Fracture mechanics based approach to fatigue analysis of welded joints

Engineering Failure Analysis 49 (2015) 67–78

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Fracture mechanics based approach to fatigue analysis of welded joints Andrea Carpinteri, Camilla Ronchei, Daniela Scorza, Sabrina Vantadori ⇑ Department of Civil-Environmental Engineering & Architecture, University of Parma, Parco Area delle Scienze 181/A, 43124 Parma, Italy

a r t i c l e

i n f o

Article history: Received 3 November 2014 Received in revised form 30 December 2014 Accepted 31 December 2014 Available online 12 January 2015 Keywords: T-joint Cyclic bending Surface crack Stress-intensity factor Francis turbine runner

a b s t r a c t Decades of operating experience related to welded T-joint connections in metallic structures have shown that fatigue cracks generally develop at welding due to both material heterogeneity (mismatch) and stress concentration. In the present paper, the fatigue behaviour of a metallic welded T-joint subjected to cyclic bending is analysed. A semielliptical surface crack is assumed to exist at the welding. The crack propagation is numerically examined by using the stress-intensity factor (SIF) values obtained from finite element analyses, and extensively presented in a recent work. Geometry and sizes of the welded T-joint are chosen in order to compare numerical results with experimental data available in literature, related to an actual welded joint of a common Francis turbine runner. Ó 2015 Elsevier Ltd. All rights reserved.

1. Introduction As is well known, material heterogeneity such as that represented by a mismatched welded T-joint in metallic structural components may have an important role on fracture behaviour of such components [1]. Mismatch means that base metal and weld metal have different mechanical properties in terms of yield stress, elastic modulus, Poisson’s ratio and hardening properties [2,3]. Cracks generally nucleate at welding, that is, in a zone characterised by both material heterogeneity and high stress gradients due to the welding geometry. Several criteria are available in the literature for fracture assessment of welded joints [4–6]. In presence of cyclic loading, such cracks can grow up to failure of the structural components [7,8]. Unexpected failure of turbine blades in hydroeletric plants [9] can occur either very early in service life or after twenty years of operation [10,11]. Such a difference in terms of turbine life is due to the fact that the integrity of the turbines is strongly dependent on design, manufacture and material properties [12,13]. The aforementioned turbine failure is produced by cyclic service load [14,15], represented by either start-up and shutdown operations or transverse blade vibrations, the latter caused by the interaction of the guide vane wakes with the runner blades. In the present paper, the fatigue crack growth simulation for a metallic welded T-joint is carried out by employing a model [16] based on the Paris law [17]. The geometry, sizes and loading here analysed are chosen in order to numerically simulate some experimental fatigue tests available in the literature [10], related to a common Francis hydraulic turbine runner welded joints with hot-formed steel blades welded to both band and crown (Fig. 1).

⇑ Corresponding author. Tel.: +39 0521 905962. E-mail address: [email protected] (S. Vantadori). http://dx.doi.org/10.1016/j.engfailanal.2014.12.021 1350-6307/Ó 2015 Elsevier Ltd. All rights reserved.

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A. Carpinteri et al. / Engineering Failure Analysis 49 (2015) 67–78

Nomenclature a; b semi-axes of the ellipse BmðbnÞ polynomial coefficients of the power series expansion for stress distribution rbn D blade width F trasversal force acting on the blade In error index on the predicted value of n Ia error index on the predicted value of a K IðbnÞ ; K IðbnÞ SIF and dimensionless SIF, respectively, for stress distribution rbn K IðmÞ ; K IðmÞ SIF and dimensionless SIF, respectively, for elementary stress distribution rIðmÞ K tðbÞ stress concentration factor for bending lF arm of the force F, with respect to the XY plate Nf fatigue life N0 number of loading cycles to crack initiation R stress ratio t blade thickness t1 band/crown thickness XYZ global coordinate system w local coordinate axis, with its origin A at the most internal point on the crack front a crack aspect ratio DN number of cycles of each bending loading block producing crack propagation during experimental beach marking procedure g ¼ w=a normalised coordinate, with its origin at the most internal point on the crack front rbn ; rbn Mode I stress distribution and dimensionless Mode I stress distribution, respectively, at the location of the highest stress concentration for the uncracked T-joint rIðmÞ m-th elementary stress distribution rref ðbÞ nominal surface stress under bending rref ðbÞ; max maximum nominal surface stress under cyclic bending rref ðmÞ reference stress for the m-th elementary stress distribution rIðmÞ q radius of the transition arc between blade and band (or crown) n relative crack depth

