Research
Report number: 2016:27 ISSN: 2000-0456
Validation of fatigue fracture
mechanics approaches
2016:27
Author: Magnus Dahlberg Dave Hannes
SSM perspective
Background
In a previous study (SSM research report 2015:38) fatigue experiments were performed on welded austenitic stainless steel piping components. The fatigue experiments offer an opportunity to evaluate fatigue flaw tolerance assessments used in industry, which are based on the fracture mechanical approach implemented in ProSACC.
Objective
The present study aims to validate the used fatigue flaw tolerance approaches by comparing experimental results obtained for the total fatigue life of the considered piping component and the computed fatigue life estimate. Safe and reliable long term operation (LTO) of the plant has to be demonstrated when NPPs approach the end of their design service life time, and this process includes amongst others the evaluation of fatigue resistance of components.
Results
The study indicates that an ASME inspired flaw tolerance approach causes extensive conservatism, implying that the propagation fatigue life at most represents 10% of the total fatigue life. A best-estimate flaw tolerance approach on the other hand presents a significant reduction of conserva-tism, which indicates that fatigue initiation represents a negligible con-tribution to the total fatigue life. The estimated 90% prediction limits of the best-estimate approach show good agreement with the experimental results. Overall conservatism of the fatigue flaw tolerance approach is preserved by assuming a relatively large initial flaw size and neglecting effects from inelastic material behaviour, sequence effects for variable amplitude loads and crack closure effects.
The results support the use of flaw tolerance approaches for demonstrating reliability of a component.
Project information
Contact person SSM: Fredrik Forsberg Reference: SSM2015-3855
2016:27
Author:
Date: September 2016
Report number: 2016:27 ISSN: 2000-0456
Magnus Dahlberg, Dave Hannes
Inspecta Technology AB, Stockholm, Sweden
Validation of fatigue fracture
mechanics approaches
This report concerns a study which has been conducted for the Swedish Radiation Safety Authority, SSM. The conclusions and
view-Title: Validation of fatigue fracture mechanics approaches
Author: Dave Hannes, Magnus Dahlberg
Inspecta Technology AB, Stockholm Sweden
Date: 2016-08-26
Summary
Fatigue experiments have previously been performed on welded austenitic stainless steel piping components subjected to both constant and variable amplitude loads. The results are reported in Evaluation of fatigue in austenitic stainless steel pipe
components – SSM 2015:38 and offer the opportunity to evaluate fatigue flaw
tolerance assessments used in industry and based on a fracture mechanical approach implemented in ProSACC (version 2.1, rev 2). The current investigation aims at validation of the used flaw tolerance approaches by comparing experimental results obtained for the total fatigue life of the considered piping component and the computed fatigue life estimates. More specifically, the conservatism of the approach is evaluated and a sensitivity analysis is performed to determine to which extent the uncertainty of selected input parameters contributes to the variation in estimated fatigue life.
The study indicates that a standard flaw tolerance approach inspired on ASME yields extensive conservatism, implying that the propagation fatigue life at most represents 10% of the total fatigue life, whereas a best-estimate flaw tolerance approach presents a significant reduction of conservatism, which indicates that fatigue initiation represents a negligible contribution to the total fatigue life for the performed fatigue experiments. The estimated 90% prediction limits of the best-estimate approach show good agreement with the experimental results.
The different assessments contain some potential sources for non-conservatism, such as uncertainties or approximations of the actual local stress field near the weld joint or even application of LEFM to potentially short cracks. Overall conservatism of the fatigue flaw tolerance approach is however preserved by postulating relatively large initial flaws and conservative assumptions regarding the fatigue growth law and determination of the fatigue crack driving force, which ensures increased fatigue crack growth rates. The sensitivity analysis highlights that the variation in the estimated fatigue life is best reduced by limiting or controlling the variation of the load, which may be accomplished by means of accurate load measurement or monitoring programs. To a lesser extent the variation in the fatigue growth law parameters also contributes to the variation in the estimated fatigue life. The results support the use of flaw tolerance approaches for demonstrating reliability of a component using fracture mechanics methods, although the selection of input data was observed to significantly affect the overall degree of conservatism for the obtained fatigue life estimate. The performed work has contributed to verification of flaw tolerance approaches used in industry, which will facilitate the choice of optimal and safe control intervals for components subjected to fatigue loads.
Sammanfattning
Utmattningsprov har utförts på svetsade austenitiska rostfria rör som utsattes
för både konstant och variabel amplitud belastning. Resultaten redovisades i
Evaluation of fatigue in austenitic stainless steel pipe components - SSM
2015: 38 och möjliggör en utvärdering av de brottmekaniska analyser som
används inom industrin och som grundar sig på en brottmekanisk metodik
implementerad i ProSACC (version 2.1, rev 2). Den aktuella studien syftar
till att validera analyserna genom jämförelse av beräknade livslängder med
experimentella för de betraktade rören. Metodens konservatism utvärderas
och bidraget av osäkerheten i utvalda inparametrar till den totala variationen
i beräknade livslängd bestäms med hjälp av en känslighetsanalys.
Studien visar att brottmekanisk analys enligt ASME medför omfattande
konservatism. En analys i stället baserat på en best-estimate visar en
betydande minskning av konservatismen och indikerar att
utmattningsinitieringen utgör ett försumbart bidrag till den totala
utmattningslivslängden för de utförda experimenten. Det uppskattade 90%
prediktionsintervallet för best-estimate analysen visar god
överensstämmelse med de experimentella resultaten.
De genomförda analyserna enligt ASME uppvisar övergripande
konservatism, trots osäkerheter kring det lokala spänningsfältet vid
svetsfogen eller tillämpning av LEFM till potentiellt korta sprickor.
Konservatism säkerställts genom att postulera relativt stora initiala defekter
och konservativa antaganden för tillväxtlagen och skademekanismens
drivkraft. Känslighetsanalysen belyser att variationen i beräknad livslängd
minskas mest genom att kontrollera lastens variation, vilket kan
åstadkommas med hjälp av noggrann lastmätning eller
övervakningsprogram. I mindre utsträckning bidrar också variationen i
tillväxtlagens parametrar till variationen i beräknad livslängd.
Resultaten stöder användning av brottmekanik för att visa tillförlitlighet hos
en komponent, även om valet av indata påverkar nivån på konservatism i de
uppskattade utmattningslivslängder. Studien har bidragit till verifiering av
de skadetålighetsanalyser som används inom industrin och därmed
underlättar valet av optimala och säkra kontrollintervall för komponenter
som utsätts för utmattningslaster.
Con
1 CONC 2 NOME 3 INTRO BAC 3.1 PER 3.2 OBJ 3.3 4 PROB GEO 4.1 MA 4.2 LOA 4.3 5 THEOR FLA 5.1 5.1.1 5.1.2 5.1.3 5.1.4 COM 5.2 5.2.1 5.2.2 5.2.3 6 RESUL EST 6.1 SEN 6.2 7 DISCU 8 CONC 9 RECOM 10 ACKNO 11 REFERtents
LUSIVE SUMM ENCLATURE .. ODUCTION .... CKGROUND ... RFORMED PIPING JECTIVES ... LEM DESCRIP OMETRY ... ATERIAL ... AD ... RY AND METH AW TOLERANCE A Load mode Equivalent Propagatio Crack closu MPARISON BETW Fatigue life Experimen Sensitivity LTS ... TIMATE OF EXPEC NSITIVITY ANALYS USSION ... LUSIONS ... MMENDATIO OWLEDGEME RENCES ...s 201
MARY ... ... ... ... G COMPONENT F ... PTION ... ... ... ... HODS ... APPROACH ... el ... t strain amplit on fatigue life ure effects ... WEEN EXPERIMEN e consumed b ntal or total fa analysis appl ... CTED PROPAGATI SIS ... ... ... ONS ... ENT ... ...16:27
... ... ... ... FATIGUE TESTS .. ... ... ... ... ... ... ... ... tude measure e ... ... NTS AND FRACTU by propagation atigue life, Nexplied to propag ... ON FATIGUE LIFE ... ... ... ... ... ...
