International Journal of Fatigue 31 (2009) 1882–1888

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International Journal of Fatigue

journal homepage: www.elsevier .com/locate / i j fa t igue

A reliability assessment method for structural metallic component withinherent flaws based on finite element analysis and probabilistic fracturemechanics model

Bo Wu a,b,*, Anglika Brückner-Foit b, Qiang Li a, Lu Chen a, Jinbiao Fu a, Chaohui Zhang a

a College of Materials Science and Engineering, Fuzhou University, University Park, 350108 Fuzhou, PR Chinab Quality and Reliability Laboratory, Institute of Materials Technology, Department of Mechanical Engineering, Universität Kassel, Mönchbergstr. 3, 34125 Kassel, Germany

a r t i c l e i n f o a b s t r a c t

Article history:Received 18 September 2008Received in revised form 2 February 2009Accepted 7 February 2009Available online 20 February 2009

Keywords:Reliability assessmentLifetime distributionFinite element analysisProbabilistic fracture mechanicsFatigue crack growth

0142-1123/$ - see front matter � 2009 Elsevier Ltd. Adoi:10.1016/j.ijfatigue.2009.02.013

* Corresponding author. Address: College of MaterFuzhou University, University Park, 350108 Fuzho38725008; fax: +86 591 22866537.

E-mail addresses: wubo@fzu.edu.cn (B. Wu), a.(A. Brückner-Foit), qli@fzu.edu.cn (Q. Li), cliivfujinbiao0114@yahoo.com.cn (J. Fu), arbor4388@sina.

Reliability assessment is an essential step to promote advanced materials and components into applica-tions. In this paper, a general reliability assessment framework was proposed to predict the lifetime dis-tribution of a structural steel component with inherent flaws. By combining materials information, finiteelement analysis of the stress field and probabilistic fracture mechanics model, the distribution of failureprobability subjected to fatigue loads was predicted. The local failure probability distributions identifythe critical regions of the component visually. Both the global failure probability and the local failureprobability distribution can be considered as essential and fundamental data in structure design and sys-tem maintenance. Focus was placed on the probabilistic fracture mechanics model and fatigue crackgrowth model.

� 2009 Elsevier Ltd. All rights reserved.

1. Introduction

Structural safety and reliability assessment is essential work fordesign and maintenance of modern engineering systems. The sys-tems range from microelectronic and bio-medical devices to largemachinery and structures, as well as civil engineering [1–4]. Forexample, in a commercial project concerning a high efficient gasturbine rotor in European Union, the full project was subdividedinto five work packages, and one of them was to develop a reliabil-ity assessment tool for the steel component made from a nano-pre-cipitate hardened high nitrogen steel. The reliability assessmenttool integrates material properties, finite element stress analysisand probabilistic fracture mechanics consideration. The probabilis-tic fracture mechanics model is generally based on the assumptionthat failure occurs due to the subcritical and catastrophic crackgrowth of crack-like defects introduced during fabrication. Suchdefects are initially present with a given probability, and are foundduring pre- and in-service inspections with a probability depend-ing on their size. The subcritical and catastrophic growth of thesedefects is governed by fracture mechanics considerations, which

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ials Science and Engineering,u, PR China. Tel.: +86 591

brueckner-Foit@uni-kassel.deerson@sina.com (L. Chen),com (C. Zhang).

may also involve material properties that are randomly distrib-uted. Cracks found by inspection determine the component to beretiring or not.

The basic ideas of the reliability assessment tool are presentedin this paper. The probabilistic fracture mechanics model is de-scribed in Section 2. In Section 3, a semi-elliptical surface crackgrowth model subjected to fatigue load is discussed and pro-grammed. The demonstration of the reliability assessment tool isgiven in Section 4 with a four-point-bending bar. A summary andoutlook are presented in Section 5.

