Scholarly article on topic 'Creep and Damage Analysis of Reactor Pressure Vessel Considering Core Meltdown Scenario'

Creep and Damage Analysis of Reactor Pressure Vessel Considering Core Meltdown Scenario Academic research paper on "Materials engineering"

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{Creep / Damage / "In-Vessel Retention" / FEM / RPV.}

Abstract of research paper on Materials engineering, author of scientific article — J.F. Mao, J.W. Zhu, S.Y. Bao, L.J. Luo, Z.L. Gao

Abstract The Fukushima accident shows that In-Vessel Retention (IVR) of molten core debris has not been appropriately assessed, and a certain pressure (up to 8.0MPa) still exists inside the reactor pressure vessel (RPV). Generally, the pressure is supposed to successfully be released, and the externally cooled lower head wall mainly experiences the temperature difference which may be more than 1000°C. Therefore, in order to make the IVR succeed, it is necessary to investigate the creep behaviour and damage distribution of the RPV under complex thermal-mechanical loadings. Accordingly, considering the unlikely core melt down scenario for a light water reactor (LWR) a possible failure mode of the reactor pressure vessel (RPV) and its failure time has to be predicted for a determination of the loadings on the containment. Due to the thickness of RPV, the high temperature gradient results in various failure modes, i.e., plastic failure and creep failure. In disclosing the failure mechanism, the finite element model has been developed simulating the thermal processes and the visco-plastic behaviours of vessel wall. An advanced model for creep damage has been established to analyze the fracture time and fracture position of a vessel with an internally heated melt pool. Before the above, the stress and strain distributions along the wall thickness are investigated by ABAQUS software. Finally, the result shows that the calculated stress outside the RPV is lower than the yield stress of the material through most thickness. It is concluded that the RPV can maintain its integrity under IVR with given time, even if there exists the internal pressure of 8MPa.

Academic research paper on topic "Creep and Damage Analysis of Reactor Pressure Vessel Considering Core Meltdown Scenario"

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Procedía Engineering 130 (2015) 1148 - 1161

Procedía Engineering

www.elsevier.com/loeate/procedia

14th International Conference on Pressure Vessel Technology

Creep and Damage Analysis ofReactor Pressure Vessel Considering Core Meltdown Scenario

J.F. Maoab *, J.W. Zhuac,S.Y. Baoab *, L.J. Luoa, Z.L. Gaoab

aInstitute of Process Equipment and Control Engineering, Zhejiang University of Technology Hangzhou, Zhejiang 310032, P. R. China

bEngineering Research Center of Process Equipment and Its Re-manufacturing, Ministry of Education, P. R. China cDepartment of Mechanical and Electrical engineering, Huzhou Vocational & Technical College Huzhou, Zhejiang 313000, P. R. China

Abstract

The Fukushima accident shows that In-Vessel Retention (IVR) of molten core debris has not been appropriately assessed, and a certain pressure (up to 8.0MPa) still exists inside the reactor pressure vessel (RPV). Generally, the pressure is supposed to successfully be released, and the externally cooled lower head wall mainly experiences the temperature difference which may be more than 1000 °C. Therefore, in order to make the IVR succeed, it is necessary to investigate the creep behaviour and damage distribution of the RPV under complex thermal-mechanical loadings. Accordingly, considering the unlikely core melt down scenario for a light water reactor (LWR) a possible failure mode of the reactor pressure vessel (RPV) and its failure time has to be predicted for a determination of the loadings on the containment. Due to the thickness of RPV, the high temperature gradient results in various failure modes, i.e., plastic failure and creep failure. In disclosing the failure mechanism, the finite element model has been developed simulating the thermal processes and the visco-plastic behaviours of vessel wall. An advanced model for creep damage has been established to analyze the fracture time and fracture position of a vessel with an internally heated melt pool. Before the above, the stress and strain distributions along the wall thickness are investigated by ABAQUS software. Finally, the result shows that the calculated stress outside the RPV is lower than the yield stress of the material through most thickness. It is concluded that the RPV can maintain its integrity under IVR with given time, even if there exists the internal pressure of 8MPa. ©2015 The Authors.PublishedbyElsevierLtd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.Org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of ICPVT-14

Keywords: Creep; Damage; In-Vessel Retention; FEM; RPV.