The specimens consisted of T-joint samples (Fig. 2(a)) made according to the standard manufacturing process for commercial turbines runners. For each specimen, the vertical plate represented the blade, and the horizontal one the band (or crown). Such plates were joined by a double fillet weld. The weld geometry at the T-joint transition is represented by a circular arc, and described by the radius, q (Fig. 2(a)). The specimens were subjected to cyclic bending. Load levels were chosen in a way that the fatigue life of the specimens ranged between 104 and 2(10)5 loading cycles. Therefore, HCF loading was taken into account, according to the definition by Schijve [18]. In such a contest, a linear elastic fracture mechanics approach is suitable for the numerical simulation of the aforementioned experimental campaign. A semi-elliptical surface crack is assumed to exist in the welded zone of the T-joint (Fig. 2(a)). Such a crack shape was that generally observed during the above campaign, and commonly occurs during welding [19,20].

blade band welding crown

Fig. 1. Francis hydraulic turbine runner.

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A. Carpinteri et al. / Engineering Failure Analysis 49 (2015) 67–78

(a)

Z

t 20 mm t1 24 mm

F

ρ

lF D X

D lF t

w 2b

50 mm 200 mm 7.5 mm

a A

ρ

Y t1

(b)

Z

Y D A

C

t

a X

(c)

b

w

Y A a

Z

σI (0) σI (1) σI (2) σI (3)

σbn

w Fig. 2. T-joint specimen: (a) actual geometry and sizes; (b) simplified geometry; (c) elementary stress distributions rI(m) (m = 0, . . ., 3) and complex stress distribution rbn, acting on the crack faces.

In order to compute the Mode I stress-intensity factor (SIF) distribution along the crack front, a numerical procedure was proposed by Carpinteri et al. [21,22], which examines a plate (Fig. 2(b)) instead of the actual joint geometry (Fig. 2(a)), evaluating an approximate Mode I stress-intensity factor. Such a procedure is briefly discussed in the following. Finally, fatigue crack growth simulation results determined through a two-parameter numerical model [16] are compared with experimental fatigue test results on T-jointed specimens [10]. The experimental data numerically simulated hereafter are different from those taken into account in Ref. [21]. The agreement is quite satisfactory, that is, the proposed numerical procedure seems to be promising for fatigue life assessment of structural components with semi-elliptical surface defects.

2. Description of the T-joint and applied loading The geometric sizes of the T-joint here examined (Fig. 2(a)) are those of the specimens employed for the experimental campaign [10]. Note that such experimental sizes are scaled by a factor equal to 2 with respect to those of a common Francis turbine runner. The two plates of the T-joint and the double fillet weld were made of the martensitic–austenitic stainless steel X4 CrNiMo 16-5, whose mechanical and fatigue properties are shown in Table 1 [10].

Table 1 Martensitic-austenitic stainless steel X4CrNiMo 16-5: mechanical and fatigue properties (values employed in the present numerical FE analysis and fatigue crack growth simulations). X4 CrNiMo 16-5 Young’s modulus Yield strength (0.2% plastic strain) Ultimate tensile strength Paris’ factor (R = 0.1) Paris’ exponent (R = 0.1)

E Rp0.2 Rm C k

(GPa) (MPa) (MPa) pffiffiffiffiffi (with KI in MPa m)

220 850 945 6.67 (10)13 3.22

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A. Carpinteri et al. / Engineering Failure Analysis 49 (2015) 67–78

During the experimental tests, a transversal force F was applied to the specimens by the hydraulic cylinder of the fatiguetesting machine (Fig. 2(a)). To carry out the fatigue analysis of the T-joint, we need to numerically evaluate the stress rbn in the Z-direction at Z = 0 level, for the uncracked T-joint under bending (Fig. 2 (a)). Details of such a computation are reported in Ref. [22], where the dimensionless stress rbn is employed:

rbn ¼ rbn =rref ðbÞ

ð1Þ

with the nominal surface stress under bending given by

rref ðbÞ ¼ ½F  lF =ðD  t3 =12Þ 

t 2

ð2Þ

The stress concentration factor under bending (Kt(b)), determined as the ratio between the maximum value of the stress

rbn and rref(b), is equal to 1.22. Details are reported in Ref. [22]. A semi-elliptical crack with semi-axes a and b is assumed to exist in the XY plane, i.e. in correspondence to the highest stress concentration zone (Fig. 2(a)). By introducing a local coordinate system, w, with its origin A at the most internal point on the crack front, the approximate expression of the dimensionless stress distribution rbn is obtained by performing a thirdorder polynomial fitting of the results determined through the above numerical computation [22]:

rbn ðwÞ ffi

M¼3 X

BmðbnÞ  gm

m¼0

¼ ½1:17537  0:20514  a þ 1:31425  ð10Þ2  a2  4:38125  ð10Þ4  a3  þ ½0:20514  a  2:62849  ð10Þ2  a2 þ 1:31437  ð10Þ3  a3  g þ ½1:31425  ð10Þ2  a2  1:31437  ð10Þ3  a3  g2 þ ½4:38125  ð10Þ4  a3  g3

