On the thermal residual microstresses in WC–Co hard metals

International Journal of Refractory Metals & Hard Materials 25 (2007) 341–344 www.elsevier.com/locate/ijrmhm

On the thermal residual microstresses in WC–Co hard metals V.T. Golovchan V. Bakul Institute for Superhard Materials of the National Academy of Sciences of Ukraine, 2 Avtozavodskaya Street, 04074 Kyiv, Ukraine Received 4 August 2006; accepted 7 August 2006

Abstract The formulae for calculation of thermal residual microstresses on the surface and in the center of a WC–Co hard metal sample are given. The theoretical results are in good agreement with experimental values measured by X-ray diffraction and neutron diffraction methods.  2006 Elsevier Ltd. All rights reserved. Keywords: WC–Co hard metals; Thermal residual microstresses; Surface and volumetric residual stresses

1. Introduction The thermal residual stresses in the phases of WC–Co hard metals arise during cooling from sintering temperature because of the pronounced difference in the thermal expansion coefficients of WC and cobalt. An analysis of early X-ray and theoretical investigations of these stresses is contained in review [1]. This topic continues to attract attention of the researchers at present as well. The effect of residual stresses on the fracture toughness of WC–Co hard metals was considered in [2]. The spatial distribution of residual microstresses in a WC–10 wt.%Co hard metal was computed using a two-dimensional finite element model in [3]. The effect of residual stresses on yield-point of the WC–Co hard metals was investigated in [4]. Neutron powder diffraction is used in [5] to study the effect of WC particle size on the thermal residual stresses in WC–Co. The aim of this note is to consider a strict theoretical algorithm for computing of the thermal residual microstresses in WC–Co hard metals and to compare the theoretical and the known experimental results. 2. The computing algorithm The thermal residual microstresses in a quasi-isotropic two-phase composite material are determined by the following formulae E-mail address: [email protected] 0263-4368/$ - see front matter  2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijrmhm.2006.08.002

rV1 ¼

3 V1

Z

T room T res:



K 1 ðT ÞK 2 ðT Þ K 1 ðT Þ  K 2 ðT Þ

 ða1 ðT Þ  a2 ðT ÞÞdT ; V1 rV2 ¼  rV1 V2

2 

1 V1 V2   KðT Þ K 1 ðT Þ K 2 ðT Þ



ð1Þ

Here Vi is the volume fraction of the ith phase, Ki is the bulk modulus of the ith phase, ai is the coefficient of the thermal expansion of the ith phase, i = 1, 2. In addition, K is the bulk modulus of the composite and Tres. is the temperature at which thermal stresses are first set up during cooling. The microstresses in (1) are the mean normal stresses r ¼ 13 ðrx þ ry þ rz Þ in any representative volume of the composite. The representative volume is small compared to the whole volume of a sample, yet large compared to the size of phase grains. The similar formulae for the thermal residual stresses on surface of a sample can be represented by the equalities Z T room 2 k 1 ðT Þk 2 ðT Þ rS1 ¼ ðaðT Þ  a1 ðT ÞV 1  a2 ðT ÞV 2 ÞdT ; V 1 T res k 2 ðT Þ  k 1 ðT Þ V1 ð2Þ rS2 ¼  rS1 V2 where 9K i li k 1 a1 ðl þ k 1 Þðk 1 a1  k 2 a2 Þ  1 1 ; a¼ ki ¼ 3K i þ 4li k k ðl1 þ k 2 Þ  l1 ðk 1  k 2 Þ V2 1 ð3Þ

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Here li is the shear modulus for the ith phase. The remaining designations are same as in (1). The stresses in (2) are the mean surface stresses r ¼ 12 ðrx þ ry Þ. The derivation of the relations (1) and (2) is based on equations of the thermoelasticity law and general scheme of volume averaging [4]. The formula (1) is correct in a case both elastic and plastic strains of a composite. The macroscopic bulk moduli K and k must be calculated from the following expressions K ¼ K 1V 1 þ K 2V 2 

V 1 V 2 ðK 1  K 2 Þ2 ; V 1 K 2 þ V 2 K 1 þ 43 lc

ð4Þ

2

V 1 V 2 ðk 1  k 2 Þ ; k ¼ k1V 1 þ k2V 2  V 1 k 2 þ V 2 k 1 þ lc

where lc is the shear modulus of the comparison body [6]. The choice of appropriate values of the parameters Tres. and lc is a main problem of the algorithm under consideration. The accuracy of theoretical evaluation of the thermal residual microstresses in some composite depends essentially on these parameters. 3. Calculation of the thermal residual microstresses in WC–Co hard metals

