STRENGTH OF PARTICULATE COMPOSITES WITH A HIGH

CONTENT OF HIGH- MELTING POINT FILLER

L.L. Mishnaevsky Jr and S. Schmauder

Universityof StuttgartStaatliche Materialprüfungsanstalt (MPA)

Pfaffenwaldring 32, D-70569 Stuttgart (Vaihingen), Germany

Abstract

Interrelations between microstructural parameters of particulate composites with ahigh content of high melting point filler and the strength and damage initiation in thecomposite are studied using methods based on the fractal theory. Effects of the pre­sence of a skeleton from fiBer grains in the composite, size of filler grains and structureof filler/matrix interface on damage initiation in a loaded composite are investigatedtheoietically. Conditions at which the skeleton is formed during sintering and optimalduration of the sintering process are determined on the basis of a probabilistic sinteringmodel. A mathematical model for the formation of fractal interfaces in liquid-phasesintering of composites is developed and the influence of interface fractality on damageinitiation in matrix is investigated.

Processing and Design Issuesin High Temperature Materials

Edited by N.S. Stoloff and R.H. JonesThe Minerals, Metals & Materials Society, 1997

311

Introduction

Liquid-phase sintered particulate composites are widely used in industry. For example,drilling and cutting tools are made from hard alloys and cermets; such materials areused inproduction of high- pressure apparatus, etc.

The strength of the coinposites influence the efficiency and service properties of toolsmade from these materials to a large extent. This paper seeks to study theoreticallyfactors influencing the str~ngth of composites, and to analyze ways to improve thecomposite strength by varying parameters of composite microstructure.

As shown in [1-3], a skeleton from filler grains is formed in sintering particulate compo­sites with a high content of high-strength filler. This skeleton determines the strengthof the composite at initial stages of loading (in particular, the load at which irreversibledeformation begins) [2,3].

The destruction of a loaded composite with a high content of filler grains proceeds asfollows: when the material contains a skeleton from filler grains, just the skeleton isdeformed firstly; microcracks are formed in the skeleton and this process leads to thefaHure of the skeleton; thereafter, the irreversible deformation of the composite (withdestroyed skeleton) begins; microcracks are formed both in the matrix and in the fillergrains; the formation of microcracks in the matrix occurs mainly in the vicinity offiller grains or at interfaces, and the greater the stress concentration caused by thegrains, the lower the critical stress at which microcracks are formed in matrix [4,5];growth and coalescence of microcracks lead to the formation of initial cracks, whichgrow and propagate as well [6] and that determine faHure of a loaded body, when theenergy of loading is large enough. This short description of the mechanism of compositedestruction is based on the results obtained and presented in [1,3,4].

Here, we consider maihly the influence of some parameters of composite structure(grain size, structure of interface) and condition of sintering (time of sintering, masstransfer, etc.). on the damage initiation in composites. It is rather evident that themore intensive is damage initiation at early stage of composite deformation, the lessthe strength of composite and time to faHure (the greater is the microcracks density,the more the probability of crack formation and rate of crack growth [6]). Thus,when studying the interrelation between structure and strength of composite, one canconsider firstly the infiuence of composite structure on the initiation of microcracks.

To model the form8;tion of co.mposite structure in sintering, one uses the methods offractal theory and theory of fractal growth. These methods al\ow to take into accountthe complex nature of the formation,of filler skeleton and fillerjmatrix interface as wenas to describe the structure of as-formed objects on the basis of modelling their growth.

In [2], a mathematical model of formation -of the skeleton, its destruction and itsinfluence on the composite strength is developed on the basis of the reliability theory.Here, this model is used in order to determine the optimal duration of sintering (atwhich all workpiece contains a skeleton from joined filler grains) and the influence ofthe size of fil)~r grains on the strength of composite. These parameters determineconditionsof beginning of irreversible deformation in a loaded composite with highcontent of filler grains.

312

Influence of Iiller grainsize on the strength of a composite

It is known that the size of Iiller grains has a great impact on the strength of particulatecomposites. Theinfluence ofthis parameter on the composite strength is determined bythe following physical mechanisms: strengthening of the soft matrix of the compositeby hard inclusions, which present'a barrier for dislocations movement; availability ofskeleton from joined Iiller grains, which determine the hardness of material at initialstages of loading [2,3]; size effect: usually, the greater are Iiller grains, the lesss theirstrength and the grater the probability of their failure; etc. One can see that theinfluence of Iiller grain size on the composite strength is rather complex and dependson a number of contradictory tendencies. Here we consider only the influence of grainsize on the compressive strength of composites with skeleton.

