Introduction
The scheme of RPV with water cooling in severe accident and
its typical failure segments.
During the severe accident (SA) causing a pressurized core meltdown,
the reactor pressure vessel (RPV) integrity may be threatened by
various thermal-mechanical loadings. In the accident event, a large
quantity of liquid corium can relocated to the lower head (LH) of
RPV. Without proper external cooling, the local overheating must lead
to the LH breach, resulting in the release of a large amount of core
material into the containment (see Fig. 1). In overcoming it,
the famous strategy of “In-vessel retention (IVR)” based on the idea
of external water cooling is widely used by most advanced nuclear
power plant (NPP) (Yue et al., 2015). Moreover, the so called “IVR”
mitigation had been certified by the authority organization as
a standard measure for preventing overheating of the core in severe
accident since 1996 (Schulz et al., 2006). Essentially, the IVR
mitigation is to provide long-term water cooling on the outer RPV
wall, so the decay heat is able to be removed without any active
actions and assistance measures (Jung et al., 2015). In the
traditional concept of IVR, there are two assumptions (S.-H. Kim
et al., 2015;
Duijvestijn and Birchley, 1999): (1) the LH
sited at the bottom of RPV is assumed to be fully submerged into water
flooding; (2) The melting pool within the RPV is depressurized.
However, the above assumptions weren't seriously challenged until the
Fukushima accident on 2011 (An et al., 2016). The accident event
showed that the structural behavior had not been appropriately
assessed, and a certain pressure (up to 8 MPa) was found to
exist at the inside (Mao et al., 2016a). Most notably, the
malfunction of water supply system had occurred during the SA,
resulting in insufficiency of cooling water on the external RPV wall
(Government of Japan, 2011). Toward this end, the present paper tries
to address the structural-related issue on whether RPV integrity can
be secured within the prescribed period, if the RPV is subjected to
various thermal-mechanical loads created during such a severe
accident.
As for the structural-related issue under the IVR condition, the
particular concern is to characterize the timing, mode, and size of
a possible RPV failure with varying parameters such as water levels,
internal pressures, since the structural safety is a requirement of
paramount important in the operation of NPP (T. H. Kim et al., 2015). During
the SA, the IVR mitigation should be robust and capable of maintaining
the RPV in a manageable state and preventing the collapse of LH (Li
et al., 2015). As mentioned previously, the depressurization of
primary system was the basic assumption in the conventional IVR
concept, so the traditional methodology takes effect only in a limit
design basis accident (Mao et al., 2016b). As a matter of fact, the
RPV failure can be affected by numerous factors, such as internal
pressure and water cooling state. Due to the intrinsic limitation, the
structural behaviors weren't fully understood for the RPV with various
water levels under pressurized melt pool. Some of the failure
mechanisms hadn't been recognized across the wall thickness using the
traditional IVR concept (Zhu et al., 2017). As indicated in previous
studies, at the low pressures, the creep failure mechanism dominates
the failure process, whereas the plasticity becomes increasingly
important at higher pressures (Duijvestijn et al., 2006). Some
researchers pointed out that the higher internal pressure gave rise to
the instant plastification, and inevitably aggravates creep
deformation (Park et al., 2015). In most accident situations, the RPV
safety should be ensured during the first 72 h, so it leaves enough
time for the operator to take some emergence measures (Mao
et al., 2016a). However, certain thermal-mechanical loading processes
(or history) may accelerate or delay RPV failure time (Tsai
et al., 2017; Tusheva et al., 2015). Furthermore, the failure mode as
well as site can be significantly different for various loading
situations. For instance, the unexpected malfunction of water supply
system leads to various cooling water levels, resulting in the larger
stored heat inventory due to the insufficiency of water cooling
(Government of Japan, 2011). In fact, the continuing decay heat
generation and the possibly high internal pressure are the acute and
long term challenge to the RPV safety. As discussed previously, the
external cooling state is the essential for ensuring the RPV
integrity. Although the early vessel failure isn't allowed in most
situations, the RPV will eventually fail unless the decay heat can be
completely removed by the water cooling (Mao et al., 2016b).
