The effect of surface texturing in soft elasto-hydrodynamic lubrication

ARTICLE IN PRESS Tribology International 42 (2009) 284– 292

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The effect of surface texturing in soft elasto-hydrodynamic lubrication A. Shinkarenko, Y. Kligerman , I. Etsion Faculty of Mechanical Engineering, Technion—Israel Institute of Technology, Haifa 32000, Israel

a r t i c l e in f o

a b s t r a c t

Article history: Received 15 October 2007 Received in revised form 2 June 2008 Accepted 17 June 2008 Available online 29 July 2008

A theoretical model is developed to study the potential use of laser surface texturing (LST) in the form of spherical micro-dimples for soft elasto-hydrodynamic lubrication (SEHL). The model consists of mutual smooth elastomeric and LST rigid surfaces moving relatively to each other in the presence of viscous lubricant. The pressure distribution in the fluid film and the elastic deformations of the elastomer are obtained from a simultaneous solution of the Reynolds equation and the equation of elasticity for the elastomer. An extensive parametric investigation is performed to identify the main important parameters of the problem, which are the aspect ratio and area density of the dimples. The parametric analysis provides optimum parameters of the surface texturing and shows that LST effectively increases load capacity and reduces friction in SEHL. & 2008 Elsevier Ltd. All rights reserved.

Keywords: Elasto-hydrodynamic lubrication Surface texturing Numerical analysis

1. Introduction Soft elasto-hydrodynamic lubrication (SEHL) is used in several mechanical components such as rotary lip seals [1,2], elastomeric reciprocating seals [3–6] and metering size press [7], etc. Until the last decade, most studies of SEHL were of experimental nature only because of the complexity of the problem. The first significant analytical models appeared in the literature only about 10 years ago, thanks to the rapid development of computing capabilities. Shi and Salant [1] developed a deterministic mixed lubrication model, simulating the interface between a moving perfectly smooth rigid surface and a stationary rough perfectly elastic lip surface. The asperity pattern on the lip surface (roughness) was modeled by a two-dimensional sinusoid. The model was applied to rotary lip seal. The authors divided the domain between the surfaces into three regions: the lubrication region, the cavitation region and the contact region. In a following paper, Shi and Salant [2] expanded their model to a more realistic condition of a quasirandom sealing surface. The results showed that interasperity cavitation appears at extremely low shaft speeds, and the flattening of asperities is significant throughout the entire range of operation. Furthermore, the shear deformation of asperities plays an important role in preventing leakage. Nikas [3,4] developed a numerical model to study the sealing performance of rectangular elastomeric seals for reciprocating piston rods used in linear hydraulic actuators. The model was used to calculate the contact pressures and film thickness as well

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E-mail address: [email protected] (Y. Kligerman). 0301-679X/$ - see front matter & 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.triboint.2008.06.008

as the leakage rates and friction for a dynamic or static contact between a seal and a reciprocating piston rod. The models [3,4] aiming at the minimization of both the leakage and the wear of the seal assumed that the elastomer obeys the Hooke’s law. This assumption was examined in a following paper by Nikas and Sayles [5], where a non-linear constitutive law (Mooney–Rivlin model) instead of the classic Hookean model of linear elasticity was used. It was shown that the differences between the results of the linear and non-linear models are usually between 0% and 5%, but can be as high as 15% depending on the operating conditions. Salant et al. [6] developed a model for a reciprocating elastomeric seal with a non-rectangular cross-section geometry. The model consists of an iterative computational procedure to analyze the coupled fluid mechanics, deformation, and contact mechanics problems. The model takes into account mixed lubrication and surface roughness effects and shows the major role that these subjects play in determining whether or not the seal leaks. Low friction and wear are main factors in a well-designed seal. Laser surface texturing (LST) is known as an effective means for achieving these goals in mechanical components such as mechanical seals [8–10], piston rings [11–13], and bearings [14,15]. The models presented in Refs. [8,11,14] assume full hydrodynamic lubrication in the gap between the relevant mating surfaces, when one of them contains thousands of micro-dimples produced by laser texturing. The classical Reynolds equation was used to evaluate the hydrodynamic pressure in the fluid film, and cavitation effects were accounted for. The viscous friction was evaluated from shear stresses in the fluid film. It was found from extensive parametric analyses that the most important LST parameter is the dimple aspect ratio (the ratio of dimple depth over dimple diameter). These models were verified

