Fracture Mechanics Based Analysis of the Fatigue Life of Defective Welded Joints

Ľubomír Gajdoš+ and Martin Šperl

Institute of Theoretical and Applied Mechanics of the Czech Academy of Sciences, Prosecká 76, 190 00 Prague 9, Czech Republic

This work was motivated by the endeavour to experimentally determine the influence of crack position on the fatigue life of weldments.Plates were cut from a pipe of X52 pipeline steel ­ 830mm in diameter and 10mm in wall thickness ­ and their contact edges were then preparedfor single-bevel butt welds. The plates were then welded by manual arc welding, and separate specimens ­ 10mm in width and 5mm inthickness ­ were cut from the weldment perpendicularly to the weld bead. The electro-spark method was then used to produce blunt crackdiscontinuities for the initiation of fatigue cracks. The cracked weldment specimens then underwent cyclic loading at reference force levelFmax = 5.5 kN and stress asymmetry ratio R = Fmin/Fmax = 0.1. The test results made it possible to quantify the effects of the size and position ofcrack-like discontinuities on weld fatigue life. [doi:10.2320/matertrans.MT-M2019258]

(Received September 10, 2019; Accepted January 8, 2020; Published April 25, 2020)

Keywords: welded structures, weld discontinuities, arc welding, carbon manganese steel, fatigue, stress intensity factor

1. Introduction

At the fundamental level, welding consists of using heat,pressure, or a combination of both to bond two pieces ofmetal together. Welding requires knowledge of basic physics,chemistry, and metallurgy principles. There are a number ofwelding techniques and a science behind each one of them.The aim of the welding process is simply to produce jointswhich possess the mechanical properties required for theirintended purpose of use. Many experiments were carried outto investigate various parameters in welding for obtainingrequired properties of the welded joints. For example, Sang-Woo Song et al.1) found that in friction stir welding ofdissimilar Al joints the weld formation and mechanicalproperties of the joints depend on both the materialarrangement and conventional welding parameters. Similarly,El-Labban and Mahmoud2) showed that mechanical proper-ties of welds of Al alloy AA6028 and its nanocomposite(reinforced with 2mass% Al2O3) can be varied by a fillermaterial.

In some cases of welding, residual tensile stresses can beinduced in welded joints as a result of shrinkage that is due tothermal contraction during solidification of the weld metal.These stresses can affect the strength and fatigue life ofwelded joints.3) Many studies have been done on fatigueproperties of welded joints. They were aimed at (i) obtainingS-N curves for various combinations of weld geometry,welding conditions and residual stresses, (ii) estimation offatigue behaviour of welded joints, and (iii) fatigue crackinitiation from weld discontinuities and propagation of thecracks. For example, Berge4) investigated fatigue strengthof transverse fillet welds in axial loading with the aim ofdetermining the effect of the plate thickness on their fatiguestrength. A power law decrease of the fatigue strength withthe plate thickness was found for plates of thickness rangingfrom 12.5mm to 80mm. Wide background information onthe fatigue lives of welded joints has been given byMaddox.5) He reviewed the behaviour of welded joints underfatigue loading, the role of significant features of welds underfatigue, factors which affect the fatigue process in welded

joints, and fatigue failure. Papers concerned with fatigueanalysis of welded joints are reviewed in the work of Fricke.6)

Chapetti and Jaureguizahar7) attempted to predict the fatiguestrength of welded joints on the basis of fracture mechanics,including the behaviour of short cracks. They usedexperimental results from literature for a comparison.Similarly, Zerbst et al.8) used fracture mechanics to determinethe fatigue strength of weldments with fatigue cracksinitiating at the weld toes. They also presented somevalidation examples which comprised different weldmenttypes, two different types of steel, different weld geometriesand different stress ratios. Another method for assessment ofthe fatigue life of welded joints was used by Mikkola et al.9)

They used the equivalent crack length method to predictthe crack propagation life of a welded joint from initialdiscontinuity size to the final crack length at fracture. Theproposed method has been successfully applied to exper-imental data. However, owing to only fatigue crackpropagation phase considered in fatigue life calculations,the method yields conservative predictions in high cyclefatigue.

