simulation of crack propagation in metal powder compaction · such as powder movement, density...

International Journal for Computational Methods in Engineering Science and Mechanics, 7:293–302, 2006Copyright c© Taylor & Francis Group, LLCISSN: 1550–2287 print / 1550–2295 onlineDOI: 10.1080/15502280600549363

Simulation of Crack Propagation in MetalPowder Compaction

S. M. Tahir and A. K. AriffinDepartment of Mechanical & Materials Engineering, Universiti Kebangsaan Malaysia, Bangi,Selangor, Malaysia

This paper presents the fracture criterion of metal powder com-pact and simulation of the crack initiation and propagation duringcold compaction process. Based on the fracture criterion of rock incompression, a displacement-based finite element model has beendeveloped to analyze fracture initiation and crack growth in ironpowder compact. Estimation of fracture toughness variation withrelative density is established in order to provide the fracture pa-rameter as compaction proceeds. A finite element model with adap-tive remeshing technique is used to accommodate changes in geom-etry during the compaction and fracture process. Friction betweencrack faces is modelled using the six-node isoparametric interfaceelements. The shear stress and relative density distributions of theiron compact with predicted crack growth are presented, where theeffects of different loading conditions are presented for comparisonpurposes.

Keywords Powder Compact, Crack Propagation, Finite Element

1. INTRODUCTIONManufacturing parts using powder metallurgy (PM) involves

four major steps: powder and lubricant mixing, compacting pow-ders into appropriate shapes in closed dies to produce green com-pacts, sintering the green compacts at elevated temperature and,finally, post-sintering secondary operations [1, 2]. Generally,it is agreed that during compaction process the metal powderundergoes three stages. Initially, at low pressure, particle slid-ing occurs leading to particle rearrangement. Secondly, elasticand plastic deformation of particles occurs through their contactareas, which leads to geometric hardening due to plastic defor-mation and void closure. Lastly, at very high pressure, materialstrain hardening causes a rapid increase in the flow resistance ofthe material. Details on cold compaction process can be foundin [3], where the numerical modelling of the complete cycle

Received 15 December 2004; accepted 31 August 2005.Address correspondence to S. M. Tahir, Department of Mechanical

& Materials Engineering, Universiti Kebangsaan, Malaysia, 43600Bangi, Selangor, Malaysia. E-mail: suraya [email protected]

(compaction, relaxation, ejection and emergence) has been de-veloped, and validated by experiments.

Modelling of fracture and crack propagations involves suit-able fracture criterion and numerical modelling to represent thecrack propagation process. Even though methods for crack de-tection in green powder metallurgy (PM) parts [4, 5] exist, todate the crack propagation during the compaction process is un-known and needed to improve the performance of parts producedby this process. In this paper a suitable fracture criterion for metalpowder during cold compaction process is outlined, taking intoaccount the mechanical behavior of metal powder under com-paction process. Finite element modelling of the crack initiationand propagation has been developed where the effects of differ-ent loading conditions on the crack initiation and propagationare presented.

2. MODELLING OF COMPACTION PROCESSIn modelling the compaction process, the macro-mechanical

modelling approach is used in this work, which provides infor-mation on the macroscopic behavior of the powder assemblysuch as powder movement, density distribution, stress state andthe shape of the compact during and after compaction. Thusthe powder medium is considered as a continuum that under-goes large elastic-plastic deformation. In order to describe theeffect of stress state on the response of the powder material, aconstitutive model based on granular material is used, since itwas found in the literature that powder behaves similarly to africtional granular material with regard to dilatancy and densifi-cation behavior [6]. Constitutive equations used in the model torepresent the powder under loading conditions can be summa-rized as below.

2.1. Elastoplastic Material Model with LargeDisplacement

In three-dimensional space �n(1 < n < 3), the stress andstrain at any point in a material body are represented with second

293

294 S. M. TAHIR AND A. K. ARIFFIN

order symmetric stress, σ , and strain, ε tensors. The total strainεij can be expressed as the sum of elastic, εe

ij, and plastic, εpij

components, which in terms of small increments is given by

dεij = dεeij + dε

pij for i, j = 1, . . . . . . , n (1)

For two-dimensional Euclidean space, where plane strain or ax-isymmetric simulations can be used to represent a large numberof components produced by the powder compaction process, theelastic stress and strain relationship can be written in the forms:

σ = Deεe or σij = Deijklε

ekl (2)

or in terms of elastic increment:

dεe = D−1e dσ (3)

De is the elasticity matrix or fourth order tensor given by

De =

E

(1 + υ)(1 − 2υ)

1 − υ υ 0 01 − υ υ 0

symmetry 1 − υ 012 (1 − 2υ)

(4)

where E is the modulus of elasticity and v is Poisson’s ratio.

