Constitutive models for the sintering of ceramic components—I. Material models

Acta metall, mater. Vol. 40, No. 8, pp. 1969-1979, 1992

0956-7151/92 $5.00 + 0.00 Copyright 7~ 1992 Pergamon Press Ltd

Printed in Great Britain. All rights reserved

CONSTITUTIVE MODELS FOR THE SINTERING OF CERAMIC COMPONENTS--I. MATERIAL MODELS Z.-Z. DU and A. C. F. C O C K S Department of Engineering, Cambridge University, Trumpington Street, Cambridge CB2 I PZ, England (Received 25 November 1991) Abstract--A number of possible two state variable material models for the sintering of fine grained ceramic compacts are described. The two state variables employed in the models relate to the relative density of the material and the mean grain size. Constitutive relationships for the strain-rate of a material element and evolution laws for the two state variables are presented which reflect the different stages of sintering. The predictions of the model are in good agreement with experimental studies involving pressureless sintering and HIPing of green compacts. R6sum&--On d6crit un certain nombre de mod+les possibles de mat6riau fi deux variables d'6tats pour le frittage de c6ramiques compactes 5. grains fins. Les deux variables d'6tat employ6es dans le mod+le sont la densit6 relative du mat6riau et la taille moyenne du grain. On pr6sente des relations constitutives pour la vitesse de d&ormation d'un ~l~ment et les lois d'6volution pour les deux variables d'6tat; ces relations refl6tent les diff6rents stades du fritage. Les pr6visions du mod+le sont et bon accord avec diverses 6tudes exp6rimentales.

Zusammenfassung--Eine Anzahl m6glicher Materialmodelle mit zwei Zustandsvariablen wird f~r feink6rnige Keramikgriinlinge beschrieben. Die beiden im Modell verwendeten Zustandsvariablen h/ingen mit der relativen Dichte des Materials und der mittleren Korngr6Be zusammen. Grundlegende Zusammenhfinge ffir die Dehnungsrate eines Materialelementes und Entwicklungsgesetze f/ir die beiden Zustandsvariablen, die die verschiedneen Sinterstadien beschreiben, werden entwickelt. Die Voraussagen des Modelles stimmen mit Experimenten zum drucklosen Sintern und heiBisostatischen Pressen von Grtinlingen gut /iberein.

1. INTRODUCTION Ceramic components for engineering applications are generally formed by sintering fine powder compacts. If the compact contains agglomerates, or if a toughening phase in the form of fibres or whiskers is introduced into the compact, then differential rates of sintering can occur resulting in the development of a residual stress field within the component. In regions of tensile residual stress sintering can stop, resulting in the presence of regions of residual porosity in the finished component, or cracks can form. The presence of these damaged regions can severely limit the strength of a component, particularly if they occur in regions of stress concentration under mechanical load. In order to fully understand how the microstructure and residual stress field develops around these inhomogeneities constitutive laws for the sintering process are required for the full range of possible stress states that could develop within the sintering compact. The purpose of this paper is to develop a set of equations that are suitable for the finite element analysis of this class of problem. Analyses of a range of representative situations are presented in an accompanying paper [1].

The general structure of the constitutive relationships developed in this paper are based on the experimental observations of Coble [2], who conducted sintering experiments on a number of alumina compacts, measuring the evolution of density and grain-size with time. He identified essentially four stages to the sintering process, which we denote as stages la, lb, 2 and 3. The major characteristics of each of these stages are as follows:

1969

Stage 1a: represents the early stages of sintering for loosely packed particles with low initial relative densities (density/density of fully dense compact) when the pores form an interconnected network through the material. As the body sinters there is no change in the mean grain size. Stage lb: takes over from stage la at a relatively low relative density of the order of 0.6. The pore structure is the same as during stage la, but densification is now accompanied by grain growth. Stage 2: occurs during the later stages of sintering when the pores pinch off forming an array of isolated pores. During this stage the

1970

DU and COCKS: THE SINTERING OF CERAMIC COMPONENTS--I

grains continue to grow as densification proceeds. The transition from stage lb to 2 generally occurs at a relative density of about 0.95. Stage 3: intervenes when the migrating grain boundaries break away from the pores leaving them isolated in mid grain. Densification virtually ceases after this point, but grain growth continues, often at an accelerated rate. The mechanisms responsible for densification in stages 1 and 2 have been described by Swinkels and Ashby [3] and Helle et al. [4]. Recently McMeeking and Kuhn [5] and Cocks [6] have obtained general constitutive relationships for deformation and densification for stages 1 and 2 respectively when grainboundary diffusion is the dominant mechanism. Lange and Kellett [7, 8] described the interaction between grain-growth and densification in stage lb when discrete necks still exist between the particles. They analyse the thermodynamics of this interaction, but kinetic relationships for this process are not yet available. Hillert [9] has modelled the process of grain growth in fully dense material and Shewmon [10] has analysed the dragging effect of an array of spherical pores on a migrating grain boundary. These analyses have been combined by Brook [11] and Yan et al. [12] to provide a set of equations for grain-growth during stage 2 of sintering, and to defne the condition under which the pores break away from the grain-boundaries, which characterises the transition from stage 2 to stage 3. Zhao and Harmer [13] have used these relationships to examine the interaction between grain-growth and densification during stage 2, taking into account the effect of the variation of pore size in the compact. An alternative approach to the development of sintering constitutive relationships is the phenomenological approach recently reviewed by Bordia and Scherer [14]. Models of this type generally assume a linear-viscous response with the shear and bulk viscosities being functions of the relative density. There is no consideration of grain-growth and no identification of the different stages of sintering, with the same set of equations used for the entire process. A model which has features of both the above approaches is that described by Hsueh et al. [15]. Like the phenomenological models described by Bordia and Scherer [14], Hsueh et al.'s [15] model is essential a linear-viscous description of the process with the viscosities expressed as functions of the relative density. A distinguishing feature of this model, however, is that micromechanical models of the creep of porous materials and models for the process of grain-growth were employed in the development of the final constitutive relationships. In formulating the material model the experimentally observed functional relationship between grain-size and relative density during pressureless sintering was employed, thus allowing the grain size to be expressed in terms of the relative density, and providing a set of

