On constrained sintering—III. Rigid inclusions

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Acra metolf. Vol. 36, No. 9, pp. 241 l-2416, 1988 Printed in Great Britain. All rights reserved

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Copyright .!; 1988 Pergwnon Press plc

OVERVIEW

NO. 70

ON CONSTRAINED SINTERING-III. RIGID INCLUSIONS R. K. BORDIA and G. W. SCHERER E. I. du Pont de Nemours & Co.. Central Research & Development Department, Experimental Station E356, Wilmington, DE 19898, U.S.A. (Received 8 Mav 1987; in rerisedform 27 October 1987) Abstract-In Part II of this series we examined constitutive equations that have been used primarily to analyze constrained sintering problems, including sintering of a matrix phase with rigid inclusions. For the latter problem, two important effects have been identified: generation of internal stresses (which could lead to formation of crack-like defects) and deviation of the densifi~tion behavior from the rule of mixtures. Currently available analyses give very different results for the stresses, which we show to be due to the choice of the constitutive laws. The analyses that give large values of internal stresses and significant slowing down of densification use constitutive laws that underestimate the shear relaxation of the densifying body, which leads to a negative value for the Poisson’s ratio. Since this has never been observed in sinter-forging experiments, it is concluded that either the internal stresses are small (as predicted by the constitutive laws given in Part I) or the basic assumptions of linearity and isotropy used in all of the analyses are incorrect. We discuss some phenomena that could be important in explaining the densification of composites. RsDans la seconde partie de cette s%e d’articles, nous avons examine les equations de base qui ont 6th utilisies jusqu’a present pour analyser les problemes de frittage sous contrainte, et parmi eux le frittage d’une phase matrice comportant des inclusions rigides. Dans ce demier car+,deux effets importants ont et& identifies: la production de contraintes intemes (qui pourraient conduire ii la formation de defauts du type fissures) et la deviation du comportement de densification par rapport a la regle des melanges. Les analyses que l’on rcncontre couramment donnent des resultats tr&s differents pour les contraintes; nous montrons que ceci est dfi au choix des lois de base. Les analyses qui conduisent a de fortes valeurs des contraintes internes et I une diminution significative de la densification utilisent des lois de base qui sousestiment la relaxation de cisaillement du solide soumis i une densification, ce qui conduit a une valeur. negative du coefficient de Poisson. Comme ceci n’a jamais ete observe dans les experiences de forgeage-frittage, nous en dMuisons que ou bien les contraintes intemes sont faibles (comme le pmvoient les lois de base propo&s dans la partie I). ou bien les hypotheses de Ii&a&C et d’isotropie utilisees dans toutes ces analyses sont incorrectes. Nous discutons quelques phenomenes probablement importants pour expliquer la densification des composites. ZtIsmilllleslfamung--In Teil II dieser Serie haben wir Grundgleichungen unte~ucht, die haupt~chlich zur Analyse des Sintems unter einschrlnkenden Redingunge, einschlieBlich des Sinterns einer Matrix mit starren Einschliissen eingesetzt werden. Rei dem Problem mit starren Einschliissen wurden xwei wichtige Effekte gefunden: Erxeugung innerer Spannungen (welche zur Bildung ri~~hnIicher Fehler filhren konnten) und Abweichung des Verdichtungsverhahens von der Mischungsregel. Die zur Zeit verfiigbaren Analysen ergeben sehr unterschiedliche Ergebnisse fur die Spannungen; wir zeigen, daB diese Unterschiede auf die Wahl der Grund~eichungen zuriickgehen. Analysen. die hohe innere S~nnun~n und ~t~chtliche Verlangsamung des Verdichtungsprozesses ergeben, benutxen Grundgleichungen, die die Scherrelaxation des Sinterkiirpers unterschltxen und damit zu einer negativen Querkontraktionsxahl fiihren. Da dieser Befund niemals in Sinterex~rimenten beobachtet worden ist, la& sich folgern, dal3 entweder die inneren Spannungen klein sind (wie mit der im Teil I abgeleiteten Grundgleichung vorausgesagt) oder die Annahmen der Linearitlt und der Isotropie bei samtlichen Analysen unrichtig sind. Wir diskutieren einige Erscheinunge, die fur die erkliirung der Verdich:ung von Verbunden wichtig sein kiinnten.

