865 applications of analytical model for characterizing

Applications of analytical model for characterizing thepear-shaped tribotest for tube hydroforming. Part 2G Ngaile* and C Yang

Department of Mechanical and Aerospace Engineering, North Carolina State University, Raleigh, North Carolina, USA

The manuscript was received on 16 November 2007 and was accepted after revision for publication on 31 March 2008.

DOI: 10.1243/09544054JEM1058

Abstract: Applications of the analytical model for characterizing the pear-shaped tribotestare presented. Details on the derivations of the analytical model can be found in Part 1 ofthis paper (published in Proceedings of the Institution of Mechanical Engineers, Journal ofEngineering Manufacture, 2008, Vol. 222). In this test, a tubular specimen is pressurized,forcing the material to flow towards the apex of a pear-shaped die. The height of thepear-shaped tube is a function of the magnitude of friction stress at the tube–die interface.The analytical model is used rapidly to establish the calibration curves for determinationof friction coefficient in the pear-shaped tribotest. The model is also used to optimizeboth process and die geometric variables to suit specific tribological needs. The paper pre-sents examples of how optimal pear-shape tribotest conditions pertaining to die geometry,tubular material properties, tube sizes, and pressure loading can be achieved via thismodel.

Keywords: tribotest, friction coefficient, tube hydroforming, closed-form equations

1 INTRODUCTION

The main purpose of a tribotest is to provide infor-mation on performance characteristics of a lubricantin question. In addition to ranking lubricants andproviding friction coefficient values for use in processmodelling, the test should accommodate a widerange of variables such as material type, interfacepressure level induced, and so on. Dependingon the complexity of the tribotest it is sometimesdifficult to obtain most of the tribological data bytaking experimental measurements. The advancesin numerical modelling via finite element analysis(FEA) have opened new avenues where additionaltribo data can be acquired. Double cup backwardextrusion tribotest is a typical example where bycombining the FEA and experiments friction coeffi-cient for the tested lubricant can be determined[1–3].

The pear-shaped tribotest is commonly used toevaluate lubricant performance for the expansion

zone in tube hydroforming. Figure 1 shows theschematic of the test. In this test the fluid is pres-surized forcing the material to flow towards theapex of the pear-shaped die. The height of thepear-shaped tube is a measure of the friction stressbetween the die and the tube [4, 5]. Upon comple-tion of the experiment the protrusion height (PH) isreadily measurable and thus lubricants can beranked instantly. Owing to the complexity of thegeometry, however, friction coefficient cannot easilybe determined. To obtain friction coefficient, it isnecessary to run finite element simulations withdifferent coefficient of friction values and selectthe simulation that matches with the experiment.As this approach is time consuming it is better toseek an analytical solution.

In Part 1 of this paper an analytical model tocharacterize the pear-shaped tribotest was pre-sented [6]. The analytical model is capable of pro-viding field variables such as pressure distributionat the interface between tube and die, effectivestress and strain distribution, and longitudinal andhoop stress/strain distribution. This analyticalmodel provides for a quick way of determining fric-tion coefficient and optimization of the tribo testconditions.

*Corresponding author: Department of Mechanical and

Aerospace Engineering, North Carolina State University, Cam-

pus Box 7910 , Rale igh , NC 27695 , USA. emai l :

[email protected]; [email protected]

JEM1058 � IMechE 2008 Proc. IMechE Vol. 222 Part B: J. Engineering Manufacture

865

2 OPTIMIZATION OF PEAR-SHAPED TRIBOTEST

In the development of the pear-shaped expansiontest, Ngaile et al. [5] used FEA to determine the bestgeometry. They simulated various die geometriesand came up with a die with vertex angle of 48�.Optimization of a pear-shaped test requires the studyof numerous parameters such as die angle, tube size,tube wall thickness, material variables, frictioncoefficient values, and pressure. Carrying out thisparametric study will require extensive simulations.The closed-form solutions established in this studydrastically reduce the time and effort in optimizingthis test.

