finite element modelling of the thermal residual

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JOURNAL OF HARD MATERIALS, VOL. 3, NO. 2, 1992

ICSHM4-PAPER 39Finite Element Modelling of the Thermal ResidualStress Distribution in a WC-IOwt%Co Alloy

R. SPIEGLER, S. SCHMADDER & H. E. EXNER

143

ABSTRACT The different thermal contraction of the hard phase and of the binderphase results in residual stresses in hard metals after cooling from sintering temperature.The spatial distribution of residual stresses is computed using a two-dimensional finiteelement model for a realistic microstructure of a WC-] Owt%Co aIloy. Distributions of thedeviatoric and the hydrostatic stress components in the individual phases are given. Theyare compared with other theoretically derived or measured stress distributions. The finiteelement results emphasize the importance of the local residual stress distributions: whilethe mean equivalent stress in the cobalt phase is calculated to be below the yield limit, themaxima are weIl above the yield limit. Furthermore, these maxima are found atinterfaces in the cobalt phase and are relaxed by local plastic deformation. Stresstriaxialities reach local peak values leading to the formation of sintering voids in thebinder phase during cooling.

1. INTRODUCTIONIn the process of manufacturing hard metals, the difference in thermal expansion ofthe carbide and binder causes residual stresses when cooling down from sintering toroom temperature [1,2]. Residual stresses originating for other reasons, e.g. by surfacegrinding [3, 4], will not be discussed in this paper.

The local tensor of residual stresses, (Jij> can be separated into a hydrostatic and adeviatoric part. The hydrostatic part is frequently characterized by the hydrostaticstress (JH:

3(JH = 1/3 I (Jkkk~land the deviatoric part by the equivalent stress according to:

3 3(Jv = (3/2 I I (Jij - c5,PH)2) 1/2i=l j=lIn general, hydrostatic and equivalent stresses inftuence the mechanical properties ofhard metals [5-8]: depending on their sign, hydrostatic stresses favour or retard voidR. SpiegIer, Th. Goldschmidt AG, Goldschmidtstr. 100, D-4300 Essen, Germany; S. Schmauder, MaxPlanck-Institut fr Metallforschung, Institut fr Werkstoffwissenschaft, Seestr. 92, D-7000 Stuttgart,Germany; H. E. Exner, Technische Hochschule Darmstadt, Petersenstr. 30, D-6100 Darmstadt, Germany.


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144 R. SPIEGLER, S. SCHMADDER & H. E. EXNERformation in the ductile binder phase, while equivalent stresses lead to elastic or plasticdeformation.

In general, the hydrostatic residual stresses are in tension in the binder phase andin compression in the carbide [1]. In technical WC-Co alloys with cobalt contentsbetween 5 and 15 wt%, the mean hydrostatic stresses "Hn the phases lie within thefollowing ranges:

O"H(CO): 1000 to 1300 MPa"H(WC): -100 to -400 MPaThese values are quoted from an earlier review article [1], where hydrostatic stressescalculated theoretically or obtained experimentally from X-ray strain measurementsare compared for the individual phases.Only little is known about the magnitude of mean equivalent stresses in hardmetals and the spatial distribution of hydrostatic and equivalent stresses in theindividual phases [2,9, 10]. In the present work a finite element approach is presentedwhich yields estimates for such stress distributions.2. FINITE ELEMENT CALCULATIONSFigure 1 shows the microstructure surrounding a crack tip in a WC-10wt%Co alloy.This microstructure was chosen because of the presence of a variety of large and smallbinder regions. Finite element (FE) modelling of this microstructure was also part of astudy dealing with the simulation of the binder crack path [11]. The FE nets used inthat study were the basis of the present calculations. Figure 2 shows the FE modelconsisting of two subnets. The subnets were connected by means of the subnettechnique [12]. Subnet 1 represents the two-phase region shown in Figure 1 andconsists of 1117 nodes and 2170 triangular elastic plane strain elements. Subnet 2represents the extended surrounding of this two-phase region taken as a continuumwith the elastic properties of the composite: it consists of 286 nodes and 472 triangularelastic plane strain elements. The FE mesh was designed with the mesh generatorPATRAN [13]. The computations were performed with the commercial FE programPERMAS [14]. The fOllowing elastic constants (Young's modulus E and Poisson'snumber v) and thermal expansion coefficients a were used [10, 15]:

Ewc = 714 GPa vwc = 0.19 awc = 5.8 X 10-6 K- JEco = 211 GPa vCo = 0.31 aco = 12 X 10-6 K-JEcomp = 595 GPa VComp = 0.22 aComp = 6.4 X 10-6 K-l

The thermal expansion coefficient of the composite was calculated using the relation[10]:aComp= L aiKi V/ L Ki Vi

where Ki are the bulk moduli and Vi are the volume fractions of the individual phases(i = Co, WC; VCo = 0.164). To calculate residual stress and strain distributions, initialstrains a6T were imposed on the structure. The temperature difference 6T waschosen to be 800 K [1].3. RESULTSResults of the residual stress calculations are given in Figures 3-6. The distributions offour quantities have been examined: Figures 3(a) and 3(b) show the spatial distribu-


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FE MODELLING OF THERMAL RESIDUAL STRESS 145

Figure 1. Microstructure surrounding a crack tip in a WC-lOwt%Co alloy (dark phase, cobalt;grey phase, WC).

tions of the hydrostatic stresses O"B and of the equivalent stresses O"y, respectively. Thedistribution of stress triaxiality y/, which is defined as:

y/ = O"H/O"yis shown in Figure 4(a). The S-value is an important quantity with wh ich a criterionfor the nucleation of voids under stress [11, 16] can be formulated and is defined as:S= O"H"/7IK

The distribution of the S-value is shown in Figure 4(b). Figures 5 and 6 depict thefrequency distributions of hydrostatic and deviatoric stresses in the individual phases.

Hydrostatic stresses in the cobalt phase are rather uniformly distributed and tensilein character (Figure 3(a)). Their values range between 900 and 1800 MPa with amean value of 1330 150 MPa (the values denote mean value and standard deviation). As can be seen in Figure 3(a), the highest values are reached in the neighbourhood of acute dihedral angles, when pockets of cobalt are surrounded by Wc. Sincethe standard deviation of the distribution is small compared with the mean value(Figure 5(a)), the hydrostatic stress state of the cobalt phase may be characterizedsatisfactorily by its mean value.

The situation is different for the hydrostatic stresses in the WC phase (Figure5(b)). They are mainly in compression with values up to -600 MPa. A comparisonwith the local distribution of Figure 3(a) shows that the maximum compressivestresses in the WC phase are found on shortest connections between cobalt domains.However, next to the WC-Co interfaces there exist small WC regions which are underhydrostatic tension (Figure 5(b)). The me an hydrostatic stress amounts to-210 110 MPa. Thus, the width of the distribution is comparable with that of thecobalt phase, while the mean value is much smaller. As a result, the stress situation inthe WC phase should be described by both the mean value and its standard deviation.

In contrast to the narrow distribution of hydrostatic stresses, the frequencydistribution of equivalent stresses in the cobalt phase shows a rather broad distribution(Figure 6(a)). The mean equivalent stress is 950 480 MPa. The calculated maxi-


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146 R. SPIEGLER, S. SCHMADDER & H. E. EXNER1

2

Figure 2. Finite element model of the microstructure of Fig. 1: subnet 1 with the phasearrangement of an actua1 microstructure is embedded in subnet 2 with the e1asticproperties ofthe composite. The cobalt phase in the two-phase region (subnet 1) is shaded.

mum values in the cobalt phase are reached near WC corners (Figure 3(b)). Theequivalent stresses in the WC phase are smaller with a mean value of 700 320 MPa.The distribution shows an asymmetrie peak at about 600 MPa (Figure 6(b)). Highequivalent stresses are less frequent than in the cobalt phase, as can be seen bycomparison of Figures 6a and 6b.

Stress triaxiality and S-value are irregularly distributed and reach locally peakvalues (Figures 4(a) and 4(b)): the maximum values of stress triaxiality and S-valuein the cobalt phase are 2.6 and 0.02, respectively.

