INSTABLE CRACK PROPAGATION AT HIGH TEMPERATURE
Nicolas Tardif (1), Michel Coret (2), Alain Combescure (2), Vincent Koundy(1)
(1) : IRSN DSR/SAGR/BPhAG, BP 17, 92262 Fontenay-aux-Roses, FRANCE
(2) : LaMCoS, INSA-Lyon, F-69621, France
{nicolas.tardif, michel.coret, alain.combescure}@insa-lyon.fr
[email protected]
ABSTRACT
This paper presents high temperature (900°C or 1000 °C) tensile tests on precracked flat 16MND5 test pi eces with a
rectangular section. Cohesive zone model was used to simulate the crack propagation in a very ductile material.
The context of these tests is the study of the rupture of the nuclear reactor bottom head submitted to an accidental mechanical
and thermal loading due to the reactor core melting during an uncontrolled nuclear reaction scenario [1, 2, 3]. The test purpose
is the crack initiation threshold determination and the propagation crack speed characterization.
2
The 16MND5 test piece had a 4 x 25 mm effective area at the 5 mm length notch. The mechanical loading was applied by a
tensile testing machine which was controlled in force. The thermal loading was applied by an inductor heating. We first
imposed a slope in temperature until the aimed one (900°C or 1000°C) and then a slope in stress until rupture. Force and
displacement were recorded. An infrared camera and thermocouples were used to measure the temperature distribution.
Finally, two methods were used to obtain the crack propagation speed. A high-speed digital camera (1000 f/s) enabled us to
find the position of the crack tip. Moreover, we applied a constant electrical current in the test piece and we measured the
voltage on both side of the crack during the propagation. The voltage evolution was correlated to the crack tip position.
To simulate the test, we modelled the crack by a cohesive zone. An elastic damageable behaviour law from Hinte [4, 5] has
been used for the cohesive elements. The 16MND5 behaviour was modelled by an elasto-plastic with kinematic hardening
behaviour [6]. First, we were interested in the Hinte parameters adjustment and then, in the plastic parameters influence on the
crack propagation. The propagation speed resulted from the competition of two phenomena. The snap-back instability,
governed by the ratio of the material stiffness and the interface one, tends to accelerate the propagation [7, 8]. On the
contrary, plastic dissipation in the material tends to reduce the propagation speed.
Introduction
The paper context is the simulation of the rupture of the reactor pressure vessel bottom head under severe accident
conditions. A severe accident causes the fuel degradation by the reactor core melting. The reactor core melting brings about
the formation of a highly calorific corium melt down on the reactor bottom head. Then, two phenomena could imply the bottom
head fracture:
•
A material behaviour alteration due to a local heating by direct contact with the corium melt.
•
An increase of pressure in the reactor due to the bottom head residual water vaporization.
Then, the crack initiation could be caused either by plastic instability or by creep.
In the case of this highly hypothetical scenario, it is important to predict the localization, the size of the crack and the time
before its beginning because these parameters have a strong influence on the evolution and consequences of the accidental
scenario. These three values depend on the crack propagation mode.
Tests were already realized about this problematic type in the case of the American reactors with fifth scale bottom head
model (LHF and OLHF [9, 10]). These tests showed the American bottom head material behaviour variability at 1000°C which
leaded to more or less ductile fracture [11].
From this established fact, a lot of tests were carried out in order to characterize the French bottom head material behaviour
(16MND5) according to its steel grades (RUPTHER [12]).
Within this framework, the tests which are described in this paper, have aimed at the cracking behaviour characterization of
one 16MND5 steel grade: RUPTHER.
So, from a very complex real situation, we were only interested in the problems of crack appearance and of its propagation.
This is why the following tests used a simple structure under controlled loading.