The authors10) investigated the fatigue crack initiation lifeof butt-welded joints in relation to (i) weld geometryparameters like weld toe radius, plate thickness, preparationangle and weld bead flank angle, and (ii) residual weldingstresses. The results of the study enabled the authors todevelop a mathematical model for prediction of the effectsof welding parameters on the fatigue life of welded joints.Similarly, Ohta et al.11) proposed a fatigue crack growthequation for welded structures on the basis of results ofinvestigation into fatigue threshold and high crack growthrate region in several types of welded joints.

After a short excursion into the literature on fatigue life ofwelds it can be concluded that when dealing with weldedstructures exposed to cyclic loading, consideration of theeffects of crack-like discontinuities potentially present in aweld is an indispensable aspect of evaluating safety ofoperation. The most reliable way of determining the impactof discontinuities in welded joints on their fatigue life wouldbe to carry out fatigue tests on specimens with naturaldiscontinuities. However, it should be taken into account thatsuch tests cannot be carried out under equal damage+Corresponding author, E-mail: gajdos@itam.cas.cz

Materials Transactions, Vol. 61, No. 5 (2020) pp. 926 to 934©2020 The Japan Institute of Metals and Materials

conditions since the natural discontinuities in differentspecimens cannot be assumed to be mutually identical. Away to overcome this problem is to use specimens withsimulated discontinuities for an assessment of the effect ofweld defects on the fatigue life of welded joint specimens.We therefore carried out an experimental study in which welddefects were simulated by artificial discontinuities (slits) withpre-defined geometry, dimensions and position with respectto the weld seam.

2. Preparation of Specimens

From a longitudinally welded pipe DN 800 (º 830/10mm,steel X52 according to API) two plates were cut out fromsections on both sides of the weld seam, the dimensions ofthe plates being ³10 © 250 © 130mm (thickness © length ©width). The contact edges of the plates were prepared for asingle bevel butt (a 1/2 V-weld). The plates were then weldedup by manual arc welding (ISO 4063-111) with a coatedelectrode in three layers. A WTU 315.34 welding rectifierwas used as the source for the welding current. The root layerwas made by an ESAB Vamberk E-B-122 electrode (inaccordance with CSN 055020 and CSN 055041) 2.5mm indiameter. This is an electrode with a basic coating specifiedfor welding root beads of pipelines. Chemical composition ofthe weld metal: C 0.08, Si 0.3, Mn 1.0; mechanical propertiesof the weld metal: Re = min 430MPa, Rm = 500­650MPa,A5 = min 20%, KCV = min 130 J/cm2 at a temperature of+20°C. An ESAB Vamberk E-B-241 electrode 3.2mm indiameter (in accordance with CSN 055098 and CSN 055050)was used for welding the filling layer and the top layer. Thisis an electrode with a basic coating. Chemical compositionof the weld metal: C 0.06, Si 0.3, Mn 1.0, Ni 1.0, Mo 0.3;and mechanical properties of the weld metal: Re = 550MPa,Rm = 620MPa, A5 = 24%, KV = 130 J at a temperature of+20°C and 40 J at a temperature of ¹50°C. All electrodesused here were hard dried at 300­350°C for 2 hours. Thebasic material to be welded was not pre-heated.

Separate specimens 5mm in thickness and 10mm in widthwere cut from the weldment in the direction perpendicular tothe weld bead, as is shown in Fig. 1.

Slits for the initiation of fatigue cracks in the welds ofthese specimens were then produced in either the face side ofthe weld or the root side of the weld in the transition zoneusing the electro-spark method. For the relative ease ofmanufacturing a starting discontinuity and positioning it atthe interface between the weld and the parent metal a single

bevel butt was used instead of a single - V butt (V-weld). Theslits were made on a VUMAVJ-4 cutting machine, using thefollowing parameters: medium ­ kerosene, wire ­ Cu 0.3mmin diameter, cutting rate - 0.5mm2/min, speed of motion ofthe wire ­ 50mm/s, peak voltage ­ 240V, pulse width ­200 µs. The width of the electro-spark slits (blunt cracks)was ³0.4mm. Supposing a rounded root of the slit, the rootradius was μ ³ 0.2mm. According to Neuber12) we can arriveat the following theoretical stress concentration factors kt: 3.6for crack depth a = 2mm; 3.5 for crack depth a = 3mm; and3.3 for crack depth a = 4mm. According to the positionand the depth of the electro-sparked blunt crack, thespecimens can be distributed into six groups characterizedby the position and the depth of the blunt crack. Thecharacteristics of the groups are shown in Table 1. Figure 2shows a metallographic photograph of specimen 3-1 (Fig. 3)with a face discontinuity.