4. DISCUSSIONThe calculated mean hydrostatic stresses are in agreement with results from X-ray


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( 0 )

0" [MPa]1: 1500: 1400: 1200:

05: -200: -300: -400

( b)

0y [MPa]1: 14002: 10003: 600

Figure 3. Distributions of (a) hydrostatic stresses (OH) and (b) equivalent stresses (Oy).

measurements on hard metals of the same composition [1]. lnterestingly, the distribu-tion of hydrostatic stress in the WC phase, as calculated in the present study, extendsto tensile stresses up to 200 MPa. This is in accordance with results of neutrondiffraction experiments of Krawitz et al. [10], which suggest both compressive andtensile stresses in the WC phase. However, Krawitz et al. found very broad distribu-tions extending over several gigapascals in contrast to the present FE results.

Kreher and Pampe [17] calculated the residual stresses in hard metals using theprinciple of 'maximum information theoretical entropy': they determined a probabilitydistribution of the internal stresses by considering the fundamental thermamechanicalequations and the boundary conditions of the measured quantities such that the


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148 R. SPIEGLER, S. SCHMADDER & H. E. EXNER( 0)

5Tl

1: 2.42: 1.63: 0.84: 05: -0.8

( b )

s1: 0.022: 0.01 1i&..: ........... ..

Figure 4. Distributions of (a) stress triaxiality (I] = aHlav) and (b) 5-value (5 = aH' 1]/10.

disorder of the stresses reaches a maximum. This leads to Gaussian probabilitydistributions of the local stresses. For a hard metal alJoy with a cobalt content of 10wt% corresponding to the alJoy composition as investigated in the present study, andfor a temperature difference D.T = 500 K, Kreher and Pompe [17] calculated a meanhydrostatic stress in the cobalt phase of 860 160 MPa and a mean hydrostatic stressin the WC phase of -190 250 MPa. The corresponding values of the present FEcalculations for the same temperature difference would be 830 100 MPa in thebinder and -135 70 MPa in the carbide phase. While the mean values of the FEresults are in reasonable agreement with the theoretical values, the widths of thedistributions differ distinctly. The difference may be attributed to the three-dimensionality of the calculations in ref. [17] in contrast to our two-dimensional FE


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(a)25

2000000 1500eTH (MPa)

5

o500

.~ 10oCl:

~ 20~'"uc~ 15cr~

(b)30

~~" 20Cl>;Jr~

.2:E10

Cl:

o-600 -400 -200 0

eT H (MPa)

Figure 5. Frequency of hydrostatic stresses (a) in the cobalt phase and (b) in the WC phase.

calculations. Furthermore, in ref. [17] Gaussian probability distributions of the localstresses are assumed, while our calculated distribution of hydrostatic stresses in thecobalt phase seems to be bimodal (Figure 5(a)). However, the bimodal nature of thehydrostatic stress distribution may be an artefact of the particular microstructurechosen. The advantage of the present FE calculation is the possibility to obtain stressdistributions in the microstructure and to calculate local S-values.

Although the me an equivalent stress in the cobalt phase is below the yield limit, themaximum stresses up to 1700 MPa would be high enough to cause local deformation inthe microstructure. Therefore, local plastic f10w is assumed to have occurred duringcooling in front of WC corners reducing the high peak values of equivalent stresses.Furthermore, equivalent stresses reach high values in the carbide (about 1800 MPa),which may lead to local deformation of carbide phase and damage of the carbideskeleton under additional external loading. lndeed, slip lines in WC crystals havefrequently been observed in transmission electron microscopy (TEM) studies [18].