* Corresponding author. Tel.: +86 0571 88320349; fax: +86 0571 88320842. E-mail address: jianfeng-mao@163.com(Mao J.F.); bsy@zjut.edu.cn(Bao S.Y.)

1877-7058 © 2015 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license

(http://creativecommons.org/licenses/by-nc-nd/4.0/).

Peer-review under responsibility of the organizing committee of ICPVT-14

doi: 10.1016/j.proeng.2015.12.283

1. Introduction

In-vessel retention (IVR) of core meltdown is recognized as a key severe accident management strategy, which has been adopted by advanced light water reactor (ALWR) and most operating nuclear power plants. However, the Fukushima accident shows that IVR of molt core debris has not been appropriately assessed, and a certain pressure (up to 8MPa) still exists inside the reactor pressure vessel (RPV). Furthermore, the accident shows that the applied IVR technology over-proportionally underestimates the failure time. Actually, some tests have already proved that the predictability of the time, mode and location of RPV failure varies with the applied pressure and temperature[l]. This topic concerning the nuclear safety has attracted a great deal of research interest. As pointed out in literature[2], the calculation on the safety margin of IVR in ALWR is significantly meaningful, and IVR analysis code in some severe accidents has been developed[3]. Therefore, a good understanding of the mechanical behavior of the RPV is urgently necessary both for severe accident assessment and for the evaluation of accident mitigation strategies.

Nomenclature

di, ái, d3 d4 material parameters in creep constitutive equation

5 creep strain ratio T temperature

AD the damage increment

£frac creep fracture strain

£p.ac plastic fracture strain

Rv the triaxiality factor

Q parameter related to gas constant

R the stress exponent in interpolation

6 polar coordinate angle along the lower header

In order to characterize the mode, time, size of a possible lower head failure (LHF) of the RPV in the core meltdown accident, many geometrical scaled experiments had been conducted under FOREVER, USNRC/SNL and CORVIS programs[4-6], one of which measured a typical failure size like a fish mouth as seen in Fig. 1. Most of the tests paid attention to the mechanical failures of the RPV under a variety of core slump scenarios. The experiments examined LHF at upmost pressure lOMPa in most case and with large through-wall temperature differentials. They also investigated RPV failure with low and moderate pressures (2~5MPa) but with high through-wall temperature gradients[7]. Consequently, the lower head failure resulted from following mechanisms: creep; plasticity (including thermo-plasticity); and melt-through. The failure location may be local or global. At low pressure but high temperature, creep damage accumulates over a wide area, resulting in probable global failure, while at high pressure but low temperature, plastic deformation plays an important role in failure process, leading to a possibility of local failure. Regarding to melt-through phenomenon, it is closely related to heat flux characteristics among the RPV surface. In fact, local melting can take place before failure if there is a highly concentrated heat flux[8]. Although failure time and site were found in good agreement with test results, large discrepancies were observed on the mode of failure: creep or plasticity[9]. The creep deformation is a visco-plastic process with time-dependent characteristics, while the plasticity is a prompt strain accumulation without time dependency. Analysis of LHF showed that the creep and plasticity accompanied with each other through the whole IVR process, and plastic strain was negligible before tertiary creep occurrence. The existing literatures[10-12] revealed that the multiaxial state of stress significantly accelerated the damage evolution as well as the interaction between creep and plasticity under the IVR-like condition, especially at tertiary creep stage due to the occurrence of necking and wall thickness reducing[2]. As for first rupture estimations, someone pointed out that one of most important mechanical parameter was the creep rupture strain[6]. Despite of the fact that the experimental investigation is very important for characterizing the failure of the RPV, the effect of temperature and strain fields on RPV damage is rather significant concerning the mechanical material behavior and resulting safety issue[13, 14]. The high temperatures (above

1200°C) always make thermocouple fail and the high expense always make the experiment impossible. Thus, the finite element method (FEM) is after all accepted as a good choice for creep and damage analysis ofRPV.