Accordingly, the RPV failure is determined as time- and temperature-dependent process. In disclosing the complexity of the structural
behaviors, several experimental programs such as FOREVER, USNRC/SNL,
CORVIS, and LIVE had been performed (Willschutz et al., 2001, 2006;
Nicolas et al., 2003; Adroguer et al., 2003; Gaus-Liu et al., 2010;
Buck et al., 2010), the results of which validated the predictability
on the failure time, mode and site by using the finite element method
(FEM). The already conducted experiment span a wide range of pressure
(0.2–10 MPa) and temperature gradient (130–1500 ∘C) (Theofanous et al., 1997). Besides, the
experimental data were also compared with the results by FEM in terms
of field parameters, such as temperature, wall enlargement, wall
thinning (Song et al., 2009). The results had revealed that the
structural behavior was governed by the multilayer failure mechanism
(Mao et al., 2016c), and the geometric discontinuity was most
vulnerable to RPV failure (Mao et al., 2016d). Actually, the RPV
integrity under SA was compromised by various failure mechanisms, such
as melt-through, plasticity, thermal-plasticity, creep, and even
necking. However, the influence of various water levels and internal
pressures on RPV integrity is scarcely found in the previous
studies. Therefore, a good understanding of structural behavior is of
vital importance for ensuring the RPV safety in the above SA
conditions.
In this paper, the main task is to numerically investigate the RPV
structural behaviors with the effect of various water levels and
internal pressures. Toward this end, the 2-D FE-models of the RPV were
established on the ABAQUS platform. In order to illustrate the effect
of various water levels on the structural behavior, three typical
water levels were considered in FEA, including low, mid and high water
levels. Due to the melting pool on the inside and external water
cooling on the outside, the high temperature gradient is formed from
150 to 1327 ∘C across the wall thickness. Accordingly, the
RPV failure is governed by several mechanisms, including creep,
plasticity and thermal-plasticity. With the thermal-mechanical
loadings, the structural behaviors of the RPVs are comparatively
investigated in terms of temperature, deformation, stress, plastic
strain, creep strain, and total damage. Due to the time- and
temperature-dependent characteristics, both creep and plastic
deformation are analyzed during the IVR condition, including the
interaction between creep and plasticity. Through vigorous
investigation, it is found that the failure site, mode and time are
greatly affected by the water levels. Furthermore, the internal
pressure plays an important role in determining the RPV failure.
Mathematical model description
Creep and plastic damage modeling
As is well known, the severe accident condition leads to the form of
high temperature gradient across the RPV wall thickness. The inner
surface of RPV is exposed to the pressurized melting pool, so the
material temperature on the inside exceeds the 0.4 time of the melting
point (1327 ∘C). Due to the high temperature on the inside, the
creep failure mechanism occurs at the corresponding site. In
describing the creep behavior of the RPV wall, the creep law (Mao
et al., 2016d) with a number of free parameters is formulated as
follows,
ε˙c=d1σd2εcd3exp-d4T,
Wherein, the d1, d2, d3, d4 are material
coefficients by fitting the creep law to test data obtained at
constant load and temperature. As for the RPV steel, its creep process
experiences primary, secondary and tertiary stages. Eq. ()
is a general form of strain hardening representation, and it has the
capability in describing the above highly-nonlinear process. However,
the requirement of the d1>0 and d3>0 must be
satisfied. As indicated in Eq. (), the material coefficients
are rate- and temperature-related parameters. In order to have
a proper parameter fitting, a newly-proposed subroutine with nonlinear
weighting method is implemented in the ABAQUS. For plasticity, the
stress–strain relationship is highly nonlinear, so the
Ramberg–Osgood (R–O) equation is adopted for describing the material
constitutive relationship,
ε=σE+ασσyn,
Wherein, σ and ε are the true stress and true
strain respectively, E is Young's modulus, σy is the yield
strength, α and n are the material constants that can be
fitted with the tensile data ranging from the yield point to the
necking point. In the FE calculation, the option of multi-linear
isotropic hardening is activated in ABAQUS for higher accuracy.
