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Fracture toughness and brittle-ductile transition controlled by grain boundary character distribution (GBCD) in polycrystals
Acta metall, mater. Vol. 38, No. 12, pp. 2507-2516, 1990
0956-7151/90 $3.00+ 0.00 Copyright © 1990 Pergamon Press plc
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F R A C T U R E TOUGHNESS A N D BRITTLE-DUCTILE TRANSITION CONTROLLED BY GRAIN B O U N D A R Y CHARACTER DISTRIBUTION (GBCD) IN POLYCRYSTALS L. C. L I M l and T. W A T A N A B E 2
~Department of Mechanical and Production Engineering, National University of Singapore, Kent Ridge, Singapore 0511, Republic of Singapore and ~Department of Materials Science, Faculty of Engineering, Tohoku University, Sendai 980, Japan (Received 11 December 1989; in revised form 16 May 1990)
Abstract--The structural dependence of intergranular fracture processes in bicrystals and polycrystals of metals and alloys is first reviewed. It is shown that even in polycrystals, grain boundary structure plays a significant role in controlling the fracture properties of the material. Next, we evaluate quantitatively the effect of different types, frequencies and configurations of grain boundaries, so-called the grain boundary character distribution (GBCD), on the toughness of a three-dimensional (3D) polycrystal. The results show that the toughness of a polycrystal increases monotonically with increasing overall fraction of fracture-resistant low-energy boundaries in the material. A brittle-ductile transition, corresponding to a change of fracture mode from predominantly intergranular with low toughness to predominantly transgranular with high toughness, is observed when the overall fraction of low-energy boundaries reaches a critical value. For a 3D polycrystal with a non-random GBCD such that the fraction of low-energy boundaries on the inclined boundary facets is maximised, a smaller critical overall fraction of low-energy boundaries is needed to bring about the brittle-ductile transition. Similar effect is also found if the grains are made elongated and aligned with the stress axis. The results are discussed in relation to the concept of grain boundary design for strong and tough polycrystals proposed by one of the present authors (T.W.). R6smn6---On fait d'abord un revue de la variation en fonction de la structure des processus de fracture intergranulaire dans les bicristaux et dans les polycristaux de m6taux et d'alliages. On montre que, m~me darts les polycristaux, la structure du joint de grain joue un r61e important en contr61ant les propri6t6s de fracture du mat6riau. De plus, on 6value quantitativement l'effet de diff6rents types, fr6quences et configurations de joints de grains, que nous appelons distribution du caract6re des joints de grains (DCJG), sur la tenacit6 d'un polycristal/t trois dimensions. Les r6sultats montrent que la tenacit6 d'un polycristal augmente d'une fa~on monotone lorsque la fraction globale des joints de basse 6nergie r6sistant fi la fracture s'616ve dans la mat6riau. Une transition fragile
1. I N T R O D U C T I O N It is well k n o w n t h a t fracture processes in polycrystalline materials occur by t r a n s g r a n u l a r a n d / o r
i n t e r g r a n u l a r fracture a n d t h a t the m o d e o f fracture can be strongly affected by material composition, microstructure a n d e n v i r o n m e n t . W h e n the fracture
2507
2508
LIM and WATANABE: BRITTLE-DUCTILE TRANSITION IN POLYCRYSTALS
mode is predominantly intergranular, a polycrystal shows brittle fracture behavior [1,2]. This has been the main weakness of many advanced, high performance structural materials, such as engineering ceramics and high-temperature intermetallic compounds [3-5], as well as structural metals and alloys embrittled by impurities and/or in the presence of corrosive environment [6-8]. In recent years it has been found that the propensity for intergranular fracture is closely related to the type and structure of grain boundaries. Low-energy boundaries are resistant to fracture while highenergy, or the so-called random, boundaries are preferential sites for crack nucleation and propagation. One of the present authors (T.W.) has recently discussed the potential for the control of intergranular fracture by grain boundary design, that is, by suitable processing to yield a certain type, frequency and configuration of grain boundaries, so-called grain boundary character distribution (GBCD), to develop strong and ductile polycrystalline materials [9-11]. The present authors have also provided a theoretical background of the subject for two-dimensional (2D) polycrystals [12]. They showed that the toughness of a 2D polycrystal is a strong function of its GBCD, increasing with the overall fraction of fractureresistant low-energy boundaries in the material. In the above work, the effects of the geometrical configuration of grain boundaries and grain shape on the toughness were also evaluated. The present paper attempts to extend the above work to real, three-dimensional (3D) polycrystals. The experimental evidence reported by various researchers on the boundary structural effect on intergranular fracture is first reviewed. Then, the toughness of 3D polycrystals is evaluated theoretically as a function of GBCD. Implications of the findings are discussed. 2. STRUCTURE-DEPENDENT INTERGRANULAR FRACTURE The dependence of grain boundary structure on intergranular fracture has been studied by many workers using bicrystals of normally brittle materials such as refractory metals and diamond cubic materials. Kobylanski and Goux [13], Brosse et al. [14] and Kurishita et al. [15, 16] have measured the fracture stress of molybdenum bicrystals with (100), (110) tilt and (110) twist boundaries, while Sato et al. [17] studied silicon bicrystals containing a (111) twist boundary of different misorientations. For semibrittle and normally ductile materials such as h.c.p. and f.c.c, metals, the dependence of boundary misorientation on fracture stress has also been studied by liquid metal embrittlement techniques. Notable in this respe_ct include work by Watanabe et al. [18] on (1010) tilt and twist bicrystals of zinc ernbrittled by liquid gallium, by Otsuki and Mizuno [19] on (110) tilt aluminium bicrystals embrittled by liquid Sn-Zn,
and the work by Kargol and Albright [20] who measured the crack extension force for intergranular fracture of (110) tilt aluminium bicrystals embrittled by liquid Hg-Ga. Table 1 gives the fracture stresses reported by the various workers for several 27coincidence boundaries and random boundaries. Several features are noteworthy in this table. First, the fracture stress or the crack extension force of coincidence boundaries (including E l/low-angle boundaries) is always several times greater than that of random boundaries. Second, even though the reported fracture stresses are found to vary somewhat from one boundary type to the other, it is interesting to note that the ratio of the fracture stress of coincidence boundaries to that of random ones (tr~/aR) increases with decreasing 27 value irrespective of test condition and material. Third, the limited data given in Table 1 have indicated a lower az/tr R for twist than for tilt boundaries. This, however, could be attributed to a much higher a R for twist boundaries, as indicated by the results of Kurishita et al. [15, 16]. Further work is needed to ascertain this important aspect. Using polycrystalline specimens having characterized grain boundaries, the effects of boundary 27 on intergranular fracture in metallic or ceramic materials and in Ni3A1 intermetallic compounds have been investigated by several workers [21-23]. It has been shown that low-angle boundaries and low-27 coincidence boundaries are resistant to intergranular fracture while high-angle random boundaries easily fail, irrespective of material, test condition and environment. The works of Lim and Raj [24] and Don and Majumdar [25] have further shown that the propensity to cavitation and fracture at higher temperatures decreases with decreasing 27 value for coincidence boundaries. The above works have confirmed that similar structural effects on intergranular fracture as observed on bicrystals are also found in polycrystals, strongly indicating that the structural effect on intergranular fracture is strong enough not to be masked by the different stress conditions experienced by the various grain boundaries in the polycrystals. Even though there has been no theoretical prediction for the upper limit of ,~ below which a coincidence boundary may not exhibit special mechanical behavior, judging from the experimental work reported in the literature, it is likely that the upper limit of ,~ may take a value of around 29, although the actual value may depend on the material composition, microstructure, test condition and environment [26]. As regards microstructural processes of fracture in polycrystalline materials, it has been shown that whether fracture occurs in typically intergranular manner or a combined mode of intergranular and transgranular fracture depends, to a large degree, on the type of grain boundary in front of the propagating crack [9]. When the main crack keeps propagating on weak random boundaries, a typical intergranular
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LIM and WATANABE: BRITTLE-DUCTILE TRANSITION IN POLYCRYSTALS Table I. Fracture stress of coincidenceboundaries Fracture stress Material Mo
Type of boundary
Test condition
Coincidence a~
Random art
a~ltrrt
Reference
(100) Sym. tilt
R.T, a
1340 MPa
75 MPa
17.8
[13]
(110) Sym. tilt
R.T.
(5o/z l)
1150 MPa
15.3
(8°/271) 800 MPa (109°/273) 570 MPa
88 MPa
9.1
[13]
6.5
(50°/2711) Mo
(100) Sym. tilt
R.T.
(110) Sym. tilt
R.T.
Mo
(110) Sym. tilt
77 K
Mo
(110) Twist
77 K
Si
(111) Twist
R,T.