Fig. 1. View oflower head during failure (a) and Post-test view ofthe failure location (b)[8].

The major objective of the present study is to numerically investigate the creep and damage behavior of the RPV under the IVR condition. As a matter of fact, the IVR condition is an accident situation, the so called 'core meltdown scenario' finally results in the failure of the RPV. Although there are some published literatures with respect to this topic, the importance of IVR is not fully recognized until Fukushima nuclear disaster happens. As pointed out above, the IVR technology was not suitably assessed before the Fukushima accident, the failure estimation may be unreasonable by previous methodology with the practical thermal-mechanical loading. In overcoming this difficulty, a 2D FE model of the RPV was developed on ABAQUS platform with newly-implemented creep and damage subprogram for the RPV material. In order to achieve the prediction accuracy, a non-linear interpolation methodology was employed in the FE calculation for various temperature and stress levels. By taking a practical test loading as a basis, the creep, plasticity and melt-through of the RPV were analyzed in detail, the transient temperature and strain fields were obtained before the structural strength analysis. Finally, the effect of the multiaxial stress on RPV damage was investigated in depth, the damage caused by creep and plasticity were discussed spatially and temporally for the RPV.

2. Mathematical modeling

2.1. Creep and damage modeling

Due to the complex heat exchange between the melting pool and the vessel wall, the temperature and stress changes very significantly in both spatial and transient ways[2]. Furthermore, the high temperature gradient induces great deformation. Creep mechanism plays a decisive role in the failure process of the RPV under the IVR condition. In describing it, an advanced approach for the numerical creep modeling has been established, the constitutive equation of which can be formulated with a number of coefficients,

è = djadl£d3 exp

In FE calculation, the nonlinearity characteristic can't be ignored, e.g. geometrical softening, material nonlinearity. Consequently, the large strain option was activated during the calculation. In fact, the tertiary creep is the most severe nonlinear stage. In satisfying the requirement, Eq. (1) has to fit for dl > 0 and d3>0 simultaneously. For failure time and range prediction, it is necessary to develop a damage criterion. Usually, the damage within the RPV is caused by the significant creep and plastic strain, so called 'ductility criterion' is widely used in judging whether the RPV is failure or not under the IVR condition. Accordingly, the damage increment AD can be modeled

by Eq. (2), which is incrementally accumulated at the end of a time step or substep. AD=0 means "no damage"' occurred in the RPV at a sub-step.

(ct,T) efac (T)

where creep fracture strain £jmc was set conservatively for each temperature level, ranging from 35% at 600°C to

65% at 1200°C. The plasticity was calculated by using the multilinear isotropic hardening option in ABAQUS. The

plastic fracture strain £p.ac was obtained from the last point of the stress-strain curve at corresponding temperature.

The creep and plastic strain components were calculated separately according to the experimentally found material behavior[15]. More importantly, RV is the triaxiality factor, considering the damage behavior in dependence on multiaxial state of stress,

Rv = -(1 + v)+ 3(1 - 2v)

í V ct.

\ aM J

where v is the Poisson's ratio, aH is the hydrostatic stress and aM is the von-Mises equivalent stress. With the linear accumulation method, the damage parameter D is the sum of the damage increament AD, as presented in the following Eq. (4),

D =ZAD1 (4)

It should be pointed out that the damage increment is computed by averaging its nodel equivalent strains for each element. If the damage parameter D reaches the value of D=l, the element of the material is setted inactive properties by using element death technique. The death element does no longer contribute to the stiffness.