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Nomenclature A c C E E ff Ff h H h0 H0 hp l1 L1 l2 L2 p P pa

dimensionless area of half of one dimple column, R1L1 initial clearance dimensionless initial clearance, c/rp Young’s modulus of elastomer SEHL stiffness index, Erp/6mU friction force per unit area dimensionless friction force, ffrp/6mU local film thickness dimensionless local film thickness, h/rp local dimple depth dimensionless local dimple depth, h0/rp maximum dimple depth elastomer length in the x1 direction dimensionless elastomer length in the X1 direction, l1/ rp elastomer length in the x2 direction dimensionless elastomer length in the X2 direction, l2/ rp local pressure dimensionless local pressure, (ppa)rp/6mU ambient pressure

experimentally [8,12,15], showing good correlation between the analytical and experimental results. An added benefit of LST was observed during experiments under starved lubrication condition [12], where the dimples served as ‘‘micro-reservoirs’’ for lubricant. An extensive review of LST state of art can be found in Ref. [16]. The laser texturing usually produces spherical dimples, but other shapes may also be produced (see e.g. Refs. [17–19]). In most cases, the dimple geometry has little influence on the tribological performance, provided the aspect ratio is preserved. Surface texturing in the form of asperities rather than dimples for SEHL application is described in Ref. [20]. Various asperity geometries were analyzed to study their effect on lip seal performance. In this case too, the optimum performance depends on the aspect ratio of the asperity and is very little affected by its actual geometry. In the present study, we investigate the effect of LST on the tribological performance (load-carrying capacity, viscous friction force, and leakage) in SEHL. The main goal is to find the LST parameters, which for known elastomer physical properties, lubricant viscosity, and given operating conditions (temperature, sliding velocity, and pressure gradient) provide the best tribological performance.

2. The model A schematic of the model is presented in Fig. 1. The model consists of an elastomeric body with a smooth untextured surface and an infinitely long rigid LST counterpart (which can be a piston rod, for example, in a hydraulic cylinder). The two bodies are sliding relatively to each other with a constant velocity, U, in the presence of a viscous lubricant. The viscous flow between the elastomer and the LST counterpart produces a hydrodynamic pressure (see e.g. Ref. [8]) tending to separate the mating surfaces and to deform the elastomer. Fig. 1(a) shows the elastomer prior to any deformation, whereas Fig. 1(b) presents the case of the deformed elastomer. The elastomer is infinitely long in the x3 direction and has finite length, l1, and thickness, l2 (see Fig. 1). The elastomer is attached to a rigid foundation and is separated from the rigid LST counterpart by a thin layer of viscous lubricant having a local thickness h(x1, x3) (see Fig. 1(b)). It should be emphasized that the distance between

r1 R1 rp Sp uni ui U w W xi Xi

d D

e eij m n snij sij ,j

285

half-side of imaginary square cell dimensionless half-side of imaginary square cell, r1/rp base dimple radius dimple area density displacement vector components dimensionless displacement vector components, uni =r p sliding velocity average pressure dimensionless load-carrying capacity, wrp/6mU Cartesian coordinates dimensionless Cartesian coordinates, xi/rp variation of the local film thickness dimensionless variation of the local film thickness, d/ rp dimple aspect ratio, hp/2rp deformation tensor components dynamic viscosity Poisson’s ratio stress tensor components dimensionless stress tensor components, snij r p =6mU. partial derivative of a variable with respect to xj

the rigid LST textured counterpart and elastomer foundation is maintained constant. It should also be noted that the moving textured surface may cause transient effects due to periodic entering and leaving of dimples into and out of the fluid film zone, which may lead to transient elastomer deformation and possible squeeze action. However, because of the relatively large number of dimples present in the fluid film zone at any given time, usually between 10 and 15 dimples depending on the elastomer length and area density of dimples, this end-effect can be neglected, as was shown in Ref. [11]. Such squeeze action was also neglected in the case of a lip seal and a rotating shaft textured with controlled microasperities [20]. In Fig. 1, the clearance c is the initial clearance between the still stationary surfaces. The value of c may be either positive or negative, depending on the loading condition prior to any relative sliding. In both cases, during sliding the surfaces of the elastomer and the counterpart are separated by a clearance h(x1, x3). Fig. 2 presents a geometrical model of the laser-textured surface. The dimples are uniformly distributed on the counterpart surface (see Fig. 2(a)) with an area density Sp. The curvature of the rigid LST rod and the elastomer can be neglected due to the small clearance (in the x2 radial direction) compared to the radius of the rod. In this case, the problem domain is infinitely long in the x3 (circumferential) direction and has finite length in the x1 (axial) direction from x1 ¼ 0 to x1 ¼ l1 (see Fig. 1(a)). Each dimple is located in the center of an imaginary square cell of sides 2r1  2r1 (see Fig. 2(b)) and has the shape of a spherical segment with a base radius rp and maximum depth hp. The dimple area density and base radius are related to the square cell side by rffiffiffiffiffi rp p r1 ¼ . (1) 2 Sp Modeling of the SEHL requires a simultaneous solution of two interrelated problems: the hydrodynamic lubrication between the sliding surfaces and the deformation of the elastomer. 2.1. The hydrodynamic lubrication model The local pressure distribution between the elastomer and the LST counterpart is obtained from a solution of the

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x2

Rigid foundation

l1 l2 Undeformed elastomer hp

pa

pa

x3

c

x1

Rigid LST counterpart

x2

Rigid foundation

Deformed elastomer

h0(x1, x3)

x3

δ(x1, x3)

c

h(x1, x3)

x1

Rigid LST counterpart

U Fig. 1. Schematic illustration of (a) the model with initially undeformed elastomer and (b) the model with final deflections of the elastomer.