1. INTRODUCTION

and Poisson’s

In Part I [I] we proposed a constitutive

equation to describe the response of a sintering body to external constraints. The material was assumed to be isotropic and to have a strain rate linearly proportional to the applied stress. The constitutive equation could be written in terms of the shear and bulk viscosities (G, and K,,, respectively) or the uniaxial viscosity (E,) 2411

ratio

(up). Several

authors

have pro-

posed expressions for G, and Kr, and these were critically examined in Part II [2]. It was shown that some of those models imphed vp < 0, in contradiction to experimental evidence. In this paper, we examine the sintering of a composite containing rigid spherical inclusions. The stresses in the sintering matrix have been treated [3-51 with widely differing results. We

BORDIA and SCHERER:

2412

OVERVIEW NO. 70-111 Part I [l] as: c = (C + U)/KP’

in the

(3)

where the mean stress is defmed in equation (2) and Kp is the bulk viscosity. The sintering potential, ,X, is the compressive hydrostatic stress that causes densification of an unconstrained sintering material [I]. For a uniform matrix (no heterogeneity), there is no stress so: i = Z/K,.

Sbherical Non-Sintering Inclusion

Fig. 1. Geometry of the problem: a spherical inclusion in a spherical shell.

show in Section 2 that the anaiyses that predict large matrix stresses are those for which Poisson’s ratio is negative. Since realistic models indicate that the stresses are small, we must find another explanation for the retardation of densification that is observed [6,7] in such composites. Some possibilities are discussed in Section 3.

2. I. Maximum stress

The problem of a non-densifying inclusion in a sintering matrix has received considerable attention f3-51. The geometry of the problem is shown in Fig. I. For this case, a compressive radial stress, a, and tensile hoop stresses, Q@(= er+,)are generated in the matrix. The stresses have their maximum values at the matrix-inclusion interface (i.e. r = a) and decrease as l/r’ in the matrix. Let u,(a) be the radial stress at the interface and as(a) be the hoop stress in the matrix at the interface. The two are related by [3]: (1 + 20,)

1 2(1 -vi)

1

_D

r

(a)

L’i ( I--v,’ >

Since a,(a) is compressive (negative), its effect is to reduce the shrinkage rate of the matrix. The circumferential (hoop) strain rate is found by writing equation (25) of Part I [l] in spherical coordinates: 60= Z/3& + &,-‘[a” - v,(u, + a,,)],

Kp = E,/[3( 1 - 2vJ.

(6)

(2)

In the inclusion, all three stress components are equal to a,(a). Therefore, the inclusion is under purely hydrostatic stresses whereas in the matrix both hy drostatic and deviatoric stresses are present. The tensile stress u,t(a) has been postulated f3-9 to be responsible for radial cracks observed [7-g] around heterogeneities. In addition, the uniform tensile hydrostatic stress u has been used to explain the significant retardation of the densification of the matrix observed experimentally [6-81, as follows. The volumetric strain rate, i, is given by equation (35) of

(7)

Since the matrix cannot contract around the perimeter of the inclusion, e,(a) = 0. Using equation (1) in equation (6), we find that for d,(a) = 0 a@(a) = -

Z(1 - 2v,) 1 +

V,(l

-

4Vi)/(l + 2Vi)’

(8)

From equation (8) it can be seen that ]~~(a)] < ]Z 1 unless vp < 0 for all values of 4. Further, using equation (8) for u&z) in equation (l), we find that the radial stress is:

(1)

where c, is the volume fraction of inclusions. It can be shown that the mean (hydrostatic) stress in the matrix is not a function of r, and is given by [S]: (I = b,(r) + 26(r)l= 3

