Tribology International 43 (2010) 1775–1785

Contents lists available at ScienceDirect

Tribology International

0301-67

doi:10.1

� Corr

E-m

URL

journal homepage: www.elsevier.com/locate/triboint

A transient 2D-finite-element approach for the simulation of mixedlubrication effects of reciprocating hydraulic rod seals

T. Schmidt a,�, M. Andre a, G. Poll b

a Corporate Sector Research and Advance Engineering (CR/APP2), Robert Bosch GmbH, Postfach 1131, D-71301 Waiblingen, Germanyb Institute of Machine Elements, Engineering Design and Tribology, University of Hanover, Postfach 6009, D-30060 Hanover, Germany

a r t i c l e i n f o

Article history:

Received 31 August 2009

Received in revised form

27 November 2009

Accepted 30 November 2009Available online 5 December 2009

Keywords:

Mixed lubrication

Transient Reynolds equation

FEA

Hydraulic seals

9X/$ - see front matter & 2009 Elsevier Ltd. A

016/j.triboint.2009.11.012

esponding author. Tel.: +49 7151 503 2359; f

ail address: timo.schmidt2@de.bosch.com (T.

: http://research.bosch.com/ (T. Schmidt).

a b s t r a c t

This paper presents a method for the computation of soft elasto-hydrodynamic lubrication (EHL) based

on the strong coupling of a non-linear finite-element model with the transient Reynolds equation for

thin fluid films. This approach allows the usage of arbitrary non-linear elastic or inelastic material

models for the finite deformations. The transient Reynolds equation is simultaneously solved within a

finite-element computation. In order to account for the effect of surface roughness in the sealing

contact, flow factors are incorporated into the transient Reynolds equation. The method is currently

restricted to planar or axisymmetric geometries.

& 2009 Elsevier Ltd. All rights reserved.

1. Introduction

Today dynamic seals used in hydraulic applications undergoextreme pressures, loads and temperatures during operation.Although the production costs of seals are low, any failure of thesecomponents can damage the hydraulic system heavily and oftenleads to excessive leakage of fluid to the environment. Therefore, areliable design and dimensioning of the sealing system ismandatory to assure safe usage during all operating states ofhydraulic components. That is mostly done based on engineeringexperience, expert knowledge and experimental and theoreticalstudies, that date at least back to the 1960s [1,2].

In the meantime several numerical approaches for thesimulation of hydrodynamic effects in soft EHL contacts havebeen developed [3]. Hereby, the main challenge in sealed contactsis the modelling of the interaction between the fluid filmmechanics and the soft deformation behaviour of the sealstructure. Thereby numerical simulations help to understandthe fundamental behaviour of seals and to study the backgroundof observed effects during testing and operation.

In literature, approaches for inverse and direct methods can befound. The inverse methods [4] assume full lubrication conditionsat all times and predict the fluid film thickness based on thecontact pressure distribution in the lubricated contact zone. Thecomputation is based on the inverse solution of the steady-stateReynolds equation. The direct methods consider the deformations

ll rights reserved.

ax: +49 7151 503 2664.

Schmidt).

due to the hydrodynamic pressure in the fluid film, additionally.However, most of the recent models assume steady-state condi-tions in the hydrodynamic fluid film and in the seal structure [5–9].Furthermore, the model of Stupkiewicz [8] is restricted to full filmlubrication and treats the contacting surfaces as perfectly smooth.

Since, experiments have shown that mixed lubrication occursover a wide range of conditions [10], Thatte and Salant [11]propose a transient model of an elastomeric hydraulic seal, wherethe fluid mechanics analysis consists of a finite volume solution ofthe transient Reynolds equation using a mass-conserving algo-rithm. Flow factors according to Patir and Cheng [12,13] areincorporated to account for the effect of the surface roughness ofthe seal. The contact mechanics analysis utilizes the Greenwood–Williamson model [14]. The deformation mechanics is solvedwithin an on-line finite-element computation at each time step.

The goal of this work is the finite-element implementation of amethod for the computation of mixed lubrication effects in softEHL contacts. The approach is based on the publication of Ongunet al. [7], in which the implementation of a hydrodynamicinterface element in ABAQUS for the simulation of mixedlubrication contacts is described. The strong coupling of structuraldeformations and hydrodynamic effects in the fluid film isprovided by the direct solution of the fluid mechanics equationswithin the user-subroutine UEL of the finite-element softwarepackage ABAQUS [15] and, thus, computational time is kept aslow as possible. In this paper the authors present threeenhancements of this approach:

1.

The squeeze terms are added to the Reynolds equation used in[7] in order to account for time-dependent effects in the fluid

Nomenclature

AðyÞ WLF shift function,Ai nodal area, mm2

A specific area, mmb;w width, mmC10 Neo–Hooke material parameter, N=mm2

C1;C2 WLF parameters,c0 integration constant for hydrostatic solution, mm2=sD1 compressibility parameter, mm2=Nd diameter, mmFfl

i ; Tfli total radial and axial nodal fluid forces, N

~Ffl

i fluid residual load vector, Nf fli ; t

fli radial and axial nodal fluid forces, N

G1;G0 initial and instantaneous shear modulus, N=mm2

gRðtÞ; kRðtÞ dimensionless relaxation and bulk modulus,gP

i Prony coefficients,h fluid film thickness, mmh0, y0 integration constants for hydrodynamic solution, mmhs

mix, psmix mixed lubrication parameters, mm, N=mm2

I1 first deviatoric strain invariant,J; Jel; Jth total, elastic and thermal volume ratio,K1;K0 initial and instantaneous bulk modulus, N=mm2

