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The relation between changes in ductility and in ductile fracture topography: Control by microvoid nucleation
~~,o meta//. vol. 31. No. IO, pp. 1517-1523. Printed in Great Britain. All rights reserved
1983 %?‘right
0
OOOI-6160/83 S3.00 +O.OO 1983 Pcrgamon Press Ltd
THE RELATION BETWEEN CHANGES IN DUCTILITY AND IN DUCTILE FRACTURE TOPOGRAPHY: CONTROL BY MICROVOID NUCLEATION A. W. THOMPSON+ Department
of Metallurgy and Materials Science, University of Cambridge, Cambridge CB2 3QZ, England (ReceiuedJor
publication
10 May 1983)
Abstract-Changes in microvoid size on ductile fracture surfaces under varying experimental conditions can arise in several ways. An analysis is developed here for the case in which an altered density of microvoid nuclei, expressed as the volume fraction of nucleation of microvoids. is the sole reason for size changes. The analysis is based on the concept that an altered nucleation density can be regarded as an alteration in the effective volume fraction of nuclei. The analysis is shown to agree with the form of available experimental data. Although more detailed experiments are nv before a complete comparison can be made, the physical rationale presented appears qualitatively correct, and selfconsistent.
R&smn&Des changements de la taille_des micmcavi&
sur les surfaces de rupture ductile peuvent se produire. de plusieurs man&es. Nous developpons ici une analyse pour le cas oti une modiication de la densit de germes de microcavitls, exprimCe comme fraction volumique de la germination de micmcav&, est la seule raison d’une variation de cette taille. L’analyse repose sur I’idCequ’une densit de germination modifiie peut 2tre consid&& comme unc modification de la fraction volumique e&ctive de gcrmes. Nous montrons que cette analyse est en bon accord avec la forme des donnCes expirimentales dent on dispose. Bicn que des expkiences plus d&aill&s soient kessaires avant de pouvoir faire une comparaison
compl&e, les raisonnements physiques p&sent&sici semblent qualitativement corrects et auto cohkrents. Zusammenfassung-Mikrohohltiumc, die auf duktilcn Bruchoberfliichen bei unterschiedlichen experimentellen Bedingungen auftreten, k&men unterschicdliche GrGBcn haben. Es wird cinc Analyst Wr den Fall entwickelt, daB eine geiinderte Dichte von Keimen der Mikrohohltiumebeschrieben als Volumbruchteil an nukleierten Mikrohohlriiumendie e.itige Ursache der Gr&niinderung ist. Diesc Analyse bemht auf dem Konzept, dal3 eine gciinderte Keimdichte angesehen werden kann als eine i(nderung im effektiven Volumanteil von Keimen. Diese Analyse stimmt mit den ver!Xigbaren experimentellen Daten i&rein. Wenn fiir einen genaueren Vergleich such genauere Untersuchungen notwendig sind, so scheint die vorgelegte physikalische Argumentation doch qualitativ richtig und konsistent.
1. INTRODUCTION
There are a number of circumstances which can cause changes in the fracture strain of a material, yet the fracture mode, microvoid coalescence (MVC), remains unchanged. For example, changes in temperature, strain rate, particle composition or shape, or hydrogen content can produce such effects. In general, these could be the result of alterations in microvoid nucleation, in microvoid growth, or in both. These phenomena are well reviewed in the literature [l-8], including discussion of the behavior caused by hydrogen [8-121. Although the general problem is rather complex [3,5], certain aspects, such as nucleation, can be treated in isolation [7, 13, 141. The present paper is addressed to nucleation of microvoids. For this limiting case, two kinds of changes in nucleation can be envisioned. At one extreme, the TPermanent address: Department of Metallurgical Engineering and Materials Science, CamegicMellon University, Pittsburgh, PA 15213, U.S.A.
density of nuclei is unchanged but the strain at which nucleation occurred is reduced. At the other extreme, the nucleation strain is approximately fixed but the density of nuclei is increased. This paper addresses the second extreme. As will be published elsewhere, it can be shown that the first extreme would in general give rise to smaller reductions in ductility, and is therefore of less interest than the second; moreover, the first extreme implies an increase in the size of microvoids or dimples on the fracture surface, while the second implies a decrease in size. In what follows, the case considered is that of a reduction in fracture strain, caused for example by reduction in temperature or by hydrogen charging, although it will be evident that ductility increases could be described in an entirely analogous manner. When ductility is reduced, but MVC continues to be the fracture mode (under the present assumption that all changes in fracture behavior are due to nucleation density phenomena), increased nucleation density means that the e$ktive density of nuclei has increased. The word “effective” is used because not
1517
THOMPSON:
151X
ON FRACTIJRE
all potential nuclei, e.g. inclusions, may in fact act as nuclei in a particular fracture event. One advantage of such a comparison of ductility chorzges is that
details of micromechanisms, for example as discussed by Goods and Brown [7], need not be essential to the analysis, and it is necessary to consider only the reason for changes in effectiveness of a particular micromechanism. The largest body of quantitative information on the microvoid fracture topography happens to comprise data on fracture hydrogen-assisted [8, 10, 15,161, although other interesting data exist [6,17-201. The hydrogen data exhibit a pattern of strong correlation between the reduction in ductility, expressed as loss in RA or reduction of area, (RA - RA”) + RA, and the reduction in microvoid fracture surface dimple size, expressed as the ratio R of dimple diameters D, as D”/D, when R < 1. (The superscript “H” here refers to the property value observed when hydrogen is present.) This correlation has been interpreted [S, 10, 161as a nucleation effect, but no quantitative rationale for such an interpretation exists [16,21]. It is the purpose of this paper to provide such a rationale. For simplicity, it is assumed below that microvoids are nucleated at second-phase particles. A similar analysis could be performed for the concept of “homogeneous” nucleation of microvoids in slip bands, slip band intersections, or cell walls [7,22-251, although that concept has not yet received the kind of detailed mechanical and energetic analysis which has been performed [7,13, 141 for particle-nucleated voids. The present assumption merely reflects the fact that in most materials, particles are the dominant nuclei [l-8, l&20,22,26]. “Nucleation density” therefore is treated below as the effective volume fraction of nuclei for microvoids. The essential concept is that a reduction in ductility implies an increase in the effective volume fraction of particles, an implication drawn from extensive experimental work
TOPOGRAPHY
itself (e.g. by partitioning [28] of elements which segregate to and weaken the bond, such as metalloid elements [20] or hydrogen), causes attainment of the interracial stress needed [I31 to de-bond (or crack) the more strongly bonded particles, thus increasing the number of particles which nucleate voids. (b) Particle shape and bond strength of all particles are equal, but size varies. The size effect in ease of nucleation is well known and has been widely discussed [7, 13, 14,26,29,30], with smaller particles being less easy to de-bond than larger ones. Again, changes in experimental conditions can act either to attain higher stresses, which could de-bond smaller particles, or to lower the bond strength of all particles, permitting smaller ones to de-bond which previously did not de-bond. (c) The size and bonding of the particles are fixed, but shape, or aspect ratio, varies. That particle shape affects ease of void nucleation is well known [e.g. Refs 4,5,3 1,321, and again, changes in experimental conditions must either raise the local stress to attain the bond strength of the more nearly equiaxed particles, or the bond strength of all particles must be lowered. The review literature [7, 13,141 indicates that of these three cases, case (c) or particle shape is the most difficult to quantify, and moreover is often expressed as a subset of (a), i.e. particles of strongly nonspherical shape among a population of spherical particles typically also exhibit different bonding to the matrix. In the analysis below, therefore, case (c) will be neglected. Moreover, since cases (a) and (b) may both be important, a combined case can be considered: (d) Particle shape is fixed (nominally spherical), but both particle size and bonding differ (see, for example, Ref. [33]). 3. ANALYTICAL
DEVELOPMENT
3.1. Basis for analysis
[4,5,18,27l.
