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The effect of particle size distributions on the CRSS of aged NiAl alloys
THE EFFECT OF PARTICLE SIZE DISTRIBUTIONS ON THE CRSS OF AGED Ni-Al ALLOYS k’. MUNJAL
and A. J. ARDELL
Materials Department. School of Engineering and Applied Science, University of California. Los Angeles. CA. U.S.A. (Recriced
24 F‘rbruary
1976)
Abstract-The effect of the width of the 7’ particle size distribution on the CRSS of Ni-AI single crystals containing nominally 6 wt.>; Al was investigated. The samples were given two-step aging treatments to produce broader unimodal distributions than those resulting from isothermal aging. The first aging treatments were done at 600 and 650°C for 458 and 69 hr, respectively, while the second were done at 625’C for various lengths of time. In all the samples the average particle size was nearly constant. but the standard deviation, b. was increased by as much as 300/,, primarily by the addition of smaller precipitates. After correcting for variations in the 7’ volume fraction introduced by these treatments, the effect of increasing D was to reduce the CRSS of the samples by approx. So& This change, which is very small, is nevertheless substantially larger than that predicted by the computer experiments of Foreman and Makin. It is suggested that the discrepancies are due to the shortcomings of modelling real precipitate dispersions by arrays of point particles. R&urn&--On a ttudie I’effet de la largeur de repartition des tailles des particules 7’ sur la CCR de monocristaux de Ni-AI contenant 6% d’aluminium en poids. On faisait subir aux echantillons un vieillissement en deux etapes afin de produire une repartition unimode plus large que celle que I’obtient par recuit isotherme. Les premiers recuits itaient de 488 et 69 h a 600 et 650°C respectivement, alors que les suivants Ctaient effectues a 625°C pendant des durees variables. La taille moyenne des particules etait a peu prts constante dans tous les ichantillons. mais la deviation standard u pouvait augmenter de 30?/,. essentiellement par I’addition de petits precipites. Apres correction pour la variation de la fraction volumique de ; ’ introduite par ces traitements, on voit que l’augmentation de a produit une diminution de la CCR d’environ 8%. Ce changement est tres petit, il est toutefois nettement plus grand que celui prevu sur ordinateur par Foreman et Makin. Nous pensons que les differences proviennent de l’insuffisance de modele qui represente une repartition de pricipitts reels par un reseau de particules ponctuelles. Zusammenfassuttg--Der EinAuD der Breite der GroDenverteilung von v’-Partikeln auf die kritische FlieBspannung wurde an Ni-AI Einkristallen mit 6 Gew.-% Al Nominalgehalt untersucht. Die Proben wurden in zwei Stufen gealtert, urn breitere Verteilungen als bei isothermer Alterung zu erhalten. Die erste Stufe fand statt bei 600 und 650°C fur 488 und 49 h. die zweite bei 62X iiber verschieden lange Zeiten. In ailen Proben war die mittlere PartikelgrijBe nahezu konstant, die Standardabweichung e aber war 30’1; gro&r. vorw.iegend wegen der vorhandenen kleineren Ausscheidungen. Es ergab sich nach einer Korrektur wegen Anderungen im Volumenbruchteil der y’-Phase, die durch die Behandlung erzeugt worden yar, da8 das vergrijBerte G die kritische FlieDspannung der Proben urn etwa 8% erniedrigte. Diese Anderung ist zwar sehr klein, jedoch betrachtlich griil3er als die von den Computerexperimenten von Foreman und Makin vorhergesagte. Es liegt nahe, daD diese Diskrepanz von der Unzulanglichkeit einer Beschreibung einer realen Verteiiung von Ausscheidungen mit punkfiirmigen Teilchen herriihrt. 1. INI’RODUCI-ION
Several years ago Foreman and Malcin [l] using a computer model, investigated the detailed movement of a dislocation line of constant line tension through an array of point obstacles randomly distributed in space. Each obstacle had the same strength, characterized by the value of the critical angle, 4, defined as the angle included between the two arms of the dislocation, pinned at the obstacle, at the instant of its release. They obtained numerical results for the critical shear stress. Ar, defined as the stress required for the steady-state movement of the dislocation through the array of obstacles, as a function of &. In the limit of weak obstacles (&,- R), the so-called Friedel relation (see. e.g. Brown and Ham [Z]) was 827
found to satisfactorily describe the relationship between Ar and $J,, whereas in the strong particle limit (I$,- 0) Ar was found to be relatively insensitive to 9,. In precipitation-hardened alloys the obstacles are not point particles but are finite, the average planar radius, (r,), being related to & Furthermore, real precipitates are never monodisperse; there is always a distribution of particle sixes and hence a distribution of 4, values. (Even for a monodisperse system there will be a distribution of r* values.) Under these circumstances, it is useful to know to what extent the stress required to move a dislocation through a real distribution of precipitates is related to the strength of the average particle in the distribution.
828
MUNJAL c&
>
AND
ARDELL:
AT INTERVALS
o = 10,000
EFFECT OF PARTICLE SIZE DISTRIBUTIONS Of a/S
+_,
OBSTACLES
+max
Fig. 1. The results of the computer experiment of Foreman and Makin [3] showing the variation of the critical shear stress, AT, as a function of the spectrum width, 4,. for an array of point obstacles having a square spectrum of breaking angles (inset).
It is usually assumed that they are equal. Foreman and Makin [3] also considered this problem in another computer experiment. They determined the CRSS of a system containing a random array of point obstacles with a square spectrum of strengths, the results of which are reproduced in Fig. 1. It can be seen here that there is an increase in strength when the spectrum width, &, increases from 0 to - ~min for a given value of the mean critical 4 bziking angle, (4,). The predicted effect of &, is much larger for larger average breaking angles (weaker particles) than it is for small values of (Q, but it is nevertheless relatively small. Since 4, can be related to the standard deviation, CJ,of the particle size distribution, changes in cr in a real alloy system can be expected to have a relatively small effect on the CRSS. This conclusion, however, had never before been tested by mechanical property measurements on real alloy systems, and was the reason for undertaking this investigation. 2. EXPERIMENTAL
PROCEDURES
We expected that the detection of small changes in Ar due to changes in Q would be successful only under restricted circumstances in which the influence of other factors could be either estimated or compensated for. The experimental approach adopted for these purposes is described below.