Zn
(10"f0)Sym. tilt (10]'0) Twist
AI
(110) Tilt
Liquid Ga 303 K Liquid Ga 303 K Liquid Sn-Zn 513K
533 MPa (9.3°/2~ I) 293 MPa
(lOO/S1)
400 MPa (110°/273) 400 MPa (50°/27 1 1) 1720 MPa (70°/273) 1440 MPa (73°/273) 300 MPa
(25°/$7) 0.86MPa (55°/-r9) 1.3 MPa (54°/279) 25 MPa (5°/El) 39 MPa (70<'/273) 23 MPa
106-120 MPa
4.4-5.0
[14]
67 MPa
4.4
[14]
6.0 6.0 280 MPa
6.1
[15]
623 MPa
2,3
[16]
140 MPa
2.1
[17]
0.24MPa
3.6
[18]
0.2ff4).33MPa
3.9~.5
[18]
4.9 MPa
5.1
[19]
8.0 4.7
(12o°/2711) AI
(110) Sym, tilt
Liquid HI~Ga
3 kN/m b (70°/273) 2.4 kN/m b
0.34.4 kN/m b
7.5-10.0
[20]
6.0-8.0
(130°/2711) a R.T. = room temperature, bCrack extension force at 3.5/~m/s.
fracture occurs resulting in the loss of ductility and brittleness of the polycrystal. In essence, the vast volume of work in the literature has shown that for materials prone to brittle intergranular fracture, be it an intrinsic effect or a result of impurity segregation and/or environment effect, high-energy random boundaries are preferential sites for crack nucleation and propagation. They can be considered as "latent cracks" similar to "pre-existing cracks" in brittle materials suggested by Griffith [27]. The size of such latent cracks must be almost the grain size. Therefore, an increase in the frequency of random boundaries may correspond to an increase in the density of latent cracks in the polycrystal. Reversely, an increase in the frequency of fractureresistant low-energy boundaries would help interrupt crack propagation along weak, random boundaries, thereby rendering the polycrystal strong and ductile. 3. GRAIN BOUNDARY CHARACTER DISTRIBUTION CONTROLLED TOUGHNESS--A THREE DIMENSIONAL MODEL 3.1. General consideration Figure 1 shows a 3D equiaxed polycrystal made up of tetrakaidecahedron-shaped grains with the loading
axis lying along one of the crystal grain axes. It shows that there are a total of seven boundary facets per grain, out of which one lies transverse (t), two longitudinal (l) and the other four inclined (i) to the stress axis. Being parallel to the stress axis, the longitudinal boundary facets will not experience large normal stresses and hence are less liable to intergranular fracture irrespective of their boundary character. In the toughness calculation to follow, it is assumed that they do not participate in the fracture process of the polycrystal. Consider the fracturing process of the polycrystal. For simplicity, we shall assume that the overall fracture plane lies approximately normal to the stress axis and that the maximum peak-to-valley height of the fracture surface is equal to or less than one-grain height, or H (hereafter referred to as Assumption I). To evaluate the toughness of a polycrystal, we need to know the fracture path taken by the polycrystal. Two common types of fracture path can be identified. The first is the "zig-zag path" formed by interlinkage of intergranular crack facets at different levels in the material by the transgranular crack facets, which themselves lie at different angles to the stress axis [Fig. l(b)]. This is the most common type of fracture path observed when the fraction of intergranular
2510
LIM and WATANABE: BRITTLE-DUCTILE TRANSITION IN POLYCRYSTALS
"LL. L,J .J
B ~
~ g
A ~
~A
N VIEW D (=3 d) SECTION
ISOMETRIC VIEW /
c~=H/D =H/3d
HI
-J-- " " A _SECTION '- - --',. 22-%
.# - -", ,
(a) SECTIONAL VIEW
x
/
\
___t
/
x___
AMAGE
6E ZONE
Cb)
(c)
INTERGRANULAR CRACK FACET TRANSGRANULAR CRACK FACET Fig. l. (a) Geometry of a 3D polycrystal. (b) shows the sectional view of a zig-zag fracture path and (c) shows that of a planar fracture path. d is the length of edge of a 3D equiaxed grain. facets on the fracture surface is comparatively high. When the fracture surface is largely transgranular, another type of fracture path may become more prevailing. In this case, the fracture surface is rela-
tively smooth with a waviness much less than that of the zig-zag path [Fig. l(c)]. This type of fracture path is hereafter referred to as the "planar path". Whether the zig-zag or the planar path will be favored will, to
2511
LIM and WATANABE: BRITTLE-DUCTILE TRANSITION IN POLYCRYSTALS a great extent, depend on which of these two paths possesses a lower resistance to fracture. Let us first calculate the toughness associated with the zig-zag path. Assuming that on loading, fracture takes place along a zone of extensive damage within which all random boundaries fracture intergranularly. Invoking Assumption I, the thickness of such a damage zone would be equal to about one-grain height, or H [Fig. l(b)]. It can be seen that within this damage zone, the "elemental fracture path link" consists of six segments: two normal (or n) type segments in which the boundaries lie transverse to the stress axis, and four shear (or s) type segments in which the boundaries lie at an angle to the stress axis [shaded in Fig. l(b)]. The two n-type segments can be further divided into n~ and n2 types. The former contains only one while the latter two transverse boundaries. Since the nrtype link segment would fracture intergranularly should the transverse boundary in it be a random boundary, its probability of fracturing intergranularly is thus given by 1 - f ~ , f being the fraction of fracture-resistant low-energy boundaries among all the transverse boundaries in the material. For the n2-type segment, it will fracture intergranularly should either one or both of the two transverse boundary facets comprising the segment be a random one. Its probability of fracturing intergranularly is thus 1 _ f 2 . Similarly, for each of the four s-type segments, the probability that it will fracture intergranularly is 1 - f ~ , f being the fraction of low-energy boundaries among all the inclined boundaries in the material. If (#T and f#i are, respectively, the toughness associated with a transgranular and intergranular crack and A t and A i respectively the projected areas of the transverse and inclined boundary facets onto a plane normal to the loading axis, it can be shown that the toughness associated with the zig-zag path is given by
~PL= {(1 --ft)At~l "3LUtAt~T-[-A3~T}/(At + A3)
+(1 --f~)At(¢l +fZFn2At~T + Ns[(l
(1)
where N~ is the number of s-type segments per "elemental fracture path link" of the polycrystal system under study. _¥s = 4 for a 3D polycrystal. 01 is the angle between the inclined boundary facet and the stress axis (Fig. 1). Ai/sin 01 is thus the physical area of the inclined boundary facet. Sin 01 can be expressed in terms of the grain aspect ratio, a, through equation (AI3) given in the Appendix. Fn,, tKowning .~, the fracture toughness value, Klc, can be estimated through the following relationship: Klc = ~/adE, where the Young's modulus E is relatively insensitive to the overall fraction of low-energy boundaries in the material.
(2)
where
A 3 = A t + N s A i . For a 3D polycrystal, 1.5. Using the value ofAi/A t given above and equation (A13) given in the Appendix, it can be shown that for a 3D polycrystal, the toughness is given by
Ai/At =
C~zz
~T
1
8{I2--f --f2+6(1--fi)x/l+]~ZJ~TT
+f,(V., + f~r. 2) + 6f~ F: t (3) for fracture occurring along the zig-zag path, and
~PL ~T - 81{ (1 - - f ) ~ T(~I +
(¢zz = {(1 -- ft)Atff~+ f t F , , At(~y
-f'~)(AUsin 01)qJ~ + fZ F~AiqJT]}/(2At+ N~A~)
Fn: and Fs in equation (1) are the correction factors which take into account of the fact when nl, n2 or s-type links fracture transgranularly, the fracture path needs not necessarily lie normal to the loading axis. Instead, they may take several possible paths making different angles with the stress axis. The actual path taken will be determined by the cracking behavior of the adjacent boundary facets. The forms of Fn,, Fn2 and Fs can be determined from the knowledge of the geometry of the polycrystal under study and of the probability of occurrence of the various transgranular crack paths. The approximate forms of Fn,, Fn2 and F~ are given in the Appendix. For the "planar path", the damage zone is limited to the plane containing the transverse boundary in the n-type segment. The elemental fracture path link in this case can be thought as consisting of two segments: one nl-type segment and one ns-type segment, as shown in Fig. l(c). The nl-type segment may fracture intergranularly with a probability given by 1 - f . The n3-type segment however, will always fracture transgranularly on a plane containing the transverse boundary in the nrtype segment. The toughness associated with such a fracture path is given by
7 +f
}
(4)
for fracture occurring along the planar path. In a real polycrystal, fracture will take place along the path of least resistance. The toughness of the polycrystal, (~(~, f ) is thus given by ~(~, f )
~T
Min. f~ZZ, ~PL~ ~
~TJ
(5)
where f is the overall fraction of fracture-resistant, low-energy grain boundaries in the polycrystal. Note that f, f , f and f (where f,, f and f are respectively the fractions of low-energy boundaries among all the transverse, inclined and longitudinal boundary facets in the polycrystal) need not be identical. Note also that the toughness (and hence the fracture toughnesst of the polycrystal is independent of the grain size. However, grain size dependence of the toughness may still arise should the GBCD of the material be grain
2512
LIM and WATANABE: BRITTLE-DUCTILE TRANSITION IN POLYCRYSTALS
size dependent, as reported recently by Grabski [28] and Watanabe [10, 11].
1.0
3.2• Random distribution
For a polycrystal with a random GBCD, the fractions of low-energy boundaries must be identical on all types of boundary facets. For a 3D polycrystal, this implies that f = f t = f i =f~.
