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Indentation fracture mechanics toughness dependence on grain size and crack size: Application to alumina and WC–Co
International Journal of Refractory Metals & Hard Materials 24 (2006) 129–134 www.elsevier.com/locate/ijrmhm
Indentation fracture mechanics toughness dependence on grain size and crack size: Application to alumina and WC–Co q Ronald W. Armstrong a
a,*
, Oana Cazacu
b
Department of Mechanical Engineering, University of Maryland, College Park, MD 20742, USA b Graduate Engineering and Research Center, University of Florida, Shalimar, FL 32579, USA Received 11 November 2004; accepted 31 March 2005
Abstract Measurements, for alumina, of the indentation fracture mechanics stress intensity, K, at different grain sizes show a reversed trend in the material toughness property: at small crack sizes, although larger than the material grain size, K increases with decrease in grain size in accordance with a Hall–Petch (H–P) type dependence of being linearly dependent on the inverse square root of the grain diameter; whereas, at larger crack sizes, K reverses itself and increases with increase in grain size. The experimental results are interpreted here in terms of approximate equations obtained from the Bilby/Cottrell/Swinden (BCS) model for critical crack growth with an associated plastic zone size, s, at the crack tip. The particular variations in s associated with the two types of K dependencies, nevertheless, give a positive H–P grain size dependence, at sensibly constant crack size, for computed pre-crack fracture stresses, though of lesser H–P slope than the reference crack-free fracture stress. By comparison, K measurements for WC–Co cemented carbide systems show lower values both at smaller WC particle sizes and, especially, at smaller mean free path, k, of the fracture-prone Co binder phase. Such result is predicted to occur if s is proportional to k. The WC–Co system is of special interest, also, because H–P dependencies for the WC and Co constituent components in different alloy compositions make an important contribution, along with the contiguity of the WC particles, to determining the corresponding material hardness properties. Ó 2005 Elsevier Ltd. All rights reserved. Keywords: Indentation fracture mechanics; Stress intensity; Toughness; Fracture stress; Hardness; Grain size; Crack size; Plastic zone size; HallPetch dependence; Bilby/Cottrell/Swinden model; Contiguity; Alumina; WC-Co composite; Cemented carbide
1. Introduction A previous description has been given of a Hall– Petch (H–P) dependence, that is, a linear dependence on the reciprocal square root of grain diameter, ‘, of indentation fracture mechanics stress intensity, K, measurements that are obtained for smaller grain sizes of alumina materials [1]. Other strength properties of aluq This article is dedicated to the research accomplishments achieved on these type considerations in the life-work of Professor Joseph Gurland along with his colleagues and students. * Corresponding author. E-mail address: [email protected] (R.W. Armstrong).
0263-4368/$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijrmhm.2005.03.010
mina materials, such as the hardness and crack-free fracture stress, show H–P dependencies although also showing appreciable scatter of the measurements [2]. The referred-to K measurements [3,4] were obtained at very small crack sizes even though being larger than the material grain sizes. At larger crack sizes, however, the K measurements showed a reversed trend in that increased values were obtained with increase in grain size. Such alternate possibilities are allowed, for example, by the Bilby/Cottrell/Swinden (BCS) fracture mechanics based analysis [5] for critical crack growth with an associated plastic zone size, s, ahead of the crack tip. If s is independent of grain size, an H–P dependence is obtained for K, as described previously for alumina [1].
130
R.W. Armstrong, O. Cazacu / International Journal of Refractory Metals & Hard Materials 24 (2006) 129–134
For the opposite case, K is predicted to be linearly dependent on the square root of grain diameter if s itself becomes equal to, or proportional to, the material grain size [6]. K measurements for the more complicated, highly-filled, cemented carbide, WC–Co, system, generally, are higher for larger WC particle sizes and, especially, for larger mean free paths of the fracture-prone Co binder phase [7]. Such measurements for the WC–Co system are of particular interest for the additional reason that its hardness properties have been quantitatively evaluated in a classic paper by Lee and Gurland [8] on the basis of H–P dependencies being determined separately for both the carbide particle sizes and the mean free paths of the Co phase, plus involvement of the carbide particle contiguities, in obtaining a rule-of-mixtures equation for the material Vickers hardness, HV as follows: H V ¼ H WC V WC C þ H m ð1 V WC CÞ;
ð1Þ
where HWC and Hm are the H–P determined hardnesses of the WC and Co binder phases employing the WC particle sizes, d, and Co binder mean free paths, k; VWC is the volume fraction of WC; and, C is the material contiguity that measures the relative number of WC–WC particle contacts. The purpose of the present study was to assess these mentioned K and HV measurements, along with other similarly available measurements, in terms of the combined equation predictions obtained from H–P and modified BCS model analyses.
1
σF
0.9
σY
0.8
σF σY
= ( 8 / π)
=
s/ c 1+ s / c
s c+ s
0.7
σF σY
0.6
σF σY
= ( 8 / π)
s c
0.5
0.4
0.3
0.2
0.1
s c 0 0
2. Application of the approximated-BCS equations to alumina measurements Fig. 1 illustrates the derived transcendental relationship between the ratio of pre-crack fracture stress, rF, and yield stress, ry, versus the square root of the ratio of crack tip plastic zone size, s, and half-internalcrack or edge-crack length, c, as obtained by Bilby, Cottrell and Swinden (BCS), albeit for a plane stress condition [5]. The named identification of ry is italicized because the modern consideration is that the relevant stress to be achieved at the pre-crack tip is the crack-free fracture stress [6]. Also shown in Fig. 1 are several approximations to the BCS (transcendental) equation. The curve for the solid line BCS equation, itself, ðrF =ry Þ ¼ ð2=pÞsec1 ½1 þ ðs=cÞ
ð2Þ
at small values of (s/c), may be expanded in (1/ cos1[1 + (s/c)]) form, with retention of the first term only in (s/c), to obtain the asymptotic linear dependence p p ðrF =ry Þ ¼ ð 8=pÞ ðs=cÞ.
