Batura A . S . , Orynyak I.V

Batura A.S., Orynyak I.V.Batura A.S., Orynyak I.V.

IPS NASUIPS NASU

Pisarenko’ Institute for Problems of Strength , Kyiv, Ukraine National Academy of Sciences of Ukraine

Pisarenko’ Institute for Problems of Strength , Kyiv, Ukraine National Academy of Sciences of Ukraine

ENGINEERING METHODS FOR STRESS INTENSITY FACTOR CALCULATION FOR 2-D AND 3-D BODIES WITH CRACKS

ENGINEERING METHODS FOR STRESS INTENSITY FACTOR CALCULATION FOR 2-D AND 3-D BODIES WITH CRACKS

IPS NASUIPS NASUWeight Function Method for plane bodiesWeight Function Method for plane bodies

x a

a

I dxxxGhSIFK0

)(),(

),( xGh - weight function,

x - the law of stress distribution,

G – geometry parameters.

xhGAxhxGh CA 00,

axxhA -1

1~)( - asymptotical (singular) part of WF,

axxhC -1~)( - correction (regular) part of WF.

Then for any specified stress law x (for example i

i a

xx

) obtain

,00

i

IC

i

IA

i

I IGAIK where .,00

a

C

i

i

IC

a

A

i

i

IA dxxhax

Idxxhax

I

i

IAI and i

ICI doesn’t depend upon geometry.

IPS NASUIPS NASUWeight Function Method for plane bodiesWeight Function Method for plane bodies

In particular, for a plane body with an edge crack

The main idea of Weight Function Methods:The main idea of Weight Function Methods:If we have the SIF solution for one particular loading we can obtain the SIF If we have the SIF solution for one particular loading we can obtain the SIF solution for any other law of loading.solution for any other law of loading.

;

1

2)(

cA

ax

a

cxh

;1151

)1(32

1)(

)(2

00

ax

ax

a

xaAxhC

.4

22

2

c

IPS NASUIPS NASUApplication of WFM for a pipesApplication of WFM for a pipes

In the circular pipe additional force N and moment M appear. Angle and displacement discontinuity can be expressed in the next form:

Crack compliance methodCrack compliance method(modification of (modification of Cheng & Finnie approachCheng & Finnie approach))

),( ,' NNMMqqEtu

),(6

,' NNMMqqE

,/6 2tMM ,/ tNN ,)()()(0

dYY iNi ,)()()(0

dYY iMi

where YN, YM – are the dimensionless SIF, induced by M and N as in the plane body,

.1

,/ 2/

E

Eta .1

,/ 2/

E

Eta



qIq qYaK NNIN YaK

MMIM YaK

- caused by loading, - caused by force,

- caused by moment.

IqIMINI KKKK Obtain result SIF :

SIF is smaller than in the case of straight plane !Using equilibrium equations for a ring and initial parameter method, get the expression for a dimensionless SIF decrease from the case of straight plane (Y0):

)),(1(6

)(9

6

1 0

/0

/

00

Y

Ep

tR

EYY

YY

M

pM where - dimensionless

pressure.p



Result plotsResult plots

Conclusion:Conclusion:Advanced SIF formula for pipes was obtained. The feature of the SIF Advanced SIF formula for pipes was obtained. The feature of the SIF decreasing at rising of the pressure was found. decreasing at rising of the pressure was found.

IPS NASUIPS NASUWeight Function Method for 3-D bodiesWeight Function Method for 3-D bodies

x

y

Q/ Q

r

R

Qx Qxy

Qy

a

b

)(

/ ,)( /

SQQ

dsyxWQSIF (1)

CQQ

AQQQQ

WGDWW /// , (2)

EAQQ

AQQ

WW ,// - elliptical crack,

EAQQ

EAQQ

AQQ X

WWW ,,/// - for semi-

elliptical crack,

EAQQ

EAQQ

EAQQ

EAQQ

AQQ XYYX

WWWWW ,,,,///// - for quarter-elliptical crack

ba

l

dl

R

ra

W

QQQQ

AQQ

/,cossin

cossin)(,

)(

)(1

)(2222

242

22

21

2

2

41

/

/

/

RrWW AQQ

CQQ

/1// -correction part

- asymptotical part for elliptical crack.