ð3Þ

where g = w/a is the normalised coordinate. In this way, the stress field in Z-direction is evaluated by using a linear FE analysis and then fitted through a polynomial of 3rd order, or analytical expressions available in the literature could be employed [23], being the stress field in notched structural components a research topic of great interest, both in elastic and plastic regime [24,25]. Note that expression (3) is suitable for SIF problems related to different structural component geometries, such as doublecurvature notched shells [26–28] and notched round bars [29]. If the crack is relatively small with respect to the geometrical sizes of the T-joint, the SIF distribution along the crack front is not significantly influenced by the geometry of the body, and principally depends on the stress field acting on the defect faces [21]. For such a reason, in order to simplify the SIF computation (briefly presented in next Section), the cracked finitethickness plate shown in Fig. 2(b) and (c) is examined. 3. SIF computation In order to make the proposed procedure for the SIF computation rather general, so that it can be applied for other cracked bodies with complex geometries different from that of the T-joint here analysed, we have examined a finite cracked plate (Fig. 2(b) and (c)) subjected to four types of Mode I elementary stress distributions acting on the defect faces:

rIðmÞ ¼ rref ðmÞ 

wm a

¼ rref ðmÞ  gm

with m ¼ 0; . . . 3 and

rref ðmÞ ¼ 1:0

ð4Þ

Three-dimensional linear FE analyses have been performed for such elementary loading conditions. The corresponding dimensionless SIFs

K IðmÞ ¼ K IðmÞ =rref ðmÞ 

pffiffiffiffiffiffiffiffiffiffi pa

ð5Þ

obtained from the displacements along the Z-direction of the wedge finite elements (such displacements are measured in correspondence to the quarter-point nodes [30,31]) are reported in Appendix A for some points along the crack front. Note that each surface flaw configuration being considered is characterised by two parameters: the crack aspect ratio, a = a/b, and the relative crack depth, n = a/t. It can be observed that, according to Eq. (4), Eq. (3) may be rewritten as follows:

rbn ðwÞ ffi

M ¼3 X

BmðbnÞ  gm ¼

m¼0

M ¼3 X

BmðbnÞ

m¼0

rref ðmÞ



rIðmÞ

ð6Þ

By recalling Eq. (6) and applying the superposition principle, the approximated dimensionless SIF for the T-joint under bending

K IðbnÞ ¼ K IðbnÞ =ðrref ðbÞ 

pffiffiffiffiffiffiffiffiffiffi p  aÞ

ð7Þ

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A. Carpinteri et al. / Engineering Failure Analysis 49 (2015) 67–78

is determined by employing the dimensionless elementary SIFs K IðmÞ reported in Appendix A:

K IðbnÞ ¼

M ¼3 X

BmðbnÞ  K IðmÞ

ð8Þ

m¼0

Note that the finite element mesh created by an in-house software developed by Vantadori et al. [22] is different from that used by Huth [10] and Carpinteri et al. [21]. Such a mesh is very accurate (in terms of FE discretisation) in the vicinity of the crack front. Further, the polynomial (Eq. (3)) employed by Vantadori et al. [22] to interpolate the stress field in the vicinity of the crack front is of 3rd order, whereas that used by Huth [10] and Carpinteri et al. [21] is of 5th order. Such a lower order polynomial is able to accurately interpolate the actual stress field, and allows us to compare the SIF results with those determined by applying the Pommier’s formula [32], since such a formula can be employed for polynomials up to 3rd degree. In Ref. [22], a quite extensive comparison in terms of K IðbnÞ has been made between the results determined through the above procedure (Eq. (8)) and using either the elementary SIFs reported in Appendix A (derived by FEM analyses) or those computed by employing Pommier’s formula [32] (derived by the body force method results). For the crack configurations examined, the relative difference (in absolute value) between the above results is lower than or equal to 8%. Note that the force F (Fig. 2(a)) produces not only Mode I normal stresses, rbn, but also Mode II shear stresses in the XY plane. Since the maximum value of the nominal shear stress is equal to 2.5% of the nominal bending stress value rref(b), the Mode II SIF is neglected in the fatigue crack propagation analyses presented in next Section. Such a result is also confirmed by the experimental campaign here examined: as a matter of fact, the specimen fracture surface observed post-mortem is quite plane. 4. Fatigue analysis of the T-joint 4.1. Experimental tests The experimental tests reported in Ref. [10] are briefly summarised in the present Section. Two types of specimen geometries were employed to investigate the fatigue behaviour of the T-joint, which differ from each other only for the shape of the welding profile. In other words, each specimen is composed by two plates (sizes reported in Fig. 2(a)), jointed together by a double fillet weld: the welding profile has a circular-arc shape for the first type of specimens (see Fig. 2(a)), whereas the welding profile for the second type is represented by a suitable B-spline curve, which reduces the stress concentration factor at the notch with respect to value (Kt(b) = 1.22) for the first type. For the two profile shapes examined in the above experimental campaign, we have verified that the corresponding stress fields along Z-direction at Z = 0 level (Fig. 2(a)) differ from each other only in a small extent and in a very little region near the external surface of the T-joint. In order to highlight the crack front evolution, a beach marking technique was adopted, where each bending loading block (that is, block of DN loading cycles that produce crack propagation) was characterised by the parameters reported in Table 2. Note that N0 is the number of loading cycles at which the first beach mark is observed, and Nf is the fatigue life. Such fatigue bending tests showed that: (a) A fatigue crack nucleated at the surface of the welded zone. (b) The beach marks of the propagating crack had a semi-elliptical shape. (c) The post-mortem fracture surface was rather plane. 4.2. Numerical model According to the experimental beach mark shape and the aspect of the post-mortem fracture surface, the two-parameter theoretical model proposed by Carpinteri [16] and based on the Paris law [17] can be applied in order to numerically