2.
CASES TO BE CONSIDERED
An increase in effective volume fraction f of particles which act as nuclei, which is assumed responsible for decreases in true strain at fracture s/, could arise from several different sources. These sources include the size, shape and bonding of nucleating particles. The corresponding postulates are listed below.
(a) The size and shape of particles are, on average, no different, but their bond strength with the matrix varies. For example, a steel may contain populations of sulfide and carbide particles, of which the latter are more strongly bonded. In this case, a change in experimental conditions which either raises the matrix flow stress (e.g. by a decrease in temperature in steels). or decreases the particle-matrix bond strength
The primary variables affecting true strain at fracture, s,, are particle diameter d, and particle volume fraction J which are connected through the initial planar particle spacing L,, as [27,34] +=(1-f)
$. J
The relation betweenSand E, has been determined experimentally for a wide variety of materials [4,5,27J and can most simply be expressed as c,= k(l/!!)
(2)
where k is an empirical constant. The value of k evidently varies with several parameters, including particle-matrix bond strength [4,5], but for nucleation control, as assumed here, the case of relatively weak bonding may be most important. In that case, the extensive data of Ed-Bon and Baldwin [27]
THOMPSON: ON FRACTURE TOPOGRAPHY exhibit a best-tit value of k = 0.0140 (correlation coefficient r = 0.967). As discussed below, data for spheroidized or pearlitic steels (4,5] must be described with a larger value of k. There has been proposed [35] a slightly different version of equation (2), i.e. B,= k (I -.f)/fi the r value for fitting this equation to the original data is similar to that for equation (2) but there is no obvious reason not to use the simpler equation (2). When fracture behavior is altered by a change in experimental conditions, a superscript x will be used; e.g. the altered fracture strain would be ET. For a particular change in the experiment, e.g. changing temperature or introducing hydrogen, x is replaced by, respectively, T or H. 3.2. Case (a): fixed particle size From equation pressed as
(2), the ratio of ductilities is ex-
ET__ f -_E/
(3)
f”
remembering that these are eflctivefvalues, that is, they are the volume fractions of particles at which nucleation actually occurs. Since particle size d is fixed, the initial spacing (prior to deformation) of particles which nucleate voids, &, is introduced from equation (I)
1519
(This usage is discussed below in some detail.) Then R = L;/L, R =
gives
C,[(~).~~‘*~
(7a)
and again, when k <
(7f-4 Equations (7a) and (7b) are written with both RA and cI terms for convenience; however, the term incorporating RA can be replaced through the relation I-RA” ~ = exp(s,I- RA
s;)
from .s[= In(l/l - RA). It should be pointed out that although values of & or R calculated from equations (5) and (7) with C, = 1 should represent the primary dependence of & or R upon changing effective volume fraction (expressed through Ed), nevertheless, a factor C, # 1 should be necessary in general to account for the relative roughness or depth of microvoids on microvoid fracture surfaces. This point has been made before [lo, 361 and is discussed below in Section 3.5. 3.3. Case (6): varying particle size
(4) where R,, = Li/L,, and a is a constant. The value of a reflects, as discussed below, the local roughness of the microvoid fracture surface when fracture surface measurements of &, are made. Such fracture surface measurements would be of the dimple size values, D, expressed as the ratio D”/D, e.g. as in Ref. [6]. Re-application of equation (2) then gives
W where C, = (l/a)‘/’ and generally 0 4 C, s 1. For materials like those of Edelson and Baldwin [27], k is small enough compared to typical values of s, and E; that a simplified equation can usually be written
This case is relatively simply treated by returning to equations (1) and (3), but retaining a variable d (i.e. d-rd”) in equation (I). Then (9) This is similar to equation (5a). In the same way (10)
as a parallel to equation (7a). Equation (8) can be used to remove the RA terms. Here again, the change from d to d” refers to the change in mean diameter of that part of the particle population which is effective as microvoid nuclei. The approximations for k KE, would be written as in equations (5b) and (7b). 3.4. Case (d): particle size and volume fraction varying
I
0
I,2
R. 2~C, 3 9
(5b)
where C, = 1 in the limit where +.s,. When fracture surface microvoids are deep, it may be appropriate to use the analysis presented earlier [IO] and introduce. the final particle spacing on the fracture surface, L,, given by L, = LI/(l - RA)“‘.
(6)
Analytically (though not necessarily quantitatively) this case is the same as case (b), above, since there is no a priori way to know how much of the change in E, is due to change in effective particle size, and how much to bringing differently-bonded particles into play. However, observation of the size and composition of the particles which are actually effective as nuclei [4, 12, 18,20,30,37] should permit this division to be made in practice. The relevant equations are still (9) and (10).
I520
THOMPSON:
ON FRACTURE
Fig. 1. Schematic cross-section of a fracture surface microvoid or dimple, defining the dimensions h and W.
3.5. Efects of fracture micro-roughness
As mentioned above, the relative depth of dimples or microvoids h and their width w, Fig. 1, should play a role in the exact relation between R,, and E,, as well as affecting the choice between & and R. For present purposes, the fracture microroughness M is defined as M = h/w.