IPS NASUIPS NASUWeight Function Method for 3-D bodiesWeight Function Method for 3-D bodies

)1(0 const

CI

AII IGDIK ,0

geometry dependentgeometry dependentloading dependentloading dependent

If 0IK is known we obtain GD , and can calculate IK

for any law of loading.

,)(~

)(~)(~,

C

AI

I

IKGD

where IK~ - is a known SIF for any law of loading.

Similarly to the 2-D case,

So

.,)()(

// S

CQQ

CI

S

AQQ

AI dsWIdsWI

SIF along crack front (angle), homogeneous loadingSIF along crack front (angle), homogeneous loading

IPS NASUIPS NASUCheck of the PWFM accuracy for

semi-elliptic cracks

Check of the PWFM accuracy for semi-elliptic cracks

a/l=0.2 (a/t=0.8)

0,2

0,4

0,6

0,8

1

1,2

1,4

1,6

1,8

2

0 20 40 60 80 100

Angle, degree

Tension by the PWFM Tension by Raju-Newman

Bending by the PWFM Banding by Raju-Newman

a/l=0.4 (a/t=0.8)

0

0,2

0,4

0,6

0,8

1

1,2

1,4

1,6

0 20 40 60 80 100

Angle, degree


Bending by the PWFM Bending by Raju-Newman

0

90

IPS NASUIPS NASU

a/l=0.6 (a/t=0.8)

0

0,2

0,4

0,6

0,8

1

1,2

1,4

0 20 40 60 80 100

Angle, degree



a/l=1.0 (a/t=0.8)

-0,2

0

0,2

0,4

0,6

0,8

1

1,2

1,4

1,6

0 20 40 60 80 100

Angle, degree

Tension by the PWFM Tension by Raju-Newmen


a/l=2.0 (a/t=0.8)

-0,2

0

0,2

0,4

0,6

0,8

1

0 20 40 60 80 100

Angle, degree



IPS NASUIPS NASU

IPS NASUIPS NASU

Homogeneous loading

1

1,2

1,4

1,6

1,8

2

0 0,2 0,4 0,6 0,8 1 1,2

a/l

90 degree by the PWFM 90 degree by Murakami


Linear loading

0,2

0,4

0,6

0,8

1

1,2

1,4

0 0,2 0,4 0,6 0,8 1 1,2

a/l



Dependence SIF from ratio a/lDependence SIF from ratio a/l

IPS NASUIPS NASU

Quadratic loading

0

0,2

0,4

0,6

0,8

1

1,2

0 0,2 0,4 0,6 0,8 1 1,2

a/l



Cubic loading

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1

0 0,2 0,4 0,6 0,8 1 1,2

a/l



Dependence SIF from ratio a/lDependence SIF from ratio a/l

IPS NASUIPS NASUWeight Function Method for 3-D bodies.

Simplified (speed up) approach.Weight Function Method for 3-D bodies.

Simplified (speed up) approach.

The problem: The problem: triple integraltriple integral (square and contour) with (square and contour) with singularity at the edge high computation singularity at the edge high computation costcost (especially for repeating – fatigue, stress-corrosion,… – (especially for repeating – fatigue, stress-corrosion,… – calculations) !!!calculations) !!!

The solution: approximation of the stress law with The solution: approximation of the stress law with

function of the next type: , function of the next type: ,

calculation of the SIF array for each stress functioncalculation of the SIF array for each stress function

. Approximate SIF function can be . Approximate SIF function can be

build as linear combination of precalculated .build as linear combination of precalculated .

ji

ijijapprox yxfMyx,

,,

1,, ijijij Myxfyx

ijY

ijY

IPS NASUIPS NASUWeight Function Method for 3-D bodies.

Simplified (speed up) approach.Weight Function Method for 3-D bodies.

Simplified (speed up) approach.

