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SQUEEZE FILM OF CONFORMAL, LAYERED, COMPLIANT AND POROUS

CONTACTS

Conference Paper · January 2004

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SQUEEZE FILM OF CONFORMAL, LAYERED, COMPLIANT AND POROUS CONTACTS

Mircea D. PASCOVICI,. Christian RUSSU, Traian CICONE∗

1. INTRODUCTION

For humans, as for other vertebrates, the ends of bone surfaces are covered by articular cartilage, elastic and porous. The cartilage modulus of elasticity is several orders of magnitude less than that of bones. Mabuchi and Sasada [1] give values for articular cartilage modulus of elasticity between 0.05÷0.5 GPa.

For high performance prosthesis achievement, several authors have analyzed [1-3], theoretically or/and experimentally, the artificial joint replacements with compliant layers homologous with the articular cartilage functions. In 1966, Castelli et al. [4] proposed Winkler model for the analysis of thrust hydrostatic bearings with compliant layers. Later, Pascovici [5] used this model for the compliant squeeze film lubrication.

The porosity of the compliant cartilage induces also another mode of lubrication named by McCutchen “weeping” [6]. This load carrying mechanism alike with squeeze, but also different, could be studied correctly only if the variation of permeability with porosity variation, under the local effect of the load [7,8], is considered.

In a recently published paper [9], the authors have analyzed the squeeze effect for a layered contact in two distinct cases: (a) impermeable and compliant layer for a spherical contact, respectively (b) porous and deformable layer with variable permeability.

In the first case (a) the squeeze film acts between a rigid and a compliant surface. This case represents an Elasto-Hydrodynamic (EHD) mechanism. In the second case (b) the porous layer is squeezed and the lubricant imbibed in this layer is extracted. This mechanism was named Ex-Poro-Hydrodinamic (XPHD) [10].

In this paper the previous study [9] is continued by superposing both elastic and porous effects, separately analyzed previously. Hence the squeeze effect in a conformal contact consisting of a rigid body in normal motion against an elastic and porous layer is analyzed. The two surfaces are separated by a continuos liquid film which is expelled out, concomitantly, through the film and the porous layer, elastically deformed by the generated pressure in the film.

2. THE MODEL

The geometry of the contact is schematically shown in Fig. 1. A plane, rigid disc in normal (squeeze) motion against a plane, porous and deformable layer is considered. The porous layer is supported by a flat, rigid and impermeable surface. The gap between the two surfaces is full of lubricant as well as the porous layer. Therefore, a continuos thin film layer is present between the mating surfaces during the entire process of approaching.

The following assumptions are made:

1. Axisymmetry prevails;

2. Constant pressure across the film and the porous layer thicknesses;

3. The lubricant is a Newtonian liquid in isothermal and laminar flow;

∗ Polytechnic Univesity of Bucharest, Faculty of Mechanical Engineering, Dept. of Machine Elements and Tribology, Spl. Independentei 313, Bucharest 060042, Tel. (xx40)-21-4029.411, Tel & Fax. (xx40)-21-4029.581, E-mail:mircea@omtr.pub.ro

MTM 2004 Squeeze film of conformal, layered, compliant and porous contacts

Fig. 1. Geometry of the contact.

4. The flow in radial direction inside the porous layer is governed by Darcy law [8];

5. Kozeny-Carman law [7,8] is used to define permeability, φ, as a function of porosity,ε :

2

3

2

3 )1()1( σ

σεεφ −

=−

=DD (1)

where D=d2/16k is a complex constant of the porous layer structure, d is a characteristic dimension of the porous structure and k a quasi-constant value (k=5÷10, for ε=0.2…0.8);

6. Winkler model is considered for elastic behavior of the deformable (and porous) layer: Kphe = (2)

where K=h0 (1-ν2)/E is the elastic parameter. 7. The volume of the solid structure of the porous medium is invariable during the

dislocation process in normal direction. That is, the reduction of the total volume of the porous layer is due to the reduction of the volume of pores:

00p hσh =σ or 00p hεh )1()1( −=−ε (3)

where subscript 0 denotes the initial condition (before the squeeze motion).