In fact, the plastic yielding and creep deformation occurs at the
failure site simultaneously, so the “ductility failure criterion” is
used for evaluating the damage. The material damage due to the
significant plastic and creep strains is formulated by a damage
measure D, which is the result of plastic damage increment plus
creep damage increment. The increment ΔD is accumulated at the
end of a load step or sub-step of the FEA, the expression of which is
described as,
ΔD=Δεeqvcrεfraccrσ,T+ΔεeqvplεfracplTRv,
In Eq. (), the εfraccr and
εfracpl are creep and plastic rupture
strain respectively, which can be obtained by the tensile test with
constant thermal-mechanical loads for RPV material. It should be noted
that the damage behavior is in dependence on the triaxiality factor
Rv that is also incorporated in the “ductility failure
criterion”,
Rv=231+v+31-2vσHσM2,
Where, σH is the hydrostatic stress, σM is the
von-Mises equivalent stress, v is the Poisson's ratio. The presence
of is to reflect the effect of multiaxial stress state on damage
evolution of RPV component. With the linear accumulation rule, the
accumulated total damage D can be computed by the sum of each damage
increment ΔD,
D=∑i=1nStepΔDi,
Actually, when the total damage D of an element is equal to 1, the
element is deactivated by multiplying its stiffness matrix with
a factor of 10-6, so the element is correspondingly set to be
inactive using element death technique. If the ΔDi=0, the
element has “no damage increment” at the specific sub-step.
The RPV configuration and its FE-model with applied boundary
conditions.
Description of FE modeling
Due to the high nonlinearity characteristics in material properties,
geometric configuration and loading, the analytical modeling is almost
impossible for evaluating the structural behaviors. In overcoming this
problem, the two-dimensional (2-D) FE-models are developed for the RPV
with various water levels using ABAQUS platform, which is displayed in
Fig. 2. In order to reduce the computational cost, only half part is
considered in the FEM, due to its axisymmetry. Actually, the RPV
structural behavior is very complex among the LH wall, so the wall
thickness is intensively meshed. It can be seen in Fig. 2 that the
grid density is aggravated at the transition portion of spherical
segment to cylindrical segment. The mesh of LH wall is made of 20
nodes, and the element size is about 8 mm in width. After
repeated convergence computation, the total element number is
finalized to be 4200. Since the severe accident condition leads to
the coupling of thermal-mechanical structural behavior, the
thermal-coupled element type of CAX4T is employed in FEM, which is the
2-D structural 4-node iso-parametric element. The selected element
type is suitable for computing time- and temperature-dependent
deformation. It also has the capability in accounting for the creep
and plasticity simultaneously. Due to the larger deformation, it
provides more flexibility for considering the geometric nonlinearity,
e.g. large plastic or creep deformation.
As for the boundary condition, the internal pressure is applied as
a surface load on the inside of the RPV. Actually, the internal
pressure load leads to an increasing primary stress with ongoing
plastic and creep deformation. Two phenomena are directly related to
the pressure load: (i) the enlargement of the bulk; (ii) the reduction
of the wall thickness. As to thermal condition, the simulated tests
showed the temperature on the inside was maintained at approximate
melting point of 1327 ∘C, while the one on the outside was
maintained at the level of nuclear boiling (∼150 ∘C). In order to account for the most dangerous thermal
loading before the melt-through of the RPV wall, the critical heat
flux (CHF) loading is considered herein. In addition to that, the
various water levels are also considered in the FEM, due to the
possibility of cooling system malfunction during the severe
accident. As shown in Fig. 2, the low, mid and high water levels are
adopted in the comparative analysis. At the RPV wall above the water
level, the nature convection cooling is assumed in thermal boundary
condition. In the empty chamber, the heat transfer processes are
governed by the radiative transfer. For simplicity, the radiative heat
flux is converted into thermal loading, so the equivalent temperature
is obtained and applied on the cylindrical surface. Besides, the
equivalent temperature is assumed to be linearly distributed along the
cylindrical segment. It should be pointed out that the adopted RPV is
drawn from the real component of AP600-type power plant. The basic
geometric sizes are plotted in Fig. 2. For instance, the radius of
the spherical portion is 2180 mm, and the wall thickness is
165 mm. The height of cylindrical segment is big enough to
allow the enlargement of the LH, so the end effect is insignificant
for the LH. Furthermore, the wall thickness of spherical segment to
cylindrical segment is gradually increasing from 165 to
180 mm. As to constraint boundary conditions, Fig. 2 shows the
zero horizontal displacement (Ux=0) at the symmetry axis, and the
zero vertical displacement (Uy=0) at the top end of cylindrical
segment.
Temperature contours comparison for RPVs with various water
levels.
Mises stress contours for RPVs with various water levels at
pre- and post-creep stages.