2.2. Mohr-Coulomb Yield CriterionReferring to Fig. 1, one of the earliest failure laws for granular

materials like soils is based on Coulomb’s friction failure, statedas

τ f = c + σn tan φ (5)

where τ f is the shear stress of failure, σn is normal stress, whileparameters needed in this model are cohesive stress, c, and pow-der internal friction φ, which are easily derived from experimen-tal data.

On extension to deviatoric stress space, this criterion is statedas:

F = J1 sin φ +√

J2D cos θ −√

3J2D

3sin φ sin θ − c cos φ = 0

(6)

FIG. 1. Mohr-Coulomb yield surface.

where J1 and J2D are first and second deviatoric stress, respec-tively.

3. FINITE ELEMENT MODELLING OF FRACTUREIN POWDER COMPACTExtensive literature reviews on materials under compression

reveal that, generally, cracks grow in mode II, no matter whetherthe material is brittle as reported in [7–11], or ductile as reportedin [12–15]. However, crack patterns are being influenced by theamount of applied stress and friction between the crack faces.Basically, it can be concluded that cracks can grow in threedifferent manners:

i) via an open wing crack, at an angle from the original crackplane.

ii) as a closed crack, straight ahead from the original crackplane.

iii) as a combination of (i) and (ii).

As compaction proceeds powder undergoes large deforma-tion, large inhomogeneous density distribution present withinthe powder compact, while inter-particle frictions as well asfriction between powder particle and die wall exist. The combi-nation of (i) and (ii) as stated above is possible for metal powderunder compaction as the normal and shear stress acting on thepowder increases, while the density distribution keeps changing.

A displacement-based finite element model has been devel-oped to simulate the powder compaction and fracture process.Mohr-Coulomb yield criteria is used in this work, while FOR-TRAN programming language is used in developing the com-puter codes.

3.1. Geometry and Boundary Conditions of FiniteElement Model

A multi level component, in this case a rotational flangedcomponent made of iron powder, is modelled by an axisymmet-ric representation as shown in Fig. 2.

FIG. 2. Geometry and boundary conditions of a rotational flanged compo-nent.

SIMULATION OF CRACK PROPAGATION 295

TABLE 1Three different compaction sequences

Step 1 to 20 Step 20 to 40

Total TotalCompaction Punch displacement Punch displacementsequence movement (mm) movement (mm)

CS1 Bottom 7.69 Top 6.06CS2 Top 6.06 Bottom 7.69CS3 Bottom 7.69

& Top 6.06

Iron powder with material properties as listed in Tran et al.[16] is compacted by bottom and top punch movements. Threedifferent compaction sequences, as given in Table 1, have beenapplied in the compaction process.

In CS1, powder is compacted by the bottom punch movementfor the first 20 steps (step 1 to 20), followed by another 20 steps(step 21 to 40) by top punch movement. In contrast, powderis compacted by the top punch movement for the first 20 steps(step 1 to 20) in CS2, followed by the bottom punch movementfor the next 20 steps (step 21 to 40). For CS3, both top andbottom punches move simultaneously in 20-steps movement tocomplete the compaction process.

3.2. Fracture CriteriaA large number of inhomogenities due to soft and hard in-

clusions, such as voids, pores and microcracks in heterogeneousbrittle materials like rock, concrete and ceramic, provide for thecrack tip in such material a rather large process region. Thismay be the reason for the macroscopic mode II failure undercompressive loads in such materials. Similarly, during the com-paction process, inhomogenities within the powder compact andfrictional behavior of powder under compaction will obviouslybe expected to lead to failure in mode II. Since powder behavessimilarly to frictional granular materials like rock in compres-sion, fracture criteria by Quihua et al. [17] is adopted in thiswork.

