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Simulation of the firing process for ceramic products

Kai Oide*, Yasuko Mihara*, Takaya Kobayashi*, Toshiyuki Saitou**,

Manabu Umeda***, Seishiro Matsubara**** and Kenjiro Terada****

*Mechanical Design & Analysis Corporation, Tokyo, Japan

**LIXIL Corporation, R&D Advanced Technology Research Center, Tokyo, Japan

**LIXIL Corporation, Aichi, Japan

****Tohoku University, Miyagi, Japan

Abstract: The firing process of ceramic products under a specific heat curve is generally divided

into three different phases: thermal expansion, sintering, and thermal contraction phases. These

non-mechanical deformations are assumed to be unrelated, and the mechanical ones are assumed

to be represented by the viscoplastic constitutive model. The key issue is how to determine

sintering strains by calibrating the employed function forms of the densification rate with the data

obtained from the stairway thermal cycle (STC) test. In addition, the presentations of the

dependencies of the elastic and creep properties on both temperature and density are of

importance to accurately predict the overall deformation of ceramic products. All the relevant

material parameters and the coefficients of thermal expansion and contraction are identified with

the data obtained from respective mechanical experiments conducted in testing equipment for

thermomechanical analysis (TMA) under several levels of temperature. This paper shows the

calculation results for the real ceramic products, considering these constitutive equations using

the Abaqus user subroutines; UEXPAN is used for the thermal strain and the sintering strain,

CREEP for the creep strain. And the calculation result of actual sanitaryware is compared with

the experimental data.

Keywords: Ceramics, Creep, Sintering, Densification, Dried bodies, Thermal deformation.

1. Introduction

Ceramic products are one of the most difficult to manufacture in terms of predicting their final

shape and evaluating their strength in the design stage. They are manufactured through a variety of

processes, such as mixing, casting, forming, drying, grazing, and firing. Among them, the

understanding of drying and firing must be essential. In fact, after the slurry forming of ceramics

by slip casting, the evaporation of fluids in the drying process and the densification of the internal

structure in firing, respectively, cause approximately 3 and 10 percent shrinkage strains. In

addition, during these processes, a ceramic product is subjected to creep deformation due to its

own weight, which leads to the non-uniformity of its shape distortion. At present, the geometry of

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a modal form for the slurry is commonly designed in an empirical manner with intuitive prediction

of the amount of shrinkage and creep deformations during firing. The combination of the large

shrinkage and inelastic deformations makes it difficult to properly evaluate the residual stresses

causing localized cracks in the final product. As such, we proposed a set of constitutive equations

for a dried body of ceramics within the finite strain framework, in which the elastic and creep

properties as well as the densification rate depend on the temperature history of the relevant firing

process (Matsubara 2017). With the goal of conducting large-scale numerical simulations for

sanitaryware products subjected to large deformations during firing, a hypoelastic model is used to

evaluate the stress rate, and the dependencies of its elastic moduli on temperature and mass density

are newly introduced. Moreover, the proposed creep law is of a Norton’s type, and the

dependencies on temperature and mass density are originally introduced in line with those of the

elastic properties. On the other hand, the thermal expansion/contraction and sintering-induced

shrinkage deformations are treated as nonmechanical deformations. The function form of the

densification rate during sintering is borrowed from Hsueh et al. (Hsueh 1986) and Favrot et al.

(Favrot 1997), while different linear coefficients of expansion/contraction are introduced for

elevated and lowered temperature processes. In addition, based on the framework proposed by

Largiller et al. (Largiller 2011, Largiller 2012a, Largiller 2012b), the function form of the ultimate

relative density is originally proposed to represent the densification-induced (non-mechanical)

strain during sintering in accordance with the dependencies of the mechanical properties on

temperature.

The present paper describes in detail a set of constitutive equations, verifies their performance, and

discusses an example simulation with Abaqus.

2. Material behavior of ceramic products during the firing process

2.1 Mechanical behavior during the firing process

Figure 1 shows a typical time variation of temperature (called a heat curve) during the firing

process of a dried body and the resulting time variation of the relative density, which is a non-

dimensional quantity defined as the current mass density divided by the mass density of the

completion of firing. This time variation of the mass density is mainly caused by two stress-free or

non-mechanical deformations, i.e., thermal and sintering shrinkage strains.