2.2. Non-linear interpolation

The coefficients are used to adapt the creep constitutive equation at constant load and temperature. However, to achieve a good fitting with very limited set of coefficients is very difficult. If computing step time is not sufficiently small, the linear interpolation may overestimate the creep strain increment. In general, the creep constitutive equation largely depends on the stress, strain and temperature. In overcoming the over-proportional dependence on these variables, the nonlinear calculation scheme is desirable and useful for weighting coefficients fitting and damage increment. Assuming that the creep strain rate depends exponentially on the stress, the appropriate strain rate between two points can be nonlinearly interpolated as follows,

£ = -

cr, - a

CT, -CT

„-r r

CT — CT

ct, -CT

Wherein, the exponent r can be obtained with two points,

r = M>2)- ln(g, ) ln(er2)- Info)

Likewise, assuming that the creep strain rate dependency on the temperature can be described by £ ~ e q'T, the corresponding interpolation between two points is

where the parameter q can be formulated by

Infe )- ln(g, )

1/t -1/T2

Note that the parameter q varies with the strain rate and temperature as like the parameter r is strain rate and stress dependent.

2.3. Description of FE modeling

The model applied to FE calculation represents the lower head RPV in the geometrical scale of 1:10. The geometry of RPV is shown in Fig. 2. It consists of a hemispherical head and vertical cylindrical section. The cylindrical section is 0.46m high with a wall thickness of 0.08m. The inner radius of hemispherical section is 0.457m and the wall thickness 0.06m. For mechanical calculation, the 2D axisymmetric model with appropriate boundary conditions and material properties is established. The number of elements over the wall thickness was set to 8, the type of which is CAX4T used for meshing. Actually, a sufficient number of elements over the wall thickness is necessary to model the body load of heat flux which is changing along and perpendicular to the wall surface. The 2D structural 4-node isoparametric elements used are suited for creep and plasticity by taking the geometric nonlinearity into account, e.g. larger strain and deflection. As shown in Fig. 2, the internal pressures of _P=3MPa; 5MPa were applied on the internal surface of the RPV, the vertical displacement was set to zero (Uz=0) for all nodes at the top end as well as radial displacement (Ux=0) at the symmetry axis. Besides, the following loads were also considered: ®gravity(dead weight of vessel); ©heat flux density on the surface of vessel wall, which is

Fig. 2. FE-model ofthe RPV with applied boundary conditions.

from corresponding thermal solution. The comparison between empirical correction and test data in Fig. 3 shows the great agreement is achieved for heat flux along the latitude, 0. Fig. 3 shows the heat flux is unevenly distributed, and it increases with the 0. Note that there is a large concentrated heat flux between 0=60° and 0=90°, so the local melting may possibly occur before failure. As well known, the radiation heat transfer and convection at free surfaces are the mechanisms for release of the core meltdown heat. Due to the high temperatures in the RPV, the heat transfer processes are governed by radiation transfer when surface temperature is above 600 °C. As for RPV material, isotropic material behaviour is assumed, the temperature-dependence of all material properties is considered in the range of 25°C to 1300°C. According to the database of SA533B1 RPV steel from our co-operator, the mechanical properties have been generated from the fitted strain curves, including the creep data.

1800 1600 ^ 1400 1200 ■i 1000 | 800 g 600 1 400 200 0

0 8 16 24 32 40 48 56 64 72 80 88 Latitude, 6 (degree)

Fig. 3. Comparison of critical heat flux between test and empirical formula along the lower head

3. Results and Discussions

3.1. verification calculation

As mentioned before, the internal surface of RPV is exposed to the melting pool with very high temperature, so some regions suffer meltdown and the thickness changes with the time and space. Thus, the remnant thickness is presented at the failure time in Fig. 4, representing the wall thickness of the lower head along the 0 azimuth at 10° intervals of latitude. As shown in Fig. 4, the thickness profile is not constant for both pre- and post-test RPV. Although the thickness changes in the same trend, the reduction of thickness at hot focus site is greatly underestimated by the FEM. In other words, the remnant thickness of is predicted more conservatively by FEM. The significant reduction of thickness can be explained by the concentrated heat flux in corresponding region, falling in the range of 0=60° to 0=90°. So to speak, the mode and location of failure is reasonably predicted, the thickness in close proximity to the transition position is predicted to be 40mm which compares with 25mm measured in the test. Moreover, a comparison of horizontal displacement (or elongation) of the RPV lower head is presented in Fig. 5. between FEM and test data. It can be seen in Fig. 5 that the simulation by FEM is in quite good agreement with the test results for elongation. The curve of displacement can be characterized by a approximately constant slope at the initial stage, mainly due to the steady creep behavior, while the displacement is increasing very significantly later, which indicating that failure takes place at this time as observed in the test. However, the displacement by simulation increases much faster than that by test after 240 min, mainly due to the difference in the tertiary creep region.

