Constitutive models for the sintering of ceramic components—II. Sintering of inhomogeneous bodies

Acta metall, mater. Vol. 40, No. 8, pp. 1981-1994, 1992 Printed in Great Britain. All rights reserved

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

CONSTITUTIVE MODELS FOR THE SINTERING OF CERAMIC COMPONENTS--II. SINTERING OF INHOMOGENEOUS BODIES Z.-Z. D U and A. C. F. COCKS Department of Engineering, Cambridge University, Trumpington Street, Cambridge CB2 IPZ, England (Received 25 November 1991)

Abstraet--A sintering model recently developed by the authors to predict grain growth, densification and creep deformation during sintering is used to analyse the response of a sintering body containing a distribution of spherical and ellipsoidal inclusions. The analysis was performed using the finite element code ABAQUS, in which the grain size and relative density are considered as solution dependent state variables. The development of microstructure and residual stress field within the body are described in detail. For a prolate (fibre-like) inclusion a zone of high compressive mean stress develops around the tip of the inclusion, where densification and grain-growth occur at an accelerated rate. Eventually, because of the high density and large grain size the material within this zone becomes highly creep resistant and effectively forms an extension to the original inclusion. During the early stages of sintering the maximum principal stress achieves its maximum value in the matrix at the interface with the inclusion, but away from the tip. As sintering progresses the peak moves away from the interface to the edge of the creep resistant zone, i.e. to the surface of the "effective inclusion". Similar results are obtained for an oblate (disc-like) inclusion, except that the general levels of stress are lower. R4sum&-Un module de frittage r6cemment d6velopp6 par les auteurs pour pr6voir la croissance des grains, la densification et la d6formation par fluage pendant le frittage est utilis6 pour analyser la r6ponse au frittage d'un corps contenant une distribution d'inclusions sph&iques et ellipsoidales. L'analyse est effectu& par la m&hode des ~16ments finis, code ABAQUS, dans laquelle la taille du grain et la densit~ relative sont consid6r&s comme des variables d'6tat d6pendant de la solution. Le d~veloppement d'une microstructure et d'un champ de contraints r6siduelles fi l'interieur du corps est d6crit en detail. Pour un inclusion allong& (en forme de fibre) une zone de forte contrainte moyenne en compression se d&eloppe autour de l'extr~mit6 de l'inclusion, 1~. off la densification et la croissance du grain se produisent 5. une vitesse acc61&&. Finalement, 5. cause de la densit6 61ev6e et de la grande taille du grain, le mat&iau 5. l'int6rieur de cette zone devient tr~s r&istant au fluage et forme effectivement une extension 5. l'inclusion originale. Pendant les stades initiaux du frittage la contrainte principale maximale atteint sa valeur maximale dans la marice 5. l'interface avec l'inclusion, mais loin de l'extremit& Comme le processus de frittage se d~veloppe, le pic s'6carte de l'interface et se d6place vers le bord de la zone resistant au fluage, c'est a dire vers la surface de l'"inclusion effective". Des r&ultats similaires sont obtenus pour une inclusion plate &n forme de disque), mais les niveaux g6n6raux de contrainte sont plus bas.

Zusammenfassung--Mit einem von den Autoren entwickelten Sintermodell, mit dem Kornwachstum, Verdichtung und Kriechverformung w/ihrend des Sinterns vorausgesagt werden k6nnen, wird das Verhalten eines Sinterk6rpers, der eine Verteilung yon sphfirischen und ellipsoidischen Einschlfissen enth~lt, analysiert. Die Analyse wird mit dem Finit-Elementcode ABAQUS durchgeffihrt, bei dem Korngr6Be und relative Dichte als 16sungsabh/ingige Zustandsvariablen angesehen werden. Die Entwicklung der Mikrostruktur und des Restspannungsfeldes werden ausffihrlich beschrieben. Bei einem gestreckten (faserartigen) EinschluB entwickelt sich eine Zone hoher kompressiver mittlerer Spannung um die Spitze des Einschlusses herum, wo Verdichtung und Komwachstum beschleunigt ablaufen. SchlieBlich wird der Kriechwiderstand des Materials wegen der hohen Dichte und der groBen K6rner innerhalb dieser Zone so hoch, dab sich effektiv eine Verl/ingerung des ursprfinglichen Einschlusses bildet. WS,hrend der frfihen Sinterstadien erreicht die maximale Hauptspannung ihren h6chsten Wert in der Matrix an der Grenzfl/iche zum EinschluB, aber entfernt yon der Spitze. Mit fortschreitendem SinterprozeB bewegt sich das Sintermaximum yon der Grenzflfiche weg zur Kante der kriechfesten Zone, d.h. zur OberflS_chedes "effektiven Einschlusses" hin. ,~hnliche Ergebnisse ergeben sich ffir einen abgeplatteten (scheibenfihnlichen) EinschluB, auBer wenn das allgemeine Spannungsniveau niedriger ist.

1. INTRODUCTION In an a c c o m p a n y i n g paper [1] (hereafter refered to as paper 1) the general structure o f a two state variable model for the sintering o f ceramic c o m p a c t s is

described which can a c c o m o d a t e both micromechanical and empirical models o f densification and graingrowth. The two state variables within the model relate to the physically measurable quantities o f relative density (density o f c o m p a c t / d e n s i t y o f fully

1981

1982

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

compacted material) and grain size. Evolution laws for the grain size were developed from the works of Hillert [2], Brook [3] and Ashby [4]. Three different forms of equation for the deformation rate were presented, which were based either on micromechanical models of the sintering process [5-7], the semiempirical model of Hsueh et al. [8], or an empirical approach equivalent to one which is often employed in continuum damage mechanics [9]. The final form of each set of equations, obtained by fitting the experimental data of Coble [10] are presented in the following section. Du and Cocks [11] have used the model with the creep equations based on Hseuh et al.'s [8] model to analyse the sintering response of a body which contains a spherical inhomogeneity. A range of problems were analysed, with the inclusion having a different initial density, or initial grain-size, or completely different creep properties than the surrounding matrix material. The resulting computations were consistent with the analytical studies of Bordia and Scherer [12]. It was found that the most severe stresses were generated in the matrix when the inclusion was fully dense and was only allowed to deform elastically. The mean stress was always less than twice the sintering stress in magnitude and there was only a slight variation of grain-size within the matrix throughout the entire sintering process. We obtain similar results when using the mechanistic or empirical equations for the creep response in place of those proposed by Hsueh et al. [8]. As we may expect, agglomerates or reinforcement introduced into the compact are rarely spherical and it is important to determine the influence of the shape of any inhomogeneity on the response of the compact. In Sections 3 and 4 we analyse the response of a body which contains a distribution of ellipsoidal inclusions, which are either prolate (fibre-like) or oblate (disc-like). The effect of aspect ratio on the development of microstructure and the residual stress field are presented in Section 4 for each of the material models. We also vary certain parameters within the models, for example the ratio of shear to bulk viscosity or the relationships for the sintering stress, to determine the sensitivity of the predicted response to variations in the details of the models, with the aim of determining those features of the models which most critically influence the predicted response. To provide further insight into the response of this class of problem we use Eshelby's method [13, 14] in Section 3 to determine the stress distribution around a rigid ellipsoidal inclusion embedded in a linear viscous matrix, with uniform properties and uniform sintering stress. 2. THE MATERIAL MODEL In this section we describe the constitutive models of paper 1 for sintering obtained by fitting the experimental data of Coble [10], which is expressed

in terms of two internal state variables. If the total strain-rate £,7 is identified with the symmetric part of the deformation-rate tensor then the general structure of the constitutive relationships becomes _