On this empirical basis, the entire firing can be divided into three phases, as indicated by the blue

arrows: thermal dilation, sintering, and thermal contraction. These three phases constitute four

stages. In the first stage, thermal expansion dominates sintering shrinkage with minimal change to

the relative density, as sintering hardly occurs in relatively low temperature environments. In the

second stage when the temperature attains some level (in this case, 1230℃), the sintering starts,

and the resulting shrinkage is combined to thermal expansion so that overall deformation seems to

remain constant. The third stage is regarded as a period of sintering without thermal deformation,

which is accompanied by a drastic change in the internal structure of the dried body and

accordingly causes significant shrinkage. As a result, the material is rapidly densified, implying

that the relative density is precipitously increased, as shown in Figure 1. In this stage, the elastic

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properties are changed due to the significant change in the material’s internal structure, and the

creep deformation is distinguished under the relatively high-temperature environment. Sintering

gradually ceases at the end of the third stage, while the creep deformation is counteracted and is

assumed to no longer occur in the fourth or final stage, which exhibits thermal shrinkage

exclusively accompanied by a slight increase in the relative density.

Figure 1. A typical time variation in the temperature and relative density during firing.

According to these phenomena, the elastic and viscoplastic behavior of the material is significantly

affected by not only the temperature history, but also the change in the relative density that is

supposed to reflect the change in the material’s internal structure during firing. Therefore, the non-

mechanical characteristics, including thermal and sintering-induced shrinkage, and the mechanical

characteristics for elastic and viscoplastic deformations must be estimated adequately.

2.2 Kinematics

To begin with, the total deformation gradient 𝑭 is decomposed as

𝑭 = 𝑭𝑚𝑭𝑖𝑚 (1)

where 𝑭𝑚 is the mechanical component and 𝑭𝑖𝑚 is the non-mechanical component. 𝑭𝑖𝑚 is

assumed to be composed of only volumetric deformations due to a temperature change and

sintering as

𝑭𝑖𝑚 = (1 + 휀𝑡 + 휀𝑠)𝟏 (2)

where 휀𝑡 and 휀𝑠 respectively represent the amounts of thermal and sintering-induced strain. On the

other hand, the rate of deformation tensor associated with 𝑭𝑚 can be decomposed into the elastic

and viscoplastic (or creep) deformation rate tensors 𝒅𝑒 and 𝒅𝑣𝑝 as follows.

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𝒅𝑚 = sym[�̇�𝑚(𝑭𝑚)−1] = 𝒅𝑒 + 𝒅𝑣𝑝 (3)

2.3 Non-mechanical deformations

The evolutions of the thermal and sintering-induced strains, both of which are non-mechanical or

stress-free deformations, are presented.

The relationship between the time rate of change in the thermal strain and temperature is

commonly expressed as

휀̇𝑡 = 𝛼𝑇�̇� (4)

where 𝛼𝑇 is the linear coefficient of thermal expansion/contraction. Here, 𝛼𝑇 is assumed to be

constant but can be different for increased and decreased temperature processes.

Following the approach used by Largiller et al. (Largiller 2011), the time rate of change in the

sintering strain is expressed as

휀̇𝑠(𝑇) = −1

3(

�̇�

𝜌) (5)

where �̇�/𝜌 is the densification rate. Here, the sign of this equation implies the shrinkage of the

sintering-induced strain with an increase in the mass density. The concrete forms of the

constitutive function of 휀̇𝑠 will be presented in a later section.

2.4 Mechanical deformations

As the Jaumann rate of Cauchy stress 𝝈 is amenable to most commercial FE software, the

following hypo-elastic constitutive law is used to calculate the stress rate with the elastic

deformation rate tensor.

𝝈∇ = ℂ: 𝒅𝑒 (6)

Here, the Jaumann rate of Cauchy stress is defined as 𝝈∇ = �̇� − 𝒘𝝈 + 𝝈𝒘 with 𝒘 being the spin

tensor. ℂ(𝑇) is the fourth-order isotropic elasticity tensor, which is expressed with Young’s

modulus 𝐸(𝑇) and Poisson’s ratio 𝜈 as usual. The dependencies of 𝐸 on the temperature 𝑇 and

mass density will be described later, but ν is assumed to be constant.

The rate of viscoplastic deformation tensor 𝒅𝑣𝑝 = 𝒅 − 𝒅𝑒 is assumed to obey the following

associative flow rule

𝒅𝑣𝑝 = �̇�𝑣𝑝𝑵 (7)

with the flow direction 𝑵 = 𝒔/||𝒔||, where �̇�𝑣𝑝 ≥ 0 is the viscoplastic multiplier and 𝒔 is the

deviatoric component of the Cauchy stress. The concrete form of the viscoplastic multiplier will

be expressed later.