Rigid LST Counterpart

Elastomer

2r1 x2

x1 x3 x1

2r1

U x3

x1

2r1

x3 Fig. 2. A geometrical model of a laser-textured surface: (a) schematic of the model; (b) zoom-in on a group of dimples; and (c) a single column of dimples in the x1 direction.

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287

time-independent (squeeze effect is neglected) Reynolds equation for viscous incompressible Newtonian fluid in laminar flow:     q q qh 3 qp 3 qp h h . (2) þ ¼ 6mU qx1 qx3 qx1 qx1 qx3

The pressure and shear stress distributions obtained from the solution of the Reynolds equation are used as boundary conditions for the equations of elastomer deformation.

In Eq. (2), x1 and x3 are the Cartesian coordinates, p is the local pressure, m is the dynamic viscosity of the fluid, U is the relative velocity between the mating surfaces, and h is the local film thickness given as

2.2. The elastomer deformation model

hðx1 ; x3 Þ ¼ c þ h0 ðx1 ; x3 Þ þ dðx1 ; x3 Þ,

1. The elastomer is isotropic and homogeneous. 2. The deformations of the elastomer are small (geometric linearity). 3. The elastomer obeys Hooke’s law (physical linearity between stress and strain). 4. The elastomer is almost incompressible (Poisson’s ratio 0.49ono0.5). 5. Visco-elastic effects are negligible.

(3)

where d(x1, x3) is the change in the local film thickness due to deformation of the elastomer, c is the initial clearance (see Fig. 1(b)), and h0(x1, x3) is the local depth of the dimple (see Fig. 1(b)). Because of periodicity of the surface texturing in the x3 direction (see Fig. 2(b)), and the symmetry of each axial dimple column of width 2r1 about its longitudinal axis x3 ¼ 0 (see Fig. 2(c)), it is sufficient to consider the pressure distribution within just one half of a single dimple column (confined between x3 ¼ 0 and x3 ¼ r1). Thus, the boundary conditions for the Reynolds equation are as follows:

 symmetry of the pressure distribution about x3 ¼ 0, periodi-

This model is based on the following assumptions:

The time-independent equilibrium equations (when body forces are neglected), the linear strain–displacement relation, and the linear constitutive law (Hooke’s law) are given by

snij;j ¼ 0,

city, and smoothness of the pressure distribution in the x3 direction:

ij ¼ 12ðuni;j þ unj;i Þ,

qp qp ðx ; x ¼ 0Þ ¼ ðx ; x ¼ r 1 Þ ¼ 0; qx3 1 3 qx3 1 3

snij ¼

(4a)

 ambient pressure at the inlet and outlet of the slider (see Fig. 1(a)): pðx1 ¼ 0; x3 Þ ¼ pðx1 ¼ l1 ; x3 Þ ¼ pa .

(4b)

The boundary conditions Eqs. (4a) and (4b) should be complemented by a cavitation pressure in potential cavitation regions. In the present paper, the Reynolds cavitation condition was applied. The dimensionless formats of the Reynolds Eq. (2), the local film thickness Eq. (3), and the boundary conditions Eqs. (4a) and (4b) are:     q qP q qP qH H3 H3 , (5) þ ¼ qX 1 qX 3 qX 1 qX 1 qX 3 HðX 1 ; X 3 Þ ¼ C þ H0 ðX 1 ; X 3 Þ þ DðX 1 ; X 3 Þ, 

qP qP r1 ðX ; X ¼ 0Þ ¼ X1; X3 ¼ qX 3 1 3 qX 3 rp



¼ 0,

PðX 1 ¼ 0; X 3 Þ ¼ PðX 1 ¼ L1 ; X 3 Þ ¼ 0,

xi ; rp l1 L1 ¼ ; rp

hp h c ; ¼ ; C¼ , rp rp 2r p ðp  pa Þr p h0 d , H0 ¼ ; D¼ ; P¼ rp rp 6mU

(10)

! h  n  i En ij þ kk dij ; 1þn 1  2n

(

dij ¼

1;

if i ¼ j;

0;

if iaj:

(11)

In Eqs. (9)–(11), an index notation of the general form ui,j denotes partial derivative of ui with respect to xj and a repeated index denotes summation (e.g. ekk ¼ e11+e22+e33). Also snij;j is the divergence of the stress tensor snij , and eij represents the deformation tensor components. The components of the local displacement vector within the elastomer are uni , E is the Young’s modulus, and n is the Poisson’s ratio of the elastomer material. Using Eq. (10) in Eq. (11) and then substituting in Eq. (9) leads to the equation of equilibrium in terms of displacements:    n  1 n ðui;j þ unj;i Þ þ unk;k dij ¼ 0. (12) 2 1  2n j