Table 2 Beach marking technique: parameters related to the block of loading cycles that produces crack propagation, and fatigue life Nf. Specimens No.

Welding profile

R

rref ðbÞ; max (MPa)

DN

N0

Nf

1 2 3 4 5

Circular-arc B-spline B-spline B-spline B-spline

0.1 0.1 0.1 0.1 0.1

911 911 911 911 850

4000 5000 5000 3500 5000

23,600 90,500 76,000 99,500 97,000

42,000 115,600 95,000 113,000 108,800

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A. Carpinteri et al. / Engineering Failure Analysis 49 (2015) 67–78

simulate the above experimental results. As a matter of fact, such a model considers the shape assumption, that is, the surface crack under Mode I loading condition keeps a semi-elliptical shape during the whole propagation. The above model, already applied to analyse the fatigue behaviour of metallic structural components with different geometries such as double-curvature shells and round bars, is presented in detail in Refs. [16,28,33]. The SIF values required to apply the Paris law are computed as discussed in Section 3, whereas the constant parameters of such a law are reported in Table 1 for stress ratio R equal to 0.1. 4.3. Comparison between numerical and experimental results The numerical diagram of the crack aspect ratio a against the relative crack depth n is plotted in Fig. 3. The sizes of the initial crack configurations are those that we have determined by measuring the first visible beach mark appearing in each of the five specimens here examined. The numerical crack propagation is analysed up to the relative crack depth n = 0.6. It can be observed that, for n equal to about 0.4, the crack paths tend to converge to an inclined asymptote described by the following equation:

a ¼ 0:66n þ 0:79

ð9Þ

α = a/ b

CRACK ASPECT RATIO,

Experimental results are also shown in Fig. 3. The solid symbols indicate the experimental values of n and a computed by measuring the sizes of the beach marks of the propagating cracks, and interpolating them through suitable semi-elliptical curves. Note that such experimental values of n and a are slightly different from those reported in Ref. [21]. As a matter of fact, we have here fitted again each experimental beach mark since better quality images of the post-mortem fracture surfaces experimentally obtained are now available. Fig. 4 displays the loading cycle number N against the relative crack depth n for each of the above initial crack configurations. By comparing numerical and experimental results, it can be observed that the proposed model is able to correctly

1.0

0.8

0.6

No. Exp. Present [10] study

1 2 3 4 5

0.4 0.0 0.2 0.4 0.6 RELATIVE CRACK DEPTH,

ξ = a/ t

ξ = a/ t

RELATIVE CRACK DEPTH,

Fig. 3. Crack aspect ratio a against relative crack depth n. Experimental results [10] related to specimens No. 1 to No. 5 are also plotted (open and solid symbols).

0.6 0.5 0.4 0.3 0.2 0.1

No. Exp. Present [10] study

1 2 3 4 5

0.0 0.0 1.0 2.0 3.0 4.0 5.0 NUMBER OF CYCLES, N/10 4

Fig. 4. Loading cycle number N against relative crack depth n. Experimental results [10] related to specimens No. 1 to No. 5 are also plotted (open and solid symbols).