Results and discussions
The general analysis on field parameters
As illustrated in Fig. 2, the cooling water level varies with various
water supplies during severe accident situations. Actually, sufficient
flooding for external reactor vessel cooling (ERVC) isn't guaranteed
as desired in most cases. As required in NRC regulation and design
standard, the IVR-ERVC strategy should come into action successfully,
although the insufficient water cooling may occur in the severe
accident due to the malfunction of the supply system. However, the
reality may not be in the assumed way. As indicated in Fukushima
accident on 2011, the RPV structural integrity was greatly threatened
by the insufficient water cooling, and the RPV failure took place
before prescribed safety time of 72 h. Accordingly, it is important
to know the temperature distribution for the RPVs with various cooling
water levels. During the period of severe accident, the so called
“IVR-ERVC” strategy is assumed to be able to arrest the degraded
melting core within the lower head (LH), so the melting pool is formed
on the inside. Consequently, it can be seen in Fig. 3 that the
temperature on the inner surface is at approximate melting point
(1327 ∘C) of RPV material. As for the outer surface of the
LH, the temperature distribution varies with various cooling water
levels, as shown in Fig. 3. It can be learnt that the water cooling
significantly lowers the temperature on the outside, while the
temperature becomes homogeneous on the RPV wall without water
cooling. The observation of Fig. 3 reveals that the high temperature
gradient is formed from 1327 ∘C on the inside to
150 ∘C on the outside across the wall thickness for the
portion below the cooling water level. Clearly, Fig. 3 shows that the
high temperature zone is significantly decreasing with increasing the
water level. Figure 3 also indicates that the high temperature
gradient exists at the site aside around the water level. Since the
cylindrical segment is above the melting pool, the temperature of the
corresponding wall thickness is mainly affected by the thermal
radiation. From Fig. 3, it can be seen that the temperature of
cylindrical segment is much lower than the one of the LH.
In accordance with the aforementioned basic scenarios, Fig. 4 plots
the Mises stress contours for RPVs with various water levels at pre-
and post-creep stages. The comparison of Fig. 4 discloses that the
Mises stress is relaxed very significantly after a certain period of
creep, especially for the inner surface of the RPV. Clearly, it can be
seen in Fig. 4 that the Mises stress at high temperature zone is much
lower than that at the water-cooled zone. There exists
a highly-concentrated zone for the Mises stress at the site aside
around the water level. The maximum value of Mises stress is found to
be around 550 MPa at pre-creep stage. After a certain period
of creep time, the lower head (LH) suffers great deformation, as
indicated in Fig. 4. For RPV with low to moderate water level, the
necking phenomenon is found at the transition region above the water
level, while the RPV with high water level still has some
load-carrying capability in preventing necking. Close observation of
Fig. 4 reveals that the RPV with low water level is most vulnerable to
structural failure after a certain amount of creep hours. After
a period of creep, the Mises stress field is redistributed into a new
one for the three RPVs. Although the stress relaxation somehow occurs
throughout the whole region, the phenomenon of stress concentration
still exists at the site in the proximity of water level at post-creep
stage. As indicated in Fig. 4, for the RPV with low to moderate water
level, the RPV structure must undergo severe plastic and creep
deformation at the weakest link of the wall thickness, since the RPV
wall necking is ductile failure mode. It can also be inferred in
Fig. 4 that the maintenance of structural integrity highly depends on
the cooling water level under severe accident scenario.
As is well known, the yield limit of RPV material is decreasing with
increasing the temperature, which is depicted as the symbol of dash
line in Fig. 5. Taking the yield limit as a failure criterion, it can
be observed in Fig. 5 that the wall thickness of inner surface to mid
layer is almost performed in the state of plasticity, because the
equivalent stress computed by FEM exceeds the yield limit at the
corresponding site. More interestingly, there exists a core of elastic
layer within the wall thickness, as shown in Fig. 5. The elastic core
is squeezed into a very small region with lowering the cooling water
level, implying that the load-carrying capability is correspondingly
decreasing. As a matter of fact, the existence of elastic core is
essential for ensuring the RPV safety under SA condition, since the
elastic core has some load-carrying capability in preventing immediate
collapse locally. For the outer surface layer, the equivalent stress
is approaching the yield limit as it increases with decreasing the
temperature. Taking the creep effect into account, it can be observed
in Fig. 5 that the stress is relaxed very significantly within the
wall thickness of 0<t<115 mm, which is reduced to near
stress-free level for the wall thickness of T>425 ∘C.