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3. Constitutive functions for non-mechanical deformations

3.1 Relative density

The material behavior during the firing process highly depends on its density change. The relative

density is introduced as a measure of densification in this study and defined as

𝜌𝑟𝑒𝑙(𝑇(𝑡)) =𝜌(𝑇(𝑡))

𝜌𝑓 (8)

where 𝜌 and 𝜌𝑓 are the densities in the current and final stages of firing, respectively. Here, the

current density depends on thermal history 𝑇(𝑡).

To determine the function value of the relative density (Equation 8), a thermomechanical analysis

(TMA) test is carried out on a small unloaded cylindrical specimen under a specific thermal

history 𝑇(𝑡) for firing. During the test, the current height of the specimen is continually measured

as

ℎ(𝑇) = ℎ𝑠 + 𝛿ℎ(𝑇) (9)

where ℎ𝑠 is the initial height of the specimen and δℎ(𝑇) is the height change. Assuming isotropy,

the volumetric change can be calculated by the Jacobian defined as follows.

𝐽(𝑇) = (ℎ𝑠 + 𝛿ℎ(𝑇)

ℎ𝑠)

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(10)

At the same time, the current density can be evaluated as

𝜌(ℎ(𝑇)) =𝜌𝑠

𝐽(𝑇) (11)

where 𝜌𝑠 is the initial density. The substitution of Equation 10 into Equation 11 yields

𝜌(𝛿ℎ(𝑇)) = 𝜌𝑠 (ℎ𝑠

ℎ𝑠 + 𝛿ℎ(𝑇))

3

. (12)

Then, the density in the final stage of firing can be expressed as

𝜌𝑓 = 𝜌𝑠 (ℎ𝑠

ℎ𝑓)

3

(13)

where ℎ𝑓 is the final height of the specimen after the firing.

As can be seen from the above formulation, the relative density Equation 8 is defined as a function

of the height change in view of Equations 12 and 13. But as the height change is determined by

thermal history 𝑇(𝑡), which is specified with time, different heat curves provide different function

forms of the relative density.

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3.2 Function form of the sintering-induced strain rate

The following function form of the densification rate, which was proposed by Hsueh et al. (Hsueh

1986) and Favrot et al. (Favrot 1997), is used

�̇�

𝜌= Ω(𝑇)[𝜌𝑟𝑒𝑙

∞ (𝑇) − 𝜌𝑟𝑒𝑙(𝑇)]𝑛(𝑇) (14)

where Ω(𝑇) and n(𝑇) are temperature-dependent material parameters. Here, 𝜌𝑟𝑒𝑙∞ (𝑇) represents a

stationary value of the relative density under a certain isothermal environment with temperature 𝑇

and is called the ultimate relative density. Combining Equations 14 and 5, the following time-

evolution equation of the sintering-induced strain is obtained.

휀̇𝑠(𝑇, 𝛿ℎ) = −1

3Ω(𝑇)[𝜌𝑟𝑒𝑙

∞ (𝑇) − 𝜌𝑟𝑒𝑙(𝑇)]𝑛(𝑇) (15)

Incidentally, the thermal and sintering-induced strains are assumed to be mutually exclusive in this

study.

3.3 Determination of the relative density by a stairway thermal cycle test

The function form of the relative density is determined using the data obtained from a stairway

thermal cycle (STC) test, which was proposed by Gillia and Bouvard (Gillia and Bouvard 2000).

The heat curve applied in the STC test is a stepwise elevate curve, which like Figure 2, has the

advantage of necessitating only a small amount of information to determine the function form of

the sintering-induced strain rate expression in Equation 15.

Figure 2 Heat curve in the STC test

First, to the calculate densification rate from the STC test, the vertical strain is defined by the

measured displacement in the axial or vertical direction as follows.

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휀𝑍(𝑇) = ln (ℎ𝑠 + 𝛿ℎ(𝑇)

ℎ𝑠

) (16)

It can be identified with the sintering-induced strain so that 휀𝑠(𝑇) = 휀𝑍(𝑇). The time rate of

change of 휀𝑠(𝑇) can be computed from the time variation of the vertical displacement in a discrete

manner and is used to calculate the densification rate with Equation 5 as follows.

(�̇�

𝜌) = −3휀̇𝑠(𝑇) (17)

The relative density (Equation 8) can also be calculated from the STC test.

