# Fracture mechanical analysis of a thermal shock scenario for VVER-440 RPV

### Ansprechpartner: Dr. Eberhard Altstadt [e-mail]

## Abstract

This page describes the analytical work done by modelling and evaluating a thermal shock in a VVER-440 reactor pressure vessel due to an emergency case. An axial oriented semi-elliptical underclad/surface crack is assumed to be located in the core weld line. Three-dimensional finite element models are used to compute the global transient temperature and stress-strain fields. By using a three-dimensional submodel, which includes the crack, the local crack stress-strain field is obtained. The determination of the stress intensity factor, which is a measure for the loading of the crack, follows the VERLIFE code. With a subsequent postprocessing using the j-integral technique the stress intensity factors K_{I} along the crack front are obtained. The results for the underclad and surface crack are provided and compared, together with a critical discussion of the VERLIFE code.

## Problem Outline

The analysis is done for a theoretical emergency scenario, where the main reactor cooling system has failed and the emergency cooling system is the only source of cooling fluid. Additionally, an unintentional opening of a pressurizer safety valve (PRZ SV) is assumed, which will be closed again after 3600 seconds, which causes a sudden increase of the internal pressure (see Fig. 1).

**Figure 1**: Internal pressure, Pressurizer Safety Valve opening with reclosure after 3600 seconds. [Full Size]

This scenario results in two opposite cold plumes below the cold legs in the downcomer of the RPV, which suddenly cool the inside of the RPV (see Fig. 2).

**Figure 2**: Temperature after 1000 seconds. The cold plumes can be clearly distinguished from the ambient region. [Full Size]

The undercooled region includes the core weld line, which is supposed to be one of the most embrittled regions of the RPV due to neutron radiation. Additionally, weld lines are also likely locations for cracks or flaws. Therefore, the scenario postulates an axial oriented semi-elliptical underclad crack. The axial orientation is chosen because the maximum principal stress in a pressurized cylindric vessel is acting in hoop direction (see Fig. 4) and so perpendicular to the faces of the postulated crack. The undercooled inner surface of the RPV and the crack are exposed to tensile stress. Repressurizing the RPV after 3600 seconds by closing the safety valve will suddenly increase the tensile stress and is assumed to be the critical phase of the scenario.

The primary questions regarding this scenario are:

- What is the loading of the crack during this scenario?
- What are the stress intensity factors during this scenario?
- What are the allowable stress intensity factors based on critical temperatures of brittleness?
- What is the allowable critical transition temperature of the material?

The component of interest is a VVER-440/V-213 RPV, which has an inner radius of 1771 mm. The inner surface has a cladding with a thickness of 9 mm. The thickness of the base material in the cylindric part of the vessel is 140 mm. The geometry of the two opposite cold plumes is given by means of levels, whereas level means the vertical distance below the lower part of the cold leg (see Fig. 2). The geometry and the physical properties of the cold plumes were obtained from a thermohydraulic simulation, which is not part of this work.

**Figure 3**: Coolant temperature in the cold plumes and the ambient region in the downcomer. [Full Size]

There is zero heat transfer at the outside of the vessel, which is reasonable because of the existence of an outer thermal isolation. The stress free temperature of the cladded vessel is 267°C. The weld material has the same thermal-physical and tensile properties as the base material but is supposed to have residual stresses both axial and circumferential orientation. These residual stresses result from the welding process. The postulated semi-elliptical underclad crack is located at the core weld 5/6 at level 3.485 m. As already mentioned above, this weld accumulates the highest damage (embrittlement) due to neutron radiation of all welds from the RPV and is therefore the most critical one, which means that the ductile to brittle transition temperature (DBTT) for this weld line is above the DBTT for all the others. The supposed crack width is 15 mm and the width/length ratio is 0.3. The crack is orientated in axial direction.

## Problem Solution

To solve the given problem we use a three-dimensional finite element (FE) model of one quarter of the RPV. Since the region of interest (core weld 5/6) is far away from the in- and outlets, these nozzles have been neglected in the model. This model does not contain any crack so far. It is used to compute in a first run the transient spatial thermal field.

