Application of the Beremin Model for Cruciform Specimen to Determine the Fracture Probability in Case of WPS
Based on large-scale experimental tests performed on cruciform specimens made of French type RPV steel, the warm prestressing (WPS) effect for LCF (load–cool–fracture) and LCTF (load–cool–transient–fracture) load cycles is studied by finite element simulation. A modified Beremin model is used to det...
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Інститут проблем міцності ім. Г.С. Писаренко НАН України
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irk-123456789-1119642020-12-17T16:30:09Z Application of the Beremin Model for Cruciform Specimen to Determine the Fracture Probability in Case of WPS Beleznai, R. Szavai, Sz. Научно-технический раздел Based on large-scale experimental tests performed on cruciform specimens made of French type RPV steel, the warm prestressing (WPS) effect for LCF (load–cool–fracture) and LCTF (load–cool–transient–fracture) load cycles is studied by finite element simulation. A modified Beremin model is used to determine the fracture probability of the specimens. С использованием результатов экспериметальных иссследований крупногабаритных крестообразных образцов из реакторной стали французского производства выполнено конечно-элементное моделирование эффекта горячей опрессовки при циклической нагрузке по схемам: нагрузка–охлаждение–разрушение и нагрузка–охлаждение–переходный процесс–разрушение. Для оценки вероятности разрушения образцов использована модифицированная модель Беремина. Із використанням результатів експериментальних досліджень великогабаритних хрестоподібних зразків із реакторної сталі французького виробництва виконано скінченноелементне моделювання ефекту гарячого опресування за циклічного навантаження за схемами: навантаження–охолодження–руйнування і навантаження–охолодження–перехідний процес–руйнування. Для оцінки імовірності руйнування зразків використано модифіковану модель Береміна. 2013 Article Application of the Beremin Model for Cruciform Specimen to Determine the Fracture Probability in Case of WPS / R. Beleznai, Sz. Szava // Проблемы прочности. — 2013. — № 4. — С. 135-142. — Бібліогр.: 2 назв. — англ. 0556-171X http://dspace.nbuv.gov.ua/handle/123456789/111964 539.4 en Проблемы прочности Інститут проблем міцності ім. Г.С. Писаренко НАН України |
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Научно-технический раздел Научно-технический раздел Beleznai, R. Szavai, Sz. Application of the Beremin Model for Cruciform Specimen to Determine the Fracture Probability in Case of WPS Проблемы прочности |
description |
Based on large-scale experimental tests performed on cruciform specimens made of French type RPV steel, the warm prestressing (WPS) effect for LCF (load–cool–fracture) and LCTF (load–cool–transient–fracture) load cycles is studied by finite element simulation. A modified Beremin model is used to determine the fracture probability of the specimens. |
format |
Article |
author |
Beleznai, R. Szavai, Sz. |
author_facet |
Beleznai, R. Szavai, Sz. |
author_sort |
Beleznai, R. |
title |
Application of the Beremin Model for Cruciform Specimen to Determine the Fracture Probability in Case of WPS |
title_short |
Application of the Beremin Model for Cruciform Specimen to Determine the Fracture Probability in Case of WPS |
title_full |
Application of the Beremin Model for Cruciform Specimen to Determine the Fracture Probability in Case of WPS |
title_fullStr |
Application of the Beremin Model for Cruciform Specimen to Determine the Fracture Probability in Case of WPS |
title_full_unstemmed |
Application of the Beremin Model for Cruciform Specimen to Determine the Fracture Probability in Case of WPS |
title_sort |