~Kfl

ij fluid stiffness matrix, N/mmPfl

i , Tfli total nodal fluid pressure and shear stress, N

ph, pfl, ps hydrostatic, fluid and solid contact pressure, N=mm2

Q volume flow rate, mm3=s

q volume flow rate per unit circumferential length,mm2=s

Rq RMS surface roughness, mmR numerical residuals for c0 and y0, N=mm2

r rod radius, mmT fl integrated fluid friction force acting on seal surface, Nt time, su; v displacements, mm_u; _v velocities, mm/s~ufl

j fluid displacement vector, mmV leakage volume, mm3

x; y Cartesian coordinates, mma, b numerical weighting factors,g shear strain, slip,gs

crit critical frictional shear strain,ei principal nominal strains,evol volumetric strain,Z dynamic viscosity, Ns=mm2

y temperature, Kli principal stretches,ms solid friction coefficient,x reduced time, ssi principal Cauchy stresses, N=mm2

t shear stress, N=mm2

tscrit critical frictional shear stress, N=mm2

tGi relaxation times, sFp, Fsh pressure and shear flow factor,c strain energy potential, N=mm2

T. Schmidt et al. / Tribology International 43 (2010) 1775–17851776

film. Still the transient Reynolds equation is simultaneouslysolved within the finite-element computation.

2.

Furthermore, flow factors are incorporated into the transientReynolds equation to consider the surface roughness in thecontact zone.

3.

The approach allows the usage of any in ABAQUS availablematerial models for the structural part. The definition ofviscoelastic material behaviour for the seal provides a morerealistic prediction of lubrication conditions in the sealingcontact region during transient simulations.

2. Basic equations

2.1. Solid mechanics

2.1.1. Hyperelasticity

Rubber materials are used in many different types of sealingapplications. The quasi-static, long-term mechanical behaviour ofthese materials can sufficiently be described by hyperelasticmaterials models, defined in terms of a strain energy potential c.The Neo–Hookean form for compressible material behaviour isgiven by

c¼ C10ðI1�3Þþ1

D1ðJel�1Þ2; ð1Þ

where C10 and D1 are temperature-dependent material constants,which are derived from the initial shear modulus G1 ¼ 2C10 andthe initial bulk modulus K1 ¼ 2=D1. Based on the assumption ofisotropic behavior throughout the deformation history, the strainenergy potential is formulated as a function of the first deviatoricstrain invariant

I1 ¼ l2

1þl2

2þl2

3; ð2Þ

where li ¼ J�1=3li are the deviatoric stretches, li ¼ 1þei are theprincipal stretches and ei are the principal nominal strains(i¼ 1;2;3). The elastic volume ratio

Jel ¼J

Jthð3Þ

relates the total volume ratio

J¼ det

l1 0 0

0 l3 0

0 0 l3

264

375 ð4Þ

and the thermal volume ratio

Jth ¼ ð1þethÞ3; ð5Þ

where eth ¼ athðy�y0Þ is the linear thermal expansion that isobtained at the temperature y with respect to a referencetemperature y0 and the isotropic thermal expansion coefficientath. The principal Cauchy stresses are then computed by

si ¼ li@c@l i

þph ði¼ 1;2;3Þ; ð6Þ

where ph denotes the equivalent hydrostatic pressure part of thestress tensor.

2.1.2. Viscoelasticity

The linear viscoelastic material behaviour is defined by a Pronyseries expansion of the dimensionless relaxation modulus

gRðtÞ ¼ 1�XN

i ¼ 1

gPi ð1�e�t=tG

i Þ; ð7Þ

with number of Prony elements N, Prony coefficients gPi and

relaxation times tGi . By considering a relaxation test in which a

shear strain g is suddenly applied to a specimen and then held

pn = poutfl fl

i−1i+1

p1 = pinfl fl

fifl

i

solid elements

Δyi

y, v

rigid rod surface

rv

Ai

hifluid film

tifl

x, u

Fig. 2. Discrete configuration of the fluid domain.

T. Schmidt et al. / Tribology International 43 (2010) 1775–1785 1777

constant for a long time, the shear stress is given by

tðtÞ ¼ G0ðyÞ g�Z t

0

_gRðsÞgðt�sÞds

� �; ð8Þ

where the instantaneous shear modulus

G0ðyÞ ¼G1ðyÞ

1�PN

i ¼ 1 gPi

ð9Þ

is related to the temperature-dependent long-term shear mod-ulus G1. The hydrostatic pressure is derived from

phðtÞ ¼�K0ðyÞZ t

0kRðt�sÞ_evol

ðsÞds; ð10Þ

where K0ðyÞ is the temperature-dependent instantaneous bulkmodulus, kRðtÞ is the dimensionless bulk relaxation modulus andevol is the volumetric strain.

The effect of temperature on the viscoelastic behaviour isintroduced through the reduced time

xðtÞ ¼Z t

0

ds

AðyðsÞÞ; ð11Þ

in which the shift function

logðAðyÞÞ ¼�C1ðy�y0Þ

C2þðy�y0Þð12Þ

is defined in the Williams–Landell–Ferry (WLF) form. y0 is thereference temperature at which the constants C1 and C2 are given.

