Study on the relation between structural parameters and fracture strength of WC-Co cemented carbides

Materials Chemistry and Physics 62 (2000) 35±43

Study on the relation between structural parameters and fracture strength of WC-Co cemented carbides a

Binghai Liua,c,*, Yue Zhangb, Shixi Ouyangc

Institute of Technical Chemistry and Physics, East China University of Science and Technology, Shanghai 200237, China b Materials Physics Department of Beijing University of Science and Technology, Beijing 100083, China c State Key Laboratory For Synthesis and Processing of Advanced Materials of China, Wuhan University of Technology, Wuhan 4300 70, China Received 20 April 1999; accepted 25 June 1999

Abstract In this article, a directly proportional relation between average free path (M) and ductile deformation energy ( ) was proposed, and on the basis of it, a quantitative analysis was conducted for studying the effects of the structural parameters on fracture strength of WC-Co cemented carbides. The results show that, for different WC-Co cemented carbides with different cobalt contents, there exist different critical WC grain size Rc and critical free path of binder Mc. Rc and Mc act as the criteria that determine the growth behavior of crackles. When average free path of cobalt binder M < Mc, or WC grain size R < Rc, crackles will expand mainly across cobalt binder, which will result in intergranular fracture; when M > Mc or R > Rc, transgranular fracture will happen; when M ˆ Mc or R ˆ Rc, concurrence of intergranular fracture and transgranular fracture will take place. Rc and Mc will decrease with increasing of cobalt content, followed by increasing of fracture strength. The dimension of crackles in the circular ®ssure-breeding district is also a determinative factor to affect fracture strength of cemented carbides. # 2000 Elsevier Science S.A. All rights reserved. Keywords: WC-Co cemented carbides; Average free path of cobalt binder; Critical grain size critical free path of binder; Residual thermal stress

1. Introduction For WC-Co cemented carbides, the relation between fracture strength and microstructure is always an attracting question, and has absorbed much attention of researchers around the world, thus many theories have been developed, p e M e p theory [2], such as Gurland theory [1], Limmushuta theory [3] et al. These theories play important and conductive roles in elucidating the fracture behavior and mechanism of WC-Co cemented carbides. However, there also exist some limitations for these proposed theories. For instance, Gurland theory fails to provide reasonable explanation to interpret why fracture strength will decrease with decreasing of cobalt content when WC grain size is constant, furthermore, the effects of microdefects on fracture strength were also not taken into account by this theory. Limmushuta theory demonstrates the determinative effects of defects on fracture strength of WC-Co cemented carbides, but it didn't give an extensive account of the effects of structural parameters on fracture strength. *

Corresponding author.

So, a comprehensive consideration of the effects of all the structure parameters should be conducted when analyzing fracture strength of WC-Co cemented carbides. In this article, a quantitative analysis was conducted for comprehensively studying the effects of structural parameters on fracture strength of WC-Co cemented carbides. In the light of fracture mechanism and thermal elastic stress theory, a directly proportional relation between the energy of ductile deformation and average free path of cobalt binder was suggested, and the critical average free path of cobalt binder (Mc) and critical WC grain size (Rc) for determining fracture behavior of WC-Co cemented carbides was deduced. 2. Fracture strength of WC-Co cemented carbide with circular field of thermal elastic stress 2.1. Fracture strength of particle-reinforced materials As regards non-cubic polycrystalline solids or particlereinforced materials, it was believed that the failure of such materials is associated with the thermal expansion aniso-

0254-0584/00/$ ± see front matter # 2000 Elsevier Science S.A. All rights reserved. PII: S 0 2 5 4 - 0 5 8 4 ( 9 9 ) 0 0 1 5 2 - 2

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B. Liu et al. / Materials Chemistry and Physics 62 (2000) 35±43

tropy or the incompatibility of thermal expansion between the particulate phase and the matrix, accompanying spontaneous microcracking of annular ¯aws [4,5]. By taking into consideration the effects of external stress, B. Comwall analyzed multiple-particle reinforced materials and gave the fracture strength of such materials as the following [5]: "  # 1=2 1 E t ÿBP (1) C1 C2 f ˆ c c D…1 ‡ S=R†…1ÿ2 † " c ˆ 1ÿ ‡