application of the beremin model for cruciform specimen to determine the fracture probability in case of wps |
publisher |
Інститут проблем міцності ім. Г.С. Писаренко НАН України |
publishDate |
2013 |
topic_facet |
Научно-технический раздел |
url |
http://dspace.nbuv.gov.ua/handle/123456789/111964 |
citation_txt |
Application of the Beremin Model for Cruciform Specimen to Determine the Fracture Probability in Case of WPS / R. Beleznai, Sz. Szava // Проблемы прочности. — 2013. — № 4. — С. 135-142. — Бібліогр.: 2 назв. — англ. |
series |
Проблемы прочности |
work_keys_str_mv |
AT beleznair applicationofthebereminmodelforcruciformspecimentodeterminethefractureprobabilityincaseofwps AT szavaisz applicationofthebereminmodelforcruciformspecimentodeterminethefractureprobabilityincaseofwps |
first_indexed |
2025-07-08T03:04:26Z |
last_indexed |
2025-07-08T03:04:26Z |
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1837046310423756800 |
fulltext |
UDC 539.4
Application of the Beremin Model for Cruciform Specimen to Determine
the Fracture Probability in Case of WPS
R. Beleznai and Sz. Szavai
Bay Zoltan Nonprofit Ltd., Institute for Logistics and Production Systems, Miskolc,
Hungary
ÓÄÊ 539.4
Îöåíêà âåðîÿòíîñòè ðàçðóøåíèÿ êðåñòîîáðàçíûõ îáðàçöîâ â ñëó÷àå
ãîðÿ÷åé îïðåññîâêè ïî ìîäåëè Áåðåìèíà
Ð. Áåëåçíàè, Ø. Øàâàè
Áàé Çîëòàí Ëòä., Èíñòèòóò ëîãèñòèêè è ïðîèçâîäñòâåííûõ ñèñòåì, Ìèøêîëüö,
Âåíãðèÿ
Ñ èñïîëüçîâàíèåì ðåçóëüòàòîâ ýêñïåðèìåòàëüíûõ èñññëåäîâàíèé êðóïíîãàáàðèòíûõ êðåñòî-
îáðàçíûõ îáðàçöîâ èç ðåàêòîðíîé ñòàëè ôðàíöóçñêîãî ïðîèçâîäñòâà âûïîëíåíî êîíå÷íî-
ýëåìåíòíîå ìîäåëèðîâàíèå ýôôåêòà ãîðÿ÷åé îïðåññîâêè ïðè öèêëè÷åñêîé íàãðóçêå ïî ñõåìàì:
íàãðóçêà–îõëàæäåíèå–ðàçðóøåíèå è íàãðóçêà–îõëàæäåíèå–ïåðåõîäíûé ïðîöåññ–ðàçðóøåíèå.
Äëÿ îöåíêè âåðîÿòíîñòè ðàçðóøåíèÿ îáðàçöîâ èñïîëüçîâàíà ìîäèôèöèðîâàííàÿ ìîäåëü Áåðå-
ìèíà.
Êëþ÷åâûå ñëîâà: êðåñòîîáðàçíûé îáðàçåö, ìåòîä êîíå÷íûõ ýëåìåíòîâ, ìîäè-
ôèöèðîâàííàÿ ìîäåëü Áåðåìèíà.
Introduction. In the framework of NESC VII European project, large-scale
experimental tests have been performed on cruciform specimens made of French
type RPV steel under different loading and temperature conditions to study the
warm prestressing (WPS) effect. Several European industrial partners and research
institutes participate in the project (EDF, CEA, AREVA NP France, AREVA
GmbH, NRI, AEKI-KFKI, SCK-CEN, INSPECTA, IWM, TWI, SERCO, Bristol
University, etc.). The experimental tests were conducted by CEA in France. LCF
(load–cool–fracture) and LCTF (load–cool–transient–fracture) load cycles are
considered during the analyses. The purpose of this paper is to review recent
research into the WPS effect in different testing configurations by numerical
calculation simulating the experimental tests. Five specimens of similar type, but
with different crack dimensions, were tested and separate finite element (FE)
models were created for them according to the crack geometry (Table 1). The
experiments have been performed on large-scale cruciform specimens with a
semi-elliptical crack (Fig. 1).
1. Geometry Description. The thickness and length of the specimen are 98 mm
and 840 mm, respectively. The applied finite element software is the MSC.MARC
2010. The model contains 39,520 eight-node solid elements and 45,726 nodes. The
four slots and the crack are modeled with sharp edges. Only one quarter of the
© R. BELEZNAI, Sz. SZAVAI, 2013
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2013, ¹ 4 135
specimen is modeled. Type of the discretization of the support and the ball is
analyzed, and finally the supports of the specimen are modeled as elastic solids and
the sphere is defined as a rigid surface. The size of the mesh around the crack tip is
50 50� �m complying with the requirement in [1].