Moreover the tests were numerically simulated. The material behaviour was modelled by an elasto-plastic with kinematic
hardening one. A cohesive zone from Hinte [4, 5] was used to model the crack propagation. Indeed, the cohesive zone model
enabled us to simulate elastic-plastic fracture in ductile metals and enabled us to calculate the crack tip rate. We were
principally interested, first in the Hinte parameters adjustment and then in the 16MND5 plastic parameters influence on the
crack propagation.
Experimental tests
Figure 1 shows the experimental device:
Figure 1. Experimental device
As we said before, the test purpose is the crack initiation threshold characterization and the cracking propagation rate
determination of precracked flat rectangular section specimens submitted to tension at high temperature (900°C or 1000°C).
2
The test piece geometry had a 4 x 25 mm effective area at the 5 mm length notch. The mechanical loading was applied by a
tensile testing machine which was controlled in force. The thermal loading was applied by induction heating. We first imposed
a slope in temperature until the aimed one (900°C o r 1000°C) and then a slope in stress until rupture. The thermal setting rate
was 4°C/s and the mechanical one was 20kN/s.
Force and displacement were recorded by the tensile test machine sensors. An infrared camera and thermocouples were used
to measure the temperature distribution.
Two methods were eventually used to obtain the crack propagation speed:
•
A 180 ampere electrical constant current was applied in the test piece and we measured the voltage on both side of
the notch during the propagation. The crack propagation and the test piece necking increased the resistance (1), so
we obtained a voltage variation during the tests. The voltage evolution, recovered by an oscilloscope, could be
correlated to the crack tip position.
R=
ρ(T ).l
S
(1)
where ρ(T) is the resistivity, T the temperature, l the length travelled by the current, S the area crossed by the current.
•
A digital high speed camera (1000 f/s) enabled us to measure the position of the crack tip at different times of the
cracking. Then we easily deduce the instantaneous crack speed. The fracture revealed by the electrical measure
stops the camera record.
Result
Six tests were carried out using the 16MND5 RUPTHER steel grade: three at 1000°C and three at 900°C.
We were interested in the following cracking thresholds:
•
Maximum stress during the tensile loading
•
Vertical displacement (tensile testing machine cylinder) at the end of the failure.
•
Logarithmic strain at the edge of the fracture calculated as it is described in Figure 2 and equation (2).
Figure 2. Logarithmic strain at the edge of the fracture
 l − l0 

ε = ln
 l0 
(2)
Table 1 recapitulates the results. As we expected, the maximum stress is more important in the case of the tests at 900°C and
the vertical displacement, as for it, is smaller. However, the logarithmic strains are all from the same order of magnitude.
N°
B2-09
B2-10
B2-11
B2-12
B2-13
B2-14
Temperature
°C
900
900
1000
1000
900
1000
Thresholds
Maximum stress (N)
Vertical displacement (mm)
13775
4.17
12257
3.78
11377
4.04
11475
4.1
16652
3.93
10098
4.28
Table 1. Thresholds summary
Logarithmic strain
2.82
2.78
2.61
2.59
Moreover, we were interested in the crack propagation. At these temperatures, the 16MND5 behaviour is highly viscoplastic.
So, before the initiation of the crack, there were large displacements and large deformations in the test piece effective area.
During the crack propagation a necking zone around the crack tip were added to the other phenomena.
Figure 3 presents some frames of the crack propagation from which we calculated the crack speed.
notch
Inductor coil
Crack initiation
Crack propagation
Figure 3. Crack propagation frames
As it is shown in Figure 4, we observed that the crack velocities were very close between -600 and -200ms independently of
the temperature; they increased from 0.02 to 0.08m/s. Then, the cracking accelerated very quickly during the last 200ms until
20m/s at most.
Figure 4. Crack velocities
The crack rate, measured by electrical resistance, just enabled us to check the total time of cracking with regard to the optical
measure (example shown in Figure 5). Indeed, we didn’t calibrate the position of the crack tip with the voltage signal (we
should have known the coefficient of resistivity for each temperature and the real geometry of the test piece effective area at
any time to simulate it numerically).
