Numerical and Experimental Analysis of Anisotropic

Shrinkage during Sintering

Zhou Penga, Jianzhong Xiaob , Siyang Meic and Shijun Huangd

State Key Laboratory of Material Processing and Die & Mould Technology, Huazhong University of Science and Technology, Wuhan 430074, P. R. China

avivi200207@gmail.com, bjzxiao@126.com, copflashpoint@gmail.com,dsjhuang521@163.com

Keywords: sintering; anisotropic shrinkage; master sintering curve; numerical simulation

Abstract. With a view of simulating the dimensional changes during sintering of rectangle green

bodies, the thermo-mechanical behavior of zirconium powder compacts at high temperature is

investigated. In order to better describe the behavior of anisotropic shrinkage, the revised Master

Sintering Curve is modified. Finite element calculations are then carried out on the green body

according to the modified equation with the different shrinkages coefficients at the different stages

of sintering. The possible causes of the anisotropic shrinkage are explained by macro-surface

energy model. Numerical shape predictions have been compared with experimental data, which are

considered to be in good agreement.

Introduction

A standard manufacturing process for ceramic parts involves several separate stages. Sintering is

the last step of the role of different processing steps, which has shown product shaping (powder

preparation, consolidation, and densification) [1]. In order to reduce the costs of manufacturing, it is

very important to control the dimensional change during sintering, which has been widely studied in

recent years.

Since the late 1940s, the sintering theory has received great attentions, a lot of successful

approaches have been made to understand the mechanisms of shrinkage which mainly based on

three states, namely the early sintering stages [2,3], the intermediate shrinkage stages [4,5] and the

late sintering stages [6]. In the ordinary way, researchers used the concept of the master sintering

curve (MSC) to predict and control sintering [7]. The MSC is an empirical curve that provides a

characteristic measure of the densification of a material over a given density range [8].

Unfortunately, all this previous work ignored the effect of anisotropic shrinkage which can cause a

common and serious problem in ceramics production [9-12].

Though Toussaint’s model [13] considered a lot of additional effects including anisotropic

shrinkage, he based his work on thermo-elastic equation which ignored the different sintering states.

Thus, the modle is inconvenience for application.

Improved accuracy in the predictions demands not only proper characterization of the

densification of material but also constitutive models of increased realism which are capable of

predicting dimensional changes during sintering. To explore the role of the anisotropic behavior, we

employ numerical simulations of a three-stage law. This paper concentrates on the application of

sintering models to explain some of the trends observed in the anisotropic shrinkage measurements

as sintering progresses.

Experimental procedures

Raw materials. Commercially available Y-TZP (NANO) with average size between 50 and 100 nm

is used in this experiment. The nano powder is first heat treated to 100 ºC for 24 h prior to mixing

for de-agglomeration. Feedstock with 59 wt.% solid content is shown in

Table 1. The tested specimens are cubic samples (3.0mm × 4.4mm × 54mm) and injection moulded

by SZL50g Moulding Machine.

Advanced Materials Research Vols. 233-235 (2011) pp 3068-3073Online available since 2011/May/12 at www.scientific.net© (2011) Trans Tech Publications, Switzerlanddoi:10.4028/www.scientific.net/AMR.233-235.3068

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Table 1. Composition’s weight of feedstock

Feedstock Weight of powder (g) PW (g) EVA (g) SA (g)

59% 800 65.55 17.85 0.84

Debinding and sintering. After slow thermal debinding profile, the green body with very slow

heating rates (from 25 ºC to 100 ºC at 2 ºC /min; to 400 ºC at 5 ºC /min; to 25 ºC at 0.5 ºC /min) is

carried out in atmosphere environment. The slow heating rate can prevent defects such as

microcracks that are induced during the binder extraction process; while debinding to 400 ºC

provides strength to the debound sample for easy handling. Sintering with heating rate (from 25 ºC

to 400 ºC at 3 ºC /min; to 900 ºC at 5 ºC /min, and hold for 1 h, to 25 ºC at 5 ºC /min) is carried out

in atmosphere environment.

Modelling

Gillia et al.[14] used the model that considers the behavior of isotropic ceramic compacts during

sintering as a continuous process. And the total strain rate can be decomposed into two reversible

parts (shrinkage, thermal expansion) and two irreversible parts (free sintering, Newtonian

viscosity). While taking into account the mechanical aspects, some models are based on the three

different sintering stages despite the fact that the sintering mechanisms of these three stages may

not be the same. For instance, the shrinkage behavior can be quite different according to the

sintering time [15].And the situation becomes much more complicated in the case of an anisotropic

material [16]. There are different shrinkages in different sintering stages, so we can define different

coefficients of Thermal-Elasticity Model. The shrinkage in the first stage exhibited a linear

increase, and then exhibited a parabolic increase in the second stage followed by a linear increase in

the final stage.