2. The framework of reliability assessment tooland probabilistic fracture mechanics model

2.1. The framework of reliability assessment tool

The framework of the reliability assessment tool is sketched inFig. 1. The ABAQUS, PATRAN and ANSYS represent the commercialfinite element analysis software packages, and any of them can beemployed depending on the availability. The ginput.for file is aninterface program developed in the current work to extract thedata of the geometry model and the stress field from the finite ele-ment analysis, which coincides with the commercial finite elementanalysis package mentioned above, and the STAUF is a finite ele-ment analysis post-processor developed in the current work basedon the probabilistic fracture mechanics method.

Nomenclature

a the crack size, in the semi-elliptical crack, a representsthe minor semi-axis (in depth)

a0 the initial crack sizeac the critical size of cracka(c,0)(x,x) the critical initial crack size,c the major semi-axis of the semi-elliptical crack

(on surface)a0/c0 the aspect ratio of the cracks at the beginning of the

lifetimea/c the crack aspect ratio during the fatigue processd the diameter of the precipitate particlesw the width of the platet the thick of the plateA the surface of the component under considerationdA the surface elements under considerationM the average number of crack on the surface area A of the

componentM0 the mean density of surface crack on the componentn the statistically independent cracks on the surface

elements dA of the componentN the cycle numberNn the corresponding cycle numbers of crack size an

Nf the lifetime (the cycle numbers to failure)C, m the parameters of the Paris law of the fatigue crack

growthr the applied stress fieldreq the equivalent stressrmax the maximum fatigue loadrmin the minimum fatigue loadrn, sII, sIII the projections of the stress tensor on the crack planeKI, KII, KIII the stress intensity factors with the mode I, II, IIIYI, YII, YIII the geometric correction factors with the mode I, II, IIIx the orientation of the crack planeg(KI, KII, KIII) the failure criteriongc the critical value of the failure criterionKIeq the equivalent mode I stress intensity factorKIc the mode I fracture toughness of the materialx the location of the cracksfA the probability density functions of the location

Px the probability of a surface crack having the location xX the orientation of the crack relative to a pre-defined

coordinate systemfX the probability density functions of the orientationPx the probability of a surface crack having the orientation

x relative to the global coordinate systemfa0 ðaÞ the probability density function of the initial crack sizePFa the probability of a crack size exceeding the critical

valuePð1ÞF ;Q ð1ÞF the failure probability of a component containing exact

one crackPð1ÞS the survival probability of a component containing

exact one crackPðnÞF the failure probability of a component containing n

statistically independent cracksPðnÞS the survival probability of a component containing n

statistically independent cracksPS,A the survival probability of a component containing an

arbitrary number of surface cracksPF,A the failure probability of a component containing an

arbitrary number of surface cracksPn the probability of having exact n cracks on the surface A

of a component with an average number M of cracksFa0 ðaðc;0ÞÞ the cumulative probability distribution function of the

initial crack size a(c,0)

DK, DKA, DKc the stress intensity factor range in general, indeepest point A and on surface point C, respectively

KA, KC the stress intensity factor in deepest point A and onsurface point C, respectively

E(k) the second elliptical integralz the vector of all random variablesZ1, . . ., Zk the random variables of type (1)ABAQUS, PATRAN, ANSYS the commercial finite element

analysis software packagesginput.for the interface program between the finite element

analysis and finite element analysis post-processorSTAUF a finite element analysis post-processorGUI the graphical user interface

Fig. 1. The framework of the reliability assessment tool.

B. Wu et al. / International Journal of Fatigue 31 (2009) 1882–1888 1883

Generally, the procedure of the reliability assessment can besubdivided into the following steps:

(1) Finite element analysis (FEA) pre-process.(2) Finite element analysis (FEA).(3) Finite element analysis (FEA) output data extract and

rearrangement.(4) Probabilistic fracture mechanics computation.(5) Result graphical presentation.(6) Graphical user interface (GUI).

Steps (3–6) are the so-called finite element analysis post-pro-cessor. In the present paper, the emphasis is put on the probabilis-tic fracture mechanics model.

