A New Yield Function for Porous Materials

8/12/2019 A New Yield Function for Porous Materials

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Journal of Materials Processing Technology 179 (2006) 3643

A new yield function for porous materials

Lus M.M. Alves, Paulo A.F. Martins, Jorge M.C. Rodrigues

Institute Superior Tecnico, Departamento de Engenharia Mecanica, Av. Rovisco Pais, 1049-001 Lisboa, Portugal

Abstract

Densification of sintered copper under uniaxial compression is analyzed by means of three different plasticity criteriaShima and Oyane [S.

Shima, M. Oyane, Plasticity theory for porous metals, Int. J. Mech. Sci. 18 (1976) 285291], Doraivelu et al. [S.M. Doraivelu, H.L. Gegel, J.S.

Gunasekera, J.C. Malas, J.T. Morgan, J.F. Thomas, A new yield function for compressible P/M materials, Int. J. Mech. Sci. 26 (9/10) (1984)

527535] and Lee and Kim [D.N. Lee, H.S. Kim, Plasticity yield behaviour of porous metals, Powder Metall. 35 (1992) 275279]. Theoretical

predictions are compared with experimental data and served as a basis for developing a new yield function for compressible P/M materials. The

proposed yield function is not only capable of providing a better agreement with the experiments as its implementation into an existing finiteelement computer program proves to be successful for the numerical modelling of a powder forged component.

2006 Elsevier B.V. All rights reserved.

Keywords: Powder forging; Finite element method (FEM); Experimentation

1. Introduction

Recent advances in computing technology and reduction in

the associated costs are presently extending the availability of

computer programs to simulate complex P/M forming processes.

At present time computer programs have the potential to accu-rately model complex three dimensional P/M processes without

the need to take advantage of possible geometrical and material

flow simplifications.

Most of the scientific and numeric ingredients which are

necessary to develop and utilize these programs (discretization

procedures, solution techniques, contact algorithms and remesh-

ing schemes) are now nearly established but other important

ingredients, probably even more relevant than the former, to

the progress of numerical simulation, still deserve attention.

The yield function and its associated constitutive equations for

porous materials [1,2] is oneof these ingredients because it plays

a crucial role in the overall performance of the computer pro-

grams.

This paper is based on a comprehensive investigation of the

commonly used yield functions due to Shima and Oyane [3],

Doraivelu et al.[4]and Lee and Kim [5].The main objective

was to characterize and understand its limitations but also to

develop a new more accurate one if necessary.

Corresponding author. Fax: +351 21 841 90 58.E-mail address:[email protected](J.M.C. Rodrigues).

The most important drawback in the utilization of the above-

mentionedyieldfunctions is dueto thefact that theinitial relative

density does not explicitly influence the geometric hardening of

the base material and the plastic Poisson coefficient. Some of

the yield functions cannot even be used when the initial relative

density of the preforms is low.The paper deals with these problems and proposes a new

yield function for powder materials that takes into account the

initial relative density of thepreform. New testing procedures for

characterizing the mechanical behaviour of the base material are

also introduced in order to obtain a more accurate stressstrain

curve of the powder material.

The implementation of the new proposed yield function into

an existing finite element computer program is also briefly con-

sidered and assessment is provided by means a numerical and

experimental study of a powder forging process. To this end, a

laboratory controlled test comprising the forging of a sintered

copper perform was designed in such a way that material flow

behaviour, geometric profile, distribution of relative density and

forging load, could be reproduced and compared against numer-

ical estimates.

2. Theoretical background

The yield function for porous materials differs from the von

Mises function for dense materials due to the changes produced

in the volume of the porous bodies during plastic deformation.

In general, considering that the material is isotropic, and that the

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L.M.M. Alves et al. / Journal of Materials Processing Technology 179 (2006) 3 643 37

Table 1

The parametersA,B and for different porous yield functions

A B

Shima and Oyane[3] 31+(2.49/3)2(1R)1.028

(2.49/3)2 (1R)1.0281+(2.49/3)2 (1R)1.028

R5

1+(2.49/3)2 (1R)1.028

Doraivelu et al.[4] (2 +R2) (1R2 )

3 (2R2 1)

Lee and Kim[5] (2 +R2) (1R2 )

3

(RRc )2

(1Rc)2

deformation does not produce anisotropy, the yield function for

porous materials must include the first invariant I1of the stress

tensor and the second invariant of the deviatoric stress tensorJ2[1,2],

AJ2 + BI21= 20= 2R (1)The yield function for porous materials represents in the prin-

cipal stress space, the surface of an ellipsoid that is symmetrical

with respect to the three principal stress axes and has a major

axis that coincides with the mean stress axis. The parameters

A, B and are to be determined from experimental work andmust be expressed as functions of the relative density R (ratio

of the volume of the base material to the total volume of porous

material). The symbol R denotes the apparent yield stress of

the porous material which depends on the yield stress 0of the

base material and the relative density R.