Polynomial examplePolynomial example

22 1 k

ji

jiij b

x

a

yMyx

0,

,

),,(),(),()(

)(),(),(

41

DII

aM

kEKG C

ijAij

ij

ijij

The expression for dimensionless SIF functions :ijG

dSWbx

ay

a

kEI A

QQ

ji

S

Aij /

)(41

)(

)(),(

dSW

bx

ay

a

kEI C

QQ

ji

S

Cij /

)(41

)(

)(),(

IPS NASUIPS NASU

Weight Function Method for 3-D bodies.Simplified (speed up) approach.

Weight Function Method for 3-D bodies.Simplified (speed up) approach.

Polynomial examplePolynomial example For 2,0 simple expressions for Iij

A,C(α) were obtained :

0761.00515.00108.0;0

0018.01576.01369.00439.0;0

0285.03858.04071.01572.0;0

96.019.04747.02448.0;0

2303

2302

2301

2300

A

A

A

A

I

I

I

I 0327.00297.00357.00127.0;0

0703.00578.00717.00267.0;0

2.01049.01262.00452.0;0

1;0

2303

2302

2301

00

C

C

C

C

I

I

I

I

4216.00196.01513.00621.0;2

496.00236.01475.00655.0;2

6297.00420.01005.00524.0;2

9808.01753.02455.00901.0;2

2303

2302

2301

2300

A

A

A

A

I

I

I

I 1335.00227.00484.00165.0;2

205.004.00343.00081.0;2

374.00481.00278.00057.0;2

1;2

2303

2302

2301

00

C

C

C

C

I

I

I

I

Depth of crack Loading (0, j) I0j(0) I0j(π/2)

=a/t j Exact Approx. Exact Approx.

0.2 0123

1.1400.1970.0740.038

1.1400.1960.0780.040

1.0150.7150.5880.512

1.0150.7260.6050.533

0.5 0123

1.2190.2210.0850.044

1.2190.2140.0850.044

1.0500.7290.5960.515

1.0500.7420.6100.540

Semi-elliptical crack on the inner surface of the cylinder.

IPS NASUIPS NASU

Application of the peveloped methods:Software “ReactorA”Application of the peveloped methods:Software “ReactorA”

• Residual life is calculated Residual life is calculated deterministically and deterministically and probabilistically (MASTER probabilistically (MASTER CURVE approach) for CURVE approach) for various points of crack frontvarious points of crack front

• Residual life is calculated Residual life is calculated deterministically and deterministically and probabilistically (MASTER probabilistically (MASTER CURVE approach) for CURVE approach) for various points of crack frontvarious points of crack front

• This program is intended This program is intended for calculation of reactor for calculation of reactor pressure vessel residual life pressure vessel residual life and safety margin with and safety margin with respect to brittle fracturerespect to brittle fracture.

• This program is intended This program is intended for calculation of reactor for calculation of reactor pressure vessel residual life pressure vessel residual life and safety margin with and safety margin with respect to brittle fracturerespect to brittle fracture.

• User sets loading and User sets loading and temperature fields in the temperature fields in the different moments of time. different moments of time. Then material fracture Then material fracture toughness, embrittlement toughness, embrittlement parameters are also set by parameters are also set by useruser.

• User sets loading and User sets loading and temperature fields in the temperature fields in the different moments of time. different moments of time. Then material fracture Then material fracture toughness, embrittlement toughness, embrittlement parameters are also set by parameters are also set by useruser.

IPS NASUIPS NASUReactorA advantagesReactorA advantages

• The sizes of stress and temperature fields' aren't bounded• Number of time moments is bounded only by the

memory size • Cladding is taken into account • Welding seam and heat-affected area are taken into

account • Deterioration is taken into account not only as shift of

the material fracture toughness function but also as its inclination

• Original feature of the software is using of the author variant of the weight function method. It allows to set loading on the crack surface in the form of table.

• The sizes of stress and temperature fields' aren't bounded• Number of time moments is bounded only by the

memory size • Cladding is taken into account • Welding seam and heat-affected area are taken into

account • Deterioration is taken into account not only as shift of

the material fracture toughness function but also as its inclination

• Original feature of the software is using of the author variant of the weight function method. It allows to set loading on the crack surface in the form of table.