Since i represents contraction and Z is compressive, both are negative quantities. For a composite in which the heterogeneity sinters more slowly than the matrix, the volumetric strain rate in the matrix is found from equations (2) and (3):

where Ep is the uniaxial viscosity and vp is Poisson’s ratio: these quantities are related to the bulk viscosity by:

2. SiNTERiNG WITH RIGID INCLUSIONS

o,i(a) = - u,(a)

(4)

ur(a)=(l

2X(1 -2v,)(l +v,)+2v,(l

-vi) -2v,)’

(9)

The variation of u,(a) and a,(a) with ui is shown in Figs 2 and 3, respectively. The maximum magnitude of u,(a) from equation (9) is obtained for Vi= 0 and is less than ]2Z I unless vp < 0. Thus, the stresses around a rigid inclusion in a sintering matrix have bounds such that la,(a)1 < ]2C / and upper Iu&)l < I Z I unless v,, < 0. Although a negative value of Poisson’s ratio is allowed thermodynamically, it was shown in Part II [2] that all sinter-forging experiments imply vp > 0. From equations (l), (2) and (5) it can be seen that the value of or(a) determines all the important effects of composite processing. The three published analyses of this problem [3-S] give very different vatues for u,(a). Table 1 shows a comparison of the three calculations for an isolated heterogeneity in an

BORDIA and SCHERER:

Fig. 2. Radial stress in an inclusion, a,(a), normalized by twice the sintering potential, 2Z, vs volume fraction of inclusions, 4, for various values of Poisson’s ratio,o,.

OVERVIEW

NO. 70-111

properties of a porous material. The shear response was modelled by a Maxwell element and the densification response by a Kelvin-Voigt element. The choice of densification response was phenomenological since typical densification curves are very similar to the response of a Kelvin-Voigt solid to a step load. The step load was assumed to be the sintering potential, Z:. Two rate constants wp and oI. were defined for the shear relaxation rate and densification rate, respectively, as discussed in Part II [2), A non-dimensional parameter fl = wr/‘wL,characterizes the relative rates of creep and densification. and all results were obtained in terms of 8. 01CC1 implies that creep relaxation is insignificant relative to densification and conversely for /l >z1. The interfacial stress, a,(a), was calculated as a function of normalized time, f = t/T,. where '5,is the time constant for densification. The calculated stresses exhibit a very fast transient (f - 10m4) to a plateau value and then relax back to zero in time f - 1. The maximum values of the stresses and the relaxation time constant depend on 8. Small vaiues of B lead to large stresses and longer relaxation times. The maximum value of the interfaciai stress was calculated to be [3]:

II-1 or(a)

0.0

0.2

0.4

0.6

ae

1.0

Fig. 3. Circumferential stress in the matrix at the interface of an inclusion, ci&), normalized by sintering potential, Z, vs volume fraction of inclusions, ui, for various values of Poisson’s ratio, VP.

infinite matrix (i.e. oi = 0). Since a,(a) cannot be measured easily (being an internal stress in a porous material), it is difficult to check directly the validity of these calculations and to determine their ranges of appli~abiiity. In addition, it has not been clear why the predictions are so different. In the next section, we examine the model presented in Ref. [3] and show that the choice of parameters that leads to predictions of large stresses (lo,(a) ( >>I2Z 1) also implies a negative value of Poisson’s ratio; the same is true of the analysis presented in Ref. [4], which also predicts very large stresses. The Poisson’s ratio for the constitutive laws used in Ref. [q is always positive and the predicted stresses are within the limits set by equations (8) and (9).

2.2. Results of published analyses The model of Raj and Bordia [3] assumed explicit spring-dashpot elements to represent the constitutive Table I. A comparison of the calculated values of the maximal

radial stress around rigid inclusions Reference Raj and Bordia [3] Hsueh et al. [4] Scherer 151

o,(o)lr -45 -250 2

Remark for b = 0.01 a typical value from Fig. 8 absolule maximal value

‘413

z

4( I - Vi)

max= 2&T@

.