2.2. Probabilistic fracture mechanics model

Compared with the volume flaws, the surface flaws are believedto be more dangerous in metallic materials, so the surface cracksare considered as the risk inherent flaws in this work.

Probabilistic fracture mechanics [5–7] is used to describe thefailure behaviour induced by inherent cracks. Unstable propaga-tion, i.e. spontaneous failure, occurs if the crack size exceeds a cer-tain critical value ac. The random orientation of a crack in theapplied stress field r in general leads to a mixed-mode load of

1884 B. Wu et al. / International Journal of Fatigue 31 (2009) 1882–1888

the crack with the mode I, II, III stress intensity factors KI, KII andKIII, respectively.

KI ¼ rn �ffiffiffiap� YI ð1Þ

KII ¼ sII �ffiffiffiap� YII ð2Þ

KIII ¼ sIII �ffiffiffiap� YIII ð3Þ

where YI, YII and YIII denote the corresponding geometric correctionfactors, rn, sII and sIII denote the projections of the stress tensor onthe crack plane. Obviously, the values obtained for the projectionsof the stress tensor depend on the orientation x of the crack plane.

The failure of cracks subjected to the mixed-mode load is de-scribed in terms of the failure criterion g(KI,KII,KIII). Failure occursif g exceeds a critical value gc

g � gc ð4Þ

It is common practice to express the failure condition in termsof an equivalent mode I stress intensity factor defined as

gðKIeq ;0;0Þ ¼ gðKI;KII;KIIIÞ ð5Þ

The failure criterion g P gc can be reformulated in terms of themode I fracture toughness:

KIeq � KIc ð6Þ

An equivalent stress req is defined as

KIeq ¼ reqYI

ffiffiffiap

ð7Þ

req depends on the local value of the stress tensor r and on the ori-entation x. Eq. (7) is only valid if req can be considered constantalong the crack size a, i.e. for small stress gradients. In the sourcecode, different criteria for the determination of the equivalent stressreq are prepared for choice by user. From Eq. (7), the critical cracksize ac is determined by

ac ¼KIC

Yrmax

� �2

ð8Þ

where rmax is the maximum stress in the load case.Failure occurs if a, the size of a given crack, exceeds the critical

value ac. The probability of a crack size exceeding the critical valueis given by

PFa ¼Z 1

ac ðx;xÞfa0ðaÞda ð9Þ

where a is the size of the crack in certain location x and orientationx on the bulk material or component, fa0 ðaÞ denotes the probabilitydensity function of the initial crack size a0. The formula of failureprobability described in Eq. (9) can be applied in many damage pro-cess such as stress corrosion cracking, static tensile load and cyclicload. It is the one of basements to compute the failure probability offatigue process in the present study.

In the present work, only homogeneous and isotropic materialsare considered. This implies that uniform distributions are used todescribe the scatter of the location x and of the orientation x of thecracks. With the aid of the corresponding probability density func-tions fA and fX, the probability Px of a surface crack having the loca-tion x and the probability Px of a surface crack having theorientation x relative to the global coordinate system are obtainedas follows

Px ¼ fAðxÞdA ¼ 1A� dA ð10Þ

Px ¼ fXðxÞdX ¼ 12p� dX ð11Þ

where A denotes the surface of the component under consideration,X is the orientation of the crack relative to a pre-defined coordinate

system, and 2p is the normalization factor of the uniform distribu-tion of the random orientations of cracks normal to the surface.