ð3Þ
2
4
6
8
10
12
Fig. 1. Comparison of approximation relations, both at small and large ratios of plastic zone size and crack size, s/c, to the Bilby/Cottrell/ Swinden [5] transcendental equation for critical crack growth with an associated plastic zone at the crack tip: (rF/ry) = (2/p)sec1[1 + (s/c)].
Eq. (3), that applies at small (s/c) ratio, is in agreement being with a linear fracture mechanics description of rFp p directly proportional to (1/ c). At larger value of (s/c) in Fig. 1, where the BCS equation is non-linear, an approximate but close numerical representation of the transcendental curve is the top dashed curve for ðrF =ry Þ
p
½ðs=cÞ=f1 þ ðs=cÞg.
ð4Þ
Measurements for alumina of KIc, defined for plane strain, as reported by Franco et al. [3] in Table 3 of their article, were taken here at the smallest (4.91 N) and largest (245.3 N) applied loads and with a selected crack-free fracture stress, rF0 (in place of ry above) from Rice [2], also as employed earlier by Armstrong [1], first, to determine plastic zone sizes appropriate to critical crack growth on a BCS modified equation basis. This involves equating rF in the conventional fracture mechanics expression
R.W. Armstrong, O. Cazacu / International Journal of Refractory Metals & Hard Materials 24 (2006) 129–134
p rF ¼ K Ic = ðpcÞ
ð5Þ
to its counterpart in Eq. (3), with rF0 in place of ry, and K KIc, to obtain 2
s ¼ ðp=8ÞðK Ic =rF0 Þ :
ð6Þ
It is noted, also, that the crack size measurements, c, taken from Table 3 of Franco et al. correspond to penetrations from the specimen surface as a consequence of the rather shallow state of tensile stress produced by the indenter strains. Further, it should be noted that the same Eq. (6) is obtained from matching Eq. (4) with Eq. (5) if the latter equation is modified to include an effective crack length (c + 2r), as conventionally done, with r being the plastic zone radius and 2r = s [9]. Comparison of the computed s values in Table 1 with the values of c, for which the K Õs were evaluated, shows that the pre-crack size determined fracture stress, rFc(=rF above), should correspond to the (s/c) condition established for Eq. (4), at the small crack size measurements, and for Eq. (3), at the large crack size measurements, both as listed in the last column of Table 1. Also, comparison of the s and ‘ values in Table 1 shows that the plastic zone size involves about one grain diamTable 1 Modelled BCS-based pre-crack fracture stresses KIc, MPa m1/2
c, mm
rF0, MPa
s, mm
rFc, MPa
0.0012 0.0038 0.01984
3.16 3.09 2.95
0.0213 0.0225 0.0245
536 461 415
0.01365a 0.01764 0.01984
335b 305.5 277.6
536 461 415
a
a b c
2.93 3.06 3.23
0.3301 0.3239 0.3146
eter for the larger grain size material and more than ten grain diameters for the smaller grain size material. Fig. 2, then, shows relative to the higher plotted crackfree fracture stress, the modified H–P dependencies computed for the two pre-crack fracture stresses at the smaller and larger crack sizes, with the appropriate s and c values also identified in the Figure body. The influence of a larger plastic zone size at larger grain size is seen to reduce the H–P dependence and, more so, at the larger crack size.
3. The hardness properties of WC–Co cemented carbide Lee and Gurland (L&G) [8] investigated the hardness properties of cemented carbide, WC–Co, materials of differing Co volume fractions, VCo, and also as a function of the WC particle size, d, mean free path, k, of the Co binder phase, and contiguity, C, measurement of the WC particle-to-particle contacts; see Table 1 of their article. L&G reported that the total hardness results could be fitted with the contiguity-modified rule of mixtures expression given in Eq. (1). The H–P type component hardnesses of the WC and Co phases were given by L&G as H WC ¼ 1382 þ 23.1d 1=2 ; in kgf=mm2 ; and d in mm;
‘, mm
0.0012 0.0038 0.01984
131
0.01735 0.01730 0.02379
c
110.5 95.9 102.75
s = (p/8)(KIc/rF0)2. p rFc ¼ rF0 ð½s=c=f1 þ ½s=cgÞ. p p rFc ¼ ð 8=pÞrF0 ðs=cÞ.
Fig. 2. Hall–Petch type dependencies for measured crack-free [2], and computed pre-crack, fracture stresses of alumina, in the latter case, evaluated in terms of Eqs. (3) or (4) for relatively smaller or larger (s/c) ratios, respectively.
ð7Þ and H Co ¼ 304 þ 12.7k1=2 ; in the same units.
ð8Þ
An interesting feature of the L&G analysis was that, taking into account the important role played by the WC particle-to-particle contacts, as measured by C, had the effect of converting the material hardness measurements from following a weakest link, series type, hardness dependence on VCo to the stronger behavior represented by the C-modified rule-of-mixtures expression. Also, an important part of the L&G study was their noting an apparent discrimination in the hardness values between micrometer-sized and larger WC particles. Fig. 3, taken from the L&G Fig. 2 is adapted here to demonstrate graphically the excellent agreement of experimental measurements with Eq. (1). More recent hardness measurements reported by Richter and von Ruthendorf [10] for smaller WC particle sizes are added to the graph and confirm the pointed-to importance of the hardness being increased at smaller d values. In Fig. 3, the individual H–P hardness values for the component WC and Co constituents were computed from L&GÕs tabulation for three arbitrarily selected, two ‘‘xÕs’’ and one open circle, points to yield, first, the filled square points that are plotted on the terminal WC and Co axes. Then, the C-modified hardness value for each composite structure was plotted at its new coordinate position for the effective volume fraction to
132
R.W. Armstrong, O. Cazacu / International Journal of Refractory Metals & Hard Materials 24 (2006) 129–134 2400
and
2200
k H ¼ C½V WC =V Co B1=2 k WC þ ð1=V Co Þð1 CV WC Þk Co
2000
ð11Þ while employing the H–P ÔkÕ symbols for the WC and Co constituents in place of their numerical counterparts in Eq. (1). Consequently, in accordance with Eq. (9), Fig. 4a has been plotted to show the Milman et al. HV measurements, grouped according to constant VCo values of 0.1, 0.16, and 0.24, across each grade, and that are in reasonable agreement with computed measurements according to substitutions in the L&G Eqs. (1), (7) and (8). For the condition of k1/2 = 0, intercept values of HV0 were computed from Eq. (10) also employing the L&G intercept constants.
