Ceramic Processing Research

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1 Journal of Ceramic Processing Research. Vol. 10, No. 2, pp. 195~201 (2009) J O U R N A L O F Ceramic Processing Research Computer simulation of a ceramics powder compaction process and optimization of process parameters Y. T. Keum a, *, J. L. Kim b and J. K. Shim b a School of Mechanical ngineering, Hanang Universit, 17 Haengdang-dong, Seongdong-ku, Seoul 1-791, South Korea b Graduate School of Mechanical ngineering, Hanang Universit, 17 Haengdang-dong, Seongdong-ku, Seoul 1-791, South Korea In order to optimize the process parameters of ceramic powder compaction, a process simulation was first performed using a finite element method. In the finite element analsis, a quasi-random multi-particle arra was introduced for modeling the non-periodic and randoml scattered powder particles with initiall-low relative densities. Homogenization theor was emploed to find the equivalent material properties associated with various porosities. The size of Al 2 O particles, amplitude of the cclic compaction pressure, and friction coefficient among powder particles were chosen as process parameters. Finite element analsis results show that relative densit becomes bigger as the process variables become smaller. Net, a regression model was found b using the response surface method, based on the results of the finite element analsis. Then, the optimal conditions of the process parameters providing the highest relative densit were pursued b emploing a grid search method. Ke word: Ceramic powder compaction, Homogenization method, Process parameter optimization, Relative densit, Response surface method, Grid search method, Finite element analsis. Introduction The characteristics of mechanical strength, phsical, and chemical properties of pol crstalline ceramic materials are affected b the arrangement of crstal directions because the are caused b the anisotrop of single crstals associated with crstal directions. Therefore, in order to make an analtical model and to perform a computer simulation for a ceramic powder compaction process, it is important to analze the deformation behavior of material microstructures. perimental research on the ceramic powder compaction process has been made in was to allow the stud of the man qualitative characteristics of a pol crstalline ceramic material during the powder compaction process. ven the quantitative analses of some ideal models are in progress using numerical methods. However, the ideal model of compact forming is different from the real model because the real powders are not regularl arraed, and the powder densit distribution and replenishment state is not uniform. Therefore, to improve the various properties of ceramic components and the efficienc of powder compaction forming processes, the modeling of compaction forming processes, which can quantitativel analze the real compaction forming process, is recommended. Furthermore, to reduce the time and numerical cost of simulation, the relationship among the relative densit and the process *Corresponding author: Tel : Fa: mail: tkeum@hanang.ac.kr parameters influenced greatl b the relative densit during the ceramic powder compaction forming has to be considered. In addition, optimal process parameter values for maimizing the relative densit should be adopted. Looking at the research trends of ceramic powder compaction forming, an ideal model in the static and repetitive compactions for compound materials was epedd. rimentall and numericall studied b Wu et al. [1] and Jiang et al. [2]. Xin et al. [] introduced a computer simulation method using eplicit FM for analzing the powder compaction phenomena of single and compound materials. Zahlan et al. [4] presented a modeling method for real products made b powder compaction forming under repetitive loads. Zipse [5] also performed research on the FM simulation of compaction forming and sintering processes of real ceramic products. Deis and Lannutti [6] and Lu and Lannutti[7-9] measured the densit slope and pore distribution during powder compact forming using an X-ra CT and mercur pore analzer and eperimentall investigated the how initial densit slope affected powder compaction forming b emploing the X-ra CT. Hassani and Hinton [10] formulated an analtical solution of homogenization theor for various material models to be used in the finite element analsis. Also Takano et al. [11-12] presented a calculation of the elastic coefficient as a function of porosit and pore shape for compound materials and porous ceramic materials using homogenization theor. In addition, as the relative densit is one of the important factors in evaluating ceramic products, much effort to get high densit products from the beginning of compact forming 195