12000 8000 4000

50

100

Iξ, max = 9% Iα,max = 3%

No. cycles 15000

2.0

0.0 250 10.0

Specimen No.2

Present study

Exp. [10] Initial crack

50

100

Iξ, max = 13% Iα, max = 3%

4.0

Specimen No.3

Present study

Exp. [10] Initial crack

No. cycles 3500

0

100

Iξ, max = 4% Iα,max = 0%

10

200

8.0 6.0 4.0

(c) 00

2.0

0.0 250 10.0

150200

5000

8.0 6.0

(b)

00

10000

4.0

150200

10000 5000

No. cycles

6.0

(a)

00

8.0

2.0

20

0.0

10.0

Specimen No.4 Present study

Exp. [10] Initial crack

8.0 6.0 4.0

(d)

2.0 0.0

00

50 I = 8% No. cycles ξ, max Iα,max = 0%

100

5000

150200 25010.0 Specimen No.5 Present study 8.0 Exp. [10] Initial crack

6.0 4.0

(e) 0.0

5.0

10.0

15.0

CRACK LENGHT,

20.0

b [mm]

2.0

0.0 25.0

CRACK DEPTH, a [mm]

Exp. [10] Initial crack

CRACK DEPTH, a [mm]

Present study

CRACK DEPTH, a [mm]

No. cycles

10.0

Specimen No.1

CRACK DEPTH, a [mm]

Iξ, max = 14% Iα,max = 3%

73

CRACK DEPTH, a [mm]

A. Carpinteri et al. / Engineering Failure Analysis 49 (2015) 67–78

Fig. 5. Both numerical and experimental crack front evolutions for the five specimens tested.

estimate the value of N. Note that, in order to perform such a comparison, the experimental total number of loading cycles has been reduced by the number of loading cycles to crack nucleation, N0 (see Table 2), because the nucleation phenomenon is not taken into account through the above numerical model, that is, an initial surface crack is assumed to already exist in the welded zone before loading is applied. Finally, the shape evolution of the crack front at different numbers of loading cycles is displayed in Fig. 5, by juxtaposing the experimental beach marks reported in Fig. 6. The quality of the numerical results in terms of crack front evolution can be evaluated through error indexes (related to the estimated parameters n and a) computed at a given value of loading cycle number:

  nexp  n   100% In ¼  nexp 

ð10aÞ

  aexp  a   100%  Ia ¼  aexp 

ð10bÞ

where nexp and aexp characterise the experimental beach mark configuration. For each of the above initial crack configurations, both the maximum value of In and that of Ia are shown in Fig. 5. The former ranges from 4% to 14%, whereas the latter ranges from 0% to 3%. Such maximum values are quite low even if the numerical evaluations are not always conservative, as can be observed in Fig. 5 for those cases where, at a given loading cycle number, the numerical ellipse is larger than the corresponding experimental beach mark. The numerical results in terms of crack growth paths, crack growth rate and crack front evolution are quite satisfactory, although the effect of the heat-affected zone (HAZ) and changes in microstructure within HAZ are not taken into account through the present model.

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A. Carpinteri et al. / Engineering Failure Analysis 49 (2015) 67–78

(a)

Specimen No.1

(b)

Specimen No.2

(c)

Specimen No.3

(d)

Specimen No.4

(e)

Specimen No.5

Fig. 6. Experimental beach marks for the five specimens analysed.

5. Conclusions In the present paper, the fatigue behaviour of a metallic welded T-joint with a surface crack has been examined. First of all, a finite-thickness plate with a semi-elliptical surface flaw has numerically been analysed. Then, approximate values of SIF for the cracked T-joint under bending have been determined. Such SIF values have been used for fatigue crack growth simulations by also applying a two-parameter model based on the Paris law. Experimental fatigue test results on T-jointed specimens, designed to simulate Francis turbine runner welded joints, have been compared with the numerical results. The agreement is quite satisfactory, that is, the proposed numerical procedure seems to be promising for fatigue life assessment of structural components with semi-elliptical surface defects.

Acknowledgements The authors gratefully acknowledge the research support for this work provided by the Italian Ministry for University and Technological and Scientific Research (MIUR). The authors also gratefully acknowledge Dott. Huth H.J. for the documentation provided us during his stay at the Department of Civil Engineering, Environment and Architecture of the University of Parma in Italy.

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Appendix A Some of the most significant dimensionless SIF results for a finite cracked plate under elementary Mode I stress distributions, rI(m) (m = 0, . . ., 3), are shown in Tables A1–A4. In more detail, we have computed dimensionless SIFs related to crack configurations characterised by a = a/b ranging from 0.1 to 1.2 and n = a/t ranging from 0.1 to 0.8 (Fig. A1). A dimensionless coordinate f⁄ = f/b (Fig. A1) is introduced to identify a generic point on the crack front. The reported SIF values are related to points A (f⁄ = 1.00), L (f⁄ = 0.75), J (f⁄ = 0.50), N(f⁄ = 0.25) and C (f⁄ = 0.10) on the defect front. Table A1 Dimensionless SIF for the opening stress distribution rI(0) acting on the crack faces.