2. Material Description. The specimens are made of 18MND5 ferritic steel
(A533B). Temperature dependent elastic-plastic material properties with isotropic
hardening law are assumed for the calculation. Elastic-plastic analysis for
determination of the crack driving force (J-integral) is performed using
MSC.MARC 2010 FE code. The stress-strain curves used in the analyses are
shown in Fig. 2.
3. Loading Path. The loading paths, which are imposed to the specimens, are
the LCF (for Tests 1 and 2) and LCTF (Tests 3, 4, and 5) type of thermo mechanical
transient. Main parameters of these loading transients are given in Table 2. The
R. Beleznai and Sz. Szavai
136 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2013, ¹ 4
T a b l e 1
Crack Dimensions
Test No. a, mm 2c, mm
Test 1 12.0 35.0
Test 2 12.8 36.5
Test 3 12.3 36.0
Test 4 12.2 36.0
Test 5 12.2 36.0
Fig. 1. Mesh of the specimens.
loading path of the LCF tests involves the following stages: an isothermal
preloading up to WPS load (at about K J WPS_ �80 MPa m), then cooling down
under constant mechanical load and, finally, the isothermal fracture loading at a
considerably lower temperature than the preload. In case of the LCTF cycle between
the cooling and fracture phase there is a transient load, where the temperature and
the mechanical load decrease at the same time. All geometry, material and loading
data as well as characteristics were provided by the AREVA NP and CEA. The
results of the experimental tests for LCF load cycle were used for verification of
the finite element model, and benchmark analyses of the LCTF load cycle have
been performed to predict the fracture moment and failure probability.
4. Validation of the Finite Element Models. The obtained results are
compared with the available experimental test results, but the comparison is
presented only for Test 4 configuration. The load–deflexion curve is shown in Fig. 3.
The deflexion is obtained from the displacement difference between points A and B
(Fig 1.) The load–CMOD diagram contains three CMOD curves (Fig. 4), one
represents the experimental test, and the others belong to FE results. One of the FE
curves presents the CMOD curve in the midpoint (point A in Fig. 1) of the crack
Application of the Beremin Model for Cruciform Specimen ...
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2013, ¹ 4 137
T a b l e 2
Parameters of the Loading Transients for Different Load Cycles
Test No. WPS cycle Tpreload , �C Tunload , �C T fract , �C FWPS , kN Funload , kN
Test 1 LCF �33 6. �110 0. �150 1328.8 –
Test 2 LCF �32 5. �110 0. �150 1328.8 –
Test 3 LCTF �35 0. �110 0. �150 1280.0 570
Test 4 LCTF �29 5. �110 0. �150 1280.0 570
Test 5 LCTF �30 5. �110 5. �150 1280.0 70
Fig. 2. Stress–strain curves for 18MND5 material.
mouth, the other curve is the corrected result: the clip gauge has a certain width
(10 mm) and because of this the CMOD cannot be measured at the widest point of
the opened crack mouth (point A in Fig. 1). That is why 5 mm from the crack
symmetry plane the CMOD should be post-processed (point C in Fig. 1). The mesh
contains nodes only at a distance of 4.4 and 5.5 mm from the symmetry plane,
thus, CMOD is plotted in the node at a distance of 4.4 mm from the centre. The
temperature–CMOD curve is presented in Fig. 5 and the temperature–K J curves
are seen in Fig. 6, where the CMOD curve is obtained from A and C points (Fig. 1).
The curves of the FE results are in good agreement with the test curves. In case of
the load–deflexion curve the compliance and the thermal expansion of the testing
machine are taken into account. The K J values are obtained from converting J to
K J assuming plane strain conditions and using the following expression:
K
EJ
J �
�1 2�
[MPa m]. (1)
138 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2013, ¹ 4
Fig. 3. Load–deflexion curve for the cruciform specimen.
Fig. 4. Load–CMOD curve for the cruciform specimen.