(6)
With the above relationship, the toughness of a 3D polycrystal having a random GBCD can be calculated using equations (3) to (5) for different values of ~X/C~. The results are shown in Fig. 2 as a function of the overall fraction of low-energy boundaries in the material. For easy reference, we have plotted in Fig. 3 the toughness enhancement over that of a reference 3D polycrystal with f = 0.15, i.e. ogREF = ~R(~te, 0.15), where the subscript R stands for "Random" distribution and cte is the aspect ratio of an equiaxed grain, tee=0.94 for a 3D equiaxed polycrystal, f = 0.15 is the fraction of low-energy boundaries which a polycrystal would possess if all the grains in the polycrystal are randomly oriented.~" Figures 2 and 3 illustrate that for a 3D polycrystal having an equiaxed grain structure, the zig-zag path is always favoured for ~TI~REF> 2 . It is evident that the toughness of a 3D polycrystal increases monotonically with increasing f. Furthermore, the higher the value of f~r/~REF, the more effective is the toughness enhancement per increment of the fraction of low-energy boundaries in the material. Similar results are also obtained for the 2D case in the previous work [12]. 3.3. Non-random distribution
When the GBCD in a polycrystal is no longer random, different types of boundary facets in the material will possess different fractions of low-energy boundaries. However, geometry requires that, for a 3D polycrystai ft + 2f~ + 4fi = 7f
0.8
(7)
Using the above relationship in place of equation (6) and assuming that f =f~, we have calculated the toughness of a 3D polycrystal with different fractions, ft, of transverse low-energy boundaries. The results for c~r/fgREF = 5 are shown in Fig. 4. The bold curve in this figure is the toughness values for a 3D polycrystal with a random GBCD. It can be seen from Fig. 4 that an increase of the fraction of the transverse low-energy boundaries at low f level improves the toughness of the 3D polycrystal somewhat. On the other hand, at highfvalues, increasing the fraction of low-energy boundaries on both the inclined and longitudinal boundary facets becomes beneficial instead. It is also interesting to
0.4
-/y/ 0.2 ~
y
0.0
m . m PLANAR PATH ,
0.0
Zlr-ZA6 PATH
- -
J ,
i
O.2
,
,
~
0.4
,
,
0. G
,
0.8
1.0
f Fig. 2. Effect of overall fraction of low-energy boundaries, f, and ~T/ffl on the toughness of a 3D polycrystal having a random GBCD. note that at large f value and with ft < 0.6, the toughness of the polycrystal increases rapidly with increasing f, and a transition from zig-zag path to planar path occurs at f = 0.7. In other words, a brittle-ductile transition occurs with the fracture mode changing from predominantly intergranular to predominantly transgranular. From this point on, the toughness of the polycrystal stays high and is relatively insensitive to the overall fraction of lowenergy boundaries in the material. This is in contrast to the 2D case in which increasing the fraction of low-energy boundaries on the transverse boundary facets is always found to be beneficial for ft > 0.5 [12]. Another main difference between the 2D and 3D cases is that even though remarkable toughness enhancement can be achieved for the 2D case through processing which results
a 3D-RANDOM EQUIAXED Z,G-ZAGPAtH
"-~._¢.~/
I 0.2
(13
0.t,
0.5
0.6
0.7
0.S
0.9
tO
-F tStrictly speaking,f = 0.136 should one take the upper limit of Z' value to be 29 for coincidence boundaries [29, 30].
Fig. 3. Same as Fig. 2 but normalised with respect to ffREF where fg~F = c'dR(ct,, 0.15).
LIM and WATANABE: BRITTLE-DUCTILE TRANSITION IN POLYCRYSTALS ,
I
i
,
,
,
i
to
"~/'~R EF--
g
"
0
~ ~
i
l, ~~"
~j/~.2..~-~¢
/ / / X " ; t RANDOM
i
~1
5
3D NON R A N D O M
i
I
.
i
.
i
...._.._~
g
o, 0.3
i 0.4
i 0,5
i 0.6
!7: 07
/
4 ~
-
-Lo_ Ao PAT: 0.2
III/I >_.--
3
~
I
,
-
3
•
2513
RE~ 5 . 0
-
(18
1.0
PLANAR PATH
0.8
i 0.9
1.0
f
(12
0.3
a4
0.5
(16
03
0.9
f
Fig. 4. Effect of non-random GBCD.
Fig. 5. Effect of grain shape and boundary inclination.
in a high proportion of low-energy boundaries on the transverse boundary facets, the magnitude of toughness enhancement due to such is much less pronounced for the 3D case. These discrepancies can be attributed to the fact that Ai < At for the 2D case but it is the reverse for the 3D case. (Ai/A t = 0.5 and 1.5 for the 2D and 3D case respectively.) Thus, even though maximizing transgranular fracture on the transverse boundary facets produces a larger transgranular area for the 2D case, the same does not hold for the 3D case. This also explains why promoting a higher proportion of transgranular fracture on the inclined boundary facets would give rise to significant toughness improvement for the 3D case but not for the 2D case.
effected and the polycrystal would possess a high toughness value. It is evident that elongated grain structures are a very effective means of imparting high toughness to a polycrystal even at lowerfvalues, e.g. a predominantly transgranular fracture is predicted with f , ~ 0.4 for ct > 3. On the other hand for a polycrystal having an equiaxed grain structure or ~t = 1, the toughness increases only relatively slowly with increasing overall fraction of low-energy boundaries, f. Similar results have also been found for 2D polycrystals [12]. A comparison of the results for both the 2D and 3D cases has shown that the effect of grain shape and boundary inclination is more pronounced for the 3D case. This again can be attributed to the larger projected area of the inclined boundary facets in the 3D case, as discussed earlier in relation to the effect of non-random GBCD on the toughness of a 3D polycrystal.
4. EFFECT OF GRAIN SHAPE AND BOUNDARY INCLINATION The results of the previous section has shown that during fracture of an equiaxed 3D polycrystal having a random GBCD, the zig-zag fracture path often prevails. This suggests that if we could process the material such that the grain structure is elongated along the stress axis. a longer span of transgranular crack facets will then be required to link up the intergranular cracks at different levels in the material. This, in turn, would lead to further improvement in the toughness of the polycrystal, as suggested by Watanabe [31, 32]. Figure 5 shows the effect of grain shape and boundary inclination, expressed in terms of the grain aspect ratio ~, on the toughness of a 3D polycrystal having a random GBCD for c~T/ffREF = 5. For a given f value, the toughness of the polycrystal is seen to increase sharply with increasing grain aspect ratio. Most interestingly, the result also shows that if a polycrystal can be processed such that it not only has a reasonably high overall fraction of low-energy boundaries but also an elongated grain structure aligned with the stress axis, a planar path can be
5. DISCUSSION
5.1. Toughening by grain boundary design As described in the Introduction, many advanced engineering materials and structural alloys are susceptible to brittle intergranular fracture either due to the intrinsic properties of the materials or the result of impurity segregation and/or environmental effects. Recently, Watanabe [9-11, 31, 32] has suggested that the brittleness of polycrystalline materials associated with intergranular fracture can be controlled by designing grain boundaries in different ways, for instance, by manipulating the GBCD, boundary inclination, boundary chemistry and so on. In what follows, we shall discuss the various aspects of Grain Boundary Design for strong and tough materials based on the results obtained in the present work. 5. I. 1. Engineering grain boundary character distribution (GBCD). Many researchers have investigated the GBCD in polycrystais of different materials pro-
2514
LIM and WATANABE: BRITTLE-DUCTILE TRANSITION IN POLYCRYSTALS
duced by different techniques (see Refs [10] and [11] type relationship for grain size dependence of fracfor recent reviews on the subject). In general, it has ture properties from GBCD viewpoint. A similar been found that there exists a close relationship view has recently been expressed by Wyrzykowski between GBCD and the processing method for and Grabski [34] for grain size dependence of yield polycrystalline materials. stress. For polycrystals of metallic materials produced by 5.1.3. Grain boundary inclination. It is evident from ordinary thermo-mechanical processing, it has been the present work that engineering the grain shape and reported by a number of workers that the frequency boundary inclination is certainly a very effective of low-energy boundaries such as low-angle bound- way of toughness enhancement for polycrystalline aries and low-Z coincidence boundaries decreases materials. Figure 5 shows that to significantly inwith increasing annealing temperature and resultant crease the toughness of a polycrystal, the grains must grain size [10, 28, 33-35]. Grain size dependence of be orientated parallel to the stress axis and have an GBCD has been discussed in connection with the aspect ratio of 3 or higher. In fact, a similar effect has change in grain boundary energy during recrystalliz- been utilized in controlling intergranular fracture ation [36]. In f.c.c, polycrystals of metals and alloys, in polycrystalline materials through unidirectional the occurrence of a much higher frequency of S 3 and solidification [42] and directional recrystallization related coincidence boundaries has been reported [43], even though no theory has so far been proposed [25, 37, 38]. This has been attributed to the result of to explain the phenomenon. multiple twinning occurring during recrystallisation Strictly speaking, the control of boundary inclianneal of these materials. The presence of texture nation should be made in connection with the control seems to increase the frequencies of low-angle and of the frequencies of fracture-resistant low-energy low-Z coincidence boundaries [39, 40]. An inverse boundaries on different types of boundary facets, cubic root Z" dependence of the frequencies of coinci- to bring about the best toughening effect. Even dence boundaries has been observed for Fe~.5 mass though no such attempt is made in the present work, % Si ribbons and other b.c.c, polycrystals [40]. a quick reference to Figs 4 and 5 suggests that more Moreover, the Z values of those coincidence bound- pronounced beneficial effects are to be expected, aries which occurred in higher frequencies were found when both the effects of GBCD and of boundary to be related to the type of texture, that is, (100) or inclination are taken into account together. (110) texture, in polycrystalline ribbons of the 5.2. Brittle-ductile fracture transition induced by iron-silicon alloy [41]. 