COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE
EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China
©2006 Tsinghua University Press & Springer
Fatigue Damage Analysis of Reactor Vessel Model Under Repeated
Thermal Loading
M. Takagaki 1*, Y. Toi 1, T. Asayama 2
1
2
Institute of Industrial Science, University of Tokyo, 4-6-1 Komaba, Meguro-ku, Tokyo 153-8505, Japan
O-arai Research and Development Center, Japan Atomic Energy Agency, 4002 Narita-cho, O-arai-machi,
Higashi-ibaraki-gun, Ibaraki 311-1393, Japan
Email: toi@iis.u-tokyo.ac.jp
Abstract The local approach to fracture based on continuum damage mechanics and the finite element method has
been applied to the low-cycle fatigue damage and fracture behavior of a nuclear reactor vessel model subjected to
repeated thermal loading due to the alternate inflow of high-temperature and low-temperature liquid sodium. The
viscoplastic strain based on the creep plasticity isotropic hardening theory with damage evolution is used in the
analysis. The calculated results for the neibourhood of intake nozzle and the conical body have been compared with the
experimental results. The calculated results considering the difference of mechanical properties of parent metal, weld
metal and heat-affected zone have corresponded well with the experimental results for the initiation and propagation of
thermal fatigue cracks.
Key words: damage mechanics, fatigue, finite element method, local approach to fracture
INTRODUCTION
The local approach to fracture, which combines the material damage analysis with the material fracture is one of
methods available to evaluation of damage in structures[1, 2]. In this method the damage analysis is conducted by
means of the finite element method using the constitutive equation based on continuum damage mechanics. In the case
of the damage evaluation by the conventional method, such as fracture mechanics, it is necessary to carry out some
simulations for the damage evaluation. At first, the stress analysis using FEM is conducted, and the damage is
estimated by means of calculating the fracture mechanics parameters like stress intencity factor. The nodes which are
considered to reach the fracture release the connectivity, and the mesh-model is modified. The analysis of crack
propagation is conducted by repeating these procedures. Since, in the present method, stiffness reduction from the
damage is expressed by the damage parameter, the modification of the mesh-model is unnecessary. Therefore we can
conduct the calculation of the damage and the stress analysis at the same time.
Though the performance of the recent computer makes progress, there are few examples of the application to the
fatigue analysis for the structure. Therefore, in the present study, the damage evaluation based on the local approach to
fracture for the model of reator vessel under cyclic thermal loading is conducted. The analysis is carried out by the
three-dimensioal finite element method using the elasto-viscoplasticity damage constitutive equation proposed by the
authors. The results of the comparison of the analysis and the fatigue test of the reactor vessel model show good
agreement in the distribution of crack initiation and propagation.
LOCAL APPROACH TO FRACTURE
Damage evaluation methods based on the local approach to fracture are three types such as uncoupled analysis, locally
coupled analysis and fully coupled analysis.
On the uncoupled and locally coupled analyses, the damage is evaluated only at an integral point. Therefore, it is
difficult to calculate crack evolution, though it is possible to predict the crack initiation life.
⎯ 352 ⎯
In the present study, we have conducted the evaluation of crack propagation by means of fully coupled method based
on the finite element method using elasto-viscoplastic damage constitutive law. The stress analysis considering
damage by this method is performed not only at the notice point but also at all integral points. Crack generation is
regarded when one integral point has damage critical value. Crack path is expressed by a series of the elements in
which the damage variable at all integral points reaches damage critical value. If this analysis is repeatedly carried out,
we can estimate the crack propagation behavior.
ANALYSIS METHOD CONSIDERING THERMO-DAMAGE
1. The thermo-elasto-viscoplastic damage constitutive law The present method of thermo-elasto- viscoplastic
damage analysis employs the extended Lemaitre’s[3] scalar damage evolution law and the creep-plastic isotropic
hardening theory. The thermo-elasto-viscoplastic damage constitutive equation is given as follows
[ ]
{σ&} = [De ]({ε&} − {ε&vp}− {ε&T }) + D& e {ε e }
(1)
where {σ& } is the total stress rate vector, {ε&} is the total strain rate vector, {ε&e } is the elastic strain rate vector, {ε&vp } is
[ ]
the viscoplastic strain rate vector and {ε&T } is the thermal strain rate vector. Then [De ] and D& e are the elastic stiffness
matrix considering damage and temperature dependency of Young’s modulus and its rate. In this matrix, the effective
Young’s modulus E is used as follows:
E = E (1 − D )
(2)
where E is Young’s modulus and D (0 ≤ D ≤ D cr ) is the damage variable, where the states D = 0 and D = D cr
represent the undamaged and the fractured state, respectively. The yield function is defined as Eq. (3)
f = f (σ ,ε vp ,T , D )
(3)
*
The viscoplastic potential ψ vp
is defined as follows:
*
ψ vp
= f =
K σ e (1 − D ) − R − k
N +1
K
N +1
(4)
where σ e is von Mises equivalent stress, R is the non-linear isotropic hardening parameter, k is the initial yield stress
and K, N are material constants. K, k and N are assumed to be temperature-dependent. The non-linear isotropic
hardening parameter R is expressed by Eq. (5)
R = Q1 p + Q 2 [1 − exp (− bp )]
(5)
where p is the accumulated equivalent strain and Q1, Q2 and b are material constants considering temperature
dependency. The elasto-viscoplastic damage constitutive equation is obtained as in Eq.