Input Data

1) Stress field for time1) Stress field for time it

Table arbitrary sizeTable arbitrary size

IPS NASUIPS NASU3. Residual Life calculation of the NPP

pressure vessel using fracture mechanics methods

3. Residual Life calculation of the NPP pressure vessel using fracture mechanics

methods

IPS NASUIPS NASU

2) Temperature field for time2) Temperature field for time0t it

Input Data

Table arbitrary sizeTable arbitrary size

a) Axial with weld seama) Axial with weld seam

IPS NASUIPS NASU

Input Data

weld seamheat-affected zonebase materialcladdingcrack

weld seamheat-affected zonebase materialcladdingcrack

base materialcladdingcrack

base materialcladdingcrack

b) circumferentialb) circumferential

3) Crack types3) Crack types

)f(TAKcI

IPS NASUIPS NASU

4) The basic material characteristics4) The basic material characteristics

1. Arctangents 1. Arctangents 0arctan2 TTBAK

cI

2. Exponent2. Exponent

0exp TTBAKcI

Common shape of the crack growth resistance function is

for user function A takes from coordinates of first point

Common shape of the crack growth resistance function is

for user function A takes from coordinates of first point

3. User (pointed) function3. User (pointed) function

IPS NASUIPS NASU

1. Shift1. Shift

TTAKcI f

2. Inclination2. Inclination

TT

TTTAK

cI

1

1f

A

ICK

T

T

A

ICK

T

T

5) Shift and inclination conceptions 5) Shift and inclination conceptions

nn

FF YTF

ffAAT

exp

0

00

IPS NASUIPS NASU

a)Analytical forma)Analytical form

b)Table formb)Table form

6) Dependence of shift on radiation6) Dependence of shift on radiation

IPS NASUIPS NASU Results

Scenario – Break of the Steam Generator Collector WWER-1000 operated at full powerScenario – Break of the Steam Generator Collector WWER-1000 operated at full power

It is given : - stress field, - temperature field,

= 1000, 2000, 2800, 3000, 3160, 3600, 4000 sec - time points

It is given : - stress field, - temperature field,

= 1000, 2000, 2800, 3000, 3160, 3600, 4000 sec - time points

Axial crack. Half-length l - 40 мм., depth a - 50 мм.

Axial crack. Half-length l - 40 мм., depth a - 50 мм.

ii tT

it

ii t

IPS NASUIPS NASU

a) Dependences of the calculated and critical SIF from temperature for time = 3000 sec

a) Dependences of the calculated and critical SIF from temperature for time = 3000 sec

SIF for base material --//-- for weld seam

Critical SIF for base material --//-- for weld seam

--//-- for heat-affected area

SIF for base material --//-- for weld seam

Critical SIF for base material --//-- for weld seam


it

IPS NASUIPS NASU

history for basic material --//-- for weld seam critical SIF for basic material --//-- for weld seam


history for basic material --//-- for weld seam critical SIF for basic material --//-- for weld seam


b) History of the dependences calculated SIF from temperature for some points and all times intervals and

critical SIF

b) History of the dependences calculated SIF from temperature for some points and all times intervals and

critical SIF

T

IPS NASUIPS NASU

fields for chosen history pointsminimal marginmargin for time points

fields for chosen history pointsminimal marginmargin for time points

c) Table of the calculated temperature margin

for all points of crack front and time points

c) Table of the calculated temperature margin

for all points of crack front and time points

T

T

calculated temperature marginshift of the temperature by user table

shift of the temperature by analytical model

calculated temperature marginshift of the temperature by user table

shift of the temperature by analytical model

IPS NASUIPS NASUd) Figure of the calculated margind) Figure of the calculated margin