3. Experimental Work and Results

The purpose of the fatigue tests was to assess the impactof the depth of a sharp surface discontinuity (a blunt crack)on the residual life of welded joints, and the extent to whichthe position of the crack is decisive. The fatigue tests werecarried out on a high frequency RUMUL vibrophore,working on the resonance principle with a maximum forcecapacity of 20 kN. Tests were performed in the uniaxialtensile loading regime at the reference force level Fmax =5.5 kN with stress asymmetry ratio R = Fmin/Fmax = 0.1.Tensile stress corresponding to the force level 5.5 kN was·t = 110MPa. The resonance frequency was monitoredduring cyclic loading; a decrease in the resonance frequencyindicated the initiation and growth of a fatigue crack from theroot of the starting discontinuity. Cyclic loading continueduntil the depth of the crack reached such a magnitude thatFig. 1 Geometry of single bevel butt-welded plates.

Table 1 Characteristics of specimen groups.

Fig. 2 Metallographic photograph of specimen 3-1 with a face defect(a = 3mm) and a fatigue crack.

Fracture Mechanics Based Analysis of the Fatigue Life of Defective Welded Joints 927

further loading of the specimen in the resonance regime wasnot possible unless the amplitude was drastically reduced.This happened when the crack depth was approximately 70%of the width. The results of fatigue tests on the specimensin groups 1 through 6 are presented in Fig. 3. Specimenno. 5-3 was not taken into account when evaluating theresults because the electro-spark slit was wrongly positioneda few millimeters away from the weld metal - parent metalinterface, i.e. in the transition zone. Specimen no. 1-3 wasalso not taken into account because it was cycled at amaximum load level of Fmax = 7.0 kN due to a mistake madeby the operating personnel.

The authors are aware of residual welding stresses whichcould affect the results of the fatigue tests. However, at thisstage it is worth noting that the technology of manufacturingthe fatigue specimens ensured that longitudinal residualstresses in the weldment were released by cutting thinspecimens (5mm in thickness) perpendicularly to thelongitudinal weld bead. Secondly, eventual transverseresidual stresses which could affect the results of the fatiguetests were reduced due to cutting the slits for the initiation offatigue cracks. Owing to this the residual welding stresses in

the specimens were neglected when analyzing the fatiguelives of specimens on the basis of fracture mechanics.

4. Discussion of Results

4.1 Specimens in groups 1 and 2The specimens in group 1, with a blunt crack 2mm in

depth in the face side of the weld, exhibited a life in excess of106 cycles at reference force level Fmax = 5.5 kN. No fatiguecrack was initiated at the tip of the face blunt crack even afterthe application of more than one million cycles. However, acrack was initiated in the weld root where no starting bluntcrack was present. It is interesting to note that fatigue damagetook place at the tip of the face blunt crack in specimen 1-3which was cyclically loaded at the level Fmax = 7.0 kN (bya mistake of the personnel). This shows that when cyclicloading is applied at the reference force level, the fatiguestrength of a welded joint is not crucially influenced by adiscontinuity 2mm in depth artificially produced in the facepart of the weld. Conversely, the critical location for theinitiation of a fatigue crack is the weld root. In the specimensin group 2, all fatigue cracks were initiated at the tips of blunt

Fig. 3 Results of fatigue tests.

Ľ. Gajdoš and M. Šperl928

cracks situated in the weld root. The average life of thespecimens was ³105 cycles.