1800 1600 1400 HV (kg/mm2) 1200 1000 800 600 400 200 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Carbide Volume Fraction Or Effective Volume Fraction
25
Fig. 3. The Vickers hardness of WC–Co cemented carbide materials as a function of volume fraction of the Co binder phase, VCo, as reported by Lee and Gurland (L&G) [8], with discrimination between the WC particle sizes, and containing recent confirming WC particle size measurements reported by Richter and von Ruthendorf [10]; also, with comparison, for three (L&G) specimens, to Hall–Petch determined hardnesses on the terminal axes for the components and to the shifted hardness positions on a contiguity-modified, effective volume fraction, basis, as proposed also by Lee and Gurland. The hardness units are expressed in conventional kgf/mm2 for which 1.0 kgf/mm2 = 9.81 MPa.
confirm agreement with the linear form of Eq. (1) on that effective volume fraction basis. The previous abscissa coordinate positions of the same composite hardness measurements were shifted then to the effective volume fraction points to show excellent agreement in each case between the filled square computation and the measurement. 4. HV and rF0 values for the WC–Co results of Milman et al. [11] Milman et al. [11] have reported both HV and rF0 measurements made on three types of WC–Co micron and submicron sized materials, designated S, NY, and NYA grades in order of decreasing d values, also, at three volume fractions of the Co composition at each d value. Complete stereological measurements were reported, as well, of d, k, and C. An examination of the various parameters listed in the Milman et al. Table 1 shows that d is approximately proportional to k at each value of VCo across the material grades and, consequently, Eq. (1) may be written to include the H–P dependencies for WC and Co in expanded form, with d Bk, as H V H V0 þ k H ðV Co k1=2 Þ;
ð9Þ
where H V0 ¼ CV WC H 0WC þ ð1 CV WC ÞH 0Co ;
ð10Þ
H V (GPa) 20
15
10
Milman et al.
5
H = HWC VWC C + Hm (1–VWCC)
V CO λ
–1/2
a 0 0
0.2
0.4
, µ m–1/2
0.6
0.8
4.5
σF (GPa)
4 3.5
NY
3 2.5
S
2 1.5
NYA
1
b
0.5
V CO λ–1/2µm–1/2
0 0
0.1
0.2
0.3
0.4
0.5
0.6
Fig. 4. (a) The Vickers hardnesses, reported by Milman et al. [11], of three compositions of WC–Co as a function of VCok1/2 in comparison with Lee and Gurland (L&G) type estimations and, also, expectation of a linear dependence predicted in each case for d/k being approximately constant; and (b) Comparison of bend test fracture stresses reported by Milman et al. for the same three compositions, perhaps, indicating a linear fracture stress dependence largely controlled by the volume fraction of, and characteristics of, the Co binder phase.
R.W. Armstrong, O. Cazacu / International Journal of Refractory Metals & Hard Materials 24 (2006) 129–134
Another reason for employing the units for the abscissa scale in Fig. 4a stems from the frequently reported experimental observation that the fracturing behavior of these cemented carbide systems is controlled by the characteristics of the Co binder phase. For this reason, the Milman et al. measurements of rF0 measured in three-point bend tests were plotted also on the same abscissa scale in Fig. 4b. Considerable variation is shown for the various measurements, especially including within the NY and NYA grades, however, the three micron sized S-grade materials and the highest rF0 measurement for the middle composition NY grade material might possibly be interpreted to follow the linear fracture stress dependence that is shown. 5. Correlation of K measurements with a predicted k1/2 dependence Returning to the earlier mentioned observation that K in WC–Co cemented carbide systems is generally lower at smaller d and k values, such dependence follows from Eq. (6) even if an H–P dependence applies for rF0 with k being the relevant size parameter if s, in turn, is also proportional to k. Such dependence has been described for the exceptionally brittle fracture of certain steel material [6] and, also, was previously indicated by 2r being shown to be proportional to k in the pioneering measurements reported by Sigl and Fischmeister [7]. Fig. 5 shows on a positive k1/2 basis an updated graph of compiled results reported in the separate investigations of Jia, Fischer and Gallois [12] and Richter and von Ruthendorf [10] and, also, including the extensive results of Sigl and Fischmeister whose KIc measurements were obtained from pre-cracked bend tests. The relatively raised open circle points for Jia et al. corresponded to specimens fabricated with nanostructured WC particles of 70 nm. The ordinate K value, plotted 20
K, MPa m1/2 15
Sigl & Fischmeister d > 0.8 um (Jia, Fischer & Gallois) d~0.070 um (Jia, Fischer & Gallois) R&vR
10
5
λ 1/2 µm1/2
0 0
0.2
0.4
0.6
0.8
1
Fig. 5. The fracture mechanics stress intensity, K, measurements reported in several investigations [7,10,12] are shown to follow a k1/2 dependence as should apply for a Hall–Petch type stress dependence for k and with it being proportional to the plastic zone size [6].
133
at k1/2 = 0, as reported by Richter and von Ruthendorf applies for totally WC material. The results give reasonable indication of the applicability of the modified BCS analysis to such cemented carbide systems under the designated conditions.
