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Experimental and numerical study of fracture mechanisms in UO2 nuclear fuel
Engineering Failure Analysis 47 (2015) 299–311
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Engineering Failure Analysis journal homepage: www.elsevier.com/locate/engfailanal
Experimental and numerical study of fracture mechanisms in UO2 nuclear fuel J.-M. Gatt ⇑, J. Sercombe, I. Aubrun, J.-C. Ménard CEA, DEN, DEC, F-13108 St Paul Lez Durance, France
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 25 August 2013 Received in revised form 17 July 2014 Accepted 21 July 2014 Available online 28 August 2014
In this paper we study the brittle behavior of UO2 nuclear fuel. Firstly, we present the interpretation of bending tests with three different approaches to assess rupture parameters (critical stress and surface energy). Secondly, we present Vickers’ indentation tests on fresh UO2 fuel. The comparison between bending and indentation tests on fresh fuel allows us to assess the parameters necessary to derive the critical stress and the surface energy from indentation tests. Vickers’ indentation is then used to evaluate rupture parameters of irradiated fuels. At the end, we present some applications to fuel rod modeling taking into account the different rupture mechanisms. Ó 2014 Elsevier Ltd. All rights reserved.
Keywords: Fracture Bending tests Indentation tests Nuclear fuel
1. Introduction This paper deals with the rupture of UO2 fuel in a nuclear reactor. At low temperature (<900 °C) this ceramic has a brittle behavior in traction. It has been shown that radial cracks in the rim of a UO2 pellet are formed due to pellet expansion (thermal expansion and fission gas swelling), induce a stress relaxation in the surrounding cladding and thus lead to an improved behavior of the fuel rod in reactor [1]. Therefore, the knowledge of rupture parameters for irradiated UO2 fuel is very important to model and understand the behavior of the fuel rod in reactor. The rupture behavior of fresh fuel (before irradiation) has been studied mostly in the 1970s and 1980s. The rupture stress and toughness were measured [2–4] on fresh fuel and more recently the toughness on irradiated fuel has been assessed [5]. The rupture stress has never been measured on irradiated fuel because of the fuel fragmentation during irradiation which leads to pieces too small for mechanical testing. The crack models used in numerical simulations of fuel rod behavior are based on two main parameters [1,6]: the critical stress and the surface energy. The aim of this work is the development of a methodology to assess irradiated fuel rupture parameters used in fuel rod modeling. In the first part of this work, we propose some simulations of bending tests on smooth and notched specimens with two different models. In the second part, we use an approach developed in Ref. [10] to assess the critical stress used in the crack models from indentation tests. In the third part, we present some experimental results obtained from bending tests on fresh fuel and indentation tests on fresh and irradiated fuels. We use indentation test results to determine the parameters of crack models for irradiated fuel. In the fourth part, we show an application of the crack model to the simulation of fuel pellet cracking during a typical incidental in-reactor irradiation history (power ramp).
⇑ Corresponding author. Tel.: +33 442 254 267. E-mail addresses: [email protected] [email protected] (J.-C. Ménard).
(J.-M.
http://dx.doi.org/10.1016/j.engfailanal.2014.07.019 1350-6307/Ó 2014 Elsevier Ltd. All rights reserved.
Gatt),
[email protected]
(J.
Sercombe),
[email protected]
(I.
Aubrun),
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2. Assessment of rupture parameters used in the crack models To model crack propagation in a fuel pellet during a power ramp, we use a simplified smeared crack model in finite element simulations [1] (the shear stresses are not considered and the crack propagation directions are fixed). Further, in this study, we consider another type of crack model: the cohesive zone model. These models depend on two rupture parameters: the critical stress and the surface energy. The aim of this chapter is to define the relationship between these two rupture parameters and the measured quantities in bending tests performed on smooth and notched specimens. 2.1. Cohesive zone model This model rests on the relationship between the stress applied (rn) and the crack opening (un) as illustrated in Fig. 1. The behavior is easily described with these equations:
8 rc > < Kn ¼ d 1 Gc ¼ 2 rc dc > : K ¼ rc f ddc
ð1Þ
The first equation defines the slope at the origin, with rc the critical stress and d a parameter. The second equation defines the surface energy (Gc), with d a parameter. The last equation defines the second slope. From Eq. (1), the following behavior law can be obtained:
rn ¼
8 K u > < n n > :
dun dc un
0
þ1
if un < d
rc if d 6 un 6 dc
ð2Þ
if un > dc
The following condition must be added for model consistency:
Kn >
r2c
ð3Þ
2Gc
This condition is equivalent to: d < dc. The value of Kn has to be very high to avoid an important displacement before that critical stress is reached, and not too high due to numerical instabilities. In a certain range (we take Kn = 6.1014 Pa/m), the response of the model is independent of this parameter, and therefore independent of d which is very small. Thus, if we know the rupture parameters (rc, Gc) it is possible to calculate dc using the second equation of system (1). This model has been used to simulate a bending test. The size of specimen is 10 1.5 1.5 mm3. The mesh of the smooth sample is shown in Fig. 2. For this simulation we have fixed the surface energy at 30 J/m2. From a parametric study, we have found the optimal value of rc allowing us to fit test results, as shown in Fig. 3b. This figure shows a result of a bending test at a prescribed displacement rate 20 lm/min, where the time evolutions of the applied force F was measured. The maximum force is referred here as the rupture strength FR and is equal to 27 N. This force is linked to rupture stress rR = 120 MPa by the following equation:
rR ¼
3L FR 2wt2
ð4Þ
L, w and t are respectively the length, the width and the thickness of the sample. In Fig. 3a (zoom of the end part of the test), we have plotted the time evolution of the stress (rxx) at each node of the plane of symmetry of the sample (we have 11 nodes along the sample0 s thickness). We see from Fig. 3a that the curves (x = 0 and x = 0.075 mm) start to decrease when the stress
Fig. 1. Cohesive zone model.
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301
Fig. 2. Mesh of the sample and sample at the end of simulation.
110
90
σ (MPa)
70
50
30
10 0 67
72
77
82
87
92
97
t (s) Fig. 3a. Stress (ry) variation versus thickness (x) and time.