Based on the examination of Mode I and Mode II stress inten-sity factors on the arbitrary plane θ (Fig. 3), K I (θ ) and KII(θ ),varying with θ (−180◦ ≤ θ ≤ +180◦), no matter what kind ofloading condition is applied, the shear fracture criterion [17] isexpressed as:

KII max

K I max>

KIIC

KIC, KII max = KIIC at θIIC (7)

where K I max and KIi max are maximum stress intensity factorsof Mode I and Mode II, respectively, while KIC and KIIC arecritical stress intensity factors of Mode I and Mode II.

3.3. Calculation of Stress Intensity Factor (SIF)In order to determine K I max and KII max in Eq. (7), mode I and

mode II stress intensity factors in the direction of θ is defined

FIG. 3. Stress component at a point near a crack tip in the polar coordinatesystem.

as

K I (θ ) = K I cos3 θ

2+ KII

(− sin

θ

2cos2 θ

2

)(8)

KII(θ ) = K I sinθ

2cos2 θ

2+ KII cos

θ

2

(1 − 3 sin2 θ

2

)(9)

where K I and KII in Eqs. (8) and (9) are the stress intensityfactors in the original crack plane, while θ is defined as posi-tive in the anticlockwise direction. In the finite element method,these values can be calculated using the displacement correlationtechnique [18] as below:

K I = G

κ + 1

√2π

L

(4(vb − vd ) − (vc − ve

2

)(10)

KII = G

κ + 1

√2π

L

(4(ub − ud ) − (uc − ue

2

)(11)

where G is the shear modulus, L is the element length, κ is de-fined as κ = (3 − 4ν) for plane strain or axisymmetric problem,and v is the Poisson’s ratio. The u and v are the displacementcomponents in x and y directions, respectively, where the sub-scripts indicate their positions as shown in Fig. 4.

The criterion in Eq. (7) stated that a crack occurs when theratio of maximum KII (θ ) over K I (θ ) exceeds the ratio ofcritical values or fracture toughness, of KII over K I . Since nopre-crack is present in this case, the direction of maximum shearstress is used as the original crack direction in the calculationof KII (θ ) and K I (θ ) for the first crack formation. This is ac-ceptable because the same conclusions regarding the crack pathare achieved in materials under compression, by assuming thatthe crack grows along the plane of maximum shear stress andby assuming that crack follows the direction of maximum KII

[10]. Without a pre-crack in this work, the point with maximumshear stress is taken as the point where the crack starts.


FIG. 4. Nodes around the crack tip for calculation of the stress intensityfactors.

3.4. Adaptive MeshAn adaptive finite element mesh is applied to accommodate

changes in the geometry of the domain. This enables the meshrefinement to be done selectively in areas where the error in ap-proximate solution is largest. An error estimator based on stresserror norm [19] is used, where automatic remeshing is calculatedat each step during the compaction process. Crack initiation andpropagation have been implemented in the model without havingto predefine the direction of crack. Initially, three-nodes trian-gle element is used. After the first stage of remeshing, the threenodes elements are automatically changed into six nodes triangleelements.

3.5. Crack MechanismIn this work, the crack is assumed to propagate by an ex-

tremely small opening, which is quickly being closed as com-paction proceeds. In finite element modelling using advancedremeshing technique, crack propagation can be modelled byinter-element or intra-element in the mesh [20]. Since the val-ues of K I and KII are calculated on the nodes around the tip,the crack is modelled to propagate inter-element, by the split-ting mechanism of crack tip node. This is similar to the releasenode mechanism except that the direction of crack propagationis depending on direction criteria. Referring to Fig. 5, the cracktip node is split into two new nodes by adding a very small gapif the crack criteria is fulfilled.

3.6. Friction CriteriaWhen two surfaces are in contact, a criteria defining tangen-

tial stick or slip occurrence is needed. In general, friction criteria

FIG. 5. Split node model.

FIG. 6. Variation of green strength with relative density.

between two surfaces can be stated as

Ff (τ, σn) ={

<0 stick

=0 slip(12)

Coulomb friction law, which is often adopted in friction prob-lems, is used in this work, given by

Ff (τ, σn) = |τ | + µσN (13)

where τ is the friction shear stress, σN is the normal stress,which should be compressive for friction to developed, and µ isthe coefficient of friction.