#1=2 "

1

…1 ‡ S=R†  3

4…1 ‡ S=R†4 "

t ˆ 1ÿ 1ÿ "  1ÿ

1‡

2

1 4…1 ‡ S=R†2

1 1 ‡ ‰…1 ‡ S=R†2 ÿ1Š 3

1

#1=2

…1 ‡ S=R†2 #1=2 1

‡

#

1 2…1 ‡ S=R†3=2

…1 ‡ S=R†2

2EDdDT 3…1ÿV†  1=2 …1 ‡ S=R† C1 ˆ 2f…1 ‡ S=R† ‡ 2…1ÿV†=3Vg  ÿ1=2 …1 ‡ S=R† C2 ˆ tan 2f…1 ‡ S=R† ‡ 2…1ÿV†=3Vg Pˆ

where f is fracture strength, V volume fraction of dispersed phase, D diameter of grain, E elastic modulus, R radius of grain, B constant,  Poisson's ratio,
T differential temperature, S length of crackles, ductile deformation energy, P residual thermal stress,
 difference of thermal expansion between matrix and dispersed phase. Eq. (1) was developed from a model for single-phased polycrystalline materials, in which it was assumed that, owing to the thermal expansion anisotropy and the grain orientations, an annular ¯aw was formed by linking of small radial cracks on the neighboring grains around the central grain A (Fig. 1a, b) [4]. Clearly, spontaneous microcracking of annular ¯aws in such single-phase materials should result in both transgranular fracture and intergranular fracture. Although, B. Cornwall didn't point out the difference of these two fracture behaviors in his analysis, Eq. (1) de®nitely implies the effects of both transgranular fracture and intergranular fracture on the fracture strength of particlereinforced materials. WC-Co cemented carbides can be considered as particlereinforced materials [6]. The thermal expansion coef®cient of WC phase and cobalt phase are 2.9  106 Kÿ1 and 13.8  106 Kÿ1 respectively, therefore, when WC-Co cemented carbide is cooled down from a certain sintering temperature to room temperature, tremendous residual ther-

Fig. 1. Polycrystalline aggregate containing an annular Flaw. (a) Hexagonal grain under compression; (b) Opening of the crack due to thermal expansion [4].

mal stress will be produced for the incompatibility of thermal expansion of these two phases. Let us assume that spherical WC grains are dispersed in cobalt binder uniformly and separately. For the thermal expansion coef®cient of cobalt phase is larger than that of WC phase, the cobalt phase at the WC/Co interface will undergo stretching stress as shown in Fig. 2. In this case, a certain quantity of inter-linking or separated micro-crackles will be produced, as a result, a circular-shaped ®ssurebreeding district will form at the WC/Co interface. In addition, WC/Co interface is always the region where all kinds of defects concentrate, such as segregated heterogeneity, pores and crackles et al. As viewed from this, the WC/ Co interface can also be considered as the circular ®ssurebreeding district. It has been well recognized that, for polycrystalline materials, expanding behavior of crackles greatly depends on the structural parameters. Different structural parameters will result in different internal stress distribution, and as a result, the materials will show different fracture behavior and different fracture strength. In this article, the effects of

Fig. 2. Circular fissure-breeding district around WC grain resulted from residual thermal stress.