The transversely-isotropic thermo-elastic law contains thermal expansion coefficients in the

different stages during sintering and the elastic modulus, poisson ratios and shear modulus of the

green body of ceramics. We can consider thermal expansion coefficients of green body as negative

values, since it will generate shrinkage deformation. It can be measured by thermal analysis

experiments or the shrinkages after sintering. The shrinkage rates of the X, Y and Z axial direction

are xε , yε and zε . Measured at the end of free sintering tests, they are supposed to be proportional

to the variation in temperature 0 fT T T∆ = − ( 0T is the initial temperature and fT the maximal

sintering temperature) as

x xTε α=∆ (1)

y yTε α=∆ (2)

z zTε α=∆ (3)

Where xα , yα and zα are the thermal expansion coefficients of X, Y and Z axial direction.

These parameters are negative since the materials shrinks when the temperature increases.

During the sintering, the sample is subjected to a stress yσ in the Y axial direction; the induced

deformations in the three directions are calculated from the following relations:

1

y

y y TE

σ

ε α′ = + ∆ (4)

1

x y x

vT

Eε σ α′ =− + ∆ (5)

Advanced Materials Research Vols. 233-235 3069

1

z y z

vT

Eε σ α′ =− + ∆ (6)

It can be deduced that

1

y

y y

Eσ

ε ε

=′ −

(7)

x x

y y

vε ε

ε ε

′ −=−

′ − (8)

The last parameter is the shear modulus G, which cannot be identified from previous

experimental data. We estimated it as

( )1

2 1

EG

v=

+ (9)

To the cubic sample, we can deduce that

( )( )( )1 1 1 1v x y zε ε ε ε= − − − − (10)

vv

T

ε

α =∆

(11)

The shrinkage of the body is different in different sintering stages:

1 1 2 2 3 3v T T Tε α α α= ∆ + ∆ + ∆ (12)

In Eq. 12, The thermal expansion coefficients of initial stage, intermediate stage and final stage

are α1, α2 and α3, and the corresponding temperature variations are ∆T1, ∆T2 and ∆T3, respectively.

It can be measured by experiment.

Table 2 summarises the main values of the thermo-elastic parameters that have been obtained

from experimental data. The thermal expansion coefficients are found with a good precision, a few

percents, whereas the accuracy of the elastic moduli is much poorer, between 10 and 25%. Table 2. Main values used for numerical simulations

α（ºC -1） -3.64×10

-4

E（GPa） 5.87

Glt（GPa） 4.11

νt 0.30

Numerical simulation

For the present study, several finite element software, either commercial or academic, have been

devoted to the macroscopic simulation of sintering in order to predict shape changes of green body

[17]. We used 3D module of ANSYS code with the transversely-isotropic constitutive equations

previously presented (Eq. 4 , Eq. 5 and Eq. 6 ) and the material parameters listed in Table 2.

Calculations are carried out with the plane stress assumption. The contact between the cuboid

green body and the load bearing board is supposed to be frictionless. Sintering is simulated by

applying to the green body a uniform temperature of 1450 ºC. For describing horizontal sintering

the green body is assumed to be also submitted to its own weight. In these cases, the calculated

thermo-elastic deformation of the green body resulting from these conditions is assumed to be its

final deformation after the sintering cycle.

Fig. 1 compares the results before and after sintering. The size change of the green body

indicated different extents of the shrinkage in X, Y and Z axial direction. The blue part represents

3070 Fundamental of Chemical Engineering

the body after sintering, and the larger part denotes the body before sintering. In Table 3, the data

show the experimental and numerical shrinkage, and they have a maximum relative error of 5%.

The final density calculated from equations was in good agreement with the values determined

using the three-stage orthotropic thermo-elastic law.

Fig. 1. Comparison of simulation of the cubic sample before and after sintering. The blue part is

the green body after sintering, and the white part is the green body before sintering

Table 3 shows the differences between the experimental group and numerical group are always

less than 5%. The shrinkages in Y and Z axial direction are particularly better predicted (less than

2%) than in X axial direction (less than 5%). Considering the complexity of Thermal-Elasticity

model, the result could be acceptable. Table 3. Comparison between experimental measurements and numerical prediction of the sample

Axial direction Experimental shrinkage (%) Numerical shrinkage (%) Error (%)