Considering the porous material as a continuum medium,

the strain-rate components can be obtained from the associated

porous flow rule(1)as follows[7],

ij=R

R

A

2ij+ (3 A)mij

(2)

where m is the apparent mean stress and R is the apparenteffective strain rate of the porous material given by,

R=

2

Aij

2 + 13(3 A)

2v

1/2(3)

The symbol v= R/R refers to the volumetric strain-rate(ratio of the rate of variation of the relative density to the relative

density) and the apparent effective stress Ris determined by,

R=

(AJ2 + BI21 ) (4)The plastic deformation of a porous material includes two

different forms of hardening; (i) geometric hardening due to

material densification (parameter in Eq.(1))and (ii) materialhardening due to changes in the mechanical properties of the

base material (the yield stress 0 of the base material). The

relation between the apparent strain and strain-rate and those of

the base material can be obtained by combining the plastic work

rate,

w = R RVR= 0 0V0= 0 0RVR (5)with the geometric hardening parameter resulting from mate-

rial densification,

0

=

RR 0

= t

0

0dt

= t

0

R0dt (6)

Table 1summarizes the parameters A, B and of the yield

functionsthat were proposed by Shima andOyane[3], Doraivelu

et al.[4] and Lee and Kim[5].These functions are commonly

utilized in thenumerical simulation of porousmetal forming pro-

cesses and turn into the conventional von Mises yield function

for dense materials as the relative density R approaches unity.

In these limiting conditions ( = 1), the total apparent energy

of deformation becomes equal to the elastic energy of distortion

(becauseA =3andB = 0). Itis alsoimportantto notefromTable 1

that the yield function proposed by Doraivelu et al. cannot be

applied for values of the relative density Rc 1/2 = 0.71and that the yield function due to Lee and Kim makes use of

a critical relative density Rc that needs to be determined from

experiments.

The numerical simulation of P/M metal forming processes

starts from the following weak variational form expressed

entirely in terms of the arbitrary variation in the velocity u,

=

V

R RdV

S

FiuidS= 0 (7)

where V is the control volume of the porous body limited by

surfacesSu and SF, where velocity u i and traction Fi are pre-

scribed, respectively. The solution of Eq. (7) by thefiniteelementmethod requires subdivision of the control volume Vby means

ofMelements linked through Nnodal points. In each element

the velocity distribution can be represented in the vectorial form

as,

u = Nv (8)

whereN is the matrix containing the shape functions Ni of the

element and v is the nodal velocity vector. The strain rate vector

can be obtained from,

= Bv (9)whereB is the velocity strain rate matrix. Substituting Eqs.(8)

and(9)into Eq.(7)at the elemental level and assembling the

element equations with global constraints, it follows,

Mm

V(m)

RR

Kv dV

S(m)NF dS

= 0 (10)

whereK = BTDBandDis a matrix resulting from the inversion

of the porous constitutive Eq.(2).Further details on the imple-

mentation of the finite element method for P/M metal forming

processes can be found elsewhere[6,7].


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38 L.M.M. Alves et al. / Journal of Materials Processing Technology 179 (2006) 3643

Fig. 1. Scanning electron micrographs of the copper powder utilized in the experiments: (a) undeformed copper grains (magnification of 1000); (b) compacted andsintered copper grains (magnification of 4000).

3. Experimental work

3.1. Uniaxial compression test

Thematerialemployed in theexperimentswas a purecopperpowder(99.5%)

withan apparentdensityof 2.86g/cm3 andthe followingdistributionof grainsize

(obtained by means of sieve analysis): 94.92% 45m, 5.08% 45m and0% 75m.Fig. 1presents two photomicrographys of the material showingthe original (Fig. 1(a)) and the deformed powder grains (Fig. 1(b)) after being

compacted and sintered at 825 C, during 30 min in a vacuum atmosphere of0.55 mbar.

The experimental work was performed in a 500kN computer-controlled

hydraulicpress under a veryslow speed. The cylindrical compactedand sintered

specimens utilized in the uniaxial compression tests were produced in a double

effect tool, lubricated with zinc stearate, under different applied pressures in

order to ensure a good homogeneity and to obtain test specimens with different

values of the initial relative densityR0.Fig. 2presents some of these specimens

before and after being deformed.