30 25
F (N)
20 15 10 5 0 0
20
40
60
80
100
120
140
t (s) Fig. 3b. Force variation versus time.
reaches the critical stress (equal at 98 MPa in this case). A few seconds after that, the sample fails, i.e., all the stresses tend to zero. From this observation, we defined a Critical Length (CL), above which the sample fails. This CL is the length of the line constituted by the set of Gauss points where the stress decreases after the critical stress is reached. This length can be evaluated between 75 and 150 lm (the uncertainty is due to the finite element size). This evaluation shows that the crack size is great in comparison of materiel grain size and porosities size distribution (<10 lm). This means that the crack can be considered at the macroscopic scale. Furthermore, from our simulations, we found that to different surface energy and rupture stress (rR, Gc), corresponds a different critical stress rc. When we change the thickness of the specimen we find a different
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critical stress value for the same couple rc. When we change the thickness of the specimen we find a different critical stress value for the same couple (rR, Gc). This means that rc depends on the surface energy, the rupture stress and the thickness of the sample. All these dependencies are dealt in the next part of the paper. 2.2. Smeared crack model The smeared crack model used in this paper is described in [1]. This model is used in our code to simulate the fuel cracking during in reactor irradiation sequences. The greatest difference with a cohesive zone model is that we define an equivalent crack strain from the crack opening. Thus the behavior is described by a stress versus crack strain relationship instead of a stress versus displacement as it is the case in the cohesive zone model. The advantage of this model is that it can be easily written in the framework of the finite element method without introducing another type of finite element (e.g., joint elements for the cohesive zone model). The downside is that we have to introduce the length of the finite element in its formulation. According to Fig. 4, which illustrates the stress–strain relationship, we have the following equation:
Efiss ¼
1 r2c Lf 2 Gc
ð5Þ
Efiss is a crack softening modulus, and Lf the finite element length in the direction perpendicular to the crack plane. With this model, we have simulated the previously described bending test on smooth sample using a plane stress hypothesis. The mesh and the crack strains obtained from the simulation are showed in Fig. 5. The area in other colors than blue is where the stress has reached the critical stress. With this model, we obtained very similar estimates of the critical stress than with the cohesive zone model in function of the rupture stress and of the energy surface (see Fig. 12) which will be detailed latter. We have also simulated a bending test on a notched sample (28 4 4 mm3). The mesh is showed in Fig. 6. The calculated crack strains and stresses are illustrated in Figs. 7a and 7b. In Fig. 7a, the area in other colors than blue is where the stress has reached the critical stress. In Fig. 7b the evolution of the stress (ryy) versus x axis (the ligament) is given at different times. The stress rises up till time 15 s is reached. At time 15 s, the stress has reached the critical stress and starts to decrease afterwards. At times 20 s and 28 s, the stress has decreased at more nodes. Up to time 28 s, the stress is not null meaning that the crack is not fully opened. After 28 s, all the stresses tend to zero indicating that the sample is broken. The stress distribution at time 28 s gives an estimation of the critical crack length (CL), obviously greater than 100 lm. This value is great in comparison of the grain size and the porosity size distribution in commercial UO2 fuels. This means that the crack length can be considered a macroscopic quantity. 2.3. Conclusion on the crack models The finite element based interpretation of bending tests on smooth and notched samples with two crack models shows that the Critical Length is much greater than the grain size and the defect size of UO2 materials. We therefore can conclude that measurements of toughness Kc from notched samples and of rupture stress rR from smooth samples lead macroscopic quantities. They are not local measurements at the porosity scale. In the case of bending test on smooth sample, we have shown that there exist a relationship between the critical stress rc used in the crack model and experimental parameters: the rupture stress, the surface energy (toughness) and the thickness of the sample (rR, Gc, e). The goal of the next chapter is to analyze this relationship, and to find an analytical estimate of the critical stress such that finite element simulations of the bending tests can be avoided.
Fig. 4. Smeared crack model.
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303
Fig. 5. Result of simulation with smeared crack model.
Fig. 6. Mesh of a notched sample.
3. Analytical approach We now propose to use an analytical approach to interpret the bending tests on smooth specimen. This approach, introduced by Leguillon [8], allowed us to study the failure of smooth sample during a bending test according to energy and stress criteria. These two criteria read. 3.1. Stress criterion
kðxÞrmax P rc With:
ð6Þ
3FL d 2d rmax ¼ 2wt 2 ; x ¼ t and KðxÞ ¼ 1 t ¼ 1 2x, the parameters are defined in Fig. 8.
3.2. Energy criterion
pffiffi AðxÞ trmax P K c pffiffiffiffiffiffiffiffi A(x) is a function of x defined below, and K c ¼ Gc E in plane stress conditions (E is the Young0 s modulus).
ð7Þ
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crack
Fig. 7a. Simulation results (smeared crack model).
Fig. 7b. Stress variation along the ligament.
x
w t d
L Fig. 8. Notations.
Using [9], we obtain the following equations for the fracture toughness:
FR 2wt 2 K c ¼ Y pffiffi with F R ¼ rR 3L b t p ffiffi ffi 3 Lt x a Y¼ ½1:99 xð1 xÞð2:15 3:93x þ 2:7x2 Þ and x ¼ t 2ð1 þ 2xÞð1 xÞ1:5
ð8Þ
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305
The stress criterion is reached before the energy criterion in very small structures. In this case, the failure of the structure is controlled by the surface energy. On the contrary, the energy criterion is reached before the stress criterion in structures of great size. The stored energy is important. The failure is controlled by the stress. In a bending test, the failure criterion depends on the size of the sample and on the value of the rupture parameters (rc, Gc). To define the A function of Eq. (7), we use the Eq. (8), and we obtain:
AðxÞ ¼
pffiffiffi xPðxÞ
ð9Þ
ð1 þ 2xÞð1 xÞ3=2
With: PðxÞ ¼ 1:99 xð1 xÞð2:15 3:93x þ 2:7x2 Þ. The function A is plotted in Fig. 9. We have the following properties for x 2 ½0; 1: A(0) = 0. A is continuous and strictly increasing. A tends towards infinity when x tends towards 1. The failure of a smooth specimen is obtained if the two conditions (6) and (7) are verified:
(
pffiffi AðxÞ t ¼ kð^xÞ Krcc
ð10Þ
kð^xÞrR ¼ rc
^¼^ d xt is the CL. The last equation of system (10) gives the link between the rupture stress rR and the critical stress rc. The CL is obtained thanks to the following equation.