X 21.7 22.7 4.61

Y 18.67 18.3 1.98

Z 19.09 19.2 0.58

Discussion

The large defects and cracks caused by the effect of anisotropic shrinkage may be developed into a

complex shape of compact during sintering. The formation of the compact requires a

time-consuming machining involving diamond tools for reshaping which markedly increases the

production cost of ceramics. Anisotropic sintering shrinkage is observed in most of the forming

processes—e.g. injection molding, extrusion, tape casting. The possible causes of shape changes

during sintering are: ①heterogeneous green density field owing to powder-die friction or

non-uniform compression ratio during die pressing, ② structural anisotropy induced by

unidirectional die compaction, ③creep under the stresses created by the weight of the part, ④

heterogeneous temperature field in the part [18]. And Zavaliangos et al. [19] believed a number of

possible sources (including prior compaction, elongated particles, gas pore pressure, interface

porosity and crystallographic texture) of sintering anisotropy based on simulations of a

two-dimensional array of particles.

The pervious models described anisotropy shrinkage were too complex, for example, Wakai and

Akatsu used the linear sintering stress tensor and the viscosity tensor’s model to understand

anisotropic viscosities and shrinkage [20], however, uncertainty increases because that the model

contained too much variables which are inexact measured in practice, thus it is very important to

build a model which is suitable for application. During sintering, any mechanism that does not

cause the particle centers move closer will not lead to the densification. Instead, coarsening will

occur as matter is transferred from the surface to the neck region of the particles. The nondensifying

mechanisms involve: surface diffusion, lattice diffusion from the surface, and vapor transport. Even

Advanced Materials Research Vols. 233-235 3071

though these mechanisms do not lead to the densification, they do have an effect on accelerating or

alleviating the densification progress, because they cause a decrease in the neck curvature, which in

turn slows the rate of the densifying mechanisms. Accompanying the neck growth is an increase in

the compact strength. The mechanisms that lead to densification are grain boundary diffusion,

lattice diffusion from the grain boundary and plastic flow. All these possible sintering mechanisms

produce neck growth and surface energy reduce. For this reason, the rate at which the width of the

neck increases can be used to measure the rate of sintering. Furthermore, it can be used to compare

the relative rates of those mechanisms. Additionally, the rate of shrinkage of the powder compact

can be used to measure the rate of sintering for the densifying mechanisms. From the macroscopic

model, the rate of shrinkage of the cubic sample also used to measure the rate of sintering in

different stages.

For the phenomenon of shrinkage in sintering experiments, the free sintering process of ceramic

is the process of lowering its surface energy as well as free energy from the theory of diffusion. The

less specific surface area of ceramics, the less surface energy goes. The length, width, and height of

specimen after sintering are l, b and h, respectively, so the specific surface area S is as follows:

(13)

If l = b = h, the value of takes minimum value as well as the surface energy, when

the value of (l + b + h) is definite in the cuboid model. The variation of l, b and h tends to be

equivalent from the viewpoint of the trend of drop in free energy. However, as Fig. 2, the value of l

is larger than its b and the value of b is larger than its h in specimen, thus the relationship of

shrinkage rate is as follows: l b hε ε ε> > . It is identical to our experiments and numerical

simulations.

Fig. 2. The cubic model of macro-surface energy. The length, width, and height of specimen after

sintering are l, b and h, respectively, the value of l is larger than its b and the value of b is larger

than its h in specimen, thus the relationship of shrinkage rate is as follows: l b hε ε ε> > .

Conclusions

This study proposed a valuable approach for predicting the final deformation of anisotropic Y-TZP

(NANO) ceramics cubic sample during sintering. And the new predicting approach takes the

advantage of the classical method based on the three-stage Model. Numerical shape predictions and

experimental data are considered to be in good agreement. The results showed the differences

between the experimental group and numerical group are always less than 5%. The shrinkages in Y

and Z axial direction are particularly better predicted (less than 2%) than in X axial direction (less

than 5%). The possible causes of the anisotropy in shrinkage during sintering are discussed by

macro-surface energy model. The less specific surface area of ceramics, the less surface energy

goes. And the accuracy in predictions is improved by considering the additional effect of anisotropy

shrinkage.

3072 Fundamental of Chemical Engineering

Acknowledgements

We are grateful to Mrs. Wei Wei and Mr. Dai Wei in State Key Laboratory of Material Processing

and Die & Mould Technology of Huazhong University of Science and Technology for their kind

support.