V 1 V 2 ðl1  l2 Þ2 l ð9K c þ 8lc Þ l ¼ l1 V 1 þ l2 V 2  ; bc ¼ c 6ðK c þ 2lc Þ V 1 l2 þ V 2 l1 þ bc ð5Þ where Kc is the bulk modulus of the comparison body [6]. Young’s modulus is determined by the following formulae E = 9Kl/(3K + l). The equalities for the moduli of the comparison body we set in the form K c ¼ K 1 ð1  CÞ þ K 2 C;

ð6Þ

where contiguity of the carbide phase C is determined by the following expression C ¼ 1  V 0:644 Co expð0:391 VÞ

ð7Þ

Here V is the variation coefficient of WC grain size distribution [8]. All magnitudes in (1–6) with subscripts 1 and Table 1 Physical properties of constituents at room temperature [7] Property

WC

Co

l, GPa K, GPa E, GPa m a · 106, K1

301 392 719 0.194 5.2

81.5 187.3 213.5 0.31 13.4

VCo

C

Eexp (GPa)

lexp (GPa)

Etheor. (GPa)

ltheor. (GPa)

0.053 0.103 0.164 0.25 0.305 0.368

0.81 0.71 0.61 0.48 0.41 0.34

673 642 595 533 500 468

280 267 245 218 202 188

678 640 596 536 500 461

283 265 245 219 203 186

2 we refer to the cobalt binder phase and carbide phase, respectively. The results of the calculation at V = 0.6 are presented in Table 2. Thus, we have excellent agreement between theoretical and experimental values of the moduli E and l in the range of technical WC–Co hard metals. Of course, the obtained strict agreement is unexpected because of the possible experimental errors in the order of (1–2)% and more. Notice only, the computing algorithm (4–7) is very simple. We suppose the temperature dependence of the elastic constants and the coefficients of the thermal expansion in the form E1 ðT Þ ¼ 213:5  0:05ðT  T room Þ;

Relevant values of elastic moduli and linear coefficients of thermal expansion of the WC–Co constituents at room temperature are given in Table 1. At first we compute the macroscopic Young’s modulus E and shear modulus l and compare them with experimental ones [7]. The shear modulus is computed using

lc ¼ l1 ð1  CÞ þ l2 C;

Table 2 The experimental [7] and the theoretical values of elastic constants of WC–Co hard metals at room temperature

a1 ðT Þ ¼ 13:3  10

6

K

m1 ðT Þ ¼ 0:31;

1

for cobalt binder and E2 ðT Þ ¼ 719  0:05ðT  T room Þ; a2 ðT Þ ¼ 5:2  10

6

K

m2 ðT Þ ¼ 0:194;

1

for carbide phase [9]. The value of temperature Tres. we take equal to Troom + 700 K. The values for the thermal residual microstresses are calculated using Eqs. (1)–(4) and (6), (7). The results are given in Table 3. In this Table rVL and rVU are the lower and the upper bounds for volumetric stresses rVWC , rSL and rSU are the similar bounds for surface stresses rSWC . The lower bounds and the upper bounds were calculated at C = 1.0 and C = 0.0 in (6), respectively. It is of interest to compare these results with some experimental ones. The experimental data of Japanese and Swedish investigators are cited in [1]. The measured stresses in the WC agree with the calculated rSWC sufficiently close in the all range of the WC–Co under consideration, Table 4. In addition, the experimental values satisfy the inequality rSL < rexp < rSU . Notice that the agreement between rSWC and the experimental values of residual stresses in the tungsten carbide [10] is good too. The recent experimental data are contained in [5]. The authors of this paper found that for fixed Co content the residual stresses in the WC increased in magnitude with decreasing WC particle size. The stresses in the WC were calculated for KWC = 439 GPa [5]. Using KWC = 392 GPa (see Table 1) we must reduce these experimental data in 1.12 times. In that case we have for the WC–20 wt.% Co the residual stresses 0.342 GPa, 0.490 GPa, 0.590 GPa and 0.675 GPa for coarse,