Figure 2 shows the computed thermal field at the time t=1000 s. On the right part of the figure the cold plume is clearly to see. The incoming cold water leads to a general cool down of the inner surface of the RPV, but especially in the cold plum region the temperatures are up to 50 K lower than in the ambient region. This leads to elevated tensile stresses in hoop and vertical direction in the cold plum region of the inner surface of the RPV.

In a second run the mechanical solution is obtained using the thermal field as a body load. In the mechanical solution also the time dependent inner pressure, the acceleration of gravity and the initial residual stresses in the welds are considered.

**Figure 4**: Hoop stress after 1000 seconds. [Full Size]

Fig. 4 shows the computed hoop stress at the time t=1000 s. It can be seen that in general the highest hoop stresses are located in the cladding which is directly in contact with the coolant. Secondly, we identify the upper part, where the vessel wall thickness is greater than in the lower part, having in general higher hoop stress values than the lower part. And thirdly, the cold plum region in the lower part is also a region with elevated hoop stress.

**Figure 5**: Hoop stress (without cladding) after 1000 seconds. [Full Size]

In Fig. 5 the cladding is virtually removed and we have a direct view to the base and weld material. It can be seen that the weld lines have higher hoop stress values than the base material. This is due to the residual stresses, which are applied before the simulation starts. With a closer look we find that the upper weld lines are subjected to higher hoop stresses, the reason for this is the greater RPV wall thickness. But we have to keep in mind that the upper weld lines are much less damaged due to radiation than the core weld line.

**Figure 6**: Surface crack. [Full Size]

**Figure 7**: Underclad crack. [Full Size]

To consider cracks in the model we use a submodel technique. Two different crack are assumed, an underclad crack as defined in the benchmark and a surface crack as shown in Fig. 6 and 7 . Only the crack and a reasonable large surrounding is modeled. At the cut boundaries of the submodel the interpolated degree of freedom results (displacements) of the global (coarse) model are applied. The thermal field obtained in the first run and the gravity loads are used as body loads. Additionally, the pressure (see Fig. 1) is applied at the inner surface. The crack is assumed to be an underclad crack. Here arises a principal problem, since the cladding itself contains no crack and has common nodes with the base material, the crack mouth is virtually clamped close, which results in a second straight and sharp crackfront at the interface between cladding and base material.

**Figure 8**: Hoop stress around the surface crack after 1000 seconds. [Full Size]

The VERLIFE code does make any suggestions how to deal numerically with underclad cracks. Therefore a second submodel is used, which includes a surface crack which goes through the cladding. Fig. 8 shows the hoop stress at t=1000 s. Here, the crack face (dark blue) is clearly distinguishable from the rest of the model and we the note the high (red) stresses at the crack tip.

## Results

The computation of the stress intensity factors is done in a postprocessing step. It includes extensive mathematical operations, which are not explained on this page. Detailed information can be found here.

### Underclad crack

**Figure 9**: Underclad crack: Variation of the stress intensity factor as a function of time for different positions at the crack front. [Full Size]

Fig. 9 shows the variation of the stress intensity factor K_{I} computed with the j-integral technique for the deepest point of the crack (a=15 mm) and a point (a=2 mm) below the cladding-base material interface. The point at a=2 mm is located outside of the weld. Therefore it was assumed, that in this location there are no residual stresses. Contrary to that, the VERLIFE code suggests that all points of the crack front are supposed to lie entirely in the weld.

There is a slightly decrease of K_{I} at the beginning due to the pressure reduction, then an increase up to the maximum at 1000 seconds due to the thermal shock effects. It follows a decrease again, as the temperature gradient through the vessel wall decreases. The repressurization is the critical phase for this scenario, a sudden increase of K_{I} must be noted.

**Figure 10**: Variation of the stress intensity factor along the underclad crack front. [Full Size]

The variation of K_{I} along the crack front is shown in Fig. 10. The maximum of K_{I} is found at an angular position of alpha=90°. The effect of the residual stresses in the weld lines, which cause an increase of K_{I} between 60° and 120°, can clearly be seen.

