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SpringerWienNewYork
CISM COURSES AND LECTURES
Series Editors:
The RectorsGiulio Maier - Milan
Jean Salençon - PalaiseauWilhelm Schneider - Wien
The Secretary General
Executive Editor
The series presents lecture notes, monographs, edited works and
and Applied Mathematics.
This volume contains 157 illustrations
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© 2007 by CISM, UdinePrinted in ItalySPIN 12068043
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Emerging Spatial and Temporal Discretization Methods 3
Emerging Spatial and Temporal Discretization Methods 5
Emerging Spatial and Temporal Discretization Methods 7
Emerging Spatial and Temporal Discretization Methods 9
10
Emerging Spatial and Temporal Discretization Methods 11
12
Emerging Spatial and Temporal Discretization Methods 13
0 0.1 0.2 0.3 0.4 0.5 0.6 0.70
1
2
3
4
5
6
7
8
9
10
x
Con
tact
Tra
ctio
n
Normal traction, analyticalFrictional traction, analyticalNormal traction, numericalFrictional traction, numerical
Emerging Spatial and Temporal Discretization Methods 15
0 0.1 0.2 0.3 0.4 0.5 0.6 0.70
1
2
3
4
5
6
7
8
9
10
x
Con
tact
Tra
ctio
n
Normal traction, analyticalFrictional traction, analyticalNormal traction, numericalFrictional traction, numerical
16
Emerging Spatial and Temporal Discretization Methods 17
Emerging Spatial and Temporal Discretization Methods 19
0 5 10 15 20 25 30 35 40 450
0.5
1
1.5
2
2.5
3
3.5
4
4.5x 10
9
Displacement
Rea
ctio
n F
orce
20
Emerging Spatial and Temporal Discretization Methods 21
22
-1.01E+03
-8.12E+02
-6.14E+02
-4.16E+02
-2.18E+02
-1.97E+01
1.78E+02
3.76E+02
5.74E+02
7.72E+02
9.70E+02
-1.21E+03
1.17E+03
_________________ S T R E S S 3
Time = 1.00E+00
-9.56E+02
-7.62E+02
-5.67E+02
-3.73E+02
-1.78E+02
1.64E+01
2.11E+02
4.06E+02
6.00E+02
7.95E+02
9.89E+02
-1.15E+03
1.18E+03
_________________ S T R E S S 3
Time = 1.50E+03
5.31E+01
1.06E+02
1.59E+02
2.12E+02
2.65E+02
3.18E+02
3.72E+02
4.25E+02
4.78E+02
5.31E+02
5.84E+02
6.37E+02
0.00E+00
_________________ Mises Stress
Time = 1.00E+00Time = 1.00E+00
5.95E+01
1.19E+02
1.78E+02
2.38E+02
2.97E+02
3.57E+02
4.16E+02
4.76E+02
5.35E+02
5.95E+02
6.54E+02
7.14E+02
0.00E+00
_________________ Mises Stress
Time = 2.02E+02Time = 2.02E+02
Emerging Spatial and Temporal Discretization Methods 23
Emerging Spatial and Temporal Discretization Methods 25
26
Emerging Spatial and Temporal Discretization Methods 27
Emerging Spatial and Temporal Discretization Methods 29
0 5 10 15 20Time (scaled units)
0
2
6
10
12
16
Ene
rgy
(sca
led
units
)
0 5 10 15 20Time (scaled units)
0
2
6
10
12
16
Ene
rgy
(sca
led
units
)
Emerging Spatial and Temporal Discretization Methods 31
Emerging Spatial and Temporal Discretization Methods 33
Time (scaled units)
0.0
50.0
100.0
150.0
200.0
Ene
rgy
(sca
led
units
)
Time (scaled units)
0.0
50.0
100.0
150.0
200.0
Ene
rgy
(sca
led
units
)
Emerging Spatial and Temporal Discretization Methods 35
Time (milliseconds)
0.00
0.05
0.10
0.15
0.20
0.25
Potential Energy, BallTotal Energy, Ball
Emerging Spatial and Temporal Discretization Methods 37
G. Zavarise and M. Paggi
Reliability of Micromechanical Contact Models 41
G. Zavarise and M. Paggi
Reliability of Micromechanical Contact Models 43
G. Zavarise and M. Paggi
Reliability of Micromechanical Contact Models 45
G. Zavarise and M. Paggi
Reliability of Micromechanical Contact Models 47
G. Zavarise and M. Paggi
Reliability of Micromechanical Contact Models 49
50 G. Zavarise and M. Paggi
Reliability of Micromechanical Contact Models 51
52 G. Zavarise and M. Paggi
Reliability of Micromechanical Contact Models 53
G. Zavarise and M. Paggi
Reliability of Micromechanical Contact Models 55
56 G. Zavarise and M. Paggi
Reliability of Micromechanical Contact Models 57
G. Zavarise and M. Paggi
Reliability of Micromechanical Contact Models 59
60 G. Zavarise and M. Paggi
Reliability of Micromechanical Contact Models 61
62 G. Zavarise and M. Paggi
Reliability of Micromechanical Contact Models 63
G. Zavarise and M. Paggi
Reliability of Micromechanical Contact Models 65
66 G. Zavarise and M. Paggi
Reliability of Micromechanical Contact Models 67
G. Zavarise and M. Paggi
Reliability of Micromechanical Contact Models 69
70 G. Zavarise and M. Paggi
Reliability of Micromechanical Contact Models 71
72 G. Zavarise and M. Paggi
Reliability of Micromechanical Contact Models 73
G. Zavarise and M. Paggi
Reliability of Micromechanical Contact Models 75
76 G. Zavarise and M. Paggi
Reliability of Micromechanical Contact Models 77
G. Zavarise and M. Paggi
Reliability of Micromechanical Contact Models 79
G. Zavarise and M. Paggi
Reliability of Micromechanical Contact Models 81
G. Zavarise and M. Paggi
M. Brinkmeier, et al.
Modern Approaches on Rolling Contact 85
M. Brinkmeier, et al.
Modern Approaches on Rolling Contact 87
M. Brinkmeier, et al.
Modern Approaches on Rolling Contact 89
M. Brinkmeier, et al.
0 50 100 150 200 250
0
0.01
0.02
Tan
g
hom.meshinhom.meshinhom.mesh (bound. term)