B. Liu et al. / Materials Chemistry and Physics 62 (2000) 35±43

structural parameters, such as average free path of cobalt binder, WC grain size, cobalt content and dimension of defects, on the fracture strength and fracture behavior of WC-Co carbides will be discussed quantitatively. 2.2. Relation between fracture strength and average free path of cobalt binder of WC-Co cemented carbides Assuming that there exists a crack in the circular ®ssurebreeding district with the length of S as shown in Fig. 2. When an exterior stress is imposed on the material, the expanding behavior of the microcrack can be classi®ed into the following manners: Manner (l): Expanding just along the interface of WC and Co phase; Manner (2): Expanding across cobalt binder; Manner (3): Expanding through WC grains. On the matter of Manner (3), the crack spreads through WC grains, which results in transgranular fracture. In this case, Eq. (1) can be transferred into Eq. (2): "  #  1 Ewc wc 1=2 t ÿBP (2) C1 C2 f ˆ c c DA…1ÿ2 † 



2

1 3 A ÿ1 ‡ 4 1‡ 2 4A 4A 3  1=2  1=2 1 1 1 ‡ 3=2 1ÿ 2 t ˆ 1ÿ 1ÿ 2 A A 2A  1=2 A C1 ˆ 2…A ‡ 2…1ÿV†=3V†  ÿ1=2 A C2 ˆ tan 2…A ‡ 2…1ÿV†=3V†

c ˆ

1ÿ

1 A2

1=2 

1‡



where, A ˆ l ‡ S/R, Ewc is elastic modulus of WC phase, M average free path of Co binder, S average length of crackles, V volume fraction of dispersed phase, wc average surface energy of WC grains. The quantitative relation among WC grain size, length of crackles and average free path of cobalt binder can be shown as [6] Rˆ

3V…Mÿ2S† 4…1ÿV†

(3)

where R refers to average WC grain size, V refers to volume fraction of WC phase, M refers to average free path of binder and S refers to average length of crackles. Thereby, integrating Eq. (2) and Eq. (3) will derive Eq. (4): "  #  1 2Ewc wc …1ÿV† 1=2 t ÿBP (4) C1 C2 f ˆ c c 3A…1ÿ2 †V…Mÿ2S† 4S…1ÿV† f fracture strength of WC-Co where A ˆ 1 ‡ 3V…Mÿ2S† alloys, c, t, P, B, C1, C2, Ewc, gwc,  Ð same as Eq. (2). Eq. (4) implies the quantitative relation between fracture strength (f) and average free path of binder (M) providing

37

that the dimension of crackles (S) and cobalt content i.e. volume fraction of WC phase (V) are constants. In the light of Eq. (4), different fracture strength with different average free path (M) are calculated in the case of transgranular fracture. The calculated results were plotted as the six right branches of the curves in Figs. 4 and 5. As seen by them, when cobalt content and dimension of crackles keeping constant, the fracture strength (f) will decrease with increasing of average free path of binder (M). As far as crackle expanding Manner (1) and Manner (2) are concerned, i.e., crackles expand across cobalt binder or along WC/Co interface, then intergranular fracture of WCCo cemented carbides will take place. In this case, a large ductile deformation will be produced through the areas such as WC/Co interfaces and cobalt binder, as suggested by p e M e p theory [2]. p e M e p pointed out, accompanying crackles spreading, the resulting ductile deformation energy ( ) is directly proportional to cobalt content (f) [2],

ˆ Kf

(5)

where K is constant, f refers to volume fraction of cobalt phase. Eq. (5) was deduced on the prerequisite that WC grain size is constant. According to Eq. (5), it seems that the alloys with the same cobalt content will possess the same ductile deformation energy. However, for the alloys with the same cobalt content, different WC grain size will result in different average free path of cobalt binder, which indicates the different dimension of ductile transforming district at the front of micro crackles, and consequently leads to the variation of the ductile deformation energy. Therefore, Eq. (5) can not properly elucidate the factors which directly in¯uences the ductile deformation energy of the cobalt binder. It's well-recognized that enhancement of fracture strength of WC-Co composite materials can be ascribed to deformation buffering effects of ductile cobalt binder. Therefore, the change of the dimension of ductile transforming district will directly lead to variation of fracture strength of materials. Hilll [7] has performed much detailed research on this phenomenon, and suggested a conception of deformation buffering factor (pcf) to describe it. pcf ˆ

max ˆ ‰1 ‡ ln…1 ‡ R=†Š ys

where, pcf is deformation buffering factor, max, maximum nominal stress, ys yield strength, R radius of ductile deformation district,  radius of curvature of crackles. As seen by the above equation, deformation-buffering factor (pcf) will increase with increasing of dimension of ductile deformation district. As viewed from this, according to WC-Co cemented carbides, any variation of average free path of binder (M) will consequently result in the change of