The failure probability of a component containing exactly onecrack with random size, location, and orientation failing becauseof the unstable propagation of the crack is obtained by multiplyingthe probability PFa , Px, and Px and summing over all possible loca-tions and orientations of the crack

Pð1ÞF ¼1A

ZA

12p

ZX

Z 1

acðx;xÞfa0ðaÞdadXdA ð12Þ

The corresponding survival probability is

Pð1ÞS ¼ 1� Pð1ÞF ð13Þ

The survival probability PðnÞS of a component containing n statis-tically independent cracks is given by the probability of simulta-neous survival of the n cracks. Hence, PðnÞS equals the product ofthe individual survival probabilities

PðnÞS ¼ ð1� Pð1ÞF Þn ð14Þ

For the statistically independent infinitesimal surface elementsdA, the actual number n of cracks contained in a component is aPoisson distribution random variable. The probability of having ex-actly n cracks on the surface A of a component with an averagenumber M of cracks is

Pn ¼Mne�M

n!ð15Þ

The survival probability PS,A of a component containing an arbi-trary number of surface cracks is obtained by multiplying the prob-ability Pn with the corresponding survival probability PðnÞS andsumming over all possible numbers of cracks

PS;A ¼X1n¼0

Pn � PðnÞS ð16Þ

Taking into consideration the above two Eqs. (14) and (15) andthe series expansion rule for the exponential function, Eq. (16)leads to

PS;A ¼ expð�M � Pð1ÞF Þ ð17Þ

The corresponding expression for the failure probability is

PF;A ¼ 1� expð�M � Pð1ÞF Þ ð18Þ

In common practice, user inputs the value of the mean densityof surface crack M0, so the average crack number, M in surface areaA is

M ¼ A �M0 ð19Þ

and Eq. (18) is transformed into

PF;A ¼ 1� expð�M0 � A � Pð1ÞF Þ ð20Þ

During the process of fatigue loading, the value of M0 dependsstrongly on the crack initiation mechanism and the incubationtime. If the alloy contains micro-cracks as shown in Fig. 2, a certainpercentage of these cracks will start to grow immediately, whereassome will be blocked by micro-structural barriers and only start togrow after a certain number of load cycles. If the forging procedurecan be improved in such a way that no micro-cracks are introducedin manufacturing routine, the fatigue damage is mostly likely re-lated to the inclusion structure of the material. Most inclusionsare spherical. This implies that any micro-crack, which may be ini-tiated from these inclusions after a certain number of load cycles,needs a much longer incubation time. For the time being, we as-sume that all micro-cracks do start to extend once subjected to fa-tigue loading. This assumption can be changed as soon as we have

0.0

0.2

0.4

0.6

0.8

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70

=FaP

ac

Failure domain

Crack Size with LogNormal Distribution

f(a)

Crack size, a (mm)

Fig. 2. The schematic failure domain in the initial state.

B. Wu et al. / International Journal of Fatigue 31 (2009) 1882–1888 1885

more information about the defect structure of the individual alloy.In the sequential work, the phase of crack initiation has been con-sidered to integrate into the reliability assessment framework [4].Since the present paper focuses on the probabilistic fracturemechanics model, the crack initiation is not discussed in more de-tails, the interested reader can refer to the concerning literatures.

Micro-cracks may have very complicated geometry, which de-pends strongly on the initiation and crack extension mechanismsin the early stage. Even though these phenomena may be interest-ing from a theoretical point of view, a lifetime model for real com-ponents has to rely on comparatively coarse assumptions in orderto infer design rules from an incomplete database. Therefore, thefatigue damage is modelled as a population of semi-elliptical sur-face cracks, which extend in depth and length direction governedby the crack growth law. The determination of fatigue crackgrowth and failure will be discussed in Section 3 in detail.

Suppose that the initial crack sizes show a lognormal distribu-tion, the schematic failure domain of the initial state (i.e. fatigueloading cycle N = 0) is shown in Fig. 2, and the schematic failure do-main during the fatigue damage process is shown in Fig. 3.

Based on the linear elastic fracture mechanics (LEFM) theory,when considering the failure probability during the process of fati-gue loading, Eq. (9) can be transformed into

PFa ¼Z 1

aðc;0Þðx;xÞfa0 ðaÞda ð21Þ

and Eq. (12) can be transformed into

0.0

0.2

0.4

0.6

0.8

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70

mA )K(C

dN

da Δ=

a(c,0)

ac

Failure domain

Crack Size with LogNormal Distribution

f(a)

Crack size, a (mm)

Fig. 3. The schematic failure domain during the cyclic load process.