**Figure 11**: Underclad crack: Stress intensity factor as a function of the crack tip temperature and the critical stress intensity curve K_{Ic3}C3 for the maximum allowable transition temperature. [Full Size]

In Fig. 11 K_{I} is plotted as a function of the crack tip temperature together with the critical [K_{Ic}]_{3}-curve for all computed pairs of K_{I} and T. It is obvious that the repressurization causes the critical K_{I}=40.2MPam^{0.5} at a crack tip temperature T=68.9°C. The maximum value of K_{I}=43 MPam^{0.5} occurs at 1000 seconds where the crack tip temperature is T=170°C, which is an uncritical state.

### Surface crack

The results for the surface crack are in principle the same as for the underclad crack, but with elevated values of the stress intensity factors. Here also, the repressurization of the RPV causes the critical value of K_{I}=67 MPam^{0.5} at a crack tip temperature of T=68.9°C.

**Figure 12**: Variation of the stress intensity factor along the surface crack front. [Full Size]

**Figure 13**: Surface crack: Variation of the stress intensity factor as a function of time for different positions at the crack front. [Full Size]

**Figure 14**: Surface crack: Stress intensity factor as a function of the crack tip temperature and the critical stress intensity curve K_{Ic3} for the maximum allowable transition temperature. [Full Size]

## Conclusions

underclad crack | surface crack | ||||||

j-integral |
kcalc | verlife | j-integral |
kcalc | verlife | ||

T | [°C] | 68.9 |
68.9 | 68.9 | 68.9 |
68.9 | 68.9 |

K_{I}^{a} |
[MPam^{0.5}] |
40.2 |
42.9 | 53.8 | 67.0 |
68.5 | 67.6 |

T_{k}^{a} |
[°C] | 115.5 |
106.1 | 81.8 | 62.4 |
60.0 | 61.7 |

**Table 1**: Maximum allowable transition temperatures T_{k}^{a} along with the K_{I}^{a} and crack tip temperature T values for the deepest point (a=15 mm) calculated with different methods.

Table 1 shows the main results of this work, as there are the critical K_{I}^{a}(T) and T_{k}^{a} values for the underclad and the surface crack located in the most embrittled weld line of a WWER-440 RPV during a thermal shock scenario. For comparison additional values are given which are obtained using simplified approaches, as there are the kcalc procedure of the FE-code ANSYS and the VERLIFE engineering approach. Both methods are not qualified since they are valid only for elastic cases and the VERLIFE simplified engineering approach does not take into account the existence of a cladding.

For small scale yielding at the crack tip, as it is in this case, the kcalc approach delivers conservative K_{I}^{a} and T_{k}^{a} values, since it uses the normal displacements of the crack faces and the elastic properties of the material to compute K_{I}. These displacements are larger if plasticity occurs.

The VERLIFE simplified engineering approach uses the stress components normal to the crack face for K_{I} computation. These stresses are limited if plasticity occurs. Furthermore there are restrictions to the crack geometry. As the results in table ?? show this approach delivers a strongly conservative value for the underclad crack. For the surface crack the K_{I}^{a} value is only slightly conservative, compared to the one obtained with the j-integral method.

Reliable values are obtained by the j-integral method even for small scale yielding at the crack tip.

Up to here, the VERLIFE procedure is clear within the recommendations of the methods. A more critical point concerns the finite element crack modelling for the underclad crack. The VERLIFE code allows to assume an underclad crack, if the fracture properties of the austenitic cladding are known and if the cladding integrity is assured. As mentioned in section Problem Solution the common nodes of the cladding and the base material cause an artificial crack mouth clamping, which results in an underestimation of K_{I} for an underclad crack. A better approach might be to define a crack, which affects both cladding and base material, as shown in Figure 15. The postulation of a surface crack leads to more conservative results.

**Figure 15**: Crack affecting both cladding and base material. [Full Size]