38

B. Liu et al. / Materials Chemistry and Physics 62 (2000) 35±43

deformation buffering factor (pcf). Meanwhile, the intensity of stress concentration at the tip of crackles will also be changed, then variation of ductile deformation energy will appear. For the reasons mentioned above, we suggest that the determinative factor which directly dominates the ductile deformation energy is not cobalt content, but average free path of binder, and a directly proportional relation between ductile deformation energy ( ) and average free path (M) is then proposed,

ˆ K0M

(6)

where K0 is a constant, which means the unit ductile deformation energy for the unit length of free path of binder. M refers to average free path of binder. Combining Eq. (6) with Eq. (1), we can obtain the equation for calculating the intergranular fracture strength of WC-Co cemented carbides, "

#   1 2Eco K 0 …1ÿV†M 1=2 t ÿBP C1 C2 f ˆ c c 3AV…Mÿ2S†…1ÿ2 †

(7)

where, Eco is elastic modulus of cobalt binder, K0 constant. Aˆ1‡

4S…1ÿV† 3V…Mÿ2S†

The calculated results of Eq. (7) were plotted as the six left branches of curves shown in Figs. 4 and 5. As shown by them, when cobalt content (V) and length of crackles (S) keeping constant, increasing average free path (M) will result in increasing of fracture strength of materials. Moreover, the left and the right branches of the curves, which are correspondent to intergranular fracture and transgranular fracture respectively, are intersected at a certain value of Mc. It indicates that, according to the WC-Co cemented carbides with a certain fraction of cobalt phase, there exists a de®nite critical average free path Mc, which can be considered as a criterion for assessing fracture behavior, namely assessing whether transgranular fracture or intergranular fracture will take place. Furthermore, at the point of Mc, the fracture strength achieves the maximum value. 2.3. Relation between WC grain size and fracture strength of WC-Co cemented carbides In the case of transgranular fracture of WC-Co carbides, the relation between WC grain size and fracture strength of carbides can be implied by Eq. (2) as mentioned above. The calculated results were plotted in Figs. 3 and 6 as shown by the six right branches of the cures. As indicated by them, transgranular fracture strength increases with decreasing of WC grain size. If crackles expanding results in intergranular fracture, the equation of the fracture strength of WC-Co carbides can be

Fig. 3. The relation between WC grain size and the predicted fracture strength of WC-Co cemented carbides (
T ˆ800 K, B ˆ0.1, K0 ˆ 5.5  106, S ˆ 0.01 mm).

derived from Eq. (1), Eqs. (3) and (6), as shown by the following Eq. (8), "  #  1 ECo K 0 …2Rÿ2RV ‡ 3VS† 1=2 t f ˆ ÿBP C1 C2 c c 3RAV…1ÿ2 † (8) where, A ˆ 1 ‡ S/R, ECo is elastic modulus of cobalt binder, K0 the constant same as that in Eq. (6) In the light of Eq. (8), a quantitative relation between intergranular fracture strength and WC grain size was developed, and the calculated results were shown by the left branches of the curves in Figs. 3 and 6. As seen by them, intergranular fracture strength will decrease with decreasing of WC grain size. Moreover, the left and right branches of curves are intersected at a de®nite value of Rc as shown in Figs. 3 and 6, which indicates that there exists a critical WC grain size Rc for WC-Co alloys with a certain cobalt content. Rc can act as a criterion just like Mc. Furthermore, at the point of Rc, fracture strength achieves a maximum value. 3. Results and discussion 3.1. Effects of average free path and WC grain size on the fracture behavior of WC-Co cemented carbides As seen by Figs. 3 and 5, for different WC-Co alloys with different cobalt content, there exist different critical grain size of WC phase (Rc) and critical free path of binder (Mc). When M < Mc or R < Rc, the relation between fracture strength and WC grain size (or average free path of binder) appears positive-going; while M > Mc or R > Rc, the relation shows negative-going. These two different variant trends of fracture strength can be ascribed to the combined effects of