2 196 Y. T. Keum, J. L. Kim and J. K. Shim is in progress. Recentl, researches on the optimum process parameters enabling high relative densities have been made. Roughl summarizing the researches on the parameter stud of relative densit, Jiang et al. [1-14] carried out eperiments looking at the effect of the lubricant of a metal composite, the aspect ratio of the test material and the size of particles. Rendanz and Fleck [15] used a computer simulation technique to stud the compaction of randoml distributed powders. Inter-particle friction and affined motion has been studied both for the compaction response and for the resulting ield surface. Procopio and Zavaliangos [16] simulated the multi-aial compaction for a granular media to stud relative densities related to the particle shape and size as well as material properties. Consuelo and Lannutti [17] also eamined the effect of particle size during the packing. Hassanqaur and Ghadiri [18] emploed the distinct element method to simulate the bulk deformation, based on single particle properties. Briscoe and Rough [19] carried out an eperiment to see the effect of friction between powder particles and the die. Kim et al. [20] looked into the effects of repetitive pressure, repetitive velocit, and biased pressure on powder densification under repetitive compaction in room temperature. Also Shin et al. [21] presented the forming conditions to be able to improve the reproducibilit b studing the changes in the forming densit and sintering densit in normal pressure sintering. In this stud, the arbitraril-intensified powder compact is first modeled and the equivalent material propert is introduced in the finite element analsis to simulate ceramic powder compaction. In addition, the optimal values of process parameters providing the maimum relative densit are pursued on the basis of the response surface model obtained from the analsis of the eperimental region of interest chosen b the full factorial design. Forming Process Analsis Modeling Fig. 1 is a 2-dimensional view of the powder compaction device and ceramic powders before compaction. The ceramic powders are located inside the die and base punch stationar and are compacted to the desired shapes b the upper punch operating up and down. While the base punch and dies are stationar, the upper punch is movable so that repetitive loads can be applied to the powders. Generall, homogeneous or non-homogeneous powders are modeled in the finite element analsis of a ceramic powder compaction process with the assumption that unit cells are periodicall arraed in heahedron or cubic shapes. Although the powders are not periodicall arraed in a real compaction, the powders are modeled using a quasi-random multi-particle arra, which can represent the non-periodicit and arbitrariness of the powder distribution b packing the particles one b one considering the powder arra and the Fig dimensional view of powder compaction device and ceramic powders. size distribution, position, and rearrangement of powder particles. After packing the particles to give contact between particle and particle or between tool and particles, the size and arra of the particles are adjusted to make the desired relative densit. In order to analze green compacts whose initial relative densities are 0.554, and 0.615, respectivel, the size of an Al particle is fied to 15 µm. Then Al 2 O particles whose sizes are 7.5 µm, 15 µm and 22.5 µm, respectivel, are modeled. The models whose sizes of Al 2 O are 7.5 µm, 15 µm and 22.5 µm are called Model 1, Model 2 and Model, respectivel. In ceramic powder compaction forming, an important factor to control the relative densit is the friction coefficient. Generall, the friction coefficient between the dr Al 2 O powders unlubricated is and lubricated with zinc stearate is [22]. To investigate the effect of various friction coefficients on the relative densit of powder compaction, different friction coefficients between the powders are used in the analsis of the powder compaction forming process. Formulation The finite element formulation of stress fields, which is used to analze the powder compaction forming process, is written as follows [2]: ([ K] { u} { F} ) e = 0 e= 1 (1) where [K] is the stiffness matri obtained from the homogenized theor and [F] is the force vector. Homogenization theor is introduced to find the equivalent material propert from the microscopic scale model for analzing accuratel macroscopic scale porous materials such as powder ceramics. For eample, the homogenized elastic tensor, H ijkl, is written as:

3 Computer simulation of a ceramics powder compaction process and optimization of process parameters 1 H ijkl = ---- Y Y kl χ m jikl n dy (2) where ijkl is the elastic tensor and mkl is the characteristic displacement. In the governing equation of a linear elastic problem, which is derived from the homogenization theor equations, the 2-dimensional powder compact Ω is assumed to be the assembl of micro-unit cells [2]. Forming Analsis Analsis model and homogenized material propert In this stud, the compound compact consisting of three aluminum and three alumina powder particles are modeled with 2-dimensional rods b using the quasi-random multi- 197 particle arra. A rod arra model is often emploed in numerical simulations because not onl is there no big difference in the densit, although it is less dense than a real powder compact, but also it clearl shows material behaviors and contact states [1]. For the elasto-plastic finite element analsis, powder particles are modeled with 4 node elements in a plane strain state, as seem in Fig. 2. Model 1 emplos 526 nodes and 444 finite elements. Model 2 uses 695 nodes and 598 finite elements. Model has 821 nodes and 714 finite elements. The upper punch, base punch, and die are modeled with mass elements. The densit, elastic modulus, Poisson s ratio and ield strength are the material properties used in the forming analsis. The homogenized elastic modulus is also emploed. Table 1 shows the material properties of Al and Al2O used in the analsis. Fig. 2 presents the boundar conditions used in the finite element analsis. While the dies and base punch are assumed to be rigid, namel not deformable, the upper punch presses ceramic powders, taking 0 seconds up to the peak pressure P = 200 MPa. For calculating the homogenized elastic modulus associated with porosities, the unit cell structure with a cross shaped porosit, which is similar to the porosit shape a of powder compact, is modeled with 2-dimensional finite elements (see Fig. ). Then, the relative elastic modulus r is obtained using the homogenization method from the unit cell structures Material properties of Al and Al2O Material propert Al Mass densit (ρ) 2.12(10) kg/m Yield stress (YS) 28 MPa Poisson s ratio (ν) 0.45 lastic modulus () 62 GPa Homogenized elastic modulus ( ) e GPa Table 1. Finite element modeling of ceramic powders and boundar conditions for simulating the powder compaction process. Fig. 2. Fig.. h Cross pore-solid models for calculating homogenized elastic modulus associated with porosities(f). r -4.87f Al O.89(10) kg/m 00 MPa GPa e GPa 2 r -4.87f

4 Y. T. Keum, J. L. Kim and J. K. Shim 198 Fig. 4. Formed shapes and principal stresses of 6 particles during the powder compaction. associated porosities f, and the homogenized elastic modulus h is calculated b interpolating r as follows: () h = r e 4.87f. Numerical analsis The powder compact forming process model is analzed b emploing the finite element method using both a micromechanics approach and a continuum-mechanics approach. In the micro-mechanics approach, the phsical quantities such as stress and velocit can be different from those averaged in the macroscopic approach. In this method, the powder compact is regarded as an assembl of particles. Particles and gaps occup the space inside the powder compact. Particles are defined as 1 and gap 0 in the densit function. The simulation is performed using the elastic modulus for 0 seconds taking up to the peak pressure P = 200 MPa. Fig. 4 shows the formed shapes of the 6 particles during the compaction. A micro-mechanics approach shows that the relative densit graduall increases to 0.699, 0.70, 0.769, and during the compaction. In the continuum mechanics approach, the atomic structures of materials are neglected. All phsical quantities are assumed to be represented with continuous mathematical functions. Therefore, stress and strain are defined at all points in the continuum. Using the homogenized elastic modulus associated with porosities h, as done in the micro-mechanics approach, the simulation is performed for 0 seconds. Relative densities are increased to 0.748, 0.778, 0.792, and finall to Fig. 5 shows a comparison of the relative densit between FM and the eperiment done b Jiang. [2] The relative densities obtained b the continuum approach which introduces the homogenized elastic modulus associated with porosities in the analsis are closer to the eperimental ones than those of the micro-mechanics approach. The reason for the sudden rise of relative densit when the homogenized elastic modulus h is used in the continuum approach is wh the h is small, which is calculated in the initiall low relative densit of the powder compact, namel the high porosit state, but h suddenl increases as the relative densit increases. Net, the analsis is performed under the condition Comparison of relative densities obtained from FM analses and eperiment during the powder compaction. Fig. 5. et al Comparison of relative densit among friction coefficients 0.4, 0.6, and 0.8. Fig. 6. where the friction coefficient between the powders is changed to 0.4, 0.6, and 0.8 while the friction coefficient between the powders and die is fied. Fig. 6 shows the changes in relative densities with time as the friction coefficient changes. After 0 seconds reached to the peak pressure, the relative densities are 0.816, 0.827, and 0.81 when friction coefficients are 0.4, 0.6, and 0.8, respectivel.