Six-node isoparametric elements are used as interface ele-ments for friction between powder material and die wall duringthe compaction process. Details on the constitutive model offriction based on plasticity, which is incorporated into the inter-face elements, can be found in [3]. The interface elements arealso used to model friction on the crack faces in contact. Usingadaptive finite element mesh, the interface element can be in-serted automatically on the crack faces, whenever a crack startsand grows.

FIG. 7. Shear stress-strain curves at different compaction loads.


FIG. 8. Variation of KIIC with relative density.

FIG. 9. Shear stress distribution at step 10, step 18, step 19 and step 21 for CS1.

3.7. Fracture ToughnessThe fracture criterion in Eq. (7) requires values of the critical

stress intensity factors KIC and KIIC, which are the materialparameters and also called fracture toughness. Standard proce-dures exist for determination of fracture toughness for solids,such as the three-point bending test or four–point bending test.However, the fracture toughness of powder compact duringcompaction process is not as simple as solids due to continuouschanging of density and other material properties at each com-paction step. Thus variation of fracture toughness with relativedensity must be obtained in order to provide these fracture pa-rameters as compaction proceeds.

3.7.1. Mode I Fracture Toughness (KIC)Based on approximate formulas for mode I fracture toughness

(KIC) in [21], finite element models have been developed by Choiand Sankar [22] to estimate the variation of KIC as a function of


FIG. 10. Shear stress distribution at step 4 and Step 6 for CS2.

solidity of carbon foam. By assuming that the crack is parallelto one of the principal material axes, KIC can be estimated usingthe following formula:

KIC = σUS

σmax(14)

whereσUS is the strength of the material andσmax is the maximumprincipal stress.

A study on the mechanical properties of iron compact hasbeen performed by Poquillon et al. [23]. Three-point bendingtests have been carried out on iron compacts, which provide thevariation of green strength with relative density, as shown inFig. 6.

Using the variation of green strength in Fig. 6 and by assum-ing that green compact behaves as foam metal, where open andclosed porosity in green compact correspond to open and closedcells in foam, Eq. (14) hence becomes

KIC = 148.66ρ ′ − 65.627

σmax(15)

where ρ ′ is the relative density of the iron compact with respectto solid iron, and σmax is the maximum principal stress as inEq. (14).

3.7.2. Mode II Fracture Toughness (KIIC)Fracture toughness of a material is the amount of energy a

material can absorb before fracture, or energy required to create

new crack surfaces. Thus it is also equal to the area under stress-strain curve up to fracture. Experimental data by Aidah [24]is used in this work to obtain the shear stress-strain curves atdifferent compaction loads as shown in Fig. 7. These curves aresubsequently used to estimate the variation of mode II fracturetoughness (KIIC) with relative density, as depicted in Fig. 8.

From Fig. 8, the values of KIIC as compaction proceeds canbe obtained from

KIIC = 0.1174e6.5031ρ ′(16)

where ρ ′ is the relative density of iron compact as in Eq. (15).Since the crack is predicted to follow the direction of KII max,

TABLE 2Details of crack initiation and propagation for CS1 and CS2

CS1 CS2Compaction Compaction

Step θ◦ ρ ′ Step θ◦ ρ ′

9 0 0.2271 3 0 0.3546(Crack (CrackStarts) Starts)

17 +22.7883 5 +39.330718 +5.3488920 +13.74237


FIG. 11. Relative density distribution at step 10, step 18, step 19 and step 21 for CS1.

the relative density at the point with maximum KII around thecrack tip is used in the calculation of both KIC and KIIC.

4. RESULTS AND DISCUSSIONDetails of crack initiation and propagation for CS1 and CS2

are presented and compared in Table 2. The value of θ iscalculated with respect to the earlier crack direction, using thesign convention as shown in Fig. 4, while ρ ′ is the relative den-sity value at the point where the crack starts. For CS1, the crack

starts at step 9 and propagates at steps 17, 18 and 20. No fur-ther crack propagation occurs after step 20. Surprisingly forCS2, the crack starts at an early stage of compaction that is atstep 3. However, the crack propagates only once at step 5, andno further propagation occurs until compaction is completed atstep 40.