6. Discussion A linear dependence of K on k1/2, was indicated already in the results reported in the pioneering paper of Sigl and Fischmeister [7]. These authors pointed to the association of k with the plastic zone size, 2r, in the conventional fracture mechanics description when modified to include the plastic zone size addition to the crack size; see the above text following Eq. (6). Sigl and Fischmeister proposed a direct one-to-one relationship between k and 2r on the basis that k had a major influence on the fracturing energy and it, in turn, was mainly attributed in their own quantitative model estimates to dimple fracturing of a strain hardened cobalt binder phase. Their model estimates of the intergranular and transgranular fracturing of the WC phase, although representing a prominent portion of the fracture surface only accounted for on the order of 10% of the total fracturing energy. The free surface energies of the binder and WC could be neglected. Here, the k1/2 dependence of Fig. 5 is associated with the resultant equation adapted from the BCS analysis [6] as K ¼ c0 s1=2 ½r0 þ k‘1=2
ð12Þ
in which k replaces ‘ in the bracketed Hall–Petch (H–P) type stress intensity factor, as established by Lee and Gurland [8], and s is proportional to k, as established by Sigl and Fischmeister [7]. The H–P stress is taken as the crack-free fracture stress, rF0, as employed in the analysis of measured crack-free and computed precrack fracture stresses rFc shown for alumina in Fig. 2. Implicit in the experimental H–P stresses that are established either for yielding, or hardness, or fracturing, is their origin in the dislocation pile-up mechanism for generating the internal concentrations of stress for overcoming the grain boundary or interphase boundary obstacles to plastic flow or cracking. In this regard, it is interesting to point to a related modification of the H–P type analysis proposed, at least, for the hardness of WC–Co materials by Nabarro and colleagues [13]. In their model consideration, a role for WC–WC particle connections produces a direct influence of a d1/4 multiplier of the H–P k1/2 dependent factor. The factor goes a long way towards accomplishing the same numerical effect of taking the WC–WC particle contacts into account through the material contiguity, C. Alternatively, Golochan and Litoshenko [14] have given added importance to the consideration of C as a primary material
134
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parameter in the WC–Co system by establishing a proposed stereological relationship basis for C being dependent on VCo and a new parameter designated as the coefficient of variation. Beyond the agreement with measurements demonstrated in Fig. 3 for the L&G analysis, there is also independent confirmation of the analysis demonstrated in Fig. 4a. Kotoul [15] employed a model of cracking for the WC–Co system involving critical crack growth under condition of its length being restrained by a trailing multiligament zone formed by bridging Co binder connections. The developed model is analogous to, but more complicated than, that described for the BCS model in Section 2 and Fig. 1. An important outcome of the model description was numerical evaluation of a KIc dependence on k for the results of Sigl and Fischmeister that approximates very well the square root relation given for the same results, and others, in Fig. 5. A last, interesting, consideration mentioned here relates to the role of thermal strains in determining the fracturing properties of the WC–Co system. Such smaller thermal strain consideration was mentioned for alumina to be a major consideration only at very large grain sizes [1]. The mismatch of thermal expansion coefficients, however, in the WC–Co system is larger, by greater than a 4X difference, than those lesser differences caused simply by crystal anisotropy in alumina. Liu, Zhang and Ouyang [16] have, in fact, presented a theoretical model for the fracturing energy of the WC–Co system being proportional to k because of thermal strain influences. The thermal strains from the WC particles are proposed to facilitate the formation of circumferential cracks in the encompassing Co binder by lowering the Griffith fracture energy in direct proportion to VCo, and this to be proportional to k, but otherwise to give an increasing H–P type dependence on d1/2. The interesting feature then is that a proportional dependence to k for the Griffith energy corresponds to a k1/2 dependence in the fracture mechanics stress intensity, K, again as shown in Fig. 5.
References [1] Armstrong RW. Grain size dependent alumina fracture mechanics stress intensity. Int J Refract Met Hard Mater 2001;19:251–5. [2] Rice RW. Review—Ceramic tensile strength-grain size relations: grain sizes, slopes, and branch intersections. J Mater Sci 1997;32: 1671–92. [3] Franco A, Roberts SG, Warren PD. Fracture toughness, surface flaw sizes and flaw densities in Al2O3. Acta Mater 1997;45: 1009–15. [4] Muchtar A, Lim LC. Indentation fracture toughness of high purity submicron alumina. Acta Mater 1998;46:1683–90. [5] Bilby BA, Cottrell AH, Swinden KH. The spread of plastic yielding from a notch. Proc Roy Soc London 1963;A272:304–14. [6] Armstrong RW. The (cleavage) strength of pre-cracked polycrystals. Eng Fract Mech 1989;28:529–38. [7] Sigl LS, Fischmeister HF. On the fracture toughness of cemented carbides. Acta Metall 1988;36:887–97. [8] Lee H-C, Gurland J. Hardness and deformation of cemented tungsten carbide. Mater Sci Eng 1978;33:125–33. [9] Stonesifer FR, Armstrong RW. Effect of prior austenite grain size on the fracture toughness properties of A533 B steel. In: Taplin DMR, editor. Fracture 1977, 2. Canada: University Waterloo Press; 1977. p. 1–4. [10] Richter V, von Ruthendorf M. On hardness and toughness of ultrafine and nanocrystalline hard materials. Int J Refract Met Hard Mater 1999;17:141–52. [11] Milman YV, Luyckx S, Goncharuck VA, Northrop JT. Results from bending tests on submicron and micron WC–Co grades at elevated temperatures. Int J Refract Met Hard Mater 2002; 20:71–9. [12] Jia K, Fischer TE, Gallois B. Microstructure, hardness, and toughness of nanostructures and conventional WC–Co composites. Nanostruct Mater 1998;10:875–91. [13] Makhele-Lekata I, Luyckx S, Nabarro FRN. Semi-empirical relationship between the hardness, grain size and mean free path of WC–Co. Int J Refract Met Hard Mater 2001;19:245–9. [14] Golovchan VT, Litoshenko NV. On the contiguity of carbide phase in WC–Co hardmetals. Int J Refract Met Hard Mater 2003;21:241–4. [15] Kotoul M. On the shielding effect of a multiligament zone of a crack in WC–Co. Mater Sci Eng A 1997;234–236:119–22; Shielding model of fracture in WC–Co. Acta Mater 1997; 45:3363–76. [16] Liu B, Zhang Y, Ouyang S. Study on the relation between structural parameters and fracture strength of WC–Co cemented carbides. Mater Chem Phys 2000;62:35–43.