ON THE CRSS
Two Ni-AI single crystal samples. cut from the same crystal used previously [J]. containing nominally 6 wt.:/, Al were solution treated at 1015 &-5C in an atmosphere of purified He and quenched into a refrigerated saturated NaOH solution. One each was then aged at 650 and 6OO’C for a period long enough to attain the same average particle size (a/2) z 113 A of the cuboidal shaped 7’ (iVi,Al) precipitates of edge length a. This is the value of (a:,2> at peak strength in Ni-676 Al samples aged isothermally at 62YC for 175 hr [4]. The amount of aging time required to produce this value of (a/2) at the two aging temperatures* was determined by interpolation of the coarsening data of Ardell and Nicholson [5]_ The distribution of y’ particle sizes in these isothermally aged samples was expected to be the same. a conclusion which follows from the experimentally determined histograms of Ardell and Nicholson [5]_ The distribution of particle sizes was then changed by re-aging these samples at 625’C for various times. The value of (u/2) at peak strength was chosen because Ar is insensitive to (a/2) in the neighborhood of the aging peak (see Fig. 2). Thus, a small variation of the acrrage particle size resulting from the two-step aging treatments should have a minimal effect on the CRSS. The major disadvantage in selecting this particular value of (a;)) is that the insensitivity of the CRSS to changes in (a/2> could be accompanied by a similar insensitivity to changes in 6. Nevertheless, it was felt that if a smaller value of (a/2) were selected (e.g. somewhere in the linear portion of Fig. 2, where Ar x(a ,2)’ ‘1 it would have been very difficult, if not impossible. to separate the effects of changes in (a/2) from those due to changes in IJ because of the relatively large sensitivity of Ar to the average particle size in this range. In addition, it would have been considerably more difficult to measure changes in the width of the distribution for values of (~$2) in the neighborhood of 25-50 A,
“\
60 TESTING A 0 0 0
TEMP I-K) 77 191 297 373
2.1 Aging ti-eafmenf The values of cr were changed by using two-step aging treatments involving a change in temperature.
* A third sample was aged first at 675’C. However, reaging at 625’C resulted in a bimodal distribution of 7’ particle sizes [6], which was not appropriate
to this study.
Fig. 2. The increment in the CRSS. AT. YS the square root of half the mean particle edge length. (a) 2 of the y’ precipitates in a Ni-67; Al alloy isothermally- aged at 625C. Data of Munjal and ArdeU [J].
MUNJAL
AND
ARDELL:
EFFECT OF PARTICLE SIZE DISTRIBUTIONS
because errors in the measurements are frequently very large in this range of particle sizes. 2.2 Compression testing and measurement of the volume fraction Since the CRSS is theoretically dependent upon the volume fracti0n.J at the aging peak [2], the expected variations in f resulting from the two-step aging treatments were determined experimentally. This was done by measuring the Al content of the matrix after each aging treatment using a Curie point apparatus[7]. To minimize the effects of possible variations of the initial concentration of Al that can occur from one sample to another, the compression samples that were aged first at 650 and 600’C were then re-aged at 625’C for a given time in a purified He atmosphere. quenched into the refrigerated NaOH solution. tested under compression, re-aged at 635C, retested. etc. The total aging time at 625’C thus represents the accumulated value up to the time specified. The initial Al contents of the two samples were also determined by measuring their Curie points after solution treating and quenching. The composition of the sample aged first at 65O’C was actually j.SSg;, Al. while that of the sample aged first at 6OO’C was actually 5.9OY/, Al. The samples were compression tested at room temperature only. After this testing was completed, the samples were re-solution treated, isothermally aged at 625°C for 175 h and mechanically tested again at room temperature. This was done to provide control specimens with which to compare the CRSS values determined from the previous tests. By performing the experiments in this way, the danger existed that subsequent aging would be influenced by the dislocations introduced during plastic deformation. since 7’ precipitates have been observed to grow or coarsen preferentially at edge dislocations [8]. However, we believe that this process had a negligible effect on the results of subsequent CRSS measurements because the plastic strain during each test was very small ( r0.3Yb).
ON THE CRSS
829
of 7’ particles in a particular sample was determined using a Zeiss particle size analy-zer. To obtain statistically meaningful results. the diameters of between 1000 and 1200 particles were measured. 3. RESLLTS 3.1 Distribution of;:’ particles The particle size distributions in the doubly-aged samples are shown in Fig. 3. The distribution of -,J’ particles in the sample first aged at 65O’C did not appear to change substantially as t at 625’C increased. Nor did there appear to be any significant shift in the peak of the histograms as a function of t at 625’C. In Fig. 3(b). it can be seen that the histograms for the samples first aged at 6OO’C are definitely narrower than those in the sample first aged at 650°C. As t at 625’C increased, the distribution of y’ particles for these samples apparently became narrower, and the peak clearly shifted toward smaller particle sizes.
2.3 Determination of the distributiorf of 7’ particle si:es To determine the distribution of 7’ particles, polycrystalline sheet samples (also containing nominally 6% Al) were heat-treated at the same temperatures for the same amounts of time as the single crystal samples, and were examined by compression transmission electron microscopy. It should be pointed out that magnetic measurements indicated a slight difference in composition between the polycrystalline NCAl sheet and the Ni-Al single crystals (about 0.142 wt.% Al). However, this was not expected to influence the size distribution of the ^r” particles since coarsening behavior is independent of volume fraction [S, 91. Thin foils in (001) orientation taken from the polycrystalline samples were photographed in dark field using a 7’ superlattice reflection. The distribution
Fig. 3. Histograms of the 7’ particle sizes observed in samples aged first at (a) 65O’C for 69 h. and (b) 600°C for 188 h. and then re-aged at 625’C for the times (h). indicated by the numbers inset. The dashed curves represent the theoretical distribution of the LSlV theory.
SjO
MUNIAL
AXD
ARDELL:
EFFECT OF PARTICLE SIZE DISTRIBUTIONS
Table 1. The effect of the doubIe-a~ng Aging treatment
tihrl
69 hr, 650°C i-
oiA1
a,!
99.1 101.6 98.3
33.7 32.8 35.7
0.35
0.32 0.36
98.6 89.1 87.8 78.2
30.9 29.7 24.9 27.3
0.3 I 0.33 0.2s 0.35
59.5 72.5 89.5 95.5 104.0
15.7 18.0 23.1 22.8 24.2
0.26 0.25 0.26 0.25 0.23
I 4
488 hr. 6WC f
0.25 1 4 10
Isothermally aged* at 625°C (6.35% Al)
treatments on the SD., o, of the Ni-A1 alloys
w>(A)
0.25
24 48 72 96 146
ON THE CRSS
(G’(U
z,),”
0.3-1
0.32
0.25
* Data of Ardell and Nicholson fj].