In the actual model the fluid dislocated by the squeeze motion is partly expelled radially between the disc and the stationary surface (a typical squeeze process) and the rest is forced to flow radially through the porous matrix (see Figure 1). Consequently, according to the mass conservation law, we have the rate of flow of the dislocated fluid equal with two flow components:

- the hydrodynamic radial flow between the two surfaces; - the Darcy rate of flow inside the porous layer. Mathematically, this gives:

−

−=

rph

rphhurVr pa

dd

12dd

1222

32

µφ

µππ (4)

where φ is the variable permeability (function of the local deformation) of the poro-elastic layer, and ua the slipping velocity of the fluid at the porous layer surface, i.e. the radial velocity in the porous matrix.

According to Fig. 1, we have the following geometrical equations: peg0gp hhhhhhhh ++=+=+= (5)

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MTM 2004 Squeeze film of conformal, layered, compliant and porous contacts

According to the flow continuity, at any radius, r, on the upper boundary of the porous layer, the pressure in the fluid film and in the porous matrix is the same. Hence, we obtain the slipping velocity:

rpua d

dµφ

−= (6)

Combining Eqs. (1)÷(6), after some algebra, yields the differential equation of the compliant layer deformation:

( ) ( ) ( )

−++

−= XXHhXXH

XXDh

KhrrV 0

0

00

0 dd6d6 332

3

σσ

µ (7)

where the following dimensionless parameters have been introduced:

00

p

hKp

hh

X −== 1 0hhH = (8)

Equation (7) can be analytically solved assuming that the compliant layer is deformed only on the projected surface of the disc, i.e. there is no deformation outside the radius R. This gives the boundary condition: 0p hh = ( 1=X ) at Rr = (9)

After integration we obtain the implicit formulation of the compliant layer deformation, X:

( )( ) ( ) ( )

( ) ( ) ( )( ) ( )[ ]44

2

443

223

22

2

21

41

4ln

1312

31

3613 XHHh

XHXH

XHXH

XHDr

hKVR 0

000

00

00

−−−−

−−−+−

−−+−−−=−

σσσ

σσ

σµ (10)

where Rrr = is the dimensionless radius.

Now, remembering the Winkler model formulation -see Eq. (8)- we remark that Eq. (10) gives also the pressure distribution in the film.

3. PARTICULAR CASES

The proposed model has a great degree of generality, including elastic and porous effects to typical squeeze film lubrication. For this reason, the process was named ex-poro-elasto-hydrodynamic (XPEHD) lubrication.

By adequate simplifications, Eq. (10) leads to well known particular forms of equations for pressure distribution in various lubrication conditions, as follows.

• Poro-elasto-hydrodynamic (PEHD) lubrication corresponding to squeeze film in condition of porous and deformable layer, but with constant porosity, φ=const. (the porosity does not vary with the deformation). This is the case of an ideal porous layer used as approximation, by McCutchen [6]

• Ex-poro-hydrodynamic (XPHD) lubrication (also known as “lubrication by dislocation” [10,11]) which is the case of an imbibed, porous and compliant layer squeezed by the contact with the disc (there is no fluid film between the disc and the compliant layer, i.e. hg= h=0). In this case, the elasticity of the compliant layer contributes to the changes in porosity only, its load carrying capacity being neglected.

• If we neglect the effect of porosity, that is to consider zero permeability in the compliant layer (φ→0, that is D→0) in Eq. (10), we obtain the well known equation for EHD squeeze lubrication [5]:

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MTM 2004 Squeeze film of conformal, layered, compliant and porous contacts

( )[ ]gg hhrRVKK

p −+−= 4 422 1121 µ (11)

• If the elastic effect is neglected (K→0), that gives constant permeability, φ=const., (the porous matrix is rigid and, consequently, it is not affected by the fluid pressure) then the pressure distribution for poro-hydrodynamic (PHD) lubrication is obtained:

( )( ) 3

22

2613

g0g hhhrVRp++

−−=

φµ (12)

• A second level of simplification can be obtained either neglecting the elasticity (K→0) in Eq. (11) or by neglecting the porosity(φ→0) in Eq. (12). From both equations yields the well-known parabolic pressure distribution for parallel circular plate approaching a plane in parallel position -classical squeeze film lubrication problem [12]:

( )3

22 13

ghrVRp −

−=µ (13)

For the first two particular cases, the load capacity can be obtained only by numerical integration.