Indentation fracture mechanics toughness dependence on grain size and crack size: Application to alumina and WC–Co q Ronald W. Armstrong a
a,*
, Oana Cazacu
b
Department of Mechanical Engineering, University of Maryland, College Park, MD 20742, USA b Graduate Engineering and Research Center, University of Florida, Shalimar, FL 32579, USA Received 11 November 2004; accepted 31 March 2005
Abstract Measurements, for alumina, of the indentation fracture mechanics stress intensity, K, at different grain sizes show a reversed trend in the material toughness property: at small crack sizes, although larger than the material grain size, K increases with decrease in grain size in accordance with a Hall–Petch (H–P) type dependence of being linearly dependent on the inverse square root of the grain diameter; whereas, at larger crack sizes, K reverses itself and increases with increase in grain size. The experimental results are interpreted here in terms of approximate equations obtained from the Bilby/Cottrell/Swinden (BCS) model for critical crack growth with an associated plastic zone size, s, at the crack tip. The particular variations in s associated with the two types of K dependencies, nevertheless, give a positive H–P grain size dependence, at sensibly constant crack size, for computed pre-crack fracture stresses, though of lesser H–P slope than the reference crack-free fracture stress. By comparison, K measurements for WC–Co cemented carbide systems show lower values both at smaller WC particle sizes and, especially, at smaller mean free path, k, of the fracture-prone Co binder phase. Such result is predicted to occur if s is proportional to k. The WC–Co system is of special interest, also, because H–P dependencies for the WC and Co constituent components in different alloy compositions make an important contribution, along with the contiguity of the WC particles, to determining the corresponding material hardness properties. Ó 2005 Elsevier Ltd. All rights reserved. Keywords: Indentation fracture mechanics; Stress intensity; Toughness; Fracture stress; Hardness; Grain size; Crack size; Plastic zone size; HallPetch dependence; Bilby/Cottrell/Swinden model; Contiguity; Alumina; WC-Co composite; Cemented carbide
1. Introduction A previous description has been given of a Hall– Petch (H–P) dependence, that is, a linear dependence on the reciprocal square root of grain diameter, ‘, of indentation fracture mechanics stress intensity, K, measurements that are obtained for smaller grain sizes of alumina materials [1]. Other strength properties of aluq This article is dedicated to the research accomplishments achieved on these type considerations in the life-work of Professor Joseph Gurland along with his colleagues and students. * Corresponding author. E-mail address: [email protected] (R.W. Armstrong).
0263-4368/$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijrmhm.2005.03.010
mina materials, such as the hardness and crack-free fracture stress, show H–P dependencies although also showing appreciable scatter of the measurements [2]. The referred-to K measurements [3,4] were obtained at very small crack sizes even though being larger than the material grain sizes. At larger crack sizes, however, the K measurements showed a reversed trend in that increased values were obtained with increase in grain size. Such alternate possibilities are allowed, for example, by the Bilby/Cottrell/Swinden (BCS) fracture mechanics based analysis [5] for critical crack growth with an associated plastic zone size, s, ahead of the crack tip. If s is independent of grain size, an H–P dependence is obtained for K, as described previously for alumina [1].
130
R.W. Armstrong, O. Cazacu / International Journal of Refractory Metals & Hard Materials 24 (2006) 129–134
For the opposite case, K is predicted to be linearly dependent on the square root of grain diameter if s itself becomes equal to, or proportional to, the material grain size [6]. K measurements for the more complicated, highly-filled, cemented carbide, WC–Co, system, generally, are higher for larger WC particle sizes and, especially, for larger mean free paths of the fracture-prone Co binder phase [7]. Such measurements for the WC–Co system are of particular interest for the additional reason that its hardness properties have been quantitatively evaluated in a classic paper by Lee and Gurland [8] on the basis of H–P dependencies being determined separately for both the carbide particle sizes and the mean free paths of the Co phase, plus involvement of the carbide particle contiguities, in obtaining a rule-of-mixtures equation for the material Vickers hardness, HV as follows: H V ¼ H WC V WC C þ H m ð1 V WC CÞ;
ð1Þ
where HWC and Hm are the H–P determined hardnesses of the WC and Co binder phases employing the WC particle sizes, d, and Co binder mean free paths, k; VWC is the volume fraction of WC; and, C is the material contiguity that measures the relative number of WC–WC particle contacts. The purpose of the present study was to assess these mentioned K and HV measurements, along with other similarly available measurements, in terms of the combined equation predictions obtained from H–P and modified BCS model analyses.
1
σF
0.9
σY
0.8
σF σY
= ( 8 / π)
=
s/ c 1+ s / c
s c+ s
0.7
σF σY
0.6
σF σY
= ( 8 / π)
s c
0.5
0.4
0.3
0.2
0.1
s c 0 0
2. Application of the approximated-BCS equations to alumina measurements Fig. 1 illustrates the derived transcendental relationship between the ratio of pre-crack fracture stress, rF, and yield stress, ry, versus the square root of the ratio of crack tip plastic zone size, s, and half-internalcrack or edge-crack length, c, as obtained by Bilby, Cottrell and Swinden (BCS), albeit for a plane stress condition [5]. The named identification of ry is italicized because the modern consideration is that the relevant stress to be achieved at the pre-crack tip is the crack-free fracture stress [6]. Also shown in Fig. 1 are several approximations to the BCS (transcendental) equation. The curve for the solid line BCS equation, itself, ðrF =ry Þ ¼ ð2=pÞsec1 ½1 þ ðs=cÞ
ð2Þ
at small values of (s/c), may be expanded in (1/ cos1[1 + (s/c)]) form, with retention of the first term only in (s/c), to obtain the asymptotic linear dependence p p ðrF =ry Þ ¼ ð 8=pÞ ðs=cÞ.
