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Page 1: NUMERICAL INVESTIGATION OF STEADY WEAR PROCESS

NUMERICAL INVESTIGATION OF STEADY WEAR PROCESS

Páczelt István University of Miskolc,

Department of Mechanics , Miskolc, Hungary

2-nd Hungarian-Ukrainian Joint Conference on SAFETY-RELIABILITY AND RISK OF ENGINEERING

PLANTS AND COMPONENTS

KYIV, September 19-21, 2007.

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A contact problem

F F

C

G GAdmissiblecontact region

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Linear elastic contact problems Contact conditions

][)( 12nnn uguugdd u

S(1)c

S(2)c u(2)

u(1)

1st body

2nd body

n(1)

Q1

Q2

)1(nu

)2(nu

n(2)

nc

g

S(2)c

S(1)c

1st body

2nd body

)1(nτ

)1(u

)2(u

)2(nτ

Q1

Q2

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Signorini contact conditions

0uτ ,0nn

d 0, 0np Cx

d 0, 0np Gx 0dpn cSx

Friction conditions:In adhesion subregion

In slip subregion

0uτ ,0nn

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Contact stresses

,021 nnnp

u

uτττ

nnnn p 21

nnστ nn p

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Clasification of mechanical wear processes

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Factors influencing dry wear rates

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Modified Archard wear model

2,1,~

)()( ivpvppw iiiiiiii ar

bni

a

rb

ni

abni

abni uu

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Problem classification

1. Rigid body wear velocities allowed, contact area fixed- steady states present

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2.Rigid body wear velocities allowed, contact area evolving in time due to wear- quasi steady states

a

0F

A B

z

x

rv

1B

2B

R

pn=pn(a,F0)

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3.No rigid body wear velocities allowed- steady states corresponding to vanishing wear rate and contact pressure (wear shake down).

a

L

0F

EIA B C

z

x

rv

1B

2B

g

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Initial gap g= g_0 =0.05 mm, Beam side a_0=10 mm, b_0=25 mm,

lenght L=300 mm. Load F_0=10 kN, AB distance (a)

=150mm, Relative velocity v_r=50 mm/s Coefficient of Winkler foundation=

0.0000002 mm/N

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The wear parameters are: beta=0.0025, a=b=1

coefficient of friction mu=0.3 In initial state: (u1_n beam

displacement in vertical direction without body 2.)

def1, def2 are vertical displacement of body 1 and 2 in the contact.

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Initial state:

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In the time t=0.8 sec

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In the time t=1200*dt=1200*0.001=1,2 sec

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In the time t=2,4 sec

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In the time= 3,6 sec

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In the time= 7,8 sec

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In the time=12 sec

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In the time=60 secHere p_n= 1*10e-7 that is practically p_n is equal to zero.

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4. Oscillating sliding contacts (fretting process)

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Type of investigated mechanical systems

The analysis of the present investigation is referred to such class of problems when

the contact surface does not evolve in time and is specified

the wear velocity associated with rigid body motion does not vanish and is compatible with the specified boundary conditions

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[1] Páczelt I, Mróz Z. On optimal contact shapes generated by wear, Int. J. Num. Meth. Eng. 2005;63:1310-1347.

[2] Páczelt I, Mróz Z. Optimal shapes of contact interfaces due to sliding wear in the steady relative motion, Int. J. Solids Struct 2007;44:895-925.

[3] Pödra P, Andersson S. Simulating sliding wear with finite element method, Tribology Int 1999;32:71-81.

[4] Öqvist M. Numerical simulations of mild wear using updated geometry with different step size approaches, Wear 2001;49:6-11.

[5] Peigney U. Simulating wear under cyclic loading by a minimization approach, Int. J. Solids Struct 2004;41: 6783-6799.

[6] Marshek KM, Chen HH. Discretization pressure wear theory for bodies in sliding contact, J. Tribology ASME 1989; 111:95-100.

[7] Sfantos GK, Aliabadi MH. Application of BEM and optimization technique to wear problems, Int. J. Solids Struct 2006;43:3626-3642.

[8] Kim NH, Won D, Burris D, Holtkamp B, Gessel GC, Swanson P, Sawyer WG. Finite element analysis and experiments of metal/metal wear in oscillatory contacts, Wear 2005;258:1787-1793.

[9] Fouvry S. et al. An energy description of wear mechanisms and its applications to oscillating sliding contacts, Wear, 2003;255:287-298

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The generalized wear volume rate

Generalized friction dissipationpower

q

q

q

S

rnqF BdSvpD

c

/1

/1

)(

qi

i

q

S

qar

bni

i

q

S

qi

i

q AdSvpdSwWc

ii

c

/12

1

/12

1

/12

1

)~()(

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The generalized wear dissipation power