(10)

A plot of this equation is shown in Fig. 10 of Ref. f3). It can be seen that large interfacial stresses are predicted for small vaIues of Ui and #I. The densification behavior of the composite was also found to be strongly affected by p and it was shown that the hydrostatic tension generated in the matrix was of the same order as the sintering potential for small values of 8. This led to drastic reduction in the shrinkage rate of the matrix even for small volume fractions of inclusions. It was shown in Part II [ZJ that smali values of ~9 are unacceptable, because they lead to negative values of the Poisson’s ratio. The smallest admissible value of p is /I = n, where: n = 2G,/3K,, = (I - 2v,),‘(l + v,),

(11)

where G,,, &, and v, are the elastic shear modulus, bulk modulus, and Poisson’s ratio, respectively. Therefore, the high values of the stresses calculated in Ref. [3] for small /Ylare not possible (as long as v,, > 0). The calculated stress increases as b decreases, so the highest possible stress is obtained for the smallest allowed value of /I@ = n). In Ref. [3], all results for stresses and densification were obtained using v,=O.35, which corresponds to n = 219. If in equation (10) then B =n = 2/9 is substitute c,(a)/Z = 2 when vi = 0. This result is the same as that obtained from equation (9) and from the analysis presented by Scherer [5] for Vi and vp both equal to zero. Finally, it can be shown that a similar result is obtained if the elastic elements G, and & are omitted

2414

BORDIA and SCHERER:

from the spring-dashpot models (so that the constitutive equation is purely viscous, rather than viscoelastic). For this case, we define fi = Q/V and find that the allowed range of B is 2/3 < B < co in order that vp > 0. Hsueh et al. [4] predicted that stresses in the matrix could be hundreds of times larger than the sintering potential. However, we showed in Part II [2] that their model predicts that Poisson’s ratio is negative -+ - 1) until the body is almost fully dense. tvp _ This problem was analyzed in Ref. [S] using a constitutive model for which vp > 0, so the stresses were within the bounds discussed in Section 2.1. The densification kinetics of the matrix were shown to depart only slightly from the rule of mixtures, because of the small value of stresses.

OVERVIEW NO. 7@--III

ur i

1 A

B

:

=e

@

1

1

Fig. 4. Schematic of the two necks during the early stages of sintering. The neck at A is subjected to a compressive stress and the neck at B is subjected to a tensile stress.

grain boundary at the neck becomes mobile at a certain critical neck size. The grain boundary at B will In Refs [3] and [4], large values of a,(a) were become mobile at lower matrix density than would be the case if no stresses were present (because the neck calculated. These in turn led to values of the hydrostatic stress, u [equation (211, equal to 60-90% of Z at A has not grown as fast as it would without stresses). Near the inclusion, this effect could be at relatively small volume fractions of inclusions. This large value of u has been used to explain the substantial because the centers of particles are not substantial slowing down of the densification in the going to approach each other in the 6 direction matrix observed experimentally [6,7]. Here and in (i. = 0). If the grain boundary leaves the neck region then Ref. [5], it has been shown that the interfacial stresses, further densification would cease. This effect should and hence Q, cannot be very large unless vp c 0, and be easy to confirm experimentally. One could look for the latter condition is inconsistent with sinter-forging differences in grain size in the radial and the hoop results. We should, therefore, look for other expladirection near an inclusion. Similar observations nations for the retardation of densification. In this could be made for the sintering of porous films on section, we present an outline of some of the factors rigid substrates. Here the stress state is uniform and that may be important. one can get a large sampling of grains. If this effect (a) Behavior of porous body in tension and com- is important, then we expect to see larger grains pression normal to the substrate than parallel to it. Since this phenomenon involves grain boundaries, it would It has been assumed that the behavior in tension and compression is the same. However, after some occur only in crystalline materials. (ii) Even if the grain boundaries do not move, the neck growth, the neck sizes of particles near an neck at B will lose its curvature, and hence driving inclusion will be different in the radial and hoop directions so it is expected that the viscosities in the force for matter transport, at low density. Further two directions will be different. The shear and bulk densification would require transport of matter from viscosity have been assumed to be the same every- the center of the neck region B to the pore at A. This where in the matrix. However, owing to the effect could be accomplished by volume diffusion or grain mentioned above, we expect the viscosities to be boundary and surface diffusion. In any case, the different in the radial and hoop directions and to vary diffusion distance for further densification would increase and lead to slower densification rates. with distance. This phenomenon will be present in both crysSintering experiments under both tensile and compressive loads should be conducted to obtain the talline and amorphous materials. However, the deviscosities. The viscosities need to be related to neck pendence of densification rate on the characteristic distance for matter transport, d, is different for size and not just density. as density is a global parameter which averages out the variation in neck amorphous and crystalline materials (I/d for viscous flow and l/d2 or l/d3 for diffusion). We expect the size. retardation in densification rate to be more for (b) Competition between coarsening and densgying crystalline materials because of the stronger demechanisms pendence on d. (iii) Center-to-center approach of the particles for(i) Consider the situation shown in Fig. 4. The neck A is under a tensile stress a0 and the neck B is ming neck A is not allowed. However, coarsening under a compressive stress 6,. Let us assume that the mechanisms from various external surfaces deposit 3. DISCUSSION