.e

.¢

~,7- E,7+ E,7

(1)

where ~,~and ~ are the elastic and inelastic (or creep) strain-rates

~,~ = C,Tktrk/

(2)

~,~ = ~(~, p, L).

(3)

In these expressions C,Tkt is the elastic compliance matrix and 6,7 is the Jaumann rate of Cauchy stress. From the inelastic strain-rate of equation (3), which is a function of the true stress tr, p and L, we may determine the densification rate (4)

[J = -- P E~kk"

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

providing

t~ = £ ( p , L).

(5)

The form of equations (3) and (5) can change during the course of sintering as the internal structure of the material and mechanisms responsible for densification and grain growth change. From his experiments on alumina compacts Coble [10] identified four stages of sintering and the equations we present in the remainder of this section represent the best fit to these experiments obtained in paper 1. The first stage of sintering (stage 1a) is only evident for loosely packed powders (initial relative densities, p~ less than 0.6). During this stage the pores form an interconnected network through the material, and during densification the grain size remains constant. Stage lb takes over when p = 1.1 Pi and only differs from stage la in that densification is now accompanied by grain-growth. The transition to stage 2, when the porosity is closed, occurs when p ~ 0.95. When p ~0.988 densification ceases, but grain growth continues at an increasing rate (stage 3). The inelastic strain-rate during each stage can be written in the following form eo { Lo "~3 r3 , , d~=~o ~--~ j [ i c l p )sq + 3f(p)(trm-as)6,71

(6)

where ~0 is the strain-rate experienced by a fully dense material of grain size L0 at a constant uniaxial stress tr0, 6,7 is the Kroneker delta, trs is the sintering potential, o"m is the mean stress am = ]akk, S,7 are the deviatoric stresses and c ( p ) a n d f ( p ) are dimensionless functions of the relative density. The functions c ( p ) a n d f ( p ) are given in Table 1 for the different models of paper l for all 3 stages in sintering. These quantities are plotted in Fig. 1. It is evident that the different models provide a wide variation in the shape and general level of these curves. The sintering potential of equation (6) is either set equal to 1 MPa throughout sintering in accordance

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

1983

Table 1. The functions c(p) and f(p) for the differentmodels for all three stages of sintering Models Stages

Mechanistic

Hsueh et al.

Empirical

p )029 f

0.54

(1-P°)~ p ( p ~ po) 2

c

1.o8 P(1° )--~

0.54 ( 1 -

0.5p

p

526

1 p(p

-

p 12

,o0)2

p

p )029

f 2

3.2( 1 _ p)~,2 p

-,

1

C

0"54(1p

p

526

__ p )0.29 1"2(1 pS2~

1,2

p 526

(1 -- 2.5(1 -- p)2,3) 3

f c

0 1

0 1

with Brook and co-workers [8, 15] observations or Ashby's [4] relationships are employed, with a S-

Y p2(2p - 0.64) 0.06 L

(7)

0 1

The appropriate equations for the grain growth-rate are then /~ =/'~0

-

£ = £0

-

during stage 1 and

(9)

,

(10)

O's~ ~-

during stage 2, where 7 is the surface free energy per unit area. These different expressions for the sintering potential reflect the different structure of the porosity in stages 1 and 2. For the conditions employed in Coble's [10] sintering experiments Du and Cocks [1] determine that grain growth is limited by the mobility of the pores.

1 0 0 -- (a)

\ \

10-

\

...... . . . ."" . . ~ "" '>"'.'x

1

"\

~

I

0.1

I

068

06

076

-

loo

I

(b)

-

"~

"N~i

0.84

0,92

1

\ ...

\

'..

\

...

"N. "-..

1 0.6

0.68

~.

t

I

I

076

0 . 8 '°,

0.92

' '~

.,J 1

p

Fig. 1. Comparison of (a)fas a function of relative density and (b) c as a function of relative density for the different material models described in Section 2. AM 40/8--M

2'~]

(11)

where ['b is the rate of grain-growth in a fully dense material of mean grain-size L0. The values of the quantities do, £0,/~b for a0 = 7/L0, where Lo is the initial grain-size, which was typically 0.3 #m, obtained by fitting each model to the experimental data of Coble are given in Table 2(a) and (b) respectively when the sintering potential is either expressed by equations (7, 8) or set equal to 1MPa. In the analyses presented in the following sections where the initial grain-size is allowed to vary the values of do, [,0,/2~ and a o were adjusted accordingly. In the present work, the elastic Young's modulus at full density, E = 133 G P a was used throughout the whole process of sintering, and Poisson's ratio was set to 0.33. 3. A RIGID ELLIPSOIDAL INCLUSION EMBEDDED IN A LINEAR VISCOUS SOLID

\ 10

/~=£b(~°)[l-5.5(l-p)

Mechanistic Model ,, Hsueh et al's Model

~

....... Empirical Model \

C

for stage lb and 2 respectively, where/~0 is a material constant. During stage 3, when densification ceases and the grain-boundaries break away from the pores

As a precursor to the full analysis of a sintering body containing a distribution of ellipsoidal inclusions for the constitutive laws of Section 2 we consider the simpler problem of a rigid inclusion embedded in an infinite matrix which deforms according to equation (6) with, L, c(p), f(p) and a~ uniform throughout the body. In the Appendix Eshelby's method [13, 14, 16] is used to determine the stress distribution in the matrix immediately outside of the inclusion. Here we are interested in two particular stress components; the mean stress a m,

1984

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

Table 2. Values of the various material parameters for each of the models obtained by fitting Coble's experimental data [2]. (a) When the sintering potentials of equations (7, 8) are used and (b) when the sintering potential is set equal to 1 MPa (a)

(b)