Moreover, the stress on the outer surface layer is found to be relaxed
by approximate 40 %. General observation of Fig. 5 displays that
the stresses of RPV with various water levels are arranged as low
water level > mid water level > high water level.
Mises stress distributions across the wall thickness at pre-
and post-creep stages.
Effect of internal pressure on plastic strain distribution
for RPV with low water level.
Effect of internal pressure on plastic strain distribution
for RPV with mid water level.
The effect of creep on plastic strain of RPVs
As for the structural integrity assessment, it is interesting to
examine the distribution of plastic strain at the failure time. With
the effect of creep on the plastic strain, the plastic strain
distribution varies with the creep time, since the creep is
a time-dependent process. Due to the existence of the primary pressure
on the inside of the RPV, the failure time also varies with the water
levels for the RPVs. Figure 6 shows that the distribution of
equivalent plastic strain along the lower head (LH) with low water
level. In order to make a judgment on the RPV safety, the plastic
limit of εp=0.25 is taken as a basis for
comparison. It can be seen in Fig. 6 that the internal pressure plays
an important role in determining whether the RPV integrity is
maintained or not after a certain period of creep time. At the low to
moderate pressure levels (P<1 MPa), the most significant
plastic strain is concentrated at the site near the water level. With
the increase of internal pressure, the equivalent plastic strain
spreads out over a wide area above the water level of the LH. As
indicated in Fig. 6, the plastic strain is insignificantly distributed
among the LH without the internal pressure effect after 100 creep
hours, whereas the maximum plastic strain abruptly increases up to the
plastic limit of εp=0.25 after 2.3 creep hours with
the internal pressure loading of 1 MPa. Most notably, the
comparison in Fig. 6 demonstrates that the LH may fail locally at the
site in proximity of low water level as soon as 5 s under the
internal pressure loading of 3 MPa, due to its significant
plastic strain computed by FEM exceeding the plastic limit of the RPV
material. Besides, in order to illustrate the instant structural
behavior under the various internal pressure loads, Figs. 7 and 8 plot
the distributions of plastic strain for the RPV with mid and high
water level respectively. Through the comparison, it can be found that
the most significant plastic strain always occurs at the location near
water level. In other words, the corresponding site is most vulnerable
to failure, due to the unacceptable plastic yielding. Clearly, both
Figs. 7 and 8 show that the maximum plastic strain exceeds the plastic
limit with the effect of internal pressure. It can be observed that
the highly-plasticized zone is enlarged with the increase of internal
pressure. General observation of Figs. 6, 7 and 8 reveals that the
high plastic strain spreads over the high temperature zone above the
water level, especially for the condition of high internal pressure
loading. Besides, it should be noted in Figs. 6, 7 and 8 that the
failure time of the LH is different from each other. Due to the
existence of certain internal pressure, the failure time is remarkably
reduced accordingly by comparing with the one without internal
pressure. It can be learnt that the RPV fails at the time of 2.3, 4.2,
64 h for the respective situation of low, mid and high water
level under internal pressure loading of 1 MPa. If the
internal pressure is further increased up to 3 MPa, the
failure time of RPV is lowered down to 5, 33 s, 10 h
for the respective water level under the loading of
3 MPa. From this perspective, it is safe to say the RPV with
high water level has most advantage in preventing the immediate
structural failure, the reason for that can be attributed to the well
maintenance of the material strength in the outer layer of wall
thickness. As for the RPV wall thickness below the water level, the
plastic strain is insignificantly distributed, most of which is less
than εp=0.025. Under the present severe accident
conditions, it can be seen that the cooling water level determines the
failure site, and the failure time is greatly affected by the internal
pressure. Also, it can be concluded from Figs. 6, 7 and 8 that the
global failure isn't a feasible mode for the RPV under primary
pressure loading.
Effect of internal pressure on plastic strain distribution
for RPV with high water level.
Effect of creep on plastic strain distribution for RPV with
low water level.
Effect of creep on plastic strain distribution for RPV with
mid water level.