7
... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... RE MECHANICAL n, Np ... p ... gation fatigue ... E ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... FATIGUE ASSESS ... ... life ... ... ... ... ... ... ... ... ... ... 4 ... 5 ... 7 ... 7 ... 8 ... 10 ... 11 ... 11 ... 11 ... 12 ... 13 ... 13 ... 13 ... 16 ... 18 ... 19 SMENTS ... 20 ... 20 ... 20 ... 21 ... 23 ... 23 ... 26 ... 27 ... 30 ... 31 ... 31 ... 321 Conclusive Summary
The current study presents a numerical investigation of fatigue flaw tolerance approaches based on fracture mechanical analyses, which are used to estimate fatigue life and determine service or inspection intervals for components. The investigation compared estimates of fatigue life for a welded austenitic stainless steel piping component computed using ProSACC (version 2.1 rev 2) with available experimental results. The numerical analyses considered fatigue flaw tolerance assessments of both internal and external fatigue cracks with a standard conservative approach inspired on ASME and a best-estimate approach. A parametric analysis using Variation Mode and Effect Analysis (VMEA) was also performed based on the best-estimate approach. The current study resulted in the following main findings:
The conservative ASME inspired fatigue flaw tolerance approach yields
extensive conservatism.
The best-estimate flaw tolerance approach presents significantly reduced
conservatism.
The results of the best-estimate flaw tolerance approach imply a negligible
contribution of fatigue initiation to the total fatigue life.
The studied approaches contained potential sources of non-conservatism
related to the assumed load description and applicability of LEFM.
Overall conservatism of the fatigue flaw tolerance approach is preserved by
means of conservative flaw geometry, material, load and fatigue growth assumptions.
Variation of selected input data covering initial flaw geometry, growth law
and load description induced relatively large variability of the estimated fatigue life.
The VMEA indicated that the extent of the variability in the estimated
fatigue life is primarily due to the variation or uncertainty in load.
Load measurement or monitoring programs allowing for accurate load
description enable to significantly reduce the variability in the estimated fatigue life.
The estimated 90% predictions limits for the best-estimate flaw tolerance
approach contained the experimental results.
The results support the use of flaw tolerance approaches for demonstrating reliability of a component using fracture mechanics methods, although the choice of input data is shown to strongly affect the overall degree of conservatism for the obtained fatigue life estimate. The performed work has contributed to verification of flaw tolerance approaches used in industry, which will facilitate the choice of optimal and safe control intervals for components subjected to fatigue loads.
2 Nomenclature
a Crack depth
A Cross-sectional area c Sensitivity coefficient
C Fatigue growth law factor (Paris law)
C0 Modified fatigue growth law factor to avoid accounting for crack closure effects
dg Average grain size diameter
E Young’s modulus
f Transfer function used in sensitivity analysis
F Normalized through-thickness evolvement of stress concentration factor H Geometry function in stress intensity factor formulation
� Integral included in expression of η i,j Dummy indices
I Area moment of inertia of the cross-section K Stress intensity factor
Kt Stress concentration factor
l Crack length
L Moment arm
m Fatigue growth law exponent (Paris law) n Total number of (strain) cycles in a load sequence
N Fatigue life, Total number of cycles (simulated or experimental) Nexp Total number of cycles (from fatigue experiments)
Ni Total number of cycles consumed by fatigue crack initiation
Np Total number of cycles consumed by fatigue crack propagation
r Radial coordinate of cross-sectional polar coordinate system rpc Cyclic plastic zone radius
R Load ratio
RC Ratio of fatigue growth law factors C
�� Ratio of integrals �
Ri Inner radius of piping component
Rη Ratio of Basquin equation factors η
Rσ Stress ratio
s Standard deviation
t Wall thickness of piping component
u Local radial coordinate with origin at fatigue crack initiation position w Coefficient of variation
x Variable in sensitivity analysis
α Exponent in Basquin equation for experimental or total fatigue life β Exponent in Basquin equation for experimental or total fatigue life γ Ratio of bending and membrane stress
Δ∎ Range or difference
ε Strain
η Factor in Basquin equation for fatigue life consumed by propagation κ Factor in Basquin equation for experimental or total fatigue life λ Ratio estimated propagation fatigue life and total experimental fatigue life
μ Mean value
ν Poisson’s ratio
σ Pseudo-stress
τ Contribution to standard deviation of logarithmic fatigue life φ Angular coordinate of cross-sectional polar coordinate system χ Ratio m-norm and β-norm
∎a Amplitude
∎init Quantity related to fatigue crack initiation (position)
∎max, max∎ Maximum value
∎min Minimum value
∎nom Nominal value
∎th Threshold value
∎� Quantity related to logarithmic variable ∎0 Quantity related to φ = 0
∎φ Quantity including a φ-dependence or evaluated at φ.
3 Intro
Bac
3.1
Componen induced d Componen time, often ASME Bo procedure i in air for available f however no fatigue resi correction factor can f an unaccep complied w effects. Safe and re when NPP extension o the evaluat time. The factors fol replaced w component reveal to b fatigue des When fatig reliability o postulate a flaw. The f service int Section III and Pressu ASME cod for demons fracture me initial flaw mechanical ProSACC output of t on the sel variability analyses to It is there experimentoductio
ckground
nts in nuclea degradation nts are dimen n set to 40 y oiler and Pr is based on t fatigue sens for different ot include th istance, whic factor (Fen for some comptable total with the ori eliable long Ps approach or renewal of tion of fatig required fat llowing ASM with the actu
t, but showin be challengi ign methods gue usage li of a compon flaw of a gi flaw toleranc tervals. Thes [3], but alrea ure Vessel C de applies to strating fitne echanical ap w, a fatigue l analyses a (version 2.1 the fatigue fr lection of a in the result o a same prob efore of inte tal results.