The objective function in the optimization of thepear-shaped tribotest is the protrusion height differ-ence DPH obtained by imposing two friction condi-tions, as given in equation (1). The protrusionheight difference is influenced by several variablesas expressed in equation (2)

DPH ¼ PHm1 � PHm2 ð1Þ

DPH ¼ f ðp; K ; n; r; t; cÞ ð2Þwhere DPH is the change in protrusion height, mcoefficient of friction, p internal pressure, K strengthcoefficient, n strain hardening exponent, r radiusof tube, t tube wall thickness, and c pear-shapeddie vertex angle. The equations used to establishthe friction sensitivity calibration curves (PH versusp) are based on volume constancy, pressureprediction, and geometric relationship between Land PH, see Fig. 2. Details of the derivation aregiven in Part 1. For brevity some of the equationsare recapitulated below.

1. Volume constant equation

prt0 ¼Z a

0

rt0

· e� "nN� ðl�1Þp=Kbnþ1½ �f gemðu�aÞþ ðl�1Þp=Kbnþ1½ �ð Þ1=ndu

þ ðt0Þ2Kbnþ1

pm ð"Qn�"NnÞþP1i¼1

ð�2Þii!

nnþið"Qnþi�"NnþiÞ

� �

þðr�Lctgðp�aÞÞðp�aÞt0e�"Q

1nð"Qn�"NnÞþP1

i¼1

ð�1Þii!

1iþnð"Q iþn�"N iþnÞ¼ mp

Kbnþ1nt0L

8>>>>>>>>>>>><>>>>>>>>>>>>:

ð3Þ2. Pressure prediction

p¼ Kbnþ1"Qnt0e

�"Q

r�Lctgðp�aÞ ð4Þ

3. Geometrical relationship between L and PHBy trigonometric manipulation, equations (5) and (6)can be obtained

rr ¼ r�Lctgðp�aÞ ð5Þ

L¼ PH�2rcscðp�aÞ�ctgðp�aÞ ð6Þ

Combining equations (3), (4), (5), and (6) equationgroup (7) can be obtained

prt0¼R a

0 rt0e� "N

n� ðl�1Þp=Kbnþ1½ �gemðu�aÞþ ðl�1Þp=Kbnþ1½ �Þfð 1=n

du

þðt0Þ2Kbnþ1

pm ð"Qn�"NnÞþP1

i¼1

ð�2Þii!

nnþið"Qnþi�"N

nþiÞ� �


pK

m

bnþ1nt0L¼ 1

nð"Qn�"NnÞþP1

i¼1

ð�1Þii!

1iþnð"Q iþn�"N

iþnÞ

p¼ Kbnþ1"Qnt0e

�"Q

r�Lctgðp�aÞL¼ PH�2r

cscðp�aÞ�ctgðp�aÞ ð7Þ

8>>>>>>>>>>>><>>>>>>>>>>>>:From equation (7) when the geometric dimensions r, t,a, and l, material properties, K and n, and process

Fig. 1 Pear-shape tribo test

Fig. 2 Geometric parameters for pear-shape expandedtube

866 G Ngaile and C Yang

Proc. IMechE Vol. 222 Part B: J. Engineering Manufacture JEM1058 � IMechE 2008

parameters p, and m are known, then there arefour unknown variables: PH, «N, «Q, and L which canbe solved mathematically. By solving equation (7)friction sensitivity profile can be obtained. Figure 3(a)shows an example of a friction sensitivity profile fora pear die shape with a vertex angle c ¼ 48�

and 34.93mm diameter tubular specimen. Thiscurve shows protrusion height difference betweentwo friction conditions; m1 ¼ 0.05 and m2 ¼ 0.10.FEA results from DEFORM 2D software are alsoplotted. Figure 3(a) shows that the protrusionheight difference initially increases with increase inpressure, but then starts to decrease. This variationin protrusion height difference DPH with change inp implies that there is an optimal pressure associatedwith higher frictional sensitivity of the tribotest.