4.2 Specimens in groups 3 and 4An average life of ³500 000 cycles was found for the

specimens in group 3 (see Fig. 3), and all fatigue cracks wereinitiated at the tip of face blunt cracks. Unlike for thespecimens in group 1, a blunt crack 3mm in depth is here thedecisive discontinuity for the strength reliability of the weldfor the given loading regime. The specimens in group 4 withblunt cracks of the same depth as for the specimens ingroup 3, but positioned in the weld root, had a life of only³25 000 cycles. In the following text the ratio of the fatiguelife of a weld with a face blunt crack of a certain depth to thefatigue life of a weld with a root blunt crack of the samedepth will be referred to as the life ratio. A comparison of thelives of the specimens in groups 3 and 4 shows that the liferatio is approximately 20. This means that the life of a weldwith a weld root crack 3mm in depth is only 5% of theaverage life of a weld with a discontinuity of the same depthpositioned in the face part of the weld.

4.3 Specimens in groups 5 and 6The life of the specimens in groups 5 and 6 with blunt

cracks 4mm in depth is significantly less than in the case ofthe specimens in groups 1 and 2 and in groups 3 and 4. Thespecimens in group 5 with face cracks in the welds exhibitan average life of ³90 000 cycles, while the specimens ingroup 6 with root cracks in the welds exhibit an average lifeof ³9 000 cycles. The life ratio between them is therefore³10. A comparison between the life ratio for the specimensin groups 5 and 6 and the life ratio for the specimens ingroups 3 and 4 shows that the relative danger (expressed bythe life ratio) of a discontinuity 4mm in depth in a weld rootis lower than for a discontinuity 3mm in depth, but that sucha discontinuity in a weld root is approximately 10 times moredangerous than a similar discontinuity in the weld face. Thus,the life of a weld with a 4mm deep crack in the weld root isonly 10% of the life of a weld with a 4mm deep crack in theweld face.

4.4 Some notes on the experimental resultsThe weld lives for cyclic loading with maximum force

Fmax = 5.5 kN are demonstrated by the points in Fig. 4 inrelation to the position and depth of the blunt crack. Theprobable courses of the average lives of specimens in groups

1 through 6 are represented by solid lines. Figure 4 showsthat fatigue life decreases with increasing crack depth morerapidly for face cracks than for root cracks, so that the liferatio decreases with increasing crack depth. Even a smallcrack in the weld root considerably reduces the life of theweld, while a blunt crack 2mm in depth in the weld face doesnot seem to be an initiatory discontinuity for the rise of afatigue crack. Thus, in the case of a blunt crack 2mm indepth, the specimen behaves as if there were no discontinuityin the weld face. A fractographic examination of fatiguefracture surfaces was carried out to provide more detailedinsight into the significant differences in lives betweenspecimens with a weld face discontinuity and specimens witha weld root discontinuity. The fracture surface of specimens3-2 (A), 4-2 (B), and 4-2 in position 1 (C) is shown in Fig. 5.

Photo C represents the character of striations in position 1of specimen 4-2. A similar photo of striations in specimen3-2 was also taken and striation spacings from these photoswere investigated. It was found that the average striationspacing is roughly 2 © 10¹7m, both for face cracks and forroot cracks. Considering (i) the distance over which a fatiguecrack has to grow in specimens 3-2 and 4-2 to reach thefinal fracture (4mm), and (ii) the striation spacing 2 ©10¹7m, it can be calculated that a face crack grows by thelength of the striation spacing approximately in each twentysecond cycle, whilst a root crack grows by the same distancein almost each cycle. This point will be discussed shortly inparagraph 6.4.

Fig. 4 The lives of welds with blunt cracks in cyclic loading withFmax = 5.5 kN, R = 0.1.

(A) (B) (C)

Fig. 5 Fatigue fracture surface of specimen (i) 3-2 with a face crack (A), (ii) 4-2 with a root crack (B), and (iii) 4-2 (detail) in position 1(C).

Fracture Mechanics Based Analysis of the Fatigue Life of Defective Welded Joints 929

5. Fracture Mechanics Analysis of Experimental Results

5.1 BackgroundWelded joints are neither homogeneous nor isotropic,

hence fatigue crack growth is affected by the properties of theregions surrounding a weld as well as residual stresses.Reference is generally made to three basic regions. These arethe base material, the weld’s heat affected zone (HAZ) andthe weld metal. It is widely accepted that these regions mayhave differing mechanical properties. Nevertheless, for thesake of the argument we shall consider that a fatigue crackin a 1/2V weld grows through the width (the thickness of theinitial weldment) in the HAZ. As for the residual weldingstresses, these can be neglected due to the arguments made inthe previous text.

