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Nuclear Engineering and Design 240 (2010) 1355–1362

Contents lists available at ScienceDirect

Nuclear Engineering and Design

journa l homepage: www.e lsev ier .com/ locate /nucengdes

hermal frequency response studies of a hollow cylinder subject to loads ofifferent amplitude and shape

. Paffumia,∗, K.-F. Nilssona, N.G. Taylorb

Institute for Energy, European Commission Joint Research Centre, P.O. Box 2, 1755 ZG Petten, The NetherlandsInstitute for Energy, European Commission Joint Research Centre, Ispra (Va), Italy

r t i c l e i n f o

rticle history:eceived 30 November 2009eceived in revised form 29 January 2010ccepted 5 February 2010

a b s t r a c t

Thermal fatigue is an important degradation mechanism, which must be considered in life managementof nuclear plant piping systems. The analysis is very complex due to a number of complicating factors,with the determination of the load as the primary one. There is clearly a need for simplified engineeringapproaches, such as simulating the spectrum load by a constant frequency thermal load with the nominal

temperature difference between mixing fluids. The fatigue life is determined by the frequency that givesthe shortest life using fatigue curves for initiation and Paris law for crack propagation. This paper analysesthree aspects that affect the conservatism of such an approach: the thermal load shape (sinusoidal orsquare shaped), the structural boundary conditions at the edges of the modelled pipe (traction free orclamped) and the defect shape (circumferential or elliptic). Furthermore it is shown how the methodology

screent fa

can be used to determinethere will be no compone

. Introduction

During operation of light water reactors (LWR), temperatureuctuations occur in many areas such as core outlet zone, lower partf hot pool, free surface of pool, secondary circuit and water/steamnterface in steam generators. In certain conditions, these tem-erature fluctuations can lead to thermo-mechanical damage andomponent failure (IAEA, 2002; Chapuliot et al., 2005). Thermalatigue is also an important issue for liquid metal fast rectors (LMFR)here the temperatures are higher than for LWR and the proper-

ies of the coolant may infer additional problems (IAEA, 2002; Hut al., 2004).

Calculation of the thermal fatigue damage and the associatedatigue life are difficult and prone to large uncertainties due tohe complexity of the phenomena involved. The temperature fluc-uations can be local or global and induce spectrum loads at theall (Hu et al., 2004; Paffumi et al., 2008a). These resulting thermal

tresses and strain variations result in surface degradation followedy formation of surface cracks and crack growth that eventuallyay lead to component failure. In addition to the amplitude of the

hermal loads, the degradation depends also strongly on their fre-uency. At very high frequencies the thermal stresses are confinedo the surface and no deep cracks develop. The thermal stressesre low for very low frequencies as the temperature gradients

∗ Corresponding author. Tel.: +31 224 565082; fax: +31 224 565641.E-mail address: elena.paffumi@jrc.nl (E. Paffumi).

029-5493/$ – see front matter © 2010 Elsevier B.V. All rights reserved.oi:10.1016/j.nucengdes.2010.02.036

ning criteria, i.e. a lower limit of the temperature difference below whichilure due to thermal fatigue.

© 2010 Elsevier B.V. All rights reserved.

through the wall are small (Kasahara et al., 2002; Boley and Weiner,1960). Hence there is an intermediate frequency range for whichmost fatigue damage is expected. Numerical simulations of thermalstripping and high-cycle thermal fatigue of tee junctions of LWRpiping systems imply that the fastest crack initiation and propaga-tion occur for frequencies in the range 0.1–1 Hz (Hu et al., 2004;Paffumi et al., 2008a; Dahlberg et al., 2007; Buckthorpe et al., 1988;Lee et al., 1999). Much effort continues to be devoted to experi-mental studies and development of models with different levelsof complexity (Paffumi et al., 2008a, 2005a,b, 2004; Ancelet et al.,2007; Haddar et al., 2005).