100 90 80 70 60 50 40 30 20

Xx. V\ T /

—a— Pre-test thickness ---- FE result I-/ / J

- Post-test thickness i i i ^y -

10 20 30 40 50 60 70 80 90 Latitude, 8 (degree)

Fig. 4. Remanent thickness profiles at the failure time for RPV hemisphere.

2.5 2.0

c 1.5 <u £ <u o

2 1.0 Q. (0 Q

0.5 0.0

----Simulation by FEM

- Test data from literature [1] / / / / / / / /

0 60 120 180 240 300 360 420 Time (min)

Fig. 5. Comparison of displacement-time curves ofthe lower head between simulation and test.

In order to validate the temperature field, the temperature profile of external wall obtained from the experiment is compared with that predicted by ABAQUS. It should be emphasized that the internal wall temperature was not able to be measured due to the failure of thermocouples in melting pool, so the external wall temperature is applied for the comparison in Fig. 6. Clearly, it can be seen from Fig. 6 that the shape of the temperature profile measured agrees very well with that by FEM within the hemispherical section. Still, the higher temperature is observed for test data compared to the FEM in the transition position, due to the higher heat flux correspondingly. With the thickness reduction and hemispherical section enlargement, the level of melting pool is declining, and the actual temperature of cylindrical section is subsequently lower than the predicted value, which is shown in Fig. 6. Due to the internal melt convection, the hottest region of the vessel wall is located in the upper part of the hemisphere just below the surface layer of the melt. In order to further illustrate the temperature distribution, Fig. 7 presents the transient fields in the vessel wall at t=3000s; 12000s; 15000s respectively. Fig. 7 shows that the high temperatures in the hot focus site are clearly visible and therefore it is further proved that the maximum displacement occurs there as well as the thinnest remained thickness does. The maximum temperature reaches 1300°C through the wall thickness when the time is at the 15000s. Close observation of Fig. 7 discloses that no matter what stage the RPV experience, the high temperature gradient always exists, and the temperature inside is much higher than that outside during the process. There is still a maximum difference of 500 °C between the inside and outside when the RPV is approaching the failure.

a> i—

® 600

0 100 200 300 400 500 600 Distance along the wall (mm)

Fig. 6. Comparison ofthe temperature profiles on external wall between FEM and test.

Fig. 7. Comparison ofthe temperature profiles on external wall between FEM and test.

As indicated in Fig. 7, the influence of thermal conditions on the mechanical behaviour are inevitable, no matter what material the RPV is made of, the elastic, plastic and visco-plastic material properties are all temperature dependent. For instance, the temperature gradient leads to hindered thermal expansion and thereby induces thermal stresses in the RPV wall. As for membrane stress plus secondary stress, the Mises stress is adopted in the comparison with the yield stress of the material, which is shown in Fig. 8, Fig. 9 and Fig. 10. As well known, the plasticity is a prompt process, occurring only above a stress threshold, e.g. yield strength. Furthermore, it can be seen from Fig. 8 to Fig. 10 that both Mises and yield stress decreases with increasing temperatures. Although secondary stress caused by the temperature gradients can be relieved by creep deformation, the prompt high thermal stress is a threat to the RPV safety due to its inducement on the thermal cracking and plastic deformation. In order to look into the structural strength for the RPV, three typical paths (see Fig. 11) were selected in the analysis: ©Pathl is located in the transition line of hemispherical section to cylindrical section; ©Path2 is at maximum stress; @ Path3 is through the thinnest wall thickness. Fig. 8, Fig. 9 and Fig. 10 are corresponding to the Pathl, Path2 and Path 3 respectively. General observation of Fig. 8 to Fig. 10 reveals that the Mises stress is much larger than the

