On constrained sintering—II. Comparison of constitutive models

ooo1-6160/8x 53.00 + 0.00 Copyright SC,1988 Per~amcn Press plc

ACM metal!. Vol. 36. No. 9, pp. 2399-2409. 1988 Printed in Great Britain. All rights reserved

OVERVIEW

NO. 70

ON CONSTRAINED SINTERING-II. COMPARISON OF CONSTITUTIVE MODELS R. K. BORDIA and G. W. SCHERER E. I. du Pont de Nemours & Co., Central Research % Development Department, Experimental Station E356, Wilmington, DE 19898, U.S.A. (Received

8 Muy

1987; in revised

form 27 &r&r

1987)

Aketract-Several authors have proposed constitutive equations to describe bodies that sinter under constraint. We provide a critical examination of these models and show that some of them imply a negative Poisson’s ratio, in contradiction to experimental evidence. R&sum&-De nombreux auteurs ont propose des equations de base pour d&crire les solides frittes sous contrainte. Nous examinons cm modeles de fa9on critique et nous montrons que certains d’entrc eux supposent un coefficient de Poisson negatif, ce qui est contraire 1 l’exp&ence. Z~f~~Meh~re Autoren haben G~nd~eichungen zur Beschreibung von Festkiirpern vorgeschlagen, die einem Sintenrorgang unter einschmnkenden Redingungen unterworfen sind. Diese Modelle werden nun kritisch untersucht. Es wird gezeigt, daD einige dieser Beschreibungen eine negative Querkontraktionszahl erfordem, im Gegensatz zum experimentellen Befund.

1. INTRODUCTION In Part I of this series [I] we developed a constitutive equation for an isotropic viscous material for which the strain rate is linearly proportional to the stress. The equation can be written in various forms in terms of Poisson’s ratio and functions that represent the viscous response of the porous body to pure shear, hydrostatic, and uniaxial stresses. In this paper, we examine similar constitutive equations that have been used in the analysis of constrained sintering problems, such as sintering under a uniaxiat load, on a rigid substrate, or with rigid inclusions. It is shown that some of these modeis imply that Poisson’s ratio (v,) is negative, in contradiction to experimental evidence. In Part III [2] we will apply the models to the calculation of stresses in a matrix sintering around rigid inclusions. Very high stresses are erroneously predicted by the models for which Poisson’s ratio is negative. In Section 2 the response of a body during sinter forging (i.e. sintering under a uniaxial load) is used to illustrate the consequences of a negative Poisson’s ratio. In Section 3, the available constitutive models are compared, and their limitations are discussed in Section 4.

want to calculate the value of s_(t) that will maintain a constant radius &, i.e. that will give zero radial strain. A schematic of the experiment is shown in Fig. 1. The constitutive equations presented in Part I f 1J can be written in cylindrical coordinates as: i,=i,+E,‘[a,-v,(cr,+a,,)]

(1)

i,=il+E,‘[u,-v,(rr,+a,)]

(2)

and

where i, and it are the radial and axial strain rates, a,, a,, and a= are the radiai, hoop, and axial stresses, Et, is the uniaxial viscosity, and vp is Poisson’s ratio. The free strain rate, i,, is the linear contraction rate of a freely sintering body (i.e. with no externally applied load). The free strain rate is related to the sintering potential, Z:. by [I]: if = Z/3iy,,

(3)

4

2. SlNTER FORCING

i,

=o

or Rft) = R(O)

Let us assume that a porous body is sintered under a uniaxial stress 0,. The freely sintering cylinder would contract radially but a compressive axial load will cause a counteracting outward radial flow. We 2399

I

I

ttttttt

Fig. 1. Schematic of the sinter-forging experiment.

2400

BORDIA and SCHERER:

OVERVIEW NO. 70-H

where K, is the bulk viscosity, given by:

1.0

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

(4)

The free strain rate is negative (contraction) and so is the (compressive) sintering potential. Equations (1) and (2) were used to analyze sintering under a uniaxial stress [3], cr:. The radial strain rate is given by: i,=i,-v,uJE,.

0; = Ep c’,/v,.

(6)

Since ir is negative, the required stress is compressive as long as Ep and vi, are positive; however, if Poisson’s ratio is negative, a tensile load would be required to prevent radial contraction of the body. Using equations (3) and (4), equation (6) can also be written as: UJZ = (1 -2v,)/v,,

(7)

which indicates that eZ and I: have the same sign (viz., both are compressive) when 0 < vp < l/2. When vp = 0, the axial load has no effect on the radial strain rate [see equation (5)]. Sinter forging experiments [4-71 indicate, as our intuition suggests, that compressive axial loads reduce the rate of radial contraction. That is, Poisson’s ratio is experimentally observed to be positive. Therefore, the constitutive models that require v,, < 0 are inconsistent with observations. 3. COMPARISON

OF THE VARIOUS MODELS FOR SHEAR AND DENSIFICATION

In Part I [l] it was shown that for a sintering body the elastic strain is negligible, so it is sufficient to use the elastic-viscous analogy. The constitutive equation for pure shear can be written as: (8)

where So and e, are the shear stress and strain rate, respectively, and the shear viscosity of the porous material is: G, =

E,/P(l + v,)l.

(9)

The constitutive equation for hydrostatic stress (a) is: o = K#

- 3ir),

0.4

(5)

If the axial load is compressive (a; < 0), we expect the radial contraction rate to be slower (i.e. i, less negative), because the load causes outward flow. There is no radial deformation (i, = 0) if:

sii = 2G,t$.

0.6 3 ?