Given the importance of thermal fatigue for the safe and eco-nomical operation of nuclear power plants as well as other processindustries, there is a general need to improve engineering toolsand predictions techniques that capture the main characteristics ofthermal fatigue and associated experimental studies that supportmodel development and increase the understanding of the relevantphenomena (Hu et al., 2004; Paffumi et al., 2008a,b; Green, 1985;Jones and Lewis, 1994; Jones, 1997; Paffumi, 2004). The loading isthe most important unknown factor for a reliable assessment andit is very difficult to compute or measure the load spectrum for agiven situation. The sinusoidal approach (SIN-method), where thethermal loads are assumed to vary sinusoidally (Radu et al., 2007,

2008, 2009) with the frequency that gives the shortest life for agiven nominal temperature difference between the mixing fluids,is one of the simplest methods. In Europe the SIN-method has beenthe basis for the development of a “European Procedure” for ther-mal fatigue in nuclear components (Faidy, 2007). As a first step

1356 E. Paffumi et al. / Nuclear Engineering and Design 240 (2010) 1355–1362

Nomenclature

a crack depthE Young’s modulus˛ thermal expansion coefficientN, n number of cyclesK stress intensity factor�K stress intensity factor range�Keff effective stress intensity factor rangeT temperatureTm mean temperature�T temperature range variation� stress�o yield stressx distance across the wall thickness from the inner

pipe surfacet wall thickness of the cylinderri inner radius of the cylinderro outer radius of the cylinderf frequency� Poisson ratio� thermal conductivityLWR light water reactorLMFR liquid metal fast reactorNESC network for evaluation of structural componentsNULIFE nuclear plant life prediction-network of excellenceTF thermal fatigueFEA finite element analysis

tsa(ttcNpatdarcaiuttaefliahdTsibf

Table 1316L temperature dependence material data.

Temperature, ◦C 20 300 500 700Young’s modulus, MPa 192e3 170e3 153e3 137e3Poisson ratio 0.3 0.3 0.3 0.3Density, kg/m3 8000 7870 7780 7680Thermal conductivity, W/mK 14.5 18 20 23

outside toward the inner surface to capture the large thermal andstress variations at the inside. The inside of the pipe is subject to a

SIN-method sinusoidal method

he SIN-method was applied using a fatigue curve approach andurface stresses to assess crack initiation in a project under theuspices of the Network for Evaluation of Structural ComponentsNESC) (Dahlberg et al., 2007; Faidy, 2007). Crack propagation withhe SIN-method is more complicated than crack initiation sincehe stress distribution and the stress intensity factors need to beomputed as function of crack depth. In the European NetworkULIFE, a procedure based on Paris law and the SIN-method wasroposed (Paffumi and Radu, 2009). The SIN-method is generallyssumed to provide conservative estimates for both the crack ini-iation and crack propagation life. The analysis uses the nominalifference between the mixing fluids and a conservative value isssigned for the heat transfer coefficient. The mechanical stressesesult from the through-wall thermal gradients and the geometri-al constraints are therefore displacement controlled and an elasticnalysis therefore provides a conservative estimate. But the mostmportant conservatism is that the most damaging frequency issed for the loading. There are, however, also a number of assump-ions that are not necessarily conservative. The thermal variation athe pipe wall may be more abrupt than the sinusoidal function andsquare shaped function, which could be more representative, is

xpected to give higher stresses. The stress–strain analysis is per-ormed for a pipe segment which should be representative for theoaded section of the piping system but the boundary condition thats adopted for the pipe segment will affect the computed stressesnd associated fracture parameters. In this paper we investigateow the assumed shape of the thermal load and the boundary con-itions influence the crack propagation life using the ‘SIN-method’.he thermal load shape is assessed by comparing a sinusoidal and a

quared thermal load variation whereas boundary condition effects assessed by comparing pipe segment with clamped ends (upperound for the constraint) and with traction free ends (lower boundor the constraint).

Thermal expansion, 1/K 15.16e−6 18.92e−6 20.36e−6 21.28e−6Specific heat, J/kg K 480 550 580 600

2. Model description

The pipe has an inner radius is ri ∼= 120 mm and outer radius isro = 129 mm, which is a representative pipe geometry (Chapuliotet al., 2005). The material is 316L stainless steel and the physicalmaterial properties at different temperatures are given in Table 1.