(6)

The boundary conditions for the elastomer are as follows:

(7a)

 The local shear and normal stresses on the elastomer interface

(7b)

with the fluid film (see Fig. 1) are equal to the viscous shear stresses due to the Couette flow and hydrodynamic pressures, respectively. Hence (see Ref. [8]),

where the various dimensionless parameters are defined as Xi ¼

(9)

sn12 ðx1 ; x2 ¼ 0; x3 Þ ¼

H¼

(8)

and e is the aspect ratio of a dimple. The Reynolds Eq. (5) with its boundary conditions Eqs. (7a) and (7b) and the Reynolds cavitation condition was solved by a finite difference method [8]. This method allows reducing the Reynolds differential equation to a set of linear algebraic equations for the nodal values of the pressure. These were solved using successive over relaxation iterative method [21]. Shear stresses are also developed in the fluid film due to both Poiseuille and Couette flow components. However, because of the very thin film thickness, the effect of the Poiseuille flow on the shear stresses is negligible compared to that of the Couette flow.

mU a, hðx1 ; x3 Þ

sn22 ðx1 ; x2 ¼ 0; x3 Þ ¼ ðpðx1 ; x3 Þ  pa Þ,



(13a)

(13b)

where a ¼ 0 accounts for the vanishing stresses in cavitating zones and a ¼ 1 belongs to the non-cavitating zones. The upper surface of the elastomer is attached to a rigid foundation (see Fig. 1), and therefore all the displacement vector components vanish on this surface: uni ðx1 ; x2 ¼ l2 ; x3 Þ ¼ 0.

(13c)

 The symmetry and periodicity of the shear and normal stresses at the elastomer/fluid interface in the x3 direction result in similar symmetrical and periodical deformations within the entire elastomer body. Thus, the boundary conditions for the

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elastic deflections are given by 8 n u3 ðx1 ; x2 ; x3 ¼ 0Þ ¼ un3 ðx1 ; x2 ; x3 ¼ r 1 Þ ¼ 0; > > > > n n > > < qu1 ðx ; x ; x ¼ 0Þ ¼ qu1 ðx ; x ; x ¼ r Þ ¼ 0; 1 2 3 1 qx3 qx3 1 2 3 > > n n > qu2 qu2 > > > : qx ðx1 ; x2 ; x3 ¼ 0Þ ¼ qx ðx1 ; x2 ; x3 ¼ r 1 Þ ¼ 0: 3 3

(13d)

The dimensionless form of Eq. (12) and its corresponding boundary conditions Eqs. (13a)–(13d) are given in Eqs. (14) and (15a)–(15d), respectively:    n  1 ðui;j þ uj;i Þ þ uk;k dij ¼ 0, (14) 2 1  2n j

s12 ðX 1 ; X 2 ¼ 0; X 3 Þ ¼

1 a, 6HðX 1 ; X 3 Þ

(15b)

ui ðX 1 ; X 2 ¼ L2 ; X 3 Þ ¼ 0,

(15c)

8 u3 ðX 1 ; X 2 ; X 3 ¼ 0Þ ¼ u3 ðX 1 ; X 2 ; X 3 ¼ R1 Þ ¼ 0; > > > > > < qu1 ðX ; X ; X ¼ 0Þ ¼ qu1 ðX ; X ; X ¼ R Þ ¼ 0; 1 qX 3 1 2 3 qX 3 1 2 3 > > > qu2 qu2 > > ðX ; X ; X ¼ 0Þ ¼ ðX ; X ; X ¼ R1 Þ ¼ 0; : qx3 1 2 3 qx3 1 2 3

(15d)

where the various length and stress dimensions are normalized in the following forms: uni ; rp

L2 ¼

l2 ; rp

R1 ¼

r1 ; rp

(19)

(15a)

s22 ðX 1 ; X 2 ¼ 0; X 3 Þ ¼ PðX 1 ; X 3 Þ,

ui ¼

coordinate, whereas u3 is an odd function. Therefore, the full Fourier expansions are given as   8 N P kpX 3 > 0 k > u ðX ; X ; X Þ ¼ 0:5u ðX ; X Þ þ u ðX ; X Þ cos ; > 1 1 2 3 1 2 1 2 1 1 > > R1 k¼1 > > >   > N > P kpX 3 > > > u2 ðX 1 ; X 2 ; X 3 Þ ¼ 0:5u02 ðX 1 ; X 2 Þ þ uk2 ðX 1 ; X 2 Þ cos ; > > R1 > k¼1 > >   > < N P kpX 3 uk3 ðX 1 ; X 2 Þ sin ; u3 ðX 1 ; X 2 ; X 3 Þ ¼ R1 > k¼1 > >   > > N > P kpX 3 > 0 k > ; X Þ ¼ 0:5P ðX Þ þ P ðX Þ cos ; PðX > 1 3 1 1 > > R1 k¼1 > > >   > N > P 1 kpX 3 > 0 > > F k ðX 1 Þ cos : : 6HðX ; X Þa ¼ FðX 1 ; X 3 Þ ¼ 0:5F ðX 1 Þ þ R1 1 3 k¼1