(a) Plot from 0.747 % to 0.765 %

(b) Plot from 0.74 % to 1.0 %

Figure 3 Relationship between the relative density and densification rate

The relationship between the relative density and the densification rate is plotted in Figure 3. It

can be seen from this figure that the densification rate is decreased during each isothermal step.

These decreasing curves are polynominally approximated so that the intersect with the horizontal

axis, each of which corresponds to the ultimate relative density in Equation 14 at the

corresponding isothermal level of temperature. The obtained discrete points of ultimate relative

densities are plotted in Figure 4 with red squares with respect to temperature. It is confirmed from

this figure that the densification starts around 1020℃ so that the ultimate relative density rapidly

increases up to the relative density in the final stage, which is equal to 1.0. This relationship is

approximated by a hyperbolic tangent function of temperature as follows.

𝜌𝑟𝑒𝑙∞ (𝑇) = 𝑎1 tanh (

𝑇 − 𝑎2

𝑎3) + 𝑎4 (18)

With the use of the least squares method, the parameters 𝑎1, 𝑎2, 𝑎3, and 𝑎4 have been obtained and

the corresponding identified curve is depicted with a blue line in the same figure.

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To determine the remaining parameters, 𝑛(𝑇) and Ω(𝑇), in the expression of the rate of sintering-

induced strain of Equation 15, the absolute values of both sides are taken, and then their

logarithms are calculated as follows:

ln[|휀̇𝑠(𝑇)|] = ln [1

3Ω(𝑇)] + 𝑛(𝑇) ln[𝜌𝑟𝑒𝑙

∞ (𝑇) − 𝜌𝑟𝑒𝑙(𝑇)] (19)

As the function form of the ultimate relative density has been determined in Equation 18,

Equations 17 and 18 are used to obtain the relationships between ln[𝜌𝑟𝑒𝑙∞ (𝑇) − 𝜌𝑟𝑒𝑙(𝑇)] and

ln[|휀̇𝑠(𝑇)|] for all the isothermal temperature levels used in the STC test, all of which are shown

in Figure 6. As the set of the plots for each isothermal temperature level can be approximated,

whose intercept and slope respectively represent ln[Ω(𝑇)/3] and 𝑛(𝑇) in Equation 19, they can be

plotted with respect to the corresponding isothermal temperature levels, as shown in Figure 7. In

this figure, however only the results for the temperature levels higher than 930 ℃ are shown, as

the value obtained for the temperature level 900 ℃ was too high to relate with the others. These

curves can be expressed as polynomials of 𝑋(𝑇) = (𝑇Ω − 𝑇)/𝑆Ω and 𝑋(𝑇) = (𝑇n − 𝑇)/𝑆n,

respectively, as follows.

Ω(𝑇) = 𝑅Ω exp(𝑏4𝑋(𝑇)4 + 𝑏3𝑋(𝑇)3 + 𝑏2𝑋(𝑇)2 + 𝑏1𝑋(𝑇) + 𝑏0) (20)

𝑛(𝑇) = 𝑐3𝑌(𝑇)3 + 𝑐2𝑌(𝑇)2 + 𝑐1𝑌(𝑇) + 𝑐0 (21)

Here the coefficients 𝑏𝑖 , (𝑖 = 0, ⋯ ,4), 𝑅Ω, 𝑇Ω, 𝑆Ω, 𝑐𝑗(𝑗 = 0, ⋯ ,3), 𝑇𝑛, and 𝑆𝑛 can be determined by

the least squares method and the identified curves correspond to the red lines in Figure 6 and

Figure 7.

Figure 4 Identification results of the ultimate relative density

Figure 5 Estimated data of Equation 19 for all the isothermal temperature

levels

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Figure 6 Estimated data with Equation 19 for all the isothermal temperature

levels

Figure 7 Relationships of 𝒏(𝑻) and 𝛀(𝑻) with temperature

3.4 Coefficient of thermal expansion/contraction

The coefficient of thermal expansion (CTE) or contraction is estimated from the data obtained in a

nearly no-load TMA test on a specimen subjected to firing. Figure 8 shows a time variation of a

vertical true strain along with a typical heat curve. It should be noted that, due to the specification

limit of the TMA test device, a small weight must be placed on the specimen to measure the

displacement. In this study, 10g is consistently used for nearly no-load TMA tests. In addition, the

body force due to its own weight is neglected in the evaluation of CTE for simplicity.