equations in terms of a single state variable. In the following section we make use of Hsueh et al.'s [15] analysis when formulating our set of constitutive relationships, but we choose to separate the processes of densification and grain-growth, providing different evolution laws for each process. The rational for this is that we would expect a superimposed pressure to strongly influence the rate of densification, but to have virtually no effect on the process of grain growth. This view is supported by the observation that a much finer grain size is obtained after HIPing than after pressureless sintering a compact [16]. Under zero applied pressure the sintering process is driven by the sintering potential. Ashby [17] provides expressions for the sintering potential during stages 1 and 2 in terms of the relative density and grain size. The experiments of Brook and co-workers [15, 18], however, indicate that the sintering potential is of the order of 1 MPa throughout sintering. In the following section we present a general structure for constitutive relationships for the sintering of ceramic powder compacts. This structure can accept all the various models described above and accommodate the different expressions for the sintering potential. Each model, or combination of models, is fit to the experimental results of Coble [2] in Section 3 to determine the various unknown material parameters. We do not make any judgement at this stage as to the most appropriate set of equations, but in an accompanying paper [1] the full range of models are used to analyse the response of an inhomogeneous sintering body to determine those features of the material response which most critically influence the evolution of residual stress field and microstructure. 2. THE MATERIAL MODEL In this section we aim to develop a constitutive model for sintering in terms of a limited number of internal state variables. From the discussion in the above section it is evident that the most important variables are the relative density p and mean grain size L, and although the grain and pore size distributions play a role in determining the overall kinetics of the process [12, 13], we express our constitutive relationships only in terms of these two state variables. The grain and pore size distribution are simply taken into account in terms of kinetic constants which are determined experimentally. The process of sintering results in large volumetric strains, and if the compact contains heterogeneities, there is also the possibility of quite substantial deviatoric strains in parts of the component. These large strains are inelastic in character, with the elastic strains remaining small. If we identify the total strain-rate ~j with the symmetric part of the deformation-rate tensor then the general structure of the constitutive relationship becomes

~,j = ~7~ + ~j

(1)

DU and COCKS:

THE SINTERING OF CERAMIC COMPONENTS--I

where iTj and i~j are the elastic and inelastic (or creep) strain-rates •

~C,k~

~i = Cijklffkl ~- Cijklffkl ~" Cijkl~kl AV [)-~pp ffkl • c

~;,/

- Qi( a, P, L). _

'c

(2)

p

= - pi~k.

(4)

The formulation is completed by providing expressions for the grain-growth rate L

=

£ ( p , L).

Also in forging experiments on ceramic compacts it is generally observed that there is zero transverse straining during the early stages of a test as the component contracts in the direction of loading [21], which implies that

(3)

In these expressions Cijkz is the elastic compliance matrix, aij represents the true (or Cauchy) stress and 6~j is the Jaumann rate of Cauchy stress. The second term of equation (2) is included here for completeness and takes into account the change of the elastic compliance matrix with density. Equations are available for this effect in the literature [15, 19], but in creep and sintering studies it is generally observed that inclusion of the variation of compliance with time does not significantly influence the results of the analysis [14, 20], particularly for situations where the inelastic deformation is large. In our analysis we therefore assume for simplicity that Gm remains constant during sintering. From the inelastic strain-rate of equation (3), which is a function of the true stress a, p and L, we may determine the densification rate

(5)

This structure of constitutive law is applicable to both the micromechanical and phenomenological models of the sintering process. Within this framework it is possible to select governing equations for the strain-rate and grain-growth rate reflecting the changing mechanisms and the different stages of sintering. In the following sub-sections we provide explicit forms for equation (3) and equation (5) based on the works of Hsueh et al. [15], Helle et al. [4], McMeeking and Kuhn [5] and Cocks [6] and the grain-growth studies of Brook [ll] and Ashby [17].

f(P) c(p)

- = 0.5

0.54(1 - p0) 2 f(p)

-

p ( p -- po) 2

(6)

which is similar to that originally proposed by Besson et al. [21], where d0 is the strain-rate experienced by a fully dense material of grain-size L0 at a constant uniaxial stress a0, s u are the deviatoric stresses, 6u is the Kroneker delta, G is the sintering potential, am is the mean stress am = lakk, a is a dimensionless material constant reflecting the sensitivity of the deformation response to grain-size and c ( p ) a n d f ( p ) are dimensionless functions of the relative density, p. When determining exact forms for c ( p ) a n d f ( p ) we note that in the limit o f p = 1, c ( p ) = 1 a n d f (p) = 0.