B. Liu et al. / Materials Chemistry and Physics 62 (2000) 35±43

WC grain size and average free path of binder on the expanding behavior of crackles. When M > Mc or R > Rc, transgranular fracture of WC-Co carbides will occur due to expanding of crackles across WC grains, and the fracture strength should be determined by Eq. (2) or Eq. (4) as elucidated by the above discussion. When M < Mc or R < Rc, intergranular fracture will take place since crackles spread cobalt binder, and the fracture strength can be derived from Eq. (7) and Eq. (8). In case of transgranular fracture, when comparing the calculated results of Eq. (4) with that of Eq. (7) or comparing the result of Eq. (2) with that of Eq. (8), we can ®nd that transgranular fracture strength is obviously lower than that of intergranular fracture, which indicates that the deformation of WC grains is more sensitive than that of cobalt binder. The reason can be interpreted as the results of relative large value of ductile deformation energy ( ) in case of M > Mc or R > Rc. In the view of dislocation theory, the larger the WC grain size, the larger the squeezing stress which is produced by the dislocations squeezed at WC/Co interface [8,9], and the more possible the transgranular fracture appears. Therefore, according to the WC-Co cemented carbides with average free path M larger than Mc or WC grain size larger than Rc, transgranular fracture will possess much greater possibility than intergranular fracture, due to the large squeezing stress of dislocation and ductile deformation energy of binder. All above discussion pertaining to the case in which M > Mc or R > Rc can be re¯ected by the right branches of the curves in Figs. 3±6. When M < Mc or R
Fig. 4. The relation between free path of cobalt binder and the predicted fracture strength of Wc-Co cemented carbides (
T ˆ800 K, B ˆ0.1, K0 ˆ 5.5  106, Co% ˆ 12 wt.%).

39

Fig. 5. The relation between free path of cobalt binder and the predicted fracture strength of WC-Co cemented carbides (
T ˆ 800 K, B ˆ 0.1, K0 ˆ 5.5  106, S ˆ 0.01 mm).

For polycrystalline materials, Hall-Petch equation advanced an inversely proportional relation between yield strength  and grain size d1/2  ˆ 0 ‡ Kd ÿ1=2

(9)

Eq. (9) indicates that ®ne-grained structure can endow materials with higher strength, and just for this reason, Hall-Petch relation has become the elementary guiding principle for the development of ultra®ne-grained materials. According to WC-Co cemented carbides, when M < Mc or R < Rc, WC grain size is relatively by ®ne, WC grains will show a relatively high yielding strength. As a result, intergranular fracture strength will be relatively by lower than transgranular fracture strength. Therefore, crackles are

Fig. 6. The relation between WC grain size and the predicted fracture strength of WC-Co cemented carbides (
T ˆ 800 K, B ˆ 0.1, K0 ˆ 5.5  106, Co% ˆ 12 wt.%).