5 Computer simulation of a ceramics powder compaction process and optimization of process parameters Process Parameters Optimization perimental region of interest The optimization flow of process parameters of powder compaction followed in this stud is seen in Fig. 7. After the eperimental region of interest is set up b the full factorial design, the finite element analses are performed for all cases in the region. The response surface called the regression model, which is represented b the second 199 order polnomial function of process parameters, is then found from the responses obtained from the analsis results. Finall, the optimal values of process parameters providing the maimum relative densit are determined b the grid search method. The Al2O particle size, amplitude of cclic compaction, and friction coefficient are chosen as design variables and the eperimental region of interest is set up b the full factorial design, as shown in Table 2. Net, the finite element analses are performed a total of 27 times determined b the three variables and three levels. In ever finite element analsis, the relative densit and principal stress for 0, 1, 10, 100, and 1000 ccles of compaction pressure are calculated, as seen in Fig. 8, and the difference in relative densit between 1 ccle and 1000 ccles is defined as the response. Regression model The response surface method, which was introduced b Bo and Wilson [24], is to optimize approimatel the variables after representing an analsis model with eplicit functions. In this stud, the response surface method is used for optimizing the powder compact process parameters defined in the previous section. Approimatel epressing the response surface obtained from the simulations done at the eperimental points selected b the eperimental design with an objective N Process parameters and levels for optimization Levels Process parameters Al O particle size [μm] Amplitude of cclic compaction pressure[mpa] Coefficient of friction Table 2. 2 Schematic flow for optimizing process parameters of powder compaction. Fig. 7. Fig Formed shapes and principal stresses associated with pressure ccles in the case of 1 = 2 = = 0.

6 Y. T. Keum, J. L. Kim and J. K. Shim 200 function as follows: = Xβ + ε (4) Using the least square estimate about the response obtained from 27 simulations, the estimated regression model can be epressed as follows: = (5) where 1, 2, and are the process parameters defined in Table 2. The recurrence of the estimated regression model, which is evaluated through the F-test performed b variance analsis [25], is listed in Table. The regression model of q. (5) has a recurrence because of the rejection F(0.05;9,17) = 2.98 when the level of significance is 5% and F0 = (10)4. Response surface model for Al2O particle size ( 1) and coefficient of friction ( ) in the case of cclic compaction pressure amplitude 2 = 0. Fig. 10. Optimal conditions q. (5), which is derived as a result of the response surface method to optimize process parameters, can be graphicall shown in Fig. 9, Fig. 10, and Fig. 11 when the remaining third process variable has a middle level. Fig. 9 shows that the variation of relative densit increases with the Al2O particle size ( 1 direction of 1) and the amplitude of Variance analsis of regression model Degree of freedom of regression Regression sum of squares Regression mean squares Degree of freedom of error rror sum of squares Mean square error Total degree of freedom Total sum of squares Fo = Regression mean squares / Mean square error Table (10) (10) (10)4 Response surface model for amplitude of cclic compaction pressure ( 2) and coefficient of friction ( ) in the case of particle size 1 = 0. Fig. 11. cclic compaction pressure (1 direction of 2) decreases. Fig. 10 shows that the variation of relative densit when the friction coefficient becomes smaller ( 1 direction of ) is not as big as much as that when the Al2O particle size is smaller ( 1 direction of 1). Fig. 11 implies that the increase in the variation of relative densit when the amplitude of cclic compaction pressure decreases (1 direction of 2) is bigger than that when the coefficient of friction decreases ( 1 direction of ). Therefore, the response surface model eplains that the effects of the Al2O particle size and the amplitude of cclic compaction pressure on the relative densit is bigger than that of the friction coefficient in ceramic powder compaction forming. In order to find the values of design parameters associated with the maimum relative densit, the grid search method is used [26]. The optimal condition, in which the maimum relative densit is 0.990, is found when the Al2O particle size, the amplitude of cclic compaction pressure and the friction coefficient are 22.5 μm, 75 MPa and 0.110, respectivel. Response surface model for Al2O particle size ( 1) and amplitude of cclic compaction pressure ( 2) in the case of friction coefficient = 0. Fig. 9.