(10)

where i is the volumetric strain rate, it is the linear free strain rate, and K, is the bulk viscosity of the porous material, defined by equation (4). Most of the published constitutive equations for sintering materials are in the form of equations (8) and (10). They differ in the assumed dependences of G, and Kp on porosity and grain size. In this section we compare the expressions suggested by various authors for G, and Kp, and examine their implications for v,. We

0.0 0 i

Fig. 2. A plot of the density dependence of the normalized shear viscosity. For Hseueh et al. [9], the viscosity has been normalized by the viscosity at p = 0.95; all other viscosities have been normalized by the viscosity of the fully dense solid. Curve label: 1 = Raj and Bordia [8]; 2 = Hseuh e/ al. [9]; 3 = Scherer [ 131;4 = Skorokhod [17]: 5 = Rahman et al. [4]; 6 = Venkatachari and Raj [5].

find that models that give negative values of Poisson’s ratio overestimate G, relative to Kp. 3.1. Models for shear viscosity (G,) The model of Raj and Bordia [8] assumed explicit spring-dashpot elements to represent the constitutive properties of a porous material. The shear response was modelled by a Maxwell element and the densification response by a Kelvin-Voigt element. The choice of densification response was phenomenological since typical densification curves are very similar to the response of a Kelvin-Voigt solid to a step load. The step load was assumed to be the sintering potential, 2. Two rate constants o, and w, were defined for the shear relaxation rate and densification rate, respectively, as follows:

where G, and K, are the elastic shear and bulk modulus, respectively. A non-dimensional parameter, B E w&ok, characterizes the relative rates of creep and densification, and all results were obtained in terms of B. /I<< 1 implies that creep relaxation is insignificant relative to densification and conversely for p>>l. Raj and Bordia [8] assumed a constant viscosity for shear. There was no physical basis for this choice. However, this simplification made it possible for them to use the linear viscoelastic formulation. G, is independent of density and is given by: G

2 = tl

1,

(13)

where r~ is the viscosity for the fully dense material. This is plotted in Fig. 2 as a function of density. Many shortcomings of this analysis have been pointed out. Hsueh et al. [9] argued that it was unreasonable to use constant values of elastic con-

BORDIA and SCHERER: OVERVIEW

stants G, and K. (since they are functions of density) and the shear viscosity is not a constant (a function of density and grain size). The use of constant values for the parameters made it possible to use the viscoelastic analogy without making a conceptual error, as was shown in Part I [l]. The use of constant Go and K,, is not a serious error, since the elastic components only control the time constant for the very fast transient rise of the interfacial stress [IO]. In fact, if viscoelastic elements without K,, and Go are used, then the stress solution starts at the plateau value and the maximum value of the stress and the relaxation time constant are the same as the results obtained with the elastic components. Further the densification results are not at all affected by the elastic constants. The assumption of a constant shear viscosity is a serious limitation. and in applying this analysis to experimental results, a mean value of shear viscosity had to be used [6]. Hsueh et al. [9] used Coble and Kingery’s [l l] data on the properties of porous A&O, to obtain Gp as a function of density. The viscosity was assumed to depend on the instantaneous grain size and density. Using appropriate grain growth and densification laws, the viscosity was written as: Gp=qO (Q)p (1 -p)-L,

(14)

where Q,,is a reference viscosity which depends on the temperature, initial grain size, and initial density. The relative density is defined by p = p/p*, where p and pT are the bulk densities of the porous and fully dense bodies, respectively. The density dependence is made up of two functions: p” gives the dependence of G, on density for constant grain size and (1 -p)-’ accounts for the grain growth that accompanies densification. The parameters qO, p and 1 were obtained by fitting equation (14) to Cable and Kingery’s [I l] data, In Fig. 2. GP obtained from equation (14) is plotted as a function of density. The shear viscosity, G,. is normalized by the viscosity for /i = 0.95 [i.e. G,(@)/G,(0.95) is plotted as a function of C]. A close examjnation of Cable and Kingery’s [l I J data raises many questions regarding the interpretation of that data by Hsueh et al. [9] In Ref. [I I] all samples had the same grain size (and different porosities). Since equation (14) is derived with the assumption that there is grain growth, its use in fitting the data of Ref. [I I] is not justified, although the fit obtained is very good. In Ref. [I I], the samples had a grain size of 23 pm whereas in Ref. 1121, from which the densification data were taken, the initial grain size was 0.3 pm. Apparently, no correction was made for this grain size difference. Therefore, an erroneous value of q0 was obtained leading to an overestimation of the shear viscosity (relative to bulk viscosity) and a negative Poisson’s ratio. We will look at this point further in Subsection 4.1. In fact. the data of Ref. (I l] may not be relevant to the problem of interest. The experiments reported

NO. 70-11

2401

in Ref. [I I] were carried out on samples with porosity induced by the addition of different amount of naphthalene. This led to a situation in which the pores were isolated even at fairly small densities (0.5 _ 0.6). In real sintering systems, pores remain continuous up to fairly high densities (p _ 0.9). The connectivity of pores is known to affect most properties and is expected to affect the viscosity also. In particular. the unusual pore morphology in Ref. [I l] may have permitted an atypical mechanism of creep. It was suggested that because the pores are much larger than the average grain size (100 and 23 g m, respectively), grains under compressive stress might move into adjacent pores and provide local stress relief. This effect would lead to a lower shear viscosity than would be observed if the pore size were smaller [I 11. In support of this suggestion, we note that the dependence of the elastic shear modulus. G,, on the relative density is different from that of the shear viscosity, G,. Since there is no grain growth, we expect the same dependence of G,, and G, on density, both being proportional to the load-bearing fraction of the cross-sectional area. The stronger dependence of G, on p may indicate that grain motion into the pores is occurring. Scherer [ 133 proposed expressions for the viscous equivalent of Young’s modulus (E,) and Poisson’s ratio (v,) for porous materials. in terms of the viscosity of the solid phase and the relative density. They were derived using a microstructural model that has been used successfully to model the viscous sintering behavior of porous glasses [l4-161. The constitutive parameters E,, and vp are given by the following equations [ 131:

(15)

(16) where #J is the load-bearing fraction of the crosssectional area. These expressions depend only on 4; the pores are assumed to be continuous. Equations (I 5) and (16) give the correct limits: v~-+O when p -+O and v,+l/Z when p -1; E,,+O for p-+0 and E,--r3~ when 0 --) 1. G, for Scherer’s model is obtained by substituting En and I’~from equations (I 5) and (16) in equation (9): G

A.=

q

3fi 6 - 40 + (30 - 202)‘;?