Pð1ÞF ¼1A

ZA

12p

ZX

Z 1

aðc;0Þðx;xÞfa0 ðaÞdadXdA ð22Þ

where a(c,0)(x,x) is the critical initial crack size, which will reach thecritical size ac after N cycles loading. a(c,0)(x,x) depends on thelocation of the crack, its orientation, material properties and loadconditions. Predefining the cycle number of N, a(c,0)(x,x) can bedetermined by integrating the crack growth law and inserting theresult into the failure criterion, see Section 3. An efficient numericalalgorithm for the solution of this problem has been finished. Thetransformation of the failure domain simplifies the question verymuch since it is very difficult to obtain the probability densityfunction of the crack once the subcritical cracks start to grow underfatigue loading.

Considering all the random variables [8] involved in thefatigue process, the failure probability that a specific crack startsto extend in an unstable manner described in Eq. (22) can betreated as

Pð1ÞF ¼Z

failure domainfzðzÞdz ð23Þ

where z is the vector of all random variables. There are three typesof random variables

(1) Random variables with limited amount of scatter with com-paratively good database. These are the parameters of thecrack growth law (C,m), the fracture toughness (KIc), theaspect ratio of the cracks (a0/c0) at the beginning of the life-time, the applied loads (rmax,rmin) and the componentgeometry. In a probabilistic analysis, these variables can bedealt with in a crude Monte Carlo simulation.

(2) Random variables with broad scatter, good database andstrong influence on the results. These are the micro-crack’slocation x and orientation x. In this study, only homoge-neous and isotropic materials are considered. This impliesthat uniform distributions have to be used to describe thescatter of the location x and of the orientation x of thecracks. An efficient integration algorithm is needed in orderto obtain good estimates of the failure probability. In thiswork, the Gauss quadrature rule for each finite element isemployed.

(3) Random variables on which very limited information isavailable. This is the crack size distribution at the beginningof the lifetime (fa0 ðaÞ). A library of statistical models is pro-vided in the package from which the user could select a suit-able one.

Keeping these considerations in mind, the failure probabilitycan be written as

Pð1ÞF ¼Z

range of Z1 ;...Zk

fZ1 ðz1Þ � . . . fZkðzkÞ �

1A� 1p

Z p

0

ZA

�Z

failure domainfa0ðaÞdadAdXdz1 . . . dzk ð24Þ

where Z1, . . .,Zk are random variables of type (1). Solving the inner-most integral of Eq. (24) leads to

Q ð1ÞF ¼Z

range of Z1 ;...Zk

fZ1 ðz1Þ � . . . � fZkðzkÞ �

1A� 1p

Z p

0

�Z

Að1� Fa0ðaðc;0ÞÞÞdAdXdz1 . . . dzk ð25Þ

where Fa0 ðaðc;0ÞÞ is the cumulative probability distribution functionof the critical initial crack size a(c,0) which will reach the criticalcrack size ac after certain cycles of fatigue loading.

1886 B. Wu et al. / International Journal of Fatigue 31 (2009) 1882–1888

3. The semi-elliptical surface cracks growth model underfatigue loading

In this study, as mentioned in Section 2, the surface flaw is mod-elled as semi-elliptical crack, which is sketched in Fig. 4, where cand a are the major and minor semi-axis of the semi-ellipticalcrack, respectively. w and t are the width and thick of the plate,respectively. And the quantitative relationships are t� a andw� c.

Paris-Erdagon equation [9] is employed to describe the relationbetween the crack-growth rate and stress intensity factor range atdeepest point A and surface point C on the semi-elliptical surfacecrack

dadN¼ CðDKAÞm ð26Þ

dcdN¼ CðDKCÞm ð27Þ

where C and m are the material parameters during the fatigue pro-cess. c and a are the major and minor semi-axis of the semi-ellipti-cal crack, respectively. DKA and DKC represent the stress intensityfactor ranges in deepest point A and on surface point C, respectively.