Void formation in the cobalt phase of hard metals under externalloading at room


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150 R. SPIEGLER, S. SCHMAUDER & H. E. EXNER

2000

2000

1500

1000erv (MPo)

1000erv (MPo)

(0 )10

~

8~ >-u: 61> ::>CT~Cl> 4.~ Cil:: 2

0

0( b) 10~

8~ >-u:l> 6:J CT~Cl>

4~ El>l:: 20

Figure 6. Frequency of equivalent stresses (a) in the cobalt phase and (b) in the WC phase.

temperature has been suggested at stress triaxialities >3 and S-values ofO.08 [15, 16].These critical values are higher than the maximum values caused by residual stresses inthe present microstructure (I] = 2.6 and S = 0.02); however, as the yield limit ofcobalt decreases at high temperatures void nucleation during the cooling process afterthe sintering of hard metals is possible. Experimentally, there is evidence for theformation of such sintering voids. A detailed discussion on void formation underloading or under thermal stresses in the particular microstructure shown in Figure 2will be given elsewhere [16].

5. SUMMARYAccording to the FE results, the mean hydrostatic stresses in the Co and WC phase ofthe microstructure in Figure 2 are 1330 150 MPa and - 210 110 MPa, respectively. The hydrostatic stresses in the cobalt phase reach their highest values at acute


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FE MODELLING OF THERMAL RESIDUAL STRESS 151dihedral angles between WC-Co interfaces, while the highest compressive stresses inWC are reached at shortest connections between cobalt domains.

The distributions of hydrostatic stresses in the individual phases are rather narrowin contrast to the broad distributions of equivalent stresses. The mean equivalentstresses are 950 480 MPa in the cobalt phase and 700 320 MPa in the WC phase.Maximum values in the cobalt phase are found near WC corners and are high enoughto cause local plastic deformation.Stress triaxiality and S-value in the binder assume locally peak values of 2.6 and0.02, respectively. This could result in the formation of sintering voids.References[1] H.E. Exner, !in. Met. Rev., 4, 149-173 (1979).[2] A. Hara, M. Megata and S. Yazu, Powder Met. Int., 2,43-47 (1970).[3] H.E. Exner, Trans. AlME, 245,677-683 (1969).[4] D.N. French, Trans. AlME, 245,2351-2352 (1969).[5] G.G. Chell, in Advanees in Elasto-Plastie Fraeture Meehanies (L.H. Larsson, Ed.), AppliedScience, London, pp. 359-384 (1979).[6] A.K. Khaund, V.D. Krstic and P.S. Nicholson,J. Mater. Sei., 12, 2269-2273 (1977).[7] RA. Cutler and A.V. Virkar, J. Mater. Sei., 20, 3557-3573 (1985).[8] P.A. Mataga, Aeta Metall., 37,3349-3359 (1989).[9] S. hman, E. Prnama and S. Palmqvist,Jernkontorets, Ann., 151,126-159 (1967).[10] A.D. Krawitz, M.L. Crapenhoft, D.G. Reichel and R. Warren, Mater. Sei. Eng., A105/l06,275-281 (1988).[11] R Spiegler, S. Schmauder and H.F. Fischmeister, Proeeedings of the 2nd InternationalFinite Element Conferenee (INTES GmbH, Ed.), Steinkopf Druck Stuttgart, FRG, Strasbourg, France, pp. 21-41 (1990).[12] }.F.RS. Argyris and H.-P. Mlejnek, Die Methode der Finiten Elemente, Friedr. Vieweg&Sohn, Braunschweig (1986).[13] PDA Engineering, 2975 Redhill Avenue, Costa Mesa, CA 92626, USA.[14] INTES GmbH, Nobelstrae 15, D-7000 Stuttgart 80, Germany.[15] S. Schmauder, Die ModelIierung zhigkeitsbestimmender Prozesse in Mikrogefgen mit Hilfe

der Finite-Elemente-Methode, Forlsehrittsberiehte VDI, Reihe 5: Grund- und Werkstoffe Nr.146, VDI, Dsseldorf (1988).[16] R Spiegler, S. Schmauder and H.F. Fischmeister, Int. J. Plastieity (in press).[17] W. Kreher and W. Pompe, VI. Konferenz Metallkundliehe Probleme der Werkstoffentwieklung, Freiberg, GDR, 13-15 September 1989, Bergakademie Freiberg, GDR, pp. 107-117(1990).[18] S. Hagege, J. Vicens, G. Nouet and P. Delavignette, Phys. Status Solidi (a), 61,675-687(1980).

finite element modelling of the thermal residual

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