2.2. Fluid mechanics

The definition of the fluid domain is based on the transientReynolds equation [3]

@

@yFp h3

12Z@pfl

@y

� �¼

@

@y

hð _vaþ _vbÞ

2

� �þ@

@y

ð _va� _vbÞ

2RqFsh

� �

þ _ua� _ub� _va@h

@y; ð13Þ

which is given in one-dimensional form. In Fig. 1 the definitions ofthe coordinate directions and the velocity components are shown.It is assumed that the fluid is incompressible and fluid densityremains constant. To consider temperature dependence on thefluid properties, the dynamic viscosity Z¼ f ðyÞ can be defined as afunction of temperature y. In order to account for the effectof surface roughness in the sealing contact, flow factors Fp andFsh are incorporated into the Reynolds equation [12,13]. Thecomposite roughness

Rq ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR2

q;aþR2q;b

qð14Þ

y

x

housing

seal

vb

ub

vaua

rod

pfl

�

Fig. 1. Sealing system.

is calculated from the root mean square (RMS) roughness of thesolid contact partners a and b. Cavitation effects within the fluidfilm are not considered in this work.

With regard to the finite-element implementation, the numer-ical solution of the Reynolds equation will be split into hydrostatic(hs) and hydrodynamic (hd) solutions, which is explained in detailbelow.

2.2.1. Hydrostatic solution

The hydrostatic solution assumes pure pressure flow due to apressure difference Dpfl ¼ pfl

in�pflout , with pfl

inapflout , and zero

motion of both contact partners, i.e. _ua ¼ _ub ¼ _va ¼ _vb ¼ 0. Hence,Eq. (13) reduces to

@

@yFp h3

12Z@pfl

@y

� �¼ 0: ð15Þ

Integration with respect to y yields the fluid pressure gradient

@pfl

@y¼

12ZFph3

c0; ð16Þ

where c0 is an integration constant, which describes the fluidvolume flow rate per unit circumferential length due to pressuredifference Dpfl. Therefore, the nodal fluid pressure gradient ateach node i (see Fig. 2) is given by

@pfl

@y

� �i

¼12ZFp

i h3i

c0; ð17Þ

where

hi ¼ xiþui�r ð18Þ

denotes the nodal fluid film thickness, which is calculated fromthe initial x-position xi, the nodal displacement ui and the rodradius r. Because the flow factors are defined as a function of fluidfilm thickness h, the nodal pressure flow factor is Fp

i ¼FpðhiÞ.

2.2.2. Hydrodynamic solution

The hydrodynamic solution describes the case for movingcontact partners (rod and seal), but for zero pressure difference

Dpfl ¼ pflin�pfl

out ¼ 0. Integration of the Reynolds equation (13) with

respect to y yields the fluid pressure gradient

@pfl

@y¼

6ZFph3

½ð _vb� _vaÞðh�h0�RqðFsh�Fsh

0 ÞÞþ2ð _ua� _ubÞðy�y0Þ�: ð19Þ

The integration constants h0 and y0 mark the fluid gap height andy-position, where the fluid pressure gradient is zero. Therefore,

T. Schmidt et al. / Tribology International 43 (2010) 1775–17851778

both constants are connected through

h0 ¼ hðy0Þ: ð20Þ

As aforementioned for the pressure flow factor, the shear flow

factor is computed by Fshi ¼Fsh

ðhiÞ and, respectively, Fsh0 ¼Fsh

ðh0Þ

is the shear flow factor value for h0. Due to the fact that the rod isassumed to be rigid and the axial seal velocity _va is assumed to bealmost zero, the velocity definition simplifies to

_va � 0 ) _v ¼ _vb: ð21Þ

In the case of pure axial rod movement, the radial rod velocity _ub

is zero and yields

_ub ¼ 0 ) _u ¼ _ua: ð22Þ

Thus, the pressure gradient can be written as

@pfl

@y¼

6ZFph3

½ _vðh�h0�RqðFsh�Fsh

0 ÞÞþ2 _uðy�y0Þ�: ð23Þ

Accordingly, the nodal pressure gradient is defined through

@pfl

@y

� �i

¼6ZFp

i h3i

½ _vðhi�h0�RqðFshi �F

sh0 ÞÞþ2 _uiðyi�y0Þ�; ð24Þ

where

yi ¼ yiþvi ð25Þ

denotes the nodal axial position, which is calculated from the

initial y-position yi and the nodal displacement vi. Additionally,the nodal radial velocity

_ui ¼Dui

Dt¼

uiðtþDtÞ�uiðtÞ

Dtð26Þ

is computed by the difference of the nodal radial displacements ui

at times t and tþDt divided by the time increment Dt.

2.2.3. Numerical implementation

The calculation of the nodal fluid pressure

pfli ¼ pfl

1þXi

k ¼ 2

Dyk a @pfl

@y

� �k�1

þð1�aÞ @pfl

@y

� �k

� �ð27Þ

is realised by a forward Euler integration scheme along they-coordinate, where Dyk ¼ yk�yk�1 is the incremental integrationwidth. The seal surface in the finite-element model is approxi-mated by i¼ 1; . . . ;n nodes, which are used as integration points.The parameter a defines the weighting ratio between the fluidpressure gradients at nodes k�1 and k and takes values between0oao1. The fluid pressure at the first node pfl

1 ¼ pflin is given by

the system fluid pressure. As mentioned above, c0 in Eq. (15) forhydrostatic solution and y0 in Eq. (19) for hydrodynamic solutionare integration constants, which need to be calculated iteratively,so that the fluid pressure at the last node pfl

n meets the ambientpressure pfl

out (see left side in Fig. 2). Based on Eq. (27) the residualvalue

R¼ pflin�pfl

outþXn

k ¼ 2

Dyk a @pfl

@y

� �k�1

þð1�aÞ @pfl

@y

� �k

� �¼!