The stress intensity factor range DK involves the geometry fac-tors. Based on Newman–Raju Equation [10], Fett [11] proposed anextension geometry factor formula for the semi-elliptical surfacecracks, which covers the range 0 6 a/c 6 1.8. The general geometryfactor is given in Eq. (28)

YI ¼ 1:13� 0:1 �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiac� tanhð0:1Þ � a

c

� �10r !

�ffiffiffiffipp

EðkÞ 1þ 0:1 � 1� sin hð Þ2h i

� ac

� �2� cos2 hþ sin2 h

� �14

ð28Þ

where E(k) is the second elliptical integral. Now we can establishthe following system of coupled differential equations comprisingEqs. (29)–(35) to compute the semi-elliptical surface crack growthunder fatigue load [6]

KA ¼ 1:13� 0:00997 � ac

� �112

� �� r �

ffiffiffiffiffiffiffiffiffiffip � ap

EðkÞ ð29Þ

KC ¼ 1:13� 0:00997 � ac

� �112

� �� r �

ffiffiffiffiffiffiffiffiffiffip � ap

EðkÞ � 1:1 � ac

� �12 ð30Þ

Fig. 4. The semi-elliptical surface crack in a plate with thickness t and width w.

ac¼ 1:1mþ c0

a

� �mþ22

1�1:1m � a0

c0

� �mþ22

!" #� 2mþ2

ð31Þ

EðkÞffi 1þ1:464ac

� �1:65� �1

2

; when0<a=c�1 ð32Þ

EðkÞffi 1þ1:464ac

� ��1:65� �1

2

; when1<a=c<1:8 ð33Þ

maxðKA;KCÞ¼KIc ð34ÞdadN¼CðDKAÞm¼C 1:13�0:00997 � a

c

� �112

� ��Dr �

ffiffiffiffiffiffiffiffiffip �ap

EðkÞ

m

ð35Þ

where Eq. (29) represents the stress intensity factor in deepestpoint A, Eq. (30) represents the stress intensity factor on surfacepoint C, Eq. (31) represents the crack aspect ratio, Eqs. (32) and(33) are the second elliptical integrals, Eq. (34) represents failurecriterion, and Eq. (35) represents fatigue crack growth process.

The boundary conditions of Eq. (35) are given in the following

a ¼ a0jN¼0 ð36Þa ¼ anjN¼Nn

ð37Þ

When solving the system of equations containing ordinary differen-tial equation, generally, we separate the variables and convert to itsintegration form

Nn ¼Z an

a0

1C

DKAð Þ�mda

¼Z an

a0

1C� 1:13� 0:00997 � a

c

� �112

� �� Dr �

ffiffiffiffiffiffiffiffiffiffip � ap

EðkÞ

�m

da ð38Þ

Once the initial conditions ðKIc;rmax;rmin; C;m; a0; a0=c0Þ are gi-ven, solving Eqs. (29)–(35), we can get the critical crack size ac un-der the failure condition. Thereafter, we integrate the crack growthEq. (38) from a0 to an (an = ac) and get the fatigue cycles to failure.Due to the complicacy of the system of equations, the numericalsolution is employed. The relationship between the load cycles tofailure and the initial conditions can be established numericallyas follow

Nf ¼ Nf ðKIc;rmax;rmin;C;m; a0; a0=c0Þ ð39Þ

In practice, the lifetime Nf is a user prescribed parameter, andthe corresponding initial crack size a(c,0)(x,x) is what should bedetermined in order to compute the distribution of the failure

1E-3 0.01 0.1 1

1E4

a0/c

0=0.75

1E6

1E5

1E3

=50 MPaminσ=250 MPamaxσ

KIc=50 MPa(m)0.5

c=1.36E-10 m=2.25

Lif

etim

e, C

ycle

Initial crack size, a (mm)

Fig. 5. The schematic relationship between the initial crack size and lifetime.