(11)
When M-*0, microvoids are vanishingly shallow and the fracture surface tends toward being locally flat. Under such circumstances, the fracture surface approaches the requirements of a metallographic section [lo, 34,3&N] and & is the appropriate metallographic and fractographic parameter. When M increases toward unity, the fracture surface is quite unlike a metallographic section [lo, 16,36,38-40], and R may become increasingly more appropriate; moreover, the term C,, reflecting the “depth sampled” [lo, 36] by the fracture surface, may become increasingly important in equations (5) and (7). C, should be somewhat less than unity, as mentioned above, because of the overestimation of fracture surface L as M increases toward unity [lo, 16,36,40]; presumably C, oc (1 -M). There are a number of reasons [5-7,36,37,41,42] to expect M to be of magnitude 0.5, and not greatly to exceed unity except under exceptional circumstances (which would likely require additional analysis). Strictly on an empirical basis, it is also possible [40] to consider equations of the form
where C, a M. Whichever viewpoint is adopted, it is clear that there are multiple origins of C [36,40], and that further analysis is needed. As is shown below, it appears that C can be approximately materialindependent (for similar values of M) and that C, is not too much smaller than unity (or that C, is small). 4. COMPARISON
WITH EXPERIMENT
The relation between RA loss due to hydrogen, and fractographic R (i.e. D”/D) which was mentioned in the introduction is shown in Fig. 2. This relation can be represented by RA loss=(l
- R)“‘2
(13)
TOPOGRAPHY
with a correlation coefficient r = 0.989 (curve shown in Fig. 2). Also included in Fig. 2 are data [ 19,201 for quenched and tempered steels which were nof hydrogen charged. For some of those data [19], the parameter x is the change in tempering temperature from 600 to 500 or 400°C; complete MVC was observed without any complications from changes in fracture behavior near the temperature of tempered martensite embrittlement [19,20,43]. For the other data [20], x represents the segregation of phosphorus to carbide particle interfaces; data analysis is described in the Appendix. This example is particularly interesting because it represents a well defined, nonhydrogen shift in microvoid topography. These steels are, however, quite different microstructurally from the other materials in Fig. 2, and some [19] might not be appropriate for this comparison [4,5,8]; k in equation (2), for example, is significantly larger. More work is needed to clarify this point. The values of R,, or R determined from equations (5) and (7) are effectively predictions of the fractographic values of R, i.e. D”/D [8, 161 and therefore
A 304L A
.
\
309s
•I Hostelloy
.
.
l
+
Alloy 600
x 0
Alloy 625 Fe-Ni-Cr
l
Alloy
X
alloys
903
O Ni-Cr-Mo
steel
I 0.4
0.6
0.6
1.0
R
Fig. 2. Relation between RA loss due lo changes in experimental conditions. e.g. introduction of hydrogen, and microvoid or dimple size ratio R (defined in text). The curve shown is that of equation (I 3). Data from review literature [8,l6] except for Ni-Cr-Mo steel [19,20] and Appendix, alloys 600 and 625 [21]. Fe-Ni-Cr alloys [44] and alloy 903 [45]. Symbols indicate austenitic stainless steels (triangles), nominally single-phase nickel alloys (other symbols), precipitation-strengthened alloys (circles) and quenched and tempered martensitic steels (diamonds).
THOMPSON:
ON FRACTURE
TOPOGRAPHY
1521
represents a better prediction of kas than does R (as in several previous cases [6,46]), but as discussed below, such a conclusion would be premature. There are two aspects of the comparison in Fig. 3 which are unsatisfactory. One is that no explicit derivation of C has been presented, although it seems evident that the ,forn~ of equations (5) and (7) is correct even if C, were unity [or C, = zero in equation
0.6
a
Equation fbaf
9
Equation (5a1121
0
Equation (7a/121
0.6
0.6
la
Fig. 3. Predicted values of R or & from equations @a), (74 and (I 2), see Table I, compared to observed R values from Fig. 2. Line shown is a 1: I line, not a fit. Filled points from [ 19,201. Table 1. Values of C temts and forrelation coeEcients for equations (5) and (7) Equation Sa 7a 5a/12’ 7al12’
c
r
0.72 0.65 0.20 0.35
0.977 0.845 0.977 0.845
-
‘Equation (Sa) or (7a) with C, = 1 and C, as in equation (12).
should
be compared to the observed R values here designated Roes. In Fig. 3 this comparison is made, and the agreement is seen to be satisfactory. The values of C and the correlation coefficients are collected in Table 1s One could draw the conclusion from Fig. 3 and Table 1 that R,, [8, 1%21,44,45J,
(12)]. Thus, Fig. 3 and Table 1 represent the circumstances surrounding case (a), i.e. that particle size and shape are fixed but that bonding strength of at least some particles is sensitive to the change x in experimental conditions. For cases (b) and (d), one may set C, equal to unity and simply allow variation in d”/d to compensate for the overprediction of R. Alternatively, C values as in Table 1 can be applied, thus causing d”/d values to be much closer to unity. In either case, da/d is markedly material dependent. As an example, equations (9) and (lo), the analogs of equations (Sa) and (7a), were used to generate Table 2 with C, = 1. Values from the analogs of equations (Sb) and (7b) were very similar, respectively, to those of (Sa) and (7a), verifying that k KE,, ET. The second unsatisfacto~ aspect of Fig. 3 and Table 1 is that comparison with experiment can only be made for RoBs values which lie between 0.25 and 0.75. Indeed, equations (S), (7) and (12) only reduce to the presumed limit of Ross + 1 as ~3 -, e,if C, = 1 (or C, = 0). There is little that can be done to correct this problem without a more complete understanding of C or the availability of data for Ross > 0.75 or < 0.25. The qualification which should be stated here is that Fig. 3 and Table 1 are adequate descriptions for the data of Fig. 2, but may require revision when ROBS fails outside the range 0.25475. It should be pointed out that no experimental observations have been located in the literature which would permit a complete test of the foregoing analysis. Values of M and k are rarely available, nor are values off, f”, d and d”. It is to be urged that such
Table 2. Values of da/d from equations (9) and (IO) d”/d
Material’
Alloy 600 Alloy 625 304L 3095 Alloy 903 Alloy 903 Alloy 903 Hastelloy X Fe-Ni-Cr alloys Fe-Ni-Cr alloys Fe-Ni-Cr alloys Fe-Ni-Cr alloys Fe-Ni-Cr alloys NiCr-Mo steel Ni-Cr-Mo steel Ni-Cr-Mo steel
Ohservcd R
Equation (9)
Equation (IO)
0.655 0.525 0.30 0.74 0.25 0.32 0.41 0.40 0.51 0.57 0.63 0.725 0.77 0.66 0.74 0.65
0.75 0.71 0.63 0.85 0.47 0.56 0.62 0.62 0.68 0.72 0.76 0.81 0.85 0.72 0.78 0.75
0.64
‘See caption to Fig. 2 for sources of data.