The analysis is performed in two steps. First the through-walltemperature distribution is computed and the resulting stressesand strains are calculated in a subsequent stress–strain analy-sis. For an elastic case with an uncracked body and axisymmetricloading and no temperature dependence of the material proper-ties, the time dependent temperature and stress distributions fora given load can be computed analytically, e.g. (Radu et al., 2007,2008). In more complicated cases, for instance involving tempera-ture dependence of material properties, plasticity effects, crackedbody analysis or complex thermal loads or boundary conditions, afinite element analysis is required. In this paper we have adoptedthe finite element method with the commercial code ABAQUS as itgives more versatility for assessment of different parameters andmodelling conditions. The analysis is performed using axisymmet-ric 8-node elements for both the thermal and stress analysis. Fig. 1shows the finite element mesh. Symmetry conditions are imposedat the lower edge whereas the upper edge is either traction free(free end) or with zero axial displacements (fixed end) (Fig. 1a andb). The finite element mesh has a gradual refinement from the pipe’s

Fig. 1. FE model for thermal and stress analysis: mesh and the boundary conditions.Specimen completely free to expand (a) or fixed in the axial direction (b) with sym-metry condition at the symmetry line. Thermal load applied at the inner surface asshown.

ring a

p

wttthwBT0vecplttao

p1srt

at2

K

wn�sadabc2

3

sahis

3

ctt

E. Paffumi et al. / Nuclear Enginee

eriodic thermal load that is either sinusoidal or square shaped:

Tsin(t)=Tm+�T/2 · sin(2�ft)

Tsq(t)=Tm+�T/2+

⎡⎣ N/f∑

tk=1/f

�T(H(t−tk−1/f )−H(t−tk))

⎤⎦

⎫⎪⎬⎪⎭ , (1)

here H represents the Heaviside function, f is the frequency, Nhe number of cycles within the time considered and Tm the meanemperature. The temperature transferred from a fluid to the struc-ure depends on the heat transfer coefficient. The impact of theeat transfer coefficient, h, is measured by the Biot number, ht/�,here t is the wall thickness and � is the thermal conductivity. Theiot number may vary in the range 5–50 (Dahlberg et al., 2007).he influence of the heat transfer increases with the frequency. For.1 Hz the effect is very small whereas for 10 Hz the stresses mayary by a factor two for the Biot number range considered (Dahlbergt al., 2007). The effect on the fatigue life would be even higher sincerack initiation and propagation depend on the stresses through aower law relationship. In this analysis the wall temperature fol-

ows exactly the fluid temperature, which corresponds to no heatransfer loss between the fluid and the wall and to an infinite heatransfer coefficient. This is a conservative assumption but does notffect the purpose of our analysis at relatively low frequencies. Theuter surface has zero heat flux.

The loading frequency varied from 0.06 to 6 Hz and the tem-eratures �T applied for each given frequency of load were: 100,50, 200, 250 and 300 ◦C. The temperature range is applied to theurface so that the maximum temperature for each case always cor-esponds to 300 ◦C. The stress free temperature is set to the meanemperature Tm.

The stress intensity factors used for the crack propagationnalyses below were calculated from the through-wall stress dis-ributions using Handbook solutions (Marie et al., 2007; Chapuliot,000) from the general formula:

=√

�a

Q

[b0i0+b1i1

x

t+b2i2

(x

t

)2+b3i3

(x

t

)3+b4i4

(x

t

)4]

, (2)

here x is a variable indicting the distance across the wall thick-ess, t; b0, b1, b2, b3 and b4 are coefficients for the polynomial:= b0 + b1(x/t) + b2(x/t)2 + b3(x/t)3 + b4(x/t)4, which fitted to the

tress distribution through the wall thickness for 0 ≤ x ≤ a, withcrack depth; i0, i1, i2, i3 and i4 are influence coefficients which

epend on the crack/thickness ratio, a/t, on the shape of the crack,/c, and on the location along the crack front, and are given by hand-ook solutions (Marie et al., 2007; Chapuliot, 2000); Q is a shapeorrection factor, defined for a semi-elliptical crack as (Chapuliot,000): Q = 1 + 1.464(a/c)1.65, a/c < 1.

For each �T cycle, the analysis was run for 10 full cycles.

. Analysis

The thermal loads result in a bi-axial stress field at the innerurface for the free edge case. For the constrained edge case thexial stress at the inner surface will be somewhat higher than theoop stress. Before performing the simplified fatigue life analysis

t is instructive to have a closer look at the resulting stresses andtress intensity factors.

.1. Stress distributions

Fig. 2a–d shows the computed stress range for four frequen-ies, for the sinusoidal and squared thermal load functions and forhe free and constrained edge boundary conditions for tempera-ure range �T = 150 ◦C. These analyses were all performed with the

nd Design 240 (2010) 1355–1362 1357

temperature dependent properties given in Table 1 and linear inter-polation. Calculations assuming no temperature dependence werealso performed with the material properties at the mean tempera-ture value for 20 and 300 ◦C. The resulting stresses were lower thanfor temperature dependent material properties cases but the effectwas small (typically 10%).

