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12th International Conference
on Fracture
Study of crack propagation by
infrared thermography during very
high cycle fatigue regime
N. RANC*, D. WAGNER**, L. ILLOUL*,
PC. PARIS***, C. BATHIAS**
*Laboratoire PIMM, ENSAM-ParisTech, FRANCE
**LEEE, University of Paris X, FRANCE
***LAMEFIP, ENSAM-ParisTech, Bordeaux FRANCE
Introduction
• Many elements of mechanical structures can be loaded
beyond 107 cycles :
⇒ Very high cycle fatigue regime (VHCF) or gigacycle
fatigue regime
• Fracture mechanisms associated with the VHCF are
different from those known in high cycle fatigue (HCF) :
• Fatigue failure can occur at values of cycle exceed 107 cycles
and for stress level below the conventional high cycle fatigue
limit.
• In high yield strength steels the fracture does not occur on the
surface but rather internally in the materials : formation of a
fish eye
2
Experimental study of the crack
propagation in the VHCF regime
•
Experimental difficulties :
– High frequency : In gigacycle fatigue test is carried out at ultrasonic
frequency (20kHz) one cycle : 50µs
– Test duration : 14 hours for 109 cycles
– Internal crack initiation
Ö Is it possible to use pyrometry to understand and to study the fatigue
initiation in VHCF regime?
•
Advantage of pyrometry technique:
–
–
–
–
Good spatial and time resolutions
It allows to obtain field measurements on the surface of the specimen
The dissipated power is higher than at low frequency
It allows to detect the thermal effects associated with an internal crack
propagation (the fish eye formation, i.e. crack propagation , inside the material in the
gigacycle fatigue regime is related to a local increase of the temperature due to plasticity on
the crack tip.)
3
Experimental device : ultrasonic
fatigue test with measurement of
temperature fields
Experimental device : ultrasonic
fatigue test
• Ultrasonic fatigue machine
• Geometry of the specimen
R1=1.5mm
R2=10mm
L1=16mm
L2=10mm
5
Experimental device : temperature
field measurement
• Temperature measurement
by infrared pyrometry:
– MCT detector
– Wavelength range : 3.7µm – 4.9µm
– Aperture time : 100µs-1500µs
– Refresh time : 16ms – 40ms
– Spatial resolution : 0.12mm/pixel
Experimental device : test with R=0.01
6
Temperature evolution during an
ultrasonic fatigue test
•
Case of a high-strength steel :
(0.22% of C; 1.22% of Mn; 1% of
Cr; 1% of Ni); UTS = 950MPa
Exothermic Martensitic
transformation
Fracture
260
200
240
180
220
160
200
140
Temperature in °C
Temperature variation in °C
Case of another high-strength
steel : (0.22% of C; Mn, Cr, Si:
less than 1%); UTS ≈ 1700MPa
•
120
100
80
100 MPa
200 MPa
275 MPa
300 MPa
325 MPa
60
40
20
180
160
140
315MPa
120
100
80
60
40
0
20
0.0
5.0x10
7
1.0x10
8
1.5x10
Cycle
8
2.0x10
8
2.5x10
8
3.0x10
8
0.0
2.0x10
6
4.0x10
6
6.0x10
6
8.0x10
6
1.0x10
7
1.2x10
Cycle
Specimen is cooled by an air circulation 7in
order to limit its heating
7
Temperature field just before the fracture
•
R=-1; ∆σ=335MPa; NF=8.37x107 cycles (gigacycle domain)
Temperature
in °C
340
(5)
Temperature in °C
320
300
(1)
(2)
N = 8.35890x107
cycles
N = 8.36500x107
cycles
(4)
280
260
(3)
240
(1)
(2)
220
8.358x10
7
8.361x10
7
8.364x10
7
8.367x10
7
8.370x10
7
Cycle
(3)
(4)
(5)
N = 8.36968x107
cycles
N = 8.36996x107
cycles
N = 8.37000x107
cycles8
Post mortem observation of the fracture
surface
• Optical microscopy
Observation
direction
• Scanning electronic
microscopy
rc
Inclusion in the center of the
fatigue crack :
0.5mm
Propagation of a fish eye fatigue crack
⇒Average radius of the
inclusion: aint=7.6µm
⇒Eccentricity: e=rc/R1=0.81
9
Thermal model of the crack
propagation
The model of crack propagation 1
•
Paris-Hertzberg-McClintock
crack growth rate law :
⎛ ∆K eff
da
= b⎜⎜
dN
⎝E b
⎞
⎟⎟
⎠
3
b norm of the Bürgers vector
E Young modulus
•
On the threshold corner
da
=b
dN
and
∆K 0
=1
E b
11
The model of crack propagation 2
∆K eff = ∆K =
2
π
Circular
crack
a(t)
By supposing a circular crack and neglecting crack
closure:
∆σ πa
Cross section
of the
specimen