The occurrence of optimal pressure whichcorresponds to maximum protrusion height changeis caused by the material flow characteristic towardsthe apex. Figure 3(b) shows velocity profiles of

material flow along the die–tube contact region andnon-contact region for different pressure levels.These profiles were obtained from FEA for m ¼ 0.05.It can be observed from Fig. 3(b) that at pressurelevels of 60MPa and 80MPa hoop velocity betweenthe two regions exhibit linear increase from thebottom of a pear-shaped tube to the apex (P1 toP54). However, when the pressure level exceeds90MPa the hoop velocity from P1 to P54 is non-linear. That is, the rate of increase of the hoopvelocity in the contact region decreases with anincrease in pressure. At 130MPa and 140MPa, thematerial movement for a large portion of thedie–tube contact region approaches zero. As aresult, friction sensitivity decreases. Thus Fig. 3(a)shows that beyond 90MPa the increment ofprotrusion height is largely dominated by rapidthinning in the free expansion region.

To gain better insight of the characteristics of thepear-shaped expansion tribotest, a parametric studywas conducted. Table 1 shows the variables thatwere used in the model to study:

(a) the influence of vertex angle c on DPH;(b) the influence of strain hardening exponent on

DPH;(c) the influence of tube wall thickness on DPH;(d) the influence of tube radius on DPH.

2.1 Influence of vertex angle c on DPH

Parametric study was carried out under the followingconditions:

(a) varying Pi from 10 MPa to 110 MPa;(b) varying c from 30� to 60� at intervals of 6�;(c) friction coefficients m1 ¼ 0.05 and m2 ¼ 0.10;(d) K ¼ 1450 MPa and n ¼ 0.52;(e) r ¼ 28.5 mm and t ¼ 1.6 mm.Fig. 3(a) Friction sensitivity profile

Fig. 3(b) Velocity profiles of material points along the die for various pressure levels

Characterizing the pear-shaped tribotest for tube hydroforming. Part 2 867


Figure 4 depicts a family of curves for protrusionheight difference between m1 ¼ 0.05 and m2 ¼ 0.10for different vertex angles. Figure 4 shows that theprotrusion height difference initially increases withincrease in pressure, but then starts to decrease.This variation in protrusion height difference DPHwith change in Pi implies that there is an optimalpressure associated with higher frictional sensitivityof the tribotest. Figure 4 indicates that maximumDPH/friction sensitivity decreases with the increaseof vertex angle. However, Fig. 4 also shows that withdecrease in the vertex angle, the curve (DPH versusp) becomes steeper, which means that the selectabletest pressure band Pb becomes smaller. The pressureband concept is illustrated in Fig. 5. Care shouldtherefore be taken in selecting an optimal die vertexangle, which corresponds to the effective pressureband for the tribotests. As shown in Fig. 5, for adesired test pressure band Pb, a friction sensitivityprofile which is determined by the die vertex anglec, should be chosen such that the intersections atP1 and P2 are not far away from the DPH associatedwith Popt.

2.2 Influence of strain hardening exponent non DPH

There are various tubular materials used in THF.These materials have different strain hardeningvalues. Thus, it is important to determine how

strain hardening influences friction sensitivity of thepear-shaped test. Parametric study was carried outunder conditions:

(a) varying Pi from 10 MPa to 110 MPa;(b) friction coefficients m1 ¼ 0.05 and m2 ¼ 0.10;(c) K ¼ 1450 MPa, n ¼ 0.1, 0.2, 0.3, 0.4, 0.5, and 0.6;(d) r ¼ 28.5 mm, t ¼ 1.6 mm, and c ¼ 48�;(e) maximum wall thinning was set at 30 per cent.

Figure 6 shows that the protrusion height differenceincreases with an increase in strain hardening expo-nent. This implies that more care should be takenwhen evaluating lubricants using tubular materialswith lower strain hardening exponent.

2.3 Influence of tube wall thickness on DPH

Parametric study was carried out under conditionssimilar to those used for studying the influence of

Table 1 Variables used in the parametric study

Geometric variables c 30�, 36�, 42�, 48�, 54�, 60�r 14.25mm, 28.5mm, 57mmt 0.8mm, 1.6mm, 3.2mm

Process variables P 10MPa–110MPam 0.05, 0.1

Material variables K 1450 MPan 0.1, 0.2, 0.3, 0.4, 0.5, 0.6

Fig. 4 Influence of vertex angle on protrusion heightdifference between m1 ¼ 0.05 and m2 ¼ 0.10

P1 P2

Pb

Pro

trusio

n H

eig

ht

Chan

ge (

∆PH

)

Internal pressure ( Pi)

Popt

Fig. 5 Effective test pressure band as related to frictionsensitivity profile

Fig. 6 Influence of strain hardening on protrusion heightdifference between m1 ¼ 0.05 and m2 ¼ 0.10 forc ¼ 48 �



strain hardening. Additional parameters for thisstudy were:

(a) t ¼ 0.8 mm, 1.6 mm, and 3.2 mm;(b) r ¼ 28.5 mm;(c) K ¼ 1450 and n ¼ 0.52;(d) c ¼ 30�.