sij ¼

snij rp . 6mU

Substitution of Eq. (19) into Eqs. (14), (17a), and (17b), leads to the k independent sets of partial differential equations: uk1 ðX 1 ; X 2 Þ, uk2 ðX 1 ; X 2 Þ, and uk3 ðX 1 ; X 2 Þ. These new differential equations were solved numerically by finite element method. An appropriate nine-node rectangular finite element was developed to obtain uk1 ðX 1 ; X 2 Þ, uk2 ðX 1 ; X 2 Þ and uk3 ðX 1 ; X 2 Þ for each harmonic k of the shear load, Fk(X1), and normal pressures, Pk(X1), on the elastomer/ fluid interface. Once the deflections u1(X1, X2, X3), u2(X1, X2, X3), and u3(X1, X2, X3) are found, the dimensionless change of the local film thickness D(B) ¼ u2(A) at any point B (see Fig. 3) due to the deformation u(A) that moved the point A to the position A0 can be obtained from

DðX 1 þ u1 ðX 1 ; X 2 ¼ 0; X 3 Þ; X 3 þ u3 ðX 1 ; X 2 ¼ 0; X 3 ÞÞ (16)

¼ u2 ðX 1 ; X 2 ¼ 0; X 3 Þ.

(20)

Eq. (14) for the elastomer deflection requires proper deflection boundary conditions. For this purpose, the boundary conditions Eqs. (15a) and (15b) expressed in terms of stresses should be rewritten in terms of elastic deflections. Using Eqs. (10) and (11) for the Hooke’s law and strain–displacement relation, Eqs. (15a) and (15b) become

Using the solution of Eq. (20) in Eq. (6) leads to the modified fluid film profile for the next iteration of the Reynolds equation solution as described in the following text.

ðu1;2 þ u2;1 Þ X

Fig. 4 presents a flowchart of the numerical procedure for the simultaneous iterative solution of the hydrodynamic lubrication problem (see Eqs. (5)–(7)), and the deformation of the elastomer (see Eqs. (14), (15a), and (15b)). An initial guess of the local clearance is used to solve the Reynolds equation (Eq. (5)), resulting in a first approximated pressure and shear stress distributions. These distributions are used as the boundary conditions (see Eqs. (15a)–(15d)) for the set of differential equations (Eq. (14)) giving the elastomer deflections, which in turn change the local clearance or, in other words, the film profile. The modified film profile is returned to the Reynolds equation, etc. The iterative process is repeated until a desired convergence is achieved (see Fig. 4). The hydrodynamic load-carrying capacity, W, is obtained by integrating the pressure over the area A: ZZ rp 1 W¼ w, (21) PðX 1 ; X 3 Þ dX 1 dX 3 ¼ A A 6mU



u2;2 þ

2 ¼0

n 1  2n

¼

2 1 ð1 þ nÞ a, E 6HðX 1 ; X 3 Þ

 ðu1;1 þ u2;2 þ u3;3 Þ

X 2 ¼0

(17a)

1 ¼  ð1 þ nÞPðX 1 ; X 3 Þ, E (17b)

where the dimensionless parameter E, which is subsequently referred to as the SEHL stiffness index, is given by E¼

En r p . 6mU

(18)

This SEHL stiffness index gives the ratio of the elastomer stiffness (elastomer Young’s modulus, E) over the unit hydrodynamic opening force (proportional to mU/rp). Since E and mU/rp affect the elastomer deformations in opposite directions, namely increasing E or reducing mU/rp reduces these deformations, it is clear from Eq. (18) that the SEHL stiffness index E is a measure of the level of elastomer deformations. The higher is E, the smaller are the elastomer deformations. As will be shown later, the SEHL stiffness index E is the most important dimensionless parameter in the present model. Since all the variables are periodic in the x3 direction, they may be expanded into Fourier series with coordinate-depending coefficients. This reduces the level of complexity of the solution from a single 3D problem to a set of k independent 2D problems [22]. It follows from the symmetry and periodicity that the shear and normal stresses as well as the u1 and u2 components of the displacement vector are even functions with respect to the X3

2.3. The iterative procedure

where A ¼ R1L1 is the dimensionless area of one half of a dimple column and w is the dimensional average pressure over the elastomer/fluid interface. The dimensionless friction force, Ff, due to the viscous shear stresses in Couette flow is obtained by (see Ref. [13]): ZZ 1 dX 1 dX 3 rp ¼ f , (22) Ff ¼ A A 6H 6mU f where ff is the dimensional average shear-distributed load over the elastomer/fluid interface.