Modern Approaches on Rolling Contact 91
M. Brinkmeier, et al.
Modern Approaches on Rolling Contact 93
M. Brinkmeier, et al.
Modern Approaches on Rolling Contact 95
M. Brinkmeier, et al.
Modern Approaches on Rolling Contact 97
M. Brinkmeier, et al.
Modern Approaches on Rolling Contact 99
100 M. Brinkmeier, et al.
Modern Approaches on Rolling Contact 101
102 M. Brinkmeier, et al.
Modern Approaches on Rolling Contact 103
M. Brinkmeier, et al.
Modern Approaches on Rolling Contact 105
106 M. Brinkmeier, et al.
Modern Approaches on Rolling Contact 107
M. Brinkmeier, et al.
Modern Approaches on Rolling Contact 109
110 M. Brinkmeier, et al.
Modern Approaches on Rolling Contact 111
112 M. Brinkmeier, et al.
Modern Approaches on Rolling Contact 113
M. Brinkmeier, et al.
Modern Approaches on Rolling Contact 115
116 M. Brinkmeier, et al.
Modern Approaches on Rolling Contact 117
M. Brinkmeier, et al.
Modern Approaches on Rolling Contact 119
120 M. Brinkmeier, et al.
Modern Approaches on Rolling Contact 121
0 0.5 1 1.50
0.2
0.4
0.6
0.8
1
0.6
0.7
0.8
0.9
1 = 1000 1/s =2.5 1/s = 0.01 1/s
10 10 10 100
101
102
103
104
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
elasticviscoelastic
122 M. Brinkmeier, et al.
Modern Approaches on Rolling Contact 123
M. Brinkmeier, et al.
Modern Approaches on Rolling Contact 125
126 M. Brinkmeier, et al.
Modern Approaches on Rolling Contact 127
Homogenization and Multi-Scale Approaches 131
Homogenization and Multi-Scale Approaches 133
Homogenization and Multi-Scale Approaches 135
Homogenization and Multi-Scale Approaches 137
Homogenization and Multi-Scale Approaches 139
Homogenization and Multi-Scale Approaches 141
Homogenization and Multi-Scale Approaches 143
0.0001
0.001
0.01
0.1
1
10
0.001 0.01 0.1 1 10 100
cz1cz2
cz5
road track
Homogenization and Multi-Scale Approaches 145
mm
mm
mm
mm
mm
mm
0.5
0 0.2
0.6
Homogenization and Multi-Scale Approaches 147
Frequency [1/s]
Mod
ule
[N/m
m²]
1E+111E+071E+051000100.10.001
0.1
10
1
100
1000
10000
0.01
0.001
1E+15
storage modulus
loss modulus
Homogenization and Multi-Scale Approaches 149
150
Homogenization and Multi-Scale Approaches 151
152
Homogenization and Multi-Scale Approaches 153
Homogenization and Multi-Scale Approaches 155
156
Homogenization and Multi-Scale Approaches 157
0
0.05
0.1
0.15
0.25
10 100 1000 10000
0.2
2
0
0.1
0.2
0.5
0.6
10 100 1000 10000horizontal sliding velocity
0.51
51
0
0.1
0.2
0.5
0.6
10 100 1000 10000horizontal sliding velocity
0.20.5
2
Homogenization and Multi-Scale Approaches 159
0
0.1
0.2
0.5
0.6
0.7
10 100 1000 10000horizontal sliding velocity
0.20.5
1
0
0.05
0.1
0.15
0.2
10 100 1000 10000 500
0
0.1
0.2
0.5
0.6
0.7
1 10 100 1000 10000
