Fracture Mechanics and Crack Growth (Recho/Fracture Mechanics and Crack Growth) || Potential Use of...

Chapter 8

Potential Use of Crack Propagation Laws inFatigue Life Design

For a given initial defect size, the fatigue life of a structural element can bedetermined from the integration of a crack propagation law if the crack initiation lifeis determined by measurements or estimates and, on the other hand, the parametersfor the law of propagation are known. The influence of variable amplitude loadingcan be integrated using the damage accumulation laws. Also, the overload effect thatproduces plasticity at the crack tip can be integrated to evaluate the consecutive ofcrack growth retardation effect.

The influence of different parameters on the fatigue resistance of the elementstudied may be investigated using the calculation of relative fatigue life between twostructural elements of the same type.

The analytical formula of fatigue life, according to different parameters thatinfluence resistance to fatigue, allows a probabilistic analysis of the lifespan to bemade, to the extent that some of these parameters are random in nature.

These are the three types of studies that will be approached in the followingsections.

8.1. Calculation of the crack propagation fatigue life of a welded-joint

If we assume that the crack growth rate in a mechanical component is governedby Paris’ law (see equation [6.79], the calculation of fatigue propagation life(NR – NI) is carried out using equation [6.83], where the geometry of the component,

Fracture Mechanics and Crack Growth Naman Recho© 2012 ISTE Ltd. Published 2012 by ISTE Ltd.

396 Fracture Mechanics and Crack Growth

and its load are represented by (ΔK), the material quality by C and n, the initial cracksize by ai and the final crack length by af. NR can then be calculated subject to anaccurate measurement or estimate of NI. This measurement (or estimate) is directlylinked to the initial crack size considered, see equation [7.12].

8.1.1. Case of a welded cruciform joint

In this case, two types of joints can be observed:

– In the first type (load carrying weld), the force is transmitted by the weld. Wemust remember that the welded joints can, in the first type, have full penetration orcan lack penetration, see Figure 8.1a.

– In the second type (load non-carrying weld), the force is transmitted across thecontinuous plate, as shown in Figure 8.1b.

Root

Foot

Figure 8.1. Different types of welded cruciform joint

In this type of joint, two types of failure are experimentally observed under axialloading (see Figure 8.2):

– A type of failure originating at the root of the weld, from the notch formed bythe lack of penetration. This type of failure is observed in the majority of jointsbelonging to the first type with partial penetration.

– A type of failure at the toe of the weld, from a notch effect that is oftenamplified by some fault or other (undercut, groove, blister, etc.). This mode isobserved in some of the joints belonging to the first type with partial penetration,and in all of the full penetration joints, whether they are of the first or second type.

Potential Use of Crack Propagation Laws 397

Crack at the root of the weld Crack at the foot of the weld

Figure 8.2. Types of cracks in a welded cruciform joint

One simplified calculation of propagation life based on fracture mechanics hasalready been established for the cruciform joint with failure at the toe of the weld[127]. This study has allowed the expression of a simple relationship between thenumber of cycles to propagation (NR – NI) and the variation in axial loading (Pr) inthe case of fatigue stresses under constant amplitude loading.

From equation [6.83] the crack propagation fatigue life is written in thefollowing way:

nrIR PIC

NN −⋅−⋅=− 1 [8.1]

With Pr being the variation in axial force T in daN applied to the joint(see Figure 7.19); by unit width. We will obtain the nominal stress variation, Δσnom,according to Pr by the relationship:

max min/ with :nom r rP t P T TΔσ = = −

where I−is the factor that takes into account the joint’s geometry and the failure

type considered (crack at the root or toe of the weld). It also depends on n.

8.1.1.1. Propagation life from a crack at the root of the welding bead

In this case:

– it has been shown [61] that coefficients C and n of Paris’ law are statisticallylinked by the following average relationship:

52.52 10 (units daN, mm)(67)n

C −= ⋅ [8.2]


– for exponent n, the mean value of 3.6 is proposed (standard deviation ±0.8);

– factor I−

has been established by a parametric study of the stress intensity

factor. An empirical relationship of I−was able to be established in the form:

12_

( ) , t in mm12

n

tI I

+

= ⋅ [8.3]

with:

I = exp (A1 +A2.n)

where:

– A1 and A2 are two coefficients given according to geometric ratios of the jointst/t1, e/t and 2aR/t (see Figure 7.19), using graphs from Figure 8.3;

– the initial crack length is taken as being the lack of penetration (2aR) and the

final crack length as being eaR 222 + .

Figure 8.3. Coefficients A1 and A2 according to geometrical ratios

8.1.1.2. Propagation life for a crack at the toe of the weld

In this case, the statistical analysis gives the following average relationshipbetween the coefficients C and n of Paris’ law:

65.60 10 (units daN, mm)(62.2)n

C −= ⋅ [8.4]

and 3.76 is proposed as the mean value n (standard deviation ±0.87).


Factor_I is shown in Figure 8.4 according to an initial crack length at the weld

toe, ap, where the thickness of the metal sheet is t (Figure 7.19) and the exponent isn = 3.76. The final crack length is taken to be equal to af = t/2.

8.1.1.3. Comments

After calculating fatigue life NR from equation [8.1], we are obligated to estimateor measure the number of cycles to crack initiation (NI). This value is oftenestimated to be negligible [128], especially in welded cruciform joints displaying afailure type from the root of the weld. From experimental results, Yamada et al.[129] state that it can, however, represent up to 40 % of the total lifespan in certaincases, particularly for failure types from the weld toe. All of these estimates dependeffectively on what the authors consider the initial crack length to be.

Figure 8.4. The_I integral according to initial crack length

Note that while a welded joint containing a real crack is detected, for example,over the life of the joint, the calculation for the remaining fatigue life will beconsidered where NI is equal to zero.

To calculate propagation life (NR – NI) in the case of the fracture type from theweld toe run, it is necessary to fix the initial defect size at the toe of the weldthat will be considered as being an initial crack (ap) before hand. This is becauseof the importance of this parameter for the evaluation of (NR – NI).


According to measurements, some codes and authors have given values for the sizeof the defect. Much of this information is summarized below.

The DnV rule [113] recommends the average size of the initial defect to be0.1 mm. Berge et al. [130] suggest a value of 0.015 mm. This highlights the greatdiversity in the estimated values of the initial defect by many authors.

Recently, a statistical study was carried out using DnV for the number of defectsof those types indicated in Figure 8.5. The statistics relate to 829 standard weldobservations at angle or butt welds. In 504 cases, the defects were not visible (ormeasurable). They can therefore be considered to be less than 0.05 mm. For the restof the “measurable” defects, a histogram is given in Figure 8.5, from which it hasbeen deduced that 75% of defects are less than 0.05 mm. These experimental resultsare comparable with the simulated results involved in Figures 7.15 and 7.16. Inaddition, a number of authors – the majority being English and Norwegian – use avalue of 0.1 mm [131] or 0.5 mm [132] for the initial crack length in order tocalculate the fatigue life of welded structures. These values are standard and areused primarily to evaluate fatigue life with sufficient accuracy.

Figure 8.5. Relative statistics for initial defect size

8.1.1.4. Comparison with experimental results and conclusions

For both failure types considered, the results obtained with the proposed formula(equation [8.1]) generally show a good coherence with the test results [121]. Taking


into account the lack of penetration as being the initial crack length, this is a validassumption for interpreting cracks originating from the root of the weld. Incomparison with other tests results, the analytical formula put forward (with NI = 0)expresses the influence that the geometrical parameters and the adopted failure typecan have in a fairly accurate way. The following observations are given as proof:

– fatigue life increases with the increasing thickness of the weld bead and, for agiven plate thickness, t, this increase is similar whether in the formula or in theexperiment;

– fatigue life decreases with increasing plate thickness for a constant e/t ratio.This result has often been found during fatigue tests and the formula given inequation [8.1] takes this into account;

– we will notice that the load-carrying weld cruciform joints, in full penetrationor load non-carrying weld, are very sensitive to the type of welding used. Perhaps itwould be necessary to make a distinction between the diverse welding procedures bysetting them apart based on the conventional values for the initial defect size (seesection 8.3).

The calculation of fatigue life using the given formula (equation [8.1]),compared to the traditional approach that uses one single S-N curve, is the best wayto take into account the influence of certain parameters that are completely ignoredby traditional S-N curve plots. These parameters are essentially the geometry, thematerial and the failure.

Other approaches, also based on the concepts of “fracture mechanics”, wereestablished by different authors, [133], [134] notably that given by Gurney andMaddox [128], which was compared to the current approach in part of a discussionpaper presented at the International Institute of Welding [135].