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289

X2 Deformed elastomer

(X1 - X3) plane of the initial un-deformed elastomer surface

A′ u(A)

Δ(B) B

A(X1, 0, X3)

u2(A) A

u(A) – displacement vector of the elastomer at point A

A′(X1+u1, u2, X3+u3)

Δ(B) = u2(A)

B(X1+u1, 0, X3+u3) Fig. 3. Variation of the fluid film profile due to the elastomer deformations.

3. Results and discussion

START

The load-carrying capacity (Eq. (21)) and the viscous friction force (Eq. (22)) are a good measure of the effectiveness of the surface texture. Hence, a parametric analysis was performed to study the effect of the following dimensionless parameters on the load-carrying capacity and friction force:

Parameters Input: E, ν, C, ε, Sp, L1, L2

      

Initial guess of local film thickness, H, and pressure, P.

Evaluation of the hydrodynamic pressure, P (Reynolds Eq. (5)-(7))

The main goal of the parametric analysis is to identify the optimum LST parameters e and Sp (see the first two in the above list) for a certain elastomer application represented by the rest of the parameters in the above list (geometry, physical properties, and operating conditions). Two criteria of optimization are implied: maximum loadcarrying capacity and minimum friction in the fluid film. The reference case for the present parametric analysis was selected as follows:

Evaluation of the surface deformation, Δ (Elasticity Eqs. (14), (15a) -(15d) and 20))

    

Modification of the film profile (local film thickness), H (Eq. (6))

Unacceptable Error calculation for film thickness and pressure

Acceptable Evaluation of load carrying capacity (Eq. (21)) and friction force (Eq. (22))

aspect ratio, e ¼ hp/2rp; dimple area density, Sp; length of elastomer, L1 ¼ l1/rp; thickness of the elastomer, L2 ¼ l2/rp; Poisson’s ratio, n; initial clearance, C ¼ c/rp; SEHL stiffness index, E ¼ Erp/6mU.

dimple diameter, 2rp ¼ 100 mm; dimple depth, hp ¼ 10 mm; dimple area density, Sp ¼ 0.3; Young’s modulus, E ¼ 107 Pa; Poisson’s ratio, n ¼ 0.499.

A value of the Poisson’s ratio slightly less than 0.5 was used in the present solution to avoid the singularity in Eq. (11) for incompressible materials. Another option of dealing with this problem is by a different formulation of Eq. (11), e.g. the mixed formulation in Ref. [22], but this is outside the scope of the present work. Typical dimensions of the elastomer were assumed as

 elastomer length, l1 ¼ 1.5  103 m;  elastomer thickness, l2 ¼ 1 103 m. The reference lubricant viscosity was assumed m ¼ 0.05 Pa s, and typical operating conditions were assumed to be

END Fig. 4. Flowchart of the numerical procedure.

 initial clearance, c ¼ 1.5  106 m;  sliding velocity, U ¼ 2 m/s.

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The dimensionless parameters corresponding to the reference case along with their wide range of variation used for the numerical ‘‘experiment’’ are shown in Table 1. The parametric analysis is performed by changing the value of each parameter in Table 1 in its turn while maintaining a fixed value for all the others. Fig. 5 shows the profiles of the undeformed and deformed elastomer interface along with the textured profile of the rigid counterpart and its sliding direction indicated by the arrow. The results are presented for the reference case (see Table 1). The distorted shape of the dimples is due to the different scales used for the vertical and horizontal axes. The elastomer deformations consist of three components:

10 Dimensionless Local Pressure, P

290

8 6 4 2 0 0

The dimensionless pressure distribution (for the above reference case of Fig. 5) in the fluid film along the center line of the dimple column, X3 ¼ 0 (see Fig. 2(c)) is presented in Fig. 6. The pressure achieves local maxima at the local converging clearance region of each dimple and reduces to zero at the local diverging clearance region of each dimple (cavitation zones). These local maxima decrease in their value in the direction of the counter body sliding because of the diverging average clearance.

Minimum value

Reference value

Maximum value

Dimple area density, Sp Aspect ratio, e ¼ hp/2rp Initial clearance, C ¼ c/rp Length of elastomer, L1 ¼ l1/rp Thickness of elastomer, L2 ¼ l2/rp SEHL stiffness index, E ¼ Erp/ (6mU)

0.05 0.02 0.16 10 0 415

0.3 0.1 0.03 30 20 830

0.5 0.2 0.06 60 60 125000

Dimensionless Vertical Coordinate, X2

Parameter

Undeformed Elastomer Profile

0.05

Δ(X1,X3)

0.00 -0.05 -0.10 -0.15 Rigid Counterpart Profile

U

-0.20 0

5

10 15 20 25 Dimensionless Longitude Coordinate, X1

Fig. 5. Profiles of the mating surfaces.