Now consider a fatigue specimen in the grips of a fatiguemachine (upper part of Fig. 6). Because of the naturalcurvature of the specimen there will be bending stresses ·bbesides the tensile stress · t ¼ P

Bw acting on the specimenduring application of the force P. This means that thereis combined bending and tension of the specimen. Themaximum tensile component of the bending stress will be atthe internal surface of the specimen (as considering the pipe),and the maximum compressive component of the bendingstress will be at the outside surface of the specimen. Thebending moment is given by Pe where e ­ eccentricityis the distance of the specimen’s central cylindrical surfacefrom the plane passing through intersections of the centralcylindrical surface with the edges of the grips (force line). Anequivalent to this model is a single edge notch specimen pin-loaded eccentrically by a tensile force with eccentricity e(lower part of Fig. 6). The length of specimens outside thegrips (l) in tests was approximately 60mm, and the meanpipe radius was R = (D ¹ t)/2 = (830 ¹ 10)/2 = 410mm.

Knowing this data and supposing the same curvature of thespecimens as that of the pipe, the magnitude of theeccentricity e can be readily obtained:

e ¼ R�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR2 � ðl=2Þ2

qAfter substituting for R and l in this expression we can arriveat e µ 1.1mm.

The corresponding bending moment is then M = Pe =5500 © 1.1 = 6050Nmm. The section modulus is given byWb ¼ Bw2

6. After substituting for B = 5mm and w = 10mm

into this expression we can arrive at Wb = 83.3mm3. Thebending stress then will be ·b ¼ M

Wb¼ 72:6MPa.

5.2 Determination of stress intensity factorsThe basic quantity which we will need to determine before

calculating fatigue lives of the specimens is the stressintensity factor KI (opening mode). We shall proceed on thebasis of the principle of superposition, i.e. we shall determineindependently the KI factor for the tensile stress and thenthe KI factor for the bending stress. To distinguish betweenstresses acting on the specimens we shall suffix K by t fortension and by b for bending. Finally we shall sum themagnitudes of Kt and Kb to obtain the K factor for combinedloading.

A schematic representation of tensile loading and bendingloading of an SEN specimen is shown in Fig. 7.5.2.1 Stress intensity factor for tension loading

According to Tada and Paris13) the Kt factor for SENspecimens loaded over the width of the specimen (Fig. 7(A))can be expressed by eq. (1)

Kt ¼ffiffiffiffiw

p ffiffiffiffiffiffiffiffi³a

w

r· t ftða=wÞ ð1Þ

where

ftða=wÞ ¼ gða=wÞ 0:752þ 2:02a

w

� �þ 0:37 1� sin

³a

2w

� �3� �ð2Þ

The function g(a/w) in (2) is given by eq. (3):

gða=wÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2w

³atg

�³a

2w

�s

cos

�³a

2w

� ð3Þ

The accuracy of eq. (1) is better than 0.5% for any ratio a/w.5.2.2 Stress intensity factor for bend loading

Again, according to Tada and Paris13) the Kb factor forSEN specimens (Fig. 7(B)) can be expressed by eq. (4)

Kb ¼ffiffiffiffiw

p ffiffiffiffiffiffiffiffi³a

w

r·bfbða=wÞ ð4Þ

where

Fig. 6 Schematic representation of a specimen in the grips of the fatiguemachine (upper part of the figure) and substitution of the specimen by asingle edge notch (SEN) specimen off-centre pin-loaded in tension (lowerpart of the figure).

(A)

(B)

Fig. 7 Single edge crack tension specimen (A) and single edge crackbending specimen (B).

Ľ. Gajdoš and M. Šperl930

fbða=wÞ ¼ gða=wÞ 0:923þ 0:199 1� sin³a

2w

� �4� �ð5Þ

and g(a/w) is given by eq. (3).The inaccuracy of eq. (4) is less than 0.5% when bending

stress is linearly distributed across the thickness passing zeroat the centre plane of the specimen, as shown in Fig. 7(B).The authors are aware of the use of Kb factor for a crack inthe compressive stress field. This makes no sense whileno tensile stress is acting. However, there is a tensile stress·t = 110MPa which acts on the welded specimens, so thatafter adding this tensile stress to the bending stress (accordingto the principle of superposition) the total stress intensityfactor can rise above the threshold value for the growth offatigue cracks.