wl=0.1 mm
wl=0.025 mm
160
0
0.2
0.6
1
1 10 100 1000 10000 100000
simulationexperiment
Homogenization and Multi-Scale Approaches 161
P. Alart
Contact on Multiprocessor Environmen… 165
166 P. Alart
Contact on Multiprocessor Environmen… 167
P. Alart
Contact on Multiprocessor Environmen… 169
170 P. Alart
Contact on Multiprocessor Environmen… 171
172 P. Alart
Contact on Multiprocessor Environmen… 173
P. Alart
Contact on Multiprocessor Environmen… 175
176 P. Alart
Nstep
itera
tions point de bifurcation
Contact on Multiprocessor Environmen… 177
P. Alart
Contact on Multiprocessor Environmen… 179
P. Alart
Contact on Multiprocessor Environmen… 181
P. Alart
Contact on Multiprocessor Environmen… 183
r
r
r r
r r
rd
bb
aa
k+1
k+1
k+1
rc rc
k+1/2
k+1/2
k+1/2
k+1/2 t
n
drk+1
=
status =k / slip
r
r
c
t
n
rk
rk+1/2a
rk+1/2b rb
k+1
k+1ra
bk+1
ak+1
status = slip +k
wak
uak
wck uc
k
rn
rt
wkb
kub
uak
kr
rk
ubk
uak
wak
wbk
ka0=w
r
status = stick status = gap status = slip+
k
k
k
k
=
=
k k k
P. Alart
Contact on Multiprocessor Environmen… 185
P. Alart
Contact on Multiprocessor Environmen… 187
P. Alart
Contact on Multiprocessor Environmen… 189
P. Alart
Contact on Multiprocessor Environmen… 191
P. Alart
Contact on Multiprocessor Environmen… 193
P. Alart
Contact on Multiprocessor Environmen… 195
P. Alart
Contact on Multiprocessor Environmen… 197
P. Alart
0
100
200
500
G.M
.Res
. ite
ratio
ns = 0) = 0.2)
= 0.2) = 0.2)
= 0.2)
0
50
100
150
200
G.M
.Res
. ave
. ite
.
0
20
60
100
120
G.M
.Res
. ave
. ite
.
= 0) = 0.2)
= 0.2) = 0.2)
= 0.2)
shear compression
Contact on Multiprocessor Environmen… 199
200 P. Alart
Contact on Multiprocessor Environmen… 201
202 P. Alart
Contact on Multiprocessor Environmen… 203
P. Alart
Contact on Multiprocessor Environmen… 205
206 P. Alart
Contact on Multiprocessor Environmen… 207
P. Alart
Contact on Multiprocessor Environmen… 209
210 P. Alart
Contact on Multiprocessor Environmen… 211
212 P. Alart
Contact on Multiprocessor Environmen… 213
P. Alart
Contact on Multiprocessor Environmen… 215
216 P. Alart
Contact on Multiprocessor Environmen… 217
220
Numerical Soil Mechanics Experiments… 221
222
Numerical Soil Mechanics Experiments… 223
Numerical Soil Mechanics Experiments… 225
226
Numerical Soil Mechanics Experiments… 227
Numerical Soil Mechanics Experiments… 229
Numerical Soil Mechanics Experiments… 231
Numerical Soil Mechanics Experiments… 233
Numerical Soil Mechanics Experiments… 235
Numerical Soil Mechanics Experiments… 237
Numerical Soil Mechanics Experiments… 239
Numerical Soil Mechanics Experiments… 241
Numerical Soil Mechanics Experiments… 243
Numerical Soil Mechanics Experiments… 245
Numerical Soil Mechanics Experiments… 247