Figure 5. Electrical measure of cracking example
Numerical analysis
The use of a cohesive zone model enabled us to calculate the cracking rate in a ductile material and the different thresholds.
Indeed, a cohesive zone model behaves like a zip which we unbutton during the cracking. So, we can easily simulate the crack
tip propagation through the cohesive element behaviour law.
We used the elastic damageable cohesive behaviour law from Hinte [4-5] to model the cracking. The 16MND5 behaviour was
modeled by an elasto-plastic with kinematic hardening one without taking into account the viscous phenomena. We used a
simple two-dimensional quasi-static modeling with the small perturbations hypothesis (shown in Figure 6) in order to focus on
the following problematics. We were interested in the adjustment of the Hinte cohesive behaviour law parameters and then in
the influence of the 16MND5 model parameters on the crack propagation.
Figure 6. Test modeling
The Hinte behaviour law (equations 3, 4, 5) has required the setting of 8 parameters and some of them can be linked with the
fracture mechanics [4]: the critical damageable dcrit, the elastic domain Y0, the elastic damageable energy Yc which can be
compared to the energy release rate in pure mode 1 G1C, α and γ which refer to the mixed mode, the normal stiffness k1 (=kn) ,
the shear stiffness k2 (=ks), a coefficient n which refers to the more or less ductility of the material.
ti = (1 − d i ).ki .[ui ]
and
1
Yi = .ki .[ui ]
2
{
Y (t ) ≡ supτ ≤ t (Y2 (τ ))α + (γ .Y1 (τ ))α
n


Y
−
Y
0
n
 
+
d =   n + 1 Y − Y  if
c
0



1 else
(3)
}
1
α
(4)
d = dcrit
(5)
In our pure mode 1 case, the setting of kn and n is the principal problematic. Indeed, if we plot the behaviour law (Figure 7)
with the default parameters which are introduced in Table 2, we observe that the maximum normal stress t2max and the vertical
displacement at rupture ucrit are dependent on both kn and n. The higher (smaller) kn and the smaller (higher) n are, the higher
ucrit (t2max) is.
Y0
(MPa.mm)
0
Yc
(Mpa.mm)
4500
γ
0.1
α
dcrit
1
1
Table 2. Hinte default parameters
kn
(MPa/mm)
13
4E
ks
(Mpa/mm)
13
3.07E
n
0.5
Figure 7. Behaviour law from Hinte dependence on kn and n parameters
If ucrit is too important, the interface normal displacements are not negligible comparatively to the strain of the structure. The
imposed displacement is absorbed in the interface of theoretically null thickness and implies that the displacement solution is
wrong. Moreover, the crack initiation appears late and the propagation is too fast because all the cohesive elements are
damaged at the crack initiation and not just the elements near the crack tip.
On the contrary, if t2max is too important, the stress at the crack tip is too small comparatively to t2max, so the cohesive element
can’t be damaged. So the crack initiation is too late and the crack propagation is too slow. In these two cases, we don’t have
numerical convergence when we use a more accurate mesh.
So a method was used to set these two parameters. First, we chose a quite small value of n because the material is ductile. By
comparison with the rupture mechanics in pure mode 1 and with its elastic linear hypothesis, we can situate the convergence
range of kn. The stress t2max of the cohesive model have to be smaller than the stress σyyu at the middle of the cohesive
element just after the crack tip (equation 6 and Figure 8) [13].
Figure 8. Setting of kn criterion
t2 max ≤ σ yyu ≡
K1c
π .h
where K1c is the pure mode 1 critical stress intensity factor and h the length of the cohesive element.
Using equations 3, 5 and 6, we find the equation 7 [13]:
(6)
2. n +1
n
k2 ≤ k2 max
n
1 (2.n + 1)
.
.
≡
n + 1 2.Yc
(2.n) 2
≡
.
2. n +1
n
n
(2.n + 1)
.
2(n + 1)
(2.n) 2
.