Figure 7 shows a family of curves for DPH for thedifferent wall thicknesses t ¼ 0.8 mm, 1.6 mm, and3.2 mm. The three profiles in Fig. 7 are for die vertexangle of 30�. Other die vertex angles resulted in simi-lar patterns. Figure 7 shows that the influence ofthickness on protrusion height difference is insignifi-cant. The protrusion height differences for t ¼ 0.8mm, t ¼ 1.6 mm, and t ¼ 3.2 mm using a die with ver-tex angle of 30� are 1.06 mm, 1.02 mm, and 1.01 mm,respectively.

2.4 Influence of tube radius on DPH

Parametric study was carried out under conditionssimilar to those used for studying the influence ofwall thickness. Additional parameters for this studywere r ¼ 14.25 mm, 28.5 mm, 57 mm, t ¼ 1.6 mm,K¼ 1450, n¼ 0.52, and c¼ 30�. Figure 8 shows a dras-tic increase of protrusion height difference with anincrease in tube radius. The figure shows that a dievertex angle of 30� resulted in protrusion height differ-ences of 2.14mm, 1.06mm, and 0.52mmfor tube radii57mm, 28.5mm, and 14.25mm, respectively. In otherwords, the friction sensitivity increases by about 100per cent when tube radius doubles as to be expectedfrom geometrical scaling consideration. The protru-sion height difference is associated with the changein friction coefficient from m1 ¼ 0.05 to m2 ¼ 0.10.

2.5 Interface pressure loading

In a tribotest, the performance of the lubricant isalso measured upon the maximum pressure it can

withstand before failure. Furthermore, the interfacepressure distribution at the tool can be used tostudy wearing characteristic of the die. Under thepear-shaped test two distinct pressure distributionsare observed at the tube–die interface: region AB,which is the arc section of the die exhibits lower inter-face pressure, as compared to the region BC (linearsection of the die). Figure 9 shows that for the internalfluid pressure Pi ¼ 36MPa the average pressure in thearc region was 6 MPa which is about 83 per cent lowerthan 36MPa exhibited at the linear die section. At Pi ¼62MPa, the arc section experienced interface pressureof 23MPa, which is 64 per cent lower than 62MPaexhibited at the linear section.

3 PREDICTION OF FRICTION COEFFICIENT

Equation group (7) can be rearranged to obtainequation (8) where internal pressure p is normalizedwith K

prt0 ¼ R a

0 rt0e� "N

n�ðl�1ÞpKbnþ1

n o�emðu�aÞþ ðl�1Þp

Kbnþ1

h i�1=n

du

þðt0Þ2bnþ1

p=kð Þm ð"Qn�"NnÞþP1i¼1

ð�2Þii!

nnþið"Qnþi�"NnþiÞ

� �


pK

m

bnþ1nt0L¼ 1

nð"nQ�"NnÞþP1i¼1

ð�1Þii!

1iþnð"Q iþn�"N iþnÞ

p=K ¼ bnþ1"Qnt0e

�"Q

r�Lctgðp�aÞL¼ PH�2r

cscðp�aÞ�ctgðp�aÞð8Þ

8>>>>>>>>>>>>>><>>>>>>>>>>>>>>:

In equation group (8), if dimensions r, t, and a andmaterial properties K and n are known, there remainonly six unknown variables: PH, L, p, m, «N, and «Q.Because the internal pressure PI and protrusionheight PH can be obtained from experiment, theunknown variables in equation (8) are L, m, «N, and «Q.Thus, equation group (8) can be solved to obtain m.