0 ð28Þ

for iteration of the integration constants c0 and y0 is obtained.The finite-element approach needs nodal forces instead of

pressure values. Therefore, the nodal radial fluid force iscalculated through

f fli ¼ pfl

i � Ai; ð29Þ

where

Ai ¼

A � 12ðyiþ1�yiÞ if i¼ 1;

A � 12ðyiþ1�yi�1Þ if 1o ion;

A � 12ðyi�yi�1Þ if i¼ n

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

ð30Þ

is the nodal area as shown in Fig. 2. Depending on the type ofsimulation and its dimensionality, the specific area is defined by

A¼w if 2D planar;

2pr if axisymmetric;

(ð31Þ

with width w or radius r. The axial fluid shear stress acting onnode i of the seal surface is calculated through

tfli ¼ Z

@ _v

@xþ@ _u

@y

� �i

¼Z _vhi�

hi

2

@pfl

@y

� �i

�Z @ _u

@y

� �i

; ð32Þ

where the nodal radial velocity gradient is given by

@ _u

@y

� �i

¼

ð1�bÞ_uiþ1� _ui

yiþ1�yiif i¼ 1;

b_ui� _ui�1

yi�yi�1þð1�bÞ

_uiþ1� _ui

yiþ1�yiif 1o ion;

b_ui� _ui�1

yi�yi�1if i¼ n:

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

ð33Þ

Herein, the parameter b defines the weighting ratio between theradial velocity gradients at nodes i�1, i and iþ1 and takes valuesbetween 0obo1. The conversion from nodal shear stresses tonodal shear forces is performed by

tfli ¼ t

fli � Ai: ð34Þ

In Sections 2.2.1 and 2.2.2 the solutions for the hydrostatic andthe hydrodynamic cases are described in detail. To obtain the totalsolution of a problem with pressure difference Dpfla0 andmoving rod _va0, the results of both solutions need to besuperimposed node by node through

Pfli ¼ pfl;hs

i þpfl;hdi : ð35Þ

This is done for the total shear stress Tfli , the total radial force F fl

i

and the total shear force T fli , similarly. The integrated fluid shear

force acting on the seal surface is then computed by

T fl¼Xn

i ¼ 1

T fli : ð36Þ

The implementation of the above described expressions is doneby an user-defined element within the software package ABAQUSusing the internal user-subroutine interface UEL. ABAQUS alsorequires the stiffness iteration matrix for the fluid domain [15]

~Kfl

ij ¼D ~F

fl

i

D ~uflj

; ð37Þ

which is defined by the total derivative of the load vector

~Ffl

i ¼ fFfl1 ; T

fl1 ; F

fl2 ; T

fl2 ; . . . ; F

fln ; T

fln ;Rc0 ;Ry0 ;0; T fl

gT ð38Þ

at node i with respect to the displacement vector

~uflj ¼ fu1; v1;u2; v2; . . . ;un; vn; c0; y0;0; _vg

T ð39Þ

at node j. By adding c0 and y0 as additional degrees of freedom(DOF) to the displacement vector in Eq. (39) and Rc0 and Ry0 tothe residual load vector in Eq. (38), the residual expression inEq. (28) can be added to the Newton–Raphson iteration schemeand, thus, be solved by the finite-element solver directly.

ABAQUS

START

ABAQUS

UEL

Subroutine

STOP

Converged?

Yes

No

Yes

No

Total timereached?

t = 0

Assemble global stiffness matricesand residuals

Next time step t+Δt

Rod velocity v

Guesses for c0 and y0

Nodal positions hi, yi, ui, vi

Fluid properties η, pin, poutfl fl

Kij, Fifl ~~ fl

Solve system of equations inNewton-Raphson iteration step

Solve transient Reynolds equation and

calculate fluid stiffness matrix

Fig. 3. Computational procedure. hmin

l

v

sh

y

u

x

Fig. 4. Geometry of rigid slider bearing.

T. Schmidt et al. / Tribology International 43 (2010) 1775–1785 1779

The computational procedure is shown in Fig. 3. Due to thestrong, non-linear coupling of the structural deformations andthe fluid domain, it is necessary to use an iterative procedure. The

analysis starts at time t¼ 0. ABAQUS transmits the pre-defined fluidproperties, the current nodal positions, the rod velocity and theguesses for the integration constants c0 and y0 to the user-subroutineUEL. There, according to the above-mentioned expressions, thetransient Reynolds equation is solved and the fluid stiffness matrix iscalculated. Then the fluid stiffness matrix and the fluid load vectorare sent back to the ABAQUS main program, which assembles theglobal stiffness matrices and residual load vectors. Subsequently, thesystem of equations is solved by use of the Newton–Raphsoniteration scheme. If all convergence criteria are satisfied, the analysisproceeds to the next time step tþDt. Otherwise ABAQUS will repeatthe iteration process and update nodal positions, rod velocity andvalues for c0 and y0. The procedure continues until the specified totaltime is reached.

Leakage quantities are directly computed during the solutionprocess. The volume flow rate per unit circumferential length isgiven by

q¼�c0þ_v

2ðh0�RqFsh

0 Þ: ð40Þ

By use of the specific area A, the current volume flow rate

Q ¼ q �A ð41Þ

can be calculated. Finally, the total leakage (over elapsed time t)can be derived from

V ¼

Z t

0Q dt: ð42Þ

2.2.4. Verification

In this section the implementation of the user-defined fluidelement is verified. The example of a rigid slider bearing is used tocompare the numerical results and the analytical solution.