a

n

0.1

0.2

0.3

0.4

0.5

0.6

0.7

K Ið0Þ 0.1

A L J N C

1.06220 1.05070 0.93803 0.75093 0.58153

1.17890 1.15810 1.05240 0.83740 0.61226

1.37060 1.31900 1.16030 0.91747 0.67253

1.60310 1.53950 1.33860 1.03510 0.74743

1.95430 1.87050 1.60530 1.18820 0.83147

2.09810 2.05110 1.82650 1.43250 1.06720

2.36280 2.37020 2.07010 1.72690 1.28260

0.2

A L J N C

1.00890 0.99880 0.93105 0.76582 0.62497

1.08500 1.06830 0.98608 0.82131 0.66020

1.20530 1.17160 1.06720 0.89683 0.72225

1.33290 1.29530 1.18210 1.00150 0.80353

1.54540 1.50330 1.35730 1.11590 0.90039

1.75490 1.72150 1.56370 1.29380 1.04120

1.92040 1.92190 1.73490 1.52010 1.23190

0.4

A L J N C

0.89628 0.88854 0.84428 0.73532 0.66658

0.94073 0.92777 0.88091 0.78664 0.71057

1.00532 0.98225 0.92050 0.83520 0.74510

1.04430 1.02910 0.97871 0.90374 0.81444

1.13650 1.11710 1.05680 0.97003 0.88162

1.20110 1.19100 1.13690 1.05750 0.97139

1.21900 1.22140 1.18740 1.17620 1.10810

0.6

A L J N C

0.79006 0.78794 0.76693 0.71383 0.67887

0.82171 0.81257 0.78407 0.74247 0.71677

0.85460 0.84150 0.81186 0.77563 0.74070

0.87127 0.86514 0.84560 0.81893 0.79019

0.92165 0.91493 0.89079 0.85958 0.83357

0.94406 0.94188 0.92429 0.91129 0.88911

0.92777 0.93485 0.93367 0.97708 0.98326

0.8

A L J N C

0.69902 0.70228 0.67828 0.67319 0.67382

0.72455 0.72391 0.71766 0.70753 0.69973

0.74167 0.73841 0.72928 0.72815 0.72496

0.74839 0.74539 0.74692 0.75570 0.75675

0.78021 0.77805 0.77497 0.77877 0.78572

0.78668 0.78910 0.79132 0.81155 0.82408

0.75726 0.77632 0.78052 0.84595 0.88997

1.0

A L J N C

0.62234 0.63149 0.63018 0.63522 0.65673

0.64452 0.64675 0.65157 0.66539 0.68697

0.65391 0.65508 0.66124 0.68701 0.70025

0.66133 0.66203 0.67223 0.70562 0.72738

0.67653 0.67876 0.68859 0.71879 0.74602

0.67656 0.68440 0.69968 0.73517 0.76821

0.64723 0.66321 0.68794 0.76641 0.81600

1.2

A L J N C

0.55879 0.57238 0.58724 0.59966 0.63360

0.57846 0.58527 0.60030 0.63466 0.66119

0.58796 0.59195 0.60824 0.64376 0.67524

0.58657 0.59400 0.61486 0.66213 0.69552

0.59718 0.60429 0.62549 0.66904 0.70726

0.59886 0.60699 0.62948 0.68021 0.72544

0.56714 0.58149 0.61777 0.69936 0.75910

Table A2 Dimensionless SIF for the opening stress distribution rI(1) acting on the crack faces.

a

n

0.1

0.2

0.3

0.4

0.5

0.6

0.7

K Ið1Þ 0.1

A L J N C

0.42285 0.43624 0.44361 0.44400 0.41858

0.49425 0.50223 0.51190 0.49844 0.43887

0.61332 0.60666 0.58653 0.55321 0.48068

0.76471 0.75018 0.70376 0.63284 0.53214

0.99032 0.96415 0.87983 0.73831 0.59043

1.09491 1.08935 1.02676 0.89865 0.75137

1.28236 1.30732 1.20391 1.10297 0.90272

0.2

A L J N C

0.39445 0.40705 0.43242 0.44257 0.43835

0.44067 0.44952 0.46776 0.47780 0.46198

0.51610 0.51740 0.52324 0.52799 0.50404

0.60391 0.60270 0.60011 0.59739 0.55886

0.73915 0.73551 0.71533 0.67563 0.62478

0.87925 0.88037 0.85290 0.79567 0.72215

0.99894 1.0202 0.97765 0.95197 0.85592

(continued on next page)