n
ar power pl mechanism nsioned to re years, based ressure Vess he determina sitive compo materials. E he detrimenta ch was reme ), as describ mponents int fatigue us iginal fatigu term operat the end of f the operatin gue resistanc tigue analys ME, Section ual fatigue l ng complianc ng, due to .mits are exc nent may be b iven size in t ce approach se methods ady exist in t ode, Section o in-service i ess for servic pproach, whe e load and are implemen rev 2), wh racture mech appropriate i ts may be ex blem, as illus erest to vali lants (NPPs) ms. Fatigue esist fatigue on design b sel Code, S ation of cum onents using Early versio al effect of l ediated by th bed in NUR troduce very sage factor, ue design re tion (LTO) o f their desig ng license. T e of compon ses are howe n III [1]. C load history ce with the re inherent (ex ceeded, an a based on flaw the compone allows estim are current the non-man n XI [4]. Ho inspection, w ce. The flaw ere the inpu a fatigue nted in diffe hich are used hanical appro input data a xpected whe strated in the idate a fatig ) are subjec is one o loads during by analysis p ection III [ mulative fatigu g prescribed ons of the d light water r he introductio REG/CR-690 large penalt although th quirements of the plant n service li This process i nents consid ever again b Conservative observed d equirements xcessive) con alternative ap w tolerance ent and inves mation of a f tly under de ndatory Appe owever this p with procedu w tolerance a ut typically c crack grow erent comme d in the indu oach depend and modelli en applying e round-robin gue flaw to cted to vari of these m g the design procedures c 1]. The fati ue usage fact d design fati design requir reactor envir on of the env 09 [2]. This ties, which m he compone omitting env has to be de fe time, and includes amo dering the ex based on fat design loa during servic
for all comp nservatism i pproach to d methods. Su stigate the gr fatigue life o evelopment endix L of AS particular sec ures which a pproach for consists of a wth law. Su ercial softwa ustry and at ds however s ng assumpti different flaw n study prese olerance app ious service mechanisms. service life contained in igue design ctors (CUFs) igue curves rements did ronments on vironmental s correction may result in ent initially vironmental emonstrated d apply for ongst others xtended life atigue usage ads may be ce life of a ponents may in the basic demonstrate uch analyses rowth of the or necessary for ASME, SME Boiler ction of the are intended fatigue is a a postulated uch fracture are such as NPPs. The significantly ions. Large aw tolerance ented in [3]. proach with
Per
3.2
The experi seamless T joint butt w nominal w mm and 24 capping wa with a radio All specim reversed be hydraulic t situated in using custo small mem fatigue test leakage, i.e The perform amplitude ( (CA) fatigu of three dif a pip a Ga a two Each type loading typ Figure 1 (a) W condition. (b)rformed P
imental stud TP 304 LE s weld, see Fig wall thickness 4.6 mm. The as not remov ographic exa mens were fa ending loadin esting machi the bending om-built fixt mbrane stress ting pressuri e. when the in med study h (VA) loading ue tests and fferent load s ping spectrum aussian spect o-level block of loading pes, except fo Welded piping c Actual mountediping Com
y reported i stainless stee gure 1 (a). At s (t) and inn e welding jo ved. All of th amination. atigue tested ng with disp ine. The nomplane. The e tures, which s [5], see Fig ized with w nternal press ad particular g. The fatigu experiments spectra: m (VAP), ba rum (GAP), k spectrum (V was perform or the VA2 lo (a) component with d test specimen
mponent F
in [5] includ el pipes join t the vicinity ner radius (R oints were in he 28 test spe d at room te lacement con minal axial st experimental introduced gure 1 (b). T water at 70 b sure could no r focus on hi ue experimen s with variab sed on chara based on the VA2). rmed at diff oading wher close-up view in servo-hydraFatigue Te
ded 28 test s ned with a c y of the circu Ri) were esti n as-welded ecimens wer emperature (2 ntrol in a sta train was rec l set-up was a predomina The piping c bar. Fatigue o longer be s gh cycle fati nts included ble amplitud acteristic pipi e piping spec ferent severi re only one s of the circumfe ulic testing macests
specimens co ircumferenti mferential bu mated at res condition, i. re verified an 20°C) and s andard single orded with a based on a c ant bending s omponents w failure was ustained. igue (HCF) a both constan e loading us ing loads, ctrum, andties for the everity was c
(b) rential butt weld chine.
onsisting of ial single v-butt weld the
spectively 3 .e. the weld nd approved subjected to e axis servo-a strservo-ain gservo-age construction stress and a were during defined by and variable nt amplitude sing one out
considered considered.
)
Table 1 Selected fatigue results for the performed fatigue tests. Pipe
ID Load type Severity
(*)
Nexp max ε
a,nom initiation Radial
position
φinit
[cycles] [%] [°]
1 VAP Medium 575000 0.171 outside 34 2 VAP Low 2500000 0.126 inside 0 3 VAP High 217000 0.203 outside 161 4 VAP Peak 139000 0.288 outside 30 5 VAP Low 2520000 0.124 inside 17 6 VAP Medium 253000 0.173 outside 17 7 VAP High 269000 0.207 outside 8 8 VAG Medium 941000 0.136 inside -15 9 VAG Medium 1063624 0.140 outside 0 10 VAG High 126350 0.185 outside 12 11 VAG Low 3921275 0.101 inside 26 (†)13 VAG Low 5133411 0.103 - -
14 VAG High 247441 0.180 outside -21 15 CA 2.2 740735 0.085 inside 172 (†)16 CA 1.7 5269515 0.065 - - 18 CA 1.95 1027847 0.074 inside -148 19 CA 2.6 291260 0.099 outside 6 20 VA2 - 1131716 0.069 inside 8 21 VA2 - 4880396 0.069 inside -31 (†)22 VA2 - 5024628 0.068 - - 23 VA2 - 913856 0.069 inside 8 24 VA2 - 321904 0.069 inside 171 25 CA 2.8 105769 0.109 outside 8 26 CA 2.8 144230 0.115 outside 0 27 CA 1.8 1367448 0.073 outside -149 28 CA 1.7 512749 0.065 inside 9 (†)29 CA 1.7 5000000 0.068 - - (†)30 CA 1.7 5000000 0.067 - -
(*)The severity for the CA experiments corresponds to the prescribed displacement amplitude.
(†) Run-out experiment, where the number of cycles exceeded the run-out limit of 5 million cycles. The fatigue tests were stopped prior to leakage and no fatigue initiation position was identified or detected.
Selected fatigue results for the 28 considered specimens are summarized in
Table 1, where the total number of applied load cycles (Nexp) and the maximum
nominal strain amplitude in the applied load sequence (max εa,nom) are reported
from [5]. The radial (inside/outside) and circumferential position (φinit) of fatigue
initiation are taken from [6]. The circumferential position corresponds to the angular coordinate of a cross-sectional centered cylindrical coordinate system. The position of the strain gage situated in the bending plane of the specimen corresponds to a circumferential angle equal to zero. Additional information about the test specimens, experimental set-up, testing procedure, load description and/or obtained results is presented in [5, 7, 6].