Kc Að^xÞ ¼ pffiffi ¼ rR t
sffiffiffiffiffiffiffiffi EGc tr2R
ð11Þ
According to the second equation of system (10) and the definition of k(x), we have:
rc ¼ 1 2^x rR
ð12Þ
Using Eqs. (11) and (12) we can plot the curve shown in Fig. 10. In this figure, we can see that if we increase the thickness of the sample or if we decrease the surface energy then the critical stress tends towards the rupture stress. At the limit, if the surface energy is equal to zero, then the critical stress equals the rupture stress. Thus, if we know (rR, Gc), it is possible by using Eqs. (11) and (12) or Fig. 10 to deduce the critical stress used in the mod^¼^ els. To evaluate the CL we use the equation d xt (see Fig. 11). The CL evaluated by the simulation of smooth samples bending (between 75 and 150 lm) has the same order of magnitude than the one obtained from our analytical approach: 134 lm (for rR = 120 MPa and Gc = 30 J/m2). Finally, in Fig. 12, we compare the relationship between the critical stress and the rupture stress obtained from the three approaches developed in this paper (FE simulations with cohesive zone model and smeared crack model and the analytical approach). We observe a very good agreement between the three approaches which validates the proposed analytical correlation. These three approaches are consistent between them.
Fig. 9. A function.
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Fig. 10. Curve allowing assessing the critical stress from rupture stress and Kc.
180 160 140
CL (µm)
120 100 80 60 40 20 0 80
90
100
110
120
130
140
150
σR (MPa) small sample
large sample
Fig. 11. Critical length for large and small samples versus rupture stress.
4. Bending tests on fresh fuel. Experimental results We performed bending tests on smooth and notched samples. The tests on the smooth samples were used to estimate the rupture stress (rR), and the tests on the notched samples to obtain the fracture toughness Kc. Large (28 4 4 mm3) and small samples (10 1.5 1.5 mm3) of UO2 with large or small grain sizes have been used in the tests. The fuels were porous and the microstructures of the samples very different from one another. Table 1 gives the upper and lower estimated values for Kc and rc. Kc has been calculated using Eq. (8) and rc using Eqs. (11) and (12). The experimental scatter is important. The number of tests is not sufficient for a Weibull analysis. New tests are planned to complete this experimental database.
5. Analysis of Vickers’ indentation test A series of Vickers’ indentation tests were performed on the materials used in the bending tests, small and large grains UO2. In this part, we present the methodology we developed to estimate the rupture parameters, Kc and rc, from the analysis of Vickers test results and the comparison to bending test results.
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J.-M. Gatt et al. / Engineering Failure Analysis 47 (2015) 299–311 180 160
y = 0.8874x + 30.678 R² = 0.9999
σR (MPa)
140 AM
120
CZM SCM
100 80 60
40
60
80
100
120
140
160
σc (MPa) Fig. 12. Comparison of the three modeling approaches: cohesive zone model (CZM), smeared crack model (SCM) and analytical (AM).
Table 1 Experimental results: bending tests with notched sample for Kc and with smooth samples for rc. Kc min (MPa UO2 UO2 UO2 UO2
small sample, small grain small sample, large grain large sample, small grain large sample, large grain
1.63 1.98 2.08 1.8
pffiffiffiffiffi m)
Kc max (MPa
pffiffiffiffiffi m)
1.89 2.98 2.6 2.18
rc min (MPa)
rc max (MPa)
83 116 107 87
125 147 128 101
5.1. Critical stress intensity factor The indentation test consists in punching a material with an indenter by imposing a normal force. After unloading, if the load is high enough, four cracks are observed as depicted in Fig. 13. Two kinds of cracks can appear: Palmqvist cracks (P) and Median cracks (M) as shown in Fig. 13. It is generally difficult to know with certainty which mechanism is activated. Thus, we propose here to use an arbitrary simple criterion: if c > 2a we consider the mechanism M (see Fig. 13 for the notations). This criterion allowed us to obtain a good interpretation of our measurements. For M type cracks we used the following equation [10,11]:
rffiffiffiffi E P K c ¼ jM H c3=2
ð13Þ
with E the Young0 s modulus, H the hardness, P the loading and jM a constant. In the same way, Niihara [13] proposed the following equation for P type cracks:
K c ¼ jP
2=5 E P pffiffi H a ‘
ð14Þ
where jP is a constant for mechanism P. 5.2. Critical stress To obtain all the rupture parameters used in the crack models, an evaluation of the critical stress from Vickers indentation tests is essential. To reach this objective, we have considered that a hypothetical bending test is performed on a sample after indentation according to the methodology presented in reference [12]. During the bending test, after a stable propagation of
Fig. 13. (a) Diagram of indentation surface, (b) median type crack (M), (c) Palmqvist type crack (P).
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damage, the crack due to indentation becomes unstable. To interpret this test (see Fig. 14), we have considered two stress intensity factors: the first, due to the residual stresses which hold the crack open after indentation, is proportional to P/c3/2 (it pffiffiffi decreases when the crack increases), the second, due to bending, is proportional to r c (it increases when the crack increases). The overall stress intensity factor reads therefore as follows [12]:
K ¼ j0M
rffiffiffiffi pffiffiffi E P þ fM r c H c3=2
ð15Þ
with j0 M and fM two constant parameters (note that j0 M is different from jM). The crack becomes unstable when:
@K ðcL Þ ¼ 0 and K c ¼ KðcL Þ @c
ð16Þ
From Eqs. (15) and (16), we obtained the following equations:
3 0 j 2 M
rffiffiffiffi E P 1 rc þ fM pffiffiffiffiffi ¼ 0 H c5=2 2 cL L
ð17Þ
rffiffiffiffi pffiffiffiffiffi E P þ fM rc cL H cL3=2
ð18Þ
And:
K c ¼ j0M
Eqs (17) and (18) can also be written:
rc c2L ¼
3j0M fM
rffiffiffiffi E P H
ð19Þ
And:
Kc ¼ j0M cL
rffiffiffiffi E P rc þ fM pffiffiffiffiffi H cL5=2 cL
ð20Þ
From Eqs. (20) and (17), we obtained the following equations.