5.1.2. GBCD and fracture toughness of polycrystals. GBCD The results of the present work demonstrate that Perhaps, one of the most interesting findings of the toughness of a 3D polycrystal can be greatly the present work is the prediction of the change of improved by engineering the GBCD of the poly- fracture mode from predominantly intergranular crystal. For instance, Fig. 3 shows that the toughness (zig-zag path) with low toughness to predominantly of a 3D polycrystal increases monotonically with transgranular (planar path) with high toughness, increasing overall fraction of fracture-resistant low- when the overall fraction of low-energy boundaries is energy boundaries in the material. A similar effect of increased above a certain level. In other words, for GBCD on the toughness has also been previously materials prone to brittle intergranular fracture either reported for 2D polycrystals [12]. intrinsically or as a result of impurity segregation Some experimental support for the above has and/or environment effect, a "brittle-ductile fracture recently been obtained by Watanabe et al. [40]. transition" can be induced through suitably enginThey reported that the toughness of a normally eering the GBCD and the boundary inclination in the brittle high-silicon iron alloy can be drastically polycrystal. improved by rapid solidification followed by a high In the case of random GBCD (Fig. 3), even though temperature anneal, which also resulted in a high the toughness of the polycrystal was found to increase frequency of low-energy boundaries (45% of all monotonically with increasing overall fraction, f, of low-energy boundaries in the material, no abrupt boundaries). The present results and Ref. [12] also showed that brittle~luctile transition can be concluded. Figure 4, the effect of GBCD on toughness can be made more however, shows that when the material is so prosignificant by engineering the polycrystal such that a cessed that the fraction of low-energy boundaries on higher percentage of those boundary facets possess- the transverse boundary facets, ft, is less than 0.6, the ing the largest projected area normal to the stress axis brittle--ductile fracture transition may occur. For are of the low-energy type. This, for a 3D polycrystal, instance, for a material system with f¢T/f~r = 5 and means those boundary facets which lie inclined to the ft ~- 0, the transition occurs at f = 0.7. Another very stress axis (Fig. 4). effective way of inducing the brittle--ductile fracture Equations (4) and (5) further suggest a possi- transition is to make use of the combined effect of bility that the grain size dependence of toughness GBCD and boundary inclination (Fig. 5). In this of a polycrystal may arrive through that of GBCD. case, the critical overall fraction of low-energy It is thus of interest to re-examine the Hall-Petch boundaries at which the brittle-ductile transition
LIM and WATANABE: BRITTLE-DUCTILE TRANSITION IN POLYCRYSTALS occurs can be lowered to 0.5 or less. In general, with increasing (~r/ffREF, corresponding to a material with increasing susceptibility to brittle intergranular fracture (i.e. lower ~i), the critical overall fraction of low-energy boundaries required to bring about such a brittle-ductile transition also increases accordingly. From a comparison between the results for random and for non-random configurations of grain boundaries for a polycrystal with an equiaxed grain structure and ~T/ffREF= 5, we can see that the nonrandom configuration causes the brittle
2515
fracture can be induced by suitably engineering the GBCD and boundary inclination in the polycrystal. Certain non-random configurations of grain boundaries and an elongated grain structure aligned with the loading axis bring about such a transition at a smaller value of overall fraction of low-energy boundaries than for random boundary configurations. Acknowledgements--The first author (L.C.L.) wishes to acknowledge the financial support received from the National University of Singapore, through the grant No. RP 101/85. The second author (T.W.) gratefully acknowledges The Nippon Sheet Glass Foundation for a Grant-inAid for research in Materials Science. REFERENCES
1. C. L. Briant and S. K. Banerji (editor), Embrittlement of Engineering Alloys, Treatises on Mater. Sci. Tech., Vol. 25 (1983). 2. M. H. Kamdar (editor), Embrittlement by Liquid and Solid Metals. Mater. Soc. A.I.M.E. (1984). 3. B. H. Kear et al. (editors), Ordered Alloys--Physical Metallurgy and Structural Applications. Claitor's, Baton Rouge, La (1970). 4. C. C. Koch, C. T. Liu and N. S. Stoloff (editor), High-Temperature Ordered Intermetallic Alloys, MRS Symp. Proc., Vol. 39 (1984). 5. N. S. Stoloff, C. C. Koch, C. T. Liu and O. Izumi, High-Temperature Ordered Intermetallic Alloys--ll, MRS Syrup. Proc., Vol. 81 (1986). 6. R. T. Holt and W. Wallace, Int. Metall. Rev. 21, 1 (1976). 7. M. P. Seah, Phil. Trans. R. Soc. A295, 265 (1980). 8. C. L. Briant, Metallurgical Aspects of Environmental Failures, p. 49. Elsevier, Amsterdam (1985). 9. T. Watanabe, Res Mechanica 11, 47 (1984). 10. T. Watanabe, in Interracial Structure, Properties and Design (edited by M. H. Yoo, W. A. T. Clark and C. L. Briant), MRS Symp. Proc., Vol. 122, 443 (1988). 11. T. Watanabe, Materials Forum, Australia's Bicentenary Anniversary Volume, Vol. 11, 284 (1988). 12. L. C. Lira and T. Watanabe, Scripta metall. 23, 489 (1989). 13. A. Kobylanski, thesis, Univ. Paris VI (1972), with C. Goux, C.r. Acad. Sci. Paris 272, 1937 (1971). 14. J. B. Brosse, R. Fillit and M. Biscondi, Scripta metall. 15, 619 (1981). 15. H. Kurishita, A. Oishi, H. Kubo and H. Yoshinaga, J. Japan Inst. Metals 47, 546 (1983). 16. H. Kurishita, S. Kuba, H. Kubo and H. Yoshinaga, J. Japan Inst. Metals 47, 539 (1983). 17. K. Sato, H. Miyazaki, Y. Ikuhara, H. Kurishita and H. Yoshinaga, J. Japan Inst. Metals 53, 536 (1989). 18. T. Watanabe, S. Shima and S. Karashima, in Ref. [2], p. 161 (1984). 19. A.'Otsuki and M. Mizuno, Trans. Japan Inst. Metals Suppl. 27, 789 (1986). 20. J. A. Kargol and D. L. Albright, Metall. Trans. 8A, 27 (1977). 21. T. Watanabe, Metall. Trans. 14A, 531 (1983). 22. S. Lartigue and L. Priester, Trans. Japan Inst. Metals, Suppl. 27, 205 (1986). 23. S. Hanada, T. Ogura, S. Watanabe, O. Izumi and T. Masumoto, Acta metall. 34, 13 (1986). 24. L. C. Lim and R. Raj, Acta metall. 32, 1183 (1984). 25. J. Don and S. Majumdar, Acta metall. 34, 961 (1986). 26. T. Watanabe, M. Tanaka and S. Karashima, in Ref. [2], p. 183 (1984). 27. A. A. Griffith, Phil. Trans. R. Soc. A221, 163 (1920). 28. M. W. Grabski, J. Physique, Suppl. C4, 46, 567 (1985).
2516
LIM and WATANABE: BRITTLE-DUCTILE TRANSITION IN POLYCRYSTALS
29. D. H. Warrington and M. Boon, Acta metall. 23, 599 (1975). 30. M. Mori, Doctoral thesis, Tokyo Univ. (1975). 31. T. Watanabe, in Creep and Fracture of Engineering Materials and Structures (edited by B. Wilshire and R. W. Evans), p. 155. Inst. of Metals, London (1987). 32. T. Watanabe, in Chemistry and Physics of Fracture (edited by R. M. Latanision and R. H. Jones), NATO series 130, p. 492. Martinus Nijhoff, The Hague (1988). 33. T. Watanabe, J. Physique, Suppl. C4, 46, 555 (1985). 34. J. W. Wyrzykowski and M. W. Grabski, Phil. Mag. A53, 505 (1986). 35. T. Watanabe, Y. Kawamata and S. Karashima, Trans. Japan Inst. Metals Suppl. 27, 601 (1986). 36. T. Watanabe, Scripta metall. 21, 427 (1987). 37. L. C. Lim and R. Raj, Acta metall. 32, 1177 (1984). 38. P. Neumann and A. Tonnessen, in Fatigue 87(edited by R. O. Ritchie and E. A. Starke Jr), p. 3. EMAS, Cradley Heath, Warley, West Midland (1987). 39. T. Watanabe, N. Yoshikawa and S. Karashima, Proc. Int. Conf. on Texture of Materials (ICOTOM-6), p. 609. Japan Iron Steel Inst. (1981). 40. T. Watanabe, H. Fujii, H. Oikawa and K. I. Arai, Acta metall. 37, 941 (1989). 41. T. Watanabe, K. I. Arai, K. Yoshimi and H. Oikawa, Phil. Mag. Lett. 59, 47 (1989). 42. N. S. Stoloff, in Alloy and Microstructural Design (edited by J. K. Tien and G. S. Ansell), p. 65. Academic Press, New York (1976). 43. E. Hornbogen, in Fundamental Aspects of Structural Alloy Design (edited by R. I. Jaffe and B. A. Wilcox), p. 389. Plenum Press, New York (1977).
1
1
1
l)] +(ah+be)
-1
+(de+ch)
1), + 2[aa,h + (bb2+ 2b~)el [13/18 5/18 _ l l x [_sin 01 + sin 02 ] 4' 2 [13/18
5/18
where the coefficients are related to the probabilities of occurrence of the different types of transgranular crack facets and 0's the angles made by them with the stress axis. They have the explicit forms below. a 1= (1 --ft) +ft( 1 --fi2)2
(A4)
a =f~a 1 b, =ft(1 - f t ) +f~(1 -fi)2[1 - (I - f i ) 21 b2 = (1 - f t ) 2 +f2(1 - f i ) 4
(A5) (A6)
b = b2 + 2b l
(AS)
c = 2ftfi4(l _f2)2
(A9)
d = 4f2f~(1 _f~)4 + 4ft2fi(l _fi)2 x [f~ + 3f~(1 - Z ) ]
APPENDIX
Fn,, Fn~ and F~for 3D Polycrystal
e =ftf6(l --fi2)
(All)
h =f2f4(l --fi) 2 + 2f~f~(1 --f~)
(A12)
1
sin 0, -- x/1 + 9 0t2 Fnl '--1 +~{cd[(sinl 0~ - l ) + ( s i T 0 6 - 1 ) ]
I
sin 02 _ x/1 + 9 ~ 2 1
sin03
-- x/1 +¼*t 2
1 +
(A1)
( -01t l
sin 04 - /x/1
+
Ct 2
l sin 05 -- ~/1 + ~ e2
Fn:~-1 + ~ 2 t f e d I ( s i ~ s - 1 ) + ( s i T 0 6 - 1 ) 1
I
+ e\sin ( ± -031 ) J (A2)
(A13) (AI4) (A15) (A16) (AI7)
- x/1 + ~ ~2
(A18)
1 = lv/i~-2~ ~ 2 sin 07
(Al9)
1 = ~/1 sl ;~ sin 08 e is the aspect ratio of the grains (Fig. 1).