30

5.5

0.53

5.0

0.37

0.30

4.5 0.24

4.0 3.5

0.18

3.0

Sp = 0.10

2.5 2.0 0.05

0.1 Aspect Ratio, ε

0.15

0.2

Fig. 7. Dimensionless load-carrying capacity, W, vs. the aspect ratio, e, for different values of the dimple area density, Sp.

Deformed Elastomer Profile

0.10

10 15 20 25 Dimensionless Longitude Coordinate, X1

L1 = 29, C = 0.03, L2 = 20, E = 833

6.0

0 Table 1 The dimensionless parameters and their range of variation

5

Fig. 6. Dimensionless pressure distribution, P, along the center line of a single column of dimples.

Dimensionless Load Capacity, W

1. global compression of the elastomer due to the pressures in the fluid film; 2. tilt of the elastomer due to viscous shear stresses, which result in a bending moment, and hence clearance tendency to diverge in the sliding direction (the shear stresses are in the direction of the counterpart velocity); 3. the local deflections of the elastomer having a ‘‘wave’’-like appearance (see Fig. 5) with the same frequency as that of the dimples resulting from local change of the hydrodynamic pressure within the dimples.

30

As shown in Refs. [7–14], the aspect ratio, e, plays a major role in affecting the tribological performance of LST components. The effect of this parameter on the dimensionless load-carrying capacity, W, is presented in Fig. 7 for several values of the dimple area density, Sp. The aspect ratio has an optimum value that maximizes the load-carrying capacity. This is because the aspect ratio defines the average slope of the converging film thickness zone of each dimple, which is known to be the main factor affecting load capacity in slider bearings. The optimum aspect ratio changes slightly with the area density but a ratio of e ¼ 0.060.08 can be used as an optimum for the entire range of Sp. Fig. 7 also shows that the load W increases with the dimple area density, Sp, up to about Sp ¼ 0.30. Above this value, W is little affected by Sp, and in fact for very high Sp values, W will start decreasing. The existence of optimum area density can be understood by examining the two extreme cases of Sp ¼ 0 and Sp ¼ 1 (in both cases, no dimples exist). In approaching both cases, the dimples effect decreases drastically. This phenomenon was observed for the entire range of E with decreasing saturating value of Sp as E increases but with negligible effect on the saturated value of the load W. From many numerical simulations, it was found that varying the other parameters of the problem (C, L1, and L2) over their entire range (see Table 1) does not affect the saturating value of the dimple area density. As a result, the value Sp ¼ 0.3 was selected as a preferred value for the entire range of E regardless of all other parameters.

ARTICLE IN PRESS A. Shinkarenko et al. / Tribology International 42 (2009) 284–292

The effect of e on the load-carrying capacity for different values of the SEHL stiffness index, E, is presented in Fig. 8. It can be seen that the aspect ratio, e, has a very weak optimum but this optimum value depends on E. As can be seen from Fig. 8, a single value of (e)opt ¼ 0.06 can cover a wide range of the SEHL stiffness index from 420 to 25 000. For higher E values, a somewhat smaller (e)opt value is found. A higher E results in smaller deflections of the elastomer and thus produces smaller clearance, which according to the findings in Ref. [14] requires a smaller optimum aspect ratio. The smaller clearance due to higher E also increases the hydrodynamic pressure and thus the load capacity as is shown in Fig. 8. The present results obtained for high values of the SEHL stiffness index are in good agreement with those of the nominally parallel thrust bearing [14]. This is because infinitely high value of E leads to negligible deformations of the elastomer and the SEHL model approaches the case of two rigid surfaces. Fig. 9 presents some typical results of the load-carrying capacity vs. the aspect ratio for different combinations of L1, L2, and C. From many numerical simulations, it was found that the parameters L1, L2, and C hardly affect the optimum aspect ratio. On the other hand, L1, L2, and C strongly affect the load-carrying capacity. Increasing the elastomer length, L1, increases its bending stiffness, while decreasing the elastomer thickness, L2, decreases the bending moment on the elastomer resulting from the fluid film shear stresses. In both cases, the tilt of the elastomer (see Fig. 5) decreases and results in smaller divergence of the clearance

L1 = 29, L2 = 20, Sp = 0.3

L1 = 29, L2 = 20, C = 0.03, Sp = 0.3 125000 75000

25000

10

E = 420

3400

1650

833

6800

Dimensionless Load Capacity, W

Dimensionless Load Capacity, W

in the sliding direction. This in turn increases the efficiency of the surface texturing (which is the highest for parallel surfaces) and therefore increases the load-carrying capacity. Reducing the initial clearance, C (mainly to negative values), increases the load-carrying capacity, W, as would be expected. The effect of C on W is displayed in Fig. 10, which shows the dimensionless load-carrying capacity vs. the dimensionless initial clearance for different values of the SEHL index and their corresponding optimum values of the LST aspect ratio. A principal assumption of the present model is the ever existence of full hydrodynamic lubrication (no contact between the surfaces is permitted). It was found from numerous simulations that for certain negative values of the initial clearance, C, and depending on E, the local gap may disappear at least at one point. In these cases, the present model is not valid any more. These limiting negative C values are shown as the very left points of each curve in Fig. 10. They depend on the SEHL stiffness index because higher values of E correspond to smaller deflections of the elastomer surface and as a result smaller real positive clearance (film thickness). Further increase of E or decrease of C increases the number of contact points between the mating surfaces. A transition from full hydrodynamic lubrication to mixed lubrication can be obtained by plotting the maximum load W vs. E corresponding to the limiting initial clearance C values shown in Fig. 10. The results are presented in Fig. 11 showing the range of