In order to allow load recording during the test, force was applied to the

upper compression plate through one load cell based on traditional strain-gauge

technology in a full Wheatstone bridge. Displacements were measured using a

laser-based optical displacement transducer and a PC-baseddata logging systemwasused to recordand store loads anddisplacements. In addition to this, dimen-

sions and density of the compression test specimens were also measured at each

stepof deformation. Density wasmeasuredby means of a submersion technique,

using a precisionbalance with an accuracy of0.0001 g, andthe specimens hadto be previously waterproof sealed with a varnish of known density.

The compression tests were initially performed with the objective of inves-

tigating the consistency and accuracy of the aforementioned porous plasticity

criteria (Table 1)and of identifying a criterion to subsequent implementation

into an existing finite element computer program that is currently being devel-

oped within the research team. However, the study of the main features of these

criteria combined with some discrepancies that were found during its exper-

imental assessment stimulated the authors to develop a new porous plasticity

criterion. The criterion, entitled the initial density criterionIDC, considers,

Fig. 2. Compression test specimens made from compacted and sintered copper

powder before and after deformation. The labels 15 refer to specimens with

different values of the initial relative densityR0.

as the name indicates, the influence of initial relative densityR0 of the sintered

parts and will be presented in detail in the following sections.

3.2. Mechanical characterisation of the base material

The approach followed by the authors for the mechanical characteriza-

tion of the base material diverges from the commonly utilized methodology ofemploying a wrought material with identical chemical composition because the

metallurgical conditions play a crucial role in the overall mechanical response

of the materials. As shown inFig. 3the grain size of the wrought electrolytic

copper is much bigger than that of the compacted and sintered copper powder

with the same level of purity. Actually, this is one of the main advantages that

are usually attributed to the structural components produced by means of P/M

processes, because the reduced size of the powder grains is capable of ensuring

better mechanical properties whenever the overall level of porosity is reduced.

In order to overtake the inherent difficulties of obtaining the mechanical

characteristicsof thebasematerialby meansof directmethods, it wasdecidedto

develop a new indirect method based in compression tests, without friction, that

employscylindricalspecimens withdifferent valuesof theinitialrelative density.

Thecompression tests areperformed in such a waythateach incrementof stroke

Fig. 3. Micrographs showing the structure of: (a) a wrought electrolytic copper and (b) a sintered copper powder with equal purity. The magnification is equal to

400 in both cases.


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Fig.7. Powder forging of a copper flanged component. (ac)Initial, intermediate and finalgeometryof the flanged P/M component. (d) Correlation between measured

and computed geometrical profile of the cross section of the flange at three different stages of deformation.

all the porous plasticity criteria under investigation. The investigation made

use of test specimens with different initial densities and as it can be seen the

initial density criterion (IDC) is the only model that accurately reproduces the

mechanical behaviour of the sintered material for the test specimens with low

values of the initial relative densityR0 (Fig. 6(d)). In the case of the Doraivelu

et al. porous plasticity criterion it is not even possible to get an estimate for the

experimental test case performed with R0=0.686 because the critical relative

density of this criterion is higher Rc= 0.707> 0.686.

4. Powder forging of a flanged component

This section deals with the numerical modelling of the cold

powder forging of a copper flanged component and it was

included in the paper with the objective of validating the numer-

ical predictions provided by the finite element computer pro-

gram (modified in order to include the IDC porous criterion)

with the experimental measurements obtained from laboratory-

controlled conditions. Assessment is performed in terms of

material flow, geometrical profile, relative density and forging

load.

On account of rotational symmetry it was possible to carry

out the numerical simulation of the process under axisymmet-

rical modelling conditions. The initial cross section of the sin-tered preform (with an initial relative density R0= 0.79) was

discretized by means of a mesh consisting of 584 four-node

axisymmetric quadrilateral elements.Larger elements were used

to fill the body and the initial free regions of the preform whereas

the surfaces of interest (where contact is expected to occur) were

meshed with smaller elements in order to obtain more accurate

results on the filling behaviour. The contour of the dies were

approximated by means of contact-friction linear elements and

the friction factor was found to be equal to m = 0.2 in accordance

to a calibrationprocedurebased on the conventionalring-test [8].

Fig. 7 shows the geometry of the compacted and sintered pre-

forms that were used for producing the powder forged flanged

components together with deformed parts corresponding to the

intermediate and final stages of deformation. Geometry and rel-

ative density was also measured at these stages of deformation

and latter compared with the numerical estimates derived from

Fig.8. Powder forging of a copper flanged component. (a) Computedand exper-

imental evolution of the load vs. displacement; (b) computed and experimental

evolution of the average relative density vs. displacement.