(6) by the conventional incremental strain theory and Eq. (1), (3), (4).
[ ]
[ ]
⎞
∂f
∂f
∂f ∂f
∂f
1⎛
1
{σ&} = [De ]({ε&} − {ε&T }) − ⎜⎜[De ]⎧⎨ ⎫⎬⎧⎨ ⎫⎬ [De ]⎟⎟({ε&} − {ε&T }) − [De ]⎧⎨ ⎫⎬⎛⎜ T& + D& + D& e {ε e }⎞⎟ + D& e {ε e }
g
g
∂D
⎩ ∂σ ⎭⎩ ∂σ ⎭
⎩ ∂σ ⎭⎝ ∂T
⎠
T
⎝
⎠
(6)
where,
T
⎧⎪ ∂f ⎫⎪ ⎧ ∂f ⎫ ⎧ ∂f ⎫T
⎧ ∂f ⎫
g = −⎨
⎬ ⎨
⎬+⎨
⎬ De ⎨
⎬
∂
∂
∂
ε
σ
σ
⎪⎩ vp ⎪⎭ ⎩
⎭ ⎩
⎭
⎩ ∂σ ⎭
[ ]
(7)
Then, in this method, the damage evaluation is based on continuum damage mechanics. Therefore the evolution
equation of the damage variable is given by the damage potential defined as in Eq. (8) using the energy release rate Y.
⎯ 353 ⎯
The damage increment D& is obtained as in Eq. (9).
ψ D* =
S
1 ⎛ Y⎞
⎜− ⎟
s +11− D ⎝ S ⎠
s +1
(8)
∂ψ D*
D& =
∂Y
(9)
where S and s are material constants. The space-discretization on this method is conducted by Galerkin method, and
the time integration is carried out by the central difference scheme. The heat conduction analysis is also carried out by
using the finite element method.
2. Heat conduction analysis Though the heat conduction equation is based on the conventional theory, the quantity
of heat conduction in the damaged material is smaller than in the undamaged solid because spaces (voids, microcracks
etc.) exist in the damaged zone. Consequently, since the temperature distribution of the damaged part is different from
that of the undamaged part, it becomes an additional cause of growth of thermal strain. Therefore, in the present study,
the heat conductivity λeq considering the damage effect proposed by Skrzypek[4] has been used in the analysis. This
heat conductivity is defined as follows:
~
λeq = λ0 (1 − D) + dλ rad
(10)
~
where λ0 is the heat conductivity of the undamaged material and dλ rad as shown in Eq. (11) is the influence term of
thermal radiation.
⎛
~
∂D ∂ X 4 ⎞
dλ rad = σ Bε 0 ⎜⎜ 4 DT 3 +
T ⎟dX
∂T ∂ X ⎟⎠
⎝
(11)
where σ B is Stefan-Boltzmann constant, ε 0 is the thermal emissivity and T is the temperature.
FATIGUE DAMAGE ANALYSIS
The fatigue damage analysis is conducted for the model of nuclear reactor vessel made of 304 stainless steel. This
vessel is the structure which contains welding parts (Height: 1200mm, Diameter: 800mm) as shown in Fig. 1. The
targets of the analysis are the intake nozzle and the conical body. Each mesh-model is illustrated in Fig. 2 and Fig. 3.
Figure 1: Sectional view of nuclear reactor vessel
⎯ 354 ⎯
Figure 2: Mesh model in intake nozzle
Figure 3: Mesh model in conical body
Table 1 Material constants
Parent Metal
Weld Metal
HAZ
Young’s modulus E [MPa]
E = 3.75 × 10 −2 T 2 − 44.97T + 146752.0
K [MPa]
K = −0.08T + 298.17
N
N = 4.0 × 10−3 T + 2.15
Initial yield stress k [MPa]
k = −0.13T + 164.25
k = −0.13T + 229.95
k = −0.13T + 114.98
Hardening parameter Q1[MPa]
Q1 = 0.12T + 582.0
Q1 = 0.12T + 698.4
Q1 = 0.12T + 582.0
Hardening parameter Q2[MPa]
Q2 = −0.51T + 440.5
b
b = 6.29 × 10 −2 T − 3.58
S [MPa]
6.4
s
s = −1.16 × 10−4 T − 2.01
Damaget threshold ε pd
0.14
Damage criterion D cr
0.8
Stefan-Boltzmann
[W/(m2K4)]
Emissivity ε 0
cont.