The experim curve for au incre cons impr the f the reali spec high loadi ampl Continued fatig plane mult large leaka crack crack two weld The differe potential fa
Obj
3.3
The curren fatigue life flaw tolera (version 2. a sta a bes Based on th the inside a compared t stainless st number of Conservati investigate The curren prov weld mental resul ustenitic stee eased unders servatism in t roved knowl fundamental developmen stic compon ific design c hlighting the ing to obta litude (CA) t work includ gue initiation e both on the tiple adjacen e strain ampl age being th k, which occ ks. distinct fat ded piping co ent achievem atigue risks inectives
nt study focu e and determ ance approac 1 rev 2) and andard conser st-estimate a he findings r and outside w to the experi teel piping cycles consu sm of the d. nt investigatio viding increa ded austenitic ts were used el in [1]. The standing of the ASME fa edge on fatig issue of tran t of an exp nent allowin urves. e importance in reliable testing. ed a fractogr n occurring e inside and o nt fatigue cr litudes. e result of w casionally ha tigue failure omponent yie ments of the n piping com usses on frac mine service ch. The inv will include rvative appro nalysis in co reported in [6 will be studie imental outc component umed by cra considered on aims in pa sed reliabilit c steel, d to investiga e work report the ASME atigue proced gue in austen nsferability. perimental p ng for more e of using design curv raphic exami in the vicin outside of th racks for sp wall penetrati ad merged w e mechanism elding differ performed w mponents. cture mecha intervals fo vestigation w : oach similar ombination w 6], both fatig ed. Estimated come obtaine in [5], whic ack initiation fatigue fla articular at: ty in fracturate the margi ted in [5, 7] margins by dure. nitic stainles procedure fo e realistic m realistic va ves, as oppo ination [6], w nity of weld he weld joint. pecimens hav ion due to a with adjacent ms for the ent fatigue c work aimed anical analys or componen will be perfo to Appendix with a sensitiv
gue crack pro d number of ed for the stu ch will allow and consecu aw toleranc
e mechanics
ins of the des contributed w highlighting ss steel comp or fatigue te margins and riable ampli osed to usin which reveale toes near t . ving been s single domin t quasi-copla considered rack features at improving ses in order nts, followin ormed using x L in ASME vity analysis opagation ini cycles to fai udied welde w an estima utive crack p e approache s methods fo sign fatigue with: ng extensive ponents and esting of a component litude (VA) ng constant ed: the bending subjected to nant fatigue anar fatigue thin-walled s. g control of to estimate ng a fatigue g ProSACC E XI [4], s. nitiated from ilure will be ed austenitic ation of the propagation. es will be or fatigue in
4 Prob
Geo
4.1
The overal thickness ( fatigue flaw with semi-crack depth thickness. following t the finding (initiation f elliptical c accordance The postula approaches inspired by depth. For degree of c approach. T 2. A ratio a investigatio Table 2 Postu Analysis initial a/t initial a[mm]Mat
4.2
The TP 30 at room tem GPa, and a similar to 3 tolerance a where the factor rang C and grow (m) is assu 8410 [4] fo will namely inside of th effect of w therefore n agreement exponents ablem D
ometry
ll geometry t) and inner w tolerance -elliptical sh h represents The total c the paramete gs in [6], b from outside rack is assu e with the proated initial fl s by assumin y ASME, co the best-est conservatism The selected a/t = 1/10, as on given the
ulated initial flaw
]
terial
4 LE austen mperature (R a Poisson’s 316 L stainle approaches as crack growt ge, ΔK = Kma wth law expo umed to be c or crack grow y focus on fa he piping co water on the f not considere with experim at room tempDescript
of the pipi radius (Ri) d approaches ape, defined the flaw siz crack size in er definitions both an inte e) initial crac umed to pres oposed asses law geometry ng a differen onservatism timate analys m will nevert d initial ratio s suggested in small wall th w depths. S itic stainless RT), defined ratio, ν = 0 ess steel used ssume crack d d th rate (da/dN ax - Kmin, and onent m. In th constant and wth in air in atigue crack omponent oc fatigue crack ed in the cur mental findin perature are rtion
ing compon determined fo assume an d by crack d ze in the rad n the circum s in ProSAC ernal (initiat ck are invest sent an aspe ssment proce ry differs bet nt initial crac is ensured b sis a smaller rtheless also s a/t for eac n [3], was co hickness of t Standard 1/2 1.5 s steel piping at 20 °C, giv 0.3. The cyc d in [9], i.e. σ growth follo d� d� � ����� dN) is expres d two materia he current inv equal to the austenitic st growth in ai ccurred in w k growth rate rrent study. ngs from [1 reported in t nent is given or the consid initial circu depth (a) an dial directio mferential d CC (version 2 tion from in tigated. The ct ratio a/l = edure in [3].ween the sta ck depth. Fo by taking an r crack dept be maintain h analysis ar onsidered not the considere g component ven by a You
lic yield stre
σyc = 405 MP owing a Paris � ssed in term al parameters vestigation th e value presc teels, i.e. m = ir, although c water. Given e is assumed The assume 0], where fa the range 3 - n by the no dered test spe mferential p d crack leng n, i.e. throug irection is g 2.1 rev 2) [8 nside) and a postulated i = 1/6, which andard and be r the standar n extended i h is selected ned in the be re summariz t suitable for ed piping com Best-estimat 1/6 0.5 t has materia ung’s modul ength is assu Pa.The used f s law: s of the stre s: the growth he growth la cribed in AS = 3.3. The pr crack initiati
the test con d to be small ed value of m atigue crack 4. ominal wall ecimen. The planar crack gth (l). The ugh the wall given by l, 8]. Based on an external initial semi-h is also in est-estimate rd approach initial crack d. A certain est-estimate zed in Table r the current mponent. te al properties lus, E = 200 umed to be fatigue flaw (1) ess intensity h law factor aw exponent SME XI, C-resent study ion from the nditions, the l [6], and is
m is also in
Table 3 Assum Analysis C [mm/cycle The standa law factor growth law factor for c and load ra positive lo constant (2 constant gr less conser results in [ crack grow 304 austeni Equation (1 ΔKth. In th ΔKth = 0. T contribute unaffected
Loa
4.3
The fatigu contributio water press load or loa specimens the fixture machine an induced a m loads are a crack grow means of amed fatigue gro e]
ard and
best-C, as presen w description crack growth atio R = Km oad ratios. I 20°C) and pr rowth law fa rvative or sm [10]. The wo wth rates at r itic stainless 1) is usually e current inv This assump to fatigue by possible
ad
ue cracks in ons affecting surized with ad ratio R. T to a princip es is given nd the mom minor memb alternating du wth driving strain gageowth law factors Standar 2.05 10 --estimate app nted in Tabl n stipulated in h in air in au min / Kmax. Th In the curre redominant r actor in Tabl maller const ork performe oom tempera steel pipe an assumed val vestigation fa ption preserv crack growt history or se n the piping g their growt h a constant The custom-b pal bending by the alter ment arm L = brane load in uring testing loads. Durin situated in th
for the perform rd -9 proaches dif le 3. The sta n ASME XI ustenitic stee he latter dep ent investig reversed ben le 3. The be tant growth ed in [10] in ature using s nd subjected lid for ΔK ex fatigue crack ves conservat wth. Furtherm equence effec g componen th. During f internal pres built fixtures load. The be rnating verti = 300 mm. the test spec g of the pipin ng testing th he bending p
med flaw toleranc Bes 6.80 ffer by assum andard appro I, C-8410 [4] els a priori d endence is t ation the te ding loads in est-estimate a law factor ncludes the d specimens m to fatigue te xceeding a c k growth will tism, as even more, it ens cts occurring nt are subje fatigue testin ssure of 70 s allowed fur ending mom ical force a The alterna cimen. Both ng componen he nominal lane of the sp ce approaches. st-estimate 0 10-10 ming a differ oach follows ], where the depends on t though only emperature nduce R ≤ 0 approach wi based on ex determination machined from esting at R = onstant thres l be modelle n small load sures the re g in VA fatig ected to dif ng the speci bar, affectin rthermore to ment prevailin at the hydrau ating vertical bending and nts inducing strain was r pecimen [5]. rent growth the fatigue growth law temperature applied for is however 0, giving the ill assume a xperimental n of fatigue m a welded 0.1. shold value, ed assuming d cycles can esults to be gue loads. fferent load imens were ng the mean subject the ng between aulic testing l force also d membrane g the fatigue recorded by .