This approach contributes greatly to the concepts of fracture mechanics andshows a certain interest in the development of simplified rules for fatigue jointverification that, in the most complete way possible, take into account the differentparameters acting on the fatigue life of these joints.

For defects originating at the toe of the weld, the “conventional” size of theinitial crack should be made clear because this has a significant effect on fatigue lifeprediction.


8.2. Study of the influence of different parameters on fatigue life

This study involves the prediction of what we call “relative fatigue life”. It iswritten, using equation [6.83], as follows:

( )

( )∫

∫

Δ

Δ⋅

−−

⋅=*

*

**

***

*/

/

/1/1

f

i

f

i

aa

n

naa

RI

RI

R

R

Kda

Kda

NNNN

CC

NN

[8.5]

with *RN being the fatigue life of the joint (*) compared to that where RN is life, and

C*, n*, *IN , ΔK*, * *andi fa a describe the parameters governing the fatigue life of

the joint (*).

For two welded joints of the same type, made from the same base material andusing the same welding procedure, it can be assumed that C, n and NI/NR areconstant. Taking into account equations [8.1] and [8.3], and equations [7.12] and[7.13] for Σ0 = 1, the expression for relative fatigue life is written:

)21exp()21exp(

**2

1

*** nAAnAA

tt

N

Nnn

nom

nom

R

R

⋅+

⋅+⋅⎟⎟⎠

⎞⎜⎜⎝

⎛⋅

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

Δ

Δ=

−−

σσ

[8.6]

in the case of a welded cruciform joint, where the crack propagates from the root ofthe weld.

It is written:

γγ

σσ

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛⋅⎟⎟

⎠

⎞⎜⎜⎝

⎛⋅

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

Δ

Δ=

−−−

*2

1

***i

i

nn

nom

nom

R

R

a

a

tt

N

N[8.7]

in the case of a welded cruciform joint where the crack propagates from the toeweld. Here, γ is a function of n (see equation [7.13]).

A1 and A2 depend on the geometry of the joint and on the initial crack size(considered as being the lack of penetration of the weld here). They are given in theform of graphs in Figure 8.3.


Equations [8.6] and [8.7] allow the study of this joint’s two failure types and theinfluence of parameters, such as stress variation, the propagation law exponent,initial defect size, failure criterion, plate thickness or any other geometricalparameter for the fatigue life of this joint.

For example, the influence of the thickness of the metal sheet on fatigue life inthe case of a cruciform joint, in which the crack originates at the toe of the weld, iswritten:

12*

*

−+

⎟⎟⎠

⎞⎜⎜⎝

⎛=

n

R

Rtt

N

Nγ

[8.8]

for * *andnom nom i ia aΔσ Δσ= = .

These last two equalities can be justified when the two joints compared aresubject to the same variation in nominal stresses, and when the welds are performedidentically.

Expression [8.8] is concordant with the test results [118] shown in Figure 8.6.Indeed, if n is considered as being the slope of the S-N curve, which is obtained bylinear regression, n is 3.75 in both of the samples considered – one with t = 25 mmand the other with t = 38 mm. Under these conditions, equation [8.8]gives */ 1.92R RN N = . The result is comparable with 2/ * =RR NN obtained fromexperimentation.

Figure 8.6. Comparison between experimental and modelresults for two plate thicknesses


Reference [122], shows the influence that the thickness of the welding bead, e,can have on the fatigue life of the welded-cruciform joint, in which the crackpropagates from the root of the weld in equation [8.6].

Figure 8.7 shows the good relationship between this estimate and theexperimental results given by Booth [118].

Figure 8.7. Comparison between model and experimentalresults for two thicknesses of welding beads

The influence of other parameters can be estimated in these types of weldedjoints using equations [8.6] and [8.7]. Note that this type of analysis is a valuabletool by which to address issues relating to the control of crack evolution and thedesign of structures submitted to fatigue.

To calculate the influence of each parameter and compare it to the experiment, astatistical study associated with a number of assumptions is required.

8.3. Statistical characterization of the initial crack size according to the weldingprocedure

The fatigue tests carried out by Lassen [136] allow the length of the crack to bemeasured according to number of cycles. The experiments are carried out using thesame type of material, the same geometry and four different welding procedures.


The test specimen and an example of the evolution of crack length are shown inFigure 8.8.

Figure 8.8. Test specimen and example of the evolution of crack length

Approaching fatigue behavior by using linear fracture mechanics allows us tocalculate crack length a (mm) according to number of cycles N, by the integration ofa crack propagation law. Let us consider Paris’ law (see equation [6.79]), with

)Tag(πaΔσΔK ⋅⋅= which can be obtained by integration of the following:

n

Ta

Ta

n

Tag

Ta

Tad

nTC

N

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛⋅

⎟⎠⎞

⎜⎝⎛

⋅Δ⋅⎟⎠⎞

⎜⎝⎛ −⋅= ∫−

π

σ0

211 [8.9]

where:

– T= 25 mm is the thickness of the plate;

– Δσ: = 150 MPa is the stress variation;

– a0 = 0.1 mm is the initial depth of the crack under consideration;

– C and n, are coefficients of Paris’ law;

– g (a/T) is geometrical correction factor.


The values of ΔΚ used come from Gurney’s results [137]. The latter provides anaverage solution of g (a/T) and, as a result, does not take into account the localgeometry because it only has a negligible influence.

We calculate the depth of a crack (a/T) according to the number of cycles N. Ineach test where the evolution of the crack length is measured, the difference betweenNcalculated and Ntested is due to the variation in C and n, for which the most likelyvalues are found so that condition [8.10] is verified for the four welding procedures.Therefore a/T corresponds to Ncycles, taking into account the following condition:

[ ]2 is a minimumcalculated testedN Nε Σ= − [8.10]

Ncalculated is determined by using the expression of n according to C,such that the expressed condition [8.10] is ensured. An example of the correlationbetween the experimental results and those based on the method of least squares isshown in Figure 8.9.

Figure 8.9. Curves of a/t according to N

8.3.1. Crack propagation and a proposed relationship between n and C

The curve that connects the logarithm of the extent of the variation in ΔK to thelogarithm of the crack growth rate da/dN has a linear part; n and C are determinedby the secant method. It is suggested that the experimental results are used [136] forsemi-log regression on all the trials linked to four different welding procedures. Thelinear regression is written from [6.79]:


Logda

dN= Log C + n Log ΔK [8.11]

The variation in ΔK is determined from the calculation that uses the geometricalcorrection factor g(a/T), in the sense of Gurney. For each of the four weldingprocedures, we consider the mean crack growth rate da/dN. Taking into account theexperimental results, the equality [8.11] from which we deduce the relationshipbetween coefficients n and C, is estimated using the method of least squares. For agiven mean of a = a , we have:

( )( )

1

1

i i

i ia

a adadN N N

+

+

⎡ ⎤−⎧ ⎫ = ⎢ ⎥⎨ ⎬ −⎩ ⎭ ⎢ ⎥⎣ ⎦[8.12]

These relationships are derived from the processing of experimental data for thefour welding procedures, selected as part of the cruciform welded joint in Figure 8.8.The proposed model allows the relationship between n and C [138], [139] to beobtained, with a better correlation.

The following equation [8.13] offers an average relationship for each weldingprocedure and a good correlation between coefficients C and n. R is the statisticalcorrelation coefficient:

– Welding procedure – SAW (Submerged Arc Welding):

88.855 10 ; R =0.98229.39n

C−⋅= [8.13a]

– Welding procedure – FCAW (Flux-Cored Arc Welding):

82.140 10 ; R=0.99718.13n

C−⋅= [8.13b]

– Welding procedure – SMAW 57 (Shielded Metal Arc Welding):

84.092 10 ; R=0.97520.08n

C−⋅= [8.13c]


– Welding procedure – SMAW 76:

82.263 10 ; R =0.97016.23n

C−⋅= [8.13d]

If all the results for the four procedures are collected, we obtain:

86.069 10 ; R 0.963, (units: MPa and mm)24.64n

C−⋅= = [8.14]

Other relationships between n and C, for the welded joints, are found in thebibliography [122], particularly that proposed by Gurney [140]. These relationshipsare generalized and do not take into account the welding procedure.

8.3.2. Statistical approach and calculation of the initial crack depth, a0

The line for a/T according to N is used for values of a0, between 0.1 and 0.5 mm,while the measurement of a0 that is less than 0.1 mm remains questionable. Astatistical approach to the results following from { }0 ( )a f N= [141] allows us topresent a statistical “model” of initial cracks that corresponds with each of thewelding procedures. The parameters of Paris’ law – n and C – are calculated usinglinear regression. For each of the four welding procedures, it is necessary tocalculate the respective initial crack depths.