For one body

For two bodies

q>0

q

q

q

S

ar

b

n

q

q

S

nq

w CdSvpdSwpDcc

/1

/1

1

/1

)~()(

where the control parameter q usually is

qi

i

q

q

S

ini

qw CdSwpD

c

/12

1

/12

1

)(

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The relative tangential velocity on

sliding velocity at the interface

wear velocity

cS

cS

Mλ are the relative translation and rotation velocities induced by wear

,)( ,,)1(,

)1(,

)2(,

)2(, ReReRe uuuuuuu

,, Re uu wR

sRR

sR

wR ,,,

)(,

)(, , uuuuu

rΩuu sRsRsR ,,)(,

r

sR v ,uu

cwnRMF

wR

cMFwnR

u

u

nrλλu

nrλλ)(,

)(,

)(,

)(

,)(

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1e

2e

cn

PcSrv

Re

Rw w

w

11

12 eeuuuu rv

22,111,111 eenw www c

22,211,222 eenw www c

cnndcn ppp neenppp ~)( 2121

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The generalized wear dissipation power

1e

2e

cn

PcSrv

Re

Rw w

w

11

12 eeuuuu rv

Wear rate vectors: .

rλλ

rλλe

MF

MFR

RRRR ww ewew ,22,11 ,

q

qi

S

ii

qw dSD

c

/12

1

)( wp q

ii

C /12

1

Relative velocity:

.constvr

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The global equilibrium conditions forbody 1 are

cnndcn ppp neenppp ~)( 2121

0mnrm

0fnf

0

0

~

~

dSp

dSp

c

c

S

nc

S

nc

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.constvr

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Constrained minimization

Problem PW1: Min Problem PW2: Min Problem PW3: Min subject to

)( nqq pWW

)( nq

Fq

F pDD )( n

qw

qw pDD

0mnrm0fnf 00~,~ dSpdSp

cc S

nc

S

nc

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Major results of our investigation:

Question:

What kind of minimization problem generates contact pressure distribution corresponding to the steady wear state?

Answer:

Must be used: min )( n

qw

qw pDD

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Main assumption:

We shall consider only the generalized wear dissipation power and the resulting optimal pressure distribution.

It will be shown that for q=1, the optimal solution corresponds to steady state condition.

)( nq

wq

w pDD

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Congruency conditions

In stationary translation motion:

In rotation with constant angular velocity: the case of annular punch:

bbb 21

aaa 21 bbb 21

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Introducing the Lagrange multipliers and

The Lagrangian functional is

mλfλλλ MFnq

wMFnq

Dq

D bbpDpLLww

11,,

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From the stationary condition we obtain

The equations are highly nonlinear !

0qDw

L

11 qb

11

1

1

22

1

11

)tan1(~~

~~

21

qbq

q

qqarq

qqar

cMcFn

CvCvp

nrλnλ

0mnrλλm

0fnλλf

0

0

~,

~,

dSp

dSp

c

c

S

ncMF

S

ncMF

),( MF λλ

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Special case 1

the contact pressure is

the wear rate equals

the wear volume rate is

cn S

Fp 0

zF ef 00 zc en

FcF nλ

0Mλ

1~0 constv

S

Fw a

r

b

c

2~ 1

0 constSFvW bc

bar

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Special case 2: translation and rotation

kNmmMkNmmM yx 250,400 00

kNF 100 SCx=60 mm, SCy=80 mm

sec/5mmvr 2,1,5.0,1,0002.0 ba

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Special case 3: Block-on-disk wear tests

zxc een cossin

zF ef 00

cos FcF nλ

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Results

At steady wear state (q=1)

1111

10 cossincos

qb

q

qb

q

qD

n

wI

Fp

dtRI qb

q

qb

qqDw 011

111

10

0

cossincos

bqD

n

wI

Fp

1

10 cos

constI

FRRw

b

q

D

aa

w

)(cos~~

10

020121

constI

FRRw

b

q

D

aav

w

1

00201

21~~

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Contact pressure distribution for anticlockwise disk rotation

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Vertical wear rate distribution for clockwise disk rotation

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Normal contact shape for different values of friction coefficient, q=1

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Vertical contact shape for different values of friction coefficient, q=1

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Steady state normal and vertical wear rate distributions

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Special case 4: ring segment-on-disk wear tests

.

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Initial contact pressure distribution (anticlockwise rotation).

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Optimal contact vertical shape at anticlockwise rotation

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Special case 5: brake system with rotating block

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Results

at steady state (q=1)

dztzlzI cqcqx

z

z

qD

u

i

w

//)1(1

cqcqxq

Dn zlz

I

LFp

w

//)1(0

b

qD

n zI

LFp

w

/11

0

2

11

0 ~

i

ari

b

qD

Mi

w

vI

LF

zw M

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Special case 6: brake system with translating and rotating block

zxzxMxMF

yMxFR H

lz

Hee

eereee sincos

)(1)(

22)( xMMF lzH

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Results

cos

sincos Q

1)1( qbc

cq

q

qarq

qqar

cMcFn

CvCvp

1

1

22

1

11

)tan1(~~

~~

21

nrλnλ

cq

c

xMF QqZ

lz /

/1

iqarq

q

ii

qi vCqZ

12

1

~

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The non-linear equations are solved by Newton-Ralphson technique

At steady state ,

,

.