Figure 9 shows the shear stress distribution at steps 10, 18, 19and 21 for CS1, while Fig. 10 shows the shear stress distributionat step 4 and step 6 for CS2. Neglecting the sign convention,which indicates the direction of stresses, it can be seen from


FIG. 12. Relative density distribution at step 4 and step 6 for CS2.

these figures that the crack starts in the region of high shearstress distribution. As compaction proceeds, the crack propa-gates towards the region of higher shear stress, which can beseen clearly in Fig. 9(a) to Fig. 9(d). In the case of CS2, thecrack propagates only once, in the region of the highest shearstress distribution as depicted in Fig. 10(b).

As mentioned earlier, in CS2 the crack starts at an earlystage of the compaction process. It is interesting to note that

FIG. 13. Shear stress distribution at step 1, step 10 and step 20 for CS3.

even though the crack starts in the region of the highest shearstress distribution for both CS1 and CS2, the effect of relativedensity distribution on the crack initiation shows a different phe-nomenon for the two cases. Figures 11 and 12 show the relativedensity distributions when the crack starts and propagates forCS1 and CS2, respectively. From Fig. 11(a), it can be seen thatthe crack starts in the region of low relative density where themaximum relative density attained at this stage is 0.6504, while


FIG. 14. Relative density distribution at step 1, step 10 and step 20 for CS3.

the crack starts at the point with relative density of 0.2271 asstated in Table 2.

On the other hand, the crack starts at the point with relativedensity of 0.3456 in the case of CS2, while the maximum relativedensity attained at this stage is 0.4012. Therefore, in this case thecrack starts in the region with a relatively high relative density,which is in constrast to CS1. In addition, Fig. 12(a) also showsthat the crack starts exactly at the sharp corner between thesmall region with the highest relative density and the regionwith a slightly lower relative density. As compaction proceeds,Fig. 11(a) to Fig. 11(d) show that the crack propagates towardsthe region of higher relative density in the case of CS1. Sincerelative density increases as compaction pressure increases [21,22], it can also be deduced that the crack grows in the directionof higher compaction pressure. For CS2, however, a small regionwith low relative density is formed around the sharp corner ascompaction proceeds, causing the crack to propagate betweenthe region of the highest and the lower relative density as shownin Fig. 12(b).

When powder is compacted by simultaneous movement ofboth top and bottom punches as in CS3, no crack is formed withinthe powder compact until the compaction process is completed atstep 20. It can be seen from Figs. 13 and 14 that even though someregions within the powder compact have low and high shearstresses, as in Fig. 13, uniform density distributions is achievedthroughout the compaction process, as shown in Fig. 14. Hencethis compaction sequence is found to be a good sequence toavoid crack formation in this component during the compactionprocess.

5. CONCLUSIONSBased on the fracture criterion of granular material in com-

pression, a displacement-based finite element model has been

successfully developed to simulate the fracture initiation andcrack propagation in metal powder during the compaction pro-cess. Three different compaction sequences have been performedto produce a rotational flanged component made of iron pow-der. Simulations of the compaction process show that both shearstress and relative density distributions have influences on thecrack initiation and propagation process. High shear stress andinhomogeneous relative density distribution in some regionwithin the powder compact causes the crack to initiate and prop-agate as compaction proceeds. Different shear stress and rela-tive density distributions are obtained for different compactionsequences, causing the crack to initiate and propagate at dif-ferent compaction steps. The density distribution is almost uni-form throughout the compaction process when compaction isperformed by simultaneous movement of both top and bottompunches. Thus, this compaction sequence is found to be the bestsequence compared to the other two sequences in order to pro-duce a crack-free component for this case.

REFERENCES1. Chtourou, H., Guillot, M., and Gakwaya, A., “Modelling of the metal pow-

der compaction process using the cap model. Part 1: Experimental materialcharacterisation and validation,” International Journal of Solids and Struc-tures 39, 1059–1075 (2002).

2. Mori, K., Sato, Y., Shiomi, M., and Osakada, K., “Prediction of fracturegenerated by elastic recoveries of tools, in multi-level powder compactionusing finite element simulation,” Int. J. of Machine Tools & Manufacture39, 1031–1045 (1999).

3. Ariffin, A. K., and Ihsan, Mohd., “Powder compaction, finite elementmodelling and experimental validation,” PhD Thesis University of Wales,Swansea, United Kingdom (1995).