The Weibull distribution has two parameters, β and η, whose probability densityf(a0) is given by the following expression:

)(1

0 ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−⋅⎟⎟

⎠

⎞⎜⎜⎝

⎛=

− ββ

ηηηβ aExpaaf

The curves in Figure 8.10 correspond to the probability densities and tohistograms of crack length a0, for each of the four welding procedures. Notice thatthe results are consistent with Weibull’s law of two parameters.

Using fracture mechanics models of n, C and a0 allows us to improve the designto protect against fatigue in the welded joints. It provides a successful outlook forthe evolution of regulations in this domain.


Figure 8.10. Distribution of the initial crack size for welding procedure


8.4. Initiation/propagation coupled models: two phase models

These models are based on the coupling of two periods in the fatigue lifespan:initiation and propagation. They are generally semi-empirical. It is appropriate touse such models when the variation in the applied stress level is low, in the casewhere the experimental results are lacking in terms of the S-N curve. Their use isalso necessary when the period of crack initiation is large compared to that ofpropagation. Many experimental results [136], [280] show that the part where thecrack begins is significant in the welded joints, when an improvement of the weldingbead is completed. In this case, a model that only takes the propagation life intoaccount is questionable.

In what follows, we present a model developed specifically for welded joints[279].

Suppose that the total fatigue life Nt is the sum of the number of cycles toinitiation Ni, modeled by the use of local strain at the radius of the notch, see section7.2 (second approach), and the number of cycles at propagation, Np, are modeled byfracture mechanics using Paris’ simple propagation law:

Nt = Ni + Np [8.15]

The present model is calibrated to estimate the fatigue life of welded joints inwhich the crack originates at the toe of the weld, see Figure 8.11. This is comparedto two test series: one called (database 1), that is obtained with a high level ofvariation in applied stress; and the other called (database 2), where the applied stressis lower.

Figure 8.11. Schematic configuration and crack in welds at an angle


8.4.1. Propagation period

This period is modeled using linear fracture mechanics by adopting Paris’ simple law

of crack propagation, see section 6.3.2: ( ) nKCdNda Δ= with )(afaSK πΔ=Δ , in

which ΔS is the nominal variation in applied stress and )(af is the geometriccorrection factor.

By integration, the number of cycles relative to the propagation period is written:

( )∫Δ

=fa

anp

(a)faS

daC

N0

1

π[8.16]

where a0 is the initial crack length and af is the final crack length. This type ofmodeling is recommended by the regulations in reference [281].

The stress intensity factor, ΔK, can be calculated using the finite elementmethod. The simplified analytical models can also be used, they generally give goodresults. In the case of welded joints, for a particular geometry this factor is expressedas follows [282]:

( ) ( )aMaYaSafaSK kππ Δ=Δ=Δ )( [8.17]

where Y(a) is the geometric correction factor. It refers to the solution of a transversecrack in a plate of infinite dimension under tension. In the case of welded joints, thisgeometrical correction factor takes the following factors into account:

– the shape of the crack, in the case of an elliptical shape, is mainly given by therelationship between its depth, a, and its width at the surface, 2c; and

– the crack propagation from a free surface, that is to say from the weld toe.

Many solutions have been put forward for the geometric correction factor. Thetwo most commonly used are those of Newman and Raju [283] (equation [8.18])and of Gross et al. [284] (equation [8.19]).


Newman and Raju’s equation is written in a simplified manner, if we considerthe evolution of crack growth as being a and in direct tension:

( )2 4

1 2 31.65

1

1 1.464

wa aY a M M M fT Ta

c

⎡ ⎤⎛ ⎞ ⎛ ⎞= ⎢ + + ⎥⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎢ ⎥⎣ ⎦⎛ ⎞+ ⎜ ⎟

⎝ ⎠

[8.18]

with:

– 1 1.13 0.09 /cM a= − ,

– 20.89 0.54

0.2 /M

a c= −

+,

– ( )24310.5 14 1 /

0.65 /M a c

a c= − + −

+,

–3

1

1cos/ /

wfa

w T a c Tπ

=⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

.

and that of Gross et al. is written:

( )2 3 4

1.122 0.231 10.55 21.7 33.19a a a aY aT T T T

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞= ⎢ − + − + ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎢ ⎥⎣ ⎦

[8.19]

where T represents the plate thickness and w its width. These quantities are shown inFigure 8.11.

Mk(a) gives the correction corresponding to local stress concentration due to thejoint’s geometry at the site of the crack. The factor associated with stressconcentration, Mk(a), takes into account the stress distribution at the weld toe andamends the stress intensity factor to take into account the non-uniform distributionof stresses. This stress concentration factor corresponds to the ratio between the


stress intensity factor in a cracked element subject to a non-uniform field of stressesand the same factor with a uniform field.

Several studies show that the relationship between this factor, Mk(a), and thedepth of the crack takes the following form (see Figure 8.11):

( )w

k TavaM ⎟⎠⎞

⎜⎝⎛=

where v and w are constants for a particular geometry.

For all types of transverse welded joints, Hobbacher [285] puts forth twoexpressions to calculate the dimensionless coefficients v and w, according to thegeometry of each type:

( )2

0.8068 0.1554 0.00429 0.0794 tanh h hvT T T

θ⎛ ⎞ ⎛ ⎞ ⎛ ⎞= − + +⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠

( )2

0.1993 0.1839 0.00495 0.0815 tanh h hwT T T

θ⎛ ⎞ ⎛ ⎞ ⎛ ⎞= − − + +⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠

[8.20]

where h is the height of the welding bead; T is the plate thickness; and θ is the weldtoe angle.

For a welded cruciform joint, Hayes et al. [286] give analogous formulas.

Finally, the number of cycles to propagation is expressed as follows:

( )n

a

an

kp S

(a)Y(a).Ma

daC

Nf

Δ= ∫0

1

π[8.21]

This model has the advantage of great simplicity, but does not take into accountall of the possible influential parameters, such as the residual stresses or theexistence of the non-propagation threshold.


8.4.2. Initiation period

In order to predict the initial fatigue life, Ni, the local strain approach at the notchradius is based on the Coffin-Manson equation here, with Morrow’s mean stresscorrection as described in references [287], [288]:

( ) ( ) ( ) cifb

imf NN

E2'2

'2

εσσε +

−=Δ [8.22]

where Δε represents the variation in local strain and σm is the average local stresslevel. b, c, σ'f and ε'f are parameters depending on the material. The law of cyclichardening behavior of Ramberg-Osgood is elastic-plastic:

22

1n

KE′

⎟⎠⎞

⎜⎝⎛

′Δ+Δ=Δ σσε

where K' and n' are constants.

The equation [8.22] is combined with Neuber’s rule as follows:

( )ESK t

2.

Δ=ΔΔ εσ [8.23]

where ΔS is the nominal stress variable and Kt is the stress concentration factorconsidered at the weld toe, here taken from the expression given in reference [289]:

( )⎥⎥

⎦

⎤

⎢⎢

⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+=

469.0572.05121.01

ρθ TK t [8.24]

where θ is the weld toe angle in radians, ρ is the connecting radius and T is the platethickness, see Figure 8.11.

The characterization of variation in stress and local strain is shown inFigure 8.12. The nominal stress S (left) and the local stress (right) are shown by aloading (0-1) and by a loading-unloading (1-2-3). The values of Δσ, Δε and σm arethen determined by a cyclic loading.


Nombre detransitions

1 3 σm

Contrainte locale, σ

Déformation locale, ε

1, 3

2

Contrainte nominale, S

20

Δσ

Δε

Δε = ΔσE

+ 2Δσ2 ′K

⎛⎝⎜

⎞⎠⎟

1

′nε = σ

E+ 2

σ2 ′K

⎛⎝⎜

⎞⎠⎟

1

′n

Δσ. Δε =K t ΔS( )2

E

σ. ε =K t S( )2

ELocal stress, σ

Nominal stress, S

Number oftransitions

Local strain, ε

Figure 8.12. Diagram of the behavior law after stabilization of the hysteresis loop

8.4.3. S-N curve analysis from the coupled model

A coupled model based on equations [8.15], [8.16] and [8.22] was applied inreference [290] in the case of a cruciform welded joint. This, see Table 8.1, wascompared to total life Nt given by the BS regulation [281]. In this model, thetransition crack length between initiation and propagation was taken to be equal to0.1 mm. The local geometry and the mechanical characteristics of the material aregiven in reference [290].

Aside from the good coherence between the model and the regulations, it isnoted that the higher the applied stress, the shorter the initiation fatigue life.