Fig. 7 Influence of tube wall thickness on protrusionheight difference Fig. 8 Influence of tube radius on protrusion height differ-

ence, c ¼ 30�



However, m cannot be determined explicitly; acalibration chart is therefore used to estimate m.The calibration chart is composed of a family of curves(PH versus p/K) for different friction coefficient values.Figure 10 shows an example of a friction calibrationchart for a pear-shaped expansion test for tubematerial n ¼ 0.52, r ¼ 17.46mm, t ¼ 1.24 mm, and dievertex angle c ¼ 48�. By superimposing on a graphthe ratio p/K and protrusion height PH obtainedfrom pear-shaped tribotest experiments, the frictioncoefficient for a tested lubricant can be determined.Note that the friction calibration chart is dependenton material properties n, die geometry, and tube sizeand independent of K.

4 PEAR-SHAPE EXPANSION TESTS

Pear-shaped expansion tests were carried out withdifferent lubrication conditions and the calibration

charts were used to estimate friction coefficients.The experimental results were also used to validatethe analytical model.

4.1 Test set-up and experimental procedures

The experimental set-up for the pear-shaped tribot-est is shown in Figs 11 and 12. The test set-upconsists of the upper die, lower die, and two axial

Fig. 9 Tube–die interface pressure variations

Fig. 10 Friction calibration chart

Fig. 11 Hydroforming test rig

Fig. 12 Hydroforming tooling



cylinders. The upper die is connected to 150 tonhydraulic press through a 150 ton load cell. The lowerdie seats on a table. The dies are made of A2 steeland hardened to 62 HRC. Table 2 shows the testmatrix. Two sizes of stainless steel tubing (SS 304)and two lubricants were used in the tribotests. Thesmall tubing had an outer diameter of K ¼ 34.93mm and wall thickness of t ¼ 1.24 mm, while the lar-ger tubing had outer diameter of K ¼ 57.15 mm andwall thickness of t ¼ 2 mm. In order to use the samefluid pressure loading, the tube sizes were selectedsuch that the radius to thickness ratio, l, is approxi-mately equal to 14.

Two lubricants, teflon sheet, and polymer-basedlubricants were used in the test (Table 2). Thepolymeric lubricant used is based on steryl methacry-late/methyl methacrylate. Details on the compositionof the polymeric lubricant can be found in reference[7]. The specimens were cut to 205mm long andwere cleaned with acetone before applying the lubri-cant. The polymeric lubricant was applied by a brushwhile with Teflon lubricant, Teflon sheet of 0.12mmthick was wrapped around the specimen. During thetest fluid pressure was ramped linearly from 0MPato 67MPa in 30 s.

4.2 Test results and discussion

Figure 13 shows sets of specimens for smaller andlarger tubing after testing. The protrusion heightsweremeasured for all samples and the average protru-sion heights were plotted in Fig. 14 for small and largetubing. Figure 14 shows that Teflon lubricant resultedin large protrusion height of 44.6 mm followed bypolymeric lubricant with a protrusion height of44mm. The specimens formed without lubricantexhibited an average height of 43 mm. Similarly, withlarger tubing Teflon lubricant exhibited the largestprotrusion height of 70.7 mm followed by polymericlubricant with a protrusion height of 69.6mm. Thenon-lubricated specimens resulted in a protrusionheight of 68.7mm. It should be noted that plane straincondition was not fully established with large tubing(K ¼ 57.15 mm) used in the experiment because thebulge width was too short. This was attributable tothe limitations of the current tooling. However, as

seen in Fig. 13 fully plane strain condition wasachieved for small tubing.

The difference in the protrusion height betweenspecimens hydroformed with Teflon and No-lub con-ditions for smaller tubing is 1.6 mm. The large tubingresulted in a protrusion height difference of 2 mm.The larger protrusion height is as discussed earlierowing to geometric scaling. It agrees with the resultsfrom the analytical model discussed in section 2.4where it was shown that friction sensitivity increaseswith increase in tube size.

Calibration curves were used to determine the fric-tion coefficient of the tested lubricant friction.Figure 15 shows friction calibration curves withsuperimposed protrusion height values obtainedfrom experiment for the three lubrication conditions.From the calibration curves the friction coefficientsfor Teflon and polymeric lubricants were estimatedto be m ¼ 0.08 and m ¼ 0.14 respectively. The calibra-tion curves show that friction coefficients of underm ¼ 0.30 were exhibited when no lubricant was used.