5 The
Flaw
5.1
The flaw t Apart from description thickness s analysis. D approximat Note that t field in t circumferen5.1.1 Lo
A load mo present dur local throu for a given profiles are amplitude current load A cross-sec indicates th plane, see radial initia of r: � � � � Figure 2 Norm weld toe. 0 -0.5 0 0.5 1eory and
w toleranc
tolerance ap m a (crack) n, ProSACC stress distrib During calc ted with a th the implemen the circumf ntial directiooad model
odel has bee ring testing. ugh-thickness n nominal s e equivalentl σa and pseu d model. ctional cente he circumfer [5, 6] for m ation position � � �� , for ��� � � � , malized through 0.1 0.2
d Metho
ce approa
proach is pe geometry d requires bo bution as in culations the hird order p ntation in Pr ferential dir on on fatigue en adopted t It aims at pr s stress profi train amplitu ly determine udo-stress ratred polar coo rential positi more details. n; hence a lo a crack initia for a crack i h-thickness evo 2 0.3
ods
ach
erformed us definition and oth a minim nput for theese stress p polynomial f roSACC doe rection. Eff e crack growt to account f roviding the files present ude. The m ed by the dis atio Rσ = σmi ordinate syst ion of the s The stress ocal coordina ated from ins initiated from olvement of stre 0.4 0.5 sing ProSAC d selection mum and ma e implemente profiles are fit over the c es not includ
fects of st th are therefo
for the diffe necessary m at the consid inimum and tributions of in / σmax, wh tem (r, φ) is strain gage, fields will d ate u is introd side, m outside. ess concentrati 0.6 0 CC (version of a crack g aximum loc ed fracture in the cur considered c e variation o ress variati ore neglected erent load co minimum and dered initiati d maximum f the local ps hich are defi introduced, w situated in t differ depend duced, which on due to the 0.7 0.8 2.1 rev 2). growth law cal through-mechanical urrent study crack depth. of the stress ion in the d. ontributions d maximum ion position local stress seudo-stress fined by the where φ = 0 the bending ding on the h is function (2) (3) presence of a 0.9 11
During fatigue testing nominal strains were recorded, which for the fatigue flaw tolerance approaches were transformed in a nominal linear elastic pseudo-stress by means of E. The transformation from nominal to local stress or strain is performed
assuming a constant concentration factor Kt = 1.4 [5] and a through-thickness
evolvement defined by means of a function F, illustrated in Figure 2. The normalized through-thickness evolvement of the stress concentration factor at a weld toe was computed using a finite element (FE) simulation similar to the one
used for the estimation of Kt in [5]: the FE analysis assumed a weld geometry given
by a cap height of 0.5 mm and weld toe radius of 1 mm. The mesh size was approximately 0.1 mm and the simulation was performed using ANSYS 14.5 [11].
Kt and F are assumed identical for cracks starting from inside and outside. The
shape of F in Figure 2, indicates that the stress concentration mainly acts down to a depth of approximately 10% of the wall thickness, whereas a stress field close to the nominal stress field is expected to act over the remaining, major part of the wall thickness.
The predominant load is related to the alternating bending moment prevailing between the fixtures. The experimental set-up induces also a minor alternating membrane stress, which was found affecting localization of fatigue crack initiation, see [6]. The ratio of bending and membrane stress is given by the dimensionless factor γ = A L (Ri + t) / I, where A and I represent respectively the cross-sectional
area and the area moment of inertia of the cross-section for the studied piping component. For the considered pipe geometry and experimental set-up, γ = 24.2, which highlights the predominance of the bending stress. The local pseudo-stress amplitude is a function of the polar coordinates, and is assumed to be given by the following expression: ����� �� � ����� 1������ � 1����1 � �1 � � � 1 1 � 1 �� � � ��� �� ������� � (4)
where the first bracket accounts for the transfer from nominal to local stress. The second bracket considers the linear r dependency of the nominal pseudo-stress. Considering the negligible contribution of the membrane stress, the φ dependency was approximated by factorization to be included in the expression a nominal strain amplitude defined in terms of φ and the nominal strain amplitude at the strain gage ������� , i.e. at φ = 0 (and r = Ri + t):
������� � |��� �|������� (5)
For a crack initiated at φ = φinit , the local through-thickness pseudo-stress
amplitude distribution is then given by ����� ������. The developed load model is
intended for use with small angular coordinates, i.e with φinit relatively close to 0,
which was shown to be the most probable position for crack initiation [6]. At the secondary most probable initiation position, i.e. in the vicinity of φinit = 180°, the
model somewhat over-estimates the local stress amplitude, as it assumes membrane stress to act in phase with bending stress. At these locations, this minor over-estimation does however ensure conservatism of the performed flaw tolerance assessments. The through-thickness distribution of the normalized stress amplitude is illustrated in Figure 3 (a).
that started from inside the internal pressure gives an additional contribution as a constant crack face pressure. The mean load is expected to be affected by the through-thickness weld residual stress distribution, which was however not included in the current investigation. Similarly as in [3], the weld residual stress is thus assumed to be zero.
(a)
(b)
Figure 3 Through-thickness evolvement of (a) normalized stress amplitude and (b) pseudo-stress ratio for a nominal strain amplitude at the initiation position of 0.1%.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5
Initiation from inside Initiation from outside
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -0.8 -0.75 -0.7 -0.65 -0.6
Initiation from inside Initiation from outside
5.1.2 Equivalent strain amplitude measure
The load model in Equation (4) requires a scalar nominal strain amplitude, which is directly available for CA tests, but not for VA tests. For fatigue tests using VA loads, an equivalent strain amplitude measure needs therefore to be defined. For the current fracture mechanical approach an equivalent strain amplitude measure based on the fatigue growth law exponent m is selected. The equivalent measure for a load sequence consisting of n strain cycles with amplitude εa,i , is then expressed in
terms of the m-norm of the strain amplitudes. ‖��‖� � �1� � ����� � � � ��� (6) This equivalent strain amplitude measure differs in general from the β-norm strain defined in [5], as m ≠ β = 4.6. For a given type of load spectrum the ratio of the m-norm and β-m-norm, denoted χ, will however be constant. In absence of a threshold value, the magnitude of the considered load spectrum will not affect this ratio. The ratios for the considered load types are given in Table 4.
Table 4 Ratio of m-norm and β-norm strains from [5] for different load types.
Load type VAP VAG CA VA2
χ 0.842 0.881 1.000 0.974
Table 5 presents different nominal strain measures for the performed fatigue tests.
The m-norm strain at φ = 0 is denoted �������� �� and is computed using the
relevant χ in Table 4 and the β-norm strain available in [5]. The m-norm strain at the initiation position tabulated in Table 1, is denoted ������������ �� and is determined
using Equation (5). This equivalent strain measure is by definition smaller the m-norm strain at φ = 0, but no large differences are observed, as the absolute value of
the cosine of φinit is relatively close to unity. For run-out experiments, a
circumferential initiation position is not available. It was then assumed to be given by the most probable location for fatigue crack initiation, i.e. φinit = 0 [6].