Model

Eo

Lo

Lb

Mechanistic H s u e h et al. Empirical

3 . 3 5 x 10 5 2 . 6 6 x 10 - 4 2 . 2 3 × 10 5

1.5 x 10 ~o 5.3 × 10 io 1.5 x l 0 io

2.4 x l 0 9 8 . 4 8 x l 0 -9 2.4 x l 0 9

Model

~o

Lo

Lb

Mechanistic H s u e h et al. Empirical

4.53 x l0 -4 3 . 1 6 x l 0 -3 3.62 × l 0 4

7 . 4 8 x 10 lo 1.9 x 10 -1° 1.76 × 10 lo

1.20 x 10 -8 3 . 0 4 x 10 -9 2.82 x l0 9

{o)

f/e

lOS

1.0 0.5 0.2

10 z

101 t~ I

10 °

10-1 1 0 .2 1 0 "~

because of its influence on the evolution of microstructure, and the maximum principal stress ~ , which determines the development of micro-flaws during processing. For a spherical inclusion we find that o"m= 0 and at/a s = 2f/c in the matrix at the interface with the inclusion. In the models presented in Section 2 f / c is in the range 0-0.5, giving values of the maximum principal stress which are no larger than the sintering stress as noted by Bordia and Scherer [12]. In the original model of Hsueh et al. [8]f/c was of the order

,

,

10 x

10 z

(b)

0.40 0.30

f/c o.2o

~ -

0.10

-

1.o 0.5 0.2

0.00 1 0 -t

10 o Aspect ratio

101

10 z

Fig. 3. Plots of the mean stress in the matrix at the interface with the inclusion along (a) the major axis and (b) the minor axis of the ellipsoid as a function of aspect ratio for a range of values of fie.

I I I

A

10 e Aspect ratio

0.50

1 0 .2

(a)

1 0 "l

of 100, and as a result they predicted very larger stresses around the inclusion. The sign convention employed in the Appendix is shown in Fig. 2 for both the fibre-like and disc-like inclusion. In each case the ellipsoidal inclusion is formed by revolving the quarter ellipse about the X:axis, with the X2-axis representing an axis of symmetry for the solid inclusion. The mean stress in the matrix at the interface with the inclusion along the major and minor axes of the ellipsoid (position A and B respectively of Fig. 2) are plotted as a function of aspect ratio in Fig. 3 for a range of values of f / c . The maximum principal stress at each of these locations is related to the mean stress by

01 B U2

Xz (b)

I

8

I

al

a2

n

A

X2 Fig. 2. The sign convention employed in the Appendix for both (a) the fibre-like and (b) disc-like inclusions.

and acts in a direction tangential to the surface of the ellipsoid. Examination of Fig. 3 shows that the mean stress is always compressive at the point of maximum curvature, increasing in magnitude with increasing curvature. For a given peak curvature, however, the compressive mean stress for the fibre-like inclusion is greater than that for the disc-like inclusion. In each case the magnitude of the mean stress increases with increasing value of f/c. At the point of minimum curvature the mean stress is always tensile, and increases i n magnitude with increasing aspect ratio of

DU and COCKS:

THE SINTERING OF CERAMIC COMPONENTS--II

1 0

30

0

60

90

Fig. 4. The variation of the maximum principal stress around an inclusion of aspect ratio 4. the inclusion, asymptoting to a constant value as the aspect ratio goes to infinity. This asymptotic value increases with increasing value off/c. In contrast to the results at the point of maximum curvature, the tensile mean stress for the disc-like inclusion is greater than that for the fibre-like inclusion, for a given section shape. F r o m equation (12) it is evident that a~ = as for f/c = 0.5 at both A and B of Fig. 2 independent of aspect ratio. F o r large aspect ratios, however, the peak stress is very sensitive to the absolute value of f/c. F o r values of fie slightly less than 0.5 the second term of equation (12) becomes negative, and a~ is always less than as. If, however,fie is greater than 0.5 this term becomes positive and high tensile stresses can develop at the tip of the inclusion. F o r example, if al/a 2 = 100 a l i a s = 9.2 forf/c = 0.51 and - 7 . 2 for f/c = 0.49. Forf/c less than 0.5 there is a transition value for the aspect ratio below which a~ is tensile and above which it is compressive. F o r f/c = 0.2 this transition occurs at an aspect ratio of 2.57. Figure 4 shows the variation of the maximum principal stress around an inclusion of aspect of ratio 4 as a function of the angle 0 the outward normal to the inclusion makes with the Xraxis, Fig. 2. The maximum principal stress peaks at a value of 2.52 a~ when 0 = 52 ° where its direction makes an angle of 62 ° with the Xraxis. The peak value of a~ is plotted as a function of aspect ratio in Fig. 5 for a range of values off/c. F o r a given value of fie the position f/c 1.0 0.5

10 3

0.2

10 2

10 ~

10 e

1 0 -I

10 "z

t 10 "1

I 10 ° Aspect ratio

I 101

I 10 2

Fig. 5. Plot of the peak value of a I as a function of aspect ratio for a range of values of fie.

1985

of this peak stress, in terms of the angle 0, is not very sensitive to the aspect ratio; for example the value of 0 corresponding to the peak stress varies from 50 ~ to 56 ° as the aspect ratio is increased from 2 to 100 for f/c = 0.5. It is, however, much more sensitive to the value off~c, with the peak stress occurring at a value of 0 which ranges from 65 ~ to 4 8 ~for an aspect ratio of 4 as f/c is increased from 0.1 to 0.7. It is evident from Fig. 5 that the larger the ratio off/c and the sharper the shape of the inclusion, the higher the peak value of cq. Examination of Fig. 5 also reveals that the peak value of a~ for the fibre-like inclusion is always greater than that for a disc-like inclusion with the same section shape for any given value off/c, except whenf/c = 0.2 and the aspect ratio is less than or equal to 2. 4. G E N E R A L

ELLIPSOIDAL

INCLUSION

PROBLEM

In this section we consider the situation where a sintering body contains a distribution of ellipsoidal inclusions. The inclusions are only permitted to experience elastic deformation, while the surrounding matrix material also creeps and sinters according to one of the material models of Section 2. The simplified analysis of the previous section provides an indication of the effect of the ratio f/c on the stress field in the vicinity of the inclusion and how the stresses scale with respect to the sintering potential. Here we examine how the detailed forms of the q u a n t i t i e s f ( p ) and c(p), the relationship chosen for the sintering potential, and the process of graingrowth, which combine to produce non-uniform properties within the matrix, influence the stress field and evolution of microstructure. The situation we analyse here is shown in Fig. 6. A distribution of inclusions is represented using a finite cell model. The cell of Fig. 6 contains a single central inclusion, whose major axis is one tenth the diameter of the cell, and is subjected to zero applied loads. This problem was analysed using the commercial finite element code A B A Q U S [17], with the relative density and grain-size treated as solution dependent state variables. The various models of Section 2 were introduced into A B A Q U S through the user defined C R E E P subroutine. For this type of problem, where high dilatatioiaal strain-rates occur initially but the material becomes incompressible during the final stages of sintering, extreme care needs to be exercised in the selection of element type and choice of integration scheme. A range of benchmark problems were set up to determine the most suitable procedure for this class of problem. It was found that four noded isoparametric elements with selective reduced integration provided the most reliable results. A typical finite element grid for a cell which contains a fibre-like inclusion with an aspect ratio of 4 is shown in Fig. 6. In the computations presented here the matrix was assumed to have an initial density