As was shown in [2], the compressive strength of particulate composites with a hardskeleton from Iiller grains is proportional to the square root of the contiguity of theskeleton:

PrxVS (1)

(2)

where P - compressive strength of a composite with the skeleton from Iiller grains, S_ contiguity of the skeleton. The contiguity of the skeleton (which is delined as thegrain/grain interface area divided by the total surface of particles per unit volume ofthe material [1]) depends on the conditions of sintering, .and can be calculated by theformula from [2]:

S = _1_( CoW )p/w+1 _ [nd(1 + vtc)](ndv)-2W_1f (2/ + 1)]7rd2 2 + W c Coexp L L t W

where S - contiguity of the skeleton, Co = 7r4>2/4, ft - incomplete gamma-function, v ­the rate of sam pIe shrinkage insintering, wand 4> are constants of the Iiller material,which enter into the equation of diffusion-controlled growth of necks between Iillergrains in sintering [7] (Le. the diameter of a neck is proportional to 4>t1/W, where t- time [1]), d - diameter of grains, n - the amount of Iiller grains per unit volume ofsämple, tc - time of sintering, L - average distance between grains.

Using eq. (2), one can analyze an influence of grain size on the compressive strength ofcomposites. If one sUPPQses~hattte main mechanism of neck growth during sintering isvolume self-diffusion, the coefficients wand 4> are equal to 5.0, and (DdG2d2v/kbT)1/W,

respectively, where T - temperature, G2is a free energy of solid/liquid interface, v ­volume of vacancies, Dd - diffusion coefficient, kb - Boltzmann constant.

Fig. 1 shows the compressive strength of composite (which is normalized by the coef­ficient proportionality between P and VB) plotted versus size of grains. In the calcu­Iation, the following inputdata were used:DG2vwt;/wH /4kbT(W + 2) ~ 1;n(1 + vtc)/ L ~ fr.01;

DG2vft(2/w 4- 1)(nv/ L)2/w+1 /4kbT ~ 1.

It can be seen that the compressive strength is a deqeasing function of the size of Iillergrains.

313

..c.+-'0>c::Q)•...•...CI)

Q)>.Ci)

CI)

Q)•...a..Eoü

18

.16.

14

12

10

8

6

4

2

oo 0.1 0.2 0.3 0.4 0.5

Grain size0.6 0.7 0.8

Figure 1: Compressive strength of composite with skeleton plotted versus grain size

Condition of skeleton formation in sintered composites

The availability of skeleton from filler grains increases the strength of composites, espe­cially, at initial stages of loading [2,3]. The process of skeleton formation in sinteringdepends on the shrinkage rate, physieal properties of the filler material, duration ofsintering, etc. In the following, we consider conditions at which the skeleton is formedduring sintering. The processes of joining small grains and formation of aggregates(skeleton) from them can be considered as being similar to the formation of fractalclusters from small partieies (aggregation) [1,7]. The aggregation of small particlesproceeds as folIows: filler grains form relatively small aggregates, and thereupon theaggregates grow and coalesce. One can write the following equation of skeleton forma­tion in sintering:

< dM/ dt = mpt!::.M (3)

where M - ·number of filler grains which have .integrated into the skeleton in a giventime , !::.M - the average number of grains by which the value M increases due tojoining other aggregate of filler grains, Pt - the probil.bility of joining two particles perunit time, m = F/nd2 , F - exterior area of the aggregate of grains, m - the number ofgrains on exterior surface of the aggregate. If the process of cluster growth is isotropie,one can write :F = 12.7L2, where L - linear size of an aggregate from filler grains. Therelation between the number of elements in a fractal cluster and its linear size looks asfollows [8] : M = (iJd)DI, where Df - fractal dimension of the skeleton. This valuecharacterizes a density of partieies in the skeleton. Since the volume conte nt of thefiller in the composite is constant, the fractal dimension of skeleton cha;acterizes theamount of partieIes whieh are interconnected in relation to the total amount of fillerparticles.

314

The probability of joining two aggregates from filler grains (or two filler grains) perunit time cari be determined by the approximate formula from [2]:

(4)

where l = 1.24(n/t:..Mtl/a, t:..M is taken to be equal to the average number of grainsin the aggregates, Po - connectivity of skeleton (Le., the ratio between the numberof joints between filler grains and the number of particles [2]), v - shrinkage rate insintering.