The bearing geometry is shown in Fig. 4. The analy-tical solution for the fluid pressure along the y-coordinate isgiven by

pflðyÞ ¼3Z _vl

s2h

2 _ul_vsh�1

� � 2y

l�

hminþsh

2hminþsh�

hmin

sh�1

y

l�

hmin

sh�1

� �2þ

hminþsh

2hminþshþ

hmin

shþ1

hmin

shþ1

� �2

26664

37775:

ð43Þ

A finite-element model using the parameter values listed inTable 1 is set up. The lower bearing part is modelled as a rigidsurface, while the upper slider pad consists of a meshed part. Tomeet the requirements of pure rigid behaviour, all deformationsin the pad are suppressed through a rigid body definition. Fig. 5shows the comparison of the analytical and numerical solutionsfor the rigid slider bearing. The difference between both results is

0

0.5

1

1.5

2

2.5

3

0 10

flui

d pr

essu

re p

fl [

N/m

m2 ]

analyticalnumerical

y-coordinate [mm]

2 4 6 8

Fig. 5. Comparison of analytical and numerical solution for rigid slider bearing.

mixed

lubricationstate

lubricationfull

state

pmixs

ps

hmixs

h

Fig. 6. Exponential relationship of solid contact pressure ps and surface

separation h.

Table 1Parameters for rigid slider bearing model.

Parameter Value Unit

Z 10�8 N s=mm2

l 10 mm

hmin 0.001 mm

sh 0.002 mm_u �0.001 mm/s_v 100 mm/s

a 0.5 –

b 0.5 –

pflðy¼ 0Þ 0 N=mm2

pflðy¼ lÞ 0 N=mm2

T. Schmidt et al. / Tribology International 43 (2010) 1775–17851780

negligible. Hence, the proper implementation of the user-definedelement is verified.

2.3. Contact mechanics

2.3.1. Solid contact pressure

Beside the description of hydrodynamic effects within the fluiddomain, the mechanical contact behaviour in normal andtangential direction due to asperity interaction of both contactpartners needs to be described. Substantial experimental andtheoretical works on the contact behaviour of rough surfaces havebeen reported by Greenwood and Williamson [14], Johnson [16],Persson et al. [17]. In this work, a statistical approach described byan exponential relationship between solid contact pressure ps andsurface separation h is used (Fig. 6). The use of this contactpressure model for mixed lubrication simulation has beenpublished by Ongun et al. [7]. The main advantage is that thisapproach is available in ABAQUS [15] by default. Furthermore, theexponential relationship can be motivated from a physical pointof view and leads to good numerically convergence of the solidcontact problem in most cases.

According to the approach the quasi-static nodal solid contactpressure is given by

psi ¼

0 if hiZhsmix;

psmix

e1�11�

hi

hsmix

� �ðeð1�hi=hs

mixÞ�1Þ

� �if 0rhiohs

mix;

8><>: ð44Þ

where hsmix and ps

mix are constitutive parameters, that areestimated by taking into account the mechanical materialproperties and surface topographies of both contact partners orcan be identified through simple friction tests, where Stribeckcurves are measured. Thereby, hs

mix designates the limit filmheight between full and mixed lubrication region, whereas theparameter ps

mix gives the maximum contact pressure at which thesurfaces are totally tight and all asperities are completelyflattened (Fig. 6). Since the finite-element approach needs nodalforces instead of pressure values, the nodal solid forces due toasperity contacts are derived from

f si ¼ ps

i � Ai: ð45Þ

2.3.2. Frictional behaviour

For the description of the frictional behaviour between slidingsurfaces an extended version of the classical Coulomb frictionmodel is used. The classical approach is based on the assumption,that no relative motion occurs if the frictional shear stress is lessthan the critical stress

tscrit ¼ m

s � ps: ð46Þ

The coefficient of friction ms is not restricted to have a constantvalue, but can be defined as a function of the contact pressure ps,the slip rate _g or the average surface temperature ys. In finite-element codes usually a penalty regularisation procedure is used,which approximates the sticking condition of no relative motionby stiff elastic behaviour. The stiffness is chosen such that therelative motion from the position of zero shear stress is boundedby a slip value gcrit , which designates the allowable maximumelastic slip before it comes to sliding. Thus, the nodal frictionalshear stress is computed by

tsi ¼

ms � psi

gcrit

gi if jgijrgcrit ;

ms � psi if jgij4gcrit :

8><>: ð47Þ

According to Eq. (45) the nodal frictional shear force can bederived from

tsi ¼ t

si � Ai: ð48Þ

3. Results

3.1. Friction analysis

3.1.1. Simulation model

In order to show the capability of the presented mixedlubrication approach, an axisymmetric finite-element model of adynamic O-ring seal within a passive driven hydraulic cylinder

X

Y Z

X

Y Z

X

Y Z

contact zone contact zone

d2

dGd1dR

b

poutfl

poutfl

pinfl

pinfl

Fig. 7. O-ring, rod and groove geometries (a), assembled O-ring mesh (b) and deformed O-ring mesh due to fluid pressure (c).

Table 2Viscoelastic parameters for the O-ring material.

No. g Pi (–) tG

i (s)

1 0.765357482 3:81334� 10�9

2 0.071302271 3:13697� 10�8

3 0.068352334 2:58056� 10�7

4 0.021267334 2:12285� 10�6

5 0.008342972 1:74632� 10�5

6 0.005090134 0.000143658

7 0.006856138 0.001181772

8 0.006895512 0.009721619

9 0.005312413 0.079972999

10 0.003746052 0.657882226

11 0.007091882 5.411939407

12 0.000239139 44.52026064

13 0.000326593 366.2372133

14 0.008936099 3012.778778

15 0.009158911 24784.03514

Table 3Parameters for the O-ring simulation model.