Figure 8 Heat-curve and axial true strain obtained in a nearly no-load firing test

As described in Section 2.1, the heat curve in this figure can be divided into four stages according

to the type of non-mechanical deformations, and these non-mechanical deformations are assumed

to be mutually exclusive. Thus, the transition temperatures from Stages A to B and from B to C

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are, respectively, designated around the inflection point of the elevated heat curve and the point

just before the decreasing slope of the height-change curve. They are, respectively, denoted by

𝑇𝑠 and 𝑇𝑠𝑠. On the other hand, the transition from C to D is recognized as the sintering completion

time 𝑡𝑠𝑓 and can be defined as the point just before the decreasing slope of the heat curve. With

the classification of these four stages, the linear coefficients of thermal expansion/contraction in

Stages A and D, respectively, denoted by 𝛼𝑇𝑑 , 𝛼𝑇

𝑠 are calculated from the corresponding changes of

true strains with respect to the temperature changes. Here, the values of 𝑇𝑠 = 600℃, and 𝑇𝑠𝑠 =930℃ have been used.

3.5 Validation analyses for non-mechanical deformations

Two numerical analyses were performed to validate the function form of the sintering-induced

strain and the linear coefficients of thermal expansion/contraction determined above. One was to

reproduce the height change empirically observed in Figure 9, and the other was the non-

mechanical deformation analysis for a TMA test with heat curves of different rates of temperature

increase.

First, the above TMA test was simulated with a single eight-node hexahedral element set to

uniformly deform. From the heat-curves, the initial relative density 𝜌𝑟𝑒𝑙𝑠 at 𝑇𝑠 and the relative

density at the onset of sintering, 𝜌𝑟𝑒𝑙𝑠𝑠 at 𝑇𝑠𝑠 were calculated. The simulation results are shown in

Figure 9, which presents the time variation of the axial true strain in comparison with the TMA

test results. Although there is a slight difference in deformation in Stage D, which may be caused

by the evaluation error of the sintering-induced strain, it can be said that they are largely in good

agreement.

Next, another set of nearly no-load TMA tests was conducted to validate the proposed expressions

of the non-mechanical deformations. In the tests, after the temperature was increased with the

same rate of 10℃/min from 23.3 to 1000℃, three levels of rates of temperature increase, 2, 5, and

10℃/min, were given to linearly increase the temperature up to 1230℃. The FE model was

composed of a single C3D8 element having a height of 20.12 mm, a length of 4.32 mm, and a

width of 4.32 mm, which are the average sizes of three different specimens used for the tests. The

analysis condition is the same as in the above.

Figure 10 compares the analysis and test results. As shown in these graphs, the numerical results

for the rate of temperature increase, 2℃/min, exhibit some estimation errors for the sintering

strains in Stage C, although the results for 5 and 10℃/min are in good accordance with the

corresponding test results. In fact, as the sintering-induced shrinkage evaluated for 2℃/min in our

numerical simulation developed faster than reality, there seem to be some problems to solve.

Nonetheless, the axial true strains calculated for Stages A and B are consistent with the test results

and the trends of their abrupt increases have been captured. Thus, the accuracy in representations

of the non-mechanical deformations can generally be well tolerated from a practical viewpoint.

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Figure 9 Time variation of the axial true strain obtained by numerical analysis with the identified CTEs along with the experimental one

(a) Results of 2℃/min

(b) Results of 5℃/min

(c) Results of 10℃/min

Figure 10 Validation results of non-mechanical deformations

4. Constitutive functions for mechanical deformations

4.1 Elastic modulus

To determine the constitutive function of the elastic modulus of the dried body, three-point

bending tests were conducted on the beam-like specimen with a rectangular cross-section, which

has a length of supporting points of 30 mm, a width of 10.12 mm, and a height of 6.71 mm. The

test was carried out in the same TMA equipment as before, and the specimen was subjected to a

heat curve in which the temperature was increased up to 1000℃ with a temperature increase of

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10℃/min and was further increased up to 1200℃ with 2℃/min. From the load-deflection curves at

some selected levels of temperature, the elastic modulus was determined as a function of

temperature as follows.

𝐸(𝑇) = 𝑒1 − 𝑒2 tanh (𝑇 − 𝑒3

𝑒4) (22)

The application of the least squares method provided the coefficients of 𝑒1, 𝑒2, 𝑒3, and 𝑒4.