4. Hauck Eric, T., Rose Joseph, L., Song, Won-Joon, Heaney Donald, A surfacewave mediator technique for crack detection in green parts, Proceeding ofthe 2003 International Conference on Powder Metallurgy & ParticulateMaterials (2003).


5. Frederick, K., Eddy Current Testing of Powder Metal Components,www.zetec.com/documents/pwdr met insp(1).pdf (2003).

6. Gollion, J., Bouvard, D., and Stutz, P., “On the rheology of metal pow-der during cold compaction,” Proc. Int. Conf. Micromechanics of Granu-lar Media, Powder and Grains, J. Biarez and R. Gourves (eds.), 433–438(1989).

7. Melin, S., “When does a crack grow under mode II conditions?,” Int. Journalof Fracture 30, 103–114 (1986).

8. Melin, S., “Fracture From A Straight Crack Subjected to Mixed-mode Load-ing,” Int. Journal of Fracture 32, 257–263 (1987).

9. Yu, Mao Hong, Zan, Yue-Wen, Zhao, Jian, and Yoshimine, Mitsutoshi, “Aunified strength criterion for rock material,” Int. Journal of Rock Mechanics& Mining Sciences 39, 975–989 (2002).

10. Isaksson, P., and Stahle, P., “Mode II crack paths under compression in brittlesolids —A theory and experimental comparison,” Int. Journal of Solids andStructures 39, 2281–2297 (2002).

11. De Bremaecker, J. C., and Ferris, M. C., “Numerical models of shearfracture propagation,” Engineering Fracture Mechanics 71, 2161–2178(2004).

12. Liu, A. F., “Crack growth failure of aluminium plate under in-plane shear,”AIAA Journal 12, 180–185 (1974).

13. Hallback, N., “Mixed ModeI/II fracture behaviour of a high strength steel,”Int. Journal of Fracture 87, 363–388 (1998).

14. Arun Roy, Y., Narasimhan, R., and Arora, P. R., “An experimental inves-tigation of constraint effects on mixed mode fracture initiation in a ductilealuminium alloy,” Acta Mater 47, 1587–1596 (1999).

15. Isaksson, P., and Stahle, P., “A directional crack path criterion for crackgrowth in ductile materials subjected to shear and compressive loading

under plane strain conditions,” Int. Journal of Solids and Structures 40,3523–3536 (2003).

16. Tran, D. V., Lewis, R. W., Gethin, D. T., and Ariffin, A. K., “Numericalmodelling of powder compaction process: Displacement based finite ele-ment method,” Powder Metallurgy, 36(9), 257–266 (1993).

17. Rao Quihua, Sun Zongqi, Stephansson, O. Li, Chunlin, and Stillborg, B.“Shear fracture (Mode II) of brittle rock,” Int. Journal of Rock Mechanics& Mining Sciences 40, 355–375 (2003).

18. Phan, A. V., Napier, J. A. L., Gray, L. J., and Kaplan, T., “Stress intensityfactor analysis of friction sliding at discontinuity interfaces and junctions,”Computational Mechanics 32(4–6), 392–400 (2003).

19. Zienkiewicz, O. C., and Zhu, J. Z., “Error estimates and adaptive refine-ment for plate bending problems,” Int. Journal for Numerical Methods inEngineering 28, 2839–2853 (1989).

20. Bouchard, P. O., Bay, F., Chastel, Y., and Tovena, I., “Crack propagationmodelling using an advance remeshing technique,” Comput. Methods. Appl.Mech. Engineering 189, 723–724 (2000).

21. Gibson, L. J., and Ashby, M. F., Cellular Solids: Structure and Properties,Second Edition, Cambridge University Press, Cambridge, UK (1988).

22. Choi, S., and Sankar, B. V., “Fracture toughness of carbon foam,” Journalof Composite Materials 37(23), 2101–2116 (2003).

23. Poquillon, D., Baco-Carles, V., Tailhades, Ph., and Andrieu, E., “Cold com-paction of iron powders—Relations between powder morphology and me-chanical properties Part II. Bending tests: Results and analysis,” PowderTechnology 126, 75–84 (2002).

24. Jumahat, Aidah, Analysis of Thermo-Mechanical Behaviour of Warm Com-paction Process (In Malay Language), Master of Science Thesis, UniversitiKebangsaan Malaysia, Malaysia (2001).

simulation of crack propagation in metal powder compaction · such as powder movement, density...

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