Parameter Value Unit

d1 50.17 mm

d2 5.33 mm

dR ¼ 2r 50.0 mm

dG 59.4 mm

b 5.6 mm

Z 10�8 N s=mm2

a 0.5 –

b 0.5 –

pflin

1 N=mm2

pflout

0 N=mm2

ms 0.1 –

hsmix 7� 10�4 mm

psmix 15 N=mm2

Rq 2:15� 10�4 mm

T. Schmidt et al. / Tribology International 43 (2010) 1775–1785 1781

system is analysed (Fig. 7a). An EPDM rubber compound witha hardness of 75 ShA is specified for the O-ring material. Theparameters for the Neo–Hooke material model are given byC10 ¼ 1:064 N=mm2 and D1 ¼ 0:001 mm2=N. The Prony coefficientsand relaxation times for the viscoelastic material definition areshown in Table 2. The WLF-constants C1 ¼ 7:69 and C2 ¼ 157:74 Kare given for reference temperature y0 ¼ 25 3C. All parameters ofthe O-ring model, including geometric dimensions, fluidproperties and friction parameters, are illustrated in Table 3. Allsimulations are performed at y¼ 25 3C.

Rod (R) and groove (G) are both modelled as analytical rigidsurfaces. Fig. 7 b shows the assembled O-ring geometry. The O-ring is meshed with linear axisymmetric four-node elements(CAX4H). Due to the nearly incompressible behaviour of therubber material, a hybrid element formulation with an additionalDOF for the hydrostatic pressure is used. The global mean meshseed size is 0.05 mm, whereas the elements’ edge length in thecontact region is fixed to 0.01 mm. It should be noted, that thecontact zone shown in Fig. 7 b and c is not initially fixed and canchange during the simulation process.

The solution is obtained in the following simulation steps:Mounting: In the first step the seal is mounted inside the

groove using the automatic contact interference shrink featurein ABAQUS (Fig. 7 b). This is done with frictionless contactdefinitions.

Fluid pressure: During the second step the system fluidpressure is applied incrementally to the inner O-ring surface(right side in Fig. 7 c) by increasing the pressure from 0 to pfl

in

using the ABAQUS pressure penetration feature. At the same timethe user-defined fluid element is activated and the hydrostaticsolution (Section 2.2.1) is calculated for the same fluid pressureincrements. Additionally, the solid friction in the contact zone isramped from 0 to ms.

Transient analysis: While the first two steps are performedusing long-term, hyperelastic material behaviour, the thirdsimulation step is intended to use time-dependent, viscoelasticmaterial behaviour. Within the fluid element the hydrodynamicsolution (Section 2.2.2) including the squeeze effects is solved,additionally. The rod velocity is given as a function of step timeand is defined by an amplitude function in ABAQUS.

3.1.2. Friction results

In this section the results of the O-ring seal friction analysis arepresented and discussed in detail. First, the results of the mixedlubrication model for rod outstroke and instroke are shown.Accordingly, the effect of viscoelastic material behaviour and theinfluence of flow factors in the transient Reynolds equation areexplained. IHL and EHL simulation results, i.e. fluid film thicknessand fluid pressure, are compared. Finally, leakage quantities fordifferent transient EHL simulations are shown.

In Fig. 8 the output quantities of the mixed lubrication modelfor rod outstroke and rod instroke at 7100 mm=s are illustrated.In Fig. 8 a the fluid film thickness h, the fluid pressure pfl and thesolid contact pressure ps are shown. Fig. 8 b points out the fluidshear stress tfl and solid shear stress ts. It can be seen that due tothe applied system pressure pfl

in the quantity distributions differfor out- and instroke, whereas the outstroke values for fluidpressure and fluid shear stress are higher than for the inward

0

0.7

0.8

0.9

1

1.1

1.3

0.5

flui

d fi

lm th

ickn

ess

h [μ

m]

t = 0 ht = 1 ht = 5 h

t = 15 hlong term

1.2

1.5 2.5 3.531 2

relative y-coordinate [mm]

Fig. 9. Comparison of transient (viscoelastic) and long-term (quasi-static) results

at _v ¼ 500 mm=s¼ const.

0

1

2

0

flow

fac

tors

Φp , Φ

sh [

–]

Φp

Φsh

1.5

0.5

1 2 3 4 5

fluid film thickness h [μm]

Fig. 10. Pressure and shear flow factors.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.5 1.5 2.5 3.5−0.4

0

0.4

0.8

1.6

2

2.4

2.8

3.2

3.6

flui

d fi

lm th

ickn

ess

h [μ

m]

pres

sure

s pf

l , ps

[N

/mm

2]

hout

hin

poutfl

pinfl

pouts

pins

−4.5

−3

−1.5

0

1.5

3

4.5

0.5 2.5−0.15

−0.1

−0.05

0

0.05

0.1

0.15

flui

d sh

ear

stre

ss τ

fl [

10−

3 N/m

m2 ]

solid

she

ar s

tres

s τs

[N

/mm

2 ]fl

fl

s

s

1 2 3

relative y-coordinate [mm]

1.2

1 1.5 2 3 3.5

relative y-coordinate [mm]

�out

�in

�out

�in

Fig. 8. Mixed lubrication output quantities for rod outstroke and instroke at_v ¼ 7100 mm=s.

T. Schmidt et al. / Tribology International 43 (2010) 1775–17851782

motion. Hence, the solid contact pressure and solid shear stressobtained during the outward motion are smaller than for theinstroke. During outstroke the fluid pressure within the outletregion (0:5oyo1) tends to negative values. Since cavitationeffects within the fluid film are not considered in this work,negative fluid pressures can occur, which may be an indication forcavitation. Furthermore, it can be stated that for _v ¼ 7100 mm=s

the fluid shear stresses, calculated for the given modelparameters, are about two magnitudes smaller than the solidshear stresses (see axis labels in Fig. 8 b).