Table 5 Different nominal strain measures for the performed fatigue tests. Pipe
ID Load type Severity
(*) max εa,nom ฮߝୟǡ୬୭୫ ฮ ฮߝୟǡ୬୭୫ ఝ౪ฮ [%] [%] [%] 1 VAP Medium 0.171 0.061 0.051 2 VAP Low 0.126 0.045 0.045 3 VAP High 0.203 0.073 0.069 4 VAP Peak 0.288 0.096 0.083 5 VAP Low 0.124 0.044 0.042 6 VAP Medium 0.173 0.060 0.057 7 VAP High 0.207 0.072 0.072 8 VAG Medium 0.136 0.054 0.052 9 VAG Medium 0.140 0.057 0.057 10 VAG High 0.185 0.073 0.071 11 VAG Low 0.101 0.042 0.038 (†)13 VAG Low 0.103 0.041 0.041 14 VAG High 0.180 0.065 0.061 15 CA 2.2 0.085 0.085 0.084 (†)16 CA 1.7 0.065 0.065 0.065 18 CA 1.95 0.074 0.074 0.063 19 CA 2.6 0.099 0.099 0.098 20 VA2 - 0.069 0.059 0.059 21 VA2 - 0.069 0.059 0.051 (†)22 VA2 - 0.068 0.059 0.059 23 VA2 - 0.069 0.059 0.059 24 VA2 - 0.069 0.059 0.059 25 CA 2.8 0.109 0.109 0.108 26 CA 2.8 0.115 0.115 0.115 27 CA 1.8 0.073 0.073 0.062 28 CA 1.7 0.065 0.065 0.064 (†)29 CA 1.7 0.068 0.068 0.068 (†)30 CA 1.7 0.067 0.067 0.067
(*) The severity for the CA experiments corresponds to the prescribed displacement amplitude.
(†) Run-out experiment, where the number of cycles exceeded the run-out limit of 5 million cycles: φinit = 0 is
5.1.3 Propagation fatigue life
The number of cycles to failure consumed by fatigue crack propagation is obtained by integration of Equation (1), where the stress intensity factor range can be expressed in terms of the strain amplitude and a geometry function H characteristic for a given crack geometry:
�� � �� � (7)
In the current investigation two semi-elliptical crack configurations (internal and external) are considered, which will consequently have different geometry functions H. Furthermore will these functions depend on the flaw size, i.e. a and l. Integration of Equation (1), considering Equation (7), yields
� � ���� �� �
���� � ���� (8)
which is a Basquin type equation with factor η and an exponent giving by the fatigue crack growth law (Paris law) exponent m. The factor η is given by ���, where �is an integral with the initial and final crack depth as integration limits and
H-m as integrand. The expression of N in Equation (8) can be generalized to be
applicable to VA load by summation of the different strain amplitude contributions:
� � ���� �� �‖�
�‖����� � �‖��‖���� (9)
The expression includes the m-norm of the strain amplitude defined in Equation (6). The m-norm strain can thus be interpreted as the equivalent CA strain amplitude yielding identical fatigue life (number of cycles of propagation) as the original VA load sequence, for a given final crack size. The derivation assumes sequence or history effects to be negligible. Such effects are indeed not accounted for due to the assumption of ΔKth = 0.
The stress intensity factor formulations implemented in ProSACC are based on tabulated solutions, see [8], which present a range of applicability a/t ≤ 0.8. Hence, ProSACC will only allow computation of the number of cycles corresponding to propagation from the initial crack depth up till a = 0.8 t. It will be assumed that
Na=0.8t is a conservative estimate of the number of cycles to leakage starting from a
postulated initial flaw size. Continued propagation of the fatigue crack up to wall penetration, i.e. a = t, and leakage, is namely expected to occur with significantly increased fatigue crack growth rates. Additionally is linear elastic fracture mechanics (LEFM) expected to be no longer applicable during the final stages of fatigue crack growth up to wall penetration.
For given fatigue growth law parameters and final crack depth, Equation (9)
indicates that � � ��‖��‖��� is constant. This observation avoids performing
ProSACC simulations for each specimen separately, as only one simulation suffices to determine the factor η. The number of cycles to failure, N, for each specimen is then estimated by means of Equation (9) with ‖��‖� � ������������ ��,
5.1.4 Crack closure effects
In the current study, conservatism of the considered flaw tolerance approaches is ensured by not considering crack closure effects on fatigue life, as crack closure tends to reduce the crack growth driving force.
For the standard flaw tolerance approach, the fatigue growth law implementation in ProSACC is based on the formulation in ASME XI [4], where the total extent of the stress intensity factor range is used, even for R ≤ 0:
�� � ����� ���� � ���� �� � �� (10)
Consequently, crack closure effects are not considered in the definition of ΔK. The best-estimate analysis is performed using the ‘fatigue growth law defined by coefficients’ implemented in ProSACC. This implementation uses however an effective stress intensity factor range to account for crack closure effects, which for
R ≤ 0 is given by
�� � ���� (11)
In order to deal with crack closure effects in a similar way as the standard approach and avoid reduced fatigue crack growth rates, the fatigue growth law factor was modified considering Equation (1) and comparing Equations (10) and (11):
��� � �� � ��� � � � � ���� ���� (12)
The modified fatigue growth law factor will be used as input in ProSACC and is assumed common for both investigated crack geometries. The multiplicative factor 6 is derived by approximating the load ratio R with the overall mean value of the pseudo-stress ratio in Figure 3 (b), considering both crack geometries, which
resulted in approximately -0.72. This approximation was enabled as Rσ in
Figure 3 (b) is relatively constant through the thickness.
The considered flaw tolerance approaches initially consider crack closure effects differently, which is remediated by modifying the input for the best-estimate approach, i.e. using C0 from Equation (12) instead of C in Table 3.
Com
5.2
fatig
The experi the total fat two separat where the initiation to during prop Equation ( initiation an5.2.1 Fa
The fractu investigatio relation in the postula is assumed contributio commonly observed fo long or la applicable5.2.2 Ex
In [5], Nexp yielding, with α =2 estimated t the χ ratios and VAG d i.e. χ = 0. strain ampl � Through id experiment a fatiguemparison
gue asses
imental fatig tigue life of te contributio first term, o occur, and pagation of (13) allows nd propagatiatigue life c
ure mechan on are based Equation (9) ated initial fla d negligible on can be tra also neglec for short or s arge fatigue for short or sxperimenta
p was fitted w .89 and β = to be 0.42. It s for VAP an data points h 860. Equatio litude, allowi ����� � �‖ dentification, tal results wi flaw toleranbetween e
ssments
gue life, Nexpthe investiga ons: ��� Ni, refers to d the second the fatigue thus to exa ion.