4. RESULTS AND DISCUSSION

A typical squeeze lubrication case has been selected for the numerical application. A R=50mm radius disc is pressed against a porous and compliant layer of thickness h0=2mm. A continuos oil film of viscosity µ=0.02Pa⋅s is present between the mating surfaces. The compliant layer is made of elastomeric material having the following physical properties:

- modulus of elasticity, E=1MPa; - Poisson coefficient, ν=0.5; - initial compacticity, σ0=0.6; - complex constant for the porous matrix, D=10-10 m2

. First, the pressure distribution was analyzed. The implicit formulation of pressure

distribution in ex-poro-elasto-hydrodinamic lubrication –Eq. (10)- can be easily solved numerically, for a given normal velocity, V=const., using, for example, Newton-Raphson method. The corresponding load carrying capacity is obtained straightforwardly, integrating pressure distribution (e.g. using trapezoid method).

0.0

0.5

1.0

1.5

2.0

0 0.2 0.4 0.6 0.8 1

Dimensionless Radius, r

Nor

mal

ized

Pre

ssur

e, p

/pm HD & PHDXPEHD

EHD at h g =0 (at contact)

EHD

K =1.5x10-9 m3/ND =10-10 m2

σ 0 =0.6

µ =0.02 Paxsec

V= 2 mm/sh 0 =2 mmH =1.05

PEHDφ =1.778x10-11m2

Fig. 2. Pressure distribution

Typical results, in terms of normalized pressure distribution (ratio between local pressure and the mean pressure on the contact, 2/ RFpm π= ), are shown in Fig. 2. As can be seen, there

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MTM 2004 Squeeze film of conformal, layered, compliant and porous contacts

are two extreme cases: pure hydrodynamic (HD) squeeze respectively, EHD squeeze with zero film thickness (hg=0) –when the two surfaces are in contact. In the first case (HD), the normalized pressure distribution does not depend on the normal velocity of the upper disc. All the other lubrication cases including the most general case (XPEHD) have the normalized pressure between these two limit cases. For the numerical application plotted in Fig. 2 (XPEHD, PEHD and EHD), the film thickness was hg=100µm (H=1.05) and the normal velocity V=2mm/sec.

It is worth to be noted that the theoretical possible case of PHD squeeze lubrication gives the same normalized pressure distribution as pure HD lubrication. However, the rate of flow, expelled out, in PHD lubrication is greater than in HD squeeze film. One can remark that the elastic effect of the compliant layer seems to be more important for squeeze lubrication, than the effect of the porosity.

Of greater interest is the film thickness variation during the squeeze motion, under constant load. It can be obtained from Eq. (10), where the instantaneous normal speed is:

0dd h

tHV −= (14)

Thus, Eq. (10) becomes a nonhomogenous, nonlinear, differential equation which can be solved using standard predictor-corrector Euler method.

The results, for the squeeze with a normal load, W=100N, starting at the initial position defined by H=1.1 (hg0=0.2mm) are presented in Fig. 3. One can remark the great difference between the two extreme cases: pure HD lubrication, respectively, XPEHD lubrication. Hence, the elasticity and the porosity are very important for squeeze lubrication.

1.00

1.02

1.04

1.06

1.08

1.10

0 1 2 3 4 5

Time, t , [sec]

Dim

ensi

onle

ss F

ilm T

hick

ness

, H

HD

EHDPHD

XPEHD

W = 100 NR = 50mmh 0 = 2 mmh g0 = 0.2 mm

K = 1.5x10-9 m3/ND = 10-10 m2

σ 0 = 0.6

µ = 0.02 Paxsec

Fig. 3. Film thickness variation in time

5. CONCLUSIONS

An original, complex lubrication model including the effects of elasticity and porosity has been developed. The results obtained, partly analytically and partly numerically, are of great generality, which can lead, by appropriate simplifying assumptions, to classical, simple lubrication problems.