yield stress at the outer wall of the RPV, which indicates that the plastic deformation first occurs at the external. Interestingly, it is found that the Mises stress with Pmtemai^MPa is larger than that with Pintemai=5MPa at the transition line ofthe RPV for most wall thickness, the behavior of which is opposite to those at Path2 and Path 3. With internal pressure increasing up to 5MPa, the region suffered plastic deformation enlarges very significantly due to its larger Mises stress, especially for Path2 and Path3. As predicted in Fig. 10, the most dangerous situation occurs at the thinnest wall thickness under Pintemai=5MPa among the three selected paths. For this location, the plastic failure mechanism dominates the whole process, and the instability may even further take place during the necking.

1400 1200

E1000 p>

Q. 600

I- 400

\ / • Yield stress

/ ■ Mises stress

----at P,„,„„„=3MPa vi

-at P,.i„n.i=5MPa j y

J* J ^ -

500 400

200 £

10 20 30 40 Wall thickness (mm)

Fig. 8. Mises stress distribution along the path of cylinder to sphere transition line at t =12000s.

Fig. 9. Mises stress distribution along the path of max. stress at t =12000s.

Fig. 10. Mises stress distribution along the path of thinnest wall at t =12000s

Due to the internal pressure and melting pool temperature inside, the great temperature gradient causes huge thermal stress through the wall thickness, and the compressive stresses distribute among the inside wall while the tensile stresses take place in the surface layer of the outer wall. Accordingly, it can be judged from Fig. 11(a) that the external ofthe RPV is more vulnerable to damage except for the material melting inside. As shown in Fig. 11(a), the maximum value of thermal stress reaches 472.46MPa at the outside of Path2. To demonstrate the RPV deformation at the location ofthe maximum stress, Fig. 11(b) and (c) show contour plots ofthe displacement and plastic strain fields at the failure time. The effect of the moving vessel and melting location becomes additionally obvious by the scaled plots where the vessel is moved right to a opposite C-shaped profile. Correspondingly, the maximum displacement occurs at the Path2 with maximum stress, the value of which is reached at £/=24.44mm. This enlargement phenomenon can be clearly seen in Fig. 11 (b) along the horizontal direction. Since the Path2 undertakes great deformation, the plastic strain is unavoidable. By investigating the plastic stain in Fig. 11(c), one can find that the biggest concentrates on the internal wall of selected Path2 under Pmtemai— 5MPa. Close observation of Fig. 11(c) discloses that although the RPV suffers huge deformation as a whole, the percentage of plastic region is still at a low level in the structure. It can be explained that the thermal expansion, elastic strain and creep hold the rest percentage, and it further indicates that the creep should be taken into account in the failure analysis ofthe RPV.

Fig. 11. The distribution of (a) equivalent stress, (b) displacement, (c) plastic strain under P=3MPa at failure time.

Due to the thermal expansion at the beginning and accumulating plastic and viscoplastic strain later on the shape ofthe RPV, the vessel wall are changing all the time as shown in Fig. 12. Describing the long-term deformation, the most different part is the creep strain. Actually the equivalent stress can be relaxed through the whole creep process, and redistributed on the structure. Accordingly, the equivalent stress in the Fig. 12(a) with consideration of creep is much lower than that in Fig. 11(a). Contrary to the plasticity, the creep (also called as viscoplasticity) is a time dependent process, and it comes into work at elevated temperature, especially for the temperature above 500 °C, however occurring at very low stress. The in-vessel retention condition is accident situation, which can result in remarkable creep deformation as shown in Fig. 12(b). Unlike a creep test on the tension bar by keeping the applied stress constant, the equivalent stress is not constant during the RPV creep because ofthe significant reduction ofthe cross section under the IVR condition. Consequently, because of the necking later on, both the increasing creep strain and strain rate are observed, the results can be partially illustrated in Fig. 12. If the Fig. 12(b) were compared with the Fig. 11 (c), one can find that the creep strain is significantly larger than the plastic strain, and the plastic strain only account for approximately 10% ofthe creep strain. Due to the microstructural changes, e.g. microcrack, creep cavities, the creep resistance of RPV material is actually decreasing with the increase of creep time. The contour plots of Mises stress and strain in Fig. 12 shows that the volume ofthe RPV is extraordinarily enlarged at failure time. No doubt it is in the tertiary creep stage correspondingly. Furthermore, it is so dangerous ofthe RPV that one can't ignore the geometrical softening by increase of the primary stress due to the coupling of creep and plastic ongoing deformation simultaneously. Therefore, the assumption that the creep strain is dominant is not appropriate when the wall thickness reduction induces remarkable enlargement. Accordingly, the failure estimation