(A20)
sin 06
+ 4hL L \sin 02
(A10)
and
For a 3D polycrystal, Fn, and Fn2 and Fs are given by
d
(A7)
0956-7151/90 $3.00+ 0.00 Copyright © 1990 Pergamon Press plc
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F R A C T U R E TOUGHNESS A N D BRITTLE-DUCTILE TRANSITION CONTROLLED BY GRAIN B O U N D A R Y CHARACTER DISTRIBUTION (GBCD) IN POLYCRYSTALS L. C. L I M l and T. W A T A N A B E 2
~Department of Mechanical and Production Engineering, National University of Singapore, Kent Ridge, Singapore 0511, Republic of Singapore and ~Department of Materials Science, Faculty of Engineering, Tohoku University, Sendai 980, Japan (Received 11 December 1989; in revised form 16 May 1990)
Abstract--The structural dependence of intergranular fracture processes in bicrystals and polycrystals of metals and alloys is first reviewed. It is shown that even in polycrystals, grain boundary structure plays a significant role in controlling the fracture properties of the material. Next, we evaluate quantitatively the effect of different types, frequencies and configurations of grain boundaries, so-called the grain boundary character distribution (GBCD), on the toughness of a three-dimensional (3D) polycrystal. The results show that the toughness of a polycrystal increases monotonically with increasing overall fraction of fracture-resistant low-energy boundaries in the material. A brittle-ductile transition, corresponding to a change of fracture mode from predominantly intergranular with low toughness to predominantly transgranular with high toughness, is observed when the overall fraction of low-energy boundaries reaches a critical value. For a 3D polycrystal with a non-random GBCD such that the fraction of low-energy boundaries on the inclined boundary facets is maximised, a smaller critical overall fraction of low-energy boundaries is needed to bring about the brittle-ductile transition. Similar effect is also found if the grains are made elongated and aligned with the stress axis. The results are discussed in relation to the concept of grain boundary design for strong and tough polycrystals proposed by one of the present authors (T.W.). R6smn6---On fait d'abord un revue de la variation en fonction de la structure des processus de fracture intergranulaire dans les bicristaux et dans les polycristaux de m6taux et d'alliages. On montre que, m~me darts les polycristaux, la structure du joint de grain joue un r61e important en contr61ant les propri6t6s de fracture du mat6riau. De plus, on 6value quantitativement l'effet de diff6rents types, fr6quences et configurations de joints de grains, que nous appelons distribution du caract6re des joints de grains (DCJG), sur la tenacit6 d'un polycristal/t trois dimensions. Les r6sultats montrent que la tenacit6 d'un polycristal augmente d'une fa~on monotone lorsque la fraction globale des joints de basse 6nergie r6sistant fi la fracture s'616ve dans la mat6riau. Une transition fragile
1. I N T R O D U C T I O N It is well k n o w n t h a t fracture processes in polycrystalline materials occur by t r a n s g r a n u l a r a n d / o r
i n t e r g r a n u l a r fracture a n d t h a t the m o d e o f fracture can be strongly affected by material composition, microstructure a n d e n v i r o n m e n t . W h e n the fracture
2507
2508
LIM and WATANABE: BRITTLE-DUCTILE TRANSITION IN POLYCRYSTALS
mode is predominantly intergranular, a polycrystal shows brittle fracture behavior [1,2]. This has been the main weakness of many advanced, high performance structural materials, such as engineering ceramics and high-temperature intermetallic compounds [3-5], as well as structural metals and alloys embrittled by impurities and/or in the presence of corrosive environment [6-8]. In recent years it has been found that the propensity for intergranular fracture is closely related to the type and structure of grain boundaries. Low-energy boundaries are resistant to fracture while highenergy, or the so-called random, boundaries are preferential sites for crack nucleation and propagation. One of the present authors (T.W.) has recently discussed the potential for the control of intergranular fracture by grain boundary design, that is, by suitable processing to yield a certain type, frequency and configuration of grain boundaries, so-called grain boundary character distribution (GBCD), to develop strong and ductile polycrystalline materials [9-11]. The present authors have also provided a theoretical background of the subject for two-dimensional (2D) polycrystals [12]. They showed that the toughness of a 2D polycrystal is a strong function of its GBCD, increasing with the overall fraction of fractureresistant low-energy boundaries in the material. In the above work, the effects of the geometrical configuration of grain boundaries and grain shape on the toughness were also evaluated. The present paper attempts to extend the above work to real, three-dimensional (3D) polycrystals. The experimental evidence reported by various researchers on the boundary structural effect on intergranular fracture is first reviewed. Then, the toughness of 3D polycrystals is evaluated theoretically as a function of GBCD. Implications of the findings are discussed. 2. STRUCTURE-DEPENDENT INTERGRANULAR FRACTURE The dependence of grain boundary structure on intergranular fracture has been studied by many workers using bicrystals of normally brittle materials such as refractory metals and diamond cubic materials. Kobylanski and Goux [13], Brosse et al. [14] and Kurishita et al. [15, 16] have measured the fracture stress of molybdenum bicrystals with (100), (110) tilt and (110) twist boundaries, while Sato et al. [17] studied silicon bicrystals containing a (111) twist boundary of different misorientations. For semibrittle and normally ductile materials such as h.c.p. and f.c.c, metals, the dependence of boundary misorientation on fracture stress has also been studied by liquid metal embrittlement techniques. Notable in this respe_ct include work by Watanabe et al. [18] on (1010) tilt and twist bicrystals of zinc ernbrittled by liquid gallium, by Otsuki and Mizuno [19] on (110) tilt aluminium bicrystals embrittled by liquid Sn-Zn,
and the work by Kargol and Albright [20] who measured the crack extension force for intergranular fracture of (110) tilt aluminium bicrystals embrittled by liquid Hg-Ga. Table 1 gives the fracture stresses reported by the various workers for several 27coincidence boundaries and random boundaries. Several features are noteworthy in this table. First, the fracture stress or the crack extension force of coincidence boundaries (including E l/low-angle boundaries) is always several times greater than that of random boundaries. Second, even though the reported fracture stresses are found to vary somewhat from one boundary type to the other, it is interesting to note that the ratio of the fracture stress of coincidence boundaries to that of random ones (tr~/aR) increases with decreasing 27 value irrespective of test condition and material. Third, the limited data given in Table 1 have indicated a lower az/tr R for twist than for tilt boundaries. This, however, could be attributed to a much higher a R for twist boundaries, as indicated by the results of Kurishita et al. [15, 16]. Further work is needed to ascertain this important aspect. Using polycrystalline specimens having characterized grain boundaries, the effects of boundary 27 on intergranular fracture in metallic or ceramic materials and in Ni3A1 intermetallic compounds have been investigated by several workers [21-23]. It has been shown that low-angle boundaries and low-27 coincidence boundaries are resistant to intergranular fracture while high-angle random boundaries easily fail, irrespective of material, test condition and environment. The works of Lim and Raj [24] and Don and Majumdar [25] have further shown that the propensity to cavitation and fracture at higher temperatures decreases with decreasing 27 value for coincidence boundaries. The above works have confirmed that similar structural effects on intergranular fracture as observed on bicrystals are also found in polycrystals, strongly indicating that the structural effect on intergranular fracture is strong enough not to be masked by the different stress conditions experienced by the various grain boundaries in the polycrystals. Even though there has been no theoretical prediction for the upper limit of ,~ below which a coincidence boundary may not exhibit special mechanical behavior, judging from the experimental work reported in the literature, it is likely that the upper limit of ,~ may take a value of around 29, although the actual value may depend on the material composition, microstructure, test condition and environment [26]. As regards microstructural processes of fracture in polycrystalline materials, it has been shown that whether fracture occurs in typically intergranular manner or a combined mode of intergranular and transgranular fracture depends, to a large degree, on the type of grain boundary in front of the propagating crack [9]. When the main crack keeps propagating on weak random boundaries, a typical intergranular
2509
LIM and WATANABE: BRITTLE-DUCTILE TRANSITION IN POLYCRYSTALS Table I. Fracture stress of coincidenceboundaries Fracture stress Material Mo
Type of boundary
Test condition
Coincidence a~
Random art
a~ltrrt
Reference
(100) Sym. tilt
R.T, a
1340 MPa
75 MPa
17.8
[13]
(110) Sym. tilt
R.T.
(5o/z l)
1150 MPa
15.3
(8°/271) 800 MPa (109°/273) 570 MPa
88 MPa
9.1
[13]
6.5
(50°/2711) Mo
(100) Sym. tilt
R.T.
(110) Sym. tilt
R.T.
Mo
(110) Sym. tilt
77 K
Mo
(110) Twist
77 K
Si
(111) Twist
R,T.