Stress range(MPa)

Ni (cycle)TPM

Np (cycle)TPM

Nt (cycle)TPM

Ni/Nt %TPM Nt F-class

150 1.3×105 3.3×105 4.6×105 28 5.1×105

120 4.3×105 6.4×105 1.1×106 40 1.0×106

100 1.3×106 1.1×106 2.4×106 54 1.7×106

80 6.6×106 2.2×106 8.8×106 75 3.4×106

60 8.1×107 5.1×106 8.5×107 94 8.0×106

Table 8.1. Comparison between the TPM coupled model and the S-N curve of theF-class in the BS regulation [281]


Subsequently, the model is compared to the International Institute of Welding(IIW) regulations and to two series of fatigue tests:

– database 1: a series where the variation in applied stress is higher, and as aconsequence the crack initiation period is short; and

– database 2: a second series with a long crack initiation period and a lowvariation in applied stress.

The calculation of life by the coupled model (TPM − Two Phases Model) wascarried out by the FLAWS procedure (Fatigure Life Analysis of Welded Structures;see reference [291]), by considering a joint with manual arc welding and a residualstress of (400 MPa) equal to the base material’s yield stress. These conditionscorrespond to those in database 2.

Figure 8.13. S-N curve obtained by the TPM model corresponding to the mean regression,compared with the IIW standard curve

Figure 8.13 shows the comparison between the test results, the calculation andthe curve proposed by the IIW. The specific regression used in this case is developedin reference [292]. According to the TPM model, we find that the initiation life of acrack is more than 75% of the total life, while the variation in applied stress is lessthan 80 MPa. Also, the S-N curve is nonlinear on a log-log scale and accuratelypredicts fatigue life.

Compared to the IIW curve, it is clear that the TPM curve is more in line withthe experimental results in the relative “knee region” of the curve. We must alsonote that the TPM curve does not contain an endurance limit.


8.4.4. Coupled model application in the case of variable amplitude loading

According to the current regulations, the prediction of fatigue life under variableloading uses Miner’s linear rule of cumulative damage. This rule does not respond tothe physical phenomenon of fatigue, which is why a high safety margin isrecommended in the regulations. On the other hand, the slope of the S-N curvebeyond 107 cycles is chosen (2n – 1) without the confirmation of experimentalresults, which are often lacking in this zone.

Here the TPM model is adapted to take into account the accumulation of damageby treating the two periods of initiation and propagation separately without havingan endurance limit.

Miner’s linear damage accumulation rule is associated with two damagecoefficients: Da for initiation and Dp for propagation. For a given loading spectrum(Δσi – ni), the summation is performed until Da = 1, then the summation resultcontinues until Dp = 1.

,11.0

ki

aa ii

nD

N== =∑

,11.0i

pp ii k

nD

N

∞

= += =∑

This means that the recovery of the S-N curve is carried out during damageaccumulation to take into account the damage caused during the initiation period,see Figure 8.14.

The crack occurs when both damage factors Da and Dp equal 1, i.e.:

2a pD D D= + =

The association of the TPM model with this “linear bicumulative” replaces theconcept of the endurance limit in S-N curves with the concept of a long initiationperiod. A study carried out in 2004 [293] uses a similar approach to accumulate theloading cycles to fatigue. The authors suggest that a damage factor (DaF) equal tozero exists for an intact specimen.

This factor is equal to 1 when microcracks appear. From this, another factorknown as DF, is measured at the microscopic level, takes its place. The relativedamages from the initiation and propagation periods, Na and NP, respectively, arethen deduced from the following relationships:

/aF adD dN N=


( 1) /F aF pdD H D dN N= −

where H is a “Heaviside” function, ( 1)aFH D − , that is defined as follows:

0 11

1 1aF

aFaF

if DH(D )=

if D<⎧

− ⎨ ≥⎩

Figure 8.14. Modification strategy for the S-N curve used for theaccumulation of damage under variable loading

By integration of the first relationship we get the link between the variations inapplied stress and factor DaF. When DaF = 1, the number of cycles corresponds to Naand factor DF, which is equal to zero, begins to grow. The crack arises when DFincreases by microscopic crack propagation until it reaches DF = 1. This thereforeinvolves the use of “double linear accumulation” of damage.

Depending on the histogram of loading, the block sequence of variation inapplied stress ensures that the TPM model can give more or less damage. This is notpossible if a conventional S-N curve, with a slope of (2n – 1) after the “knee point”,Nc, is used, see Figure 8.14.

According to the TPM model, the blocks of variation in stress are going to showgreater damage if they appear late in the sequence. This is more consistent with thephysics of fatigue accumulation. Consequently, the prediction of fatigue life isdirectly linked to the loading sequence. This phenomenon cannot be described byconsidering a constant slope of (2n – 1).


8.5. Development of a damage model taking into account the crack growthphenomenon

The crack propagation laws represent a significant tool for adjusting a damagelaw for fatigue.

If we consider the propagation law given in equation [6.80] and put forth thehypothesis that the damage factor, D, is a function of crack length, a, according tothe following relationship:

/1

0

000 ⎟

⎟⎠

⎞⎜⎜⎝

⎛

−−

−+aaaa

)Δ(ΔΔ=Df

r [8.25]

equation [8.25] is useful in the evaluation of Δσeff as follows:

0r

r

Δ ΔΔσ Δσ

Δ Deff−

= ⋅−

[8.26]

where:

– D = Δ0 when a = a0⇒ Δσeff = Δσ (intact specimen);

– D = Δr when a = af⇒ Δσeff = ∞ (damaged specimen);

– a0 and af are the initial and final crack lengths, respectively; and

– is a factor that generally depends on the material from which the crackpropagates.

From equation [8.25], we can write:

000

0 a)a(aΔΔΔ-D

a fr

+−⎟⎟⎠

⎞⎜⎜⎝

⎛−

= [8.27]

i.e.:

NdDdaa

ΔΔΔΔΔ-D

Ndda

frr

)(10

0

1

0

0 −⋅−

⋅⎟⎟⎠

⎞⎜⎜⎝

⎛−

=−

[8.28]


Under the assumptions of linear fracture mechanics, we get the relationship:

)(afaK πσΔ=Δ [8.29]

with f(a) being the geometric correction factor.

Taking into account [8.27], f(a) = g (D), and by replacing [8.27], [8.28] and[8.29] in equation [6.80] we arrive at the following relationship:

( )

( )

10 0

0 0

00 0 0

0

r

r f

n

fr

D Δ Δ ΔdD CdN Δ Δ a a

D ΔΔσ π a a a g(D) ΔK

Δ Δ

−⎛ ⎞− −

= ⋅⎜ ⎟− −⎝ ⎠

⎧ ⎫⎧ ⎫⎛ ⎞−⎪ ⎪ ⎪ ⎪⋅ + − ⋅ −⎨ ⎨ ⎬ ⎬⎜ ⎟−⎝ ⎠⎪ ⎪ ⎪ ⎪⎩ ⎭⎩ ⎭

[8.30]

where af, a0, Δ0 and Δr are obtained from the specimen’s intact state and damagedstate.

C, n and ΔK0 are known from fatigue tests on specimens made of the samematerial.

Equation [8.30] represents the damage speed. Integration for:

0 and 0Δ D D N N≤ ≤ ≤ ≤

gives a damage law:

nD f

N= ⎛ ⎞

⎜ ⎟⎝ ⎠

, which is in a nonlinear function.

Integration for:

0 1 1

1 1 1 2

0 nnr

D D ND D N n nΔ

Δ≤ ≤ ≤ ≤≤ ≤ ≤ ≤ +


with n1 + n2 corresponding to failure, gives a damage accumulation law for twostress levels:

1 2

1 2, 0, where is a nonlinear function

n nf f

N N⎛ ⎞

= =⎜ ⎟⎝ ⎠

These integrations were performed analytically for g (D) = constant, n ≠ 1 and 2[72].

Equation [8.30] leads to a nonlinear law of damage accumulation due to theexistence of 0KΔ in the formula, but the presence of 0KΔ cannot physicallyexplain the crack growth retardation effect.

Following this, we deem the delayed effect to be due to the presence of a plasticzone at the crack tip where the material’s relative cracking parameters, namely C, nand 0KΔ , become C*, n* and *

0KΔ .

Figure 8.15. Plastic zone under variable amplitude loading

This amendment taken into consideration at the moment of change in load levelwould lead to this retardation effect. This effect occurs when the loading changesfrom a high level of stress to a low level of stress, see Figure 8.15a; it does not occurin the opposite configuration, see Figure 8.15b. This is largely due to the smallplastic zone generated in Figure 8.15b and to the crack propagating in the area underthe increased load.