As mentioned before, the time-dependent creep plays an important role
in affecting the RPV structural failure process. Also, the plastic
strain profile varies for the RPVs with various water levels, as shown
in Figs. 9, 10 and 11. As a matter of fact, the higher internal
pressure gives rise to plastic strain rather than creep strain
immediately after the load is applied on the RPV. With the increase of
the creep time, the creep strain accumulated on heat-focused zone may
be larger than the instant plastic strain. As for the RPV with low
water level, Fig. 9 shows that the plastic strain significantly
increases with the creep time, especially at the site aside around the
water level. After creep time of 6 s, it can be seen in Fig. 9
that the maximum creep strain reaches the value of εp=0.25 with the loading of 3 MPa. Figure 9 also reveals that
the creep accelerates the plastic strain evolution, and the plastic
strain become increasingly significant among the region in the
proximity of water level. At the wall thickness below the water
level, there is no significant change for the plastic strain
distribution, and the corresponding plastic strain is maintained at
a relatively low level throughout the whole creep process. The
comparison of Figs. 9, 10 and 11 discloses that the highly-plasticized
zone is squeezed into a smaller size with increasing the water
level. Accordingly, the plasticized zone is the smallest one for the
RPV with high water level, as shown in Fig. 11. Most notably, it can
be found in Figures that the evolution of plastic strain become very
sensitive to the creep, when the failure time is exceeded. In other
words, it can be inferred that the interaction between plastic and
creep strain becomes much severer after the failure time. Finally, the
RPV failure (or collapse) is triggered by the creep evolution, and is
contributed by the sum of plastic and creep strain. Again, it can be
concluded from the above analysis that the RPV must collapse or fail
locally at the site near the water level. The comparison of
Figs. 9, 10 and 11 discloses that the RPV with high water level has
the most ability in preventing RPV failure.
The effect of water levels on creep strain of RPVs
Effect of creep on plastic strain distribution for RPV with
high water level.
Comparison of creep distributions for RPV with various water
levels under P=0 MPa.
As indicated above, the creep strain makes a great contribution to the
RPV failure. In illustrating it, Fig. 12 plots the distribution of
equivalent creep strain (simply called as “CEEQ”) along the LHs
without internal pressure effect for the condition of various water
levels. It can be seen in Fig. 12 that the peak value of CEEQ occurs
at the site aside around the water level, correspondingly at the low,
mid and high angular position. The comparison of Fig. 12 shows that
the maximum creep strain is arranged as the one of low water
level > mid water level > high water level. Before the peak
value of creep strain, there exists a significant fluctuation of the
creep strain, due to the deformation incompatibility. At the
transition portion of spherical to cylindrical segment, a small peak
value of creep strain is found correspondingly for the RPV with low
and mid water level. Throughout the whole lower head, it can be found
in Fig. 12 that the maximum creep strain reaches the value of
0.6 % without consideration of the internal pressure effect. For
the RPV with low water level, the creep strain at the high temperature
segment is maintained at a relatively low level except at the
transition location of hot and soft to cool and strong wall thickness.
With the increase of internal pressure, the creep strain is
significantly increased at the failure time for the condition of
respective water level. Figures 13 and 14 plot the distributions of
CEEQ along the LH under the internal pressure of 1 and 3 MPa
respectively. Figure 13 shows that RPV with low water level suffers
the severest CEEQ at the transition site of cool to hot wall
segment. For the RPV with mid water level, the high CEEQ is
distributed widely among the transition location, whereas the CEEQ is
insignificantly distributed throughout the LH for the RPV with high
water level. Compared with Fig. 13, it can be seen in Fig. 14 that the
high CEEQ zone becomes much wider with increasing the internal
pressure. Again, both Figs. 13 and 14 display that the maximum CEEQ
occurs at the site aside around the water level. Still, the RPV with
high water level has the lowest CEEQ at a specific time. At the
internal pressure of 3 MPa, the CEEQ exceeds the value of
2.5 % for most of the LH wall thickness, indicating that the
significant creep damage for the corresponding RPVs with low to mid
water level, as shown in Fig. 14. As compared to the one in Fig. 12,
it can be observed in Figs. 13 and 14 that only one peak value of CEEQ
take place across the whole LH for the condition of
3 MPa. Furthermore, the peak value does not exist at the
transition portion of spherical segment to cylindrical segment.
Comparison of creep distributions for RPV with various water
levels under P=1 MPa.