76

A. Carpinteri et al. / Engineering Failure Analysis 49 (2015) 67–78

Table A2 (continued)

a

n

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.4

A L J N C

0.33304 0.34444 0.37417 0.40707 0.44814

0.35906 0.36777 0.39701 0.43824 0.47770

0.39914 0.40426 0.42571 0.47106 0.50160

0.43008 0.43802 0.46362 0.51500 0.54726

0.48709 0.49386 0.51563 0.56053 0.59275

0.53174 0.54357 0.56925 0.61933 0.65457

0.55194 0.57154 0.60876 0.69979 0.75045

0.6

A L J N C

0.27422 0.28656 0.32274 0.38130 0.44424

0.29188 0.30106 0.33418 0.39838 0.46919

0.31200 0.32057 0.35296 0.42019 0.48538

0.32797 0.33874 0.37533 0.44826 0.51842

0.35715 0.36837 0.40450 0.47619 0.54797

0.37379 0.38778 0.42777 0.51050 0.58631

0.37156 0.39078 0.43935 0.55467 0.65243

0.8

A L J N C

0.22442 0.23791 0.27012 0.34827 0.43312

0.23770 0.24924 0.28956 0.36761 0.44973

0.24899 0.26002 0.29906 0.38142 0.46682

0.25746 0.26889 0.31151 0.39935 0.48796

0.27429 0.28663 0.32916 0.41569 0.50790

0.28032 0.29493 0.34027 0.43691 0.53434

0.27092 0.2924 0.34013 0.46056 0.5805

1.0

A L J N C

0.18365 0.19842 0.23779 0.31922 0.4165

0.19453 0.20666 0.2488 0.33574 0.43588

0.20135 0.21354 0.25628 0.34933 0.44497

0.20766 0.21965 0.26392 0.36134 0.46295

0.21615 0.22911 0.27425 0.37114 0.47596

0.21777 0.23305 0.28124 0.38211 0.49137

0.20797 0.22655 0.27864 0.40188 0.52464

1.2

A L J N C

0.15117 0.16674 0.21025 0.29341 0.39763

0.16021 0.17331 0.21702 0.31182 0.41514

0.16562 0.17849 0.22309 0.31834 0.42471

0.16828 0.18147 0.22775 0.32947 0.4381

0.17364 0.18729 0.23455 0.33522 0.44640

0.17450 0.18896 0.23744 0.34259 0.45899

0.16444 0.18125 0.23418 0.35472 0.48229

Table A3 Dimensionless SIF for the opening stress distribution rI(2) acting on the crack faces.

a

n

0.1

0.2

0.3

0.4

0.5

0.6

0.7

K Ið2Þ 0.1

A L J N C

0.27161 0.27683 0.27845 0.29508 0.31351

0.32286 0.32430 0.32662 0.33363 0.32850

0.40929 0.40076 0.38272 0.37498 0.36017

0.52065 0.50661 0.46980 0.43474 0.39895

0.68713 0.6648 0.60085 0.51496 0.44356

0.76894 0.76115 0.71203 0.63447 0.56532

0.91100 0.92559 0.84941 0.79048 0.68206

0.2

A L J N C

0.25212 0.25673 0.26887 0.28959 0.32218

0.28521 0.28726 0.29485 0.31504 0.33994

0.34023 0.33710 0.33612 0.35221 0.37149

0.40580 0.40102 0.39392 0.40384 0.41258

0.50585 0.49925 0.47966 0.46341 0.46251

0.61138 0.60825 0.58341 0.55403 0.53698

0.70234 0.71421 0.68029 0.67302 0.63992

0.4

A L J N C

0.20973 0.25673 0.22687 0.25830 0.31955

0.22804 0.28726 0.24321 0.28001 0.34132

0.25717 0.33710 0.26498 0.30449 0.35978

0.28088 0.40102 0.29326 0.33652 0.39330

0.32312 0.49925 0.33228 0.37131 0.42780

0.35755 0.60825 0.37325 0.41592 0.47514

0.37371 0.71421 0.40493 0.47654 0.54875

0.6

A L J N C

0.16888 0.17304 0.18987 0.23554 0.31057

0.18103 0.18316 0.19835 0.24749 0.32864

0.19554 0.19715 0.21198 0.26334 0.34088

0.20799 0.21124 0.22878 0.28394 0.36537

0.22947 0.23304 0.25049 0.30529 0.38791

0.24293 0.24854 0.26888 0.33123 0.41737

0.24228 0.25205 0.27938 0.36434 0.46807

0.8

A L J N C

0.13443 0.13934 0.15373 0.20985 0.29864

0.14331 0.14696 0.16619 0.22268 0.31051

0.15148 0.15477 0.17337 0.23279 0.32321

0.15819 0.16195 0.18294 0.24595 0.33888

0.17045 0.1748 0.19595 0.25869 0.35420

0.17595 0.18193 0.20513 0.27466 0.37444

0.17010 0.18100 0.20697 0.29259 0.40969

1.0

A L J N C

0.10654 0.11223 0.13064 0.18782 0.28395

0.11363 0.11775 0.13787 0.19855 0.29768

0.11857 0.12266 0.14328 0.20801 0.30453

0.12336 0.12755 0.14922 0.21682 0.31773

0.12968 0.13450 0.15693 0.2247 0.32787

0.13184 0.13820 0.16267 0.23328 0.33982

0.12565 0.13453 0.16231 0.24756 0.36503

77

A. Carpinteri et al. / Engineering Failure Analysis 49 (2015) 67–78 Table A3 (continued)

a 1.2

n A L J N C

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.08469 0.09084 0.11136 0.16866 0.26851