The values of (a/2) and (7 obtained by analysis of the histograms in Fig. 3 are summarized ln Table 1. The (u/2> values for the doubly-aged samples were found to be slightly smaller than the calculated value of (a/2> in samples isothermally aged at 625°C for 175 h (113 A). For the samples first aged at 65O”C, the values of IJ and (n/2> remained essentially constant, whereas for the samples first aged at 600°C the values of G and (q/2> decreased as t at 62X increased. However, the values of a/(u/2> are nearly constant for the second series of samples, suggesting that the observed changes in 0 are basically due to scaling factors. The values of 0 in Table 1 were compared with those for isothermally aged samples in order to determine how the heat treatments used here affected them. For this purpose, the histograms of Ardell and Nicholson t5J, obtained from a Ni-6.357: AI alloy isothermally aged at 625°C for various times, were
/f: /d / i I
rl 1
I
“L -0
1
I
25
50
75
100
125
150
175
a/2 (AI
Fig. 4. Histogram of the 7‘ particle sizes obtained from a sample of Ni-6.35% Al isothe~~ly aged at 62X for 96 h. Data of Ardell and Nicholson [YJ.
analyzed to obtain values of o/(a/2>, which are theoretically time-independent [IO, 1I]. These values are also shown in Table 1, where it is seen that the two-step heat-treatments increased the values of aJ(a/2> by about 30% over the value normally obtained on isothermal aging. It is important to note the increase in c resulting from the two-step aging treatments is due primarily to the addition of small particles. This may be verified by su~rim~sing the theoretical distribution of the Lifshitz-Slyozov-Wagner theory on the histograms in Fig. 3 (dashed curves), and comparing them with the histograms measured by Ardell and Nicholson, one of which is shown in Fig. 4. In isothermally aged alloys the agreement between the theoretical distribution and the ex~rim~~1 histograms is excellent at smalf. particie sizes. It is evident on comparing Figs. 3 and 4 that the discrepancy at small sixes is much larger for the doubly-aged alloys used in this study.
The results of the compression tests are shown in Fig. 5(a). The contribution to strengthening due to the ‘J’ precipitates alone (AT) was obtained by subtracting the room temperature value of rc of the solid solution (41 MN/m’) [6] from the values of the CRSS. There is an increase in the As value as t at 62jSC increases for the samples first aged at 650°C; whereas in the sample first aged at 6OO’C, As shows a slight drop as t at 625°C increases. The variation off with t in these samples is shown in Fig. 5(b). It is apparent that this variation could possibly account for the entire variation of Ar. To test this possibility, the values of Ar in Fig. 5(a) were compensated by plotting As/f II2 as a function of t. This method of plotting is justified by the theoretical proportionality between AT andf”’ in the vicinity of the aging peak [Z]. The results are shown in Fig. 5(c), where it is evident that the AT/f iI2 values remain essentially constant within experimental error (estimated as &37, in this experiment}. The horizontal dashed line in Fig. 5(c) shows the average value of AT/~‘/* for the two samples used
MUNJAL
AND
ARDELL:
EFFECT OF PARTICLE SIZE DISTRIBUTIONS
a’
in this study after re-solution treating and isothermally aging them (see also Table 2).* On comparing these results with those of the two-step aging treatments we concluded that increasing the value of 0 [or (a/(u/2),,] by about 3O”a results in a decrease in the Ar/fil’ values of about By/,. In other words, the addition of smaller particles to a lowering of the CRSS.
to increase
(a)
ON THE CRSS
-A-A-
LA
i
0 leads
erw k
4. DISCUSSION
In this section we attempt to rationalize the observations in Fig. 5 with results of Foreman and Makin’s computer experiments [3]. For this purpose, the critical breaking angles (4,) for each size interval of Fig. 3 were calculated using equations originally derived by Ham[12] for the shearing of finite ordered particles by dislocations initially edge in character. The details of the breaking angle calculations are discussed in the Appendix, and the results are shown in Fig. 6. It should be noted that these breaking angle histograms do not include the effect of the distribution of particle radii that one observes on the slip plane even when the particles are monodisperse. This effect will, of course, broaden the histograms in Fig. 6, which thus underestimate the width of the distribution of breaking angles. Using these data, the mean critical breaking angle, (4,) and the standard deviation of &, cr,+,were calculated. Table 3 summarizes these results and also shows values of the spectrum width, &, which were calculated for an array of obstacles with a square spectrum of strength having the same a, values as the histograms in Fig. 6. (It is easy to show that when the distribution of +c is
--A
7-
4-
31
I
--__ _______ -De
I
_____ ____ ___-
9 -LO
5.86
Table 3. Effect of aging treatment
2’ s
Initial aging treatment 0650°C, 69 hr 46oo’C, 488 hr
200
a I
I
I
IO
t,
hr
Fig. 5. The variation of: (a) the increment of the CRSS due to ‘J’ precipitates, AT; (b) the volume fraction of 7’. f: (c) AT/~“‘, with re-aging time, t, at 62X for the Ni-Al alloys initially aged at 600°C for 488 h and 65O’C for 69 h. The dashed horizontal line in (c) represents the average value of AK/~“’ of three samples aged isothermally at 625’C for 175 h.
AS 1 (MN/m-)
f
AT/f ‘.’ (MN/m’)
(AT/~’ ‘)av (MN/m’)
70 68 74
0.048 0.050 0.055
320 304 316
313
on the mean critical breaking angle, (4,). width. &
and the spectrum
Aging treatment
nhr)
(A)
tiA)
69 hr. 65OT
0.25
+
1
99.1 101.6 98.3
34.7 32.8 35.7
82.0 80.3 82.9
22.8 21.8 23.8
79. I 75.5 82.6
98.6 s9.1 87.8 78.2
30.9 29.7 24.9 27.3
82.0 88.4 88.9 95.9
19.9 19.8 16.5 18.9
68.8 68.5 57.1 65.3
4 488 hr, 6OOC +
0.25 1 4 10
____
IA--
Table 2. The values of AT/~‘!? obtained after re-solution treating and isothermally aging the Ni-Al samples at 625°C for 175 hr
5.88 5.90
t
9
* The value of AT obtained after re-solution treating the sample first aged at 675°C is also included in Fig. 5(c) and Table 2.