consumed
nical fatigu d on LEFM ). The estim aw and final compared ansferred to cted due to small fatigue cracks [12] small flaws.al or total fa
with a Basqu ���� = 4.6. Additi t can be note nd VAG is les have a comm on (14) can ing direct co ‖��‖� � � �� � , the factor ith Np is of im nce approacexperime
p, reported in ated piping c ��� ��� � o the numb d term, Np, re crack up till amine the sby propag
ue analyses and aim at mate does howpropagation to the total Ni. Early f the signific e cracks, wh ]. Furthermo
atigue life,
uin relation u � ��‖��‖� ionally the c ed in Table 4 ss than 5%. I mon χ ratio e then be ref omparison wi � �� ��� �‖� � κ is then es mportance in ch and allownts and fr
n Table 1, fr component a �� ber of cycles epresents the l leakage of subdivisionation, N
p performed t estimating wever not in n for a > 0.8t l fatigue lif fatigue crack antly increa en compared ore is LEFMN
exp using the VA ��� coefficient othat the rela It can thus b equal to thei formulated in ith Np given i �‖����� � stimated to b n the investig ws the stud
acture me
rom [5], corr nd can be su s necessary e fatigue life f the piping of total fati d during t Np, using t clude propag . The latter c fe, whereas k propagation sed crack g d to the grow M expected P and VAG of variation o ative differen e assumed th r weighted m n terms of t in Equation ( �‖��‖���� be 1.44. Com gation of the y of the coechanical
rresponds to ubdivided in (13) for fatigue e consumed component. igue life in the current the Basquin gation up to contribution the former n is though growth rates wth rates of d to be not data points, (14) of Nexp was nce between hat the VAP mean value, the m-norm (9): (15) mparison of e margins of onservatism5.2.3 Sensitivity analysis applied to propagation fatigue life
The standard fatigue flaw tolerance approach using recommendations from ASME is expected to yield a conservative estimate of the propagation fatigue life. However no information is available about the variation of the result due to uncertainty in the input parameters. The sensitivity analysis based on a best-estimate analysis, aims at studying the variation in best-estimated fatigue life and investigating which properties of the fracture mechanical assessment have a dominant effect on the estimated fatigue life variation. This parametric procedure is known as Variation Mode and Effect Analysis (VMEA) [13].
The procedure is based on a linearization of a transfer function. To reduce effects of non-linearities, the sensitivity analysis considers the logarithmic fatigue life. For the current investigation four assumed independent logarithmic stochastic variables, �� ��, are considered:
�� � � �����‖��‖��� �� � � �� � � �� �� (16)
where f denotes the transfer function and the variables xj cover the effects on
fatigue life of load (‖��‖�), material (C) and initial crack geometry (a and l). The
effects of other parameters such as for instance the nominal pipe geometry dimensions or the fatigue growth law exponent have not been directly included in the current study. A more explicit expression of the transfer function is obtained by taking the natural logarithm of Equation (9) yielding,
�� � � �� ���‖��‖�� � �� � � ���� ��� � � �� ��� (17)
The expected values (μ’) and standard deviations (s’) of the selected logarithmic variables are considered known. They can namely be expressed as follows for a given variable xj: ��� � ��� ��� �� ���� �� �� (18) ��� � � �� �� � ��� ��� ���� �� �� � ��� (19)
where wj denoted the coefficient of variation of variable xj.
Under the assumption that the input variables are independent, the standard deviation of the logarithmic fatigue life is approximated by means of the Gauss approximation formula: ��� � � ��� � �� ������ �� � � ��� � �� ���� �� � ��� � �� ��� � ��� (20)
where cj denotes the sensitivity coefficient belonging to �� ��. Equation (20) yields
thus an estimation of the coefficient of variation of the propagation fatigue life, i.e.
wN. The sensitivity coefficients are the partial derivatives of the transfer function
with respect to the input variables taken around the expected values of the input values, which for a given variable xj gives:
��� ��� ���
��
Considering Equation (17), analytical expressions of these sensitivity coefficients can be derived for the load or equivalent nominal strain amplitude and the fatigue crack growth law factor C:
����‖��‖��� � �� (22)
��� �� ��� (23)
However for the two remaining variables related to the initial crack geometry no analytical solutions are directly available. The sensitivity coefficients are then estimated numerically by a central difference approximation using ProSACC. Knowing the standard deviation of the logarithmic fatigue life by means of Equation (20), allows the determination of prediction limits. The 90% prediction limits will be determined assuming a normal distribution of the logarithmic fatigue life, and aims at illustrating the variation in estimated fatigue life due to variation in selected input parameters for the best-estimate flaw tolerance approach.
6 Res
Est
6.1
The mean (9). As the considered the results Note that t amplitudes fatigue cra starting fro is explaine flaw toleran C, see Tab sizes are respectively estimate ap size, than differences wall thickn The relativ initial flaw considering contributio where ��, factor C an 0.33. The r then given the initial f computed i the differen difference the differen predominan Table 6 Estim geometries. T Crack geom or initiation location Internal Externalsults
imate of e
fatigue life e exponent flaw toleran for the perfo the reported s expressed in ack growth. om an interna ed by the dif nce approach le 3, and the postulated. y to 0.9 an pproach used the standard s explain the ness of the pi ve effect on w size betwee g the diffe ons) and the d ��ln � �� and �� r nd the initia relative cont by - ln RC / flaw size is c in Table 6 fo nce in fatigu in expected nce in postu nt role. mate of η factor The relative cont metry Flaw to Standa 2.45 1.61expected p
consumed b m is kept c nce approach ormed simula values are n %. A large The results al flaw is slo fference in l hes are relate e extent of si The standa nd 1.9 mm d a lower fat d conservati larger numb ping compon expected fat en performe erence in l definition of �� � ln ��� refer respecti l flaw size d tribution of C / ln Rη, wher computed as or both consid ue growth law propagation ulated initial in Equation (9) tributions of C a η lerance assess rd Best-est 5 64. 1 52.propagatio
by propagatio constant, onlhes and crac ations with a
intended to er factor η in in Table 6 ower when co oad, see Fig ed to the diff imulated fati ard and be of simulated tigue growth ive approach ber of cycles nent using th tigue life of ed flaw toler logarithmic η introduced � ln��� � � � ively to the r dependent in C to the diff
reas the com s 1+ ln RC / l dered crack w factor con n fatigue life flaw size. T ), for different s and the initial fla
Rη sment timate .7 26.41 .5 32.6
on fatigue
on, Np, is es ly the factor ck geometrie a nominal stra be used in ndicates long indicate tha ompared to t gure 3 (a). T ferences in fa igue growth, st-estimate d fatigue cr h law factor h based on s necessary to he best-estim the differenc rance assessm fatigue life d in (8): � ln ��� ln � ratios of facto ntegral �. B ference in lo mplementary ln Rη. The re geometries. ntributes to a e, whereas th The latter costudied flaw tole aw size to the ch Re C � ln �� ln �� 1 34% 32%
e life
stimated usin r η will diff es. Table 6 s ain amplitud conjunction ger fatigue lif at fatigue cr the external f The differenc atigue growth as different analyses co rack growth. and smaller ASME XI o propagate mate approach ces in C and ments, are es e (to obtai �� or η, fatigue Based on Tab garithmic fa relative con elative contri The results i pproximately he remaining ontribution p erance approac hange in ln(η) ar elative contribut initial ln �� ln �� � 66 68 ng Equation fer between summarizes de of 0.01%. with strain fe or slower rack growth flaw, which ces between h law factor t initial flaw orresponded . The best-initial flaw [4]. These through the h. d postulated stimated by in additive (24) growth law able 3, RC = atigue life is ntribution of ibutions are indicate that y 1/3 of the g 2/3 is due plays thus aches and crack re included. tions a and l � ��ln �ln �� � 6% 8%
Figure 4 illustrates the different estimates of fatigue life and related prediction limits for both the conservative standard and best-estimate approaches obtained for the two investigated crack geometries. The estimate of the mean total fatigue life
based on experimental results (Nexp - mean) is defined by Equation (15) and
represented with a solid black curve. The dashed black curve (Nexp - 90%)
corresponds to the lower 90% prediction limit or design curve derived in [5]. Experimental data points relevant for the considered crack geometry are included in Figure 4 using ฮߝୟǡ୬୭୫ఝ౪ ฮ
in Table 5 as equivalent strain measure. The different
symbols used for the experimental data points refer to the different load types. The mean expected fatigue life consumed by propagation obtained with the standard
approach (Np - ASME) and best-estimate approach (Np - VMEA - mean), are
respectively represented with a magenta and cyan solid curve. These mean curves are defined by Equation (9) and the η factors in Table 6.