Table 2 Test matrix

Tube material LubricantsNumber ofspecimens

SS 304 K ¼ 34.93mmt ¼ 1.24mm

Teflon sheet 2Polymer 2No-lub 2

K ¼ 57.15mmt ¼ 2mm

Teflon sheet 2Polymer 2No-lub 2

Teflon Polymer No-Lub

Tube size

Tube size

Φ=57.15mm

Φ=34.93mm

Fig. 13 Pear-shaped hydroformed tubes



5 CONCLUSIONS

Potential applications of the analytical model forcharacterizing the pear-shaped tribotest were pre-sented. Besides facilitating the determination of fric-tion coefficient for tested lubricants, the analyticalmodel has proved to be a useful tool in rapid optimi-zation of pear-shaped tribotest conditions.

The analytical model also provides a clear pictureof the local pressure loading distribution on thetube–die interface. This tribo-information can beused in studying wear characteristics of hydrofor-ming dies as a function of lubricant type, pressurelevels, and pressure gradients at the tube–die inter-faces.

Hydroforming experiments carried out with twolubricants have demonstrated how friction coeffi-cient values for tested lubricants can be estimatedusing the established calibration curves. The experi-ments also demonstrated that the friction sensitivity

of the pear-shaped tribotest increases with increasein the diameter of tubular specimens. This is animportant consideration, particularly when evaluat-ing lubricants using tubular materials with a lowerstrain hardening exponent.

ACKNOWLEDGEMENT

The authors would like to acknowledge the NationalScience Foundation, through which this work wasfunded under Project No. DMI-0448885.

REFERENCES

1 Buschhausen, A., Weinmann, K., Lee, J. Y., and Altan, T.Evaluation of lubrication and friction in cold forgingusing a double backward-extrusion process. J. Mater.Process. Technol., 1992, 33, 95–108.

2 Schrader, T., Shirgaokar, M., and Altan, T. A critical eva-luation of the double cup extrusion test for selection ofcold forging lubricants. J. Mater. Process. Technol., 2007,189, 36–44.

3 Nakamura, T., Bay, N., and Zhang, Z. L. FEM simulationof a friction testing based on combined forward conicalcan-backward straight can extrusion. ASME, J. Tribol.,1998, 120, 716–723.

4 Ngaile, G., Jaeger, S., and Altan, T. Lubrication in tubehydroforming (THF). Part II. Performance evaluation oflubricants using LDH test and pear-shaped tube expan-sion test. J. Mater. Process. Technol., 2004, 146, 116–123.

5 Ngaile, G., Jaeger, S., and Altan, T. Lubrication in tubehydroforming (THF). Part I: Lubrication mechanismsand development of model tests to evaluate lubricantsand die coatings in the transition and expansion zones.J. Mater. Process. Technol., 2004, 146, 108–115.

6 Ngaile, G. and Yang, C. Analytical model for characteriz-ing the pear-shaped tribotest for tube hydroforming. Part1. Proc. IMechE, Part B: J. Engineering Manufacture, 2008,222(B7), 849–864.

7 Ngaile, G., Cochran, J., and Stark, D. Formulation ofpolymer-based lubricant for metal forming. Proc.IMechE, Part B: J. Engineering Manufacture, 2007, 221,559–569.

Fig. 14 Influence of lubricants on protrusion height for tube size K ¼ 34.93 mm, and K ¼ 57.15 mm

Fig. 15 Estimated friction coefficient for Teflon and poly-mer lubricants



APPENDIX

Notation

K strength coefficientL linear section lengthn strain hardening exponentPH protrusion height of deformed tubeDPH protrusion height changePi/p internal pressurer outer radius of tube

rr corner radius of tubet deformed tube thicknesst0 initial tube thickness

a centre angle of arc section«N hoop strain at the intersection between arc

and linear sections«Q hoop strain at the free expansion zonel ratio of outer radius to tube thicknessm friction coefficientf outer diameter of tubec vertex pear-shaped die angle



865 applications of analytical model for characterizing

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