0.63 0.36 0.76 0.38 0.50 0.54 0.44 0.58 0.63 :: 0:80 0.69 0.72 0.70
THOMPSON:
I522
ON FRACTURE
data be collected in order to examine more fully the assumptions and conclusions reached herein. 5. DISCUSSION
AND CONCLUSIONS
The following three points have been made in the
course of this analysis. (1) A simple empirical prediction of D”/D, i.e. Roes, can be made from RA loss data using equation (13); nearly as good an empirical prediction can be made using the ratio &T/E,,as described below. (2) An analysis for case (a), that of effective f varying only because of bonding effects, and giving rise to R,, or R changes through equations (5) and (7), was the basis for an experimental comparison. Figure 3 shows the result. The same analysis was extended to the other cases, (b) and (d). (3) Cases (b) and (d), those in which effective f varied due to changes in effective particle size d, or in both bonding effects and in d, were found to be analytically (though not in general quantitatively) the same; one description of those cases is equations (9) and (lo), which in turn were used to calculate the results shown in Table 2. Some additional comment is appropriate on the results shown in Fig. 3 and Table 1. The correction terms C, and C, have been treated as though they are constants, independent of iU or of other variables such as sf, &j/s,, or (for most of the data in Fig. 3) hydrogen content or location. It is unlikely that this constancy is valid in general, though the pattern of Fig. 3 suggests that C, and C, may be nearly constant. This may, for example, mean that M is similar for all the data collected here: a qualitatively reasonable suggestion on the basis of the scanning electron fractography for these alloys, but one for which there is as yet no quantitative support. In low-alloy steels, M has been shown to vary with stress state [42], as would be expected; the data in Fig. 2 are for tensile specimens of varying neck depth and thus of varying stress state [47,48], which must be one source of variable M. Although the quantitative fractographic analysis necessary to rationalize C, or C, is not yet complete [lo, 16,36,40], it does appear that a multiplicative constant, i.e. C, , is a more realistic approach to the incorporation of A4 into the analysis. In the absence of M data, and recognizing as above, that M may not vary widely among the materials in Fig. 3. no effort has been made to determine the role of M. However, setting C, = (E;/E,)‘~ provides (as do other, similar relations) as good a fit as do the values in Table 1. In addition, it should be mentioned that observed values of R in at least some cases are a marked function of measurement location on the fracture surface, especially distance from the specimen surface [l&44]. Thus mechanical variables such as neck shape, state of stress, and work-hardening rate [29,47-501 may be important in addition to neck
TOPOGRAPHY
depth in some cases, although many of the results used in Fig. 3 were reported to be independent of fracture surface location. It should be pointed out that the nucleation strain
for microvoids is assumed essentially constant and thus plays no role in the present analysis (although it can, 6f course, be used in comprehensive models [41,50] of ductile fracture). Thus the concern which has occasionally been expressed [51,52] about the development of ductile fracture topography late in the stress-strain history of a tensile specimen is not germane to the present analysis. Detailed studies of void nucleation and growth as a function of plastic strain [12, SO]are necessary to account explicitly for such effects, when it is important to do so. Despite the incomplete knowledge of C and M, it is evident that an analysis of fractographic changes, i.e. R, or R as predictors of Ross = D”/D, derived entirely from changes in microvoid nucleation alone, can account for the experimental data in Fig. 2. To demonstrate that the analysis is quantitatively correct would at least require more detailed experimental data, as described in Section 4. It appears appropriate at this point, however, to conclude that the earlier suggestion [8,10,16] of nucleation phenomena as giving rise to the curve of Fig. 2 is reasonable, since the analysis above appears qualitatively correct and self-consistent. Since an analysis for R under conditions of fixed nucleation strain and density but varying microvoid growth rates has already been presented [IO], it only remains to treat the case of simultaneous changes in both nucleation and growth [41,53], as evidently occurs for 7075 aluminum and several spheroid&d plain carbon steels [lo, 161. appreciate helpful discussions with J. F. Knott. M. F. Ashby and G. C. Smith, and provision of facilities by Professor R. W. K. Honeyeombe, FRS. This work was conducted during the tenureship of a Science and Acknowledgements-I
Engineering Research Council Grant, and with partial support from U.S. National Science Foundation Grant DMR 81-19540, while the author was Overseas Fellow, Churchill College, Cambridge and Visiting Scientist in the Department of Metallurgy and Materials Science. REFERENCES 1. W. A. Backofen, in Fracture of Engineering Materials, pp. 107-126. Am. Sot. Metals, Metals Park, OH (1964). 2. H. C. Rogers, in Fundamentals of Deformation Processing (edited by W. A. Backofen et al.), pp. 199-255, Syracuse University Press, Syracuse (1964). 3. A. R. Rosenfield, Metall. Reviews 13, 29 (1968). 4. T. Gladman, B. Holmes and I. D. McIvor, in Efict of Second-phase Particles on the Mechanical Properties of Steel, pp. 68-78, Inst. Metals, London (1971). 5. J. F. Knott, Fundamentals of Fracture Mechanics,
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9. IO.
I I. 12. 13. 14. IS. 16.
ON FRACTURE
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28. H. F. Fischmeister, E. Navara and K. E. Easterling, Metals Sci. J. 6, 211 (1972). 29. R. H. van Stone. R. H. Merchant and J. R. Low. in Fatigue and Fracke
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TOPOGRAPHY
I523
38. J. L. Chermant and M. Coster, J. Muter. Sci. 14, 509 ( 1979). 39. E. E. Underwood and S. B. Chakrabortty, in Fracmgruphy und Marcriuls Science (edited by L. N. Gilbcrtson and R. D. Zipp), ASTM STP 733, pp. 337-354. ASTM, Philadelphia (1981). 40. S. M. El-Soudani. Fundumenra1.s uf Quanritariue Fracrogruphy, Ph.D. Dissertation, Univ. of Cambridge ( 1980). 41. L. M. Brown and J. D. Embury, in The Microstructure and Design of AIIOJU (Proc. ICSMA 3). Vol. I. pp. 164-169. Inst. Metals, London (1973). 42. C. P. You and J. F. Knott. To be published. 43. J. E. King and J. F. Knott, Metal Sci. 15, I (1981). 44. A. W. Thompson and J. A. Brooks, Metall. Truns. A 6A, 1431 (1975).
45. C. G. Rhodes and A. W. Thompson, Merall. Trans. A 8A, 949 (1977). 46. T. Inoue and S. Kinoshita, in The Microstructure and Design of Alloys (Proc. ICSMA 3) Vol. I, pp. 159-163. Inst.- Metals, London (I 973). 47. A. S. Argon. J. Im and A. Needleman. Mefall. Trans. A 6A, 8l-5 (i975). 48. J. C. Earl and D. K. Brown, Engng Fruc. Mech. 8, 599 (1976).
49. A. C. MacKenzie, J. W. Hancock and D. K. Brown, Engng Fract. Mech. 9, 167 (1977).
50. G. LeRoy, J. D. Embury. G. Edward and M. F. Ashby, Actn metall. 29, 1509 (1981).
51. T. D. Lee, T. Goldenberg and J. P. Hirth, Mefall. Trans. A lOA, 199 (1979). 52. R. A. Oriani and P. H. Josephic, Acta melall. 27, 997 (1979).
53. W. Roberts, B. Lehtinen and K. E. Easterling, Acra metall. 24, 745 (1976).
APPENDIX
In Hippsley and Druce’s paper (201, dimple size data are presented as the percentage of the fracture surface occupied by dimples within intervals of dimple diameter. This is the form in which the data were acquired on a Quantimet. The mean dimple diameter D is given by
I187 (1978).
31. A. S. Argon, Trans. A.S.M.E.
Ser. H (J. Engng Mater.
Techn.) 98, 60 (1976).
32. J. R. Fisher and J. Gurland, Metal Sci. 15, 185 (1981). 33. M. Fujita, Y. Kawabe and N. Nishimoto, Truns. natn Res. inst. Metals 23, 149 (1981).