Due to the stress free condition at Tm, the stresses are symmetricwith respect to the zero stress.

As expected the maximum stress range occurs at the free sur-face and it is almost unaffected by the frequency for the sinusoidalload cases, whereas it decreases slightly with the frequency for thesquared load cases. The stress ranges are somewhat higher for thesquared shape thermal load in particular for the lower frequencies.This means that, in a fatigue curve approach, for the same �T a crackcan initiate with fewer cycles with a square thermal loads (Radu etal., 2009; Faidy, 2007). It is also quite clear that the stress gradientclose to the free surface increases with the frequency. The stressesbecome very low after a short distance from the inner wall for fre-quencies above 1 Hz, whereas the stress range at deeper depthsincreases with lower frequency. The stress ranges at deeper depthsare generally higher for the fixed end case than for the free endcase. For the sinusoidal load and low frequency the stress range ishowever higher at the outer surface with the free end. The timeneeded for the thermal load to cross the thickness of the cylinderby conduction is around 1.75 s (Paffumi, 2004), which correspondsto a frequency of about 0.6 Hz. The high frequency 6 Hz has a periodof about 0.16 s, which is quite small compared to the time it takesto conduct the heat through the thickness and in that case therewill only be large stress gradients and stresses close to the free sur-face. The low frequency, 0.06 Hz, corresponds to a period of 16.6 s,which is much longer than 1.75 s. If the frequency is sufficiently lowthe stress range will be constant through the thickness, because ofthermal homogenisation (Kasahara et al., 2002; Boley and Weiner,1960), and in fact it would be zero for the free edge.

3.2. Stress intensity factor

Crack growth in this paper is assumed to be governed by Parislaw. The crack growth rate is related to the stress intensity factorrange (�K) for the postulated crack geometry. The �K for the elasticcases were determined for the different amplitudes of load and fre-quencies using the handbook solutions (Chapuliot, 2000) (Eq. (2))directly and the time dependent stress distributions (Paffumi et al.,2008a). The results in this section are all derived for a circumfer-ential crack. As the mean temperature is stress free, the computedstress intensity factors will be both positive and negative. Whenapplying Paris law only the range when the crack is open is used(the effective stress intensity range �Keff ). In absence of plasticresidual stresses this means that �Keff = Kmax (Itatani et al., 2001).

The effective stress intensity factor range is plotted for five dif-ferent frequencies in Fig. 3a–d, for the two load shape functionsand pipe end conditions for the temperature range �T = 150 ◦C. Theshape of the applied thermal load function and the boundary condi-tion have a much larger impact on the stress intensity factor rangethan on the stress range. The stress intensity factor range increasewith lower frequencies. For crack lengths below 0.5 mm this fre-quency effect is small whereas for deeper cracks the stress intensityfactor range depends strongly on the frequency but also on theboundary conditions and the shape of the thermal load function. Forboth the free and fixed edge cases, the squared thermal load func-tion gives stress intensity factor ranges that are consistently higher

than for the sinusoidal. It can also be noted that for high frequenciesthere is a crack depth for which the effective stress intensity factorhas a peak value. As the frequency decreases the crack depth withthe peak value increases and the peak becomes less pronounced.For the fixed end conditions the stress intensity range increases

1358 E. Paffumi et al. / Nuclear Engineering and Design 240 (2010) 1355–1362

Fig. 2. Range of axial stress across the wall thickness for �T equal to 150 ◦C and different loading frequencies: (a) sinusoidal load and top free boundary conditions, (b)sinusoidal load and top fixed boundary conditions, (c) square load and top free boundary conditions, and (d) square load and top fixed boundary conditions.

Fig. 3. Stress intensity factor range as function of crack depth for �T equal to 150 ◦C and different loading frequencies: (a) sinusoidal load and top free boundary conditions,(b) sinusoidal load and top fixed boundary conditions, (c) square load and top free boundary conditions, and (d) square load and top fixed boundary conditions.