0

/

/1

FdztQqZ

lz cq

c

xMFz

z

u

i

0

/

/1

)( MdztlzQqZ

lzx

cq

c

xMFz

z

u

i

iar

ii vqZ

2

1

~1 bc 1q

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Numerical results:pressure distribution

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Pressure distribution

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Wear velocity distribution

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Wear velocity distribution

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Special case 7: Automative Braking system

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Drum brake

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Model of drum brake

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mpRllllpm nxzxznyc 0sincoscossin)( erp

yy MLF eem 000

00 Mmm c

Equilibrium equation, wear rate vector

----------------------------------------------

A

ll zxRc

1sincoscos en

xzzx

y

yR RlRl

Aee

er

ere

cossin

100

202

0 cossin RlRlA zx

RRR w ew

cos

wwR

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Pressure and wear rate

At steady wear state (q=1)

cq

qD

n mI

LFp

w

/10 tan1

dtRmmIcqq

D

u

i

w 0

/1)(tan1

cbq

b

qD

a

ii m

I

LFRw

w

i/0

0

2

1

tan1)(~

b

zxqD

b

qD

n llI

LFA

I

LFp

ww

/1

1

0/1

1

0)sincos(cos

ARI

LFw i

w

a

b

qDi

iR 01

02

1

~

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Optimal contact pressure distribution at anticlockwise drum rotation

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Optimal contact pressure distribution:clockwise drum rotation

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Wear rate distributions in the steady state: q=1, (anticlockwise drum rotation)

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Wear rate distributions in the steady state: q=1, (clockwise drum rotation)

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Special case 8: Cylindrical punch rotation

with respect to symmetry axis .

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Contact pressure, Lagrangian functional

zxdyzxdcc eeeeeeenn sincoscossin~21

cnp nppp ~21

F

q

S

qab

nid

ni

qD bdSrppL

c

i

w

1~

cos

)sin(cos/1

2

1

cqc

q

dcqa

qD

n ReI

Fp

w

/1

/0

0 cossincossin

qD

caqcqc

bq

dcqacaqq

D wwIdRReI /

0//1

0/ cossincossin2

~

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At steady state (q=1)

Contact pressure

Vertical wear rate

bba

qD

n ReI

Fp

w

/1/01

0 cossin

constI

Fwww a

b

qD

vvv

w

10

2121

~~

vF w

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Distribution of contact pressure

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Vertical gap at steady state

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Stress state in body B1

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If

then the contact surface will be a plate

0R 0

ba

qD

ba

qD

n rI

FRe

I

Fp

ww

/10/

010 sin

wwvF

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Contact pressure for different wear parameters

20 30 40 50 60 70 80 90 100 110 1200

100

200

300

400

500

600

700

800

900

1000

R [mm]

p [M

Pa]

with p=8 order finite elements

a=1, b=0.5 (..), b=1 (+), b=2 (-)

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Contact shape in the steady state

20 30 40 50 60 70 80 90 100 110 1200

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

R [mm]

shap

e [m

m]

with p=8 order finite elements

a=1, b=0.5 (..), b=1 (+), b=2 (-)

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Initial shape calculated for

1,2 ab MPap 100~

The wear process for the system

1,1 ab 6102.0

srad /10 MPap 50~

Time step: st 025.0

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DeltaV= V(i)-V(i-1) - (V(i-1)-V(i-2)), i=2,3,4,….

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The wear shape in the different time step

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Effect of heat generation

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Shape in steady wear state a) without temperature, b) with influence of heat generation

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Special case 9: disk brake

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Model of disk brake

F0

b

ab

Dn r

I

Fp

w

0

drrIe

i

w

r

r

b

a

D

)1(

)(

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Special case 10: Nuclear fuel fretting

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Procedure to solve wear problem in the industry.

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Kim: Tribology International, 36 (2006), p.1305-1319

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M. Helmi Attia: Tribology International, 39 (2006), p. 1294-1304.

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Heat exchanger tube

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System approach to the fretting wear process

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Conclusions

1.The present lecture provides a uniform approach to the analysis of steady wear regimes developing in the case of sliding relative motion of contacting bodies.

2. Usually, the steady state may be attained experimentally or in the numerical analysis by integrating the wear rate in the transient wear period.

3. A fundamental assumption is now introduced, namely, at the steady state the wear rate vector is collinear with the rigid body wear velocity of body 1.

4. The minimum of the generalized wear dissipation power for q = 1 generates steady state regimes.

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Conclusions:5. The optimal solution corresponds to steady state

condition. Thus, this condition can be specified directly from formulae for contact pressure and equilibrium equations instead of integration of the wear rule for whole transient wear process until the steady state is reached.

6. The specification of steady wear states is of engineering importance as it allows for optimal shape design of contacting interfaces in order to avoid the transient run-in periods.

7. Different numerical examples demonstrate usefulness of the proposed principle and corresponding numerical methods.

8. High accuracy solution may be reached using the p-version of finite elements for the contact problems combined with the positioning technique and the special remeshing.

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Closure

Thank you very much for your kind attention!