(17)

A plot of equation (17) is given in Fig. 2. Skorokhod [ 171 suggested expressions for effective shear and bulk viscosities for porous material in terms of the viscosity of the solid phase and porosity. These constitutive laws have been used to analyze sintering problems under a variety of loading conditions: hot pressing, upset hot pressing, hot isostatic pressing, hot rolling, etc. [ 181. Ermeev and Gudov [19] attempted to relate Skorokhod’s constitutive laws to

BORDIA and SCHERER:

2402

OVERVIEW

NO. 70-11

microscopic models. They calculated the effective viscosities for typical porous structures, including one pore or an array of pores on grain boundaries or within grains. Using this approach, they have related Skorokhod’s viscosities to the transport properties of materials. We are not aware of any work which uses Skorokhod’s laws to analyze the problem of heterogeneous densification. The shear modulus for Skorokhod’s analysis, which is shown in Fig. 2, is given by:

Sinter-forging experiments have generated a body of data from which the shear and bulk viscosities have been obtained for a wide variety of materials. Rahaman et al. [4] have experimentally determined the following relationship for shear viscosity, G,:

A% X=lexp[-2u0(l

-B)]

where d is the instantaneous grain size and A depends on the transport properties of the material?; the constant n, is evaluated experimentally, and a value of 2.0 was obtained for CdO. A plot of G,A/d3 as a function of p is shown in Fig. 2. Equation (19) was derived by using Coble’s creep mechanism for densification (since the grain size dependence was cdnsistent with this mechanism) and assuming an exponential form for the load bearing fraction of the cross-sectional area, 4. Finally, Venkatachari and Raj [5] found that the following equation best described the shear viscosity, G,, in sinter-forging experiments on A&O,:

(20) where B, is a constant which depends only on temperature, and d is the instantaneous value of the grain size. The density dependence (p) is based on the premise that 4 is proportional to the relative density [20]. The density dependence given by equation (20) is plotted in Fig. 2. From Fig. 2. it can be seen that different sensitivity of G, to @is assumed in each of the available models: however, none of these is fundamentally incorrect. In the region of interest (say p = 0.5-l .O) the difference between the various models can be up to a factor of 10. However, for the models of Scherer, Skorokhod, Rahaman et al. and Venkatachari and Raj, the difference is less than a factor of 2. It is not clear whether experiments would have enough sensitivity to clearly establish one model over the other. The available experiment work is rather limited. In Refs tThe values of G, obtained from equations (19) and (20) are different by a factor of 2/3 from those obtained from equation (9) of Ref. [4] and equation (11) of Ref. [S], respectively, because of the different definition of shear stress used here.

Fig. 3. Plot of the inverse of normalized bulk viscosity. All viscosities have been normalized by the viscosity at the onset of sintering & = 0.5). Curve labels same as for Fig. 2.

[4] and [5], sinter-forging was used to get G,, yet the density dependences of equations (19) and (20) are different. This difference could be due to the different stress levels used in the two studies, or possible differences in particle morphology. Thus, there is clearly need for more experimental work. To the best of our knowledge, Ref. [I I] is the only experimental study in which Gp has been obtained in pure shear and, as was discussed earlier, the microstructure in that study is not representative of a typical sintering microstructure. The analytical models (e.g. Ref. [ 131) assume particular pore morphologies. and the effect of this assumption on G, should be investigated. In Subsection 4.2, some recommendations are made about the needed experimental work. 3.2. Models for bulk viscosity (K,) For the sake of comparison, we plot the inverse of as F- 1. Kp is normalized by its value at the onset of sintering, and the initial relative density, A, is assumed to be 0.50. We define: Kp, since K,+m

(&)-’

=&(0.5)/&,(P).

(21)

Plots of equation (21) for all the available models are compared in Fig. 3. Raj and Bordia [8] represented densification by a Kelvin element. The densification rate from this model can be explicitly obtained as a function of density. For PO= 0.5, (R,)-’ = -ln(P)/ln(2).

(22)

The densification model used in Ref. [8] was phenomenological and later experimental work [6] showed that two Kelvin elements in series were needed to represent densification completely. There is some indication [6,21] that the two time constants (corresponding to two Kelvin elements) represent the intermediate and final stage of sintering. This observation should be checked, since it has the potential of establishing a simple phenomenological densification law. The basic problem in Raj and Bordia’s analysis [8] stems from the use of linear viscoelasticity for sin-

BORDIA and SCHERER:

2403

OVERVIEW NO. 70-11 (1

2.5 f%‘)-’

2.0

1.0

(RJ

0.5

~)2.67 5fl.67

(2%

’

=

3 - 28 - (3p - 2P’)‘,? (26)

2ij

For Skorokhod’s model [17], I$, is given by:

-_

0.0’

-

‘@(o

For Scherer’s model [13], Kp is obtained by substituting equations (15) and (16) into equation (4). The following expression is obtained:

1.5 ” c

=

’

’

’

’

’

’

’

’

’

’

1.0

4rlB3

(27)

Kp=m. From this, the normalized

0.6

bulk viscosity is:

F

fR,)-’

= y.