8 qffiffiffi E P < K c ¼ 4j0 M H c3=2 L pffiffiffiffiffi : K c ¼ 4f3M rc cL
ð21Þ
These two equations show that 1/4 of Kc is due to indentation test and 3/4 to bending test. Using (21), we found the following relation for crack type M:
rc
rffiffiffiffi rffiffiffiffi E P E P ¼3 ¼ 3hM H c2 fM H c 2
j0M
ð22Þ
Using the same approach for cracks of type P, we found:
rc ¼
2=5
j0M E fM
H
2=5 P E P ¼ hP a‘ H a‘
ð23Þ
The parameters hM and hP are constants measured experimentally. Thus, Eqs. (13) and (14) for Kc and (22) and (23) for the critical stress allowed us to assess the rupture parameters from Vickers’ indentation tests. All the parameters can be
Fig. 14. (a) Median type crack induced by indentation and (b) crack extend under combined action of residual stress and tensile stress.
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considered as macroscopic quantities since the length of the cracks formed by indentation (20 lm) is greater than the typical size of the pores in the grains (<1 lm). 5.3. Determination of the constants The indentation tests were carried out on the same material as the one used for the bending tests. The goal of this determination was not to have a good accuracy on the values (the number of available tests is not adequate for that), but to show the relative behavior of the different fuels. From the estimated measurements of the fracture toughness in the bending tests, we determined the Vickers’ indentation constants associated to Kc. The goal of this determination, based on the values reported in Table 1, is to find the variation of Kc measured on the different samples (bending test on notched samples), as shown in Fig. 15. From this process, we found the following values:
jM ¼ 0:0432 and jP ¼ 0:00445 From the bending test (smooth samples), the critical stress can be evaluated from the rupture stress and Gc, using Fig. 10. We determined the constants (hM, hP) associated with critical stress (Eqs. (22) and (23)) to find the relative variation of rc evaluated from bending tests on the different fuels. This process led to following values:
hM ¼ 0:003585 and hP ¼ 0:000896 The choice of the criteria (if c > 2a we consider the mechanism M) to determine the activated mechanism is very important. This process shows a posteriori that it was a good choice. Fig. 15 compares the results obtained from the bending tests and the Vickers’ indentation tests. We can conclude that: We have a significant scattering in indentation tests on UO2. The P and M mechanisms are activated for the small grain size UO2, while the M mechanism only is activated in the case of the large grain size UO2. The order of magnitude of the different parameters and their relative evolutions with material characteristics and sample sizes are in good agreement. 6. Irradiation effects A series of Vickers’ indentation tests have been performed on commercial UO2 irradiated during various times in reactor. The test results were interpreted with Eqs. (13) and (14) to estimate the fracture toughness Kc and Eqs. (22) and (23) to evaluate the critical stress. Note that the constants in these equations used are those obtained from the tests performed on non-irradiated (fresh) fuel samples. In Figs. 16 and 17, we show a comparison of the rupture parameters deduced from the Vickers tests performed on fresh and irradiated fuels.
3.5
160 150
3.0 140 130
(MPa)
2.0
σc
Kc (MPa.m0,5)
2.5
1.5
120 110 100 90
1.0
80
0.5 70 60
0.0 0
1
2
indentation mean bending min
3
indentation min bending max
4
5
0
1
2
3
4
5
indentation max indentation
bending min
bending max
Fig. 15. Comparison between indentation and bending tests (1: small UO2 sample, 2: small UO2 sample with large grain size, 3: large UO2 sample and 4: large UO2 sample with large grain size).
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J.-M. Gatt et al. / Engineering Failure Analysis 47 (2015) 299–311 3.0
Kc (MPa.m1/2)
2.5
2.0
1.5
1.0
0.5
0.0 small grain size
irradiated small grain size
large grain size
irradiated large grain size
Fig. 16. Comparison of Kc for fresh and irradiated fuels.
160 140 120
σc (MPa)
100 80 60 40 20 0 smallgrain size
irradiated small grain size
large grain size
irradiated large grain size
Fig. 17. Comparison of critical stress for fresh and irradiated fuel.
Cladding Fuel pellet
Simulated radial cracks
Fig. 18. Example of calculated fuel pellet radial cracking with smeared crack model during in-reactor irradiation.
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We can see that the irradiation-induced variation in the critical stress rc is small compared to that of the fracture toughness Kc. We can conclude that the critical stress intensity factor is the main parameter that differentiates fresh and irradiated fuels. 7. Applications These parameters have been used in the multi-dimensional fuel code ALCYONE [14,6] to model UO2 cracking during reactor normal and off-normal loading conditions. In brief, ALCYONE describes the thermal–mechanical behavior of cylindrical fuel pellets (typical dimensions are 8 mm diameter and 13 mm height) stacked in a Zircaloy cladding (external diameter 9 mm). During in-reactor irradiation, pellet thermal expansion and cladding creep due to the external coolant pressure (150 bars in a Pressurized Water Reactor) leads to the closing of the initial pellet-clad gap (80 lm). In case of power transients, stresses in the cladding due to this mechanical interaction can increase considerably. The release at the same time of corrosive products from the irradiated fuel pellets can lead to the failure of the cladding by a Stress Corrosion Cracking (SCC) mechanism [15]. Fuel radial cracking can relax considerably the stresses at the pellet clad interface and must therefore be accounted for when modelling in-reactor fuel behavior. Fig. 18 gives an illustration of the calculated fuel pellet radial cracking in a 2D plane strain simulation of a power transient using a smeared crack model. The elements in blue in the pellet are those where the dissipated energy exceeds the surface energy Gc of the crack model. Due to the high thermal gradient that exists between the pellet center (2000 °C) and its periphery (600 °C), high hoop tensile stresses appear at the pellet periphery. When the critical stress is exceeded, radial cracks are initiated and propagate more or less rapidly inside the pellet depending on the surface energy. The crack network is consistent with experimental observations of fuel pellets after power transients, see Fig. 17. 8. Conclusions In this paper, we have presented a methodology based on Vickers indentation tests to assess the rupture parameters of irradiated fuels. This approach is based on the estimation of rupture parameters from bending and indentation tests performed on fresh fuels. Firstly, we have shown the link between simulation results and bending test results in order to define the scale of measured parameters (rupture stress and toughness) and the link between the measured rupture force and the rupture parameters introduced in the crack models. Secondly, after a validation of our numerical simulations thanks to an analytical approach, we have presented an interpretation of Vickers tests on fresh fuel and used this methodology to derive rupture parameters for irradiated fuel. We have shown that the toughness is the main parameter to differentiate the rupture of fresh and irradiated fuels. Finally, we have presented a finite element simulation of in-reactor pellet cladding interaction using a smeared crack approach to model pellet cracking. Acknowledgements The authors thank EDF and AREVA NP for their technical and financial support to fuel mechanical behavior studies. References [1] [2] [3] [4] [5] [6] [8] [9] [10] [11] [12] [13] [14] [15]
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Engineering Failure Analysis journal homepage: www.elsevier.com/locate/engfailanal
Experimental and numerical study of fracture mechanisms in UO2 nuclear fuel J.-M. Gatt ⇑, J. Sercombe, I. Aubrun, J.-C. Ménard CEA, DEN, DEC, F-13108 St Paul Lez Durance, France
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 25 August 2013 Received in revised form 17 July 2014 Accepted 21 July 2014 Available online 28 August 2014
In this paper we study the brittle behavior of UO2 nuclear fuel. Firstly, we present the interpretation of bending tests with three different approaches to assess rupture parameters (critical stress and surface energy). Secondly, we present Vickers’ indentation tests on fresh UO2 fuel. The comparison between bending and indentation tests on fresh fuel allows us to assess the parameters necessary to derive the critical stress and the surface energy from indentation tests. Vickers’ indentation is then used to evaluate rupture parameters of irradiated fuels. At the end, we present some applications to fuel rod modeling taking into account the different rupture mechanisms. Ó 2014 Elsevier Ltd. All rights reserved.