(8)

where P0 is the initial relative density of the compact, which is typically 0.64 for a dense random packing of spherical particles. If we now assume that equation (7) holds during stage 1, we obtain 1.08(1 - po) 2

c(p) -

p(p

po) 2

--

(9)

This same functional form for c ( p ) has recently been proposed by McMeeking and Kuhn [5], although in their formalism the constant 1.08 is replaced by 2.7, giving f ( p ) / c ( p ) = 0.2. The analysis of Helle et al. [4] gives 3.2(1 _p)12 =

2.1. Inelastic strain rates

•c ~o/Lo'~a 3 ~ij=~o~)[~c(p)so+3f(P)(am--G)~u]

(7)

during these early stages provided a m>>aS. In this section we present three material models with the form of equation (6) which satisfy equation (7) during stage 1. In an accompanying paper [1] we examine how the different structures of these equations influence the predicted response of an inhomogeneous sintering body. 2.1.1. The mechanistic model. The first set of equations we present here are based on theoretical models of sintering for situations where deformation and densification are determined by the rate of diffusional transport in the compact [4 6]. Helle et al. [4] provide expressions f o r f ( p ) for both stage 1 and stage 2. During stage 1

f(p)

All the models referred to earlier predict inelastic strain-rates which can be written in the following general form

1971

(10)

P

during stage 2. We have adjusted the numerical constant from that originally proposed by Helle et al. [4] to provide a smooth transition from stage 1 to stage 2. Thus equations (8) and (10) provide the same value o f f ( p ) when p = 0.95. Cocks [6] has analysed the shear deformation of a compact containing a regular array of spherical pores, which approximates stage 2 of sintering. He finds that e ( p ) is well approximated by c(p)=

1 [1 - 2 . 5 ( 1

- p)2'~ l

(11)

2.1.2. Hsueh et al.'s model. Hsueh et al. [15] assume that the same functional forms o f f ( p ) and c ( p ) can be used throughout stages 1 and 2. They propose that c(p)=p

lz

(12)

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DU and COCKS: THE SINTERING OF CERAMIC COMPONENTS--I

and b(1 - p ) "

f(p) = - -

P

(13)

where b and u are numerical constants to be determined by experiment. In their original analysis Hsueh et al. [15] combined these expressions for deformation with suitable expressions for graingrowth and determined the various parameters and material constants in their model by fitting the experimental sintering data of Coble [2] and creep data of Coble and Kingery [22] who performed torsional creep tests on alumina samples over a range of densities. The sintering tests were used to determine the dilatational contributation to the strain-rate and the creep tests to determine the shear component. There is an internal inconsistency in this approach in that the combined sintering equations assume that grain-size is a unique function of the relative density. The creep experiments of Coble and Kingery [22] were, however, designed to separate the influence of these effects on the creep properties, by manufacturing specimens with a range of relative densities, but having the same grain size. As a result Hsueh et al. [15] underestimated the effect of porosity on c(p) compared with its effect on f (p), predicting values of f(p)/c(p) that are some two orders of magnitude larger than that predicted by equation (7). Appropriate values of b, g0 and ct are determined in Section 3 by fitting the dilatational part of the constitutive relationship to the experimental results of Coble [2]. The shear properties are evaluated by simply requiring equation (7) to hold when p = 0.64. 2.1.3. An empirical model. Rather than base the constitutive relationships for sintering on Coble's experiments [2], a possible functional form can be obtained by examining the creep experiments of Coble and Kingery [22] to determine the shear contribution and invoking the restrictions on f ( p ) and c(p) to determine appropriate forms for the dilatational component. We should, however, do this with caution. The micromechanical models of Section 2.1.1 demonstrate that the form of the constitutive equation is not only a function of the relative density but also depends on the morphology of the pores. Cocks and Searle [23] have also demonstrated that the relative sizes of the pores and grains can significantly influence the properties of a material and the structure of the constitutive laws. We have noted that the form that the porosity takes is different during stages 1 and 2 and the pores are always smaller than the grains. In Coble and Kingery's experiments [22] the pores were introduced by adding naphthalene to the compact, which was burnt away during firing. The resulting pores were roughly spherical and much larger than the grain size. The resulting form of equation we present here are, therefore, at best, a first approximation.

Following an approach that is often used to evaluate the response of creep damaging materials (24) we assume that

c(p) = p -h

(14)

during both stages 1 and 2, where h is a material constant which can be determined by fitting equation (6) to Coble and Kingery's [22] data. Doing this gives h = 5.26. We now assume that equation (7) holds throughout stage 1 so that

p-h f(p) = -~-.

(15)

During stage 2, however, f ( p ) must gradually decrease from that given by equation (15) to zero when p = 1. We therefore assume that, during stage 2, b(1 - p ) " f(p)=

2p5.26

(16)

where ~ and b are determined by fitting Coble's sintering data [2] and from the requirement that equation (16) is consistent with equation (15) when p = 0.95. During stage 3 when the pores have broken away from the grain-boundaries we assume that there is no further densification and 3~o f Lo'~" for each of the models.