Zn
(10"f0)Sym. tilt (10]'0) Twist
AI
(110) Tilt
Liquid Ga 303 K Liquid Ga 303 K Liquid Sn-Zn 513K
533 MPa (9.3°/2~ I) 293 MPa
(lOO/S1)
400 MPa (110°/273) 400 MPa (50°/27 1 1) 1720 MPa (70°/273) 1440 MPa (73°/273) 300 MPa
(25°/$7) 0.86MPa (55°/-r9) 1.3 MPa (54°/279) 25 MPa (5°/El) 39 MPa (70<'/273) 23 MPa
106-120 MPa
4.4-5.0
[14]
67 MPa
4.4
[14]
6.0 6.0 280 MPa
6.1
[15]
623 MPa
2,3
[16]
140 MPa
2.1
[17]
0.24MPa
3.6
[18]
0.2ff4).33MPa
3.9~.5
[18]
4.9 MPa
5.1
[19]
8.0 4.7
(12o°/2711) AI
(110) Sym, tilt
Liquid HI~Ga
3 kN/m b (70°/273) 2.4 kN/m b
0.34.4 kN/m b
7.5-10.0
[20]
6.0-8.0
(130°/2711) a R.T. = room temperature, bCrack extension force at 3.5/~m/s.
fracture occurs resulting in the loss of ductility and brittleness of the polycrystal. In essence, the vast volume of work in the literature has shown that for materials prone to brittle intergranular fracture, be it an intrinsic effect or a result of impurity segregation and/or environment effect, high-energy random boundaries are preferential sites for crack nucleation and propagation. They can be considered as "latent cracks" similar to "pre-existing cracks" in brittle materials suggested by Griffith [27]. The size of such latent cracks must be almost the grain size. Therefore, an increase in the frequency of random boundaries may correspond to an increase in the density of latent cracks in the polycrystal. Reversely, an increase in the frequency of fractureresistant low-energy boundaries would help interrupt crack propagation along weak, random boundaries, thereby rendering the polycrystal strong and ductile. 3. GRAIN BOUNDARY CHARACTER DISTRIBUTION CONTROLLED TOUGHNESS--A THREE DIMENSIONAL MODEL 3.1. General consideration Figure 1 shows a 3D equiaxed polycrystal made up of tetrakaidecahedron-shaped grains with the loading
axis lying along one of the crystal grain axes. It shows that there are a total of seven boundary facets per grain, out of which one lies transverse (t), two longitudinal (l) and the other four inclined (i) to the stress axis. Being parallel to the stress axis, the longitudinal boundary facets will not experience large normal stresses and hence are less liable to intergranular fracture irrespective of their boundary character. In the toughness calculation to follow, it is assumed that they do not participate in the fracture process of the polycrystal. Consider the fracturing process of the polycrystal. For simplicity, we shall assume that the overall fracture plane lies approximately normal to the stress axis and that the maximum peak-to-valley height of the fracture surface is equal to or less than one-grain height, or H (hereafter referred to as Assumption I). To evaluate the toughness of a polycrystal, we need to know the fracture path taken by the polycrystal. Two common types of fracture path can be identified. The first is the "zig-zag path" formed by interlinkage of intergranular crack facets at different levels in the material by the transgranular crack facets, which themselves lie at different angles to the stress axis [Fig. l(b)]. This is the most common type of fracture path observed when the fraction of intergranular
2510
LIM and WATANABE: BRITTLE-DUCTILE TRANSITION IN POLYCRYSTALS
"LL. L,J .J
B ~
~ g
A ~
~A
N VIEW D (=3 d) SECTION
ISOMETRIC VIEW /
c~=H/D =H/3d
HI
-J-- " " A _SECTION '- - --',. 22-%
.# - -", ,
(a) SECTIONAL VIEW
x
/
\
___t
/
x___
AMAGE
6E ZONE
Cb)
(c)
INTERGRANULAR CRACK FACET TRANSGRANULAR CRACK FACET Fig. l. (a) Geometry of a 3D polycrystal. (b) shows the sectional view of a zig-zag fracture path and (c) shows that of a planar fracture path. d is the length of edge of a 3D equiaxed grain. facets on the fracture surface is comparatively high. When the fracture surface is largely transgranular, another type of fracture path may become more prevailing. In this case, the fracture surface is rela-
tively smooth with a waviness much less than that of the zig-zag path [Fig. l(c)]. This type of fracture path is hereafter referred to as the "planar path". Whether the zig-zag or the planar path will be favored will, to
2511
LIM and WATANABE: BRITTLE-DUCTILE TRANSITION IN POLYCRYSTALS a great extent, depend on which of these two paths possesses a lower resistance to fracture. Let us first calculate the toughness associated with the zig-zag path. Assuming that on loading, fracture takes place along a zone of extensive damage within which all random boundaries fracture intergranularly. Invoking Assumption I, the thickness of such a damage zone would be equal to about one-grain height, or H [Fig. l(b)]. It can be seen that within this damage zone, the "elemental fracture path link" consists of six segments: two normal (or n) type segments in which the boundaries lie transverse to the stress axis, and four shear (or s) type segments in which the boundaries lie at an angle to the stress axis [shaded in Fig. l(b)]. The two n-type segments can be further divided into n~ and n2 types. The former contains only one while the latter two transverse boundaries. Since the nrtype link segment would fracture intergranularly should the transverse boundary in it be a random boundary, its probability of fracturing intergranularly is thus given by 1 - f ~ , f being the fraction of fracture-resistant low-energy boundaries among all the transverse boundaries in the material. For the n2-type segment, it will fracture intergranularly should either one or both of the two transverse boundary facets comprising the segment be a random one. Its probability of fracturing intergranularly is thus 1 _ f 2 . Similarly, for each of the four s-type segments, the probability that it will fracture intergranularly is 1 - f ~ , f being the fraction of low-energy boundaries among all the inclined boundaries in the material. If (#T and f#i are, respectively, the toughness associated with a transgranular and intergranular crack and A t and A i respectively the projected areas of the transverse and inclined boundary facets onto a plane normal to the loading axis, it can be shown that the toughness associated with the zig-zag path is given by
~PL= {(1 --ft)At~l "3LUtAt~T-[-A3~T}/(At + A3)
+(1 --f~)At(¢l +fZFn2At~T + Ns[(l
(1)
where N~ is the number of s-type segments per "elemental fracture path link" of the polycrystal system under study. _¥s = 4 for a 3D polycrystal. 01 is the angle between the inclined boundary facet and the stress axis (Fig. 1). Ai/sin 01 is thus the physical area of the inclined boundary facet. Sin 01 can be expressed in terms of the grain aspect ratio, a, through equation (AI3) given in the Appendix. Fn,, tKowning .~, the fracture toughness value, Klc, can be estimated through the following relationship: Klc = ~/adE, where the Young's modulus E is relatively insensitive to the overall fraction of low-energy boundaries in the material.
(2)
where
A 3 = A t + N s A i . For a 3D polycrystal, 1.5. Using the value ofAi/A t given above and equation (A13) given in the Appendix, it can be shown that for a 3D polycrystal, the toughness is given by
Ai/At =
C~zz
~T
1
8{I2--f --f2+6(1--fi)x/l+]~ZJ~TT
+f,(V., + f~r. 2) + 6f~ F: t (3) for fracture occurring along the zig-zag path, and
~PL ~T - 81{ (1 - - f ) ~ T(~I +
(¢zz = {(1 -- ft)Atff~+ f t F , , At(~y
-f'~)(AUsin 01)qJ~ + fZ F~AiqJT]}/(2At+ N~A~)
Fn: and Fs in equation (1) are the correction factors which take into account of the fact when nl, n2 or s-type links fracture transgranularly, the fracture path needs not necessarily lie normal to the loading axis. Instead, they may take several possible paths making different angles with the stress axis. The actual path taken will be determined by the cracking behavior of the adjacent boundary facets. The forms of Fn,, Fn2 and Fs can be determined from the knowledge of the geometry of the polycrystal under study and of the probability of occurrence of the various transgranular crack paths. The approximate forms of Fn,, Fn2 and F~ are given in the Appendix. For the "planar path", the damage zone is limited to the plane containing the transverse boundary in the n-type segment. The elemental fracture path link in this case can be thought as consisting of two segments: one nl-type segment and one ns-type segment, as shown in Fig. l(c). The nl-type segment may fracture intergranularly with a probability given by 1 - f . The n3-type segment however, will always fracture transgranularly on a plane containing the transverse boundary in the nrtype segment. The toughness associated with such a fracture path is given by
7 +f
}
(4)
for fracture occurring along the planar path. In a real polycrystal, fracture will take place along the path of least resistance. The toughness of the polycrystal, (~(~, f ) is thus given by ~(~, f )
~T
Min. f~ZZ, ~PL~ ~
~TJ
(5)
where f is the overall fraction of fracture-resistant, low-energy grain boundaries in the polycrystal. Note that f, f , f and f (where f,, f and f are respectively the fractions of low-energy boundaries among all the transverse, inclined and longitudinal boundary facets in the polycrystal) need not be identical. Note also that the toughness (and hence the fracture toughnesst of the polycrystal is independent of the grain size. However, grain size dependence of the toughness may still arise should the GBCD of the material be grain
2512
LIM and WATANABE: BRITTLE-DUCTILE TRANSITION IN POLYCRYSTALS
size dependent, as reported recently by Grabski [28] and Watanabe [10, 11].
1.0
3.2• Random distribution
For a polycrystal with a random GBCD, the fractions of low-energy boundaries must be identical on all types of boundary facets. For a 3D polycrystal, this implies that f = f t = f i =f~.