judged only by creep strain is non-conservative. Besides, it is clear in Fig. 12 that the enlargement of the RPV is increasing with the increase of the internal pressure. It should be noted from Fig. 12 that the wall thinning and stretching in the hot focus site is obviously visible. Additionally, the wall is horizontally thinned by 25% and vertically stretched by roughly 16%.

Fig. 12. (a) Mises stress, (b) Creep strain field of the RPV at the failure time.

3.3. Failure and damage analysis

So far, the most interesting question for pre-test analysis is the vessel failure time and location, although they are affected by numerous factors. As indicated above, there are three typical failure criterions used in the assessment of structural integrity: ® Stress criterion; ©Strain criterion; ©Damage criterion. However, usual creep tests are load controlled. Consequently, the true stress is not constant and not easy to measure during the test. Accordingly, the strain and damage criterions are widely used in the failure assessment of the RPV under the IVR condition. As already mentioned above, the creep strain predicted by FEM is compared with the creep rupture strain assumed Efi-ac=60% at all stresses and temperatures, as suggested by uniaxial creep test. If we look at the damage increment Eq. (2) in the section of mathematical modelling, the strain criterion can be obtained by setting the triaxiality factor Rv=l. This reduction of damage parameter indicates that the strain criterion don't take multiaxial state of stress into account. In reality, the RPV failure behaves in a multiaxial manner, especially during the necking process. In illustrating the multiaxial state of stress through the creep time, Fig. 13 demonstrates the changes of the Rv for both large and small deformation nodes. As displayed in Fig. 13, the comparison shows that Rv of the large deformation node increases continuously, while for the small deformation node, the same increases first and then tends to decrease. In addition, Rv of the large one is always much higher than that of the small one. The discrepancy between

the two becomes much higher with increasing creep time. The damage contribution by the Rv is not negligible, because the stresses distributed among the cylinder to sphere transition are complex, and then the wall thinning severely results in multiaxial state of stress. Fig. 14 indirectly shows the multiaxial factor Rv accelerates the damage evolution by consideration of multiaxial indicator anloeq. The curve of OHlaeq=0.33 is corresponding to the Rv=l, suggesting the uniaxial state of stress, while increasing anloeq induces the increase of the creep damage. According to the comparison between strain and damage criterions, it can be inferred that the strain criterion may underestimate the creep failure simply due to its neglect on multiaxial effect. Besides, examination of Fig. 14 reveals that the damage gradually increases at initial phase and sharply increases at the later phase with the creep strain, so it further proves that the tertiary creep is very dangerous for the RPV.

re 4.0

0 4500 9000 13500 18000

Creep time (s)

Fig. 13. The plot ofmultiaxial factor history during creep time for two typical points.

2! 0.25

£ 0.20

S 0.15

S 0.10

0.05 •

0.00 ■_._i_._i_._i_,_i_._

0 0.002 0.004 0.006 0.008 0.010 0.012 Strain

Fig. 14. The effect ofmultiaxial indicator on creep damage with the strain alteration.