Geometry of the circular crack
The crack growth rate law is thus written:
⎛a⎞
⎛ ∆K ⎞
da
⎟⎟ = b⎜⎜ ⎟⎟
= b⎜⎜
dN
⎝ a0 ⎠
⎝ ∆K 0 ⎠
3
1.8
2
After integration between a0 (t=0) and
a (time t):
a (t ) =
a0
⎛
t ⎞
⎜⎜1 − ⎟⎟
⎝ tc ⎠
2
with
2 a0
tc =
bf
1.6
Crack radius [mm]
3
2.0
1.4
1.2
1.0
The crack reaches the
specimen surface
0.8
0.6
tf=3.7s
0.4
0.2
0.0
0.0
0.5
1.0
1.5
2.0
Time [s]
2.5
3.0
3.5
4.0
Evolution of the crack radius12
(a0=8µm; e=0.8; tc=4.4s)
Calculation of the dissipated energy
during crack growth
•
The reverse plastic zone size
in plane strain:
K
Kmax
∆K
ry
1 ∆K 2
rR = =
4 6π (2σ y ) 2
t
ry
rR<0.13µm
•
Energy dissipated per unit
length of crack:
E = ηr
2
R
σy
K max
2πx
Plastic
zone
(radius ry)
∆K
2πx
Reverse
plastic zone
(radius rR)
−σy
rR
σy −
∆K
2πx
13
Calculation of the dissipated energy
during crack growth
Evolution of the dissipated
energy versus time:
∆K 4
2
E = ηrR = η 2 2 4
24 π σ y
36π σ
4
4
y
=η
9000
8000
7000
a02 ∆σ 4
6000
⎛
t⎞
36π 4σ y4 ⎜⎜1 − ⎟⎟
⎝ tc ⎠
4
Evolution of the dissipated
power versus time:
P = Ef =
P0
⎛
t⎞
⎜⎜1 − ⎟⎟
⎝ tc ⎠
4
with
tc =
P/P0
E =η
a2
10000
5000
4000
3000
tf=3.7s
2000
1000
2 a0
bf
P0 is a constant which is identified from
experimental data
0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
Time [s]
Evolution of dissipated power
14
(a0=8µm; e=0.8; tc=4.4s)
Modeling of the thermal problem
• Thermal model assumptions:
– The specimen is modeled by a cylinder with a radius
of R1=1.5mm
– For symmetry reasons, only one half of the specimen
is consider
– rR is very small (about 0.13µm) : the fatigue crack is
modeled by a dissipated power along the crack tip (P
is the dissipated power per unit length)
15
Heat transfer equation
• Heat transfer equation:
ρC
∂T P(t )
=
δ (rc − a(t ))δ ( z ) + λ∆T
∂t
2
• Initial condition
ℓ=5R1
T (t = 0) = T0
a(t)
• Boundary condition
(adiabatic conditions)
∂T
∂T
∂T
( z = 0) = 0
(
z
=
l
)
=
0
(r = R1 ) = 0
∂z
∂z
∂r
A
rc
Maximum surface
temperature 16
Numerical simulation of the
temperature field 1
• The thermal problem is solve numerically with a
finite element method:
– Linear finite element
– The integration scheme is implicit
– The mesh is more refined in the zone close to the
plan of crack propagation : 100 elements on the
specimen radius
• Parameters of the simulation:
– Initial crack radius: a0=8µm
– Eccentricity: e=0.8
– Characteristic time: tc=4.4s
17
Identification of the dissipated
power P0
•
Minimization of least squares method :
340
P0=2.9Wm-1
320
Temperature [°C]
Minimization of least squares of
the difference between the
maximum temperature measured
in experiments and the
calculated temperature:
Experiment
Model
300
280
260
Crack
initiation
240
220
8.358x10
7
8.361x10
7
8.364x10
7
8.367x10
7
8.370x10
Cycle number
18
7
Numerical simulation associated to
the experimental configuration
Temperature in °C
340
366
Temperature in °C
351
(7)
320
336
(6)
300
321
280
260
307
(5)
Crack
initiation
292
(4)
277
(3)
(2)
(1)
240
262
248
220
7
8.362x10
8.364x10
7
8.366x10
7
Cycle
8.368x10
7
8.370x10
7
(7) N = 73594
(2)
(1)
(3)
(4)
(5)
(6)
69301 cycles;
67786
70530
71553
72342
73174
t = 3.68s;
3.46s; a = 0.270mm
3.39s;
3.53s;
3.58s;
3.62s;
3.66s;
0.164mm
0.142mm
0.188mm
0.210mm
0.233mm
0.256mm
19
Experimental determination of the
crack initiation
•
Experimental crack
initiation criterion:
increase of 0.07◦C of the
filtered temperature
(a≈0.02mm)
Crack propagation:
between 7x104 and 9x104
cycles
⇒ very small part of
the life of the
specimen
240
Experimetal result
Filtered curve
238
Temperature [°C]
•
236
234
7x104 – 9x104 cycles
Prediction of the
crack initiation
232
230
8.358x10
7
8.361x10
7
8.364x10
7
8.367x10
7
8.370x10
Cycle number
20
7
Conclusion
• Experimental approach : temperature
measurement during ultrasonic fatigue test
• Modeling of the thermal effect associated to the
crack propagation :
– Crack propagation
– Dissipated power during crack growth
– Modeling of the thermal problem and numerical
solution
• Results
– Good correlation with the experiment
– Good estimation of the crack propagation duration
21