(6)
With the above relationship, the toughness of a 3D polycrystal having a random GBCD can be calculated using equations (3) to (5) for different values of ~X/C~. The results are shown in Fig. 2 as a function of the overall fraction of low-energy boundaries in the material. For easy reference, we have plotted in Fig. 3 the toughness enhancement over that of a reference 3D polycrystal with f = 0.15, i.e. ogREF = ~R(~te, 0.15), where the subscript R stands for "Random" distribution and cte is the aspect ratio of an equiaxed grain, tee=0.94 for a 3D equiaxed polycrystal, f = 0.15 is the fraction of low-energy boundaries which a polycrystal would possess if all the grains in the polycrystal are randomly oriented.~" Figures 2 and 3 illustrate that for a 3D polycrystal having an equiaxed grain structure, the zig-zag path is always favoured for ~TI~REF> 2 . It is evident that the toughness of a 3D polycrystal increases monotonically with increasing f. Furthermore, the higher the value of f~r/~REF, the more effective is the toughness enhancement per increment of the fraction of low-energy boundaries in the material. Similar results are also obtained for the 2D case in the previous work [12]. 3.3. Non-random distribution
When the GBCD in a polycrystal is no longer random, different types of boundary facets in the material will possess different fractions of low-energy boundaries. However, geometry requires that, for a 3D polycrystai ft + 2f~ + 4fi = 7f
0.8
(7)
Using the above relationship in place of equation (6) and assuming that f =f~, we have calculated the toughness of a 3D polycrystal with different fractions, ft, of transverse low-energy boundaries. The results for c~r/fgREF = 5 are shown in Fig. 4. The bold curve in this figure is the toughness values for a 3D polycrystal with a random GBCD. It can be seen from Fig. 4 that an increase of the fraction of the transverse low-energy boundaries at low f level improves the toughness of the 3D polycrystal somewhat. On the other hand, at highfvalues, increasing the fraction of low-energy boundaries on both the inclined and longitudinal boundary facets becomes beneficial instead. It is also interesting to
0.4
-/y/ 0.2 ~
y
0.0
m . m PLANAR PATH ,
0.0
Zlr-ZA6 PATH
- -
J ,
i
O.2
,
,
~
0.4
,
,
0. G
,
0.8
1.0
f Fig. 2. Effect of overall fraction of low-energy boundaries, f, and ~T/ffl on the toughness of a 3D polycrystal having a random GBCD. note that at large f value and with ft < 0.6, the toughness of the polycrystal increases rapidly with increasing f, and a transition from zig-zag path to planar path occurs at f = 0.7. In other words, a brittle-ductile transition occurs with the fracture mode changing from predominantly intergranular to predominantly transgranular. From this point on, the toughness of the polycrystal stays high and is relatively insensitive to the overall fraction of lowenergy boundaries in the material. This is in contrast to the 2D case in which increasing the fraction of low-energy boundaries on the transverse boundary facets is always found to be beneficial for ft > 0.5 [12]. Another main difference between the 2D and 3D cases is that even though remarkable toughness enhancement can be achieved for the 2D case through processing which results
a 3D-RANDOM EQUIAXED Z,G-ZAGPAtH
"-~._¢.~/
I 0.2
(13
0.t,
0.5
0.6
0.7
0.S
0.9
tO
-F tStrictly speaking,f = 0.136 should one take the upper limit of Z' value to be 29 for coincidence boundaries [29, 30].
Fig. 3. Same as Fig. 2 but normalised with respect to ffREF where fg~F = c'dR(ct,, 0.15).
LIM and WATANABE: BRITTLE-DUCTILE TRANSITION IN POLYCRYSTALS ,
I
i
,
,
,
i
to
"~/'~R EF--
g
"
0
~ ~
i
l, ~~"
~j/~.2..~-~¢
/ / / X " ; t RANDOM
i
~1
5
3D NON R A N D O M
i
I
.
i
.
i
...._.._~
g
o, 0.3
i 0.4
i 0,5
i 0.6
!7: 07
/
4 ~
-
-Lo_ Ao PAT: 0.2
III/I >_.--
3
~
I
,
-
3
•
2513
RE~ 5 . 0
-
(18
1.0
PLANAR PATH
0.8
i 0.9
1.0
f
(12
0.3
a4
0.5
(16
03
0.9
f
Fig. 4. Effect of non-random GBCD.
Fig. 5. Effect of grain shape and boundary inclination.
in a high proportion of low-energy boundaries on the transverse boundary facets, the magnitude of toughness enhancement due to such is much less pronounced for the 3D case. These discrepancies can be attributed to the fact that Ai < At for the 2D case but it is the reverse for the 3D case. (Ai/A t = 0.5 and 1.5 for the 2D and 3D case respectively.) Thus, even though maximizing transgranular fracture on the transverse boundary facets produces a larger transgranular area for the 2D case, the same does not hold for the 3D case. This also explains why promoting a higher proportion of transgranular fracture on the inclined boundary facets would give rise to significant toughness improvement for the 3D case but not for the 2D case.
effected and the polycrystal would possess a high toughness value. It is evident that elongated grain structures are a very effective means of imparting high toughness to a polycrystal even at lowerfvalues, e.g. a predominantly transgranular fracture is predicted with f , ~ 0.4 for ct > 3. On the other hand for a polycrystal having an equiaxed grain structure or ~t = 1, the toughness increases only relatively slowly with increasing overall fraction of low-energy boundaries, f. Similar results have also been found for 2D polycrystals [12]. A comparison of the results for both the 2D and 3D cases has shown that the effect of grain shape and boundary inclination is more pronounced for the 3D case. This again can be attributed to the larger projected area of the inclined boundary facets in the 3D case, as discussed earlier in relation to the effect of non-random GBCD on the toughness of a 3D polycrystal.
4. EFFECT OF GRAIN SHAPE AND BOUNDARY INCLINATION The results of the previous section has shown that during fracture of an equiaxed 3D polycrystal having a random GBCD, the zig-zag fracture path often prevails. This suggests that if we could process the material such that the grain structure is elongated along the stress axis. a longer span of transgranular crack facets will then be required to link up the intergranular cracks at different levels in the material. This, in turn, would lead to further improvement in the toughness of the polycrystal, as suggested by Watanabe [31, 32]. Figure 5 shows the effect of grain shape and boundary inclination, expressed in terms of the grain aspect ratio ~, on the toughness of a 3D polycrystal having a random GBCD for c~T/ffREF = 5. For a given f value, the toughness of the polycrystal is seen to increase sharply with increasing grain aspect ratio. Most interestingly, the result also shows that if a polycrystal can be processed such that it not only has a reasonably high overall fraction of low-energy boundaries but also an elongated grain structure aligned with the stress axis, a planar path can be
5. DISCUSSION
5.1. Toughening by grain boundary design As described in the Introduction, many advanced engineering materials and structural alloys are susceptible to brittle intergranular fracture either due to the intrinsic properties of the materials or the result of impurity segregation and/or environmental effects. Recently, Watanabe [9-11, 31, 32] has suggested that the brittleness of polycrystalline materials associated with intergranular fracture can be controlled by designing grain boundaries in different ways, for instance, by manipulating the GBCD, boundary inclination, boundary chemistry and so on. In what follows, we shall discuss the various aspects of Grain Boundary Design for strong and tough materials based on the results obtained in the present work. 5. I. 1. Engineering grain boundary character distribution (GBCD). Many researchers have investigated the GBCD in polycrystais of different materials pro-
2514
LIM and WATANABE: BRITTLE-DUCTILE TRANSITION IN POLYCRYSTALS
duced by different techniques (see Refs [10] and [11] type relationship for grain size dependence of fracfor recent reviews on the subject). In general, it has ture properties from GBCD viewpoint. A similar been found that there exists a close relationship view has recently been expressed by Wyrzykowski between GBCD and the processing method for and Grabski [34] for grain size dependence of yield polycrystalline materials. stress. For polycrystals of metallic materials produced by 5.1.3. Grain boundary inclination. It is evident from ordinary thermo-mechanical processing, it has been the present work that engineering the grain shape and reported by a number of workers that the frequency boundary inclination is certainly a very effective of low-energy boundaries such as low-angle bound- way of toughness enhancement for polycrystalline aries and low-Z coincidence boundaries decreases materials. Figure 5 shows that to significantly inwith increasing annealing temperature and resultant crease the toughness of a polycrystal, the grains must grain size [10, 28, 33-35]. Grain size dependence of be orientated parallel to the stress axis and have an GBCD has been discussed in connection with the aspect ratio of 3 or higher. In fact, a similar effect has change in grain boundary energy during recrystalliz- been utilized in controlling intergranular fracture ation [36]. In f.c.c, polycrystals of metals and alloys, in polycrystalline materials through unidirectional the occurrence of a much higher frequency of S 3 and solidification [42] and directional recrystallization related coincidence boundaries has been reported [43], even though no theory has so far been proposed [25, 37, 38]. This has been attributed to the result of to explain the phenomenon. multiple twinning occurring during recrystallisation Strictly speaking, the control of boundary inclianneal of these materials. The presence of texture nation should be made in connection with the control seems to increase the frequencies of low-angle and of the frequencies of fracture-resistant low-energy low-Z coincidence boundaries [39, 40]. An inverse boundaries on different types of boundary facets, cubic root Z" dependence of the frequencies of coinci- to bring about the best toughening effect. Even dence boundaries has been observed for Fe~.5 mass though no such attempt is made in the present work, % Si ribbons and other b.c.c, polycrystals [40]. a quick reference to Figs 4 and 5 suggests that more Moreover, the Z values of those coincidence bound- pronounced beneficial effects are to be expected, aries which occurred in higher frequencies were found when both the effects of GBCD and of boundary to be related to the type of texture, that is, (100) or inclination are taken into account together. (110) texture, in polycrystalline ribbons of the 5.2. Brittle-ductile fracture transition induced by iron-silicon alloy [41]. 