Rahaman er al. [4] have used the following equation for the bulk viscosity: Iyp = (d’P/A)exp[-a*(1 0.0

0.4

0.8

1.2

I.6

2.0

T Fig. 4. The response of the Kelvin-Voigt element to loading and partial unloading.

- /?)I.

(29)

The A in equation (29) is the same as in equation (19) and a, is 2.0 from their experiments on CdO. The normalized viscosity is given by (RJ-‘=O.l8exp[-2(1

-p)]/p.

(30)

Finally, from Ref. [S] (Venkatachari and Raj) the following expression is obtained for Kp: tering and from the use of a Kelvin-Voigt element to represent densification. Upon unloading, the strain in a Kelvin-Voigt element is recovered anelastically, and this is not physically reasonable for a body undergoing densi~cation. This point is illustrated in Fig. 4, where the sintering body is subjected to the stress-time cycle shown in the upper part and the strain-time response is shown in the lower part. Upon removal of the load, the body is predicted to expand, which is unrealistic. In Refs [S] and (221, unloading was not considered, so this implication of the model was not discovered. Further, the use of this model requires that the driving force for densification (Z: + 6, where CTis the hydrostatic component of the applied stress) he related to the anelastic modulus, E(,. by: (Z: + ci)/K= = tn(&J.

(23)

This improbable coupling occurs because the asymp totic value of the densification strain is given by the right hand side of equation (23). The bulk viscosity for the model used by Hsueh et al. [9] is given by:

STY (1 -i&Y Kp = (1

_py+7

’

(24)

where t is a coupled measure of grain size and grain growth kinetics during sintering and y is a constant. From equation (24), using y = 1.67 (from Ref. IS]), the following expression is obtained for normalized bulk viscosity:

K,=

- &ht(l

-D)+OS@@

a

The normalized value of (%)-’

(KJ’

=

i-2)].

(31)

from equation (31) is

-0.14@

ln(1 -ii)+O.Sp(p

$2)’

(32)

All the curves in Fig. 3 are very similar in their density dependence, the maximum variation at any density being only a factor of 2. This observation is interesting considering that widely different assumptions have gone into the derivation of these equations. For example, Scherer’s expression [equation (26)] was developed for viscous materials and hence does not have grain growth built into it. The grain growth factor has also heen taken out of Rahaman et al.‘s, and Venkatachari and Raj’s expressions. Hsueh er al.‘s expression is a fit to ex~~mental data for Al,O, and does incorporate grain growth. Raj and Bordia’s expression also has the grain growth implicitly included. 3.3. Poisson’s ratio Figure 5 is a composite pIot in which the Poisson’s ratios given by the various models are plotted as functions of density. For Raj and Bordia’s model [8], we can explicitly show that Poisson’s ratio becomes negative if /I c n, where:

2G -- I - 2v,

n==@-

1+v,

(33)

2404

BORDIA and SCHERER:

OVERVIEW

NO. 70-JJ

s.

(Vp)p,a,au =

-1.00 0.0

0.2

0.4

1.0

0.6

F

Fig. 5. Plot of the Poisson’s ratio as a function of 9. For Hsueh et a/. (Ref. [9]), and Raj and Bordia (Ref. [S]), a value of p0 = 0.5 has been used. Curve labels same as Fig. 2; I(a) is for /I = I/9 (/I n).

(37)

From equation (37) it is apparent that the plateau value of Poisson’s ratio becomes negative for p
where v, is the elastic Poisson’s ratio. Poisson’s ratio for a porous material, vp, is related to the shear and bulk viscosities by [I]: 3% - 2G,

(34)

vp = 2(3Kr + G,)’

According to the VE analogy 123,241, the same equation relates the Laplace transforms of vr, G, and Kp of a VE material. Substituting the transformed functions for Kp and Gr from the spring-dashpot analogs into equation (34) and inverting the result, we get the following expression for the timedependent Poisson’s ratio (see Appendix II):

3&L - B)

+

[(2 + n)(28 + n) ]exp[

-(~~~~~I

Thus, from equation (38), of p’ and & and this value (36) to get vp as a function Cs,= 0.5, vp is plotted for (#I < n) and fl= I (/I > n). for B < n.

(38)

fis obtained as a function is substituted in equation of 8. In Fig. 5, assuming two values of #?: fi = l/9 As expected. vp is negative

VP

(35)

for where I = t/r,, T, is the time constant densification and a = K,JK,,; for most sintering systems, a u lo-’ [S]. Equation (35) gives the correct limiting values: v,,(O)equals the elastic Poisson’s ratio [v, = (3& - 2G~)/2(3~ + Go)] and vp(co) = I/2 (incompressible solid). A plot of equation (35) is shown in Fig. 6 for different values of b. The solution consists of a short term (Fig. 6a) and a long term (Fig. 6b) response. The Poisson’s ratio starts at the elastic value, vp = v,,, quickly reaches a plateau, then increases to 112 in times of the order of 7 for fl> 0. Since for most of the sintering cycle the long term solution is of importance, we will look at it in a little more detail. The long term solution (f 51 1) from equation (35) is given by:

i (a)

=P

-a75

8’ -1.00

0.0

’

’ 1.0

’

’

2.0

’

’

3.0

’

’

4.0

’

6.0

T @I

v,(f) = ;-[2(2;+n)]exP[ The plateau value is obtained equation (36) and is given by:

-~I.

(36)

by setting 7~. 0 in

Fig. 6. Poisson’s ratio v,(T) for the model used in Ref. [8] as a function of time: (a) short term behavior; note that v,, is negative if /l < n. (b) long term behavior; Y, increases as time (and hence density) increases.