40

B. Liu et al. / Materials Chemistry and Physics 62 (2000) 35±43

inclined to spread across binder phase or along WC/Co interface rather than across WC grains. On this condition, the fracture of WC-Co carbides is characterized as intergranular fracture, which is determined by Eq. (7) or Eq. (8). The calculated results show that fracture strength increases with increasing of WC grain size or average free path of binder, just as indicated by the left branches of the curves in Figs. 3±6 3.2. Effects of cobalt content, average free path of cobalt binder and WC grain size on fracture strength of WCCo cemented carbides As shown by Figs. 3 and 5, when the length of crackles (S) and WC grains size are all constant, fracture strength of WC-Co alloy will increase with increasing of cobalt content. During transgranular fracture, ®rstly, crackles will spread through WC grains, then continue to expand across cobalt binder, and consequently result in the failure of WC-Co alloys. In this case, if WC-Co alloys have same average WC grain size and dimension of crackles, WC grains will exhibit the same yielding strength. Therefore, the higher the cobalt content, the larger the average free path of binder and the larger the ductile deformation energy, which means a larger barricade for crackles to overcome. Therefore, transgranular fracture strength of the materials will increase with increasing cobalt content, This trend is implied by the three right branches of curves in Fig. 3. On the other hand, when average free path (M) and the dimension of crackles (S) are constant, increasing cobalt content will result in decreasing of average WC grain size (R), which indicates increasing of yielding strength of WC grains. Therefore, transgranular fracture strength will consequently increase with increasing cobalt content, just as shown by the three right branches of curves in Fig. 5. In the case of intergranular fracture, increasing cobalt content also results in increasing of intergranular fracture strength, which is implied by the three left branches of curves both in Figs. 3 and 5. When the length of crackles (S) and average WC grain size are constant, a higher cobalt content indicates a larger average free path of binder and a higher ductile deformation energy, which consequently endow the alloys with a higher intergranular fracture strength, as shown by the three left branches of the curves in Fig. 3. On the other hand, when the average free path of cobalt binder and dimension of crackles keeping constant, increasing cobalt content indicates decreasing of WC grain size, which means the increasing of the number of WC grains in unit area on which exterior stress is imposed. It's well recognized that the interface of polycrystalline materials can provide coordinating effects, namely can coordinate the non-uniform deformation between cobalt binder and WC grains. Therefore, decreasing of WC grain size will de®nitely enhance the coordinat-

ing effects of interface on deformation, and weaken the difference of deformation between binder and WC grains. As a result, intergranular fracture strength will increase with increasing of cobalt content, which is implied by the three left branches of curves in Fig. 5. As far as the effect of WC grain size on fracture strength of WC-Co alloys is concerned, Fig. 3 shows two different variant trends of fracture strength accompanying with variation of WC grain size. According to the three right branches of curves in Fig. 3, i.e. in the case of transgranular fracture, the relation between fracture strength and WC grain size is consistent with HallPetch Law, while in the case of intergranular fracture, the three left branches of curves shows a reverse relation against Hall-Petch Law. For the alloy with a de®nite cobalt content, decreasing of WC grain size means decreasing of average free path of cobalt binder and decreasing of ductile deformation energy as implied by Eqs. (3) and (6). Therefore, the intergranular fracture strength will decrease with decreasing of WC grain size i.e. a reverse Hall-Petch Law appears. On the other hand, in the case of transgranular fracture, fracture strength will increase with decreasing of WC grain size for the increasing of yield strength of WC grains, which well accords with Hall-Petch Law. For the reasons mentioned above, simply ®ning WC grain size despite the effects of other structural parameters can not ameliorate the fracture strength of WC-Co alloys. Enhancement of fracture strength should be based on a comprehensive consideration of the effects of all structural parameters. The effects of average free path of cobalt binder on fracture strength of WC-Co alloys are shown in Figs. 4 and 5. In the case of transgranular fracture, when cobalt content keeping constant, decreasing average free path means decreasing of WC grain size, which leads to increasing of transgranular fracture strength, as shown by the right branches of the curves in Figs. 4 and 5. In the case of intergranular fracture, decreasing of average free path indicates reduction of ductile deformation energy of cobalt binder, and thus results in decreasing of intergranular fracture strength of alloys, as implied by the left branches of the curves in Figs. 4 and 5. 3.3. Effect of dimension of crackles in the circular fissure-breeding district on the fracture strength of WC-Co cemented carbides As shown by Figs. 4 and 6, according to WC-Co cemented carbide with a de®nite cobalt content and average free path of binder, whether transgranular or intergranular fracture occurs, fracture strength will remarkably decrease with increasing of dimension of crackles for the tremendous stress concentration, it's well consistent with the view of Limmushuta theory [3] and Grif®ths fracture theory.