Keywords: Fracture Bending tests Indentation tests Nuclear fuel
1. Introduction This paper deals with the rupture of UO2 fuel in a nuclear reactor. At low temperature (<900 °C) this ceramic has a brittle behavior in traction. It has been shown that radial cracks in the rim of a UO2 pellet are formed due to pellet expansion (thermal expansion and fission gas swelling), induce a stress relaxation in the surrounding cladding and thus lead to an improved behavior of the fuel rod in reactor [1]. Therefore, the knowledge of rupture parameters for irradiated UO2 fuel is very important to model and understand the behavior of the fuel rod in reactor. The rupture behavior of fresh fuel (before irradiation) has been studied mostly in the 1970s and 1980s. The rupture stress and toughness were measured [2–4] on fresh fuel and more recently the toughness on irradiated fuel has been assessed [5]. The rupture stress has never been measured on irradiated fuel because of the fuel fragmentation during irradiation which leads to pieces too small for mechanical testing. The crack models used in numerical simulations of fuel rod behavior are based on two main parameters [1,6]: the critical stress and the surface energy. The aim of this work is the development of a methodology to assess irradiated fuel rupture parameters used in fuel rod modeling. In the first part of this work, we propose some simulations of bending tests on smooth and notched specimens with two different models. In the second part, we use an approach developed in Ref. [10] to assess the critical stress used in the crack models from indentation tests. In the third part, we present some experimental results obtained from bending tests on fresh fuel and indentation tests on fresh and irradiated fuels. We use indentation test results to determine the parameters of crack models for irradiated fuel. In the fourth part, we show an application of the crack model to the simulation of fuel pellet cracking during a typical incidental in-reactor irradiation history (power ramp).
⇑ Corresponding author. Tel.: +33 442 254 267. E-mail addresses: [email protected] [email protected] (J.-C. Ménard).
(J.-M.
http://dx.doi.org/10.1016/j.engfailanal.2014.07.019 1350-6307/Ó 2014 Elsevier Ltd. All rights reserved.
Gatt),
[email protected]
(J.
Sercombe),
[email protected]
(I.
Aubrun),
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2. Assessment of rupture parameters used in the crack models To model crack propagation in a fuel pellet during a power ramp, we use a simplified smeared crack model in finite element simulations [1] (the shear stresses are not considered and the crack propagation directions are fixed). Further, in this study, we consider another type of crack model: the cohesive zone model. These models depend on two rupture parameters: the critical stress and the surface energy. The aim of this chapter is to define the relationship between these two rupture parameters and the measured quantities in bending tests performed on smooth and notched specimens. 2.1. Cohesive zone model This model rests on the relationship between the stress applied (rn) and the crack opening (un) as illustrated in Fig. 1. The behavior is easily described with these equations:
8 rc > < Kn ¼ d 1 Gc ¼ 2 rc dc > : K ¼ rc f ddc
ð1Þ
The first equation defines the slope at the origin, with rc the critical stress and d a parameter. The second equation defines the surface energy (Gc), with d a parameter. The last equation defines the second slope. From Eq. (1), the following behavior law can be obtained:
rn ¼
8 K u > < n n > :
dun dc un
0
þ1
if un < d
rc if d 6 un 6 dc
ð2Þ
if un > dc
The following condition must be added for model consistency:
Kn >
r2c
ð3Þ
2Gc
This condition is equivalent to: d < dc. The value of Kn has to be very high to avoid an important displacement before that critical stress is reached, and not too high due to numerical instabilities. In a certain range (we take Kn = 6.1014 Pa/m), the response of the model is independent of this parameter, and therefore independent of d which is very small. Thus, if we know the rupture parameters (rc, Gc) it is possible to calculate dc using the second equation of system (1). This model has been used to simulate a bending test. The size of specimen is 10 1.5 1.5 mm3. The mesh of the smooth sample is shown in Fig. 2. For this simulation we have fixed the surface energy at 30 J/m2. From a parametric study, we have found the optimal value of rc allowing us to fit test results, as shown in Fig. 3b. This figure shows a result of a bending test at a prescribed displacement rate 20 lm/min, where the time evolutions of the applied force F was measured. The maximum force is referred here as the rupture strength FR and is equal to 27 N. This force is linked to rupture stress rR = 120 MPa by the following equation:
rR ¼
3L FR 2wt2
ð4Þ
L, w and t are respectively the length, the width and the thickness of the sample. In Fig. 3a (zoom of the end part of the test), we have plotted the time evolution of the stress (rxx) at each node of the plane of symmetry of the sample (we have 11 nodes along the sample0 s thickness). We see from Fig. 3a that the curves (x = 0 and x = 0.075 mm) start to decrease when the stress
Fig. 1. Cohesive zone model.
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301
Fig. 2. Mesh of the sample and sample at the end of simulation.
110
90
σ (MPa)
70
50
30
10 0 67
72
77
82
87
92
97
t (s) Fig. 3a. Stress (ry) variation versus thickness (x) and time.