34. E. E. Underwood, in Quuntitative Microscopy (edited by R. T. DeHofi’ and F. N. Rhines), pp. 77-127. McGraw-Hill, New York (1968). 35. J. Gurland and J. Plateau, Trans. Am. Sot. Metals 56,
where ni = number of dimples in the ith interval. Further, ni is given by F, + Ai, where F, = area fraction on fracture surface of the ith interval and Ai = area of an individual dimple, since the number of dimples per unit area is l/A,. The dimples are roughly circular in plan view but fill the fracture surface, so A, can be taken to be nb:/4. Thus
442 (1963).
36. D. J. Widgery and J. F. Knott, Mefal Sci. 12, 8 (1978). 37. I. G. Palmer and G. C. Smith, in Oxide Dispersion Strengthening (edited by G. S. Ansell, T. D. Cooper and F. V. Lenel), pp. 253-290, Gordon & Breach, New York (1968).
For the reported data [20], b for the “UE” material was 0.330 pm, and b for the “E30” material was 0.215 pm, an R value of 0.65.
1983 %?‘right
0
OOOI-6160/83 S3.00 +O.OO 1983 Pcrgamon Press Ltd
THE RELATION BETWEEN CHANGES IN DUCTILITY AND IN DUCTILE FRACTURE TOPOGRAPHY: CONTROL BY MICROVOID NUCLEATION A. W. THOMPSON+ Department
of Metallurgy and Materials Science, University of Cambridge, Cambridge CB2 3QZ, England (ReceiuedJor
publication
10 May 1983)
Abstract-Changes in microvoid size on ductile fracture surfaces under varying experimental conditions can arise in several ways. An analysis is developed here for the case in which an altered density of microvoid nuclei, expressed as the volume fraction of nucleation of microvoids. is the sole reason for size changes. The analysis is based on the concept that an altered nucleation density can be regarded as an alteration in the effective volume fraction of nuclei. The analysis is shown to agree with the form of available experimental data. Although more detailed experiments are nv before a complete comparison can be made, the physical rationale presented appears qualitatively correct, and selfconsistent.
R&smn&Des changements de la taille_des micmcavi&
sur les surfaces de rupture ductile peuvent se produire. de plusieurs man&es. Nous developpons ici une analyse pour le cas oti une modiication de la densit de germes de microcavitls, exprimCe comme fraction volumique de la germination de micmcav&, est la seule raison d’une variation de cette taille. L’analyse repose sur I’idCequ’une densit de germination modifiie peut 2tre consid&& comme unc modification de la fraction volumique e&ctive de gcrmes. Nous montrons que cette analyse est en bon accord avec la forme des donnCes expirimentales dent on dispose. Bicn que des expkiences plus d&aill&s soient kessaires avant de pouvoir faire une comparaison
compl&e, les raisonnements physiques p&sent&sici semblent qualitativement corrects et auto cohkrents. Zusammenfassung-Mikrohohltiumc, die auf duktilcn Bruchoberfliichen bei unterschiedlichen experimentellen Bedingungen auftreten, k&men unterschicdliche GrGBcn haben. Es wird cinc Analyst Wr den Fall entwickelt, daB eine geiinderte Dichte von Keimen der Mikrohohltiumebeschrieben als Volumbruchteil an nukleierten Mikrohohlriiumendie e.itige Ursache der Gr&niinderung ist. Diesc Analyse bemht auf dem Konzept, dal3 eine gciinderte Keimdichte angesehen werden kann als eine i(nderung im effektiven Volumanteil von Keimen. Diese Analyse stimmt mit den ver!Xigbaren experimentellen Daten i&rein. Wenn fiir einen genaueren Vergleich such genauere Untersuchungen notwendig sind, so scheint die vorgelegte physikalische Argumentation doch qualitativ richtig und konsistent.
1. INTRODUCTION
There are a number of circumstances which can cause changes in the fracture strain of a material, yet the fracture mode, microvoid coalescence (MVC), remains unchanged. For example, changes in temperature, strain rate, particle composition or shape, or hydrogen content can produce such effects. In general, these could be the result of alterations in microvoid nucleation, in microvoid growth, or in both. These phenomena are well reviewed in the literature [l-8], including discussion of the behavior caused by hydrogen [8-121. Although the general problem is rather complex [3,5], certain aspects, such as nucleation, can be treated in isolation [7, 13, 141. The present paper is addressed to nucleation of microvoids. For this limiting case, two kinds of changes in nucleation can be envisioned. At one extreme, the TPermanent address: Department of Metallurgical Engineering and Materials Science, CamegicMellon University, Pittsburgh, PA 15213, U.S.A.
density of nuclei is unchanged but the strain at which nucleation occurred is reduced. At the other extreme, the nucleation strain is approximately fixed but the density of nuclei is increased. This paper addresses the second extreme. As will be published elsewhere, it can be shown that the first extreme would in general give rise to smaller reductions in ductility, and is therefore of less interest than the second; moreover, the first extreme implies an increase in the size of microvoids or dimples on the fracture surface, while the second implies a decrease in size. In what follows, the case considered is that of a reduction in fracture strain, caused for example by reduction in temperature or by hydrogen charging, although it will be evident that ductility increases could be described in an entirely analogous manner. When ductility is reduced, but MVC continues to be the fracture mode (under the present assumption that all changes in fracture behavior are due to nucleation density phenomena), increased nucleation density means that the e$ktive density of nuclei has increased. The word “effective” is used because not
1517
THOMPSON:
151X
ON FRACTIJRE
all potential nuclei, e.g. inclusions, may in fact act as nuclei in a particular fracture event. One advantage of such a comparison of ductility chorzges is that
details of micromechanisms, for example as discussed by Goods and Brown [7], need not be essential to the analysis, and it is necessary to consider only the reason for changes in effectiveness of a particular micromechanism. The largest body of quantitative information on the microvoid fracture topography happens to comprise data on fracture hydrogen-assisted [8, 10, 15,161, although other interesting data exist [6,17-201. The hydrogen data exhibit a pattern of strong correlation between the reduction in ductility, expressed as loss in RA or reduction of area, (RA - RA”) + RA, and the reduction in microvoid fracture surface dimple size, expressed as the ratio R of dimple diameters D, as D”/D, when R < 1. (The superscript “H” here refers to the property value observed when hydrogen is present.) This correlation has been interpreted [S, 10, 161as a nucleation effect, but no quantitative rationale for such an interpretation exists [16,21]. It is the purpose of this paper to provide such a rationale. For simplicity, it is assumed below that microvoids are nucleated at second-phase particles. A similar analysis could be performed for the concept of “homogeneous” nucleation of microvoids in slip bands, slip band intersections, or cell walls [7,22-251, although that concept has not yet received the kind of detailed mechanical and energetic analysis which has been performed [7,13, 141 for particle-nucleated voids. The present assumption merely reflects the fact that in most materials, particles are the dominant nuclei [l-8, l&20,22,26]. “Nucleation density” therefore is treated below as the effective volume fraction of nuclei for microvoids. The essential concept is that a reduction in ductility implies an increase in the effective volume fraction of particles, an implication drawn from extensive experimental work
TOPOGRAPHY
itself (e.g. by partitioning [28] of elements which segregate to and weaken the bond, such as metalloid elements [20] or hydrogen), causes attainment of the interracial stress needed [I31 to de-bond (or crack) the more strongly bonded particles, thus increasing the number of particles which nucleate voids. (b) Particle shape and bond strength of all particles are equal, but size varies. The size effect in ease of nucleation is well known and has been widely discussed [7, 13, 14,26,29,30], with smaller particles being less easy to de-bond than larger ones. Again, changes in experimental conditions can act either to attain higher stresses, which could de-bond smaller particles, or to lower the bond strength of all particles, permitting smaller ones to de-bond which previously did not de-bond. (c) The size and bonding of the particles are fixed, but shape, or aspect ratio, varies. That particle shape affects ease of void nucleation is well known [e.g. Refs 4,5,3 1,321, and again, changes in experimental conditions must either raise the local stress to attain the bond strength of the more nearly equiaxed particles, or the bond strength of all particles must be lowered. The review literature [7, 13,141 indicates that of these three cases, case (c) or particle shape is the most difficult to quantify, and moreover is often expressed as a subset of (a), i.e. particles of strongly nonspherical shape among a population of spherical particles typically also exhibit different bonding to the matrix. In the analysis below, therefore, case (c) will be neglected. Moreover, since cases (a) and (b) may both be important, a combined case can be considered: (d) Particle shape is fixed (nominally spherical), but both particle size and bonding differ (see, for example, Ref. [33]). 3. ANALYTICAL
DEVELOPMENT
3.1. Basis for analysis
[4,5,18,27l.