BORDIA

and SCHERER:

Table I. Values of the important parameters from Ref. [9] for A&O, of A =0.5 at 1500 c Numerical values 190.5 s I .67 IOOGPas 0.5 1.67 -l.OMPa

z

-

For the analysis of Hsueh et al. [9], vp is derived in Appendix III and is given by:

co-S(P 1 vp = 2c, +f(p)’ where co is a constant defined by:

(independent

of density)

3rZy c” = -

w

2tlo

and

f(D)=(P)p-‘(1 -p)‘+~-A (1 -boy

.

(41)

Using the values of 7, C, q. and the exponents y, I and p reported in Ref. [9] and reproduced in Table 1, we obtain the curve shown in Fig. 5 for Do= 0.5. The Poisson’s ratio for Scherer’s model is given by equation (16). The G, and K, for Skorokhod’s analysis [17] are given by equations (18) and (27). Substituting these in equation (34) leads to the following expression for the Poisson’s ratio: 3p - 1 VP= 7. 1+3p

(42)

OVERVIEW

2405

NO. 70-H

ties is negative for small values of /I. For /? > II. the Poisson’s ratio is always positive. In Ref. [8], b was allowed to vary from 0 to co, but based on the analysis in this paper we conclude that /I must lie between n and 00. The small value of fi reported in Ref. [6] from sinter-forging experiments could be due to the following reasons: (a) application of the load right at the beginning could lead to anisotropy in the axial and radial directions; (b) large deformation occurring during densification is represented by one viscoelastic law (since viscoelasticity is not appropriate for large deformation, the results obtained by fitting the data could be incorrect); (c) most importantly, /l (as defined in Ref. [6]) is the ratio of the bulk viscosity at the onset of sintering to the average shear viscosity. Since the shear viscosity increases by two orders of magnitude during densification, use of the average value leads to a small value of fl. The experimental data of Ref. [6] will be reanalyzed using viscous constitutive laws to obtain G, and K,,: the results will be reported soon. Hsueh er ul.‘s analysis [9] is the only one which uses constitutive laws derived from experimental data and yet gives a negative Poisson’s ratio in the range of densities in which the experiments were conducted. The reason for this is the use of two different sets of experiments (Refs [1 l] and [ 121) to obtain G, and K,, . The two experiments had widely different grain sizes and, in our opinion, this is the main reason for the gross overestimation of G,. In what follows, we attempt to calculate the factor by which G, has been overestimated. The Poisson’s ratio for Ref. [9] is given by equation (39). Figure 5 shows that vp remains very close to - 1 until p w 0.99 and then sharply increases to l/2 for p = 1. For PO= 0.5. using the exponents given in Ref. [9], f(p) from equation (41) is given by:

For Rahaman et al.‘s analysis [4], expressions for G, and Kp from equations (19) and (29) are substituted in equation (34) to give: 3exp[u,(l -p)]-2 ‘b=2{1 +3exp[a,(l -p)]}

(43)

where a, is equal to 2 for CdO. Finally, Poisson’s ratio for Venkatachari and Raj’s model [5] is obtained by using equation (20) for G,, equation (31) for Kp, and substituting them in equation (34). The expression for B, is obtained from equation (15) in Ref. [5] and E, from the expression for Coble’s creep (251. The result is: 2l[ln(l -P)+0.5p(p +2)]+4p2 VP=42[1n(l -p)+O.Sp(p+2)]-4p2’

From equation (45),f(p) equals 2.25 for p = 0.5 and monotonically decreases to 0.163 for p = 0.95 (see Fig. 7). For up to be positive, equation (39) indicates that c, must be greater than .f(p). It is shown in Appendix III that co >_/“((6)is also required to give a compressive stress (rr:) when i, = 0 for the sinterforging experiment shown in Fig. 1. However, using the numerical values of 7, Z, y and q. quoted in Ref. [9] (see Table l), co is equal to 0.005. In Ref. [9], 7 and q. were given in terms of initial grain size and temperature dependent coefficients:

(44)

4. DISCUSSION

4. I. Poisson j. rulio Figure 5 most clearly brings out the differences between the various models. For the model used by Raj and Bordia [8], the Poisson’s ratio at low densi-

7 =

d:/B

(46)

and qo=Cdo”(l

-PO)’

(47)

where do is the initial grain size, M is an exponent of the order of three, B is a function of temperature and initial density, and C is a function of temperature.

2406

BORDIA and SCHERER:

OVERVIEW NO. 70-R

Poisson’s ratio for this model at low densities is due to extrapolation (since the analysis is phenomenological and was developed to fit experimental data at high densities). The Poisson’s ratio from Rahaman et al.3 analysis [4] gives a positive value of the Poisson’s ratio. However, this is the only analysis for which vp decreases as density increases (v,-l/8 as p-+1). Physically this seems unreasonable because it would imply that the uniaxial compressive stress necessary ’ ’ ’ ’ ’ ’ ’ ’ ’ to give zero radial strain would increase as a function 0.50 0.90 0.70 as0 0.90 of time [see equation (6)j. This unusual result occurs b because in their analysis G, has a stronger deFig. 7. A plot of c, and f(p) from equations (40) and (41) pendence on the load bearing fraction of the area as functions of 6, when & = 0.5. than $ [see equations (19) and (2911. Venkatachari and Raj’s [S] expression for Poisson’s ratio was obtained from sinter-forging experiments in From Appendix IV, based on the data in Ref. [l 11, which the shear and densification rate were measured simultaneously. The bulk viscosity for densification the value of no is 780 GPa-s, not 100 GPa-s (as reported in Ref. [9]). If this value of ‘to is used, then was obtained by assuming the sintering of an idealc, = 6.1 x IO-‘, which is negligible compared tof@), ized array of periodically spaced pores on a single so equation (39) gives vp * - 1. The grain size in Ref. boundary and using the solution for diffusional growth of cavities [26]. It is expected that this model [1 l] (from which q,, was calculated) was 23 pm. For densification, data from Ref. 1121 were used. The would approximate the pore morphology during the initial particle size in the latter experiments was final stage of sintering. In Ref. [5] the density ranged 0.3 pm, so it is reasonable to assume that the grain from 0.7 to 0.95 and in this range, the densification size at the onset of isothermal sintering was h 1 pm. law gave a reasonable fit to experimental data (within Taking account of this factor of 20 difference in grain a factor of 2). Thus the Poisson’s ratio obtained from size, the estimated value of c, is _ 5 [using equations the constitutive laws of Ref. [S] should be used only (40), (46) and (47)]; we would expect this value if the at high densities. From Fig. 5, indeed, for $ > 0.7 the same grain size had been used in both experiments. values of the Poisson’s ratio are reasonable: however, As can be seen from equation (39) and Fig. 7, this negative values of V~are obtained due to the inapvalue of c, will give a positive Poisson’s ratio. plicability of the densification law at low densities. The analysis presented in the previous paragraph is 4.2. Exper~ental deter~inut~on of the constit~ti~e only for illustrative purposes. It has been assumed laws that shear in Ref. [ 1l] is by grain boundary diffusion. Further, it has been assumed that the mechanism In this paper, we have emphasized the need for remains the same when temperature is increased from experimental determination of the constitutive par1275°C to 1500°C and the grain size is changed from ameters. So few data exists on the creep of porous 23 to 1 pm. The calculation performed in this section materials that creep data from an atypical microemphasizes the need for simultaneous measurements structure (1 I] had to be used in Ref. [9]. In the recent of shear and densification in one experiment. past, sinter-forging experiments have been used to As shown in Fig. 5, Scherer’s model [13] gives a obtain shear and densification behavior simulpositive value of the Poisson’s ratio over the entire taneously ]4-7]. This experiment can give all the density range. The model was developed for viscous parameters (G,, KP and 0.4), the analysis gives plained in Part III [2]. Therefore, it is recommended positive values of Poisson’s ratio. In addition, for that in sinter-forging experiments loads be applied at J? > 0.5, the Poisson’s ratio from Skorokhod is very different densities and the axial and radial strain rates close to that from Scherer. This would imply that for measured soon after the load application, SO that the higher density samples the Poisson’s ratio is not very microstructure is not biased. The experiments should sensitive to pore morphoIogy. The negative value of be carried out at several different densities and at

O.ooII

1

BORDIA and SCHERER:

each density data are needed for several values of the stresses to check the assumption of linearity. It is essential to obtain the grain size of the test specimens, and indications of anisotropy should be sought. It is inadvisable to take the shear viscosity and densification data from different experiments because of the probable differences in microstructural details. 5. CONCLUSIONS Constitutive equations have been proposed by several authors to describe the behavior of materials sintering under constraints: Close examination of the various models reveals that several of them imply a negative Poisson’s ratio, which is allowed on thermodynamic grounds but is inconsistent with experimental observations. We have suggested experiments that will provide the data needed to establish a satisfactory model. In Part III [2], we examine the published analyses for the stresses in a sintering matrix containing rigid inclusions. The models that predict large stresses are those for which Poisson’s ratio is negative. REFERENCES I.

2. 3. 4. 5.

R. K. Bordia and G. W. Scherer. Acta mefall. 36, 2393 (1988). R. K. Bordia and G. W. Scherer. Acta merall. 34, 2411 (1988). G. W. Scherer. J. Am. Ceram. Sot. 69, C206 (1986). M. N. Rahman, L. C. De Jonghe and R. J. Brook, J. Am. Ceram. Sot. 69;53 (1986). K. R. Venkatachari and R. Raj. J. Am. Ceram. Sot. 69,

OVERVIEW NO. 7&-II APPENDIX

(1988).

I. M. N. Rahman. L. C. De Jonge. G. W. Scherer and R. J. Brook. J. Am. Cerum. SW. 70. 766 (1987). 8. R. Raj and R. K. Bordia. Arra merail. 32, ‘1003(1984). 9. C. H. Hsueh. A. G. Evans and R. M. Cannon, Acra merall. 34, 927 (1986).

IO. R. K. Bordia. unpublrshed work. II. R. L. Coble and W. D. Kingery. J. Am. Ceram. Sot. 39, 377 (1956). 12. R. L. Cable. .I. appl. Phj,.v. 32. 793 (1961). 13. G. W. Scherer, .I. non-crj~t. Solirls 34, 239 (1979). 14. G. W. Scherer. J. Am. Ceram. Sot. 60, 236 (1977). 15. G. W. Scherer and D. L. Bachman. J. Am. Cerum. Sot. 60, 239 (1977).

16. M. D. Sachs and T. Y. Tseng. J. Am. Ceram. Sot. 67, 532 (1984). 17. V. V. Skorokhod, Poroshk. Metall. 2, 14 (1961). 18. V. M. Gorokov. M. S. Kovalchenko and 0. V. Roman, Poroshk. Meroll. 241, 8 (1983).

19. V. S. Eremeev and A. 1. Gukhov. Poroshk. MefaN. 183,

4 CO d da e, m n p, i

S!l f A

B

(1985).