2.2. The sintering potential Here we adopt two different approaches for determination of the sintering potential of equation [6]. Brook and co-workers [15, 18] propose that the sintering potential is independent of grain-size and relative density and is equal to 1 MPa throughout sintering. Ashby [17], however, proposes that the sintering potential is a function of p and L and that different functional relationships are required during stages 1 and 2 due to the different structure of the porosity. During stage 1, when there is interconnected porosity, discrete necks exist between the particles. Ashby [17] provides an expression for as for an array of spherical particles of uniform size

67p2(ZP --Po~ ~'=7 \ l-po J

(18)

where ~ is the surface free energy per unit area and P0 is the initial relative density of the compact, typically 0.64 for dense random packing. In this paper we wish to develop a set of equations that is suitable for randomly shaped grains and for situations where the compact has been cold pressed, leading to spreading of the contact zones prior to sintering. For these more general situations we need to be careful how we interpret P0 in equation (18). This point is discussed more fully in Section 3 where we use equation (18) when fitting the experimental data of Coble [2].

DU and COCKS:

THE SINTERING OF CERAMIC COMPONENTS--I

In stage 2, the pores become isolated at fourgrain junctions. It would be possible to determine the exact shape of these pores, but for simplicity we assume that they are spherical, with one pore at each corner of a tetrakaidecahedron shaped grain, then [17] 27 a~ = - -

(19)

r

where r = [(1 -p)/6]L'3L/2 is the mean pore radius. During grain growth, the pores are dragged along by the migrating grain boundaries and coalesce, resulting in an increase of the mean pore size, whereas densification results in a reduction in the pore size. The above equations take this effect into account and are consistent with Kingery and Francois' [25] observation that pore size is proportional to the grain size at a given density.

on the mechanism of material transport with /3 = 4 for the most common situation of surface diffusion controlled mobility. If fib Mb = - a0

(25)

where fib is the velocity of a grain boundary in a fully dense material under a driving pressure a0, then combining this expression with equations (19)-(21) and (23)-(25), we obtain the grain growth rate rib L/~ -

7u (26)

1 + f(p, L/Lo)aoL o

where

l (Ly

f (p, L/Lo) =-~l \--~oj

2

(1- P)'~'3bUp

and

2.3. Grain growth-rate In this subsection, we first apply Brook's [11] and Ashby's [17] analyses of grain-growth to the second stage of sintering, identifying the conditions for that transition to stage 3. The results are then extended to stage lb. During stage 2 the pores are dragged along at the same velocity as the migrating grain-boundaries. Brook [11] shows that if F b is the driving pressure for grain growth and M b and Mp are the mobilities of the boundaries and pores respectively the grains grow at a rate

L

1973

MbMp F b Mp + MbU

(20)

C1 = 2 ~6~3 ~---. 24 The above equations are expressed in terms of the average grain size L. The transition to stage 3, however, occurs when the largest grains break away from their attached pores. These grains are then able to spread quickly through the compact leading to abnormal grain growth. Hillert [9] demonstrates that during normal grain growth the largest grains are twice the average grain size and the driving force for the growth of these grains is twice that given by equation (22). The force exerted by the largest grains on their pores [equation (22)] is then given by

where N is the number of pores per unit area of grain boundary, which is given by Ashby [17] as 24

N = 7cL2

(21)

and the force on each pore is

Mb f P- Mp + Mb NFb"

(22)

Hillert [9] postulates a steady state process and obtains the driving pressure for the rate of growth of the mean grain-size 7b F b = -L

(23)

where 7b is the grain-boundary energy per unit area. Shewmon [113] has determined the mobility of the pores for a range of mechanisms of material transport around the pore. For each mechanism he analyses

Mp

L~o \ r j

fP=f

~(o,L/Lo)+I

where C 2 = ~ / 12.

(24)

where tip is the velocity of a pore of radius L0 at a driving force aoL2o, and/~ is a constant that depends

The maximum force that a pore can exert on a grain boundary is the Zener pinning force [26]. For a pore on a planar boundary this force is f p,X = zrr7b

(28)

but a pore at a four grain junction exerts a greater pinning force, because a greater area of grain boundary is created when the pore moves away from the boundary. We take this into account by introducing a factor C3 into the above equation

f ~x = C37zr7b

(29)

where C 3 is in the range 1-2. Iffp given by equation (27) is greater t h a n f ~,x then the grain boundaries will break away from the pores and stage 3 of sintering will start. During stage 3 the grain boundaries will effectively cut through a random array of pores. The pore force is now limited to f~ax of equation (28). Following Ashby [17], we

1974

DU and COCKS: THE SINTERING OF CERAMIC COMPONENTS--I (a)

grain boundaries, the above equations can be simplified considerably, equation (26) then becomes

0.8

'

/

/~ = L 0

~

0.4

, 10 °

I 10 2

I

, 10 4

,

(33)

where

/// Experimental Data / ~¢ - - - Mechanistic Model ./'/" Hsueh et. al.'s Model ~----Empirical Model

0.6

(1 - p)-~/3

/-~0 - C1 Up ?b

aoLo and the force on a pore [equation (27)] is

, 10 6

7C

t(s)

fp = ~ 7bL.