4.2 Viscoplastic multiplier

The constitutive function of the viscoplastic multiplier �̇�𝑣𝑝 using a viscoplastic flow rule in

Equation 7 is proposed as

�̇�𝑣𝑝 = √3

2𝐶1‖𝜎𝑒𝑞‖

𝐶2𝑡𝐶3 [tanh (

𝑇 − 𝑇0

𝑊) + 1.0] exp(−𝐶4𝜌𝑟𝑒𝑙) (23)

which follows the standard creep low of Norton’s type. Here, 𝜎𝑒𝑞 = √3𝒔: 𝒔/2 is the von Mises

equivalent stress, and 𝐶1, 𝐶2, 𝐶3, 𝐶4, 𝑇0, and 𝑊 are the material parameters to be identified with

reference to the following experimental data.

To identify the creep parameters, a set of compressive-constant-load TMA tests were conducted

for the specimens subjected to firing by applying different constant loading levels and different

patterns of temperature histories. Specifically, three constant load levels with weights of 10, 50,

and 100 g, and three patterns of heat curves were applied in combination; i.e., nine creep tests

were carried out. The specimens were rectangular parallelepiped having an average height of

10.087 mm, an average length of 3.966 mm, and an average width of 4.024 mm and expected to

uniformly deform in the vertical direction.

The method of differential evolution (DE) is used to identify the creep parameters from the data

obtained above; see Storn and Price (Storn 1997) and Ali et al. (Ali 2013) for its optimization

algorithm.

5. Finite element analysis of actual sanitaryware

To demonstrate the capability of the set of constitutive functions with the identified material

parameters, finite element analysis (FEA) of an actual sanitaryware product subjected to firing

using Abaqus is performed. There are several ways to implement the user-defined constitutive

functions into general purpose FE software. Initially, the models were implemented into the

UMAT, and its performance has been confirmed with the FE model of a specimen composed of

several elements. However, when such a UMAT program is applied to numerical simulations for

an actual sanitaryware product, the following issues must be recognized: to analyze with an

adequate (neither too fine nor too coarse) time increment over the temperature range beyond

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1000℃; to stably compute large displacements with the change of strains over 10 percent; and to

consider frictional contact between the floor surface and the base of a sanitaryware product, which

deforms with time. Therefore, other user-defined functions—UEXPAN for the thermal strain and

the sintering strain, and CREEP for the creep strain—are used instead.

Figure 11 shows the FEM model used for the analysis, which is composed of 80,000 C3D10M

elements with 150,000 nodes. For this simulation, the C3D10M element was used because

tetrahedral element is required for the complicated geometry, a second-order element is required

for high accuracy, and a modified tetrahedral element is required for working well in contact. This

model is the deformed state obtained by the preliminary numerical analysis for the shrinkage

deformation due to drying and thus approximately 3 percent smaller than that the actual plaster

mold. The model was located on a rigid floor, and its own weight was applied in the Z-direction in

the figure. The friction coefficient between the floor surface and the base of the sanitaryware

product was set at 0.65. The heat curve was used as the temperature history, and its distribution is

uniformly applied to the whole domain for the model, as the time rate of temperature change is

moderate.

The final stage of the model obtained by the numerical analysis is shown in Figure 12, in which

contour colors indicate the magnitude of the minimum principal strain. It can be observed from the

figure that the compressive strains develop up to approximately 10 percent according to the

location of the product. The main factors causing the locally developed compressive strains are the

deflection due to the model’s own weight and the constraint on the contractive strain due to the

friction with the floor surface. The maximum level of the principal stress is approximately 0.6

MPa, which does not permit us to say that there is out of the potential of cracking during the firing

process. Figure 13 compared the calculated displacements with those measured at some points of

the product in the final state of firing. Although some errors of up to 3 percent are observed

according to the measurement points, most are within 1–2 percent.

Figure 11 FE model of a sanitaryware product

Figure 12 Analysis results in the final stage of firing

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Figure 13 Errors in the dimension of the analysis results in comparison with the measured data in the final stage of firing, which was taken out of the furnace and

cooled down to a normal temperature

6. Conclusion

This paper shows the calculation results for the actual ceramic product considering the set of

constitutive equations using the Abaqus user subroutines; UEXPAN was used for the thermal

strain and the sintering strain, CREEP for the creep strain. And the calculation result of actual

sanitaryware was compared with the experimental data.

7. References

1. S. Matsubara, K. Tarada, T. Kobayashi, T. Saitou, M. Umeda, Y. Mihara, Kai Oide, H. Shin,

Y. Katsuda, "A set of constitutive functions for dried body to predict entire deformation

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