As mentioned above, the effect of viscoelastic materialbehaviour on the formation of the fluid film is studied. Thesimulation results of a transient and a long-term analysis areshown in Fig. 9. The long-term results are computed for puresteady-state conditions, i.e. hyperelastic material behaviour andsteady-state Reynolds equation. The transient solution is obtainedby using viscoelastic material definition and solving the transientReynolds equation for the fluid film. Both analyses are performedwith the same procedure: the rod velocity increases from 0 to500 mm/s and is then held constant for a long time. It can be seenthat for the transient analysis the fluid film thickness h reachesmaximum values at time t¼ 0 h, but then decreases for timest¼ 1;5;15 h due to the viscoelastic relaxations in the sealstructure. For t-1 the transient solution tends towards thelong-term curve. In this example the difference of the fluid filmthickness between long-term and transient solutions is at most0:15mm. Hence, the viscoelastic material behaviour in seals has asignificant influence on the calculation of leakage quantities,which will be shown later.

The influence of flow factors incorporated in the transientReynolds equation is the second effect to be analysed in this work.Therefore, the pressure flow factor and shear flow factordistributions shown in Fig. 10 are used. These are derived fromnumerical flow simulations of a turned rod surface using a non-commercial software code. Since the calculation of flow factors isnot subject of this work, the authors refer to [12,13,18,19].The comparison of the simulation results is given in Fig. 11, wherethe dashed lines represent the analysis including flow factors. It isobvious that the fluid pressure within the mixed lubricationregion is higher, if flow factors are active. Hence, the fluid filmthickness is higher, but solid contact pressure and solid shearstress are lower than for the analysis without flow factordefinition. This can be explained by the fact that the flowfactors reduce the nodal fluid film thickness to an effectiveheight by considering the influence of surface roughness. Though,for the same rod velocity and equal conditions the fluid pressurein the effectively reduced fluid film must be higher.

Since the implementation of the user-defined fluid elementhas been verified for rigid contact partners in Section 2.2.4, theresults of EHL analysis for the O-ring example are now comparedto calculations based on the inverse hydrodynamic lubrication

−500

−250

0

250

500

0 10

rodv

eloc

ity v

[m

m/s

]˙

time t [s]2 4 6 8

Fig. 13. Rod velocity amplitude.

00.10.20.30.40.50.60.70.80.9

11.11.2

1 1.5 2.5

0

2.733.3

flui

d fi

lm th

ickn

ess

h [μ

m]

00.10.20.30.40.50.60.70.80.9

11.11.2

flui

d fi

lm th

ickn

ess

h [μ

m]

pres

sure

s pf

l , ps

[N

/mm

2 ]

hIHL

hEHL

pIHLfl

pEHLfl

pEHLs

hIHL

hEHL

pIHLfl

pEHLfl

pEHLs

relative y-coordinate [mm]

2 3

1 1.5 2.5

relative y-coordinate [mm]

2 3

2.42.11.81.51.20.90.60.3

−0.3

0

2.733.3

pres

sure

s pf

l , ps

[N

/mm

2 ]

2.42.11.81.51.20.90.60.3

−0.3

Fig. 12. Comparison of IHL and EHL results for _v ¼ 100 mm=s (a) and_v ¼ 500 mm=s (b).

0

1.5

3

4.5

1.5 2.5 3.50

0.05

0.1

0.15

flui

d sh

ear

stre

ss �

fl [

10−

3 N/m

m2 ]

solid

she

ar s

tres

s τs

[N

/mm

2 ]

fl

s

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.5 1.5 2.5 3.5−0.4

0

0.4

0.8

1.2

1.6

2

2.4

2.8

3.2

3.6

flui

d fi

lm th

ickn

es s

h [μ

m]

pres

sure

s pf

l , ps

[N

/mm

2 ]

relative y-coordinate [mm]

h

hΦ

pfl

pΦfl

ps

pΦs

1 2 3

relative y-coordinate [mm]

0.5 1 2 3

�fl

�Φ�s

�Φ

Fig. 11. Effect of flow factors in mixed lubrication state.

T. Schmidt et al. / Tribology International 43 (2010) 1775–1785 1783

(IHL) theory [4]. Due to the fact that the IHL method used in thispaper predicts the fluid film thickness based on the contactpressure distribution in the lubricated contact zone, the usedalgorithm is restricted to zero boundary pressures. Hence, thecomparison is performed for pfl

in ¼ pflout ¼ 0 N=mm2, i.e. the defor-

mation state given in Fig. 7 b. Furthermore, the IHL theoryassumes quasi-static material behaviour and steady-state condi-tions in the fluid film. Therefore, the EHL results are obtained forpure hyperelastic material behaviour and the steady-state solu-tion of the Reynolds equation. The results are given in Fig. 12,where Fig. 12 a shows the fluid film thickness, fluid pressureand solid contact pressure for the mixed lubrication state at_v ¼ 100 mm=s and Fig. 12 b the same quantities for full lubrica-tion conditions at _v ¼ 500 mm=s. Within the full lubrication stateIHL and EHL quantities coincide, whereas for _v ¼ 100 mm=sthe difference between estimated fluid film thickness of IHLcalculation and computed EHL film thickness is obvious. Thedifference is explained by the IHL assumption of full lubricationconditions for all rod velocities, what is known to be unrealisticfor many sealing applications.

The transient simulations are performed using the sinusoidal rodvelocity amplitude as shown in Fig. 13. The time signals of volumeflow rate Q and leakage volume V for different simulations arepresented in Fig. 14. The explanations of the used subscripts are:

�

EHL: Calculations using the introduced mixed lubricationapproach. � he: Hyperelastic material behaviour. � ve: Viscoelastic material behaviour. � sq: Solution of the transient Reynolds equation (in all other

cases the steady-state Reynolds equation is solved).