B. Liu et al. / Materials Chemistry and Physics 62 (2000) 35±43

3.4. Effects of structural parameters on critical WC grain size Rc and critical free path of binder Mc On the basis of the above discussion, the critical WC grain size and critical free path of binder can be considered as the criteria by which fracture behavior of WC-Co cemented carbides can be assessed. Furthermore, for the WC-Co alloy with a de®nite Co content, when WC grain size is equal to Rc or average free path of binder is equal to Mc, the alloy will possesses the highest fracture strength. Meanwhile, the fracture behavior of the material are correspondent to the concurrence of transgranular fracture and intergranular fracture with 50% occupied by each. As shown by Figs. 3 and 5, Mc and Rc will decrease with increasing of cobalt content. The effects of cobalt content on the value of Rc is more apparent than the effects of cobalt content on Mc. The reason for this can be ascribed to the less variation of M than that of R with change of cobalt content as implied by Eq. (3). 3.5. Comparison between the calculated results and experimental results Fig. 7 illustrates the variation of measured fracture strength with WC grain size. As shown by it, the measured fracture strength against WC grain size plots shape as the right branches in Figs. 3 and 6, in which case only transgranular fracture takes place as mentioned in the previous Section 3.2. Furthermore, the critical WC grain size Rc and critical free path of cobalt binder Mc can't be observed from Fig. 7. Inspection of the curves in Fig. 7 shows that the variant tendencies of measured fracture strength are different from

Fig. 7. The relation between average WC grain size (R) and the measured fracture strength (f) of WC-Co cemented carbides.

41

that shown by the right branches in Figs. 3 and 6. As illustrated by Fig. 7, although the measured fracture strength trends to increase with the increasing of grain size, the increasing tendency becomes weakened and the fracture strength appears to be constant or even lower in some cases, after average size of WC grains decreases to 1.0 mm or so. It seems that there exists a critical condition that lays a barrier for further improving of fracture strength. This is always the case encountered in our experiments. Certainly, the preexisted and later resulted micro defects resulted from the processes of shaping and sintering may lead to great variations of the measured fracture strength. However, with the same raw materials and the same manufacturing processes, the effects of the micro defects should be under comparable conditions, therefore, in case of transgranular fracture, the measured fracture strength should trend to increase constantly as the way shown by the right branches in Figs. 3 and 6. Considering this fact, here it is appropriate to conclude that there really exists a critical WC grain size (Rc) for a real WC-Co cemented carbide, which can alter the fracture behavior of the materials when WC grain size decreases to a certain value. As mentioned above, when average WC grain size R decreases to Rc or lower, the transgranular fracture will transferred to intergranular fracture, and hence the determinative factor of fracture strength will not be the yielding strength of WC grains but the deformation energy of cobalt binder. In this case, as mentioned in Section 3.2, further improving fracture strength will become very dif®cult for the decreasing of average free path of cobalt binder, i.e. the decreasing of the dimension of ductile deformation district. Such kind of fracture behavior transferring is also well suggested by the SEM fractographs shown in Fig. 8a, b. As indicated by them, `A' areas in both fractographs appears apparent intergranular fracture feature, shaped as somewhat dimple-shaped holes, much like the dimples that are always developed when ductile fracture of metals occurs. These dimple-shaped holes are the relicts left by ®ne WC grains when intergranular fracture developed along WC/Co interfaces, while the large WC grains in `B' areas in both fractographs appear clear fan-shaped or river-shaped veins, being typical of transgranular fracture. Furthermore, the size of the transgranular fractured grains are usually larger than 1.0 mm in diameter. While in the case of intergranular fracture, the size of WC grains is generally smaller than 1.0 mm in diameter. For these reasons, a strong justi®cation can be given for predicting the existence of the critical WC grain size Rc, which de®nitely alters the fracture behavior of WC-Co cemented carbides. According to the three curves in Fig. 7, although the detailed values of Rc for the three cemented carbides can not be obtained, a conclusion can be inferred from Fig. 7 is that the critical grain size (Rc) are all less than 1.0 mm, which is also suggested by the fractographs in Fig. 8a, b as mentioned above. Obviously, the values of Rc are substantially different from the predicated values as suggested