(34)

(b)

Equating this with equation (29) and noting equation (19) gives the condition for the onset of stage 3

10 -s

f#" °f"

L(m)

p = 1 - 0.028C 33

10 .6 ental Data - - - Mechanistic Model --Hsueh et. al.'s Model ........... Empirical Model

~...~"

10-7

t lO °

I 10 z

I

t(s)

I 10 4

I

I 10 6

Fig. I. (a) Comparison of constitutive densification law for different material models with experimental data of Coble [2] when the sintering potentials of equations (18) and (19) are used. (b) Comparison of constitutive grain-growth law for different material models with experimental data of Coble [2] when the sintering potentials of equations (18) and (19) are used.

(35)

where, for C3 = 1, equation (35) becomes p = 0.972. If f (p, L/Lo) is not much greater than 1, then the force on the individual pores is less than that given by equation (34) and stage 3 starts at a larger value of p than predicted by equation (35). The interaction between grain-growth and densification during stage lb is described by Lange and Kellett [7, 8], but kinetic equations are not yet available for this process. The situation is, however, analogous to stage 2 with grain-growth requiring material transport around the surface of the pores. As a first approximation of this process we follow the (a)

assume that Hillert's [9] equation for normal grain growth still holds in this limit, then £ = Mb(F b - N r f ~ ~X)

(30)

where N r is the number of pores per unit area on a planar boundary migrating through a random dispersion of pores

1

P 0.8

J

0.6 .f

. ..~J .J J

/:Y/

Experimental Data - - - Mechanistic Model Hsueh et al's Model . . . . . . . . Empirical Model

I

C4(1 _ p)l/3 Nr ~

L2

(31)

I

0.4

I 10 2

1

where 65/3 C4 ~ - . 7c

t

I 10 4

I

I 10 6

(b) 10 -s

Substituting equation (31) into equation (30) and noting equation (23) gives L = /~b(-~) [ 1 - C 5 ( 1 -

I

p)2/3]

(32)

L(m)

10 .6 ~

where

n

i

64/3

c5 = ~ -

10 .7

I 1

and - L0 a0"

For situations w h e r e f (p, L/Lo) >> 1, i.e. when the mobility of the pores is much slower than that of the

t 102

s

f f41tmental Data t i c Model Hsueh et al's Model ........... Empirical Model t ) I J 104 106 t(s)

Fig. 2. (a) Comparison of constitutive densiflcation law for different material models with experimental data of Coble [2] when the sintering potential is set equal to ! MPa. (b) Comparison of constitutive grain-growth law for different material models with experimental data of Coble [2] when the sintering potential is set equal to I MPa.

DU and COCKS:

THE SINTERING OF CERAMIC COMPONENTS--I

1975

Table 1. Values of the various material parameters for each of the models obtained by fitting Coble's experimental data [2] when the sintering potentials of equations (18) and (19) are used Model Mechanistic Hsueh et al. Empirical

g0

L0

C7

P0

b

c~

3.35 x 10 -~ 2.66 x 10 -4 2.23 x 10 -s

1.5 x 10-I° 5.3 x 10-~° 1.5 x 10-10

1,0 1.02 1.0

0.32 0.64 0.64

-0.54 2.4

-0.29 0.29

same analysis given above for stage 2, but for the purpose of analysis consider the porosity to consist of an interconnected network of cylindrical pores. I f f p is the force per unit length of cylindrical pore then for a given mechanism of material transport around the pore

Mp= f4p

-(Lo'~-I ]

(36)

L 0o-0 \ r ] where r is the mean radius of the pore. 1

r = - (1 7Z

p)l/ZL.

(37)

N in equation (20) is now the number of pores per unit length of grain boundary 1

U = L"

(38)

The grains then grow at a rate given by

(,0)

~b - Z

/~ -

~b

(39)

1 + f~ (p, L/Lo) aoLo

where 1 fib ( L y - 2 ( 1

f, (p, L/Lo) = C66ftp \Lo/

_

p)'a~)_

(40)

and C 6 = ~ ~ 1 For f l ( p , L/Lo) >> 1, equation (39) becomes /~=CTL o

(l-p)-

:

(41)

where C7 = C6/C1 and equals 1.35 for/3 = 4. We now have equations for densification and grain growth during all stages of sintering. 3. ANALYSIS OF EXPERIMENTAL DATA In this section we determine the material parameters of the models described in the previous section for the conditions employed by Coble [2] in his sintering experiments on fine grained compacts of alumina at 1480°C. When fitting the material parameters for each model to the experimental data of Coble [2], the sintering potential is either expressed by equations (18) and(19) or set equal to 1 MPa throughout sintering. The variation of density and grain-size during the course of a sintering experiment are shown in Figs l(a, b) and 2(a, b) respectively for the cases when the different sintering potential expressions are employed. Before attempting to fit this data, we make a number of assumptions about the