ð3Þ
2
4
6
8
10
12
Fig. 1. Comparison of approximation relations, both at small and large ratios of plastic zone size and crack size, s/c, to the Bilby/Cottrell/ Swinden [5] transcendental equation for critical crack growth with an associated plastic zone at the crack tip: (rF/ry) = (2/p)sec1[1 + (s/c)].
Eq. (3), that applies at small (s/c) ratio, is in agreement being with a linear fracture mechanics description of rFp p directly proportional to (1/ c). At larger value of (s/c) in Fig. 1, where the BCS equation is non-linear, an approximate but close numerical representation of the transcendental curve is the top dashed curve for ðrF =ry Þ
p
½ðs=cÞ=f1 þ ðs=cÞg.
ð4Þ
Measurements for alumina of KIc, defined for plane strain, as reported by Franco et al. [3] in Table 3 of their article, were taken here at the smallest (4.91 N) and largest (245.3 N) applied loads and with a selected crack-free fracture stress, rF0 (in place of ry above) from Rice [2], also as employed earlier by Armstrong [1], first, to determine plastic zone sizes appropriate to critical crack growth on a BCS modified equation basis. This involves equating rF in the conventional fracture mechanics expression
R.W. Armstrong, O. Cazacu / International Journal of Refractory Metals & Hard Materials 24 (2006) 129–134
p rF ¼ K Ic = ðpcÞ
ð5Þ
to its counterpart in Eq. (3), with rF0 in place of ry, and K KIc, to obtain 2
s ¼ ðp=8ÞðK Ic =rF0 Þ :
ð6Þ
It is noted, also, that the crack size measurements, c, taken from Table 3 of Franco et al. correspond to penetrations from the specimen surface as a consequence of the rather shallow state of tensile stress produced by the indenter strains. Further, it should be noted that the same Eq. (6) is obtained from matching Eq. (4) with Eq. (5) if the latter equation is modified to include an effective crack length (c + 2r), as conventionally done, with r being the plastic zone radius and 2r = s [9]. Comparison of the computed s values in Table 1 with the values of c, for which the K Õs were evaluated, shows that the pre-crack size determined fracture stress, rFc(=rF above), should correspond to the (s/c) condition established for Eq. (4), at the small crack size measurements, and for Eq. (3), at the large crack size measurements, both as listed in the last column of Table 1. Also, comparison of the s and ‘ values in Table 1 shows that the plastic zone size involves about one grain diamTable 1 Modelled BCS-based pre-crack fracture stresses KIc, MPa m1/2
c, mm
rF0, MPa
s, mm
rFc, MPa
0.0012 0.0038 0.01984
3.16 3.09 2.95
0.0213 0.0225 0.0245
536 461 415
0.01365a 0.01764 0.01984
335b 305.5 277.6
536 461 415
a
a b c
2.93 3.06 3.23
0.3301 0.3239 0.3146
eter for the larger grain size material and more than ten grain diameters for the smaller grain size material. Fig. 2, then, shows relative to the higher plotted crackfree fracture stress, the modified H–P dependencies computed for the two pre-crack fracture stresses at the smaller and larger crack sizes, with the appropriate s and c values also identified in the Figure body. The influence of a larger plastic zone size at larger grain size is seen to reduce the H–P dependence and, more so, at the larger crack size.
3. The hardness properties of WC–Co cemented carbide Lee and Gurland (L&G) [8] investigated the hardness properties of cemented carbide, WC–Co, materials of differing Co volume fractions, VCo, and also as a function of the WC particle size, d, mean free path, k, of the Co binder phase, and contiguity, C, measurement of the WC particle-to-particle contacts; see Table 1 of their article. L&G reported that the total hardness results could be fitted with the contiguity-modified rule of mixtures expression given in Eq. (1). The H–P type component hardnesses of the WC and Co phases were given by L&G as H WC ¼ 1382 þ 23.1d 1=2 ; in kgf=mm2 ; and d in mm;
‘, mm
0.0012 0.0038 0.01984
131
0.01735 0.01730 0.02379
c
110.5 95.9 102.75
s = (p/8)(KIc/rF0)2. p rFc ¼ rF0 ð½s=c=f1 þ ½s=cgÞ. p p rFc ¼ ð 8=pÞrF0 ðs=cÞ.
Fig. 2. Hall–Petch type dependencies for measured crack-free [2], and computed pre-crack, fracture stresses of alumina, in the latter case, evaluated in terms of Eqs. (3) or (4) for relatively smaller or larger (s/c) ratios, respectively.
ð7Þ and H Co ¼ 304 þ 12.7k1=2 ; in the same units.
ð8Þ
An interesting feature of the L&G analysis was that, taking into account the important role played by the WC particle-to-particle contacts, as measured by C, had the effect of converting the material hardness measurements from following a weakest link, series type, hardness dependence on VCo to the stronger behavior represented by the C-modified rule-of-mixtures expression. Also, an important part of the L&G study was their noting an apparent discrimination in the hardness values between micrometer-sized and larger WC particles. Fig. 3, taken from the L&G Fig. 2 is adapted here to demonstrate graphically the excellent agreement of experimental measurements with Eq. (1). More recent hardness measurements reported by Richter and von Ruthendorf [10] for smaller WC particle sizes are added to the graph and confirm the pointed-to importance of the hardness being increased at smaller d values. In Fig. 3, the individual H–P hardness values for the component WC and Co constituents were computed from L&GÕs tabulation for three arbitrarily selected, two ‘‘xÕs’’ and one open circle, points to yield, first, the filled square points that are plotted on the terminal WC and Co axes. Then, the C-modified hardness value for each composite structure was plotted at its new coordinate position for the effective volume fraction to
132
R.W. Armstrong, O. Cazacu / International Journal of Refractory Metals & Hard Materials 24 (2006) 129–134 2400
and
2200
k H ¼ C½V WC =V Co B1=2 k WC þ ð1=V Co Þð1 CV WC Þk Co
2000
ð11Þ while employing the H–P ÔkÕ symbols for the WC and Co constituents in place of their numerical counterparts in Eq. (1). Consequently, in accordance with Eq. (9), Fig. 4a has been plotted to show the Milman et al. HV measurements, grouped according to constant VCo values of 0.1, 0.16, and 0.24, across each grade, and that are in reasonable agreement with computed measurements according to substitutions in the L&G Eqs. (1), (7) and (8). For the condition of k1/2 = 0, intercept values of HV0 were computed from Eq. (10) also employing the L&G intercept constants.