5.67 × 10 −8
0.6
⎯ 355 ⎯
The model of intake nozzle has 1650 3-dimensional isoparametric elements and 2751 nodes, and the conical body has
1212 elements and 2136 nodes. The present analysis takes into consideration the difference of mechanical properties of
the parent metal, the weld metal and the heat-affected zone (HAZ), because the mechanical property of the welded parts
has serious effects on the damage initiation and propagation. Material parameters considering the temperature
dependency and constants are shown in Table 1. The thermal loading condition for one cycle is assumed to be the
alternate inflow of high-temperature (600 oC) and low-temperature (250 oC) liquid sodium from the upper intake
nozzle. Each inflow time is 900sec. The cyclic loading number is 1002 cycles.
In the present analysis, the elasto-viscoplastic damage analysis and heat conducted analysis are performed step by step.
The results of heat conduction at the intake nozzle are as follows. The distributions of temperature at the inflow of high
and low temperature liquid sodium are shown in Fig. 4, respectively. Clearly, we can find that the change of
temperature at the part of nozzle is greater than in the body. Then the distributions of equivalent stress are illustrated in
Fig. 5. High equivalent stress takes place at the base and the inner surface of the nozzle. As shown in Fig. 6, the
distribution of damage concentrates on the inner surface of the nozzle where high stress occurrs. The obtained results
are in good agreement with the experiment.
Figure 4: Distribution of temperature in intake nozzle
Figure 5: Distribution of equivalent stress in intake nozzle
Figure 6: Distribution of damage in intake nozzle
⎯ 356 ⎯
For the analysis of conical body, Fig. 7 shows the distributions of temperature in inflow of high-temperature and
low-temperature liquid sodium. Obviously the temperature at the lower head and the outlet nozzle varies largely due to
the thermal fatigue loading. The part of vessel stand is subjected to lower temperature than the other parts because the
bottom of stand holds constant temperature (100 oC). Fig. 8 illustrates the distribution of equivalent stress. The region
of the maximum equivalent stress is around the point connecting the body and the stand in the case of the inflow of
low-temperature liquid sodium.
Figure 7: Distribution of temperature at conical body
Figure 8: Distribution of temperature at conical body
Fig. 9 shows the analytical and experimental results for the distribution of damage. Both has corresponded well with each
other. Fig. 10 shows the result of damage distribution calculated without considering the mechanical property of welding
zone in order to observe the effect of weld. It is observed that the generated macro-cracks hardly propagates in the
direction of thickness when the damage analysis does not take into account the mechanical property of welding area.
Figure 9: Distribution of damage at conical body
⎯ 357 ⎯
Figure 10: Distribution of damage at conical body without the welding zone
In the damage estimation of conical body, the evident macro-crack is generated and propagated. The history of damage
propagation is shown in Fig. 11. Crack grows rapidly at about 1000 cycles and stops propagating at around 1400
cycles. From these results, it is understood that the damage state of the vessel does not necessarily lead to the final
fracture.
Figure 11: History of crack propagation in conical body
CONCLUSIONS
In the present study, we have proposed the finite element method based on continuum damage mechanics that can
solve the elasto-viscoplasticity damage and heat conduction problems in order to analyze the faigue damage problems
caused by repeated thermal loading. Computational studies of thermal fatigue problem in nuclear reactor vessel have
been conducted. The validity of the analysishas been shown by the comparison between the numerical analysis and
experiment. In the damage estimation, the generated macro-cracks hardly propagates in the direction of thickness
when the damage analysis does not take into account the mechanical property of welding area. When we conduct the
damage estimation for the structures with welding area, it is necessary to make the model considering the parent metal,
weld metal and HAZ to obtain accurate solutions. We have carried out numerical analyses using the fully coupled
approach of local approach to fracture based on continuum damage mechanics and estimated crack initiation and
propagation. Though high performance computer is required for the analysis by means of fully coupled approach, the
present study has shown that it is possible to conduct the fatigue damage simulation for practical structures.
REFERENCES
1. Toi Y, Lee JM. Thermal elasto-viscoplastic behavior of structural members in hot-dip galvanization. Int. J. of
Damage Mechanics, 2002; 11(2): 171-185.
2. Lemaitre J. Local approach of fracture. Engineering Fracture Mechanics, 1982; 25: 523-537.
3. Lemaitre J. A Course on Damage Mechanics. 2nd ed., Springer, 1996.
4. Skrzypek J, Ganczarski A. Modeling of Material Damage and Failure of Structures (Theory and Applications).
Springer, 1999.
⎯ 358 ⎯