controlling mechanisms of deformation and graingrowth to limit the number of parameters we need to fit to the experimental data, We assume that throughout the sintering process deformation occurs by Coble creep, as a result a in equations (6) and (17) is equal to 3, and that during stages lb and 2 the mobility of the pores is limited by surface diffusion, giving/3 = 4. During the course of Coble's [2] experiments the grain size increased from 0.3 to 3.5/~m over 12 h. The starting relative density of the compacts was 0.47 and stage la was observed to last until a relative density of 0.52, with the transition to stage 2 occurring when p = 0.95. Taking Ashby's data [17] for the pore and grain-boundary mobilities at 1480°C we find that f(p, L/Lo) and fl(P, L/Lo) given by equations (26) and (40) respectively are in the range 20-100 for p = 0.6 to 0.988. Since these quantities are >>1 during stages lb and 2 we employ equations (41) and (33) for the respective grain-growth rates. In the analysis of grain growth in the previous section a number of dimensionless constants, C~ to C7, were introduced. The values of these constants were determined by assuming idealized pore and grain structures. The particular constants that are of interest to us in this section are C3 of equation (35) and C7 of equation (41). Since we are not dealing with an idealized situation we will treat these constants as quantities to be determined experimentally, although their values should be of the order of magnitude of those determined above i.e. C3 ~ 1 and C7 ~ 1.35. In the present analysis, if the final density observed experimentally by Coble [2] is 0.988, C, obtained from equation (35) is 1.32. We now combine the grain-growth rate expressions of equations (41) and (33) with the strain-rate expressions for the three different models of Section 2.1 and determine the various material parameters by fitting the combined models to the experimental data of Coble [2]. For each combination go,/~0 and (;7 must be determined. When using the mechanism based equations of Section 2.1.1 a value of the initial density P0 needs to be selected. These equations were developed for the situation where the initial compact consists of a dense random packing of spherical particles with an initial relative density of 0.64. where the particles are in point contact. The initial relative density of 0.47 in Coble's experiments [2] suggests that the starting powder consisted of particles of irregular size and shape. Also the samples were pressed in a die at a pressure of 160 MPa prior to sintering to form the test sample. Die pressure of

1976

DU and COCKS:

THE SINTERING OF CERAMIC COMPONENTS--I

Table 2. Values of the various material parameters for each of the models obtained by fitting Coble's experimental data [2] when the sintering potential is set equal to 1 MPa

Model Mechanistic Hsueh et al. Empirical

i0 4.53 x 10-4 3.16 x 10 3 3.62 x 10-4

L0 7.48 x 10 io 1.9 x 10-~° 1.76 x 10 io

48 MPa produced samples with an initial relative density of 0.42. It is evident that some spreading of the contacts occurs during cold pressing and when using the mechanistic equations an apparent initial density, P0, for point contact which is less than 0.42 should be used. The initial shape of the density/time curve is very sensitive to this choice of P0, and we select the value that provides the best fit of the experimental data during the early stages of sintering. In the other two models P0 only appears in the expression for the sintering potential during stage 1, equation (19). The predictions of these models are much less sensitive to the assumed value of P0. To limit the number of parameters we fit to experimental data we simply set P0 = 0.64 in equation (19) when using these models. Hsueh et al.'s [15] model and the empirical model contain one additional parameter, ct, which is to be obtained from the experimental results. This quantity influences the shape of the density/time and grainsize/time plots throughout an entire experiment for Hsueh et aL's model [15], while only the portion of these plots corresponding to stage 2 are affected in the empirical model. The quantity b is then determined from the consistency requirements discussed in Section 2.1. Experimental measurements of density and grain size made during the course of sintering experiments are generally plotted as functions of time with equations fitted directly to this base data. As the equations for grain growth developed above contain the current density, and the equations for densification contain the current grain-size, the process of grain growth and densification are coupled. It is therefore not possible to produce independent predictions of density and grain-size as functions of time to facilitate the determination of the unknown par-

C7 1.0 1.02 1.0

Po 0.32 0.64 0.64

b -0.54 2.4

a -0.29 0.29

ameters. These were therefore determined by a process of trial and error, with adjustments made to each parameter at the end of each iteration to bring the resulting density/time and grain-size/time plots closer to the experimentally observed relationships. Values of the various material parameters are given in Table 1 and Table 2 respectively when the sintering potential is either expressed by equations (18) and (19) or set to 1 MPa, when a 0 = y/Lo and L0 is taken as the initial grain-size of the material, which was typically 0.3 #m. When extending these models to other temperatures we assume that the activation energy for creep deformation is 419, while that for pore mobility is 500 k J/mole. The resulting variations of relative density and grain-size with time for each model are compared with the experimental results in Figs 1 and 2. It was possible to obtain a much better fit of the experimental data using Hsueh et al.'s model [15] for the creep rate for both cases when different expressions for the sintering potential were used. This is not surprising since this model contains four independent parameters for the entire process, three rate parameters and one which determines the shape of the density/time plot, while the shape of this plot is largely determined a priori with the other models, with the choice of P0 only influencing the shape during the early stages of sintering for the mechanistic model and ct only influencing the shape during the latter stages of sintering for the empirical model. If, however, we examine the variation o f f ( p ) / c ( p ) with p for the material models and compare this with the experimental results of Besson et al. [21], Fig. 3, we find that Hsueh et al.'s [15] model provides the worst correlation. Clearly each model has its advantages and disadvantages, but they all provide a reasonable fit of the experimental data. In an accompanying paper [1] we examine how the differences 20

0.6

L/Lo 15

f/c 0.4

10 4

0.2

- - - M ~ i c

Model

Hsueh. et. al.'s Model

o

...........E ~ . I

5

N\~

;

Mo~

0

0.4

0.64

0.76

P

0.88

1

Fig. 3. Comparison o f f / c as a function of relative density for different material models with experimental data of Besson et al. [21].

20

~4o

~]~ i 0.6

i 0.8

1

P Fig. 4. Plot of normalized grain-size as a function of relative density for a range of different applied hydrostatic pressures, am/ao, for Hsueh et aL's model [15] when the sintering potentials of equations (18) and (19) are used.