1800 1600 1400 HV (kg/mm2) 1200 1000 800 600 400 200 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Carbide Volume Fraction Or Effective Volume Fraction
25
Fig. 3. The Vickers hardness of WC–Co cemented carbide materials as a function of volume fraction of the Co binder phase, VCo, as reported by Lee and Gurland (L&G) [8], with discrimination between the WC particle sizes, and containing recent confirming WC particle size measurements reported by Richter and von Ruthendorf [10]; also, with comparison, for three (L&G) specimens, to Hall–Petch determined hardnesses on the terminal axes for the components and to the shifted hardness positions on a contiguity-modified, effective volume fraction, basis, as proposed also by Lee and Gurland. The hardness units are expressed in conventional kgf/mm2 for which 1.0 kgf/mm2 = 9.81 MPa.
confirm agreement with the linear form of Eq. (1) on that effective volume fraction basis. The previous abscissa coordinate positions of the same composite hardness measurements were shifted then to the effective volume fraction points to show excellent agreement in each case between the filled square computation and the measurement. 4. HV and rF0 values for the WC–Co results of Milman et al. [11] Milman et al. [11] have reported both HV and rF0 measurements made on three types of WC–Co micron and submicron sized materials, designated S, NY, and NYA grades in order of decreasing d values, also, at three volume fractions of the Co composition at each d value. Complete stereological measurements were reported, as well, of d, k, and C. An examination of the various parameters listed in the Milman et al. Table 1 shows that d is approximately proportional to k at each value of VCo across the material grades and, consequently, Eq. (1) may be written to include the H–P dependencies for WC and Co in expanded form, with d Bk, as H V H V0 þ k H ðV Co k1=2 Þ;
ð9Þ
where H V0 ¼ CV WC H 0WC þ ð1 CV WC ÞH 0Co ;
ð10Þ
H V (GPa) 20
15
10
Milman et al.
5
H = HWC VWC C + Hm (1–VWCC)
V CO λ
–1/2
a 0 0
0.2
0.4
, µ m–1/2
0.6
0.8
4.5
σF (GPa)
4 3.5
NY
3 2.5
S
2 1.5
NYA
1
b
0.5
V CO λ–1/2µm–1/2
0 0
0.1
0.2
0.3
0.4
0.5
0.6
Fig. 4. (a) The Vickers hardnesses, reported by Milman et al. [11], of three compositions of WC–Co as a function of VCok1/2 in comparison with Lee and Gurland (L&G) type estimations and, also, expectation of a linear dependence predicted in each case for d/k being approximately constant; and (b) Comparison of bend test fracture stresses reported by Milman et al. for the same three compositions, perhaps, indicating a linear fracture stress dependence largely controlled by the volume fraction of, and characteristics of, the Co binder phase.
R.W. Armstrong, O. Cazacu / International Journal of Refractory Metals & Hard Materials 24 (2006) 129–134
Another reason for employing the units for the abscissa scale in Fig. 4a stems from the frequently reported experimental observation that the fracturing behavior of these cemented carbide systems is controlled by the characteristics of the Co binder phase. For this reason, the Milman et al. measurements of rF0 measured in three-point bend tests were plotted also on the same abscissa scale in Fig. 4b. Considerable variation is shown for the various measurements, especially including within the NY and NYA grades, however, the three micron sized S-grade materials and the highest rF0 measurement for the middle composition NY grade material might possibly be interpreted to follow the linear fracture stress dependence that is shown. 5. Correlation of K measurements with a predicted k1/2 dependence Returning to the earlier mentioned observation that K in WC–Co cemented carbide systems is generally lower at smaller d and k values, such dependence follows from Eq. (6) even if an H–P dependence applies for rF0 with k being the relevant size parameter if s, in turn, is also proportional to k. Such dependence has been described for the exceptionally brittle fracture of certain steel material [6] and, also, was previously indicated by 2r being shown to be proportional to k in the pioneering measurements reported by Sigl and Fischmeister [7]. Fig. 5 shows on a positive k1/2 basis an updated graph of compiled results reported in the separate investigations of Jia, Fischer and Gallois [12] and Richter and von Ruthendorf [10] and, also, including the extensive results of Sigl and Fischmeister whose KIc measurements were obtained from pre-cracked bend tests. The relatively raised open circle points for Jia et al. corresponded to specimens fabricated with nanostructured WC particles of 70 nm. The ordinate K value, plotted 20
K, MPa m1/2 15
Sigl & Fischmeister d > 0.8 um (Jia, Fischer & Gallois) d~0.070 um (Jia, Fischer & Gallois) R&vR
10
5
λ 1/2 µm1/2
0 0
0.2
0.4
0.6
0.8
1
Fig. 5. The fracture mechanics stress intensity, K, measurements reported in several investigations [7,10,12] are shown to follow a k1/2 dependence as should apply for a Hall–Petch type stress dependence for k and with it being proportional to the plastic zone size [6].
133
at k1/2 = 0, as reported by Richter and von Ruthendorf applies for totally WC material. The results give reasonable indication of the applicability of the modified BCS analysis to such cemented carbide systems under the designated conditions.