In order to visualize the distribution of creep strain across the
whole LH, Fig. 15 plots the equivalent creep strain (simply called
“CEEQ”) contours for RPV with various cooling water levels and
internal pressures. The overall observation of Fig. 15 discloses that
the maximum creep strain occurs at the location of wall thickness in
proximity of water level. It should be noted in Fig. 15 that the LH
creep deformation under the low to mid pressure is not accompanied by
the essential vessel wall thinning, and its failure is caused by the
accumulated creep strain during the IVR process. However, under the
high internal pressure (3 MPa), the accumulated CEEQ results
in the significant deformation, even necking phenomenon for the RPV
with low to mid water level, as shown in Fig. 15c. The severest CEEQ
reaches the value of approximate 1.90 at the failure site after
6 s creep time. From this perspective, the existence of
internal pressure is very detrimental to the RPV safety during the
severe accident. Combined with the above analysis on the plasticity,
it can be concluded that the plastic and creep strains are interacted
with each other. The results indicate that the location of failure
zone caused by plastic strain coincide with the one by creep
strain. Again, it can be found in Fig. 15 that the RPV with low water
level is most vulnerable to structural failure under the condition of
severe accident, since the concentration of CEEQ at the failure site
is the severest among the three cases. Close observation of Fig. 15
reveals that the significant CEEQ takes place on the inner surface of
the LH at low pressure, whereas the higher CEEQ occurs on the outer
surface at mid pressure, and finally the higher CEEQ zone is
interlinked across the wall thickness of the failure site at high
pressure. This phenomenon can be attributed to the variance of
deformation incompatibility with the effect of various internal
pressures. The variance of higher CEEQ distribution will inevitably
lead to the corresponding failure initiation. At a constant low
pressure, the failure initiates at the inner surface, while with
further increase of internal pressure, the outer surface tends to
undergo more creep deformation. Actually, it can be inferred in
Fig. 15 that the creep effects dominate the LH failure process at low
to moderate pressures, and instant plasticity becomes increasingly
important at higher pressures.
Comparison of creep distributions for RPV with various water
levels under P=3 MPa.
Comparison of creep strain contours for RPVs with various
cooling water levels and internal pressures.
The evaluation on RPV structural failure behaviors
As indicated above, the final RPV failure (or collapse) is caused by
the combination of plastic and creep deformation. In order to
illustrate the failure process, Fig. 16 plots the relationship of the
maximum deformation vs. internal pressure for the RPV with various
water levels. It can be seen in Fig. 16 that the deformation at
post-creep stage is shown as a nonlinear function of the internal
pressure. Clearly, for the condition of low to moderate internal
pressure, the bulge of LH is abruptly increasing at the internal
pressure of approximate 2.25 MPa, whereas the LH with high
water level significantly stretches outward at the site near the water
level for the condition of more than 10 MPa. Through careful
observation, it can be found that the maximum deformation occurs at
the failure site in proximity of corresponding water level. Further
observation of Fig. 16 shows that the max. deformation is increasing
very fast at the internal pressure of less than 1 MPa, and
then it stably increases up to a certain value at the internal
pressure from 1 to 2.25 MPa. Furthermore, Fig. 16 demonstrates
that the max. deformation is arranged as the one with high water
level > the one with mid water level > the one with low water
level, throughout the spectrum of internal pressure. In fact, the max.
deformation of the LH is contributed by the thermal expansion, plastic
yielding and creep accumulation. As well known, the thermal expansion
is a constant value with the fixed temperature distribution, so the
relationship of max. deformation vs. internal pressure is dominated by
the plastic and creep deformation at the corresponding site.
Relationship of deformation vs. internal pressure for RPVs
with various water levels.
Comparison of damage contours for RPV with various water levels and
internal pressures.