As mentioned above, the most valuable prediction is the failure location and time. In achieving it, the contour plots of the damage parameter D calculated by Eq. (2) is shown in Fig. 15 with respective step time. It can be seen from Fig. 15 that the region of the maximum damage focuses to a small area at the vessel external surface. Actually, the maximum damage is located at the thinnest wall thickness, which is corresponding to the Path3 in Fig. 11(a). In the damage calculation, both plastic and creep strain contribute to the damage development simultaneously, and the failure take place when the damage reaches almost unity, corresponding to a maximum local total strain (plastic plus creep) of £frac=45%. Consequently, the failure strain predicted by damage criterion is much lower than that predicted by strain criterion without consideration of the Rv. Note that the maximum damage area is in accordance with the hot focus site, so it can be logically inferred that the damage is much more sensitive to high temperature than the

internal pressure in this specific case. The most endangered zone exhibits the highest creep strain rates, so the fastest creep deformation may further accelerates the creep damage process. Contrary to the hot focus site, Fig. 15 shows that the lower portion of the vessel head exhibits a higher strength due to the lower heat flux and stress in this zone. As a matter of fact, the damage criterion is much more accurate in predicting the failure of the RPV. An in-depth investigation of total damage is listed in Table 1 for hot focus site at /=619s, including plastic and creep damage. According to Eq. (2), linear accumulation method (LAM) is adopted in both strain and damage criterion. The comparison in Table 1 reveals at least two results: ©Creep damage is dominant in total damage; ©The uniaxial strain criterion significantly underestimates the total damage.

Fig. 15. Contourplot ofthe damage parameter D forthe RPV at transient fields.

Table 1. The comparison of damage allocation ofthe RPV with two failure criterions at t=619s

Damage accumulation Plastic damage Creep damage Total damage

Strain criterion 0.03 0.31 0.34

Damage criterion 0.06 0.63 0.69

4. Conclusions

In the event of a severe core meltdown accident, the temperature of core melting pool reaches as high as approximate 1300 °C with a possible internal pressure of 2~10MPa, this thermo- mechanical loads easily lead to a rupture of the lower head of the RPV. It can be leant from the paper that the RPV failure can occur as a result of the following processes: ©creep; ©plasticity; ©melt-through. In further analyzing the failure of the RPV, an advanced numerical creep and damage parameter model is developed. Subsequently, the 2D axisymmetric FE model is quite well validated by some test data from published literatures. The numerical approaches presented in current study have shown their capacity to characterize the failure process fairly well. Through FE calculation, the transient temperature field of the vessel wall is evaluated as well as stress field, and the effect of the creep on deformation of the vessel wall is investigated. Moreover, the failure time and position are determined by use of material damage. In order to assess the damage, two failure criterions are applied to the RPV under IVR condition. One is strain criterion, and the other damage criterion. Finally, the main results can be summarized as follows:

(1) Creep effect dominates the failure of the RPV under the IVR condition. The creep strain mainly contributes to the total deformation, accounting for 90% of the total, while the plastic strain can be negligible before tertiary creep stage. At low pressure, the creep mechanism leads to global failure with strain criterion, and plastic effect become increasingly important at higher pressure, possibly inducing local failure to a large extent. Melt-through can take place at very low pressure if a large heat flux is concentrated on a small area.

(2) The failure position is in the zone of the thinnest wall thickness close to the highest temperature region in which there is some melt-through phenomenon observed. Furthermore, the most endangered zone exhibits the highest creep rates. It is located at the hot focus site corresponding to the concentrated heat flux region.

(3) The creep deformation of cylinder to sphere transition position results in a wall thickness reduction, which further accelerates the creep damage. The reduction percentage of wall thickness reaches approximate 30%, which is very detrimental to the structural safety. At low pressure level, the effect of high temperature is more dangerous than that of the pressure for the RPV.

(4) The multiaxial state of stress sharply accelerates the total damage accumulation, leading to a local failure of the RPV at thinnest wall. The multiaxial effect plays an important role in the damage distribution during the necking. The comparison between strain and damage criterion shows that the damage predicted by the latter is much larger than the one by strain criterion.

Acknowledgements

This work supported by Zhejiang Provincial Natural Science Foundation of China under Grant No. LQ15E050007, National Natural Science Foundation of China (Grant No.51505425), ZJUT Teaching Reform Project (No.JXY201404).

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