IPS NASUIPS NASU

New geometry for axial crackNew geometry for axial crack

Calculated temperature marginCalculated temperature margin

Half length l - 60мм Depth a - 40 ммHalf length l - 60мм Depth a - 40 мм

Results for other crack geometries

New geometry for axial crackNew geometry for axial crack


IPS NASUIPS NASU

Calculated temperature marginCalculated temperature margin


New geometry for circumferential crackNew geometry for circumferential crack

IPS NASUIPS NASU

calculated temperature margincalculated temperature margin

IPS NASUIPS NASU

1. Failure probability calculation for structural element 1. Failure probability calculation for structural element

bIi

f KTK

KK

B

BP i

imin0

min

0exp1

2. Failure probability calculation for crack2. Failure probability calculation for crack

N

iiff PP

1, )1(1

3. Calculation parameters 3. Calculation parameters

))(019,0exp(7731 00 xTTTK

4. In addition4. In addition

Кmin , K0(Т), В0, b - arbitrarily

Pf = 63,2% Кmin = 20 В0 = 25 мм b = 4

Implementation MASTER CURVE Conception

Implementation MASTER CURVE Conception

For time T =0 failure probability equal 1.07*10-05For time T =0 failure probability equal 1.07*10-05

IPS NASUIPS NASU

Time point t4 = 3000 sec

Axial crack half length l - 40 мм., depth a - 50 мм.

Time point t4 = 3000 sec

Axial crack half length l - 40 мм., depth a - 50 мм.

50

60

70

80

90

100

110

0 20 40 60 80 100 120 140 160 180

Angle, degree

K1

SIF dependences on angleSIF dependences on angle

Result for main scenario

Dependences of logarithm probability on TDependences of logarithm probability on T

IPS NASUIPS NASU

ln(Pf) from deltaT

-14

-12

-10

-8

-6

-4

-2

0

0 50 100 150 200

deltaT

ln(P

f)

IPS NASUIPS NASUProbability density for T = 50Probability density for T = 50

IPS NASUIPS NASU

Application of the developed methods:Software “WFM”Application of the developed methods:Software “WFM”

• SIF, grow of the crack dimensions in time and endurance are SIF, grow of the crack dimensions in time and endurance are calculated. “Until specified depth” or “until specified count of cycles” calculated. “Until specified depth” or “until specified count of cycles” modes are presented. modes are presented.

• SIF, grow of the crack dimensions in time and endurance are SIF, grow of the crack dimensions in time and endurance are calculated. “Until specified depth” or “until specified count of cycles” calculated. “Until specified depth” or “until specified count of cycles” modes are presented. modes are presented.

• This program is intended This program is intended

for for SIFSIF calculation for calculation for different (1-D and 2-D) types different (1-D and 2-D) types of cracks and for endurance of cracks and for endurance estimation with using estimation with using different fatigue and stress-different fatigue and stress-corrosion laws.corrosion laws.

• This program is intended This program is intended

for for SIFSIF calculation for calculation for different (1-D and 2-D) types different (1-D and 2-D) types of cracks and for endurance of cracks and for endurance estimation with using estimation with using different fatigue and stress-different fatigue and stress-corrosion laws.corrosion laws.

• User sets “maximum”, User sets “maximum”, “minimum” and “corrosion” “minimum” and “corrosion” loading fields. loading fields.

• User sets “maximum”, User sets “maximum”, “minimum” and “corrosion” “minimum” and “corrosion” loading fields. loading fields.

1. Damages

2. Cracks

IPS NASUIPS NASUWFM: implemented types

of damages and cracksWFM: implemented types

of damages and cracks

IPS NASUIPS NASUWFM: example of result windowWFM: example of result window

• Input and output data can be exchanged with clipboard. . • Input and output data can be exchanged with clipboard. .

IPS NASUIPS NASU CONCLUSIONCONCLUSION

1. Efficient methods of stress intensity factor (SIF)calculation are developed.

2. The computer software which reflected all modern requirements for brittle strength analysis of Reactor Pressure Vessel is created.

1. Efficient methods of stress intensity factor (SIF)calculation are developed.

2. The computer software which reflected all modern requirements for brittle strength analysis of Reactor Pressure Vessel is created.

Documents

Batura A . S . , Orynyak I.V