1986

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

h ) / -

02

d

/

lOG 1

X2 Fig. 6. A typical finite element grid for a cell which contains a fibre-like inclusion with an aspect ratio of 4.

of 0.7 and an initial grain size 1 pm, and the quantity P0 in each model was set equal to 0.64. To smooth the computational process, a transition period (0.975 ~

_

.c

.c

E~t - - k~~/stage2 -'t- (1 - - k)Egstage3

(13)

/~'t = k/~'stage2 q- ( l - - k)[,stage3

(14)

where k = e -[(p-Oq75)/(pc-O975)]n, in order to provide a gradually transition from stage 2 to stage 3. Here we choose n = 2 and Pc = 0.98. The structure of the constitutive models naturally provides a smooth transition from stage 1 to stage 2, apart from a slight difference in the sintering potential at the transition, but this does not cause any numerical difficulties. When examining the numerical results, however, at values of p which correspond to this transition we normalise the stresses with respect to the average of the two sintering potentials. Initially we describe the response of the fibre like inclusion problem of Fig. 6 for the situation where the matrix deforms according to the mechanistic equations of Section 2, with the sintering potential varying according to Ashby's relationships of equations (7, 8). This allows us to identify the major features of the development of the residual stress field and microstructure within a component. We then present general plots which compare the results of the computations for the different models, allowing the influence of the different structure of these relationships to be readily assessed. During stage 1 of sinteringf/c = 0.5 for the mechanistic equations, with this ratio decreasing during

stage 2 and asymptoting to a value of 0 at full density. Forf/c = 0.5 and uniform creep properties the analysis of the previous section predicts a mean stress of - 20-s and a maximum principal stress of 0-~at the tip of the inclusion of Fig. 6. The variation of the stress 0-22 along the .~k"1 axis of Fig. 2 away from the tip of the inclusion is shown in Fig. 7 for different times during sintering, corresponding to the instances when Ptip 0.8, 0.95 and 0.975 at the tip and when a zone has developed around the tip within which densification has ceased, i.e. the material locally is in stage 3 of sintering. The variation of mean stress along the X~ axis at these same times is shown in Fig. 8. When Ptip=0.8, L = 1.023pm locally and for ? = 1 Jm -2 the local value of 0-s is l0 MPa. Examination of Figs 7 and 8 reveals that 0"22= 1.23 0"~ and 0"m= --2.45 0"sat the tip of the inclusion. These values =

30 .-'... 10

~.~

......

....... ..... ......

n -10

:

P'~ip 0.8 0.95 - - - 0.975 . . . . . . O. 9 8 8

:

b -30

: -230MPa -50

I 0.1

I 0.2

I 0.5

I 0.4

0.5

Xl/a~ Fig. 7. The variation of the stress 0"22along the Xraxis away from the tip of the inclusion at different times during sintering.

DU and COCKS: THE SINTERING OF CERAMIC COMPONENTS--II 10-

-10 n

S

-50 /"

bE -50

---.....

.'

0.8 095 0975 0988

:'

-70

: - 260MPa

-90

~'

0

I

01

02

t

I

03

04

I 05

Xl/a~

Fig. 8. The variation of the mean stress ~m along the X~-axis away from the tip of the inclusion at different times during sintering. are about 23% greater than predicted by the analysis of Section 3. This difference arises from the variation of creep properties in the matrix which results from the different local grain-sizes and relative densities. U p to this stage there is minimal grain growth with L being of the order of 1/~m everywhere. There is already, however, quite a large variation of density within the body, which ranges from 0.8 at the tip of the inclusion to 0.74 in the bulk of the body. The maximum principal stress has its maximum value at the surface of the inclusion where the outward normal makes an angle 0 = 55 ° with the Xl-axis. The density and grain-size at this location are 0.765 and 1.02/~m giving a local value of the sintering potential of 8.5 MPa. To avoid confusion we choose, however, to normalize all stresses with respect to the sintering potential at the tip of the inclusion. Using this normalization the maximum principal stress is 2.3 ~rS, whose direction makes an angle of - 5 9 ° with respect to the Xt-axis. The high compressive mean stress generated at the tip of the inclusion accelerates the rate of densification locally, which results in a reduction of the pinning effect of the pores, leading to an acceleration of the grain-growth rate compared with that in the surrounding material. By the time p has increased to 0.95 (the transition from stage 1 to stage 2) at the tip of the inclusion the local grain size has increased to 1.36/~m, giving a local value of the sintering potential of 13.9 MPa. (It is important to note that although the local grain-size is greater than that in the surrounding material it is less than that obtained in a pressureless sintering experiment which has densified to p = 0.95. As a result the sintering potential is greater than that in a sintering experiment at a comparable density; the value of ~ = 13.9 MPa compares with a~ = 7.98 MPa during pressureless sintering when p =0.95.) There is now a much wider variation of relative density and grain-size within the body. The m a x i m u m principal stress az2 scales with the sintering potential in the same way as before (~22= 1.21 ~s), but the compressive mean stress is considerably increased to a value of o-m = - - 3 . 5 3 as as