E. Paffumi et al. / Nuclear Engineering and Design 240 (2010) 1355–1362 1359

F tempa to 2 H

mypti

3

omfi3ti

t

N

wdT

D

ti

fsdcaisrpt

frequency for different load cases and boundary conditions. In par-ticular if we neglect the temperature dependence of the materialparameters, which in any case is small as mentioned above, thenthe crack propagation life can be computed from the result for one

ig. 4. Fatigue crack growth lifetime in hours in function of frequency for differentnd for a temperature range equal to �T = 150 ◦C and different shape of the load up

onotonically with crack depth for the lower frequencies. The anal-ses in Fig. 3 were performed with temperature dependent materialroperties. Analysis with no temperature dependence and proper-ies at the mean value of the temperature range 20–300 ◦C resultedn about 10% lower values for the �K range.

.3. Lifetime assessment

The crack growth rate is assumed to follow Paris law:

da

dN= C0 · (�KI)

n. (3)

For austenitic steels the ASME standard (ASME, 2004b) rec-mmends n = 3.3 and C0 = C·S for crack growth rates in units ofm/cycle and �KI in units of MPa

√m. The parameter C accounts

or temperature effects by C = 10˛ but the temperature dependences rather small in this temperature range (˛ equal to −8.4520 at00 ◦C and −8.6075 at 100 ◦C and C differs by a factor 1.4). S is usedo account for R-ratio effects; S = 1.0 when R ≤ 0, which is the casen the present assessment.

The number of cycles Np required for a crack to advance from aio af follows directly from integration of Eq. (3):

p(�T, f ) =∫ af

ai

da

C0(�KI(�T, f ; a))n . (4)

Eq. (4) is integrated from an initial crack depth, ai = 0.9 mm,hich corresponds to 10% of the wall thickness, to a final crackepth af = 7.2 mm, which corresponds to 80% of the wall thickness.he corresponding crack growth in time (hours) is given by

cg(�T, f ) = Np(�T, f )3600 · f

. (5)

The frequency at which the shortest life is obtained is referredo as the critical crack growth frequency, fcr (Radu et al., 2009) ands found by minimizing Eq. (5).

Fig. 4a depicts the fatigue crack growth lifetime in hours asunction of the frequency for different temperature ranges for theinusoidal load and the free edge condition using the temperatureependence of the material properties. Fig. 4b shows the fatiguerack growth lifetime in hours versus the frequency for �T = 150 ◦Cnd for the two boundary conditions and two load cases. If there

s no temperature dependence of the material properties then thetress intensity range is directly proportional to the temperatureange (�K ∝ �T) and consequently the crack growth rate da/dN isroportional to �Tn, the propagation life, Dcg, scales with �T−n andhe critical frequency is independent of �T. The different curves in

erature ranges and sinusoidal load up to 1 Hz for the free boundary conditions (a)z, for both free and fixed boundary conditions (b).

Fig. 4a scale almost perfectly with �T−n and we also note that thecritical frequency is temperature independent, indicating that thematerial property dependence is very weak for the elastic analy-sis. Fig. 4b allows us to compare the crack propagation life for thetwo end conditions of the pipe and the two thermal load functions.The crack propagation life is shorter for the square load than forthe sinusoidal load and the fixed end condition gives a shorter lifethan the free end condition. The critical frequency is about 0.2 Hzfor the sinusoidal and 0.3 Hz for the square load. It should also benoted that the minimum in the crack propagation curves are rela-tively shallow and the crack propagation life does not differ morethan a factor 2 in the frequency range between 0.1 and 1 Hz forthe different cases. The critical frequency depends on the mate-rial parameters, such as the heat conductivity, but the differenceis small between different steels. In fact 0.2 Hz was also found for304L (Radu et al., 2009).

Crack propagation life curves versus the temperature amplitudecan be computed directly from the propagation life curve at a given

Fig. 5. Fatigue life curve versus �T for the two shapes of the loading function,sinusoidal and square and for the corresponding critical frequency 0.2 and 0.3 Hz.Specimen free to expand in the axial direction.

1360 E. Paffumi et al. / Nuclear Engineering and Design 240 (2010) 1355–1362

shold

t

D

actcibstosa

sr(id(wc�ipwipibtetTfitaiftfofpt

form a single fatigue crack that is much longer than the standard6:1 aspect ratio flaws used by ASME Section XI for damage tolerancecalculations. In ASME Section XI Appendix L flaw tolerance assess-ments, a value of 2c/a = 30 ratio is proposed for thermal fatigue to

Fig. 6. Fatigue life curve versus �T with and without considering the thre

emperature range, �T*, by a simple scaling procedure:

cg(�T, f ) = Dcg(�T∗, f ) ·(

�T∗

�T

)n

(6)

Fig. 5 shows the resulting fatigue life curves for the squarednd sinusoidal load cases and the two end conditions at theritical frequencies 0.2 and 0.3 Hz, respectively. It can be noted thathe propagation life is shorter for the square load function and thelamped boundary condition reduces the lifetime further, but theres not a dramatic difference in the crack propagation life for the twooundary conditions. Most of the life is spent in the propagation ofhort cracks. As seen in Fig. 3, the difference in �Keff between thewo end conditions increases with crack depth. Thus the final stagef crack propagation prior to component failure would be muchhorter for the fixed end, which should be considered in a safetyssessment.