30 25
F (N)
20 15 10 5 0 0
20
40
60
80
100
120
140
t (s) Fig. 3b. Force variation versus time.
reaches the critical stress (equal at 98 MPa in this case). A few seconds after that, the sample fails, i.e., all the stresses tend to zero. From this observation, we defined a Critical Length (CL), above which the sample fails. This CL is the length of the line constituted by the set of Gauss points where the stress decreases after the critical stress is reached. This length can be evaluated between 75 and 150 lm (the uncertainty is due to the finite element size). This evaluation shows that the crack size is great in comparison of materiel grain size and porosities size distribution (<10 lm). This means that the crack can be considered at the macroscopic scale. Furthermore, from our simulations, we found that to different surface energy and rupture stress (rR, Gc), corresponds a different critical stress rc. When we change the thickness of the specimen we find a different
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critical stress value for the same couple rc. When we change the thickness of the specimen we find a different critical stress value for the same couple (rR, Gc). This means that rc depends on the surface energy, the rupture stress and the thickness of the sample. All these dependencies are dealt in the next part of the paper. 2.2. Smeared crack model The smeared crack model used in this paper is described in [1]. This model is used in our code to simulate the fuel cracking during in reactor irradiation sequences. The greatest difference with a cohesive zone model is that we define an equivalent crack strain from the crack opening. Thus the behavior is described by a stress versus crack strain relationship instead of a stress versus displacement as it is the case in the cohesive zone model. The advantage of this model is that it can be easily written in the framework of the finite element method without introducing another type of finite element (e.g., joint elements for the cohesive zone model). The downside is that we have to introduce the length of the finite element in its formulation. According to Fig. 4, which illustrates the stress–strain relationship, we have the following equation:
Efiss ¼
1 r2c Lf 2 Gc
ð5Þ
Efiss is a crack softening modulus, and Lf the finite element length in the direction perpendicular to the crack plane. With this model, we have simulated the previously described bending test on smooth sample using a plane stress hypothesis. The mesh and the crack strains obtained from the simulation are showed in Fig. 5. The area in other colors than blue is where the stress has reached the critical stress. With this model, we obtained very similar estimates of the critical stress than with the cohesive zone model in function of the rupture stress and of the energy surface (see Fig. 12) which will be detailed latter. We have also simulated a bending test on a notched sample (28 4 4 mm3). The mesh is showed in Fig. 6. The calculated crack strains and stresses are illustrated in Figs. 7a and 7b. In Fig. 7a, the area in other colors than blue is where the stress has reached the critical stress. In Fig. 7b the evolution of the stress (ryy) versus x axis (the ligament) is given at different times. The stress rises up till time 15 s is reached. At time 15 s, the stress has reached the critical stress and starts to decrease afterwards. At times 20 s and 28 s, the stress has decreased at more nodes. Up to time 28 s, the stress is not null meaning that the crack is not fully opened. After 28 s, all the stresses tend to zero indicating that the sample is broken. The stress distribution at time 28 s gives an estimation of the critical crack length (CL), obviously greater than 100 lm. This value is great in comparison of the grain size and the porosity size distribution in commercial UO2 fuels. This means that the crack length can be considered a macroscopic quantity. 2.3. Conclusion on the crack models The finite element based interpretation of bending tests on smooth and notched samples with two crack models shows that the Critical Length is much greater than the grain size and the defect size of UO2 materials. We therefore can conclude that measurements of toughness Kc from notched samples and of rupture stress rR from smooth samples lead macroscopic quantities. They are not local measurements at the porosity scale. In the case of bending test on smooth sample, we have shown that there exist a relationship between the critical stress rc used in the crack model and experimental parameters: the rupture stress, the surface energy (toughness) and the thickness of the sample (rR, Gc, e). The goal of the next chapter is to analyze this relationship, and to find an analytical estimate of the critical stress such that finite element simulations of the bending tests can be avoided.
Fig. 4. Smeared crack model.
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303
Fig. 5. Result of simulation with smeared crack model.
Fig. 6. Mesh of a notched sample.
3. Analytical approach We now propose to use an analytical approach to interpret the bending tests on smooth specimen. This approach, introduced by Leguillon [8], allowed us to study the failure of smooth sample during a bending test according to energy and stress criteria. These two criteria read. 3.1. Stress criterion
kðxÞrmax P rc With:
ð6Þ
3FL d 2d rmax ¼ 2wt 2 ; x ¼ t and KðxÞ ¼ 1 t ¼ 1 2x, the parameters are defined in Fig. 8.
3.2. Energy criterion
pffiffi AðxÞ trmax P K c pffiffiffiffiffiffiffiffi A(x) is a function of x defined below, and K c ¼ Gc E in plane stress conditions (E is the Young0 s modulus).
ð7Þ
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crack
Fig. 7a. Simulation results (smeared crack model).
Fig. 7b. Stress variation along the ligament.
x
w t d
L Fig. 8. Notations.
Using [9], we obtain the following equations for the fracture toughness:
FR 2wt 2 K c ¼ Y pffiffi with F R ¼ rR 3L b t p ffiffi ffi 3 Lt x a Y¼ ½1:99 xð1 xÞð2:15 3:93x þ 2:7x2 Þ and x ¼ t 2ð1 þ 2xÞð1 xÞ1:5
ð8Þ
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305
The stress criterion is reached before the energy criterion in very small structures. In this case, the failure of the structure is controlled by the surface energy. On the contrary, the energy criterion is reached before the stress criterion in structures of great size. The stored energy is important. The failure is controlled by the stress. In a bending test, the failure criterion depends on the size of the sample and on the value of the rupture parameters (rc, Gc). To define the A function of Eq. (7), we use the Eq. (8), and we obtain:
AðxÞ ¼
pffiffiffi xPðxÞ
ð9Þ
ð1 þ 2xÞð1 xÞ3=2
With: PðxÞ ¼ 1:99 xð1 xÞð2:15 3:93x þ 2:7x2 Þ. The function A is plotted in Fig. 9. We have the following properties for x 2 ½0; 1: A(0) = 0. A is continuous and strictly increasing. A tends towards infinity when x tends towards 1. The failure of a smooth specimen is obtained if the two conditions (6) and (7) are verified:
(
pffiffi AðxÞ t ¼ kð^xÞ Krcc
ð10Þ
kð^xÞrR ¼ rc
^¼^ d xt is the CL. The last equation of system (10) gives the link between the rupture stress rR and the critical stress rc. The CL is obtained thanks to the following equation.
