Numerical Simulation and Experimental Verification of Dry Pressed MgTiO 3 Ceramic Body during Pressureless Sintering

To clarify the densification law of dry pressed MgTiO 3 ceramic body during pressureless sintering, SOVS model modified with creep characteristics was embedded into finite element software Abaqus. The selected model can effectively express the grain boundary characteristics and densification mechanism. The change law of relative density, shrinkage rate, sintering stress and grain size of MgTiO 3 cylindrical specimens were investigated by the above numerical simulation method. It showed that the average relative density of ceramic body rose from 60% to 97% and the shrinkage rate resepectively reached 17.28% and 11.99% in axial and radial direction. The average grain size increased from 1μm to 6 μm. In order to verify the accuracy of the simulation results, corresponding sintering experiments on cylindrical specimens were carried out to obtain actual sintering densities and shrinkage rates. It showed that the errors of relative density and shrinkage is below 5% and 2%. Grain growth trend was also basically consistent with the simulation results. After that, the above numerical simulation method was applied into the prediction of fabricating MgTiO 3 filter with complex structure. Therefore, the present work provided a reliable numerical simulation method to predict the densification behavior of MgTiO 3 ceramics during the pressureless sintering process, which was helpful to design and fabricate microwave dielectric products.

main problems in MgTiO3 filter production [3~4]. It can reduce the number of trial and error and production cost to analyze the ceramic sintering process using numerical simulation method.
The powder model for numerical simulation has been developed for more than ten years [6]. Ashby combined pore movement and surface tension to establish the constitutive model basis for powder sintering [7]. Coble further studied the change rules of pore and grain microstructure and optimized the sintering process [8]. Riedel adopted hexagonal grain as the microscopic model and deduced a series of constitutive models [9]. Cocks proposed the theory of creep and grain boundary diffusion [10]. The rheological model known as SOVS model proposed by Skorohod and modified by Olevsky has been mainly applied [11]. Kraft summarized the rheological theory and simulated the solid-phase sintering of SiC ceramics [12].
Recently, SOVS model has been developed and applied. WANG simulated the solid phase sintering of alumina ceramics using the improved SOVS model, and the simulation results were consistent with the actual sintering [13][14]. Hong simulated the sintering of iron powder under the condition of fully considering the influence of heat conduction, thermal convection, thermal radiation and thermophysical parameters [15]. Anastasiia study the field assisted sintering of silicon germanium alloys by a finite element method [16]. YANG simulated the sintering of Li metal at different temperatures and considered the three stages of sintering [17].
Previous studies have proved the feasibility of SOVS model, which has the advantages of simple parameters and easy measurement. But most of the existing researches only focus on deformation, the relative density changes and grain growth during simulation are rarely studied. Furthermore, the previous studies mostly ignored the thermal expansion and creep behavior. Thermal expansion affects the deformation at the initial stage of sintering [13]. In addition, there is no research about the simulation of MgTiO3 wave dielectric ceramics. Therefore, it is necessary to use SOVS model modified with creep characteristics to simulate the sintering densification process of MgTiO3 filter with complex structure.
In this study, SOVS model modified with creep characteristics was embedded into finite element software to simulate pressureless sintering process of MgTiO3 in Abaqus. The changing law of relative density, shrinkage, sintering stress and grain growth along sintering process was analyzed and experimentally verificated. The densification behavior of MgTiO3 wave filter was also predicted and verificated during the pressureless sintering process, which was helpful to structure design and practical production of MgTiO3 ceramic components.

Route planning
The flow chart of this work is shown in Figure 1.