7 Computer simulation of a ceramics powder compaction process and optimization of process parameters 201 Conclusions Multi-scale modeling and finite element analsis were performed to predict the relative densit of ceramic powder compaction forming and the optimization of process parameters was carried out to maimize the relative densit. Through the forming analsis and optimization, the following conclusions are made: (1) The modeling of non-periodicit and arbitraries of ceramic powder particles can be successfull achieved b introducing a quasi-random multi-particle arra. (2) The ceramic powder compaction forming process can be accuratel simulated b a multi-scale model and equivalent material properties obtained b emploing the homogenization method. () The relative densit becomes bigger with the size of the Al 2 O particles, the amplitude of the cclic compaction pressure, and coefficient of friction are smaller. (4) The regression model enables to predict the relative densit associated with compaction conditions and to provide the optimal conditions of process parameters related to the highest densit of the green ceramic compact. Acknowledgements This work was supported b the research fund of Hanang Universit (HY-2006-I). The authors appreciate Prof. R.H.Wagoner for discussions leading to these results. References 1. W. Wu, G. Jiang, R.H. Wagoner and G.S. Daehn, Acta mater. 48 (2000) G. Jiang, W. Wu, G.S. Daehn and R.H. Wagoner, Acta mater. 48 (2000) X.J. Xin, P. Jaaraman, G. Jiang, R.H. Wagoner and G.S. Daehn, Metallurgical and Materials Transactions A, A (2002) N. Zahlan, D.T. Knight, A. Backhouse and G.A. Leiper, Powder Technolog 114 (2001) H. Zipse, Journal of uropean Ceramic Societ 17 (1997) T.A. Deis and J.J. Lannutti, Journal of the American Ceramic Societ 81[5] (1998) P.K. Lu and J.J. Lannutti, Journal of the American Ceramic Societ 8[] (2000) P.K. Lu and J.J. Lannutti, Journal of the American Ceramic Societ 8[10] (2000) P.K. Lu and J.J. Lannutti, Journal of the American Ceramic Societ 8[6] (2000) B. Hassani and. Hinton, Comput. Str. 69 (1998) N. Takano, Y. Uetsuji, Y. Kashiwagi and M. Zako, Model. Simul. Mater. Sci. ng. 7 (1999) N. Takano, M. Zako and M. Ishizono, J. Comput.-Aided Mater. Design. 7 (2000) G. Jiang, G.S. Daehn, J.J. Lannutti, Y. Fu and R.H. Wagoner, Acta mater, 49 (2001) G. Jiang, G.S. Daehn and R.H. Wagoner, Scripta mater, 44 (2001) P. Rendanz, N.A. Fleck, Acta Materialia, 49 (2001) A.T. Procopio and A. Zavaliangos, Mechanics and Phsics of Solids, 5 (2005) M.K. Consuelo and J.J. Lannutti, Journal of the American Ceramic Societ, 8[9] (2000) A. Hassanpour and M. Ghadiri, Powder Technolog, 141 (2004) B.J. Briscoe and S.L Rough, Powder Technolog, 99 (1998) K.T. Kim, G.S. Son and J. Suh, Journal of Korean Ceramic Societ, 29[2] (1992) D.W. Shin, G.D. Kim, S.S. Park, C.S. Lim and S.W. Lee, Journal of Korean Association of Crstal Growth, 8[2] (1998) S. Turenne and P.. Mongeon, Advances in Powder Metallurg & Particulate Materials-1995 (1995) Y.T. Keum and J.W. Oh, Modelling and Simulation in Materials Science and ngineering, Institute of Phsics Publishing, 1 (2005) G..P. Bo and K.B. Wilson, Journal of the Roal Statistical Societ, Series B., 1 (1951) p R.. Wapole, R.H. Mers, S.L. Mers and k. Ye, in Probabilit & Statistics for ngineers & Scientists 8th ed (Pearson education, 2006). 26. Chen, D. Liu, X. Zhou, Y. Yang, X. Lu, L. Wang and X.J. Xin, Journal of Applied Polmer Science, 76[4] (2000)