Kc Að^xÞ ¼ pffiffi ¼ rR t
sffiffiffiffiffiffiffiffi EGc tr2R
ð11Þ
According to the second equation of system (10) and the definition of k(x), we have:
rc ¼ 1 2^x rR
ð12Þ
Using Eqs. (11) and (12) we can plot the curve shown in Fig. 10. In this figure, we can see that if we increase the thickness of the sample or if we decrease the surface energy then the critical stress tends towards the rupture stress. At the limit, if the surface energy is equal to zero, then the critical stress equals the rupture stress. Thus, if we know (rR, Gc), it is possible by using Eqs. (11) and (12) or Fig. 10 to deduce the critical stress used in the mod^¼^ els. To evaluate the CL we use the equation d xt (see Fig. 11). The CL evaluated by the simulation of smooth samples bending (between 75 and 150 lm) has the same order of magnitude than the one obtained from our analytical approach: 134 lm (for rR = 120 MPa and Gc = 30 J/m2). Finally, in Fig. 12, we compare the relationship between the critical stress and the rupture stress obtained from the three approaches developed in this paper (FE simulations with cohesive zone model and smeared crack model and the analytical approach). We observe a very good agreement between the three approaches which validates the proposed analytical correlation. These three approaches are consistent between them.
Fig. 9. A function.
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Fig. 10. Curve allowing assessing the critical stress from rupture stress and Kc.
180 160 140
CL (µm)
120 100 80 60 40 20 0 80
90
100
110
120
130
140
150
σR (MPa) small sample
large sample
Fig. 11. Critical length for large and small samples versus rupture stress.
4. Bending tests on fresh fuel. Experimental results We performed bending tests on smooth and notched samples. The tests on the smooth samples were used to estimate the rupture stress (rR), and the tests on the notched samples to obtain the fracture toughness Kc. Large (28 4 4 mm3) and small samples (10 1.5 1.5 mm3) of UO2 with large or small grain sizes have been used in the tests. The fuels were porous and the microstructures of the samples very different from one another. Table 1 gives the upper and lower estimated values for Kc and rc. Kc has been calculated using Eq. (8) and rc using Eqs. (11) and (12). The experimental scatter is important. The number of tests is not sufficient for a Weibull analysis. New tests are planned to complete this experimental database.
5. Analysis of Vickers’ indentation test A series of Vickers’ indentation tests were performed on the materials used in the bending tests, small and large grains UO2. In this part, we present the methodology we developed to estimate the rupture parameters, Kc and rc, from the analysis of Vickers test results and the comparison to bending test results.
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y = 0.8874x + 30.678 R² = 0.9999
σR (MPa)
140 AM
120
CZM SCM
100 80 60
40
60
80
100
120
140
160
σc (MPa) Fig. 12. Comparison of the three modeling approaches: cohesive zone model (CZM), smeared crack model (SCM) and analytical (AM).
Table 1 Experimental results: bending tests with notched sample for Kc and with smooth samples for rc. Kc min (MPa UO2 UO2 UO2 UO2
small sample, small grain small sample, large grain large sample, small grain large sample, large grain
1.63 1.98 2.08 1.8
pffiffiffiffiffi m)
Kc max (MPa
pffiffiffiffiffi m)
1.89 2.98 2.6 2.18
rc min (MPa)
rc max (MPa)
83 116 107 87
125 147 128 101
5.1. Critical stress intensity factor The indentation test consists in punching a material with an indenter by imposing a normal force. After unloading, if the load is high enough, four cracks are observed as depicted in Fig. 13. Two kinds of cracks can appear: Palmqvist cracks (P) and Median cracks (M) as shown in Fig. 13. It is generally difficult to know with certainty which mechanism is activated. Thus, we propose here to use an arbitrary simple criterion: if c > 2a we consider the mechanism M (see Fig. 13 for the notations). This criterion allowed us to obtain a good interpretation of our measurements. For M type cracks we used the following equation [10,11]:
rffiffiffiffi E P K c ¼ jM H c3=2
ð13Þ
with E the Young0 s modulus, H the hardness, P the loading and jM a constant. In the same way, Niihara [13] proposed the following equation for P type cracks:
K c ¼ jP
2=5 E P pffiffi H a ‘
ð14Þ
where jP is a constant for mechanism P. 5.2. Critical stress To obtain all the rupture parameters used in the crack models, an evaluation of the critical stress from Vickers indentation tests is essential. To reach this objective, we have considered that a hypothetical bending test is performed on a sample after indentation according to the methodology presented in reference [12]. During the bending test, after a stable propagation of
Fig. 13. (a) Diagram of indentation surface, (b) median type crack (M), (c) Palmqvist type crack (P).
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damage, the crack due to indentation becomes unstable. To interpret this test (see Fig. 14), we have considered two stress intensity factors: the first, due to the residual stresses which hold the crack open after indentation, is proportional to P/c3/2 (it pffiffiffi decreases when the crack increases), the second, due to bending, is proportional to r c (it increases when the crack increases). The overall stress intensity factor reads therefore as follows [12]:
K ¼ j0M
rffiffiffiffi pffiffiffi E P þ fM r c H c3=2
ð15Þ
with j0 M and fM two constant parameters (note that j0 M is different from jM). The crack becomes unstable when:
@K ðcL Þ ¼ 0 and K c ¼ KðcL Þ @c
ð16Þ
From Eqs. (15) and (16), we obtained the following equations:
3 0 j 2 M
rffiffiffiffi E P 1 rc þ fM pffiffiffiffiffi ¼ 0 H c5=2 2 cL L
ð17Þ
rffiffiffiffi pffiffiffiffiffi E P þ fM rc cL H cL3=2
ð18Þ
And:
K c ¼ j0M
Eqs (17) and (18) can also be written:
rc c2L ¼
3j0M fM
rffiffiffiffi E P H
ð19Þ
And:
Kc ¼ j0M cL
rffiffiffiffi E P rc þ fM pffiffiffiffiffi H cL5=2 cL
ð20Þ
From Eqs. (20) and (17), we obtained the following equations.
8 qffiffiffi E P < K c ¼ 4j0 M H c3=2 L pffiffiffiffiffi : K c ¼ 4f3M rc cL
ð21Þ
These two equations show that 1/4 of Kc is due to indentation test and 3/4 to bending test. Using (21), we found the following relation for crack type M:
rc
rffiffiffiffi rffiffiffiffi E P E P ¼3 ¼ 3hM H c2 fM H c 2
j0M
ð22Þ
Using the same approach for cracks of type P, we found:
rc ¼
2=5
j0M E fM
H
2=5 P E P ¼ hP a‘ H a‘
ð23Þ
The parameters hM and hP are constants measured experimentally. Thus, Eqs. (13) and (14) for Kc and (22) and (23) for the critical stress allowed us to assess the rupture parameters from Vickers’ indentation tests. All the parameters can be
Fig. 14. (a) Median type crack induced by indentation and (b) crack extend under combined action of residual stress and tensile stress.