As a matter of interest we can compare the values of thestress intensity factor, as determined by the above equations,with those published by Gross and Srawley.14) Thecomparison is made in Table 2 for SEN specimens withwidth w = 10mm, thickness B = 5mm and force P =5 500N for the following ratios: e/w = ¹0.2;¹0.1; 0.0;0.1; 0.2 and a/w = 0.1; 0.2; 0.3; 0.4.

The authors14) used a boundary value collocationprocedure in conjuction with the Williams stress function todetermine the stress intensity factor Kb for single edge cracksof various depths in specimens subjected to pure bending andthen they superposed the values of the stress intensity factorKt for tension loaded SEN specimens, as determined from theresults of authors.15) As follows from Table 2, our resultscompare quite well with the results of the work14) althoughthey are consistently lower by several tenths of MPa

ffiffiffiffim

p.

6. Determining Fatigue Life

6.1 The procedure usedNow we shall attempt to determine the fatigue lives of

welded specimens with sharp slits according to Fig. 3 andwill compare them with experimental results. In principle,the lives of all groups of specimens are given by the sum ofthe number of cycles needed to initiate a fatigue crack in theslit root, Ni, and the number of cycles needed to propagatethe crack to the critical (final) depth, Np. If we denote thetotal number of stress cycles to fracture by Nf , it can bewritten:

Nf ¼ Ni þ Np ð6ÞThe number of cycles Ni can be determined for a certaingeometry of a particular notched specimen and a certain

range of stress intensity factor "K acting on the specimen,assuming that the notch behaves as a sharp crack. Jack andPrice16) found that the number of cycles Ni at a particularstress was independent of the root radius μ of the notch (ora slit) up to a value of about 0.25mm. This critical value μowas independent of notch depth a and stress ·. Tests onspecimens which had been pre-cracked, heat treated and re-tested showed that Ni was the same for a sharp crack as for anotch with μ < μo, providing the depth a was the same. Fora notch with μ ¯ μo the number of cycles Ni was found to bea function of the stress intensity factor range "K only. Theresults presented in the work16) in graphical form can betransformed into eq. (7a) or eq. (7b), depending on the unitschosen.

logNi ¼ 8:760564� 4:11438 logð�KÞ (7a)

logNi ¼ 8:591455� 4:11438 logð�KÞ (7b)

For a notch with μ > μo the number of cycles Ni dependedon both the range of the stress intensity factor "K and theroot radius μ. The graphical form of the results16) can betransformed into eq. (8a) or eq. (8b), depending on the unitschosen.

logNi ¼ 9:75458� 3:99568 log�Kffiffiffiμ

p (8a)

logNi ¼ 12:397011� 3:99568 log�Kffiffiffiμ

p (8b)

In eqs. (7a) and (8a) "K is in [MPaffiffiffiffim

p] and μ is in [mm]; in

(7b) and (8b) "K is in [ksiffiffiffiffiin

p] and μ is in [in].

The results of Jack and Price’ investigation into theinitiation of a crack at the tip of a sharp notch related toloading specimens in tension. However, in our analysis wealso used the results for combined tension and bending byconsidering the total range of the stress intensity factor "K asa sum of the stress intensity factor range for tensile loading"Kt and the stress intensity factor range for bending loading"Kb. Taking into account that the radius μ of the slits was³0.2mm, we only used eq. (7a).

The number of cycles for crack propagation Np can bedetermined on the basis of the Paris law

da

dN¼ Cð�Kef Þm ð9Þ

where "Kef, effective stress intensity factor range, is givenby (10)

�Kef ¼�K

ð1� RÞ£ ð10Þ

Table 2 Magnitudes of the stress intensity factor K [MPaffiffiffiffim

p] for SEN specimens for various values of the load-eccentricity ratio e/w and

the relative crack length a/w.