BORDIA

and SCHERER:

matter on this neck and thereby reduce the driving force for densification (in the situation shown in Fig. 4, densification occurs by transport of matter from the grain boundary at neck B to the pore at neck B or A). Most of these mechanisms of retardation depend on interactions between grain boundaries and pores. This could account for the differences in the sintering behavior of a composite with a crystalline or a viscous matrix. Lange [lo] presented a model of densification of composites based on the idea that the linear strain required for complete densification increases with the spacing between the inclusions. Sintering will be inhibited after the matrix between closely spaced inclusions densifies. With one adjustable parameter, he was able to obtain a good fit between the experimental data of Ref. [6] and his analysis. However, the inclusions were required to form a rather sparse contiguous network and this seems physically improbable. Most importantly, the model is purely geometric and cannot account for the difference observed between crystalline and viscous material. There is some sketchy experimental evidence suggesting that coarsening mechanisms are important. First, in the experiments on TiO,-A&O,, [7] it was observed that the ratio of the shrinkage rate of the composite to the shrinkage rate of pure TiO, monotonically decreased as the density increased. Based on stress arguments, this is not what we would expect. The stresses are maximal at the early stages of sintering and hence the reduction in densification rate should be greatest then. However, if the competition between coarsening and densifying mechanisms is important, then the reduction in the densification rate will become progressively worse (as observed experimentally). The second experimental evidence is from sinter-forging experiments on TiO,-A&O, composites [8]. The composites were sintered under an applied compressive stress in an effort to overcome the internal (tensile) stresses by external compressive stresses. However, it was found that the technique did not achieve the desired improvement in densification unless the stresses were applied during the heating up stage of the sintering cycle. If the stress was applied at a temperature of 1023 K (the temperature at which pure TiOz started densifying) densification accelerated. The advantage of applied stress was considerably reduced if it was applied after reaching the isothermal soak temperature (1273 K). This observation also lends support to the importance of coarsening mechanisms; i.e. the additional stresses have to be applied before the coarsening mechanisms make the matrix unsinterable. It should be emphasized that coarsening would lead to anisotropy in the microstructure around the inclusion (differing in the radial and circumferential directions), and also a microstructure that depends on the distance from the inclusion. At present there is no experimental evidence of such features.