R. Beleznai and Sz. Szavai
Figure 7 presents the variation of the crack driving force K J along the crack
front for two load levels (primary loading and fracture load). Here, �0 is the
location on the free surface and � �90 at the deepest point of the crack. It is
observed that the maximum K J occurs at a location of around � �11 75. for
Test 1 and � �12 01. for Test 2 degrees along the crack front.
5. Modified Beremin Model. For determination of the fracture moment and
failure probability the modified Beremin model was used [1]. Under this complex
loading path, the so-called active plastic zone, which is varying during loading,
should be determined. The total equivalent plastic strain criterion was used for
defining the active plastic zone during unloading and fracture stage similarly as in
[2]. This criterion is different from the one described in [1]. The plastic zone
includes points where the
p �0 002. (
p is the total equivalent plastic strain). To
determine the cumulative probability of failure, the following expression was used
P
u
u
dV
Vr
u p u u
m
� � �
�
��
�
�
��
� �
1
1 0
1
exp max
( )
( ( )){ , � ( ) }
�
� � 0�
�
�
�
�
�
�
�
�, (2)
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2013, ¹ 4 139
Fig. 5. Temperature–CMOD curve for the cruciform specimen.
Fig. 6. Temperature–KJ curve for the cruciform specimen.
Application of the Beremin Model for Cruciform Specimen ...
where �1( )u is the maximum principal stress at instant t, p u( ) is the cumulated
plastic deformation, and � �u ( ) is the cleavage characteristic stress, which can
depend on the temperature �.
The active plastic zone criterion for unloading and reloading stages is
p actual p_ _ ,� �0 0 (3)
where
p_ 0 is
p at the end of the preloading stage and
p actual_ is the
p at
the actual point of the unloading or fracture stage. The applied Beremin parameters
are given in Table 3.
The software is developed in FORTRAN language for selection of the
elements in the process zone, which contains Gauss points according to the active
plastic zone criterion. With this software, the Weibull stress and the cumulative
failure probability are determined using the above mentioned expressions. Table 4
provides a summary about the K J values at different stages of the loading paths
and the predicted failure probabilities. Based on the obtained results, the failure
probability–load curves are plotted (Fig. 8). In case of the Test 5, the fracture
probability is determined without knowing the experimental fracture force. The
purpose of this analysis was to estimate the application of the modified Beremin
model to determine the failure probability.
To determine the fracture moment, the failure probability is calculated in
several points during reloading, and fracture is assumed to occur where the failure
probability is higher than 0.99. Fracture was predicted at a load of F �1230 kN, at
140 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2013, ¹ 4
Fig. 7. Value of KJ along the elliptical curve of the crack front at different load levels: (1) F �
�1280 kN; (2) F �1453 6. kN.
T a b l e 3
The Beremin Parameters
m 23.43
� , �C �150 �100 �50 �20 20
�u , MPa 2736 2007 3161 3421 4200
R. Beleznai and Sz. Szavai
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2013, ¹ 4 141
T a b l e 4
Values of K
J
and Pr Relating to the Crack Deepest Point
Test No. WPS cycle KJ WPS_ ,
MPa m
KJ unload_ ,
MPa m
KJ fract_ ,
MPa m
Pr
Test 1 LCF 81.97 – 92.20 0.9999
Test 2 LCF 82.94 – 92.76 0.9999
Test 3 LCTF 79.11 ~ .30 90 79.88 0.1369
Test 4 LCTF 79.34 ~ .30 09 89.97 0.9999
Test 5 LCTF 80.22 0 77.02 0.9998
Fig. 8. Failure probability–load curves.
Application of the Beremin Model for Cruciform Specimen ...
the temperature T �� �150 C. The vertical line in the diagram for Test 5 (Fig. 8)
shows the fracture load from the experimental test (F fract �1578 kN).
