Three-dimensional finite element analysis ofpowder compaction process for formingcylinder block of hydraulic pump

M. C. Lee1, S. H. Chung2, J. H. Cho3, S. T. Chung2, Y. S. Kwon2, H. J. Kim4 andM. S. Joun*5

A three-dimensional finite element analysis of a powder compaction process was undertaken to

determine the optimum manufacturing conditions for the complex cylinder block found in the

hydraulic pump of an excavator. A porous material model was used to ascertain the material

behaviour. The finite element predictions for both the density distribution and compaction load

were in good agreement with experimental results.

Keywords: Finite element analysis, Powder metallurgy, Hydraulic cylinder block, Density distribution, Compaction load

IntroductionPowder metallurgy is an effective means of manufactur-ing near net shape products of complicated parts.1 Itoffers many advantages, including high productivity andlow production costs. However, it cannot be used toform long and thin shapes since severe non-uniformitiesin the density distribution can arise from the inter-granular friction within narrow cavities that, in turn, canlead to a deterioration in the mechanical properties ofthe final product. This non-uniform distribution indensity is the most serious problem in terms of boththe performance reliability and the resulting shape ofpowder metallurgy products. Any heterogeneity indensity can generate cracks during the die compactionand cause non-uniform shrinkage or fractures duringsintering.

Consequently, considerable research has been devotedto enhancing the density uniformity in powder metal-lurgy products. The density distribution depends notonly on the process geometries, but also on the processconditions, including the frictional conditions and themoving histories of the dies, punches and cores. Theperformances of the dies, punches and cores need to beoptimised in order to ensure the quality of near netshape powder metallurgy products, particularly whenthe products possess long and/or thin walls.

Computer simulation technology can be very helpfulwhen optimising successful process designs for compli-cated powder metallurgy products with long and/or thinwalls. In recent years, computer aided engineering

(CAE) systems have been applied in a variety ofmanufacturing processes, including forging, castingand injection moulding, to reduce the number of designtry-outs and optimise the process designs. However,CAE technology has rarely been used for the processdesigns of powder metallurgy products, despite variousattempts to do so.2–10

It is important to model the material behaviour inpowder metallurgy to relate the stress and hydrostaticpressure to the strain or strain rate and the density.Traditionally, porous material, granular material, or capplasticity models have been used for powder metallurgi-cal studies. The porous material model is derived byapplying the von Mises yield criterion to fully densematerials such that these materials behave isotropically.Although such a model is simple, it can result in errorswhen the tensile stress is not negligible. In contrast, thegranular material model incorporates the intrinsicintergranular frictional and non-isotropic characteristicsof the powder. However, the theory involved is relativelycomplex and difficult to program, whether for develop-ment or application purposes. Also, the relevantmaterial constants are not easy to obtain.

Researchers including Khoei et al.4–6 and Rossi et al.7

have used the cap plasticity in conjunction with the finiteelement method to simulate powder compaction pro-cesses. Other researchers8,9 have used the cap plasticitymodel supplied commercially by ABAQUS to explorepowder compaction. However, all applications to datehave used simple process geometries that demand littlefrom the present state of the art technology that wouldbe required to form the more complex metallurgicalproducts currently being manufactured. Recently, Khoeiet al.4 applied the cap plasticity model to three-dimensional (3D) simulations of powder metallurgyprocesses with relatively simple 3D geometries. Theircomparisons between two-dimensional and 3D applica-tions, as well as between experiments and predictions,showed that the cap plasticity model and its relatedfinite element methodology can lead to errors that

1Gyeongsang National University, Jinju 660-701, Korea2Cetatech, Inc., Kyungnam 664-953, Korea3Graduate School, Gyeongsang National University, Jinju 660-701, Korea4Department of Automotive Engineering, Jinju National University, Jinju660-758, Korea5School of Mechanical and Aerospace Engineering, RRC/Aircraft PartsTechnology, Gyeongsang National University, Jinju 660-701, Korea

*Corresponding author, email msjoun@gsnu.ac.kr

� 2008 Institute of Materials, Minerals and MiningPublished by Maney on behalf of the InstituteReceived 27 June 2007; accepted 21 September JulyDOI 10.1179/174329008X277424 Powder Metallurgy 2008 VOL 51 NO 1 89

are not negligible from the viewpoint of process designengineers.