Alloy composition (wt.% Al)
831
832
MUNJAL
AND
ARDELL:
EFFECT OF PARTICLE SIZE DISTRIBLTIONS
ON THE CRSS
0 10
160
160
140
120
60
100
60
40
20
0
O,KEGREES)
Fig. 6. Histograms of the critical breaking angles, &, for the samples aged first at (a) 650X for 69 h and (b) 600°C for 488 h and then re-aged at 625°C for the times (h). The values of 4, were calculated from the histograms in Fig. 4 using the procedures described in the Appendix.
square, 4, = 2J3 G+). This was done so that the present results could be compared somewhat more directly with the predictions of Foreman and Makin’s computer experiments [3]. To estimate the changes in (4,) and & brought about by the two-step aging treatments, these parameters were calculated for several values of (n/2) in the size range 8%105A from the distributions of Ardell and Nicholson [S], which are characteristic of isothermally aged Ni-Al alloys. The values of (I$~) were found to vary from 84 to 73”, while the values of & varied from 59 to 70”. On comparing these figures with those seen in Table 3, it is apparent that the two-step aging treatments increase the values of 4, by a very small amount over the values normally obtained after isothermal aging. Although it is somewhat difficult to compare the values of 4, in isothermally aged samples with those of the doubly-aged samples on a one-to-one basis (e.g. for samples having the same value of (a/2)), it appears that the change in &,, brought about by our two-step aging treatment was approx. 17% for particles in the range 85 < (a/Z) < 105A. Now, Fig. 1 shows that when (4,) is in the neighborhood of 3~18, changes in b, have almost no in-
fluence on the values of the CRSS. However, the data show that the 17% increase in the values of & brought about by the two-step aging treatments are accompanied by an 8% decrease in the CRSS. Even though this is a small effect, it is much larger than predicted by the computer experiments of Foreman and Makin. The finite character of real precipitates could be responsible for this result. The motion of dislocations through an array of finite obstacles should differ from that through an array of point obstacles, because the interaction between dislocations and precipitates in a real alloy system occurs over a range of distances. The introduction of finiteness also affects the average obstacle spacing along the dislocation, which is an important parameter governing the CRSS, particularly when the precipitates are small and relatively weak. Although it is premature to claim that finiteness alone could acco&tt for the small discrepancy that exists behveen theory and experiment, it is the most obvious source. There is clearly a need for computer experiments on arrays of finite obstacles. Finally, our results confirm what all investigations of precipitation hardening have always taken for granted: the influence of u is so small that we need
MUNJAL
ASD
ARDELL:
EFFECT OF PARTICLE SIZE DISTRIBUTIONS
not worry about the shape of the particle size distri-
bution in any quantitative comparison between theory and experiment. The distributions are adequately characterized by the particle of average size, which is justifiably equated to the single particle size that is encountered in the typical idealized theoretical situation. dci;nowlrdgement-Financial support for this research was provided by the U.S. Energy Research and Development Administration under contract No. E(O4-3)-34, P.A. 172.
REFERENCES 1. A. J. E. Foreman and M. J. Makin. Phil. Maq. 14, 911 (1966). 2. L. M. Brown and R. K. Ham, Srrengthening Methods in Crvstals. edited bv A. Kelly and R. B. Nicholson. p. 9. Wiley, New Ydrk (1971).3. A. J. E. Foreman and M. J. Makin. Can. J. Phqs. 45, 737 (1967). 4. V. Munjal and A. J. Ardell. Acta Met. 23, 513 (1975). 3. A. J. Ardell and R. B. Nicholson, J. Php. Chem. Solids 27, 1793 (1966). 6. V. Munjal. Ph.D. Dissertation, University of California, Los Angeles (1975). 7. A. J. Ardell, Acta Met. 16. 511 (1965). 8. A. J. Ardeil and R. B. Nicholson. rlcta &fer. 14, 1295 (1966). 9. D. J. Chellman and A. J. Ardell, rlcra iMet. 22. 577 (1974). 10. I. M. Lifshitz and V. V. Slyozov, J. Phys. Chem. Solids 19, 35 (1961). 11. C. Wagner, Z. Elektrochem. 65, 581 (1961). 12. R. K. Ham. Trans. Janan Inst. bfetals. Suool. . . 9. 52 (1968). ’ 13. S. M. Selby, Standard Math. Tables. 15th ed., p. 85, Chem. Rubber Co., Cleveland (1967).
APPENDIX Calculation of breaking angles Ham [12] has derived an equation which can be used to calculate the breaking angles for edge and screw dislocations in a random array of finite ordered particles. He assumed that the (spherical) precipitates intersect the slip plane with a mean planar radius (r,), and that their strength is defined in terms of the critical breaking angle 4, (0 5 4, 1; a) required to release the dislocation from the obstacle. For the internal segment of an edge dislocation the angle /I, defined by 2fl = n - 4. is given by
(1 -
sin’/I + -
wsinp_ (7 Y
r,b)(Xr,))(l 2AV
- v)
o = 7
833
ON THE CRSS
The cubic equation (A-3) is of the type Y’ i a’.Y + h’ = 0
(A-4
the solutions of which can be found in standard texts (e.g. Selby [13]). The biggest difficulty in solving for b is choosing the right value of ?r to substitute into equation (A-3). Two approaches are possible: (I) We can select the experimentally measured values of AT and apply certain corrections in order to estimate the stress on the lead dislocation of the pair that shears the 7’ particles: (2) We can calculate the value of r, theoretically. which is sort of a backward way of doing it. since an approximate solution of equation (A-3) is required in the first place. However, the second of these two approaches was adopted because it was more convenient. In this case. r, was evaluated from the equations [2]
[(~)(~)“‘i
F<(r)
<;
(A_ja)
47
(r> > -, l-r;
(A-5b)
where the line tension, T, for edge dislocations is given by T=(’
- 2v),4, I-V
Equation (A-5,) is valid, strictly speaking. in the limit of large 4, where the effective obstacle spacing along the dislocation is equal to the Friedel spacing, Lp [2]. Equation (A-5b) becomes valid for strong particles. for which $-0 and the obstacle spacings approaches the square lattice spacing, L(L’ is equal to the average area per obstacle). For the values offin our samples, we used. following Brown and Ham [I]. the average value T= 3Gb’/Sx in evaluating 7, from equation (A-5,). However, the variation of A with (r) was incorporated into the values of A used to solve equation (A-3) for the larger particle regime where equation (A-5b) is valid. Using 7 = 167 mJ m-l and the same room temperature values of the other parameters (G, b and fi as those used by Munjal and Ardell[4]. values of j3 were determined for different values of (r) (=(a/Z)). For the larger particles we used the value A = Z(r,) = a(r)/2 The resulting variation of 4, with (r) is shown in Fig. 7. We point out that the values of rJ2 - yf/26 are entirely consistent with the theoretical values of As calculated by Munjal and Ardell[4] for the purpose of comparing theory with experiment.