A first difference between the mean curves for Nexp and Np is the difference in
slope, which is due to the different exponents of the Basquin equations, i.e. m ≠ β.
The estimate of Np using the standard approach inspired on ASME XI [4] is always
situated below the fitted estimate of Nexp, for the considered equivalent strain
amplitudes. This observation is valid for both considered crack geometries and illustrates the extensive conservatism of the standard approach. Using a flaw tolerance approach based on the best-estimate approach preserves conservatism for equivalent strain amplitude less than 0.05 %. For the smaller equivalent strain amplitudes a significant reduction of conservatism is obtained. For larger equivalent strain amplitudes the mean estimate of propagation fatigue life exceeds
the fitted estimate of Nexp, resulting in non-conservatism. It can however be noted
that the extent of conservatism for the best-estimate approach will strongly depend on the slope of the mean curve, i.e on the fatigue growth law exponent m which was assumed equal to the one of the standard approach. Larger exponents are expected to increase conservatism of the best-estimate approach.
When the Np estimates are assumed to represent the total number of cycles of
fatigue life consumed by propagation, then one can compute the ratio λ = Np / Nexp,
for the different crack geometries and flaw tolerance approaches, see Figure 5. The ratio λ indicates then the portion of the total fatigue life consumed by propagation, whereas 1- λ would inform about the portion of the total fatigue life consumed by initiation.
For the standard approach, the total fatigue life consumed by actual crack propagation is less than 10% for the smaller equivalent strain amplitudes, whereas it is approximately less than 20% for the larger equivalent strain amplitudes. For the smallest considered equivalent strain amplitudes the standard flaw tolerance approach predicts that almost the entire total fatigue life is consumed by fatigue crack initiation. For the best-estimate approach, λ is larger as it predicts a larger portion of the fatigue life to be consumed by propagation of the flaw. For equivalent strain amplitudes exceeding approximately 0.05 %, λ approaches unity, which can be interpreted as a negligible contribution of fatigue crack initiation to the total fatigue life. An increase in λ is observed for increasing equivalent strain amplitudes, i.e. fatigue crack initiation represents a smaller part of the total fatigue life when larger loads are applied. These results indicate that a larger portion of the total fatigue life is consumed by initiation for the smaller equivalent strain amplitudes than for the larger ones.
(a)
(b)
Figure 4 Equivalent strain amplitude vs number of cycles for (a) an internal fatigue flaw and (b) an external fatigue flaw.
Figure 5 Equivalent strain amplitude vs portion of total fatigue life consumed by propagation.
103 104 105 106 107 108 0.01 0.02 0.04 0.06 0.08 0.1 0.2 Nexp - mean Nexp - 90% Np - ASME Np - VMEA - mean Np - VMEA - 90% VAP VAG CA VA2 103 104 105 106 107 108 0.01 0.02 0.04 0.06 0.08 0.1 0.2 Nexp - mean Nexp - 90% Np - ASME Np - VMEA - mean Np - VMEA - 90% VAP VAG CA 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.01 0.02 0.04 0.06 0.080.1 0.2
Initiation from inside - Standard Initiation from outside - Standard Initiation from inside - Best-estimate Initiation from outside - Best-estimate
Sen
6.2
The best-es VMEA, wh propagation variables contributio fatigue lif considered were deter variables ar a reduction sensitivity growth law Compariso contributio propagation dimensions the equiva respectively will give t fatigue life coefficient An increas the applied relatively l the load to become pre flaw toleran Although t between th 0.821. This dimensions 90% predic curves in F tolerance a fatigue life noted that study in [5 relatively analysis ba Table 7 Contr Variables j xj 1 ‖�� 2 C 3 ansitivity an
stimate appr hich resulted n fatigue li and their r ons of each v fe for the t crack geom rmined num re negative, n of the fatig coefficient, w exponent. on of τj² in ons of the co n fatigue life s contribute alent strain y 30 and 65% the largest c e. For the cu of variation e of this coe d load will large sensitivthe total var edominant. C nce approach the sensitivi he considere s is due to th s to the varia ction limits ( Figure 4. Th analysis may e with a fact the 90% pr 5], which giv accurately p ased on a pos ributions to the c Coe xj ‖� C a
nalysis
roach was al d in an estim fe. The est respective c variable to th two conside metry differ b merically. A which indica gue life. The which is di n Table 7 onsidered va e, see Equati with about 5 amplitude %. Conseque contribution urrent study n for the equifficient of va induce a si vity coefficie riance of the Consequently h to reduce th ity coefficie d crack geo he relatively ance of logar (Np - VMEA he considere y thus resul tor of almos ediction lim ves support f predict the stulated initia coefficient of va fficient of variat wj 0.2 0.45 0.5 so used as b mate of the st timate is ba coefficients he total varia ered crack by the sensiti All sensitivit ates that an i equivalent s irectly relate for each v ariables to th ion (20). The 5%, whereas present con ently the var
to the varia y, based on ivalent strain ariation in ap gnificantly i ent. In such logarithmic y, accurate lo he variability ents for the ometries, the negligible c rithmic fatigu A - mean) we ed variation lt in a varia t 4 starting mits include t for the best-e
total fatigue al fatigue flaw ariation of propa tion cj -3. -1 -0.2 basis for a se tandard devi ased on the of variatio ance of the l geometries. ivity coeffic ty coefficien increase of th strain amplit ed to the m ariable allo he total varia e results ind s the fatigue nsiderably l iation in equ ation of the controlled fa n amplitude pplications w increased wN a case, the propagation oad descripti y of the estim fatigue flaw e estimates o ontributions ue life. Base ere determin in input for ation of the from the me the data poin estimate flaw e life with w. agation fatigue li Internal flaw j τj² 3 0.436 1 0.203 271 0.018 ensitivity ana ation of the considerati on. Table 7 logarithmic p The result ients for a a nts of the he variable w tude presents agnitude of ws estimati ance of the icate that the e growth law larger contri uivalent strain estimated p atigue exper was assume with more un N, which is relative con n fatigue life ion is of imp mated fatigue w size varia of wN are id of the fatigu d on the esti ed, see the c r the best-est estimated p ean estimate nts of the ex w tolerance a a fracture fe. Exter cj -3.3 -1 -0.258 alysis using logarithmic ion of four 7 lists the propagation ts for each and l, which considered will result in s the largest the fatigue ing relative logarithmic e crack size w factor and ributions of n amplitude propagation riments, the ed to be 0.2. ncertainty in due to the ntribution of will rapidly portance in a e life. ables differ dentical, i.e. ue flaw size imate of wN, cyan dashed stimate flaw propagation e. It may be xperimental approach to mechanical rnal flaw τj² 0.436 0.203 0.017
![Table 4 Ratio of m-norm and β-norm strains from [5] for different load types.](https://thumb-eu.123doks.com/thumbv2/5dokorg/3338756.18455/22.892.169.725.528.584/table-ratio-norm-norm-strains-different-load-types.webp)