Interesting research by Lee and Kim10 has shown thatfinite element predictions made using the Shima-Oyanemodel,11 a special type of model for porous materials,12

agree relatively well with experimental results comparedto other models, including the Fleck and Gurson modelthat used tuned flow stresses,13 the Cam-Clay model14

and the modified Drucker-Prager cap model.15 Theresults of Lee and Kim show that the Shima-Oyanemodel has distinct merits, even though it underestimatesthe experimental results obtained in the low densityregion. By contrast, very few researchers have appliedthe porous material model to simulate powder com-paction, despite its theoretical and numerical simplicityand rigidity.2,3 Chung et al.16 have demonstrated thatpowder compaction process simulation technology canbe used to resolve real problems, such as optimising thecompaction process to form cemented carbide cuttingtools. Mori17 used the porous material model tosimulate powder compaction and used the predictedresults, including the product shape and the statevariables, to determine the required sintering using afinite element method.

In this paper, the porous material model is applied in acomputer simulation of the powder compaction processto form a cylinder block with a complex geometry.Tetrahedral elements are employed to enhance thegeneral applicability of this approach in three dimen-sions in order to examine the complexities of the powdercompaction processes. The calculated mean densitiesand compaction loads are compared with experimentalresults.

Three-dimensional finite elementanalysis of compaction process

Problem descriptionFigure 1 shows an exploded schematic representation ofa hydraulic pump assembly for an excavator. Thecylinder block is one of the most critical parts of theassembly and has been traditionally manufactured bycostly and time consuming machining after casting.

The size of the cylinder block for the excavator is solarge and its wall so thin compared to its height (heightto wall thickness ratio59?6) that manufacturing it usingpowder metallurgy technology presents great difficulties.However, process optimisation through computer simu-lation offers a potential method to achieve a successfulprocess design. To this end, a finite element method wasapplied to develop a powder metallurgy process to allowthe authors to form the hydraulic cylinder block free ofdesign failures with a near optimal density distribution.

A ferrous powder consisting of iron powder with1 wt-%S, 0?5 wt-%P, and 1 wt-% acrowax was used tofabricate the cylinder block. The pycnometer density ofthe ferrous mixed powder with a zero porosity binderwas 7?410 g cm23.

Material properties and flow behaviourFollowing the Lee and Kim’s work10 on various materialmodels through die compaction by single action press-ing, the Shima and Oyane model11 was employed tocharacterise the flow behaviour of the powder, which isgiven by

W~3J

0

2

s2m

za(1{r)c J21

9s2m

{rm (1)

where W is the yield function, J0

2~12

s0

ijs0

ij is the second

deviatoric stress invariant, J1~sii is the first stressinvariant, and sm and r are the flow stress of matrixmaterial and relative density respectively. In equa-tion (1) a, c, and m are material constants and the flowstress sm is assumed to be

sm~azb-

e nm (2)

where a, b, and n are material constants and-

em is theeffective plastic strain of matrix material.

To acquire the material properties explained above, asimple and effective uniaxial die compaction test wasconducted, as developed by Kwon et al.3 The pressurerelative density curves necessary for obtaining thematerial constants are shown in Fig. 2. Following thedetailed procedure3 to determine them from the pressurerelative density curves in Fig. 2, the following materialconstants were obtained

1 Schematic diagram of hydraulic pump 2 Finding material parameters by simple cylindrical die

compaction

Lee et al. Three-dimensional finite element analysis of powder compaction process

90 Powder Metallurgy 2008 VOL 51 NO 1

a~6:2000, c~1:0280, m~9:9709

a~453:3100, b~8221:5000, n~4:6914

Not that the same experiment was used to determine thecoefficient of Coulomb friction, i.e. m50?03.