Table 3The reliability assessment for the four-point-bending bar under cyclic load.

Load case 1 2

Load valuermax ;MPa 250 250rmin MPa 50 50

B. Wu et al. / International Journal of Fatigue 31 (2009) 1882–1888 1887

probability. The iterative step has to be involved in the numericalprogram, so the efficient algorithm is very important to improvethe computing time.

The relationship between the initial crack size and lifetime dur-ing the fatigue process is shown in Fig. 5 schematically.

Our calculation show that the aspect ratio a/c of the crack whenfailure tend to be constant at 0.90 if the initial crack is considerablysmaller than the critical crack size. Once the corresponding criticalinitial crack size a(c,0)(x,x) is determined, the distribution of thefailure probability can be computed easily.

4. The demonstration of the reliability assessment tool

In the primary period, the four-point-bending bar is employedto verify the usability of the reliability assessment tool, althoughit is designed to compute any complex geometry and load case

F F

Fig. 6. The sketch of the four-point-bending bar.

Table 1The parameters of the initial cracks.

Aspect ratio(a0/c0)

Crack density (M0)Numbers/mm2

Crack size

Distributiontype

Mean(mm)

Variance

0.75 50 Lognormaldistribution

0.10 0.0016

Table 2The fatigue crack growth parameter of material (mean).

KIc (MPa �ffiffiffiffiffimp

) C m

50 1.36�10�10 2.25

Fig. 7. The FEM model, displacement and Von Mises stress distribution of the four-point-bending bar in the finite element analysis under unit load. The type of finiteelement is C3D8.

[6]. The sketch of the four-point-bending bar is shown in Fig. 6.The failure probability distributions are predicted under the initialconditions shown in Tables 1 and 2.

The finite element model, displacement and Von Mises stressdistribution of the four-point-bending bar under unit compressstress are shown in Fig. 7. The failure probability distributionsare predicted under different load cycles; see Table 3, Figs. 8 and9. The local failure probability distributions allow us to identify

Lifetime (Nf, cycles, prescribed) 40,000 50,000Local failure probability distribution Fig. 8 Fig. 9Global failure probability, PF,A 0.0064 0.38045Global survival probability, PS,A 0.99352 0.61955

Fig. 8. The local failure probability distribution of the four-point-bending barsubjected to 40,000 cyclic loads, and the global failure probability is 0.0064.

Fig. 9. The local failure probability distribution of the four-point-bending barsubjected to 50,000 cyclic loads, and the global failure probability is 0.38045.

1888 B. Wu et al. / International Journal of Fatigue 31 (2009) 1882–1888

the critical regions of the component visually, although there is nofatigue data to compare for the similar four-point-bending bar ofthe considered alloy. Both the global failure probability and the lo-cal failure probability distribution can be considered as the essen-tial and fundamental data in structure design and systemmaintenance.

5. Summary and outlook

A general reliability assessment framework was proposed. Thelifetime distribution of the bulk metallic material or componentcan be determined by combining materials information, finite ele-ment analysis and probabilistic fracture mechanics model. Theprobabilistic fracture mechanics model and the fatigue crackgrowth model were analysed and programmed. The global failureprobability and local failure probability distribution of a four-point-bending bar under fatigue loading were assessed as demon-stration. However, the crack initiation and short crack growth havenot yet been fully integrated into the current probabilistic fracturemechanics model. And the Monte Carlo simulation should be con-sideration further to deal with the first type random variables onwhich with limited amount of scatter and with comparatively gooddatabase. Last but not least, the exact assessment value dependsstrongly on the accumulation of material database.

Acknowledgments

This study was initially performed under the European Unionfinancial support through Grant NNE5/2001/375 (Nanorotor),which involved six European partners. Numerous helpful and clar-

ifying discussions have been made. The Science Foundations inFuzhou University through Project Nos. 826212, 2007F3045, and2007-XQ-03 are gratefully acknowledged to continue the interest-ing research topic.

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