The Modified SOVS model
Skorohod-Olevsky Viscous Sintering model which is known as the SOVS model, is a continuous mechanical models based on the theory of porous plasticity and nonlinear viscoplastic deformation. In general, the SOVS constitutive relation of nonlinear porous materials can be expressed as: Result analysis and optimization strain rate tensor, e is trace of strain rate tensor, and ij  is Kronecker coefficient.  and  are the standardized shear viscosity and volume viscosity respectively. PL is the sintering stress,  is the shear viscosity of solid, and their definitions are as follows: Where  is porosity, S  is specific surface energy, G is average grain size, V Q is surface activation energy, R is gas constant, T is absolute temperature, and 0  is the initial shear viscosity of solid.
With the progress of sintering, grain boundary diffuses with temperature, so the grain size also increases. The growth rule described by Guillaume Bernard-Granger is as follows [16] : Where, G0 is the initial average grain size, which can be measured through the analysis of SEM images by image-Pro Plus software. k1 is a constant that does not change with temperature, 0  is the initial relative density, and  is the relative density that changes constantly during sintering.
Cocks defined the bulk expansion strain increment sw   and shear creep strain increment cr   during sintering, which were defined as follows: Where, P is equivalent hydrostatic pressure, and q is mises equivalent deviator stress.

parameter acquisition
Owens two-liquid method was used to test specific surface energy of ceramics.
bbV-1000 high temperature viscometer was used to test the viscosity-temperature curve of MgTiO3 ceramics. The experimental results are synthesized into arrhenius function (5). NETZSCH LFA 457 MicroFlash laser thermal conductivity meter was used to test the thermal diffusion coefficient of MgTiO3 ceramics (sample diameter: 12.5-12.7mm, thickness: 1-3mm). The specific values are shown in Table 1.

Table1
FORTRAN-based Abaqus user subroutine can implement the constitutive model and parameter input. In this study, CREEP subroutine was adopted. Variables can be defined in subroutines, such as relative density, grain size, and so on.

The experiment
Before sintering, MgTiO3 is formed by dry pressed. So the relative density distribution of the green body after dry pressed should be imported. This can be done by defining the initial relative density in the subroutine. Cylindrical sample with the radius of 5mm and the height of 10mm were prepared by pressing as standard samples.
In the standard group, the powder particle size was 0.1mm, and the initial temperature was 20℃. The maximum sintering temperature reached 1350℃ after 4.5h of continuous heating, and the holding time was 3h, and then the powder was naturally cooled to room temperature. Take 1/4 of the sample as the research object.

Simulation results
The simulation results of relative density are shown in figure 2. Before the sintering, the relative density of each part of the dry pressed body was different. The average value is about 0.6, but the relative density around the top surface of the cylinder is relatively high, while the relative density around the bottom surface of the cylinder is relatively low. After sintering, the relative density of the center of the top surface is relatively low, while that of the center of the bottom surface is relatively high. The overall relative density rises from 0.6 to about 0.97.

FIG. 2 Sintering simulation results of standard cylindrical sample
In figure 2, three points with large differences in relative density were selected, and their relative density change curves were shown in figure 3. In the early heating process, the relative density did not change. When the temperature is close to the sintering temperature (1350℃), the relative density begins to increase. After holding for a certain time, the relative density does not change and reaches the maximum value. For point A, B and C, before sintering, point A has the highest relative density and point C has the lowest. But after sintering, point A is the lowest and point C is the highest. The reason is that after pressing, the top surface, namely point A, contacts with the punch, so the pressing force is slightly higher than the rest surface, so the relative density is also higher. In simulation, the displacement along the Z direction is limited at the bottom of the sample, and the gravity load is applied. Therefore, the sintering stress is higher and the relative density is also higher. sharply, and the shrinkage rate is positive with the rapid growth rate. In this stage, the creep mechanism of sintering plays a leading role. Under the action of sintering stress, the sample begins to shrink, and the relative density also increases rapidly in this stage. Finaly, the axial shrinkage rate was gradually greater than the radial rate, and finally reached 17.28%, while the radial rate was 11.99%. The reason is that axial gravity is added in the simulation process, and the axial shrinkage rate is higher under the action of gravity and sintering stress. Figure 5 shows the stress cloud map of each period of sintering. The external stress was higher than the internal stress at the beginning of sintering, which was the residual stress generated by pressing. As the temperature rises to 1350℃, the stress distribution is different, increasing from top to bottom, and finally forming stress concentration at the bottom center. The reason is that the constraint on the bottom surface to limit its displacement makes it lose the degree of freedom in the Z direction.