(A-1)
where r, is the effective stress on the dislocation, 7 is the antiphase boundary energy, b is the Burgers vector, Y is Poisson’s ratio .md
where A and r0 are outer and inner cut-off distances, respectively, and G is the shear modulus of the matrix. Taking v = l/3 and (r,) = (n/4)(r), equation (A-l) reduces to sin3P + sin fl -
(A-3)
60
-
40
-
20
-
.o. 0
tl’l’l’l/I’(tm 20 40
I;0 60
80
100 a/2
120
140
180
J 200
IAl
Fig. 7. Calculated values of the critical breaking angle. 9, as a function of the 7’ particle size. a 2.
and A. J. ARDELL
Materials Department. School of Engineering and Applied Science, University of California. Los Angeles. CA. U.S.A. (Recriced
24 F‘rbruary
1976)
Abstract-The effect of the width of the 7’ particle size distribution on the CRSS of Ni-AI single crystals containing nominally 6 wt.>; Al was investigated. The samples were given two-step aging treatments to produce broader unimodal distributions than those resulting from isothermal aging. The first aging treatments were done at 600 and 650°C for 458 and 69 hr, respectively, while the second were done at 625’C for various lengths of time. In all the samples the average particle size was nearly constant. but the standard deviation, b. was increased by as much as 300/,, primarily by the addition of smaller precipitates. After correcting for variations in the 7’ volume fraction introduced by these treatments, the effect of increasing D was to reduce the CRSS of the samples by approx. So& This change, which is very small, is nevertheless substantially larger than that predicted by the computer experiments of Foreman and Makin. It is suggested that the discrepancies are due to the shortcomings of modelling real precipitate dispersions by arrays of point particles. R&urn&--On a ttudie I’effet de la largeur de repartition des tailles des particules 7’ sur la CCR de monocristaux de Ni-AI contenant 6% d’aluminium en poids. On faisait subir aux echantillons un vieillissement en deux etapes afin de produire une repartition unimode plus large que celle que I’obtient par recuit isotherme. Les premiers recuits itaient de 488 et 69 h a 600 et 650°C respectivement, alors que les suivants Ctaient effectues a 625°C pendant des durees variables. La taille moyenne des particules etait a peu prts constante dans tous les ichantillons. mais la deviation standard u pouvait augmenter de 30?/,. essentiellement par I’addition de petits precipites. Apres correction pour la variation de la fraction volumique de ; ’ introduite par ces traitements, on voit que l’augmentation de a produit une diminution de la CCR d’environ 8%. Ce changement est tres petit, il est toutefois nettement plus grand que celui prevu sur ordinateur par Foreman et Makin. Nous pensons que les differences proviennent de l’insuffisance de modele qui represente une repartition de pricipitts reels par un reseau de particules ponctuelles. Zusammenfassuttg--Der EinAuD der Breite der GroDenverteilung von v’-Partikeln auf die kritische FlieBspannung wurde an Ni-AI Einkristallen mit 6 Gew.-% Al Nominalgehalt untersucht. Die Proben wurden in zwei Stufen gealtert, urn breitere Verteilungen als bei isothermer Alterung zu erhalten. Die erste Stufe fand statt bei 600 und 650°C fur 488 und 49 h. die zweite bei 62X iiber verschieden lange Zeiten. In ailen Proben war die mittlere PartikelgrijBe nahezu konstant, die Standardabweichung e aber war 30’1; gro&r. vorw.iegend wegen der vorhandenen kleineren Ausscheidungen. Es ergab sich nach einer Korrektur wegen Anderungen im Volumenbruchteil der y’-Phase, die durch die Behandlung erzeugt worden yar, da8 das vergrijBerte G die kritische FlieDspannung der Proben urn etwa 8% erniedrigte. Diese Anderung ist zwar sehr klein, jedoch betrachtlich griil3er als die von den Computerexperimenten von Foreman und Makin vorhergesagte. Es liegt nahe, daD diese Diskrepanz von der Unzulanglichkeit einer Beschreibung einer realen Verteiiung von Ausscheidungen mit punkfiirmigen Teilchen herriihrt. 1. INI’RODUCI-ION
Several years ago Foreman and Malcin [l] using a computer model, investigated the detailed movement of a dislocation line of constant line tension through an array of point obstacles randomly distributed in space. Each obstacle had the same strength, characterized by the value of the critical angle, 4, defined as the angle included between the two arms of the dislocation, pinned at the obstacle, at the instant of its release. They obtained numerical results for the critical shear stress. Ar, defined as the stress required for the steady-state movement of the dislocation through the array of obstacles, as a function of &. In the limit of weak obstacles (&,- R), the so-called Friedel relation (see. e.g. Brown and Ham [Z]) was 827
found to satisfactorily describe the relationship between Ar and $J,, whereas in the strong particle limit (I$,- 0) Ar was found to be relatively insensitive to 9,. In precipitation-hardened alloys the obstacles are not point particles but are finite, the average planar radius, (r,), being related to & Furthermore, real precipitates are never monodisperse; there is always a distribution of particle sixes and hence a distribution of 4, values. (Even for a monodisperse system there will be a distribution of r* values.) Under these circumstances, it is useful to know to what extent the stress required to move a dislocation through a real distribution of precipitates is related to the strength of the average particle in the distribution.
828
MUNJAL c&
>
AND
ARDELL:
AT INTERVALS
o = 10,000
EFFECT OF PARTICLE SIZE DISTRIBUTIONS Of a/S
+_,
OBSTACLES
+max
Fig. 1. The results of the computer experiment of Foreman and Makin [3] showing the variation of the critical shear stress, AT, as a function of the spectrum width, 4,. for an array of point obstacles having a square spectrum of breaking angles (inset).
It is usually assumed that they are equal. Foreman and Makin [3] also considered this problem in another computer experiment. They determined the CRSS of a system containing a random array of point obstacles with a square spectrum of strengths, the results of which are reproduced in Fig. 1. It can be seen here that there is an increase in strength when the spectrum width, &, increases from 0 to - ~min for a given value of the mean critical 4 bziking angle, (4,). The predicted effect of &, is much larger for larger average breaking angles (weaker particles) than it is for small values of (Q, but it is nevertheless relatively small. Since 4, can be related to the standard deviation, CJ,of the particle size distribution, changes in cr in a real alloy system can be expected to have a relatively small effect on the CRSS. This conclusion, however, had never before been tested by mechanical property measurements on real alloy systems, and was the reason for undertaking this investigation. 2. EXPERIMENTAL
PROCEDURES
We expected that the detection of small changes in Ar due to changes in Q would be successful only under restricted circumstances in which the influence of other factors could be either estimated or compensated for. The experimental approach adopted for these purposes is described below.