Fig. 6a shows the resulting crack propagation curves for the sinu-oidal load with free end condition for different frequencies in theange 0.06–6 Hz. In constant amplitude tests there is often a �K�Kth) below which the crack propagation is small enough (typ-cally 10−8 mm/cycle) to be assumed zero. Such threshold valueepends on the R-ratio but 5 MPa

√m is a typical value for R = 0

ASME, 2004a,b). If the thermal load is sufficiently low then �Kill fall below this value and the crack would arrest. This would

orrespond to a threshold value for the temperature difference,Tth. It was shown in Fig. 3 that �K for a given �T decreases with

ncreasing frequency and that there is a certain depth at which theeak �K is attained. From Fig. 3 it is possible to estimate the �T athich the peak values of �K equal the threshold value by the scal-

ng �K(�T) = �K(�T∗) · (�T/�T∗). Fig. 6b shows the same crackropagation curves as in Fig. 6a, but indicating the correspond-

ng threshold �T. A relation between �Tth and crack depth cane directly obtained from the �K curves. Fig. 7 depicts this rela-ionship for the sinusoidal load case for the free and constraineddge for 0.06, 0.3, 1 and 6 Hz and shows the maximum allowableemperature range for the growth of a defect versus crack depth.he �T needed to reach the �Kth increases with frequency and forxed end conditions, smaller �Ts are needed to fulfill this condi-ion. In the NESC Thermal Fatigue Procedure (Dahlberg et al., 2007)screening criterion for the temperature difference between mix-

ng fluids was adopted below which there would be no thermalatigue failure. This value was set to 80 ◦C and was based on indus-rial experience. The results in Fig. 7 suggest a value around 50 ◦C

or a critical frequency of 0.3 Hz. The local temperature variationf the fluid at the pipe wall is always lower that the nominal dif-erence between the mixing fluids. In Dahlberg et al. (2007) it wasroposed that 80% of the nominal temperature would still be onhe conservative side. Moreover the actual temperature variation

�Kth equal to 5 MPa√

m. Specimen free to expand in the axial direction.

in the pipe wall is still lower due to the heat transfer coefficient,than in the analysis where a perfect heat transfer is assumed. Thusa computed value of 50 ◦C with the conservative assumptions inthe analysis is well in line with the suggested threshold of 80 ◦C. Ofcourse a threshold value for the temperature range assumes thatthere is a threshold value for stress intensity factor range. Physicallya threshold value is related to crack tip closure, which is affected byresidual crack tip stresses and oxidation of crack surfaces. For con-stant amplitude there are ample data to support a threshold valuefor crack arrest, although the actual value depends on for instancethe R-ratio. For variable amplitude loading the picture is less clearand there are data that suggest that there is no threshold value(Ohji et al., 1988). A threshold value may also depend on the envi-ronment where for instance the threshold may increase at elevatedtemperatures due to formation of oxidation products on the cracksurfaces (Nishikawa et al., 1987).

In the analyses above the cracks were circumferential. Thisis a conservative estimate. Service experience as summarised in(Gosselin et al., 2007) showed that typically a large number offatigue cracks initiate at multiple sites and then link together to

Fig. 7. Threshold temperature range values as a function of crack depth for differentstriping frequencies and �Kth equal to 5 MPa

√m. Specimen free and fixed in the

axial direction.

E. Paffumi et al. / Nuclear Engineering and Design 240 (2010) 1355–1362 1361

Fig. 8. Stress intensity factor range as function of crack depth for two different crack asmaximum temperature range of 150 ◦C and different frequencies. Specimen free to expan

Ffaa

ar

FrTsoTiftfcaeswa(ctn

assumptions of these present analyses.

ig. 9. Temperature range values as a function of the life in hours for the criticalrequency 0.2 Hz for the two shapes of the crack, fully circumferential and c = 16a,nd the two boundary conditions; sample free to expand or fully constrained in thexial direction.

ccount for coalescence of small cracks from multiple initiation andepresentative for thermal fatigue in mixing conditions.