With the application of the associated flow rule toequation (1), the stress–strain rate relationship isobtained

sij~-

s:-

e

2

A(r)

:eijz

1

3½3{A(r)�:-

ekkdij

� �(3)

where sij,:eij,

:-

ekk, and dij are the stress tensor, deviatoric

strain rate tensor, volumetric strain rate and theKronecker delta respectively, and A~9=½3za(1{r)c�.The effective stress

-

s and the effective strain rate:-

e inequation (3) can be defined as

-

s~A(r)

2s0ijs

0ijz

3{A(r)

3skksll

� �1=2

(4)

:-

e~2

A(r)

:e0ij:e0ijz

1

3 3{A(r)½ �:ekk

:ell

� �1=2

(5)

Finite element formulation of powdercompaction processThe boundary value problem of die compactionprocesses can be formulated using the constitutiverelationship derived in the section on ‘Material proper-ties and flow behaviour’. In a metallurgical powdercompaction analysis, the boundary value problemrequires the authors to find the velocity field vi thatsatisfies the material denoted as domain V withboundary S. The boundary S can be divided into the

velocity prescribed boundary Svi, where the velocity is

given as vi~-

vi, the traction prescribed boundary Sti,

where the stress vector is given as t(n)i ~

-

t(n)i , and the die

workpiece interface Sc, where the non-penetrationcondition, vn~-

vDn , must be maintained when the inter-

face is in compression. The boundary value problem canhence be formulated as

(i) equation of equilibrium

sij,jzfi~0 (6)

(ii) (ii) stress–strain rate relationship

sij~{pdijzs0ij (7)

s0ij~2

A-

s:-

e

:e0ij (8)

p~{1

3(3{A)-

s:-

e

:ekk (9)

(iii) boundary conditions

sijnj~-

t(n)i on Sti

(10)

vi~-

vi on Svi(11)

sn~{j vn{-

vDn z

d

Dt

� �on Sc (12)

st~{msng(vt) on Sc (13)

where fi in equation (6) is the body force, j inequation (12) is the penalty constant, which is relativelylarge, and d and Dt in equation (13) are the normaldistance from the material point to the die surface andthe time increment of the current solution steprespectively. With a large penalty constant, one canapproximately satisfy the velocity prescribed on bound-ary Sc, i.e. the non-penetration condition will bemaintained if the converged solution of the normalstress is several orders of magnitude less than the penaltyconstant. In equation (13), g(vt) is a function thatreflects the frictional effect of the relative velocitybetween the tools and the material. Following Chen

3 Finite element model of cylinder block

4 Loading schedule of process

Lee et al. Three-dimensional finite element analysis of powder compaction process

Powder Metallurgy 2008 VOL 51 NO 1 91

et al.,18 the following function was employed

g(vt)~{2

ptan{1 vt{-

vDt

a

� �(14)

where vt and-

vDt are the tangential velocity components

of the material and die respectively, and a is a small

positive constant relative to-

vDt

�� ��.The weak form corresponding to the above boundary

value problem can be formulated asðV

s0ijv0ijdV{

ðV

pviidV{

ðV

fividV{X

i

ðSti

-

t(n)i vidS

z

ðSc

j vn{-

vDn z

d

Dt

� �vndS{

ðSc

mj vn{-

vDn z

d

Dt

� �g(vt)vtdS~0 (15)

where vij is defined as vij~1=2 vi,jzvj,i

� and v

0

ij is the

deviatoric tensor of vij. In the weak form, the weightingfunction vi is arbitrary except that it vanishes onSvi and vn~0 on Sc. The subscripts n and t in equa-tion (15) denote the normal and tangential componentsrespectively.

Finite element analysis of cylinder block powdercompaction processThe cylinder block formed in this study was 67 mm high,which is larger than that of common powder metallurgyproducts. The near tenfold ratio of its height to thicknessratio was also large. It is therefore very important notonly to maintain density uniformity from top to bottomreducing the compaction load, but also to maintain asufficient average density for preventing oil leakage. It isthus essential to optimise the design process to ensure thatthe highest mean density is achieved as uniformly aspossible. To this end, the powder compaction process wasanalysed using the finite element method.