23. R. M. Christensen. Theory of Viscoelasticiry, An Inrroducfion. 2nd Edn. Academic Press. New York (1982). 24. G. W. Scherer, Relaxarion in Glass and Comiosires. Wiley-Interscience, New York (1986). 25. R. M. Cannon, W. H. Rhodes and A. H. Heuer, .I. Am. Cerum. Sot. 63, 46 (1980). 26. R. Raj. H. M. Shih and H. H. Johnson, Ser. Merall. 11, 839 (1977).

experimentally obtained coefficient (equation (19). a in Ref. [4]) dimensionless coefficient, independent of density [equation (39)] instantaneous grain size of the sintering body [equation (19)] initial grain size of the sintering body [equation (46)] shear strain (equation (2) of Ref. [I]. 6 in Ref. [4]; c, in Ref. [5]; C’,,in Ref. [ 171) an exponent of the order of three [equation (46)] dimensionless constant [equation (33)] coefficients which give dependence of shear viscosity on density [equation (l4)] shear stress (equation (1) of Ref. [I], o, in Refs [4. 51; uI, in Ref. [ 171) normalized time (f = taoI;) constant, depends on temperature and transport properties of the sintering body (equation (19). Ki in Ref. 14)) a function of temperature, transport properties and initial density [equation (4611

B,

constant, depends on temperature and transport properties of the material [equation (3 I)]

B,

constant, depends on temperature and transport properties of the sintering body [equation (20)] a function of temperature, and transport properties [equation (47)] uniaxial viscosity and Poisson’s ratio respectively of a porous body (equation (I), also see Ref. [I]) shear viscosity of porous body (equation (8). (also see Ref. [I]). G in Ref. [8]; r) in Ref. [9]) elastic bulk and shear moduli respectively see Ref. [I] bulk viscosity of porous body (equation (4). (also see Ref. [I]), K in Ref.

c EP’“P

G,

Kw G, K, (&-’ a

B L

Cr c,. c,

31 (1978).

20. R. L. Cable. J. uppl. Phw. 41, 4798 (1970). 21. C. P. Cameron and Risdi Raj, private communication 22. R. K. Bordia and R. Raj. .I. Am. Ceram. Sot. 68, 287

I

Nomenclature

499 (1986).

6. R. K. Bordia and R. Rat. J. Am. Cerum. Sot. 71. 302

2407

9

[81) inverse of normalized bulk viscosity [equation (2l)] ratio of the anelastic bulk modulus to the elastic bulk modulus for the porous body (see Ref. [8]). a = K,/rC, ratio of the characteristic shear rate to the densification rate of the porous body (B = q/q) volumetric strain (equation (4) of Ref. [I], c, in Ref. (51: Ap in Ref. [8]; U,, in Ref. [l7]) linear free strain [equation (I)] axial and radial strains in the cylindrical coordinates [equations (I) and (2)] load bearing fraction of the crosssectional area (equation (I 5). (I /cj~)in Ref. [4]) coefficient which gives the dependence of densification rate on density (equation (24). y = l//I, /I was used in Ref.

191) shear viscosity of the fully dense body (equation (13). also see Ref. [I]), 9%in Ref. 181;rrr in Ref. 191;‘I, in Ref. 117) anelastic bulk viscosity of the porous body [equation (I 2)]

2408

BORDIA and SCHERER. reference viscosity from Ref. [9] [equation (47)J elastic Poisson’s ratio (equation (33). 1’ in Ref. [9]) time dependent Poisson’s ratio for the sintering body [equation (35)] relative density of the sintering body: the instantaneous density is normalized by the density of the fully dense body (v = p/p,], p in Refs [4-6,8]: d in Ref. [I 31) relative density at the onset of isothermal sintering (equation (23). ps in Ref. [8]) mean (hydrostatic) stress (equation (3) of Ref. [I], -p in Ref. [5]; u, in Ref. [9]; ui, in Ref. [ 171) principal stress components in cylindrical coordinates [equation (I)] characteristic time constant for sintering [equation (24)] characteristic shear relaxation rate of the porous body (see Ref. [8]), equation (11) characteristic densification rate of the porous body [equation (12)] sintering potential (equation (33) of Ref. [I], -pO in Refs [5,6, 8,221; -I: in Refs [4,9]) partial derivative with respect to time

‘lo “0 v,(f) P

PO u

u,, a, and on

% wk 8

superscript “dot”

OVERVIEW

NO. 70-11

Equation (AII.5) is easily inverted and the time-dependent Poisson’s ratio is given by equation (35) in the text. (a) Axial stress required lo give zero radial strain rare (i) Elastic solution. The geometry of the problem is shown in Fig. I. Under a uniaxial stress. cr,. the radial strain is given by: X(1 - 2v,)

V”U>

(All.61 e,=------E,Eil The first term is due to the sintering potential Z and is equivalent to the free strain within the framework of the analysis presented in Ref. [S]. To obtain the stress that will give zero radial strain, we simply set c, = 0 and solve for u: from equation (A11.6): Z(l - 2r,) UZ= ___ (All.7) VII Substituting (3K,, - 2G,)/[2(3K0 + G,,)] for r,,. the following expression for u, is obtained. (AII.8) (ii) Time-dependent solurian. The time-dependent solution is obtained by taking the Laplace transform of equation (AII.8) and making the substitutions given in equations (All.]) and (AII.2). The initial condition is:

0.r


z(f)=i z, t > 0 so that

APPENDIX

II

The time-dependent Poisson’s ratio is obtained by using the viscoelastic analogy. The moduli in equation (34) are replaced by their Laplace transformed functions, which are obtained from the spring-dashpot analogs shown in Ref. [8]. Thus, Kp is replaced by:

K,,b +aw,) s+w,(l+a)’ and G, is replaced by: SK*(S) =

Use of equations (AII.1). (AII.2) and (AII.9) in equation (AII.8) leads to (All.10) where

(AIL I)

a2=

(AII.2) s + w, With these substitutions the following expression is obtained: sG *(s) = -

v;(s)

=

a,s’+azs

(AII.9)

Z*(s)=;

The Analysis Used by Raj and Bordia (Rcf [8]) (a) Poisson’s ratio

+a,

(AII.3)

a,s (s? + a,s + ah)’

where a,=l-n a2 = w, - nwk a, = au, wp

(

5

>

(B-n).