V.T. Golovchan / International Journal of Refractory Metals & Hard Materials 25 (2007) 341–344

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Table 3 The calculated values of the thermal residual microstresses, GPa VCo

rVCo

rVWC

rSCo

rSWC

rVL

rVU

rSL

rSU

0.053 0.103 0.164 0.25 0.305 0.368

1.89 1.76 1.62 1.41 1.27 1.12

0.11 0.20 0.32 0.47 0.56 0.65

1.06 0.99 0.90 0.79 0.71 0.62

0.06 0.11 0.18 0.26 0.31 0.36

0.111 0.22 0.36 0.57 0.71 0.88

0.058 0.11 0.18 0.29 0.35 0.43

0.061 0.13 0.21 0.33 0.41 0.51

0.03 0.059 0.096 0.15 0.19 0.23

Table 4 The calculated and the experimental [1] values of the residual stresses in the carbide phase of some WC–Co hard metals Cobalt content (wt.%)

rWC (exp) (GPa) (approx.)

rSWC (GPa)

7 10 16 30

0.15 0.17 0.28 0.40

0.13 0.18 0.26 0.41

medium, fine and ultrafine sample grade, respectively. The calculated rVWC ¼ 0:56 GPa for this hard metal corresponds to its microstructure with dWC = 1.6, 2.3 lm (see Table 2 and [7]). And what is more, the above experimental values of the residual stresses lie practically in the range [rVL , rVU ]. Thus, the agreement between the theoretical and experimental results is very good. The experimental values of rWC for WC–10 wt.% Co [5] correspond to calculated ones worse, these stresses for fine and ultrafine sample grade are less than rVL . Notice at last that the thermal residual stresses rVWC and rSWC in WC–17 wt.% Co are equal to 0.494 GPa and 0.276 GPa, respectively. The measured by neutron diffraction method rWC is equal to 0.501 GPa [11]. 4. Discussion As input data the above theoretical algorithm (1–7) contains the elastic moduli K, l and the coefficients of the thermal expansion of both phases of the WC–Co composite, and also the temperature Tres. and the variation coefficient V. In common, for an evaluation of residual stresses the numerical values of eight magnitudes together with VCo should be given. Most uncertain among all these parameters is the Tres. It is the temperature above which the thermal stresses disappear. For the Tres. a range between 400 and 1200 C was discussed in earlier works [1]. It is quite probable, that the Tres. can depend on cobalt content and chemical composition of a binder. The residual microstresses depend essentially on the contiguity of carbide phase C. This factor defines the degree to which a continue skeleton of the carbide phase exists in a hard metal. According to the experimental data [12 and 13] the contiguity C increases with decreasing of dWC at fixed VCo. Indicated in Table 3 the lower rVL and the upper rVU bounds correspond to the composite with carbide matrix and the composite with cobalt matrix,

respectively. Any theoretical or experimental values of volumetric residual stresses in the WC should satisfy the inequality rVL < r < rVU provided that used by us at evaluations the values of parameters correspond to a considered hard metal. The experimental data for the WC–20 wt.% Co [5] satisfy this inequality. One of reasons why there is a discordance between calculated and experimental data for WC–10 wt.% Co [5] can be related to the value Tres. = Troom + 700 K chosen by us. It is necessary also to mean that neutron diffraction method is accurate only for high Co contents, when contiguity is low [14]. 5. Conclusions The above theoretical findings may be summarized as follows: (1) The thermal residual microstresses in centre of a large WC–Co hard metal piece are more than the stresses on its surface approximately in 1.8 times. (2) The measured by X-ray residual stresses correspond to the surface stresses. (3) The measured by neutron diffraction residual stresses correspond to the volumetric stresses. (4) The magnitude of the residual stress in the WC increases with decreasing the dimensionless quantity d WC V Co (or with increasing contiguity, C ¼ 1  dlWC , lCo Co V WC [14]) for constant Co content. (5) The above theoretical algorithm allows reasonably to estimate the thermal residual stresses in a WC–Co hard metal provided that all input data are known. We mean the average volumetric r ¼ 13 ðrx þ ry þ rz Þ and surface r ¼ 12 ðrx þ ry Þ residual microstresses on representative volume which contains a great many of carbide grains. The macroscopic properties of the hard metal in the representative volume coincide with the properties of a whole hard metal sample.

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