Figure 3 shows the initially filled powder shape andthe die configurations employed for the finite element

a initial condition; b 14?7% reduction; c 29?5% reduction; d 44?2% reduction5 Relative density distribution

Lee et al. Three-dimensional finite element analysis of powder compaction process

92 Powder Metallurgy 2008 VOL 51 NO 1

analysis of the selected process design to evaluate theoptimal density distribution. It is noted that the processgeometries were taken upside down for better view andunderstanding of the process.

The coefficient of Coulomb friction, 0?03, wasobtained from the same experiment (see Fig. 2) used todetermine the material constants.3 The loading scheduleof the process is shown in Fig. 4. Given the symmetry ofthe final product, only one-ninth of the total processgeometry was analysed. The mesh system shown inFig. 3 was composed of 22 644 tetrahedral elementswith 5082 nodes.19

The final solution was obtained after 300 steps.Figure 5 shows the predicted relative density distribu-tions at selected solution steps and Fig. 6 shows therelative density distribution of the final product. Theinitial relative density was 0?500. The results in Fig. 6show that the density was quite uniform in the earlystages of the powder compaction process, and becameless uniform as the stroke increased. The maximum andminimum values of the predicted mass density of theproduct were 6?880 and 6?760 g cm23, respectively. Thepredicted mean mass density of the compacted cylinderblock, composed of both metal powder and binder, was6?810 g cm23. Figure 7 shows the variation of thecompaction load with the stroke; the maximum compac-tion load was 2870 kN.

Experimental results and comparisonwith predictionsA cylinder block was fabricated using a 5000 kNhydraulic powder metallurgy press with a three stepdie set consisting of one upper step and two lower steps.The initial height of the uncompacted material was110 mm, and the final height of the product was 67 mm.The powder material was compacted by increasing thecompaction load up to 2500 kN until the mean massdensity of the powder mixed with the binder reached6?800 g cm23. After the powder was compacted, the

material was sintered at 1220uC in a continuous furnace.The final sintered product is shown in Fig. 8. Thedimensional change due to sintering was less than 0?3%.

The sintered product was divided into several pieces,as shown in Fig. 9, and the global density of each piecewas measured individually by water densimetry. Eachpiece was coated before the density measurements werecarried out. The experimental and predicted densities,given in Table 1, differed by less than 0?4%.

The experimental and predicted compaction loads arecompared in Fig. 7. The compaction load predicted bythe finite element method with the porous materialmodel was 12–16% higher than that obtained experi-mentally. This difference was acceptable and demon-strated the validity of the proposed approach.

ConclusionsA 3D finite element analysis of a powder compactionprocess allowed the authors to optimise the design

6 Density distribution at final stroke

7 Predicted and experimental compaction loads

8 Sintered product after powder compaction

Lee et al. Three-dimensional finite element analysis of powder compaction process

Powder Metallurgy 2008 VOL 51 NO 1 93

process used to manufacture the complicated cylinderblock found in a hydraulic pump of an excavator. Thebehaviour of the powder material was modelled using aporous material model. The predicted density distribu-tion and compaction loads were in good quantitativeagreement with the experimental results.

The proposed finite element approach offers a viablemeans of achieving the process optimisation required touse powder metallurgy to fabricate complex products,such as the cylinder block described here. Such a

fabrication technique is necessary for reducing themanufacturing costs and increasing the product quality

Acknowledgements

This work was supported by the second stage BK21project and the programme for the training of graduatestudents in regional innovation of MOCIE of Korea.

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9 Sampled pieces of final sintered product

Table 1 Comparison of measured and predicted densities

Section Predicted a Measured b

Relative error, %

(a–b)/a6100

1 6.88 6.86 0.292 6.81 6.83 20.293 6.76 6.79 20.444 6.80 6.75 0.74Average 6.81 6.81 0.00

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94 Powder Metallurgy 2008 VOL 51 NO 1