UWk?P al=-----.

_

I-n

The expressions

for a, and a2 were obtained with the approximation a<< I. Due to the small value of CI,the time dependent-solution can be broken up into a short term and a long term solution [8,22]. The short term solution is given by:

u,(f) -““(l+r&)exp[--s]} n B

a4= 2+n

(AlI.4) where n = 2G,,/3K0 and z = K,/KO. Expressions given in equation (AII.4) have been obtained with the approximation z c
S-S,

;.

A,

1

s - .rz

(AIIS)

(All.1 1)

_

(AlI.12)

where r = I/T, and *a is the time constant for densification ( = I/so,). This solution is valid for T 5 a. At i = 0, equation. (AII. 12) leads to the elastic solution. For a sintering body, the interest is primarily in the long term solution (i.e. f 5 I), which is:

F =($-~p(-&),

(AI1.13)

where r is the normalized time and n is defined by equation (33). It is clear from equation (AII. 13) that if/i is less than n, the unacceptable conclusion is that to get zero radial strain, u; and Z must be of opposite sign (i.e. u._must be tensile). This means that the Poisson’s effect is negarioe for /I
BORDIA and SCHERER:

III

APPENDIX The Analysis (a) Conslirutive

Used by Hsueh er al. (Rt$

191)

OVERVIEW

Table 2. The calculated values of the reference virositv. n,. for Ref. (91. A value 0f p = 0.5 and i * 1.67 has been used kom Table 1 G,/lO’MPa-min (from Fig. 5 of

laws

For densification. from equation (10) of Ref. (91,

=[;;f(--“-‘J(l +a/z’)

P

(AIII. 1)

where o is the hydrostatic component of applied stress. The coefficient y is defined as l/b where fi is an exponent defined in Ref. [9]. and pr, p0 and p are the final, initial, and instantaneous values of density, respectively. Equation (AIII.1) can be rewritten as:

=;[!g-+

2409

NO. 7&-II

P

Ref. [9])

0.5 0.6 0.7 0.8 0.9

2.15 4.68 8.32 17.30 59.80

~o~IO”MPa-min

~ /jp (1 -P)

(calculated from equaclon A1V.I)

2.25 3.58 6.25 13.15 44.37

I .22 1.30 1.33 I.31 1.31

(AIII.2)

where G, is given by equation (AIII.4). Subtracting equation (AIII.8) from equation (AIII.7), we obtain:

where, p is the normalized density defined as p = p/pf. For shear deformation, from equation (16) of Ref. [9]. the viscosity is given by:

(1 + a,;3Z) - 2 p (AIII.9)

;

+am

(AIII.3)

G, = ‘lo(piprY (1 -P/P,)-“. This can be rewritten as:

%=qo [ (1

DP +)A

6.

1

(AIII.4)

where ‘lo is a reference viscosity and p and 1 are exponents defined in Ref. [9]. (b) Poisson 5 rario From equation (AIII.2) the following expression is obtained for the bulk viscosity, Kp: K

=

_TZYD(1 -PO)’

P

(1

-3r>+j(l 2[q#(l

-&)‘(l -p)‘+‘-

-i)“-2t@P(l

-p_)‘+:l

3rZyP(l - &)Y(1 - /?)i]

(c) Axial stress required to give zero radial slain rate If a uniaxial stress, uz, is applied on a cylindrical compact, then the mean stress is u = u-/3. Using the definition of the shear stress and shear St&in, and the constitutive law relating the two, we obtain the following: = -[r;;;;~;;Y][I+~]

(AIII.7)

and c, - c, = :

(I.

2%

AM

we--B

Calculation

IV

q/t~” in Rq[ [9]

The viscosity data taken from Fig. 5 of Ref. [9] are shown in Table 2. The relationship between G, and q0 is:

PP (1 -P) Using values of p = 0.5 and i. = I .67, q. is obtained from G,=qo-.

equation (AIV.1) and values of G, given in Table 2. These values are also shown in Table 2. From this. the average value of ‘lo at 1275°C is found to be:

This is equivalent to equation (39) of the text.

-;

(AIII.10)

where c0 and J(p) are given by equations (40) and (41) of the text. For u,/Z to be positive (i.e. uz to be compressive), co must be greater thanf(p). c, andf(P) are plotted in Fig. 7 for PO= 0.5 using the values of the parameters from Ref. [9]. It can be seen that only for p > - 0.99 is the condition co >f(p) satisfied.

APPENDIX

(AIII.6)

c’,+2&=

-z =_, 3&B) 1 co -f(P)

(AIIIS)

_py+7

Substituting equations (AIII.4) and (AIII.5) into equation (34) and simplifying, the following expression is obtained for I’~:

VP=

Equation (AIII.9) can be solved for u, necessary to give c, = 0:

(AIII.8)

(AIV.2) (%)l”er.$e= 1.3 x lO”MPa.min. This can be extrapolated to the value at 1500°C using the following equation: ‘loa T exptQ/RT).

(AVI.3)

Using Q =418 kJ/mol K. q,,= 14OOGPa.s at 15oo’C is obtained. However, according to Ref. [25], from which the activation energy was taken. the appropriate activation energy is 481 kJ/mol K and not 418 (probably a typographical error in Ref. [9]). Using this value of Q, the value of lo at 1500°C is ‘lo= 780 GPa’s.