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considered as macroscopic quantities since the length of the cracks formed by indentation (20 lm) is greater than the typical size of the pores in the grains (<1 lm). 5.3. Determination of the constants The indentation tests were carried out on the same material as the one used for the bending tests. The goal of this determination was not to have a good accuracy on the values (the number of available tests is not adequate for that), but to show the relative behavior of the different fuels. From the estimated measurements of the fracture toughness in the bending tests, we determined the Vickers’ indentation constants associated to Kc. The goal of this determination, based on the values reported in Table 1, is to find the variation of Kc measured on the different samples (bending test on notched samples), as shown in Fig. 15. From this process, we found the following values:
jM ¼ 0:0432 and jP ¼ 0:00445 From the bending test (smooth samples), the critical stress can be evaluated from the rupture stress and Gc, using Fig. 10. We determined the constants (hM, hP) associated with critical stress (Eqs. (22) and (23)) to find the relative variation of rc evaluated from bending tests on the different fuels. This process led to following values:
hM ¼ 0:003585 and hP ¼ 0:000896 The choice of the criteria (if c > 2a we consider the mechanism M) to determine the activated mechanism is very important. This process shows a posteriori that it was a good choice. Fig. 15 compares the results obtained from the bending tests and the Vickers’ indentation tests. We can conclude that: We have a significant scattering in indentation tests on UO2. The P and M mechanisms are activated for the small grain size UO2, while the M mechanism only is activated in the case of the large grain size UO2. The order of magnitude of the different parameters and their relative evolutions with material characteristics and sample sizes are in good agreement. 6. Irradiation effects A series of Vickers’ indentation tests have been performed on commercial UO2 irradiated during various times in reactor. The test results were interpreted with Eqs. (13) and (14) to estimate the fracture toughness Kc and Eqs. (22) and (23) to evaluate the critical stress. Note that the constants in these equations used are those obtained from the tests performed on non-irradiated (fresh) fuel samples. In Figs. 16 and 17, we show a comparison of the rupture parameters deduced from the Vickers tests performed on fresh and irradiated fuels.
3.5
160 150
3.0 140 130
(MPa)
2.0
σc
Kc (MPa.m0,5)
2.5
1.5
120 110 100 90
1.0
80
0.5 70 60
0.0 0
1
2
indentation mean bending min
3
indentation min bending max
4
5
0
1
2
3
4
5
indentation max indentation
bending min
bending max
Fig. 15. Comparison between indentation and bending tests (1: small UO2 sample, 2: small UO2 sample with large grain size, 3: large UO2 sample and 4: large UO2 sample with large grain size).
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Kc (MPa.m1/2)
2.5
2.0
1.5
1.0
0.5
0.0 small grain size
irradiated small grain size
large grain size
irradiated large grain size
Fig. 16. Comparison of Kc for fresh and irradiated fuels.
160 140 120
σc (MPa)
100 80 60 40 20 0 smallgrain size
irradiated small grain size
large grain size
irradiated large grain size
Fig. 17. Comparison of critical stress for fresh and irradiated fuel.
Cladding Fuel pellet
Simulated radial cracks
Fig. 18. Example of calculated fuel pellet radial cracking with smeared crack model during in-reactor irradiation.
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We can see that the irradiation-induced variation in the critical stress rc is small compared to that of the fracture toughness Kc. We can conclude that the critical stress intensity factor is the main parameter that differentiates fresh and irradiated fuels. 7. Applications These parameters have been used in the multi-dimensional fuel code ALCYONE [14,6] to model UO2 cracking during reactor normal and off-normal loading conditions. In brief, ALCYONE describes the thermal–mechanical behavior of cylindrical fuel pellets (typical dimensions are 8 mm diameter and 13 mm height) stacked in a Zircaloy cladding (external diameter 9 mm). During in-reactor irradiation, pellet thermal expansion and cladding creep due to the external coolant pressure (150 bars in a Pressurized Water Reactor) leads to the closing of the initial pellet-clad gap (80 lm). In case of power transients, stresses in the cladding due to this mechanical interaction can increase considerably. The release at the same time of corrosive products from the irradiated fuel pellets can lead to the failure of the cladding by a Stress Corrosion Cracking (SCC) mechanism [15]. Fuel radial cracking can relax considerably the stresses at the pellet clad interface and must therefore be accounted for when modelling in-reactor fuel behavior. Fig. 18 gives an illustration of the calculated fuel pellet radial cracking in a 2D plane strain simulation of a power transient using a smeared crack model. The elements in blue in the pellet are those where the dissipated energy exceeds the surface energy Gc of the crack model. Due to the high thermal gradient that exists between the pellet center (2000 °C) and its periphery (600 °C), high hoop tensile stresses appear at the pellet periphery. When the critical stress is exceeded, radial cracks are initiated and propagate more or less rapidly inside the pellet depending on the surface energy. The crack network is consistent with experimental observations of fuel pellets after power transients, see Fig. 17. 8. Conclusions In this paper, we have presented a methodology based on Vickers indentation tests to assess the rupture parameters of irradiated fuels. This approach is based on the estimation of rupture parameters from bending and indentation tests performed on fresh fuels. Firstly, we have shown the link between simulation results and bending test results in order to define the scale of measured parameters (rupture stress and toughness) and the link between the measured rupture force and the rupture parameters introduced in the crack models. Secondly, after a validation of our numerical simulations thanks to an analytical approach, we have presented an interpretation of Vickers tests on fresh fuel and used this methodology to derive rupture parameters for irradiated fuel. We have shown that the toughness is the main parameter to differentiate the rupture of fresh and irradiated fuels. Finally, we have presented a finite element simulation of in-reactor pellet cladding interaction using a smeared crack approach to model pellet cracking. Acknowledgements The authors thank EDF and AREVA NP for their technical and financial support to fuel mechanical behavior studies. References [1] [2] [3] [4] [5] [6] [8] [9] [10] [11] [12] [13] [14] [15]
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