cavity Radius A three-dimensional spherical cavity of radius r was placed inside a three-dimensional hexagonal array of grains. The situation is illustrated schematically in elevational and sectional form in Fig. Al. Parallel sections were cut through the cavity in the xy plane, with the first section at the cavity diameter through KA in Fig. Al(a) and subsequent sections displaced in the z direction. The section at KA is illustrated in Fig. Al(b), where 0 is the center of the section and OA is the cavity radius r. To estimate the total number of grain boundaries intersected by the cavity, N, the complete hexagonal array was drawn in Fig. Al(b) and the number of grain boundaries intersecting the circumference of the circle was counted and labelled N(0). Then, with 0 as center, a second circle was drawn on the hexagonal array with a radius of XB where, as shown in Fig. Al(b), OX is equal to the height of a hexagon. This circle represents the section LX’B’ in Fig. Al(a), and the number of grain boundary intersections around the circumference gave a value for N(2). By continuing this procedure for sections at consecutive hexagon heights up to the top of the spherical cavity, it was

I

The Validity of the Approximation Used to Derive Equation (18) Equation (18) was derived from equation (17) by using the approximation [1 - x(d/r)2]“Z N 1, where x = 3/16, 3/4, 3, etc. Physically, this approximation appears to imply that the decrease in the radii of the sections above the diametrical plane of the spherical cavity has been neglected. To check the significance of this approximation, it is necessary to estimate the location of the sectional plane in the sphere lying a distance of yh above the diametrical section and having a radius of r, = r/2. From Fig. 6(b), it follows that rf = r2 - (yh)2

(A])

so that, with r, = r/2 and h2 = 3d2/16, y = 2(r/d).

+ rn) < nr < 2(r, + r2 +

It follows from equation (A3) that the approximation used in the analysis is within a factor of 2 of the correct value, so that equation (20) is also correct to within a factor of 2. This is reasonable in view of the uncertainties in the various parameters involved in the superplastic diffusion growth process.

(A2)

Equation (A2) is equivalent to the earlier relationship of n r 2(r/d), representing the total number of sections above the diametrical plane in a hemisphere of the cavity. Thus, only the top section of the spherical cavity has a radius equal to one-half of the spherical cavity, and the intermediate

(b)

u

Fig. Al. Schematic illustration of a three-dimensional spherical cavity contained in a three-dimensional hexagonal array of grains: (a) the method of taking parallel sections in the xy plane, and (b) the section at KA.

CHOKSHI

and LANGDON:

DIFFUSIONAL

CAVITY GROWTH 102

Table AI. Calculated variation of N with the ratio r/d r&l

N

0.25 0.65

4 16 24 36 60 204 403 619 800 1331

I .oo 1.15

1.25 2.50 3.50 4.50 5.15 6.50

I

IN SUPERPLASTICITY t

’

1101

’ c ““$1

IOr

e i LO-

i < MM 0

possible to estimate the numbers of intersections as w(4), N(6), etc. Since a cavity may intersect a grain at two gram boundaries, it was necessary also to inchzde the numbers of intersections at the mid-sections, such as K’O’A’ in Fig. Al(a). These numbers, labelled N(l), N(3), N(5), etc. were obtained by linear interpolation from the even numbered sections on either side. The summation from N(1) to N(last) gives the number of intersections in one hemisphere of the cavity, so that the total number of intersections for the spherical cavity is given by N = N(O) + 2[N(lf + N(2) -t- N(3) + . + N(iast)],

(A4)

This procedure was adopted for a range of cavity radii varying from 0.25 to 6.5 d, and the resultant values of N are presented in Table Al. The variation of r/d with N is shown in Fig. A2, and it is clear that, for the significant values of

fb’,*o

I IO

I 102

L I ,r,,,,l 103

104

N Fig. A2 The relationship between r/d and N for cavity radii from 0.25 to 6.5 d.

N when rjd & 0.5, all of the datum points he on a straight line with a slope of OS. The equation of the Line in Fig. A2 is given by N = 32(r/d)2 (AS) and this is identical to equation (20) derived earlier by making some approximations. Thus, there is excellent agreement between the approximate procedure outlined in section 3.2 and the rigid geometric procedure described herein.