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B. Liu et al. / Materials Chemistry and Physics 62 (2000) 35±43

Fig. 8. SEM photographs of the fracture surface of WC-Co cemented carbides. (a) The WC-10 wt.% Co cemented carbide; (b) The WC-12 wt.% Co cemented carbides.

by Figs. 3 and 6. The reason for this can be ascribed to the difference between the assumptions involved in the model and the real conditions of the produced bulk materials. First, the model assumed that, accompanying microcracking, only one kind of fracture (either intergranular or transgranular fracture) can occur. However, as far as a real cemented carbide is concerned, the failure of the material virtually includes both the intergranular fracture and transgranular fracture. In case of mixed fracture, the transgranular fracture is dominant when WC grains size R > Rc, while intergranular fracture becomes dominant when R < Rc. The tendency is well corroborated by Figs. 8a, b. Fig. 8a shows a standard-sized WC-l0 wt.% Co cemented carbide whose average particle size is more than 4 mm. Apparently, the fractograph (Fig. 8a) suggests that the fracture behavior of such material is dominated by transgranular fracture. Whereas, for the ®ne-grained WC12 wt.% cemented carbide shown in Fig. 8b whose average grain size is less than 1 mm, mainly intergranular fracture has taken place throughout the fracture surface. Based on the reasons mentioned above, the mixed fracture behavior altered the variant tendency of measured fracture strength with WC grain size, and consequently made the critical WC grain size undetectable. Second, the non-uniform dispersion of WC grains in cobalt phase, the dimension of micro defects and other aspects involved in real materials are also different from the conditions assumed by the model, which are therefore the other contribution factors to the difference between the measured values and the predicted results. On the other hand, because of the dif®culties in detecting the detailed values of average free path M, it's unavailable to establish the relation between the measured fracture strength and average free path of cobalt binder (M). However, it is the common sense that the average free path of cobalt binder will decrease with the decreasing of average WC grain size when cobalt content is constant. Therefore, the plots of the measured fracture strength (f) against average free path of binder (M) should be analogous to the curves shown in Fig. 7.

As far as the effects of cobalt content on the measured fracture strength is concerned, as shown by Fig. 7, apparently, under otherwise identical conditions, the higher the cobalt content, the higher the measured fracture strength, which is well consistent with the calculated results. 4. Conclusion 1. For WC-Co cemented carbides, the incompatibility of thermal expansion between WC and Co phase results in circular ®ssure-breeding district. The dimension of cracks in this district conducts great effects on the fracture strength of WC-Co alloys. 2. The free path of cobalt binder is the determinative factor that dominates the ductile deformation energy of cobalt binder. The ductile deformation energy of cobalt binder is directly proportional to the free path of binder. 3. For the WC-Co cemented carbides with different cobalt content, there exist different critical WC grain size Rc and critical free path of cobalt binder Mc. Mc and Rc decrease with increasing of cobalt content. When M < Mc (R < Rc), the fracture behavior of WC-Co alloy is featured as intergranular fracture, and fracture strength will increase with increasing of cobalt content and WC grain size. While M > Mc (R > Rc), transgranular fracture will occur, and fracture strength will decrease with increasing of WC grain size and free path of cobalt binder. When M ˆ Mc (or R ˆ Rc), intergranular fracture and transgranular fracture will concur with 50% chance occupied by each, and fracture strength will reach the maximum value. 4. Fracture strength of WC-Co alloys will increase with increasing of cobalt content and reduction of dimension of crackles. 5. The variant tendency of the measured fracture strength implied that there exists a critical condition that lays the barrier for further improving fracture strength. That the experimental results doesn't suggest the detailed value of critical WC grain size (Rc) lies in the difference

B. Liu et al. / Materials Chemistry and Physics 62 (2000) 35±43

between the real conditions for real bulk materials and the assumptions involved in the model. Acknowledgements The authors wish to express their gratitude to the National Nature Science Foundation of China (No. 59872004) for ®nancial support of this work. The instruction of Prof. Wuyang Chu of Beijing University of Science and Technology is highly appreciated.

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