DU and COCKS:

1

10

p

~

4

20

0

0.8 100 0.6 0.4

10°

I

I

i

I

102 t(s) 104

1977

THE SINTERING OF CERAMIC COMPONENTS--I

I

I

106

Fig. 5. Plot of relative density as a function of time for different values of (L0/g0) x l0 7 using Hsueh et al.'s creep equations [15] when the sintering potentials of equations (18) and (19) are used.

between the different models influences the predicted evolution of microstructure and residual stress field in sintered components. So far we have only examined stages 1 and 2 of sintering. The models, however, all reduce to the same structure during state 3, equations (17) and (32), and /2b is the only remaining parameter to be determined. When determining /2b in equation (32), only limited experimental data for abnormal grain growth during sintering is available for magnesia doped alumina from Coble's experiments (2). However, the grain growth rate for pure and MgO-doped alumina powder during HIPing at 1672°C was studied by Monaham and Halloran [27]. They found that the grain growth rate for MgO-doped alumina was lower by a factor of about 25 in comparison with that for pure alumina at 1672~C. After the temperature difference is taken into account using Ashby's [17] data for the activation energies for surface diffusion and boundary mobility, we determine that/2b in equation (32) is about 16 times /2o in equation (33) at the temperature of 1480°C for pure alumina. It is instructive to examine the predictions of the models beyond the set of data used to determine the various parameters. In the remainder of this section we examine the extension of the model based on Hsueh et al.'s [15] equations to a range of different loading situations. Similar results and conclusions are obtained for the other two models. Figure 4 shows the effect of a superimposed pressure on the relationship between grain-size and relative density for this model when the sintering potentials of equations (18) and (19) were used. In formulating the models it was assumed that the application of an applied stress only affects the grain growth rate indirectly through its effect on the densification process. This was confirmed by Besson et al. [28], who carried out post-HIP tests on nearly densified samples. They found that pressure has no effect on grain growth if no simultaneous deformation occurs. As a result the application of hydrostatic pressure results in an increase in the rate of densification, while at a given density the rate of grain-growth remains unchanged. The grain-size at the end of stage 2 therefore

decreases with increasing applied pressure. Examination of Fig. 4 also reveals that for all situations grain growth is more extensive in stage 2, with limited grain-growth during stage lb. These results are broadly consistent with the experimental results of Uematsu et al. [29] who observed that the application of an applied pressure considerably reduces the extent of grain growth at a given density. They also note that the rate of grain growth decreased with increasing applied pressure in the latter stages of HIPing. For the range of pressures and temperature used in their HIPing experiments they observed, however, that the extent of grain-growth was only weakly dependent on the applied pressure for most of the time, although the extent of this dependence was not quantified. Figure 4 possibly predicts a stronger dependence on pressure than their observations suggest. We discuss this point further in the next section. When fitting Coble's [2] data it was found that the ratio of/20 to E0 were 4.5 x 10 6 m 1, 2.0 x 10 6 m 1 and 6.7 x 10 6m t respectively for the different models when the sintering potentials of equations (18) and (19) were used. The effect of changing this ratio by varying/20 is examined in Figs 5 and 6 for Hsueh et al.'s [15] model for pressureless sintering where the curves in each of these figures have only been extended to the end of stage 2. As one would expect increasing /20, increases the extent of grain growth at a given density and thus reduces the rate of densification (Fig. 4). An interesting observation, however, is that at short times the extent of grain-growth at a given instant increases with increasing/2o, as one would expect, while at longer times the opposite is true. This is a direct result of the coupling between grain-growth and densification. The rates of grain-growth and densification are each dependent on L and p. For small values of/20, grain-growth is slow initially and densification can therefore proceed at a fast rate, but as soon as stage 2 is reached the pinning effect of the pores on the migrating grain-boundaries is significantly reduced, and the grains can grow much faster. Since stage 2 is reached earlier for low values of/20, the acceleration of grain-growth rate due to this effect 1 0 "s

20 40 100

/

L(m)

1 0 .6

1 0 .7

I

10°

I

10z

I

t(s)

I

104

I

I

106

Fig. 6. Plot of grain-size as a function of time for different values of (£0/E'0) x 107 using Hsueh et al.'s creep equations [15] when the sintering potentials of equations (18) and (19) are used.

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DU and COCKS:

THE SINTERING OF CERAMIC COMPONENTS--I

sets in earlier and results in a significant increase in the total extent of grain growth at longer times. 4. DISCUSSION In this paper we have set out to obtain a set of constitutive laws for the sintering of fine grained ceramics that are suitable for the finite element analysis of inhomogeneous sintering bodies. The models described in Section 2 make use of the grain-growth studies of Brook [11] and Ashby [17] and the creep and densification models of Hsueh et al. [15], Helle et al. [4], McMeeking and Kuhn [5] and Cocks [6], with the material parameters in the models determined by fitting Coble's [2] experimental data for the evolution of grain-size and relative density with time. These models all provide a reasonable fit of the data over the range of conditions employed in Coble's [2] experiments (Figs 1 and 2), although more work is required to assess the relative merits of the different models proposed in this study. There are a number of areas, however, where the models could be improved. The equations for creep deformation and grain-growth effectively assume that these processes are controlled by the rate of grain-boundary diffusion within the compact. Besson et al. [21], however, demonstrate that at low stresses creep of fine grained materials is controlled by an interface reaction process, with the creep and densification rates being proportional to the stress squared and inversely proportional to the grain-size. Apart from the semi-empirical model of Besson et al. [21] this mechanism has received very little attention and we do not include it in our model at this stage. We have also assumed that there is a sharp transition between each stage. In practice, however, there will be a gradual transition from stage lb to 2 where the transition from open to closed porosity occurs over a range of densities. Similarly the transition to stage 3 occurs gradually over a period of time as the grain boundaries progressively break away from their attached pores. Also we have assumed that densification stops completely during stage 3, but as the grain boundaries sweep through the array of pores, a proportion of the pores are attached to the boundaries at a given instant and these can continue to sinter by the same mechanism as in stage 2. There are, however, two complicating features to this process: firstly the mean spacing of the pores on the grain boundaries increases during stage 3 since the majority are distributed within the grains, but, more importantly, the pores are only attached to a boundary for a short period of time before it breaks away and becomes attached to another set of pores. The sintering of these pores is therefore a transient process, since they are not attached for a sufficient period to allow a steady state to develop. To avoid analysing this situation, and since the rate of sintering is likely to be very slow, we

have simply assumed that densification ceases at the start of stage 3, although grain-growth can proceed at an accelerated rate. We have also ignored the effect of any trapped gas in the pores during stage 2 of sintering. As densification proceeds the pressure exerted by the trapped gas can become significant, slowing down or even preventing further densification of the compact. There are two contributions to this change of internal pressure: the first arises directly from the shrinking volume of the void and leads to an increase of pressure; while the second arises from the rate at which the ceramic material can absorb the trapped gas, leading to a reduction in pore pressure. The net effect is very sensitive to the atmosphere and temperature used during sintering. We inherently assume here that the alumina can readily absorb any trapped gas, although a more complete model must take this effect into account. There is strong evidence, both theoretical [7, 8] and experimental [30], for the dependence of grain-growth on densification during stage lb under pressureless sintering. If grain-growth depended exclusively on densification then we would expect the same functional relationship between density and grain-size to hold during HIPing. Uematsu et al. [29], however, observed that the extent of grain-growth at a given density during HIPing is much less than that under pressureless sintering, although over the range of pressures and temperatures employed in their HIPing experiments there was a unique relationship between density and grain-size during state lb. The present model, which assumes that pressure only affects the rate of densification, does not provide such a unique relationship. If we were, however, to follow Hsueh et al. [15] and assume that there was always the same functional relationship between grain-growth and density then we would obtain the same extent of grain-growth during HIPing and pressureless sintering. Clearly the reality of the situation lies somewhere between these two extreme assumptions, and further research is required to provide a more detailed description of grain-growth during stage lb, but since grain-growth is limited during this stage we would not expect the predictions of the present model to be significantly in error. There is, however, general agreement in the literature [11-13,17] that the equations adopted here for grain-growth during stage 2 provide an adequate description of the process. Despite the reservations expressed above the model described in this paper exhibits the correct general features of the sintering process. When analysing the response of compacts containing inhomogeneities the simple structure adopted here will permit the interaction between densification and grain-growth in these more complex situations to be readily assessed, without being complicated by the effect of the different mechanisms of deformation and graingrowth.

DU and COCKS:

THE SINTERING OF CERAMIC COMPONENTS--I

Acknowledgement--This research was supported by the

Science Engineering Research Council under Grant No. GR/F/0546.6. REFERENCES

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1979

16. E. Dorre and H. Hubner, Alumina, Processing, Properties, and Applications. Springer, Berlin (1984). 17. M. F. Ashby, Background Reading, H I P 6.0. Univ. of Cambridge (1990). 18. M. N. Rahaman, L. C. De Jonghe and R. J. Brook, J. Am. Ceram. Soe. 69, 53 (1986). 19. A. C. F. Cocks, J. Mech. Phys. Solids 37, 693 (1989). 20. M. Rides, A. C. F. Cocks and D. R. Hayhurst. J. appl. Mech. 56, 493 (1989). 21. J. Besson, M. Abouaf, F. Mazerolle and P. Suquet, I U T A M Symp. on Creep in Structures, Krakow, Poland (1990). 22. R. L. Coble and W. D. Kingery, J. Am. Ceram. Soc. 39, 377 (1956). 23. A. C. F. Cocks and A. A. Searle, Acta metall, mater. 38, 2493 (1990). 24. F. A. Leckie and D. R. Hayhurst, Proc. R. Soc. (1975). 25. W. D. Kingery and B. J. Francois, J. Am. Ceram. So~. 48, 546 (1965). 26. C. Zener, quoted by C. S. Smith, Trans. Am. Inst. Min. Engrs 175, 15 (1948). 27. R. D. Monahan and J. W. Halloran, J. Am. Ceram. Soc. 62, 564 (1979). 28. J. Besson and M. Abouaf, Proc. Secondlnt. Con/~ on Hot lsostatic Pressing--Theory and Application, Gaithersburg (1989). 29. K. Uematsu, K. Itakura, N. Uchida, K. Saito, A. Miyamoto and T. Miyashita, J. Am. Ceram. Soc. 73, 74 (1990). 30. T. K. Gupta, J. Am. Ceram. Soc. 55, 276 (1972).