From knowing n*, C* and *0KΔ , we can use equation [8.30] with n = n*, C = C*

and 0KΔ = *0KΔ when the crack propagates in the plastic zone. A preliminary

calculation for determining the shape of the plastic zone is essential (see section4.1.6).

8.5.1. Numerical determination of the number of cycles according to crack lengthor vice versa

Considering the propagation law given in equation [6.80], the stress intensityfactor varies linearly following the crack length a between two values, ai and ai+1,

i.e. . iK kΔ Δσ Δ= with .i i ik aΔ α β= + .

On replacing in [6.80] and by integrating 1j ja a a +≤ ≤ and n ≠ 1 and 2, we get:

[ ]

{ } { }

1

1 11 0 0

1(1 )

( ) ( )

j

j

NN

in n

i j i i j i

NC n

a K a K

Δσ α

Δσ α β Δ Δσ α β Δ

+

− −+

= ⋅− ⋅ ⋅

⎡ ⎤+ − − + −⎢ ⎥⎣ ⎦

[8.31]

The values of αi and βi change when there is a change in segment.

If we integrate 0 ,fa a a≤ ≤ we get the fatigue life, NR, as follows:

( ){ } { }1 11 0 0

1( 1)

) ( )

Ri j

n ni j i i j i

NC n

a K a K

Δσ α

Δσ α β Δ Δσ α β Δ− −

+

= ⋅− ⋅ ⋅

⎡ ⎤+ − − + −⎢ ⎥⎣ ⎦

∑∑

For every Δσ given, equation [8.31] allows one of the parameters Nj+1, Nj, aj+1 oraj, to be calculated if the other three are known, and subject to determiningparameters C, n, ΔK0, αi and βi.

Equation [8.31] makes it possible to combine multiple levels of stress variation.This takes into account the plastic zone in terms of retardation effect, which willdepend on the values of n, C and ΔK.


8.6. Taking into account the presence of residual welding stresses on crackpropagation

The residual stresses due to welding are tensile stresses in the vicinity of thecrack. The distribution of residual stresses changes according to crack length. In thestudy by Yong-Bok Lee [294], the distribution of residual stresses is estimated by anumerical method based on two parameters: σmax, the maximum level of residualstresses; and b, the width of the residual tensile stress zone. With regard to thedistribution of residual stresses, a numerical procedure using the energy release rateG is proposed to study the influence of residual stresses on crack propagation.

8.6.1. Distribution of residual stresses

From an initial residual stress field, there is first tension, then compression in theplane of cracking, and then the welding residual stresses relax when the crackpropagates up to around a third of the width of the specimen.

The following formula, relative to the distribution of residual stresses, isproposed by Masubuchi and Martin [295]:

2)(21

2max )(1)( b

y

res ebyy

−×⎥⎦⎤

⎢⎣⎡ −= σσ [8.32]

with:

– σmax: the maximum level of residual stresses;

– b: the width of the residual tensile stress zone; and

– σmax and b changing according to crack length.

From the work of Yong-Bok Lee [294], we can establish two equations of σmaxand b on the basis of a linear regression. These two equations are written in thefollowing way:

max max 0

0

13.360.5

ab b aσ σ= −⎧⎨ = +⎩

[8.33]


with:

– a: crack length;

– σmax0: maximum residual stresses in the case of no crack; and

– b0: width of the tensile residual stress zone when there is no crack.

σmax0 and b0 in equation [8.33] can be measured experimentally.

For the specimen of steel (SS330) shown in Figure 8.16a [294], σmax0 = 256 MPaand b0 = 17 mm, Using formula [8.32], Figure 8.16b gives us a redistribution ofresidual stresses to the right of the weld in the case of no crack and with differentcrack lengths. The maximum value for residual tensile stress is found to be near tothe weld when there is no crack. This value decreases according to crack length. Theresidual stresses become very weak when the crack reaches a third of the width ofthe specimen.

Figure 8.16. Distribution of residual stresses in a welded sample (dimensions in mm)

For a welded steel cruciform joint (Figure 8.17a), a prediction can be madeaccording to σmax and b (equation [8.33]): they vary in the same way as the samplesin Figure 8.16a. Let us suppose that:

max 0

0

1006.5

MPab mmσ =⎧⎨ =⎩


Then, the residual stress distribution is calculated in Figure 8.17b. This figureshows the same trend in the variation of residual stress as Figure 8.16b.

(a) (b)

Figure 8.17. Distribution of residual stresses in a weldedcruciform joint (dimensions in mm)

8.6.2.Method for calculating the energy release rate, G

We use parameter G, the energy release rate, to study crack propagation. Takinginto account residual stresses, G is made up of two parts: G due to exterior loading;and G due to welding residual stresses. To calculate G due to welding residualstresses, we adopt the approach in Figure 8.18, according to which the stressintensity factor K of a crack in an infinite medium under remote loading isequivalent to that determined from the local stress field far from the load where thecrack is found. For more details, see section 4.1.5.4 relative to the Buckner integral,which covers this type of approach. In this way, with a residual stress field in thecase of there being no crack, we can apply this parameter to the crack’s lips toestimate the influence of residual stresses on the energy release rate. At the sametime, the redistribution of residual stresses is considered in the calculations.

Let us study crack propagation of a welded cruciform joint, see Figure 8.17a. Ina numerical simulation, crack propagation is introduced by a familiar calculation inwhich crack length is progressively increased. Using the distribution of weldedresidual stresses shown in Figure 8.17b, we calculate the energy release rate G step-by-step, according to crack length (with ∆a = 1 mm). According to the approachabove, we directly apply the residual stresses due to welding on the lips of the crack.


In other words, for each step we apply the residual stresses of the previous step, seeFigure 8.19. For example, when crack length is an-1, residual stresses exist at thefront of the crack tip; there are no longer residual stresses elsewhere because thecrack opens up completely. Then the crack propagates, ∆a. In the following step, weapply residual stresses on ∆a at the crack’s lip. The distribution of welded residualstresses changes when crack length increases. Therefore, the applied residualstresses are different at each step.

Figure 8.18. The superposition principle, see section 4.1.5.4

Figure 8.19. Diagram showing the application of load on the lips of the crack

8.6.3. Numerical simulation

Numerical simulation is carried out on a welded-steel cruciform joint (seeFigure 8.17a) in which the crack propagates at the weld toe.

The external load applied introduces a stress σG of about 100 MPa at the weldtoe. This is the same level as the maximum residual stress for the weld σmax, seeFigure 8.17b. The determination of σG is carried out according to the Eurocode-3rules [299]: σG is the value extrapolated at the weld toe from two values – one at0.4 T (where T is the plate thickness) and the other at 1 T, see Figure 8.20a. These


two values are calculated by using the finite element method. According to thedistribution of residual stresses given in Figure 8.19, Figure 8.20b gives us themaximum residual stress at the weld toe.

(a) Geometric stress σG (b) Residual stresses

Figure 8.20. Distribution of stresses at the weld toe

Figure 8.21. The energy release rate due to residual stresses, Gres

First of all we calculate Gres, the energy release rate due to welding residualstresses in the case without remote loading. According to the results of Gres (seeFigure 8.21), note that Gres decreases (except in the first step) when the crackpropagates. At the same time, crack growth has an effect on the increase in Gres andon the decrease in residual stresses. The combination of these two effects leads to adecrease in Gres. When the crack propagates to 6.5 mm (about a third of thethickness of the sheet), Gres becomes zero because the welded residual stresses


slowly relax. In this case, the residual stresses are no longer an influence on crackpropagation.

To analyze the influence of residual stresses on crack propagation, we apply theexternal load in mode I. Figure 8.22 shows the numerical results where G0 representsthe energy release rate due to external loading, i.e. the crack is subjected to remoteloading without residual stresses. Gtotal represents the total energy release rate withthe residual stresses due to welding and remote loading.

Figure 8.22. Influence of residual welding stresses on G

The residual stresses are always assumed to be in tension at the crack tip, whichis why the Gtotal curve is higher than that of G0 curve in Figure 8.22. The differencebetween G0 and Gtotal decreases and the curves of G0 and Gtotal converge after theresidual stresses disappear at a = 6.5 mm.

8.6.4. The influence of welded residual stresses on crack growth rate

To analyze the influence of welded residual stresses on crack growth rate, weconsider Paris’s law:

nKCdNda )(Δ=

with:

21 ν−= GEK


in the case of plane strain, see equation [4.92].

In the welded case, Paris’s law is modified and written as follows [294]:

1

n

eff

da C KdN R

Δ=−

[8.34]

where Reff is the fixed stress ratio:

res

reseff KK

KKR

++

=max

min .