2.
CASES TO BE CONSIDERED
An increase in effective volume fraction f of particles which act as nuclei, which is assumed responsible for decreases in true strain at fracture s/, could arise from several different sources. These sources include the size, shape and bonding of nucleating particles. The corresponding postulates are listed below.
(a) The size and shape of particles are, on average, no different, but their bond strength with the matrix varies. For example, a steel may contain populations of sulfide and carbide particles, of which the latter are more strongly bonded. In this case, a change in experimental conditions which either raises the matrix flow stress (e.g. by a decrease in temperature in steels). or decreases the particle-matrix bond strength
The primary variables affecting true strain at fracture, s,, are particle diameter d, and particle volume fraction J which are connected through the initial planar particle spacing L,, as [27,34] +=(1-f)
$. J
The relation betweenSand E, has been determined experimentally for a wide variety of materials [4,5,27J and can most simply be expressed as c,= k(l/!!)
(2)
where k is an empirical constant. The value of k evidently varies with several parameters, including particle-matrix bond strength [4,5], but for nucleation control, as assumed here, the case of relatively weak bonding may be most important. In that case, the extensive data of Ed-Bon and Baldwin [27]
THOMPSON: ON FRACTURE TOPOGRAPHY exhibit a best-tit value of k = 0.0140 (correlation coefficient r = 0.967). As discussed below, data for spheroidized or pearlitic steels (4,5] must be described with a larger value of k. There has been proposed [35] a slightly different version of equation (2), i.e. B,= k (I -.f)/fi the r value for fitting this equation to the original data is similar to that for equation (2) but there is no obvious reason not to use the simpler equation (2). When fracture behavior is altered by a change in experimental conditions, a superscript x will be used; e.g. the altered fracture strain would be ET. For a particular change in the experiment, e.g. changing temperature or introducing hydrogen, x is replaced by, respectively, T or H. 3.2. Case (a): fixed particle size From equation pressed as
(2), the ratio of ductilities is ex-
ET__ f -_E/
(3)
f”
remembering that these are eflctivefvalues, that is, they are the volume fractions of particles at which nucleation actually occurs. Since particle size d is fixed, the initial spacing (prior to deformation) of particles which nucleate voids, &, is introduced from equation (I)
1519
(This usage is discussed below in some detail.) Then R = L;/L, R =
gives
C,[(~).~~‘*~
(7a)
and again, when k <
(7f-4 Equations (7a) and (7b) are written with both RA and cI terms for convenience; however, the term incorporating RA can be replaced through the relation I-RA” ~ = exp(s,I- RA
s;)
from .s[= In(l/l - RA). It should be pointed out that although values of & or R calculated from equations (5) and (7) with C, = 1 should represent the primary dependence of & or R upon changing effective volume fraction (expressed through Ed), nevertheless, a factor C, # 1 should be necessary in general to account for the relative roughness or depth of microvoids on microvoid fracture surfaces. This point has been made before [lo, 361 and is discussed below in Section 3.5. 3.3. Case (6): varying particle size
(4) where R,, = Li/L,, and a is a constant. The value of a reflects, as discussed below, the local roughness of the microvoid fracture surface when fracture surface measurements of &, are made. Such fracture surface measurements would be of the dimple size values, D, expressed as the ratio D”/D, e.g. as in Ref. [6]. Re-application of equation (2) then gives
W where C, = (l/a)‘/’ and generally 0 4 C, s 1. For materials like those of Edelson and Baldwin [27], k is small enough compared to typical values of s, and E; that a simplified equation can usually be written
This case is relatively simply treated by returning to equations (1) and (3), but retaining a variable d (i.e. d-rd”) in equation (I). Then (9) This is similar to equation (5a). In the same way (10)
as a parallel to equation (7a). Equation (8) can be used to remove the RA terms. Here again, the change from d to d” refers to the change in mean diameter of that part of the particle population which is effective as microvoid nuclei. The approximations for k KE, would be written as in equations (5b) and (7b). 3.4. Case (d): particle size and volume fraction varying
I
0
I,2
R. 2~C, 3 9
(5b)
where C, = 1 in the limit where +.s,. When fracture surface microvoids are deep, it may be appropriate to use the analysis presented earlier [IO] and introduce. the final particle spacing on the fracture surface, L,, given by L, = LI/(l - RA)“‘.
(6)
Analytically (though not necessarily quantitatively) this case is the same as case (b), above, since there is no a priori way to know how much of the change in E, is due to change in effective particle size, and how much to bringing differently-bonded particles into play. However, observation of the size and composition of the particles which are actually effective as nuclei [4, 12, 18,20,30,37] should permit this division to be made in practice. The relevant equations are still (9) and (10).
I520
THOMPSON:
ON FRACTURE
Fig. 1. Schematic cross-section of a fracture surface microvoid or dimple, defining the dimensions h and W.
3.5. Efects of fracture micro-roughness
As mentioned above, the relative depth of dimples or microvoids h and their width w, Fig. 1, should play a role in the exact relation between R,, and E,, as well as affecting the choice between & and R. For present purposes, the fracture microroughness M is defined as M = h/w.