Unfortunately, a complete validation of the model, based on experiments, is difficult due to the lack of consistent data regarding the porosity, permeability and elasticity of typical materials.

The present study, focused on the squeeze effects both in fluid film and inside the porous layer can be considered as a complementary study of the previously published studies focused on the squeeze effects after the complete squeeze of the fluid film (after the contact of the mating surfaces). Consequently, it seems necessary for the future, to complete the study of complex squeeze effects by coupling the two separate cases.

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MTM 2004 Squeeze film of conformal, layered, compliant and porous contacts

ACKNOWLEDGEMENT

Financial support for the work described in this paper was provided by National Council for Academic Scientific Research (CNCSIS) under Grant No. 463/2003.

NOMENCLATURE

D – complex constant of the porous matrix; d – typical dimension of the porous matrix; E – Young’s modulus of compliant layer; H – thickness ratio, h/h0; h – film or deformed layer thickness; k – constant in Kozeny-Carman equation; K – Winkler constant, h0(1-ν2)/E; p – pressure; R – radius of apparent area, or the disc; r – radial coordinate; r – dimensionless radial coordinate, r/R; t – time; ua– slipping velocity;

V – normal speed; W – load; X – dimensionless thickness of the

deformed layer, hp/h0; ε – porosity; φ – permeability D(1-σ)3/σ2; µ – viscosity; σ – compacticity, εσ −= 1

Subscript

0 – initial e – elastic p – porous

BIBLIOGRAPHY

[1.] Mabuchi K, Sasada T. - Numerical analysis of elastohydrodynamic squeeze film lubrication of total hip protheses. Wear, 140 (1990), p. 1-16.

[2.] Gaman I.D, Higginson G.R, Norman R.: Fluid entrapment by a soft surface layer. Wear, 28(1974), p. 345-352.

[3.] McClure G., Jin ZM, Fisher J, Tighe BJ.: Determination of lubricating film thickness for permeable hydrogel and non-permeable polyurethane layers bounded to a rigid substrate with particular reference to cushion form hip joint replacements. Proc.Instn.Mech.Engrs., Vol. 210(1996), Part H, p. 89-93.

[4.] Castelli V, Rightmire GK, Fuller DD.: On the analytical and experimental investigation of a hydrostatic axisymmetric compliant – Surface Thrust Bearing, Report No. 5, Lubrication Research Laboratory, Columbia University, New York, 1966, p. 24-25.

[5.] Pascovici M.D.: The study of lubricant squeeze film in elastohydrodynamic conditions. S.C.M.A, 1982, Tom 41(1), p. 139-145 (in Romanian).

[6.] McCutchen, C.W.: The Frictional Properties of Animal Joints, Wear, 5, 1962,pp. 1-17. [7.] Ghaddar, C.K.: On the Permeability of Unidirectional Fibrous Media; A Parallel

Computational Approach, Phys. Fluids 7(11), 1995, pp.2563-2586. [8.] Scheidegger, A.E.: The Physics of Flow through Porous Media”, University of Toronto

Press, 1974, p.141. [9.] Pascovici, M.D., Cicone, T.: Squeeze-film of unconformal, compliant and layered

contacts, Tribology International, Vol. 36(2003), No.11, p. 791-799 [10.] Pascovici, M.D.: Lubrication by Dislocation: A New Mechanism for Load Carrying

Capacity, 2nd World Tribology Congress, Viena, 2001, p. 418. [11.] Pascovici, M.D.: Squeeze-Film of Unconformal, Compliant and Layered Contacts,

NORTRIB 2002, The 10th Nordic Symp. on Tribology, Stockholm June 9-12, 2002.

[12.] Khonsari MM, Booser ER.: APPLIED TRIBOLOGY BEARING DESIGN AND LUBRICATION, John Wiley&Sons, Inc., New York, 2001, p. 268-271.

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