1000

100

0

0.05

0.1

0.15

E = 83300, ε = 0.03

100

1

E = 8330, ε = 0.04

10

E = 833, ε = 0.06

1 -0.16

0.2

Aspect Ratio, ε Fig. 8. Dimensionless load-carrying capacity, W, vs. the aspect ratio, e, for different values of the SEHL stiffness index, E.

7.0

-0.12

-0.04 0 0.04 -0.08 Dimensionless Initial Clearance, C

0.08

Fig. 10. Dimensionless load-carrying capacity, W, vs. the dimensionless initial clearance, C, for different values of the SEHL stiffness index, E.

Sp = 0.3, E = 833

6.5

350

L1 = 45, L2 = 20, C = 0.03

L1 = 39, L2 = 20, Sp = 0.3

6.0

Maximum Dimensionless Load Carrying Capacity, W

Dimensionless Load Capacity, W

291

5.5 5.0 4.5 L1 = 29, L2 = 20, C = 0.01

4.0 3.5

L1 = 29, L2 = 20, C = 0.03

3.0

L1 = 29, L2 = 50, C = 0.03

2.5

300 250 200 150 100

Mixed Lubrication

50 Hydrodynamic Lubrication

2.0 0

0.05

0.1 Aspect Ratio, 

0.15

0.2

Fig. 9. Dimensionless load-carrying capacity, W, vs. the aspect ratio, e, for different values of L1, L2, and C.

0 1.E+02

1.E+03 1.E+04 SEHL Stiffness Index, E

1.E+05

Fig. 11. Maximum dimensionless load-carrying capacity, W, vs. the SEHL stiffness index, E.

ARTICLE IN PRESS 292

A. Shinkarenko et al. / Tribology International 42 (2009) 284–292

The validity of the present model, which is based on the assumption of full hydrodynamic lubrication, was analyzed in terms of the limiting SEHL stiffness index and associated loadcarrying capacity, and the maximum load causing transition to mixed lubrication was identified.

Dimensionless Friction Force, Ff

2.70 L1 = 29, L2 = 20, Sp = 0.3, E = 833

2.50 2.30 2.10 1.90

7

1.70 Acknowledgments

1.50

5

1.30 1.10

W=3

0.90 0.70 0

0.05

0.1 Aspect Ratio, ε

0.15

0.2

Fig. 12. Dimensionless friction force, Ff, vs. the aspect ratio, e, for different values of the dimensionless load-carrying capacity, W.

validity of the present model. At any given E, there is a maximum value of W above which the present model becomes invalid. So far, the optimization of the LST parameters involved maximizing the load-carrying capacity. It is expected that such optimization will also minimize the friction force at a given load. This is because with a given load the optimum LST results in more efficient hydrodynamic pressure generation, and hence in the highest film thickness and the lowest shear stresses at the fluid–elastomer interface. This argument is indeed demonstrated in Fig. 12 that shows typical behavior of the dimensionless friction force, Ff, vs. the dimple aspect ratio for different values of the loadcarrying capacity. It is observed from Fig. 12 that with E ¼ 833 the optimum aspect ratio for minimum friction is in the range e ¼ 0.08–0.1, similar to the optimum e range for maximum loadcarrying capacity with the same E ¼ 833 as shown in Figs. 7–9. Namely, the same optimum LST parameters provide both maximum load and minimum friction. Fig. 12 also shows that higher load-carrying capacity is associated with higher friction force as would be expected from a thinner fluid film.

4. Conclusion A theoretical model was developed to analyze the potential of LST in soft elasto-hydrodynamic lubrication. It was found that texturing of the rigid counterpart generates a load-carrying capacity that can be maximized by selecting a preferred dimple area density, Sp, and an optimum dimple aspect ratio, e. The optimum parameters for maximum load also minimize the friction force. The optimum values of the LST parameters were found from an intensive parametric investigation. It was found that the dimple radius does not affect the tribological performance of SEHL. The best value of the dimple area density, Sp, is almost independent of all the other parameters of the problem and is about Sp ¼ 0.3. The optimum aspect ratio depends exclusively on the SEHL stiffness index, E. As E changes from 420 to 6  105, the optimum aspect ratio (e)opt varies from 0.1 to 0.02, respectively. Further increase of E does not affect the optimum aspect ratio, which remains 0.02.

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