ON THE CRSS
Two Ni-AI single crystal samples. cut from the same crystal used previously [J]. containing nominally 6 wt.:/, Al were solution treated at 1015 &-5C in an atmosphere of purified He and quenched into a refrigerated saturated NaOH solution. One each was then aged at 650 and 6OO’C for a period long enough to attain the same average particle size (a/2) z 113 A of the cuboidal shaped 7’ (iVi,Al) precipitates of edge length a. This is the value of (a:,2> at peak strength in Ni-676 Al samples aged isothermally at 62YC for 175 hr [4]. The amount of aging time required to produce this value of (a/2) at the two aging temperatures* was determined by interpolation of the coarsening data of Ardell and Nicholson [5]_ The distribution of y’ particle sizes in these isothermally aged samples was expected to be the same. a conclusion which follows from the experimentally determined histograms of Ardell and Nicholson [5]_ The distribution of particle sizes was then changed by re-aging these samples at 625’C for various times. The value of (u/2) at peak strength was chosen because Ar is insensitive to (a/2) in the neighborhood of the aging peak (see Fig. 2). Thus, a small variation of the acrrage particle size resulting from the two-step aging treatments should have a minimal effect on the CRSS. The major disadvantage in selecting this particular value of (a;)) is that the insensitivity of the CRSS to changes in (a/2> could be accompanied by a similar insensitivity to changes in 6. Nevertheless, it was felt that if a smaller value of (a/2) were selected (e.g. somewhere in the linear portion of Fig. 2, where Ar x(a ,2)’ ‘1 it would have been very difficult, if not impossible. to separate the effects of changes in (a/2) from those due to changes in IJ because of the relatively large sensitivity of Ar to the average particle size in this range. In addition, it would have been considerably more difficult to measure changes in the width of the distribution for values of (~$2) in the neighborhood of 25-50 A,
“\
60 TESTING A 0 0 0
TEMP I-K) 77 191 297 373
2.1 Aging ti-eafmenf The values of cr were changed by using two-step aging treatments involving a change in temperature.
* A third sample was aged first at 675’C. However, reaging at 625’C resulted in a bimodal distribution of 7’ particle sizes [6], which was not appropriate
to this study.
Fig. 2. The increment in the CRSS. AT. YS the square root of half the mean particle edge length. (a) 2 of the y’ precipitates in a Ni-67; Al alloy isothermally- aged at 625C. Data of Munjal and ArdeU [J].
MUNJAL
AND
ARDELL:
EFFECT OF PARTICLE SIZE DISTRIBUTIONS
because errors in the measurements are frequently very large in this range of particle sizes. 2.2 Compression testing and measurement of the volume fraction Since the CRSS is theoretically dependent upon the volume fracti0n.J at the aging peak [2], the expected variations in f resulting from the two-step aging treatments were determined experimentally. This was done by measuring the Al content of the matrix after each aging treatment using a Curie point apparatus[7]. To minimize the effects of possible variations of the initial concentration of Al that can occur from one sample to another, the compression samples that were aged first at 650 and 600’C were then re-aged at 625’C for a given time in a purified He atmosphere. quenched into the refrigerated NaOH solution. tested under compression, re-aged at 635C, retested. etc. The total aging time at 625’C thus represents the accumulated value up to the time specified. The initial Al contents of the two samples were also determined by measuring their Curie points after solution treating and quenching. The composition of the sample aged first at 65O’C was actually j.SSg;, Al. while that of the sample aged first at 6OO’C was actually 5.9OY/, Al. The samples were compression tested at room temperature only. After this testing was completed, the samples were re-solution treated, isothermally aged at 625°C for 175 h and mechanically tested again at room temperature. This was done to provide control specimens with which to compare the CRSS values determined from the previous tests. By performing the experiments in this way, the danger existed that subsequent aging would be influenced by the dislocations introduced during plastic deformation. since 7’ precipitates have been observed to grow or coarsen preferentially at edge dislocations [8]. However, we believe that this process had a negligible effect on the results of subsequent CRSS measurements because the plastic strain during each test was very small ( r0.3Yb).
ON THE CRSS
829
of 7’ particles in a particular sample was determined using a Zeiss particle size analy-zer. To obtain statistically meaningful results. the diameters of between 1000 and 1200 particles were measured. 3. RESLLTS 3.1 Distribution of;:’ particles The particle size distributions in the doubly-aged samples are shown in Fig. 3. The distribution of -,J’ particles in the sample first aged at 65O’C did not appear to change substantially as t at 625’C increased. Nor did there appear to be any significant shift in the peak of the histograms as a function of t at 625’C. In Fig. 3(b). it can be seen that the histograms for the samples first aged at 6OO’C are definitely narrower than those in the sample first aged at 650°C. As t at 625’C increased, the distribution of y’ particles for these samples apparently became narrower, and the peak clearly shifted toward smaller particle sizes.
2.3 Determination of the distributiorf of 7’ particle si:es To determine the distribution of 7’ particles, polycrystalline sheet samples (also containing nominally 6% Al) were heat-treated at the same temperatures for the same amounts of time as the single crystal samples, and were examined by compression transmission electron microscopy. It should be pointed out that magnetic measurements indicated a slight difference in composition between the polycrystalline NCAl sheet and the Ni-Al single crystals (about 0.142 wt.% Al). However, this was not expected to influence the size distribution of the ^r” particles since coarsening behavior is independent of volume fraction [S, 91. Thin foils in (001) orientation taken from the polycrystalline samples were photographed in dark field using a 7’ superlattice reflection. The distribution
Fig. 3. Histograms of the 7’ particle sizes observed in samples aged first at (a) 65O’C for 69 h. and (b) 600°C for 188 h. and then re-aged at 625’C for the times (h). indicated by the numbers inset. The dashed curves represent the theoretical distribution of the LSlV theory.
SjO
MUNIAL
AXD
ARDELL:
EFFECT OF PARTICLE SIZE DISTRIBUTIONS
Table 1. The effect of the doubIe-a~ng Aging treatment
tihrl
69 hr, 650°C i-
oiA1
a,!
99.1 101.6 98.3
33.7 32.8 35.7
0.35
0.32 0.36
98.6 89.1 87.8 78.2
30.9 29.7 24.9 27.3
0.3 I 0.33 0.2s 0.35
59.5 72.5 89.5 95.5 104.0
15.7 18.0 23.1 22.8 24.2
0.26 0.25 0.26 0.25 0.23
I 4
488 hr. 6WC f
0.25 1 4 10
Isothermally aged* at 625°C (6.35% Al)
treatments on the SD., o, of the Ni-A1 alloys
w>(A)
0.25
24 48 72 96 146
ON THE CRSS
(G’(U
z,),”
0.3-1
0.32
0.25
* Data of Ardell and Nicholson fj].