(11)
When M-*0, microvoids are vanishingly shallow and the fracture surface tends toward being locally flat. Under such circumstances, the fracture surface approaches the requirements of a metallographic section [lo, 34,3&N] and & is the appropriate metallographic and fractographic parameter. When M increases toward unity, the fracture surface is quite unlike a metallographic section [lo, 16,36,38-40], and R may become increasingly more appropriate; moreover, the term C,, reflecting the “depth sampled” [lo, 36] by the fracture surface, may become increasingly important in equations (5) and (7). C, should be somewhat less than unity, as mentioned above, because of the overestimation of fracture surface L as M increases toward unity [lo, 16,36,40]; presumably C, oc (1 -M). There are a number of reasons [5-7,36,37,41,42] to expect M to be of magnitude 0.5, and not greatly to exceed unity except under exceptional circumstances (which would likely require additional analysis). Strictly on an empirical basis, it is also possible [40] to consider equations of the form
where C, a M. Whichever viewpoint is adopted, it is clear that there are multiple origins of C [36,40], and that further analysis is needed. As is shown below, it appears that C can be approximately materialindependent (for similar values of M) and that C, is not too much smaller than unity (or that C, is small). 4. COMPARISON
WITH EXPERIMENT
The relation between RA loss due to hydrogen, and fractographic R (i.e. D”/D) which was mentioned in the introduction is shown in Fig. 2. This relation can be represented by RA loss=(l
- R)“‘2
(13)
TOPOGRAPHY
with a correlation coefficient r = 0.989 (curve shown in Fig. 2). Also included in Fig. 2 are data [ 19,201 for quenched and tempered steels which were nof hydrogen charged. For some of those data [19], the parameter x is the change in tempering temperature from 600 to 500 or 400°C; complete MVC was observed without any complications from changes in fracture behavior near the temperature of tempered martensite embrittlement [19,20,43]. For the other data [20], x represents the segregation of phosphorus to carbide particle interfaces; data analysis is described in the Appendix. This example is particularly interesting because it represents a well defined, nonhydrogen shift in microvoid topography. These steels are, however, quite different microstructurally from the other materials in Fig. 2, and some [19] might not be appropriate for this comparison [4,5,8]; k in equation (2), for example, is significantly larger. More work is needed to clarify this point. The values of R,, or R determined from equations (5) and (7) are effectively predictions of the fractographic values of R, i.e. D”/D [8, 161 and therefore
A 304L A
.
\
309s
•I Hostelloy
.
.
l
+
Alloy 600
x 0
Alloy 625 Fe-Ni-Cr
l
Alloy
X
alloys
903
O Ni-Cr-Mo
steel
I 0.4
0.6
0.6
1.0
R
Fig. 2. Relation between RA loss due lo changes in experimental conditions. e.g. introduction of hydrogen, and microvoid or dimple size ratio R (defined in text). The curve shown is that of equation (I 3). Data from review literature [8,l6] except for Ni-Cr-Mo steel [19,20] and Appendix, alloys 600 and 625 [21]. Fe-Ni-Cr alloys [44] and alloy 903 [45]. Symbols indicate austenitic stainless steels (triangles), nominally single-phase nickel alloys (other symbols), precipitation-strengthened alloys (circles) and quenched and tempered martensitic steels (diamonds).
THOMPSON:
ON FRACTURE
TOPOGRAPHY
1521
represents a better prediction of kas than does R (as in several previous cases [6,46]), but as discussed below, such a conclusion would be premature. There are two aspects of the comparison in Fig. 3 which are unsatisfactory. One is that no explicit derivation of C has been presented, although it seems evident that the ,forn~ of equations (5) and (7) is correct even if C, were unity [or C, = zero in equation
0.6
a
Equation fbaf
9
Equation (5a1121
0
Equation (7a/121
0.6
0.6
la
Fig. 3. Predicted values of R or & from equations @a), (74 and (I 2), see Table I, compared to observed R values from Fig. 2. Line shown is a 1: I line, not a fit. Filled points from [ 19,201. Table 1. Values of C temts and forrelation coeEcients for equations (5) and (7) Equation Sa 7a 5a/12’ 7al12’
c
r
0.72 0.65 0.20 0.35
0.977 0.845 0.977 0.845
-
‘Equation (Sa) or (7a) with C, = 1 and C, as in equation (12).
should
be compared to the observed R values here designated Roes. In Fig. 3 this comparison is made, and the agreement is seen to be satisfactory. The values of C and the correlation coefficients are collected in Table 1s One could draw the conclusion from Fig. 3 and Table 1 that R,, [8, 1%21,44,45J,
(12)]. Thus, Fig. 3 and Table 1 represent the circumstances surrounding case (a), i.e. that particle size and shape are fixed but that bonding strength of at least some particles is sensitive to the change x in experimental conditions. For cases (b) and (d), one may set C, equal to unity and simply allow variation in d”/d to compensate for the overprediction of R. Alternatively, C values as in Table 1 can be applied, thus causing d”/d values to be much closer to unity. In either case, da/d is markedly material dependent. As an example, equations (9) and (lo), the analogs of equations (Sa) and (7a), were used to generate Table 2 with C, = 1. Values from the analogs of equations (Sb) and (7b) were very similar, respectively, to those of (Sa) and (7a), verifying that k KE,, ET. The second unsatisfacto~ aspect of Fig. 3 and Table 1 is that comparison with experiment can only be made for RoBs values which lie between 0.25 and 0.75. Indeed, equations (S), (7) and (12) only reduce to the presumed limit of Ross + 1 as ~3 -, e,if C, = 1 (or C, = 0). There is little that can be done to correct this problem without a more complete understanding of C or the availability of data for Ross > 0.75 or < 0.25. The qualification which should be stated here is that Fig. 3 and Table 1 are adequate descriptions for the data of Fig. 2, but may require revision when ROBS fails outside the range 0.25475. It should be pointed out that no experimental observations have been located in the literature which would permit a complete test of the foregoing analysis. Values of M and k are rarely available, nor are values off, f”, d and d”. It is to be urged that such
Table 2. Values of da/d from equations (9) and (IO) d”/d
Material’
Alloy 600 Alloy 625 304L 3095 Alloy 903 Alloy 903 Alloy 903 Hastelloy X Fe-Ni-Cr alloys Fe-Ni-Cr alloys Fe-Ni-Cr alloys Fe-Ni-Cr alloys Fe-Ni-Cr alloys NiCr-Mo steel Ni-Cr-Mo steel Ni-Cr-Mo steel
Ohservcd R
Equation (9)
Equation (IO)
0.655 0.525 0.30 0.74 0.25 0.32 0.41 0.40 0.51 0.57 0.63 0.725 0.77 0.66 0.74 0.65
0.75 0.71 0.63 0.85 0.47 0.56 0.62 0.62 0.68 0.72 0.76 0.81 0.85 0.72 0.78 0.75
0.64
‘See caption to Fig. 2 for sources of data.