1987

a result of the variation of material properties. This results in a further increase of the local densification and grain-growth rates compared to the surrounding material. The maximum principal stress has its peak value at the same location as when P t i p = 0 . 8 , but its value is now 2.29 as, and it now acts in a direction which makes an angle of - 5 8 '~ with respect to the X~-axis. By the time P~ip= 0.975 L = 1.61 ffm and as = 15.45 MPa (again this can be compared with a s = 7.22 MPa at the same density during pressureless sintering). This value of p is well into stage 2 of sintering a n d f / c has reduced to 0.408. The maximum principal stress is now compressive at the tip of the inclusion, a22 = - 1 . 7 3 a~. This stress increases sharply over a short distance until a tensile stress of order a~ is achieved. Meanwhile the mean stress has considerably increased in magnitude to am = - 6 . 1 6 a s. The maximum principal stress has its maximum value at the same location as when Ptip = 0.8 and 0.95, and it acts in approximately the same direction, - 5 7 with respect to the X~-axis, but its value is now 2.11 a~. It is instructive to examine the full variation of relative density, grain size, mean stress and maximum principal stress within the body at this stage of the sintering process. These are shown as contour plots in Fig. 9. The degree of densification and extent of grain growth in the bulk of the body are now much less than at the tip of the inclusion. There are two effects which compete with each other to produce a high compressive mean stress at the tip of the inclusion as the body sinters. The high density and large grain-size combine to increase the shear and bulk viscosities locally. If the material locally is to deform at the same rate as when the creep properties do not vary through the body, then higher levels of stress are required at the tip of the inclusion. This effect probably accounts for the increase of stress observed when p = 0.8 and p = 0.95 at the tip of the inclusion. If the viscosities at this location are much larger than those in the bulk of the body, then the surrounding material essentially sees a highly creep resistant material, similar to that which forms the inclusion. The material surrounding the tip of the inclusion can then be thought of as forming an extension to the inclusion and increasing its aspect ratio. Within the inclusion, and thus within this creep resistant zone, a net compressive stress field develops as the surrounding material densities and deforms at a faster rate. Immediately outside of this zone the mean stress is still compressive, but a tensile maxim u m principal stress develops. It is this latter explanation which better describes the situation when p = 0.975 at the tip of the original inclusion. As sintering progresses the creep resistant zone spreads out from the tip of the inclusion. Figure 10 shows the contour for p = 0.975, which represent the position of the transition from material sintering in stage 2 to that which is just entering the transition stage towards stage 3, at various times during sintering.

1988

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

(b)

1

L/L o 1.21

2

1.26

3

1.31

4

1.36

5

1.41

6

146

7

1 51

8

156

9

1.62

P

3

1

0.83

2

o. 8 4 9

3

0.869

4

0.889

5 6

9

0.909 0.928

7

0948

8

0.968

9

2

0.975 2

0"~ ( max1/0"$ (d)

(C)

4

- O'rn/O"s

2

1

-1.748

2

- 1.262

I

-0.144

3

-0,777

2

0.643

4

-0,291

3

1.431 2.220

i

4

0.194 0.680

5

3. 004

6

3794

5

7

4.583

4

8

5 373

9

6.163

~

"

~

5

1.165 8

1.651 2.136

Fig. 9. Variation of (a) the relative density, (b) the grain size, (c) the mean stress and (d) the maximum principal stress within the sintering body when Ptip = 0.975.

When this zone is small tr[ is a maximum at the surface of the inclusion at a position where 0 = 55 °. As the creep resistant zone grows the position of maximum th moves towards the zone along the surface of the inclusion, until eventually it is coincident with the zone boundary. With further expansion of the creep resistant zone the position of maximum trx translates around the boundary towards the X1axis. The position of maximum a~ for different sizes of creep resistant zone is shown in Fig. 10. The sintering potential at these locations gradually decreases with increasing time due to the increasing grain size. The magnitude of the maximum principal stress however depends on two factors which compete with each other. These are the increase of the aspect ratio of the "effective inclusion", which produces a higher maximum principal stress and the decrease of the ratio of f/c in the surrounding material, which tends to reduce the level of the maximum

principal stress. For the present problem, the maximum principal stress gradually decreases in overall magnitude, reflecting the fact that the variation of the maximum principal stress is more sensitive to the change off/c than to the aspect ratio of the "effective inclusion". The same problem has been solved for the other models described in Section 3. Computations have also been performed for situations where the ratiof/c for a model has been arbitrarily changed. These involve reducing f/c in the Hsueh et al. equation by a factor of 2 and that in the mechanistic model by a factor of 2.15 to yield a value of f/c = 0.23 during stage 1. Computations have also been performed for inclusions of different aspect ratio in the range 0.25 to 4. By examining the results of these different calculations we can determine the influence of the detailed forms o f f ( p ) and c(p), the ratio of these two quantities, and the functional form for the

DU and COCKS: THE SINTERING OF CERAMIC COMPONENTS--II t/(s)

\o 2

I

9 394E3

2

1 317E4

3

15 7 0 E 4

4

1 891 E4

5

2 298E4

6

2 811E4

"o

Fig. 10. The position of the boundary to the creep resistant zone at different times during sintering. The dots represents the position of the peak value of a Lfor the different positions of the zone boundary.

sintering potential, on the evolution of microstructure and residual stress field within the body. The results of these computations are summarized in Fig. 11 where the notation used in these figures is defined in Table 3. In this table, for a problem with an inclusion of aspect ratio 1, as the results for a given value of f i e obtained from different materials models are closely similar, one type of symbol is used to represent the results for the different models. The variation of ffm/as at the tip of the inclusion and the maximum value of a I/as in the body when Pt~p= 0.8 are shown in Fig. 1l(a) and (b) respectively as a function o f f / c for the fibre like inclusion, where a~ is the value of the sintering stress at the tip of the inclusion. The solid line corresponds to the predictions from the analysis of Section 3. In each case the computations lie close to the analytical results, with the magnitudes of am and a~ for the situation where a S is taken to have a constant value of 1 MPa being consistently greater than when the expressions of equations (7) and (8) are employed. As noted earlier, the material in the vicinity of the tip of the inclusion is constrained to deform and densify at a rate that is compatible with the deformation of the surrounding material, and at a similar rate to when the creep properties and sintering stress are uniform through the body. As a result of the increase of the shear and bulk viscosities local to the tip of the inclusion the stresses must increase in this region. The increase of sintering potential which occurs at the tip of the inclusion compared with that in the bulk of the body