The values of (a/2) and (7 obtained by analysis of the histograms in Fig. 3 are summarized ln Table 1. The (u/2> values for the doubly-aged samples were found to be slightly smaller than the calculated value of (a/2> in samples isothermally aged at 625°C for 175 h (113 A). For the samples first aged at 65O”C, the values of IJ and (n/2> remained essentially constant, whereas for the samples first aged at 600°C the values of G and (q/2> decreased as t at 62X increased. However, the values of a/(u/2> are nearly constant for the second series of samples, suggesting that the observed changes in 0 are basically due to scaling factors. The values of 0 in Table 1 were compared with those for isothermally aged samples in order to determine how the heat treatments used here affected them. For this purpose, the histograms of Ardell and Nicholson t5J, obtained from a Ni-6.357: AI alloy isothermally aged at 625°C for various times, were
/f: /d / i I
rl 1
I
“L -0
1
I
25
50
75
100
125
150
175
a/2 (AI
Fig. 4. Histogram of the 7‘ particle sizes obtained from a sample of Ni-6.35% Al isothe~~ly aged at 62X for 96 h. Data of Ardell and Nicholson [YJ.
analyzed to obtain values of o/(a/2>, which are theoretically time-independent [IO, 1I]. These values are also shown in Table 1, where it is seen that the two-step heat-treatments increased the values of aJ(a/2> by about 30% over the value normally obtained on isothermal aging. It is important to note the increase in c resulting from the two-step aging treatments is due primarily to the addition of small particles. This may be verified by su~rim~sing the theoretical distribution of the Lifshitz-Slyozov-Wagner theory on the histograms in Fig. 3 (dashed curves), and comparing them with the histograms measured by Ardell and Nicholson, one of which is shown in Fig. 4. In isothermally aged alloys the agreement between the theoretical distribution and the ex~rim~~1 histograms is excellent at smalf. particie sizes. It is evident on comparing Figs. 3 and 4 that the discrepancy at small sixes is much larger for the doubly-aged alloys used in this study.
The results of the compression tests are shown in Fig. 5(a). The contribution to strengthening due to the ‘J’ precipitates alone (AT) was obtained by subtracting the room temperature value of rc of the solid solution (41 MN/m’) [6] from the values of the CRSS. There is an increase in the As value as t at 62jSC increases for the samples first aged at 650°C; whereas in the sample first aged at 6OO’C, As shows a slight drop as t at 625°C increases. The variation off with t in these samples is shown in Fig. 5(b). It is apparent that this variation could possibly account for the entire variation of Ar. To test this possibility, the values of Ar in Fig. 5(a) were compensated by plotting As/f II2 as a function of t. This method of plotting is justified by the theoretical proportionality between AT andf”’ in the vicinity of the aging peak [Z]. The results are shown in Fig. 5(c), where it is evident that the AT/f iI2 values remain essentially constant within experimental error (estimated as &37, in this experiment}. The horizontal dashed line in Fig. 5(c) shows the average value of AT/~‘/* for the two samples used
MUNJAL
AND
ARDELL:
EFFECT OF PARTICLE SIZE DISTRIBUTIONS
a’
in this study after re-solution treating and isothermally aging them (see also Table 2).* On comparing these results with those of the two-step aging treatments we concluded that increasing the value of 0 [or (a/(u/2),,] by about 3O”a results in a decrease in the Ar/fil’ values of about By/,. In other words, the addition of smaller particles to a lowering of the CRSS.
to increase
(a)
ON THE CRSS
-A-A-
LA
i
0 leads
erw k
4. DISCUSSION
In this section we attempt to rationalize the observations in Fig. 5 with results of Foreman and Makin’s computer experiments [3]. For this purpose, the critical breaking angles (4,) for each size interval of Fig. 3 were calculated using equations originally derived by Ham[12] for the shearing of finite ordered particles by dislocations initially edge in character. The details of the breaking angle calculations are discussed in the Appendix, and the results are shown in Fig. 6. It should be noted that these breaking angle histograms do not include the effect of the distribution of particle radii that one observes on the slip plane even when the particles are monodisperse. This effect will, of course, broaden the histograms in Fig. 6, which thus underestimate the width of the distribution of breaking angles. Using these data, the mean critical breaking angle, (4,) and the standard deviation of &, cr,+,were calculated. Table 3 summarizes these results and also shows values of the spectrum width, &, which were calculated for an array of obstacles with a square spectrum of strength having the same a, values as the histograms in Fig. 6. (It is easy to show that when the distribution of +c is
--A
7-
4-
31
I
--__ _______ -De
I
_____ ____ ___-
9 -LO
5.86
Table 3. Effect of aging treatment
2’ s
Initial aging treatment 0650°C, 69 hr 46oo’C, 488 hr
200
a I
I
I
IO
t,
hr
Fig. 5. The variation of: (a) the increment of the CRSS due to ‘J’ precipitates, AT; (b) the volume fraction of 7’. f: (c) AT/~“‘, with re-aging time, t, at 62X for the Ni-Al alloys initially aged at 600°C for 488 h and 65O’C for 69 h. The dashed horizontal line in (c) represents the average value of AK/~“’ of three samples aged isothermally at 625’C for 175 h.
AS 1 (MN/m-)
f
AT/f ‘.’ (MN/m’)
(AT/~’ ‘)av (MN/m’)
70 68 74
0.048 0.050 0.055
320 304 316
313
on the mean critical breaking angle, (4,). width. &
and the spectrum
Aging treatment
nhr)
tiA)
69 hr. 65OT
0.25
+
1
99.1 101.6 98.3
34.7 32.8 35.7
82.0 80.3 82.9
22.8 21.8 23.8
79. I 75.5 82.6
98.6 s9.1 87.8 78.2
30.9 29.7 24.9 27.3
82.0 88.4 88.9 95.9
19.9 19.8 16.5 18.9
68.8 68.5 57.1 65.3
4 488 hr, 6OOC +
0.25 1 4 10
____
IA--
Table 2. The values of AT/~‘!? obtained after re-solution treating and isothermally aging the Ni-Al samples at 625°C for 175 hr
5.88 5.90
t
9
* The value of AT obtained after re-solution treating the sample first aged at 675°C is also included in Fig. 5(c) and Table 2.