OVERVIEW NO. 70-111

2415

4. CONCLUSIONS In this series of papers, a critique has been presented of several constitutive laws proposed for porous materials. These laws are needed for the following broad categories of problems: densification stresses (e.g. hot pressing, under applied sinter-forging, hot rolling) and heterogeneous or constrained densification. In recent literature, the constitutive laws have been used mainly for analyzing the problem of heterogeneous densification. In the earlier work, a linear viscoelastic formulation was used, but we have shown that a sintering body is not linearly viscoelastic. However, it is adequate to consider the material to be purely viscous, since the elastic strains are so small that they can be neglected and the elastic-viscous analogy can be used. Heterogeneities in a densifying porous compact have two important effects: development of internal stresses (which could lead to formation of crack-like defects) and modification of the densification behavior. For the case when the matrix sinters at a rate much faster than the inclusion, very different theoretical results have been reported. The analyses that predict very high stresses have been shown to lead to negative values of the Poisson’s ratio. Since a negative Poisson’s ratio has never been observed in sinter-forging experiments. we conclude that internal stresses are not very large. The negative Poisson’s ratio is due to an overestimation of the shear viscosity relative to the bulk viscosity. The experimentally obtained constitutive laws give positive values of Poisson’s ratio. However. the density dependences of the bulk and shear viscosities are found to be different. We have calculated uppper bounds on the magnitudes of the internal stresses in matrices containing nonsintering inclusions. The tensile hoop stress can at most be equal to the sintering potential. Tensile stresses of this magnitude can generate internal flaws simply by the diffusive growth of pores. These small stresses. however. do not by themselves explain the observed reduction in the densification rate of polycrystalline matrices. It is speculated here that the reduction in the densification rate could be a result of subtle interactions between thedensifying and coarsening mechanisms in an inhomogeneous powder compact. Such mechanisms, which depend on the presence of grain boundaries, would not affect composites with glassy matrices, for which the rule of mixtures is in fact obeyed. Several experiments have been suggested to check the relative importance of some of the factors that have been suggested to be important. In addition, we have described the type of experiments that need to be done to establish the constitutive laws of the porous material. REFERENCES I. R. K. Bordia and G. W. Scherer, Acfu metal/. 36, 2393 (1988).

BORDIA and SCHERER:

2416

2. R. K. Bordia and G. W. Scherer, Acta metall. 36, 2399 (1988). 3. R. Raj and R. K. Bordia, Acta metall. 32, 1003 (1984). 4. C. H. Hsueh, A. G. Evans and R. M. Cannon, Acta metall. 34, 927 (1986).

5. G. W. Scherer, J. Am. Ceram. Sot. 70, 719 (1987). 6. L. C. DeJonghe, M. N. Rahaman and C. H. Hsueh, Acta metall. 34, 1467 (1986). 7. R. K. Bordia and R. Raj, J. Am. Ceram. Sot. 71, 302 (1988). 8. R. K. Bordia and R. Raj, Adv. Ceram. Mater. 3, 122 (1988). 9. F. F. Lange and M. Metcalf, J. Am. Ceram. Sot. 66,398 (1983). IO. F. F. Lange, J. Mater. Rex 2, 59 (1987).

APPENDIX Nomenclature a

inclusion radius

OVERVIEW

NO. ‘IO-111

a non-dimensional parameter for the matrix [equation (1 I)] volume fraction of the inclusion [equation (1). r in L’i Ref. [3]) Ep, vp uniaxial viscosity and Poisson’s ratio respectively of the porous body [equation (6) (also see Ref. [I])] Kp bulk viscosity of the porous body [equation (3) (also see Ref. [I])] ratio of the characteristic shear rate to the B densification rate of the matrix [equation (IO) (also see Ref. [2])] volumetric strain rate [equation (3) (also see Ref. [I])] ca hoop strain rate in the matrix [equation (6)] mean stress (equation (2), u, in Ref. [4]) u u,(a) radial stress in the matrix at the matrix-inclusion interface (equation (I), ua in Ref. [3]; u in Ref. [4]) radial and hoop stress components in the matrix (see Fig. 1) [equation (l), (a!‘, uf) in Ref. [4]: (u,, u,,,,,) in Ref. [5]) hoop stress in the matrix at the matrix-inclusion u0(a) interface (equation (I), u. in Ref. [3]) z sintering potential (equation (3) (also see Ref. [I]), -p,, in Ref. [3]; - Z in Ref. [4]) n