Here, Kmin and Kmax are the minimum and maximum factors for stress intensity.

For the as-welded steel joints, the material constants n and C are those from thestatistical analysis, given in equation [8.34]:

3.67n = ,65.6 10

62.2nC

−×=

According to the results of the energy release rate, the number of cycles, N, andthe crack growth rate, da/dN, can be deduced. In the welded case, da/dN iscalculated and modified by Paris’s law (equation [8.34]).

Figure 8.23 shows the increase in crack length according to the number ofcycles. In our example, the welded residual stresses (see Figure 8.17b) are undertension at the crack tip. We observe an increase in crack growth that is linked to thetensile residual stresses perpendicular to the crack. Note that there is a slowing downof crack growth in response to increasing compression stresses.

On the other hand, even though the lifespan is different (see Figure 8.23),the crack growth rate, da/dN, is almost constant according to ΔK calculated, whetherwith or without welding, especially in the case where ΔK takes into account residualstresses and their evolution according to crack length.


Figure 8.23. Simulation of crack growth rate (a-N curves)

8.7. Consideration of initial crack length under variable amplitude loading

The method developed in this section is based on fracture mechanics. It consistsof determining an initial crack length from the results of classical fatigue tests atconstant amplitude in terms of the Si – NRi pair of values.

For variable amplitude loading, fatigue life is calculated by applying the fracturemechanics approach in which the initial crack length is previously defined, seesection 7.2.2.1. It is therefore necessary to establish a network of S-N curves, eachcorresponding to a specific and variable amplitude.

The welded cruciform joint subjected to axial loading (see Figure 8.24) isstudied. The crack is assumed to start at the weld toe.

For this type of joint, initiation life is approximately 5% of the total lifespan[129].

Figure 8.24. Geometry of the welded cruciform joint studied


8.7.1.Method description

According to the Eurocode-3 regulation, each type of detail corresponds to anS-N curve allowing an estimation of fatigue life. This curve is made up of threeparts, each one having its own gradient, see Figure 8.25.

Figure 8.25. S-N curve according to Eurocode-3 [299]

The method is applied through the following four stages, which are explained insections 8.7.1.1 to 8.7.1.4.

8.7.1.1. Stage 1: describing experimental results

The experimental results from references [296] and [297] come from fatiguetests on welded cruciform joints subjected to three loading spectrums (p = 1, 2/3 and1/3). Each spectrum corresponds to programmed block loading where p representsthe overload frequency index that determines the sequence and the levels ofvariation in applied stress with R = Smin/Smax = 0 with Smax = SM. The histogramrelating to each p value is assumed to be Gaussian truncated. The factor p = 1corresponds to constant amplitude loading. Here, p = 1/3 means that one third of thestress variation (S) in the loading spectrum is equal to the stress variation SM.

We consider a sample of Si – Ni from these experimental results.

8.7.1.2. Stage 2: determining the initial crack size

Using the coupled Si – Ni values with constant amplitude loading, the initialcrack size is estimated using the following process.


For each pair of values Ni = NRi and Si = Δσi, we determine a0i fromequation [8.16] where the stress intensity factor (K) is determined by the finiteelement method. The analysis is performed in linear elastic medium in a state ofplane strain.

The values for a0i are then processed statistically to deduce a0 mean value ( 0a ) andthe confidence intervals.

8.7.1.3. Stage 3: determining life under variable amplitude loading

We consider ( 0a ) as being a local characteristic value. This same value is thenused for variable amplitude loading. Fatigue life, NR, is determined fromequation [8.16] after applying amplitude loading cycle-by-cycle, depending on theload in question, by adapting the integral limits in this calculation.

8.7.1.4. Stage 4: establishing S-N curves for variable amplitude loading

An S-N curve is established on the basis of linear damage accumulation (Miner’srule). Figure 8.26 shows the three S-N curves established using the facturemechanics approach applied cycle-by-cycle [298].

Given the conformity between the numerical results and the following approach,we can consider that the assumption of a0 being a local characteristic is verified.

Eurocode-3 recommends a change in slope at 5.106 cycles, see Figure 8.25. Ourresults show that the change in slope occurs at 106 cycles. Any quantitativeconclusion seems premature, however, given the limited number of results availableand their significant dispersion. A specific statistical treatment of the results(coupled Si – NRi values) for NR> 106 cycles appears to be necessary. This treatmentwill lead to a proposed slope for each process of variable loading where p < 1.

The confrontation between the classically considered fatigue approach(reglementary S-N curves and test results in terms of Δσi – NRi) and the fracturemechanics modeling of a fatigue crack growth can in effect form a basis allowingthe fatigue life of a welded component to be estimated, subject to any variableamplitude loading.


Figure 8.26. Comparison between experimental data (points) and the network of curvescalculated using fracture mechanics for a loading pathway (p = 1/3, p = 2/3 and p = 1)

This confrontation leads to the establishment of a system that allows the residualfatigue life of structures in service to be appreciated. It also allows us to improve thecontrol techniques we are using. It represents an alternative and complimentarysolution to the direct use of S-N curves. This provides a fruitful outlook for theregulations in this domain.

8.8. Propagation of short cracks in the presence of a stress gradient

In this section we look at the influence of the notch on the crack-growth behaviorof a crack arising from the notch tip. Here we provide a systematic comparison interms of the stress intensity factor (SIF) between a crack originating from the tip anda real one that has already been established. This study allows a better understandingof the fatigue behavior of welded joints because they have the same type ofgeometrical singularity to varying degrees, according to the notch angle and themixed loading mode. The study of the welded cruciform joint provides varioussuggestions for improving the fatigue design of angle-welded joints at specificangles. In the future, taking into account the degree of singularity in a crackpropagation law, we should be able to modify the law of fatigue damageaccumulation.

The calculation of the lifespan of welded structures is based on the value of thegeometrical stress factor σG calculated at the weld toe. According to Eurocode-3[299] it is obtained by extrapolating a line passing through two points belonging tothe stress gradient of the upper layer of the metal plate (see Figure 8.20a).


The difficulty lies in having to the section of the welding bead (inertial orgeometrical effect) alone take into account and not the local effect, which is stillcalled a microgeometrical effect [301]. In section 7.1, we showed that the limitbetween the overall effect and the inertial effect was in terms of distance from theweld toe.

Many studies have been devoted to the analysis of stresses with strongsingularities, such as cracks. In real structures, however, those with numerous weaksingularities (V-shaped notches) or non-singular stresses (holes, cavities, grooves,etc.) are also crack initiators that eventually lead to the final break. In general, theusual criteria are not adapted to predict rupture or crack initiation.

The notch effect in welded joints introduces two types of singularity: one ofwhich is linked to the coupling radius (Figure 8.27); and the other is linked to thenotch angle (Figure 8.28).

σ σ

Figure 8.27. Notch radius

t

t

2α

F Fθ

a

Figure 8.28. Notch angle

Taking into account the degree of singularity, a crack propagation law shouldmodify the fatigue damage cumulation law. The main objective is therefore toanalyze the stress fields near the weld toe by means of numerical, analytical and


experimental results (asymptomatic development) in order to predict the crack’scritical loading and the path of the crack to optimize the failure criteria.

As reported in section 4.4.2 with equation [4.218], the stress field is controlledby λ, which is the crack’s degree of singularity. Moreover, the crack growth rate of acrack initiated at the tip of the V-shaped notch is dependent on the degree ofsingularity of the notch. This dependence is limited to crack length (l0), which isrelatively small at the notch tip. The length of this crack can be determinedaccording to the mechanical characteristics of the material and the degree ofsingularity of the notch. Fatigue life associated with this small crack represents animportant part in total fatigue life.

Figure 8.29 and Table 8.2 give values for (l0) obtained for a linearly elasticmaterial [302]. This length (l0) varies according to the mixed loading applied,measured by the angle γ, and according to the notch angle, w. The experimentalresults obtained by brittle fracture test in PMMA [302] show a curved cracking pathfrom the notch tip with a crack extension angle, θ. From the length (l0), we find theapproximate crack extension angle corresponding to the case where a real crackwould replace the V-shaped notch.

Figure 8.29. Crack path from the tip of the notch

When we wish to determine the fatigue life of a sample by using a crackpropagation law, the calculation of the SIF is necessary. This calculation dependsdirectly on the stress field near the notch tip. It is completely different from thatclassically calculated for a crack tip.


Table 8.2. Length and crack-extension angle of the crack affected bynotch angle w and a mixed load set affected by angle γ

To determine the SIF of the crack initiated at the V-notch tip, we use theboundary element method. This is a very efficient calculation for modeling thesingular stress field at the tip of the V-shaped notch, like that of a crack. Theamplitudes and degrees of singularity are then calculated very efficiently [178], seesection 4.4.2.