Substituting eq. (4) into eq. (3), one can find the time of sintering which is necessaryfor the formation of a skeleton from filler grains in a workpiece with linear size Lm :

(5)

Correlating eqs. (4) and (5) one can conclude that the time needed for the formationof a skeleton depends on the rate of shrinkage, size of filler grains and their densityin the workpiece. One can note as well that the time of skeleton formation does notdepend strongly on the workpiece size.

Eq.(5) allows us to determine the duration of sintering which is enough for formationof skeleton from interconnected filler grains in all sintered body. The availability of thisskeleton contribute to the strength of composite, especially at initial stages of loading.

Redistribution of grain sizes during sintetihg and interface structure

In previous sections, tlie conditions of skeleton formation from filler grains and itsmaximal strength were considered. Yet, the strength of a composite is determined bythe micro- and macrocracking not only in the filler grains, but in the matrix as well,especially, after failure of the skeleton. The initiation of microcracks in a particulatecbmposite which does notcontain a skeleton (or after the failure of skeleton) proceedsmainly at the filler/matrix interface, sites of high stress concentration in the matrix[4]. The stress concentration in the matrix depends on shape of filler grains, whichis determined in turn by conditions of the liquid-phase sintering. Surface layers, and,consequently, shapes of filler particles are formed by the dissolvation/precipitation ofliller material in liquid phase. These processes cause redistribution of sizes of fillergrains as well: the small filler grains dissolve in the liquid phase and then the dissolvedmaterial diffuses through the liquid phase andis deposited on the large filler grains[1, 7]. This is caused by the well-known dependence of the solubility of material onthe radius of surface of a body which dissolves [9]. Such redistribution of the fillermaterial leads to the increase in 'the average size of the filler grains. One can write thefollowing equation for the growth of filler grains during sintering: da/dt = F(a), whereF(a) is an increasing function of a, a is the average size of the filler grains, t - time.Suppose that the function F(a) can be present~d ~ apower function F(a) = man,where m and n are some constants which characterize the intensity of grain growth.Consider mainly a group of relatively large filler grains (which grow during the masstransfer in sintering) since just large filler grains make higher stress concentrationsin the matrix when the composite is loaded. If one accepts that a fractal object isformed due to the precipitation of dissolved filler material on the filler grains, one can

315

determine the fractal dimension of new formed surfaces as weIl. The following relationbetween the linear size of a grain and the number of particles which have aggregatedand formed the grain can be written: da/dN = 'lj;(a, N), where N - the number ofparticles which have formed the cluster by the mechanism of the random joining ofparticles, 'lj; - some function.Having detennined the function 'lj; and integrating theequation of grains growth, one can find. the fractal dimension D of the cluster as aindex power in the formula: N<x aD (this method of determination of the fractaldimension of growing objects was developed by Thrkevich and Scher [10, 11]). Thevalue 1/ = dN / dt is the content of the dissolved filler material in the liquid phase.Here, the value 1/ is supposed to be constant. After some rearrangements, one derives:da/dN <X (l/1/)an• By definition of the fractal dimension of a cluster, one deduces:N<x anH, and D = n+ 1.

These relations mean that the greater is the rate of diffusion controlled mass transferin the liquid-phase sintering, the greater the fractal dimension of large grain surfaces(grain/filler interfaces) in the as-formed composite. It is of interest to compare thisconclusion with the results by Tanaka [12], which investigated interrelations betweenconditions of cold work in pure iron and the fractal dimension of grain boundaries.

Damage initiation in matrix in the vicinity of filler grains

Fracture of particulate composites begins with the initiation and coalescence of micro­cracks in the matrix. Let us consider now conditions of microcracks initiation in thematrix in the vicinity of filler grains. The local stress in anypoint of the compositecan be determined by the formula like: (7 = K (70, where (70 is the applied stress, and Kis the stress concentration factor in point. If the grain is elliptically shaped, the stress

concentration factor K is a linear function of ;;rp, where a is the linear size of thegrain, pis a radius of the hill on the grain surface [11]. Consider also the more reali­stic case when the grain (inclusion) has a surface like shown in Fig. 2. The followingsymbols were used: a - angle betwen two radii from the grain centre to the ends of aconditional hill on the grain surface, !1R - the distance between the ends of the hill, a- the size of a grain ).