References

[1] J.C. Agarwal: Adv. Ceram. Mater. Vol. 1 (1986), p. 32

[2] J. Frenkel: J. Phys (USSR). Vol. 9 (1945), p. 385

[3] G.C. Kuczynski: Met. Trans. Vol. 185 (1949), p. 896

[4] R.L. Coble: J. Appl. Phys. Vol. 32 (1961), p. 787

[5] W.D. Kingery and M. Berg: J. Appl. Phys. Vol. 26 (1955), p. 1205

[6] J.K. Mackenzie and R. Shuttleworth: Proc. Phys. Soc. Sec. B. Vol. 62 (1949), p. 833

[7] H. Su and D.L. Johnson: J. Am. Ceram. Soc. Vol. 79 (1996), p. 3211

[8] W.Q. Shao, S.O. Chen, D. Li, H.S. Cao, Y.C. Zhang and S.S. Zhang: Sci. Sinter. Vol. 40

(2008), p. 251

[9] J.S. Sung, K. D. Kood and J.H. Park: J. Am. Ceram. Soc. Vol. 82 (1999), p. 537

[10] P.M. Raj and W.R. Cannon: J. Am. Ceram. Soc. Vol. 82 (1999), p. 2619

[11] D.S. Park and C.W. Kim: J. Mater. Sci. Vol. 34 (1999), p. 5827

[12] A. Shui and N. Uchida: Powder. Technol. Vol. 127 (2002), p. 9

[13] F. Toussaint and D. Bouvard: J. Mater. Process. Tech. Vol. 147 (2004), p. 72

[14] O. Gillia and D. Bouvard: Adv. Powder. Metall. Part. Mater. Vol. 2 (1996), p. 739

[15] S. Kiani, J. Pan and J.A. Yeomans: J. Am. Ceram. Soc. Vol. 89 (2006), p. 3393

[16] A. Jagota, P.R. Dawson and J.T. Jenkins: Mech. Mater. Vol. 7 (1988), p. 255

[17] H.G. Kim, O. Gillia, P. Doremus and D. Bouvard: Int. J. Mech. Sci. Vol. 44 (2002), p. 2523

[18] O. Lame, D. Bouvard and H. Wiedemann: Powder. Metall. Vol. 45 (2002), p. 181

[19] A. Zavaliangos, J.M. Missiaen and D. Bouvard: Sci. Sinter. Vol. 38 (2006), p. 13

[20] F. Wakai and T. Akatsu: Acta. Mater. Vol. 58 (2010), p. 1921

Advanced Materials Research Vols. 233-235 3073

Fundamental of Chemical Engineering 10.4028/www.scientific.net/AMR.233-235 Numerical and Experimental Analysis of Anisotropic Shrinkage during Sintering 10.4028/www.scientific.net/AMR.233-235.3068

DOI References

[4] R.L. Coble: J. Appl. Phys. Vol. 32 (1961), p.787.

http://dx.doi.org/10.1063/1.1736107 [5] W.D. Kingery and M. Berg: J. Appl. Phys. Vol. 26 (1955), p.1205.

http://dx.doi.org/10.1063/1.1721874 [6] J.K. Mackenzie and R. Shuttleworth: Proc. Phys. Soc. Sec. B. Vol. 62 (1949), p.833.

http://dx.doi.org/10.1088/0370-1301/62/12/310 [7] H. Su and D.L. Johnson: J. Am. Ceram. Soc. Vol. 79 (1996), p.3211.

http://dx.doi.org/10.1111/j.1151-2916.1996.tb08097.x [8] W.Q. Shao, S.O. Chen, D. Li, H.S. Cao, Y.C. Zhang and S.S. Zhang: Sci. Sinter. Vol. 40 (2008), p.251.

http://dx.doi.org/10.2298/SOS0803251S [9] J.S. Sung, K. D. Kood and J.H. Park: J. Am. Ceram. Soc. Vol. 82 (1999), p.537.

http://dx.doi.org/10.1111/j.1151-2916.1999.tb01798.x [10] P.M. Raj and W.R. Cannon: J. Am. Ceram. Soc. Vol. 82 (1999), p.2619.

http://dx.doi.org/10.1111/j.1151-2916.1999.tb02132.x [11] D.S. Park and C.W. Kim: J. Mater. Sci. Vol. 34 (1999), p.5827.

http://dx.doi.org/10.1023/A:1004770520830 [12] A. Shui and N. Uchida: Powder. Technol. Vol. 127 (2002), p.9.

http://dx.doi.org/10.1016/S0032-5910(02)00004-9 [13] F. Toussaint and D. Bouvard: J. Mater. Process. Tech. Vol. 147 (2004), p.72.

http://dx.doi.org/10.1016/j.jmatprotec.2003.11.019 [15] S. Kiani, J. Pan and J.A. Yeomans: J. Am. Ceram. Soc. Vol. 89 (2006), p.3393.

http://dx.doi.org/10.1111/j.1551-2916.2006.01252.x [17] H.G. Kim, O. Gillia, P. Doremus and D. Bouvard: Int. J. Mech. Sci. Vol. 44 (2002), p.2523.

http://dx.doi.org/10.1016/S0020-7403(02)00189-3