0.63 0.36 0.76 0.38 0.50 0.54 0.44 0.58 0.63 :: 0:80 0.69 0.72 0.70
THOMPSON:
I522
ON FRACTURE
data be collected in order to examine more fully the assumptions and conclusions reached herein. 5. DISCUSSION
AND CONCLUSIONS
The following three points have been made in the
course of this analysis. (1) A simple empirical prediction of D”/D, i.e. Roes, can be made from RA loss data using equation (13); nearly as good an empirical prediction can be made using the ratio &T/E,,as described below. (2) An analysis for case (a), that of effective f varying only because of bonding effects, and giving rise to R,, or R changes through equations (5) and (7), was the basis for an experimental comparison. Figure 3 shows the result. The same analysis was extended to the other cases, (b) and (d). (3) Cases (b) and (d), those in which effective f varied due to changes in effective particle size d, or in both bonding effects and in d, were found to be analytically (though not in general quantitatively) the same; one description of those cases is equations (9) and (lo), which in turn were used to calculate the results shown in Table 2. Some additional comment is appropriate on the results shown in Fig. 3 and Table 1. The correction terms C, and C, have been treated as though they are constants, independent of iU or of other variables such as sf, &j/s,, or (for most of the data in Fig. 3) hydrogen content or location. It is unlikely that this constancy is valid in general, though the pattern of Fig. 3 suggests that C, and C, may be nearly constant. This may, for example, mean that M is similar for all the data collected here: a qualitatively reasonable suggestion on the basis of the scanning electron fractography for these alloys, but one for which there is as yet no quantitative support. In low-alloy steels, M has been shown to vary with stress state [42], as would be expected; the data in Fig. 2 are for tensile specimens of varying neck depth and thus of varying stress state [47,48], which must be one source of variable M. Although the quantitative fractographic analysis necessary to rationalize C, or C, is not yet complete [lo, 16,36,40], it does appear that a multiplicative constant, i.e. C, , is a more realistic approach to the incorporation of A4 into the analysis. In the absence of M data, and recognizing as above, that M may not vary widely among the materials in Fig. 3. no effort has been made to determine the role of M. However, setting C, = (E;/E,)‘~ provides (as do other, similar relations) as good a fit as do the values in Table 1. In addition, it should be mentioned that observed values of R in at least some cases are a marked function of measurement location on the fracture surface, especially distance from the specimen surface [l&44]. Thus mechanical variables such as neck shape, state of stress, and work-hardening rate [29,47-501 may be important in addition to neck
TOPOGRAPHY
depth in some cases, although many of the results used in Fig. 3 were reported to be independent of fracture surface location. It should be pointed out that the nucleation strain
for microvoids is assumed essentially constant and thus plays no role in the present analysis (although it can, 6f course, be used in comprehensive models [41,50] of ductile fracture). Thus the concern which has occasionally been expressed [51,52] about the development of ductile fracture topography late in the stress-strain history of a tensile specimen is not germane to the present analysis. Detailed studies of void nucleation and growth as a function of plastic strain [12, SO]are necessary to account explicitly for such effects, when it is important to do so. Despite the incomplete knowledge of C and M, it is evident that an analysis of fractographic changes, i.e. R, or R as predictors of Ross = D”/D, derived entirely from changes in microvoid nucleation alone, can account for the experimental data in Fig. 2. To demonstrate that the analysis is quantitatively correct would at least require more detailed experimental data, as described in Section 4. It appears appropriate at this point, however, to conclude that the earlier suggestion [8,10,16] of nucleation phenomena as giving rise to the curve of Fig. 2 is reasonable, since the analysis above appears qualitatively correct and self-consistent. Since an analysis for R under conditions of fixed nucleation strain and density but varying microvoid growth rates has already been presented [IO], it only remains to treat the case of simultaneous changes in both nucleation and growth [41,53], as evidently occurs for 7075 aluminum and several spheroid&d plain carbon steels [lo, 161. appreciate helpful discussions with J. F. Knott. M. F. Ashby and G. C. Smith, and provision of facilities by Professor R. W. K. Honeyeombe, FRS. This work was conducted during the tenureship of a Science and Acknowledgements-I
Engineering Research Council Grant, and with partial support from U.S. National Science Foundation Grant DMR 81-19540, while the author was Overseas Fellow, Churchill College, Cambridge and Visiting Scientist in the Department of Metallurgy and Materials Science. REFERENCES 1. W. A. Backofen, in Fracture of Engineering Materials, pp. 107-126. Am. Sot. Metals, Metals Park, OH (1964). 2. H. C. Rogers, in Fundamentals of Deformation Processing (edited by W. A. Backofen et al.), pp. 199-255, Syracuse University Press, Syracuse (1964). 3. A. R. Rosenfield, Metall. Reviews 13, 29 (1968). 4. T. Gladman, B. Holmes and I. D. McIvor, in Efict of Second-phase Particles on the Mechanical Properties of Steel, pp. 68-78, Inst. Metals, London (1971). 5. J. F. Knott, Fundamentals of Fracture Mechanics,
Chap. 8. Butterworths,
London (1973).
6. D. Broek, Int. metall. Rev. 19, 135 (1974). I. S. H. Goods and L. M. Brown, Acta metall.
27, I (1979). 8. A. W. Thompson, in E&et of Hydrogen on Behavior oj Materials (edited by A. W. Thompson and 1. M.
THOMPSON:
9. IO.
I I. 12. 13. 14. IS. 16.
ON FRACTURE
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TOPOGRAPHY
I523
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APPENDIX
In Hippsley and Druce’s paper (201, dimple size data are presented as the percentage of the fracture surface occupied by dimples within intervals of dimple diameter. This is the form in which the data were acquired on a Quantimet. The mean dimple diameter D is given by
I187 (1978).
31. A. S. Argon, Trans. A.S.M.E.
Ser. H (J. Engng Mater.
Techn.) 98, 60 (1976).
32. J. R. Fisher and J. Gurland, Metal Sci. 15, 185 (1981). 33. M. Fujita, Y. Kawabe and N. Nishimoto, Truns. natn Res. inst. Metals 23, 149 (1981).
34. E. E. Underwood, in Quuntitative Microscopy (edited by R. T. DeHofi’ and F. N. Rhines), pp. 77-127. McGraw-Hill, New York (1968). 35. J. Gurland and J. Plateau, Trans. Am. Sot. Metals 56,
where ni = number of dimples in the ith interval. Further, ni is given by F, + Ai, where F, = area fraction on fracture surface of the ith interval and Ai = area of an individual dimple, since the number of dimples per unit area is l/A,. The dimples are roughly circular in plan view but fill the fracture surface, so A, can be taken to be nb:/4. Thus
442 (1963).
36. D. J. Widgery and J. F. Knott, Mefal Sci. 12, 8 (1978). 37. I. G. Palmer and G. C. Smith, in Oxide Dispersion Strengthening (edited by G. S. Ansell, T. D. Cooper and F. V. Lenel), pp. 253-290, Gordon & Breach, New York (1968).
For the reported data [20], b for the “UE” material was 0.330 pm, and b for the “E30” material was 0.215 pm, an R value of 0.65.