In this case, any "hill" on this rand-o~ Burface concentrates stresses and a microcrackcan be initiated at any site in which the stress concentration factor exceeds some critical

value. The linear relation between K and ;;rp is appröpriate for this case as weil, butthe values p and a depend on the random characteristics of the grain surface. Considerthe dependence of these characteristics of the geometry of grain (a and p) on the grainsurface structure. If the surface of filler grain is formed during liquid-phase sintering,and determined by the solution/ diffusion/deposition of the filler material, it meansthat the grains are fractal clusters [8] and the surface of the grains is fractal. Supposethat the deposition of the filler material and the grains growth proceed isotropically.Then, the relation between the length of its surface Sand the distance between itsends !1R for some "hills" looks as follqws: S<x !1RD, where D- the fractal dimensionof the surface of grain. If one approximates this hill by a sEmii~ircle,one can obtaina/ p = 1 - cos(a/2), where S/!1R = (7r /180)a/ sin(a/2), ais the angle shown in Fig.2.After some rearrangements one can derive a following approximate relation:

(6)

316

Figure 2: Filler grain with fractal surface

From the definition of the stress concentration factor and eq. (6), one can see that thestress concentration factor and the value [g(al +a)/Nh](D-l)/2 are related. Here al andaare the sizes of the elliptical grain, Nh is the number of "hills" on the grain surfaceand g is a characteristic of the ellipsoid al shape of the grain.

Thus, the fractal dimension 6f the filler/matrix interface and the stress concentrationfactor in the matrix ofthe composite are related: the greater the fractal dimensionof thefiller grains surface, the greater the stress concentration in the matrix of thecomposite. It is clear that the more the stress concentration in the matrix of composite,the greater the damage in the composite, and, consequently, the less its strength [6].Thus, it can be concluded that the greater is the fractal dimension of the filler/matrixinterface in particulate composites, the less the strength of the composites.

Conclusions

The influence of sintering conditions on the structureand damage formation in liquidphase sintered particulate composites is investigated theoretically on the basis of themethods of fractal theory. Conditions of formation of skeleton from fiBer grains insintered composites are determined.

It is shown that the conditions of the diffusion mass transfer in liquid phase sinteringinfluence the interface structure and strength of composites: the greater is the rate ofmass transfer of fiBer material in sintering, the greater th.e fractal dimension of filler/matrix interface and the more intensive the damage initiation in composite matrix un­der loading.

317

AcknowledgementThe author (L.L.MJ is grateful to the Alexander von Humboldt Foundation for thepossibility to carry 6utthe researchproject at the University of Stuttgart, MPA (Ger­many).

References

1. P.S. Kisly et al , Physical and chemical foundations of production of high-meltingsuperhard materials, (Naukova Dumka, Kiev, 1986), 53-173 .

2. L. Mishnaevsky Jr, "A New Approach to the Analysis of Strength of Matrix Compo­sites with High Content of Hard Filler", J. of Appl. Comp. Matls, Vol.1, pp.317-324,1995

3. S.B. Luyckx, "Contiguity and the fracture process of WC-Co alloys, in: Advancesin Fracture Research", Proc. Int. Conr. Fracture-5, ed. D.Francois, (pergamon Press,1981) Vol.2

4. T.J. Chuang et al, "Creep rupture of metal-ceramic particulate composites" , Proc.Int.Conf. Fracture - 7, eds. S. Salama et al (Pergamon Press, 1989) Vol.4, pp.2965- 2975

5. S. Schmauder, Die Modellierung zähigkeit bestimmender Prozesse in Mikrogefügenmit Hilfe der Finite-Elemente-Methode, VDI, Reihe 5, Nr. 146, (VDI-Verlag, Düsseldorf,1988)

6. L.L. Mishnaevsky Jr., "Damage evolution in brittle materials", Materials AgeinggruiComponent Life Extension, ed.. V.Bicego et al (EMAS, London, 1995) Vo1.2, pp.1135­1143

7. W.D. Kingery, Introductio'n ioto Ceramics, (Wiley, NY, 1960)

8. B.M. Smirnov, Physics of Fractal Aggregates (Nauka, Moscow, 1991)

9. R.A. Swalin, Thermodynamics of Solids, (John Wiley and Sons, 1968)

10. L. Turkevich and H. Scher,Fractals in Physics, eds. L. Pietronero and E. Tosatti,(North-Holland, NY, 1Q86)

11. L.L. Mishnaevsky Jr, "Str~cture of Ip.terface in Sintered Compositesand its In­fiuence on the Strength of Composites", Proceeding' EUROMAT-95 (4th EuropeanConference on Advanced Materials and Processes, Venicej Padua, 1995) (AIG, 1995)

12. M. Tanaka, "Effects of Cold Work on the Fractal Dimension of Grain Boundariesin Pure Iron", Z. r. Metallk., 87, NoA, (1996), pp.310-314

318