As mentioned before, the final RPV failure is caused by the plastic
and creep damage. In order to visualize the total damage distribution,
Fig. 17 plots the contour of total damage for the RPVs with various
water levels under various internal pressures. As can be seen from
Fig. 17, the bulge of the LH is observed for all the mentioned
cases. It stretches outward along both horizontal and vertical
direction, and the displacement along the vertical direction is much
larger than the one along the horizontal direction. With the increase
of internal pressure, the LH stretches out more and more
significantly, especially for the RPV with low water level. For the
condition of low to moderate pressure, the maximum displacement of the
vessel wall has reached up to 50 mm at the failure
time. Clearly, the sagging of the deformed LH could be several times
larger than the initial undeformed LH, and the deformed LH shape is
appeared as an “egg shell” after a period of creep time. It is clear
to see from Fig. 17 that the sagging is associated with severe necking
phenomenon at the failure site, especially for the RPV with low water
level. During the creep period, the wall thinning is observed for all
the mentioned cases. For instance, the most significant reduction of
wall thickness is reached up to 25 % at creep time of
2.23 h for the RPV with low water level under internal
pressure of 1 MPa. For the case with low pressure and high
water level, the damage is highly concentrated on each side of wall
thickness near water level. It can be seen in Fig. 17 that the damage
in the middle layer of the wall thickness is much lower than the one
on surface layer at the initial creep stage, due to the existence of
elastic core. However, with the aggravation of creep damage, the
development of plastic damage must be accelerated, so the total damage
evolves much faster with increasing creep time. It can be inferred
that the creep and plastic damages are interacted with each
other. Accordingly, time- and temperature-dependent analyses are
necessary for assessing the RPV safety under severe accident
condition. Consequently, the nonlinearities, large displacement, large
strain and follower damage, must be taken into account in the finite
element analysis (FEA). Furthermore, one can note that the calculated
sagging and cumulated damage is greatly affected by the internal
pressure and external water level. With the effect of internal
pressure, the RPV failure time significantly decreases. As for the RPV
with low water level, the failure time is >100 h,
2.23 h, 6 s corresponding to condition of 0.1, 1,
3 MPa, since the highest damage exceeds 1. Moreover, the
observation of Fig. 17 reveals that the localized failure is the
feasible failure mode, and the RPV instability takes place at the
transition portion of cool to hot wall thickness. Due to the large
elongation aside around the failure site, the failure process presents
ductile behavior, so the final failure is caused by the combination of
ductile tension and shear. Accordingly, the instant global collapse is
impossible immediately after the thermal-mechanical loading is applied
on the LH.
Conclusions
In the current paper, the RPV with various water levels is chosen for
study, due to the possibility of water cooling system malfunction
under severe accident of core meltdown. As indicated in some severe
accidents, a certain pressure (up to 8.0 MPa) may exist inside
the RPV, so the scenario of pressurized core meltdown is considered in
the FEM. In order to illustrate the structural behavior and its
failure, the thermal-elastic-visco-plastic FE models are developed in
the platform of ABAQUS with consideration of highly nonlinear
characteristics. Due to the presence of the melting pool (∼1327 ∘C) at the inside and external cooling water (∼150 ∘C), the most dangerous thermal loading is considered and
applied on RPV wall as the type of critical heat flux (CHF) loading
before the material melting of the wall thickness. Actually, the
severe accident condition leads to the formation of high temperature
gradient, so the RPV structural behavior spans a wide range of failure
mechanisms, such as creep deformation and plastic yielding. Through
vigorous FE investigation, the main results can be summarized as
follows:
Due to the existence of various water levels, the temperature fields are
quite different from each other, resulting in significant variation of
structural failure site, time and mode for the RPVs. It is found that the
increase of cooling water level improves the RPV structural safety, that is
to say, the corresponding IVR strategy is enhanced.
The failure site takes place at the location in the proximity of cooling
water level, since the total damage is highly concentrated at the site. At
the low pressure (≤1 MPa), the RPV tends to globally stretch outwards,
whereas the localized ductile failure is the failure mode at higher pressure
(≥3 MPa), especially for the RPV with low to moderate water level.
Due to the existence of elastic core in the midpoint of wall thickness
for some cases, the damage distributed around aside each surface of RPV wall
may well exceed the one at midpoint. Further analysis shows that the plastic
damage at failure time accounts for approximate 80 % of total damage for
the cases with low water level, while the evolution of total damage is
significantly accelerated by creep.
The stress on the inner surface is unanimously relaxed to a near-zero
level at failure time, while the stress (strain) field is varying during the
IVR period. Due to the incompatible deformation, the RPV wall at around the
water level suffers severe stress triaxiality, which exacerbates the RPV
failure process. Through the analysis, it is found that the existence of
internal pressure at any level is a huge threat to RPV integrity, so the
pressure release is necessary to IVR strategy.
The RPV enlargement is observed throughout the whole failure process,
the wall thickness is simultaneously thinning all over the time, especially
for the failure site. With the increase of internal pressure, the vertical
bulge of RPV is much larger than the horizontal bulge. Furthermore, the most
significant bulge is found at the failure site for the RPV with low water
level.