Alloy composition (wt.% Al)
831
832
MUNJAL
AND
ARDELL:
EFFECT OF PARTICLE SIZE DISTRIBLTIONS
ON THE CRSS
0 10
160
160
140
120
60
100
60
40
20
0
O,KEGREES)
Fig. 6. Histograms of the critical breaking angles, &, for the samples aged first at (a) 650X for 69 h and (b) 600°C for 488 h and then re-aged at 625°C for the times (h). The values of 4, were calculated from the histograms in Fig. 4 using the procedures described in the Appendix.
square, 4, = 2J3 G+). This was done so that the present results could be compared somewhat more directly with the predictions of Foreman and Makin’s computer experiments [3]. To estimate the changes in (4,) and & brought about by the two-step aging treatments, these parameters were calculated for several values of (n/2) in the size range 8%105A from the distributions of Ardell and Nicholson [S], which are characteristic of isothermally aged Ni-Al alloys. The values of (I$~) were found to vary from 84 to 73”, while the values of & varied from 59 to 70”. On comparing these figures with those seen in Table 3, it is apparent that the two-step aging treatments increase the values of 4, by a very small amount over the values normally obtained after isothermal aging. Although it is somewhat difficult to compare the values of 4, in isothermally aged samples with those of the doubly-aged samples on a one-to-one basis (e.g. for samples having the same value of (a/2)), it appears that the change in &,, brought about by our two-step aging treatment was approx. 17% for particles in the range 85 < (a/Z) < 105A. Now, Fig. 1 shows that when (4,) is in the neighborhood of 3~18, changes in b, have almost no in-
fluence on the values of the CRSS. However, the data show that the 17% increase in the values of & brought about by the two-step aging treatments are accompanied by an 8% decrease in the CRSS. Even though this is a small effect, it is much larger than predicted by the computer experiments of Foreman and Makin. The finite character of real precipitates could be responsible for this result. The motion of dislocations through an array of finite obstacles should differ from that through an array of point obstacles, because the interaction between dislocations and precipitates in a real alloy system occurs over a range of distances. The introduction of finiteness also affects the average obstacle spacing along the dislocation, which is an important parameter governing the CRSS, particularly when the precipitates are small and relatively weak. Although it is premature to claim that finiteness alone could acco&tt for the small discrepancy that exists behveen theory and experiment, it is the most obvious source. There is clearly a need for computer experiments on arrays of finite obstacles. Finally, our results confirm what all investigations of precipitation hardening have always taken for granted: the influence of u is so small that we need
MUNJAL
ASD
ARDELL:
EFFECT OF PARTICLE SIZE DISTRIBUTIONS
not worry about the shape of the particle size distri-
bution in any quantitative comparison between theory and experiment. The distributions are adequately characterized by the particle of average size, which is justifiably equated to the single particle size that is encountered in the typical idealized theoretical situation. dci;nowlrdgement-Financial support for this research was provided by the U.S. Energy Research and Development Administration under contract No. E(O4-3)-34, P.A. 172.
REFERENCES 1. A. J. E. Foreman and M. J. Makin. Phil. Maq. 14, 911 (1966). 2. L. M. Brown and R. K. Ham, Srrengthening Methods in Crvstals. edited bv A. Kelly and R. B. Nicholson. p. 9. Wiley, New Ydrk (1971).3. A. J. E. Foreman and M. J. Makin. Can. J. Phqs. 45, 737 (1967). 4. V. Munjal and A. J. Ardell. Acta Met. 23, 513 (1975). 3. A. J. Ardell and R. B. Nicholson, J. Php. Chem. Solids 27, 1793 (1966). 6. V. Munjal. Ph.D. Dissertation, University of California, Los Angeles (1975). 7. A. J. Ardell, Acta Met. 16. 511 (1965). 8. A. J. Ardeil and R. B. Nicholson. rlcta &fer. 14, 1295 (1966). 9. D. J. Chellman and A. J. Ardell, rlcra iMet. 22. 577 (1974). 10. I. M. Lifshitz and V. V. Slyozov, J. Phys. Chem. Solids 19, 35 (1961). 11. C. Wagner, Z. Elektrochem. 65, 581 (1961). 12. R. K. Ham. Trans. Janan Inst. bfetals. Suool. . . 9. 52 (1968). ’ 13. S. M. Selby, Standard Math. Tables. 15th ed., p. 85, Chem. Rubber Co., Cleveland (1967).
APPENDIX Calculation of breaking angles Ham [12] has derived an equation which can be used to calculate the breaking angles for edge and screw dislocations in a random array of finite ordered particles. He assumed that the (spherical) precipitates intersect the slip plane with a mean planar radius (r,), and that their strength is defined in terms of the critical breaking angle 4, (0 5 4, 1; a) required to release the dislocation from the obstacle. For the internal segment of an edge dislocation the angle /I, defined by 2fl = n - 4. is given by
(1 -
sin’/I + -
wsinp_ (7 Y
r,b)(Xr,))(l 2AV
- v)
o = 7
833
ON THE CRSS
The cubic equation (A-3) is of the type Y’ i a’.Y + h’ = 0
(A-4
the solutions of which can be found in standard texts (e.g. Selby [13]). The biggest difficulty in solving for b is choosing the right value of ?r to substitute into equation (A-3). Two approaches are possible: (I) We can select the experimentally measured values of AT and apply certain corrections in order to estimate the stress on the lead dislocation of the pair that shears the 7’ particles: (2) We can calculate the value of r, theoretically. which is sort of a backward way of doing it. since an approximate solution of equation (A-3) is required in the first place. However, the second of these two approaches was adopted because it was more convenient. In this case. r, was evaluated from the equations [2]
[(~)(~)“‘i
F<(r)
<;
(A_ja)
47
(r> > -, l-r;
(A-5b)
where the line tension, T, for edge dislocations is given by T=(’
- 2v),4, I-V
Equation (A-5,) is valid, strictly speaking. in the limit of large 4, where the effective obstacle spacing along the dislocation is equal to the Friedel spacing, Lp [2]. Equation (A-5b) becomes valid for strong particles. for which $-0 and the obstacle spacings approaches the square lattice spacing, L(L’ is equal to the average area per obstacle). For the values offin our samples, we used. following Brown and Ham [I]. the average value T= 3Gb’/Sx in evaluating 7, from equation (A-5,). However, the variation of A with (r) was incorporated into the values of A used to solve equation (A-3) for the larger particle regime where equation (A-5b) is valid. Using 7 = 167 mJ m-l and the same room temperature values of the other parameters (G, b and fi as those used by Munjal and Ardell[4]. values of j3 were determined for different values of (r) (=(a/Z)). For the larger particles we used the value A = Z(r,) = a(r)/2 The resulting variation of 4, with (r) is shown in Fig. 7. We point out that the values of rJ2 - yf/26 are entirely consistent with the theoretical values of As calculated by Munjal and Ardell[4] for the purpose of comparing theory with experiment.
(A-1)
where r, is the effective stress on the dislocation, 7 is the antiphase boundary energy, b is the Burgers vector, Y is Poisson’s ratio .md
where A and r0 are outer and inner cut-off distances, respectively, and G is the shear modulus of the matrix. Taking v = l/3 and (r,) = (n/4)(r), equation (A-l) reduces to sin3P + sin fl -
(A-3)
60
-
40
-
20
-
.o. 0
tl’l’l’l/I’(tm 20 40
I;0 60
80
100 a/2
120
140
180
J 200
IAl
Fig. 7. Calculated values of the critical breaking angle. 9, as a function of the 7’ particle size. a 2.