Fracture Mechanics Based Analysis of the Fatigue Life of Defective Welded Joints 931

In eq. (10) the stress intensity factor range "K is given bythe difference Kmax ¹ Kmin; here R stands for the cycleasymmetry ratio, given by the ratio Kmin/Kmax, and £ is aconstant, the magnitude of which is approximately 0.7 formost steels. In all fatigue tests the cycle asymmetry ratio wasR = 0.1.

6.2 Stress intensity factor range "K6.2.1 A root crack

�K ¼ Kmax �Kmin ¼ ð1� RÞKmax

Kmax ¼ Kt þKb ¼ffiffiffiffiw

p ffiffiffiffiffiffiffiffi³a

w

rð· t ft þ ·b fbÞ

hence

�K ¼ ð1� RÞ ffiffiffiffiw

p ffiffiffiffiffiffiffiffi³a

w

rð· t ft þ ·b fbÞ ð11Þ

6.2.2 A face crack

�K ¼ Kmax �Kmin ¼ ð1� RÞKmax

Kmax ¼ Kt �Kb ¼ffiffiffiffiw

p ffiffiffiffiffiffiffiffi³a

w

rð· t ft � ·b fbÞ

�K ¼ ð1� RÞ ffiffiffiffiw

p ffiffiffiffiffiffiffiffi³a

w

rð· t ft � ·b fbÞ ð12Þ

As mentioned earlier, only loading cycles characterized bya higher stress intensity factor range "K than that of thethreshold value "Kthres can drive a crack ahead. In thecalculations, the threshold value "Kthres = 7MPa

ffiffiffiffim

pwas

considered.

6.3 Paris law parametersIt is very likely that the fatigue cracks in the welded

specimens under investigation propagated through the heataffected zone. To comply with this assumption we have usedParis law parameters for HAZ in X52 steel welds publishedby Zahiri et al.17) The magnitudes of these parameters wereas follows: C = 1.13 © 10¹9 and m = 3.25. Using these Parislaw parameters we determined the number of stress cycles Np

needed for a fatigue crack to propagate to a limit depth of0.7 © wall thickness = 0.7 © 10 = 7mm.Calculation procedure:

Calculation of "K according to eq. (11) or eq. (12)

Calculation of "Kef according to eq. (10)

Calculation of Ni by eq. (7a) +++ (notch root radiusμ = 0.2mm)

Calculation of Np from eq. (9)

By integrating eq. (9) we get

Np ¼Zacrao

da

Cð�Kef Þmð13Þ

An approximate solution to eq. (13) was obtained by theRiemann sum:

Np ¼Xni¼1

�a

Cð�Kef ; iÞmð14Þ

where

�a ¼ acr � aon

ð15Þ

acr ¼ 7mm C ¼ 1:13� 10�9 ao ¼ 2; 3; 4mm m ¼ 3:25

In the calculations the number n was chosen to be 500. Theresults of the calculations are presented in Table 3. To get abetter idea of the results obtained, diagrams were constructedof the fatigue lives in logarithmic scale in relation to crackdepth. They are presented in Fig. 8 for face cracks and inFig. 9 for root cracks. Experimental fatigue lives arerepresented here by diamonds and calculated lives arerepresented by triangles.

For 2mm deep slits at the face part of the weld the fatiguelives in Table 3 are denoted as >106 which is in keepingwith both the experimental data and the calculated results.However, in Fig. 8 the results are plotted as 2 © 106 in order

Table 3 Calculated fatigue lives.

Fig. 8 Representation of lives Nf for face cracks.

Fig. 9 Representation of lives Nf for root cracks.

Ľ. Gajdoš and M. Šperl932

to assign them a concrete value. As follows from Fig. 8 andFig. 9, the calculated results fall within the range ofexperimental points with the exception of the specimenswith 4mm deep root slits. In that case the calculated lifefell beyond the lower bound of the extent of experimentalresults. The results represented in Fig. 8 and Fig. 9 showgood agreement between predicted (calculated) fatigue livesof specimens and experimental lives.