8.8.1. Parametric study of a sample in mode I opening of a notch

In technical literature [303], [304], the lifespan of a fracture in a samplecontaining a crack and another containing a V-shaped notch have been compared.The influence of a notch on a crack emanating from the notch tip, however, islimited by a certain distance. In other words, within this distance, the SIF of a crackoriginating from the notch tip has a value that is notably different to that obtainedfrom a crack of the same length (see Figure 8.30). These two values will be the samebeyond this distance.

In this section, we will determine the characteristic crack length for the twosamples in Figure 8.30, which have identical FIC values.

Let us consider a unit containing a V-shaped notch subject to a loading fatigue ofΔσ = 0.1 MPa (see Figure 8.30a), with w = 40 mm and h = 200 mm. The notch angleand depth are 2β and l, respectively. 0l is the crack length emanating from the notchtip. Figure 8.30b shows a crack with the length 0a l l= + . When the crack originatingfrom the notch tip has the same SIF as crack length a, the characteristic crack lengthac will be obtained.


Figure 8.30. (a) Crack at the tip of the notch (sample-1); and(b) crack without notch (sample-2)

Figure 8.31. Characteristic crack length ac according tothe angle of the V-shaped notch

We note that the SIF varies greatly for small cracks stemming from the notch.When the crack goes beyond the characteristic length ac, the SIF values convergetowards the SIF value of a crack without a notch. The characteristic crack length, ac,varies according to both notch angle and depth. Figure 8.31 shows that ac increaseswith notch angle 2β for a given notch depth, l. In the same way, ac increases withnotch depth for a given notch angle.

When the characteristic length, ac, is determined, the crack growth rate from thenotch can be calculated.


We must remember that ac corresponds to the crack length where the SIF isidentical for a sample containing a crack with a notch (sample-1) and that containinga crack without a notch (sample-2).

8.8.2. Application in the case of a welded joint

Figure 8.32 shows the geometry of a welded cruciform joint [305] subject tofatigue ( 0.1MPaΔσ = ) with a sheet thickness, t, of l = 50 mm. The height of thewelding bead is h = 10 mm and its width, b, varies to form the contact angle2 110 ,120 ,135 ,150 ,160 .β = ° ° ° ° °

The weld toe is considered to be a V-shaped notch with a zero radius. The cracklength initiated from the notch is noted as a. The direction of crack propagation isassumed to be perpendicular to the metal sheet, see Figure 8.32. In this case, loadingmode II does not generate a singular stress field. Only the SIF in mode I has beencalculated in the presence of a notch by a specific method [178].

Figure 8.32. Geometry of a welded cruciform joint

For reasons of simplicity, the mechanical characteristics of the base metal andwelding bead have been considered to be identical in these calculations. Figure 8.33shows the SIF values obtained. Note that the notch effect is very significant for2 135β ≥ ° , and especially when the crack is small.


0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.20.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50ΔK

I(Nmm

-3/2)

a (mm)

0 1 2 3 4 5 6 7 8 9 100.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

2β=110 2β=120 2β=1352β=150 2β=160 2β(Unit:Deg)

Figure 8.33. SIF of cracks emanating from V-notch tip

8.8.3. Conclusion and future extensions

This study has been carried out to show the influence of the stress gradient(introduced by the V-shaped notch effect) on the crack growth rate. The comparisonbetween a cracked specimen and a specimen cracked from the V-shaped notch, interms of relative life, shows that:

– relative fatigue life /V CR RN N increases with notch angle. This increase

becomes very significant beyond 2 130β ≥ ° ;

– for a given notch depth, the ratio /V CR RN N decreases with initial crack length.

The greater the importance of the notch angle, however, the greater the influence ofthe initial crack length on the ratio /V C

R RN N ; and

– for a given initial crack length, the ratio /V CR RN N increases with notch depth.

The numerical results show that an increase in the contact angle of the weldingbead leads to a significant increase in the fatigue life of welded joints.


For reasons of simplicity, in the calculations relative to the welded joint, theconnection radius was considered to be zero and the intrinsic constants of thematerial in the propagation law were chosen to match those of the base material. Thefuture extensions of this study are going to deal with the influence of the connectionradius and the mechanical characteristics of the heat-affected zone on fatigue life.

The calculation of fatigue life in welded joints will eventually coverthree distinct phases of number of cycles:

– the phase of crack initiation 0va at the notch tip;

– the phase of crack propagation 0va until the length of the crack ai, for whichthe notch no longer influences the calculation for the singular stress field; and

– finally the propagation phase, for which the notch no longer has any influence.

Then Paris’s classical propagation law can be used from ai to aR, crack length atfailure.

In the second stage, a specific calculation for singularity is needed. The crackpropagation law is then meant to be controlled by variation in energy release rate atthe notch ( vGΔ ). The following equation will be used to evaluate the lifespan ofthis:

( ) ( )∫∫Δ

+Δ

+=R

i

i

vv

a

am

I

a

am

vvR

Kda

CGda

CNN 11

0

0 [8.35]

8.9. Probabilistic approach to crack propagation fatigue life: reliability–failure

Certain parameters, on which crack propagation fatigue life depends, are meantto be of a random nature. This naturally leads to the development of a probabilisticapproach that allows the uncertainties of whichever calculation of life to be known.Crack propagation fatigue life is expressed by the function:

R IN N N= −

Function N depends on a number of random variables and on variables that aresupposed to be deterministic. The probabilistic problem returns when studyingprobability where the function N is greater than a limited value NC set a priori, i.e.


(N > NC), from which it is concluded that every random aspect of this function isdirectly linked to the deterministic formula.

Depending on the component studied, the development of fatigue life modelsbased on fracture mechanics will allow relationships such as equation [6.83] to bewritten.

Many models of this type have been developed (see equations [8.6]and [8.7]). These models take the following form:

( , , , , , )CR I nom i fN N f a a G MΔσ− = [8.36]

with:

– G: geometric parameters;

– M: material parameters; and

– C: parameters relative to the boundary conditions;

which, under the form of relative fatigue life, would allow the distribution law NR (or*/R RN N ) to be known from the knowledge of the probability density laws for the

supposed random parameters in the relationship [8.36]. This is determined by theuse of simulation methods [142], Cornell-type linearization methods, level IIHasofer-Lind methods [143] or even integral damage indicator methods [144].

It is also possible to perform the integral of the propagation law for

I iN < N N, a a a< < < and obtain the models written in one of the following forms:

i

,

( , a , a, G, M, )

( , , , G, M, )

CC

I nom

nom i I

N N f

a g a N N

Δσ

Δσ

− =

=[8.37]

Applying the probabilistic approach to one of these two equations is of greatinterest when studying the reliability of resistance to fatigue in metallic structures.

If the example of the welded cruciform joint is considered with a failure modefrom the weld toe, equation [8.7] gives an analytical expression of relative life inthis joint. The variable */R RN N N= can be determined from knowing the

variables n, T = t/t*, A = */i ia a and *nomS /= nomΔσ Δσ . These are random


variables. Knowing their statistical distribution laws (see the probability densities)allows the law of distribution, N, to be determined and therefore the relativeuncertainties concerned with fatigue life between the two states of crack growth in awelded joint.

Figure 8.34. Law of distribution of random variables n and ai

The distribution law for the random variable n is given in Figure 8.34a [121]. Itis Gaussian, with a mean of 3.76 and standard deviation of ±0.87. According to thedistribution law of random variable A, we will consider the related statistical data inreference [121]. From the histogram in Figure 8.34b, a Rayleigh law is establishedwith a mean of 0.089 mm and a standard deviation of ±0.088 mm.

The other two variables T and S, on which N depend, can be considereddeterministic values due to their weak dispersion. Nevertheless, their introduction asrandom variables is not a problem. In the current numerical application, we willconsider them to be reliable and equal to 1. This assumes a constant stress variationand a constant thickness of the welded joint:

12n

nN S T Aγ γ− −−= ⋅ ⋅ [8.38]

with γ given as being a function of n (see equation [7.13]).


This relationship can be represented approximately by a linear function by usingthe development of Taylor’s series around the mean values A and n , base variablesA and n, and ignoring the nonlinear terms (Cornell’s linearization method). Wetherefore have:

( )( ) ( )( )n n A An AN NN N ∂ ∂∂ ∂

≅ + − + − [8.39]

with N = the mean of N calculated from A and n ;

,n AN N∂ ∂∂ ∂

= the partially derived means of N, namely:

( ), ( )n AN N∂ ∂∂ ∂

calculated from A and n .