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42 L.M.M. Alves et al. / Journal of Materials Processing Technology 179 (2006) 3643

finite elements. As seen fromFig. 7(d) the agreement between

the theoretical and experimental profiles is a very good.

Further assessment between computed and experimental

results can be found in Fig. 8, where the estimates of the

forging load and average relative density are plotted against

the correspondent experimental values. In case of the forging

load the agreement is excellent. Two different stages can be

identified; an initial stage where the evolution of the load is

Fig. 9. Computed and experimental distribution of the relative density. (ab) Computed distribution of the relative density at an intermediate stage of deformation

and at the final stage of deformation. (c) Micrographies showing the porosity in selected regions of the cross section of a component at the final stage of deformation.

Labels 115 refer to positions indicated in (b).


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smooth and similar to what is generally obtained for head-

ing operations with flat punches (although some radial flow

into the bottom of the preform begins at the early stages of

deformation), and a final stage where the forging load increases

sharply. The later starts when the die cavity commences to be

filled.

In what concerns the average relative density there is also

a good agreement between theory and experimentation but the

number of measurements and the average (integrator) charac-

teristics of the parameter are not adequate to conclude about the

overall quality of the finite element distribution of the relative

densityRinside the workpiece.

The above-mentioned limitations stimulated the authors to

perform a set of micrographies in selected regions of the cross-

section of the final flanged component and to compare theoreti-

calpredictionswith experimental observations. A closer analysis

ofFig. 9 enables to conclude that there is a good agreement

between the theoretical and experimental distribution of density

inside the workpiece and proves that the initial density criterion

IDC can be successfully applied in the numerical modelling ofP/M metal forming processes.

5. Conclusions

The porous plasticity criteria of Shima and Oyane, Doraivelu

et al. and Lee and Kim present difficulties in modelling the

mechanical behaviour of sintered parts with low values of the

initial relative density. This is due to the influence of the initial

relative density in the geometric hardening of the base material

and in the plastic Poisson coefficient.

The initial density criterion (IDC) that was proposed by

the authors takes this influence into account and proved tobe able to provide better estimates of the apparent field vari-

ables during the uniaxial compression of sintered copper powder

specimens.

A new indirect method for determining the stressstrain

behaviour of the base material was also proposed as an alter-

native to the commonly utilised approximate procedures based

on the utilization of wrought materials with similar chemical

compositions.

The implementation of the initial density criterion (IDC)

in a finite element computer program that is currently being

developed by the authors proved to be successful in the numer-

ical modelling of a powder forged component. Assessment is

provided by comparing finite element predictions of the mate-

rial flow, geometrical profile, relative density and forging load

against experimental measurements obtained from laboratory-

controlled conditions.

References

[1] H.A. Kuhn, C.L. Downey, Deformation characteristics and plasticity the-

ory of sintered powder materials, Int. J. Powder Metall. 7 (1971) 15

25.

[2] R.J. Green,A plasticity theoryfor porous solids, Int. J. Mech. Sci. 14 (1972)

215224.[3] S. Shima, M. Oyane, Plasticity theory for porous metals, Int. J. Mech. Sci

18 (1976) 285291.

[4] S.M. Doraivelu, H.L. Gegel, J.S. Gunasekera, J.C. Malas, J.T. Morgan, J.F.

Thomas, A newyield functionfor compressible P/M materials., Int. J. Mech.

Sci. 26 (9/10) (1984) 527535.

[5] D.N. Lee, H.S. Kim, Plasticity yield behaviour of porous metals, Powder

Metall. 35 (1992) 275279.

[6] M.L. Alves, J.M.C. Rodrigues, P.A.F. Martins, Simulation of three-

dimensional bulk forming processes by finite element flow formulation,

Model. Simul. Mater. Sci. Eng. 11 (2003) 803821.

[7] P.A.F. Martins, M.J.M. Barata Marques, A general three-dimensional finite

element approach for porous and dense metal-forming processes, Proc. Inst.

Mech. Eng. 205 (1991) 257263.

[8] L.M.M. Alves, Deformacao plastica na massa de sinterizados metalicos:

modelacaonumericaeanaliseexperimental, Tese de Doutoramentoem Enga

Mecanica, IST, 2004.

[9] G.M. Zhdanovich,Theory of Compacting of Metal Powders(Teorize presso-

vaniya metzllichaskikli poroshkov), 1969, pp. 1262.

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A New Yield Function for Porous Materials