1989

for the computations which employ equations (7) and (8) provides an increasing driving force for densification above that which is available when a~ = 1 MPa everywhere, and as a result the mean stress need not increase by as much to ensure compatibility with the surrounding material. Similarly if the viscosities are less sensitive to changes in density then we would expect that lower stresses would be required to maintain compatibility. Examination of Fig. 1 shows that both f ( p ) and c ( p ) vary more rapidly for the mechanistic model than for the empirical and Hsueh et al. [8] models, with the latter model providing viscosities which are relatively insensitive to density in these early stages of sintering. As a result we find that for the same sintering potential the magnitude of the stresses for the empirical and Hsueh et al. [8] models are less than those for the mechanistic model at a given value o f f / c . For example, for a problem with an inclusion of aspect ratio 4, i f f / c = 0.5, a~~as = 2.31 and a m / a s = - 2 . 4 5 for the mechanistic model and al/as=2.18 and a m / a s = - 1 . 7 1 for the empirical model when the sintering potential of equations (7, 8) are employed. If the sintering potential is set to I MPa, we find, ax/as=2.95 and am/a~= -3.31 for the mechanistic model and aL/as=2.84 and am/a s = - 2 . 4 5 for the empirical model. In fact when using the sintering potential of equations (7) and (8) the magnitude of the stresses obtained using the empirical and Hsueh et al. [8] equation are lower than predicted by the analysis of Section 3, indicating that the local increase of sintering potential has more than compensated for the increase of viscosity. As one would expect this effect is more pronounced when using Hsueh et al.'s [8] relationships. When Ptip= 0.95 there is a wider discrepancy between the analysis of Section 3 and the computations for the mean stress, Fig. 1l(c), (d), as a result of the much wider variation of creep properties within the body, but the general trend and relative order of the models in terms of the magnitude of the stresses local to the inclusion remains the same. For a given value o f f / c , the maximum principal stresses are however, still close to the analytical prediction, with the greatest variation occurring when employing a constant sintering potential of 1 MPa, reflecting the fact that the maximum principal stresses are more sensitive to the value of f / c than other factors. Examination of Fig. 1l(c) also reveals that in contrast to the results for/gtip - - 0 . 8 , the magnitude of the mean stress obtained using the empirical and Hsueh et al.'s models becomes larger than that predicted analytically in Section 3 when the sintering potential of equations (7, 8) are employed. This is due to the fact that stage 2, in which grain size increases at a more significant rate, has been reached locally. As a result, the increase of viscosity becomes the dominant factor in determining the local densification rate, requiring a larger increase in the magnitude of the mean stress in order to maintain compatibility with the deformation of the surrounding material.

1990

DU and COCKS:

THE SINTERING OF CERAMIC COMPONENTS--II

(a)

(b)

0.5 F

Aspect ratio

Pttp=0.8

Aspect ratio

pttp:0.8

4

4

1 -0.5

--

2

3

"- -1.5

~ ~

-2.5

2

2

4 •

1

II

1

®

-3.5 0.2

I

0.4

0.6

0.8

0.2

f/c

0.4

0.6

f/c

0.8

(d)

(c) Ptlp =0-9

o

o

o

Aspect ratio 4

P'tP=0"95

Aspect ratio

1

i

-1 4

o

~

E

2

-3

-5 0

I

I

0.2

0.4

ID

I

I

0.6

0.8

I

0.2

Ue (e)

0.6

f/c

0.8

(f)

e

•

-6

Aspect ratio

Aspect ratio

Ptlp=0.97S

-

-2

0.4

4

pap=0.975

4

3

°

E

e

E

2

2

1 * ®

-10 -14 0

i 0.2

i 0.4

1 i 0.6

J 0.8

B

I

0 0.2

0.4

0.6

f/c

f/c

0.8

Fig. I 1. (a, c, e) O'm/O" s at the tip and (b, d, f) the peak value of trl/trs in the body for different material models compared with the analytical results of Section 3 for the fibre-like inclusion. The symbols are defined in Table 3.

By the time Ptip --- 0.975 this effect is even m o r e p r o n o u n c e d a n d a high net compressive stress develops at the tip o f the inclusion, Fig. l l(e, f). The m a x i m u m principal stresses will still very close to the analytical predictions w h e n the sintering potential o f e q u a t i o n s (7, 8) are used. If the sintering potential is, however, set equal to 1 M P a , the difference between

Initial f/c

0.5 0.23

the analytical a n d the numerical results is m u c h wider in c o m p a r i s o n with those situations w h e n p~p = 0.8 a n d 0.95. Similar results are o b t a i n e d for the disc shaped inclusion with the same trend in the d e v e l o p m e n t of microstructure. These results are s u m m a r i z e d in Fig. 12. In c o m p a r i s o n with those for the fibre-like

Table 3. Notation used in Figs 11 and 12 Models Mechanistic Hsueh et aL Aspect ratio Aspect ratio Sintering 4 2 1 4 2 1 potential (0.25) (0.5) Equations (7, 8) • • <> ~9 ~ O 1 MPa • [] <> Equations (7, 8) C) [] C, @ ~ O 1 MPa (3 [] 0

Empirical Aspect ratio 4 2 1 ~ ~

[] []

O

DU and COCKS:

THE SINTERING OF CERAMIC COMPONENTS--II

(a)

1991

(b)

0.4

3.0

p.p=0.S

POp=0.8

Aspect ratio 0.25

Aspect ratio

0

o

~

1

2.0 -0.4

~ ' - ' - ~ - " ~

•

--

0.5

1.0 -0.8

~

-1.2 0

0.25

t

t

t

t

0.2

0.4

0.6

0.8

0.0 0.2

f/c

(c)

f/c

0.6

0.8

(d)

0.5

Ptlp=0.9s

0

o

3.0

Aspect ratio

PtlV=0.95

Aspect ratio 0.25

1 0.5

2.0

-0.5

bF

0.4

0.25

-1

×

1

E 1.0

-1.5 -2 o

t

t

0.2

0.4

L 0.6

i

I

0.0

0.8

0.2

0.4

f/c

f/e

(e)

Aspect ratio

<>

~

1 0.5

-0.5

0.25 -1.5

-2.5

O -3.5 0

0.8

(f)

_ P tiP=0"975

0.5

0.6

i

i

i

i

0.2

0.4

0.6

0.8

f/c

ptip=0.975

Aspect ratio 0.25

0.5 1

I

0 0

0.2

0.4

0.6

0.8

f/c

Fig. 12. (a,c,e) O'm/ffs a t the tip and (b,d,f) the peak value of az/aS in the body for the different material models compared with the analytical results of Section 3 for disc-like inclusion. The symbols are defined in Table 3. inclusion, it is found that the magnitude of the mean stress generated at the point of m a x i m u m curvature for the disc-like inclusion are less than those for the fibre-like inclusion. As a result, a much wider variation of creep properties is observed in the matrix surrounding a fibre-like inclusion. The peak values of the maximum principal stress for the fibre-like inclusion with an aspect ratio of 4 is always greater than those for the disc-like inclusion with an aspect ratio of 0.25. I f f / c > 0.3 the results for the fibre-like inclusion of aspect ratio 2 are greater than those for the disc-like inclusion of aspect ratio 0.5, while if f/c < 0 . 3 the opposite is true. F o r example, when f / c = 0 . 5 and p = 0 . 8 , we find a~/a~= 1.19 for the fibre-like inclusion with aspect ratio of 2 and a z / a s = 1.11 for the disc like inclusion with aspect ratio of 0.5; however, when f i e = 0.3, and

p = 0.8, we find a l / a s = 0.516 and ai/as = 0.546 for the fibre-like and disc-like inclusions respectively. These results are consistent with the analytical predictions of Section 3. 5. DISCUSSION In the present paper we have examined how the detailed structure of the constitutive relationships for the sintering of ceramic compacts influence the evolution of microstructure and the development of residual stress field within a body that contains a distribution of ellipsoidal inclusions. The major aim of this study was to determine those features of the material that most influence the response of a sintering component. At the present time there is a lack of agreement in the literature about the exact form of