FIG. 4 Size and shrinkage change curve
Therefore, under the action of sintering stress, the stress accumulates downward and causes stress concentration.In the cooling stage of sintering, the stress concentration area is gradually eliminated, and the stress inside the sample block tends to be average.
The surrounding area of the lower surface of the sample is slightly higher than other areas under the action of boundary conditions. Figure 6 shows the displacement (deformation) cloud diagram of the material after sintering. This diagram indicates the deformation direction of sintering densification, that is, shrinkage deformation is carried out from top to bottom and from outside to inside.  Figure 7 shows the simulated grain growth diagram. It can be seen that grain growth is closely related to the change of relative density. When the relative density is less than 80%, the grain growth is slow.When the relative density is greater than 90%, the grain growth is accelerated. Figure 8 shows the statistical graph of grain size after simulation. It can be seen from the figure that the size distribution is relatively concentrated, and the grain size increases from 0.001mm at the initial setting to 0.005867mm on average. boundaries is small, so the grain growth rate is slow.When the density is higher, the grain boundary is more, so the grain grows faster.

1) Relative density
The dielectric property and quality factor of MgTiO3 depend on its density. So the change of relative density is the focus of the research. Figure 9 shows the cylinder samples which was equally cut into three parts along the height direction. The relative densities of the three parts were measured and compared with the simulated values.
The results were shown in Table 2. The distribution of the relative density of the simulated values is consistent with the actual measurement, and the errors of the three parts are all within 5%. The error is caused by the difference between the material defined by the simulation and the actual material. In the simulation, we mainly defined the thermodynamic properties of materials. The actual material is more complex, which contain a small amount of additives such as burning AIDS.  Figure 10 shows the shrinkage curve of simulation and actual sintering without considering gravity. The results of SOVS simulation are generally consistent with the trend of actual sintering.The difference is that the thermal expansion effect of the material is actually less pronounced than in the simulation. The sintering temperature of MgTiO3 is set at 1350℃, but the actual maximum sintering temperature is only 1266.3℃. The reason is that the actual powder contains a small amount of impurities, which will be ignored in the simulation. The final shrinkage rate of both numerical simulation and practical sintering is about 14%. Table 3 shows the shrinkage ratio between simulated values and actual values. After sintering, both the simulated size and the actual size shrink, and the Z direction shrink more. The error is within 2%.

FIG. 10 Comparison curve of shrinkage change
Table3 Figure 11 is the SEM diagram of the sample. Before sintering, the gap between grains is large and the grains are independent. Three points in the sample were selected for SEM observatione. After sintering, sintered neck was formed between grains and grain boundary was formed. The gap between grains are eliminated and the grains gradually form into a whole. In addition, grain size was significantly increased.   Figure 13 shows the MgTiO3 ceramic fiter with complex structure, which has more blind holes and grooves. The sintering model was used to simulate the densification process. The relative density at the bottom of the blind hole and the bottom of the groove are large, and the maximum value approaches 100%. The upper edge and groove edge of the blind hole are low, and the circular arc of groove edge is weak, with relative density only about 75%. Figure 14 shows the grain growth diagram. In the areas with high relative density, the gap between grains is small and the diffusion resistance is large, so the growth of grains is slow.

conclusion
In this paper, pressureless sintering simulation of MgTiO3 ceramic body is studied, by embedding SOVS model modified with creep characteristics into finite element software Abaqus. The relative density, shrinkage, grain growth and sintering stress of were analyzed and experimentally vericated througth cylindrical sample and complicated wave filter. It was concluded as below: (1) Simulation results show that the average relative density rises from 60% to about 97%. The shrinkage rate resepectively reached 17.28% and 11.99% in axial and radial direction. The average grain size increased from 1um to 6um.
(2) Sintering experiments were carried out to obtain actual sintering densities and shrinkage rates. It showed that the errors of relative density and shrinkage is below 5% and 2%. Grain growth was also basically consistent with the simulation results.
(3) The method is used to simulate the filter with complex structure. The relative density and grain size were analyzed, which provides guidance for its practical production. The simulation results of complex shapes are verified, and the simulation error is within 2%.