5.1.2. GBCD and fracture toughness of polycrystals. GBCD The results of the present work demonstrate that Perhaps, one of the most interesting findings of the toughness of a 3D polycrystal can be greatly the present work is the prediction of the change of improved by engineering the GBCD of the poly- fracture mode from predominantly intergranular crystal. For instance, Fig. 3 shows that the toughness (zig-zag path) with low toughness to predominantly of a 3D polycrystal increases monotonically with transgranular (planar path) with high toughness, increasing overall fraction of fracture-resistant low- when the overall fraction of low-energy boundaries is energy boundaries in the material. A similar effect of increased above a certain level. In other words, for GBCD on the toughness has also been previously materials prone to brittle intergranular fracture either reported for 2D polycrystals [12]. intrinsically or as a result of impurity segregation Some experimental support for the above has and/or environment effect, a "brittle-ductile fracture recently been obtained by Watanabe et al. [40]. transition" can be induced through suitably enginThey reported that the toughness of a normally eering the GBCD and the boundary inclination in the brittle high-silicon iron alloy can be drastically polycrystal. improved by rapid solidification followed by a high In the case of random GBCD (Fig. 3), even though temperature anneal, which also resulted in a high the toughness of the polycrystal was found to increase frequency of low-energy boundaries (45% of all monotonically with increasing overall fraction, f, of low-energy boundaries in the material, no abrupt boundaries). The present results and Ref. [12] also showed that brittle~luctile transition can be concluded. Figure 4, the effect of GBCD on toughness can be made more however, shows that when the material is so prosignificant by engineering the polycrystal such that a cessed that the fraction of low-energy boundaries on higher percentage of those boundary facets possess- the transverse boundary facets, ft, is less than 0.6, the ing the largest projected area normal to the stress axis brittle--ductile fracture transition may occur. For are of the low-energy type. This, for a 3D polycrystal, instance, for a material system with f¢T/f~r = 5 and means those boundary facets which lie inclined to the ft ~- 0, the transition occurs at f = 0.7. Another very stress axis (Fig. 4). effective way of inducing the brittle--ductile fracture Equations (4) and (5) further suggest a possi- transition is to make use of the combined effect of bility that the grain size dependence of toughness GBCD and boundary inclination (Fig. 5). In this of a polycrystal may arrive through that of GBCD. case, the critical overall fraction of low-energy It is thus of interest to re-examine the Hall-Petch boundaries at which the brittle-ductile transition
LIM and WATANABE: BRITTLE-DUCTILE TRANSITION IN POLYCRYSTALS occurs can be lowered to 0.5 or less. In general, with increasing (~r/ffREF, corresponding to a material with increasing susceptibility to brittle intergranular fracture (i.e. lower ~i), the critical overall fraction of low-energy boundaries required to bring about such a brittle-ductile transition also increases accordingly. From a comparison between the results for random and for non-random configurations of grain boundaries for a polycrystal with an equiaxed grain structure and ~T/ffREF= 5, we can see that the nonrandom configuration causes the brittle
2515
fracture can be induced by suitably engineering the GBCD and boundary inclination in the polycrystal. Certain non-random configurations of grain boundaries and an elongated grain structure aligned with the loading axis bring about such a transition at a smaller value of overall fraction of low-energy boundaries than for random boundary configurations. Acknowledgements--The first author (L.C.L.) wishes to acknowledge the financial support received from the National University of Singapore, through the grant No. RP 101/85. The second author (T.W.) gratefully acknowledges The Nippon Sheet Glass Foundation for a Grant-inAid for research in Materials Science. REFERENCES
1. C. L. Briant and S. K. Banerji (editor), Embrittlement of Engineering Alloys, Treatises on Mater. Sci. Tech., Vol. 25 (1983). 2. M. H. Kamdar (editor), Embrittlement by Liquid and Solid Metals. Mater. Soc. A.I.M.E. (1984). 3. B. H. Kear et al. (editors), Ordered Alloys--Physical Metallurgy and Structural Applications. Claitor's, Baton Rouge, La (1970). 4. C. C. Koch, C. T. Liu and N. S. Stoloff (editor), High-Temperature Ordered Intermetallic Alloys, MRS Symp. Proc., Vol. 39 (1984). 5. N. S. Stoloff, C. C. Koch, C. T. Liu and O. Izumi, High-Temperature Ordered Intermetallic Alloys--ll, MRS Syrup. Proc., Vol. 81 (1986). 6. R. T. Holt and W. Wallace, Int. Metall. Rev. 21, 1 (1976). 7. M. P. Seah, Phil. Trans. R. Soc. A295, 265 (1980). 8. C. L. Briant, Metallurgical Aspects of Environmental Failures, p. 49. Elsevier, Amsterdam (1985). 9. T. Watanabe, Res Mechanica 11, 47 (1984). 10. T. Watanabe, in Interracial Structure, Properties and Design (edited by M. H. Yoo, W. A. T. Clark and C. L. Briant), MRS Symp. Proc., Vol. 122, 443 (1988). 11. T. Watanabe, Materials Forum, Australia's Bicentenary Anniversary Volume, Vol. 11, 284 (1988). 12. L. C. Lira and T. Watanabe, Scripta metall. 23, 489 (1989). 13. A. Kobylanski, thesis, Univ. Paris VI (1972), with C. Goux, C.r. Acad. Sci. Paris 272, 1937 (1971). 14. J. B. Brosse, R. Fillit and M. Biscondi, Scripta metall. 15, 619 (1981). 15. H. Kurishita, A. Oishi, H. Kubo and H. Yoshinaga, J. Japan Inst. Metals 47, 546 (1983). 16. H. Kurishita, S. Kuba, H. Kubo and H. Yoshinaga, J. Japan Inst. Metals 47, 539 (1983). 17. K. Sato, H. Miyazaki, Y. Ikuhara, H. Kurishita and H. Yoshinaga, J. Japan Inst. Metals 53, 536 (1989). 18. T. Watanabe, S. Shima and S. Karashima, in Ref. [2], p. 161 (1984). 19. A.'Otsuki and M. Mizuno, Trans. Japan Inst. Metals Suppl. 27, 789 (1986). 20. J. A. Kargol and D. L. Albright, Metall. Trans. 8A, 27 (1977). 21. T. Watanabe, Metall. Trans. 14A, 531 (1983). 22. S. Lartigue and L. Priester, Trans. Japan Inst. Metals, Suppl. 27, 205 (1986). 23. S. Hanada, T. Ogura, S. Watanabe, O. Izumi and T. Masumoto, Acta metall. 34, 13 (1986). 24. L. C. Lim and R. Raj, Acta metall. 32, 1183 (1984). 25. J. Don and S. Majumdar, Acta metall. 34, 961 (1986). 26. T. Watanabe, M. Tanaka and S. Karashima, in Ref. [2], p. 183 (1984). 27. A. A. Griffith, Phil. Trans. R. Soc. A221, 163 (1920). 28. M. W. Grabski, J. Physique, Suppl. C4, 46, 567 (1985).
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LIM and WATANABE: BRITTLE-DUCTILE TRANSITION IN POLYCRYSTALS
29. D. H. Warrington and M. Boon, Acta metall. 23, 599 (1975). 30. M. Mori, Doctoral thesis, Tokyo Univ. (1975). 31. T. Watanabe, in Creep and Fracture of Engineering Materials and Structures (edited by B. Wilshire and R. W. Evans), p. 155. Inst. of Metals, London (1987). 32. T. Watanabe, in Chemistry and Physics of Fracture (edited by R. M. Latanision and R. H. Jones), NATO series 130, p. 492. Martinus Nijhoff, The Hague (1988). 33. T. Watanabe, J. Physique, Suppl. C4, 46, 555 (1985). 34. J. W. Wyrzykowski and M. W. Grabski, Phil. Mag. A53, 505 (1986). 35. T. Watanabe, Y. Kawamata and S. Karashima, Trans. Japan Inst. Metals Suppl. 27, 601 (1986). 36. T. Watanabe, Scripta metall. 21, 427 (1987). 37. L. C. Lim and R. Raj, Acta metall. 32, 1177 (1984). 38. P. Neumann and A. Tonnessen, in Fatigue 87(edited by R. O. Ritchie and E. A. Starke Jr), p. 3. EMAS, Cradley Heath, Warley, West Midland (1987). 39. T. Watanabe, N. Yoshikawa and S. Karashima, Proc. Int. Conf. on Texture of Materials (ICOTOM-6), p. 609. Japan Iron Steel Inst. (1981). 40. T. Watanabe, H. Fujii, H. Oikawa and K. I. Arai, Acta metall. 37, 941 (1989). 41. T. Watanabe, K. I. Arai, K. Yoshimi and H. Oikawa, Phil. Mag. Lett. 59, 47 (1989). 42. N. S. Stoloff, in Alloy and Microstructural Design (edited by J. K. Tien and G. S. Ansell), p. 65. Academic Press, New York (1976). 43. E. Hornbogen, in Fundamental Aspects of Structural Alloy Design (edited by R. I. Jaffe and B. A. Wilcox), p. 389. Plenum Press, New York (1977).
1
1
1
l)] +(ah+be)
-1
+(de+ch)
1), + 2[aa,h + (bb2+ 2b~)el [13/18 5/18 _ l l x [_sin 01 + sin 02 ] 4' 2 [13/18
5/18
where the coefficients are related to the probabilities of occurrence of the different types of transgranular crack facets and 0's the angles made by them with the stress axis. They have the explicit forms below. a 1= (1 --ft) +ft( 1 --fi2)2
(A4)
a =f~a 1 b, =ft(1 - f t ) +f~(1 -fi)2[1 - (I - f i ) 21 b2 = (1 - f t ) 2 +f2(1 - f i ) 4
(A5) (A6)
b = b2 + 2b l
(AS)
c = 2ftfi4(l _f2)2
(A9)
d = 4f2f~(1 _f~)4 + 4ft2fi(l _fi)2 x [f~ + 3f~(1 - Z ) ]
APPENDIX
Fn,, Fn~ and F~for 3D Polycrystal
e =ftf6(l --fi2)
(All)
h =f2f4(l --fi) 2 + 2f~f~(1 --f~)
(A12)
1
sin 0, -- x/1 + 9 0t2 Fnl '--1 +~{cd[(sinl 0~ - l ) + ( s i T 0 6 - 1 ) ]
I
sin 02 _ x/1 + 9 ~ 2 1
sin03
-- x/1 +¼*t 2
1 +
(A1)
( -01t l
sin 04 - /x/1
+
Ct 2
l sin 05 -- ~/1 + ~ e2
Fn:~-1 + ~ 2 t f e d I ( s i ~ s - 1 ) + ( s i T 0 6 - 1 ) 1
I
+ e\sin ( ± -031 ) J (A2)
(A13) (AI4) (A15) (A16) (AI7)
- x/1 + ~ ~2
(A18)
1 = lv/i~-2~ ~ 2 sin 07
(Al9)
1 = ~/1 sl ;~ sin 08 e is the aspect ratio of the grains (Fig. 1).
(A20)
sin 06
+ 4hL L \sin 02
(A10)
and
For a 3D polycrystal, Fn, and Fn2 and Fs are given by
d
(A7)