1992

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

the quantitites f ( p ) and c(p) as defined in equation (6), and the detailed relationship for the sintering potential. In this paper we have examined three different forms for the function f ( p ) and c(p), Table 1, whose ratio can change significantly during the course of sintering, particularly during stage 2. We have also arbitrarily adjusted these functions to provide solutions for a range of values of f/c. The sintering potential has either been set equal to 1 MPa, in agreement with the observations of Brook and co-workers [8, 15], or the relationships of equations (7) and (8) have been employed. In Section 3 Eshelby's method [13, 14] was used to analyse the response of a body whose creep properties and sintering potential are uniform through the body. It was found that the general level of stress within the body scales with the sintering potential. The mean stress achieves its peak compressive value at the point of maximum curvature, with this being a function of the aspect ratio of the inclusion and the ratio f/c, as shown in Figs 3 and 5. The maximum principal stress, however, achieves its peak value away from the point of maximum curvature. Again the magnitude of this stress is a function of f/c and aspect ratio, Fig. 3(b). When examining the results from the detailed computation for the full range of models employed in this paper, we find that the simple analysis of Section 3 provides a useful framework for presenting these results, Figs l l and 12. During the early stages of sintering when there is a relatively small variation of density and grain-size, and therefore o f f ( p ) and c(p) and the sintering potential, t~s, in the body, we find that when stresses are normalised with respect to the sintering potential at the tip of the inclusion that the trends and magnitude of stresses observed in Section 3 are reproduced. As sintering progresses the compressive mean stress at the tip of an inclusion promotes further densification and grain-growth, and, as a result, the material becomes more creep resistant and a higher mean stress needs to be developed in order to maintain a compatible displacement rate with the surrounding less creep resistant material. The magnitude of the resulting stress increase depends on two principal factors. If the sintering potential varies according to equations (7) and (8) then a higher sintering potential develops at the tip of the inclusion, which promotes densification and not such a high increase of mechanical stress is required. The sharper the decrease o f f ( p ) and c(p) the higher the increase in stress that is required to maintain compatibility. The steepest drop i n f ( p ) and c(p) occurs for the mechanistic model, Fig. l, and we find that the highest stresses are obtained when this model is combined with a uniform sintering potential of 1 MPa, Fig. 11. Throughout this process, however, the position and magnitude of the peak maximum principal stress, when normalised with respect to the sintering stress at the tip, remains largely unaltered. Eventually the material surrounding the tip of the

inclusion becomes so creep resistant that a zone is formed which essentially forms an extension to the inclusion and within which a net compressive stress field develops. This zone grows with time and the point of peak maximum principal stress moves from the surface of the inclusion to the edge of this zone, Fig. 10. From the set of problems analysed in this paper it is evident that the expression for the sintering potential plays the major role in determining the magnitude of the stresses within the body. The expressions of equations (7) and (8) produce much higher values of the sintering potential than the experimentally determined value of 1 MPa [8, 15] during pressureless sintering. The difference is further enhanced in the presence of an inhomogeneity as a high mean stress develops in the vicinity of the inhomogeneity and a given density is achieved at a lower grain size. It would be tempting to always employ a sintering potential of 1 MPa, but care needs to be exercised in extending the results of Brook and co-workers to such an extent that one assumes that these limited studies are fully representative of the full range of situation that are likely to occur in practice. Also, when extracting a value for the sintering potential from the experimental data a number of assumptions were made about the structure of the constitutive relationships for the material response, which are not always consistent with the models used in this paper. For example Hsueh et al. [8] employed a model which assumes that the grain-size is a unique function of relative density in their analysis of a set of HIP experiments conducted over a range of applied pressures. Also in the experiments of Rahaman et al. [15] on CdO powder compacts the grain size ranged from 2 to 4 # m as the relative density increased from 0.5 to 0.9. Experimentally determined values of as, which were in the range 1-2 MPa, are in fact much closer to equation (7), which predicts t7s in the range 0.75-4 MPa, for these conditions. The next most significant effect is the ratio f/c, with the magnitude of the various stresses generally increasing with increasing value of f/c, Figs 11 and 12. This result is consistent with that of Bordia and Scherer [12] for spherical inclusions. As fie decreases during sintering, and the incompatible strain-field, which arises from variations of densification rate within the compact can be more readily accomodated by shear deformation of the matrix, the magnitude of both the peak mean stress and peak maximum principal stress decrease. The detailed forms of the functions f ( p ) and c(p) appear to be less important in determining the response of the body, although higher stresses are generated in the vicinity of the inclusion for those models in which f ( p ) and c(p) change most steeply with relative density. In the analyses presented in this paper we have assumed that the material remains continuous throughout the sintering process. In practice cracks

DU and COCKS:

THE SINTERING OF CERAMIC COMPONENTS--II

can develop in regions of high tensile principal stress, particularly d u r i n g the early stages o f sintering w h e n the material is particularly weak in tension. A t the present time we can not provide a n y i n f o r m a t i o n , however, a b o u t the f o r m a t i o n or p r o p a g a t i o n o f these cracks, o t h e r t h a n identify sites where these cracks are likely to form a n d the direction they are m o s t likely to p r o p a g a t e in. F o r a n ellipsoidal fibrelike inclusion of aspect ratio 4 the m a x i m u m principal stress occurs at a position where the o u t w a r d n o r m a l to the inclusion m a k e s a n angle 0 = 55 ° with respect to the X~ axis a n d acts in a direction which m a k e s a n angle of - 57 ° with respect to the X~ axis. W e would therefore expect a crack to first form at this location a n d to p r o p a g a t e initially away from the inclusion along a line which m a k e s a n angle of 33 ° with respect to the XL axis. W i t h i n the i n h o m o g e n e o u s body we observe a wide range o f different stress states r a n g i n g from pure shear to situations where the m e a n stress dominates. Also elements o f material a n d stress systems invariably rotate d u r i n g densification. We m i g h t therefore expect a certain texture to develop within the material with contacts between particles which are n o r m a l to a high compressive principal stress spreading m u c h m o r e significantly t h a n a contact plane which experiences a tensile n o r m a l stress, which could in fact shrink in size with the particles eventually separating. The models used here, which employ two scalar state variables can, at best, provide a first a p p r o x i m a t i o n o f the stress state a n d evolution of microstructure, b u t at the present time there is very limited i n f o r m a t i o n , b o t h experimental a n d theoretical, in the form of mechanistic models, on which to base a higher order model.