Conclusions. Elastic-plastic analyses of cruciform specimens made of RPV
steel with a semi-elliptical crack were performed using isotropic hardening material
law. Verification of the simulation was done by comparing different curves
obtained from the experimental tests, such as load–deflexion curve, load–CMOD
curve and CMOD–temperature curve. The cumulative probability of failure was
determined by the modified Beremin model for all five experimental tests using a
different criterion for the definition of the active plastic zone. In case of the Test 5
analysis, the fracture is predicted using the modified Beremin model. The method
provides a lower value for the failure probability than the obtained experimental
fracture load. Development of the method is currently in progress to obtain more
accurate prediction.
Ð å ç þ ì å
²ç âèêîðèñòàííÿì ðåçóëüòàò³â åêñïåðèìåíòàëüíèõ äîñë³äæåíü âåëèêîãàáàðèò-
íèõ õðåñòîïîä³áíèõ çðàçê³â ³ç ðåàêòîðíî¿ ñòàë³ ôðàíöóçüêîãî âèðîáíèöòâà
âèêîíàíî ñê³í÷åííîåëåìåíòíå ìîäåëþâàííÿ åôåêòó ãàðÿ÷îãî îïðåñóâàííÿ çà
öèêë³÷íîãî íàâàíòàæåííÿ çà ñõåìàìè: íàâàíòàæåííÿ–îõîëîäæåííÿ–ðóéíó-
âàííÿ ³ íàâàíòàæåííÿ–îõîëîäæåííÿ–ïåðåõ³äíèé ïðîöåñ–ðóéíóâàííÿ. Äëÿ
îö³íêè ³ìîâ³ðíîñò³ ðóéíóâàííÿ çðàçê³â âèêîðèñòàíî ìîäèô³êîâàíó ìîäåëü
Áåðåì³íà.
1. W. Lefevre, G. Barbier, R. Masson, and G. Rousselier, “A modified Beremin
model to simulate the warm pre-stress effect,” Nucl. Eng. Design, 216, 27–42
(2002).
2. R. Beleznai, G. B. Lenkey, and D. Lauerova, “Calibration of three-parameter
Weibull stress model for 15Kh2NMFA RPV steel,” J. Nucl. Mater., 406,
119–126 (2010).
Received 22. 05. 2012
142 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2013, ¹ 4
R. Beleznai and Sz. Szavai
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/NLD (Gebruik deze instellingen om Adobe PDF-documenten te maken voor kwaliteitsafdrukken op desktopprinters en proofers. De gemaakte PDF-documenten kunnen worden geopend met Acrobat en Adobe Reader 5.0 en hoger.)
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/ENU (Use these settings to create Adobe PDF documents for quality printing on desktop printers and proofers. Created PDF documents can be opened with Acrobat and Adobe Reader 5.0 and later.)
>>
/Namespace [
(Adobe)
(Common)
(1.0)
]
/OtherNamespaces [
<<
/AsReaderSpreads false
/CropImagesToFrames true
/ErrorControl /WarnAndContinue
/FlattenerIgnoreSpreadOverrides false
/IncludeGuidesGrids false
/IncludeNonPrinting false
/IncludeSlug false
/Namespace [
(Adobe)
(InDesign)
(4.0)
]
/OmitPlacedBitmaps false
/OmitPlacedEPS false
/OmitPlacedPDF false
/SimulateOverprint /Legacy
>>
<<
/AddBleedMarks false
/AddColorBars false
/AddCropMarks false
/AddPageInfo false
/AddRegMarks false
/ConvertColors /NoConversion
/DestinationProfileName ()
/DestinationProfileSelector /NA
/Downsample16BitImages true
/FlattenerPreset <<
/PresetSelector /MediumResolution
>>
/FormElements false
/GenerateStructure true
/IncludeBookmarks false
/IncludeHyperlinks false
/IncludeInteractive false
/IncludeLayers false
/IncludeProfiles true
/MultimediaHandling /UseObjectSettings
/Namespace [
(Adobe)
(CreativeSuite)
(2.0)
]
/PDFXOutputIntentProfileSelector /NA
/PreserveEditing true
/UntaggedCMYKHandling /LeaveUntagged
/UntaggedRGBHandling /LeaveUntagged
/UseDocumentBleed false
>>
]
>> setdistillerparams
<<
/HWResolution [2400 2400]
/PageSize [612.000 792.000]
>> setpagedevice
|