0.09042 0.09514 0.11592 0.18019 0.28077

0.09419 0.09872 0.12014 0.18505 0.28787

0.09643 0.10139 0.12389 0.19285 0.29765

0.10040 0.10567 0.12896 0.19773 0.30422

0.10180 0.10763 0.13179 0.20352 0.31387

0.09553 0.10317 0.13067 0.21230 0.33144

Table A4 Dimensionless SIF for the opening stress distribution rI(3) acting on the crack faces.

a

n

0.1

0.2

0.3

0.4

0.5

0.6

0.7

K Ið3Þ 0.1

A L J N C

0.20180 0.20492 0.20368 0.21563 0.24359

0.24180 0.24200 0.24084 0.24509 0.25537

0.30971 0.30229 0.28558 0.27819 0.28066

0.39765 0.38594 0.35466 0.32583 0.31153

0.52951 0.51136 0.45890 0.39038 0.34765

0.59621 0.58946 0.54852 0.48596 0.44540

0.71014 0.72117 0.66000 0.61195 0.54036

0.2

A L J N C

0.18691 0.55383 0.19580 0.20947 0.24673

0.21275 0.55383 0.21630 0.22935 0.26097

0.25612 0.55383 0.24903 0.25872 0.28603

0.30834 0.55383 0.29525 0.29974 0.31876

0.38772 0.55383 0.36348 0.34779 0.35892

0.47203 0.55383 0.44670 0.42062 0.41922

0.54500 0.55383 0.52534 0.51655 0.50276

0.4

A L J N C

0.15450 0.15638 0.16325 0.18300 0.23912

0.16868 0.16915 0.17595 0.19954 0.25609

0.19163 0.19023 0.19335 0.21895 0.27118

0.21076 0.21092 0.21593 0.24411 0.29745

0.24432 0.24393 0.24713 0.27225 0.32527

0.27200 0.27453 0.28024 0.30832 0.36376

0.28508 0.29261 0.30626 0.35692 0.42335

0.6

A L J N C

0.12316 0.12542 0.13460 0.16381 0.22884

0.13248 0.13321 0.14129 0.17300 0.24277

0.14390 0.14418 0.15195 0.18533 0.25261

0.15401 0.15558 0.16540 0.20159 0.27191

0.17104 0.17289 0.18272 0.21887 0.29019

0.18196 0.18550 0.19782 0.23986 0.31423

0.18158 0.18853 0.20681 0.26635 0.35522

0.8

A L J N C

0.09680 0.09963 0.10720 0.14342 0.21763

0.10353 0.10541 0.11639 0.15292 0.22674

0.10997 0.11155 0.12208 0.16081 0.23670

0.11546 0.11745 0.12981 0.17119 0.24908

0.12513 0.12759 0.14017 0.18162 0.26157

0.12968 0.13348 0.14787 0.19456 0.27805

0.12521 0.13287 0.14992 0.20903 0.30643

1.0

A L J N C

0.07559 0.07900 0.08945 0.12615 0.20495

0.08089 0.08315 0.09484 0.13400 0.21534

0.08480 0.08700 0.09903 0.14112 0.22081

0.08865 0.09099 0.10387 0.14809 0.23113

0.09368 0.09652 0.11004 0.15465 0.23949

0.09555 0.09962 0.11486 0.16182 0.24935

0.09079 0.09689 0.11504 0.17302 0.26952

1.2

A L J N C

0.05911 0.06285 0.07477 0.11130 0.19218

0.06334 0.06606 0.07821 0.11949 0.20136

0.06629 0.06882 0.08141 0.12329 0.20690

0.06816 0.07108 0.08451 0.12928 0.21452

0.07130 0.07448 0.08857 0.13346 0.22000

0.07250 0.07618 0.09111 0.13834 0.22788

0.06772 0.07286 0.09059 0.14525 0.24188

Y X

t

N

L A P J

a

C

ζ

b

Fig. A1. Dimensionless coordinate f and points on the crack front being investigated in terms of SIFs, in the case of elementary stress distributions.

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A. Carpinteri et al. / Engineering Failure Analysis 49 (2015) 67–78

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