6.4 A note to the relation between striation spacing andcrack growth rate

First it should be said how we arrived at the stating inparagraph 4.4 that a face crack grew by the length of thestriation spacing s = 2 © 10¹7m approximately in eachtwenty second cycle whilst a root crack in almost eachcycle. This was done on the basis of two assumptions: (i)striation spacing keeps constant over the whole distance froma0 = 3mm to acr = 7mm; (ii) the number of cycles Np forpropagation of a crack from the length a0 = 3mm to acr =7mm can be expressed as the difference between theexperimental life Nf and the number of cycles for initiationof a crack Ni, calculated according to eq. (7a), i.e. Np =Nf ¹ Ni. By dividing Np by the number of striations at thedistance (acr ¹ a0) we can obtain 22.8 cycles for the striationspacing s = 2 © 10¹7m for the specimen 3-2 and 1.3 cyclefor the specimen 4-2.

An idea that a fatigue crack should grow cycle-by-cycleat low growth rates is not now a universal view for that anevidence has been presented which shows that in Al alloysand steels the growth of fatigue crack is intermittent in naturebelow a growth rate of 2 © 10¹7m/cycle. Among paperswhich support this view the paper of McEvily andMatsunaga18) and also the paper by Nedbal et al.19) can becited. In the former publication the authors refer to thediagram in Fig. 10 which is based upon tests conducted byRoven and Nes.20)

Striation markings can be observed here over the growthrate range of 10¹10 to 10¹7m/cycle. An occurrence of idlecycles at low fatigue crack growth rate follows from thediagram. The latter paper19) provides an undoubtfullexperimental evidence on idle cycles which have only alatent effect on the crack tip and they do not directlycontribute to the crack growth.

7. Conclusions

This work investigated the effects of technologicaldiscontinuities in welds on the fatigue life of welded jointsin structures. Weld discontinuities were simulated by bluntcracks produced by the electro-spark method. Due to thetechnology used in manufacturing the specimens from theweldment the residual welding stresses were largely reducedso that their effect on the fatigue life of the specimens wasnegligible. Because the specimens were made from a curvedweldment they kept their natural curvature given by the meanradius of the pipe. This curvature caused additional bendingof the specimens during tensile loading. A single edge notch(SEN) specimen pin-loaded eccentrically by a tensile forceis equivalent to a curved specimen. The curvature of thespecimens can account for (i) increased fatigue lives wheninitiation slits are situated in the compression part of aspecimen and (ii) for reduced lives when initiation slits aresituated in the tension part of a specimen. Because the weldwas accomplished in such a way that the weld root appearedon the inside surface of the curved plates to be welded, theroot was situated in the tension part of the specimen andthe face of the weld was situated in the compression part ofthe specimen. Application of the fracture mechanics approachto the curved welded specimens subjected to cyclic loadingmade it possible to calculate the fatigue lives of specimenswith initiation slits in either the face part of the weld or theroot part of the weld. The calculated lives compared wellwith experimental results.

It was shown that application of fracture mechanicsprinciples can predict reasonably well the fatigue life ofwelded joints subjected to cyclic loading providing there areno residual welding stresses in the welded joint and the crackpropagates through a “single” structure of the weld (e.g. heataffected zone, weld metal) characterized by known Paris lawconstants. In engineering practice the cause for differences inexperimentally observed fatigue lives of welded joints andlives calculated on the basis of fracture mechanics should belooked for in residual welding stresses.

The results of the paper can be generalized to some extent.The level of residual welding stress in the weldment can bedetermined by a simple method which requires only Paris lawparameters to be determined experimentally on stress relievedspecimens for the weld microstructure through which a fatiguecrack is supposed to be propagated. First, a fatigue crack ofa certain length a0 is initiated by stress cycling. After that afatigue crack propagation test follows to increase the cracklength a0 by a distance "a applying the stress amplitude ·aand the cyclic stress ratio R = ·min/·max. The number of cyclesneeded to extend the crack length a0 by a distance "a isdenoted by Nexp. By substitution of Paris law parameters(C,m) to integral (13) with accounting "K relations withunknown residual welding stress ·res it is possible to determinesuch value of ·res that the integral (13) obtains the value Nexp.

Acknowledgements

The authors are grateful to the Technological Agency ofthe Czech Republic for supporting this work (Project No.:TE02000162).Fig. 10 Striation spacing vs. measured crack propagation rate for steel.

Fracture Mechanics Based Analysis of the Fatigue Life of Defective Welded Joints 933

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