The first-order reliability method [143] allows, the distribution function N to bedetermined from equation [8.39], then the probability density. Figure 8.35 shows theresults of the application of this method in the case of a welded cruciform joint witha crack from the weld toe. These results are in terms of probability density of N, inthe case where T = S = 1 and when A and n are two independent random variables.

This application allows fatigue life in the probabilistic form to be determinedaccording to the different deterministic and random variables on which it depends.There are three interesting lines of development in this type of probabilisticapproach:

– further study of a deterministic model (like equation [8.38]) of the jointstudied;

– collection of the maximum measurement results for different random variablesfound in the deterministic model; and

– performance of the probabilistic method used (linearization, simulation, etc.).


Figure 8.35. Statistical distribution law */ RR NN

The methods of reliable analyses of structures have made great progress in thelast few years, taking into account the uncertainties during the design, assessmentand maintenance of installations. Their application to welded joints subjected tofatigue is particularly fruitful. For example, estimating the residual fatigue life of astructure or a construction detail subjected to a repeated charge, is based on aprobabilistic calculation estimating behavior in terms of security (or failure) andserviceability. Such an approach (see section 6.3.5) needs:

– a description of loading and resistances;

– a physical description of limit states; and

– the definition of the overload criteria of these states.

Different reliable approaches to fatigue can be employed to evaluate the failureof a welded joint. In the following section, we are going to follow the reliablefatigue design in a welded cruciform joint subjected to overload from the outcomeof Lassen’s trials [306]. The deterministic model used is developed in section 8.4and split up into two parts: one taking into account crack initiation; and the othercrack propagation. The reliable model used is that based on the damage indicatorintegral developed in section 6.3.5.2. Three points are covered:

– establishment of a deterministic model for fatigue;

– evaluation of failure probability over time; and

– a study of reliable stability with regards to fatigue life.


8.9.1. Modeling of crack retardation effect due to overloading

We are going to consider modeling the crack retardation phenomenon linked tooverloading. This model, seen in Figure 6.34, can be split into two phases:

– The first phase takes into account the effect of the dependence of the loadingcycle during the application of peak loading; it is joined to the calculation ofminimal crack growth rate (da/dN)min and to crack length at this minimum amin.Minimum crack growth rate is calculated as follows:

( ) nTCR,pic KKKCdNda Δ−−=⎟

⎠⎞

⎜⎝⎛

maxmin

[8.40]

where KCR,pic is the stress intensity factor due to residual compression stresses at thecrack tip following the application of peak loading. The method allowing thecalculation of this parameter is defined by Darcis [238]. For the calculation of amin,this distance is equivalent to one-quarter of the depth of the monotonic plastic zoneduring overload (in the sense of Irwin).

– The second phase takes into account the effect of the dependence of crackgrowth. This is started by determining the length of the crack affected by delay, aD(considered as being equal to twice the depth of the monotonic plastic zone duringoverload (in the sense of Irwin). It also represents the return to initial propagationspeed (before applying excess load), which is equal to:

( )( ) nDeff aKCdNda

,Δ= [8.41]

where ΔKeff,D(a) is the effective variation of the stress intensity factor, whoseevolution depends on the increase in crack propagation until it returns to the initialcrack growth rate.

So total fatigue life is expressed as the sum of the four periods, namely:

– the crack initiation period;

– the period between the initial crack length and the application of an overload;

– life going from the crack length attained at the time of overload until the end ofthe zone affected by the delay through a minimal crack growth rate; and

– finally the period going from the zone affected by delay until the crack.


8.9.2. Evolution of the probability of failure

This analysis is based on the limit state function expressed on the basis of thefatigue model proposed as the sum of four periods. In this way, the failure criterioncan be expressed as follows: NT < N(t), where NT is total fatigue life and N(t) is lifeat t.

From this failure criterion, the limit state function is expressed as follows:

( )

( )

( ) ( )( ) ( )

( )( )( )

( ) ( )( )

min

0

min

( )

( )

11 1

11 1

pic pic

picth

pic D C

pic pic Dth

a a arés

i nn na a effk ai a a a

résn n na a a aeff,D k a

R a da daNC C' KY a M a a E

g X N(t)R a dada

C' CK a Y a M a a E

Δσ

Δσ

Δπ Δσ

Δ π Δσ

+

∞

+

∞+ +

⎛ ⎞−⎜ ⎟+ + +⎜ ⎟⎡ ⎤⎜ ⎟⎣ ⎦⎜ ⎟= −⎜ ⎟

−⎜ ⎟+⎜ ⎟

⎡ ⎤⎜ ⎟⎜ ⎟⎣ ⎦⎝ ⎠

∫ ∫

∫ ∫

[8.42]

where g(Xi) is the limit state function and Xi represents the random variable vector ofthis equation.

All of the parameters in the expression of this limit state function are consideredas being random variables. There are 25 variables and they are defined in greaterdetail by Darcis [238].

Figure 8.36. Evolution of the probability of failure over time


Expression [8.42] allows the probability of failure to be calculated at differentinstances from an approximation based on a first-order reliability methodsimulation. Figure 8.36a represents the evolution of the probability of failure overtime.

The evolution of the probability of failure over time, see Figure 8.36a, providesevidence for the importance of the crack retardation phenomenon and its beneficialeffect on the fatigue life of a welded joint.

8.9.3. Study of sensitivity in terms of reliability

A sensitivity study is carried out for each of the variables in the model developedabove. For this to be done, four series of calculations, associated with four values ofthe variation coefficient for each random variable, are carried out [238]. Thesecalculations are aimed at two important elements:

– fatigue life, for which the variation coefficients and means of the variablesvary;

– reliability, where the means remain deterministic and the variation coefficientsare the only ones that vary.

In this way, these sensitivity studies have allowed us to keep only 11 randomvariables of the 25 previously proposed, namely:

– the overload ratio, Rpic;

– the mean stress variation, E[Δσ];

– the exponent n of Paris’s law;

– the angle of the weld bead, θ;

– the yield strength;

– the stress intensity factor at the opening, Kopen;

– initial crack length, a0;

– thickness of the metal sheet of the welded joint, b;

– the maximum value of welded residual stresses, σmax,0;

– the length of the crack at time of overload, apic; and

– the length of the crack affected by the retardation effect.


From this new proposition, we can recalculate the evolution of the probability offailure using only these 11 random variables. This evolution is rewritten inFigure 8.36b and, as expected, the differences are negligible.

This mechanical reliability study is of great interest, for example, in theevolution of the probability of failure over time associated with a sensitivity studyrelating to fatigue life. This allows a simplification of the probabilistic model so thatwe can see which are the most important statistical characteristics, and finally so thatwe can find the predominant periods in fatigue life.

This study groups together a large number of modeled phenomena, however,which remain very complex and need to be compared to experimental results beforevalidation, correction and going further into such reliable approaches.

8.9.4. Inspection and reliability/failure

“Provisional schedules” are used at the design stage and depend on plannedinspections. They are based on a tree of events (see Figure 8.37a) that is used tore-assess the probability of failure according to inspection time, the measurementscarried out and the choice of repair that follows. Thus, the evolution of theprobability of failure over time is analyzed using the conditional probabilityallowing the next inspection time to be reached, and that from the initial stage.Figure 8.37b provides an illustration of such a prediction.

The reliable calculations in such methods are based on the use ofinitiation/propagation coupled models but not integrating the effect of a one-offevent, like an overload Npic.

When an overload is applied, two distinct cases are possible:

– The crack initiation phase is not stopped (Ni > Npic). In this case we can assumethat the effect of applying an overload before crack propagation is beneficial andleads to the non-rupture.

– Or, the crack initiation phase is terminated (Ni < Npic). In this case a certaincrack length is reached.

Looking at the second case, if we consider a single overload Npic, whereNi < Npic <N2 is the period of reference, a number, nr, of crack lengths is possible atthe time of the application of an overload. These lengths are indicated byk = 1, 2, …, nr, where k = 1 corresponds to the smallest crack length visualized andk = nr corresponds to the element being ruptured. A tree of events for such a strategy


is illustrated in Figure 8.38. The total number of crack length pathways that can beattained at the time of the application that is overloaded is nr.

Figure 8.37. (a) Tree of events; and (b) evolution of the reliability index

Figure 8.38. Tree of events corresponding to the introduction of an overload

In such a strategy, the reliable calculations are generally carried out usingsimulation methods (Monte-Carlo type) rather than using transformation methods(first-order reliability method/second-order reliability method). This is because thesepresuppose a limited number of classes of crack lengths attainable (here, by Npic) andneed the expression of complex conditional events [238]. This type of reliableanalysis can be applied to any other one-off event during the lifespan of a weldedjoint or mechanical component.

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