Stress intensity factor range versus crack depth are shown inig. 8 for a circumferential and a semi-elliptical crack with aspectatio c = 16a, where 2c is the length and a the depth of the crack.he stress intensity factors in Fig. 8 have been calculated for ainusoidal load at the surface with maximum temperature rangef 150 ◦C, different frequencies and the two boundary conditions.he fully circumferential crack gives a higher value for the stressntensity range but the difference is not very large (maximum aactor 1.2 at deepest cracks and higher �K). The same is valid forhe square loading function not reported here. Fig. 9 compares theatigue crack growth lifetime in hours for different �Ts for the tworack aspect ratio, fully circumferential and c = 16a, for both freend fixed end using the sinusoidal approach. The fully circumfer-ntial shape is slightly more conservative than the semi-ellipticalhape with the ratio c = 16a. The aspect ratio generally increasesith crack depth and the value 2c/a = 30 is typical for a deep crack

nd moreover the crack growth rate increases with the aspect ratio

Paffumi et al., 2008a). Thus taking a fixed aspect ratio of 30 gives aonservative life prediction. If a very low aspect ratio, representa-ive for a shallow crack, were adopted, then the analysis could beon-conservative (Paffumi et al., 2008a,b).

pect ratios: c = 16a and fully circumferential cracks. Thermal sinusoidal load withd in the axial direction (a) and fixed (b).

4. Conclusions

The assessment of fatigue crack growth due to turbulent mix-ing presents significant challenges, in particular to determine thethermal loading spectrum. The sinusoidal method is a simplifiedapproach for addressing this problem, in which the entire spec-trum is replaced by a sine-wave variation of the temperature atthe inner pipe surface. Such estimates are generally intended to beconservative but not unduly conservative.

In general any constant shape function could be consideredin this engineering approach. In this work sinusoidal and squarethermal loading functions of different frequencies and amplitudeshave been considered together with the two boundary condi-tions, cylinder free to expand in the axial direction or completelyrestrained. The stress variation at the surface is directly relatedto the crack initiation estimate, while the stress intensity factordistributions are correlated to the crack growth life and applyingdifferent loading functions. A perfect heat transfer from fluid tothe pipe has been assumed. This is a conservative assumption butdoes not affect the results of our analysis at relatively low frequen-cies.

• The loading functions shapes and boundary conditions influ-ence more the crack growth behaviour than the crack initiationand the square load gives a shorter life than the sinusoidalload (typically a factor 2). In case of a rapid transient vari-ation of the temperature the square load may be a moreappropriate representation. The boundary conditions havealso an influence on the crack growth life. Over-conservativeestimates may be obtained for the completely restrainedcomponent.

• The elastic analyses were performed considering the tempera-ture dependences of the material properties. Their effect on thelife estimates is negligible and hence can be neglected in anengineering approach.

• A threshold of 5 MPa√

m in Paris law suggests a possible screen-ing criteria of 50 ◦C for the temperature difference below whichthere is no crack growth, in line with the 80 ◦C suggested byindustrial experience high cycles thermal fatigue in turbulentmixing (Dahlberg et al., 2007), considering the conservative

• A circumferential crack gives a conservative, but not unduly con-servative, estimate of the crack propagation life with respect tothe ASTM recommendation of a semi-elliptical crack with aspectratio equal to 30.

1 ring a

•

R

A

A

A

BB

C

C

D

F

G

G

H

H

I

I

elastic thermal stress components in a hollow cylinder under sinusoidal tran-sient thermal loading. International Journal of Pressure Vessels and Piping 85,885–893.

362 E. Paffumi et al. / Nuclear Enginee

The critical frequency is typically 0.2–0.4 Hz and the crackpropagation life is only weakly affected by critical frequenciesin the range 0.1–1 Hz. The crack propagation life is also weaklyaffected by the material temperature dependence. The two obser-vations suggest that the fatigue life as function of the temperaturerange, �T, can be derived from one analysis with a fixed fre-quency, e.g. 0.3 Hz and one temperature range, �T*. The completecurve is then obtained by simple scaling the crack propagation lifeDcg(�T, f ) = Dcg(�T∗, f ) · (�T∗/�T)n.

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