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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.
A contact problem
F F
C
G GAdmissiblecontact region
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
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
Contact stresses
,021 nnnp
u
uτττ
nnnn p 21
nnστ nn p
Clasification of mechanical wear processes
Factors influencing dry wear rates
Modified Archard wear model
2,1,~
)()( ivpvppw iiiiiiii ar
bni
a
rb
ni
abni
abni uu
Problem classification
1. Rigid body wear velocities allowed, contact area fixed- steady states present
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)
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
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
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.
Initial state:
In the time t=0.8 sec
In the time t=1200*dt=1200*0.001=1,2 sec
In the time t=2,4 sec
In the time= 3,6 sec
In the time= 7,8 sec
In the time=12 sec
In the time=60 secHere p_n= 1*10e-7 that is practically p_n is equal to zero.
4. Oscillating sliding contacts (fretting process)
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
[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
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
)~()(
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
)(
The relative tangential velocity on
sliding velocity at the interface
wear velocity
cS
cS
Fλ
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)(,
)(,
)(,
)(
,)(
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
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
The global equilibrium conditions forbody 1 are
cnndcn ppp neenppp ~)( 2121
0mnrm
0fnf
0
0
~
~
dSp
dSp
c
c
S
nc
S
nc
.constvr
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
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
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
Congruency conditions
In stationary translation motion:
In rotation with constant angular velocity: the case of annular punch:
bbb 21
aaa 21 bbb 21
Introducing the Lagrange multipliers and
The Lagrangian functional is
Fλ
Mλ
mλfλλλ MFnq
wMFnq
Dq
D bbpDpLLww
11,,
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
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
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
Special case 3: Block-on-disk wear tests
zxc een cossin
zF ef 00
cos FcF nλ
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~~
Contact pressure distribution for anticlockwise disk rotation
Vertical wear rate distribution for clockwise disk rotation
Normal contact shape for different values of friction coefficient, q=1
Vertical contact shape for different values of friction coefficient, q=1
Steady state normal and vertical wear rate distributions
Special case 4: ring segment-on-disk wear tests
.
Initial contact pressure distribution (anticlockwise rotation).
Optimal contact vertical shape at anticlockwise rotation
Special case 5: brake system with rotating block
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
Special case 6: brake system with translating and rotating block
zxzxMxMF
yMxFR H
lz
Hee
eereee sincos
)(1)(
22)( xMMF lzH
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
~
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
Numerical results:pressure distribution
Pressure distribution
Wear velocity distribution
Wear velocity distribution
Special case 7: Automative Braking system
Drum brake
Model of drum brake
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
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
~
Optimal contact pressure distribution at anticlockwise drum rotation
Optimal contact pressure distribution:clockwise drum rotation
Wear rate distributions in the steady state: q=1, (anticlockwise drum rotation)
Wear rate distributions in the steady state: q=1, (clockwise drum rotation)
Special case 8: Cylindrical punch rotation
with respect to symmetry axis .
Contact pressure, Lagrangian functional
zxdyzxdcc eeeeeeenn sincoscossin~21
cnp nppp ~21
fλ
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
~
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
Distribution of contact pressure
Vertical gap at steady state
Stress state in body B1
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
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 (-)
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 (-)
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
DeltaV= V(i)-V(i-1) - (V(i-1)-V(i-2)), i=2,3,4,….
The wear shape in the different time step
Effect of heat generation
Shape in steady wear state a) without temperature, b) with influence of heat generation
Special case 9: disk brake
Model of disk brake
F0
b
ab
Dn r
I
Fp
w
0
drrIe
i
w
r
r
b
a
D
)1(
)(
Special case 10: Nuclear fuel fretting
Procedure to solve wear problem in the industry.
Kim: Tribology International, 36 (2006), p.1305-1319
M. Helmi Attia: Tribology International, 39 (2006), p. 1294-1304.
Heat exchanger tube
System approach to the fretting wear process
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.
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.
Closure
Thank you very much for your kind attention!