6. Discussion A linear dependence of K on k1/2, was indicated already in the results reported in the pioneering paper of Sigl and Fischmeister [7]. These authors pointed to the association of k with the plastic zone size, 2r, in the conventional fracture mechanics description when modified to include the plastic zone size addition to the crack size; see the above text following Eq. (6). Sigl and Fischmeister proposed a direct one-to-one relationship between k and 2r on the basis that k had a major influence on the fracturing energy and it, in turn, was mainly attributed in their own quantitative model estimates to dimple fracturing of a strain hardened cobalt binder phase. Their model estimates of the intergranular and transgranular fracturing of the WC phase, although representing a prominent portion of the fracture surface only accounted for on the order of 10% of the total fracturing energy. The free surface energies of the binder and WC could be neglected. Here, the k1/2 dependence of Fig. 5 is associated with the resultant equation adapted from the BCS analysis [6] as K ¼ c0 s1=2 ½r0 þ k‘1=2
ð12Þ
in which k replaces ‘ in the bracketed Hall–Petch (H–P) type stress intensity factor, as established by Lee and Gurland [8], and s is proportional to k, as established by Sigl and Fischmeister [7]. The H–P stress is taken as the crack-free fracture stress, rF0, as employed in the analysis of measured crack-free and computed precrack fracture stresses rFc shown for alumina in Fig. 2. Implicit in the experimental H–P stresses that are established either for yielding, or hardness, or fracturing, is their origin in the dislocation pile-up mechanism for generating the internal concentrations of stress for overcoming the grain boundary or interphase boundary obstacles to plastic flow or cracking. In this regard, it is interesting to point to a related modification of the H–P type analysis proposed, at least, for the hardness of WC–Co materials by Nabarro and colleagues [13]. In their model consideration, a role for WC–WC particle connections produces a direct influence of a d1/4 multiplier of the H–P k1/2 dependent factor. The factor goes a long way towards accomplishing the same numerical effect of taking the WC–WC particle contacts into account through the material contiguity, C. Alternatively, Golochan and Litoshenko [14] have given added importance to the consideration of C as a primary material
134
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parameter in the WC–Co system by establishing a proposed stereological relationship basis for C being dependent on VCo and a new parameter designated as the coefficient of variation. Beyond the agreement with measurements demonstrated in Fig. 3 for the L&G analysis, there is also independent confirmation of the analysis demonstrated in Fig. 4a. Kotoul [15] employed a model of cracking for the WC–Co system involving critical crack growth under condition of its length being restrained by a trailing multiligament zone formed by bridging Co binder connections. The developed model is analogous to, but more complicated than, that described for the BCS model in Section 2 and Fig. 1. An important outcome of the model description was numerical evaluation of a KIc dependence on k for the results of Sigl and Fischmeister that approximates very well the square root relation given for the same results, and others, in Fig. 5. A last, interesting, consideration mentioned here relates to the role of thermal strains in determining the fracturing properties of the WC–Co system. Such smaller thermal strain consideration was mentioned for alumina to be a major consideration only at very large grain sizes [1]. The mismatch of thermal expansion coefficients, however, in the WC–Co system is larger, by greater than a 4X difference, than those lesser differences caused simply by crystal anisotropy in alumina. Liu, Zhang and Ouyang [16] have, in fact, presented a theoretical model for the fracturing energy of the WC–Co system being proportional to k because of thermal strain influences. The thermal strains from the WC particles are proposed to facilitate the formation of circumferential cracks in the encompassing Co binder by lowering the Griffith fracture energy in direct proportion to VCo, and this to be proportional to k, but otherwise to give an increasing H–P type dependence on d1/2. The interesting feature then is that a proportional dependence to k for the Griffith energy corresponds to a k1/2 dependence in the fracture mechanics stress intensity, K, again as shown in Fig. 5.
References [1] Armstrong RW. Grain size dependent alumina fracture mechanics stress intensity. Int J Refract Met Hard Mater 2001;19:251–5. [2] Rice RW. Review—Ceramic tensile strength-grain size relations: grain sizes, slopes, and branch intersections. J Mater Sci 1997;32: 1671–92. [3] Franco A, Roberts SG, Warren PD. Fracture toughness, surface flaw sizes and flaw densities in Al2O3. Acta Mater 1997;45: 1009–15. [4] Muchtar A, Lim LC. Indentation fracture toughness of high purity submicron alumina. Acta Mater 1998;46:1683–90. [5] Bilby BA, Cottrell AH, Swinden KH. The spread of plastic yielding from a notch. Proc Roy Soc London 1963;A272:304–14. [6] Armstrong RW. The (cleavage) strength of pre-cracked polycrystals. Eng Fract Mech 1989;28:529–38. [7] Sigl LS, Fischmeister HF. On the fracture toughness of cemented carbides. Acta Metall 1988;36:887–97. [8] Lee H-C, Gurland J. Hardness and deformation of cemented tungsten carbide. Mater Sci Eng 1978;33:125–33. [9] Stonesifer FR, Armstrong RW. Effect of prior austenite grain size on the fracture toughness properties of A533 B steel. In: Taplin DMR, editor. Fracture 1977, 2. Canada: University Waterloo Press; 1977. p. 1–4. [10] Richter V, von Ruthendorf M. On hardness and toughness of ultrafine and nanocrystalline hard materials. Int J Refract Met Hard Mater 1999;17:141–52. [11] Milman YV, Luyckx S, Goncharuck VA, Northrop JT. Results from bending tests on submicron and micron WC–Co grades at elevated temperatures. Int J Refract Met Hard Mater 2002; 20:71–9. [12] Jia K, Fischer TE, Gallois B. Microstructure, hardness, and toughness of nanostructures and conventional WC–Co composites. Nanostruct Mater 1998;10:875–91. [13] Makhele-Lekata I, Luyckx S, Nabarro FRN. Semi-empirical relationship between the hardness, grain size and mean free path of WC–Co. Int J Refract Met Hard Mater 2001;19:245–9. [14] Golovchan VT, Litoshenko NV. On the contiguity of carbide phase in WC–Co hardmetals. Int J Refract Met Hard Mater 2003;21:241–4. [15] Kotoul M. On the shielding effect of a multiligament zone of a crack in WC–Co. Mater Sci Eng A 1997;234–236:119–22; Shielding model of fracture in WC–Co. Acta Mater 1997; 45:3363–76. [16] Liu B, Zhang Y, Ouyang S. Study on the relation between structural parameters and fracture strength of WC–Co cemented carbides. Mater Chem Phys 2000;62:35–43.