1993

12. R. K. Bordia and G. W. Scherer, Acta metall. 36, 2393 (1988). 13. J. D. Eshelby, Proc. R. Soc. A 241, 376 (1957). 14. J. D. Eshelby, Proc. R. Soc. A 362, 561 (1959). 15. M. N. Rahaman, L. C. De Jonghe and R. J. Brook, J. Am. Ceram. Soc. 69, 53 (1986). 16. T. Mura, Micromechanics o f defects in solids. Martinus Nijhoff, The Hague (1982). 17. A B A Q U S User Manual. Hibbitt, Karlsson and Sorensen, (1988). APPENDIX

E s h e l b y ' s M e t h o d A p p l i e d to the P r o b l e m o f Section 3 We start by considering the problem of an infinite matrix which deforms according to the relationship d° = ~o [3c(p)s° + 3/(P) (trm -- as)f'J]

(AI)

which contains an ellipsoidal inclusion with principal half axes at, a 2 and a 3 in the x~, x z and x 3 directions respectively, which creeps at a rate it •i E° = ~o0 [~3 c(p )s°.q_3f(,o)ffmg~)

(A2)

]

where g[,~d 0. The body is subjected to zero remote stresses, but the matrix wants to densify under the action of the sintering stress a,. If we were to cut the inclusion from the matrix then the body would creep at a uniform rate {7.

d,j = 3f(p) ~ do 60 = - 3 6o'

(A3)

We treat this strain-rate as a matrix transformation strain-i'ate or, equivalently, we can consider the situation where an inclusion embedded in a matrix with constitutive law ~ij = ~o [3 c(p)SO + 3f(p)a mg~]

(A4)

experiences a transformation strain-rate Acknowledgements--This research was supported by the Science Engineering Research Council under Grant No. GR/F/0546.6. Thanks are due to Hibbitt, Karlsson and Sorensen, Inc. for access to ABAQUS under academic license•

•T E,j = ~~; g0"

Eshelby [13, 14] demonstrates that if the transforming inclusion has the same creep properties as the matrix the total strain-rate experienced by the inclusion is d ~ = Sok,~ ~

REFERENCES

1. Z. Z. Du and A. C. F. Cocks, Acta metall, mater. 40, 1969 (1992). 2. M. Hillert, Acta metall. 13, 227 (1965). 3+ R. J. Brook, J. Am. Ceram Soc. 52, 56 (1969). 4. M. F. Ashby, Background Reading, H I P 6.0, Univ. of Cambridge (1990). 5. A. S. Helle, K. E. Easterling and M. F. Ashby, Acta metall. 26, 2163 (1985). 6. R. M. McMeeking and L. T. Kuhn, Acta metall, mater. 40, 961 (1992). 7. A. C. F. Cocks, to appear• 8. C. H. Hsueh, A. G. Evans, R. M. Cannon and R. J. Brook, Acta metall. 34, 927 (1986). 9. F. A. Leckie and D. R. Hayhurst, Proc. R. Soc. (1975)• 10. R. L. Coble, J. appl. Phys. 32, 793 (1963). I 1. Z. Z. Du and A. C. F. Cocks, Prof. 6th U.K. A B A Q U S Group Meeting, Manchester (1990)•

(A5)

(A6)

where d~ is the transformation strain-rate, d~ = (~/3) g,j, and Sukt is Eshelby's tensor which is only a function of geometry and Poisson's ratio v

2\ 1 +f/c ]

(A7)

with the strain-rate at a point immediately outside the inclusion given by do = 'g~J+ 3 6,j where 1 "T , Eiknknj "T "g'J= "~ + 1 -- v , Eklnkntnlnj--l--2v

.'r

1 l+v

r

,~fknkn

i

t

f ( 1 - ~ ) 'Ek'nkn'go -- 3 1 -- v ~ (n'nl -- ~ 6o)

(A8)

1994

D U and COCKS:

THE SINTERING OF CERAMIC COMPONENTS--II Mura [16] provides the necessary expressions for Sqkl for both prolate and oblate spheroids. For a general ellipsoidal inclusion [16]

and 1 1 + v ~r _ 1 - 2v '~n~nj. =e¢--3 l--v 1--v

In these experiments ~ ~kk, and '~u = i o - (~/3) 6 Uand [] is the outward normal to the surface of the inclusion. Eshelby [l 3, 14] further demonstrates that if the inclusion has different properties to the matrix the stresses immediately outside of the inclusion can be obtained from the solution of an equivalent transformation problem in which the matrix and inclusion have the same creep properties. If " E• iTU Is the transformation strain experienced by the inclusion for the actual problem, equation (AS), and ~ is the total strain in the inclusion after transformation then the stress field outside of the inclusion is the same as that for the equivalent problem with transformation and total strainrates ~ , g,~ provided =

-c

_

"ic

3 1 - 2v Sire - 87t(1 - v) a~In q 8~(1 - v) II'

a~l12

1 - 2v 8n(1 - v) l l '

3 Sl133 - 8if(1 - - v ) a]It3

1 - 2v 8n(1 - - v ) ll'

1 S1122 =

S1212 --

8•(1

-- V)

a~+a~l 1-2v 16n(1 - v) n-k 1 6 ~ v) (l~

(A12)

All other non-zero components are obtained by the cyclic permutation of (1,2,3). The components which can not be obtained by the cyclic permutation are zero; for instance, Sin2 = Si223 = Sn32 = 0. For a polate spheroid (a I > a 2 = a3), /i of equation (A12) are given by

and

•i~_ 0 . ~ _ E,j) + E~

(A9) Ii = 4 n - 212,

provided f and c are the same for the inclusion and matrix. In the limit as ~ - , 0 the above equations when combined with equation (A5) give '¢ __ "iT __ E~--E o-~

~ 6

-J .¢ _

21 n,

122=133= I23

3122 = 4n/a~ - 123 - (12 - 1.)/(a~ - a~), 123 = n/a~ - (12 - ll)/4(a ~ - a~).

~.

(AI0)

The resulting transformation strain-rate for the equivalent problem is then •"r

31. = 4 ~ / a ~ -

II2=(12-Ii)/(a~-a~),

-I ~

~ij = SijklEkl- Sijkl "3 ij

(A11)

where S,~ is the inverse of the Eshelby tensor and the stresses outside the inclusion are given by equation (A8).

(A13)

For an oblate spheroid (a~ < a2 = a3) the above equations are still valid, except

2.a,o cos_, a3 _O,(l_ey:2t a3\ aU J"

12=13-(a~-a~)3/2 t

(A14)

The stresses immediately outside the inclusion obtained by combining equations (A3), (AS), ( A l l ) and (A12) with (A13) for the prolate spheroid and (A14) for the oblate spheroid are presented in Section 3.