� U: Simulations using flow factor definition.

It is obvious that the solution for hyperelastic material behaviour(Vhe

EHL) gives lower fluid leakage than the computations consider-ing the viscoelastic material definition. The explanation for this isgiven in Fig. 9, where the fluid film thickness reaches highervalues for the viscoelastic material behaviour and, thus, the fluidtransport increases compared to hyperelastic behaviour. Further-more, for this specific example and rod velocity amplitude theinfluence of the squeeze effects in the fluid film on leakagecalculation is insignificant (Vve;sq

EHL ). However, the effect of flowfactors on the leakage quantities is obvious (Vve;sq;F

EHL ). The ratio of

O−ring

−30

−20

−10

0

10

20

30

40

50

0 10

volu

me

flow

rate

Q [

mm

3 /s]

QEHLhe

QEHLve

QEHLve,sq

QEHLve,sq,Φ

0

20

40

60

80

100

120

0 10

leak

age

volu

me

V [

mm

3 ]

VEHLhe

VEHLve

VEHLve,sq

VEHLve,sq,Φ

2 4 6 8time t [s]

time t [s]2 4 6 8

Fig. 14. Comparison of IHL and EHL leakage results.

T. Schmidt et al. / Tribology International 43 (2010) 1775–17851784

outward to inward fluid transport increases, so that the netleakage at the end of the cycle increases, too.

The overall computation time for this transient O-ring exampleis about 2 h on a single CPU machine (Intel XEON EM64T, 3.4 GHz,8 GB RAM), what proofs the high efficiency of this computationalapproach.

geometryinitial O−ring

rod

Fig. 15. Initial (a), deformed (b) and worn (c) O-ring geometry.

4. Conclusions

In this work the finite-element approach of a user-defined fluidelement for the simulation of mixed lubrication effects inreciprocating hydraulic rod seals is presented. The definition ofthe fluid domain is based on the transient Reynolds equation andis directly implemented within the commercial finite-elementsoftware package ABAQUS using the internal user-subroutineinterface UEL. Hence, it allows a strong coupling between thefinite deformations in the seal structure and the transienthydrodynamic effects in the fluid film. Therefore, the applicationof any arbitrary, non-linear material model provided by ABAQUSis enabled. In this work hyperelastic and viscoelastic materialmodels are used to describe the mechanical behaviour of therubber seal. In order to account for the effect of surface rough-ness in the sealing contact, flow factors are incorporated intothe transient Reynolds equation. The realisation of the mixedlubrication behaviour within the contact region is given by anexponential relationship between solid contact pressure and fluidfilm thickness. The quantities for fluid volume flow rate and

current leakage volume are calculated during the transientsimulations, automatically. The method is currently restricted toplanar or axisymmetric geometries.

In order to show the capability of the presented mixedlubrication approach, an axisymmetric finite-element model of adynamic O-ring seal within a passive driven hydraulic cylindersystem is analysed. First, the mixed lubrication output quantitiesfor rod outstroke and instroke are shown. Accordingly, the effectof viscoelastic material behaviour and the influence of flow factorsin the transient Reynolds equation are explained. IHL and EHLsimulation results, i.e. fluid film thickness and fluid pressure, arecompared. It is shown that within full lubrication state IHL andEHL results coincide, whereas for lower rod velocities and mixedlubrication conditions the estimated fluid film thickness of IHLcalculation differs from the computed EHL film thickness. Withinthe transient EHL results the solution for hyperelastic materialbehaviour gives lower fluid leakage than the computationsconsidering the viscoelastic material definition. The fluid filmthickness reaches higher values with viscoelastic materialbehaviour and, thus, the fluid transport increases compared tohyperelastic behaviour. The influence of the squeeze effects in thefluid film on leakage calculation is insignificant for this specificexample and rod velocity amplitude. However, the effect of flowfactors and their impact on the leakage quantities is shown.

Future work will focus on the validation of coupled mixedlubrication and wear simulations for real hydraulic sealingsystems (e.g. Stepseal). Therefore, the identification of mixedlubrication parameters based on Stribeck curve measurementsand the determination of wear model parameters based on resultsfrom endurance tests is required.

Based on the above-mentioned transient friction analysis, wearsimulations utilising an arbitrary-Lagrangian–Eulerian (ALE) algo-rithm within ABAQUS will be performed. Adapted from Archard’slaw the local abrasive wear due to asperity contacts during mixedlubrication conditions is computed. Within the user-subroutineinterface UMESHMOTION the calculated nodal ablation depthis applied to each boundary contact node by adaptive meshconstraints within an Eulerian simulation step. The exampleillustrated in Fig. 15 shows the qualitative performance of the

T. Schmidt et al. / Tribology International 43 (2010) 1775–1785 1785

combined mixed lubrication and wear simulation. The initialmeshed O-ring geometry is given in Fig. 15 a. In Fig. 15 b theO-ring is mounted inside the groove and the rod is movedaccording to an user-specified velocity amplitude. Afterdisassembling the O-ring, the volume loss and the worn contactsurface can be seen (Fig. 15 c). As explained above, during thetransient simulation procedure the incremental nodal ablationdepths are calculated for each time step and applied to the contactnodes using adaptive mesh constraints. The nodes inside the meshgeometry are automatically moved using an adaptive meshsweeping algorithm to obtain a regular mesh without distortedelements.

Actually, experimental validation of the mixed lubricationapproach utilising light induced fluorescence (LIF) measurementsof fluid film thickness in the sealing zone is in progress.Furthermore, the implementation of cavitation effects and theenhancement of the fluid element for three-dimensional simula-tions will be subject of future work.

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