K12c
π .h
(7)
ET
π .h
where ET is the apparent modulus of the 16MND5 material.
Then, we were able to find the size of the range with a crack initiation criterion by a variation of kn. However, we had to
introduce a very small viscosity in the Hinte cohesive zone model in order to regularize easily unstable behaviour during crack
nucleation [7] (equation 8). These instabilities are generally caused by an elastic energy release (elastic unloading) due to
stress redistributions higher than the dissipation of the damage process. They appeared as snap-backs on a forcedisplacement curve that is simultaneous decrease of the force and the displacement [14, 8].
n
 n Y − Y0 
∂d
+
= KK . 
−d
∂t
 n + 1 Yc − Y0 


M
if
d < d crit
(8)
We recapitulate all the default values of the modeling that we used to study the crack propagation dependence on the plastic
parameters of the 16MND5 model in Table 3:
Y0
(MPa.mm)
0
Yc
(Mpa.mm)
31.36
γ
0.1
HINTE
α
1
VISCOHINTE
kn
n
KK
M
(Mpa/mm)
(1/s)
1
160000
0.1
100000
1
Table 3. Modeling parameters
dcrit
ELASTIC PLASTIC
E
σy
H
(Mpa)
(Mpa)
(Mpa)
28700
150
E/20
Figure 9 shows the crack propagation and the force-displacement curves for different values of the material yield point σy.
Figure 9. Crack propagation and force-displacement curves for different values of σy
On Figure 9, we can see that, at the beginning of the crack propagation, all works as we expected in plastic energy dissipation
logic. Indeed the more the material ductile is (a small value of σy), the later the crack initiation is and the slower the
propagation is. A small value of the yield point implies a fast passing in the plastic domain. The energy due to the imposed
displacements is dissipated more rapidly by plasticity. So the cohesive elements dissipate a smaller part of energy than in the
case of an elastic material.
However, after 5mm crack propagation, a snap-back instability reverse the tendency. The criterion which enables us to
observe a snap-back instability is the separation of the Hinte and viscohinte curves for each temperature. We can observe that
the more plastic the material is, the earlier the snap-back instability appears during the crack propagation. It implies that the
propagation is faster. Indeed, the snap-back instability depends on a parameter which specifies the stiffness of the material
compared with that of the interface [7]. If this parameter is too small, a snap-back instability happens and accelerate the
propagation. In our case, the more we are in the plastic domain, the smaller the apparent modulus is and so, the smaller the
stiffness of the material is. As for it, the interface stiffness remains the same. So, the more the material ductile is, the sooner
comes the snap-back.
So, the crack propagation results from the competition of two phenomena which both depend of the plasticity. The snap-back
instability, due to the fall of the apparent modulus when the material is in the plastic domain, tends to accelerate the crack
propagation. On the contrary, the plastic energy dissipation in the material tends to reduce the propagation speed.
Conclusions
To conclude, we studied the crack propagation behaviour in a 16MND5 steel grade at high temperature. Tests enabled us to
obtain values of the different thresholds of the crack propagation and also its rate at 900°C and 1000 °C.
These tests have enabled us to simulate the crack propagation behaviour with a cohesive zone model. In a first time, we were
interested in the Hinte model parameters adjustment with a simple two dimensional quasi-static modeling with small
perturbations hypothesis in order to solve discretization convergence problem. Then, we studied the influence of the elastoplastic with kinematic hardening behaviour modeling of the 16MND5 on the crack propagation. The crack propagation resulted
from the competition of two phenomena. The snap-back instability, governed by the ratio of the material stiffness and the
interface one, tends to accelerate the propagation. On the contrary, the plastic dissipation in the material tends to reduce the
propagation speed.
So, the tests simulation remains at its beginning. We have to take into account the viscous phenomena, the large
displacements and the large deformations induced by the 16MND5 behaviour at these temperatures. The formation of a
necking zone around the crack tip also implies a 3-dimensionnal modeling.
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